From c63c9dc59bb7a918f9a6ffb41a12866ee34d7010 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 13 Feb 2023 12:45:03 +0000
Subject: [PATCH 001/316] key combinatorial lemma for binomial theorem

---
 CHANGELOG.md                                  |  15 +++
 src/Data/Nat/Combinatorics.agda               | 124 +++++++++++++++++-
 src/Data/Nat/Combinatorics/Specification.agda |   4 +
 src/Data/Nat/Properties.agda                  |  14 +-
 4 files changed, 154 insertions(+), 3 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index edb69c4b8f..0edbc0d2ef 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2069,11 +2069,14 @@ Other minor changes
   <-step        : m < n → m < 1 + n
   m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
 
+  n<ᵇ1+n : ∀ n → (n <ᵇ suc n) ≡ true
+
   z<′s : zero <′ suc n
   s<′s : m <′ n → suc m <′ suc n
   <⇒<′ : m < n → m <′ n
   <′⇒< : m <′ n → m < n
 
+  1!≡1    : 1 ! ≡ 1
   1≤n!    : 1 ≤ n !
   _!≢0    : NonZero (n !)
   _!*_!≢0 : NonZero (m ! * n !)
@@ -2112,6 +2115,18 @@ Other minor changes
   allUpTo? : (P? : Decidable P) → ∀ v → Dec (∀ {n} → n < v → P n)
   ```
 
+* Added new proofs in `Data.Nat.Combinatorics`:
+  ```agda
+  nP1≡n                   : n P 1 ≡ n
+  nC1≡n                   : n C 1 ≡ n
+  nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
+  ```
+
+* Added new proofs in `Data.Nat.Combinatorics.Specification`:
+  ```agda
+  k![n∸k]!∣n! : k ≤ n →  k ! * (n ∸ k) ! ∣ n !
+  ```
+
 * Added new proofs in `Data.Nat.DivMod`:
   ```agda
   m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index e60390c725..f14af0deb9 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -12,11 +12,13 @@ open import Data.Nat.Base
 open import Data.Nat.DivMod
 open import Data.Nat.Divisibility
 open import Data.Nat.Properties
+open import Relation.Binary.Definitions
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong; subst)
 
 import Data.Nat.Combinatorics.Base          as Base
 import Data.Nat.Combinatorics.Specification as Specification
+import Algebra.Properties.CommutativeSemigroup as CommSemigroupProperties
 
 open ≤-Reasoning
 
@@ -34,7 +36,7 @@ open Base public
 -- Properties of _P_
 
 open Specification public
-  using (nPk≡n!/[n∸k]!; k>n⇒nPk≡0; [n∸k]!k!∣n!)
+  using (nPk≡n!/[n∸k]!; k>n⇒nPk≡0; [n∸k]!k!∣n!; k![n∸k]!∣n!)
 
 nPn≡n! : ∀ n → n P n ≡ n !
 nPn≡n! n = begin-equality
@@ -46,6 +48,15 @@ nPn≡n! n = begin-equality
   where instance
     _ = (n ∸ n) !≢0
 
+nP1≡n : ∀ n → n P 1 ≡ n
+nP1≡n zero    = k>n⇒nPk≡0 (z<s {n = zero})
+nP1≡n (suc n) = begin-equality 
+  (suc n) P 1        ≡⟨ nPk≡n!/[n∸k]! (s≤s (z≤n {n})) ⟩
+  (suc n) ! / n !    ≡⟨ m*n/n≡m (suc n) (n !) ⟩
+  suc n              ∎
+  where instance
+    _ = n !≢0
+
 ------------------------------------------------------------------------
 -- Properties of _C_
 
@@ -85,3 +96,114 @@ nCn≡1 n = begin-equality
   1              ∎
   where instance
     _ = n !≢0
+
+nC1≡n : ∀ n → n C 1 ≡ n
+nC1≡n zero    = refl
+nC1≡n n@(suc n-1) = begin-equality
+  n C 1          ≡⟨ nCk≡nPk/k!  (s≤s (z≤n {n-1})) ⟩
+  (n P 1) / 1 !  ≡⟨ /-congˡ {o = 1 !} (nP1≡n n) ⟩
+  n / 1 !        ≡⟨ /-congʳ {m = n} 1!≡1 ⟩
+  n / 1          ≡⟨ n/1≡n n ⟩
+  n              ∎
+
+
+------------------------------------------------------------------------
+-- Arithmetic of (n C k)
+
+nCk+nC[k+1]≡[n+1]C[k+1] : ∀ n k → n C k + n C (suc k) ≡ suc n C suc k
+nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
+{- case k>n, in which case 1+k>1+n>n -}
+... | tri> _ _ k>n = begin-equality
+  n C k + n C (suc k) ≡⟨ cong (_+ (n C (suc k))) (k>n⇒nCk≡0 k>n) ⟩
+  0 + n C (suc k)     ≡⟨⟩
+  n C (suc k)         ≡⟨ k>n⇒nCk≡0 (m<n⇒m<1+n k>n) ⟩
+  0                   ≡˘⟨ k>n⇒nCk≡0 (s<s k>n) ⟩
+  suc n C suc k       ∎
+{- case k≡n, in which case 1+k≡1+n>n -}
+... | tri≈ _ k≡n _ rewrite k≡n = begin-equality
+  n C n + n C (suc n) ≡⟨ cong (n C n +_) (k>n⇒nCk≡0 (n<1+n n)) ⟩
+  n C n + 0           ≡⟨ +-identityʳ (n C n) ⟩
+  n C n               ≡⟨ nCn≡1 n ⟩
+  1                   ≡˘⟨ nCn≡1 (suc n) ⟩
+  suc n C suc n       ∎
+{- case k<n, in which case 1+k<1+n and there's arithmetic to perform -}
+... | tri< k<n _ _ = begin-equality
+  n C k + n C (suc k)
+    ≡⟨ cong (n C k +_) (nCk≡n!/k![n-k]! k<n)  ⟩
+  n C k + (n! / d[k+1])
+    ≡˘⟨ cong (n C k +_) (m*n/m*o≡n/o (n ∸ k) n! d[k+1]) ⟩
+  n C k + [n-k]*n!/[n-k]*d[k+1]
+    ≡⟨ cong (_+ [n-k]*n!/[n-k]*d[k+1]) (nCk≡n!/k![n-k]! k≤n) ⟩
+  n! / d[k] + _
+    ≡˘⟨ cong (_+ [n-k]*n!/[n-k]*d[k+1]) (m*n/m*o≡n/o (suc k) n! d[k]) ⟩
+  (suc k * n!) / [k+1]*d[k] + _
+    ≡⟨ cong (((suc k * n!) / [k+1]*d[k]) +_) (/-congʳ [n-k]*d[k+1]≡[k+1]*d[k]) ⟩
+  (suc k * n!) / [k+1]*d[k] + ((n ∸ k) * n! / [k+1]*d[k])
+    ≡˘⟨ +-distrib-/-∣ˡ _ (*-monoʳ-∣ (suc k) (k![n∸k]!∣n! k≤n)) ⟩
+  ((suc k) * n! + (n ∸ k) * n!) / [k+1]*d[k]
+    ≡˘⟨ cong (_/ [k+1]*d[k]) (*-distribʳ-+ (n !) (suc k) (n ∸ k)) ⟩
+  ((suc k + (n ∸ k)) * n !) / [k+1]*d[k]
+    ≡⟨ cong (λ z → z * n ! / [k+1]*d[k]) [k+1]+[n-k]≡[n+1] ⟩
+  ((suc n) * n !) / [k+1]*d[k]
+    ≡˘⟨ /-congʳ (*-assoc (suc k) (k !) ((n ∸ k) !)) ⟩
+  ((suc n) * n !) / (((suc k) * k !) * (n ∸ k) !)
+    ≡⟨⟩
+  suc n ! / (suc k ! * (suc n ∸ suc k) !)
+    ≡˘⟨ nCk≡n!/k![n-k]! [k+1]≤[n+1] ⟩
+  suc n C suc k ∎ 
+  where
+    k≤n : k ≤ n
+    k≤n = <⇒≤ k<n
+
+    [k+1]≤[n+1] : suc k ≤ suc n
+    [k+1]≤[n+1] = s≤s k≤n
+
+    [k+1]+[n-k]≡[n+1] : (suc k) + (n ∸ k) ≡ suc n
+    [k+1]+[n-k]≡[n+1] = m+[n∸m]≡n {suc k} [k+1]≤[n+1]
+
+    [n-k]                   = n ∸ k
+    [n-k-1]                 = n ∸ suc k
+    
+    [n-k-1]+1≡[n-k]         : suc [n-k-1] ≡ [n-k]
+    [n-k-1]+1≡[n-k]         = [m-n-1]+1≡[m-n] k<n
+    
+    n!                      = n !
+    k!                      = k !
+    [k+1]!                  = (suc k) !
+    [n-k]!                  = [n-k] !
+    [n-k-1]!                = [n-k-1] !
+
+    [n-k]*[n-k-1]!≡[n-k]!   : [n-k] * [n-k-1]! ≡ [n-k]! 
+    [n-k]*[n-k-1]!≡[n-k]!   = begin-equality
+      [n-k] * [n-k-1]!          ≡˘⟨ cong (_* [n-k-1]!) [n-k-1]+1≡[n-k] ⟩
+      (suc [n-k-1]) * [n-k-1]!  ≡⟨ cong _! [n-k-1]+1≡[n-k] ⟩
+      [n-k]!                    ∎
+      
+    d[k]                    = k! * [n-k]!
+    [k+1]*d[k]              = (suc k) * d[k]
+    d[k+1]                  = [k+1]! * [n-k-1]!
+    [n-k]*d[k]              = [n-k] * d[k]
+    [n-k]*d[k+1]            = [n-k] * d[k+1]
+    n!/[n-k]*d[k+1]         = n ! / [n-k]*d[k+1]
+    [n-k]*n!/[n-k]*d[k+1]   = [n-k] * n! / [n-k]*d[k+1]
+    [n-k]*n!/[k+1]*d[k]     = [n-k] * n! / [k+1]*d[k]
+    
+    [n-k]*d[k+1]≡[k+1]*d[k] : [n-k]*d[k+1] ≡ [k+1]*d[k]
+    [n-k]*d[k+1]≡[k+1]*d[k] = begin-equality
+      [n-k]*d[k+1]
+        ≡⟨ x∙yz≈y∙xz [n-k] [k+1]! [n-k-1]! ⟩
+      [k+1]! * ([n-k] * [n-k-1]!)
+        ≡⟨ *-assoc (suc k) k! ([n-k] * [n-k-1]!) ⟩
+      (suc k) * (k! * ([n-k] * [n-k-1]!))
+        ≡⟨ cong ((suc k) *_) (cong (k! *_) [n-k]*[n-k-1]!≡[n-k]!) ⟩
+      [k+1]*d[k]
+        ∎ where open CommSemigroupProperties *-commutativeSemigroup
+
+    instance
+      [k+1]!*[n-k]!≢0 = (suc k) !* [n-k] !≢0
+      d[k]≢0          = k !* [n-k] !≢0
+      d[k+1]≢0        = (suc k) !* (n ∸ suc k) !≢0
+      [k+1]*d[k]≢0    = m*n≢0 (suc k) d[k]
+      [n-k]≢0         = ≢-nonZero (m>n⇒m∸n≢0 k<n)
+      [n-k]*d[k]≢0    = m*n≢0 [n-k] d[k]
+      [n-k]*d[k+1]≢0  = m*n≢0 [n-k] d[k+1]
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index 533a565d23..90d7551499 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -98,6 +98,10 @@ k!∣nP′k n@{suc n-1} k@{suc k-1} k≤n@(s≤s k-1≤n-1) with k-1 ≟ n-1
   (subst (k ! ∣_) (nP′k≡n!/[n∸k]! k≤n) (k!∣nP′k k≤n))
   where instance _ = (n ∸ k) !≢0
 
+k![n∸k]!∣n! : ∀ {n k} → k ≤ n →  k ! * (n ∸ k) ! ∣ n !
+k![n∸k]!∣n! {n} {k} k≤n = subst (_∣ n !) (*-comm ((n ∸ k) !) (k !)) ([n∸k]!k!∣n! k≤n)
+
+
 ------------------------------------------------------------------------
 -- Properties of _P_
 
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 0426d94d14..652b20615c 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1,4 +1,4 @@
-------------------------------------------------------------------------
+-----------------------------------------------------------------------
 -- The Agda standard library
 --
 -- A bunch of properties about natural number operations
@@ -121,6 +121,10 @@ m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 <ᵇ-reflects-< : ∀ m n → Reflects (m < n) (m <ᵇ n)
 <ᵇ-reflects-< m n = fromEquivalence (<ᵇ⇒< m n) <⇒<ᵇ
 
+n<ᵇ1+n : ∀ n → (n <ᵇ suc n) ≡ true
+n<ᵇ1+n zero = refl
+n<ᵇ1+n (suc n) = n<ᵇ1+n n
+
 ------------------------------------------------------------------------
 -- Properties of _≤ᵇ_
 ------------------------------------------------------------------------
@@ -1558,6 +1562,9 @@ m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
   (m + n) ∸ o          ≡⟨ +-∸-assoc m o≤n ⟩
   m + (n ∸ o)          ∎
 
+[m-n-1]+1≡[m-n] : ∀ {m n} (n<m : n < m) → suc (m ∸ suc n) ≡ m ∸ n
+[m-n-1]+1≡[m-n] n<m = sym (+-∸-assoc 1 n<m)
+
 m≤n+m∸n : ∀ m n → m ≤ n + (m ∸ n)
 m≤n+m∸n zero    n       = z≤n
 m≤n+m∸n (suc m) zero    = ≤-refl
@@ -1918,7 +1925,10 @@ n≡⌈n+n/2⌉ (suc n′@(suc n)) =
   cong suc (trans (n≡⌈n+n/2⌉ _) (cong ⌈_/2⌉ (sym (+-suc n n′))))
 
 ------------------------------------------------------------------------
--- Properties of !_
+-- Properties of _!
+
+1!≡1 : 1 ! ≡ 1
+1!≡1 = *-identityˡ 1
 
 1≤n! : ∀ n → 1 ≤ n !
 1≤n! zero    = ≤-refl

From 21196c79f06e5c91c6130daec2b60e07b1bcfb49 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 13 Feb 2023 12:46:55 +0000
Subject: [PATCH 002/316] fix whitespace

---
 src/Data/Nat/Combinatorics.agda | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index f14af0deb9..c1f5183094 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -50,7 +50,7 @@ nPn≡n! n = begin-equality
 
 nP1≡n : ∀ n → n P 1 ≡ n
 nP1≡n zero    = k>n⇒nPk≡0 (z<s {n = zero})
-nP1≡n (suc n) = begin-equality 
+nP1≡n (suc n) = begin-equality
   (suc n) P 1        ≡⟨ nPk≡n!/[n∸k]! (s≤s (z≤n {n})) ⟩
   (suc n) ! / n !    ≡⟨ m*n/n≡m (suc n) (n !) ⟩
   suc n              ∎
@@ -150,7 +150,7 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
     ≡⟨⟩
   suc n ! / (suc k ! * (suc n ∸ suc k) !)
     ≡˘⟨ nCk≡n!/k![n-k]! [k+1]≤[n+1] ⟩
-  suc n C suc k ∎ 
+  suc n C suc k ∎
   where
     k≤n : k ≤ n
     k≤n = <⇒≤ k<n
@@ -163,22 +163,22 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
 
     [n-k]                   = n ∸ k
     [n-k-1]                 = n ∸ suc k
-    
+
     [n-k-1]+1≡[n-k]         : suc [n-k-1] ≡ [n-k]
     [n-k-1]+1≡[n-k]         = [m-n-1]+1≡[m-n] k<n
-    
+
     n!                      = n !
     k!                      = k !
     [k+1]!                  = (suc k) !
     [n-k]!                  = [n-k] !
     [n-k-1]!                = [n-k-1] !
 
-    [n-k]*[n-k-1]!≡[n-k]!   : [n-k] * [n-k-1]! ≡ [n-k]! 
+    [n-k]*[n-k-1]!≡[n-k]!   : [n-k] * [n-k-1]! ≡ [n-k]!
     [n-k]*[n-k-1]!≡[n-k]!   = begin-equality
       [n-k] * [n-k-1]!          ≡˘⟨ cong (_* [n-k-1]!) [n-k-1]+1≡[n-k] ⟩
       (suc [n-k-1]) * [n-k-1]!  ≡⟨ cong _! [n-k-1]+1≡[n-k] ⟩
       [n-k]!                    ∎
-      
+
     d[k]                    = k! * [n-k]!
     [k+1]*d[k]              = (suc k) * d[k]
     d[k+1]                  = [k+1]! * [n-k-1]!
@@ -187,7 +187,7 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
     n!/[n-k]*d[k+1]         = n ! / [n-k]*d[k+1]
     [n-k]*n!/[n-k]*d[k+1]   = [n-k] * n! / [n-k]*d[k+1]
     [n-k]*n!/[k+1]*d[k]     = [n-k] * n! / [k+1]*d[k]
-    
+
     [n-k]*d[k+1]≡[k+1]*d[k] : [n-k]*d[k+1] ≡ [k+1]*d[k]
     [n-k]*d[k+1]≡[k+1]*d[k] = begin-equality
       [n-k]*d[k+1]

From bfe6ff6ded84523846b4579743d52dd5e0bcb821 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 13 Feb 2023 12:53:58 +0000
Subject: [PATCH 003/316] typo: dropped comment character

---
 src/Data/Nat/Properties.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 652b20615c..85bf21ceda 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1,4 +1,4 @@
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Agda standard library
 --
 -- A bunch of properties about natural number operations

From 9c2da6b1fbf54bd19342ab56e88487bc6442e5cd Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Tue, 14 Feb 2023 09:45:45 +0000
Subject: [PATCH 004/316] cosmetics

---
 src/Data/Nat/Combinatorics.agda | 34 +++++++++++++++++----------------
 1 file changed, 18 insertions(+), 16 deletions(-)

diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index c1f5183094..4e6331965d 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -50,12 +50,12 @@ nPn≡n! n = begin-equality
 
 nP1≡n : ∀ n → n P 1 ≡ n
 nP1≡n zero    = k>n⇒nPk≡0 (z<s {n = zero})
-nP1≡n (suc n) = begin-equality
-  (suc n) P 1        ≡⟨ nPk≡n!/[n∸k]! (s≤s (z≤n {n})) ⟩
-  (suc n) ! / n !    ≡⟨ m*n/n≡m (suc n) (n !) ⟩
-  suc n              ∎
+nP1≡n n@(suc n-1) = begin-equality
+  n P 1        ≡⟨ nPk≡n!/[n∸k]! (s≤s (z≤n {n-1})) ⟩
+  n ! / n-1 !  ≡⟨ m*n/n≡m n (n-1 !) ⟩
+  n            ∎
   where instance
-    _ = n !≢0
+    _ = n-1 !≢0
 
 ------------------------------------------------------------------------
 -- Properties of _C_
@@ -152,14 +152,14 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
     ≡˘⟨ nCk≡n!/k![n-k]! [k+1]≤[n+1] ⟩
   suc n C suc k ∎
   where
-    k≤n : k ≤ n
-    k≤n = <⇒≤ k<n
+    k≤n                     : k ≤ n
+    k≤n                     = <⇒≤ k<n
 
-    [k+1]≤[n+1] : suc k ≤ suc n
-    [k+1]≤[n+1] = s≤s k≤n
+    [k+1]≤[n+1]             : suc k ≤ suc n
+    [k+1]≤[n+1]             = s≤s k≤n
 
-    [k+1]+[n-k]≡[n+1] : (suc k) + (n ∸ k) ≡ suc n
-    [k+1]+[n-k]≡[n+1] = m+[n∸m]≡n {suc k} [k+1]≤[n+1]
+    [k+1]+[n-k]≡[n+1]       : (suc k) + (n ∸ k) ≡ suc n
+    [k+1]+[n-k]≡[n+1]       = m+[n∸m]≡n {suc k} [k+1]≤[n+1]
 
     [n-k]                   = n ∸ k
     [n-k-1]                 = n ∸ suc k
@@ -175,9 +175,11 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
 
     [n-k]*[n-k-1]!≡[n-k]!   : [n-k] * [n-k-1]! ≡ [n-k]!
     [n-k]*[n-k-1]!≡[n-k]!   = begin-equality
-      [n-k] * [n-k-1]!          ≡˘⟨ cong (_* [n-k-1]!) [n-k-1]+1≡[n-k] ⟩
-      (suc [n-k-1]) * [n-k-1]!  ≡⟨ cong _! [n-k-1]+1≡[n-k] ⟩
-      [n-k]!                    ∎
+      [n-k] * [n-k-1]!
+        ≡˘⟨ cong (_* [n-k-1]!) [n-k-1]+1≡[n-k] ⟩
+      (suc [n-k-1]) * [n-k-1]!
+        ≡⟨ cong _! [n-k-1]+1≡[n-k] ⟩
+      [n-k]!                ∎
 
     d[k]                    = k! * [n-k]!
     [k+1]*d[k]              = (suc k) * d[k]
@@ -196,8 +198,8 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
         ≡⟨ *-assoc (suc k) k! ([n-k] * [n-k-1]!) ⟩
       (suc k) * (k! * ([n-k] * [n-k-1]!))
         ≡⟨ cong ((suc k) *_) (cong (k! *_) [n-k]*[n-k-1]!≡[n-k]!) ⟩
-      [k+1]*d[k]
-        ∎ where open CommSemigroupProperties *-commutativeSemigroup
+      [k+1]*d[k]             ∎
+      where open CommSemigroupProperties *-commutativeSemigroup
 
     instance
       [k+1]!*[n-k]!≢0 = (suc k) !* [n-k] !≢0

From 45c8c392a2a039261462e62185d6ebd0d713268d Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Tue, 14 Feb 2023 09:53:20 +0000
Subject: [PATCH 005/316] =?UTF-8?q?removed,=20and=20remoed=20need=20for=20?=
 =?UTF-8?q?`1\!=E2=89=A11`?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                    | 1 -
 src/Data/Nat/Combinatorics.agda | 5 ++---
 src/Data/Nat/Properties.agda    | 3 ---
 3 files changed, 2 insertions(+), 7 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0edbc0d2ef..231d10fe7b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2076,7 +2076,6 @@ Other minor changes
   <⇒<′ : m < n → m <′ n
   <′⇒< : m <′ n → m < n
 
-  1!≡1    : 1 ! ≡ 1
   1≤n!    : 1 ≤ n !
   _!≢0    : NonZero (n !)
   _!*_!≢0 : NonZero (m ! * n !)
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index 4e6331965d..68fceabd0c 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -101,9 +101,8 @@ nC1≡n : ∀ n → n C 1 ≡ n
 nC1≡n zero    = refl
 nC1≡n n@(suc n-1) = begin-equality
   n C 1          ≡⟨ nCk≡nPk/k!  (s≤s (z≤n {n-1})) ⟩
-  (n P 1) / 1 !  ≡⟨ /-congˡ {o = 1 !} (nP1≡n n) ⟩
-  n / 1 !        ≡⟨ /-congʳ {m = n} 1!≡1 ⟩
-  n / 1          ≡⟨ n/1≡n n ⟩
+  (n P 1) / 1 !  ≡⟨ n/1≡n (n P 1) ⟩
+  n P 1          ≡⟨ nP1≡n n ⟩
   n              ∎
 
 
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 85bf21ceda..05318b224a 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1927,9 +1927,6 @@ n≡⌈n+n/2⌉ (suc n′@(suc n)) =
 ------------------------------------------------------------------------
 -- Properties of _!
 
-1!≡1 : 1 ! ≡ 1
-1!≡1 = *-identityˡ 1
-
 1≤n! : ∀ n → 1 ≤ n !
 1≤n! zero    = ≤-refl
 1≤n! (suc n) = *-mono-≤ (m≤m+n 1 n) (1≤n! n)

From b89e8be825757850fad0a80b98a2456d658be72b Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Thu, 23 Feb 2023 16:12:48 +0000
Subject: [PATCH 006/316] moved lemma

---
 CHANGELOG.md                                  | 6 +-----
 src/Data/Nat/Combinatorics.agda               | 5 ++++-
 src/Data/Nat/Combinatorics/Specification.agda | 3 ---
 3 files changed, 5 insertions(+), 9 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 231d10fe7b..95d40d1bff 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2116,16 +2116,12 @@ Other minor changes
 
 * Added new proofs in `Data.Nat.Combinatorics`:
   ```agda
+  k![n∸k]!∣n!              : k ≤ n →  k ! * (n ∸ k) ! ∣ n !
   nP1≡n                   : n P 1 ≡ n
   nC1≡n                   : n C 1 ≡ n
   nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
   ```
 
-* Added new proofs in `Data.Nat.Combinatorics.Specification`:
-  ```agda
-  k![n∸k]!∣n! : k ≤ n →  k ! * (n ∸ k) ! ∣ n !
-  ```
-
 * Added new proofs in `Data.Nat.DivMod`:
   ```agda
   m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index 68fceabd0c..31c1346e9d 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -36,7 +36,7 @@ open Base public
 -- Properties of _P_
 
 open Specification public
-  using (nPk≡n!/[n∸k]!; k>n⇒nPk≡0; [n∸k]!k!∣n!; k![n∸k]!∣n!)
+  using (nPk≡n!/[n∸k]!; k>n⇒nPk≡0; [n∸k]!k!∣n!)
 
 nPn≡n! : ∀ n → n P n ≡ n !
 nPn≡n! n = begin-equality
@@ -109,6 +109,9 @@ nC1≡n n@(suc n-1) = begin-equality
 ------------------------------------------------------------------------
 -- Arithmetic of (n C k)
 
+k![n∸k]!∣n! : ∀ {n k} → k ≤ n →  k ! * (n ∸ k) ! ∣ n !
+k![n∸k]!∣n! {n} {k} k≤n = subst (_∣ n !) (*-comm ((n ∸ k) !) (k !)) ([n∸k]!k!∣n! k≤n)
+
 nCk+nC[k+1]≡[n+1]C[k+1] : ∀ n k → n C k + n C (suc k) ≡ suc n C suc k
 nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
 {- case k>n, in which case 1+k>1+n>n -}
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index 90d7551499..e655484b60 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -98,9 +98,6 @@ k!∣nP′k n@{suc n-1} k@{suc k-1} k≤n@(s≤s k-1≤n-1) with k-1 ≟ n-1
   (subst (k ! ∣_) (nP′k≡n!/[n∸k]! k≤n) (k!∣nP′k k≤n))
   where instance _ = (n ∸ k) !≢0
 
-k![n∸k]!∣n! : ∀ {n k} → k ≤ n →  k ! * (n ∸ k) ! ∣ n !
-k![n∸k]!∣n! {n} {k} k≤n = subst (_∣ n !) (*-comm ((n ∸ k) !) (k !)) ([n∸k]!k!∣n! k≤n)
-
 
 ------------------------------------------------------------------------
 -- Properties of _P_

From 2c62e4f67f0a55dd4c6cfd6b188005cef35eebaf Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Thu, 23 Feb 2023 17:20:07 +0000
Subject: [PATCH 007/316] fixed merge commit; lifted out lemmas; tweaked

---
 CHANGELOG.md                                  |  6 +-
 src/Data/Nat/Combinatorics.agda               | 82 ++++++++++++-------
 src/Data/Nat/Combinatorics/Specification.agda |  1 -
 src/Data/Nat/Properties.agda                  |  9 +-
 4 files changed, 55 insertions(+), 43 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 95d40d1bff..5fc1854115 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2069,8 +2069,6 @@ Other minor changes
   <-step        : m < n → m < 1 + n
   m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
 
-  n<ᵇ1+n : ∀ n → (n <ᵇ suc n) ≡ true
-
   z<′s : zero <′ suc n
   s<′s : m <′ n → suc m <′ suc n
   <⇒<′ : m < n → m <′ n
@@ -2116,7 +2114,9 @@ Other minor changes
 
 * Added new proofs in `Data.Nat.Combinatorics`:
   ```agda
-  k![n∸k]!∣n!              : k ≤ n →  k ! * (n ∸ k) ! ∣ n !
+  [n-k]*[n-k-1]!≡[n-k]!   : k < n → (n ∸ k) * (n ∸ suc k) ! ≡ (n ∸ k) !
+  [n-k]*d[k+1]≡[k+1]*d[k] : k < n → (n ∸ k) * ((suc k) ! * (n ∸ suc k) !) ≡ (suc k) * (k ! * (n ∸ k) !)
+  k![n∸k]!∣n!              : k ≤ n → k ! * (n ∸ k) ! ∣ n !
   nP1≡n                   : n P 1 ≡ n
   nC1≡n                   : n C 1 ≡ n
   nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index 31c1346e9d..6db600699c 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -12,7 +12,7 @@ open import Data.Nat.Base
 open import Data.Nat.DivMod
 open import Data.Nat.Divisibility
 open import Data.Nat.Properties
-open import Relation.Binary.Definitions
+open import Relation.Binary.Definitions 
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong; subst)
 
@@ -49,11 +49,11 @@ nPn≡n! n = begin-equality
     _ = (n ∸ n) !≢0
 
 nP1≡n : ∀ n → n P 1 ≡ n
-nP1≡n zero    = k>n⇒nPk≡0 (z<s {n = zero})
+nP1≡n zero        = refl
 nP1≡n n@(suc n-1) = begin-equality
-  n P 1        ≡⟨ nPk≡n!/[n∸k]! (s≤s (z≤n {n-1})) ⟩
-  n ! / n-1 !  ≡⟨ m*n/n≡m n (n-1 !) ⟩
-  n            ∎
+  n P 1          ≡⟨ nPk≡n!/[n∸k]! (s≤s (z≤n {n-1})) ⟩
+  n ! / n-1 !    ≡⟨ m*n/n≡m n (n-1 !) ⟩
+  n              ∎
   where instance
     _ = n-1 !≢0
 
@@ -109,7 +109,51 @@ nC1≡n n@(suc n-1) = begin-equality
 ------------------------------------------------------------------------
 -- Arithmetic of (n C k)
 
-k![n∸k]!∣n! : ∀ {n k} → k ≤ n →  k ! * (n ∸ k) ! ∣ n !
+module _ {n k} (k<n : k < n) where
+
+  private
+
+    [n-k]            = n ∸ k
+    [n-k-1]          = n ∸ suc k
+
+    [n-k]!           = [n-k] !
+    [n-k-1]!         = [n-k-1] !
+  
+    [n-k]≡1+[n-k-1]  : [n-k] ≡ suc [n-k-1]
+    [n-k]≡1+[n-k-1]  = +-∸-assoc 1 k<n
+
+
+  [n-k]*[n-k-1]!≡[n-k]! : [n-k] * [n-k-1]! ≡ [n-k]!
+  [n-k]*[n-k-1]!≡[n-k]! = begin-equality
+      [n-k] * [n-k-1]!
+        ≡⟨ cong (_* [n-k-1]!) [n-k]≡1+[n-k-1] ⟩
+      (suc [n-k-1]) * [n-k-1]!
+        ≡˘⟨ cong _! [n-k]≡1+[n-k-1] ⟩
+      [n-k]!                ∎
+
+  private
+
+    n!           = n !
+    k!           = k !
+    [k+1]!       = (suc k) !
+
+    d[k]         = k! * [n-k]!
+    [k+1]*d[k]   = (suc k) * d[k]
+    d[k+1]       = [k+1]! * [n-k-1]!
+    [n-k]*d[k+1] = [n-k] * d[k+1]
+
+  [n-k]*d[k+1]≡[k+1]*d[k] : [n-k]*d[k+1] ≡ [k+1]*d[k]
+  [n-k]*d[k+1]≡[k+1]*d[k] = begin-equality
+    [n-k]*d[k+1]
+      ≡⟨ x∙yz≈y∙xz [n-k] [k+1]! [n-k-1]! ⟩
+    [k+1]! * ([n-k] * [n-k-1]!)
+      ≡⟨ *-assoc (suc k) k! ([n-k] * [n-k-1]!) ⟩
+    (suc k) * (k! * ([n-k] * [n-k-1]!))
+       ≡⟨ cong ((suc k) *_) (cong (k! *_) [n-k]*[n-k-1]!≡[n-k]!) ⟩
+    [k+1]*d[k]             ∎
+    where open CommSemigroupProperties *-commutativeSemigroup
+
+k![n∸k]!∣n! : ∀ {n k} → k ≤ n → k ! * (n ∸ k) ! ∣ n !
 k![n∸k]!∣n! {n} {k} k≤n = subst (_∣ n !) (*-comm ((n ∸ k) !) (k !)) ([n∸k]!k!∣n! k≤n)
 
 nCk+nC[k+1]≡[n+1]C[k+1] : ∀ n k → n C k + n C (suc k) ≡ suc n C suc k
@@ -139,7 +183,7 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
   n! / d[k] + _
     ≡˘⟨ cong (_+ [n-k]*n!/[n-k]*d[k+1]) (m*n/m*o≡n/o (suc k) n! d[k]) ⟩
   (suc k * n!) / [k+1]*d[k] + _
-    ≡⟨ cong (((suc k * n!) / [k+1]*d[k]) +_) (/-congʳ [n-k]*d[k+1]≡[k+1]*d[k]) ⟩
+    ≡⟨ cong (((suc k * n!) / [k+1]*d[k]) +_) (/-congʳ ([n-k]*d[k+1]≡[k+1]*d[k] k<n)) ⟩
   (suc k * n!) / [k+1]*d[k] + ((n ∸ k) * n! / [k+1]*d[k])
     ≡˘⟨ +-distrib-/-∣ˡ _ (*-monoʳ-∣ (suc k) (k![n∸k]!∣n! k≤n)) ⟩
   ((suc k) * n! + (n ∸ k) * n!) / [k+1]*d[k]
@@ -166,48 +210,24 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
     [n-k]                   = n ∸ k
     [n-k-1]                 = n ∸ suc k
 
-    [n-k-1]+1≡[n-k]         : suc [n-k-1] ≡ [n-k]
-    [n-k-1]+1≡[n-k]         = [m-n-1]+1≡[m-n] k<n
-
     n!                      = n !
     k!                      = k !
     [k+1]!                  = (suc k) !
     [n-k]!                  = [n-k] !
     [n-k-1]!                = [n-k-1] !
 
-    [n-k]*[n-k-1]!≡[n-k]!   : [n-k] * [n-k-1]! ≡ [n-k]!
-    [n-k]*[n-k-1]!≡[n-k]!   = begin-equality
-      [n-k] * [n-k-1]!
-        ≡˘⟨ cong (_* [n-k-1]!) [n-k-1]+1≡[n-k] ⟩
-      (suc [n-k-1]) * [n-k-1]!
-        ≡⟨ cong _! [n-k-1]+1≡[n-k] ⟩
-      [n-k]!                ∎
-
     d[k]                    = k! * [n-k]!
     [k+1]*d[k]              = (suc k) * d[k]
     d[k+1]                  = [k+1]! * [n-k-1]!
-    [n-k]*d[k]              = [n-k] * d[k]
     [n-k]*d[k+1]            = [n-k] * d[k+1]
     n!/[n-k]*d[k+1]         = n ! / [n-k]*d[k+1]
     [n-k]*n!/[n-k]*d[k+1]   = [n-k] * n! / [n-k]*d[k+1]
     [n-k]*n!/[k+1]*d[k]     = [n-k] * n! / [k+1]*d[k]
 
-    [n-k]*d[k+1]≡[k+1]*d[k] : [n-k]*d[k+1] ≡ [k+1]*d[k]
-    [n-k]*d[k+1]≡[k+1]*d[k] = begin-equality
-      [n-k]*d[k+1]
-        ≡⟨ x∙yz≈y∙xz [n-k] [k+1]! [n-k-1]! ⟩
-      [k+1]! * ([n-k] * [n-k-1]!)
-        ≡⟨ *-assoc (suc k) k! ([n-k] * [n-k-1]!) ⟩
-      (suc k) * (k! * ([n-k] * [n-k-1]!))
-        ≡⟨ cong ((suc k) *_) (cong (k! *_) [n-k]*[n-k-1]!≡[n-k]!) ⟩
-      [k+1]*d[k]             ∎
-      where open CommSemigroupProperties *-commutativeSemigroup
-
     instance
       [k+1]!*[n-k]!≢0 = (suc k) !* [n-k] !≢0
       d[k]≢0          = k !* [n-k] !≢0
       d[k+1]≢0        = (suc k) !* (n ∸ suc k) !≢0
       [k+1]*d[k]≢0    = m*n≢0 (suc k) d[k]
       [n-k]≢0         = ≢-nonZero (m>n⇒m∸n≢0 k<n)
-      [n-k]*d[k]≢0    = m*n≢0 [n-k] d[k]
       [n-k]*d[k+1]≢0  = m*n≢0 [n-k] d[k+1]
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index e655484b60..533a565d23 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -98,7 +98,6 @@ k!∣nP′k n@{suc n-1} k@{suc k-1} k≤n@(s≤s k-1≤n-1) with k-1 ≟ n-1
   (subst (k ! ∣_) (nP′k≡n!/[n∸k]! k≤n) (k!∣nP′k k≤n))
   where instance _ = (n ∸ k) !≢0
 
-
 ------------------------------------------------------------------------
 -- Properties of _P_
 
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 05318b224a..0426d94d14 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -121,10 +121,6 @@ m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 <ᵇ-reflects-< : ∀ m n → Reflects (m < n) (m <ᵇ n)
 <ᵇ-reflects-< m n = fromEquivalence (<ᵇ⇒< m n) <⇒<ᵇ
 
-n<ᵇ1+n : ∀ n → (n <ᵇ suc n) ≡ true
-n<ᵇ1+n zero = refl
-n<ᵇ1+n (suc n) = n<ᵇ1+n n
-
 ------------------------------------------------------------------------
 -- Properties of _≤ᵇ_
 ------------------------------------------------------------------------
@@ -1562,9 +1558,6 @@ m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
   (m + n) ∸ o          ≡⟨ +-∸-assoc m o≤n ⟩
   m + (n ∸ o)          ∎
 
-[m-n-1]+1≡[m-n] : ∀ {m n} (n<m : n < m) → suc (m ∸ suc n) ≡ m ∸ n
-[m-n-1]+1≡[m-n] n<m = sym (+-∸-assoc 1 n<m)
-
 m≤n+m∸n : ∀ m n → m ≤ n + (m ∸ n)
 m≤n+m∸n zero    n       = z≤n
 m≤n+m∸n (suc m) zero    = ≤-refl
@@ -1925,7 +1918,7 @@ n≡⌈n+n/2⌉ (suc n′@(suc n)) =
   cong suc (trans (n≡⌈n+n/2⌉ _) (cong ⌈_/2⌉ (sym (+-suc n n′))))
 
 ------------------------------------------------------------------------
--- Properties of _!
+-- Properties of !_
 
 1≤n! : ∀ n → 1 ≤ n !
 1≤n! zero    = ≤-refl

From a868395724cd05b09145d3926317b90b74b1f1ba Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 1 Mar 2023 17:37:47 +0900
Subject: [PATCH 008/316] Update workflow version

---
 .github/workflows/ci-ubuntu.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index 2b08c7d6e6..85e9fe1365 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -74,7 +74,7 @@ jobs:
           if [[ '${{ github.ref }}' == 'refs/heads/master' \
              || '${{ github.base_ref }}' == 'master' ]]; then
             # Pick Agda version for master
-            echo "AGDA_COMMIT=tags/v2.6.2" >> $GITHUB_ENV;
+            echo "AGDA_COMMIT=tags/v2.6.3" >> $GITHUB_ENV;
             echo "AGDA_HTML_DIR=html" >> $GITHUB_ENV
           elif [[ '${{ github.ref }}' == 'refs/heads/experimental' \
                || '${{ github.base_ref }}' == 'experimental' ]]; then

From f34ffc50157bf49bb20dd5b6b8d73033b7a46720 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 2 Mar 2023 15:07:00 +0900
Subject: [PATCH 009/316] Fix missing annotations in Data.Fin.Properties

---
 src/Data/Fin/Properties.agda | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 2f6e36eabd..9089102b29 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -5,6 +5,7 @@
 -- properties (or other properties not available in Data.Fin)
 ------------------------------------------------------------------------
 
+{-# OPTIONS --cubical-compatible --safe #-}
 {-# OPTIONS --warn=noUserWarning #-} -- for deprecated _≺_ (issue #1726)
 
 module Data.Fin.Properties where

From c07ea1aaa0322d628dce2d5f572739446ac5043e Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 17 Mar 2023 13:47:02 +0900
Subject: [PATCH 010/316] Fixed failing reflection test

---
 tests/reflection/assumption/Main.agda | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

diff --git a/tests/reflection/assumption/Main.agda b/tests/reflection/assumption/Main.agda
index d7f30d3118..cf0d21b514 100644
--- a/tests/reflection/assumption/Main.agda
+++ b/tests/reflection/assumption/Main.agda
@@ -7,6 +7,7 @@ open import Data.Bool.Base using (if_then_else_)
 open import Data.List.Base using (List; []; _∷_; concatMap; map)
 open import Data.Maybe.Base using (Maybe; nothing; just)
 open import Data.Nat.Base using (ℕ; suc)
+open import Data.Product.Base using (_×_; _,_)
 open import Level using (Level)
 open import Data.Unit.Base using (⊤)
 open import Function.Base using (case_of_; _$_)
@@ -39,11 +40,11 @@ private
 -- before being compared to the type of the most local variable. The goal
 -- indeed lives below that variable's binding site!
 
-searchEntry : ℕ → Type → List Type → Maybe ℕ
+searchEntry : ℕ → Type → List (String × Arg Type) → Maybe ℕ
 searchEntry n ty [] = nothing
-searchEntry n ty (e ∷ es) = do
+searchEntry n ty ((_ , e) ∷ es) = do
   ty ← strengthen ty
-  if does (ty ≟ e)
+  if does (ty ≟ unArg e)
     then just n
     else searchEntry (suc n) ty es
 
@@ -54,8 +55,8 @@ macro
     goal ← inferType hole
     debugPrint "" 10
       (strErr "Context : "
-       ∷ concatMap (λ where (arg info ty) → strErr "\n  " ∷ termErr ty ∷ []) asss)
-    let res = searchEntry 0 goal (map unArg asss)
+       ∷ concatMap (λ where (_ , arg info ty) → strErr "\n  " ∷ termErr ty ∷ []) asss)
+    let res = searchEntry 0 goal asss
     case res of λ where
       nothing → typeError (strErr "Couldn't find an assumption of type: " ∷ termErr goal ∷ [])
       (just idx) → unify hole (var idx [])

From 95700b8b54f554f0d4f4969972c26fabb92ff90d Mon Sep 17 00:00:00 2001
From: Harrison Grodin <hgrodin@andrew.cmu.edu>
Date: Mon, 20 Mar 2023 13:45:10 -0400
Subject: [PATCH 011/316] Add monotonicity laws for `Nat._^_`

---
 src/Data/Nat/Properties.agda | 18 ++++++++++++++++++
 1 file changed, 18 insertions(+)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 0426d94d14..b177412e00 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1046,6 +1046,24 @@ m^n≡1⇒n≡0∨m≡1 m (suc n) eq = inj₂ (m*n≡1⇒m≡1 m (m ^ n) eq)
 m^n≢0 : ∀ m n .{{_ : NonZero m}} → NonZero (m ^ n)
 m^n≢0 m n = ≢-nonZero (≢-nonZero⁻¹ m ∘′ m^n≡0⇒m≡0 m n)
 
+^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
+^-monoˡ-≤ zero m≤o = s≤s z≤n
+^-monoˡ-≤ (suc n) {m} {o} m≤o = *-mono-≤ m≤o (^-monoˡ-≤ n m≤o)
+
+^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
+^-monoʳ-≤ m {n} {o} z≤n = n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{m^n≢0 m o}})
+^-monoʳ-≤ m (s≤s n≤o) = *-monoʳ-≤ m (^-monoʳ-≤ m n≤o)
+
+^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
+^-monoˡ-< (suc zero) m<o = *-monoˡ-< 1 m<o
+^-monoˡ-< (suc (suc n)) m<o = *-mono-< m<o (^-monoˡ-< (suc n) m<o)
+
+^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_
+^-monoʳ-< m 1<m {.zero} {suc o} z<s =
+  *-mono-≤ 1<m (n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{(m^n≢0 m o {{>-nonZero (<-trans z<s 1<m)}})}}))
+^-monoʳ-< m 1<m {suc n} (s<s n≤o) =
+  *-monoʳ-< m {{ >-nonZero (<-trans z<s 1<m) }} (^-monoʳ-< m 1<m n≤o)
+
 2^n>0 : ∀ (n : ℕ) → 2 ^ n > 0
 2^n>0 zero = z<s
 2^n>0 (suc n) = ≤-trans (2^n>0 n) (m≤m+n (2 ^ n) ((2 ^ n) + zero))

From db22021a214dfe4afadb5c1e3036d8c15c13cb48 Mon Sep 17 00:00:00 2001
From: Harrison Grodin <hgrodin@andrew.cmu.edu>
Date: Mon, 20 Mar 2023 13:48:04 -0400
Subject: [PATCH 012/316] Update CHANGELOG

---
 CHANGELOG.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5fc1854115..a43a87b867 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2083,6 +2083,11 @@ Other minor changes
 
   n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
 
+  ^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
+  ^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
+  ^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
+  ^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_
+
   2^n>0 : 2 ^ n > 0
 
   n≡⌊n+n/2⌋ : n ≡ ⌊ n + n /2⌋

From a8bb3074f5a8d7f87a3acdd888fb2ae054c43f67 Mon Sep 17 00:00:00 2001
From: Harrison Grodin <hgrodin@andrew.cmu.edu>
Date: Thu, 23 Mar 2023 22:38:57 -0400
Subject: [PATCH 013/316] Improve style

---
 src/Data/Nat/Properties.agda | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index b177412e00..10f523e966 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1048,7 +1048,7 @@ m^n≢0 m n = ≢-nonZero (≢-nonZero⁻¹ m ∘′ m^n≡0⇒m≡0 m n)
 
 ^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
 ^-monoˡ-≤ zero m≤o = s≤s z≤n
-^-monoˡ-≤ (suc n) {m} {o} m≤o = *-mono-≤ m≤o (^-monoˡ-≤ n m≤o)
+^-monoˡ-≤ (suc n) m≤o = *-mono-≤ m≤o (^-monoˡ-≤ n m≤o)
 
 ^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
 ^-monoʳ-≤ m {n} {o} z≤n = n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{m^n≢0 m o}})
@@ -1056,12 +1056,12 @@ m^n≢0 m n = ≢-nonZero (≢-nonZero⁻¹ m ∘′ m^n≡0⇒m≡0 m n)
 
 ^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
 ^-monoˡ-< (suc zero) m<o = *-monoˡ-< 1 m<o
-^-monoˡ-< (suc (suc n)) m<o = *-mono-< m<o (^-monoˡ-< (suc n) m<o)
+^-monoˡ-< (suc n@(suc _)) m<o = *-mono-< m<o (^-monoˡ-< n m<o)
 
 ^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_
 ^-monoʳ-< m 1<m {.zero} {suc o} z<s =
   *-mono-≤ 1<m (n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{(m^n≢0 m o {{>-nonZero (<-trans z<s 1<m)}})}}))
-^-monoʳ-< m 1<m {suc n} (s<s n≤o) =
+^-monoʳ-< m 1<m {suc _} (s<s n≤o) =
   *-monoʳ-< m {{ >-nonZero (<-trans z<s 1<m) }} (^-monoʳ-< m 1<m n≤o)
 
 2^n>0 : ∀ (n : ℕ) → 2 ^ n > 0

From 3b0669e3d1168f8a7177178432a28c3181730226 Mon Sep 17 00:00:00 2001
From: Harrison Grodin <hgrodin@andrew.cmu.edu>
Date: Mon, 27 Mar 2023 15:44:09 -0400
Subject: [PATCH 014/316] =?UTF-8?q?Clean=20up=20`^-mono=CA=B3-<`=20and=20`?=
 =?UTF-8?q?2^n>0`=20by=20adding=20`m^n>0`?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
---
 CHANGELOG.md                 |  5 +++--
 src/Data/Nat/Properties.agda | 20 ++++++++++----------
 2 files changed, 13 insertions(+), 12 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index a43a87b867..5ec865051b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2083,13 +2083,14 @@ Other minor changes
 
   n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
 
+  m^n>0 : ∀ m .{{_ : NonZero m}} n → m ^ n > 0
+  2^n>0 : 2 ^ n > 0
+
   ^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
   ^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
   ^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
   ^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_
 
-  2^n>0 : 2 ^ n > 0
-
   n≡⌊n+n/2⌋ : n ≡ ⌊ n + n /2⌋
   n≡⌈n+n/2⌉ : n ≡ ⌈ n + n /2⌉
 
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 10f523e966..3ddb4b2f71 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1046,27 +1046,27 @@ m^n≡1⇒n≡0∨m≡1 m (suc n) eq = inj₂ (m*n≡1⇒m≡1 m (m ^ n) eq)
 m^n≢0 : ∀ m n .{{_ : NonZero m}} → NonZero (m ^ n)
 m^n≢0 m n = ≢-nonZero (≢-nonZero⁻¹ m ∘′ m^n≡0⇒m≡0 m n)
 
+m^n>0 : ∀ m .{{_ : NonZero m}} n → m ^ n > 0
+m^n>0 m n = >-nonZero⁻¹ (m ^ n) {{m^n≢0 m n}}
+
+2^n>0 : ∀ (n : ℕ) → 2 ^ n > 0
+2^n>0 = m^n>0 2
+
 ^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
 ^-monoˡ-≤ zero m≤o = s≤s z≤n
 ^-monoˡ-≤ (suc n) m≤o = *-mono-≤ m≤o (^-monoˡ-≤ n m≤o)
 
 ^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
-^-monoʳ-≤ m {n} {o} z≤n = n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{m^n≢0 m o}})
+^-monoʳ-≤ m {_} {o} z≤n = n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{m^n≢0 m o}})
 ^-monoʳ-≤ m (s≤s n≤o) = *-monoʳ-≤ m (^-monoʳ-≤ m n≤o)
 
 ^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
-^-monoˡ-< (suc zero) m<o = *-monoˡ-< 1 m<o
+^-monoˡ-< (suc zero)      m<o = *-monoˡ-< 1 m<o
 ^-monoˡ-< (suc n@(suc _)) m<o = *-mono-< m<o (^-monoˡ-< n m<o)
 
 ^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_
-^-monoʳ-< m 1<m {.zero} {suc o} z<s =
-  *-mono-≤ 1<m (n≢0⇒n>0 (≢-nonZero⁻¹ (m ^ o) {{(m^n≢0 m o {{>-nonZero (<-trans z<s 1<m)}})}}))
-^-monoʳ-< m 1<m {suc _} (s<s n≤o) =
-  *-monoʳ-< m {{ >-nonZero (<-trans z<s 1<m) }} (^-monoʳ-< m 1<m n≤o)
-
-2^n>0 : ∀ (n : ℕ) → 2 ^ n > 0
-2^n>0 zero = z<s
-2^n>0 (suc n) = ≤-trans (2^n>0 n) (m≤m+n (2 ^ n) ((2 ^ n) + zero))
+^-monoʳ-< m@(suc _) 1<m {zero}  {suc o} z<s       = *-mono-≤ 1<m (m^n>0 m o)
+^-monoʳ-< m@(suc _) 1<m {suc n} {suc o} (s<s n<o) = *-monoʳ-< m (^-monoʳ-< m 1<m n<o)
 
 ------------------------------------------------------------------------
 -- Properties of _⊓_ and _⊔_

From efa58ed96974bc34ce8b45f459b96ced7cd16ddd Mon Sep 17 00:00:00 2001
From: Harrison Grodin <hgrodin@andrew.cmu.edu>
Date: Mon, 27 Mar 2023 15:48:40 -0400
Subject: [PATCH 015/316] Remove `2^n>0`

---
 CHANGELOG.md                     | 1 -
 src/Data/Nat/Logarithm/Core.agda | 4 ++--
 src/Data/Nat/Properties.agda     | 3 ---
 3 files changed, 2 insertions(+), 6 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5ec865051b..75fc5f1393 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2084,7 +2084,6 @@ Other minor changes
   n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
 
   m^n>0 : ∀ m .{{_ : NonZero m}} n → m ^ n > 0
-  2^n>0 : 2 ^ n > 0
 
   ^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
   ^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
diff --git a/src/Data/Nat/Logarithm/Core.agda b/src/Data/Nat/Logarithm/Core.agda
index 63385c89c1..08321f5a0a 100644
--- a/src/Data/Nat/Logarithm/Core.agda
+++ b/src/Data/Nat/Logarithm/Core.agda
@@ -72,7 +72,7 @@ open import Data.Unit
 ⌊log2⌋2^n≡n : ∀ n {acc} → ⌊log2⌋ (2 ^ n) acc ≡ n
 ⌊log2⌋2^n≡n zero    = refl
 ⌊log2⌋2^n≡n (suc n) = begin
-  ⌊log2⌋ ((2 ^ n) + ((2 ^ n) + zero)) _ ≡⟨ ⌊log2⌋2*b≡1+⌊log2⌋b (2 ^ n) {{>-nonZero (2^n>0 n)}} ⟩
+  ⌊log2⌋ ((2 ^ n) + ((2 ^ n) + zero)) _ ≡⟨ ⌊log2⌋2*b≡1+⌊log2⌋b (2 ^ n) {{m^n≢0 2 n}} ⟩
   1 + ⌊log2⌋ (2 ^ n) (<-wellFounded _)  ≡⟨ cong suc (⌊log2⌋2^n≡n n) ⟩
   suc n                                 ∎
   where open ≡-Reasoning
@@ -116,7 +116,7 @@ open import Data.Unit
 ⌈log2⌉2^n≡n : ∀ n {acc} → ⌈log2⌉ (2 ^ n) acc ≡ n
 ⌈log2⌉2^n≡n zero    = refl
 ⌈log2⌉2^n≡n (suc n) = begin
-  ⌈log2⌉ ((2 ^ n) + ((2 ^ n) + zero)) _ ≡⟨ ⌈log2⌉2*n≡1+⌈log2⌉n (2 ^ n) {{>-nonZero (2^n>0 n)}} ⟩
+  ⌈log2⌉ ((2 ^ n) + ((2 ^ n) + zero)) _ ≡⟨ ⌈log2⌉2*n≡1+⌈log2⌉n (2 ^ n) {{m^n≢0 2 n}} ⟩
   1 + ⌈log2⌉ (2 ^ n) (<-wellFounded _)  ≡⟨ cong suc (⌈log2⌉2^n≡n n) ⟩
   suc n                                 ∎
   where open ≡-Reasoning
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 3ddb4b2f71..3d38c77d07 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1049,9 +1049,6 @@ m^n≢0 m n = ≢-nonZero (≢-nonZero⁻¹ m ∘′ m^n≡0⇒m≡0 m n)
 m^n>0 : ∀ m .{{_ : NonZero m}} n → m ^ n > 0
 m^n>0 m n = >-nonZero⁻¹ (m ^ n) {{m^n≢0 m n}}
 
-2^n>0 : ∀ (n : ℕ) → 2 ^ n > 0
-2^n>0 = m^n>0 2
-
 ^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
 ^-monoˡ-≤ zero m≤o = s≤s z≤n
 ^-monoˡ-≤ (suc n) m≤o = *-mono-≤ m≤o (^-monoˡ-≤ n m≤o)

From b814e054e0f86365a37857852b96a6f0a83f1d8f Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Fri, 14 Apr 2023 16:30:46 +0200
Subject: [PATCH 016/316] =?UTF-8?q?Move=20suc=E2=89=A20=20slightly=20and?=
 =?UTF-8?q?=20add=20suc-injective=20to=20Data.Nat.Binary.Properties?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                        |  4 ++++
 src/Data/Nat/Binary/Properties.agda | 17 ++++++++++++-----
 2 files changed, 16 insertions(+), 5 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 75fc5f1393..8d3612bcbf 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2040,6 +2040,10 @@ Other minor changes
   +-*-rawSemiring     : RawSemiring 0ℓ 0ℓ
   ```
 
+* Added a new proof to `Data.Nat.Binary.Properties`:
+  ```agda
+  suc-injective : Injective _≡_ _≡_ suc
+  ```
 
 * Added new definitions and proofs to `Data.Nat.Primality`:
   ```agda
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 5cf86313c3..3ea094ec00 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -1309,6 +1309,18 @@ double-suc x = begin
 -- Properties of suc
 ------------------------------------------------------------------------
 
+suc≢0 : suc x ≢ zero
+suc≢0 {zero}     ()
+suc≢0 {2[1+ _ ]} ()
+suc≢0 {1+[2 _ ]} ()
+
+suc-injective : Injective _≡_ _≡_ suc
+suc-injective {zero} {zero} p = refl
+suc-injective {zero} {2[1+ y ]} p = contradiction 1+[2 p ]-injective (suc≢0 ∘ sym)
+suc-injective {2[1+ x ]} {zero} p = contradiction 1+[2 p ]-injective suc≢0
+suc-injective {2[1+ x ]} {2[1+ y ]} p = cong 2[1+_] (suc-injective 1+[2 p ]-injective)
+suc-injective {1+[2 x ]} {1+[2 y ]} refl = refl
+
 2[1+_]-double-suc : 2[1+_] ≗ double ∘ suc
 2[1+_]-double-suc zero     = refl
 2[1+_]-double-suc 2[1+ x ] = cong 2[1+_] (2[1+_]-double-suc x)
@@ -1325,11 +1337,6 @@ double-suc x = begin
   suc (double 1+[2 x ])   ∎
   where open ≡-Reasoning;  2x = double x
 
-suc≢0 : suc x ≢ zero
-suc≢0 {zero}     ()
-suc≢0 {2[1+ _ ]} ()
-suc≢0 {1+[2 _ ]} ()
-
 x+suc[y]≡suc[x]+y : ∀ x y → x + suc y ≡ suc x + y
 x+suc[y]≡suc[x]+y x y = begin
   x + suc y     ≡⟨ +-comm x _ ⟩

From c9b38bf9a9ee9a5f689ada1e608a0b099bdca10d Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Fri, 14 Apr 2023 16:38:37 +0200
Subject: [PATCH 017/316] Add conical properties for and and or

---
 CHANGELOG.md                  |  6 ++++++
 src/Data/Bool/Properties.agda | 18 ++++++++++++++++++
 2 files changed, 24 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8d3612bcbf..c3a824b156 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1835,6 +1835,12 @@ Other minor changes
 * Added new proofs in `Data.Bool.Properties`:
   ```agda
   <-wellFounded : WellFounded _<_
+  ∨-conicalˡ : LeftConical false _∨_
+  ∨-conicalʳ : RightConical false _∨_
+  ∨-conical : Conical false _∨_
+  ∧-conicalˡ : LeftConical true _∧_
+  ∧-conicalʳ : RightConical true _∧_
+  ∧-conical : Conical true _∧_
   ```
 
 * Added new functions in `Data.Fin.Base`:
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index beb99befd8..fe232885eb 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -273,6 +273,15 @@ true  <? _     = no  (λ())
 ∨-sel false y = inj₂ refl
 ∨-sel true y  = inj₁ refl
 
+∨-conicalˡ : LeftConical false _∨_
+∨-conicalˡ false false _ = refl
+
+∨-conicalʳ : RightConical false _∨_
+∨-conicalʳ false false _ = refl
+
+∨-conical : Conical false _∨_
+∨-conical = ∨-conicalˡ , ∨-conicalʳ
+
 ∨-isMagma : IsMagma _∨_
 ∨-isMagma = record
   { isEquivalence = isEquivalence
@@ -397,6 +406,15 @@ true  <? _     = no  (λ())
 ∧-sel false y = inj₁ refl
 ∧-sel true y  = inj₂ refl
 
+∧-conicalˡ : LeftConical true _∧_
+∧-conicalˡ true true _ = refl
+
+∧-conicalʳ : RightConical true _∧_
+∧-conicalʳ true true _ = refl
+
+∧-conical : Conical true _∧_
+∧-conical = ∧-conicalˡ , ∧-conicalʳ
+
 ∧-distribˡ-∨ : _∧_ DistributesOverˡ _∨_
 ∧-distribˡ-∨ true  y z = refl
 ∧-distribˡ-∨ false y z = refl

From e473bea6a57647a76974c2ef4aae4a910654b7b4 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 17 Apr 2023 14:36:02 +0100
Subject: [PATCH 018/316] Revert "Fix two whitespace violations (#1922)"

This reverts commit efc0230019937a694c87db0e3b49e45cc56a238d.
---
 CHANGELOG/v1.7.2.md                             | 2 +-
 src/Data/List/Membership/Setoid/Properties.agda | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG/v1.7.2.md b/CHANGELOG/v1.7.2.md
index 8ddea63b4d..0183ec654f 100644
--- a/CHANGELOG/v1.7.2.md
+++ b/CHANGELOG/v1.7.2.md
@@ -5,5 +5,5 @@ The library has been tested using Agda 2.6.3.
 
 * In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
   the `--without-K` flag now use the `--cubical-compatible` flag instead.
-
+  
 * Updated the code using `primFloatToWord64` - the library API has remained unchanged.
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 919bf19ba6..df260f2a17 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -122,7 +122,7 @@ module _ (S : Setoid c ℓ) where
                     length (mapWith∈ xs f) ≡ length xs
   length-mapWith∈ []       = P.refl
   length-mapWith∈ (x ∷ xs) = P.cong suc (length-mapWith∈ xs)
-
+  
   mapWith∈-id : ∀ xs → mapWith∈ xs (λ {x} _ → x) ≡ xs
   mapWith∈-id []       = P.refl
   mapWith∈-id (x ∷ xs) = P.cong (x ∷_) (mapWith∈-id xs)

From 01d338e5a5fb3d6b9a91d839df36eb4b2bb98a65 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 17 Apr 2023 14:40:09 +0100
Subject: [PATCH 019/316] Revert "punchOut preserves ordering (#1913)"

This reverts commit 692b6fa0a5b667b8117efa5d4ab278d680a7c79f.
---
 CHANGELOG.md                 |  5 -----
 src/Data/Fin/Properties.agda | 26 --------------------------
 2 files changed, 31 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index c3a824b156..e9138f2585 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1899,11 +1899,6 @@ Other minor changes
 
   ℕ-ℕ≡toℕ‿ℕ-         : n ℕ-ℕ i ≡ toℕ (n ℕ- i)
 
-  punchIn-mono-≤     : ∀ i (j k : Fin n) → j ≤ k → punchIn i j ≤ punchIn i k
-  punchIn-cancel-≤   : ∀ i (j k : Fin n) → punchIn i j ≤ punchIn i k → j ≤ k
-  punchOut-mono-≤    : (i≢j : i ≢ j) (i≢k : i ≢ k) → j ≤ k → punchOut i≢j ≤ punchOut i≢k
-  punchOut-cancel-≤  : (i≢j : i ≢ j) (i≢k : i ≢ k) → punchOut i≢j ≤ punchOut i≢k → j ≤ k
-
   lower₁-injective   : lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
   pinch-injective    : suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k
 
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 9089102b29..232a44a7bb 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -802,16 +802,6 @@ punchIn-injective (suc i) (suc j) (suc k) ↑j+1≡↑k+1 =
 punchInᵢ≢i : ∀ i (j : Fin n) → punchIn i j ≢ i
 punchInᵢ≢i (suc i) (suc j) = punchInᵢ≢i i j ∘ suc-injective
 
-punchIn-mono-≤ : ∀ i (j k : Fin n) → j ≤ k → punchIn i j ≤ punchIn i k
-punchIn-mono-≤ zero    _        _      j≤k       = s≤s j≤k
-punchIn-mono-≤ (suc _) zero    _       z≤n       = z≤n
-punchIn-mono-≤ (suc i) (suc j) (suc k) (s≤s j≤k) = s≤s (punchIn-mono-≤ i j k j≤k)
-
-punchIn-cancel-≤ : ∀ i (j k : Fin n) → punchIn i j ≤ punchIn i k → j ≤ k
-punchIn-cancel-≤ zero    _       _       (s≤s j≤k)   = j≤k
-punchIn-cancel-≤ (suc _) zero    _       z≤n         = z≤n
-punchIn-cancel-≤ (suc i) (suc j) (suc k) (s≤s ↑j≤↑k) = s≤s (punchIn-cancel-≤ i j k ↑j≤↑k)
-
 ------------------------------------------------------------------------
 -- punchOut
 ------------------------------------------------------------------------
@@ -845,22 +835,6 @@ punchOut-injective {suc n} {suc i}  {zero}  {zero}  _   _    _    = refl
 punchOut-injective {suc n} {suc i}  {suc j} {suc k} i≢j i≢k pⱼ≡pₖ =
   cong suc (punchOut-injective (i≢j ∘ cong suc) (i≢k ∘ cong suc) (suc-injective pⱼ≡pₖ))
 
-punchOut-mono-≤ : ∀ {i j k : Fin (suc n)} (i≢j : i ≢ j) (i≢k : i ≢ k) →
-                  j ≤ k → punchOut i≢j ≤ punchOut i≢k
-punchOut-mono-≤ {_    } {zero } {zero } {_    } 0≢0 _   z≤n       = contradiction refl 0≢0
-punchOut-mono-≤ {_    } {zero } {suc _} {suc _} _   _   (s≤s j≤k) = j≤k
-punchOut-mono-≤ {suc _} {suc _} {zero } {_    } _   _   z≤n       = z≤n
-punchOut-mono-≤ {suc _} {suc _} {suc _} {suc _} i≢j i≢k (s≤s j≤k) = s≤s (punchOut-mono-≤ (i≢j ∘ cong suc) (i≢k ∘ cong suc) j≤k)
-
-punchOut-cancel-≤ : ∀ {i j k : Fin (suc n)} (i≢j : i ≢ j) (i≢k : i ≢ k) →
-                    punchOut i≢j ≤ punchOut i≢k → j ≤ k
-punchOut-cancel-≤ {_    } {zero } {zero } {_    } 0≢0 _   _           = contradiction refl 0≢0
-punchOut-cancel-≤ {_    } {zero } {suc _} {zero } _   0≢0 _           = contradiction refl 0≢0
-punchOut-cancel-≤ {suc _} {zero } {suc _} {suc _} _   _   pⱼ≤pₖ       = s≤s pⱼ≤pₖ
-punchOut-cancel-≤ {_    } {suc _} {zero } {_    } _   _   _           = z≤n
-punchOut-cancel-≤ {suc _} {suc _} {suc _} {zero } _   _   ()
-punchOut-cancel-≤ {suc _} {suc _} {suc _} {suc _} i≢j i≢k (s≤s pⱼ≤pₖ) = s≤s (punchOut-cancel-≤ (i≢j ∘ cong suc) (i≢k ∘ cong suc) pⱼ≤pₖ)
-
 punchIn-punchOut : ∀ {i j : Fin (suc n)} (i≢j : i ≢ j) →
                    punchIn i (punchOut i≢j) ≡ j
 punchIn-punchOut {_}     {zero}   {zero}  0≢0 = contradiction refl 0≢0

From 6aea1751cbbbe3b6103116e6fcaf8652b2262264 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 17 Apr 2023 14:44:40 +0100
Subject: [PATCH 020/316] Revert "opposite of a `Ring` [clean version of pr
 #1900] (#1910)"

This reverts commit 7772dee7f74e35d398ac960a60633787ed716c4f.
---
 CHANGELOG.md                       |  35 ----
 src/Algebra/Construct/Flip/Op.agda | 269 +++++------------------------
 2 files changed, 39 insertions(+), 265 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e9138f2585..fe243a6110 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1664,41 +1664,6 @@ Other minor changes
   moufangLoop : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
  ```
 
-* Added new functions and proofs to `Algebra.Construct.Flip.Op`:
-  ```agda
-  zero : Zero ≈ ε ∙ → Zero ≈ ε (flip ∙)
-  distributes : (≈ DistributesOver ∙) + → (≈ DistributesOver (flip ∙)) +
-  isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero + * 0# 1# →
-                                      IsSemiringWithoutAnnihilatingZero + (flip *) 0# 1#
-  isSemiring : IsSemiring + * 0# 1# → IsSemiring + (flip *) 0# 1#
-  isCommutativeSemiring : IsCommutativeSemiring + * 0# 1# →
-                          IsCommutativeSemiring + (flip *) 0# 1#
-  isCancellativeCommutativeSemiring : IsCancellativeCommutativeSemiring + * 0# 1# →
-                                      IsCancellativeCommutativeSemiring + (flip *) 0# 1#
-  isIdempotentSemiring : IsIdempotentSemiring + * 0# 1# →
-                         IsIdempotentSemiring + (flip *) 0# 1#
-  isQuasiring : IsQuasiring + * 0# 1# → IsQuasiring + (flip *) 0# 1#
-  isRingWithoutOne : IsRingWithoutOne + * - 0# → IsRingWithoutOne + (flip *) - 0#
-  isNonAssociativeRing : IsNonAssociativeRing + * - 0# 1# →
-                         IsNonAssociativeRing + (flip *) - 0# 1#
-  isRing : IsRing ≈ + * - 0# 1# → IsRing ≈ + (flip *) - 0# 1#
-  isNearring : IsNearring + * 0# 1# - → IsNearring + (flip *) 0# 1# -
-  isCommutativeRing : IsCommutativeRing + * - 0# 1# →
-                      IsCommutativeRing + (flip *) - 0# 1#
-  semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero a ℓ →
-                                    SemiringWithoutAnnihilatingZero a ℓ
-  commutativeSemiring : CommutativeSemiring a ℓ → CommutativeSemiring a ℓ
-  cancellativeCommutativeSemiring : CancellativeCommutativeSemiring a ℓ →
-                                  CancellativeCommutativeSemiring a ℓ
-  idempotentSemiring : IdempotentSemiring a ℓ → IdempotentSemiring a ℓ
-  quasiring : Quasiring a ℓ → Quasiring a ℓ
-  ringWithoutOne : RingWithoutOne a ℓ → RingWithoutOne a ℓ
-  nonAssociativeRing : NonAssociativeRing a ℓ → NonAssociativeRing a ℓ
-  nearring : Nearring a ℓ → Nearring a ℓ
-  ring : Ring a ℓ → Ring a ℓ
-  commutativeRing : CommutativeRing a ℓ → CommutativeRing a ℓ
-  ```
-
 * Added new definition to `Algebra.Definitions`:
   ```agda
   SelfInverse : Op₁ A → Set _
diff --git a/src/Algebra/Construct/Flip/Op.agda b/src/Algebra/Construct/Flip/Op.agda
index f5d86e7714..74ae5730bc 100644
--- a/src/Algebra/Construct/Flip/Op.agda
+++ b/src/Algebra/Construct/Flip/Op.agda
@@ -9,11 +9,7 @@
 
 module Algebra.Construct.Flip.Op where
 
-open import Algebra.Core
-open import Algebra.Bundles
-import Algebra.Definitions as Def
-import Algebra.Structures as Str
-import Algebra.Consequences.Setoid as Consequences
+open import Algebra
 import Data.Product as Prod
 import Data.Sum as Sum
 open import Function.Base using (flip)
@@ -37,272 +33,136 @@ preserves₂ : (∼ ≈ ≋ : Rel A ℓ) →
              ∙ Preserves₂ ∼ ⟶ ≈ ⟶ ≋ → (flip ∙) Preserves₂ ≈ ⟶ ∼ ⟶ ≋
 preserves₂ _ _ _ pres = flip pres
 
-module ∙-Properties (≈ : Rel A ℓ) (∙ : Op₂ A) where
+module _ (≈ : Rel A ℓ) (∙ : Op₂ A) where
 
-  open Def ≈
-
-  associative : Symmetric ≈ → Associative ∙ → Associative (flip ∙)
+  associative : Symmetric ≈ → Associative ≈ ∙ → Associative ≈ (flip ∙)
   associative sym assoc x y z = sym (assoc z y x)
 
-  identity : Identity ε ∙ → Identity ε (flip ∙)
+  identity : Identity ≈ ε ∙ → Identity ≈ ε (flip ∙)
   identity id = Prod.swap id
 
-  commutative : Commutative ∙ → Commutative (flip ∙)
+  commutative : Commutative ≈ ∙ → Commutative ≈ (flip ∙)
   commutative comm = flip comm
 
-  selective : Selective ∙ → Selective (flip ∙)
+  selective : Selective ≈ ∙ → Selective ≈ (flip ∙)
   selective sel x y = Sum.swap (sel y x)
 
-  idempotent : Idempotent ∙ → Idempotent (flip ∙)
+  idempotent : Idempotent ≈ ∙ → Idempotent ≈ (flip ∙)
   idempotent idem = idem
 
-  inverse : Inverse ε ⁻¹ ∙ → Inverse ε ⁻¹ (flip ∙)
+  inverse : Inverse ≈ ε ⁻¹ ∙ → Inverse ≈ ε ⁻¹ (flip ∙)
   inverse inv = Prod.swap inv
 
-  zero : Zero ε ∙ → Zero ε (flip ∙)
-  zero zer = Prod.swap zer
-
-module *-Properties (≈ : Rel A ℓ) (* + : Op₂ A) where
-
-  open Def ≈
-
-  distributes : * DistributesOver + → (flip *) DistributesOver +
-  distributes distrib = Prod.swap distrib
-
 ------------------------------------------------------------------------
 -- Structures
 
 module _ {≈ : Rel A ℓ} {∙ : Op₂ A} where
 
-  open Def ≈
-  open Str ≈
-  open ∙-Properties ≈ ∙
-
-  isMagma : IsMagma ∙ → IsMagma (flip ∙)
+  isMagma : IsMagma ≈ ∙ → IsMagma ≈ (flip ∙)
   isMagma m = record
     { isEquivalence = isEquivalence
     ; ∙-cong        = preserves₂ ≈ ≈ ≈ ∙-cong
     }
     where open IsMagma m
 
-  isSelectiveMagma : IsSelectiveMagma ∙ → IsSelectiveMagma (flip ∙)
+  isSelectiveMagma : IsSelectiveMagma ≈ ∙ → IsSelectiveMagma ≈ (flip ∙)
   isSelectiveMagma m = record
     { isMagma = isMagma m.isMagma
-    ; sel     = selective m.sel
+    ; sel     = selective ≈ ∙ m.sel
     }
     where module m = IsSelectiveMagma m
 
-  isCommutativeMagma : IsCommutativeMagma ∙ → IsCommutativeMagma (flip ∙)
+  isCommutativeMagma : IsCommutativeMagma ≈ ∙ → IsCommutativeMagma ≈ (flip ∙)
   isCommutativeMagma m = record
     { isMagma = isMagma m.isMagma
-    ; comm    = commutative m.comm
+    ; comm    = commutative ≈ ∙ m.comm
     }
     where module m = IsCommutativeMagma m
 
-  isSemigroup : IsSemigroup ∙ → IsSemigroup (flip ∙)
+  isSemigroup : IsSemigroup ≈ ∙ → IsSemigroup ≈ (flip ∙)
   isSemigroup s = record
     { isMagma = isMagma s.isMagma
-    ; assoc   = associative s.sym s.assoc
+    ; assoc   = associative ≈ ∙ s.sym s.assoc
     }
     where module s = IsSemigroup s
 
-  isBand : IsBand ∙ → IsBand (flip ∙)
+  isBand : IsBand ≈ ∙ → IsBand ≈ (flip ∙)
   isBand b = record
     { isSemigroup = isSemigroup b.isSemigroup
     ; idem        = b.idem
     }
     where module b = IsBand b
 
-  isCommutativeSemigroup : IsCommutativeSemigroup ∙ →
-                           IsCommutativeSemigroup (flip ∙)
+  isCommutativeSemigroup : IsCommutativeSemigroup ≈ ∙ →
+                           IsCommutativeSemigroup ≈ (flip ∙)
   isCommutativeSemigroup s = record
     { isSemigroup = isSemigroup s.isSemigroup
-    ; comm        = commutative s.comm
+    ; comm        = commutative ≈ ∙ s.comm
     }
     where module s = IsCommutativeSemigroup s
 
-  isUnitalMagma : IsUnitalMagma ∙ ε → IsUnitalMagma (flip ∙) ε
+  isUnitalMagma : IsUnitalMagma ≈ ∙ ε → IsUnitalMagma ≈ (flip ∙) ε
   isUnitalMagma m = record
     { isMagma  = isMagma m.isMagma
-    ; identity = identity m.identity
+    ; identity = identity ≈ ∙ m.identity
     }
     where module m = IsUnitalMagma m
 
-  isMonoid : IsMonoid ∙ ε → IsMonoid (flip ∙) ε
+  isMonoid : IsMonoid ≈ ∙ ε → IsMonoid ≈ (flip ∙) ε
   isMonoid m = record
     { isSemigroup = isSemigroup m.isSemigroup
-    ; identity    = identity m.identity
+    ; identity    = identity ≈ ∙ m.identity
     }
     where module m = IsMonoid m
 
-  isCommutativeMonoid : IsCommutativeMonoid ∙ ε →
-                        IsCommutativeMonoid (flip ∙) ε
+  isCommutativeMonoid : IsCommutativeMonoid ≈ ∙ ε →
+                        IsCommutativeMonoid ≈ (flip ∙) ε
   isCommutativeMonoid m = record
     { isMonoid = isMonoid m.isMonoid
-    ; comm     = commutative m.comm
+    ; comm     = commutative ≈ ∙ m.comm
     }
     where module m = IsCommutativeMonoid m
 
-  isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid ∙ ε →
-                                  IsIdempotentCommutativeMonoid (flip ∙) ε
+  isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid ≈ ∙ ε →
+                                  IsIdempotentCommutativeMonoid ≈ (flip ∙) ε
   isIdempotentCommutativeMonoid m = record
     { isCommutativeMonoid = isCommutativeMonoid m.isCommutativeMonoid
-    ; idem                = idempotent m.idem
+    ; idem                = idempotent ≈ ∙ m.idem
     }
     where module m = IsIdempotentCommutativeMonoid m
 
-  isInvertibleMagma : IsInvertibleMagma ∙ ε ⁻¹ →
-                      IsInvertibleMagma (flip ∙) ε ⁻¹
+  isInvertibleMagma : IsInvertibleMagma ≈ ∙ ε ⁻¹ →
+                      IsInvertibleMagma ≈ (flip ∙) ε ⁻¹
   isInvertibleMagma m = record
     { isMagma = isMagma m.isMagma
-    ; inverse = inverse m.inverse
+    ; inverse = inverse ≈ ∙ m.inverse
     ; ⁻¹-cong = m.⁻¹-cong
     }
     where module m = IsInvertibleMagma m
 
-  isInvertibleUnitalMagma : IsInvertibleUnitalMagma ∙ ε ⁻¹ →
-                            IsInvertibleUnitalMagma (flip ∙) ε ⁻¹
+  isInvertibleUnitalMagma : IsInvertibleUnitalMagma ≈ ∙ ε ⁻¹ →
+                            IsInvertibleUnitalMagma ≈ (flip ∙) ε ⁻¹
   isInvertibleUnitalMagma m = record
     { isInvertibleMagma = isInvertibleMagma m.isInvertibleMagma
-    ; identity          = identity m.identity
+    ; identity          = identity ≈ ∙ m.identity
     }
     where module m = IsInvertibleUnitalMagma m
 
-  isGroup : IsGroup ∙ ε ⁻¹ → IsGroup (flip ∙) ε ⁻¹
+  isGroup : IsGroup ≈ ∙ ε ⁻¹ → IsGroup ≈ (flip ∙) ε ⁻¹
   isGroup g = record
     { isMonoid = isMonoid g.isMonoid
-    ; inverse  = inverse g.inverse
+    ; inverse  = inverse ≈ ∙ g.inverse
     ; ⁻¹-cong  = g.⁻¹-cong
     }
     where module g = IsGroup g
 
-  isAbelianGroup : IsAbelianGroup ∙ ε ⁻¹ → IsAbelianGroup (flip ∙) ε ⁻¹
+  isAbelianGroup : IsAbelianGroup ≈ ∙ ε ⁻¹ → IsAbelianGroup ≈ (flip ∙) ε ⁻¹
   isAbelianGroup g = record
     { isGroup = isGroup g.isGroup
-    ; comm    = commutative g.comm
+    ; comm    = commutative ≈ ∙ g.comm
     }
     where module g = IsAbelianGroup g
 
-module _ {≈ : Rel A ℓ} {+ * : Op₂ A} {0# 1# : A} where
-
-  open Str ≈
-  open ∙-Properties ≈ *
-  open *-Properties ≈ * +
-
-  isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero + * 0# 1# →
-                                      IsSemiringWithoutAnnihilatingZero + (flip *) 0# 1#
-  isSemiringWithoutAnnihilatingZero r = record
-    { +-isCommutativeMonoid = r.+-isCommutativeMonoid
-    ; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
-    ; *-assoc = associative r.sym r.*-assoc
-    ; *-identity = identity r.*-identity
-    ; distrib = distributes r.distrib
-    }
-    where module r = IsSemiringWithoutAnnihilatingZero r
-
-  isSemiring : IsSemiring + * 0# 1# → IsSemiring + (flip *) 0# 1#
-  isSemiring r = record
-    { isSemiringWithoutAnnihilatingZero = isSemiringWithoutAnnihilatingZero r.isSemiringWithoutAnnihilatingZero
-    ; zero = zero r.zero
-    }
-    where module r = IsSemiring r
-
-  isCommutativeSemiring : IsCommutativeSemiring + * 0# 1# →
-                          IsCommutativeSemiring + (flip *) 0# 1#
-  isCommutativeSemiring r = record
-    { isSemiring = isSemiring r.isSemiring
-    ; *-comm = commutative r.*-comm
-    }
-    where module r = IsCommutativeSemiring r
-
-  isCancellativeCommutativeSemiring : IsCancellativeCommutativeSemiring + * 0# 1# →
-                                      IsCancellativeCommutativeSemiring + (flip *) 0# 1#
-  isCancellativeCommutativeSemiring r = record
-    { isCommutativeSemiring = isCommutativeSemiring r.isCommutativeSemiring
-    ; *-cancelˡ-nonZero = Consequences.comm+almostCancelˡ⇒almostCancelʳ r.setoid r.*-comm r.*-cancelˡ-nonZero
-    }
-    where module r = IsCancellativeCommutativeSemiring r
-
-  isIdempotentSemiring : IsIdempotentSemiring + * 0# 1# →
-                         IsIdempotentSemiring + (flip *) 0# 1#
-  isIdempotentSemiring r = record
-    { isSemiring = isSemiring r.isSemiring
-    ; +-idem = r.+-idem
-    }
-    where module r = IsIdempotentSemiring r
-
-  isQuasiring : IsQuasiring + * 0# 1# → IsQuasiring + (flip *) 0# 1#
-  isQuasiring r = record
-    { +-isMonoid = r.+-isMonoid
-    ; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
-    ; *-assoc = associative r.sym r.*-assoc
-    ; *-identity = identity r.*-identity
-    ; distrib = distributes r.distrib
-    ; zero = zero r.zero
-    }
-    where module r = IsQuasiring r
-
-module _ {≈ : Rel A ℓ} {+ * : Op₂ A} { - : Op₁ A} {0# : A} where
-
-  open Str ≈
-  open ∙-Properties ≈ *
-  open *-Properties ≈ * +
-
-  isRingWithoutOne : IsRingWithoutOne + * - 0# → IsRingWithoutOne + (flip *) - 0#
-  isRingWithoutOne r = record
-    { +-isAbelianGroup = r.+-isAbelianGroup
-    ; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
-    ; *-assoc = associative r.sym r.*-assoc
-    ; distrib = distributes r.distrib
-    ; zero = zero r.zero
-    }
-    where module r = IsRingWithoutOne r
-
-module _ {≈ : Rel A ℓ} {+ * : Op₂ A} { - : Op₁ A} {0# 1# : A} where
-
-  open Str ≈
-  open ∙-Properties ≈ *
-  open *-Properties ≈ * +
-
-  isNonAssociativeRing : IsNonAssociativeRing + * - 0# 1# →
-                         IsNonAssociativeRing + (flip *) - 0# 1#
-  isNonAssociativeRing r = record
-    { +-isAbelianGroup = r.+-isAbelianGroup
-    ; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
-    ; identity = identity r.identity
-    ; distrib = distributes r.distrib
-    ; zero = zero r.zero
-    }
-    where module r = IsNonAssociativeRing r
-
-  isNearring : IsNearring + * 0# 1# - → IsNearring + (flip *) 0# 1# -
-  isNearring r = record
-    { isQuasiring = isQuasiring r.isQuasiring
-    ; +-inverse = r.+-inverse
-    ; ⁻¹-cong = r.⁻¹-cong
-    }
-    where module r = IsNearring r
-
-  isRing : IsRing + * - 0# 1# → IsRing + (flip *) - 0# 1#
-  isRing r = record
-    { +-isAbelianGroup = r.+-isAbelianGroup
-    ; *-cong = preserves₂ ≈ ≈ ≈ r.*-cong
-    ; *-assoc = associative r.sym r.*-assoc
-    ; *-identity = identity r.*-identity
-    ; distrib = distributes r.distrib
-    ; zero = zero r.zero
-    }
-    where module r = IsRing r
-
-  isCommutativeRing : IsCommutativeRing + * - 0# 1# →
-                      IsCommutativeRing + (flip *) - 0# 1#
-  isCommutativeRing r = record
-    { isRing = isRing r.isRing
-    ; *-comm = commutative r.*-comm
-    }
-    where module r = IsCommutativeRing r
-
-
 ------------------------------------------------------------------------
 -- Bundles
 
@@ -379,54 +239,3 @@ group g = record { isGroup = isGroup g.isGroup }
 abelianGroup : AbelianGroup a ℓ → AbelianGroup a ℓ
 abelianGroup g = record { isAbelianGroup = isAbelianGroup g.isAbelianGroup }
   where module g = AbelianGroup g
-
-semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero a ℓ →
-                                  SemiringWithoutAnnihilatingZero a ℓ
-semiringWithoutAnnihilatingZero r = record
-  { isSemiringWithoutAnnihilatingZero = isSemiringWithoutAnnihilatingZero r.isSemiringWithoutAnnihilatingZero }
-  where module r = SemiringWithoutAnnihilatingZero r
-
-semiring : Semiring a ℓ → Semiring a ℓ
-semiring r = record { isSemiring = isSemiring r.isSemiring }
-  where module r = Semiring r
-
-commutativeSemiring : CommutativeSemiring a ℓ → CommutativeSemiring a ℓ
-commutativeSemiring r = record
-  { isCommutativeSemiring = isCommutativeSemiring r.isCommutativeSemiring }
-  where module r = CommutativeSemiring r
-
-cancellativeCommutativeSemiring : CancellativeCommutativeSemiring a ℓ →
-                                  CancellativeCommutativeSemiring a ℓ
-cancellativeCommutativeSemiring r = record
-  { isCancellativeCommutativeSemiring = isCancellativeCommutativeSemiring r.isCancellativeCommutativeSemiring }
-  where module r = CancellativeCommutativeSemiring r
-
-idempotentSemiring : IdempotentSemiring a ℓ → IdempotentSemiring a ℓ
-idempotentSemiring r = record
-  { isIdempotentSemiring = isIdempotentSemiring r.isIdempotentSemiring }
-  where module r = IdempotentSemiring r
-
-quasiring : Quasiring a ℓ → Quasiring a ℓ
-quasiring r = record { isQuasiring = isQuasiring r.isQuasiring }
-  where module r = Quasiring r
-
-ringWithoutOne : RingWithoutOne a ℓ → RingWithoutOne a ℓ
-ringWithoutOne r = record { isRingWithoutOne = isRingWithoutOne r.isRingWithoutOne }
-  where module r = RingWithoutOne r
-
-nonAssociativeRing : NonAssociativeRing a ℓ → NonAssociativeRing a ℓ
-nonAssociativeRing r = record
-  { isNonAssociativeRing = isNonAssociativeRing r.isNonAssociativeRing }
-  where module r = NonAssociativeRing r
-
-nearring : Nearring a ℓ → Nearring a ℓ
-nearring r = record { isNearring = isNearring r.isNearring }
-  where module r = Nearring r
-
-ring : Ring a ℓ → Ring a ℓ
-ring r = record { isRing = isRing r.isRing }
-  where module r = Ring r
-
-commutativeRing : CommutativeRing a ℓ → CommutativeRing a ℓ
-commutativeRing r = record { isCommutativeRing = isCommutativeRing r.isCommutativeRing }
-  where module r = CommutativeRing r

From 2f1ff01f6738cd67015fb524c9fad04aa1e383a3 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 17 Apr 2023 14:45:31 +0100
Subject: [PATCH 021/316] Revert "Wellfounded proof for sum relations (#1920)"

This reverts commit da073f1719cd14b36d0568544219310a998d267c.
---
 CHANGELOG.md                                |  9 ---------
 src/Data/Sum/Relation/Binary/LeftOrder.agda | 15 ---------------
 src/Data/Sum/Relation/Binary/Pointwise.agda | 14 --------------
 3 files changed, 38 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index fe243a6110..76ca38cd42 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2291,15 +2291,6 @@ Other minor changes
     map f (map g h) ∘ assocʳ ≗ assocʳ ∘ map (map f g) h
   ```
 
-* Adden new proof in `Data.Sum.Relation.Binary.LeftOrder` :
-  ```
-  ⊎-<-wellFounded : WellFounded ∼₁ → WellFounded ∼₂ → WellFounded (∼₁ ⊎-< ∼₂)
-  ```
-* Adden new proof in `Data.Sum.Relation.Binary.Pointwise` :
-  ```
-  ⊎-wellFounded : WellFounded ∼₁ → WellFounded ∼₂ → WellFounded (Pointwise ∼₁ ∼₂)
-  ```
-
 * Made `Map` public in `Data.Tree.AVL.IndexedMap`
 
 * Added new definitions in `Data.Vec.Base`:
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index a5d0327af2..063ba49bb7 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -14,7 +14,6 @@ open import Data.Sum.Relation.Binary.Pointwise as PW
 open import Data.Product
 open import Data.Empty
 open import Function
-open import Induction.WellFounded
 open import Level
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
@@ -79,20 +78,6 @@ module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
   ⊎-<-decidable dec₁ dec₂ (inj₂ x) (inj₁ y) = no λ()
   ⊎-<-decidable dec₁ dec₂ (inj₂ x) (inj₂ y) = Dec.map′ ₂∼₂ drop-inj₂ (dec₂ x y)
 
-  ⊎-<-wellFounded : WellFounded ∼₁ → WellFounded ∼₂ → WellFounded (∼₁ ⊎-< ∼₂)
-  ⊎-<-wellFounded wf₁ wf₂ x = acc (⊎-<-acc x)
-    where
-    ⊎-<-acc₁ : ∀ {x} → Acc ∼₁ x → WfRec (∼₁ ⊎-< ∼₂) (Acc (∼₁ ⊎-< ∼₂)) (inj₁ x)
-    ⊎-<-acc₁ (acc rec) (inj₁ y) (₁∼₁ x∼₁y) = acc (⊎-<-acc₁ (rec y x∼₁y))
-
-    ⊎-<-acc₂ : ∀ {x} → Acc ∼₂ x → WfRec (∼₁ ⊎-< ∼₂) (Acc (∼₁ ⊎-< ∼₂)) (inj₂ x)
-    ⊎-<-acc₂ (acc rec) (inj₁ y) ₁∼₂ = acc (⊎-<-acc₁ (wf₁ y))
-    ⊎-<-acc₂ (acc rec) (inj₂ y) (₂∼₂ x∼₂y) = acc (⊎-<-acc₂ (rec y x∼₂y))
-
-    ⊎-<-acc  : ∀ x → WfRec (∼₁ ⊎-< ∼₂) (Acc (∼₁ ⊎-< ∼₂)) x
-    ⊎-<-acc (inj₁ x) = ⊎-<-acc₁ (wf₁ x)
-    ⊎-<-acc (inj₂ x) = ⊎-<-acc₂ (wf₂ x)
-
 module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
          {ℓ₁ ℓ₂} {∼₁ : Rel A₁ ℓ₁} {≈₁ : Rel A₁ ℓ₂}
          {ℓ₃ ℓ₄} {∼₂ : Rel A₂ ℓ₃} {≈₂ : Rel A₂ ℓ₄}
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index ac196828c6..d5ec85b943 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -11,7 +11,6 @@ module Data.Sum.Relation.Binary.Pointwise where
 open import Data.Product using (_,_)
 open import Data.Sum.Base as Sum
 open import Data.Sum.Properties
-open import Induction.WellFounded
 open import Level using (_⊔_)
 open import Function.Base using (_∘_; id)
 open import Function.Inverse using (Inverse)
@@ -75,19 +74,6 @@ module _ {a₁ a₂ ℓ₁ ℓ₂} {A₁ : Set a₁} {A₂ : Set a₂}
   ⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₁ y) = no λ()
   ⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₂ y) = Dec.map′ inj₂ drop-inj₂ (x ≟₂ y)
 
-  ⊎-wellFounded : WellFounded ∼₁ → WellFounded ∼₂ → WellFounded (Pointwise ∼₁ ∼₂)
-  ⊎-wellFounded wf₁ wf₂ x = acc (⊎-acc x)
-    where
-    ⊎-acc₁ : ∀ {x} → Acc ∼₁ x → WfRec (Pointwise ∼₁ ∼₂) (Acc (Pointwise ∼₁ ∼₂)) (inj₁ x)
-    ⊎-acc₁ (acc rec) (inj₁ y) (inj₁ x∼₁y) = acc (⊎-acc₁ (rec y x∼₁y))
-
-    ⊎-acc₂ : ∀ {x} → Acc ∼₂ x → WfRec (Pointwise ∼₁ ∼₂) (Acc (Pointwise ∼₁ ∼₂)) (inj₂ x)
-    ⊎-acc₂ (acc rec) (inj₂ y) (inj₂ x∼₂y) = acc (⊎-acc₂ (rec y x∼₂y))
-    ⊎-acc  : ∀ x → WfRec (Pointwise ∼₁ ∼₂) (Acc (Pointwise ∼₁ ∼₂)) x
-
-    ⊎-acc (inj₁ x) = ⊎-acc₁ (wf₁ x)
-    ⊎-acc (inj₂ x) = ⊎-acc₂ (wf₂ x)
-
 module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
          {ℓ₁ ℓ₂} {∼₁ : Rel A₁ ℓ₁} {≈₁ : Rel A₁ ℓ₂}
          {ℓ₃ ℓ₄} {∼₂ : Rel A₂ ℓ₃} {≈₂ : Rel A₂ ℓ₄} where

From 3f7c5ebc3e4f0737e6aeae896c080c57d3b45f68 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 17 Apr 2023 15:00:25 +0100
Subject: [PATCH 022/316] Revert "moved lemma"

This reverts commit 9e7e36cc26a805fa4638a92bcdb883dd8c3a63a8.
---
 CHANGELOG.md | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 76ca38cd42..2f03afa471 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2097,6 +2097,11 @@ Other minor changes
   nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
   ```
 
+* Added new proofs in `Data.Nat.Combinatorics.Specification`:
+  ```agda
+  k![n∸k]!∣n! : k ≤ n →  k ! * (n ∸ k) ! ∣ n !
+  ```
+
 * Added new proofs in `Data.Nat.DivMod`:
   ```agda
   m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n

From f839326ac7745e3858c77229927b878f41682b26 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 17 Apr 2023 15:01:24 +0100
Subject: [PATCH 023/316] hopefully fixed now

---
 CHANGELOG.md | 5 -----
 1 file changed, 5 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2f03afa471..76ca38cd42 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2097,11 +2097,6 @@ Other minor changes
   nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
   ```
 
-* Added new proofs in `Data.Nat.Combinatorics.Specification`:
-  ```agda
-  k![n∸k]!∣n! : k ≤ n →  k ! * (n ∸ k) ! ∣ n !
-  ```
-
 * Added new proofs in `Data.Nat.DivMod`:
   ```agda
   m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n

From 2231b25a8887c060d4181f9624e9fda99650589e Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Mon, 17 Apr 2023 15:21:48 +0100
Subject: [PATCH 024/316] recommitted the whitespace change; tree should be
 clean now?

---
 CHANGELOG/v1.7.2.md                             | 2 +-
 src/Data/List/Membership/Setoid/Properties.agda | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG/v1.7.2.md b/CHANGELOG/v1.7.2.md
index 0183ec654f..8ddea63b4d 100644
--- a/CHANGELOG/v1.7.2.md
+++ b/CHANGELOG/v1.7.2.md
@@ -5,5 +5,5 @@ The library has been tested using Agda 2.6.3.
 
 * In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
   the `--without-K` flag now use the `--cubical-compatible` flag instead.
-  
+
 * Updated the code using `primFloatToWord64` - the library API has remained unchanged.
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index df260f2a17..919bf19ba6 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -122,7 +122,7 @@ module _ (S : Setoid c ℓ) where
                     length (mapWith∈ xs f) ≡ length xs
   length-mapWith∈ []       = P.refl
   length-mapWith∈ (x ∷ xs) = P.cong suc (length-mapWith∈ xs)
-  
+
   mapWith∈-id : ∀ xs → mapWith∈ xs (λ {x} _ → x) ≡ xs
   mapWith∈-id []       = P.refl
   mapWith∈-id (x ∷ xs) = P.cong (x ∷_) (mapWith∈-id xs)

From 4f52e121163b6a285f6362174ca492cc52908d6e Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Tue, 18 Apr 2023 15:46:36 +0100
Subject: [PATCH 025/316] removed redundant use of `with`

---
 src/Algebra/Consequences/Base.agda | 4 +---
 1 file changed, 1 insertion(+), 3 deletions(-)

diff --git a/src/Algebra/Consequences/Base.agda b/src/Algebra/Consequences/Base.agda
index 1a3e105604..446a82d868 100644
--- a/src/Algebra/Consequences/Base.agda
+++ b/src/Algebra/Consequences/Base.agda
@@ -19,9 +19,7 @@ open import Relation.Binary.Definitions using (Reflexive)
 module _ {ℓ} {_•_ : Op₂ A} (_≈_ : Rel A ℓ) where
 
   sel⇒idem : Selective _≈_ _•_ → Idempotent _≈_ _•_
-  sel⇒idem sel x with sel x x
-  ... | inj₁ x•x≈x = x•x≈x
-  ... | inj₂ x•x≈x = x•x≈x
+  sel⇒idem sel x = reduce (sel x x)
 
 module _ {ℓ} {f : Op₁ A} (_≈_ : Rel A ℓ) where
 

From 5d056c1f02e41426bdb7ae41a4aa08397bc82eb6 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Thu, 20 Apr 2023 17:46:55 +0100
Subject: [PATCH 026/316] some additional simplifications

---
 src/Data/List/Properties.agda | 22 ++++++++++------------
 1 file changed, 10 insertions(+), 12 deletions(-)

diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index e9ddbe6387..7592e39478 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -497,8 +497,8 @@ module _ {P : Pred A p} {f : A → A → A} where
   foldr-forcesᵇ : (∀ x y → P (f x y) → P x × P y) →
                   ∀ e xs → P (foldr f e xs) → All P xs
   foldr-forcesᵇ _      _ []       _     = []
-  foldr-forcesᵇ forces _ (x ∷ xs) Pfold with forces _ _ Pfold
-  ... | (px , pfxs) = px ∷ foldr-forcesᵇ forces _ xs pfxs
+  foldr-forcesᵇ forces _ (x ∷ xs) Pfold =
+    let px , pfxs = forces _ _ Pfold in px ∷ foldr-forcesᵇ forces _ xs pfxs
 
   foldr-preservesᵇ : (∀ {x y} → P x → P y → P (f x y)) →
                      ∀ {e xs} → P e → All P xs → P (foldr f e xs)
@@ -627,11 +627,10 @@ scanr-defn : ∀ (f : A → B → B) (e : B) →
              scanr f e ≗ map (foldr f e) ∘ tails
 scanr-defn f e []             = refl
 scanr-defn f e (x ∷ [])       = refl
-scanr-defn f e (x ∷ y ∷ xs)
-  with scanr f e (y ∷ xs) | scanr-defn f e (y ∷ xs)
-... | []     | ()
-... | z ∷ zs | eq with ∷-injective eq
-...   | z≡fy⦇f⦈xs , _ = cong₂ (λ z → f x z ∷_) z≡fy⦇f⦈xs eq
+scanr-defn f e (x ∷ y∷xs@(_ ∷ _))
+  with eq ← scanr-defn f e y∷xs
+  with z ∷ zs ← scanr f e y∷xs
+  = let z≡fy⦇f⦈xs , _ = ∷-injective eq in cong₂ (λ z → f x z ∷_) z≡fy⦇f⦈xs eq
 
 ------------------------------------------------------------------------
 -- scanl
@@ -778,8 +777,7 @@ take++drop (suc n) (x ∷ xs) = cong (x ∷_) (take++drop n xs)
 splitAt-defn : ∀ n → splitAt {A = A} n ≗ < take n , drop n >
 splitAt-defn zero    xs       = refl
 splitAt-defn (suc n) []       = refl
-splitAt-defn (suc n) (x ∷ xs) with splitAt n xs | splitAt-defn n xs
-... | (ys , zs) | ih = cong (Prod.map (x ∷_) id) ih
+splitAt-defn (suc n) (x ∷ xs) = cong (Prod.map (x ∷_) id) (splitAt-defn n xs)
 
 ------------------------------------------------------------------------
 -- takeWhile, dropWhile, and span
@@ -912,9 +910,9 @@ module _ {P : Pred A p} (P? : Decidable P) where
   length-partition : ∀ xs → (let (ys , zs) = partition P? xs) →
                      length ys ≤ length xs × length zs ≤ length xs
   length-partition []       = z≤n , z≤n
-  length-partition (x ∷ xs) with does (P? x) | length-partition xs
-  ...  | true  | rec = Prod.map s≤s m≤n⇒m≤1+n rec
-  ...  | false | rec = Prod.map m≤n⇒m≤1+n s≤s rec
+  length-partition (x ∷ xs) with does (P? x)
+  ...  | true  = Prod.map s≤s m≤n⇒m≤1+n (length-partition xs)
+  ...  | false = Prod.map m≤n⇒m≤1+n s≤s (length-partition xs)
 
 ------------------------------------------------------------------------
 -- _ʳ++_

From 719a50f9dc0646a584cbcf4396f3996877e9671a Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Sat, 22 Apr 2023 14:15:27 +0100
Subject: [PATCH 027/316] half of the adjunction

---
 src/Data/Nat/Properties.agda | 12 ++++++++++++
 1 file changed, 12 insertions(+)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 3d38c77d07..b5f5f45738 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1473,9 +1473,21 @@ pred[m∸n]≡m∸[1+n] (suc m) zero    = refl
 pred[m∸n]≡m∸[1+n] zero (suc n)    = refl
 pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
 
+------------------------------------------------------------------------
+-- Properties of _∸_ and suc
+
+m≤n⇒[1+m]m∸n≡1+[m∸n] : ∀ {m n} → m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
+m≤n⇒[1+m]m∸n≡1+[m∸n] z≤n                                       = refl
+m≤n⇒[1+m]m∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]m∸n≡1+[m∸n] le = refl
+
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _≤_/_<_
 
+m+n≤p⇒m≤p∸n : ∀ m n p → m + n ≤ p → m ≤ p ∸ n
+m+n≤p⇒m≤p∸n zero    n p       le                 = z≤n 
+m+n≤p⇒m≤p∸n (suc m) n (suc p) (s≤s le)
+  rewrite m≤n⇒[1+m]m∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤p⇒m≤p∸n m n p le)
+
 m∸n≤m : ∀ m n → m ∸ n ≤ m
 m∸n≤m n       zero    = ≤-refl
 m∸n≤m zero    (suc n) = ≤-refl

From 80ac1bca7cc27084c7a93857c870042859e6ba42 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Sat, 22 Apr 2023 17:50:00 +0100
Subject: [PATCH 028/316] other half of the adjunction; NB side condition

---
 src/Data/Nat/Base.agda       |  2 +-
 src/Data/Nat/Properties.agda | 18 +++++++++++++-----
 2 files changed, 14 insertions(+), 6 deletions(-)

diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index 1e2db666b8..3c10279aab 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -123,7 +123,7 @@ instance
 -- Arithmetic
 
 open import Agda.Builtin.Nat public
-  using (_+_; _*_) renaming (_-_ to _∸_)
+  using (_+_; _*_) renaming (_-_ to _∸_) 
 
 open import Agda.Builtin.Nat
   using (div-helper; mod-helper)
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index b5f5f45738..347dc69c7f 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1474,11 +1474,15 @@ pred[m∸n]≡m∸[1+n] zero (suc n)    = refl
 pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
 
 ------------------------------------------------------------------------
--- Properties of _∸_ and suc
+-- Properties of _∸_ and zero, suc
 
-m≤n⇒[1+m]m∸n≡1+[m∸n] : ∀ {m n} → m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
-m≤n⇒[1+m]m∸n≡1+[m∸n] z≤n                                       = refl
-m≤n⇒[1+m]m∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]m∸n≡1+[m∸n] le = refl
+0∸n≡n :  ∀ n → 0 ∸ n ≡ 0
+0∸n≡n zero    = refl
+0∸n≡n (suc _) = refl
+
+m≤n⇒[1+m]∸n≡1+[m∸n] : ∀ {m n} → m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
+m≤n⇒[1+m]∸n≡1+[m∸n] z≤n                                       = refl
+m≤n⇒[1+m]∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]∸n≡1+[m∸n] le = refl
 
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _≤_/_<_
@@ -1486,7 +1490,11 @@ m≤n⇒[1+m]m∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]m∸n≡1+[m∸n]
 m+n≤p⇒m≤p∸n : ∀ m n p → m + n ≤ p → m ≤ p ∸ n
 m+n≤p⇒m≤p∸n zero    n p       le                 = z≤n 
 m+n≤p⇒m≤p∸n (suc m) n (suc p) (s≤s le)
-  rewrite m≤n⇒[1+m]m∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤p⇒m≤p∸n m n p le)
+  rewrite m≤n⇒[1+m]∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤p⇒m≤p∸n m n p le)
+
+m≤p∸n⇒m+n≤p : ∀ m {n p} (n≤p : n ≤ p) → m ≤ p ∸ n → m + n ≤ p
+m≤p∸n⇒m+n≤p m         z≤n       le rewrite +-identityʳ m = le
+m≤p∸n⇒m+n≤p m {suc n} (s≤s n≤p) le rewrite +-suc m n = s≤s (m≤p∸n⇒m+n≤p m n≤p le)
 
 m∸n≤m : ∀ m n → m ∸ n ≤ m
 m∸n≤m n       zero    = ≤-refl

From e73e5f77ca99598fff916cb75e552a1db1878b04 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Sun, 23 Apr 2023 10:00:16 +0100
Subject: [PATCH 029/316] `CHANGELOG`

---
 CHANGELOG.md                 |  4 ++++
 src/Data/Nat/Properties.agda | 15 ++++-----------
 2 files changed, 8 insertions(+), 11 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 76ca38cd42..44c565b327 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2044,6 +2044,10 @@ Other minor changes
   <⇒<′ : m < n → m <′ n
   <′⇒< : m <′ n → m < n
 
+  m≤n⇒[1+m]∸n≡1+[m∸n] : m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
+  m+n≤p⇒m≤p∸n         : m + n ≤ p → m ≤ p ∸ n
+  m≤p∸n⇒m+n≤p         : n ≤ p → m ≤ p ∸ n → m + n ≤ p
+
   1≤n!    : 1 ≤ n !
   _!≢0    : NonZero (n !)
   _!*_!≢0 : NonZero (m ! * n !)
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 347dc69c7f..47d8ab04ab 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1464,6 +1464,10 @@ n∸n≡0 : ∀ n → n ∸ n ≡ 0
 n∸n≡0 zero    = refl
 n∸n≡0 (suc n) = n∸n≡0 n
 
+m≤n⇒[1+m]∸n≡1+[m∸n] : ∀ {m n} → m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
+m≤n⇒[1+m]∸n≡1+[m∸n] z≤n                                      = refl
+m≤n⇒[1+m]∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]∸n≡1+[m∸n] le = refl
+
 ------------------------------------------------------------------------
 -- Properties of _∸_ and pred
 
@@ -1473,17 +1477,6 @@ pred[m∸n]≡m∸[1+n] (suc m) zero    = refl
 pred[m∸n]≡m∸[1+n] zero (suc n)    = refl
 pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
 
-------------------------------------------------------------------------
--- Properties of _∸_ and zero, suc
-
-0∸n≡n :  ∀ n → 0 ∸ n ≡ 0
-0∸n≡n zero    = refl
-0∸n≡n (suc _) = refl
-
-m≤n⇒[1+m]∸n≡1+[m∸n] : ∀ {m n} → m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
-m≤n⇒[1+m]∸n≡1+[m∸n] z≤n                                       = refl
-m≤n⇒[1+m]∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]∸n≡1+[m∸n] le = refl
-
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _≤_/_<_
 

From b182f63e81e6cfbe70dc5e34145d259c446a5d24 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Sun, 23 Apr 2023 19:24:22 +0100
Subject: [PATCH 030/316] irrefutable `with` bindings for recursive
 calls/inductive hypotheses

---
 src/Data/List/Properties.agda | 52 +++++++++++++++++------------------
 1 file changed, 26 insertions(+), 26 deletions(-)

diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 7592e39478..d4008fce07 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -130,15 +130,15 @@ module _ (f : A → Maybe B) where
 
   mapMaybe-concatMap : mapMaybe f ≗ concatMap (fromMaybe ∘ f)
   mapMaybe-concatMap [] = refl
-  mapMaybe-concatMap (x ∷ xs) with f x
-  ... | just y  = cong (y ∷_) (mapMaybe-concatMap xs)
-  ... | nothing = mapMaybe-concatMap xs
+  mapMaybe-concatMap (x ∷ xs) with ih ← mapMaybe-concatMap xs | f x
+  ... | just y  = cong (y ∷_) ih
+  ... | nothing = ih
 
   length-mapMaybe : ∀ xs → length (mapMaybe f xs) ≤ length xs
   length-mapMaybe []       = z≤n
-  length-mapMaybe (x ∷ xs) with f x
-  ... | just y  = s≤s (length-mapMaybe xs)
-  ... | nothing = m≤n⇒m≤1+n (length-mapMaybe xs)
+  length-mapMaybe (x ∷ xs) with ih ← length-mapMaybe xs | f x
+  ... | just y  = s≤s ih
+  ... | nothing = m≤n⇒m≤1+n ih
 
 ------------------------------------------------------------------------
 -- _++_
@@ -803,9 +803,9 @@ module _ {P : Pred A p} (P? : Decidable P) where
 
   length-filter : ∀ xs → length (filter P? xs) ≤ length xs
   length-filter []       = z≤n
-  length-filter (x ∷ xs) with does (P? x)
-  ... | false = m≤n⇒m≤1+n (length-filter xs)
-  ... | true  = s≤s (length-filter xs)
+  length-filter (x ∷ xs) with ih ← length-filter xs | does (P? x)
+  ... | false = m≤n⇒m≤1+n ih
+  ... | true  = s≤s ih
 
   filter-all : ∀ {xs} → All P xs → filter P? xs ≡ xs
   filter-all {[]}     []         = refl
@@ -817,9 +817,9 @@ module _ {P : Pred A p} (P? : Decidable P) where
   filter-notAll (x ∷ xs) (here ¬px) with P? x
   ... | false because _ = s≤s (length-filter xs)
   ... | yes          px = contradiction px ¬px
-  filter-notAll (x ∷ xs) (there any) with does (P? x)
-  ... | false = m≤n⇒m≤1+n (filter-notAll xs any)
-  ... | true  = s≤s (filter-notAll xs any)
+  filter-notAll (x ∷ xs) (there any) with ih ← filter-notAll xs any | does (P? x)
+  ... | false = m≤n⇒m≤1+n ih
+  ... | true  = s≤s ih
 
   filter-some : ∀ {xs} → Any P xs → 0 < length (filter P? xs)
   filter-some {x ∷ xs} (here px)   with P? x
@@ -860,9 +860,9 @@ module _ {P : Pred A p} (P? : Decidable P) where
 
   filter-++ : ∀ xs ys → filter P? (xs ++ ys) ≡ filter P? xs ++ filter P? ys
   filter-++ []       ys = refl
-  filter-++ (x ∷ xs) ys with does (P? x)
-  ... | true  = cong (x ∷_) (filter-++ xs ys)
-  ... | false = filter-++ xs ys
+  filter-++ (x ∷ xs) ys with ih ← filter-++ xs ys | does (P? x)
+  ... | true  = cong (x ∷_) ih
+  ... | false = ih
 
 ------------------------------------------------------------------------
 -- derun and deduplicate
@@ -872,9 +872,9 @@ module _ {R : Rel A p} (R? : B.Decidable R) where
   length-derun : ∀ xs → length (derun R? xs) ≤ length xs
   length-derun [] = ≤-refl
   length-derun (x ∷ []) = ≤-refl
-  length-derun (x ∷ y ∷ xs) with does (R? x y) | length-derun (y ∷ xs)
-  ... | true  | r = m≤n⇒m≤1+n r
-  ... | false | r = s≤s r
+  length-derun (x ∷ y ∷ xs) with ih ← length-derun (y ∷ xs) | does (R? x y) 
+  ... | true  = m≤n⇒m≤1+n ih
+  ... | false = s≤s ih
 
   length-deduplicate : ∀ xs → length (deduplicate R? xs) ≤ length xs
   length-deduplicate [] = z≤n
@@ -903,16 +903,16 @@ module _ {P : Pred A p} (P? : Decidable P) where
 
   partition-defn : partition P? ≗ < filter P? , filter (∁? P?) >
   partition-defn []       = refl
-  partition-defn (x ∷ xs) with does (P? x)
-  ...  | true  = cong (Prod.map (x ∷_) id) (partition-defn xs)
-  ...  | false = cong (Prod.map id (x ∷_)) (partition-defn xs)
+  partition-defn (x ∷ xs) with ih ← partition-defn xs | does (P? x)
+  ...  | true  = cong (Prod.map (x ∷_) id) ih
+  ...  | false = cong (Prod.map id (x ∷_)) ih
 
   length-partition : ∀ xs → (let (ys , zs) = partition P? xs) →
                      length ys ≤ length xs × length zs ≤ length xs
   length-partition []       = z≤n , z≤n
-  length-partition (x ∷ xs) with does (P? x)
-  ...  | true  = Prod.map s≤s m≤n⇒m≤1+n (length-partition xs)
-  ...  | false = Prod.map m≤n⇒m≤1+n s≤s (length-partition xs)
+  length-partition (x ∷ xs) with ih ← length-partition xs | does (P? x)
+  ...  | true  = Prod.map s≤s m≤n⇒m≤1+n ih
+  ...  | false = Prod.map m≤n⇒m≤1+n s≤s ih
 
 ------------------------------------------------------------------------
 -- _ʳ++_
@@ -1061,8 +1061,8 @@ module _ {x y : A} where
 
   ∷ʳ-injective : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
   ∷ʳ-injective []          []          refl = (refl , refl)
-  ∷ʳ-injective (x ∷ xs)    (y  ∷ ys)   eq   with ∷-injective eq
-  ... | refl , eq′ = Prod.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
+  ∷ʳ-injective (x ∷ xs)    (y  ∷ ys)   eq   with refl , eq′  ← ∷-injective eq
+    = Prod.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
   ∷ʳ-injective []          (_ ∷ _ ∷ _) ()
   ∷ʳ-injective (_ ∷ _ ∷ _) []          ()
 

From 160e8420f3029a4d224e651cc098f00b998276f9 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Sun, 23 Apr 2023 19:58:19 +0100
Subject: [PATCH 031/316] fix whitespace

---
 src/Data/Nat/Base.agda       | 2 +-
 src/Data/Nat/Properties.agda | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index 3c10279aab..1e2db666b8 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -123,7 +123,7 @@ instance
 -- Arithmetic
 
 open import Agda.Builtin.Nat public
-  using (_+_; _*_) renaming (_-_ to _∸_) 
+  using (_+_; _*_) renaming (_-_ to _∸_)
 
 open import Agda.Builtin.Nat
   using (div-helper; mod-helper)
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 47d8ab04ab..e6d1c24a74 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1481,7 +1481,7 @@ pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
 -- Properties of _∸_ and _≤_/_<_
 
 m+n≤p⇒m≤p∸n : ∀ m n p → m + n ≤ p → m ≤ p ∸ n
-m+n≤p⇒m≤p∸n zero    n p       le                 = z≤n 
+m+n≤p⇒m≤p∸n zero    n p       le                 = z≤n
 m+n≤p⇒m≤p∸n (suc m) n (suc p) (s≤s le)
   rewrite m≤n⇒[1+m]∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤p⇒m≤p∸n m n p le)
 

From c7b6031f27fcbd3928e6a010a7325f5559da5023 Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Sun, 23 Apr 2023 20:00:36 +0100
Subject: [PATCH 032/316] more whitespace fixes

---
 src/Data/Nat/Properties.agda | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index e6d1c24a74..a164931e7c 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1465,7 +1465,7 @@ n∸n≡0 zero    = refl
 n∸n≡0 (suc n) = n∸n≡0 n
 
 m≤n⇒[1+m]∸n≡1+[m∸n] : ∀ {m n} → m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
-m≤n⇒[1+m]∸n≡1+[m∸n] z≤n                                      = refl
+m≤n⇒[1+m]∸n≡1+[m∸n] z≤n                                     = refl
 m≤n⇒[1+m]∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]∸n≡1+[m∸n] le = refl
 
 ------------------------------------------------------------------------
@@ -1481,7 +1481,7 @@ pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
 -- Properties of _∸_ and _≤_/_<_
 
 m+n≤p⇒m≤p∸n : ∀ m n p → m + n ≤ p → m ≤ p ∸ n
-m+n≤p⇒m≤p∸n zero    n p       le                 = z≤n
+m+n≤p⇒m≤p∸n zero    n p       le               = z≤n
 m+n≤p⇒m≤p∸n (suc m) n (suc p) (s≤s le)
   rewrite m≤n⇒[1+m]∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤p⇒m≤p∸n m n p le)
 

From c05de5721a12fa40aab11af25531d6decb2afcef Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Mon, 24 Apr 2023 08:18:10 +0100
Subject: [PATCH 033/316] corrected `p` -> `o`

---
 src/Data/Nat/Properties.agda | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index a164931e7c..93c8712eb5 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1480,14 +1480,14 @@ pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _≤_/_<_
 
-m+n≤p⇒m≤p∸n : ∀ m n p → m + n ≤ p → m ≤ p ∸ n
-m+n≤p⇒m≤p∸n zero    n p       le               = z≤n
-m+n≤p⇒m≤p∸n (suc m) n (suc p) (s≤s le)
-  rewrite m≤n⇒[1+m]∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤p⇒m≤p∸n m n p le)
-
-m≤p∸n⇒m+n≤p : ∀ m {n p} (n≤p : n ≤ p) → m ≤ p ∸ n → m + n ≤ p
-m≤p∸n⇒m+n≤p m         z≤n       le rewrite +-identityʳ m = le
-m≤p∸n⇒m+n≤p m {suc n} (s≤s n≤p) le rewrite +-suc m n = s≤s (m≤p∸n⇒m+n≤p m n≤p le)
+m+n≤o⇒m≤o∸n : ∀ m n o → m + n ≤ o → m ≤ o ∸ n
+m+n≤o⇒m≤o∸n zero    n o       le               = z≤n
+m+n≤o⇒m≤o∸n (suc m) n (suc o) (s≤s le)
+  rewrite m≤n⇒[1+m]∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m n o le)
+
+m≤o∸n⇒m+n≤o : ∀ m {n o} (n≤o : n ≤ o) → m ≤ o ∸ n → m + n ≤ o
+m≤o∸n⇒m+n≤o m         z≤n       le rewrite +-identityʳ m = le
+m≤o∸n⇒m+n≤o m {suc n} (s≤s n≤o) le rewrite +-suc m n = s≤s (m≤o∸n⇒m+n≤o m n≤o le)
 
 m∸n≤m : ∀ m n → m ∸ n ≤ m
 m∸n≤m n       zero    = ≤-refl

From 4dbf96dca7cf0d1645cd78e6a5ae69a5b562da0b Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Mon, 24 Apr 2023 11:08:49 +0100
Subject: [PATCH 034/316] removed redudant additional lemma; relocated
 adjointness proofs

---
 src/Data/Nat/Properties.agda | 22 +++++++++-------------
 1 file changed, 9 insertions(+), 13 deletions(-)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 93c8712eb5..6ff1b1f38b 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1464,10 +1464,6 @@ n∸n≡0 : ∀ n → n ∸ n ≡ 0
 n∸n≡0 zero    = refl
 n∸n≡0 (suc n) = n∸n≡0 n
 
-m≤n⇒[1+m]∸n≡1+[m∸n] : ∀ {m n} → m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
-m≤n⇒[1+m]∸n≡1+[m∸n] z≤n                                     = refl
-m≤n⇒[1+m]∸n≡1+[m∸n] (s≤s le) rewrite m≤n⇒[1+m]∸n≡1+[m∸n] le = refl
-
 ------------------------------------------------------------------------
 -- Properties of _∸_ and pred
 
@@ -1480,15 +1476,6 @@ pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _≤_/_<_
 
-m+n≤o⇒m≤o∸n : ∀ m n o → m + n ≤ o → m ≤ o ∸ n
-m+n≤o⇒m≤o∸n zero    n o       le               = z≤n
-m+n≤o⇒m≤o∸n (suc m) n (suc o) (s≤s le)
-  rewrite m≤n⇒[1+m]∸n≡1+[m∸n] (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m n o le)
-
-m≤o∸n⇒m+n≤o : ∀ m {n o} (n≤o : n ≤ o) → m ≤ o ∸ n → m + n ≤ o
-m≤o∸n⇒m+n≤o m         z≤n       le rewrite +-identityʳ m = le
-m≤o∸n⇒m+n≤o m {suc n} (s≤s n≤o) le rewrite +-suc m n = s≤s (m≤o∸n⇒m+n≤o m n≤o le)
-
 m∸n≤m : ∀ m n → m ∸ n ≤ m
 m∸n≤m n       zero    = ≤-refl
 m∸n≤m zero    (suc n) = ≤-refl
@@ -1586,6 +1573,15 @@ m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
   (m + n) ∸ o          ≡⟨ +-∸-assoc m o≤n ⟩
   m + (n ∸ o)          ∎
 
+m+n≤o⇒m≤o∸n : ∀ m n o → m + n ≤ o → m ≤ o ∸ n
+m+n≤o⇒m≤o∸n zero    n o       le       = z≤n
+m+n≤o⇒m≤o∸n (suc m) n (suc o) (s≤s le)
+  rewrite +-∸-assoc 1 (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m n o le)
+
+m≤o∸n⇒m+n≤o : ∀ m {n o} (n≤o : n ≤ o) → m ≤ o ∸ n → m + n ≤ o
+m≤o∸n⇒m+n≤o m         z≤n       le rewrite +-identityʳ m = le
+m≤o∸n⇒m+n≤o m {suc n} (s≤s n≤o) le rewrite +-suc m n = s≤s (m≤o∸n⇒m+n≤o m n≤o le)
+
 m≤n+m∸n : ∀ m n → m ≤ n + (m ∸ n)
 m≤n+m∸n zero    n       = z≤n
 m≤n+m∸n (suc m) zero    = ≤-refl

From e5f41787eded77233365ff79bcabff1dcbe64fdf Mon Sep 17 00:00:00 2001
From: jamesmckinna <j.mckinna@hw.ac.uk>
Date: Mon, 24 Apr 2023 11:22:21 +0100
Subject: [PATCH 035/316] `CHANGELOG`

---
 CHANGELOG.md | 1 -
 1 file changed, 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 44c565b327..957ea761d7 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2044,7 +2044,6 @@ Other minor changes
   <⇒<′ : m < n → m <′ n
   <′⇒< : m <′ n → m < n
 
-  m≤n⇒[1+m]∸n≡1+[m∸n] : m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
   m+n≤p⇒m≤p∸n         : m + n ≤ p → m ≤ p ∸ n
   m≤p∸n⇒m+n≤p         : n ≤ p → m ≤ p ∸ n → m + n ≤ p
 

From 1c510d4d085d06ddf733ad5c807541ae508ffeb9 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 28 Apr 2023 16:20:29 +0900
Subject: [PATCH 036/316] Add citation file to repo

---
 CITATION.cff            |  8 ++++++++
 notes/release-guide.txt | 16 +++++++---------
 2 files changed, 15 insertions(+), 9 deletions(-)
 create mode 100644 CITATION.cff

diff --git a/CITATION.cff b/CITATION.cff
new file mode 100644
index 0000000000..37d6ffec0e
--- /dev/null
+++ b/CITATION.cff
@@ -0,0 +1,8 @@
+cff-version: 1.2.0
+message: "If you use this software, please cite it as below."
+authors:
+- name: "The Agda Community"
+title: "Agda Standard Library"
+version: 1.7.2
+date-released: 2023-2-1
+url: "https://github.com/agda/agda-stdlib"
\ No newline at end of file
diff --git a/notes/release-guide.txt b/notes/release-guide.txt
index 56fb2458cc..2f3f9d862f 100644
--- a/notes/release-guide.txt
+++ b/notes/release-guide.txt
@@ -5,15 +5,13 @@ procedure should be followed:
 
 * Update `README.agda` by replacing 'development version' by 'version X.Y' in the title.
 
-* Update `README.md`
-
-* Update `agda-stdlib-utils.cabal` version to `X.Y`.
-
-* Update the version in standard-library.agda-lib to `X.Y`
-
-* Update `notes/installation-guide.txt`
-
-* Update `CHANGELOG.md`.
+* Update the version to `X.Y` in:
+  - `agda-stdlib-utils.cabal`
+  - `standard-library.agda-lib`
+  - `CITATION.cff`
+  - `CHANGELOG.md`
+  - `README.md`
+  - `notes/installation-guide.txt`
 
 * Update the copyright year range in the LICENSE file, if necessary.
 

From 6338d2a6398a364e16dbbd3692e3de03421fb93d Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 2 May 2023 13:24:47 +0100
Subject: [PATCH 037/316] Restore deleted `*.Categorical.*` as deprecated
 modules (#1946)

* restored `Categorical` modules towards deprecation

* updated `CHANGELOG`

* updated `GenerateEverything`
---
 CHANGELOG.md                                  | 40 ++++++++++++++++++-
 GenerateEverything.hs                         |  5 +++
 src/Codata/Sized/Colist/Categorical.agda      | 17 ++++++++
 src/Codata/Sized/Covec/Categorical.agda       | 17 ++++++++
 src/Codata/Sized/Delay/Categorical.agda       | 17 ++++++++
 src/Codata/Sized/Stream/Categorical.agda      | 17 ++++++++
 src/Data/List/Categorical.agda                | 17 ++++++++
 src/Data/List/Categorical/Transformer.agda    | 17 ++++++++
 src/Data/List/NonEmpty/Categorical.agda       | 17 ++++++++
 .../NonEmpty/Categorical/Transformer.agda     | 17 ++++++++
 src/Data/Maybe/Categorical.agda               | 17 ++++++++
 src/Data/Maybe/Categorical/Transformer.agda   | 17 ++++++++
 src/Data/Product/Categorical/Examples.agda    | 17 ++++++++
 src/Data/Product/Categorical/Left.agda        | 17 ++++++++
 src/Data/Product/Categorical/Left/Base.agda   | 17 ++++++++
 src/Data/Product/Categorical/Right.agda       | 17 ++++++++
 src/Data/Product/Categorical/Right/Base.agda  | 17 ++++++++
 src/Data/Sum/Categorical/Examples.agda        | 17 ++++++++
 src/Data/Sum/Categorical/Left.agda            | 17 ++++++++
 .../Sum/Categorical/Left/Transformer.agda     | 17 ++++++++
 src/Data/Sum/Categorical/Right.agda           | 17 ++++++++
 .../Sum/Categorical/Right/Transformer.agda    | 17 ++++++++
 src/Data/These/Categorical/Left.agda          | 17 ++++++++
 src/Data/These/Categorical/Left/Base.agda     | 17 ++++++++
 src/Data/These/Categorical/Right.agda         | 17 ++++++++
 src/Data/These/Categorical/Right/Base.agda    | 17 ++++++++
 src/Data/Vec/Categorical.agda                 | 17 ++++++++
 src/Data/Vec/Categorical/Transformer.agda     | 17 ++++++++
 src/Data/Vec/Recursive/Categorical.agda       | 17 ++++++++
 src/Function/Identity/Categorical.agda        | 17 ++++++++
 src/IO/Categorical.agda                       | 17 ++++++++
 src/Reflection/TCM/Categorical.agda           | 17 ++++++++
 32 files changed, 553 insertions(+), 2 deletions(-)
 create mode 100644 src/Codata/Sized/Colist/Categorical.agda
 create mode 100644 src/Codata/Sized/Covec/Categorical.agda
 create mode 100644 src/Codata/Sized/Delay/Categorical.agda
 create mode 100644 src/Codata/Sized/Stream/Categorical.agda
 create mode 100644 src/Data/List/Categorical.agda
 create mode 100644 src/Data/List/Categorical/Transformer.agda
 create mode 100644 src/Data/List/NonEmpty/Categorical.agda
 create mode 100644 src/Data/List/NonEmpty/Categorical/Transformer.agda
 create mode 100644 src/Data/Maybe/Categorical.agda
 create mode 100644 src/Data/Maybe/Categorical/Transformer.agda
 create mode 100644 src/Data/Product/Categorical/Examples.agda
 create mode 100644 src/Data/Product/Categorical/Left.agda
 create mode 100644 src/Data/Product/Categorical/Left/Base.agda
 create mode 100644 src/Data/Product/Categorical/Right.agda
 create mode 100644 src/Data/Product/Categorical/Right/Base.agda
 create mode 100644 src/Data/Sum/Categorical/Examples.agda
 create mode 100644 src/Data/Sum/Categorical/Left.agda
 create mode 100644 src/Data/Sum/Categorical/Left/Transformer.agda
 create mode 100644 src/Data/Sum/Categorical/Right.agda
 create mode 100644 src/Data/Sum/Categorical/Right/Transformer.agda
 create mode 100644 src/Data/These/Categorical/Left.agda
 create mode 100644 src/Data/These/Categorical/Left/Base.agda
 create mode 100644 src/Data/These/Categorical/Right.agda
 create mode 100644 src/Data/These/Categorical/Right/Base.agda
 create mode 100644 src/Data/Vec/Categorical.agda
 create mode 100644 src/Data/Vec/Categorical/Transformer.agda
 create mode 100644 src/Data/Vec/Recursive/Categorical.agda
 create mode 100644 src/Function/Identity/Categorical.agda
 create mode 100644 src/IO/Categorical.agda
 create mode 100644 src/Reflection/TCM/Categorical.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 957ea761d7..2abb4c3c57 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -88,8 +88,7 @@ Non-backwards compatible changes
   such as `Monad`, `Applicative`, `Functor`, etc, under `Category.*`, as this
   obstructs the importing of the `agda-categories` development into the Standard Library,
   and moreover needlessly restricts the applicability of categorical concepts to this
-  (highly specific) mode of use. Correspondingly, modules grouped under `*.Categorical.*`
-  which exploited these structures for effectful programming have been renamed `*.Effectful`.
+  (highly specific) mode of use. Correspondingly, client modules grouped under `*.Categorical.*` which exploit such structure for effectful programming have been renamed `*.Effectful`, with the originals being deprecated.
 
 ### Improvements to pretty printing and regexes
 
@@ -871,6 +870,43 @@ Deprecated modules
   Algebra.Morphism.LatticeMonomorphism         ↦ Algebra.Lattice.Morphism.LatticeMonomorphism
   ```
 
+### Moving `*.Catgeorical.*` files
+
+* As discussed above the following files have been moved:
+  ```agda
+  Codata.Sized.Colist.Categorical            ↦ Codata.Sized.Colist.Effectful
+  Codata.Sized.Covec.Categorical             ↦ Codata.Sized.Covec.Effectful
+  Codata.Sized.Delay.Categorical             ↦ Codata.Sized.Delay.Effectful
+  Codata.Sized.Stream.Categorical            ↦ Codata.Sized.Stream.Effectful
+  Data.List.Categorical                      ↦ Data.List.Effectful
+  Data.List.Categorical.Transformer          ↦ Data.List.Effectful.Transformer
+  Data.List.NonEmpty.Categorical             ↦ Data.List.NonEmpty.Effectful
+  Data.List.NonEmpty.Categorical.Transformer ↦ Data.List.NonEmpty.Effectful.Transformer
+  Data.Maybe.Categorical                     ↦ Data.Maybe.Effectful
+  Data.Maybe.Categorical.Transformer         ↦ Data.Maybe.Effectful.Transformer
+  Data.Product.Categorical.Examples          ↦ Data.Product.Effectful.Examples
+  Data.Product.Categorical.Left              ↦ Data.Product.Effectful.Left
+  Data.Product.Categorical.Left.Base         ↦ Data.Product.Effectful.Left.Base
+  Data.Product.Categorical.Right             ↦ Data.Product.Effectful.Right
+  Data.Product.Categorical.Right.Base        ↦ Data.Product.Effectful.Right.Base
+  Data.Sum.Categorical.Examples              ↦ Data.Sum.Effectful.Examples
+  Data.Sum.Categorical.Left                  ↦ Data.Sum.Effectful.Left
+  Data.Sum.Categorical.Left.Transformer      ↦ Data.Sum.Effectful.Left.Transformer
+  Data.Sum.Categorical.Right                 ↦ Data.Sum.Effectful.Right
+  Data.Sum.Categorical.Right.Transformer     ↦ Data.Sum.Effectful.Right.Transformer
+  Data.These.Categorical.Examples            ↦ Data.These.Effectful.Examples
+  Data.These.Categorical.Left                ↦ Data.These.Effectful.Left
+  Data.These.Categorical.Left.Base           ↦ Data.These.Effectful.Left.Base
+  Data.These.Categorical.Right               ↦ Data.These.Effectful.Right
+  Data.These.Categorical.Right.Base          ↦ Data.These.Effectful.Right.Base
+  Data.Vec.Categorical                       ↦ Data.Vec.Effectful
+  Data.Vec.Categorical.Transformer           ↦ Data.Vec.Effectful.Transformer
+  Data.Vec.Recursive.Categorical             ↦ Data.Vec.Recursive.Effectful
+  Function.Identity.Categorical              ↦ Function.Identity.Effectful
+  IO.Categorical                             ↦ IO.Effectful
+  Reflection.TCM.Categorical                 ↦ Reflection.TCM.Effectful
+  ```
+
 ### Moving `Relation.Binary.Properties.XLattice` files
 
 * The following files have been moved:
diff --git a/GenerateEverything.hs b/GenerateEverything.hs
index d513da0ab8..ee49679f3a 100644
--- a/GenerateEverything.hs
+++ b/GenerateEverything.hs
@@ -53,6 +53,7 @@ unsafeModules = map modToFile
   , "Foreign.Haskell.Pair"
   , "IO"
   , "IO.Base"
+  , "IO.Categorical"
   , "IO.Infinite"
   , "IO.Instances"
   , "IO.Effectful"
@@ -121,6 +122,7 @@ sizedTypesModules = map modToFile
   , "Codata.Sized.Cofin.Literals"
   , "Codata.Sized.Colist"
   , "Codata.Sized.Colist.Bisimilarity"
+  , "Codata.Sized.Colist.Categorical"
   , "Codata.Sized.Colist.Effectful"
   , "Codata.Sized.Colist.Properties"
   , "Codata.Sized.Conat"
@@ -129,6 +131,7 @@ sizedTypesModules = map modToFile
   , "Codata.Sized.Conat.Properties"
   , "Codata.Sized.Covec"
   , "Codata.Sized.Covec.Bisimilarity"
+  , "Codata.Sized.Covec.Categorical"
   , "Codata.Sized.Covec.Effectful"
   , "Codata.Sized.Covec.Instances"
   , "Codata.Sized.Covec.Properties"
@@ -136,6 +139,7 @@ sizedTypesModules = map modToFile
   , "Codata.Sized.Cowriter.Bisimilarity"
   , "Codata.Sized.Delay"
   , "Codata.Sized.Delay.Bisimilarity"
+  , "Codata.Sized.Delay.Categorical"
   , "Codata.Sized.Delay.Effectful"
   , "Codata.Sized.Delay.Properties"
   , "Codata.Sized.M"
@@ -143,6 +147,7 @@ sizedTypesModules = map modToFile
   , "Codata.Sized.M.Properties"
   , "Codata.Sized.Stream"
   , "Codata.Sized.Stream.Bisimilarity"
+  , "Codata.Sized.Stream.Categorical"
   , "Codata.Sized.Stream.Effectful"
   , "Codata.Sized.Stream.Instances"
   , "Codata.Sized.Stream.Properties"
diff --git a/src/Codata/Sized/Colist/Categorical.agda b/src/Codata/Sized/Colist/Categorical.agda
new file mode 100644
index 0000000000..6cee36ca22
--- /dev/null
+++ b/src/Codata/Sized/Colist/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Codata.Sized.Colist.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --sized-types #-}
+
+module Codata.Sized.Colist.Categorical where
+
+open import Codata.Sized.Colist.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Codata.Sized.Colist.Categorical was deprecated in v2.0.
+Use Codata.Sized.Colist.Effectful instead."
+#-}
diff --git a/src/Codata/Sized/Covec/Categorical.agda b/src/Codata/Sized/Covec/Categorical.agda
new file mode 100644
index 0000000000..0da702ff2c
--- /dev/null
+++ b/src/Codata/Sized/Covec/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Codata.Sized.Covec.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --sized-types #-}
+
+module Codata.Sized.Covec.Categorical where
+
+open import Codata.Sized.Covec.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Codata.Sized.Covec.Categorical was deprecated in v2.0.
+Use Codata.Sized.Covec.Effectful instead."
+#-}
diff --git a/src/Codata/Sized/Delay/Categorical.agda b/src/Codata/Sized/Delay/Categorical.agda
new file mode 100644
index 0000000000..2aa88bb896
--- /dev/null
+++ b/src/Codata/Sized/Delay/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Codata.Sized.Delay.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --sized-types #-}
+
+module Codata.Sized.Delay.Categorical where
+
+open import Codata.Sized.Delay.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Codata.Sized.Delay.Categorical was deprecated in v2.0.
+Use Codata.Sized.Delay.Effectful instead."
+#-}
diff --git a/src/Codata/Sized/Stream/Categorical.agda b/src/Codata/Sized/Stream/Categorical.agda
new file mode 100644
index 0000000000..597d7b7bfc
--- /dev/null
+++ b/src/Codata/Sized/Stream/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Codata.Sized.Stream.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --sized-types #-}
+
+module Codata.Sized.Stream.Categorical where
+
+open import Codata.Sized.Stream.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Codata.Sized.Stream.Categorical was deprecated in v2.0.
+Use Codata.Sized.Stream.Effectful instead."
+#-}
diff --git a/src/Data/List/Categorical.agda b/src/Data/List/Categorical.agda
new file mode 100644
index 0000000000..dc3391f1c4
--- /dev/null
+++ b/src/Data/List/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.List.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.Categorical where
+
+open import Data.List.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Data.List.Categorical was deprecated in v2.0.
+Use Data.List.Effectful instead."
+#-}
diff --git a/src/Data/List/Categorical/Transformer.agda b/src/Data/List/Categorical/Transformer.agda
new file mode 100644
index 0000000000..75b83f14f3
--- /dev/null
+++ b/src/Data/List/Categorical/Transformer.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.List.Effectful.Transformer` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.Categorical.Transformer where
+
+open import Data.List.Effectful.Transformer public
+
+{-# WARNING_ON_IMPORT
+"Data.List.Categorical.Transformer was deprecated in v2.0.
+Use Data.List.Effectful.Transformer instead."
+#-}
diff --git a/src/Data/List/NonEmpty/Categorical.agda b/src/Data/List/NonEmpty/Categorical.agda
new file mode 100644
index 0000000000..442c1a0df1
--- /dev/null
+++ b/src/Data/List/NonEmpty/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.List.NonEmpty.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.NonEmpty.Categorical where
+
+open import Data.List.NonEmpty.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Data.List.NonEmpty.Categorical was deprecated in v2.0.
+Use Data.List.NonEmpty.Effectful instead."
+#-}
diff --git a/src/Data/List/NonEmpty/Categorical/Transformer.agda b/src/Data/List/NonEmpty/Categorical/Transformer.agda
new file mode 100644
index 0000000000..7842b65054
--- /dev/null
+++ b/src/Data/List/NonEmpty/Categorical/Transformer.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.List.NonEmpty.Effectful.Transformer` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.NonEmpty.Categorical.Transformer where
+
+open import Data.List.NonEmpty.Effectful.Transformer public
+
+{-# WARNING_ON_IMPORT
+"Data.List.NonEmpty.Categorical.Transformer was deprecated in v2.0.
+Use Data.List.NonEmpty.Effectful.Transformer instead."
+#-}
diff --git a/src/Data/Maybe/Categorical.agda b/src/Data/Maybe/Categorical.agda
new file mode 100644
index 0000000000..e6ec7a7a2c
--- /dev/null
+++ b/src/Data/Maybe/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Maybe.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Maybe.Categorical where
+
+open import Data.Maybe.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Data.Maybe.Categorical was deprecated in v2.0.
+Use Data.Maybe.Effectful instead."
+#-}
diff --git a/src/Data/Maybe/Categorical/Transformer.agda b/src/Data/Maybe/Categorical/Transformer.agda
new file mode 100644
index 0000000000..aca111a00a
--- /dev/null
+++ b/src/Data/Maybe/Categorical/Transformer.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Maybe.Effectful.Transformer` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Maybe.Categorical.Transformer where
+
+open import Data.Maybe.Effectful.Transformer public
+
+{-# WARNING_ON_IMPORT
+"Data.Maybe.Categorical.Transformer was deprecated in v2.0.
+Use Data.Maybe.Effectful.Transformer instead."
+#-}
diff --git a/src/Data/Product/Categorical/Examples.agda b/src/Data/Product/Categorical/Examples.agda
new file mode 100644
index 0000000000..e832d67b19
--- /dev/null
+++ b/src/Data/Product/Categorical/Examples.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Product.Categorical.Examples` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Product.Categorical.Examples where
+
+open import Data.Product.Effectful.Examples public
+
+{-# WARNING_ON_IMPORT
+"Data.Product.Categorical.Examples was deprecated in v2.0.
+Use Data.Product.Effectful.Examples instead."
+#-}
diff --git a/src/Data/Product/Categorical/Left.agda b/src/Data/Product/Categorical/Left.agda
new file mode 100644
index 0000000000..de68a8575e
--- /dev/null
+++ b/src/Data/Product/Categorical/Left.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Product.Categorical.Left` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Product.Categorical.Left where
+
+open import Data.Product.Effectful.Left public
+
+{-# WARNING_ON_IMPORT
+"Data.Product.Categorical.Left was deprecated in v2.0.
+Use Data.Product.Effectful.Left instead."
+#-}
diff --git a/src/Data/Product/Categorical/Left/Base.agda b/src/Data/Product/Categorical/Left/Base.agda
new file mode 100644
index 0000000000..ffa7fed34c
--- /dev/null
+++ b/src/Data/Product/Categorical/Left/Base.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Product.Categorical.Left.Base` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Product.Categorical.Left.Base where
+
+open import Data.Product.Effectful.Left.Base public
+
+{-# WARNING_ON_IMPORT
+"Data.Product.Categorical.Left.Base was deprecated in v2.0.
+Use Data.Product.Effectful.Left.Base instead."
+#-}
diff --git a/src/Data/Product/Categorical/Right.agda b/src/Data/Product/Categorical/Right.agda
new file mode 100644
index 0000000000..bdd34822e3
--- /dev/null
+++ b/src/Data/Product/Categorical/Right.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Product.Categorical.Right` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Product.Categorical.Right where
+
+open import Data.Product.Effectful.Right public
+
+{-# WARNING_ON_IMPORT
+"Data.Product.Categorical.Right was deprecated in v2.0.
+Use Data.Product.Effectful.Right instead."
+#-}
diff --git a/src/Data/Product/Categorical/Right/Base.agda b/src/Data/Product/Categorical/Right/Base.agda
new file mode 100644
index 0000000000..556009642a
--- /dev/null
+++ b/src/Data/Product/Categorical/Right/Base.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Product.Categorical.Right.Base` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Product.Categorical.Right.Base where
+
+open import Data.Product.Effectful.Right.Base public
+
+{-# WARNING_ON_IMPORT
+"Data.Product.Categorical.Right.Base was deprecated in v2.0.
+Use Data.Product.Effectful.Right.Base instead."
+#-}
diff --git a/src/Data/Sum/Categorical/Examples.agda b/src/Data/Sum/Categorical/Examples.agda
new file mode 100644
index 0000000000..5a74842598
--- /dev/null
+++ b/src/Data/Sum/Categorical/Examples.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Sum.Categorical.Examples` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Sum.Categorical.Examples where
+
+open import Data.Sum.Effectful.Examples public
+
+{-# WARNING_ON_IMPORT
+"Data.Sum.Categorical.Examples was deprecated in v2.0.
+Use Data.Sum.Effectful.Examples instead."
+#-}
diff --git a/src/Data/Sum/Categorical/Left.agda b/src/Data/Sum/Categorical/Left.agda
new file mode 100644
index 0000000000..b1a26c995d
--- /dev/null
+++ b/src/Data/Sum/Categorical/Left.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Sum.Categorical.Left` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Sum.Categorical.Left where
+
+open import Data.Sum.Effectful.Left public
+
+{-# WARNING_ON_IMPORT
+"Data.Sum.Categorical.Left was deprecated in v2.0.
+Use Data.Sum.Effectful.Left instead."
+#-}
diff --git a/src/Data/Sum/Categorical/Left/Transformer.agda b/src/Data/Sum/Categorical/Left/Transformer.agda
new file mode 100644
index 0000000000..de39522246
--- /dev/null
+++ b/src/Data/Sum/Categorical/Left/Transformer.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Sum.Categorical.Left.Transformer` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Sum.Categorical.Left.Transformer where
+
+open import Data.Sum.Effectful.Left.Transformer public
+
+{-# WARNING_ON_IMPORT
+"Data.Sum.Categorical.Left.Transformer was deprecated in v2.0.
+Use Data.Sum.Effectful.Left.Transformer instead."
+#-}
diff --git a/src/Data/Sum/Categorical/Right.agda b/src/Data/Sum/Categorical/Right.agda
new file mode 100644
index 0000000000..724d6a5b3a
--- /dev/null
+++ b/src/Data/Sum/Categorical/Right.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Sum.Categorical.Right` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Sum.Categorical.Right where
+
+open import Data.Sum.Effectful.Right public
+
+{-# WARNING_ON_IMPORT
+"Data.Sum.Categorical.Right was deprecated in v2.0.
+Use Data.Sum.Effectful.Right instead."
+#-}
diff --git a/src/Data/Sum/Categorical/Right/Transformer.agda b/src/Data/Sum/Categorical/Right/Transformer.agda
new file mode 100644
index 0000000000..52fbb16f11
--- /dev/null
+++ b/src/Data/Sum/Categorical/Right/Transformer.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Sum.Categorical.Right.Transformer` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Sum.Categorical.Right.Transformer where
+
+open import Data.Sum.Effectful.Right.Transformer public
+
+{-# WARNING_ON_IMPORT
+"Data.Sum.Categorical.Right.Transformer was deprecated in v2.0.
+Use Data.Sum.Effectful.Right.Transformer instead."
+#-}
diff --git a/src/Data/These/Categorical/Left.agda b/src/Data/These/Categorical/Left.agda
new file mode 100644
index 0000000000..73576b3bd0
--- /dev/null
+++ b/src/Data/These/Categorical/Left.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.These.Categorical.Left` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.These.Categorical.Left where
+
+open import Data.These.Effectful.Left public
+
+{-# WARNING_ON_IMPORT
+"Data.These.Categorical.Left was deprecated in v2.0.
+Use Data.These.Effectful.Left instead."
+#-}
diff --git a/src/Data/These/Categorical/Left/Base.agda b/src/Data/These/Categorical/Left/Base.agda
new file mode 100644
index 0000000000..7c7e3add7d
--- /dev/null
+++ b/src/Data/These/Categorical/Left/Base.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.These.Categorical.Left.Base` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.These.Categorical.Left.Base where
+
+open import Data.These.Effectful.Left.Base public
+
+{-# WARNING_ON_IMPORT
+"Data.These.Categorical.Left.Base was deprecated in v2.0.
+Use Data.These.Effectful.Left.Base instead."
+#-}
diff --git a/src/Data/These/Categorical/Right.agda b/src/Data/These/Categorical/Right.agda
new file mode 100644
index 0000000000..2e42e8d4a8
--- /dev/null
+++ b/src/Data/These/Categorical/Right.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.These.Categorical.Right` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.These.Categorical.Right where
+
+open import Data.These.Effectful.Right public
+
+{-# WARNING_ON_IMPORT
+"Data.These.Categorical.Right was deprecated in v2.0.
+Use Data.These.Effectful.Right instead."
+#-}
diff --git a/src/Data/These/Categorical/Right/Base.agda b/src/Data/These/Categorical/Right/Base.agda
new file mode 100644
index 0000000000..05acbcf139
--- /dev/null
+++ b/src/Data/These/Categorical/Right/Base.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.These.Categorical.Right.Base` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.These.Categorical.Right.Base where
+
+open import Data.These.Effectful.Right.Base public
+
+{-# WARNING_ON_IMPORT
+"Data.These.Categorical.Right.Base was deprecated in v2.0.
+Use Data.These.Effectful.Right.Base instead."
+#-}
diff --git a/src/Data/Vec/Categorical.agda b/src/Data/Vec/Categorical.agda
new file mode 100644
index 0000000000..aaad325cac
--- /dev/null
+++ b/src/Data/Vec/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Vec.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Categorical where
+
+open import Data.Vec.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Data.Vec.Categorical was deprecated in v2.0.
+Use Data.Vec.Effectful instead."
+#-}
diff --git a/src/Data/Vec/Categorical/Transformer.agda b/src/Data/Vec/Categorical/Transformer.agda
new file mode 100644
index 0000000000..9d90b030f7
--- /dev/null
+++ b/src/Data/Vec/Categorical/Transformer.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Vec.Effectful.Transformer` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Categorical.Transformer where
+
+open import Data.Vec.Effectful.Transformer public
+
+{-# WARNING_ON_IMPORT
+"Data.Vec.Categorical.Transformer was deprecated in v2.0.
+Use Data.Vec.Effectful.Transformer instead."
+#-}
diff --git a/src/Data/Vec/Recursive/Categorical.agda b/src/Data/Vec/Recursive/Categorical.agda
new file mode 100644
index 0000000000..1dba5fd7b0
--- /dev/null
+++ b/src/Data/Vec/Recursive/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Data.Vec.Recursive.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Recursive.Categorical where
+
+open import Data.Vec.Recursive.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Data.Vec.Recursive.Categorical was deprecated in v2.0.
+Use Data.Vec.Recursive.Effectful instead."
+#-}
diff --git a/src/Function/Identity/Categorical.agda b/src/Function/Identity/Categorical.agda
new file mode 100644
index 0000000000..7fbdec019e
--- /dev/null
+++ b/src/Function/Identity/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Function.Identity.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Identity.Categorical where
+
+open import Function.Identity.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Function.Identity.Categorical was deprecated in v2.0.
+Use Function.Identity.Effectful instead."
+#-}
diff --git a/src/IO/Categorical.agda b/src/IO/Categorical.agda
new file mode 100644
index 0000000000..9e264a5283
--- /dev/null
+++ b/src/IO/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `IO.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --guardedness #-}
+
+module IO.Categorical where
+
+open import IO.Effectful public
+
+{-# WARNING_ON_IMPORT
+"IO.Categorical was deprecated in v2.0.
+Use IO.Effectful instead."
+#-}
diff --git a/src/Reflection/TCM/Categorical.agda b/src/Reflection/TCM/Categorical.agda
new file mode 100644
index 0000000000..f1c5d76319
--- /dev/null
+++ b/src/Reflection/TCM/Categorical.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Reflection.TCM.Effectful` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Reflection.TCM.Categorical where
+
+open import Reflection.TCM.Effectful public
+
+{-# WARNING_ON_IMPORT
+"Reflection.TCM.Categorical was deprecated in v2.0.
+Use Reflection.TCM.Effectful instead."
+#-}

From 96e4d8fb85ed189a3037f2d23568b53164b8ff45 Mon Sep 17 00:00:00 2001
From: James McKinna <J.McKinna@hw.ac.uk>
Date: Tue, 2 May 2023 15:30:07 +0100
Subject: [PATCH 038/316] cosmetic: two whitespace violations

---
 src/Data/List/Properties.agda   | 2 +-
 src/Data/Nat/Combinatorics.agda | 4 ++--
 2 files changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index d4008fce07..d18d8562ce 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -872,7 +872,7 @@ module _ {R : Rel A p} (R? : B.Decidable R) where
   length-derun : ∀ xs → length (derun R? xs) ≤ length xs
   length-derun [] = ≤-refl
   length-derun (x ∷ []) = ≤-refl
-  length-derun (x ∷ y ∷ xs) with ih ← length-derun (y ∷ xs) | does (R? x y) 
+  length-derun (x ∷ y ∷ xs) with ih ← length-derun (y ∷ xs) | does (R? x y)
   ... | true  = m≤n⇒m≤1+n ih
   ... | false = s≤s ih
 
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index 6db600699c..850cd01857 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -12,7 +12,7 @@ open import Data.Nat.Base
 open import Data.Nat.DivMod
 open import Data.Nat.Divisibility
 open import Data.Nat.Properties
-open import Relation.Binary.Definitions 
+open import Relation.Binary.Definitions
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong; subst)
 
@@ -118,7 +118,7 @@ module _ {n k} (k<n : k < n) where
 
     [n-k]!           = [n-k] !
     [n-k-1]!         = [n-k-1] !
-  
+
     [n-k]≡1+[n-k-1]  : [n-k] ≡ suc [n-k-1]
     [n-k]≡1+[n-k-1]  = +-∸-assoc 1 k<n
 

From 51c0a6437fa8cc16c54333387246cb9294462d9d Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 5 May 2023 11:51:09 +0900
Subject: [PATCH 039/316] Fix date format

---
 CITATION.cff | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/CITATION.cff b/CITATION.cff
index 37d6ffec0e..416e595b98 100644
--- a/CITATION.cff
+++ b/CITATION.cff
@@ -4,5 +4,5 @@ authors:
 - name: "The Agda Community"
 title: "Agda Standard Library"
 version: 1.7.2
-date-released: 2023-2-1
+date-released: 2023-02-01
 url: "https://github.com/agda/agda-stdlib"
\ No newline at end of file

From 9c5112b609307f062d971f4e41f0704eecfe5266 Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Sat, 6 May 2023 02:53:16 -0300
Subject: [PATCH 040/316] Added reverse permutation

---
 CHANGELOG.md                                         |  5 +++++
 .../Binary/Permutation/Propositional/Properties.agda | 12 ++++++++++++
 2 files changed, 17 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2abb4c3c57..0e9cd648ab 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3037,3 +3037,8 @@ This is a full list of proofs that have changed form to use irrelevant instance
   ```agda
   <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
   ```
+
+* Added new function to `Data.List.Relation.Binary.Permutation.Propositional.Properties`
+  ```agda
+  ↭-reverse : (xs : List A) → reverse xs ↭ xs
+  ```
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index c9a8048592..700fc277ff 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -324,6 +324,18 @@ drop-∷ = drop-mid [] []
 ++↭ʳ++ []       ys = ↭-refl
 ++↭ʳ++ (x ∷ xs) ys = ↭-trans (↭-sym (shift x xs ys)) (++↭ʳ++ xs (x ∷ ys))
 
+------------------------------------------------------------------------
+-- reverse
+
+↭-reverse : (xs : List A) → reverse xs ↭ xs
+↭-reverse [] = ↭-refl
+↭-reverse (x ∷ xs) = begin
+  reverse (x ∷ xs) ≡⟨ Lₚ.unfold-reverse x xs ⟩
+  reverse xs ∷ʳ x ↭˘⟨ ∷↭∷ʳ x (reverse xs) ⟩
+  x ∷ reverse xs   ↭⟨ prep x (↭-reverse xs) ⟩
+  x ∷ xs ∎
+  where open PermutationReasoning
+
 ------------------------------------------------------------------------
 -- merge
 

From 14567c002c15a8154af0ee892f87286dfe48c0f0 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 15 May 2023 03:09:34 +0200
Subject: [PATCH 041/316] Deprecate `inspect` (#1930)

---
 CHANGELOG.md                                  | 14 +++++-
 README.agda                                   | 16 +++---
 README/Inspect.agda                           |  7 ++-
 src/Codata/Sized/Colist/Properties.agda       |  6 +--
 src/Data/Fin/Properties.agda                  | 20 ++++++--
 src/Data/List/Properties.agda                 |  6 +--
 .../Permutation/Propositional/Properties.agda |  2 +-
 .../Binary/Permutation/Setoid/Properties.agda |  2 +-
 .../List/Relation/Unary/Any/Properties.agda   |  8 +--
 src/Data/Rational/Properties.agda             | 24 ++++-----
 .../Vec/Relation/Unary/Any/Properties.agda    | 10 ++--
 .../Binary/PropositionalEquality.agda         | 49 ++++++++++++-------
 12 files changed, 99 insertions(+), 65 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0e9cd648ab..52b2a6762d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,7 +1,7 @@
 Version 2.0-dev
 ===============
 
-The library has been tested using Agda 2.6.2.
+The library has been tested using Agda 2.6.3.
 
 Highlights
 ----------
@@ -1334,6 +1334,15 @@ Deprecated names
 * This fixes the fact we had picked the wrong name originally. The erased modality
   corresponds to @0 whereas the irrelevance one corresponds to `.`.
 
+### Deprecated `Relation.Binary.PropositionalEquality.inspect`
+    in favour of `with ... in ...` syntax (issue #1580; PRs #1630, #1930)
+
+* In `Relation.Binary.PropositionalEquality`
+  both the record type `Reveal_·_is_`
+  and its principal mode of use, `inspect`,
+  have been deprecated in favour of the new `with ... in ...` syntax.
+  See the documentation of [with-abstraction equality](https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality)
+
 New modules
 -----------
 
@@ -1891,6 +1900,9 @@ Other minor changes
   toℕ-cancel-≤       : toℕ i ℕ.≤ toℕ j → i ≤ j
   toℕ-cancel-<       : toℕ i ℕ.< toℕ j → i < j
 
+  splitAt⁻¹-↑ˡ       : splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i
+  splitAt⁻¹-↑ʳ       : splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i
+
   toℕ-combine        : toℕ (combine i j) ≡ k ℕ.* toℕ i ℕ.+ toℕ j
   combine-injectiveˡ : combine i j ≡ combine k l → i ≡ k
   combine-injectiveʳ : combine i j ≡ combine k l → j ≡ l
diff --git a/README.agda b/README.agda
index 2d87cb56d2..6fee0aeffa 100644
--- a/README.agda
+++ b/README.agda
@@ -12,13 +12,14 @@ module README where
 -- Liang-Ting Chen, Dominique Devriese, Dan Doel, Érdi Gergő,
 -- Zack Grannan, Helmut Grohne, Simon Foster, Liyang Hu, Jason Hu,
 -- Patrik Jansson, Alan Jeffrey, Wen Kokke, Evgeny Kotelnikov,
--- Sergei Meshveliani, Eric Mertens, Darin Morrison, Guilhem Moulin,
--- Shin-Cheng Mu, Ulf Norell, Noriyuki Ohkawa, Nicolas Pouillard,
--- Andrés Sicard-Ramírez, Lex van der Stoep, Sandro Stucki, Milo Turner,
--- Noam Zeilberger and other anonymous contributors.
+-- James McKinna, Sergei Meshveliani, Eric Mertens, Darin Morrison,
+-- Guilhem Moulin, Shin-Cheng Mu, Ulf Norell, Noriyuki Ohkawa,
+-- Nicolas Pouillard, Andrés Sicard-Ramírez, Lex van der Stoep,
+-- Sandro Stucki, Milo Turner, Noam Zeilberger
+-- and other anonymous contributors.
 ------------------------------------------------------------------------
 
--- This version of the library has been tested using Agda 2.6.2.
+-- This version of the library has been tested using Agda 2.6.3.
 
 -- The library comes with a .agda-lib file, for use with the library
 -- management system.
@@ -236,11 +237,6 @@ import README.Debug.Trace
 
 import README.Nary
 
--- Explaining the inspect idiom: use case, equivalent handwritten
--- auxiliary definitions, and implementation details.
-
-import README.Inspect
-
 -- Explaining how to use the automatic solvers
 
 import README.Tactic.MonoidSolver
diff --git a/README/Inspect.agda b/README/Inspect.agda
index e10877c44c..cff492d008 100644
--- a/README/Inspect.agda
+++ b/README/Inspect.agda
@@ -1,14 +1,17 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Explaining how to use the inspect idiom and elaborating on the way
--- it is implemented in the standard library.
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module README.Inspect where
 
+{-# WARNING_ON_IMPORT
+"README.Inspect was deprecated in v2.0."
+#-}
+
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Product
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index 3de56a756a..dea697b088 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -144,10 +144,10 @@ module _ (cons : C → B → C) (alg : A → Maybe (A × B)) where
 
   scanl-unfold : ∀ nil a → i ⊢ scanl cons nil (unfold alg a)
                              ≈ nil ∷ (λ where .force → unfold alg′ (a , nil))
-  scanl-unfold nil a with alg a | Eq.inspect alg a
-  ... | nothing       | [ eq ] = Eq.refl ∷ λ { .force →
+  scanl-unfold nil a with alg a in eq
+  ... | nothing      = Eq.refl ∷ λ { .force →
     sym (fromEq (unfold-nothing (Maybeₚ.map-nothing eq))) }
-  ... | just (a′ , b) | [ eq ] = Eq.refl ∷ λ { .force → begin
+  ... | just (a′ , b) = Eq.refl ∷ λ { .force → begin
     scanl cons (cons nil b) (unfold alg a′)
      ≈⟨ scanl-unfold (cons nil b) a′ ⟩
     (cons nil b ∷ _)
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 232a44a7bb..d7a963f9fd 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -562,10 +562,20 @@ splitAt-↑ˡ : ∀ m i n → splitAt m (i ↑ˡ n) ≡ inj₁ i
 splitAt-↑ˡ (suc m) zero    n = refl
 splitAt-↑ˡ (suc m) (suc i) n rewrite splitAt-↑ˡ m i n = refl
 
+splitAt⁻¹-↑ˡ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i
+splitAt⁻¹-↑ˡ {suc m} {n} {0F} {.0F} refl = refl
+splitAt⁻¹-↑ˡ {suc m} {n} {suc i} {j} eq with splitAt m i in splitAt[m][i]≡inj₁[j]
+... | inj₁ k with refl ← eq = cong suc (splitAt⁻¹-↑ˡ {i = i} {j = k} splitAt[m][i]≡inj₁[j])
+
 splitAt-↑ʳ : ∀ m n i → splitAt m (m ↑ʳ i) ≡ inj₂ {B = Fin n} i
 splitAt-↑ʳ zero    n i = refl
 splitAt-↑ʳ (suc m) n i rewrite splitAt-↑ʳ m n i = refl
 
+splitAt⁻¹-↑ʳ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i
+splitAt⁻¹-↑ʳ {zero}  {n} {i} {j} refl = refl
+splitAt⁻¹-↑ʳ {suc m} {n} {suc i} {j} eq with splitAt m i in splitAt[m][i]≡inj₂[k]
+... | inj₂ k with refl ← eq = cong suc (splitAt⁻¹-↑ʳ {i = i} {j = k} splitAt[m][i]≡inj₂[k])
+
 splitAt-join : ∀ m n i → splitAt m (join m n i) ≡ i
 splitAt-join m n (inj₁ x) = splitAt-↑ˡ m x n
 splitAt-join m n (inj₂ y) = splitAt-↑ʳ m n y
@@ -612,13 +622,13 @@ remQuot-combine {suc n} {k} (suc i) j rewrite splitAt-↑ʳ k   (n ℕ.* k) (com
   cong (Data.Product.map₁ suc) (remQuot-combine i j)
 
 combine-remQuot : ∀ {n} k (i : Fin (n ℕ.* k)) → uncurry combine (remQuot {n} k i) ≡ i
-combine-remQuot {suc n} k i with splitAt k i | P.inspect (splitAt k) i
-... | inj₁ j | P.[ eq ] = begin
+combine-remQuot {suc n} k i with splitAt k i in eq
+... | inj₁ j = begin
   join k (n ℕ.* k) (inj₁ j)      ≡˘⟨ cong (join k (n ℕ.* k)) eq ⟩
   join k (n ℕ.* k) (splitAt k i) ≡⟨ join-splitAt k (n ℕ.* k) i ⟩
   i                              ∎
   where open ≡-Reasoning
-... | inj₂ j | P.[ eq ] = begin
+... | inj₂ j = begin
   k ↑ʳ (uncurry combine (remQuot {n} k j)) ≡⟨ cong (k ↑ʳ_) (combine-remQuot {n} k j) ⟩
   join k (n ℕ.* k) (inj₂ j)                ≡˘⟨ cong (join k (n ℕ.* k)) eq ⟩
   join k (n ℕ.* k) (splitAt k i)           ≡⟨ join-splitAt k (n ℕ.* k) i ⟩
@@ -679,8 +689,8 @@ combine-injective i j k l cᵢⱼ≡cₖₗ =
   combine-injectiveʳ i j k l cᵢⱼ≡cₖₗ
 
 combine-surjective : ∀ (i : Fin (m ℕ.* n)) → ∃₂ λ j k → combine j k ≡ i
-combine-surjective {m} {n} i with remQuot {m} n i | P.inspect (remQuot {m} n) i
-... | j , k | P.[ eq ] = j , k , (begin
+combine-surjective {m} {n} i with remQuot {m} n i in eq
+... | j , k = j , k , (begin
   combine j k                       ≡˘⟨ uncurry (cong₂ combine) (,-injective eq) ⟩
   uncurry combine (remQuot {m} n i) ≡⟨ combine-remQuot {m} n i ⟩
   i                                 ∎)
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index d18d8562ce..312ef2e7a8 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -854,9 +854,9 @@ module _ {P : Pred A p} (P? : Decidable P) where
 
   filter-idem : filter P? ∘ filter P? ≗ filter P?
   filter-idem []       = refl
-  filter-idem (x ∷ xs) with does (P? x) | inspect does (P? x)
-  ... | false | _                   = filter-idem xs
-  ... | true  | P.[ eq ] rewrite eq = cong (x ∷_) (filter-idem xs)
+  filter-idem (x ∷ xs) with does (P? x) in eq
+  ... | false            = filter-idem xs
+  ... | true  rewrite eq = cong (x ∷_) (filter-idem xs)
 
   filter-++ : ∀ xs ys → filter P? (xs ++ ys) ≡ filter P? xs ++ filter P? ys
   filter-++ []       ys = refl
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index 700fc277ff..f5a87cb067 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -29,7 +29,7 @@ open import Level using (Level)
 open import Relation.Unary using (Pred)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality as ≡
-  using (_≡_ ; refl ; cong; cong₂; _≢_; inspect)
+  using (_≡_ ; refl ; cong; cong₂; _≢_)
 open import Relation.Nullary
 
 open PermutationReasoning
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index 79ccfaadef..3483323844 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -37,7 +37,7 @@ open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
 open import Relation.Binary.PropositionalEquality as ≡
-  using (_≡_ ; refl; sym; cong; cong₂; subst; _≢_; inspect)
+  using (_≡_ ; refl; sym; cong; cong₂; subst; _≢_)
 open import Relation.Nullary.Decidable using (yes; no; does)
 open import Relation.Nullary.Negation using (contradiction)
 
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index abfc8f93f8..6ac93bc05e 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -42,7 +42,7 @@ open import Function.Related as Related using (Kind; Related; SK-sym)
 open import Level using (Level)
 open import Relation.Binary as B hiding (_⇔_)
 open import Relation.Binary.PropositionalEquality as P
-  using (_≡_; refl; inspect)
+  using (_≡_; refl)
 open import Relation.Unary as U
   using (Pred; _⟨×⟩_; _⟨→⟩_) renaming (_⊆_ to _⋐_)
 open import Relation.Nullary using (¬_; _because_; does; ofʸ; ofⁿ; yes; no)
@@ -172,9 +172,9 @@ any⁺ p (there {x = x} pxs) with p x
 ... | false = any⁺ p pxs
 
 any⁻ : ∀ (p : A → Bool) xs → T (any p xs) → Any (T ∘ p) xs
-any⁻ p (x ∷ xs) px∷xs with p x | inspect p x
-... | true  | P.[ eq ] = here (Equivalence.from T-≡ ⟨$⟩ eq)
-... | false | _        = there (any⁻ p xs px∷xs)
+any⁻ p (x ∷ xs) px∷xs with p x in eq
+... | true  = here (Equivalence.from T-≡ ⟨$⟩ eq)
+... | false = there (any⁻ p xs px∷xs)
 
 any⇔ : ∀ {p : A → Bool} → Any (T ∘ p) xs ⇔ T (any p xs)
 any⇔ = equivalence (any⁺ _) (any⁻ _ _)
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index b9d18fab34..f564ffe835 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -1304,24 +1304,24 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 ------------------------------------------------------------------------
 
 p≤q⇒p⊔q≡q : p ≤ q → p ⊔ q ≡ q
-p≤q⇒p⊔q≡q {p@record{}} {q@record{}} p≤q with p ≤ᵇ q | inspect (p ≤ᵇ_) q
-... | true  | _       = refl
-... | false | [ p≰q ] = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())
+p≤q⇒p⊔q≡q {p@record{}} {q@record{}} p≤q with p ≤ᵇ q in p≰q
+... | true  = refl
+... | false = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())
 
 p≥q⇒p⊔q≡p : p ≥ q → p ⊔ q ≡ p
-p≥q⇒p⊔q≡p {p@record{}} {q@record{}} p≥q with p ≤ᵇ q | inspect (p ≤ᵇ_) q
-... | true  | [ p≤q ] = ≤-antisym p≥q (≤ᵇ⇒≤ (subst T (sym p≤q) _))
-... | false | [ p≤q ] = refl
+p≥q⇒p⊔q≡p {p@record{}} {q@record{}} p≥q with p ≤ᵇ q in p≤q
+... | true  = ≤-antisym p≥q (≤ᵇ⇒≤ (subst T (sym p≤q) _))
+... | false = refl
 
 p≤q⇒p⊓q≡p : p ≤ q → p ⊓ q ≡ p
-p≤q⇒p⊓q≡p {p@record{}} {q@record{}} p≤q with p ≤ᵇ q | inspect (p ≤ᵇ_) q
-... | true  | _       = refl
-... | false | [ p≰q ] = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())
+p≤q⇒p⊓q≡p {p@record{}} {q@record{}} p≤q with p ≤ᵇ q in p≰q
+... | true  = refl
+... | false = contradiction (≤⇒≤ᵇ p≤q) (subst (¬_ ∘ T) (sym p≰q) λ())
 
 p≥q⇒p⊓q≡q : p ≥ q → p ⊓ q ≡ q
-p≥q⇒p⊓q≡q {p@record{}} {q@record{}} p≥q with p ≤ᵇ q | inspect (p ≤ᵇ_) q
-... | true  | [ p≤q ] = ≤-antisym (≤ᵇ⇒≤ (subst T (sym p≤q) _)) p≥q
-... | false | [ p≤q ] = refl
+p≥q⇒p⊓q≡q {p@record{}} {q@record{}} p≥q with p ≤ᵇ q in p≤q
+... | true  = ≤-antisym (≤ᵇ⇒≤ (subst T (sym p≤q) _)) p≥q
+... | false = refl
 
 ⊓-operator : MinOperator ≤-totalPreorder
 ⊓-operator = record
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index 060f917782..5b3eab5fa1 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -333,12 +333,12 @@ module _ {P : Pred A p} where
 
   concat⁺∘concat⁻ : ∀ {n m} (xss : Vec (Vec A n) m) (p : Any P (concat xss)) →
                    concat⁺ (concat⁻ xss p) ≡ p
-  concat⁺∘concat⁻ (xs ∷ xss) p  with ++⁻ xs p | P.inspect (++⁻ xs) p
-  ... | inj₁ pxs | P.[ p=inj₁ ]
-    = P.trans (P.cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (P.sym p=inj₁))
+  concat⁺∘concat⁻ (xs ∷ xss) p  with ++⁻ xs p in eq
+  ... | inj₁ pxs
+    = P.trans (P.cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (P.sym eq))
     $ ++⁺∘++⁻ xs p
-  ... | inj₂ pxss | P.[ p=inj₂ ] rewrite concat⁺∘concat⁻ xss pxss
-    = P.trans (P.cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (P.sym p=inj₂))
+  ... | inj₂ pxss rewrite concat⁺∘concat⁻ xss pxss
+    = P.trans (P.cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (P.sym eq))
     $ ++⁺∘++⁻ xs p
 
   concat⁻∘concat⁺ : ∀ {n m} {xss : Vec (Vec A n) m} (p : Any (Any P) xss) →
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index c9e2d102fd..4764635449 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -60,24 +60,6 @@ _≗_ {A = A} {B = B} = Setoid._≈_ (A →-setoid B)
          (A → Setoid.Carrier B) → setoid A ⟶ B
 →-to-⟶ = :→-to-Π
 
-------------------------------------------------------------------------
--- Inspect
-
--- Inspect can be used when you want to pattern match on the result r
--- of some expression e, and you also need to "remember" that r ≡ e.
-
--- See README.Inspect for an explanation of how/why to use this.
-
-record Reveal_·_is_ {A : Set a} {B : A → Set b}
-                    (f : (x : A) → B x) (x : A) (y : B x) :
-                    Set (a ⊔ b) where
-  constructor [_]
-  field eq : f x ≡ y
-
-inspect : ∀ {A : Set a} {B : A → Set b}
-          (f : (x : A) → B x) (x : A) → Reveal f · x is f x
-inspect f x = [ refl ]
-
 ------------------------------------------------------------------------
 -- Propositionality
 
@@ -120,3 +102,34 @@ module _ (_≟_ : DecidableEquality A) {x y : A} where
 
   ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y
   ≢-≟-identity = dec-no (x ≟ y)
+
+
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+record Reveal_·_is_ {A : Set a} {B : A → Set b}
+                    (f : (x : A) → B x) (x : A) (y : B x) :
+                    Set (a ⊔ b) where
+  constructor [_]
+  field eq : f x ≡ y
+
+inspect : ∀ {A : Set a} {B : A → Set b}
+          (f : (x : A) → B x) (x : A) → Reveal f · x is f x
+inspect f x = [ refl ]
+
+{-# WARNING_ON_USAGE Reveal_·_is_
+"Warning: Reveal_·_is_ was deprecated in v2.0.
+Please use new `with ... in` syntax described at https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality instead."
+#-}
+{-# WARNING_ON_USAGE inspect
+"Warning: inspect was deprecated in v2.0.
+Please use new `with ... in` syntax described at https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality instead."
+#-}
+

From 24a1b4891c82e38addd66fd273cc15e4ba423a8d Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Mon, 15 May 2023 19:07:04 +0200
Subject: [PATCH 042/316] Bump GHC to 9.6 for agda-stdlib-utils

---
 .github/haskell-ci.patch         |  4 +--
 .github/workflows/haskell-ci.yml | 45 ++++++++++++++++++++------------
 agda-stdlib-utils.cabal          |  7 ++---
 3 files changed, 35 insertions(+), 21 deletions(-)

diff --git a/.github/haskell-ci.patch b/.github/haskell-ci.patch
index cffc543501..c451df30df 100644
--- a/.github/haskell-ci.patch
+++ b/.github/haskell-ci.patch
@@ -8,7 +8,7 @@
 +      - .github/workflows/haskell-ci.yml
 +      - agda-stdlib-utils.cabal
 +      - cabal.haskell-ci
-+      - *.hs
++      - "*.hs"
    pull_request:
      branches:
        - master
@@ -17,7 +17,7 @@
 +      - .github/workflows/haskell-ci.yml
 +      - agda-stdlib-utils.cabal
 +      - cabal.haskell-ci
-+      - *.hs
++      - "*.hs"
  jobs:
    linux:
      name: Haskell-CI - Linux - ${{ matrix.compiler }}
diff --git a/.github/workflows/haskell-ci.yml b/.github/workflows/haskell-ci.yml
index 2fd1064fae..0aa1080f44 100644
--- a/.github/workflows/haskell-ci.yml
+++ b/.github/workflows/haskell-ci.yml
@@ -8,9 +8,9 @@
 #
 # For more information, see https://github.com/haskell-CI/haskell-ci
 #
-# version: 0.15.20230217
+# version: 0.16.1
 #
-# REGENDATA ("0.15.20230217",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
+# REGENDATA ("0.16.1",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
 #
 name: Haskell-CI
 on:
@@ -18,10 +18,20 @@ on:
     branches:
       - master
       - experimental
+    paths:
+      - .github/workflows/haskell-ci.yml
+      - agda-stdlib-utils.cabal
+      - cabal.haskell-ci
+      - "*.hs"
   pull_request:
     branches:
       - master
       - experimental
+    paths:
+      - .github/workflows/haskell-ci.yml
+      - agda-stdlib-utils.cabal
+      - cabal.haskell-ci
+      - "*.hs"
 jobs:
   linux:
     name: Haskell-CI - Linux - ${{ matrix.compiler }}
@@ -34,14 +44,19 @@ jobs:
     strategy:
       matrix:
         include:
-          - compiler: ghc-9.4.4
+          - compiler: ghc-9.6.1
             compilerKind: ghc
-            compilerVersion: 9.4.4
+            compilerVersion: 9.6.1
             setup-method: ghcup
             allow-failure: false
-          - compiler: ghc-9.2.6
+          - compiler: ghc-9.4.5
             compilerKind: ghc
-            compilerVersion: 9.2.6
+            compilerVersion: 9.4.5
+            setup-method: ghcup
+            allow-failure: false
+          - compiler: ghc-9.2.7
+            compilerKind: ghc
+            compilerVersion: 9.2.7
             setup-method: ghcup
             allow-failure: false
           - compiler: ghc-9.0.2
@@ -72,20 +87,18 @@ jobs:
           apt-get install -y --no-install-recommends gnupg ca-certificates dirmngr curl git software-properties-common libtinfo5
           if [ "${{ matrix.setup-method }}" = ghcup ]; then
             mkdir -p "$HOME/.ghcup/bin"
-            curl -sL https://downloads.haskell.org/ghcup/0.1.18.0/x86_64-linux-ghcup-0.1.18.0 > "$HOME/.ghcup/bin/ghcup"
+            curl -sL https://downloads.haskell.org/ghcup/0.1.19.2/x86_64-linux-ghcup-0.1.19.2 > "$HOME/.ghcup/bin/ghcup"
             chmod a+x "$HOME/.ghcup/bin/ghcup"
-            "$HOME/.ghcup/bin/ghcup" config add-release-channel https://raw.githubusercontent.com/haskell/ghcup-metadata/master/ghcup-prereleases-0.0.7.yaml;
             "$HOME/.ghcup/bin/ghcup" install ghc "$HCVER" || (cat "$HOME"/.ghcup/logs/*.* && false)
-            "$HOME/.ghcup/bin/ghcup" install cabal 3.9.0.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
+            "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
           else
             apt-add-repository -y 'ppa:hvr/ghc'
             apt-get update
             apt-get install -y "$HCNAME"
             mkdir -p "$HOME/.ghcup/bin"
-            curl -sL https://downloads.haskell.org/ghcup/0.1.18.0/x86_64-linux-ghcup-0.1.18.0 > "$HOME/.ghcup/bin/ghcup"
+            curl -sL https://downloads.haskell.org/ghcup/0.1.19.2/x86_64-linux-ghcup-0.1.19.2 > "$HOME/.ghcup/bin/ghcup"
             chmod a+x "$HOME/.ghcup/bin/ghcup"
-            "$HOME/.ghcup/bin/ghcup" config add-release-channel https://raw.githubusercontent.com/haskell/ghcup-metadata/master/ghcup-prereleases-0.0.7.yaml;
-            "$HOME/.ghcup/bin/ghcup" install cabal 3.9.0.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
+            "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
           fi
         env:
           HCKIND: ${{ matrix.compilerKind }}
@@ -103,13 +116,13 @@ jobs:
             echo "HC=$HC" >> "$GITHUB_ENV"
             echo "HCPKG=$HOME/.ghcup/bin/$HCKIND-pkg-$HCVER" >> "$GITHUB_ENV"
             echo "HADDOCK=$HOME/.ghcup/bin/haddock-$HCVER" >> "$GITHUB_ENV"
-            echo "CABAL=$HOME/.ghcup/bin/cabal-3.9.0.0 -vnormal+nowrap" >> "$GITHUB_ENV"
+            echo "CABAL=$HOME/.ghcup/bin/cabal-3.10.1.0 -vnormal+nowrap" >> "$GITHUB_ENV"
           else
             HC=$HCDIR/bin/$HCKIND
             echo "HC=$HC" >> "$GITHUB_ENV"
             echo "HCPKG=$HCDIR/bin/$HCKIND-pkg" >> "$GITHUB_ENV"
             echo "HADDOCK=$HCDIR/bin/haddock" >> "$GITHUB_ENV"
-            echo "CABAL=$HOME/.ghcup/bin/cabal-3.9.0.0 -vnormal+nowrap" >> "$GITHUB_ENV"
+            echo "CABAL=$HOME/.ghcup/bin/cabal-3.10.1.0 -vnormal+nowrap" >> "$GITHUB_ENV"
           fi
 
           HCNUMVER=$(${HC} --numeric-version|perl -ne '/^(\d+)\.(\d+)\.(\d+)(\.(\d+))?$/; print(10000 * $1 + 100 * $2 + ($3 == 0 ? $5 != 1 : $3))')
@@ -161,8 +174,8 @@ jobs:
       - name: install cabal-plan
         run: |
           mkdir -p $HOME/.cabal/bin
-          curl -sL https://github.com/haskell-hvr/cabal-plan/releases/download/v0.6.2.0/cabal-plan-0.6.2.0-x86_64-linux.xz > cabal-plan.xz
-          echo 'de73600b1836d3f55e32d80385acc055fd97f60eaa0ab68a755302685f5d81bc  cabal-plan.xz' | sha256sum -c -
+          curl -sL https://github.com/haskell-hvr/cabal-plan/releases/download/v0.7.3.0/cabal-plan-0.7.3.0-x86_64-linux.xz > cabal-plan.xz
+          echo 'f62ccb2971567a5f638f2005ad3173dba14693a45154c1508645c52289714cb2  cabal-plan.xz' | sha256sum -c -
           xz -d < cabal-plan.xz > $HOME/.cabal/bin/cabal-plan
           rm -f cabal-plan.xz
           chmod a+x $HOME/.cabal/bin/cabal-plan
diff --git a/agda-stdlib-utils.cabal b/agda-stdlib-utils.cabal
index 880494db15..7d93147747 100644
--- a/agda-stdlib-utils.cabal
+++ b/agda-stdlib-utils.cabal
@@ -6,8 +6,9 @@ description:     Helper programs for setting up the Agda standard library.
 license:         MIT
 
 tested-with:
-  GHC == 9.4.4
-  GHC == 9.2.6
+  GHC == 9.6.1
+  GHC == 9.4.5
+  GHC == 9.2.7
   GHC == 9.0.2
   GHC == 8.10.7
   GHC == 8.8.4
@@ -21,7 +22,7 @@ common common-build-parameters
     PatternSynonyms
 
   build-depends:
-      base          >= 4.12.0.0 && < 4.18
+      base          >= 4.12.0.0 && < 4.19
     , filemanip     >= 0.3.6.2  && < 0.4
 
 executable GenerateEverything

From 0fec2c6c051694d40c5cbe7eab056818984ddf3d Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Sat, 10 Jun 2023 08:14:38 +0000
Subject: [PATCH 043/316] Add apartness reasoning (#1973)

---
 CHANGELOG.md                                  |  12 ++
 .../Binary/Reasoning/Base/Apartness.agda      | 106 ++++++++++++++++++
 2 files changed, 118 insertions(+)
 create mode 100644 src/Relation/Binary/Reasoning/Base/Apartness.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 52b2a6762d..62e7f38907 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3054,3 +3054,15 @@ This is a full list of proofs that have changed form to use irrelevant instance
   ```agda
   ↭-reverse : (xs : List A) → reverse xs ↭ xs
   ```
+
+* Added new file `Relation.Binary.Reasoning.Base.Apartness`
+
+This is how to use it:
+  ```agda
+  _ : a # d
+  _ = begin-apartness
+    a ≈⟨ a≈b ⟩
+    b #⟨ b#c ⟩
+    c ≈⟨ c≈d ⟩
+    d ∎
+  ```
diff --git a/src/Relation/Binary/Reasoning/Base/Apartness.agda b/src/Relation/Binary/Reasoning/Base/Apartness.agda
new file mode 100644
index 0000000000..7a52568664
--- /dev/null
+++ b/src/Relation/Binary/Reasoning/Base/Apartness.agda
@@ -0,0 +1,106 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The basic code for equational reasoning with three relations:
+-- equality and apartness
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary
+
+module Relation.Binary.Reasoning.Base.Apartness {a ℓ₁ ℓ₂} {A : Set a}
+  {_≈_ : Rel A ℓ₁} {_#_ : Rel A ℓ₂}
+  (≈-equiv : IsEquivalence _≈_)
+  (#-trans : Transitive _#_) (#-sym   : Symmetric _#_)
+  (#-≈-trans : Trans _#_ _≈_ _#_) (≈-#-trans : Trans _≈_ _#_ _#_)
+  where
+
+open import Level using (Level; _⊔_)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym)
+open import Relation.Nullary.Decidable using (Dec; yes; no)
+open import Relation.Nullary.Decidable using (True; toWitness)
+
+open IsEquivalence ≈-equiv
+  renaming
+  ( refl  to ≈-refl
+  ; sym   to ≈-sym
+  ; trans to ≈-trans
+  )
+
+infix 4 _IsRelatedTo_
+
+data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
+  nothing   :                 x IsRelatedTo y
+  apartness : (x#y : x # y) → x IsRelatedTo y
+  equals    : (x≈y : x ≈ y) → x IsRelatedTo y
+
+data IsApartness {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
+  isApartness : ∀ x#y → IsApartness (apartness x#y)
+
+IsApartness? : ∀ {x y} (x#y : x IsRelatedTo y) → Dec (IsApartness x#y)
+IsApartness? nothing         = no λ()
+IsApartness? (apartness x#y) = yes (isApartness x#y)
+IsApartness? (equals x≈y)    = no (λ ())
+
+extractApartness : ∀ {x y} {x#y : x IsRelatedTo y} → IsApartness x#y → x # y
+extractApartness (isApartness x#y) = x#y
+
+data IsEquality {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
+  isEquality : ∀ x≈y → IsEquality (equals x≈y)
+
+IsEquality? : ∀ {x y} (x≲y : x IsRelatedTo y) → Dec (IsEquality x≲y)
+IsEquality? nothing       = no λ()
+IsEquality? (apartness _) = no λ()
+IsEquality? (equals  x≈y) = yes (isEquality x≈y)
+
+extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y
+extractEquality (isEquality x≈y) = x≈y
+
+infix  1 begin-apartness_ begin-equality_
+infixr 2 step-≈ step-≈˘ step-≡ step-≡˘ step-# step-#˘ _≡⟨⟩_
+infix  3 _∎
+
+begin-apartness_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsApartness? r)} → x # y
+begin-apartness_ _ {s} = extractApartness (toWitness s)
+
+begin-equality_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsEquality? r)} → x ≈ y
+begin-equality_ _ {s} = extractEquality (toWitness s)
+
+step-# : ∀ (x : A) {y z} → y IsRelatedTo z → x # y → x IsRelatedTo z
+step-# x nothing  _          = nothing
+step-# x (apartness y#z) x#y = nothing
+step-# x (equals    y≈z) x#y = apartness (#-≈-trans x#y y≈z)
+
+step-#˘ : ∀ (x : A) {y z} → y IsRelatedTo z → y # x → x IsRelatedTo z
+step-#˘ x y-z y#x = step-# x y-z (#-sym y#x)
+
+step-≈  : ∀ (x : A) {y z} → y IsRelatedTo z → x ≈ y → x IsRelatedTo z
+step-≈ x nothing         x≈y = nothing
+step-≈ x (apartness y#z) x≈y = apartness (≈-#-trans x≈y y#z)
+step-≈ x (equals    y≈z) x≈y = equals    (≈-trans   x≈y y≈z)
+
+step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z
+step-≈˘ x y∼z x≈y = step-≈ x y∼z (≈-sym x≈y)
+
+step-≡ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z
+step-≡ x nothing _            = nothing
+step-≡ x (apartness x#y) refl = apartness x#y
+step-≡ x (equals    x≈y) refl = equals    x≈y
+
+step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z
+step-≡˘ x y∼z x≡y = step-≡ x y∼z (sym x≡y)
+
+_≡⟨⟩_ : ∀ (x : A) {y} → x IsRelatedTo y → x IsRelatedTo y
+x ≡⟨⟩ x-y = x-y
+
+_∎ : ∀ x → x IsRelatedTo x
+x ∎ = equals ≈-refl
+
+syntax step-#  x y∼z x#y = x #⟨  x#y ⟩ y∼z
+syntax step-#˘ x y∼z x#y = x #˘⟨ x#y ⟩ y∼z
+syntax step-≈  x y∼z x≈y = x ≈⟨  x≈y ⟩ y∼z
+syntax step-≈˘ x y∼z y≈x = x ≈˘⟨ y≈x ⟩ y∼z
+syntax step-≡  x y∼z x≡y = x ≡⟨  x≡y ⟩ y∼z
+syntax step-≡˘ x y∼z y≡x = x ≡˘⟨ y≡x ⟩ y∼z

From fb29b2f063bb6ae6db967af4b514487a99aa1324 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 10 Jun 2023 09:23:33 +0100
Subject: [PATCH 044/316] Addd middle four exchange law (#1953)

---
 CHANGELOG.md                                | 24 +++++++++++++++
 src/Algebra/Consequences/Propositional.agda | 24 ++++++++++++++-
 src/Algebra/Consequences/Setoid.agda        | 34 +++++++++++++++++++++
 src/Algebra/Definitions.agda                |  4 +++
 4 files changed, 85 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 62e7f38907..d5ba31beaf 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1646,8 +1646,30 @@ Other minor changes
                                      Involutive _≈_ f
   ```
 
+* Added new proofs to `Algebra.Consequences.Propositional`:
+  ```agda
+  comm+assoc⇒middleFour     : Commutative _•_ →
+                              Associative _•_ →
+                              _•_ MiddleFourExchange _•_
+  identity+middleFour⇒assoc : Identity e _•_ →
+                              _•_ MiddleFourExchange _•_ →
+                              Associative _•_
+  identity+middleFour⇒comm  : Identity e _+_ →
+                              _•_ MiddleFourExchange _+_ →
+                              Commutative _•_
+  ```
+
 * Added new proofs to `Algebra.Consequences.Setoid`:
   ```agda
+  comm+assoc⇒middleFour     : Congruent₂ _•_ → Commutative _•_ → Associative _•_ →
+                              _•_ MiddleFourExchange _•_
+  identity+middleFour⇒assoc : Congruent₂ _•_ → Identity e _•_ →
+                              _•_ MiddleFourExchange _•_ →
+                              Associative _•_
+  identity+middleFour⇒comm  : Congruent₂ _•_ → Identity e _+_ →
+                              _•_ MiddleFourExchange _+_ →
+                              Commutative _•_
+
   involutive⇒surjective  : Involutive f  → Surjective f
   selfInverse⇒involutive : SelfInverse f → Involutive f
   selfInverse⇒congruent  : SelfInverse f → Congruent f
@@ -1711,6 +1733,8 @@ Other minor changes
 
 * Added new definition to `Algebra.Definitions`:
   ```agda
+  _MiddleFourExchange_ : Op₂ A → Op₂ A → Set _
+
   SelfInverse : Op₁ A → Set _
   
   LeftDividesˡ  : Op₂ A → Op₂ A → Set _
diff --git a/src/Algebra/Consequences/Propositional.agda b/src/Algebra/Consequences/Propositional.agda
index 05278fea34..7c3030f4a6 100644
--- a/src/Algebra/Consequences/Propositional.agda
+++ b/src/Algebra/Consequences/Propositional.agda
@@ -24,7 +24,10 @@ import Algebra.Consequences.Setoid (setoid A) as Base
 
 open Base public
   hiding
-  ( assoc+distribʳ+idʳ+invʳ⇒zeˡ
+  ( comm+assoc⇒middleFour
+  ; identity+middleFour⇒assoc
+  ; identity+middleFour⇒comm
+  ; assoc+distribʳ+idʳ+invʳ⇒zeˡ
   ; assoc+distribˡ+idʳ+invʳ⇒zeʳ
   ; assoc+id+invʳ⇒invˡ-unique
   ; assoc+id+invˡ⇒invʳ-unique
@@ -90,6 +93,25 @@ module _ {_•_ : Op₂ A} where
   sel⇒idem : Selective _•_ → Idempotent _•_
   sel⇒idem = Base.sel⇒idem _≡_
 
+------------------------------------------------------------------------
+-- Middle-Four Exchange
+
+module _ {_•_ : Op₂ A} where
+
+  comm+assoc⇒middleFour : Commutative _•_ → Associative _•_ →
+                          _•_ MiddleFourExchange _•_
+  comm+assoc⇒middleFour = Base.comm+assoc⇒middleFour (cong₂ _•_)
+
+  identity+middleFour⇒assoc : {e : A} → Identity e _•_ →
+                              _•_ MiddleFourExchange _•_ →
+                              Associative _•_
+  identity+middleFour⇒assoc = Base.identity+middleFour⇒assoc (cong₂ _•_)
+
+  identity+middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
+                             _•_ MiddleFourExchange _+_ →
+                             Commutative _•_
+  identity+middleFour⇒comm = Base.identity+middleFour⇒comm (cong₂ _•_)
+
 ------------------------------------------------------------------------
 -- Without Loss of Generality
 
diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index c51a6ee8b1..77badd1b90 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -29,6 +29,40 @@ open import Relation.Unary using (Pred)
 
 open import Algebra.Consequences.Base public
 
+------------------------------------------------------------------------
+-- MiddleFourExchange
+
+module _ {_•_ : Op₂ A} (cong : Congruent₂ _•_) where
+
+  comm+assoc⇒middleFour : Commutative _•_ → Associative _•_ →
+                          _•_ MiddleFourExchange _•_
+  comm+assoc⇒middleFour comm assoc w x y z = begin
+    (w • x) • (y • z) ≈⟨ assoc w x (y • z) ⟩
+    w • (x • (y • z)) ≈⟨ cong refl (sym (assoc x y z)) ⟩
+    w • ((x • y) • z) ≈⟨ cong refl (cong (comm x y) refl) ⟩
+    w • ((y • x) • z) ≈⟨ cong refl (assoc y x z) ⟩
+    w • (y • (x • z)) ≈⟨ sym (assoc w y (x • z)) ⟩
+    (w • y) • (x • z) ∎
+
+  identity+middleFour⇒assoc : {e : A} → Identity e _•_ →
+                              _•_ MiddleFourExchange _•_ →
+                              Associative _•_
+  identity+middleFour⇒assoc {e} (identityˡ , identityʳ) middleFour x y z = begin
+    (x • y) • z       ≈⟨ cong refl (sym (identityˡ z)) ⟩
+    (x • y) • (e • z) ≈⟨ middleFour x y e z ⟩
+    (x • e) • (y • z) ≈⟨ cong (identityʳ x) refl ⟩
+    x • (y • z)       ∎
+
+  identity+middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
+                             _•_ MiddleFourExchange _+_ →
+                             Commutative _•_
+  identity+middleFour⇒comm {_+_} {e} (identityˡ , identityʳ) middleFour x y
+    = begin
+    x • y             ≈⟨ sym (cong (identityˡ x) (identityʳ y)) ⟩
+    (e + x) • (y + e) ≈⟨ middleFour e x y e ⟩
+    (e + y) • (x + e) ≈⟨ cong (identityˡ y) (identityʳ x) ⟩
+    y • x             ∎
+
 ------------------------------------------------------------------------
 -- Involutive/SelfInverse functions
 
diff --git a/src/Algebra/Definitions.agda b/src/Algebra/Definitions.agda
index e9e082e7e1..c0820fafb7 100644
--- a/src/Algebra/Definitions.agda
+++ b/src/Algebra/Definitions.agda
@@ -108,6 +108,10 @@ _*_ DistributesOverʳ _+_ =
 _DistributesOver_ : Op₂ A → Op₂ A → Set _
 * DistributesOver + = (* DistributesOverˡ +) × (* DistributesOverʳ +)
 
+_MiddleFourExchange_ : Op₂ A → Op₂ A → Set _
+_*_ MiddleFourExchange _+_ = 
+  ∀ w x y z → ((w + x) * (y + z)) ≈ ((w + y) * (x + z))
+
 _IdempotentOn_ : Op₂ A → A → Set _
 _∙_ IdempotentOn x = (x ∙ x) ≈ x
 

From fd20e62fe88a424c65639698d9cf19e0a5d84413 Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Wed, 14 Jun 2023 20:41:30 -0400
Subject: [PATCH 045/316] Add take-[] and drop-[] to `Data.List.Properties`

---
 CHANGELOG.md                  |  6 ++++++
 src/Data/List/Properties.agda | 11 +++++++++++
 2 files changed, 17 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d5ba31beaf..8d64ba03c3 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2784,6 +2784,12 @@ Other minor changes
   foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
   foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
   ```
+  
+ * Added new lemmas in `Data.List.Propreties`:
+  ```
+  take-[] : ∀ m → take  m [] ≡ []
+  drop-[] : ∀ m → drop  m [] ≡ []
+  ```
 
 NonZero/Positive/Negative changes
 ---------------------------------
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 312ef2e7a8..58d5b39680 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -758,6 +758,11 @@ length-take zero    xs       = refl
 length-take (suc n) []       = refl
 length-take (suc n) (x ∷ xs) = cong suc (length-take n xs)
 
+-- Taking from an empty list does nothing.
+take-[] : ∀ m → take {A = A} m [] ≡ []
+take-[] zero = refl
+take-[] (suc m) = refl
+
 ------------------------------------------------------------------------
 -- drop
 
@@ -766,6 +771,12 @@ length-drop zero    xs       = refl
 length-drop (suc n) []       = refl
 length-drop (suc n) (x ∷ xs) = length-drop n xs
 
+-- Dropping from an empty list does nothing.
+drop-[] : ∀ m → drop {A = A} m [] ≡ []
+drop-[] zero = refl
+drop-[] (suc m) = refl
+
+
 take++drop : ∀ n (xs : List A) → take n xs ++ drop n xs ≡ xs
 take++drop zero    xs       = refl
 take++drop (suc n) []       = refl

From d3e8a84d6191b30a516dc4f9b35dcd613bed9451 Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Thu, 15 Jun 2023 00:48:17 +0000
Subject: [PATCH 046/316] =?UTF-8?q?Added=20cong=20and=20map=20to=20vec=20?=
 =?UTF-8?q?=E2=89=8B=20equality=20(#1956)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                                          | 10 ++++++++++
 src/Data/Vec/Relation/Binary/Equality/Setoid.agda     |  7 +++++++
 src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda | 10 ++++++++++
 3 files changed, 27 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8d64ba03c3..a2e132acff 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3080,6 +3080,16 @@ This is a full list of proofs that have changed form to use irrelevant instance
   <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
   ```
 
+* Added new function to `Data.Vec.Relation.Binary.Pointwise.Inductive`
+  ```agda
+  cong-[_]≔ : Pointwise _∼_ xs ys → Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)
+  ```
+
+* Added new function to `Data.Vec.Relation.Binary.Equality.Setoid`
+  ```agda
+  map-[]≔ : map f (xs [ i ]≔ p) ≋ map f xs [ i ]≔ f p
+  ```
+
 * Added new function to `Data.List.Relation.Binary.Permutation.Propositional.Properties`
   ```agda
   ↭-reverse : (xs : List A) → reverse xs ↭ xs
diff --git a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
index 86ed09b93d..116b104829 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
@@ -12,6 +12,7 @@ module Data.Vec.Relation.Binary.Equality.Setoid
   {a ℓ} (S : Setoid a ℓ) where
 
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
+open import Data.Fin using (zero; suc)
 open import Data.Vec.Base
 open import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
   using (Pointwise)
@@ -79,6 +80,12 @@ map-++ : ∀ {b m n} {B : Set b}
 map-++ f []       = ≋-refl
 map-++ f (x ∷ xs) = refl ∷ map-++ f xs
 
+map-[]≔ : ∀ {b n} {B : Set b}
+          (f : B → A) (xs : Vec B n) i p →
+          map f (xs [ i ]≔ p) ≋ map f xs [ i ]≔ f p
+map-[]≔ f (x ∷ xs) zero    p = refl ∷ ≋-refl
+map-[]≔ f (x ∷ xs) (suc i) p = refl ∷ map-[]≔ f xs i p
+
 ------------------------------------------------------------------------
 -- concat
 
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
index 54a85d7115..e435e2681b 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
@@ -217,6 +217,16 @@ module _ {_∼_ : REL A B ℓ} where
   tabulate⁻ (f₀∼g₀ ∷ _)   zero    = f₀∼g₀
   tabulate⁻ (_     ∷ f∼g) (suc i) = tabulate⁻ f∼g i
 
+------------------------------------------------------------------------
+-- cong
+
+module _ {_∼_ : Rel A ℓ} (refl : Reflexive _∼_) where
+  cong-[_]≔ : ∀ {n} i p {xs} {ys} →
+              Pointwise _∼_ {n} xs ys →
+              Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)
+  cong-[ zero ]≔  p (_   ∷ eqn) = refl ∷ eqn
+  cong-[ suc i ]≔ p (x∼y ∷ eqn) = x∼y  ∷ cong-[ i ]≔ p eqn
+
 ------------------------------------------------------------------------
 -- Degenerate pointwise relations
 

From 0376427c9a0605d0c6de64819922c475b51439ec Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sun, 18 Jun 2023 21:25:51 -0400
Subject: [PATCH 047/316] Fixities for `Data` and `Codata` (#1985)

---
 CHANGELOG.md                                  | 29 +++++++++++++++++++
 src/Algebra/Definitions.agda                  |  2 +-
 src/Codata/Sized/Conat.agda                   |  4 ++-
 src/Data/List/Kleene/Base.agda                |  4 +++
 src/Data/List/Membership/Propositional.agda   |  2 ++
 src/Data/List/Reflection.agda                 |  2 ++
 .../Binary/Equality/DecPropositional.agda     |  2 ++
 .../Relation/Binary/Prefix/Heterogeneous.agda |  2 ++
 .../Relation/Binary/Suffix/Heterogeneous.agda |  2 ++
 src/Data/List/Relation/Ternary/Appending.agda |  2 ++
 .../Ternary/Appending/Propositional.agda      |  2 ++
 src/Data/List/Relation/Unary/Any.agda         |  2 ++
 src/Data/Parity/Base.agda                     |  2 ++
 .../Function/NonDependent/Propositional.agda  |  4 +++
 .../Product/Function/NonDependent/Setoid.agda |  6 ++++
 src/Data/Sum/Function/Propositional.agda      |  4 +++
 src/Data/Sum/Function/Setoid.agda             |  5 ++++
 src/Data/Vec/Membership/Setoid.agda           |  2 ++
 src/Data/Vec/Reflection.agda                  |  2 ++
 19 files changed, 78 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index a2e132acff..2dd7726462 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -29,6 +29,35 @@ Highlights
 Bug-fixes
 ---------
 
+* The following operators were missing a fixity declaration, which has now
+  been fixed -
+  ```
+  infix  4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
+  infix  6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
+  infixl 4 _+ _*                                            (Data.List.Kleene.Base)
+  infixr 4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
+  infix  4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
+  infix  4 _≢∈_                                             (Data.List.Membership.Propositional)
+  infixr 5 _`∷_                                             (Data.List.Reflection)
+  infix  4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
+  infixr 5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
+  infixr 5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
+  infixr 5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
+  infixr 5 _∷=_                                             (Data.List.Relation.Unary.Any)
+  infixr 5 _++_                                             (Data.List.Ternary.Appending)
+  infixr 2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
+  infixr 2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
+  infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
+  infixr 2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
+  infixr 1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
+  infixr 1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
+  infixr 1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
+  infixr 1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
+  infix  8 _⁻¹                                              (Data.Parity.Base)
+  infixr 5 _`∷_                                             (Data.Vec.Reflection)
+  infixr 5 _∷=_                                             (Data.Vec.Membership.Setoid)
+  ```
+
 * In `System.Exit`, the `ExitFailure` constructor is now carrying an integer
   rather than a natural. The previous binding was incorrectly assuming that
   all exit codes where non-negative.
diff --git a/src/Algebra/Definitions.agda b/src/Algebra/Definitions.agda
index c0820fafb7..e64fc4f30b 100644
--- a/src/Algebra/Definitions.agda
+++ b/src/Algebra/Definitions.agda
@@ -109,7 +109,7 @@ _DistributesOver_ : Op₂ A → Op₂ A → Set _
 * DistributesOver + = (* DistributesOverˡ +) × (* DistributesOverʳ +)
 
 _MiddleFourExchange_ : Op₂ A → Op₂ A → Set _
-_*_ MiddleFourExchange _+_ = 
+_*_ MiddleFourExchange _+_ =
   ∀ w x y z → ((w + x) * (y + z)) ≈ ((w + y) * (x + z))
 
 _IdempotentOn_ : Op₂ A → A → Set _
diff --git a/src/Codata/Sized/Conat.agda b/src/Codata/Sized/Conat.agda
index 8711e26199..d57e3e56e7 100644
--- a/src/Codata/Sized/Conat.agda
+++ b/src/Codata/Sized/Conat.agda
@@ -35,7 +35,7 @@ pred : ∀ {i} {j : Size< i} → Conat i → Conat j
 pred zero    = zero
 pred (suc n) = n .force
 
-infixl 6 _∸_ _+_
+infixl 6 _∸_ _+_ _ℕ+_ _+ℕ_
 infixl 7 _*_
 
 _∸_ : Conat ∞ → ℕ → Conat ∞
@@ -91,6 +91,8 @@ toℕ (suc n) = suc (toℕ n)
 ------------------------------------------------------------------------
 -- Order wrt to Nat
 
+infix 4 _ℕ<_ _ℕ≤infinity _ℕ≤_
+
 data _ℕ≤_ : ℕ → Conat ∞ → Set where
   zℕ≤n : ∀ {n} → zero ℕ≤ n
   sℕ≤s : ∀ {k n} → k ℕ≤ n .force → suc k ℕ≤ suc n
diff --git a/src/Data/List/Kleene/Base.agda b/src/Data/List/Kleene/Base.agda
index 906af96ea1..c8476318b7 100644
--- a/src/Data/List/Kleene/Base.agda
+++ b/src/Data/List/Kleene/Base.agda
@@ -40,6 +40,7 @@ private
 --   simplify certain proofs.
 
 infixr 5 _&_ ∹_
+infixl 4 _+ _*
 
 record _+ {a} (A : Set a) : Set a
 data _* {a} (A : Set a) : Set a
@@ -101,6 +102,7 @@ module _ (f : B → A → B) where
 -- Concatenation
 
 module Concat where
+  infixr 4 _++++_ _+++*_ _*+++_ _*++*_
   _++++_ : A + → A + → A +
   _+++*_ : A + → A * → A +
   _*+++_ : A * → A + → A +
@@ -270,6 +272,8 @@ module _ (f : A → Maybe B → B) where
 ------------------------------------------------------------------------
 -- Indexing
 
+infix 4 _[_]* _[_]+
+
 _[_]* : A * → ℕ → Maybe A
 _[_]+ : A + → ℕ → Maybe A
 
diff --git a/src/Data/List/Membership/Propositional.agda b/src/Data/List/Membership/Propositional.agda
index 84ce10c8ba..4a74db3d04 100644
--- a/src/Data/List/Membership/Propositional.agda
+++ b/src/Data/List/Membership/Propositional.agda
@@ -22,6 +22,8 @@ open SetoidMembership (setoid A) public hiding (lose)
 ------------------------------------------------------------------------
 -- Different members
 
+infix 4 _≢∈_
+
 _≢∈_ : ∀ {x y : A} {xs} → x ∈ xs → y ∈ xs → Set _
 _≢∈_ x∈xs y∈xs = ∀ x≡y → subst (_∈ _) x≡y x∈xs ≢ y∈xs
 
diff --git a/src/Data/List/Reflection.agda b/src/Data/List/Reflection.agda
index 20063430c4..3d4ab919a1 100644
--- a/src/Data/List/Reflection.agda
+++ b/src/Data/List/Reflection.agda
@@ -24,6 +24,8 @@ open import Reflection.AST.Argument
 `[] : Term
 `[] = con (quote List.[]) (2 ⋯⟅∷⟆ [])
 
+infixr 5 _`∷_
+
 _`∷_ : Term → Term → Term
 x `∷ xs = con (quote List._∷_) (2 ⋯⟅∷⟆ x ⟨∷⟩ xs ⟨∷⟩ [])
 
diff --git a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
index 6410ef418c..1697020ed3 100644
--- a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
@@ -32,5 +32,7 @@ open DecSetoidEq (decSetoid _≟_) public
 ------------------------------------------------------------------------
 -- Additional proofs
 
+infix 4 _≡?_
+
 _≡?_ : Decidable (_≡_ {A = List A})
 _≡?_ = ≡-dec _≟_
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
index 648663fc95..bf107dfcc3 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
@@ -45,6 +45,8 @@ module _ {a b r s} {A : Set a} {B : Set b} {R : REL A B r} {S : REL A B s} where
 
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
+  infixr 5 _++ᵖ_
+
   _++ᵖ_ : ∀ {as bs} → Prefix R as bs → ∀ suf → Prefix R as (bs List.++ suf)
   []       ++ᵖ suf = []
   (r ∷ rs) ++ᵖ suf = r ∷ (rs ++ᵖ suf)
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
index 1d84c07b50..984b857c59 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
@@ -31,6 +31,8 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   tail (here (_ ∷ rs)) = there (here rs)
   tail (there x) = there (tail x)
 
+  infixr 5 _++ˢ_
+
   _++ˢ_ : ∀ pre {as bs} → Suffix R as bs → Suffix R as (pre List.++ bs)
   []       ++ˢ rs = rs
   (x ∷ xs) ++ˢ rs = there (xs ++ˢ rs)
diff --git a/src/Data/List/Relation/Ternary/Appending.agda b/src/Data/List/Relation/Ternary/Appending.agda
index 025364f122..75e50d15af 100644
--- a/src/Data/List/Relation/Ternary/Appending.agda
+++ b/src/Data/List/Relation/Ternary/Appending.agda
@@ -38,6 +38,8 @@ module _ (L : REL A C l) (R : REL B C r) where
 ------------------------------------------------------------------------
 -- Functions manipulating Appending
 
+infixr 5 _++_
+
 _++_ : ∀ {cs₁ cs₂ : List C} → Pointwise L as cs₁ → Pointwise R bs cs₂ →
        Appending L R as bs (cs₁ List.++ cs₂)
 []       ++ rs = []++ rs
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional.agda b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
index 46cb27f75e..6149e2a236 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
@@ -32,6 +32,8 @@ open General public
 ------------------------------------------------------------------------
 -- Definition
 
+infixr 5 _++_ _++[]
+
 _++_ : (as bs : List A) → Appending as bs (as List.++ bs)
 as ++ bs = Pw.≡⇒Pointwise-≡ refl General.++ Pw.≡⇒Pointwise-≡ refl
 
diff --git a/src/Data/List/Relation/Unary/Any.agda b/src/Data/List/Relation/Unary/Any.agda
index 58bdc3a5fa..19af845152 100644
--- a/src/Data/List/Relation/Unary/Any.agda
+++ b/src/Data/List/Relation/Unary/Any.agda
@@ -62,6 +62,8 @@ index (there pxs) = suc (index pxs)
 lookup : {P : Pred A p} → Any P xs → A
 lookup {xs = xs} p = List.lookup xs (index p)
 
+infixr 5 _∷=_
+
 _∷=_ : {P : Pred A p} → Any P xs → A → List A
 _∷=_ {xs = xs} x∈xs v = xs List.[ index x∈xs ]∷= v
 
diff --git a/src/Data/Parity/Base.agda b/src/Data/Parity/Base.agda
index 6f9e85f9a0..0ac22a9c7a 100644
--- a/src/Data/Parity/Base.agda
+++ b/src/Data/Parity/Base.agda
@@ -26,6 +26,8 @@ data Parity : Set where
 
 -- The opposite parity.
 
+infix 8 _⁻¹
+
 _⁻¹ : Parity → Parity
 1ℙ ⁻¹ = 0ℙ
 0ℙ ⁻¹ = 1ℙ
diff --git a/src/Data/Product/Function/NonDependent/Propositional.agda b/src/Data/Product/Function/NonDependent/Propositional.agda
index f5b4973184..e686cbad6e 100644
--- a/src/Data/Product/Function/NonDependent/Propositional.agda
+++ b/src/Data/Product/Function/NonDependent/Propositional.agda
@@ -27,6 +27,8 @@ open import Function.Surjection as Surj using (_↠_; module Surjection)
 
 module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
 
+  infixr 2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_
+
   _×-⇔_ : A ⇔ B → C ⇔ D → (A × C) ⇔ (B × D)
   _×-⇔_ A⇔B C⇔D =
     Inverse.equivalence Pointwise-≡↔≡           ⟨∘⟩
@@ -64,6 +66,8 @@ module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
 
 module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
 
+  infixr 2 _×-cong_
+
   _×-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A × C) ∼[ k ] (B × D)
   _×-cong_ {implication}         = λ f g →      map        f         g
   _×-cong_ {reverse-implication} = λ f g → lam (map (app-← f) (app-← g))
diff --git a/src/Data/Product/Function/NonDependent/Setoid.agda b/src/Data/Product/Function/NonDependent/Setoid.agda
index 830e25864b..eb1a369710 100644
--- a/src/Data/Product/Function/NonDependent/Setoid.agda
+++ b/src/Data/Product/Function/NonDependent/Setoid.agda
@@ -33,6 +33,8 @@ module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
           {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
           where
 
+  infixr 2 _×-⟶_
+
   _×-⟶_ : (A ⟶ B) → (C ⟶ D) → (A ×ₛ C) ⟶ (B ×ₛ D)
   _×-⟶_ f g = record
     { _⟨$⟩_ = fg
@@ -76,6 +78,8 @@ module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
           {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
           where
 
+  infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_
+
   _×-equivalence_ : Equivalence A B → Equivalence C D →
                     Equivalence (A ×ₛ C) (B ×ₛ D)
   _×-equivalence_ A⇔B C⇔D = record
@@ -110,6 +114,8 @@ module _ {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
   {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
   where
 
+  infixr 2 _×-surjection_ _×-inverse_
+
   _×-surjection_ : Surjection A B → Surjection C D →
                    Surjection (A ×ₛ C) (B ×ₛ D)
   A↠B ×-surjection C↠D = record
diff --git a/src/Data/Sum/Function/Propositional.agda b/src/Data/Sum/Function/Propositional.agda
index e754ee2356..a22aa64c4a 100644
--- a/src/Data/Sum/Function/Propositional.agda
+++ b/src/Data/Sum/Function/Propositional.agda
@@ -24,6 +24,8 @@ open import Function.Surjection as Surj using (_↠_; module Surjection)
 
 module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
 
+  infixr 1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_
+
   _⊎-⇔_ : A ⇔ B → C ⇔ D → (A ⊎ C) ⇔ (B ⊎ D)
   _⊎-⇔_ A⇔B C⇔D =
     Inverse.equivalence (Pointwise-≡↔≡ B D) ⟨∘⟩
@@ -61,6 +63,8 @@ module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
 
 module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
 
+  infixr 1 _⊎-cong_
+
   _⊎-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A ⊎ C) ∼[ k ] (B ⊎ D)
   _⊎-cong_ {implication}         = map
   _⊎-cong_ {reverse-implication} = λ f g → lam (map (app-← f) (app-← g))
diff --git a/src/Data/Sum/Function/Setoid.agda b/src/Data/Sum/Function/Setoid.agda
index e024dc0939..277d5feb98 100644
--- a/src/Data/Sum/Function/Setoid.agda
+++ b/src/Data/Sum/Function/Setoid.agda
@@ -32,6 +32,8 @@ module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
           {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
           where
 
+  infixr 1 _⊎-⟶_
+
   _⊎-⟶_ : (A ⟶ B) → (C ⟶ D) → (A ⊎ₛ C) ⟶ (B ⊎ₛ D)
   _⊎-⟶_ f g = record
     { _⟨$⟩_ = fg
@@ -77,6 +79,8 @@ module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
           {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
           where
 
+  infixr 1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_
+
   _⊎-equivalence_ : Equivalence A B → Equivalence C D →
                     Equivalence (A ⊎ₛ C) (B ⊎ₛ D)
   A⇔B ⊎-equivalence C⇔D = record
@@ -119,6 +123,7 @@ module _ {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
          {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
          where
 
+  infixr 1 _⊎-surjection_ _⊎-inverse_
   _⊎-surjection_ : Surjection A B → Surjection C D →
                    Surjection (A ⊎ₛ C) (B ⊎ₛ D)
   A↠B ⊎-surjection C↠D = record
diff --git a/src/Data/Vec/Membership/Setoid.agda b/src/Data/Vec/Membership/Setoid.agda
index 42b4b6ffd0..1151aecf6f 100644
--- a/src/Data/Vec/Membership/Setoid.agda
+++ b/src/Data/Vec/Membership/Setoid.agda
@@ -39,6 +39,8 @@ mapWith∈ : ∀ {b} {B : Set b} {n}
 mapWith∈ []       f = []
 mapWith∈ (x ∷ xs) f = f (here refl) ∷ mapWith∈ xs (f ∘ there)
 
+infixr 5 _∷=_
+
 _∷=_ : ∀ {n} {xs : Vec A n} {x} → x ∈ xs → A → Vec A n
 _∷=_ {xs = xs} x∈xs v = xs Vec.[ index x∈xs ]≔ v
 
diff --git a/src/Data/Vec/Reflection.agda b/src/Data/Vec/Reflection.agda
index 91d5579c94..8b0fa7799f 100644
--- a/src/Data/Vec/Reflection.agda
+++ b/src/Data/Vec/Reflection.agda
@@ -25,6 +25,8 @@ open import Reflection.AST.Argument
 `[] : Term
 `[] = con (quote []) (2 ⋯⟅∷⟆ List.[])
 
+infixr 5 _`∷_
+
 _`∷_ : Term → Term → Term
 _`∷_ x xs = con (quote _∷_) (3 ⋯⟅∷⟆ x ⟨∷⟩ xs ⟨∷⟩ List.[])
 

From 662374e3777a850d1a87f0fadb5a98d8645e807c Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sun, 18 Jun 2023 21:28:56 -0400
Subject: [PATCH 048/316] Get rid of redundant dependencies in
 `System.Console.ANSI` (#1994)

---
 src/Data/Nat/Show.agda | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/Data/Nat/Show.agda b/src/Data/Nat/Show.agda
index 454bc95b76..9f0fdf2bb5 100644
--- a/src/Data/Nat/Show.agda
+++ b/src/Data/Nat/Show.agda
@@ -17,7 +17,7 @@ open import Data.Maybe.Base as Maybe using (Maybe; nothing; _<∣>_; when)
 import Data.Maybe.Effectful as Maybe
 open import Data.Nat
 open import Data.Product using (proj₁)
-open import Data.String as String using (String)
+open import Data.String.Base using (toList; fromList; String)
 open import Function.Base
 open import Relation.Nullary.Decidable using (True)
 
@@ -28,7 +28,7 @@ readMaybe : ∀ base {base≤16 : True (base ≤? 16)} → String → Maybe ℕ
 readMaybe _ "" = nothing
 readMaybe base = Maybe.map convert
               ∘′ TraversableA.mapA Maybe.applicative readDigit
-              ∘′ String.toList
+              ∘′ toList
 
   where
 
@@ -62,7 +62,7 @@ toDecimalChars : ℕ → List Char
 toDecimalChars = List.map toDigitChar ∘′ toNatDigits 10
 
 show : ℕ → String
-show = String.fromList ∘ toDecimalChars
+show = fromList ∘ toDecimalChars
 
 -- Arbitrary base betwen 2 & 16.
 -- Warning: when compiled the time complexity of `showInBase b n` is
@@ -81,5 +81,5 @@ showInBase : (base : ℕ)
              {base≥2 : True (2 ≤? base)}
              {base≤16 : True (base ≤? 16)} →
              ℕ → String
-showInBase base {base≥2} {base≤16} = String.fromList
+showInBase base {base≥2} {base≤16} = fromList
                                    ∘ charsInBase base {base≥2} {base≤16}

From d9476c4d746325ca56ba921165471f7fba74c680 Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Sun, 18 Jun 2023 21:33:04 -0400
Subject: [PATCH 049/316] Move location of `take-[]` and `drop-[]` in
 CHANGELOG.md (#1991)

---
 CHANGELOG.md | 9 +++------
 1 file changed, 3 insertions(+), 6 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2dd7726462..628862540d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2082,6 +2082,9 @@ Other minor changes
 
   length-isMagmaHomomorphism : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
   length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
+ 
+  take-[] : ∀ m → take  m [] ≡ []
+  drop-[] : ∀ m → drop  m [] ≡ []
   ```
 
 * Added new patterns and definitions to `Data.Nat.Base`:
@@ -2813,12 +2816,6 @@ Other minor changes
   foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
   foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
   ```
-  
- * Added new lemmas in `Data.List.Propreties`:
-  ```
-  take-[] : ∀ m → take  m [] ≡ []
-  drop-[] : ∀ m → drop  m [] ≡ []
-  ```
 
 NonZero/Positive/Negative changes
 ---------------------------------

From be4a71dfcc4d18f99e6b3f91811329ee084ba322 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Mon, 19 Jun 2023 10:10:41 -0400
Subject: [PATCH 050/316] =?UTF-8?q?Migrate=20`=5F=E2=86=94=CC=87=5F`=20to?=
 =?UTF-8?q?=20the=20new=20Function=20hierarchy=20+=20more=20arrows=20for?=
 =?UTF-8?q?=20indexed=20sets=20(#1982)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                      |  6 ++++
 src/Function/Indexed/Bundles.agda | 50 +++++++++++++++++++++++++++++++
 2 files changed, 56 insertions(+)
 create mode 100644 src/Function/Indexed/Bundles.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 628862540d..ad581d93fa 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1624,6 +1624,12 @@ New modules
   ```
   Algebra.Properties.KleeneAlgebra
   ```
+
+* Relations on indexed sets
+  ```
+  Function.Indexed.Bundles
+  ```
+
 Other minor changes
 -------------------
 
diff --git a/src/Function/Indexed/Bundles.agda b/src/Function/Indexed/Bundles.agda
new file mode 100644
index 0000000000..033cabd426
--- /dev/null
+++ b/src/Function/Indexed/Bundles.agda
@@ -0,0 +1,50 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Operations on Relations for Indexed sets
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Indexed.Bundles where
+
+open import Relation.Unary using (Pred)
+open import Function.Bundles using (_⟶_; _↣_; _↠_; _⤖_; _⇔_; _↩_; _↪_; _↩↪_; _↔_)
+open import Relation.Binary hiding (_⇔_)
+open import Level using (Level)
+
+private
+  variable
+    a b ℓ₁ ℓ₂ : Level
+
+------------------------------------------------------------------------
+-- Bundles specialised for lifting relations to indexed sets
+------------------------------------------------------------------------
+
+infix 3 _⟶ᵢ_ _↣ᵢ_ _↠ᵢ_ _⤖ᵢ_ _⇔ᵢ_ _↩ᵢ_ _↪ᵢ_ _↩↪ᵢ_ _↔ᵢ_
+_⟶ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ⟶ᵢ B = ∀ {i} → A i ⟶ B i
+
+_↣ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ↣ᵢ B = ∀ {i} → A i ↣ B i
+
+_↠ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ↠ᵢ B = ∀ {i} → A i ↠ B i
+
+_⤖ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ⤖ᵢ B = ∀ {i} → A i ⤖ B i
+
+_⇔ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ⇔ᵢ B = ∀ {i} → A i ⇔ B i
+
+_↩ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ↩ᵢ B = ∀ {i} → A i ↩ B i
+
+_↪ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ↪ᵢ B = ∀ {i} → A i ↪ B i
+
+_↩↪ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ↩↪ᵢ B = ∀ {i} → A i ↩↪ B i
+
+_↔ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+A ↔ᵢ B = ∀ {i} → A i ↔ B i

From 41bcad30f76a7ea43939ec6c904e380d776414f9 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Mon, 19 Jun 2023 10:11:09 -0400
Subject: [PATCH 051/316] Fixities for `Function`, `Data`, and `Reflection`
 (#1987)

---
 CHANGELOG.md                                  | 22 ++++++++++++++++++-
 .../Relation/Binary/Equality/Setoid.agda      |  2 ++
 src/Data/Vec/N-ary.agda                       |  2 ++
 src/Function/Endomorphism/Propositional.agda  |  2 ++
 src/Function/Endomorphism/Setoid.agda         |  2 ++
 src/Function/HalfAdjointEquivalence.agda      |  2 ++
 src/Function/Metric/Bundles.agda              |  7 ++++++
 src/Function/Metric/Nat/Bundles.agda          |  6 +++++
 src/Function/Related.agda                     |  2 ++
 src/Reflection/AST/Argument/Information.agda  |  2 ++
 src/Reflection/AST/Argument/Modality.agda     |  2 ++
 src/Reflection/AST/Argument/Quantity.agda     |  2 ++
 src/Reflection/AST/Argument/Relevance.agda    |  2 ++
 src/Reflection/AST/Argument/Visibility.agda   |  2 ++
 src/Reflection/AST/Definition.agda            |  2 ++
 src/Reflection/AST/Literal.agda               |  2 ++
 src/Reflection/AST/Meta.agda                  |  3 ++-
 src/Reflection/AST/Name.agda                  |  4 ++--
 src/Reflection/AST/Term.agda                  |  2 +-
 19 files changed, 65 insertions(+), 5 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ad581d93fa..e33d919313 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -31,7 +31,27 @@ Bug-fixes
 
 * The following operators were missing a fixity declaration, which has now
   been fixed -
-  ```
+   ```
+  infix  -1 _$ⁿ_          (Data.Vec.N-ary)
+  infix  4 _≋_            (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
+  infix  4 _≟_            (Reflection.AST.Definition)
+  infix  4 _≡ᵇ_           (Reflection.AST.Literal)
+  infix  4 _≈?_ _≟_ _≈_   (Reflection.AST.Meta)
+  infix  4 _≈?_ _≟_ _≈_   (Reflection.AST.Name)
+  infix  4 _≟-Telescope_  (Reflection.AST.Term)
+  infix  4 _≟_            (Reflection.AST.Argument.Information)
+  infix  4 _≟_            (Reflection.AST.Argument.Modality)
+  infix  4 _≟_            (Reflection.AST.Argument.Quantity)
+  infix  4 _≟_            (Reflection.AST.Argument.Relevance)
+  infix  4 _≟_            (Reflection.AST.Argument.Visibility)
+  infixr 8 _^_            (Function.Endomorphism.Propositional)
+  infixr 8 _^_            (Function.Endomorphism.Setoid)
+  infix  4 _≃_            (Function.HalfAdjointEquivalence)
+  infix  4 _≈_ _≈ᵢ_ _≤_   (Function.Metric.Bundles)
+  infixl 6 _∙_            (Function.Metric.Bundles)
+  infix  4 _≈_            (Function.Metric.Nat.Bundles)
+  infix  3 _←_ _↢_        (Function.Related)
+
   infix  4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
   infix  6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
   infixl 4 _+ _*                                            (Data.List.Kleene.Base)
diff --git a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
index 932ac19c95..8b29e710f8 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
@@ -24,6 +24,8 @@ open Setoid S renaming (Carrier to A)
 -- Definition
 ------------------------------------------------------------------------
 
+infix 4 _≋_
+
 _≋_ : ∀ {n} → Vector A n → Vector A n → Set ℓ
 _≋_ = Pointwise _≈_
 
diff --git a/src/Data/Vec/N-ary.agda b/src/Data/Vec/N-ary.agda
index 02b6e704ba..7e10c5567c 100644
--- a/src/Data/Vec/N-ary.agda
+++ b/src/Data/Vec/N-ary.agda
@@ -45,6 +45,8 @@ curryⁿ : ∀ {n} → (Vec A n → B) → N-ary n A B
 curryⁿ {n = zero}  f = f []
 curryⁿ {n = suc n} f = λ x → curryⁿ (f ∘ _∷_ x)
 
+infix -1 _$ⁿ_
+
 _$ⁿ_ : ∀ {n} → N-ary n A B → (Vec A n → B)
 f $ⁿ []       = f
 f $ⁿ (x ∷ xs) = f x $ⁿ xs
diff --git a/src/Function/Endomorphism/Propositional.agda b/src/Function/Endomorphism/Propositional.agda
index 32a5522318..8a78f69496 100644
--- a/src/Function/Endomorphism/Propositional.agda
+++ b/src/Function/Endomorphism/Propositional.agda
@@ -43,6 +43,8 @@ toSetoidEndo f = record
 ------------------------------------------------------------------------
 -- N-th composition
 
+infixr 8 _^_
+
 _^_ : Endo → ℕ → Endo
 f ^ zero  = id
 f ^ suc n = f ∘′ (f ^ n)
diff --git a/src/Function/Endomorphism/Setoid.agda b/src/Function/Endomorphism/Setoid.agda
index 9c3d18e7f4..b02976db88 100644
--- a/src/Function/Endomorphism/Setoid.agda
+++ b/src/Function/Endomorphism/Setoid.agda
@@ -35,6 +35,8 @@ private
 Endo : Set _
 Endo = S ⟶ S
 
+infixr 8 _^_
+
 _^_ : Endo → ℕ → Endo
 f ^ zero  = id
 f ^ suc n = f ∘ (f ^ n)
diff --git a/src/Function/HalfAdjointEquivalence.agda b/src/Function/HalfAdjointEquivalence.agda
index f600dfe759..59b11d5f69 100644
--- a/src/Function/HalfAdjointEquivalence.agda
+++ b/src/Function/HalfAdjointEquivalence.agda
@@ -16,6 +16,8 @@ open import Relation.Binary.PropositionalEquality
 
 -- Half adjoint equivalences (see the HoTT book).
 
+infix 4 _≃_
+
 record _≃_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
   field
     to               : A → B
diff --git a/src/Function/Metric/Bundles.agda b/src/Function/Metric/Bundles.agda
index 7acf81de13..ad3a29c333 100644
--- a/src/Function/Metric/Bundles.agda
+++ b/src/Function/Metric/Bundles.agda
@@ -22,6 +22,7 @@ open import Function.Metric.Core
 
 record ProtoMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
                  : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where
+  infix 4 _≈_ _≈ᵢ_ _≤_
   field
     Carrier       : Set a
     Image         : Set i
@@ -39,6 +40,7 @@ record ProtoMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
 
 record PreMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
                : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where
+  infix 4 _≈_ _≈ᵢ_ _≤_
   field
     Carrier     : Set a
     Image       : Set i
@@ -61,6 +63,8 @@ record PreMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
 
 record QuasiSemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
                      : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where
+
+  infix 4 _≈_ _≈ᵢ_ _≤_
   field
     Carrier           : Set a
     Image             : Set i
@@ -86,6 +90,7 @@ record QuasiSemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
 
 record SemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
                 : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where
+  infix 4 _≈_ _≈ᵢ_ _≤_
   field
     Carrier      : Set a
     Image        : Set i
@@ -120,6 +125,8 @@ record SemiMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
 
 record GeneralMetric (a i ℓ₁ ℓ₂ ℓ₃ : Level)
                    : Set (suc (a ⊔ i ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃)) where
+  infix 4 _≈_ _≈ᵢ_ _≤_
+  infixl 6 _∙_
   field
     Carrier         : Set a
     Image           : Set i
diff --git a/src/Function/Metric/Nat/Bundles.agda b/src/Function/Metric/Nat/Bundles.agda
index 3e406d7611..e0983246eb 100644
--- a/src/Function/Metric/Nat/Bundles.agda
+++ b/src/Function/Metric/Nat/Bundles.agda
@@ -28,6 +28,7 @@ open import Function.Metric.Bundles as Base
 -- Proto-metric
 
 record ProtoMetric a ℓ : Set (suc (a ⊔ ℓ)) where
+  infix 4 _≈_
   field
     Carrier       : Set a
     _≈_           : Rel Carrier ℓ
@@ -40,6 +41,7 @@ record ProtoMetric a ℓ : Set (suc (a ⊔ ℓ)) where
 -- PreMetric
 
 record PreMetric a ℓ : Set (suc (a ⊔ ℓ)) where
+  infix 4 _≈_
   field
     Carrier     : Set a
     _≈_         : Rel Carrier ℓ
@@ -57,6 +59,7 @@ record PreMetric a ℓ : Set (suc (a ⊔ ℓ)) where
 -- QuasiSemiMetric
 
 record QuasiSemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where
+  infix 4 _≈_
   field
     Carrier           : Set a
     _≈_               : Rel Carrier ℓ
@@ -77,6 +80,7 @@ record QuasiSemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where
 -- SemiMetric
 
 record SemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where
+  infix 4 _≈_
   field
     Carrier      : Set a
     _≈_          : Rel Carrier ℓ
@@ -97,6 +101,7 @@ record SemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where
 -- Metrics
 
 record Metric a ℓ : Set (suc (a ⊔ ℓ)) where
+  infix 4 _≈_
   field
     Carrier  : Set a
     _≈_      : Rel Carrier ℓ
@@ -117,6 +122,7 @@ record Metric a ℓ : Set (suc (a ⊔ ℓ)) where
 -- UltraMetrics
 
 record UltraMetric a ℓ : Set (suc (a ⊔ ℓ)) where
+  infix 4 _≈_
   field
     Carrier       : Set a
     _≈_           : Rel Carrier ℓ
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index 0746b3c14a..fa114d077c 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -53,6 +53,8 @@ open R public using
 -- (which implies that Agda can deduce the universe code from an
 -- expression matching any of the right-hand sides).
 
+infix 3 _←_ _↢_
+
 record _←_ {a b} (A : Set a) (B : Set b) : Set (a ⊔ b) where
   constructor lam
   field app-← : B → A
diff --git a/src/Reflection/AST/Argument/Information.agda b/src/Reflection/AST/Argument/Information.agda
index 3e97598d46..fe09976fb9 100644
--- a/src/Reflection/AST/Argument/Information.agda
+++ b/src/Reflection/AST/Argument/Information.agda
@@ -48,6 +48,8 @@ arg-info-injective₂ refl = refl
 arg-info-injective : arg-info v m ≡ arg-info v′ m′ → v ≡ v′ × m ≡ m′
 arg-info-injective = < arg-info-injective₁ , arg-info-injective₂ >
 
+infix 4 _≟_
+
 _≟_ : DecidableEquality ArgInfo
 arg-info v m ≟ arg-info v′ m′ =
   map′
diff --git a/src/Reflection/AST/Argument/Modality.agda b/src/Reflection/AST/Argument/Modality.agda
index 1e39783349..95f15b2264 100644
--- a/src/Reflection/AST/Argument/Modality.agda
+++ b/src/Reflection/AST/Argument/Modality.agda
@@ -48,6 +48,8 @@ modality-injective₂ refl = refl
 modality-injective : modality r q ≡ modality r′ q′ → r ≡ r′ × q ≡ q′
 modality-injective = < modality-injective₁ , modality-injective₂ >
 
+infix 4 _≟_
+
 _≟_ : DecidableEquality Modality
 modality r q ≟ modality r′ q′ =
   map′
diff --git a/src/Reflection/AST/Argument/Quantity.agda b/src/Reflection/AST/Argument/Quantity.agda
index 3942a667a1..0c53a521c1 100644
--- a/src/Reflection/AST/Argument/Quantity.agda
+++ b/src/Reflection/AST/Argument/Quantity.agda
@@ -21,6 +21,8 @@ open Quantity public
 ------------------------------------------------------------------------
 -- Decidable equality
 
+infix 4 _≟_
+
 _≟_ : DecidableEquality Quantity
 quantity-ω ≟ quantity-ω = yes refl
 quantity-0 ≟ quantity-0 = yes refl
diff --git a/src/Reflection/AST/Argument/Relevance.agda b/src/Reflection/AST/Argument/Relevance.agda
index 1ec1606453..6cdce1d96f 100644
--- a/src/Reflection/AST/Argument/Relevance.agda
+++ b/src/Reflection/AST/Argument/Relevance.agda
@@ -21,6 +21,8 @@ open Relevance public
 ------------------------------------------------------------------------
 -- Decidable equality
 
+infix 4 _≟_
+
 _≟_ : DecidableEquality Relevance
 relevant   ≟ relevant   = yes refl
 irrelevant ≟ irrelevant = yes refl
diff --git a/src/Reflection/AST/Argument/Visibility.agda b/src/Reflection/AST/Argument/Visibility.agda
index 93654a018c..4f94b24fe3 100644
--- a/src/Reflection/AST/Argument/Visibility.agda
+++ b/src/Reflection/AST/Argument/Visibility.agda
@@ -21,6 +21,8 @@ open Visibility public
 ------------------------------------------------------------------------
 -- Decidable equality
 
+infix 4 _≟_
+
 _≟_ : DecidableEquality Visibility
 visible   ≟ visible   = yes refl
 hidden    ≟ hidden    = yes refl
diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index 7cb403fb49..5b6dace7cd 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -59,6 +59,8 @@ record′-injective = < record′-injective₁ , record′-injective₂ >
 constructor′-injective : ∀ {c c′} → constructor′ c ≡ constructor′ c′ → c ≡ c′
 constructor′-injective refl = refl
 
+infix 4 _≟_
+
 _≟_ : DecidableEquality Definition
 function cs       ≟ function cs′        =
   map′ (cong function) function-injective (cs Term.≟-Clauses cs′)
diff --git a/src/Reflection/AST/Literal.agda b/src/Reflection/AST/Literal.agda
index 280ba13010..33ceed9ca5 100644
--- a/src/Reflection/AST/Literal.agda
+++ b/src/Reflection/AST/Literal.agda
@@ -104,5 +104,7 @@ meta x ≟ string x₁ = no (λ ())
 meta x ≟ name x₁ = no (λ ())
 meta x ≟ meta x₁ = map′ (cong meta) meta-injective (x Meta.≟ x₁)
 
+infix 4 _≡ᵇ_
+
 _≡ᵇ_ : Literal → Literal → Bool
 l ≡ᵇ l′ = isYes (l ≟ l′)
diff --git a/src/Reflection/AST/Meta.agda b/src/Reflection/AST/Meta.agda
index b5db1fad1a..59ca4ae4db 100644
--- a/src/Reflection/AST/Meta.agda
+++ b/src/Reflection/AST/Meta.agda
@@ -23,12 +23,13 @@ open import Agda.Builtin.Reflection.Properties public
 
 -- Equality of metas is decidable.
 
+infix 4 _≈?_ _≟_ _≈_
+
 _≈_ : Rel Meta _
 _≈_ = _≡_ on toℕ
 
 _≈?_ : Decidable _≈_
 _≈?_ = On.decidable toℕ _≡_ ℕₚ._≟_
 
-infix 4 _≟_
 _≟_ : DecidableEquality Meta
 m ≟ n = map′ (toℕ-injective _ _) (cong toℕ) (m ≈? n)
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index 50c6ee7878..203521330c 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -37,11 +37,11 @@ Names = List Name
 -- Decidable equality for names
 ----------------------------------------------------------------------
 
+infix 4 _≈?_ _≟_ _≈_
+
 _≈_ : Rel Name _
 _≈_ = _≡_ on toWords
 
-infix 4 _≈?_ _≟_
-
 _≈?_ : Decidable _≈_
 _≈?_ = decidable toWords _≡_ (Prodₚ.≡-dec Wₚ._≟_ Wₚ._≟_)
 
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index fb0f5bde61..183f839015 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -128,7 +128,7 @@ absurd-clause-injective = < absurd-clause-injective₁ , absurd-clause-injective
 
 infix 4 _≟-AbsTerm_ _≟-AbsType_ _≟-ArgTerm_ _≟-ArgType_ _≟-Args_
         _≟-Clause_ _≟-Clauses_ _≟_
-        _≟-Sort_ _≟-Pattern_ _≟-Patterns_
+        _≟-Sort_ _≟-Pattern_ _≟-Patterns_ _≟-Telescope_
 
 _≟-AbsTerm_  : DecidableEquality (Abs Term)
 _≟-AbsType_  : DecidableEquality (Abs Type)

From 8dd8c59c6c33809b8b3fb26b44d0a2ca339c7f91 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Mon, 19 Jun 2023 22:29:36 -0400
Subject: [PATCH 052/316] More fixities for `Data` and `Codata` (#1989)

---
 CHANGELOG.md                                  | 119 +++++++++++-------
 src/Codata/Guarded/Stream.agda                |   4 +
 .../Stream/Relation/Binary/Pointwise.agda     |   4 +
 src/Codata/Musical/Colist.agda                |   2 +
 src/Codata/Musical/Colist/Bisimilarity.agda   |   2 +
 .../Musical/Colist/Relation/Unary/All.agda    |   2 +
 src/Codata/Musical/Conat.agda                 |   2 +
 src/Codata/Sized/Colist.agda                  |   2 +
 src/Codata/Sized/Colist/Bisimilarity.agda     |   2 +
 src/Codata/Sized/Conat/Properties.agda        |   2 +
 src/Codata/Sized/Covec.agda                   |   2 +
 src/Codata/Sized/Covec/Bisimilarity.agda      |   1 +
 src/Codata/Sized/Cowriter.agda                |   4 +
 src/Codata/Sized/Cowriter/Bisimilarity.agda   |   2 +
 src/Codata/Sized/Stream.agda                  |   2 +
 src/Codata/Sized/Stream/Bisimilarity.agda     |   2 +
 src/Data/Container/Core.agda                  |   2 +
 src/Data/Container/FreeMonad.agda             |   3 +
 src/Data/Container/Indexed.agda               |   4 +
 src/Data/Float/Base.agda                      |   2 +
 src/Data/Nat/Base.agda                        |   6 +-
 src/Data/Nat/Properties.agda                  |   2 +
 .../Rational/Unnormalised/Properties.agda     |   2 +
 .../AVL/Map/Membership/Propositional.agda     |   2 +
 src/Induction/WellFounded.agda                |   1 +
 25 files changed, 131 insertions(+), 47 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e33d919313..13f0b0f20f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -31,51 +31,80 @@ Bug-fixes
 
 * The following operators were missing a fixity declaration, which has now
   been fixed -
-   ```
-  infix  -1 _$ⁿ_          (Data.Vec.N-ary)
-  infix  4 _≋_            (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
-  infix  4 _≟_            (Reflection.AST.Definition)
-  infix  4 _≡ᵇ_           (Reflection.AST.Literal)
-  infix  4 _≈?_ _≟_ _≈_   (Reflection.AST.Meta)
-  infix  4 _≈?_ _≟_ _≈_   (Reflection.AST.Name)
-  infix  4 _≟-Telescope_  (Reflection.AST.Term)
-  infix  4 _≟_            (Reflection.AST.Argument.Information)
-  infix  4 _≟_            (Reflection.AST.Argument.Modality)
-  infix  4 _≟_            (Reflection.AST.Argument.Quantity)
-  infix  4 _≟_            (Reflection.AST.Argument.Relevance)
-  infix  4 _≟_            (Reflection.AST.Argument.Visibility)
-  infixr 8 _^_            (Function.Endomorphism.Propositional)
-  infixr 8 _^_            (Function.Endomorphism.Setoid)
-  infix  4 _≃_            (Function.HalfAdjointEquivalence)
-  infix  4 _≈_ _≈ᵢ_ _≤_   (Function.Metric.Bundles)
-  infixl 6 _∙_            (Function.Metric.Bundles)
-  infix  4 _≈_            (Function.Metric.Nat.Bundles)
-  infix  3 _←_ _↢_        (Function.Related)
-
-  infix  4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
-  infix  6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
-  infixl 4 _+ _*                                            (Data.List.Kleene.Base)
-  infixr 4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
-  infix  4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
-  infix  4 _≢∈_                                             (Data.List.Membership.Propositional)
-  infixr 5 _`∷_                                             (Data.List.Reflection)
-  infix  4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
-  infixr 5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
-  infixr 5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
-  infixr 5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
-  infixr 5 _∷=_                                             (Data.List.Relation.Unary.Any)
-  infixr 5 _++_                                             (Data.List.Ternary.Appending)
-  infixr 2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
-  infixr 2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
-  infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
-  infixr 2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
-  infixr 1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
-  infixr 1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
-  infixr 1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
-  infixr 1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
-  infix  8 _⁻¹                                              (Data.Parity.Base)
-  infixr 5 _`∷_                                             (Data.Vec.Reflection)
-  infixr 5 _∷=_                                             (Data.Vec.Membership.Setoid)
+  ```
+  infixr  5 _∷_                                              (Codata.Guarded.Stream)
+  infix   4 _[_]                                             (Codata.Guarded.Stream)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Binary.Pointwise)
+  infix   4 _≈∞_                                             (Codata.Guarded.Stream.Relation.Binary.Pointwise)
+  infixr  5 _∷_                                              (Codata.Musical.Colist)
+  infix   4 _≈_                                              (Codata.Musical.Conat)
+  infixr  5 _∷_                                              (Codata.Musical.Colist.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Musical.Colist.Relation.Unary.All)
+  infixr  5 _∷_                                              (Codata.Sized.Colist)
+  infixr  5 _∷_                                              (Codata.Sized.Covec)
+  infixr  5 _∷_                                              (Codata.Sized.Cowriter)
+  infixl  1 _>>=_                                            (Codata.Sized.Cowriter)
+  infixr  5 _∷_                                              (Codata.Sized.Stream)
+  infixr  5 _∷_                                              (Codata.Sized.Colist.Bisimilarity)
+  infix   4 _ℕ≤?_                                            (Codata.Sized.Conat.Properties)
+  infixr  5 _∷_                                              (Codata.Sized.Covec.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Sized.Cowriter.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Sized.Stream.Bisimilarity)
+  infixr  8 _⇒_ _⊸_                                          (Data.Container.Core)
+  infixr -1 _<$>_ _<*>_                                      (Data.Container.FreeMonad)
+  infixl  1 _>>=_                                            (Data.Container.FreeMonad)
+  infix   5 _▷_                                              (Data.Container.Indexed)
+  infix   4 _≈_                                              (Data.Float.Base)
+  infixl  7 _⊓′_                                             (Data.Nat.Base)
+  infixl  6 _⊔′_                                             (Data.Nat.Base)
+  infixr  8 _^_                                              (Data.Nat.Base)
+  infix   4 _!≢0 _!*_!≢0                                     (Data.Nat.Properties)
+  infix   4 _≃?_                                             (Data.Rational.Unnormalised.Properties)
+  infix   4 _≈ₖᵥ_                                            (Data.Tree.AVL.Map.Membership.Propositional)
+  infix   4 _<_                                              (Induction.WellFounded)
+  infix  -1 _$ⁿ_                                             (Data.Vec.N-ary)
+  infix   4 _≋_                                              (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
+  infix   4 _≟_                                              (Reflection.AST.Definition)
+  infix   4 _≡ᵇ_                                             (Reflection.AST.Literal)
+  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Meta)
+  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Name)
+  infix   4 _≟-Telescope_                                    (Reflection.AST.Term)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Information)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Modality)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Quantity)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Relevance)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Visibility)
+  infixr  8 _^_                                              (Function.Endomorphism.Propositional)
+  infixr  8 _^_                                              (Function.Endomorphism.Setoid)
+  infix   4 _≃_                                              (Function.HalfAdjointEquivalence)
+  infix   4 _≈_ _≈ᵢ_ _≤_                                     (Function.Metric.Bundles)
+  infixl  6 _∙_                                              (Function.Metric.Bundles)
+  infix   4 _≈_                                              (Function.Metric.Nat.Bundles)
+  infix   3 _←_ _↢_                                          (Function.Related)
+  infix   4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
+  infix   6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
+  infixl  4 _+ _*                                            (Data.List.Kleene.Base)
+  infixr  4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
+  infix   4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
+  infix   4 _≢∈_                                             (Data.List.Membership.Propositional)
+  infixr  5 _`∷_                                             (Data.List.Reflection)
+  infix   4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
+  infixr  5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
+  infixr  5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
+  infixr  5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
+  infixr  5 _∷=_                                             (Data.List.Relation.Unary.Any)
+  infixr  5 _++_                                             (Data.List.Ternary.Appending)
+  infixr  2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
+  infixr  2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
+  infixr  2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
+  infixr  2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
+  infixr  1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
+  infixr  1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
+  infixr  1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
+  infixr  1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
+  infix   8 _⁻¹                                              (Data.Parity.Base)
+  infixr  5 _`∷_                                             (Data.Vec.Reflection)
+  infixr  5 _∷=_                                             (Data.Vec.Membership.Setoid)
   ```
 
 * In `System.Exit`, the `ExitFailure` constructor is now carrying an integer
diff --git a/src/Codata/Guarded/Stream.agda b/src/Codata/Guarded/Stream.agda
index 88c5c5fec5..6024525cb4 100644
--- a/src/Codata/Guarded/Stream.agda
+++ b/src/Codata/Guarded/Stream.agda
@@ -28,6 +28,8 @@ private
 ------------------------------------------------------------------------
 -- Type
 
+infixr 5 _∷_
+
 record Stream (A : Set a) : Set a where
   coinductive
   constructor _∷_
@@ -69,6 +71,8 @@ lookup : Stream A → ℕ → A
 lookup xs zero    = head xs
 lookup xs (suc n) = lookup (tail xs) n
 
+infix 4 _[_]
+
 _[_] : Stream A → ℕ → A
 _[_] = lookup
 
diff --git a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
index 8f829297a0..c2d44f8145 100644
--- a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
+++ b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
@@ -25,6 +25,8 @@ private
 ------------------------------------------------------------------------
 -- Bisimilarity
 
+infixr 5 _∷_
+
 record Pointwise (_∼_ : REL A B ℓ) (as : Stream A) (bs : Stream B) : Set ℓ where
   coinductive
   constructor _∷_
@@ -191,4 +193,6 @@ module pw-Reasoning (S : Setoid a ℓ) where
 module ≈-Reasoning {a} {A : Set a} where
 
   open pw-Reasoning (P.setoid A) public
+
+  infix 4 _≈∞_
   _≈∞_ = `Pointwise∞
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index c218a7f023..2ec01be70f 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -215,6 +215,8 @@ data Finite {A : Set a} : Colist A → Set a where
   []  : Finite []
   _∷_ : ∀ x {xs} (fin : Finite (♭ xs)) → Finite (x ∷ xs)
 
+infixr 5 _∷_
+
 module Finite-injective where
 
  ∷-injective : ∀ {x : A} {xs p q} → (Finite (x ∷ xs) ∋ x ∷ p) ≡ x ∷ q → p ≡ q
diff --git a/src/Codata/Musical/Colist/Bisimilarity.agda b/src/Codata/Musical/Colist/Bisimilarity.agda
index 8bcf5e5169..64a0c37d9c 100644
--- a/src/Codata/Musical/Colist/Bisimilarity.agda
+++ b/src/Codata/Musical/Colist/Bisimilarity.agda
@@ -27,6 +27,8 @@ data _≈_ {A : Set a} : Rel (Colist A) a where
   []  :                                       []     ≈ []
   _∷_ : ∀ x {xs ys} (xs≈ : ∞ (♭ xs ≈ ♭ ys)) → x ∷ xs ≈ x ∷ ys
 
+infixr 5 _∷_
+
 -- The equality relation forms a setoid.
 
 setoid : Set a → Setoid _ _
diff --git a/src/Codata/Musical/Colist/Relation/Unary/All.agda b/src/Codata/Musical/Colist/Relation/Unary/All.agda
index df997252c3..ce937dc2b6 100644
--- a/src/Codata/Musical/Colist/Relation/Unary/All.agda
+++ b/src/Codata/Musical/Colist/Relation/Unary/All.agda
@@ -23,3 +23,5 @@ private
 data All {A : Set a} (P : Pred A p) : Pred (Colist A) (a ⊔ p) where
   []  : All P []
   _∷_ : ∀ {x xs} (px : P x) (pxs : ∞ (All P (♭ xs))) → All P (x ∷ xs)
+
+infixr 5 _∷_
diff --git a/src/Codata/Musical/Conat.agda b/src/Codata/Musical/Conat.agda
index 2d598fd75c..f8c1490a40 100644
--- a/src/Codata/Musical/Conat.agda
+++ b/src/Codata/Musical/Conat.agda
@@ -34,6 +34,8 @@ fromℕ-injective {suc m} {suc n} eq = P.cong suc (fromℕ-injective (P.cong pre
 ------------------------------------------------------------------------
 -- Equality
 
+infix 4 _≈_
+
 data _≈_ : Coℕ → Coℕ → Set where
   zero :                                 zero  ≈ zero
   suc  : ∀ {m n} (m≈n : ∞ (♭ m ≈ ♭ n)) → suc m ≈ suc n
diff --git a/src/Codata/Sized/Colist.agda b/src/Codata/Sized/Colist.agda
index c090565c33..4f916fbc68 100644
--- a/src/Codata/Sized/Colist.agda
+++ b/src/Codata/Sized/Colist.agda
@@ -42,6 +42,8 @@ data Colist (A : Set a) (i : Size) : Set a where
   []  : Colist A i
   _∷_ : A → Thunk (Colist A) i → Colist A i
 
+infixr 5 _∷_
+
 ------------------------------------------------------------------------
 -- Relationship to Cowriter.
 
diff --git a/src/Codata/Sized/Colist/Bisimilarity.agda b/src/Codata/Sized/Colist/Bisimilarity.agda
index 9e626a5315..0a9f76306b 100644
--- a/src/Codata/Sized/Colist/Bisimilarity.agda
+++ b/src/Codata/Sized/Colist/Bisimilarity.agda
@@ -32,6 +32,8 @@ data Bisim {A : Set a} {B : Set b} (R : REL A B r) (i : Size) :
   _∷_ : ∀ {x y xs ys} → R x y → Thunk^R (Bisim R) i xs ys →
         Bisim R i (x ∷ xs) (y ∷ ys)
 
+infixr 5 _∷_
+
 module _ {R : Rel A r} where
 
  reflexive : Reflexive R → Reflexive (Bisim R i)
diff --git a/src/Codata/Sized/Conat/Properties.agda b/src/Codata/Sized/Conat/Properties.agda
index b2649fb38a..43d5662a9a 100644
--- a/src/Codata/Sized/Conat/Properties.agda
+++ b/src/Codata/Sized/Conat/Properties.agda
@@ -29,6 +29,8 @@ private
 sℕ≤s⁻¹ : ∀ {m n} → suc m ℕ≤ suc n → m ℕ≤ n .force
 sℕ≤s⁻¹ (sℕ≤s p) = p
 
+infix 4 _ℕ≤?_
+
 _ℕ≤?_ : Decidable _ℕ≤_
 zero  ℕ≤? n     = yes zℕ≤n
 suc m ℕ≤? zero  = no (λ ())
diff --git a/src/Codata/Sized/Covec.agda b/src/Codata/Sized/Covec.agda
index 85b7a9e63e..01e01f2df0 100644
--- a/src/Codata/Sized/Covec.agda
+++ b/src/Codata/Sized/Covec.agda
@@ -29,6 +29,8 @@ data Covec (A : Set a) (i : Size) : Conat ∞ → Set a where
   []  : Covec A i zero
   _∷_ : ∀ {n} → A → Thunk (λ i → Covec A i (n .force)) i → Covec A i (suc n)
 
+infixr 5 _∷_
+
 head : ∀ {n i} → Covec A i (suc n) → A
 head (x ∷ _) = x
 
diff --git a/src/Codata/Sized/Covec/Bisimilarity.agda b/src/Codata/Sized/Covec/Bisimilarity.agda
index 3047816b44..cbe55ecbf7 100644
--- a/src/Codata/Sized/Covec/Bisimilarity.agda
+++ b/src/Codata/Sized/Covec/Bisimilarity.agda
@@ -22,6 +22,7 @@ data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) (i : Size) :
   _∷_ : ∀ {x y m n xs ys} → R x y → Thunk^R (λ i → Bisim R i (m .force) (n .force)) i xs ys →
         Bisim R i (suc m) (suc n) (x ∷ xs) (y ∷ ys)
 
+infixr 5 _∷_
 
 module _ {a r} {A : Set a} {R : A → A → Set r} where
 
diff --git a/src/Codata/Sized/Cowriter.agda b/src/Codata/Sized/Cowriter.agda
index 409fbe1f71..2a6ef961ec 100644
--- a/src/Codata/Sized/Cowriter.agda
+++ b/src/Codata/Sized/Cowriter.agda
@@ -39,6 +39,8 @@ data Cowriter (W : Set w) (A : Set a) (i : Size) : Set (a L.⊔ w) where
   [_] : A → Cowriter W A i
   _∷_ : W → Thunk (Cowriter W A) i → Cowriter W A i
 
+infixr 5 _∷_
+
 ------------------------------------------------------------------------
 -- Relationship to Delay.
 
@@ -101,6 +103,8 @@ ap : ∀ {i} → Cowriter W (A → X) i → Cowriter W A i → Cowriter W X i
 ap [ f ]    ca = map₂ f ca
 ap (w ∷ cf) ca = w ∷ λ where .force → ap (cf .force) ca
 
+infixl 1 _>>=_
+
 _>>=_ : ∀ {i} → Cowriter W A i → (A → Cowriter W X i) → Cowriter W X i
 [ a ]    >>= f = f a
 (w ∷ ca) >>= f = w ∷ λ where .force → ca .force >>= f
diff --git a/src/Codata/Sized/Cowriter/Bisimilarity.agda b/src/Codata/Sized/Cowriter/Bisimilarity.agda
index 823b7897ae..846101b538 100644
--- a/src/Codata/Sized/Cowriter/Bisimilarity.agda
+++ b/src/Codata/Sized/Cowriter/Bisimilarity.agda
@@ -33,6 +33,8 @@ data Bisim {V : Set v} {W : Set w} {A : Set a} {B : Set b}
   _∷_ : ∀ {x y xs ys} → R x y → Thunk^R (Bisim R S) i xs ys →
         Bisim R S i (x ∷ xs) (y ∷ ys)
 
+infixr 5 _∷_
+
 module _ {R : Rel W r} {S : Rel A s}
          (refl^R : Reflexive R) (refl^S : Reflexive S) where
 
diff --git a/src/Codata/Sized/Stream.agda b/src/Codata/Sized/Stream.agda
index 53e42aa87d..85cd886b5f 100644
--- a/src/Codata/Sized/Stream.agda
+++ b/src/Codata/Sized/Stream.agda
@@ -34,6 +34,8 @@ private
 data Stream (A : Set a) (i : Size) : Set a where
   _∷_ : A → Thunk (Stream A) i → Stream A i
 
+infixr 5 _∷_
+
 ------------------------------------------------------------------------
 -- Creating streams
 
diff --git a/src/Codata/Sized/Stream/Bisimilarity.agda b/src/Codata/Sized/Stream/Bisimilarity.agda
index 1675ec016a..737e73c306 100644
--- a/src/Codata/Sized/Stream/Bisimilarity.agda
+++ b/src/Codata/Sized/Stream/Bisimilarity.agda
@@ -30,6 +30,8 @@ data Bisim {A : Set a} {B : Set b} (R : REL A B r) i :
   _∷_ : ∀ {x y xs ys} → R x y → Thunk^R (Bisim R) i xs ys →
         Bisim R i (x ∷ xs) (y ∷ ys)
 
+infixr 5 _∷_
+
 module _ {R : Rel A r} where
 
  reflexive : Reflexive R → Reflexive (Bisim R i)
diff --git a/src/Data/Container/Core.agda b/src/Data/Container/Core.agda
index 9d07031d20..22b26955d8 100644
--- a/src/Data/Container/Core.agda
+++ b/src/Data/Container/Core.agda
@@ -38,6 +38,8 @@ map f = Prod.map₂ (f ∘_)
 
 -- Representation of container morphisms.
 
+infixr 8 _⇒_ _⊸_
+
 record _⇒_ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂)
            : Set (s₁ ⊔ s₂ ⊔ p₁ ⊔ p₂) where
   constructor _▷_
diff --git a/src/Data/Container/FreeMonad.agda b/src/Data/Container/FreeMonad.agda
index d1853782c9..1d206a4b87 100644
--- a/src/Data/Container/FreeMonad.agda
+++ b/src/Data/Container/FreeMonad.agda
@@ -103,6 +103,9 @@ module _ {P : Set ℓ}
  foldr : C ⋆ X → P
  foldr = induction (Function.const P) algP (algI ∘ -,_ ∘ □.proof)
 
+infixr -1 _<$>_ _<*>_
+infixl 1 _>>=_
+
 _<$>_ : (X → Y) → C ⋆ X → C ⋆ Y
 f <$> t = foldr (pure ∘ f) impure t
 
diff --git a/src/Data/Container/Indexed.agda b/src/Data/Container/Indexed.agda
index 752f55c2a7..a5b17d2aab 100644
--- a/src/Data/Container/Indexed.agda
+++ b/src/Data/Container/Indexed.agda
@@ -31,6 +31,8 @@ open Container public
 -- Abbreviation for the commonly used level one version of indexed
 -- containers.
 
+infix 5 _▷_
+
 _▷_ : Set → Set → Set₁
 I ▷ O = Container I O zero zero
 
@@ -103,6 +105,8 @@ module _ {i₁ i₂ o₁ o₂}
 
 module _ {i o c r} {I : Set i} {O : Set o} where
 
+  infixr 8 _⇒_ _⊸_ _⇒C_
+
   _⇒_ : B.Rel (Container I O c r) _
   C₁ ⇒ C₂ = C₁ ⇒[ ⟨id⟩ / ⟨id⟩ ] C₂
 
diff --git a/src/Data/Float/Base.agda b/src/Data/Float/Base.agda
index edc8b4943f..d09a5a811b 100644
--- a/src/Data/Float/Base.agda
+++ b/src/Data/Float/Base.agda
@@ -65,5 +65,7 @@ open import Agda.Builtin.Float public
   ; primFloatATanh             to atanh
   )
 
+infix 4 _≈_
+
 _≈_ : Rel Float _
 _≈_ = _≡_ on Maybe.map Word.toℕ ∘ toWord
diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index 1e2db666b8..4920e7751e 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -132,8 +132,8 @@ pred : ℕ → ℕ
 pred n = n ∸ 1
 
 infix  8 _!
-infixl 7 _⊓_ _/_ _%_
-infixl 6 _+⋎_ _⊔_
+infixl 7 _⊓_ _⊓′_ _/_ _%_
+infixl 6 _+⋎_ _⊔_ _⊔′_
 
 -- Argument-swapping addition. Used by Data.Vec._⋎_.
 
@@ -192,6 +192,8 @@ parity (suc (suc n)) = parity n
 
 -- Naïve exponentiation
 
+infixr 8 _^_
+
 _^_ : ℕ → ℕ → ℕ
 x ^ zero  = 1
 x ^ suc n = x * x ^ n
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 6ff1b1f38b..27725b98ad 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1948,6 +1948,8 @@ n≡⌈n+n/2⌉ (suc n′@(suc n)) =
 1≤n! zero    = ≤-refl
 1≤n! (suc n) = *-mono-≤ (m≤m+n 1 n) (1≤n! n)
 
+infix 4 _!≢0 _!*_!≢0
+
 _!≢0 : ∀ n → NonZero (n !)
 n !≢0 = >-nonZero (1≤n! n)
 
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index d1eb0fe609..6f0d490412 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -91,6 +91,8 @@ drop-*≡* (*≡* eq) = eq
      ↥ z ℤ.* ↧ x ℤ.* ↧ y ∎))
   where open ≡-Reasoning
 
+infix 4 _≃?_
+
 _≃?_ : Decidable _≃_
 p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* ↧ p)
 
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional.agda b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
index a353ddf4ad..7e5d297025 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
@@ -49,6 +49,8 @@ private
     x x′ y y′ : V
     kx : Key × V
 
+infix 4 _≈ₖᵥ_
+
 _≈ₖᵥ_ : Rel (Key × V) _
 _≈ₖᵥ_ = _≈_ ×ᴿ _≡_
 
diff --git a/src/Induction/WellFounded.agda b/src/Induction/WellFounded.agda
index 5c88529d27..d580452f0f 100644
--- a/src/Induction/WellFounded.agda
+++ b/src/Induction/WellFounded.agda
@@ -184,6 +184,7 @@ module Lexicographic {A : Set a} {B : A → Set b}
                      (RelA : Rel A ℓ₁)
                      (RelB : ∀ x → Rel (B x) ℓ₂) where
 
+  infix 4 _<_
   data _<_ : Rel (Σ A B) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     left  : ∀ {x₁ y₁ x₂ y₂} (x₁<x₂ : RelA   x₁ x₂) → (x₁ , y₁) < (x₂ , y₂)
     right : ∀ {x y₁ y₂}     (y₁<y₂ : RelB x y₁ y₂) → (x  , y₁) < (x  , y₂)

From f2dc172b9042b37452ba948ce30f8019965dc77a Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Mon, 19 Jun 2023 22:30:14 -0400
Subject: [PATCH 053/316] Rename `Relation.Binary.Construct.(Flip/Converse)`
 (#1979)

---
 CHANGELOG.md                                  |   8 +
 src/Algebra/Construct/NaturalChoice/Base.agda |   2 +-
 src/Algebra/Construct/NaturalChoice/Max.agda  |   2 +-
 .../Construct/NaturalChoice/MaxOp.agda        |   2 +-
 src/Data/Fin/Induction.agda                   |   8 +-
 src/Relation/Binary/Construct/Converse.agda   | 192 +----------------
 src/Relation/Binary/Construct/Flip.Agda       |  17 ++
 .../Binary/Construct/Flip/EqAndOrd.agda       | 195 ++++++++++++++++++
 .../Construct/{Flip.agda => Flip/Ord.agda}    |   6 +-
 .../Binary/Properties/DecTotalOrder.agda      |   4 +-
 src/Relation/Binary/Properties/Preorder.agda  |   6 +-
 .../Binary/Properties/StrictPartialOrder.agda |   4 +-
 .../Binary/Properties/TotalOrder.agda         |   4 +-
 13 files changed, 246 insertions(+), 204 deletions(-)
 create mode 100644 src/Relation/Binary/Construct/Flip.Agda
 create mode 100644 src/Relation/Binary/Construct/Flip/EqAndOrd.agda
 rename src/Relation/Binary/Construct/{Flip.agda => Flip/Ord.agda} (97%)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 13f0b0f20f..bb45ab0a49 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -918,6 +918,14 @@ Major improvements
 Deprecated modules
 ------------------
 
+### Moving `Relation.Binary.Construct.(Converse/Flip)`
+
+* The following files have been moved:
+  ```agda
+  Relation.Binary.Construct.Converse               ↦ Relation.Binary.Construct.Flip.EqAndOrd
+  Relation.Binary.Construct.Flip                   ↦ Relation.Binary.Construct.Flip.Ord
+  ```
+
 ### Deprecation of old function hierarchy
 
 * The module `Function.Related` has been deprecated in favour of `Function.Related.Propositional`
diff --git a/src/Algebra/Construct/NaturalChoice/Base.agda b/src/Algebra/Construct/NaturalChoice/Base.agda
index 901f7a5413..3ebd9b924f 100644
--- a/src/Algebra/Construct/NaturalChoice/Base.agda
+++ b/src/Algebra/Construct/NaturalChoice/Base.agda
@@ -11,7 +11,7 @@ open import Algebra.Core
 open import Level as L hiding (_⊔_)
 open import Function.Base using (flip)
 open import Relation.Binary
-open import Relation.Binary.Construct.Converse using ()
+open import Relation.Binary.Construct.Flip.EqAndOrd using ()
   renaming (totalPreorder to flipOrder)
 import Relation.Binary.Properties.TotalOrder as TotalOrderProperties
 
diff --git a/src/Algebra/Construct/NaturalChoice/Max.agda b/src/Algebra/Construct/NaturalChoice/Max.agda
index 592b887f6b..212a52602f 100644
--- a/src/Algebra/Construct/NaturalChoice/Max.agda
+++ b/src/Algebra/Construct/NaturalChoice/Max.agda
@@ -14,7 +14,7 @@ module Algebra.Construct.NaturalChoice.Max
 open import Algebra.Core
 open import Algebra.Definitions
 open import Algebra.Construct.NaturalChoice.Base
-open import Relation.Binary.Construct.Converse using ()
+open import Relation.Binary.Construct.Flip.EqAndOrd using ()
   renaming (totalOrder to flip)
 
 open TotalOrder totalOrder renaming (Carrier to A)
diff --git a/src/Algebra/Construct/NaturalChoice/MaxOp.agda b/src/Algebra/Construct/NaturalChoice/MaxOp.agda
index 28accb65b3..d6c7de61ba 100644
--- a/src/Algebra/Construct/NaturalChoice/MaxOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MaxOp.agda
@@ -12,7 +12,7 @@ open import Algebra.Construct.NaturalChoice.Base
 import Algebra.Construct.NaturalChoice.MinOp as MinOp
 open import Function.Base using (flip)
 open import Relation.Binary
-open import Relation.Binary.Construct.Converse using ()
+open import Relation.Binary.Construct.Flip.EqAndOrd using ()
   renaming (totalPreorder to flipOrder)
 
 module Algebra.Construct.NaturalChoice.MaxOp
diff --git a/src/Data/Fin/Induction.agda b/src/Data/Fin/Induction.agda
index acd098be75..3c9bf54546 100644
--- a/src/Data/Fin/Induction.agda
+++ b/src/Data/Fin/Induction.agda
@@ -23,8 +23,8 @@ open import Induction.WellFounded as WF
 open import Level using (Level)
 open import Relation.Binary
   using (Rel; Decidable; IsPartialOrder; IsStrictPartialOrder; StrictPartialOrder)
-import Relation.Binary.Construct.Converse as Converse
-import Relation.Binary.Construct.Flip as Flip
+import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
+import Relation.Binary.Construct.Flip.Ord as Ord
 import Relation.Binary.Construct.NonStrictToStrict as ToStrict
 import Relation.Binary.Construct.On as On
 open import Relation.Binary.Definitions using (Tri; tri<; tri≈; tri>)
@@ -123,7 +123,7 @@ module _ {_≈_ : Rel (Fin n) ℓ} where
     pigeon : (xs : Vec (Fin n) n) → Linked (flip _⊏_) (i ∷ xs) → WellFounded _⊏_
     pigeon xs i∷xs↑ =
       let (i₁ , i₂ , i₁<i₂ , xs[i₁]≡xs[i₂]) = pigeonhole (n<1+n n) (Vec.lookup (i ∷ xs)) in
-      let xs[i₁]⊏xs[i₂] = Linkedₚ.lookup⁺ (Flip.transitive _⊏_ ⊏.trans) {xs = i ∷ xs} i∷xs↑ i₁<i₂ in
+      let xs[i₁]⊏xs[i₂] = Linkedₚ.lookup⁺ (Ord.transitive _⊏_ ⊏.trans) {xs = i ∷ xs} i∷xs↑ i₁<i₂ in
       let xs[i₁]⊏xs[i₁] = ⊏.<-respʳ-≈ (⊏.Eq.reflexive xs[i₁]≡xs[i₂]) xs[i₁]⊏xs[i₂] in
       contradiction xs[i₁]⊏xs[i₁] (⊏.irrefl ⊏.Eq.refl)
 
@@ -137,7 +137,7 @@ module _ {_≈_ : Rel (Fin n) ℓ} where
 
   spo-noetherian : ∀ {r} {_⊏_ : Rel (Fin n) r} →
                    IsStrictPartialOrder _≈_ _⊏_ → WellFounded (flip _⊏_)
-  spo-noetherian isSPO = spo-wellFounded (Converse.isStrictPartialOrder isSPO)
+  spo-noetherian isSPO = spo-wellFounded (EqAndOrd.isStrictPartialOrder isSPO)
 
   po-noetherian : ∀ {r} {_⊑_ : Rel (Fin n) r} → IsPartialOrder _≈_ _⊑_ →
                   WellFounded (flip (ToStrict._<_ _≈_ _⊑_))
diff --git a/src/Relation/Binary/Construct/Converse.agda b/src/Relation/Binary/Construct/Converse.agda
index e750283521..c1e2d177a7 100644
--- a/src/Relation/Binary/Construct/Converse.agda
+++ b/src/Relation/Binary/Construct/Converse.agda
@@ -1,195 +1,17 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Many properties which hold for `∼` also hold for `flip ∼`. Unlike
--- the module `Relation.Binary.Construct.Flip` this module does not
--- flip the underlying equality.
+-- This module is DEPRECATED. Please use
+-- `Relation.Binary.Construct.Flip.EqAndOrd` instead.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
-
 module Relation.Binary.Construct.Converse where
 
-open import Data.Product
-open import Function.Base using (flip; _∘_)
-open import Level using (Level)
-
-private
-  variable
-    a b p ℓ ℓ₁ ℓ₂ : Level
-    A B : Set a
-    ≈ ∼ ≤ < : Rel A ℓ
-
-------------------------------------------------------------------------
--- Properties
-
-module _ (∼ : Rel A ℓ) where
-
-  refl : Reflexive ∼ → Reflexive (flip ∼)
-  refl refl = refl
-
-  sym : Symmetric ∼ → Symmetric (flip ∼)
-  sym sym = sym
-
-  trans : Transitive ∼ → Transitive (flip ∼)
-  trans trans = flip trans
-
-  asym : Asymmetric ∼ → Asymmetric (flip ∼)
-  asym asym = asym
-
-  total : Total ∼ → Total (flip ∼)
-  total total x y = total y x
-
-  resp : ∀ {p} (P : A → Set p) → Symmetric ∼ →
-             P Respects ∼ → P Respects (flip ∼)
-  resp _ sym resp ∼ = resp (sym ∼)
-
-  max : ∀ {⊥} → Minimum ∼ ⊥ → Maximum (flip ∼) ⊥
-  max min = min
-
-  min : ∀ {⊤} → Maximum ∼ ⊤ → Minimum (flip ∼) ⊤
-  min max = max
-
-module _ {≈ : Rel A ℓ₁} (∼ : Rel A ℓ₂) where
-
-  reflexive : Symmetric ≈ → (≈ ⇒ ∼) → (≈ ⇒ flip ∼)
-  reflexive sym impl = impl ∘ sym
-
-  irrefl : Symmetric ≈ → Irreflexive ≈ ∼ → Irreflexive ≈ (flip ∼)
-  irrefl sym irrefl x≈y y∼x = irrefl (sym x≈y) y∼x
-
-  antisym : Antisymmetric ≈ ∼ → Antisymmetric ≈ (flip ∼)
-  antisym antisym = flip antisym
-
-  compare : Trichotomous ≈ ∼ → Trichotomous ≈ (flip ∼)
-  compare cmp x y with cmp x y
-  ... | tri< x<y x≉y y≮x = tri> y≮x x≉y x<y
-  ... | tri≈ x≮y x≈y y≮x = tri≈ y≮x x≈y x≮y
-  ... | tri> x≮y x≉y y<x = tri< y<x x≉y x≮y
-
-module _ (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) where
-
-  resp₂ : ∼₁ Respects₂ ∼₂ → (flip ∼₁) Respects₂ ∼₂
-  resp₂ (resp₁ , resp₂) = resp₂ , resp₁
-
-module _ (∼ : REL A B ℓ) where
-
-  dec : Decidable ∼ → Decidable (flip ∼)
-  dec dec = flip dec
-
-------------------------------------------------------------------------
--- Structures
-
-isEquivalence : IsEquivalence ≈ → IsEquivalence (flip ≈)
-isEquivalence {≈ = ≈} eq = record
-  { refl  = refl  ≈ Eq.refl
-  ; sym   = sym   ≈ Eq.sym
-  ; trans = trans ≈ Eq.trans
-  } where module Eq = IsEquivalence eq
-
-isDecEquivalence : IsDecEquivalence ≈ → IsDecEquivalence (flip ≈)
-isDecEquivalence {≈ = ≈} eq = record
-  { isEquivalence = isEquivalence Dec.isEquivalence
-  ; _≟_           = dec ≈ Dec._≟_
-  } where module Dec = IsDecEquivalence eq
-
-isPreorder : IsPreorder ≈ ∼ → IsPreorder ≈ (flip ∼)
-isPreorder {≈ = ≈} {∼ = ∼} O = record
-  { isEquivalence = O.isEquivalence
-  ; reflexive     = reflexive ∼ O.Eq.sym O.reflexive
-  ; trans         = trans ∼ O.trans
-  } where module O = IsPreorder O
-
-isTotalPreorder : IsTotalPreorder ≈ ∼ → IsTotalPreorder ≈ (flip ∼)
-isTotalPreorder O = record
-  { isPreorder = isPreorder O.isPreorder
-  ; total      = total _ O.total
-  } where module O = IsTotalPreorder O
-
-isPartialOrder : IsPartialOrder ≈ ≤ → IsPartialOrder ≈ (flip ≤)
-isPartialOrder {≤ = ≤} O = record
-  { isPreorder = isPreorder O.isPreorder
-  ; antisym    = antisym ≤ O.antisym
-  } where module O = IsPartialOrder O
-
-isTotalOrder : IsTotalOrder ≈ ≤ → IsTotalOrder ≈ (flip ≤)
-isTotalOrder O = record
-  { isPartialOrder = isPartialOrder O.isPartialOrder
-  ; total          = total _ O.total
-  } where module O = IsTotalOrder O
-
-isDecTotalOrder : IsDecTotalOrder ≈ ≤ → IsDecTotalOrder ≈ (flip ≤)
-isDecTotalOrder O = record
-  { isTotalOrder = isTotalOrder O.isTotalOrder
-  ; _≟_          = O._≟_
-  ; _≤?_         = dec _ O._≤?_
-  } where module O = IsDecTotalOrder O
-
-isStrictPartialOrder : IsStrictPartialOrder ≈ < →
-                       IsStrictPartialOrder ≈ (flip <)
-isStrictPartialOrder {< = <} O = record
-  { isEquivalence = O.isEquivalence
-  ; irrefl        = irrefl < O.Eq.sym O.irrefl
-  ; trans         = trans < O.trans
-  ; <-resp-≈      = resp₂ _ _ O.<-resp-≈
-  } where module O = IsStrictPartialOrder O
-
-isStrictTotalOrder : IsStrictTotalOrder ≈ < →
-                     IsStrictTotalOrder ≈ (flip <)
-isStrictTotalOrder {< = <} O = record
-  { isEquivalence = O.isEquivalence
-  ; trans         = trans < O.trans
-  ; compare       = compare _ O.compare
-  } where module O = IsStrictTotalOrder O
-
-------------------------------------------------------------------------
--- Bundles
-
-setoid : Setoid a ℓ → Setoid a ℓ
-setoid S = record
-  { isEquivalence = isEquivalence S.isEquivalence
-  } where module S = Setoid S
-
-decSetoid : DecSetoid a ℓ → DecSetoid a ℓ
-decSetoid S = record
-  { isDecEquivalence = isDecEquivalence S.isDecEquivalence
-  } where module S = DecSetoid S
-
-preorder : Preorder a ℓ₁ ℓ₂ → Preorder a ℓ₁ ℓ₂
-preorder O = record
-  { isPreorder = isPreorder O.isPreorder
-  } where module O = Preorder O
-
-totalPreorder : TotalPreorder a ℓ₁ ℓ₂ → TotalPreorder a ℓ₁ ℓ₂
-totalPreorder O = record
-  { isTotalPreorder = isTotalPreorder O.isTotalPreorder
-  } where module O = TotalPreorder O
-
-poset : Poset a ℓ₁ ℓ₂ → Poset a ℓ₁ ℓ₂
-poset O = record
-  { isPartialOrder = isPartialOrder O.isPartialOrder
-  } where module O = Poset O
-
-totalOrder : TotalOrder a ℓ₁ ℓ₂ → TotalOrder a ℓ₁ ℓ₂
-totalOrder O = record
-  { isTotalOrder = isTotalOrder O.isTotalOrder
-  } where module O = TotalOrder O
-
-decTotalOrder : DecTotalOrder a ℓ₁ ℓ₂ → DecTotalOrder a ℓ₁ ℓ₂
-decTotalOrder O = record
-  { isDecTotalOrder = isDecTotalOrder O.isDecTotalOrder
-  } where module O = DecTotalOrder O
-
-strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ →
-                     StrictPartialOrder a ℓ₁ ℓ₂
-strictPartialOrder O = record
-  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder
-  } where module O = StrictPartialOrder O
+open import Relation.Binary.Construct.Flip.EqAndOrd public
 
-strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂ →
-                   StrictTotalOrder a ℓ₁ ℓ₂
-strictTotalOrder O = record
-  { isStrictTotalOrder = isStrictTotalOrder O.isStrictTotalOrder
-  } where module O = StrictTotalOrder O
+{-# WARNING_ON_IMPORT
+"Relation.Binary.Construct.Converse was deprecated in v2.0.
+Use Relation.Binary.Construct.Flip.EqAndOrd instead."
+#-}
diff --git a/src/Relation/Binary/Construct/Flip.Agda b/src/Relation/Binary/Construct/Flip.Agda
new file mode 100644
index 0000000000..cbae1e8867
--- /dev/null
+++ b/src/Relation/Binary/Construct/Flip.Agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- This module is DEPRECATED. Please use
+-- `Relation.Binary.Construct.Flip.Ord` instead.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Relation.Binary.Construct.Flip where
+
+open import Relation.Binary.Construct.Flip.Ord public
+
+{-# WARNING_ON_IMPORT
+"Relation.Binary.Construct.Flip was deprecated in v2.0.
+Use Relation.Binary.Construct.Flip.Ord instead."
+#-}
diff --git a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
new file mode 100644
index 0000000000..1d3e9430d9
--- /dev/null
+++ b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
@@ -0,0 +1,195 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Many properties which hold for `∼` also hold for `flip ∼`. Unlike
+-- the module `Relation.Binary.Construct.Flip.Ord` this module does not
+-- flip the underlying equality.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary
+
+module Relation.Binary.Construct.Flip.EqAndOrd where
+
+open import Data.Product
+open import Function.Base using (flip; _∘_)
+open import Level using (Level)
+
+private
+  variable
+    a b p ℓ ℓ₁ ℓ₂ : Level
+    A B : Set a
+    ≈ ∼ ≤ < : Rel A ℓ
+
+------------------------------------------------------------------------
+-- Properties
+
+module _ (∼ : Rel A ℓ) where
+
+  refl : Reflexive ∼ → Reflexive (flip ∼)
+  refl refl = refl
+
+  sym : Symmetric ∼ → Symmetric (flip ∼)
+  sym sym = sym
+
+  trans : Transitive ∼ → Transitive (flip ∼)
+  trans trans = flip trans
+
+  asym : Asymmetric ∼ → Asymmetric (flip ∼)
+  asym asym = asym
+
+  total : Total ∼ → Total (flip ∼)
+  total total x y = total y x
+
+  resp : ∀ {p} (P : A → Set p) → Symmetric ∼ →
+             P Respects ∼ → P Respects (flip ∼)
+  resp _ sym resp ∼ = resp (sym ∼)
+
+  max : ∀ {⊥} → Minimum ∼ ⊥ → Maximum (flip ∼) ⊥
+  max min = min
+
+  min : ∀ {⊤} → Maximum ∼ ⊤ → Minimum (flip ∼) ⊤
+  min max = max
+
+module _ {≈ : Rel A ℓ₁} (∼ : Rel A ℓ₂) where
+
+  reflexive : Symmetric ≈ → (≈ ⇒ ∼) → (≈ ⇒ flip ∼)
+  reflexive sym impl = impl ∘ sym
+
+  irrefl : Symmetric ≈ → Irreflexive ≈ ∼ → Irreflexive ≈ (flip ∼)
+  irrefl sym irrefl x≈y y∼x = irrefl (sym x≈y) y∼x
+
+  antisym : Antisymmetric ≈ ∼ → Antisymmetric ≈ (flip ∼)
+  antisym antisym = flip antisym
+
+  compare : Trichotomous ≈ ∼ → Trichotomous ≈ (flip ∼)
+  compare cmp x y with cmp x y
+  ... | tri< x<y x≉y y≮x = tri> y≮x x≉y x<y
+  ... | tri≈ x≮y x≈y y≮x = tri≈ y≮x x≈y x≮y
+  ... | tri> x≮y x≉y y<x = tri< y<x x≉y x≮y
+
+module _ (∼₁ : Rel A ℓ₁) (∼₂ : Rel A ℓ₂) where
+
+  resp₂ : ∼₁ Respects₂ ∼₂ → (flip ∼₁) Respects₂ ∼₂
+  resp₂ (resp₁ , resp₂) = resp₂ , resp₁
+
+module _ (∼ : REL A B ℓ) where
+
+  dec : Decidable ∼ → Decidable (flip ∼)
+  dec dec = flip dec
+
+------------------------------------------------------------------------
+-- Structures
+
+isEquivalence : IsEquivalence ≈ → IsEquivalence (flip ≈)
+isEquivalence {≈ = ≈} eq = record
+  { refl  = refl  ≈ Eq.refl
+  ; sym   = sym   ≈ Eq.sym
+  ; trans = trans ≈ Eq.trans
+  } where module Eq = IsEquivalence eq
+
+isDecEquivalence : IsDecEquivalence ≈ → IsDecEquivalence (flip ≈)
+isDecEquivalence {≈ = ≈} eq = record
+  { isEquivalence = isEquivalence Dec.isEquivalence
+  ; _≟_           = dec ≈ Dec._≟_
+  } where module Dec = IsDecEquivalence eq
+
+isPreorder : IsPreorder ≈ ∼ → IsPreorder ≈ (flip ∼)
+isPreorder {≈ = ≈} {∼ = ∼} O = record
+  { isEquivalence = O.isEquivalence
+  ; reflexive     = reflexive ∼ O.Eq.sym O.reflexive
+  ; trans         = trans ∼ O.trans
+  } where module O = IsPreorder O
+
+isTotalPreorder : IsTotalPreorder ≈ ∼ → IsTotalPreorder ≈ (flip ∼)
+isTotalPreorder O = record
+  { isPreorder = isPreorder O.isPreorder
+  ; total      = total _ O.total
+  } where module O = IsTotalPreorder O
+
+isPartialOrder : IsPartialOrder ≈ ≤ → IsPartialOrder ≈ (flip ≤)
+isPartialOrder {≤ = ≤} O = record
+  { isPreorder = isPreorder O.isPreorder
+  ; antisym    = antisym ≤ O.antisym
+  } where module O = IsPartialOrder O
+
+isTotalOrder : IsTotalOrder ≈ ≤ → IsTotalOrder ≈ (flip ≤)
+isTotalOrder O = record
+  { isPartialOrder = isPartialOrder O.isPartialOrder
+  ; total          = total _ O.total
+  } where module O = IsTotalOrder O
+
+isDecTotalOrder : IsDecTotalOrder ≈ ≤ → IsDecTotalOrder ≈ (flip ≤)
+isDecTotalOrder O = record
+  { isTotalOrder = isTotalOrder O.isTotalOrder
+  ; _≟_          = O._≟_
+  ; _≤?_         = dec _ O._≤?_
+  } where module O = IsDecTotalOrder O
+
+isStrictPartialOrder : IsStrictPartialOrder ≈ < →
+                       IsStrictPartialOrder ≈ (flip <)
+isStrictPartialOrder {< = <} O = record
+  { isEquivalence = O.isEquivalence
+  ; irrefl        = irrefl < O.Eq.sym O.irrefl
+  ; trans         = trans < O.trans
+  ; <-resp-≈      = resp₂ _ _ O.<-resp-≈
+  } where module O = IsStrictPartialOrder O
+
+isStrictTotalOrder : IsStrictTotalOrder ≈ < →
+                     IsStrictTotalOrder ≈ (flip <)
+isStrictTotalOrder {< = <} O = record
+  { isEquivalence = O.isEquivalence
+  ; trans         = trans < O.trans
+  ; compare       = compare _ O.compare
+  } where module O = IsStrictTotalOrder O
+
+------------------------------------------------------------------------
+-- Bundles
+
+setoid : Setoid a ℓ → Setoid a ℓ
+setoid S = record
+  { isEquivalence = isEquivalence S.isEquivalence
+  } where module S = Setoid S
+
+decSetoid : DecSetoid a ℓ → DecSetoid a ℓ
+decSetoid S = record
+  { isDecEquivalence = isDecEquivalence S.isDecEquivalence
+  } where module S = DecSetoid S
+
+preorder : Preorder a ℓ₁ ℓ₂ → Preorder a ℓ₁ ℓ₂
+preorder O = record
+  { isPreorder = isPreorder O.isPreorder
+  } where module O = Preorder O
+
+totalPreorder : TotalPreorder a ℓ₁ ℓ₂ → TotalPreorder a ℓ₁ ℓ₂
+totalPreorder O = record
+  { isTotalPreorder = isTotalPreorder O.isTotalPreorder
+  } where module O = TotalPreorder O
+
+poset : Poset a ℓ₁ ℓ₂ → Poset a ℓ₁ ℓ₂
+poset O = record
+  { isPartialOrder = isPartialOrder O.isPartialOrder
+  } where module O = Poset O
+
+totalOrder : TotalOrder a ℓ₁ ℓ₂ → TotalOrder a ℓ₁ ℓ₂
+totalOrder O = record
+  { isTotalOrder = isTotalOrder O.isTotalOrder
+  } where module O = TotalOrder O
+
+decTotalOrder : DecTotalOrder a ℓ₁ ℓ₂ → DecTotalOrder a ℓ₁ ℓ₂
+decTotalOrder O = record
+  { isDecTotalOrder = isDecTotalOrder O.isDecTotalOrder
+  } where module O = DecTotalOrder O
+
+strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ →
+                     StrictPartialOrder a ℓ₁ ℓ₂
+strictPartialOrder O = record
+  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder
+  } where module O = StrictPartialOrder O
+
+strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂ →
+                   StrictTotalOrder a ℓ₁ ℓ₂
+strictTotalOrder O = record
+  { isStrictTotalOrder = isStrictTotalOrder O.isStrictTotalOrder
+  } where module O = StrictTotalOrder O
diff --git a/src/Relation/Binary/Construct/Flip.agda b/src/Relation/Binary/Construct/Flip/Ord.agda
similarity index 97%
rename from src/Relation/Binary/Construct/Flip.agda
rename to src/Relation/Binary/Construct/Flip/Ord.agda
index 2f7145be9e..ecfc4b6df4 100644
--- a/src/Relation/Binary/Construct/Flip.agda
+++ b/src/Relation/Binary/Construct/Flip/Ord.agda
@@ -2,15 +2,15 @@
 -- The Agda standard library
 --
 -- Many properties which hold for `∼` also hold for `flip ∼`. Unlike
--- the module `Relation.Binary.Construct.Converse` this module flips
--- both the relation and the underlying equality.
+-- the module `Relation.Binary.Construct.Flip.EqAndOrd` this module
+-- flips both the relation and the underlying equality.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary
 
-module Relation.Binary.Construct.Flip where
+module Relation.Binary.Construct.Flip.Ord where
 
 open import Data.Product
 open import Function.Base using (flip; _∘_)
diff --git a/src/Relation/Binary/Properties/DecTotalOrder.agda b/src/Relation/Binary/Properties/DecTotalOrder.agda
index 3a84300690..386bb2189b 100644
--- a/src/Relation/Binary/Properties/DecTotalOrder.agda
+++ b/src/Relation/Binary/Properties/DecTotalOrder.agda
@@ -13,7 +13,7 @@ module Relation.Binary.Properties.DecTotalOrder
 
 open DecTotalOrder DT hiding (trans)
 
-import Relation.Binary.Construct.Converse as Converse
+import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict
 import Relation.Binary.Properties.TotalOrder totalOrder as TotalOrderProperties
 open import Relation.Nullary.Negation using (¬_)
@@ -38,7 +38,7 @@ open TotalOrderProperties public
   )
 
 ≥-isDecTotalOrder : IsDecTotalOrder _≈_ _≥_
-≥-isDecTotalOrder = Converse.isDecTotalOrder isDecTotalOrder
+≥-isDecTotalOrder = EqAndOrd.isDecTotalOrder isDecTotalOrder
 
 ≥-decTotalOrder : DecTotalOrder _ _ _
 ≥-decTotalOrder = record
diff --git a/src/Relation/Binary/Properties/Preorder.agda b/src/Relation/Binary/Properties/Preorder.agda
index a6a8c196a0..1a215b639c 100644
--- a/src/Relation/Binary/Properties/Preorder.agda
+++ b/src/Relation/Binary/Properties/Preorder.agda
@@ -13,7 +13,7 @@ module Relation.Binary.Properties.Preorder
 
 open import Function.Base
 open import Data.Product as Prod
-import Relation.Binary.Construct.Converse as Converse
+import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 
 open Preorder P
 
@@ -21,10 +21,10 @@ open Preorder P
 -- The inverse relation is also a preorder.
 
 converse-isPreorder : IsPreorder _≈_ (flip _∼_)
-converse-isPreorder = Converse.isPreorder isPreorder
+converse-isPreorder = EqAndOrd.isPreorder isPreorder
 
 converse-preorder : Preorder p₁ p₂ p₃
-converse-preorder = Converse.preorder P
+converse-preorder = EqAndOrd.preorder P
 
 ------------------------------------------------------------------------
 -- For every preorder there is an induced equivalence
diff --git a/src/Relation/Binary/Properties/StrictPartialOrder.agda b/src/Relation/Binary/Properties/StrictPartialOrder.agda
index d907fd75d4..7b6043af77 100644
--- a/src/Relation/Binary/Properties/StrictPartialOrder.agda
+++ b/src/Relation/Binary/Properties/StrictPartialOrder.agda
@@ -11,7 +11,7 @@ open import Relation.Binary
 module Relation.Binary.Properties.StrictPartialOrder
        {s₁ s₂ s₃} (SPO : StrictPartialOrder s₁ s₂ s₃) where
 
-import Relation.Binary.Construct.Converse as Converse
+import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 import Relation.Binary.Construct.StrictToNonStrict
 
 open Relation.Binary.StrictPartialOrder SPO
@@ -20,7 +20,7 @@ open Relation.Binary.StrictPartialOrder SPO
 -- The inverse relation is also a strict partial order.
 
 >-strictPartialOrder : StrictPartialOrder s₁ s₂ s₃
->-strictPartialOrder = Converse.strictPartialOrder SPO
+>-strictPartialOrder = EqAndOrd.strictPartialOrder SPO
 
 open StrictPartialOrder >-strictPartialOrder public
   using ()
diff --git a/src/Relation/Binary/Properties/TotalOrder.agda b/src/Relation/Binary/Properties/TotalOrder.agda
index 58964e71a3..da4f316892 100644
--- a/src/Relation/Binary/Properties/TotalOrder.agda
+++ b/src/Relation/Binary/Properties/TotalOrder.agda
@@ -15,7 +15,7 @@ open TotalOrder T
 
 open import Data.Product using (proj₁)
 open import Data.Sum.Base using (inj₁; inj₂)
-import Relation.Binary.Construct.Converse as Converse
+import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict
 import Relation.Binary.Properties.Poset poset as PosetProperties
 open import Relation.Binary.Consequences
@@ -51,7 +51,7 @@ open PosetProperties public
   )
 
 ≥-isTotalOrder : IsTotalOrder _≈_ _≥_
-≥-isTotalOrder = Converse.isTotalOrder isTotalOrder
+≥-isTotalOrder = EqAndOrd.isTotalOrder isTotalOrder
 
 ≥-totalOrder : TotalOrder _ _ _
 ≥-totalOrder = record

From 16dfcda6fc75e0ad16927fccc9d83d0b33f31093 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Tue, 20 Jun 2023 04:46:12 +0200
Subject: [PATCH 054/316] New property merge-is-superlist (#1950)

---
 CHANGELOG.md                                  | 61 +++++++++++--------
 .../Binary/Sublist/Setoid/Properties.agda     | 29 ++++++++-
 .../Unary/Sorted/TotalOrder/Properties.agda   | 30 ++++++---
 3 files changed, 84 insertions(+), 36 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index bb45ab0a49..25bc7bd058 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -606,7 +606,7 @@ Non-backwards compatible changes
 * It was very difficult to use the `Relation.Nullary` modules, as `Relation.Nullary`
   contained the basic definitions of negation, decidability etc., and the operations and
   proofs were smeared over `Relation.Nullary.(Negation/Product/Sum/Implication etc.)`.
-  
+
 * In order to fix this:
   - the definition of `Dec` and `recompute` have been moved to `Relation.Nullary.Decidable.Core`
   - the definition of `Reflects` has been moved to `Relation.Nullary.Reflects`
@@ -623,11 +623,11 @@ Non-backwards compatible changes
   have been deprecated and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`
   however all their contents is re-exported by `Relation.Nullary` which is the easiest way to access
   it now.
-  
+
 * In order to facilitate this reorganisation the following breaking moves have occured:
   - `¬?` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Decidable.Core`
   - `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Reflects`.
-  - `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved from `Relation.Nullary.Negation` 
+  - `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved from `Relation.Nullary.Negation`
 	to `Relation.Nullary.Decidable`.
   - `fromDec` and `toDec` have been mvoed from `Data.Sum.Base` to `Data.Sum`.
 
@@ -773,7 +773,7 @@ Non-backwards compatible changes
   ```
   NB. It is not possible to rename or deprecate `syntax` declarations, so Agda will
   only issue a "Could not parse the application `begin ...` when scope checking"
-  warning if the old combinators are used. 
+  warning if the old combinators are used.
 
 * The types of the proofs `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` in
   `Data.Rational(.Unnormalised).Properties` have been switched, as the previous
@@ -1073,7 +1073,7 @@ Deprecated names
 
 * In `Data.Fin.Induction`:
   ```
-  ≺-Rec 
+  ≺-Rec
   ≺-wellFounded
   ≺-recBuilder
   ≺-rec
@@ -1082,7 +1082,7 @@ Deprecated names
   As with Issue #1726 above: the deprecation of relation `_≺_` means that these definitions
   associated with wf-recursion are deprecated in favour of their `_<_` counterparts.
   But it's not quite sensible to say that these definiton should be *renamed* to *anything*,
-  least of all those counterparts... the type confusion would be intolerable. 
+  least of all those counterparts... the type confusion would be intolerable.
 
 * In `Data.Fin.Properties`:
   ```
@@ -1163,14 +1163,14 @@ Deprecated names
 
   ^-semigroup-morphism ↦ ^-isMagmaHomomorphism
   ^-monoid-morphism    ↦ ^-isMonoidHomomorphism
-  
+
   pos-distrib-* ↦ pos-*
   pos-+-commute ↦ pos-+
   abs-*-commute ↦ abs-*
-  
+
   +-isAbelianGroup ↦ +-0-isAbelianGroup
   ```
-  
+
 * In `Data.List.Properties`:
   ```agda
   map-id₂         ↦  map-id-local
@@ -1661,7 +1661,7 @@ New modules
   ```
   Algebra.Properties.Quasigroup
   ```
-  
+
 * Properties of MiddleBolLoop
   ```
   Algebra.Properties.MiddleBolLoop
@@ -1828,7 +1828,7 @@ Other minor changes
   _MiddleFourExchange_ : Op₂ A → Op₂ A → Set _
 
   SelfInverse : Op₁ A → Set _
-  
+
   LeftDividesˡ  : Op₂ A → Op₂ A → Set _
   LeftDividesʳ  : Op₂ A → Op₂ A → Set _
   RightDividesˡ : Op₂ A → Op₂ A → Set _
@@ -1864,7 +1864,7 @@ Other minor changes
   _^ᵗ_     : A → ℕ → A
   ```
 
-* `Algebra.Properties.Magma.Divisibility` now re-exports operations 
+* `Algebra.Properties.Magma.Divisibility` now re-exports operations
   `_∣ˡ_`, `_∤ˡ_`, `_∣ʳ_`, `_∤ʳ_` from `Algebra.Definitions.Magma`.
 
 * Added new proofs to `Algebra.Properties.CommutativeSemigroup`:
@@ -2046,7 +2046,7 @@ Other minor changes
 * Added new functions in `Data.Integer.Base`:
   ```
   _^_ : ℤ → ℕ → ℤ
- 
+
   +-0-rawGroup  : Rawgroup 0ℓ 0ℓ
 
   *-rawMagma    : RawMagma 0ℓ 0ℓ
@@ -2164,7 +2164,7 @@ Other minor changes
   _! : ℕ → ℕ
 
   parity : ℕ → Parity
-  
+
   +-rawMagma          : RawMagma 0ℓ 0ℓ
   +-0-rawMonoid       : RawMonoid 0ℓ 0ℓ
   *-rawMagma          : RawMagma 0ℓ 0ℓ
@@ -2409,10 +2409,10 @@ Other minor changes
 * Added new proof to `Data.Product.Relation.Binary.Lex.Strict`
   ```agda
   ×-respectsʳ : Transitive _≈₁_ →
-                _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_ 
+                _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_
   ×-respectsˡ : Symmetric _≈₁_ → Transitive _≈₁_ →
-                 _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_ 
-  ×-wellFounded' : Symmetric  _≈₁_ → Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ → 
+                 _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_
+  ×-wellFounded' : Symmetric  _≈₁_ → Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ →
                    WellFounded _<₁_ → WellFounded _<₂_ → WellFounded _<ₗₑₓ_
   ```
 
@@ -2565,7 +2565,7 @@ Other minor changes
                 ∀ {m n} → _Respectsˡ_ (_<_ {m} {n}) _≋_
   <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ →
                 ∀ {m n} → _Respectsʳ_ (_<_ {m} {n}) _≋_
-  <-wellFounded : Symmetric _≈_ →  Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ → 
+  <-wellFounded : Symmetric _≈_ →  Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ →
                   ∀ {n} → WellFounded (_<_ {n})
 ```
 
@@ -3184,14 +3184,21 @@ This is a full list of proofs that have changed form to use irrelevant instance
   ↭-reverse : (xs : List A) → reverse xs ↭ xs
   ```
 
-* Added new file `Relation.Binary.Reasoning.Base.Apartness`
-
-This is how to use it:
+* Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`
+  and `Data.List.Relation.Unary.Sorted.TotalOrder.Properties`.
   ```agda
-  _ : a # d
-  _ = begin-apartness
-    a ≈⟨ a≈b ⟩
-    b #⟨ b#c ⟩
-    c ≈⟨ c≈d ⟩
-    d ∎
+  ⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys
+  ⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys
   ```
+
+* Added new file `Relation.Binary.Reasoning.Base.Apartness`
+
+  This is how to use it:
+    ```agda
+    _ : a # d
+    _ = begin-apartness
+      a ≈⟨ a≈b ⟩
+      b #⟨ b#c ⟩
+      c ≈⟨ c≈d ⟩
+      d ∎
+    ```
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
index dd7638bc5f..5600339c7b 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid; _⇒_; _Preserves_⟶_)
+open import Relation.Binary using (Rel; Setoid; _⇒_; _Preserves_⟶_)
 
 module Data.List.Relation.Binary.Sublist.Setoid.Properties
   {c ℓ} (S : Setoid c ℓ) where
@@ -20,10 +20,12 @@ open import Data.Product using (∃; _,_; proj₂)
 open import Function.Base
 open import Function.Bundles using (_⇔_; _⤖_)
 open import Level
+open import Relation.Binary using () renaming (Decidable to Decidable₂)
 open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.Structures using (IsDecTotalOrder)
 open import Relation.Unary using (Pred; Decidable; Irrelevant)
 open import Relation.Nullary.Negation using (¬_)
-open import Relation.Nullary.Decidable using (¬?)
+open import Relation.Nullary.Decidable using (¬?; yes; no)
 
 import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
 import Data.List.Relation.Binary.Sublist.Setoid as SetoidSublist
@@ -203,6 +205,29 @@ module _ {as bs : List A} where
   reverse⁻ : reverse as ⊆ reverse bs → as ⊆ bs
   reverse⁻ = HeteroProperties.reverse⁻
 
+------------------------------------------------------------------------
+-- merge
+
+module _ {ℓ′} {_≤_ : Rel A ℓ′} (_≤?_ : Decidable₂ _≤_) where
+
+  ⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys
+  ⊆-mergeˡ []       ys = minimum ys
+  ⊆-mergeˡ (x ∷ xs) [] = ⊆-refl
+  ⊆-mergeˡ (x ∷ xs) (y ∷ ys)
+   with x ≤? y  | ⊆-mergeˡ xs (y ∷ ys)
+                      | ⊆-mergeˡ (x ∷ xs) ys
+  ... | yes x≤y | rec | _   = ≈-refl ∷ rec
+  ... | no  x≰y | _   | rec = y ∷ʳ rec
+
+  ⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys
+  ⊆-mergeʳ [] ys =  ⊆-refl
+  ⊆-mergeʳ (x ∷ xs) [] = minimum (merge _≤?_ (x ∷ xs) [])
+  ⊆-mergeʳ (x ∷ xs) (y ∷ ys)
+   with x ≤? y  | ⊆-mergeʳ xs (y ∷ ys)
+                      | ⊆-mergeʳ (x ∷ xs) ys
+  ... | yes x≤y | rec | _   = x ∷ʳ rec
+  ... | no  x≰y | _   | rec = ≈-refl ∷ rec
+
 ------------------------------------------------------------------------
 -- Inversion lemmas
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda b/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda
index 69654ef242..89582d2f88 100644
--- a/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda
+++ b/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda
@@ -14,15 +14,20 @@ open import Data.List.Relation.Unary.AllPairs using (AllPairs)
 open import Data.List.Relation.Unary.Linked as Linked
   using (Linked; []; [-]; _∷_; _∷′_; head′; tail)
 import Data.List.Relation.Unary.Linked.Properties as Linked
+import Data.List.Relation.Binary.Sublist.Setoid as Sublist
+import Data.List.Relation.Binary.Sublist.Setoid.Properties as SublistProperties
 open import Data.List.Relation.Unary.Sorted.TotalOrder hiding (head)
+
 open import Data.Maybe.Base using (just; nothing)
 open import Data.Maybe.Relation.Binary.Connected using (Connected; just)
 open import Data.Nat.Base using (ℕ; zero; suc; _<_)
+
 open import Level using (Level)
 open import Relation.Binary hiding (Decidable)
 import Relation.Binary.Properties.TotalOrder as TotalOrderProperties
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (yes; no)
+
 private
   variable
     a b p ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level
@@ -101,7 +106,7 @@ module _ (O : TotalOrder a ℓ₁ ℓ₂) where
 -- merge
 
 module _ (DO : DecTotalOrder a ℓ₁ ℓ₂) where
-  open DecTotalOrder DO renaming (totalOrder to O)
+  open DecTotalOrder DO using (_≤_; _≤?_) renaming (totalOrder to O)
   open TotalOrderProperties O using (≰⇒≥)
 
   private
@@ -109,22 +114,33 @@ module _ (DO : DecTotalOrder a ℓ₁ ℓ₂) where
                 Connected _≤_ (just v) (head xs) →
                 Connected _≤_ (just v) (head ys) →
                 Connected _≤_ (just v) (head (merge _≤?_ xs ys))
-    merge-con {xs = []}     {[]}     cxs cys = cys
-    merge-con {xs = []}     {y ∷ ys} cxs cys = cys
+    merge-con {xs = []}              cxs cys = cys
     merge-con {xs = x ∷ xs} {[]}     cxs cys = cxs
     merge-con {xs = x ∷ xs} {y ∷ ys} cxs cys with x ≤? y
     ... | yes x≤y = cxs
     ... | no  x≰y = cys
 
   merge⁺ : ∀ {xs ys} → Sorted O xs → Sorted O ys → Sorted O (merge _≤?_ xs ys)
-  merge⁺ {[]}     {[]}     rxs rys = []
-  merge⁺ {[]}     {x ∷ ys} rxs rys = rys
+  merge⁺ {[]}              rxs rys = rys
   merge⁺ {x ∷ xs} {[]}     rxs rys = rxs
-  merge⁺ {x ∷ xs} {y ∷ ys} rxs rys with x ≤? y |
-    merge⁺ (Linked.tail rxs) rys | merge⁺ rxs (Linked.tail rys)
+  merge⁺ {x ∷ xs} {y ∷ ys} rxs rys
+   with x ≤? y  | merge⁺ (Linked.tail rxs) rys
+                      | merge⁺ rxs (Linked.tail rys)
   ... | yes x≤y | rec | _   = merge-con (head′ rxs)      (just x≤y)  ∷′ rec
   ... | no  x≰y | _   | rec = merge-con (just (≰⇒≥ x≰y)) (head′ rys) ∷′ rec
 
+  -- Reexport ⊆-mergeˡʳ
+
+  S = Preorder.Eq.setoid (DecTotalOrder.preorder DO)
+  open Sublist S using (_⊆_)
+  module SP = SublistProperties S
+
+  ⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys
+  ⊆-mergeˡ = SP.⊆-mergeˡ _≤?_
+
+  ⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys
+  ⊆-mergeʳ = SP.⊆-mergeʳ _≤?_
+
 ------------------------------------------------------------------------
 -- filter
 

From 99651670763f11a90ea66706d9ce60b096305a36 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Tue, 20 Jun 2023 08:58:17 +0200
Subject: [PATCH 055/316] Rename Flip.Agda to Flip.agda (re #1979)

---
 src/Relation/Binary/Construct/{Flip.Agda => Flip.agda} | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 rename src/Relation/Binary/Construct/{Flip.Agda => Flip.agda} (100%)

diff --git a/src/Relation/Binary/Construct/Flip.Agda b/src/Relation/Binary/Construct/Flip.agda
similarity index 100%
rename from src/Relation/Binary/Construct/Flip.Agda
rename to src/Relation/Binary/Construct/Flip.agda

From 6b3c7ac58b84f703ae2f5bd140cdbd6a290d8b96 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Tue, 20 Jun 2023 09:10:24 +0200
Subject: [PATCH 056/316] Bump Haskell-CI to GHCs 9.6.2 and 9.2.8

---
 .github/workflows/haskell-ci.yml | 12 ++++++------
 agda-stdlib-utils.cabal          |  4 ++--
 stack-9.0.2.yaml                 |  6 +-----
 stack-9.2.8.yaml                 |  6 ++++++
 4 files changed, 15 insertions(+), 13 deletions(-)
 create mode 100644 stack-9.2.8.yaml

diff --git a/.github/workflows/haskell-ci.yml b/.github/workflows/haskell-ci.yml
index 0aa1080f44..f491e32dc6 100644
--- a/.github/workflows/haskell-ci.yml
+++ b/.github/workflows/haskell-ci.yml
@@ -8,9 +8,9 @@
 #
 # For more information, see https://github.com/haskell-CI/haskell-ci
 #
-# version: 0.16.1
+# version: 0.16.4
 #
-# REGENDATA ("0.16.1",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
+# REGENDATA ("0.16.4",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
 #
 name: Haskell-CI
 on:
@@ -44,9 +44,9 @@ jobs:
     strategy:
       matrix:
         include:
-          - compiler: ghc-9.6.1
+          - compiler: ghc-9.6.2
             compilerKind: ghc
-            compilerVersion: 9.6.1
+            compilerVersion: 9.6.2
             setup-method: ghcup
             allow-failure: false
           - compiler: ghc-9.4.5
@@ -54,9 +54,9 @@ jobs:
             compilerVersion: 9.4.5
             setup-method: ghcup
             allow-failure: false
-          - compiler: ghc-9.2.7
+          - compiler: ghc-9.2.8
             compilerKind: ghc
-            compilerVersion: 9.2.7
+            compilerVersion: 9.2.8
             setup-method: ghcup
             allow-failure: false
           - compiler: ghc-9.0.2
diff --git a/agda-stdlib-utils.cabal b/agda-stdlib-utils.cabal
index 7d93147747..5ca3bd7055 100644
--- a/agda-stdlib-utils.cabal
+++ b/agda-stdlib-utils.cabal
@@ -6,9 +6,9 @@ description:     Helper programs for setting up the Agda standard library.
 license:         MIT
 
 tested-with:
-  GHC == 9.6.1
+  GHC == 9.6.2
   GHC == 9.4.5
-  GHC == 9.2.7
+  GHC == 9.2.8
   GHC == 9.0.2
   GHC == 8.10.7
   GHC == 8.8.4
diff --git a/stack-9.0.2.yaml b/stack-9.0.2.yaml
index 76871b383c..22f6eadaf6 100644
--- a/stack-9.0.2.yaml
+++ b/stack-9.0.2.yaml
@@ -1,10 +1,6 @@
-resolver: nightly-2022-01-27
+resolver: lts-19.33
 compiler: ghc-9.0.2
 compiler-check: match-exact
 
-# extra-deps:
-#   - filemanip-0.3.6.3
-#   - unix-compat-0.5.2
-
 packages:
 - '.'
diff --git a/stack-9.2.8.yaml b/stack-9.2.8.yaml
new file mode 100644
index 0000000000..3df1c335fb
--- /dev/null
+++ b/stack-9.2.8.yaml
@@ -0,0 +1,6 @@
+resolver: lts-20.26
+compiler: ghc-9.2.8
+compiler-check: match-exact
+
+packages:
+- '.'

From 38079f484bd1a70f0295d806f360473dca181d1b Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Fri, 23 Jun 2023 09:01:25 +0200
Subject: [PATCH 057/316] stack.yaml files for GHCs 9.4.5 and 9.6.2

---
 stack-9.4.5.yaml | 6 ++++++
 stack-9.6.2.yaml | 6 ++++++
 2 files changed, 12 insertions(+)
 create mode 100644 stack-9.4.5.yaml
 create mode 100644 stack-9.6.2.yaml

diff --git a/stack-9.4.5.yaml b/stack-9.4.5.yaml
new file mode 100644
index 0000000000..67331480d1
--- /dev/null
+++ b/stack-9.4.5.yaml
@@ -0,0 +1,6 @@
+resolver: lts-21.0
+compiler: ghc-9.4.5
+compiler-check: match-exact
+
+packages:
+- '.'
diff --git a/stack-9.6.2.yaml b/stack-9.6.2.yaml
new file mode 100644
index 0000000000..59f58a5299
--- /dev/null
+++ b/stack-9.6.2.yaml
@@ -0,0 +1,6 @@
+resolver: nightly-2023-06-23
+compiler: ghc-9.6.2
+compiler-check: match-exact
+
+packages:
+- '.'

From e06c1627e35fe8fc8c42fdd20bfc622d773db9b3 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Tue, 27 Jun 2023 09:03:50 +0100
Subject: [PATCH 058/316] [ re #1993 ] Simplifying the dependency graph (#1995)

---
 src/Algebra/Consequences/Setoid.agda          |  2 +-
 src/Algebra/Construct/LexProduct/Base.agda    |  4 ++--
 .../Construct/NaturalChoice/MinMaxOp.agda     |  2 +-
 .../Construct/NaturalChoice/MinOp.agda        |  2 +-
 src/Algebra/Definitions.agda                  | 10 ++++-----
 src/Algebra/Definitions/RawMagma.agda         |  2 +-
 .../Lattice/Properties/BooleanAlgebra.agda    |  4 ++--
 src/Algebra/Lattice/Properties/Lattice.agda   |  2 +-
 .../Lattice/Properties/Semilattice.agda       |  7 ++----
 src/Algebra/Lattice/Structures.agda           |  2 +-
 .../Properties/CommutativeSemigroup.agda      |  2 +-
 src/Algebra/Properties/Group.agda             |  4 ++--
 src/Algebra/Properties/Ring.agda              |  1 -
 src/Algebra/Properties/Semigroup.agda         |  2 +-
 src/Algebra/Structures.agda                   |  2 +-
 src/Algebra/Structures/Biased.agda            |  2 +-
 src/Codata/Sized/Cowriter.agda                |  4 ++--
 src/Codata/Sized/Stream.agda                  |  6 ++---
 src/Data/Bool/Properties.agda                 | 10 ++++-----
 src/Data/Digit.agda                           | 15 ++++++-------
 src/Data/Fin/Base.agda                        |  2 +-
 src/Data/Fin/Properties.agda                  |  9 ++++----
 src/Data/List/Base.agda                       |  2 +-
 src/Data/List/Fresh.agda                      |  2 +-
 src/Data/List/Fresh/NonEmpty.agda             |  6 ++---
 src/Data/List/Kleene/Base.agda                | 11 +++++-----
 src/Data/List/Membership/Setoid.agda          |  2 +-
 src/Data/List/NonEmpty/Base.agda              |  2 +-
 src/Data/List/Properties.agda                 |  3 ++-
 src/Data/List/Relation/Binary/Lex/Core.agda   |  2 +-
 .../List/Relation/Binary/Pointwise/Base.agda  |  4 ++--
 src/Data/List/Relation/Unary/All.agda         |  6 ++---
 src/Data/List/Relation/Unary/Any.agda         |  2 +-
 src/Data/Maybe/Base.agda                      |  2 +-
 src/Data/Nat/DivMod/Core.agda                 |  2 +-
 src/Data/Nat/Divisibility.agda                | 10 ++++-----
 src/Data/Nat/Induction.agda                   |  3 +--
 src/Data/Nat/Properties.agda                  |  4 ++--
 src/Data/Nat/Show.agda                        |  8 +++----
 src/Data/Product.agda                         | 16 +-------------
 src/Data/Product/Algebra.agda                 |  4 ++--
 src/Data/Product/Base.agda                    | 22 ++++++++++++++++++-
 src/Data/Product/Effectful/Left.agda          |  2 +-
 src/Data/Product/Effectful/Left/Base.agda     |  2 +-
 src/Data/Product/Effectful/Right.agda         |  2 +-
 src/Data/Product/Effectful/Right/Base.agda    |  2 +-
 src/Data/Product/Nary/NonDependent.agda       |  8 +++----
 src/Data/Product/Properties.agda              |  5 +++--
 src/Data/Product/Relation/Unary/All.agda      |  6 ++---
 src/Data/String/Base.agda                     | 13 +++++------
 src/Data/String/Unsafe.agda                   |  2 +-
 src/Data/Sum/Algebra.agda                     |  2 +-
 src/Data/Unit/Polymorphic/Properties.agda     |  2 +-
 src/Data/Vec/Base.agda                        |  4 ++--
 src/Data/Vec/Bounded/Base.agda                | 11 +++++-----
 src/Data/Vec/Functional.agda                  |  4 ++--
 src/Data/Vec/N-ary.agda                       | 11 +++++-----
 src/Data/Vec/Recursive.agda                   | 12 +++++-----
 src/Data/W.agda                               |  4 ++--
 src/Effect/Applicative.agda                   |  2 +-
 src/Effect/Monad/State/Transformer/Base.agda  |  2 +-
 src/Effect/Monad/Writer/Transformer/Base.agda |  2 +-
 src/Function/Bundles.agda                     |  2 +-
 src/Function/Consequences.agda                |  7 +++---
 src/Function/Construct/Composition.agda       |  4 ++--
 src/Function/Construct/Identity.agda          |  4 ++--
 src/Function/Construct/Symmetry.agda          |  2 +-
 src/Function/Definitions.agda                 |  5 ++---
 src/Function/Definitions/Core1.agda           |  2 +-
 src/Function/Definitions/Core2.agda           |  4 ++--
 src/Function/Equality.agda                    |  2 +-
 src/Function/Metric/Definitions.agda          |  2 +-
 src/Function/Nary/NonDependent/Base.agda      |  2 +-
 src/Function/Properties/Inverse.agda          |  2 +-
 src/Function/Structures.agda                  |  3 +--
 src/Induction/WellFounded.agda                | 11 +++++-----
 src/Relation/Binary/Consequences.agda         |  2 +-
 .../Binary/Construct/Flip/EqAndOrd.agda       |  2 +-
 .../Binary/Construct/NaturalOrder/Left.agda   |  2 +-
 .../Binary/Construct/NaturalOrder/Right.agda  |  2 +-
 src/Relation/Binary/Core.agda                 |  2 +-
 src/Relation/Binary/Definitions.agda          |  2 +-
 src/Relation/Binary/Lattice/Definitions.agda  |  6 ++---
 src/Relation/Binary/Lattice/Structures.agda   |  2 +-
 src/Relation/Binary/Morphism/Structures.agda  |  6 ++---
 src/Relation/Binary/Properties/Preorder.agda  |  4 ++--
 src/Relation/Binary/Properties/Setoid.agda    |  5 ++---
 .../Binary/Properties/TotalOrder.agda         |  4 +---
 .../Binary/PropositionalEquality.agda         |  6 ++---
 .../Binary/PropositionalEquality/Core.agda    |  2 +-
 .../PropositionalEquality/Properties.agda     |  2 +-
 .../Binary/Reasoning/Base/Double.agda         |  6 ++---
 .../Binary/Reasoning/Base/Triple.agda         |  6 ++---
 src/Relation/Binary/Reflection.agda           | 12 +++++-----
 src/Relation/Binary/Structures.agda           |  2 +-
 src/Relation/Nullary/Decidable.agda           |  2 +-
 src/Relation/Nullary/Decidable/Core.agda      |  2 +-
 src/Relation/Nullary/Negation.agda            | 17 +++++++-------
 src/Relation/Unary.agda                       | 10 ++++-----
 src/Relation/Unary/Closure/Base.agda          |  2 +-
 src/Relation/Unary/Properties.agda            |  4 ++--
 src/Tactic/RingSolver.agda                    |  2 +-
 .../RingSolver/Core/Polynomial/Base.agda      |  6 ++---
 src/Text/Pretty.agda                          | 10 ++++-----
 src/Text/Pretty/Core.agda                     |  2 +-
 src/Text/Printf.agda                          | 13 ++---------
 src/Text/Regex/Base.agda                      |  7 ++----
 107 files changed, 243 insertions(+), 268 deletions(-)

diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index 77badd1b90..07b75806c2 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -15,7 +15,7 @@ open Setoid S renaming (Carrier to A)
 open import Algebra.Core
 open import Algebra.Definitions _≈_
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_$_)
 import Function.Definitions as FunDefs
 import Relation.Binary.Consequences as Bin
diff --git a/src/Algebra/Construct/LexProduct/Base.agda b/src/Algebra/Construct/LexProduct/Base.agda
index 6076d1afc3..85686fd20c 100644
--- a/src/Algebra/Construct/LexProduct/Base.agda
+++ b/src/Algebra/Construct/LexProduct/Base.agda
@@ -8,10 +8,10 @@
 
 open import Algebra.Core using (Op₂)
 open import Data.Bool.Base using (true; false)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Nullary.Decidable using (does; yes; no)
+open import Relation.Nullary.Decidable.Core using (does; yes; no)
 
 module Algebra.Construct.LexProduct.Base
   {a b ℓ} {A : Set a} {B : Set b}
diff --git a/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda b/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
index 4b7d0e20e8..a0b7a945f2 100644
--- a/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
@@ -11,7 +11,7 @@ open import Algebra.Core
 open import Algebra.Bundles
 open import Algebra.Construct.NaturalChoice.Base
 open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id; _∘_; flip)
 open import Relation.Binary
 open import Relation.Binary.Consequences
diff --git a/src/Algebra/Construct/NaturalChoice/MinOp.agda b/src/Algebra/Construct/NaturalChoice/MinOp.agda
index 37ed331c7e..563e1662f1 100644
--- a/src/Algebra/Construct/NaturalChoice/MinOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MinOp.agda
@@ -11,7 +11,7 @@ open import Algebra.Core
 open import Algebra.Bundles
 open import Algebra.Construct.NaturalChoice.Base
 open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id; _∘_)
 open import Relation.Binary
 open import Relation.Binary.Consequences
diff --git a/src/Algebra/Definitions.agda b/src/Algebra/Definitions.agda
index e64fc4f30b..ae0566a4bf 100644
--- a/src/Algebra/Definitions.agda
+++ b/src/Algebra/Definitions.agda
@@ -15,17 +15,17 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Core
-open import Relation.Nullary.Negation using (¬_)
+open import Relation.Binary.Core using (Rel; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Nullary.Negation.Core using (¬_)
 
 module Algebra.Definitions
   {a ℓ} {A : Set a}   -- The underlying set
   (_≈_ : Rel A ℓ)     -- The underlying equality
   where
 
-open import Algebra.Core
-open import Data.Product
-open import Data.Sum.Base
+open import Algebra.Core using (Op₁; Op₂)
+open import Data.Product.Base using (_×_; ∃-syntax)
+open import Data.Sum.Base using (_⊎_)
 
 ------------------------------------------------------------------------
 -- Properties of operations
diff --git a/src/Algebra/Definitions/RawMagma.agda b/src/Algebra/Definitions/RawMagma.agda
index 5189e02fb8..dc96fedc76 100644
--- a/src/Algebra/Definitions/RawMagma.agda
+++ b/src/Algebra/Definitions/RawMagma.agda
@@ -11,7 +11,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Bundles using (RawMagma)
-open import Data.Product using (_×_; ∃)
+open import Data.Product.Base using (_×_; ∃)
 open import Level using (_⊔_)
 open import Relation.Binary.Core
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
index 85295c9366..cf90b022b9 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
@@ -23,9 +23,9 @@ open import Algebra.Bundles
 open import Algebra.Lattice.Structures _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
 open import Relation.Binary
-open import Function.Base
+open import Function.Base using (id; _$_; _⟨_⟩_)
 open import Function.Bundles using (_⇔_; module Equivalence)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 
 ------------------------------------------------------------------------
 -- Export properties from distributive lattices
diff --git a/src/Algebra/Lattice/Properties/Lattice.agda b/src/Algebra/Lattice/Properties/Lattice.agda
index 747541e49c..b5908f57b1 100644
--- a/src/Algebra/Lattice/Properties/Lattice.agda
+++ b/src/Algebra/Lattice/Properties/Lattice.agda
@@ -11,7 +11,7 @@ import Algebra.Lattice.Properties.Semilattice as SemilatticeProperties
 open import Relation.Binary
 import Relation.Binary.Lattice as R
 open import Function.Base
-open import Data.Product using (_,_; swap)
+open import Data.Product.Base using (_,_; swap)
 
 module Algebra.Lattice.Properties.Lattice
   {l₁ l₂} (L : Lattice l₁ l₂) where
diff --git a/src/Algebra/Lattice/Properties/Semilattice.agda b/src/Algebra/Lattice/Properties/Semilattice.agda
index 4b8fc114fc..38554aedfa 100644
--- a/src/Algebra/Lattice/Properties/Semilattice.agda
+++ b/src/Algebra/Lattice/Properties/Semilattice.agda
@@ -6,11 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra.Lattice
-open import Algebra.Structures
-open import Function
-open import Data.Product
-open import Relation.Binary
+open import Algebra.Lattice.Bundles using (Semilattice)
+open import Relation.Binary.Bundles using (Poset)
 import Relation.Binary.Lattice as B
 import Relation.Binary.Properties.Poset as PosetProperties
 
diff --git a/src/Algebra/Lattice/Structures.agda b/src/Algebra/Lattice/Structures.agda
index 8e80c4a87b..b02787eee0 100644
--- a/src/Algebra/Lattice/Structures.agda
+++ b/src/Algebra/Lattice/Structures.agda
@@ -14,7 +14,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Core
-open import Data.Product using (proj₁; proj₂)
+open import Data.Product.Base using (proj₁; proj₂)
 open import Level using (_⊔_)
 open import Relation.Binary using (Rel; Setoid; IsEquivalence)
 
diff --git a/src/Algebra/Properties/CommutativeSemigroup.agda b/src/Algebra/Properties/CommutativeSemigroup.agda
index 25d1ac0152..e09253ece7 100644
--- a/src/Algebra/Properties/CommutativeSemigroup.agda
+++ b/src/Algebra/Properties/CommutativeSemigroup.agda
@@ -16,7 +16,7 @@ open CommutativeSemigroup CS
 
 open import Algebra.Definitions _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Data.Product
+open import Data.Product.Base using (_,_)
 
 ------------------------------------------------------------------------------
 -- Re-export the contents of semigroup
diff --git a/src/Algebra/Properties/Group.agda b/src/Algebra/Properties/Group.agda
index 469245b11c..7cdf88c9fe 100644
--- a/src/Algebra/Properties/Group.agda
+++ b/src/Algebra/Properties/Group.agda
@@ -13,8 +13,8 @@ module Algebra.Properties.Group {g₁ g₂} (G : Group g₁ g₂) where
 open Group G
 open import Algebra.Definitions _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Function
-open import Data.Product
+open import Function.Base using (_$_; _⟨_⟩_)
+open import Data.Product.Base using (_,_; proj₂)
 
 ε⁻¹≈ε : ε ⁻¹ ≈ ε
 ε⁻¹≈ε = begin
diff --git a/src/Algebra/Properties/Ring.agda b/src/Algebra/Properties/Ring.agda
index e310202a37..22dabbac29 100644
--- a/src/Algebra/Properties/Ring.agda
+++ b/src/Algebra/Properties/Ring.agda
@@ -16,7 +16,6 @@ import Algebra.Properties.AbelianGroup as AbelianGroupProperties
 open import Function.Base using (_$_)
 open import Relation.Binary.Reasoning.Setoid setoid
 open import Algebra.Definitions _≈_
-open import Data.Product
 
 ------------------------------------------------------------------------
 -- Export properties of abelian groups
diff --git a/src/Algebra/Properties/Semigroup.agda b/src/Algebra/Properties/Semigroup.agda
index 4147b612e2..22aab4e66e 100644
--- a/src/Algebra/Properties/Semigroup.agda
+++ b/src/Algebra/Properties/Semigroup.agda
@@ -12,7 +12,7 @@ module Algebra.Properties.Semigroup {a ℓ} (S : Semigroup a ℓ) where
 
 open Semigroup S
 open import Algebra.Definitions _≈_
-open import Data.Product
+open import Data.Product.Base using (_,_)
 
 x∙yz≈xy∙z : ∀ x y z → x ∙ (y ∙ z) ≈ (x ∙ y) ∙ z
 x∙yz≈xy∙z x y z = sym (assoc x y z)
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index 0a904276db..eb9d49b2c9 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -22,7 +22,7 @@ module Algebra.Structures
 open import Algebra.Core
 open import Algebra.Definitions _≈_
 import Algebra.Consequences.Setoid as Consequences
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Level using (_⊔_)
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Structures/Biased.agda b/src/Algebra/Structures/Biased.agda
index c03e1236eb..0aa549c66c 100644
--- a/src/Algebra/Structures/Biased.agda
+++ b/src/Algebra/Structures/Biased.agda
@@ -9,7 +9,7 @@
 
 open import Algebra.Core
 open import Algebra.Consequences.Setoid
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Level using (_⊔_)
 open import Relation.Binary using (Rel; Setoid; IsEquivalence)
 
diff --git a/src/Codata/Sized/Cowriter.agda b/src/Codata/Sized/Cowriter.agda
index 2a6ef961ec..7206c83c86 100644
--- a/src/Codata/Sized/Cowriter.agda
+++ b/src/Codata/Sized/Cowriter.agda
@@ -16,9 +16,9 @@ open import Codata.Sized.Delay using (Delay; later; now)
 open import Codata.Sized.Stream as Stream using (Stream; _∷_)
 open import Data.Unit.Base
 open import Data.List.Base using (List; []; _∷_)
-open import Data.List.NonEmpty using (List⁺; _∷_)
+open import Data.List.NonEmpty.Base using (List⁺; _∷_)
 open import Data.Nat.Base as Nat using (ℕ; zero; suc)
-open import Data.Product as Prod using (_×_; _,_)
+open import Data.Product.Base as Prod using (_×_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Vec.Bounded.Base as Vec≤ using (Vec≤; _,_)
diff --git a/src/Codata/Sized/Stream.agda b/src/Codata/Sized/Stream.agda
index 85cd886b5f..4b7d6bd37d 100644
--- a/src/Codata/Sized/Stream.agda
+++ b/src/Codata/Sized/Stream.agda
@@ -15,10 +15,10 @@ open import Data.Nat.Base
 open import Data.List.Base using (List; []; _∷_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_; _∷⁺_)
 open import Data.Vec.Base using (Vec; []; _∷_)
-open import Data.Product as P hiding (map)
-open import Function.Base
+open import Data.Product.Base as P using (Σ; _×_; _,_; <_,_>; proj₁; proj₂)
+open import Function.Base using (id; _∘_)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index fe232885eb..ef963f6511 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -13,9 +13,9 @@ open import Algebra.Lattice.Bundles
 import Algebra.Lattice.Properties.BooleanAlgebra as BooleanAlgebraProperties
 open import Data.Bool.Base
 open import Data.Empty
-open import Data.Product
-open import Data.Sum.Base
-open import Function.Base
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
+open import Function.Base using (_⟨_⟩_; const; id)
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence
   using (_⇔_; equivalence; module Equivalence)
@@ -23,8 +23,8 @@ open import Induction.WellFounded using (WellFounded; Acc; acc)
 open import Level using (Level; 0ℓ)
 open import Relation.Binary hiding (_⇔_)
 open import Relation.Binary.PropositionalEquality hiding ([_])
-open import Relation.Nullary using (ofʸ; ofⁿ; does; proof; yes; no)
-open import Relation.Nullary.Decidable using (True)
+open import Relation.Nullary.Reflects using (ofʸ; ofⁿ)
+open import Relation.Nullary.Decidable.Core using (True; does; proof; yes; no)
 import Relation.Unary as U
 
 open import Algebra.Definitions {A = Bool} _≡_
diff --git a/src/Data/Digit.agda b/src/Data/Digit.agda
index 9c47fd544b..7e7c1c7555 100644
--- a/src/Data/Digit.agda
+++ b/src/Data/Digit.agda
@@ -9,21 +9,20 @@
 module Data.Digit where
 
 open import Data.Nat.Base
-open import Data.Nat.Properties
-open import Data.Nat.Solver
+open import Data.Nat.Properties using (_≤?_; _<?_; ≤⇒≤′; module ≤-Reasoning; m≤m+n)
+open import Data.Nat.Solver using (module +-*-Solver)
 open import Data.Fin.Base as Fin using (Fin; zero; suc; toℕ)
 open import Data.Bool.Base using (Bool; true; false)
-open import Data.Char using (Char)
+open import Data.Char.Base using (Char)
 open import Data.List.Base
-open import Data.Product
+open import Data.Product.Base using (∃; _,_)
 open import Data.Vec.Base as Vec using (Vec; _∷_; [])
 open import Data.Nat.DivMod
 open import Data.Nat.Induction
-open import Relation.Nullary.Decidable using (does)
-open import Relation.Nullary.Decidable
-open import Relation.Binary using (Decidable)
+open import Relation.Nullary.Decidable using (True; does; toWitness)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
-open import Function
+open import Function.Base using (_$_)
 
 ------------------------------------------------------------------------
 -- Digits
diff --git a/src/Data/Fin/Base.agda b/src/Data/Fin/Base.agda
index 20a2636216..b591301ac0 100644
--- a/src/Data/Fin/Base.agda
+++ b/src/Data/Fin/Base.agda
@@ -14,7 +14,7 @@ module Data.Fin.Base where
 open import Data.Bool.Base using (Bool; true; false; T; not)
 open import Data.Empty using (⊥-elim)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; z≤n; s≤s; z<s; s<s; _^_)
-open import Data.Product as Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (id; _∘_; _on_; flip)
 open import Level using (0ℓ)
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index d7a963f9fd..e181a291e9 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -18,11 +18,13 @@ open import Data.Bool.Base using (Bool; true; false; not; _∧_; _∨_)
 open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Fin.Base
 open import Data.Fin.Patterns
-open import Data.Nat.Base as ℕ using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; _∸_; _^_)
+open import Data.Nat.Base as ℕ
+  using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; _∸_; _^_)
 import Data.Nat.Properties as ℕₚ
 open import Data.Nat.Solver
 open import Data.Unit using (⊤; tt)
-open import Data.Product using (Σ-syntax; ∃; ∃₂; ∄; _×_; _,_; map; proj₁; proj₂; uncurry; <_,_>)
+open import Data.Product.Base as Prod
+  using (∃; ∃₂; _×_; _,_; map; proj₁; proj₂; uncurry; <_,_>)
 open import Data.Product.Properties using (,-injective)
 open import Data.Product.Algebra using (×-cong)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)
@@ -619,7 +621,7 @@ splitAt-≥ (suc m) (suc i) (s≤s i≥m) = cong (Sum.map suc id) (splitAt-≥ m
 remQuot-combine : ∀ {n k} (i : Fin n) j → remQuot k (combine i j) ≡ (i , j)
 remQuot-combine {suc n} {k} zero    j rewrite splitAt-↑ˡ k j (n ℕ.* k) = refl
 remQuot-combine {suc n} {k} (suc i) j rewrite splitAt-↑ʳ k   (n ℕ.* k) (combine i j) =
-  cong (Data.Product.map₁ suc) (remQuot-combine i j)
+  cong (Prod.map₁ suc) (remQuot-combine i j)
 
 combine-remQuot : ∀ {n} k (i : Fin (n ℕ.* k)) → uncurry combine (remQuot {n} k i) ≡ i
 combine-remQuot {suc n} k i with splitAt k i in eq
@@ -1158,4 +1160,3 @@ Please use <⇒<′ instead."
 "Warning: <′⇒≺ was deprecated in v2.0.
 Please use <′⇒< instead."
 #-}
-
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 48a5840bf4..2b25ece003 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -17,7 +17,7 @@ open import Data.Bool.Base as Bool
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _*_ ; _≤_ ; s≤s)
-open import Data.Product as Prod using (_×_; _,_; map₁; map₂′)
+open import Data.Product.Base as Prod using (_×_; _,_; map₁; map₂′)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Function.Base using (id; _∘_ ; _∘′_; _∘₂_; const; flip)
diff --git a/src/Data/List/Fresh.agda b/src/Data/List/Fresh.agda
index 8665267dd5..72c372003a 100644
--- a/src/Data/List/Fresh.agda
+++ b/src/Data/List/Fresh.agda
@@ -16,7 +16,7 @@ module Data.List.Fresh where
 open import Level using (Level; _⊔_)
 open import Data.Bool.Base using (true; false)
 open import Data.Unit.Polymorphic.Base using (⊤)
-open import Data.Product using (∃; _×_; _,_; -,_; proj₁; proj₂)
+open import Data.Product.Base using (∃; _×_; _,_; -,_; proj₁; proj₂)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.AllPairs using (AllPairs; []; _∷_)
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
diff --git a/src/Data/List/Fresh/NonEmpty.agda b/src/Data/List/Fresh/NonEmpty.agda
index 1ed72bd9d4..e4b55cb11b 100644
--- a/src/Data/List/Fresh/NonEmpty.agda
+++ b/src/Data/List/Fresh/NonEmpty.agda
@@ -11,9 +11,9 @@ module Data.List.Fresh.NonEmpty where
 open import Level using (Level; _⊔_)
 open import Data.List.Fresh as List# using (List#; []; cons; fresh)
 open import Data.Maybe.Base using (Maybe; nothing; just)
-open import Data.Nat using (ℕ; suc)
-open import Data.Product using (_×_; _,_)
-open import Relation.Binary as B using (Rel)
+open import Data.Nat.Base using (ℕ; suc)
+open import Data.Product.Base using (_×_; _,_)
+open import Relation.Binary.Core using (Rel)
 
 private
   variable
diff --git a/src/Data/List/Kleene/Base.agda b/src/Data/List/Kleene/Base.agda
index c8476318b7..8eb83ba864 100644
--- a/src/Data/List/Kleene/Base.agda
+++ b/src/Data/List/Kleene/Base.agda
@@ -8,11 +8,12 @@
 
 module Data.List.Kleene.Base where
 
-open import Data.Product as Product using (_×_; _,_; map₂; map₁; proj₁; proj₂)
-open import Data.Nat     as ℕ       using (ℕ; suc; zero)
-open import Data.Maybe   as Maybe   using (Maybe; just; nothing)
-open import Data.Sum     as Sum     using (_⊎_; inj₁; inj₂)
-open import Level        as Level   using (Level)
+open import Data.Product.Base as Product
+  using (_×_; _,_; map₂; map₁; proj₁; proj₂)
+open import Data.Nat.Base as ℕ using (ℕ; suc; zero)
+open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
+open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
+open import Level using (Level)
 
 open import Algebra.Core using (Op₂)
 open import Function.Base
diff --git a/src/Data/List/Membership/Setoid.agda b/src/Data/List/Membership/Setoid.agda
index ca48b933d1..047620041d 100644
--- a/src/Data/List/Membership/Setoid.agda
+++ b/src/Data/List/Membership/Setoid.agda
@@ -14,7 +14,7 @@ open import Function.Base using (_∘_; id; flip)
 open import Data.List.Base as List using (List; []; _∷_; length; lookup)
 open import Data.List.Relation.Unary.Any
   using (Any; index; map; here; there)
-open import Data.Product as Prod using (∃; _×_; _,_)
+open import Data.Product.Base as Prod using (∃; _×_; _,_)
 open import Relation.Unary using (Pred)
 open import Relation.Nullary.Negation using (¬_)
 
diff --git a/src/Data/List/NonEmpty/Base.agda b/src/Data/List/NonEmpty/Base.agda
index 3cc83a3447..320cacd4b6 100644
--- a/src/Data/List/NonEmpty/Base.agda
+++ b/src/Data/List/NonEmpty/Base.agda
@@ -14,7 +14,7 @@ open import Data.Bool.Properties using (T?)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.Maybe.Base using (Maybe ; nothing; just)
 open import Data.Nat.Base as ℕ
-open import Data.Product as Prod using (∃; _×_; proj₁; proj₂; _,_; -,_)
+open import Data.Product.Base as Prod using (∃; _×_; proj₁; proj₂; _,_; -,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 58d5b39680..9df56d4781 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -25,7 +25,8 @@ open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.Properties
-open import Data.Product as Prod hiding (map; zip)
+open import Data.Product.Base as Prod
+  using (_×_; _,_; uncurry; uncurry′; proj₁; proj₂; <_,_>)
 import Data.Product.Relation.Unary.All as Prod using (All)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
diff --git a/src/Data/List/Relation/Binary/Lex/Core.agda b/src/Data/List/Relation/Binary/Lex/Core.agda
index 3c90bf3c9f..96549e013b 100644
--- a/src/Data/List/Relation/Binary/Lex/Core.agda
+++ b/src/Data/List/Relation/Binary/Lex/Core.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Binary.Lex.Core where
 
 open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Unit.Base using (⊤; tt)
-open import Data.Product using (_×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Data.List.Base using (List; []; _∷_)
 open import Function.Base using (_∘_; flip; id)
 open import Level using (Level; _⊔_)
diff --git a/src/Data/List/Relation/Binary/Pointwise/Base.agda b/src/Data/List/Relation/Binary/Pointwise/Base.agda
index 8e24dcd2dc..e547aea169 100644
--- a/src/Data/List/Relation/Binary/Pointwise/Base.agda
+++ b/src/Data/List/Relation/Binary/Pointwise/Base.agda
@@ -8,9 +8,9 @@
 
 module Data.List.Relation.Binary.Pointwise.Base where
 
-open import Data.Product using (_×_; <_,_>)
+open import Data.Product.Base using (_×_; <_,_>)
 open import Data.List.Base using (List; []; _∷_)
-open import Level
+open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (REL; _⇒_)
 
 private
diff --git a/src/Data/List/Relation/Unary/All.agda b/src/Data/List/Relation/Unary/All.agda
index 674a8e33d5..090af7e6b6 100644
--- a/src/Data/List/Relation/Unary/All.agda
+++ b/src/Data/List/Relation/Unary/All.agda
@@ -15,11 +15,11 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional renaming (_∈_ to _∈ₚ_)
 import Data.List.Membership.Setoid as SetoidMembership
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
   using (∃; -,_; _×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
-open import Function
-open import Level
+open import Function.Base using (_∘_; _∘′_; id; const)
+open import Level using (Level; _⊔_)
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Unary hiding (_∈_)
diff --git a/src/Data/List/Relation/Unary/Any.agda b/src/Data/List/Relation/Unary/Any.agda
index 19af845152..eb010e2a39 100644
--- a/src/Data/List/Relation/Unary/Any.agda
+++ b/src/Data/List/Relation/Unary/Any.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Unary.Any where
 open import Data.Empty
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List; []; [_]; _∷_)
-open import Data.Product as Prod using (∃; _,_)
+open import Data.Product.Base as Prod using (∃; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary using (¬_; yes; no; _⊎-dec_)
diff --git a/src/Data/Maybe/Base.agda b/src/Data/Maybe/Base.agda
index 0cb8c473bf..b0ae001f46 100644
--- a/src/Data/Maybe/Base.agda
+++ b/src/Data/Maybe/Base.agda
@@ -14,7 +14,7 @@ open import Level
 open import Data.Bool.Base using (Bool; true; false; not)
 open import Data.Unit.Base using (⊤)
 open import Data.These.Base using (These; this; that; these)
-open import Data.Product as Prod using (_×_; _,_)
+open import Data.Product.Base as Prod using (_×_; _,_)
 open import Function.Base
 open import Relation.Nullary.Reflects
 open import Relation.Nullary.Decidable.Core
diff --git a/src/Data/Nat/DivMod/Core.agda b/src/Data/Nat/DivMod/Core.agda
index 02b1c917c6..fed1a1ae0d 100644
--- a/src/Data/Nat/DivMod/Core.agda
+++ b/src/Data/Nat/DivMod/Core.agda
@@ -14,7 +14,7 @@ open import Agda.Builtin.Nat using ()
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index 07a80bcdd9..c202d740ba 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -12,14 +12,12 @@ open import Algebra
 open import Data.Nat.Base
 open import Data.Nat.DivMod
 open import Data.Nat.Properties
-open import Data.Product
-open import Data.Unit using (tt)
-open import Function.Base
+open import Data.Unit.Base using (tt)
+open import Function.Base using (_∘′_; _$_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (0ℓ)
-open import Relation.Nullary.Decidable using (yes; no)
-open import Relation.Nullary.Decidable as Dec using (False)
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Decidable as Dec using (False; yes; no)
+open import Relation.Nullary.Negation.Core using (contradiction)
 open import Relation.Binary
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Binary.PropositionalEquality as PropEq
diff --git a/src/Data/Nat/Induction.agda b/src/Data/Nat/Induction.agda
index ea68692a51..209f9f60bb 100644
--- a/src/Data/Nat/Induction.agda
+++ b/src/Data/Nat/Induction.agda
@@ -10,9 +10,8 @@ module Data.Nat.Induction where
 
 open import Data.Nat.Base
 open import Data.Nat.Properties using (<⇒<′)
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Unit.Polymorphic.Base
-open import Function.Base
 open import Induction
 open import Induction.WellFounded as WF
 open import Level using (Level)
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 27725b98ad..ac8a4dcaa5 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -23,7 +23,7 @@ open import Data.Bool.Base using (Bool; false; true; T)
 open import Data.Bool.Properties using (T?)
 open import Data.Empty using (⊥)
 open import Data.Nat.Base
-open import Data.Product using (∄; ∃; _×_; _,_)
+open import Data.Product.Base using (∃; _×_; _,_)
 open import Data.Sum.Base as Sum
 open import Data.Unit using (tt)
 open import Function.Base
@@ -2172,7 +2172,7 @@ module _ {p} {P : Pred ℕ p} (P? : U.Decidable P) where
   ... | _      | yes (n , n<v , Pn) = yes (n , m≤n⇒m≤1+n n<v , Pn)
   ... | no ¬Pv | no ¬Pn<v           = no ¬Pn<1+v
     where
-    ¬Pn<1+v : ∄ λ n → n < suc v × P n
+    ¬Pn<1+v : (∃ λ n → n < suc v × P n) → ⊥
     ¬Pn<1+v (n , s≤s n≤v , Pn) with n ≟ v
     ... | yes refl = ¬Pv Pn
     ... | no  n≢v  = ¬Pn<v (n , ≤∧≢⇒< n≤v n≢v , Pn)
diff --git a/src/Data/Nat/Show.agda b/src/Data/Nat/Show.agda
index 9f0fdf2bb5..1777aca9d3 100644
--- a/src/Data/Nat/Show.agda
+++ b/src/Data/Nat/Show.agda
@@ -16,9 +16,9 @@ open import Data.List.Effectful using (module TraversableA)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; _<∣>_; when)
 import Data.Maybe.Effectful as Maybe
 open import Data.Nat
-open import Data.Product using (proj₁)
-open import Data.String.Base using (toList; fromList; String)
-open import Function.Base
+open import Data.Product.Base using (proj₁)
+open import Data.String.Base as String using (String; fromList; toList)
+open import Function.Base using (_∘′_; _∘_)
 open import Relation.Nullary.Decidable using (True)
 
 ------------------------------------------------------------------------
@@ -62,7 +62,7 @@ toDecimalChars : ℕ → List Char
 toDecimalChars = List.map toDigitChar ∘′ toNatDigits 10
 
 show : ℕ → String
-show = fromList ∘ toDecimalChars
+show = fromList ∘′ toDecimalChars
 
 -- Arbitrary base betwen 2 & 16.
 -- Warning: when compiled the time complexity of `showInBase b n` is
diff --git a/src/Data/Product.agda b/src/Data/Product.agda
index 08c0aa0c7f..02bf53a97b 100644
--- a/src/Data/Product.agda
+++ b/src/Data/Product.agda
@@ -23,18 +23,11 @@ private
 open import Data.Product.Base public
 
 ------------------------------------------------------------------------
--- Existential quantifiers
-
-∃ : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)
-∃ = Σ _
+-- Negation of existential quantifier
 
 ∄ : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)
 ∄ P = ¬ ∃ P
 
-∃₂ : ∀ {A : Set a} {B : A → Set b}
-     (C : (x : A) → B x → Set c) → Set (a ⊔ b ⊔ c)
-∃₂ C = ∃ λ a → ∃ λ b → C a b
-
 -- Unique existence (parametrised by an underlying equality).
 
 ∃! : {A : Set a} → (A → A → Set ℓ) → (A → Set b) → Set (a ⊔ b ⊔ ℓ)
@@ -42,13 +35,6 @@ open import Data.Product.Base public
 
 -- Syntax
 
-infix 2 ∃-syntax
-
-∃-syntax : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)
-∃-syntax = ∃
-
-syntax ∃-syntax (λ x → B) = ∃[ x ] B
-
 infix 2 ∄-syntax
 
 ∄-syntax : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)
diff --git a/src/Data/Product/Algebra.agda b/src/Data/Product/Algebra.agda
index f11045d778..c64057dcc0 100644
--- a/src/Data/Product/Algebra.agda
+++ b/src/Data/Product/Algebra.agda
@@ -11,9 +11,9 @@ module Data.Product.Algebra where
 open import Algebra
 open import Data.Bool.Base using (true; false)
 open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
-open import Data.Product
+open import Data.Product.Base
 open import Data.Product.Properties
-open import Data.Sum as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
+open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sum.Algebra
 open import Data.Unit.Polymorphic using (⊤; tt)
 open import Function.Base using (_∘′_)
diff --git a/src/Data/Product/Base.agda b/src/Data/Product/Base.agda
index 7ee638392c..02f7e52697 100644
--- a/src/Data/Product/Base.agda
+++ b/src/Data/Product/Base.agda
@@ -9,7 +9,7 @@
 module Data.Product.Base where
 
 open import Function.Base
-open import Level
+open import Level using (Level; _⊔_)
 
 private
   variable
@@ -31,6 +31,19 @@ open import Agda.Builtin.Sigma public
 module Σ = Agda.Builtin.Sigma.Σ
   renaming (fst to proj₁; snd to proj₂)
 
+------------------------------------------------------------------------
+-- Existential quantifiers
+
+∃ : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)
+∃ = Σ _
+
+∃₂ : ∀ {A : Set a} {B : A → Set b}
+     (C : (x : A) → B x → Set c) → Set (a ⊔ b ⊔ c)
+∃₂ C = ∃ λ a → ∃ λ b → C a b
+
+------------------------------------------------------------------------
+-- Syntaxes
+
 -- The syntax declaration below is attached to Σ-syntax, to make it
 -- easy to import Σ without the special syntax.
 
@@ -41,6 +54,13 @@ infix 2 Σ-syntax
 
 syntax Σ-syntax A (λ x → B) = Σ[ x ∈ A ] B
 
+infix 2 ∃-syntax
+
+∃-syntax : ∀ {A : Set a} → (A → Set b) → Set (a ⊔ b)
+∃-syntax = ∃
+
+syntax ∃-syntax (λ x → B) = ∃[ x ] B
+
 ------------------------------------------------------------------------
 -- Definition of non-dependent products
 
diff --git a/src/Data/Product/Effectful/Left.agda b/src/Data/Product/Effectful/Left.agda
index 66362621ee..279555b9c9 100644
--- a/src/Data/Product/Effectful/Left.agda
+++ b/src/Data/Product/Effectful/Left.agda
@@ -18,7 +18,7 @@ open import Level
 module Data.Product.Effectful.Left
   {a e} (A : RawMonoid a e) (b : Level) where
 
-open import Data.Product
+open import Data.Product.Base
 import Data.Product.Effectful.Left.Base as Base
 open import Effect.Applicative using (RawApplicative)
 open import Effect.Monad using (RawMonad; RawMonadT; mkRawMonad)
diff --git a/src/Data/Product/Effectful/Left/Base.agda b/src/Data/Product/Effectful/Left/Base.agda
index d5b0b42c92..e3f7edef69 100644
--- a/src/Data/Product/Effectful/Left/Base.agda
+++ b/src/Data/Product/Effectful/Left/Base.agda
@@ -17,7 +17,7 @@ open import Level
 module Data.Product.Effectful.Left.Base
   {a} (A : Set a) (b : Level) where
 
-open import Data.Product using (_×_; map₂; proj₁; proj₂; <_,_>)
+open import Data.Product.Base using (_×_; map₂; proj₁; proj₂; <_,_>)
 open import Effect.Functor using (RawFunctor)
 open import Effect.Comonad using (RawComonad)
 
diff --git a/src/Data/Product/Effectful/Right.agda b/src/Data/Product/Effectful/Right.agda
index 640b4e908d..fd5cb9b423 100644
--- a/src/Data/Product/Effectful/Right.agda
+++ b/src/Data/Product/Effectful/Right.agda
@@ -18,7 +18,7 @@ open import Level
 module Data.Product.Effectful.Right
   (a : Level) {b e} (B : RawMonoid b e) where
 
-open import Data.Product
+open import Data.Product.Base
 import Data.Product.Effectful.Right.Base as Base
 open import Effect.Applicative using (RawApplicative)
 open import Effect.Monad using (RawMonad; RawMonadT; mkRawMonad)
diff --git a/src/Data/Product/Effectful/Right/Base.agda b/src/Data/Product/Effectful/Right/Base.agda
index 273185bf8e..556483f66f 100644
--- a/src/Data/Product/Effectful/Right/Base.agda
+++ b/src/Data/Product/Effectful/Right/Base.agda
@@ -17,7 +17,7 @@ open import Level
 module Data.Product.Effectful.Right.Base
   {b} (B : Set b) (a : Level) where
 
-open import Data.Product using (_×_; map₁; proj₁; proj₂; <_,_>)
+open import Data.Product.Base using (_×_; map₁; proj₁; proj₂; <_,_>)
 open import Effect.Functor using (RawFunctor)
 open import Effect.Comonad using (RawComonad)
 
diff --git a/src/Data/Product/Nary/NonDependent.agda b/src/Data/Product/Nary/NonDependent.agda
index 1d5678e00c..cb421f16b6 100644
--- a/src/Data/Product/Nary/NonDependent.agda
+++ b/src/Data/Product/Nary/NonDependent.agda
@@ -16,15 +16,15 @@ module Data.Product.Nary.NonDependent where
 
 open import Level as L using (Level; _⊔_; Lift; 0ℓ)
 open import Agda.Builtin.Unit
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
 import Data.Product.Properties as Prodₚ
 open import Data.Sum.Base using (_⊎_)
 open import Data.Nat.Base using (ℕ; zero; suc; pred)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Function.Base using (const; _∘′_; _∘_)
-open import Relation.Nullary.Decidable using (Dec; yes; no; _×-dec_)
-open import Relation.Binary using (Rel)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong₂)
+open import Relation.Nullary.Decidable.Core using (Dec; yes; no; _×-dec_)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 open import Function.Nary.NonDependent.Base
 
diff --git a/src/Data/Product/Properties.agda b/src/Data/Product/Properties.agda
index fa4909e955..1cc3c2383e 100644
--- a/src/Data/Product/Properties.agda
+++ b/src/Data/Product/Properties.agda
@@ -9,8 +9,9 @@
 module Data.Product.Properties where
 
 open import Axiom.UniquenessOfIdentityProofs
-open import Data.Product
-open import Function
+open import Data.Product.Base
+open import Function.Base using (_∋_; _∘_; id)
+open import Function.Bundles using (_↔_; mk↔′)
 open import Level using (Level)
 open import Relation.Binary using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/Product/Relation/Unary/All.agda b/src/Data/Product/Relation/Unary/All.agda
index 8c1b67dc50..1257de57c2 100644
--- a/src/Data/Product/Relation/Unary/All.agda
+++ b/src/Data/Product/Relation/Unary/All.agda
@@ -8,10 +8,8 @@
 
 module Data.Product.Relation.Unary.All where
 
-open import Level
-open import Data.Product
-open import Function.Base
-open import Relation.Unary
+open import Level using (Level; _⊔_)
+open import Data.Product.Base using (_×_; _,_)
 
 private
   variable
diff --git a/src/Data/String/Base.agda b/src/Data/String/Base.agda
index b3556026db..1a56376f59 100644
--- a/src/Data/String/Base.agda
+++ b/src/Data/String/Base.agda
@@ -8,7 +8,6 @@
 
 module Data.String.Base where
 
-open import Level using (zero)
 open import Data.Bool.Base using (Bool; true; false)
 open import Data.Char.Base as Char using (Char)
 open import Data.List.Base as List using (List; [_]; _∷_; [])
@@ -17,13 +16,13 @@ open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise)
 open import Data.List.Relation.Binary.Lex.Core using (Lex-<; Lex-≤)
 open import Data.Maybe.Base as Maybe using (Maybe)
 open import Data.Nat.Base using (ℕ; _∸_; ⌊_/2⌋; ⌈_/2⌉)
-open import Data.Product using (proj₁; proj₂)
+open import Data.Product.Base using (proj₁; proj₂)
 open import Function.Base using (_on_; _∘′_; _∘_)
-open import Level using (Level)
+open import Level using (Level; 0ℓ)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Unary using (Pred; Decidable)
-open import Relation.Nullary.Decidable using (does)
+open import Relation.Nullary.Decidable.Core using (does)
 
 ------------------------------------------------------------------------
 -- From Agda.Builtin: type and renamed primitives
@@ -47,17 +46,17 @@ open String public using ( String )
 -- Pointwise equality on Strings
 
 infix 4 _≈_
-_≈_ : Rel String zero
+_≈_ : Rel String 0ℓ
 _≈_ = Pointwise _≡_ on toList
 
 -- Lexicographic ordering on Strings
 
 infix 4 _<_
-_<_ : Rel String zero
+_<_ : Rel String 0ℓ
 _<_ = Lex-< _≡_ Char._<_ on toList
 
 infix 4 _≤_
-_≤_ : Rel String zero
+_≤_ : Rel String 0ℓ
 _≤_ = Lex-≤ _≡_ Char._<_ on toList
 
 ------------------------------------------------------------------------
diff --git a/src/Data/String/Unsafe.agda b/src/Data/String/Unsafe.agda
index 91a62a61d1..9c001365a3 100644
--- a/src/Data/String/Unsafe.agda
+++ b/src/Data/String/Unsafe.agda
@@ -12,7 +12,7 @@ import Data.List.Base as List
 import Data.List.Properties as Listₚ
 open import Data.Maybe.Base using (maybe′)
 open import Data.Nat.Base using (zero; suc; _+_)
-open import Data.Product using (proj₂)
+open import Data.Product.Base using (proj₂)
 open import Data.String.Base
 open import Function.Base using (_∘′_)
 
diff --git a/src/Data/Sum/Algebra.agda b/src/Data/Sum/Algebra.agda
index 717bc347c1..f69988942b 100644
--- a/src/Data/Sum/Algebra.agda
+++ b/src/Data/Sum/Algebra.agda
@@ -10,7 +10,7 @@ module Data.Sum.Algebra where
 
 open import Algebra
 open import Data.Empty.Polymorphic using (⊥)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base
 open import Data.Sum.Properties
 open import Data.Unit.Polymorphic using (⊤; tt)
diff --git a/src/Data/Unit/Polymorphic/Properties.agda b/src/Data/Unit/Polymorphic/Properties.agda
index 8513229ba1..dce385f19e 100644
--- a/src/Data/Unit/Polymorphic/Properties.agda
+++ b/src/Data/Unit/Polymorphic/Properties.agda
@@ -11,7 +11,7 @@ module Data.Unit.Polymorphic.Properties where
 
 open import Level
 open import Function.Bundles using (_↔_; mk↔)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (inj₁)
 open import Data.Unit.Base renaming (⊤ to ⊤*)
 open import Data.Unit.Polymorphic.Base using (⊤; tt)
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 911766496a..9f6d86b792 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -12,12 +12,12 @@ open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Nat.Base
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List)
-open import Data.Product as Prod using (∃; ∃₂; _×_; _,_)
+open import Data.Product.Base as Prod using (∃; ∃₂; _×_; _,_)
 open import Data.These.Base as These using (These; this; that; these)
 open import Function.Base using (const; _∘′_; id; _∘_)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
-open import Relation.Nullary.Decidable using (does)
+open import Relation.Nullary.Decidable.Core using (does)
 open import Relation.Unary using (Pred; Decidable)
 
 private
diff --git a/src/Data/Vec/Bounded/Base.agda b/src/Data/Vec/Bounded/Base.agda
index 9c874ec89e..17b33ef7be 100644
--- a/src/Data/Vec/Bounded/Base.agda
+++ b/src/Data/Vec/Bounded/Base.agda
@@ -8,7 +8,6 @@
 
 module Data.Vec.Bounded.Base where
 
-open import Level using (Level)
 open import Data.Nat.Base
 import Data.Nat.Properties as ℕₚ
 open import Data.List.Base as List using (List)
@@ -16,13 +15,13 @@ open import Data.List.Extrema ℕₚ.≤-totalOrder
 open import Data.List.Relation.Unary.All as All using (All)
 import Data.List.Relation.Unary.All.Properties as Allₚ
 open import Data.List.Membership.Propositional using (mapWith∈)
-open import Data.Product using (∃; _×_; _,_; proj₁; proj₂)
+open import Data.Product.Base using (∃; _×_; _,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.These.Base as These using (These)
-open import Function
-open import Relation.Nullary
-open import Relation.Unary
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+open import Function.Base using (_∘_; id; _$_)
+open import Level using (Level)
+open import Relation.Nullary.Decidable.Core using (recompute)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/Vec/Functional.agda b/src/Data/Vec/Functional.agda
index 26e8f32160..091886bef5 100644
--- a/src/Data/Vec/Functional.agda
+++ b/src/Data/Vec/Functional.agda
@@ -18,10 +18,10 @@ module Data.Vec.Functional where
 open import Data.Fin.Base
 open import Data.List.Base as L using (List)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; NonZero; pred)
-open import Data.Product using (Σ; ∃; _×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Product.Base using (Σ; ∃; _×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Vec.Base as V using (Vec)
-open import Function.Base
+open import Function.Base using (_∘_; const; flip; _ˢ_; id)
 open import Level using (Level)
 
 infixr 5 _∷_ _++_
diff --git a/src/Data/Vec/N-ary.agda b/src/Data/Vec/N-ary.agda
index 7e10c5567c..197552d753 100644
--- a/src/Data/Vec/N-ary.agda
+++ b/src/Data/Vec/N-ary.agda
@@ -11,14 +11,13 @@ module Data.Vec.N-ary where
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Function.Bundles using (_↔_; Inverse; mk↔′)
 open import Data.Nat.Base hiding (_⊔_)
-open import Data.Product as Prod
-open import Data.Vec.Base
-open import Function.Base
+open import Data.Product.Base as Prod using (∃; _,_)
+open import Data.Vec.Base using (Vec; []; _∷_; head; tail)
+open import Function.Base using (_∘_; id; flip; constᵣ)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
-open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality
-open import Relation.Nullary.Decidable
+open import Relation.Binary.Core using (REL)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 private
   variable
diff --git a/src/Data/Vec/Recursive.agda b/src/Data/Vec/Recursive.agda
index 765db00991..0ecc16f687 100644
--- a/src/Data/Vec/Recursive.agda
+++ b/src/Data/Vec/Recursive.agda
@@ -16,15 +16,12 @@
 
 module Data.Vec.Recursive where
 
-open import Level using (Level; lift)
-open import Function.Bundles using (mk↔′)
-open import Function.Properties.Inverse using (↔-isEquivalence; ↔-refl; ↔-sym; ↔-trans)
 open import Data.Nat.Base as Nat using (ℕ; zero; suc; NonZero; pred)
 open import Data.Nat.Properties using (+-comm; *-comm)
 open import Data.Empty.Polymorphic
 open import Data.Fin.Base as Fin using (Fin; zero; suc)
 open import Data.Fin.Properties using (1↔⊤; *↔×)
-open import Data.Product as Prod using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Prod using (_×_; _,_; proj₁; proj₂)
 open import Data.Product.Algebra using (×-cong)
 open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
@@ -32,8 +29,11 @@ open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Unit.Polymorphic.Properties using (⊤↔⊤*)
 open import Data.Vec.Base as Vec using (Vec; _∷_)
 open import Data.Vec.N-ary using (N-ary)
-open import Function
-open import Relation.Unary
+open import Function.Base using (_∘′_; _∘_; id)
+open import Function.Bundles using (_↔_; mk↔′; mk↔)
+open import Function.Properties.Inverse using (↔-isEquivalence; ↔-refl; ↔-sym; ↔-trans)
+open import Level using (Level; lift)
+open import Relation.Unary using (IUniversal; Universal; _⇒_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; sym; trans; cong; subst)
 open import Relation.Binary.Structures using (IsEquivalence)
 
diff --git a/src/Data/W.agda b/src/Data/W.agda
index 3f2b35909f..159c0883a5 100644
--- a/src/Data/W.agda
+++ b/src/Data/W.agda
@@ -8,9 +8,9 @@
 
 module Data.W where
 
-open import Level
+open import Level using (Level; _⊔_)
 open import Function.Base using (_$_; _∘_; const)
-open import Data.Product using (_,_; -,_; proj₂)
+open import Data.Product.Base using (_,_; -,_; proj₂)
 open import Data.Container.Core using (Container; ⟦_⟧; Shape; Position; _⇒_; ⟪_⟫)
 open import Data.Container.Relation.Unary.All using (□; all)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Effect/Applicative.agda b/src/Effect/Applicative.agda
index 51f7d21368..729966752c 100644
--- a/src/Effect/Applicative.agda
+++ b/src/Effect/Applicative.agda
@@ -12,7 +12,7 @@
 module Effect.Applicative where
 
 open import Data.Bool.Base using (Bool; true; false)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Unit.Polymorphic.Base using (⊤)
 
 open import Effect.Choice using (RawChoice)
diff --git a/src/Effect/Monad/State/Transformer/Base.agda b/src/Effect/Monad/State/Transformer/Base.agda
index 0680a2207b..3b5d60fd6e 100644
--- a/src/Effect/Monad/State/Transformer/Base.agda
+++ b/src/Effect/Monad/State/Transformer/Base.agda
@@ -9,7 +9,7 @@
 
 module Effect.Monad.State.Transformer.Base where
 
-open import Data.Product using (_×_; proj₁; proj₂)
+open import Data.Product.Base using (_×_; proj₁; proj₂)
 open import Data.Unit.Polymorphic.Base using (⊤)
 open import Function.Base using (_∘′_; const; id)
 open import Level using (Level; suc; _⊔_)
diff --git a/src/Effect/Monad/Writer/Transformer/Base.agda b/src/Effect/Monad/Writer/Transformer/Base.agda
index bf3c254fc3..4f910b101d 100644
--- a/src/Effect/Monad/Writer/Transformer/Base.agda
+++ b/src/Effect/Monad/Writer/Transformer/Base.agda
@@ -9,7 +9,7 @@
 module Effect.Monad.Writer.Transformer.Base where
 
 open import Algebra using (RawMonoid)
-open import Data.Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Data.Unit.Polymorphic using (⊤; tt)
 open import Function.Base using (id; _∘′_)
 open import Level using (Level; suc; _⊔_)
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index 6284182845..d113f14858 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -23,7 +23,7 @@ open import Function.Base using (_∘_)
 import Function.Definitions as FunctionDefinitions
 import Function.Structures as FunctionStructures
 open import Level using (Level; _⊔_; suc)
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Relation.Binary hiding (_⇔_)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_)
diff --git a/src/Function/Consequences.agda b/src/Function/Consequences.agda
index 2b7603deea..e8c6a0f498 100644
--- a/src/Function/Consequences.agda
+++ b/src/Function/Consequences.agda
@@ -8,13 +8,12 @@
 
 module Function.Consequences where
 
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Definitions
-open import Level
+open import Level using (Level)
 open import Relation.Binary
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
-open import Relation.Nullary.Negation using (¬_)
-open import Relation.Nullary.Negation.Core using (contraposition)
+open import Relation.Nullary.Negation.Core using (¬_; contraposition)
 
 private
   variable
diff --git a/src/Function/Construct/Composition.agda b/src/Function/Construct/Composition.agda
index 242456d212..a18b4b87b1 100644
--- a/src/Function/Construct/Composition.agda
+++ b/src/Function/Construct/Composition.agda
@@ -8,10 +8,10 @@
 
 module Function.Construct.Composition where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function
 open import Level using (Level)
-open import Relation.Binary as B hiding (_⇔_; IsEquivalence)
+open import Relation.Binary hiding (_⇔_; IsEquivalence)
 
 private
   variable
diff --git a/src/Function/Construct/Identity.agda b/src/Function/Construct/Identity.agda
index 1b69165dda..32660bee09 100644
--- a/src/Function/Construct/Identity.agda
+++ b/src/Function/Construct/Identity.agda
@@ -8,12 +8,12 @@
 
 module Function.Construct.Identity where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 open import Function.Bundles
 import Function.Definitions as Definitions
 import Function.Structures as Structures
-open import Level
+open import Level using (Level)
 open import Relation.Binary as B hiding (_⇔_; IsEquivalence)
 open import Relation.Binary.PropositionalEquality using (_≡_; setoid)
 
diff --git a/src/Function/Construct/Symmetry.agda b/src/Function/Construct/Symmetry.agda
index ea86704a57..86fa273c57 100644
--- a/src/Function/Construct/Symmetry.agda
+++ b/src/Function/Construct/Symmetry.agda
@@ -8,7 +8,7 @@
 
 module Function.Construct.Symmetry where
 
-open import Data.Product using (_,_; swap; proj₁; proj₂)
+open import Data.Product.Base using (_,_; swap; proj₁; proj₂)
 open import Function
 open import Level using (Level)
 open import Relation.Binary hiding (_⇔_)
diff --git a/src/Function/Definitions.agda b/src/Function/Definitions.agda
index ac805d9472..bfeff3c05c 100644
--- a/src/Function/Definitions.agda
+++ b/src/Function/Definitions.agda
@@ -8,7 +8,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 
 module Function.Definitions
   {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
@@ -16,10 +16,9 @@ module Function.Definitions
   (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain
   where
 
-open import Data.Product using (∃; _×_)
+open import Data.Product.Base using (_×_)
 import Function.Definitions.Core1 as Core₁
 import Function.Definitions.Core2 as Core₂
-open import Function.Base
 open import Level using (_⊔_)
 
 ------------------------------------------------------------------------
diff --git a/src/Function/Definitions/Core1.agda b/src/Function/Definitions/Core1.agda
index c81e841af9..b11ca26fea 100644
--- a/src/Function/Definitions/Core1.agda
+++ b/src/Function/Definitions/Core1.agda
@@ -9,7 +9,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 
 module Function.Definitions.Core1
   {a b ℓ₁} {A : Set a} {B : Set b} (_≈₁_ : Rel A ℓ₁)
diff --git a/src/Function/Definitions/Core2.agda b/src/Function/Definitions/Core2.agda
index 313f331063..74973d8433 100644
--- a/src/Function/Definitions/Core2.agda
+++ b/src/Function/Definitions/Core2.agda
@@ -9,13 +9,13 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 
 module Function.Definitions.Core2
   {a b ℓ₂} {A : Set a} {B : Set b} (_≈₂_ : Rel B ℓ₂)
   where
 
-open import Data.Product using (∃)
+open import Data.Product.Base using (∃)
 open import Level using (Level; _⊔_)
 
 ------------------------------------------------------------------------
diff --git a/src/Function/Equality.agda b/src/Function/Equality.agda
index 97bb752ceb..bb32bca8cd 100644
--- a/src/Function/Equality.agda
+++ b/src/Function/Equality.agda
@@ -14,7 +14,7 @@
 module Function.Equality where
 
 import Function.Base as Fun
-open import Level
+open import Level using (Level; _⊔_)
 open import Relation.Binary using (Setoid)
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid; _=[_]⇒_)
diff --git a/src/Function/Metric/Definitions.agda b/src/Function/Metric/Definitions.agda
index 6c6d471aef..f0be9cb724 100644
--- a/src/Function/Metric/Definitions.agda
+++ b/src/Function/Metric/Definitions.agda
@@ -11,7 +11,7 @@
 module Function.Metric.Definitions where
 
 open import Algebra.Core using (Op₂)
-open import Data.Product using (∃)
+open import Data.Product.Base using (∃)
 open import Function.Metric.Core using (DistanceFunction)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)
diff --git a/src/Function/Nary/NonDependent/Base.agda b/src/Function/Nary/NonDependent/Base.agda
index 72e093b3a7..efb24aec57 100644
--- a/src/Function/Nary/NonDependent/Base.agda
+++ b/src/Function/Nary/NonDependent/Base.agda
@@ -16,7 +16,7 @@ module Function.Nary.NonDependent.Base where
 
 open import Level using (Level; 0ℓ; _⊔_)
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Unit.Polymorphic.Base
 open import Function.Base using (_∘′_; _$′_; const; flip)
 
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index 6310a8479f..508a8664be 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -10,7 +10,7 @@
 module Function.Properties.Inverse where
 
 open import Axiom.Extensionality.Propositional using (Extensionality)
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Bundles
 open import Level using (Level)
 open import Relation.Binary using (Setoid; IsEquivalence)
diff --git a/src/Function/Structures.agda b/src/Function/Structures.agda
index 31004c2db4..770dab600e 100644
--- a/src/Function/Structures.agda
+++ b/src/Function/Structures.agda
@@ -15,8 +15,7 @@ module Function.Structures {a b ℓ₁ ℓ₂}
   {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain
   where
 
-open import Data.Product using (∃; _×_; _,_)
-open import Function.Base
+open import Data.Product.Base using (_,_)
 open import Function.Definitions
 open import Level using (_⊔_)
 
diff --git a/src/Induction/WellFounded.agda b/src/Induction/WellFounded.agda
index d580452f0f..c1e206a0af 100644
--- a/src/Induction/WellFounded.agda
+++ b/src/Induction/WellFounded.agda
@@ -6,15 +6,14 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
-
 module Induction.WellFounded where
 
-open import Data.Product
-open import Function
+open import Data.Product.Base using (Σ; _,_; proj₁)
+open import Function.Base using (_on_)
 open import Induction
-open import Level
-open import Relation.Binary.PropositionalEquality hiding (trans)
+open import Level using (Level; _⊔_)
+open import Relation.Binary
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Unary
 
 private
diff --git a/src/Relation/Binary/Consequences.agda b/src/Relation/Binary/Consequences.agda
index ff82600c03..cd560ec995 100644
--- a/src/Relation/Binary/Consequences.agda
+++ b/src/Relation/Binary/Consequences.agda
@@ -10,7 +10,7 @@ module Relation.Binary.Consequences where
 
 open import Data.Maybe.Base using (just; nothing; decToMaybe)
 open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Empty.Irrelevant using (⊥-elim)
 open import Function.Base using (_∘_; _∘₂_; _$_; flip)
 open import Level using (Level)
diff --git a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
index 1d3e9430d9..c3c4dc2a18 100644
--- a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
+++ b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
@@ -12,7 +12,7 @@ open import Relation.Binary
 
 module Relation.Binary.Construct.Flip.EqAndOrd where
 
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (flip; _∘_)
 open import Level using (Level)
 
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Left.agda b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
index db1d0f5aff..e0ab59e9ca 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Left.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
@@ -8,7 +8,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Core
-open import Data.Product using (_,_; _×_)
+open import Data.Product.Base using (_,_; _×_)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Relation.Binary
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Right.agda b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
index 7c4a6ab119..5278f0b116 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Right.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
@@ -8,7 +8,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Core using (Op₂)
-open import Data.Product using (_,_; _×_)
+open import Data.Product.Base using (_,_; _×_)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Relation.Binary
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Relation/Binary/Core.agda b/src/Relation/Binary/Core.agda
index d9c19173a1..079bf296cc 100644
--- a/src/Relation/Binary/Core.agda
+++ b/src/Relation/Binary/Core.agda
@@ -10,7 +10,7 @@
 
 module Relation.Binary.Core where
 
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Function.Base using (_on_)
 open import Level using (Level; _⊔_; suc)
 
diff --git a/src/Relation/Binary/Definitions.agda b/src/Relation/Binary/Definitions.agda
index a44f2d058e..2ed315191b 100644
--- a/src/Relation/Binary/Definitions.agda
+++ b/src/Relation/Binary/Definitions.agda
@@ -13,7 +13,7 @@ module Relation.Binary.Definitions where
 open import Agda.Builtin.Equality using (_≡_)
 
 open import Data.Maybe.Base using (Maybe)
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_on_; flip)
 open import Level
diff --git a/src/Relation/Binary/Lattice/Definitions.agda b/src/Relation/Binary/Lattice/Definitions.agda
index 61102f19f9..2cdc32a1c3 100644
--- a/src/Relation/Binary/Lattice/Definitions.agda
+++ b/src/Relation/Binary/Lattice/Definitions.agda
@@ -12,10 +12,10 @@
 module Relation.Binary.Lattice.Definitions where
 
 open import Algebra.Core
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Function.Base using (flip)
-open import Relation.Binary
-open import Level
+open import Relation.Binary.Core using (Rel)
+open import Level using (Level)
 
 private
   variable
diff --git a/src/Relation/Binary/Lattice/Structures.agda b/src/Relation/Binary/Lattice/Structures.agda
index 10178af05d..07dbfad6a5 100644
--- a/src/Relation/Binary/Lattice/Structures.agda
+++ b/src/Relation/Binary/Lattice/Structures.agda
@@ -19,7 +19,7 @@ module Relation.Binary.Lattice.Structures
 
 open import Algebra.Core
 open import Algebra.Definitions
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Level using (suc; _⊔_)
 
 open import Relation.Binary.Lattice.Definitions
diff --git a/src/Relation/Binary/Morphism/Structures.agda b/src/Relation/Binary/Morphism/Structures.agda
index 39a333e1b7..86eceac6e4 100644
--- a/src/Relation/Binary/Morphism/Structures.agda
+++ b/src/Relation/Binary/Morphism/Structures.agda
@@ -6,15 +6,15 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Core
+open import Relation.Binary.Core using (Rel)
 
 module Relation.Binary.Morphism.Structures
   {a b} {A : Set a} {B : Set b}
   where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Definitions
-open import Level
+open import Level using (Level; _⊔_)
 open import Relation.Binary.Morphism.Definitions A B
 
 private
diff --git a/src/Relation/Binary/Properties/Preorder.agda b/src/Relation/Binary/Properties/Preorder.agda
index 1a215b639c..e09bd0f3f1 100644
--- a/src/Relation/Binary/Properties/Preorder.agda
+++ b/src/Relation/Binary/Properties/Preorder.agda
@@ -11,8 +11,8 @@ open import Relation.Binary
 module Relation.Binary.Properties.Preorder
   {p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where
 
-open import Function.Base
-open import Data.Product as Prod
+open import Function.Base using (flip)
+open import Data.Product.Base as Prod using (_×_; _,_; swap)
 import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 
 open Preorder P
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index 137641d49b..cca28fb5ad 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -6,9 +6,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; id; _$_; flip)
-open import Relation.Nullary.Negation using (¬_)
+open import Relation.Nullary.Negation.Core using (¬_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary
 
@@ -16,7 +16,6 @@ module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where
 
 open Setoid S
 
-
 ------------------------------------------------------------------------------
 -- Every setoid is a preorder and partial order with respect to propositional
 -- equality
diff --git a/src/Relation/Binary/Properties/TotalOrder.agda b/src/Relation/Binary/Properties/TotalOrder.agda
index da4f316892..18e7095985 100644
--- a/src/Relation/Binary/Properties/TotalOrder.agda
+++ b/src/Relation/Binary/Properties/TotalOrder.agda
@@ -13,14 +13,12 @@ module Relation.Binary.Properties.TotalOrder
 
 open TotalOrder T
 
-open import Data.Product using (proj₁)
+open import Data.Product.Base using (proj₁)
 open import Data.Sum.Base using (inj₁; inj₂)
 import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict
 import Relation.Binary.Properties.Poset poset as PosetProperties
 open import Relation.Binary.Consequences
-open import Relation.Nullary.Negation using (¬_)
-open import Relation.Nullary.Negation using (contradiction)
 
 ------------------------------------------------------------------------
 -- Total orders are almost decidable total orders
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 4764635449..37c891312b 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -13,10 +13,9 @@ open import Axiom.UniquenessOfIdentityProofs
 open import Function.Base using (id; _∘_)
 open import Function.Equality using (Π; _⟶_; ≡-setoid)
 open import Level using (Level; _⊔_)
-open import Data.Product using (∃)
+open import Data.Product.Base using (∃)
 
-open import Relation.Nullary.Decidable using (yes; no)
-open import Relation.Nullary.Decidable
+open import Relation.Nullary.Decidable using (yes; no; dec-yes-irr; dec-no)
 open import Relation.Binary
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
@@ -132,4 +131,3 @@ Please use new `with ... in` syntax described at https://agda.readthedocs.io/en/
 "Warning: inspect was deprecated in v2.0.
 Please use new `with ... in` syntax described at https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality instead."
 #-}
-
diff --git a/src/Relation/Binary/PropositionalEquality/Core.agda b/src/Relation/Binary/PropositionalEquality/Core.agda
index 83c406b8aa..c0c4030b83 100644
--- a/src/Relation/Binary/PropositionalEquality/Core.agda
+++ b/src/Relation/Binary/PropositionalEquality/Core.agda
@@ -11,7 +11,7 @@
 
 module Relation.Binary.PropositionalEquality.Core where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_)
 open import Level
 open import Relation.Binary.Core
diff --git a/src/Relation/Binary/PropositionalEquality/Properties.agda b/src/Relation/Binary/PropositionalEquality/Properties.agda
index 0953c68cbd..ad936254ff 100644
--- a/src/Relation/Binary/PropositionalEquality/Properties.agda
+++ b/src/Relation/Binary/PropositionalEquality/Properties.agda
@@ -13,7 +13,7 @@
 module Relation.Binary.PropositionalEquality.Properties where
 
 open import Function.Base using (id; _∘_)
-open import Level
+open import Level using (Level)
 open import Relation.Binary
 import Relation.Binary.Properties.Setoid as Setoid
 open import Relation.Binary.PropositionalEquality.Core
diff --git a/src/Relation/Binary/Reasoning/Base/Double.agda b/src/Relation/Binary/Reasoning/Base/Double.agda
index f40a32bc9b..07761c5144 100644
--- a/src/Relation/Binary/Reasoning/Base/Double.agda
+++ b/src/Relation/Binary/Reasoning/Base/Double.agda
@@ -16,13 +16,13 @@ module Relation.Binary.Reasoning.Base.Double {a ℓ₁ ℓ₂} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} (isPreorder : IsPreorder _≈_ _∼_)
   where
 
-open import Data.Product using (proj₁; proj₂)
+open import Data.Product.Base using (proj₁; proj₂)
 open import Level using (Level; _⊔_; Lift; lift)
 open import Function.Base using (case_of_; id)
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; sym)
-open import Relation.Nullary.Decidable using (Dec; yes; no)
-open import Relation.Nullary.Decidable using (True; toWitness)
+open import Relation.Nullary.Decidable.Core
+  using (Dec; yes; no; True; toWitness)
 
 open IsPreorder isPreorder
 
diff --git a/src/Relation/Binary/Reasoning/Base/Triple.agda b/src/Relation/Binary/Reasoning/Base/Triple.agda
index 8e5ba6e159..c18aa9bf94 100644
--- a/src/Relation/Binary/Reasoning/Base/Triple.agda
+++ b/src/Relation/Binary/Reasoning/Base/Triple.agda
@@ -19,13 +19,13 @@ module Relation.Binary.Reasoning.Base.Triple {a ℓ₁ ℓ₂ ℓ₃} {A : Set a
   (<-≤-trans : Trans _<_ _≤_ _<_) (≤-<-trans : Trans _≤_ _<_ _<_)
   where
 
-open import Data.Product using (proj₁; proj₂)
+open import Data.Product.Base using (proj₁; proj₂)
 open import Function.Base using (case_of_; id)
 open import Level using (Level; _⊔_; Lift; lift)
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; sym)
-open import Relation.Nullary.Decidable using (Dec; yes; no)
-open import Relation.Nullary.Decidable using (True; toWitness)
+open import Relation.Nullary.Decidable.Core
+  using (Dec; yes; no; True; toWitness)
 
 open IsPreorder isPreorder
   renaming
diff --git a/src/Relation/Binary/Reflection.agda b/src/Relation/Binary/Reflection.agda
index 8b682a0407..5a9ec838fb 100644
--- a/src/Relation/Binary/Reflection.agda
+++ b/src/Relation/Binary/Reflection.agda
@@ -7,12 +7,12 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Fin.Base
-open import Data.Nat.Base
-open import Data.Vec.Base as Vec
-open import Function.Base
+open import Data.Fin.Base using (Fin)
+open import Data.Nat.Base using (ℕ)
+open import Data.Vec.Base as Vec using (Vec; allFin)
+open import Function.Base using (id; _⟨_⟩_)
 open import Function.Bundles using (module Equivalence)
-open import Level
+open import Level using (Level)
 open import Relation.Binary
 import Relation.Binary.PropositionalEquality as P
 
@@ -41,7 +41,7 @@ module Relation.Binary.Reflection
          where
 
 open import Data.Vec.N-ary
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 import Relation.Binary.Reasoning.Setoid as Eq
 
 open Setoid Sem
diff --git a/src/Relation/Binary/Structures.agda b/src/Relation/Binary/Structures.agda
index 058ca5135c..965e41691b 100644
--- a/src/Relation/Binary/Structures.agda
+++ b/src/Relation/Binary/Structures.agda
@@ -15,7 +15,7 @@ module Relation.Binary.Structures
   (_≈_ : Rel A ℓ)   -- The underlying equality relation
   where
 
-open import Data.Product using (proj₁; proj₂; _,_)
+open import Data.Product.Base using (proj₁; proj₂; _,_)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary.Negation.Core using (¬_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
diff --git a/src/Relation/Nullary/Decidable.agda b/src/Relation/Nullary/Decidable.agda
index a8970d9df5..b41a410391 100644
--- a/src/Relation/Nullary/Decidable.agda
+++ b/src/Relation/Nullary/Decidable.agda
@@ -11,7 +11,7 @@ module Relation.Nullary.Decidable where
 open import Level using (Level)
 open import Data.Bool.Base using (true; false; if_then_else_)
 open import Data.Empty using (⊥-elim)
-open import Data.Product as Prod hiding (map)
+open import Data.Product.Base as Prod hiding (map)
 open import Data.Sum.Base as Sum hiding (map)
 open import Function.Base
 open import Function.Bundles using
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index c8d92f9375..e06dc32239 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -16,7 +16,7 @@ open import Data.Bool.Base using (Bool; false; true; not; T; _∧_; _∨_)
 open import Data.Unit.Base using (⊤)
 open import Data.Empty using (⊥)
 open import Data.Empty.Irrelevant using (⊥-elim)
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_∘_; const; _$_; flip)
 open import Relation.Nullary.Reflects
diff --git a/src/Relation/Nullary/Negation.agda b/src/Relation/Nullary/Negation.agda
index 6f663e73ac..5111db897f 100644
--- a/src/Relation/Nullary/Negation.agda
+++ b/src/Relation/Nullary/Negation.agda
@@ -8,16 +8,15 @@
 
 module Relation.Nullary.Negation where
 
-open import Effect.Monad
+open import Effect.Monad using (RawMonad; mkRawMonad)
 open import Data.Bool.Base using (Bool; false; true; if_then_else_; not)
-open import Data.Empty
-open import Data.Product as Prod
+open import Data.Empty using (⊥-elim)
+open import Data.Product.Base as Prod using (_,_; Σ; Σ-syntax; ∃; curry; uncurry)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
-open import Function.Base
-open import Level
-open import Relation.Nullary.Negation.Core
-open import Relation.Nullary.Decidable.Core
-open import Relation.Unary
+open import Function.Base using (flip; _∘_; const; _∘′_)
+open import Level using (Level)
+open import Relation.Nullary.Decidable.Core using (Dec; yes; no; excluded-middle)
+open import Relation.Unary using (Universal)
 
 private
   variable
@@ -89,7 +88,7 @@ independence-of-premise : {R : Q → Set r} →
 independence-of-premise {P = P} q f = ¬¬-map helper excluded-middle
   where
   helper : Dec P → _
-  helper (yes p) = Prod.map id const (f p)
+  helper (yes p) = Prod.map₂ const (f p)
   helper (no ¬p) = (q , ⊥-elim ∘′ ¬p)
 
 -- The independence of premise rule for binary sums.
diff --git a/src/Relation/Unary.agda b/src/Relation/Unary.agda
index 76c064836b..f6d7cb13b0 100644
--- a/src/Relation/Unary.agda
+++ b/src/Relation/Unary.agda
@@ -8,12 +8,12 @@
 
 module Relation.Unary where
 
-open import Data.Empty
+open import Data.Empty using (⊥)
 open import Data.Unit.Base using (⊤)
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_; Σ-syntax; ∃; uncurry; swap)
 open import Data.Sum.Base using (_⊎_; [_,_])
-open import Function.Base
-open import Level
+open import Function.Base using (_∘_; _|>_)
+open import Level using (Level; _⊔_; 0ℓ; suc; Lift)
 open import Relation.Nullary.Decidable.Core using (Dec; True)
 open import Relation.Nullary.Negation.Core using (¬_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
@@ -285,7 +285,7 @@ _⟨→⟩_ : Pred A ℓ₁ → Pred B ℓ₂ → Pred (A → B) _
 -- Product.
 
 _⟨·⟩_ : (P : Pred A ℓ₁) (Q : Pred B ℓ₂) →
-        (P ⟨×⟩ (P ⟨→⟩ Q)) ⊆ Q ∘ uncurry (flip _$_)
+        (P ⟨×⟩ (P ⟨→⟩ Q)) ⊆ Q ∘ uncurry _|>_
 (P ⟨·⟩ Q) (p , f) = f p
 
 -- Converse.
diff --git a/src/Relation/Unary/Closure/Base.agda b/src/Relation/Unary/Closure/Base.agda
index 5a210104b4..992e87df65 100644
--- a/src/Relation/Unary/Closure/Base.agda
+++ b/src/Relation/Unary/Closure/Base.agda
@@ -11,7 +11,7 @@ open import Relation.Binary
 module Relation.Unary.Closure.Base {a b} {A : Set a} (R : Rel A b) where
 
 open import Level
-open import Data.Product using (Σ-syntax; _×_; _,_; -,_)
+open import Data.Product.Base using (Σ-syntax; _×_; _,_; -,_)
 open import Function.Base using (flip)
 open import Relation.Unary using (Pred)
 
diff --git a/src/Relation/Unary/Properties.agda b/src/Relation/Unary/Properties.agda
index e0b059f0b4..f30a9e59b4 100644
--- a/src/Relation/Unary/Properties.agda
+++ b/src/Relation/Unary/Properties.agda
@@ -8,10 +8,10 @@
 
 module Relation.Unary.Properties where
 
-open import Data.Product as Product using (_×_; _,_; swap; proj₁; zip′)
+open import Data.Product.Base as Product using (_×_; _,_; swap; proj₁; zip′)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Unit.Base using (tt)
-open import Level
+open import Level using (Level)
 open import Relation.Binary.Core as Binary
 open import Relation.Binary.Definitions hiding (Decidable; Universal; Irrelevant)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
diff --git a/src/Tactic/RingSolver.agda b/src/Tactic/RingSolver.agda
index 3ae3451e68..6441342e40 100644
--- a/src/Tactic/RingSolver.agda
+++ b/src/Tactic/RingSolver.agda
@@ -18,7 +18,7 @@ open import Data.Nat.Base              using (ℕ; suc; zero; _<ᵇ_)
 open import Data.Bool.Base             using (Bool; if_then_else_; true; false)
 open import Data.Unit.Base             using (⊤)
 open import Data.String.Base as String using (String; _++_; parens)
-open import Data.Product               using (_,_; proj₁)
+open import Data.Product.Base          using (_,_; proj₁)
 open import Function.Base
 open import Relation.Nullary.Decidable
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Base.agda b/src/Tactic/RingSolver/Core/Polynomial/Base.agda
index d762794b5e..6cfece2d45 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Base.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Base.agda
@@ -50,15 +50,15 @@ module Tactic.RingSolver.Core.Polynomial.Base
 
 open RawCoeff coeffs
 
-open import Data.Bool              using (Bool; true; false; T)
+open import Data.Bool.Base         using (Bool; true; false; T)
 open import Data.Empty             using (⊥)
 open import Data.Fin.Base as Fin        using (Fin; zero; suc)
 open import Data.List.Kleene
 open import Data.Nat.Base as ℕ          using (ℕ; suc; zero; _≤′_; compare; ≤′-refl; ≤′-step; _<′_)
 open import Data.Nat.Properties    using (z≤′n; ≤′-trans)
 open import Data.Nat.Induction
-open import Data.Product           using (_×_; _,_; map₁; curry; uncurry)
-open import Data.Unit              using (⊤; tt)
+open import Data.Product.Base      using (_×_; _,_; map₁; curry; uncurry)
+open import Data.Unit.Base         using (⊤; tt)
 open import Function.Base
 open import Relation.Nullary       using (¬_; Dec; yes; no)
 
diff --git a/src/Text/Pretty.agda b/src/Text/Pretty.agda
index 08c60701bf..0bdfe0cf2f 100644
--- a/src/Text/Pretty.agda
+++ b/src/Text/Pretty.agda
@@ -13,14 +13,14 @@ open import Data.Nat.Base using (ℕ)
 module Text.Pretty (width : ℕ) where
 
 import Level
-open import Data.Char.Base     using (Char)
+open import Data.Char.Base using (Char)
 open import Data.List.Base
   using (List; _∷_; []; [_]; uncons; _++_; map; filter)
 open import Data.List.NonEmpty as List⁺ using (foldr₁)
-open import Data.Maybe.Base    using (maybe′)
-open import Data.Product       using (uncurry)
-open import Data.String.Base   using (String; fromList; replicate)
-open import Function.Base
+open import Data.Maybe.Base using (maybe′)
+open import Data.Product.Base using (uncurry)
+open import Data.String.Base using (String; fromList; replicate)
+open import Function.Base using (_∘_; _∘′_; _$_)
 
 open import Effect.Monad using (RawMonad)
 import Data.List.Effectful as Cat
diff --git a/src/Text/Pretty/Core.agda b/src/Text/Pretty/Core.agda
index f546bc5962..a97db1dda2 100644
--- a/src/Text/Pretty/Core.agda
+++ b/src/Text/Pretty/Core.agda
@@ -17,7 +17,7 @@ open import Data.Irrelevant as Irrelevant using (Irrelevant) hiding (module Irre
 open import Data.List.Base as List   using (List; []; _∷_)
 open import Data.Nat.Base            using (ℕ; zero; suc; _+_; _⊔_; _≤_; z≤n)
 open import Data.Nat.Properties
-open import Data.Product as Prod using (_×_; _,_; uncurry; proj₁; proj₂)
+open import Data.Product.Base as Prod using (_×_; _,_; uncurry; proj₁; proj₂)
 import Data.Product.Relation.Unary.All as Allᴾ
 
 open import Data.Tree.Binary as Tree using (Tree; leaf; node; #nodes; mapₙ)
diff --git a/src/Text/Printf.agda b/src/Text/Printf.agda
index 7397f908f0..057b2718b8 100644
--- a/src/Text/Printf.agda
+++ b/src/Text/Printf.agda
@@ -8,17 +8,8 @@
 
 module Text.Printf where
 
-open import Level using (0ℓ; Lift)
-open import Data.List.Base as List using (List; []; _∷_)
-open import Data.Nat.Base using (ℕ)
-open import Data.Product
-open import Data.Product.Nary.NonDependent
-open import Data.String.Base
-open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
-open import Data.Unit using (⊤)
-open import Function.Nary.NonDependent
-open import Function
-open import Function.Strict
+open import Data.String.Base using (String; fromChar; concat)
+open import Function.Base using (id)
 
 import Data.Char.Base    as Cₛ
 import Data.Integer.Show as ℤₛ
diff --git a/src/Text/Regex/Base.agda b/src/Text/Regex/Base.agda
index 7fe7d56cbf..839bad4181 100644
--- a/src/Text/Regex/Base.agda
+++ b/src/Text/Regex/Base.agda
@@ -16,12 +16,9 @@ open import Data.Bool.Base using (Bool)
 open import Data.List.Base as L using (List; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any)
 open import Data.List.Relation.Unary.All using (All)
-open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
+open import Data.Sum.Base using (_⊎_)
 
-open import Relation.Nullary.Negation using (¬_)
-open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Unary using (Pred)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Nullary.Negation.Core using (¬_)
 
 open Preorder P using (_≈_) renaming (Carrier to A; _∼_ to _≤_)
 open import Data.List.Relation.Ternary.Appending.Propositional {A = A}

From 7b352f6f2ba446b12207a094f451375cb6e8b594 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Tue, 27 Jun 2023 04:08:30 -0400
Subject: [PATCH 059/316] Fixities for `Relation` and `Algebra` (#1992)

---
 CHANGELOG.md                                              | 7 +++++++
 src/Algebra/Solver/Ring.agda                              | 2 ++
 src/Relation/Binary/Bundles.agda                          | 1 +
 src/Relation/Binary/Construct/Add/Infimum/Strict.agda     | 2 ++
 src/Relation/Binary/Construct/Add/Point/Equality.agda     | 2 ++
 src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda | 2 ++
 src/Relation/Binary/Construct/Closure/Transitive.agda     | 2 ++
 src/Relation/Binary/Construct/Intersection.agda           | 2 ++
 8 files changed, 20 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 25bc7bd058..c6f541483d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -105,6 +105,13 @@ Bug-fixes
   infix   8 _⁻¹                                              (Data.Parity.Base)
   infixr  5 _`∷_                                             (Data.Vec.Reflection)
   infixr  5 _∷=_                                             (Data.Vec.Membership.Setoid)
+  infix   4 _≟H_ _≟N_                                        (Algebra.Solver.Ring)
+  infix   4 _≈_                                              (Relation.Binary.Bundles)
+  infixl  6 _∩_                                              (Relation.Binary.Construct.Intersection)
+  infix   4 _<₋_                                             (Relation.Binary.Construct.Add.Infimum.Strict)
+  infix   4 _≈∙_                                             (Relation.Binary.Construct.Add.Point.Equality)
+  infix   4 _≤⁺_ _≤⊤⁺                                        (Relation.Binary.Construct.Add.Supremum.NonStrict)
+  infixr  5 _∷ʳ_                                             (Relation.Binary.Construct.Closure.Transitive)
   ```
 
 * In `System.Exit`, the `ExitFailure` constructor is now carrying an integer
diff --git a/src/Algebra/Solver/Ring.agda b/src/Algebra/Solver/Ring.agda
index 669cbe8d25..a1ea98ead3 100644
--- a/src/Algebra/Solver/Ring.agda
+++ b/src/Algebra/Solver/Ring.agda
@@ -176,6 +176,8 @@ mutual
 
   -- Equality is weakly decidable.
 
+  infix 4 _≟H_ _≟N_
+
   _≟H_ : ∀ {n} → WeaklyDecidable (_≈H_ {n = n})
   ∅           ≟H ∅           = just ∅
   ∅           ≟H (_ *x+ _)   = nothing
diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index 498e1f95ef..5a0621629f 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -21,6 +21,7 @@ open import Relation.Binary.Structures
 ------------------------------------------------------------------------
 
 record PartialSetoid a ℓ : Set (suc (a ⊔ ℓ)) where
+  infix 4 _≈_
   field
     Carrier              : Set a
     _≈_                  : Rel Carrier ℓ
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
index 06ac74899b..45463af482 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
@@ -28,6 +28,8 @@ import Relation.Nullary.Decidable as Dec
 ------------------------------------------------------------------------
 -- Definition
 
+infix 4 _<₋_
+
 data _<₋_ : Rel (A ₋) (a ⊔ ℓ) where
   ⊥₋<[_] : (l : A)           → ⊥₋    <₋ [ l ]
   [_]    : {k l : A} → k < l → [ k ] <₋ [ l ]
diff --git a/src/Relation/Binary/Construct/Add/Point/Equality.agda b/src/Relation/Binary/Construct/Add/Point/Equality.agda
index 2968754a9f..a0b01a3d2b 100644
--- a/src/Relation/Binary/Construct/Add/Point/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Point/Equality.agda
@@ -24,6 +24,8 @@ import Relation.Nullary.Decidable as Dec
 ------------------------------------------------------------------------
 -- Definition
 
+infix 4 _≈∙_
+
 data _≈∙_ : Rel (Pointed A) (a ⊔ ℓ) where
   ∙≈∙ :                     ∙     ≈∙ ∙
   [_] : {k l : A} → k ≈ l → [ k ] ≈∙ [ l ]
diff --git a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
index 68346824a2..be30c7973c 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
@@ -26,6 +26,8 @@ import Relation.Binary.Construct.Add.Supremum.Equality as Equality
 ------------------------------------------------------------------------
 -- Definition
 
+infix 4 _≤⁺_ _≤⊤⁺
+
 data _≤⁺_ : Rel (A ⁺) (a ⊔ ℓ) where
   [_]  : {k l : A} → k ≤ l → [ k ] ≤⁺ [ l ]
   _≤⊤⁺ : (k : A ⁺)         → k     ≤⁺ ⊤⁺
diff --git a/src/Relation/Binary/Construct/Closure/Transitive.agda b/src/Relation/Binary/Construct/Closure/Transitive.agda
index 85ab19e3dc..1699dc0cb6 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive.agda
@@ -39,6 +39,8 @@ module _ {_∼_ : Rel A ℓ} where
   private
     _∼⁺_ = TransClosure _∼_
 
+  infixr 5 _∷ʳ_
+
   _∷ʳ_ : ∀ {x y z} → (x∼⁺y : x ∼⁺ y) (y∼z : y ∼ z) → x ∼⁺ z
   [ x∼y ]      ∷ʳ y∼z = x∼y ∷ [ y∼z ]
   (x∼y ∷ x∼⁺y) ∷ʳ y∼z = x∼y ∷ (x∼⁺y ∷ʳ y∼z)
diff --git a/src/Relation/Binary/Construct/Intersection.agda b/src/Relation/Binary/Construct/Intersection.agda
index f74dfb3de9..17845536ca 100644
--- a/src/Relation/Binary/Construct/Intersection.agda
+++ b/src/Relation/Binary/Construct/Intersection.agda
@@ -24,6 +24,8 @@ private
 ------------------------------------------------------------------------
 -- Definition
 
+infixl 6 _∩_
+
 _∩_ : REL A B ℓ₁ → REL A B ℓ₂ → REL A B (ℓ₁ ⊔ ℓ₂)
 L ∩ R = λ i j → L i j × R i j
 

From 6bd4afb1c03a30bc1344b478467f09abe8d49203 Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Tue, 27 Jun 2023 04:11:31 -0400
Subject: [PATCH 060/316] Add new proof `take-all` to `Data.List.Properties` 
 (#1983)

---
 CHANGELOG.md                  | 3 ++-
 src/Data/List/Properties.agda | 6 ++++++
 2 files changed, 8 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index c6f541483d..b7b0f3f225 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2152,7 +2152,8 @@ Other minor changes
 
   length-isMagmaHomomorphism : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
   length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
- 
+  
+  take-all : n ≥ length xs → take n xs ≡ xs     
   take-[] : ∀ m → take  m [] ≡ []
   drop-[] : ∀ m → drop  m [] ≡ []
   ```
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 9df56d4781..08e78cd5a6 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -759,6 +759,12 @@ length-take zero    xs       = refl
 length-take (suc n) []       = refl
 length-take (suc n) (x ∷ xs) = cong suc (length-take n xs)
 
+-- If you take at least as many elements from a list as it has, you get the whole list.
+take-all :(n : ℕ) (xs : List A) → n ≥ length xs → take n xs ≡ xs
+take-all zero [] _ = refl
+take-all (suc _) [] _ = refl
+take-all (suc n) (x ∷ xs) (s≤s pf) = cong (x ∷_) (take-all n xs pf)
+
 -- Taking from an empty list does nothing.
 take-[] : ∀ m → take {A = A} m [] ≡ []
 take-[] zero = refl

From e10c5e2842db3581342dd15d8247ad9c3457118d Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Tue, 27 Jun 2023 22:24:28 -0400
Subject: [PATCH 061/316] New functions `take-drop-1` and `take-one-more` added
 to Data.List.Properties (#1984)

---
 CHANGELOG.md                  |  8 +++++++-
 src/Data/List/Properties.agda | 23 +++++++++++++++++++++++
 2 files changed, 30 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b7b0f3f225..6d692fdb4a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2152,8 +2152,14 @@ Other minor changes
 
   length-isMagmaHomomorphism : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
   length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
+
+  take-suc : (o : Fin (length xs)) → let m = toℕ o in take (suc m) xs ≡ take m xs ∷ʳ lookup xs o
+  take-suc-tabulate : (f : Fin n → A) (o : Fin n) → let m = toℕ o in take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f o
+  drop-take-suc : (o : Fin (length xs)) → let m = toℕ o in drop m (take (suc m) xs) ≡ [ lookup xs o ]
+  drop-take-suc-tabulate : (f : Fin n → A) (o : Fin n) → let m = toℕ o in drop m (take (suc m) (tabulate f)) ≡ [ f o ]
   
   take-all : n ≥ length xs → take n xs ≡ xs     
+  
   take-[] : ∀ m → take  m [] ≡ []
   drop-[] : ∀ m → drop  m [] ≡ []
   ```
@@ -2887,7 +2893,7 @@ Other minor changes
   foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
   foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
   ```
-
+  
 NonZero/Positive/Negative changes
 ---------------------------------
 
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 08e78cd5a6..c4cf722ec6 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -30,6 +30,7 @@ open import Data.Product.Base as Prod
 import Data.Product.Relation.Unary.All as Prod using (All)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
+open import Data.Fin.Properties using (toℕ-cast)
 open import Function
 open import Level using (Level)
 open import Relation.Binary as B using (DecidableEquality)
@@ -759,6 +760,16 @@ length-take zero    xs       = refl
 length-take (suc n) []       = refl
 length-take (suc n) (x ∷ xs) = cong suc (length-take n xs)
 
+take-suc : (xs : List A) (i : Fin (length xs)) → let m = toℕ i in
+           take (suc m) xs ≡ take m xs ∷ʳ lookup xs i
+take-suc (x ∷ xs) zero    = refl
+take-suc (x ∷ xs) (suc i) = cong (x ∷_) (take-suc xs i)
+
+take-suc-tabulate : ∀ {n} (f : Fin n → A) (i : Fin n) → let m = toℕ i in
+                    take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f i
+take-suc-tabulate f i rewrite sym (toℕ-cast (sym (length-tabulate f)) i) | sym (lookup-tabulate f i)
+  = take-suc (tabulate f) (cast _ i)
+
 -- If you take at least as many elements from a list as it has, you get the whole list.
 take-all :(n : ℕ) (xs : List A) → n ≥ length xs → take n xs ≡ xs
 take-all zero [] _ = refl
@@ -789,6 +800,16 @@ take++drop zero    xs       = refl
 take++drop (suc n) []       = refl
 take++drop (suc n) (x ∷ xs) = cong (x ∷_) (take++drop n xs)
 
+drop-take-suc : (xs : List A) (i : Fin (length xs)) → let m = toℕ i in
+           drop m (take (suc m) xs) ≡ [ lookup xs i ]
+drop-take-suc (x ∷ xs) zero    = refl
+drop-take-suc (x ∷ xs) (suc i) = drop-take-suc xs i
+
+drop-take-suc-tabulate : ∀ {n} (f : Fin n → A) (i : Fin n) → let m = toℕ i in
+                  drop m (take (suc m) (tabulate f)) ≡ [ f i ]
+drop-take-suc-tabulate f i rewrite sym (toℕ-cast (sym (length-tabulate f)) i) | sym (lookup-tabulate f i)
+  = drop-take-suc (tabulate f) (cast _ i)
+
 ------------------------------------------------------------------------
 -- splitAt
 
@@ -1093,6 +1114,8 @@ module _ {x y : A} where
 ∷ʳ-++ : ∀ (xs : List A) (a : A) (ys : List A) → xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
 ∷ʳ-++ xs a ys = ++-assoc xs [ a ] ys
 
+
+
 ------------------------------------------------------------------------
 -- DEPRECATED
 ------------------------------------------------------------------------

From e4b8c0e16934b6276b385e29dff66fda18da2341 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Wed, 28 Jun 2023 01:01:24 -0400
Subject: [PATCH 062/316] Cut down `Function` imports (#2006)

---
 src/Algebra/Construct/NaturalChoice/Min.agda         |  2 +-
 src/Algebra/Morphism.agda                            |  2 +-
 src/Algebra/Ordered/Structures.agda                  |  2 +-
 src/Algebra/Solver/Ring/AlmostCommutativeRing.agda   |  2 +-
 src/Data/Container/Combinator.agda                   |  2 +-
 src/Data/Container/Combinator/Properties.agda        |  2 +-
 src/Data/Container/Indexed/Combinator.agda           |  2 +-
 src/Data/Container/Indexed/FreeMonad.agda            |  2 +-
 src/Data/Container/Indexed/WithK.agda                |  2 +-
 src/Data/Container/Morphism.agda                     |  2 +-
 src/Data/Container/Morphism/Properties.agda          |  2 +-
 src/Data/Container/Properties.agda                   |  2 +-
 src/Data/Container/Relation/Unary/All.agda           |  2 +-
 src/Data/Digit/Properties.agda                       |  2 +-
 src/Data/Fin/Permutation/Transposition/List.agda     |  2 +-
 src/Data/Fin/Substitution/Lemmas.agda                |  6 +++---
 src/Data/Fin/Substitution/List.agda                  |  2 +-
 src/Data/Float/Properties.agda                       |  2 +-
 src/Data/List/Relation/Unary/Grouped/Properties.agda |  3 ++-
 src/Data/Nat/Binary/Subtraction.agda                 |  2 +-
 src/Data/Nat/DivMod.agda                             |  2 +-
 src/Data/Nat/Show/Properties.agda                    |  2 +-
 src/Data/Parity/Properties.agda                      |  3 ++-
 src/Data/Sign/Properties.agda                        |  3 ++-
 src/Data/Star/Environment.agda                       |  2 +-
 src/Data/Tree/AVL/Value.agda                         |  2 +-
 src/Data/Tree/Binary/Properties.agda                 |  2 +-
 src/Data/Trie/NonEmpty.agda                          |  2 +-
 src/Data/Vec/Effectful.agda                          |  2 +-
 src/Data/Vec/Relation/Binary/Lex/NonStrict.agda      |  2 +-
 src/Data/Vec/Relation/Unary/AllPairs.agda            |  2 +-
 src/Data/Vec/Relation/Unary/AllPairs/Properties.agda |  2 +-
 src/Function/Injection.agda                          |  2 +-
 src/Function/Metric/Nat/Bundles.agda                 |  2 +-
 src/Function/Metric/Nat/Structures.agda              |  2 +-
 src/Function/Metric/Rational/Bundles.agda            |  2 +-
 src/Function/Metric/Rational/Structures.agda         |  2 +-
 src/Function/Reasoning.agda                          |  2 +-
 src/Reflection/AST/Meta.agda                         |  4 ++--
 src/Reflection/AST/Name.agda                         |  8 ++++----
 src/Reflection/AST/Traversal.agda                    | 12 ++++++------
 src/Reflection/External.agda                         |  2 +-
 src/Relation/Binary/OrderMorphism.agda               |  2 +-
 src/Relation/Unary/Algebra.agda                      |  2 +-
 src/Tactic/Cong.agda                                 |  2 +-
 .../RingSolver/Core/AlmostCommutativeRing.agda       |  2 +-
 tests/data/appending/Main.agda                       |  2 +-
 tests/data/colist/Main.agda                          |  2 +-
 48 files changed, 62 insertions(+), 59 deletions(-)

diff --git a/src/Algebra/Construct/NaturalChoice/Min.agda b/src/Algebra/Construct/NaturalChoice/Min.agda
index 1184f4fe30..89bd2eefb8 100644
--- a/src/Algebra/Construct/NaturalChoice/Min.agda
+++ b/src/Algebra/Construct/NaturalChoice/Min.agda
@@ -11,7 +11,7 @@ open import Algebra.Bundles
 open import Algebra.Construct.NaturalChoice.Base
 open import Data.Sum using (inj₁; inj₂; [_,_])
 open import Data.Product using (_,_)
-open import Function using (id)
+open import Function.Base using (id)
 open import Relation.Binary
 import Algebra.Construct.NaturalChoice.MinOp as MinOp
 
diff --git a/src/Algebra/Morphism.agda b/src/Algebra/Morphism.agda
index fbcb2b0d12..59a6f22e23 100644
--- a/src/Algebra/Morphism.agda
+++ b/src/Algebra/Morphism.agda
@@ -11,7 +11,7 @@ module Algebra.Morphism where
 import Algebra.Morphism.Definitions as MorphismDefinitions
 open import Algebra
 import Algebra.Properties.Group as GroupP
-open import Function hiding (Morphism)
+open import Function.Base
 open import Level
 open import Relation.Binary
 import Relation.Binary.Reasoning.Setoid as EqR
diff --git a/src/Algebra/Ordered/Structures.agda b/src/Algebra/Ordered/Structures.agda
index 1c46f007f7..ab211cc497 100644
--- a/src/Algebra/Ordered/Structures.agda
+++ b/src/Algebra/Ordered/Structures.agda
@@ -22,7 +22,7 @@ open import Algebra.Core
 open import Algebra.Definitions _≈_
 open import Algebra.Structures _≈_
 open import Data.Product using (proj₁; proj₂)
-open import Function using (flip)
+open import Function.Base using (flip)
 open import Level using (_⊔_)
 open import Relation.Binary.Definitions using (Transitive; Monotonic₁; Monotonic₂)
 open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
diff --git a/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda b/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
index 47cda5c4f9..2bcfe92901 100644
--- a/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
+++ b/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
@@ -14,7 +14,7 @@ open import Algebra.Structures
 open import Algebra.Definitions
 import Algebra.Morphism as Morphism
 import Algebra.Morphism.Definitions as MorphismDefinitions
-open import Function hiding (Morphism)
+open import Function.Base using (id)
 open import Level
 open import Relation.Binary
 
diff --git a/src/Data/Container/Combinator.agda b/src/Data/Container/Combinator.agda
index e70d3e5ef4..c68536427b 100644
--- a/src/Data/Container/Combinator.agda
+++ b/src/Data/Container/Combinator.agda
@@ -13,7 +13,7 @@ open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
 open import Data.Product as P using (_,_; <_,_>; proj₁; proj₂; ∃)
 open import Data.Sum.Base as S using ([_,_]′)
 open import Data.Unit.Polymorphic.Base using (⊤)
-import Function as F
+import Function.Base as F
 
 open import Data.Container.Core
 open import Data.Container.Relation.Unary.Any
diff --git a/src/Data/Container/Combinator/Properties.agda b/src/Data/Container/Combinator/Properties.agda
index 5799b857f8..b93d6c7a7a 100644
--- a/src/Data/Container/Combinator/Properties.agda
+++ b/src/Data/Container/Combinator/Properties.agda
@@ -15,7 +15,7 @@ open import Data.Container.Relation.Unary.Any
 open import Data.Empty using (⊥-elim)
 open import Data.Product as Prod using (∃; _,_; proj₁; proj₂; <_,_>; uncurry; curry)
 open import Data.Sum.Base as S using (inj₁; inj₂; [_,_]′; [_,_])
-open import Function as F using (_∘′_)
+open import Function.Base as F using (_∘′_)
 open import Function.Inverse as Inv using (_↔_; inverse; module Inverse)
 open import Level using (_⊔_; lower)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
diff --git a/src/Data/Container/Indexed/Combinator.agda b/src/Data/Container/Indexed/Combinator.agda
index 78e5d213d2..bc99e83f96 100644
--- a/src/Data/Container/Indexed/Combinator.agda
+++ b/src/Data/Container/Indexed/Combinator.agda
@@ -15,7 +15,7 @@ open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Product as Prod hiding (Σ) renaming (_×_ to _⟨×⟩_)
 open import Data.Sum renaming (_⊎_ to _⟨⊎⟩_)
 open import Data.Sum.Relation.Unary.All as All using (All)
-open import Function as F hiding (id; const) renaming (_∘_ to _⟨∘⟩_)
+open import Function.Base as F hiding (id; const) renaming (_∘_ to _⟨∘⟩_)
 open import Function.Inverse using (_↔̇_; inverse)
 open import Level
 open import Relation.Unary using (Pred; _⊆_; _∪_; _∩_; ⋃; ⋂)
diff --git a/src/Data/Container/Indexed/FreeMonad.agda b/src/Data/Container/Indexed/FreeMonad.agda
index 1f7d63428d..f2fe0d2aab 100644
--- a/src/Data/Container/Indexed/FreeMonad.agda
+++ b/src/Data/Container/Indexed/FreeMonad.agda
@@ -9,7 +9,7 @@
 module Data.Container.Indexed.FreeMonad where
 
 open import Level
-open import Function hiding (const)
+open import Function.Base hiding (const)
 open import Effect.Monad.Predicate
 open import Data.Container.Indexed
 open import Data.Container.Indexed.Combinator hiding (id; _∘_)
diff --git a/src/Data/Container/Indexed/WithK.agda b/src/Data/Container/Indexed/WithK.agda
index 2e1a7b7426..565b14fcdd 100644
--- a/src/Data/Container/Indexed/WithK.agda
+++ b/src/Data/Container/Indexed/WithK.agda
@@ -16,7 +16,7 @@ module Data.Container.Indexed.WithK where
 open import Axiom.Extensionality.Heterogeneous using (Extensionality)
 open import Data.Container.Indexed hiding (module PlainMorphism)
 open import Data.Product as Prod hiding (map)
-open import Function renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
+open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
 open import Level
 open import Relation.Unary using (Pred; _⊆_)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
diff --git a/src/Data/Container/Morphism.agda b/src/Data/Container/Morphism.agda
index 6494e5a3b8..0e26d48199 100644
--- a/src/Data/Container/Morphism.agda
+++ b/src/Data/Container/Morphism.agda
@@ -9,7 +9,7 @@
 module Data.Container.Morphism where
 
 open import Data.Container.Core
-import Function as F
+import Function.Base as F
 
 module _ {s p} (C : Container s p) where
 
diff --git a/src/Data/Container/Morphism/Properties.agda b/src/Data/Container/Morphism/Properties.agda
index 16b3dd9333..0859cfe0be 100644
--- a/src/Data/Container/Morphism/Properties.agda
+++ b/src/Data/Container/Morphism/Properties.agda
@@ -9,7 +9,7 @@
 module Data.Container.Morphism.Properties where
 
 open import Level using (_⊔_; suc)
-open import Function as F using (_$_)
+open import Function.Base as F using (_$_)
 open import Data.Product using (∃; proj₁; proj₂; _,_)
 open import Relation.Binary using (Setoid)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
diff --git a/src/Data/Container/Properties.agda b/src/Data/Container/Properties.agda
index 193f7cf702..9ee24ad878 100644
--- a/src/Data/Container/Properties.agda
+++ b/src/Data/Container/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.Container.Properties where
 
-import Function as F
+import Function.Base as F
 open import Relation.Binary
 
 open import Data.Container.Core
diff --git a/src/Data/Container/Relation/Unary/All.agda b/src/Data/Container/Relation/Unary/All.agda
index 2a6f858c87..65d593eea4 100644
--- a/src/Data/Container/Relation/Unary/All.agda
+++ b/src/Data/Container/Relation/Unary/All.agda
@@ -11,7 +11,7 @@ module Data.Container.Relation.Unary.All where
 open import Level using (_⊔_)
 open import Relation.Unary using (Pred; _⊆_)
 open import Data.Product as Prod using (_,_; proj₁; proj₂; ∃)
-open import Function as F
+open import Function.Base using (_∘′_; id)
 
 open import Data.Container.Core hiding (map)
 import Data.Container.Morphism as M
diff --git a/src/Data/Digit/Properties.agda b/src/Data/Digit/Properties.agda
index 0ba2cab34f..355467ec39 100644
--- a/src/Data/Digit/Properties.agda
+++ b/src/Data/Digit/Properties.agda
@@ -17,7 +17,7 @@ import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Uniqueₚ
 open import Data.Vec.Relation.Unary.AllPairs using (allPairs?)
 open import Relation.Nullary.Decidable using (True; from-yes; ¬?)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
-open import Function using (_∘_)
+open import Function.Base using (_∘_)
 
 module Data.Digit.Properties where
 
diff --git a/src/Data/Fin/Permutation/Transposition/List.agda b/src/Data/Fin/Permutation/Transposition/List.agda
index 42505de541..4ed0583372 100644
--- a/src/Data/Fin/Permutation/Transposition/List.agda
+++ b/src/Data/Fin/Permutation/Transposition/List.agda
@@ -15,7 +15,7 @@ import Data.Fin.Permutation.Components as PC
 open import Data.List using (List; []; _∷_; map)
 open import Data.Nat.Base using (ℕ; suc; zero)
 open import Data.Product using (_×_; _,_)
-open import Function using (_∘_)
+open import Function.Base using (_∘_)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; sym; cong; module ≡-Reasoning)
 open ≡-Reasoning
diff --git a/src/Data/Fin/Substitution/Lemmas.agda b/src/Data/Fin/Substitution/Lemmas.agda
index 018ed86f29..3747f21018 100644
--- a/src/Data/Fin/Substitution/Lemmas.agda
+++ b/src/Data/Fin/Substitution/Lemmas.agda
@@ -14,7 +14,7 @@ open import Data.Nat hiding (_⊔_; _/_)
 open import Data.Fin.Base using (Fin; zero; suc; lift)
 open import Data.Vec.Base
 import Data.Vec.Properties as VecProp
-open import Function as Fun using (_∘_; _$_)
+open import Function.Base as Fun using (_∘_; _$_; flip)
 open import Relation.Binary.PropositionalEquality as PropEq
   using (_≡_; refl; sym; cong; cong₂)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
@@ -222,7 +222,7 @@ record Lemmas₃ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
            ∀ k t → t /✶ ρs₁ ↑✶ k ≡ t /✶ ρs₂ ↑✶ k
   /✶-↑✶′ ρs₁ ρs₂ hyp = /✶-↑✶ ρs₁ ρs₂ (λ k x → begin
     var x /✶ ρs₁ ↑✶ k        ≡⟨ sym (lookup-⨀ x (ρs₁ ↑✶ k)) ⟩
-    lookup (⨀ (ρs₁ ↑✶ k)) x  ≡⟨ cong (Fun.flip lookup x) (hyp k) ⟩
+    lookup (⨀ (ρs₁ ↑✶ k)) x  ≡⟨ cong (flip lookup x) (hyp k) ⟩
     lookup (⨀ (ρs₂ ↑✶ k)) x  ≡⟨ lookup-⨀ x (ρs₂ ↑✶ k) ⟩
     var x /✶ ρs₂ ↑✶ k        ∎)
 
@@ -341,7 +341,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
             var (suc x) / ρ ↑ ≡ var x / ρ / wk
   suc-/-↑ {ρ = ρ} x = begin
     var (suc x) / ρ ↑        ≡⟨ var-/ ⟩
-    lookup (map weaken ρ) x  ≡⟨ cong (Fun.flip lookup x) (map-weaken {ρ = ρ}) ⟩
+    lookup (map weaken ρ) x  ≡⟨ cong (flip lookup x) (map-weaken {ρ = ρ}) ⟩
     lookup (ρ ⊙ wk)       x  ≡⟨ lookup-⊙ x {ρ₁ = ρ} ⟩
     lookup ρ x / wk          ≡⟨ cong₂ _/_ (sym var-/) refl ⟩
     var x / ρ / wk           ∎
diff --git a/src/Data/Fin/Substitution/List.agda b/src/Data/Fin/Substitution/List.agda
index 8927b3b21f..63a7e39279 100644
--- a/src/Data/Fin/Substitution/List.agda
+++ b/src/Data/Fin/Substitution/List.agda
@@ -17,7 +17,7 @@ module Data.Fin.Substitution.List {ℓ} {T : ℕ → Set ℓ} (lemmas₄ : Lemma
 open import Data.List.Base
 open import Data.List.Properties
 open import Data.Fin.Substitution
-import Function as Fun
+import Function.Base as Fun
 open import Relation.Binary.PropositionalEquality
 open ≡-Reasoning
 
diff --git a/src/Data/Float/Properties.agda b/src/Data/Float/Properties.agda
index b2b2fc2c9e..b2caebb616 100644
--- a/src/Data/Float/Properties.agda
+++ b/src/Data/Float/Properties.agda
@@ -15,7 +15,7 @@ import Data.Maybe.Properties as Mₚ
 import Data.Nat.Properties as Nₚ
 import Data.Word.Base as Word
 import Data.Word.Properties as Wₚ
-open import Function using (_∘_)
+open import Function.Base using (_∘_)
 open import Relation.Nullary.Decidable as RN using (map′)
 open import Relation.Binary
 import Relation.Binary.Construct.On as On
diff --git a/src/Data/List/Relation/Unary/Grouped/Properties.agda b/src/Data/List/Relation/Unary/Grouped/Properties.agda
index 189dba033c..a8a72e6cec 100644
--- a/src/Data/List/Relation/Unary/Grouped/Properties.agda
+++ b/src/Data/List/Relation/Unary/Grouped/Properties.agda
@@ -14,7 +14,8 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
 open import Data.List.Relation.Unary.Grouped
-open import Function using (_∘_; _⇔_; Equivalence)
+open import Function.Base using (_∘_)
+open import Function.Bundles using (module Equivalence; _⇔_)
 open import Relation.Binary as B using (REL; Rel)
 open import Relation.Unary as U using (Pred)
 open import Relation.Nullary using (¬_; does; yes; no)
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index 67eebc88f6..8b095b137f 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -19,7 +19,7 @@ import Data.Nat.Properties as ℕₚ
 open import Data.Product using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum using (inj₁; inj₂)
 open import Data.Vec using ([]; _∷_)
-open import Function using (_∘_; _$_)
+open import Function.Base using (_∘_; _$_)
 open import Level using (0ℓ)
 open import Relation.Binary
   using (Tri; tri<; tri≈; tri>; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index f2b2d8657e..b4c225d6a4 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -17,7 +17,7 @@ open import Data.Nat.DivMod.Core
 open import Data.Nat.Divisibility.Core
 open import Data.Nat.Induction
 open import Data.Nat.Properties
-open import Function using (_$_)
+open import Function.Base using (_$_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (False; toWitnessFalse)
diff --git a/src/Data/Nat/Show/Properties.agda b/src/Data/Nat/Show/Properties.agda
index 739a00c2e4..87622b79bf 100644
--- a/src/Data/Nat/Show/Properties.agda
+++ b/src/Data/Nat/Show/Properties.agda
@@ -8,7 +8,7 @@
 
 open import Data.Digit using (showDigit; toDigits)
 open import Data.Digit.Properties using (toDigits-injective; showDigit-injective)
-open import Function using (_∘_)
+open import Function.Base using (_∘_)
 import Data.List.Properties as Listₚ
 open import Data.Nat.Base using (ℕ)
 open import Data.Nat.Properties using (_≤?_)
diff --git a/src/Data/Parity/Properties.agda b/src/Data/Parity/Properties.agda
index ed25a7b2fe..b9fb8a5eb7 100644
--- a/src/Data/Parity/Properties.agda
+++ b/src/Data/Parity/Properties.agda
@@ -14,7 +14,8 @@ open import Data.Nat.Base as ℕ using (zero; suc; parity)
 open import Data.Parity.Base as ℙ using (Parity; 0ℙ; 1ℙ; _⁻¹; toSign; fromSign)
 open import Data.Product using (_,_)
 open import Data.Sign.Base as 𝕊
-open import Function hiding (Inverse)
+open import Function.Base using (_$_; id)
+open import Function.Definitions using (Injective; Surjective)
 open import Level using (0ℓ)
 open import Relation.Binary
   using (Decidable; DecidableEquality; Setoid; DecSetoid; IsDecEquivalence)
diff --git a/src/Data/Sign/Properties.agda b/src/Data/Sign/Properties.agda
index ccaa33f0d1..5ca9a36e95 100644
--- a/src/Data/Sign/Properties.agda
+++ b/src/Data/Sign/Properties.agda
@@ -12,7 +12,8 @@ open import Algebra.Bundles
 open import Data.Empty
 open import Data.Sign.Base
 open import Data.Product using (_,_)
-open import Function hiding (Inverse)
+open import Function.Base using (_$_; id)
+open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
 open import Relation.Binary
   using (Decidable; DecidableEquality; Setoid; DecSetoid; IsDecEquivalence)
diff --git a/src/Data/Star/Environment.agda b/src/Data/Star/Environment.agda
index 65fcbd4818..c7ccb18ef4 100644
--- a/src/Data/Star/Environment.agda
+++ b/src/Data/Star/Environment.agda
@@ -13,7 +13,7 @@ open import Data.Star.List
 open import Data.Star.Decoration
 open import Data.Star.Pointer as Pointer hiding (lookup)
 open import Data.Unit
-open import Function hiding (_∋_)
+open import Function.Base hiding (_∋_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 
diff --git a/src/Data/Tree/AVL/Value.agda b/src/Data/Tree/AVL/Value.agda
index 91633bfb6d..4f8543b6ba 100644
--- a/src/Data/Tree/AVL/Value.agda
+++ b/src/Data/Tree/AVL/Value.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Value {a ℓ} (S : Setoid a ℓ) where
 
 open import Data.Product using (Σ; _,_)
 open import Level using (suc; _⊔_)
-import Function as F
+import Function.Base as F
 open Setoid S renaming (Carrier to Key)
 
 -----------------------------------------------------------------------
diff --git a/src/Data/Tree/Binary/Properties.agda b/src/Data/Tree/Binary/Properties.agda
index c3dc189f8d..b523b1e6f1 100644
--- a/src/Data/Tree/Binary/Properties.agda
+++ b/src/Data/Tree/Binary/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.Tree.Binary.Properties where
 
-open import Function using (_∘_)
+open import Function.Base using (_∘_)
 open import Function.Nary.NonDependent using (congₙ)
 open import Level using (Level)
 open import Data.Nat.Base using (suc; _+_)
diff --git a/src/Data/Trie/NonEmpty.agda b/src/Data/Trie/NonEmpty.agda
index 6de3c53313..118a0db1bd 100644
--- a/src/Data/Trie/NonEmpty.agda
+++ b/src/Data/Trie/NonEmpty.agda
@@ -18,7 +18,7 @@ open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; [_]; concatMap)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′) hiding (module Maybe)
 open import Data.These as These using (These; this; that; these)
-open import Function as F
+open import Function.Base as F
 import Function.Identity.Effectful as Identity
 open import Relation.Unary using (_⇒_; IUniversal)
 
diff --git a/src/Data/Vec/Effectful.agda b/src/Data/Vec/Effectful.agda
index aa1f6dcba7..5d8e438537 100644
--- a/src/Data/Vec/Effectful.agda
+++ b/src/Data/Vec/Effectful.agda
@@ -16,7 +16,7 @@ open import Effect.Applicative as App using (RawApplicative)
 open import Effect.Functor as Fun using (RawFunctor)
 open import Effect.Monad using (RawMonad; RawMonadT; mkRawMonad)
 import Function.Identity.Effectful as Id
-open import Function hiding (Morphism)
+open import Function.Base
 open import Level using (Level)
 
 private
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index 409732dc0b..877d345045 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -19,7 +19,7 @@ open import Data.Vec using (Vec; []; _∷_)
 import Data.Vec.Relation.Binary.Lex.Strict as Strict
 open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
   using (Pointwise; []; _∷_; head; tail)
-open import Function using (id)
+open import Function.Base using (id)
 open import Relation.Binary
 import Relation.Binary.Construct.NonStrictToStrict as Conv
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
diff --git a/src/Data/Vec/Relation/Unary/AllPairs.agda b/src/Data/Vec/Relation/Unary/AllPairs.agda
index 7c3eabcb83..0fee880769 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs.agda
@@ -15,7 +15,7 @@ open import Data.Nat.Base using (suc)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.Product as Prod using (_,_; _×_; uncurry; <_,_>)
-open import Function using (id; _∘_)
+open import Function.Base using (id; _∘_)
 open import Level using (_⊔_)
 open import Relation.Binary as B using (Rel; _⇒_)
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
diff --git a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
index 7876928cc7..339a884a3b 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
@@ -17,7 +17,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base using (Fin)
 open import Data.Fin.Properties using (suc-injective)
 open import Data.Nat.Base using (zero; suc; _+_)
-open import Function using (_∘_)
+open import Function.Base using (_∘_)
 open import Level using (Level)
 open import Relation.Binary using (Rel)
 open import Relation.Binary.PropositionalEquality using (_≢_)
diff --git a/src/Function/Injection.agda b/src/Function/Injection.agda
index bfa03a1ff1..4f746c2beb 100644
--- a/src/Function/Injection.agda
+++ b/src/Function/Injection.agda
@@ -13,7 +13,7 @@
 
 module Function.Injection where
 
-open import Function as Fun using () renaming (_∘_ to _⟨∘⟩_)
+open import Function.Base as Fun using () renaming (_∘_ to _⟨∘⟩_)
 open import Level
 open import Relation.Binary
 open import Function.Equality as F
diff --git a/src/Function/Metric/Nat/Bundles.agda b/src/Function/Metric/Nat/Bundles.agda
index e0983246eb..57ff945ee0 100644
--- a/src/Function/Metric/Nat/Bundles.agda
+++ b/src/Function/Metric/Nat/Bundles.agda
@@ -13,7 +13,7 @@
 module Function.Metric.Nat.Bundles where
 
 open import Data.Nat.Base hiding (suc; _⊔_)
-open import Function using (const)
+open import Function.Base using (const)
 open import Level using (Level; suc; _⊔_)
 open import Relation.Binary.Core
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Function/Metric/Nat/Structures.agda b/src/Function/Metric/Nat/Structures.agda
index d601cad35f..648e7a4adc 100644
--- a/src/Function/Metric/Nat/Structures.agda
+++ b/src/Function/Metric/Nat/Structures.agda
@@ -9,7 +9,7 @@
 module Function.Metric.Nat.Structures where
 
 open import Data.Nat.Base hiding (suc)
-open import Function using (const)
+open import Function.Base using (const)
 open import Level using (Level; suc)
 open import Relation.Binary hiding (Symmetric)
 open import Relation.Binary.PropositionalEquality using (_≡_)
diff --git a/src/Function/Metric/Rational/Bundles.agda b/src/Function/Metric/Rational/Bundles.agda
index 89067d2718..8bbc9f85d3 100644
--- a/src/Function/Metric/Rational/Bundles.agda
+++ b/src/Function/Metric/Rational/Bundles.agda
@@ -11,7 +11,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.Rational.Base hiding (_⊔_)
-open import Function using (const)
+open import Function.Base using (const)
 open import Level using (Level; suc; _⊔_)
 open import Relation.Binary.Core
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Function/Metric/Rational/Structures.agda b/src/Function/Metric/Rational/Structures.agda
index 44339d296b..824f90b912 100644
--- a/src/Function/Metric/Rational/Structures.agda
+++ b/src/Function/Metric/Rational/Structures.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.Rational.Base
-open import Function using (const)
+open import Function.Base using (const)
 open import Level using (Level; suc)
 open import Relation.Binary hiding (Symmetric)
 open import Relation.Binary.PropositionalEquality using (_≡_)
diff --git a/src/Function/Reasoning.agda b/src/Function/Reasoning.agda
index 09747c0541..ea39d0b52e 100644
--- a/src/Function/Reasoning.agda
+++ b/src/Function/Reasoning.agda
@@ -19,4 +19,4 @@ infixl 0 ∋-syntax
 syntax ∋-syntax A a = a ∶ A
 
 -- Export pipeline functions
-open import Function public using (_|>_; _|>′_)
+open import Function.Base public using (_|>_; _|>′_)
diff --git a/src/Reflection/AST/Meta.agda b/src/Reflection/AST/Meta.agda
index 59ca4ae4db..93584f573c 100644
--- a/src/Reflection/AST/Meta.agda
+++ b/src/Reflection/AST/Meta.agda
@@ -8,8 +8,8 @@
 
 module Reflection.AST.Meta where
 
-import Data.Nat.Properties as ℕₚ                using (_≟_)
-open import Function                             using (_on_)
+import Data.Nat.Properties as ℕₚ                 using (_≟_)
+open import Function.Base                        using (_on_)
 open import Relation.Nullary.Decidable           using (map′)
 open import Relation.Binary                      using (Rel; Decidable; DecidableEquality)
 import Relation.Binary.Construct.On as On
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index 203521330c..1baa63029f 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -8,10 +8,10 @@
 
 module Reflection.AST.Name where
 
-open import Data.List.Base              using (List)
-import Data.Product.Properties as Prodₚ using (≡-dec)
-import Data.Word.Properties as Wₚ       using (_≟_)
-open import Function                    using (_on_)
+open import Data.List.Base                        using (List)
+import Data.Product.Properties as Prodₚ           using (≡-dec)
+import Data.Word.Properties as Wₚ                 using (_≟_)
+open import Function.Base                         using (_on_)
 open import Relation.Nullary.Decidable            using (map′)
 open import Relation.Binary                       using (Rel; Decidable; DecidableEquality)
 open import Relation.Binary.Construct.On          using (decidable)
diff --git a/src/Reflection/AST/Traversal.agda b/src/Reflection/AST/Traversal.agda
index de28c8db2a..336935783e 100644
--- a/src/Reflection/AST/Traversal.agda
+++ b/src/Reflection/AST/Traversal.agda
@@ -11,12 +11,12 @@ open import Effect.Applicative using (RawApplicative)
 module Reflection.AST.Traversal
   {F : Set → Set} (AppF : RawApplicative F) where
 
-open import Data.Nat     using (ℕ; zero; suc; _+_)
-open import Data.List    using (List; []; _∷_; _++_; reverse; length)
-open import Data.Product using (_×_; _,_)
-open import Data.String  using (String)
-open import Function     using (_∘_)
-open import Reflection   hiding (pure)
+open import Data.Nat          using (ℕ; zero; suc; _+_)
+open import Data.List         using (List; []; _∷_; _++_; reverse; length)
+open import Data.Product      using (_×_; _,_)
+open import Data.String       using (String)
+open import Function.Base     using (_∘_)
+open import Reflection        hiding (pure)
 
 open RawApplicative AppF
 
diff --git a/src/Reflection/External.agda b/src/Reflection/External.agda
index 3206ee2145..ba4e076426 100644
--- a/src/Reflection/External.agda
+++ b/src/Reflection/External.agda
@@ -16,7 +16,7 @@ open import Data.Product using (_×_; _,_)
 open import Data.String.Base as String using (String; _++_)
 open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Unit.Base using (⊤; tt)
-open import Function using (case_of_; _$_; _∘_)
+open import Function.Base using (case_of_; _$_; _∘_)
 open import Reflection hiding (name)
 
 -- Type aliases for the various strings.
diff --git a/src/Relation/Binary/OrderMorphism.agda b/src/Relation/Binary/OrderMorphism.agda
index 9e2cb84ba3..fbf3781c78 100644
--- a/src/Relation/Binary/OrderMorphism.agda
+++ b/src/Relation/Binary/OrderMorphism.agda
@@ -16,7 +16,7 @@ Use Relation.Binary.Reasoning.Morphism instead."
 
 open import Relation.Binary
 open Poset
-import Function as F
+import Function.Base as F
 open import Level
 
 record _⇒-Poset_ {p₁ p₂ p₃ p₄ p₅ p₆}
diff --git a/src/Relation/Unary/Algebra.agda b/src/Relation/Unary/Algebra.agda
index f302cf7272..2d121be0fa 100644
--- a/src/Relation/Unary/Algebra.agda
+++ b/src/Relation/Unary/Algebra.agda
@@ -17,7 +17,7 @@ open import Data.Empty.Polymorphic using (⊥-elim)
 open import Data.Product as Product using (_,_; proj₁; proj₂; <_,_>; curry; uncurry)
 open import Data.Sum as Sum using (inj₁; inj₂; [_,_])
 open import Data.Unit.Polymorphic using (tt)
-open import Function using (id; const; _∘_)
+open import Function.Base using (id; const; _∘_)
 open import Level
 open import Relation.Unary hiding (∅; U)
 open import Relation.Unary.Polymorphic using (∅; U)
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 5a545a4c77..25e21193be 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -24,7 +24,7 @@
 
 module Tactic.Cong where
 
-open import Function using (_$_)
+open import Function.Base using (_$_)
 
 open import Data.Bool.Base            using (true; false; if_then_else_; _∧_)
 open import Data.Char.Base   as Char  using (toℕ)
diff --git a/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda b/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda
index 84fe184f39..2817b21cf4 100644
--- a/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda
+++ b/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda
@@ -15,7 +15,7 @@ open import Algebra.Structures using (IsCommutativeSemiring)
 open import Algebra.Definitions
 open import Algebra.Bundles using (RawRing; CommutativeRing; CommutativeSemiring)
 import Algebra.Morphism as Morphism
-open import Function hiding (Morphism)
+open import Function.Base using (id)
 open import Level
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 
diff --git a/tests/data/appending/Main.agda b/tests/data/appending/Main.agda
index 93ece5eda0..b5fc0a48a6 100644
--- a/tests/data/appending/Main.agda
+++ b/tests/data/appending/Main.agda
@@ -7,7 +7,7 @@ open import Data.List using (replicate)
 open import Data.String using (toList; fromList)
 
 open import IO
-open import Function using (_$_)
+open import Function.Base using (_$_)
 
 open import TakeWhile
 
diff --git a/tests/data/colist/Main.agda b/tests/data/colist/Main.agda
index a50b5b85fe..d47585dd6e 100644
--- a/tests/data/colist/Main.agda
+++ b/tests/data/colist/Main.agda
@@ -11,7 +11,7 @@ import Data.Fin.Show as Fin
 open import Data.String.Base using (String; _++_; parens)
 open import Data.Unit.Polymorphic.Base using (⊤)
 open import Codata.Sized.Thunk
-open import Function using (_$_; _∘_; id)
+open import Function.Base using (_$_; _∘_; id)
 open import Relation.Nullary
 
 open import Codata.Musical.Notation

From 5b8c2889a758da0057474f8ea146f105ddec456a Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 29 Jun 2023 23:28:24 -0400
Subject: [PATCH 063/316] Ignore `_build` correctly (#2010)

---
 .gitignore | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.gitignore b/.gitignore
index 99f960e19f..1724ffd72e 100644
--- a/.gitignore
+++ b/.gitignore
@@ -13,7 +13,7 @@
 .*.swp
 .#*
 \#*
-./_build/*
+_build
 .DS_Store
 .vscode/*
 dist

From 9c05244c8d3042f36f5c1829634881a8f20959d2 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 29 Jun 2023 23:29:04 -0400
Subject: [PATCH 064/316] `Data.Product` imports to `Data.Product.Base` (#2003)

---
 README/Data/Container/FreeMonad.agda                          | 2 +-
 src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda  | 2 +-
 src/Algebra/Construct/Add/Identity.agda                       | 2 +-
 src/Algebra/Construct/Flip/Op.agda                            | 2 +-
 src/Algebra/Construct/LexProduct/Inner.agda                   | 2 +-
 src/Algebra/Construct/LiftedChoice.agda                       | 2 +-
 src/Algebra/Construct/NaturalChoice/Min.agda                  | 2 +-
 src/Algebra/Construct/Subst/Equality.agda                     | 2 +-
 src/Algebra/Lattice/Morphism/Construct/Identity.agda          | 2 +-
 src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda         | 2 +-
 src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda | 2 +-
 src/Algebra/Lattice/Structures/Biased.agda                    | 2 +-
 src/Algebra/Module/Morphism/Construct/Identity.agda           | 2 +-
 src/Algebra/Module/Structures.agda                            | 2 +-
 src/Algebra/Morphism/Consequences.agda                        | 2 +-
 src/Algebra/Morphism/Construct/Identity.agda                  | 2 +-
 src/Algebra/Morphism/MonoidMonomorphism.agda                  | 2 +-
 src/Algebra/Ordered/Structures.agda                           | 2 +-
 src/Algebra/Properties/BooleanAlgebra.agda                    | 2 +-
 src/Algebra/Properties/CommutativeMagma/Divisibility.agda     | 2 +-
 src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda | 2 +-
 src/Algebra/Properties/Lattice.agda                           | 2 +-
 src/Algebra/Properties/Monoid/Divisibility.agda               | 2 +-
 src/Algebra/Properties/Semigroup/Divisibility.agda            | 2 +-
 src/Algebra/Properties/Semiring/Divisibility.agda             | 2 +-
 src/Algebra/Solver/CommutativeMonoid.agda                     | 2 +-
 src/Algebra/Solver/IdempotentCommutativeMonoid.agda           | 2 +-
 src/Algebra/Solver/Ring/NaturalCoefficients.agda              | 2 +-
 28 files changed, 28 insertions(+), 28 deletions(-)

diff --git a/README/Data/Container/FreeMonad.agda b/README/Data/Container/FreeMonad.agda
index 7c4460609e..54f0935937 100644
--- a/README/Data/Container/FreeMonad.agda
+++ b/README/Data/Container/FreeMonad.agda
@@ -16,7 +16,7 @@ open import Data.Unit
 open import Data.Bool.Base using (Bool; true)
 open import Data.Nat
 open import Data.Sum using (inj₁; inj₂)
-open import Data.Product renaming (_×_ to _⟨×⟩_)
+open import Data.Product.Base renaming (_×_ to _⟨×⟩_)
 open import Data.Container using (Container; _▷_)
 open import Data.Container.Combinator
 open import Data.Container.FreeMonad as FreeMonad
diff --git a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
index 7ac5b3e58a..1f04cc2dd0 100644
--- a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
+++ b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
@@ -11,7 +11,7 @@ open import Algebra.Apartness.Bundles using (HeytingCommutativeRing)
 module Algebra.Apartness.Properties.HeytingCommutativeRing
   {c ℓ₁ ℓ₂} (HCR : HeytingCommutativeRing c ℓ₁ ℓ₂) where
 
-open import Data.Product using (_,_; proj₂)
+open import Data.Product.Base using (_,_; proj₂)
 open import Algebra using (CommutativeRing; RightIdentity)
 
 open HeytingCommutativeRing HCR
diff --git a/src/Algebra/Construct/Add/Identity.agda b/src/Algebra/Construct/Add/Identity.agda
index 750c69523e..dc82a1c67b 100644
--- a/src/Algebra/Construct/Add/Identity.agda
+++ b/src/Algebra/Construct/Add/Identity.agda
@@ -14,7 +14,7 @@ open import Algebra.Core using (Op₂)
 open import Algebra.Definitions
 open import Algebra.Structures
 open import Relation.Binary.Construct.Add.Point.Equality renaming (_≈∙_ to lift≈)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core
 open import Relation.Binary.Definitions
diff --git a/src/Algebra/Construct/Flip/Op.agda b/src/Algebra/Construct/Flip/Op.agda
index 74ae5730bc..cc84ad59c9 100644
--- a/src/Algebra/Construct/Flip/Op.agda
+++ b/src/Algebra/Construct/Flip/Op.agda
@@ -10,7 +10,7 @@
 module Algebra.Construct.Flip.Op where
 
 open import Algebra
-import Data.Product as Prod
+import Data.Product.Base as Prod
 import Data.Sum as Sum
 open import Function.Base using (flip)
 open import Level using (Level)
diff --git a/src/Algebra/Construct/LexProduct/Inner.agda b/src/Algebra/Construct/LexProduct/Inner.agda
index 718bde2610..d235c1b8fd 100644
--- a/src/Algebra/Construct/LexProduct/Inner.agda
+++ b/src/Algebra/Construct/LexProduct/Inner.agda
@@ -8,7 +8,7 @@
 
 open import Algebra
 open import Data.Bool.Base using (false; true)
-open import Data.Product using (_×_; _,_; swap; map; uncurry′)
+open import Data.Product.Base using (_×_; _,_; swap; map; uncurry′)
 open import Function.Base using (_∘_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Definitions using (Decidable)
diff --git a/src/Algebra/Construct/LiftedChoice.agda b/src/Algebra/Construct/LiftedChoice.agda
index 357d2e725a..83cf2110c3 100644
--- a/src/Algebra/Construct/LiftedChoice.agda
+++ b/src/Algebra/Construct/LiftedChoice.agda
@@ -14,7 +14,7 @@ open import Algebra.Consequences.Base
 open import Relation.Binary
 open import Relation.Nullary using (¬_; yes; no)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
 open import Relation.Unary using (Pred)
diff --git a/src/Algebra/Construct/NaturalChoice/Min.agda b/src/Algebra/Construct/NaturalChoice/Min.agda
index 89bd2eefb8..100dac2193 100644
--- a/src/Algebra/Construct/NaturalChoice/Min.agda
+++ b/src/Algebra/Construct/NaturalChoice/Min.agda
@@ -10,7 +10,7 @@ open import Algebra.Core
 open import Algebra.Bundles
 open import Algebra.Construct.NaturalChoice.Base
 open import Data.Sum using (inj₁; inj₂; [_,_])
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 open import Relation.Binary
 import Algebra.Construct.NaturalChoice.MinOp as MinOp
diff --git a/src/Algebra/Construct/Subst/Equality.agda b/src/Algebra/Construct/Subst/Equality.agda
index 697a29cd73..636b1d8004 100644
--- a/src/Algebra/Construct/Subst/Equality.agda
+++ b/src/Algebra/Construct/Subst/Equality.agda
@@ -9,7 +9,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
 open import Relation.Binary.Core
 
 module Algebra.Construct.Subst.Equality
diff --git a/src/Algebra/Lattice/Morphism/Construct/Identity.agda b/src/Algebra/Lattice/Morphism/Construct/Identity.agda
index 9b5993696f..8531643540 100644
--- a/src/Algebra/Lattice/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Lattice/Morphism/Construct/Identity.agda
@@ -11,7 +11,7 @@ module Algebra.Lattice.Morphism.Construct.Identity where
 open import Algebra.Lattice.Bundles
 open import Algebra.Lattice.Morphism.Structures
   using ( module LatticeMorphisms )
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 open import Level using (Level)
 open import Relation.Binary.Morphism.Construct.Identity using (isRelHomomorphism)
diff --git a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
index bfed8d01ec..c2c8301409 100644
--- a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
+++ b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
@@ -15,7 +15,7 @@ open import Algebra.Lattice.Morphism.Structures
 import Algebra.Consequences.Setoid as Consequences
 import Algebra.Morphism.MagmaMonomorphism as MagmaMonomorphisms
 import Algebra.Lattice.Properties.Lattice as LatticeProperties
-open import Data.Product using (_,_; map)
+open import Data.Product.Base using (_,_; map)
 open import Relation.Binary
 import Relation.Binary.Morphism.RelMonomorphism as RelMonomorphisms
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
index 587a5dcc2c..9cdad57edc 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
@@ -18,7 +18,7 @@ open import Effect.Applicative as Applicative
 open import Effect.Monad
 open import Data.Fin.Base using (Fin)
 open import Data.Nat.Base
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec)
 import Data.Vec.Effectful as VecCat
 import Function.Identity.Effectful as IdCat
diff --git a/src/Algebra/Lattice/Structures/Biased.agda b/src/Algebra/Lattice/Structures/Biased.agda
index 66a93df06d..fefe3bef67 100644
--- a/src/Algebra/Lattice/Structures/Biased.agda
+++ b/src/Algebra/Lattice/Structures/Biased.agda
@@ -14,7 +14,7 @@
 
 open import Algebra.Core
 open import Algebra.Consequences.Setoid
-open import Data.Product using (proj₁; proj₂)
+open import Data.Product.Base using (proj₁; proj₂)
 open import Level using (_⊔_)
 open import Relation.Binary using (Rel; Setoid; IsEquivalence)
 
diff --git a/src/Algebra/Module/Morphism/Construct/Identity.agda b/src/Algebra/Module/Morphism/Construct/Identity.agda
index f5e5d73b05..90f6940a16 100644
--- a/src/Algebra/Module/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Module/Morphism/Construct/Identity.agda
@@ -21,7 +21,7 @@ open import Algebra.Module.Morphism.Structures
         ; module ModuleMorphisms
         )
 open import Algebra.Morphism.Construct.Identity
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 open import Level using (Level)
 
diff --git a/src/Algebra/Module/Structures.agda b/src/Algebra/Module/Structures.agda
index 20269f13a6..4a0e40c17c 100644
--- a/src/Algebra/Module/Structures.agda
+++ b/src/Algebra/Module/Structures.agda
@@ -16,7 +16,7 @@ open import Algebra.Module.Core
 import Algebra.Definitions as Defs
 open import Algebra.Module.Definitions
 open import Algebra.Structures
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Level using (Level; _⊔_)
 
 private
diff --git a/src/Algebra/Morphism/Consequences.agda b/src/Algebra/Morphism/Consequences.agda
index d5bc6a8076..ac7d6de2c5 100644
--- a/src/Algebra/Morphism/Consequences.agda
+++ b/src/Algebra/Morphism/Consequences.agda
@@ -10,7 +10,7 @@ module Algebra.Morphism.Consequences where
 
 open import Algebra using (Magma)
 open import Algebra.Morphism.Definitions
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id; _∘_)
 open import Function.Definitions
 import Relation.Binary.Reasoning.Setoid as EqR
diff --git a/src/Algebra/Morphism/Construct/Identity.agda b/src/Algebra/Morphism/Construct/Identity.agda
index 5ea2571de0..5529ec3c01 100644
--- a/src/Algebra/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Morphism/Construct/Identity.agda
@@ -20,7 +20,7 @@ open import Algebra.Morphism.Structures
         ; module QuasigroupMorphisms
         ; module LoopMorphisms
         )
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 open import Level using (Level)
 open import Relation.Binary.Morphism.Construct.Identity using (isRelHomomorphism)
diff --git a/src/Algebra/Morphism/MonoidMonomorphism.agda b/src/Algebra/Morphism/MonoidMonomorphism.agda
index 5f2d85d86a..4a512b2687 100644
--- a/src/Algebra/Morphism/MonoidMonomorphism.agda
+++ b/src/Algebra/Morphism/MonoidMonomorphism.agda
@@ -24,7 +24,7 @@ open RawMonoid M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; 
 
 open import Algebra.Definitions
 open import Algebra.Structures
-open import Data.Product using (map)
+open import Data.Product.Base using (map)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Ordered/Structures.agda b/src/Algebra/Ordered/Structures.agda
index ab211cc497..638d8783f0 100644
--- a/src/Algebra/Ordered/Structures.agda
+++ b/src/Algebra/Ordered/Structures.agda
@@ -21,7 +21,7 @@ module Algebra.Ordered.Structures
 open import Algebra.Core
 open import Algebra.Definitions _≈_
 open import Algebra.Structures _≈_
-open import Data.Product using (proj₁; proj₂)
+open import Data.Product.Base using (proj₁; proj₂)
 open import Function.Base using (flip)
 open import Level using (_⊔_)
 open import Relation.Binary.Definitions using (Transitive; Monotonic₁; Monotonic₂)
diff --git a/src/Algebra/Properties/BooleanAlgebra.agda b/src/Algebra/Properties/BooleanAlgebra.agda
index f53815b934..2e1f148701 100644
--- a/src/Algebra/Properties/BooleanAlgebra.agda
+++ b/src/Algebra/Properties/BooleanAlgebra.agda
@@ -30,7 +30,7 @@ open import Algebra.Structures _≈_
 open import Relation.Binary
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence using (_⇔_; module Equivalence)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 
 ------------------------------------------------------------------------
 -- DEPRECATED NAMES
diff --git a/src/Algebra/Properties/CommutativeMagma/Divisibility.agda b/src/Algebra/Properties/CommutativeMagma/Divisibility.agda
index 14ad404a37..27f6ba56a5 100644
--- a/src/Algebra/Properties/CommutativeMagma/Divisibility.agda
+++ b/src/Algebra/Properties/CommutativeMagma/Divisibility.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra using (CommutativeMagma)
-open import Data.Product using (_×_; _,_; map)
+open import Data.Product.Base using (_×_; _,_; map)
 
 module Algebra.Properties.CommutativeMagma.Divisibility
   {a ℓ} (CM : CommutativeMagma a ℓ)
diff --git a/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda b/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
index c17510e530..66bec51115 100644
--- a/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
+++ b/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra using (CommutativeSemigroup)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 import Relation.Binary.Reasoning.Setoid as EqReasoning
 
 module Algebra.Properties.CommutativeSemigroup.Divisibility
diff --git a/src/Algebra/Properties/Lattice.agda b/src/Algebra/Properties/Lattice.agda
index 3e9e078f3f..ed5bfd00dc 100644
--- a/src/Algebra/Properties/Lattice.agda
+++ b/src/Algebra/Properties/Lattice.agda
@@ -11,7 +11,7 @@ open import Algebra.Lattice.Bundles
 open import Relation.Binary
 open import Function.Base
 open import Function.Bundles using (module Equivalence; _⇔_)
-open import Data.Product using (_,_; swap)
+open import Data.Product.Base using (_,_; swap)
 
 module Algebra.Properties.Lattice {l₁ l₂} (L : Lattice l₁ l₂) where
 
diff --git a/src/Algebra/Properties/Monoid/Divisibility.agda b/src/Algebra/Properties/Monoid/Divisibility.agda
index 5653312fd8..f2c8c1f20f 100644
--- a/src/Algebra/Properties/Monoid/Divisibility.agda
+++ b/src/Algebra/Properties/Monoid/Divisibility.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra using (Monoid)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Relation.Binary
 
 module Algebra.Properties.Monoid.Divisibility
diff --git a/src/Algebra/Properties/Semigroup/Divisibility.agda b/src/Algebra/Properties/Semigroup/Divisibility.agda
index d7a6154a32..76eaaa6edf 100644
--- a/src/Algebra/Properties/Semigroup/Divisibility.agda
+++ b/src/Algebra/Properties/Semigroup/Divisibility.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra using (Semigroup)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Relation.Binary.Definitions using (Transitive)
 
 module Algebra.Properties.Semigroup.Divisibility
diff --git a/src/Algebra/Properties/Semiring/Divisibility.agda b/src/Algebra/Properties/Semiring/Divisibility.agda
index a1b2ff2e9b..54a352d782 100644
--- a/src/Algebra/Properties/Semiring/Divisibility.agda
+++ b/src/Algebra/Properties/Semiring/Divisibility.agda
@@ -8,7 +8,7 @@
 
 open import Algebra using (Semiring)
 import Algebra.Properties.Monoid.Divisibility as MonoidDivisibility
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 
 module Algebra.Properties.Semiring.Divisibility
diff --git a/src/Algebra/Solver/CommutativeMonoid.agda b/src/Algebra/Solver/CommutativeMonoid.agda
index 69d1a67b0b..623beaeebd 100644
--- a/src/Algebra/Solver/CommutativeMonoid.agda
+++ b/src/Algebra/Solver/CommutativeMonoid.agda
@@ -17,7 +17,7 @@ open import Data.Maybe.Base as Maybe
   using (Maybe; decToMaybe; From-just; from-just)
 open import Data.Nat as ℕ using (ℕ; zero; suc; _+_)
 open import Data.Nat.GeneralisedArithmetic using (fold)
-open import Data.Product using (_×_; uncurry)
+open import Data.Product.Base using (_×_; uncurry)
 open import Data.Vec.Base using (Vec; []; _∷_; lookup; replicate)
 
 open import Function.Base using (_∘_)
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
index a0704ba95b..fdba81a838 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
@@ -15,7 +15,7 @@ open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Maybe.Base as Maybe
   using (Maybe; decToMaybe; From-just; from-just)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_)
-open import Data.Product using (_×_; uncurry)
+open import Data.Product.Base using (_×_; uncurry)
 open import Data.Vec.Base using (Vec; []; _∷_; lookup; replicate)
 
 open import Function.Base using (_∘_)
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients.agda b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
index 495e8cfa66..95af2b0a33 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
@@ -12,7 +12,7 @@ import Algebra.Properties.Semiring.Mult as SemiringMultiplication
 open import Data.Maybe.Base using (Maybe; just; nothing; map)
 open import Algebra.Solver.Ring.AlmostCommutativeRing
 open import Data.Nat.Base as ℕ
-open import Data.Product using (module Σ)
+open import Data.Product.Base using (module Σ)
 open import Function.Base using (id)
 open import Relation.Binary.PropositionalEquality using (_≡_)
 

From c7f6db7fafb1179514be09c64f89e75863943757 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 29 Jun 2023 23:29:50 -0400
Subject: [PATCH 065/316] Simplify `Data.List` imports to `Data.List.Base`
 (#2007)

---
 README/Data/List/Relation/Binary/Pointwise.agda  |  2 +-
 README/Data/List/Relation/Binary/Subset.agda     |  2 +-
 README/Data/Tree/AVL.agda                        |  2 +-
 README/Data/Trie/NonDependent.agda               |  2 +-
 README/Tactic/RingSolver.agda                    |  2 +-
 README/Text/Regex.agda                           |  2 +-
 src/Data/Fin/Permutation/Transposition/List.agda |  2 +-
 src/Data/List/Countdown.agda                     |  2 +-
 .../Binary/Sublist/Heterogeneous/Solver.agda     |  2 +-
 .../List/Relation/Unary/AllPairs/Properties.agda |  2 +-
 src/Data/List/Relation/Unary/Grouped.agda        |  2 +-
 src/Data/Tree/Binary/Zipper.agda                 |  2 +-
 src/Data/Vec/Functional/Properties.agda          |  2 +-
 src/Data/Vec/Reflection.agda                     |  2 +-
 src/Reflection/AST/Show.agda                     |  2 +-
 src/Reflection/AST/Traversal.agda                |  2 +-
 src/Tactic/MonoidSolver.agda                     | 10 +++++-----
 src/Tactic/RingSolver/Core/NatSet.agda           |  6 +++---
 .../Core/Polynomial/Homomorphism/Addition.agda   |  2 +-
 .../Core/Polynomial/Homomorphism/Lemmas.agda     |  2 +-
 .../RingSolver/Core/Polynomial/Semantics.agda    | 10 +++++-----
 tests/data/appending/Main.agda                   |  2 +-
 tests/data/appending/TakeWhile.agda              |  2 +-
 tests/data/list/Main.agda                        |  2 +-
 tests/data/trie/Main.agda                        | 16 ++++++++--------
 25 files changed, 42 insertions(+), 42 deletions(-)

diff --git a/README/Data/List/Relation/Binary/Pointwise.agda b/README/Data/List/Relation/Binary/Pointwise.agda
index 4932413ad4..da804f12c9 100644
--- a/README/Data/List/Relation/Binary/Pointwise.agda
+++ b/README/Data/List/Relation/Binary/Pointwise.agda
@@ -7,7 +7,7 @@
 module README.Data.List.Relation.Binary.Pointwise where
 
 open import Data.Nat using (ℕ; _<_; s≤s; z≤n)
-open import Data.List using (List; []; _∷_; length)
+open import Data.List.Base using (List; []; _∷_; length)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong; setoid)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/README/Data/List/Relation/Binary/Subset.agda b/README/Data/List/Relation/Binary/Subset.agda
index 44ba3fa053..f6ef7690a1 100644
--- a/README/Data/List/Relation/Binary/Subset.agda
+++ b/README/Data/List/Relation/Binary/Subset.agda
@@ -4,7 +4,7 @@
 -- Documentation for subset relations over `List`s
 ------------------------------------------------------------------------
 
-open import Data.List using (List; _∷_; [])
+open import Data.List.Base using (List; _∷_; [])
 open import Data.List.Membership.Propositional.Properties
   using (∈-++⁺ˡ; ∈-insert)
 open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_)
diff --git a/README/Data/Tree/AVL.agda b/README/Data/Tree/AVL.agda
index 1d14058ff8..7994afed41 100644
--- a/README/Data/Tree/AVL.agda
+++ b/README/Data/Tree/AVL.agda
@@ -60,7 +60,7 @@ t₃ = delete 2 t₂
 
 -- Conversion of a list of key-value mappings to a tree.
 
-open import Data.List using (_∷_; [])
+open import Data.List.Base using (_∷_; [])
 
 t₄ : Tree
 t₄ = fromList ((2 , v₂) ∷ (1 , v₁) ∷ [])
diff --git a/README/Data/Trie/NonDependent.agda b/README/Data/Trie/NonDependent.agda
index 09825c30c3..2387a71242 100644
--- a/README/Data/Trie/NonDependent.agda
+++ b/README/Data/Trie/NonDependent.agda
@@ -53,7 +53,7 @@ open import Data.Unit
 open import Data.Bool
 open import Data.Char        as Char
 import Data.Char.Properties  as Char
-open import Data.List        as List using (List; []; _∷_)
+open import Data.List.Base   as List using (List; []; _∷_)
 open import Data.List.Fresh  as List# using (List#; []; _∷#_)
 open import Data.Maybe       as Maybe
 open import Data.Product     as Prod
diff --git a/README/Tactic/RingSolver.agda b/README/Tactic/RingSolver.agda
index cc1db1ad21..fdbea87bbf 100644
--- a/README/Tactic/RingSolver.agda
+++ b/README/Tactic/RingSolver.agda
@@ -28,7 +28,7 @@ instance
 ------------------------------------------------------------------------------
 -- Imports!
 
-open import Data.List as List using (List; _∷_; [])
+open import Data.List.Base as List using (List; _∷_; [])
 open import Function
 open import Relation.Binary.PropositionalEquality
   using (subst; cong; _≡_; module ≡-Reasoning)
diff --git a/README/Text/Regex.agda b/README/Text/Regex.agda
index 5b4ce805c0..17d3fe45f9 100644
--- a/README/Text/Regex.agda
+++ b/README/Text/Regex.agda
@@ -9,7 +9,7 @@
 module README.Text.Regex where
 
 open import Data.Bool using (true; false)
-open import Data.List using (_∷_; [])
+open import Data.List.Base using (_∷_; [])
 open import Data.String
 open import Function.Base using () renaming (_$′_ to _$_)
 open import Relation.Nullary.Decidable using (yes)
diff --git a/src/Data/Fin/Permutation/Transposition/List.agda b/src/Data/Fin/Permutation/Transposition/List.agda
index 4ed0583372..07feee0020 100644
--- a/src/Data/Fin/Permutation/Transposition/List.agda
+++ b/src/Data/Fin/Permutation/Transposition/List.agda
@@ -12,7 +12,7 @@ open import Data.Fin.Base
 open import Data.Fin.Patterns using (0F)
 open import Data.Fin.Permutation as P hiding (lift₀)
 import Data.Fin.Permutation.Components as PC
-open import Data.List using (List; []; _∷_; map)
+open import Data.List.Base using (List; []; _∷_; map)
 open import Data.Nat.Base using (ℕ; suc; zero)
 open import Data.Product using (_×_; _,_)
 open import Function.Base using (_∘_)
diff --git a/src/Data/List/Countdown.agda b/src/Data/List/Countdown.agda
index 1ae0efc9b8..fde7439ebf 100644
--- a/src/Data/List/Countdown.agda
+++ b/src/Data/List/Countdown.agda
@@ -20,7 +20,7 @@ open import Function.Base
 open import Function.Bundles
   using (Injection; module Injection)
 open import Data.Bool.Base using (true; false)
-open import Data.List hiding (lookup)
+open import Data.List.Base hiding (lookup)
 open import Data.List.Relation.Unary.Any as Any using (here; there)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.Product
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index 9066338e8a..2fe2447efb 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -24,7 +24,7 @@ open import Data.Maybe.Base as M
 open import Data.Nat.Base as Nat using (ℕ)
 open import Data.Product
 open import Data.Vec.Base as Vec using (Vec ; lookup)
-open import Data.List hiding (lookup)
+open import Data.List.Base hiding (lookup)
 open import Data.List.Properties
 open import Data.List.Relation.Binary.Sublist.Heterogeneous
   hiding (lookup)
diff --git a/src/Data/List/Relation/Unary/AllPairs/Properties.agda b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
index 391fee8f8b..009ac3ce75 100644
--- a/src/Data/List/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.List.Relation.Unary.AllPairs.Properties where
 
-open import Data.List hiding (any)
+open import Data.List.Base hiding (any)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
diff --git a/src/Data/List/Relation/Unary/Grouped.agda b/src/Data/List/Relation/Unary/Grouped.agda
index eceb6a80d0..51f3f28a17 100644
--- a/src/Data/List/Relation/Unary/Grouped.agda
+++ b/src/Data/List/Relation/Unary/Grouped.agda
@@ -8,7 +8,7 @@
 
 module Data.List.Relation.Unary.Grouped where
 
-open import Data.List using (List; []; _∷_; map)
+open import Data.List.Base using (List; []; _∷_; map)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_; all?)
 open import Data.Sum using (_⊎_; inj₁; inj₂)
 open import Data.Product using (_×_; _,_)
diff --git a/src/Data/Tree/Binary/Zipper.agda b/src/Data/Tree/Binary/Zipper.agda
index 76b2d9fd8e..0e5f4b8d05 100644
--- a/src/Data/Tree/Binary/Zipper.agda
+++ b/src/Data/Tree/Binary/Zipper.agda
@@ -10,7 +10,7 @@ module Data.Tree.Binary.Zipper where
 
 open import Level using (Level; _⊔_)
 open import Data.Tree.Binary as BT using (Tree; node; leaf)
-open import Data.List as List using (List; []; _∷_; sum; _++_; [_])
+open import Data.List.Base as List using (List; []; _∷_; sum; _++_; [_])
 open import Data.Maybe using (Maybe; nothing; just)
 open import Data.Nat.Base using (ℕ; suc; _+_)
 open import Function
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index 0d2d08da83..9c4486be47 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Fin.Base
 open import Data.Nat as ℕ
 import Data.Nat.Properties as ℕₚ
 open import Data.Product as Product using (_×_; _,_; proj₁; proj₂)
-open import Data.List as L using (List)
+open import Data.List.Base as L using (List)
 import Data.List.Properties as Lₚ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.Vec as V using (Vec)
diff --git a/src/Data/Vec/Reflection.agda b/src/Data/Vec/Reflection.agda
index 8b0fa7799f..0ea6473ff2 100644
--- a/src/Data/Vec/Reflection.agda
+++ b/src/Data/Vec/Reflection.agda
@@ -8,7 +8,7 @@
 
 module Data.Vec.Reflection where
 
-import Data.List as List
+import Data.List.Base as List
 open import Data.Vec.Base
 open import Reflection.AST.Term
 open import Reflection.AST.Argument
diff --git a/src/Reflection/AST/Show.agda b/src/Reflection/AST/Show.agda
index ffe97352f4..f0291cf447 100644
--- a/src/Reflection/AST/Show.agda
+++ b/src/Reflection/AST/Show.agda
@@ -13,7 +13,7 @@ module Reflection.AST.Show where
 
 import Data.Char as Char   using (show)
 import Data.Float as Float using (show)
-open import Data.List hiding (_++_; intersperse)
+open import Data.List.Base hiding (_++_; intersperse)
 import Data.Nat.Show as ℕ using (show)
 open import Data.Product   using (_×_; _,_)
 open import Data.String as String
diff --git a/src/Reflection/AST/Traversal.agda b/src/Reflection/AST/Traversal.agda
index 336935783e..c311567c7f 100644
--- a/src/Reflection/AST/Traversal.agda
+++ b/src/Reflection/AST/Traversal.agda
@@ -12,7 +12,7 @@ module Reflection.AST.Traversal
   {F : Set → Set} (AppF : RawApplicative F) where
 
 open import Data.Nat          using (ℕ; zero; suc; _+_)
-open import Data.List         using (List; []; _∷_; _++_; reverse; length)
+open import Data.List.Base    using (List; []; _∷_; _++_; reverse; length)
 open import Data.Product      using (_×_; _,_)
 open import Data.String       using (String)
 open import Function.Base     using (_∘_)
diff --git a/src/Tactic/MonoidSolver.agda b/src/Tactic/MonoidSolver.agda
index e34d770cde..fcf193244b 100644
--- a/src/Tactic/MonoidSolver.agda
+++ b/src/Tactic/MonoidSolver.agda
@@ -75,11 +75,11 @@ module Tactic.MonoidSolver where
 open import Algebra
 open import Function
 
-open import Data.Bool    as Bool    using (Bool; _∨_; if_then_else_)
-open import Data.Maybe   as Maybe   using (Maybe; just; nothing; maybe)
-open import Data.List    as List    using (List; _∷_; [])
-open import Data.Nat     as ℕ       using (ℕ; suc; zero)
-open import Data.Product as Product using (_×_; _,_)
+open import Data.Bool         as Bool    using (Bool; _∨_; if_then_else_)
+open import Data.Maybe        as Maybe   using (Maybe; just; nothing; maybe)
+open import Data.List.Base    as List    using (List; _∷_; [])
+open import Data.Nat          as ℕ       using (ℕ; suc; zero)
+open import Data.Product      as Product using (_×_; _,_)
 
 open import Reflection.AST
 open import Reflection.AST.Term
diff --git a/src/Tactic/RingSolver/Core/NatSet.agda b/src/Tactic/RingSolver/Core/NatSet.agda
index 485539792e..7ebbcebc22 100644
--- a/src/Tactic/RingSolver/Core/NatSet.agda
+++ b/src/Tactic/RingSolver/Core/NatSet.agda
@@ -36,10 +36,10 @@
 
 module Tactic.RingSolver.Core.NatSet where
 
-open import Data.Nat   as ℕ     using (ℕ; suc; zero)
-open import Data.List  as List  using (List; _∷_; [])
+open import Data.Nat        as ℕ     using (ℕ; suc; zero)
+open import Data.List.Base  as List  using (List; _∷_; [])
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
-open import Data.Bool  as Bool  using (Bool)
+open import Data.Bool       as Bool  using (Bool)
 open import Function
 open import Relation.Binary.PropositionalEquality
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
index 990d6a46ed..d04ecbc131 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
@@ -16,7 +16,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Addition
 open import Data.Nat            as ℕ  using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
 open import Data.Nat.Properties as ℕₚ using (≤′-trans)
 open import Data.Product              using (_,_; _×_; proj₂)
-open import Data.List                 using (_∷_; [])
+open import Data.List.Base            using (_∷_; [])
 open import Data.List.Kleene
 open import Data.Vec                  using (Vec)
 open import Function
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index f0c5dfcd76..ce20b1e381 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -18,7 +18,7 @@ open import Data.Nat.Base as ℕ                         using (ℕ; suc; zero;
 open import Data.Nat.Properties as ℕₚ                  using (≤′-trans)
 open import Data.Vec.Base as Vec                       using (Vec; _∷_)
 open import Data.Fin                                   using (Fin; zero; suc)
-open import Data.List                                  using (_∷_; [])
+open import Data.List.Base                             using (_∷_; [])
 open import Data.Unit using (tt)
 open import Data.List.Kleene
 open import Data.Product                               using (_,_; proj₁; proj₂; map₁; _×_)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
index 648563112d..8cde662620 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
@@ -13,11 +13,11 @@ module Tactic.RingSolver.Core.Polynomial.Semantics
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Nat     using (ℕ; suc; zero; _≤′_; ≤′-step; ≤′-refl)
-open import Data.Vec     using (Vec; []; _∷_; uncons)
-open import Data.List    using ([]; _∷_)
-open import Data.Product using (_,_; _×_)
-open import Data.List.Kleene using (_+; _*; ∹_; _&_; [])
+open import Data.Nat          using (ℕ; suc; zero; _≤′_; ≤′-step; ≤′-refl)
+open import Data.Vec          using (Vec; []; _∷_; uncons)
+open import Data.List.Base    using ([]; _∷_)
+open import Data.Product      using (_,_; _×_)
+open import Data.List.Kleene  using (_+; _*; ∹_; _&_; [])
 
 open Homomorphism homo hiding (_^_)
 open import Tactic.RingSolver.Core.Polynomial.Base from
diff --git a/tests/data/appending/Main.agda b/tests/data/appending/Main.agda
index b5fc0a48a6..2015b0bc9a 100644
--- a/tests/data/appending/Main.agda
+++ b/tests/data/appending/Main.agda
@@ -3,7 +3,7 @@
 module Main where
 
 
-open import Data.List using (replicate)
+open import Data.List.Base using (replicate)
 open import Data.String using (toList; fromList)
 
 open import IO
diff --git a/tests/data/appending/TakeWhile.agda b/tests/data/appending/TakeWhile.agda
index 9030fb3c0c..5a885c3fbe 100644
--- a/tests/data/appending/TakeWhile.agda
+++ b/tests/data/appending/TakeWhile.agda
@@ -3,7 +3,7 @@
 module TakeWhile where
 
 open import Level
-open import Data.List hiding (takeWhile)
+open import Data.List.Base hiding (takeWhile)
 open import Data.List.Relation.Unary.All as List using ([]; _∷_)
 open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_; refl)
 open import Data.List.Relation.Ternary.Appending.Propositional
diff --git a/tests/data/list/Main.agda b/tests/data/list/Main.agda
index f2b11de8ae..f7f12cd63d 100644
--- a/tests/data/list/Main.agda
+++ b/tests/data/list/Main.agda
@@ -3,7 +3,7 @@
 module Main where
 
 open import Level
-open import Data.List as List using (List; _∷_; []; _++_; reverse)
+open import Data.List.Base as List using (List; _∷_; []; _++_; reverse)
 open import Data.List.Zipper
 import Data.List.Sort as Sort
 open import Data.Maybe.Base
diff --git a/tests/data/trie/Main.agda b/tests/data/trie/Main.agda
index 4eab3cfd40..9dfac5cb26 100644
--- a/tests/data/trie/Main.agda
+++ b/tests/data/trie/Main.agda
@@ -7,14 +7,14 @@ module Main where
 open import Level
 open import Data.Unit
 open import Data.Bool
-open import Data.Char        as Char hiding (show)
-import Data.Char.Properties  as Char
-open import Data.List        as List using (List; []; _∷_)
-open import Data.List.Fresh  as List# using (List#; []; _∷#_)
-open import Data.Maybe       as Maybe
-open import Data.Product     as Prod
-open import Data.String      as String using (String; unlines; _++_)
-open import Data.These       as These
+open import Data.Char         as Char hiding (show)
+import Data.Char.Properties   as Char
+open import Data.List.Base    as List using (List; []; _∷_)
+open import Data.List.Fresh   as List# using (List#; []; _∷#_)
+open import Data.Maybe        as Maybe
+open import Data.Product      as Prod
+open import Data.String       as String using (String; unlines; _++_)
+open import Data.These        as These
 
 open import Function.Base using (case_of_; _$_; _∘′_; id; _on_)
 open import Relation.Nary

From 78a871c8b335eb78788444c03352c370175e4987 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 29 Jun 2023 23:30:28 -0400
Subject: [PATCH 066/316] Add more fixities (#2008)

---
 CHANGELOG.md                                         |  8 ++++++++
 src/Codata/Guarded/Stream/Relation/Unary/All.agda    |  2 ++
 src/Foreign/Haskell/List/NonEmpty.agda               |  2 ++
 src/Function/Metric/Rational/Bundles.agda            | 12 ++++++++++++
 src/Level/Literals.agda                              |  2 ++
 src/Relation/Binary/Construct/Union.agda             |  2 ++
 .../HeterogeneousEquality/Quotients/Examples.agda    |  2 ++
 src/Relation/Binary/Indexed/Homogeneous/Bundles.agda |  3 +++
 src/Text/Regex/Search.agda                           |  2 ++
 9 files changed, 35 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6d692fdb4a..5891740538 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -112,6 +112,14 @@ Bug-fixes
   infix   4 _≈∙_                                             (Relation.Binary.Construct.Add.Point.Equality)
   infix   4 _≤⁺_ _≤⊤⁺                                        (Relation.Binary.Construct.Add.Supremum.NonStrict)
   infixr  5 _∷ʳ_                                             (Relation.Binary.Construct.Closure.Transitive)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Unary.All)
+  infixr  5 _∷_                                              (Foreign.Haskell.List.NonEmpty)
+  infix   4 _≈_                                              (Function.Metric.Rational.Bundles)
+  infixl  6 _ℕ+_                                             (Level.Literals)
+  infixr  6 _∪_                                              (Relation.Binary.Construct.Union)
+  infixl  6 _+ℤ_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
+  infix   4 _≉_ _≈ᵢ_ _≤ᵢ_                                    (Relation.Binary.Indexed.Homogeneous.Bundles)
+  infixr  5 _∷ᴹ_ _∷⁻¹ᴹ_                                      (Text.Regex.Search)
   ```
 
 * In `System.Exit`, the `ExitFailure` constructor is now carrying an integer
diff --git a/src/Codata/Guarded/Stream/Relation/Unary/All.agda b/src/Codata/Guarded/Stream/Relation/Unary/All.agda
index c69ff9124b..ad87a7e589 100644
--- a/src/Codata/Guarded/Stream/Relation/Unary/All.agda
+++ b/src/Codata/Guarded/Stream/Relation/Unary/All.agda
@@ -20,6 +20,8 @@ private
     P Q R : Pred A p
     xs : Stream A
 
+infixr 5 _∷_
+
 record All (P : Pred A ℓ) (as : Stream A) : Set ℓ where
   coinductive
   constructor _∷_
diff --git a/src/Foreign/Haskell/List/NonEmpty.agda b/src/Foreign/Haskell/List/NonEmpty.agda
index a36a1fc415..93e01e561f 100644
--- a/src/Foreign/Haskell/List/NonEmpty.agda
+++ b/src/Foreign/Haskell/List/NonEmpty.agda
@@ -25,5 +25,7 @@ private
 data NonEmpty (A : Set a) : Set a where
   _∷_ : A → List A → NonEmpty A
 
+infixr 5 _∷_
+
 {-# FOREIGN GHC type AgdaNonEmpty l a = NE.NonEmpty a #-}
 {-# COMPILE GHC NonEmpty = data AgdaNonEmpty ((NE.:|)) #-}
diff --git a/src/Function/Metric/Rational/Bundles.agda b/src/Function/Metric/Rational/Bundles.agda
index 8bbc9f85d3..f578a8a8a2 100644
--- a/src/Function/Metric/Rational/Bundles.agda
+++ b/src/Function/Metric/Rational/Bundles.agda
@@ -34,6 +34,8 @@ record ProtoMetric a ℓ : Set (suc (a ⊔ ℓ)) where
     d             : DistanceFunction Carrier
     isProtoMetric : IsProtoMetric _≈_ d
 
+  infix 4 _≈_
+
   open IsProtoMetric isProtoMetric public
 
 ------------------------------------------------------------------------
@@ -46,6 +48,8 @@ record PreMetric a ℓ : Set (suc (a ⊔ ℓ)) where
     d           : DistanceFunction Carrier
     isPreMetric : IsPreMetric _≈_ d
 
+  infix 4 _≈_
+
   open IsPreMetric isPreMetric public
 
   protoMetric : ProtoMetric a ℓ
@@ -63,6 +67,8 @@ record QuasiSemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where
     d                 : DistanceFunction Carrier
     isQuasiSemiMetric : IsQuasiSemiMetric _≈_ d
 
+  infix 4 _≈_
+
   open IsQuasiSemiMetric isQuasiSemiMetric public
 
   preMetric : PreMetric a ℓ
@@ -83,6 +89,8 @@ record SemiMetric a ℓ : Set (suc (a ⊔ ℓ)) where
     d            : DistanceFunction Carrier
     isSemiMetric : IsSemiMetric _≈_ d
 
+  infix 4 _≈_
+
   open IsSemiMetric isSemiMetric public
 
   quasiSemiMetric : QuasiSemiMetric a ℓ
@@ -103,6 +111,8 @@ record Metric a ℓ : Set (suc (a ⊔ ℓ)) where
     d        : DistanceFunction Carrier
     isMetric : IsMetric _≈_ d
 
+  infix 4 _≈_
+
   open IsMetric isMetric public
 
   semiMetric : SemiMetric a ℓ
@@ -123,6 +133,8 @@ record UltraMetric a ℓ : Set (suc (a ⊔ ℓ)) where
     d             : DistanceFunction Carrier
     isUltraMetric : IsUltraMetric _≈_ d
 
+  infix 4 _≈_
+
   open IsUltraMetric isUltraMetric public
 
   semiMetric : SemiMetric a ℓ
diff --git a/src/Level/Literals.agda b/src/Level/Literals.agda
index 8e3eebb4e5..d1ac0f7ac4 100644
--- a/src/Level/Literals.agda
+++ b/src/Level/Literals.agda
@@ -15,6 +15,8 @@ open import Level using (Level; 0ℓ)
 
 -- Increase a Level by a number of sucs.
 
+infixl 6 _ℕ+_
+
 _ℕ+_ : ℕ → Level → Level
 zero  ℕ+ ℓ = ℓ
 suc n ℕ+ ℓ = Level.suc (n ℕ+ ℓ)
diff --git a/src/Relation/Binary/Construct/Union.agda b/src/Relation/Binary/Construct/Union.agda
index 3abd3ffe42..2b70ce8500 100644
--- a/src/Relation/Binary/Construct/Union.agda
+++ b/src/Relation/Binary/Construct/Union.agda
@@ -24,6 +24,8 @@ private
 ------------------------------------------------------------------------
 -- Definition
 
+infixr 6 _∪_
+
 _∪_ : REL A B ℓ₁ → REL A B ℓ₂ → REL A B (ℓ₁ ⊔ ℓ₂)
 L ∪ R = λ i j → L i j ⊎ R i j
 
diff --git a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
index c7bddf4a5a..fb1db6b9fb 100644
--- a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
+++ b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
@@ -82,6 +82,8 @@ module Integers (quot : Quotients 0ℓ 0ℓ) where
   module _ (ext : ∀ {a b} {A : Set a} {B₁ B₂ : A → Set b} {f₁ : ∀ a → B₁ a}
                   {f₂ : ∀ a → B₂ a} → (∀ a → f₁ a ≅ f₂ a) → f₁ ≅ f₂) where
 
+    infixl 6 _+ℤ_
+
     _+ℤ_ : ℤ → ℤ → ℤ
     _+ℤ_ = Properties₂.lift₂ ext Int Int (λ i j → abs (i +² j))
          $ λ {a} {b} {c} p p′ → compat-abs (+²-cong {a} {b} {c} p p′)
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
index 670949c138..fd83521783 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
@@ -39,6 +39,8 @@ record IndexedSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where
   Carrier : Set _
   Carrier = ∀ i → Carrierᵢ i
 
+  infix 4 _≉_
+
   _≈_ : B.Rel Carrier _
   _≈_ = Lift Carrierᵢ _≈ᵢ_
 
@@ -103,6 +105,7 @@ record IndexedPreorder {i} (I : Set i) c ℓ₁ ℓ₂ :
 
 record IndexedPoset {i} (I : Set i) c ℓ₁ ℓ₂ :
                     Set (suc (i ⊔ c ⊔ ℓ₁ ⊔ ℓ₂)) where
+  infix 4 _≈ᵢ_ _≤ᵢ_
   field
     Carrierᵢ        : I → Set c
     _≈ᵢ_            : IRel Carrierᵢ ℓ₁
diff --git a/src/Text/Regex/Search.agda b/src/Text/Regex/Search.agda
index af10c2784a..c802768e38 100644
--- a/src/Text/Regex/Search.agda
+++ b/src/Text/Regex/Search.agda
@@ -87,6 +87,8 @@ module Prefix where
   []⁻¹ᴹ : ∀ {e} → Match (Prefix _≡_) [] e → [] ∈ e
   []⁻¹ᴹ (mkMatch .[] p []) = p
 
+  infixr 5 _∷ᴹ_ _∷⁻¹ᴹ_
+
   _∷ᴹ_ : ∀ {xs e} x → Match (Prefix _≡_) xs (eat x e) → Match (Prefix _≡_) (x ∷ xs) e
   x ∷ᴹ (mkMatch ys ys∈e\x ys≤xs) = mkMatch (x ∷ ys) (eat-sound x _ ys∈e\x) (refl ∷ ys≤xs)
 

From 2fb47f528ca33050e5a9f4abf2b8a11afe655468 Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Thu, 6 Jul 2023 09:34:35 +0200
Subject: [PATCH 067/316] Eliminate many propositional equality imports (#2002)

* Eliminate many propositional equality imports

* Fix merge conflict in Data.Bool.Properties

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

---------

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
---
 src/Algebra/Construct/LiftedChoice.agda       |  2 +-
 src/Algebra/Operations/CommutativeMonoid.agda |  2 +-
 .../CommutativeMonoid/Mult/TCOptimised.agda   |  1 -
 .../Properties/CommutativeMonoid/Sum.agda     |  2 +-
 src/Algebra/Properties/Monoid/Mult.agda       |  2 +-
 .../Properties/Monoid/Mult/TCOptimised.agda   |  2 +-
 src/Algebra/Properties/Semiring/Exp.agda      |  2 +-
 src/Algebra/Solver/CommutativeMonoid.agda     |  2 +-
 .../Solver/CommutativeMonoid/Example.agda     |  2 +-
 src/Algebra/Solver/Monoid.agda                |  2 +-
 src/Algebra/Solver/Ring.agda                  |  2 +-
 .../Solver/Ring/NaturalCoefficients.agda      |  2 +-
 .../Ring/NaturalCoefficients/Default.agda     |  2 +-
 src/Codata/Guarded/Stream/Properties.agda     |  2 +-
 .../Stream/Relation/Binary/Pointwise.agda     |  3 ++-
 src/Codata/Musical/Cofin.agda                 |  2 +-
 src/Codata/Musical/Colist/Infinite-merge.agda |  2 +-
 src/Codata/Musical/Conat.agda                 |  2 +-
 src/Codata/Musical/Covec.agda                 |  2 +-
 src/Codata/Musical/Stream.agda                |  2 +-
 src/Codata/Sized/Colist.agda                  |  2 +-
 src/Codata/Sized/Colist/Bisimilarity.agda     |  3 ++-
 src/Codata/Sized/Colist/Properties.agda       |  2 +-
 src/Codata/Sized/Covec/Bisimilarity.agda      |  2 +-
 src/Codata/Sized/Covec/Properties.agda        |  2 +-
 src/Codata/Sized/Cowriter/Bisimilarity.agda   |  3 ++-
 src/Codata/Sized/Delay/Bisimilarity.agda      |  2 +-
 src/Codata/Sized/Delay/Properties.agda        |  2 +-
 src/Codata/Sized/M/Bisimilarity.agda          |  2 +-
 src/Codata/Sized/M/Properties.agda            |  3 ++-
 src/Codata/Sized/Stream/Bisimilarity.agda     |  3 ++-
 src/Codata/Sized/Stream/Properties.agda       |  2 +-
 src/Data/AVL/Indexed/WithK.agda               |  2 +-
 src/Data/AVL/IndexedMap.agda                  |  2 +-
 src/Data/Bool.agda                            |  3 +--
 src/Data/Bool/Properties.agda                 |  3 ++-
 src/Data/Char/Properties.agda                 |  3 ++-
 src/Data/Container/Indexed/WithK.agda         |  2 +-
 .../Container/Relation/Binary/Pointwise.agda  |  2 +-
 .../Relation/Binary/Pointwise/Properties.agda |  2 +-
 src/Data/Digit.agda                           |  2 +-
 src/Data/Digit/Properties.agda                |  2 +-
 src/Data/Fin/Substitution/Example.agda        |  2 +-
 src/Data/Fin/Substitution/Lemmas.agda         |  2 +-
 src/Data/Graph/Acyclic.agda                   |  2 +-
 src/Data/Integer/Coprimality.agda             |  2 +-
 src/Data/List/Countdown.agda                  |  3 ++-
 src/Data/List/Extrema/Core.agda               |  2 +-
 src/Data/List/Extrema/Nat.agda                |  2 +-
 .../Fresh/Membership/Setoid/Properties.agda   |  2 +-
 .../Fresh/Relation/Unary/All/Properties.agda  |  2 +-
 .../Fresh/Relation/Unary/Any/Properties.agda  |  2 +-
 .../List/Membership/DecPropositional.agda     |  3 ++-
 src/Data/List/Membership/Propositional.agda   |  3 ++-
 .../Propositional/Properties/Core.agda        |  2 +-
 .../Propositional/Properties/WithK.agda       |  2 +-
 .../List/Membership/Setoid/Properties.agda    |  2 +-
 src/Data/List/NonEmpty.agda                   | 16 +-------------
 .../Binary/Disjoint/Propositional.agda        |  2 +-
 .../Binary/Equality/Propositional.agda        |  3 ++-
 .../Infix/Heterogeneous/Properties.agda       |  2 +-
 .../Binary/Permutation/Propositional.agda     |  2 +-
 .../Permutation/Propositional/Properties.agda |  2 +-
 .../Relation/Binary/Permutation/Setoid.agda   |  2 +-
 .../Binary/Permutation/Setoid/Properties.agda |  2 +-
 src/Data/List/Relation/Binary/Pointwise.agda  |  3 ++-
 .../Relation/Binary/Pointwise/Properties.agda |  2 +-
 .../Prefix/Heterogeneous/Properties.agda      |  2 +-
 .../Sublist/DecPropositional/Solver.agda      |  4 ++--
 .../Binary/Sublist/Heterogeneous.agda         |  2 +-
 .../Sublist/Heterogeneous/Properties.agda     |  2 +-
 .../Sublist/Propositional/Disjoint.agda       |  2 +-
 .../Example/UniqueBoundVariables.agda         |  2 +-
 .../List/Relation/Binary/Sublist/Setoid.agda  |  2 +-
 .../Binary/Sublist/Setoid/Properties.agda     |  2 +-
 .../Relation/Binary/Subset/Propositional.agda |  2 +-
 .../Suffix/Heterogeneous/Properties.agda      |  2 +-
 src/Data/List/Relation/Ternary/Appending.agda |  2 +-
 .../Ternary/Appending/Properties.agda         |  2 +-
 .../Appending/Propositional/Properties.agda   |  3 ++-
 .../Ternary/Appending/Setoid/Properties.agda  |  2 +-
 .../List/Relation/Ternary/Interleaving.agda   |  2 +-
 .../Ternary/Interleaving/Propositional.agda   |  3 ++-
 .../Propositional/Properties.agda             |  2 +-
 src/Data/List/Relation/Unary/All.agda         |  3 ++-
 .../Relation/Unary/AllPairs/Properties.agda   |  2 +-
 .../List/Relation/Unary/Any/Properties.agda   |  2 +-
 .../Unary/Enumerates/Setoid/Properties.agda   |  2 +-
 .../Relation/Unary/Linked/Properties.agda     |  2 +-
 src/Data/List/Relation/Unary/Sufficient.agda  |  2 +-
 .../Unary/Unique/DecPropositional.agda        |  2 +-
 .../Unique/DecPropositional/Properties.agda   |  2 +-
 .../Relation/Unary/Unique/Propositional.agda  |  2 +-
 .../Unary/Unique/Setoid/Properties.agda       |  2 +-
 src/Data/Maybe/Relation/Binary/Connected.agda |  2 +-
 src/Data/Maybe/Relation/Binary/Pointwise.agda |  2 +-
 src/Data/Maybe/Relation/Unary/All.agda        |  2 +-
 src/Data/Maybe/Relation/Unary/Any.agda        |  2 +-
 src/Data/Nat/Coprimality.agda                 |  2 +-
 src/Data/Nat/Divisibility.agda                |  3 ++-
 src/Data/Nat/GCD.agda                         |  2 +-
 src/Data/Nat/LCM.agda                         |  2 +-
 src/Data/Nat/Logarithm.agda                   |  2 +-
 src/Data/Nat/Show/Properties.agda             |  2 +-
 src/Data/Product/Effectful/Examples.agda      |  2 +-
 .../Product/Function/Dependent/Setoid.agda    |  3 ++-
 .../Function/Dependent/Setoid/WithK.agda      |  3 ++-
 .../Product/Relation/Binary/Lex/Strict.agda   |  2 +-
 .../Relation/Binary/Pointwise/Dependent.agda  |  2 +-
 .../Binary/Pointwise/Dependent/WithK.agda     |  3 ++-
 .../Binary/Pointwise/NonDependent.agda        |  3 ++-
 src/Data/String.agda                          |  2 +-
 src/Data/String/Properties.agda               |  2 +-
 src/Data/String/Unsafe.agda                   |  2 +-
 src/Data/Sum/Relation/Binary/LeftOrder.agda   |  2 +-
 src/Data/Sum/Relation/Binary/Pointwise.agda   |  3 ++-
 src/Data/Tree/AVL/Indexed.agda                |  2 +-
 .../Tree/AVL/Indexed/Relation/Unary/All.agda  |  2 +-
 .../Relation/Unary/Any/Properties.agda        |  2 +-
 src/Data/Tree/AVL/Indexed/WithK.agda          |  2 +-
 src/Data/Tree/AVL/IndexedMap.agda             |  2 +-
 src/Data/Tree/AVL/Key.agda                    |  2 +-
 src/Data/Tree/AVL/NonEmpty/Propositional.agda |  2 +-
 src/Data/Tree/AVL/Sets/Membership.agda        |  1 -
 .../Tree/AVL/Sets/Membership/Properties.agda  |  2 +-
 .../Relation/Binary/Equality/Setoid.agda      |  2 +-
 .../Relation/Binary/Pointwise/Properties.agda |  2 +-
 src/Data/Vec/Membership/DecPropositional.agda |  3 ++-
 src/Data/Vec/Membership/Propositional.agda    |  3 ++-
 .../Membership/Propositional/Properties.agda  |  2 +-
 src/Data/Vec/Properties/WithK.agda            |  2 +-
 src/Data/Vec/Recursive/Properties.agda        |  2 +-
 .../Vec/Relation/Binary/Equality/Setoid.agda  |  1 -
 src/Data/Vec/Relation/Binary/Lex/Core.agda    |  2 +-
 .../Vec/Relation/Binary/Lex/NonStrict.agda    |  1 -
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  2 +-
 .../Binary/Pointwise/Extensional.agda         |  2 +-
 .../Relation/Binary/Pointwise/Inductive.agda  |  2 +-
 src/Data/Vec/Relation/Unary/All.agda          |  2 +-
 .../Relation/Unary/AllPairs/Properties.agda   |  2 +-
 .../Vec/Relation/Unary/Any/Properties.agda    |  2 +-
 .../Vec/Relation/Unary/Linked/Properties.agda |  2 +-
 .../Relation/Unary/Unique/Propositional.agda  |  2 +-
 .../Unary/Unique/Setoid/Properties.agda       |  2 +-
 src/Data/W/Sized.agda                         |  2 +-
 src/Effect/Monad/Partiality.agda              |  3 ++-
 src/Effect/Monad/Partiality/All.agda          |  2 +-
 src/Function/Construct/Identity.agda          |  3 ++-
 src/Function/Endomorphism/Propositional.agda  |  3 ++-
 src/Function/Metric/Nat/Structures.agda       |  2 +-
 src/Function/Metric/Rational/Structures.agda  |  2 +-
 src/Function/Properties/Bijection.agda        |  2 +-
 src/Function/Properties/Equivalence.agda      |  2 +-
 src/Function/Properties/Inverse.agda          |  3 ++-
 src/Function/Related.agda                     |  3 ++-
 src/Function/Related/Propositional.agda       |  3 ++-
 src/Function/Related/TypeIsomorphisms.agda    |  2 +-
 src/Reflection/AST/Abstraction.agda           | 10 ++++-----
 src/Reflection/AST/Argument.agda              | 10 ++++-----
 src/Reflection/AST/Argument/Information.agda  |  8 +++----
 src/Reflection/AST/Argument/Modality.agda     |  8 +++----
 src/Reflection/AST/Argument/Quantity.agda     |  6 ++---
 src/Reflection/AST/Argument/Relevance.agda    |  6 ++---
 src/Reflection/AST/Argument/Visibility.agda   |  6 ++---
 src/Reflection/AST/Definition.agda            | 12 +++++-----
 src/Reflection/AST/Literal.agda               |  8 +++----
 src/Reflection/AST/Meta.agda                  | 10 ++++-----
 src/Reflection/AST/Name.agda                  | 16 +++++++-------
 src/Reflection/AST/Term.agda                  |  6 ++---
 .../Construct/Add/Extrema/NonStrict.agda      |  2 +-
 .../Construct/Add/Infimum/NonStrict.agda      |  3 ++-
 .../Binary/Construct/Add/Infimum/Strict.agda  |  3 ++-
 .../Binary/Construct/Add/Point/Equality.agda  |  2 +-
 .../Construct/Add/Supremum/NonStrict.agda     |  3 ++-
 .../Binary/Construct/Add/Supremum/Strict.agda |  3 ++-
 .../Closure/Reflexive/Properties.agda         |  3 ++-
 .../Closure/Reflexive/Properties/WithK.agda   |  2 +-
 .../ReflexiveTransitive/Properties.agda       |  3 ++-
 .../Binary/Construct/Closure/Transitive.agda  |  2 +-
 .../Construct/Closure/Transitive/WithK.agda   |  2 +-
 .../Binary/HeterogeneousEquality.agda         |  3 ++-
 .../Quotients/Examples.agda                   |  2 +-
 .../Binary/Indexed/Homogeneous/Bundles.agda   |  2 +-
 .../Indexed/Homogeneous/Structures.agda       |  2 +-
 .../Binary/Reasoning/MultiSetoid.agda         |  2 +-
 src/Relation/Binary/Reflection.agda           |  2 +-
 src/Relation/Nary.agda                        |  2 +-
 .../Nullary/Construct/Add/Extrema.agda        |  2 +-
 src/Relation/Nullary/Universe.agda            |  4 ++--
 src/Tactic/Cong.agda                          |  2 +-
 .../Polynomial/Homomorphism/Addition.agda     |  2 +-
 .../Homomorphism/Exponentiation.agda          |  2 +-
 .../Core/Polynomial/Homomorphism/Lemmas.agda  | 22 +++++++++----------
 src/Text/Regex/Derivative/Brzozowski.agda     |  2 +-
 src/Text/Regex/Properties.agda                |  1 -
 src/Text/Regex/Search.agda                    |  2 +-
 src/Text/Regex/SmartConstructors.agda         |  2 +-
 src/Text/Regex/String/Unsafe.agda             |  2 +-
 198 files changed, 282 insertions(+), 266 deletions(-)

diff --git a/src/Algebra/Construct/LiftedChoice.agda b/src/Algebra/Construct/LiftedChoice.agda
index 83cf2110c3..91d3f1d546 100644
--- a/src/Algebra/Construct/LiftedChoice.agda
+++ b/src/Algebra/Construct/LiftedChoice.agda
@@ -16,7 +16,7 @@ open import Relation.Nullary using (¬_; yes; no)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Product.Base using (_×_; _,_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Unary using (Pred)
 
 import Relation.Binary.Reasoning.Setoid as EqReasoning
diff --git a/src/Algebra/Operations/CommutativeMonoid.agda b/src/Algebra/Operations/CommutativeMonoid.agda
index 3c94a94419..52d72af156 100644
--- a/src/Algebra/Operations/CommutativeMonoid.agda
+++ b/src/Algebra/Operations/CommutativeMonoid.agda
@@ -12,7 +12,7 @@ open import Data.Fin.Base using (Fin; zero)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 module Algebra.Operations.CommutativeMonoid
   {s₁ s₂} (CM : CommutativeMonoid s₁ s₂)
diff --git a/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda b/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda
index b61a3e987e..8985fb21cb 100644
--- a/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda
+++ b/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda
@@ -10,7 +10,6 @@
 open import Algebra.Bundles using (CommutativeMonoid)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
 
 module Algebra.Properties.CommutativeMonoid.Mult.TCOptimised
   {a ℓ} (M : CommutativeMonoid a ℓ) where
diff --git a/src/Algebra/Properties/CommutativeMonoid/Sum.agda b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
index f70b965130..6cb417bc69 100644
--- a/src/Algebra/Properties/CommutativeMonoid/Sum.agda
+++ b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
@@ -14,7 +14,7 @@ open import Data.Fin.Permutation as Perm using (Permutation; _⟨$⟩ˡ_; _⟨$
 open import Data.Fin.Patterns using (0F)
 open import Data.Vec.Functional
 open import Function.Base using (_∘_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Negation using (contradiction)
 
 module Algebra.Properties.CommutativeMonoid.Sum
diff --git a/src/Algebra/Properties/Monoid/Mult.agda b/src/Algebra/Properties/Monoid/Mult.agda
index f74404e886..1a9e346585 100644
--- a/src/Algebra/Properties/Monoid/Mult.agda
+++ b/src/Algebra/Properties/Monoid/Mult.agda
@@ -9,7 +9,7 @@
 open import Algebra.Bundles using (Monoid)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; NonZero)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 module Algebra.Properties.Monoid.Mult {a ℓ} (M : Monoid a ℓ) where
 
diff --git a/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda b/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
index 86c4ac566c..ded5f1339e 100644
--- a/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
+++ b/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
@@ -10,7 +10,7 @@
 open import Algebra.Bundles using (Monoid)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 module Algebra.Properties.Monoid.Mult.TCOptimised
   {a ℓ} (M : Monoid a ℓ) where
diff --git a/src/Algebra/Properties/Semiring/Exp.agda b/src/Algebra/Properties/Semiring/Exp.agda
index df467b1f66..1d50298ca5 100644
--- a/src/Algebra/Properties/Semiring/Exp.agda
+++ b/src/Algebra/Properties/Semiring/Exp.agda
@@ -9,7 +9,7 @@
 open import Algebra
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Data.Nat.Properties as ℕ
 
 module Algebra.Properties.Semiring.Exp
diff --git a/src/Algebra/Solver/CommutativeMonoid.agda b/src/Algebra/Solver/CommutativeMonoid.agda
index 623beaeebd..7e4b14b18c 100644
--- a/src/Algebra/Solver/CommutativeMonoid.agda
+++ b/src/Algebra/Solver/CommutativeMonoid.agda
@@ -27,7 +27,7 @@ import Relation.Binary.Reflection as Reflection
 import Relation.Nullary.Decidable as Dec
 import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 
-open import Relation.Binary.PropositionalEquality as P using (_≡_; decSetoid)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Decidable using (Dec)
 
 open CommutativeMonoid M
diff --git a/src/Algebra/Solver/CommutativeMonoid/Example.agda b/src/Algebra/Solver/CommutativeMonoid/Example.agda
index 83351df044..e59f5b1e1f 100644
--- a/src/Algebra/Solver/CommutativeMonoid/Example.agda
+++ b/src/Algebra/Solver/CommutativeMonoid/Example.agda
@@ -8,7 +8,7 @@
 
 module Algebra.Solver.CommutativeMonoid.Example where
 
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 open import Data.Bool.Base using (_∨_)
 open import Data.Bool.Properties using (∨-commutativeMonoid)
diff --git a/src/Algebra/Solver/Monoid.agda b/src/Algebra/Solver/Monoid.agda
index 3006374347..bf066f7745 100644
--- a/src/Algebra/Solver/Monoid.agda
+++ b/src/Algebra/Solver/Monoid.agda
@@ -22,7 +22,7 @@ open import Data.Vec.Base using (Vec; lookup)
 open import Function.Base using (_∘_; _$_)
 open import Relation.Binary using (Decidable)
 
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.Reflection
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Algebra/Solver/Ring.agda b/src/Algebra/Solver/Ring.agda
index a1ea98ead3..e103b7a70e 100644
--- a/src/Algebra/Solver/Ring.agda
+++ b/src/Algebra/Solver/Ring.agda
@@ -42,7 +42,7 @@ open import Algebra.Properties.Semiring.Exp semiring
 open import Relation.Binary
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Binary.Reasoning.Setoid setoid
-import Relation.Binary.PropositionalEquality as PropEq
+import Relation.Binary.PropositionalEquality.Core as PropEq
 import Relation.Binary.Reflection as Reflection
 
 open import Data.Nat.Base using (ℕ; suc; zero)
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients.agda b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
index 95af2b0a33..fdd0a65e6b 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
@@ -14,7 +14,7 @@ open import Algebra.Solver.Ring.AlmostCommutativeRing
 open import Data.Nat.Base as ℕ
 open import Data.Product.Base using (module Σ)
 open import Function.Base using (id)
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 module Algebra.Solver.Ring.NaturalCoefficients
   {r₁ r₂} (R : CommutativeSemiring r₁ r₂)
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
index df01719174..84003922a0 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
@@ -20,7 +20,7 @@ import Algebra.Properties.Semiring.Mult as SemiringMultiplication
 open import Data.Maybe.Base using (Maybe; map)
 open import Data.Nat using (_≟_)
 open import Relation.Binary.Consequences using (dec⇒weaklyDec)
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Core as P
 
 open CommutativeSemiring R
 open SemiringMultiplication semiring
diff --git a/src/Codata/Guarded/Stream/Properties.agda b/src/Codata/Guarded/Stream/Properties.agda
index 3e4bd83dcb..29ea54bb1d 100644
--- a/src/Codata/Guarded/Stream/Properties.agda
+++ b/src/Codata/Guarded/Stream/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Product as Prod using (_×_; _,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec; _∷_)
 open import Function.Base using (const; flip; id; _∘′_; _$′_; _⟨_⟩_; _∘₂′_)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; cong; cong₂)
 
 private
   variable
diff --git a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
index c2d44f8145..3f610599a5 100644
--- a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
+++ b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
@@ -13,7 +13,8 @@ open import Data.Nat.Base using (ℕ; zero; suc)
 open import Function.Base using (_∘_; _on_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 private
   variable
diff --git a/src/Codata/Musical/Cofin.agda b/src/Codata/Musical/Cofin.agda
index 082a6848dd..bc901ef194 100644
--- a/src/Codata/Musical/Cofin.agda
+++ b/src/Codata/Musical/Cofin.agda
@@ -12,7 +12,7 @@ open import Codata.Musical.Notation
 open import Codata.Musical.Conat as Conat using (Coℕ; suc; ∞ℕ)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.Fin.Base using (Fin; zero; suc)
-open import Relation.Binary.PropositionalEquality using (_≡_ ; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_ ; refl)
 open import Function.Base using (_∋_)
 
 ------------------------------------------------------------------------
diff --git a/src/Codata/Musical/Colist/Infinite-merge.agda b/src/Codata/Musical/Colist/Infinite-merge.agda
index 9c859bcf26..677024808a 100644
--- a/src/Codata/Musical/Colist/Infinite-merge.agda
+++ b/src/Codata/Musical/Colist/Infinite-merge.agda
@@ -23,7 +23,7 @@ open import Function.Inverse as Inv using (_↔_; Inverse; inverse)
 import Function.Related as Related
 open import Function.Related.TypeIsomorphisms
 import Induction.WellFounded as WF
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.Construct.On as On
 
 ------------------------------------------------------------------------
diff --git a/src/Codata/Musical/Conat.agda b/src/Codata/Musical/Conat.agda
index f8c1490a40..d8e76d90be 100644
--- a/src/Codata/Musical/Conat.agda
+++ b/src/Codata/Musical/Conat.agda
@@ -12,7 +12,7 @@ open import Codata.Musical.Notation
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Function.Base using (_∋_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
 -- Re-exporting the type and basic operations
diff --git a/src/Codata/Musical/Covec.agda b/src/Codata/Musical/Covec.agda
index 48cc1910e8..88b15c070b 100644
--- a/src/Codata/Musical/Covec.agda
+++ b/src/Codata/Musical/Covec.agda
@@ -18,7 +18,7 @@ open import Data.Product using (_,_)
 open import Function.Base using (_∋_)
 open import Level using (Level)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 private
   variable
diff --git a/src/Codata/Musical/Stream.agda b/src/Codata/Musical/Stream.agda
index 9b00ba510f..a51f8f41b5 100644
--- a/src/Codata/Musical/Stream.agda
+++ b/src/Codata/Musical/Stream.agda
@@ -14,7 +14,7 @@ open import Codata.Musical.Colist using (Colist; []; _∷_)
 open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 private
   variable
diff --git a/src/Codata/Sized/Colist.agda b/src/Codata/Sized/Colist.agda
index 4f916fbc68..e57a5b632e 100644
--- a/src/Codata/Sized/Colist.agda
+++ b/src/Codata/Sized/Colist.agda
@@ -27,7 +27,7 @@ open import Codata.Sized.Cowriter as CW using (Cowriter; _∷_)
 open import Codata.Sized.Delay as Delay using (Delay ; now ; later)
 open import Codata.Sized.Stream using (Stream ; _∷_)
 
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
   variable
diff --git a/src/Codata/Sized/Colist/Bisimilarity.agda b/src/Codata/Sized/Colist/Bisimilarity.agda
index 0a9f76306b..9efbcebbca 100644
--- a/src/Codata/Sized/Colist/Bisimilarity.agda
+++ b/src/Codata/Sized/Colist/Bisimilarity.agda
@@ -16,7 +16,8 @@ open import Data.List.Base using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
 open import Data.List.NonEmpty as List⁺  using (List⁺; _∷_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as Eq
 
 private
   variable
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index dea697b088..de5925e482 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -31,7 +31,7 @@ open import Data.Product as Prod using (_×_; _,_; uncurry)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Function.Base
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; [_])
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 
 private
   variable
diff --git a/src/Codata/Sized/Covec/Bisimilarity.agda b/src/Codata/Sized/Covec/Bisimilarity.agda
index cbe55ecbf7..7610c41946 100644
--- a/src/Codata/Sized/Covec/Bisimilarity.agda
+++ b/src/Codata/Sized/Covec/Bisimilarity.agda
@@ -14,7 +14,7 @@ open import Codata.Sized.Thunk
 open import Codata.Sized.Conat hiding (_⊔_)
 open import Codata.Sized.Covec
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 
 data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) (i : Size) :
            ∀ m n (xs : Covec A ∞ m) (ys : Covec B ∞ n) → Set (r ⊔ a ⊔ b) where
diff --git a/src/Codata/Sized/Covec/Properties.agda b/src/Codata/Sized/Covec/Properties.agda
index 139eba6029..08122efe0b 100644
--- a/src/Codata/Sized/Covec/Properties.agda
+++ b/src/Codata/Sized/Covec/Properties.agda
@@ -14,7 +14,7 @@ open import Codata.Sized.Conat
 open import Codata.Sized.Covec
 open import Codata.Sized.Covec.Bisimilarity
 open import Function.Base using (id; _∘_)
-open import Relation.Binary.PropositionalEquality as Eq
+open import Relation.Binary.PropositionalEquality.Core as Eq
 
 -- Functor laws
 
diff --git a/src/Codata/Sized/Cowriter/Bisimilarity.agda b/src/Codata/Sized/Cowriter/Bisimilarity.agda
index 846101b538..84ab0375c4 100644
--- a/src/Codata/Sized/Cowriter/Bisimilarity.agda
+++ b/src/Codata/Sized/Cowriter/Bisimilarity.agda
@@ -13,7 +13,8 @@ open import Size
 open import Codata.Sized.Thunk
 open import Codata.Sized.Cowriter
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as Eq
 
 private
   variable
diff --git a/src/Codata/Sized/Delay/Bisimilarity.agda b/src/Codata/Sized/Delay/Bisimilarity.agda
index 1143f15263..4d9f702acc 100644
--- a/src/Codata/Sized/Delay/Bisimilarity.agda
+++ b/src/Codata/Sized/Delay/Bisimilarity.agda
@@ -13,7 +13,7 @@ open import Codata.Sized.Thunk
 open import Codata.Sized.Delay
 open import Level
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 
 data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) i :
            (xs : Delay A ∞) (ys : Delay B ∞) → Set (a ⊔ b ⊔ r) where
diff --git a/src/Codata/Sized/Delay/Properties.agda b/src/Codata/Sized/Delay/Properties.agda
index 5db7bafe7f..e491178c71 100644
--- a/src/Codata/Sized/Delay/Properties.agda
+++ b/src/Codata/Sized/Delay/Properties.agda
@@ -17,7 +17,7 @@ open import Codata.Sized.Conat.Bisimilarity as Coℕ using (zero ; suc)
 open import Codata.Sized.Delay
 open import Codata.Sized.Delay.Bisimilarity
 open import Function.Base using (id; _∘′_)
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 
 module _ {a} {A : Set a} where
 
diff --git a/src/Codata/Sized/M/Bisimilarity.agda b/src/Codata/Sized/M/Bisimilarity.agda
index 1b8cbd3188..3a60a91d16 100644
--- a/src/Codata/Sized/M/Bisimilarity.agda
+++ b/src/Codata/Sized/M/Bisimilarity.agda
@@ -17,7 +17,7 @@ open import Data.Container.Relation.Binary.Pointwise using (Pointwise; _,_)
 open import Data.Product using (_,_)
 open import Function.Base using (_∋_)
 open import Relation.Binary
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Core as P
 
 data Bisim {s p} (C : Container s p) (i : Size) : Rel (M C ∞) (s ⊔ p) where
   inf : ∀ {t u} → Pointwise C (Thunk^R (Bisim C) i) t u → Bisim C i (inf t) (inf u)
diff --git a/src/Codata/Sized/M/Properties.agda b/src/Codata/Sized/M/Properties.agda
index 3b26955e60..4161f4d1f1 100644
--- a/src/Codata/Sized/M/Properties.agda
+++ b/src/Codata/Sized/M/Properties.agda
@@ -18,7 +18,8 @@ import Data.Container.Morphism as Mp
 open import Data.Product as Prod using (_,_)
 open import Data.Product.Properties hiding (map-cong)
 open import Function.Base using (_$′_; _∘′_)
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Properties as P
 
 open import Data.Container.Relation.Binary.Pointwise using (_,_)
 import Data.Container.Relation.Binary.Equality.Setoid as EqSetoid
diff --git a/src/Codata/Sized/Stream/Bisimilarity.agda b/src/Codata/Sized/Stream/Bisimilarity.agda
index 737e73c306..74419c7809 100644
--- a/src/Codata/Sized/Stream/Bisimilarity.agda
+++ b/src/Codata/Sized/Stream/Bisimilarity.agda
@@ -15,7 +15,8 @@ open import Level
 open import Data.List.NonEmpty as List⁺ using (_∷_)
 open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as Eq
 
 private
   variable
diff --git a/src/Codata/Sized/Stream/Properties.agda b/src/Codata/Sized/Stream/Properties.agda
index 539a6b7b20..51b0719c7d 100644
--- a/src/Codata/Sized/Stream/Properties.agda
+++ b/src/Codata/Sized/Stream/Properties.agda
@@ -24,7 +24,7 @@ open import Data.Product as Prod using (_,_)
 open import Data.Vec.Base as Vec using (_∷_)
 
 open import Function.Base using (id; _$_; _∘′_; const)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≢_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; _≢_)
 
 private
   variable
diff --git a/src/Data/AVL/Indexed/WithK.agda b/src/Data/AVL/Indexed/WithK.agda
index 4a5de070f0..15da148fa2 100644
--- a/src/Data/AVL/Indexed/WithK.agda
+++ b/src/Data/AVL/Indexed/WithK.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --with-K --safe #-}
 
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; subst)
 
 module Data.AVL.Indexed.WithK
        {k r} (Key : Set k) {_<_ : Rel Key r}
diff --git a/src/Data/AVL/IndexedMap.agda b/src/Data/AVL/IndexedMap.agda
index 64a0d5c36a..25bccdee5f 100644
--- a/src/Data/AVL/IndexedMap.agda
+++ b/src/Data/AVL/IndexedMap.agda
@@ -8,7 +8,7 @@
 
 open import Data.Product as Prod
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality using (_≡_; cong; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
 import Data.Tree.AVL.Value
 
 module Data.AVL.IndexedMap
diff --git a/src/Data/Bool.agda b/src/Data/Bool.agda
index 909b659f22..5fdbcc45e0 100644
--- a/src/Data/Bool.agda
+++ b/src/Data/Bool.agda
@@ -10,8 +10,7 @@ module Data.Bool where
 
 open import Relation.Nullary
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as PropEq
-  using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- The boolean type and some operations
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index ef963f6511..56448d8e2b 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -22,7 +22,8 @@ open import Function.Equivalence
 open import Induction.WellFounded using (WellFounded; Acc; acc)
 open import Level using (Level; 0ℓ)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality hiding ([_])
+open import Relation.Binary.PropositionalEquality.Core
+open import Relation.Binary.PropositionalEquality.Properties
 open import Relation.Nullary.Reflects using (ofʸ; ofⁿ)
 open import Relation.Nullary.Decidable.Core using (True; does; proof; yes; no)
 import Relation.Unary as U
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index 1db34c4f0b..2bbf63ab17 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -22,8 +22,9 @@ import Relation.Binary.Construct.On as On
 import Relation.Binary.Construct.Subst.Equality as Subst
 import Relation.Binary.Construct.Closure.Reflexive as Refl
 import Relation.Binary.Construct.Closure.Reflexive.Properties as Reflₚ
-open import Relation.Binary.PropositionalEquality as PropEq
+open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; _≢_; refl; cong; sym; trans; subst)
+import Relation.Binary.PropositionalEquality.Properties as PropEq
 
 ------------------------------------------------------------------------
 -- Primitive properties
diff --git a/src/Data/Container/Indexed/WithK.agda b/src/Data/Container/Indexed/WithK.agda
index 565b14fcdd..ca5f18205d 100644
--- a/src/Data/Container/Indexed/WithK.agda
+++ b/src/Data/Container/Indexed/WithK.agda
@@ -19,7 +19,7 @@ open import Data.Product as Prod hiding (map)
 open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
 open import Level
 open import Relation.Unary using (Pred; _⊆_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
 open import Relation.Binary.HeterogeneousEquality as H using (_≅_; refl)
 open import Relation.Binary.Indexed.Heterogeneous
 
diff --git a/src/Data/Container/Relation/Binary/Pointwise.agda b/src/Data/Container/Relation/Binary/Pointwise.agda
index 8a022a072c..2b0b966b3d 100644
--- a/src/Data/Container/Relation/Binary/Pointwise.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise.agda
@@ -12,7 +12,7 @@ open import Data.Product using (_,_; Σ-syntax; -,_; proj₁; proj₂)
 open import Function
 open import Level using (_⊔_)
 open import Relation.Binary using (REL; _⇒_)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; subst; cong)
 
 open import Data.Container.Core using (Container; ⟦_⟧)
diff --git a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
index 24e48a74a3..3092037901 100644
--- a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
@@ -14,7 +14,7 @@ open import Data.Container.Relation.Binary.Pointwise
 open import Data.Product using (_,_; Σ-syntax; -,_)
 open import Level using (_⊔_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; subst; cong)
 
 module _ {s p x r} {X : Set x} (C : Container s p) (R : Rel X r) where
diff --git a/src/Data/Digit.agda b/src/Data/Digit.agda
index 7e7c1c7555..60557b142a 100644
--- a/src/Data/Digit.agda
+++ b/src/Data/Digit.agda
@@ -21,7 +21,7 @@ open import Data.Nat.DivMod
 open import Data.Nat.Induction
 open import Relation.Nullary.Decidable using (True; does; toWitness)
 open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
 open import Function.Base using (_$_)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Digit/Properties.agda b/src/Data/Digit/Properties.agda
index 355467ec39..37ead69a28 100644
--- a/src/Data/Digit/Properties.agda
+++ b/src/Data/Digit/Properties.agda
@@ -16,7 +16,7 @@ open import Data.Vec.Relation.Unary.Unique.Propositional using (Unique)
 import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Uniqueₚ
 open import Data.Vec.Relation.Unary.AllPairs using (allPairs?)
 open import Relation.Nullary.Decidable using (True; from-yes; ¬?)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Function.Base using (_∘_)
 
 module Data.Digit.Properties where
diff --git a/src/Data/Fin/Substitution/Example.agda b/src/Data/Fin/Substitution/Example.agda
index f4b42438e5..8b3c903a7c 100644
--- a/src/Data/Fin/Substitution/Example.agda
+++ b/src/Data/Fin/Substitution/Example.agda
@@ -14,7 +14,7 @@ open import Data.Fin.Substitution.Lemmas
 open import Data.Nat.Base hiding (_/_)
 open import Data.Fin.Base using (Fin)
 open import Data.Vec.Base
-open import Relation.Binary.PropositionalEquality as PropEq
+open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; refl; sym; cong; cong₂)
 open PropEq.≡-Reasoning
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
diff --git a/src/Data/Fin/Substitution/Lemmas.agda b/src/Data/Fin/Substitution/Lemmas.agda
index 3747f21018..6109f28726 100644
--- a/src/Data/Fin/Substitution/Lemmas.agda
+++ b/src/Data/Fin/Substitution/Lemmas.agda
@@ -15,7 +15,7 @@ open import Data.Fin.Base using (Fin; zero; suc; lift)
 open import Data.Vec.Base
 import Data.Vec.Properties as VecProp
 open import Function.Base as Fun using (_∘_; _$_; flip)
-open import Relation.Binary.PropositionalEquality as PropEq
+open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; refl; sym; cong; cong₂)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
   using (Star; ε; _◅_; _▻_)
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index 440424206a..d7b298f0ef 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -28,7 +28,7 @@ open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Function
 open import Relation.Nullary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
 -- A lemma
diff --git a/src/Data/Integer/Coprimality.agda b/src/Data/Integer/Coprimality.agda
index 35df9a1457..c3cfafcbff 100644
--- a/src/Data/Integer/Coprimality.agda
+++ b/src/Data/Integer/Coprimality.agda
@@ -16,7 +16,7 @@ import Data.Nat.Divisibility as ℕ
 open import Function.Base using (_on_)
 open import Level using (0ℓ)
 open import Relation.Binary using (Rel; Decidable; Symmetric)
-open import Relation.Binary.PropositionalEquality using (subst)
+open import Relation.Binary.PropositionalEquality.Core using (subst)
 
 ------------------------------------------------------------------------
 -- Definition
diff --git a/src/Data/List/Countdown.agda b/src/Data/List/Countdown.agda
index fde7439ebf..f1ef43ba8e 100644
--- a/src/Data/List/Countdown.agda
+++ b/src/Data/List/Countdown.agda
@@ -29,9 +29,10 @@ open import Data.Sum.Properties
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary
 open import Relation.Nullary.Decidable using (dec-true; dec-false)
-open import Relation.Binary.PropositionalEquality as PropEq
+open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; _≢_; refl; cong)
 open PropEq.≡-Reasoning
+import Relation.Binary.PropositionalEquality.Properties as PropEq
 
 private
   open module D = DecSetoid D
diff --git a/src/Data/List/Extrema/Core.agda b/src/Data/List/Extrema/Core.agda
index aa0b3f8da9..1fb59ae06d 100644
--- a/src/Data/List/Extrema/Core.agda
+++ b/src/Data/List/Extrema/Core.agda
@@ -18,7 +18,7 @@ import Algebra.Construct.NaturalChoice.Max as Max
 open import Data.Product using (_×_; _,_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 open import Algebra.Construct.LiftedChoice
 
diff --git a/src/Data/List/Extrema/Nat.agda b/src/Data/List/Extrema/Nat.agda
index d1d8f3f4ac..6e2899b8e6 100644
--- a/src/Data/List/Extrema/Nat.agda
+++ b/src/Data/List/Extrema/Nat.agda
@@ -20,7 +20,7 @@ open import Data.List.Relation.Unary.Any as Any using (Any)
 open import Data.List.Relation.Unary.All as All using (All)
 open import Data.Product using (_×_; _,_; uncurry′)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality using (_≢_)
+open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 
 private
   variable
diff --git a/src/Data/List/Fresh/Membership/Setoid/Properties.agda b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
index 52be3b0d60..0515b84efe 100644
--- a/src/Data/List/Fresh/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
@@ -21,7 +21,7 @@ open import Function.Base using (id; _∘′_; _$_)
 open import Relation.Nullary
 open import Relation.Unary as U using (Pred)
 import Relation.Binary as B
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Core as P
 open import Relation.Nary
 
 open import Data.List.Fresh
diff --git a/src/Data/List/Fresh/Relation/Unary/All/Properties.agda b/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
index cbad777b96..f2c47657b5 100644
--- a/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
@@ -16,7 +16,7 @@ open import Function.Base using (_∘′_)
 open import Relation.Nullary
 open import Relation.Unary  as U
 open import Relation.Binary as B using (Rel)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 open import Data.List.Fresh using (List#; []; cons; _∷#_; _#_)
 open import Data.List.Fresh.Relation.Unary.All
diff --git a/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda b/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
index f185a2ef32..09ec3d5333 100644
--- a/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
@@ -20,7 +20,7 @@ open import Relation.Nullary
 open import Relation.Unary  as U using (Pred)
 open import Relation.Binary as B using (Rel)
 open import Relation.Nary
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 open import Data.List.Fresh
 open import Data.List.Fresh.Relation.Unary.All
diff --git a/src/Data/List/Membership/DecPropositional.agda b/src/Data/List/Membership/DecPropositional.agda
index 4380ff8ca2..a59ce42c67 100644
--- a/src/Data/List/Membership/DecPropositional.agda
+++ b/src/Data/List/Membership/DecPropositional.agda
@@ -7,7 +7,8 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary using (Decidable)
-open import Relation.Binary.PropositionalEquality using (_≡_; decSetoid)
+open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.List.Membership.DecPropositional
   {a} {A : Set a} (_≟_ : Decidable (_≡_ {A = A})) where
diff --git a/src/Data/List/Membership/Propositional.agda b/src/Data/List/Membership/Propositional.agda
index 4a74db3d04..c1ae08761e 100644
--- a/src/Data/List/Membership/Propositional.agda
+++ b/src/Data/List/Membership/Propositional.agda
@@ -10,7 +10,8 @@
 module Data.List.Membership.Propositional {a} {A : Set a} where
 
 open import Data.List.Relation.Unary.Any using (Any)
-open import Relation.Binary.PropositionalEquality using (_≡_; _≢_; setoid; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; subst)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 import Data.List.Membership.Setoid as SetoidMembership
 
diff --git a/src/Data/List/Membership/Propositional/Properties/Core.agda b/src/Data/List/Membership/Propositional/Properties/Core.agda
index 55a03f454c..85ad85aab0 100644
--- a/src/Data/List/Membership/Propositional/Properties/Core.agda
+++ b/src/Data/List/Membership/Propositional/Properties/Core.agda
@@ -20,7 +20,7 @@ open import Data.List.Membership.Propositional
 open import Data.Product as Prod
   using (_,_; proj₁; proj₂; uncurry′; ∃; _×_)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
 open import Relation.Unary using (Pred; _⊆_)
 
diff --git a/src/Data/List/Membership/Propositional/Properties/WithK.agda b/src/Data/List/Membership/Propositional/Properties/WithK.agda
index 5909ec3e72..60a86ee5e3 100644
--- a/src/Data/List/Membership/Propositional/Properties/WithK.agda
+++ b/src/Data/List/Membership/Propositional/Properties/WithK.agda
@@ -14,7 +14,7 @@ open import Data.List.Relation.Unary.Unique.Propositional
 open import Data.List.Membership.Propositional
 import Data.List.Membership.Setoid.Properties as Membershipₛ
 open import Relation.Unary using (Irrelevant)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties as P
 open import Relation.Binary.PropositionalEquality.WithK
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 919bf19ba6..47768eed76 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -27,7 +27,7 @@ open import Function.Base using (_$_; flip; _∘_; _∘′_; id)
 open import Function.Inverse using (_↔_)
 open import Level using (Level)
 open import Relation.Binary as B hiding (Decidable)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Unary as U using (Decidable; Pred)
 open import Relation.Nullary using (¬_; does; _because_; yes; no)
 open import Relation.Nullary.Reflects using (invert)
diff --git a/src/Data/List/NonEmpty.agda b/src/Data/List/NonEmpty.agda
index 9e9f189fb5..5ce7ff4be2 100644
--- a/src/Data/List/NonEmpty.agda
+++ b/src/Data/List/NonEmpty.agda
@@ -9,21 +9,7 @@
 module Data.List.NonEmpty where
 
 open import Level using (Level)
-open import Effect.Monad
-open import Data.Bool.Base using (Bool; false; true; not; T)
-open import Data.Bool.Properties
-open import Data.List.Base as List using (List; []; _∷_)
-open import Data.Maybe.Base using (Maybe ; nothing; just)
-open import Data.Nat.Base as ℕ
-open import Data.Product as Prod using (∃; _×_; proj₁; proj₂; _,_; -,_)
-open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
-open import Data.These.Base as These using (These; this; that; these)
-open import Data.Unit.Base using (tt)
-open import Data.Vec.Base as Vec using (Vec; []; _∷_)
-open import Function.Base
-open import Function.Bundles using () renaming (module Equivalence to Eq)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≢_; refl)
-open import Relation.Nullary.Decidable using (isYes)
+open import Data.List.Base as List using (List)
 
 ------------------------------------------------------------------------
 -- Re-export basic type and operations
diff --git a/src/Data/List/Relation/Binary/Disjoint/Propositional.agda b/src/Data/List/Relation/Binary/Disjoint/Propositional.agda
index ecae6c7610..6b78c64c85 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Propositional.agda
@@ -11,7 +11,7 @@ open import Relation.Binary
 module Data.List.Relation.Binary.Disjoint.Propositional
   {a} {A : Set a} where
 
-open import Relation.Binary.PropositionalEquality using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 open import Data.List.Relation.Binary.Disjoint.Setoid as DisjointUnique
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Equality/Propositional.agda b/src/Data/List/Relation/Binary/Equality/Propositional.agda
index f263778211..9ecb2cfd6b 100644
--- a/src/Data/List/Relation/Binary/Equality/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Equality/Propositional.agda
@@ -16,7 +16,8 @@ module Data.List.Relation.Binary.Equality.Propositional {a} {A : Set a} where
 
 open import Data.List.Base
 import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 ------------------------------------------------------------------------
 -- Re-export everything from setoid equality
diff --git a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
index 9366b5a027..f9610d5422 100644
--- a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
@@ -21,7 +21,7 @@ open import Relation.Nullary.Decidable using (yes; no; does; map′; _⊎-dec_)
 open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary using (REL; _⇒_; Decidable; Trans; Antisym)
-open import Relation.Binary.PropositionalEquality using (_≢_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≢_; refl; cong)
 
 open import Data.List.Relation.Binary.Pointwise as Pointwise using (Pointwise)
 open import Data.List.Relation.Binary.Infix.Heterogeneous
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional.agda b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
index 939491ecc4..5d9068755b 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Binary.Permutation.Propositional
 
 open import Data.List.Base using (List; []; _∷_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.Reasoning.Setoid as EqReasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index f5a87cb067..135d5a778d 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -28,7 +28,7 @@ open import Function.Inverse as Inv using (inverse)
 open import Level using (Level)
 open import Relation.Unary using (Pred)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as ≡
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_ ; refl ; cong; cong₂; _≢_)
 open import Relation.Nullary
 
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid.agda b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
index e98cc6315c..5a147d8008 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
@@ -17,7 +17,7 @@ import Data.List.Relation.Binary.Pointwise as Pointwise
 open import Data.List.Relation.Binary.Equality.Setoid S
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Level using (_⊔_)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 private
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index 3483323844..24b8bee2cf 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -36,7 +36,7 @@ open import Function.Inverse as Inv using (inverse)
 open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
-open import Relation.Binary.PropositionalEquality as ≡
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_ ; refl; sym; cong; cong₂; subst; _≢_)
 open import Relation.Nullary.Decidable using (yes; no; does)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/List/Relation/Binary/Pointwise.agda b/src/Data/List/Relation/Binary/Pointwise.agda
index cadfaf0846..0852085a37 100644
--- a/src/Data/List/Relation/Binary/Pointwise.agda
+++ b/src/Data/List/Relation/Binary/Pointwise.agda
@@ -26,7 +26,8 @@ open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec using (map′)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary renaming (Rel to Rel₂)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 private
   variable
diff --git a/src/Data/List/Relation/Binary/Pointwise/Properties.agda b/src/Data/List/Relation/Binary/Pointwise/Properties.agda
index 61470fdc76..207f0fe70d 100644
--- a/src/Data/List/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/List/Relation/Binary/Pointwise/Properties.agda
@@ -13,7 +13,7 @@ open import Data.List.Base using (List; []; _∷_)
 open import Level
 open import Relation.Binary.Core using (REL; _⇒_)
 open import Relation.Binary.Definitions
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Core as P
 open import Relation.Nullary using (yes; no; _×-dec_)
 import Relation.Nullary.Decidable as Dec
 
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
index eea13df3b6..da5a26e6e0 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
@@ -26,7 +26,7 @@ open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no; _because_)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≢_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; _≢_)
 
 private
   variable
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
index 6ca8768c43..6385ff0960 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
@@ -8,12 +8,12 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary using (Decidable)
-open import Agda.Builtin.Equality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 module Data.List.Relation.Binary.Sublist.DecPropositional.Solver
        {a} {A : Set a} (_≟_ : Decidable {A = A} _≡_)
        where
 
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Properties as P
 
 open import Data.List.Relation.Binary.Sublist.DecSetoid.Solver (P.decSetoid _≟_) public
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
index 854a6a1e8f..698dd789b0 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
@@ -12,7 +12,7 @@ open import Data.List.Base using (List; []; _∷_; [_])
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Level using (_⊔_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Unary using (Pred)
 
 module Data.List.Relation.Binary.Sublist.Heterogeneous
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index 7d13343130..f11e85258c 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -34,7 +34,7 @@ open import Relation.Nullary using (Dec; does; _because_; yes; no; ¬_)
 open import Relation.Nullary.Decidable as Dec using (¬?)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
 -- Injectivity of constructors
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda
index edcf2b565b..a6a33cc43a 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Binary.Sublist.Propositional.Disjoint
 
 open import Data.List.Base using (List)
 open import Data.List.Relation.Binary.Sublist.Propositional
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 ------------------------------------------------------------------------
 -- A Union where the triangles commute is a
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
index b8b8745b68..9a681ff916 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
@@ -10,7 +10,7 @@
 
 module Data.List.Relation.Binary.Sublist.Propositional.Example.UniqueBoundVariables (Base : Set) where
 
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; cong; subst; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; sym; cong; subst; module ≡-Reasoning)
 open ≡-Reasoning
 
 open import Data.List.Base using (List; []; _∷_; [_])
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid.agda b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
index 7a7b62c673..8915308cf2 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
@@ -26,7 +26,7 @@ import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
 open import Data.Product using (∃; ∃₂; _×_; _,_; proj₂)
 
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary using (¬_; Dec; yes; no)
 
 open Setoid S renaming (Carrier to A)
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
index 5600339c7b..b9c3d21968 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
@@ -21,7 +21,7 @@ open import Function.Base
 open import Function.Bundles using (_⇔_; _⤖_)
 open import Level
 open import Relation.Binary using () renaming (Decidable to Decidable₂)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Binary.Structures using (IsDecTotalOrder)
 open import Relation.Unary using (Pred; Decidable; Irrelevant)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional.agda b/src/Data/List/Relation/Binary/Subset/Propositional.agda
index d97fc73114..c6ecf5e5fa 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Binary.Subset.Propositional
   {a} {A : Set a} where
 
 import Data.List.Relation.Binary.Subset.Setoid as SetoidSubset
-open import Relation.Binary.PropositionalEquality using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 ------------------------------------------------------------------------
 -- Re-export parameterised definitions from setoid sublists
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
index 736389bb75..05b7725d31 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
@@ -26,7 +26,7 @@ open import Relation.Unary as U using (Pred)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Binary as B
   using (REL; Rel; Trans; Antisym; Irrelevant; _⇒_)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; sym; subst; subst₂)
 
 import Data.List.Properties as Listₚ
diff --git a/src/Data/List/Relation/Ternary/Appending.agda b/src/Data/List/Relation/Ternary/Appending.agda
index 75e50d15af..0d1f935414 100644
--- a/src/Data/List/Relation/Ternary/Appending.agda
+++ b/src/Data/List/Relation/Ternary/Appending.agda
@@ -13,7 +13,7 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
 open import Data.Product as Prod using (∃₂; _×_; _,_; -,_)
 open import Relation.Binary using (REL)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 private
   variable
diff --git a/src/Data/List/Relation/Ternary/Appending/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Properties.agda
index 11db3cf16e..4508a7f719 100644
--- a/src/Data/List/Relation/Ternary/Appending/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Properties.agda
@@ -13,7 +13,7 @@ open import Data.List.Relation.Ternary.Appending
 open import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise; []; _∷_)
 open import Level using (Level)
 open import Relation.Binary using (REL; Rel; Trans)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
index e4ecefddae..757d888012 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
@@ -14,7 +14,8 @@ import Data.List.Relation.Binary.Pointwise as Pw
 open import Data.List.Relation.Binary.Equality.Propositional using (_≋_)
 open import Data.List.Relation.Ternary.Appending.Propositional {A = A}
 open import Function.Base using (_∘′_)
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; setoid)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 import Data.List.Relation.Ternary.Appending.Setoid.Properties (setoid A)
   as Appendingₚ
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
index 60e3e2f4ad..0530b8b965 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
@@ -15,7 +15,7 @@ import Data.List.Properties as Listₚ
 open import Data.List.Relation.Binary.Pointwise using (Pointwise; [])
 import Data.List.Relation.Ternary.Appending.Properties as Appendingₚ
 open import Data.Product using (_,_)
-open import Relation.Binary.PropositionalEquality using (refl)
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 open import Data.List.Relation.Ternary.Appending.Setoid S
 module S = Setoid S; open S renaming (Carrier to A) using (_≈_)
diff --git a/src/Data/List/Relation/Ternary/Interleaving.agda b/src/Data/List/Relation/Ternary/Interleaving.agda
index cfc84e52c4..62206daf2c 100644
--- a/src/Data/List/Relation/Ternary/Interleaving.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving.agda
@@ -16,7 +16,7 @@ open import Data.Product as Prod using (∃; ∃₂; _×_; uncurry; _,_; -,_; pr
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
 -- Definition
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Propositional.agda b/src/Data/List/Relation/Ternary/Interleaving/Propositional.agda
index c1ac5c9adf..55515cd447 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Propositional.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Propositional.agda
@@ -12,7 +12,8 @@ open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.Relation.Binary.Permutation.Propositional as Perm using (_↭_)
 open import Data.List.Relation.Binary.Permutation.Propositional.Properties using (shift)
 import Data.List.Relation.Ternary.Interleaving.Setoid as General
-open import Relation.Binary.PropositionalEquality using (setoid; refl)
+open import Relation.Binary.PropositionalEquality.Core using (refl)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 open Perm.PermutationReasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Propositional/Properties.agda b/src/Data/List/Relation/Ternary/Interleaving/Propositional/Properties.agda
index b8328e48e8..c4253a7253 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Propositional/Properties.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Ternary.Interleaving.Propositional.Properties
 
 import Data.List.Relation.Ternary.Interleaving.Setoid.Properties
   as SetoidProperties
-open import Relation.Binary.PropositionalEquality using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 ------------------------------------------------------------------------
 -- Re-exporting existing properties
diff --git a/src/Data/List/Relation/Unary/All.agda b/src/Data/List/Relation/Unary/All.agda
index 090af7e6b6..52d2baa819 100644
--- a/src/Data/List/Relation/Unary/All.agda
+++ b/src/Data/List/Relation/Unary/All.agda
@@ -24,7 +24,8 @@ open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Unary hiding (_∈_)
 open import Relation.Binary using (Setoid; _Respects_)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Properties as P
 
 private
   variable
diff --git a/src/Data/List/Relation/Unary/AllPairs/Properties.agda b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
index 009ac3ce75..037279cfe1 100644
--- a/src/Data/List/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Nat.Properties using (≤-refl; m<n⇒m<1+n)
 open import Function.Base using (_∘_; flip)
 open import Level using (Level)
 open import Relation.Binary using (Rel; DecSetoid)
-open import Relation.Binary.PropositionalEquality using (_≢_)
+open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (does)
 
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index 6ac93bc05e..6a0020d269 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -41,7 +41,7 @@ open import Function.Inverse as Inv using (_↔_; inverse; Inverse)
 open import Function.Related as Related using (Kind; Related; SK-sym)
 open import Level using (Level)
 open import Relation.Binary as B hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
 open import Relation.Unary as U
   using (Pred; _⟨×⟩_; _⟨→⟩_) renaming (_⊆_ to _⋐_)
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index b9eabc66b3..dc0114238a 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function
 open import Level
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.Properties.Setoid using (respʳ-flip)
 
 module Data.List.Relation.Unary.Enumerates.Setoid.Properties where
diff --git a/src/Data/List/Relation/Unary/Linked/Properties.agda b/src/Data/List/Relation/Unary/Linked/Properties.agda
index 7106ad3e1e..d969986a55 100644
--- a/src/Data/List/Relation/Unary/Linked/Properties.agda
+++ b/src/Data/List/Relation/Unary/Linked/Properties.agda
@@ -23,7 +23,7 @@ open import Data.Maybe.Relation.Binary.Connected
 open import Level using (Level)
 open import Function.Base using (_∘_; flip; _on_)
 open import Relation.Binary using (Rel; Transitive; DecSetoid)
-open import Relation.Binary.PropositionalEquality using (_≢_)
+open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (yes; no; does)
 
diff --git a/src/Data/List/Relation/Unary/Sufficient.agda b/src/Data/List/Relation/Unary/Sufficient.agda
index be7bf76261..35d63add25 100644
--- a/src/Data/List/Relation/Unary/Sufficient.agda
+++ b/src/Data/List/Relation/Unary/Sufficient.agda
@@ -18,7 +18,7 @@ module Data.List.Relation.Unary.Sufficient where
 
 open import Level using (Level; _⊔_)
 open import Data.List.Base using (List; []; _∷_; [_]; _++_)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/List/Relation/Unary/Unique/DecPropositional.agda b/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
index 19ba99025f..3f552b6fa9 100644
--- a/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary using (DecSetoid; DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (decSetoid)
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.List.Relation.Unary.Unique.DecPropositional
   {a} {A : Set a} (_≟_ : DecidableEquality A) where
diff --git a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
index 7512f05246..6d28ed56d6 100644
--- a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
@@ -11,7 +11,7 @@ import Data.List.Relation.Unary.Unique.DecSetoid.Properties as Setoid
 open import Data.List.Relation.Unary.All.Properties using (all-filter)
 open import Level
 open import Relation.Binary using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (decSetoid)
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.List.Relation.Unary.Unique.DecPropositional.Properties
   {a} {A : Set a} (_≟_ : DecidableEquality A) where
diff --git a/src/Data/List/Relation/Unary/Unique/Propositional.agda b/src/Data/List/Relation/Unary/Unique/Propositional.agda
index 4e3b600d2e..2c32f0da46 100644
--- a/src/Data/List/Relation/Unary/Unique/Propositional.agda
+++ b/src/Data/List/Relation/Unary/Unique/Propositional.agda
@@ -8,7 +8,7 @@
 
 module Data.List.Relation.Unary.Unique.Propositional {a} {A : Set a} where
 
-open import Relation.Binary.PropositionalEquality using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 open import Data.List.Relation.Unary.Unique.Setoid as SetoidUnique
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
index a7413994ba..33b66d2bd1 100644
--- a/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
@@ -24,7 +24,7 @@ open import Data.Nat.Base using (_<_)
 open import Function.Base using (_∘_; id)
 open import Level using (Level)
 open import Relation.Binary using (Rel; Setoid)
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Negation using (contraposition)
diff --git a/src/Data/Maybe/Relation/Binary/Connected.agda b/src/Data/Maybe/Relation/Binary/Connected.agda
index e2fea581af..33bff58154 100644
--- a/src/Data/Maybe/Relation/Binary/Connected.agda
+++ b/src/Data/Maybe/Relation/Binary/Connected.agda
@@ -13,7 +13,7 @@ open import Data.Product
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 
diff --git a/src/Data/Maybe/Relation/Binary/Pointwise.agda b/src/Data/Maybe/Relation/Binary/Pointwise.agda
index 389c1fde6e..3ae5ad421e 100644
--- a/src/Data/Maybe/Relation/Binary/Pointwise.agda
+++ b/src/Data/Maybe/Relation/Binary/Pointwise.agda
@@ -13,7 +13,7 @@ open import Data.Product
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 
diff --git a/src/Data/Maybe/Relation/Unary/All.agda b/src/Data/Maybe/Relation/Unary/All.agda
index 0c2c3faecd..a3818c38b2 100644
--- a/src/Data/Maybe/Relation/Unary/All.agda
+++ b/src/Data/Maybe/Relation/Unary/All.agda
@@ -16,7 +16,7 @@ open import Data.Product as Prod using (_,_)
 open import Function.Base using (id; _∘′_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
-open import Relation.Binary.PropositionalEquality as P using (_≡_; cong)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; cong)
 open import Relation.Unary
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Data/Maybe/Relation/Unary/Any.agda b/src/Data/Maybe/Relation/Unary/Any.agda
index 61a06e840a..ea81316c77 100644
--- a/src/Data/Maybe/Relation/Unary/Any.agda
+++ b/src/Data/Maybe/Relation/Unary/Any.agda
@@ -13,7 +13,7 @@ open import Data.Product as Prod using (∃; _,_; -,_)
 open import Function.Base using (id)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
-open import Relation.Binary.PropositionalEquality as P using (_≡_; cong)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; cong)
 open import Relation.Unary
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 6f126a80d6..13701b486c 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -21,7 +21,7 @@ open import Data.Nat.DivMod
 open import Data.Product as Prod
 open import Function
 open import Level using (0ℓ)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; trans; cong; subst; module ≡-Reasoning)
 open import Relation.Nullary as Nullary hiding (recompute)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index c202d740ba..26cd81bf27 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -20,8 +20,9 @@ open import Relation.Nullary.Decidable as Dec using (False; yes; no)
 open import Relation.Nullary.Negation.Core using (contradiction)
 open import Relation.Binary
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
-open import Relation.Binary.PropositionalEquality as PropEq
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst)
+import Relation.Binary.PropositionalEquality.Properties as PropEq
 
 ------------------------------------------------------------------------
 -- Definition
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index fe90cc0282..9e21f6f812 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -21,7 +21,7 @@ open import Function
 open import Induction using (build)
 open import Induction.Lexicographic using (_⊗_; [_⊗_])
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; subst; cong)
 open import Relation.Nullary.Decidable using (Dec)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/Nat/LCM.agda b/src/Data/Nat/LCM.agda
index 9b8932785a..5ce1c7cd2a 100644
--- a/src/Data/Nat/LCM.agda
+++ b/src/Data/Nat/LCM.agda
@@ -18,7 +18,7 @@ open import Data.Nat.GCD
 open import Data.Product
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl; sym; trans; cong; cong₂; module ≡-Reasoning)
 open import Relation.Binary
 open import Relation.Nullary.Decidable using (False; fromWitnessFalse)
diff --git a/src/Data/Nat/Logarithm.agda b/src/Data/Nat/Logarithm.agda
index 4e08abcf4d..461fa540e6 100644
--- a/src/Data/Nat/Logarithm.agda
+++ b/src/Data/Nat/Logarithm.agda
@@ -11,7 +11,7 @@ module Data.Nat.Logarithm where
 open import Data.Nat
 open import Data.Nat.Induction using (<-wellFounded)
 open import Data.Nat.Logarithm.Core
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 ------------------------------------------------------------------------
 -- Logarithm base 2
diff --git a/src/Data/Nat/Show/Properties.agda b/src/Data/Nat/Show/Properties.agda
index 87622b79bf..564264946c 100644
--- a/src/Data/Nat/Show/Properties.agda
+++ b/src/Data/Nat/Show/Properties.agda
@@ -13,7 +13,7 @@ import Data.List.Properties as Listₚ
 open import Data.Nat.Base using (ℕ)
 open import Data.Nat.Properties using (_≤?_)
 open import Relation.Nullary.Decidable using (True)
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Data.Nat.Show using (charsInBase)
 
 module Data.Nat.Show.Properties where
diff --git a/src/Data/Product/Effectful/Examples.agda b/src/Data/Product/Effectful/Examples.agda
index 90488642f6..daf05ce431 100644
--- a/src/Data/Product/Effectful/Examples.agda
+++ b/src/Data/Product/Effectful/Examples.agda
@@ -19,7 +19,7 @@ open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function
 import Function.Identity.Effectful as Id
 open import Relation.Binary using (Rel)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- Examples
diff --git a/src/Data/Product/Function/Dependent/Setoid.agda b/src/Data/Product/Function/Dependent/Setoid.agda
index f6629e048a..cc6d4a12fc 100644
--- a/src/Data/Product/Function/Dependent/Setoid.agda
+++ b/src/Data/Product/Function/Dependent/Setoid.agda
@@ -28,7 +28,8 @@ open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
   using (_atₛ_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 ------------------------------------------------------------------------
 -- Properties related to "relatedness"
diff --git a/src/Data/Product/Function/Dependent/Setoid/WithK.agda b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
index a343e22b43..c9468fb939 100644
--- a/src/Data/Product/Function/Dependent/Setoid/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
@@ -30,7 +30,8 @@ open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
   using (_atₛ_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 ------------------------------------------------------------------------
 -- Combinator for Inverse
diff --git a/src/Data/Product/Relation/Binary/Lex/Strict.agda b/src/Data/Product/Relation/Binary/Lex/Strict.agda
index 5c7c117a4b..d0a09fa4ac 100644
--- a/src/Data/Product/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/Strict.agda
@@ -22,7 +22,7 @@ open import Level
 open import Relation.Nullary.Decidable
 open import Relation.Binary
 open import Relation.Binary.Consequences
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
index 5f6953071a..96f7211ce7 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
@@ -15,7 +15,7 @@ open import Relation.Binary as B
   using (_⇒_; Setoid; IsEquivalence)
 open import Relation.Binary.Indexed.Heterogeneous as I
   using (IREL; IRel; IndexedSetoid; IsIndexedEquivalence)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
 -- Pointwise lifting
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
index c1228b5a91..13edd9aa66 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
@@ -16,7 +16,8 @@ open import Function.Inverse using (Inverse)
 open import Relation.Binary using (_⇒_)
 open import Relation.Binary.HeterogeneousEquality as H using (_≅_)
 open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 ------------------------------------------------------------------------
 -- The propositional equality setoid over sigma types can be
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index 1bbf5981e1..fc77109043 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -27,7 +27,8 @@ open import Function.Surjection as Surj
   using (Surjection; _↠_; module Surjection)
 open import Relation.Nullary.Decidable using (_×-dec_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 module _ {a₁ a₂ ℓ₁ ℓ₂} {A₁ : Set a₁} {A₂ : Set a₂} where
 
diff --git a/src/Data/String.agda b/src/Data/String.agda
index 98814b1909..fffe1ac300 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -23,7 +23,7 @@ open import Data.Char.Base as Char using (Char)
 import Data.Char.Properties as Char using (_≟_)
 open import Function
 open import Relation.Binary using (Rel)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Nullary.Decidable using (does)
 open import Relation.Unary using (Pred; Decidable)
 
diff --git a/src/Data/String/Properties.agda b/src/Data/String/Properties.agda
index e0f47fa175..b7ed9a93d6 100644
--- a/src/Data/String/Properties.agda
+++ b/src/Data/String/Properties.agda
@@ -20,7 +20,7 @@ open import Relation.Nullary.Decidable using (map′; isYes)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core
 import Relation.Binary.Construct.On as On
-import Relation.Binary.PropositionalEquality as PropEq
+import Relation.Binary.PropositionalEquality.Properties as PropEq
 
 ------------------------------------------------------------------------
 -- Primitive properties
diff --git a/src/Data/String/Unsafe.agda b/src/Data/String/Unsafe.agda
index 9c001365a3..ff1ff28408 100644
--- a/src/Data/String/Unsafe.agda
+++ b/src/Data/String/Unsafe.agda
@@ -16,7 +16,7 @@ open import Data.Product.Base using (proj₂)
 open import Data.String.Base
 open import Function.Base using (_∘′_)
 
-open import Relation.Binary.PropositionalEquality; open ≡-Reasoning
+open import Relation.Binary.PropositionalEquality.Core; open ≡-Reasoning
 open import Relation.Binary.PropositionalEquality.TrustMe using (trustMe)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index 063ba49bb7..34c676bcbb 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -18,7 +18,7 @@ open import Level
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ----------------------------------------------------------------------
 -- Definition
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index d5ec85b943..23d10172c9 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -17,7 +17,8 @@ open import Function.Inverse using (Inverse)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 ----------------------------------------------------------------------
 -- Definition
diff --git a/src/Data/Tree/AVL/Indexed.agda b/src/Data/Tree/AVL/Indexed.agda
index 8cd4fc49bd..16ebdff70e 100644
--- a/src/Data/Tree/AVL/Indexed.agda
+++ b/src/Data/Tree/AVL/Indexed.agda
@@ -20,7 +20,7 @@ open import Data.DifferenceList using (DiffList; []; _∷_; _++_)
 open import Function.Base as F hiding (const)
 open import Relation.Unary
 open import Relation.Binary using (_Respects_; Tri; tri<; tri≈; tri>)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
index e37c7f9b34..5add64a3cc 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
@@ -20,7 +20,7 @@ open import Function.Base
 open import Function.Nary.NonDependent using (congₙ)
 open import Level using (Level; _⊔_)
 
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Nullary.Decidable using (Dec; yes; map′; _×-dec_)
 open import Relation.Unary using (Pred; Decidable; Universal; Irrelevant; _⊆_; _∪_)
 
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
index 8b5b195d42..f2dec21b8c 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
@@ -22,7 +22,7 @@ open import Function.Base as F
 open import Level using (Level)
 
 open import Relation.Binary using (_Respects_; tri<; tri≈; tri>)
-open import Relation.Binary.PropositionalEquality using (_≡_) renaming (refl to ≡-refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_) renaming (refl to ≡-refl)
 open import Relation.Nullary using (¬_; Dec; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Unary using (Pred; _∩_)
diff --git a/src/Data/Tree/AVL/Indexed/WithK.agda b/src/Data/Tree/AVL/Indexed/WithK.agda
index 2ef4ec3544..7467a3813f 100644
--- a/src/Data/Tree/AVL/Indexed/WithK.agda
+++ b/src/Data/Tree/AVL/Indexed/WithK.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --with-K --safe #-}
 
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; subst)
 
 module Data.Tree.AVL.Indexed.WithK
        {k r} (Key : Set k) {_<_ : Rel Key r}
diff --git a/src/Data/Tree/AVL/IndexedMap.agda b/src/Data/Tree/AVL/IndexedMap.agda
index 21413e0478..cb16cd6220 100644
--- a/src/Data/Tree/AVL/IndexedMap.agda
+++ b/src/Data/Tree/AVL/IndexedMap.agda
@@ -8,7 +8,7 @@
 
 open import Data.Product as Prod
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality using (_≡_; cong; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
 import Data.Tree.AVL.Value
 
 module Data.Tree.AVL.IndexedMap
diff --git a/src/Data/Tree/AVL/Key.agda b/src/Data/Tree/AVL/Key.agda
index dc7e77824d..9a2b7e8521 100644
--- a/src/Data/Tree/AVL/Key.agda
+++ b/src/Data/Tree/AVL/Key.agda
@@ -17,7 +17,7 @@ open import Level
 open import Data.Empty
 open import Data.Unit
 open import Data.Product
-open import Relation.Binary.PropositionalEquality using (_≡_ ; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_ ; refl)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Construct.Add.Extrema
   as AddExtremaToSet using (_±)
diff --git a/src/Data/Tree/AVL/NonEmpty/Propositional.agda b/src/Data/Tree/AVL/NonEmpty/Propositional.agda
index f537b1c673..82ea9125fd 100644
--- a/src/Data/Tree/AVL/NonEmpty/Propositional.agda
+++ b/src/Data/Tree/AVL/NonEmpty/Propositional.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary using (Rel; IsStrictTotalOrder; StrictTotalOrder)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; subst)
 
 module Data.Tree.AVL.NonEmpty.Propositional
   {k r} {Key : Set k} {_<_ : Rel Key r}
diff --git a/src/Data/Tree/AVL/Sets/Membership.agda b/src/Data/Tree/AVL/Sets/Membership.agda
index 947036500c..53e0aea9f9 100644
--- a/src/Data/Tree/AVL/Sets/Membership.agda
+++ b/src/Data/Tree/AVL/Sets/Membership.agda
@@ -18,7 +18,6 @@ open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
 open import Function.Base using (_∘_; _∘′_; const)
 
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
 open import Relation.Nullary using (¬_; yes; no; Reflects)
 open import Relation.Nullary.Reflects using (fromEquivalence)
 
diff --git a/src/Data/Tree/AVL/Sets/Membership/Properties.agda b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
index 7c1f27a1d1..75fc1e9242 100644
--- a/src/Data/Tree/AVL/Sets/Membership/Properties.agda
+++ b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
@@ -18,7 +18,7 @@ open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
 open import Function.Base using (_∘_; _∘′_; const)
 
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Nullary using (¬_; yes; no; Reflects)
 open import Relation.Nullary.Reflects using (fromEquivalence)
 
diff --git a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
index 8b29e710f8..603582bf02 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
@@ -13,7 +13,7 @@ open import Data.Vec.Functional.Relation.Binary.Pointwise
 import Data.Vec.Functional.Relation.Binary.Pointwise.Properties as PW
 open import Level using (Level)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 module Data.Vec.Functional.Relation.Binary.Equality.Setoid
   {a ℓ} (S : Setoid a ℓ) where
diff --git a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
index c0ae709b0f..0aa81b07a7 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Vec.Functional.Relation.Binary.Pointwise
 open import Function.Base
 open import Level using (Level)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 private
   variable
diff --git a/src/Data/Vec/Membership/DecPropositional.agda b/src/Data/Vec/Membership/DecPropositional.agda
index 0b20505b4b..e1b100f0af 100644
--- a/src/Data/Vec/Membership/DecPropositional.agda
+++ b/src/Data/Vec/Membership/DecPropositional.agda
@@ -7,7 +7,8 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary using (Decidable)
-open import Relation.Binary.PropositionalEquality using (_≡_; decSetoid)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.Vec.Membership.DecPropositional
   {a} {A : Set a} (_≟_ : Decidable (_≡_ {A = A})) where
diff --git a/src/Data/Vec/Membership/Propositional.agda b/src/Data/Vec/Membership/Propositional.agda
index d76cedd1d3..21213e679d 100644
--- a/src/Data/Vec/Membership/Propositional.agda
+++ b/src/Data/Vec/Membership/Propositional.agda
@@ -11,7 +11,8 @@ module Data.Vec.Membership.Propositional {a} {A : Set a} where
 
 open import Data.Vec.Base using (Vec)
 open import Data.Vec.Relation.Unary.Any using (Any)
-open import Relation.Binary.PropositionalEquality using (setoid; subst)
+open import Relation.Binary.PropositionalEquality.Core using (subst)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 import Data.Vec.Membership.Setoid as SetoidMembership
 
diff --git a/src/Data/Vec/Membership/Propositional/Properties.agda b/src/Data/Vec/Membership/Propositional/Properties.agda
index 8efef380d4..983e70ae63 100644
--- a/src/Data/Vec/Membership/Propositional/Properties.agda
+++ b/src/Data/Vec/Membership/Propositional/Properties.agda
@@ -21,7 +21,7 @@ open import Data.List.Membership.Propositional
 open import Level using (Level)
 open import Function.Base using (_∘_; id)
 open import Relation.Unary using (Pred)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 private
   variable
diff --git a/src/Data/Vec/Properties/WithK.agda b/src/Data/Vec/Properties/WithK.agda
index 8d3bd95aee..7ca8cc2c91 100644
--- a/src/Data/Vec/Properties/WithK.agda
+++ b/src/Data/Vec/Properties/WithK.agda
@@ -12,7 +12,7 @@ module Data.Vec.Properties.WithK where
 open import Data.Nat.Base
 open import Data.Nat.Properties using (+-assoc)
 open import Data.Vec.Base
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
 open import Relation.Binary.HeterogeneousEquality as H using (_≅_; refl)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Recursive/Properties.agda b/src/Data/Vec/Recursive/Properties.agda
index 745681a617..d557808cee 100644
--- a/src/Data/Vec/Recursive/Properties.agda
+++ b/src/Data/Vec/Recursive/Properties.agda
@@ -14,7 +14,7 @@ open import Data.Product
 open import Data.Vec.Recursive
 open import Data.Vec.Base using (Vec; _∷_)
 open import Function.Inverse using (_↔_; inverse)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
 open ≡-Reasoning
 
 private
diff --git a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
index 116b104829..d40bd24998 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
@@ -19,7 +19,6 @@ open import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
 open import Function.Base
 open import Level using (_⊔_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
 
 open Setoid S renaming (Carrier to A)
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index 4ac943401e..a3ff214a06 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -19,7 +19,7 @@ open import Function.Base using (flip)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl; cong)
 import Relation.Nullary as Nullary
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _×-dec_; _⊎-dec_)
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index 877d345045..d3c24ab42b 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -22,7 +22,6 @@ open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 open import Function.Base using (id)
 open import Relation.Binary
 import Relation.Binary.Construct.NonStrictToStrict as Conv
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
 open import Relation.Nullary hiding (Irrelevant)
 open import Level using (Level; _⊔_)
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 7e4052476b..a1f11fb1b1 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -27,7 +27,7 @@ open import Relation.Nullary using (yes; no; ¬_)
 open import Relation.Binary
 open import Relation.Binary.Consequences
 open import Relation.Binary.Construct.On as On using (wellFounded)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Level using (Level; _⊔_)
 
 private
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
index ff6fd79833..c7b2b3d7ea 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
@@ -19,7 +19,7 @@ open import Function.Bundles using (module Equivalence; _⇔_; mk⇔)
 open import Function.Properties.Equivalence using (⇔-setoid)
 open import Level using (Level; _⊔_; 0ℓ)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Construct.Closure.Transitive as Plus
   hiding (equivalent; map)
 open import Relation.Nullary
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
index e435e2681b..011c2b731b 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
@@ -17,7 +17,7 @@ open import Level using (Level; _⊔_)
 open import Function.Base using (_∘_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Decidable using (yes; no; _×-dec_; map′)
 open import Relation.Unary using (Pred)
 
diff --git a/src/Data/Vec/Relation/Unary/All.agda b/src/Data/Vec/Relation/Unary/All.agda
index 9ba434368c..2eda26c71e 100644
--- a/src/Data/Vec/Relation/Unary/All.agda
+++ b/src/Data/Vec/Relation/Unary/All.agda
@@ -20,7 +20,7 @@ open import Level using (Level; _⊔_)
 open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no)
 open import Relation.Unary hiding (_∈_)
 open import Relation.Binary using (Setoid; _Respects_)
-open import Relation.Binary.PropositionalEquality as P using (subst)
+open import Relation.Binary.PropositionalEquality.Core as P using (subst)
 
 private
   variable
diff --git a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
index 339a884a3b..672c9fa485 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Nat.Base using (zero; suc; _+_)
 open import Function.Base using (_∘_)
 open import Level using (Level)
 open import Relation.Binary using (Rel)
-open import Relation.Binary.PropositionalEquality using (_≢_)
+open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 
 private
   variable
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index 5b3eab5fa1..1c9f171d7b 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -29,7 +29,7 @@ open import Level using (Level)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Unary hiding (_∈_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_; refl)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/Vec/Relation/Unary/Linked/Properties.agda b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
index a9604e1e6a..340b6b81a2 100644
--- a/src/Data/Vec/Relation/Unary/Linked/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Nat.Properties using (<-pred)
 open import Level using (Level)
 open import Function.Base using (_∘_; flip; _on_)
 open import Relation.Binary using (Rel; Transitive; DecSetoid)
-open import Relation.Binary.PropositionalEquality using (_≢_)
+open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (yes; no; does)
 
diff --git a/src/Data/Vec/Relation/Unary/Unique/Propositional.agda b/src/Data/Vec/Relation/Unary/Unique/Propositional.agda
index 836f7f5b67..20b042ef17 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Propositional.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Propositional.agda
@@ -8,7 +8,7 @@
 
 module Data.Vec.Relation.Unary.Unique.Propositional {a} {A : Set a} where
 
-open import Relation.Binary.PropositionalEquality using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 open import Data.Vec.Relation.Unary.Unique.Setoid as SetoidUnique
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
index 66371b89b5..4c433c04f6 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
@@ -19,7 +19,7 @@ open import Data.Nat.Base using (ℕ; _+_)
 open import Function.Base using (_∘_; id)
 open import Level using (Level)
 open import Relation.Binary using (Rel; Setoid)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Negation using (contradiction; contraposition)
 
 private
diff --git a/src/Data/W/Sized.agda b/src/Data/W/Sized.agda
index c16b21828e..72d742ff4b 100644
--- a/src/Data/W/Sized.agda
+++ b/src/Data/W/Sized.agda
@@ -15,7 +15,7 @@ open import Data.Product using (_,_; -,_; proj₂)
 open import Data.Container.Core as Container using (Container; ⟦_⟧; Shape; Position; _⇒_; ⟪_⟫)
 open import Data.Container.Relation.Unary.All using (□; all)
 open import Relation.Nullary.Negation using (¬_)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
   variable
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index 03f82db8bb..0751cb5b15 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -21,7 +21,8 @@ open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
 open import Relation.Binary as B hiding (Rel; _⇔_)
 import Relation.Binary.Properties.Setoid as SetoidProperties
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 open import Relation.Nullary
 open import Relation.Nullary.Decidable hiding (map)
 open import Relation.Nullary.Negation
diff --git a/src/Effect/Monad/Partiality/All.agda b/src/Effect/Monad/Partiality/All.agda
index 9bc5d4bc7e..7e90400a85 100644
--- a/src/Effect/Monad/Partiality/All.agda
+++ b/src/Effect/Monad/Partiality/All.agda
@@ -14,7 +14,7 @@ open import Codata.Musical.Notation
 open import Function
 open import Level
 open import Relation.Binary using (_Respects_; IsEquivalence)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 open Partiality._⊥
 open Partiality.Equality using (Rel)
diff --git a/src/Function/Construct/Identity.agda b/src/Function/Construct/Identity.agda
index 32660bee09..bec0ecd3d6 100644
--- a/src/Function/Construct/Identity.agda
+++ b/src/Function/Construct/Identity.agda
@@ -15,7 +15,8 @@ import Function.Definitions as Definitions
 import Function.Structures as Structures
 open import Level using (Level)
 open import Relation.Binary as B hiding (_⇔_; IsEquivalence)
-open import Relation.Binary.PropositionalEquality using (_≡_; setoid)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 private
   variable
diff --git a/src/Function/Endomorphism/Propositional.agda b/src/Function/Endomorphism/Propositional.agda
index 8a78f69496..d14128edfe 100644
--- a/src/Function/Endomorphism/Propositional.agda
+++ b/src/Function/Endomorphism/Propositional.agda
@@ -21,7 +21,8 @@ open import Data.Product using (_,_)
 open import Function
 open import Function.Equality using (_⟨$⟩_)
 open import Relation.Binary using (_Preserves_⟶_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 import Function.Endomorphism.Setoid (P.setoid A) as Setoid
 
diff --git a/src/Function/Metric/Nat/Structures.agda b/src/Function/Metric/Nat/Structures.agda
index 648e7a4adc..839dc7b974 100644
--- a/src/Function/Metric/Nat/Structures.agda
+++ b/src/Function/Metric/Nat/Structures.agda
@@ -12,7 +12,7 @@ open import Data.Nat.Base hiding (suc)
 open import Function.Base using (const)
 open import Level using (Level; suc)
 open import Relation.Binary hiding (Symmetric)
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 open import Function.Metric.Nat.Core
 open import Function.Metric.Nat.Definitions
diff --git a/src/Function/Metric/Rational/Structures.agda b/src/Function/Metric/Rational/Structures.agda
index 824f90b912..daa2b71a60 100644
--- a/src/Function/Metric/Rational/Structures.agda
+++ b/src/Function/Metric/Rational/Structures.agda
@@ -10,7 +10,7 @@ open import Data.Rational.Base
 open import Function.Base using (const)
 open import Level using (Level; suc)
 open import Relation.Binary hiding (Symmetric)
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 open import Function.Metric.Rational.Core
 open import Function.Metric.Rational.Definitions
diff --git a/src/Function/Properties/Bijection.agda b/src/Function/Properties/Bijection.agda
index e484098774..c8abd7db19 100644
--- a/src/Function/Properties/Bijection.agda
+++ b/src/Function/Properties/Bijection.agda
@@ -11,7 +11,7 @@ module Function.Properties.Bijection where
 open import Function.Bundles
 open import Level using (Level)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P using (setoid)
+import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 open import Data.Product using (_,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
diff --git a/src/Function/Properties/Equivalence.agda b/src/Function/Properties/Equivalence.agda
index ced34d9bd3..066316c01b 100644
--- a/src/Function/Properties/Equivalence.agda
+++ b/src/Function/Properties/Equivalence.agda
@@ -12,7 +12,7 @@ module Function.Properties.Equivalence where
 open import Function.Bundles using (Equivalence; _⇔_)
 open import Level
 open import Relation.Binary using (Setoid; IsEquivalence)
-import Relation.Binary.PropositionalEquality as Eq using (setoid)
+import Relation.Binary.PropositionalEquality.Properties as Eq
 
 import Function.Construct.Identity as Identity
 import Function.Construct.Symmetry as Symmetry
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index 508a8664be..6ef0b45af0 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -14,7 +14,8 @@ open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Bundles
 open import Level using (Level)
 open import Relation.Binary using (Setoid; IsEquivalence)
-open import Relation.Binary.PropositionalEquality as P using (setoid)
+import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 open import Function.Consequences
 
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index fa114d077c..ca06dceffb 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -18,7 +18,8 @@ open import Function.Inverse     as Inv     using (Inverse; _↔_)
 open import Function.LeftInverse as LeftInv using (LeftInverse)
 open import Function.Surjection  as Surj    using (Surjection)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 open import Data.Product using (_,_; proj₁; proj₂; <_,_>)
 
 import Function.Related.Propositional as R
diff --git a/src/Function/Related/Propositional.agda b/src/Function/Related/Propositional.agda
index 0ef8830265..5d62c0718b 100644
--- a/src/Function/Related/Propositional.agda
+++ b/src/Function/Related/Propositional.agda
@@ -13,7 +13,8 @@ open import Relation.Binary
   using (Sym; Reflexive; Trans; IsEquivalence; Setoid; IsPreorder; Preorder)
 open import Function.Bundles
 open import Function.Base
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 open import Function.Properties.Surjection   using (↠⇒↪; ↠⇒⇔)
 open import Function.Properties.RightInverse using (↪⇒↠)
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index cf151d9d3c..58cf6bab88 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -28,7 +28,7 @@ open import Function.Equivalence as Eq using (_⇔_; Equivalence)
 open import Function.Inverse as Inv using (_↔_; Inverse; inverse)
 open import Function.Related
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ)
 import Relation.Nullary.Indexed as I
diff --git a/src/Reflection/AST/Abstraction.agda b/src/Reflection/AST/Abstraction.agda
index 9ed64b8835..793c729aec 100644
--- a/src/Reflection/AST/Abstraction.agda
+++ b/src/Reflection/AST/Abstraction.agda
@@ -8,12 +8,12 @@
 
 module Reflection.AST.Abstraction where
 
-open import Data.Product                          using (_×_; <_,_>; uncurry)
-open import Data.String as String                 using (String)
+open import Data.Product                               using (_×_; <_,_>; uncurry)
+open import Data.String as String                      using (String)
 open import Level
-open import Relation.Nullary.Decidable            using (Dec; map′; _×-dec_)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong₂)
+open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 private
   variable
diff --git a/src/Reflection/AST/Argument.agda b/src/Reflection/AST/Argument.agda
index 1b0884b615..0feb2deddd 100644
--- a/src/Reflection/AST/Argument.agda
+++ b/src/Reflection/AST/Argument.agda
@@ -8,11 +8,11 @@
 
 module Reflection.AST.Argument where
 
-open import Data.List.Base as List                using (List; []; _∷_)
-open import Data.Product                          using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable            using (Dec; map′; _×-dec_)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong₂)
+open import Data.List.Base as List                     using (List; []; _∷_)
+open import Data.Product                               using (_×_; <_,_>; uncurry)
+open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 open import Level
 
 open import Reflection.AST.Argument.Visibility
diff --git a/src/Reflection/AST/Argument/Information.agda b/src/Reflection/AST/Argument/Information.agda
index fe09976fb9..e0f5be24a5 100644
--- a/src/Reflection/AST/Argument/Information.agda
+++ b/src/Reflection/AST/Argument/Information.agda
@@ -8,10 +8,10 @@
 
 module Reflection.AST.Argument.Information where
 
-open import Data.Product.Base                     using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable            using (map′; _×-dec_)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong₂)
+open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
+open import Relation.Nullary.Decidable                 using (map′; _×-dec_)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 open import Reflection.AST.Argument.Modality as Modality using (Modality)
 open import Reflection.AST.Argument.Visibility as Visibility using (Visibility)
diff --git a/src/Reflection/AST/Argument/Modality.agda b/src/Reflection/AST/Argument/Modality.agda
index 95f15b2264..02e481af58 100644
--- a/src/Reflection/AST/Argument/Modality.agda
+++ b/src/Reflection/AST/Argument/Modality.agda
@@ -8,10 +8,10 @@
 
 module Reflection.AST.Argument.Modality where
 
-open import Data.Product.Base                     using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable            using (map′; _×-dec_)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong₂)
+open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
+open import Relation.Nullary.Decidable                 using (map′; _×-dec_)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 open import Reflection.AST.Argument.Relevance as Relevance using (Relevance)
 open import Reflection.AST.Argument.Quantity as Quantity using (Quantity)
diff --git a/src/Reflection/AST/Argument/Quantity.agda b/src/Reflection/AST/Argument/Quantity.agda
index 0c53a521c1..5b3b98e092 100644
--- a/src/Reflection/AST/Argument/Quantity.agda
+++ b/src/Reflection/AST/Argument/Quantity.agda
@@ -8,9 +8,9 @@
 
 module Reflection.AST.Argument.Quantity where
 
-open import Relation.Nullary                      using (yes; no)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (refl)
+open import Relation.Nullary                           using (yes; no)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 ------------------------------------------------------------------------
 -- Re-exporting the builtins publicly
diff --git a/src/Reflection/AST/Argument/Relevance.agda b/src/Reflection/AST/Argument/Relevance.agda
index 6cdce1d96f..03ba0f4f0a 100644
--- a/src/Reflection/AST/Argument/Relevance.agda
+++ b/src/Reflection/AST/Argument/Relevance.agda
@@ -8,9 +8,9 @@
 
 module Reflection.AST.Argument.Relevance where
 
-open import Relation.Nullary                      using (yes; no)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (refl)
+open import Relation.Nullary                           using (yes; no)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 ------------------------------------------------------------------------
 -- Re-exporting the builtins publicly
diff --git a/src/Reflection/AST/Argument/Visibility.agda b/src/Reflection/AST/Argument/Visibility.agda
index 4f94b24fe3..b5e9236043 100644
--- a/src/Reflection/AST/Argument/Visibility.agda
+++ b/src/Reflection/AST/Argument/Visibility.agda
@@ -8,9 +8,9 @@
 
 module Reflection.AST.Argument.Visibility where
 
-open import Relation.Nullary                      using (yes; no)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (refl)
+open import Relation.Nullary                           using (yes; no)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 ------------------------------------------------------------------------
 -- Re-exporting the builtins publicly
diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index 5b6dace7cd..7ccabb6f51 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -8,12 +8,12 @@
 
 module Reflection.AST.Definition where
 
-import Data.List.Properties as Listₚ              using (≡-dec)
-import Data.Nat.Properties as ℕₚ                 using (_≟_)
-open import Data.Product                          using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable            using (map′; _×-dec_; yes; no)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; cong₂)
+import Data.List.Properties as Listₚ                   using (≡-dec)
+import Data.Nat.Properties as ℕₚ                       using (_≟_)
+open import Data.Product                               using (_×_; <_,_>; uncurry)
+open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 
 import Reflection.AST.Argument as Arg
 import Reflection.AST.Name     as Name
diff --git a/src/Reflection/AST/Literal.agda b/src/Reflection/AST/Literal.agda
index 33ceed9ca5..ff8835236a 100644
--- a/src/Reflection/AST/Literal.agda
+++ b/src/Reflection/AST/Literal.agda
@@ -16,10 +16,10 @@ import Data.String as String using (_≟_)
 import Data.Word as Word     using (_≟_)
 import Reflection.AST.Meta as Meta
 import Reflection.AST.Name as Name
-open import Relation.Nullary                      using (yes; no)
-open import Relation.Nullary.Decidable            using (map′; isYes)
-open import Relation.Binary                       using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Nullary                           using (yes; no)
+open import Relation.Nullary.Decidable                 using (map′; isYes)
+open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 ------------------------------------------------------------------------
 -- Re-exporting the builtin type and constructors
diff --git a/src/Reflection/AST/Meta.agda b/src/Reflection/AST/Meta.agda
index 93584f573c..56bda50d31 100644
--- a/src/Reflection/AST/Meta.agda
+++ b/src/Reflection/AST/Meta.agda
@@ -8,12 +8,12 @@
 
 module Reflection.AST.Meta where
 
-import Data.Nat.Properties as ℕₚ                 using (_≟_)
-open import Function.Base                        using (_on_)
-open import Relation.Nullary.Decidable           using (map′)
-open import Relation.Binary                      using (Rel; Decidable; DecidableEquality)
+import Data.Nat.Properties as ℕₚ                       using (_≟_)
+open import Function.Base                              using (_on_)
+open import Relation.Nullary.Decidable                 using (map′)
+open import Relation.Binary                            using (Rel; Decidable; DecidableEquality)
 import Relation.Binary.Construct.On as On
-open import Relation.Binary.PropositionalEquality using (_≡_; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 
 open import Agda.Builtin.Reflection public
   using (Meta) renaming (primMetaToNat to toℕ; primMetaEquality to _≡ᵇ_)
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index 1baa63029f..a46940e0a5 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -8,14 +8,14 @@
 
 module Reflection.AST.Name where
 
-open import Data.List.Base                        using (List)
-import Data.Product.Properties as Prodₚ           using (≡-dec)
-import Data.Word.Properties as Wₚ                 using (_≟_)
-open import Function.Base                         using (_on_)
-open import Relation.Nullary.Decidable            using (map′)
-open import Relation.Binary                       using (Rel; Decidable; DecidableEquality)
-open import Relation.Binary.Construct.On          using (decidable)
-open import Relation.Binary.PropositionalEquality using (_≡_; cong)
+open import Data.List.Base              using (List)
+import Data.Product.Properties as Prodₚ using (≡-dec)
+import Data.Word.Properties as Wₚ       using (_≟_)
+open import Function.Base               using (_on_)
+open import Relation.Nullary.Decidable                 using (map′)
+open import Relation.Binary                            using (Rel; Decidable; DecidableEquality)
+open import Relation.Binary.Construct.On               using (decidable)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 
 ----------------------------------------------------------------------
 -- Re-export built-ins
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index 183f839015..cfa7e70cb1 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -15,9 +15,9 @@ open import Data.Product            using (_×_; _,_; <_,_>; uncurry; map₁)
 open import Data.Product.Properties using (,-injective)
 open import Data.Maybe.Base         using (Maybe; just; nothing)
 open import Data.String as String   using (String)
-open import Relation.Nullary.Decidable            using (map′; _×-dec_; yes; no)
-open import Relation.Binary                       using (Decidable; DecidableEquality)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong; cong₂)
+open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)
+open import Relation.Binary                            using (Decidable; DecidableEquality)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 
 open import Reflection.AST.Abstraction
 open import Reflection.AST.Argument
diff --git a/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda b/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
index 93a04ce62a..e160bb3b85 100644
--- a/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
@@ -17,7 +17,7 @@ module Relation.Binary.Construct.Add.Extrema.NonStrict
 open import Function.Base
 open import Relation.Nullary.Construct.Add.Extrema
 import Relation.Nullary.Construct.Add.Infimum as I
-open import Relation.Binary.PropositionalEquality using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 import Relation.Binary.Construct.Add.Infimum.NonStrict as AddInfimum
 import Relation.Binary.Construct.Add.Supremum.NonStrict as AddSupremum
 import Relation.Binary.Construct.Add.Extrema.Equality as Equality
diff --git a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
index c6c313c98c..b6cd5e4bf6 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
@@ -16,8 +16,9 @@ module Relation.Binary.Construct.Add.Infimum.NonStrict
 
 open import Level using (_⊔_)
 open import Data.Sum.Base as Sum
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Construct.Add.Infimum.Equality as Equality
 open import Relation.Nullary hiding (Irrelevant)
 open import Relation.Nullary.Construct.Add.Infimum
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
index 45463af482..80a0727fc1 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
@@ -17,8 +17,9 @@ module Relation.Binary.Construct.Add.Infimum.Strict
 open import Level using (_⊔_)
 open import Data.Product
 open import Function.Base
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Construct.Add.Infimum.Equality as Equality
 import Relation.Binary.Construct.Add.Infimum.NonStrict as NonStrict
 open import Relation.Nullary hiding (Irrelevant)
diff --git a/src/Relation/Binary/Construct/Add/Point/Equality.agda b/src/Relation/Binary/Construct/Add/Point/Equality.agda
index a0b01a3d2b..1715d6f2d1 100644
--- a/src/Relation/Binary/Construct/Add/Point/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Point/Equality.agda
@@ -16,7 +16,7 @@ module Relation.Binary.Construct.Add.Point.Equality
 
 open import Level using (_⊔_)
 open import Function.Base
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Core as P
 open import Relation.Nullary hiding (Irrelevant)
 open import Relation.Nullary.Construct.Add.Point
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
index be30c7973c..a509871377 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
@@ -18,8 +18,9 @@ open import Level using (_⊔_)
 open import Data.Sum.Base as Sum
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as P
 open import Relation.Nullary.Construct.Add.Supremum
 import Relation.Binary.Construct.Add.Supremum.Equality as Equality
 
diff --git a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
index 46415e6895..0bbefc05fa 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
@@ -19,8 +19,9 @@ open import Data.Product
 open import Function.Base
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as P
 open import Relation.Nullary.Construct.Add.Supremum
 import Relation.Binary.Construct.Add.Supremum.Equality as Equality
 import Relation.Binary.Construct.Add.Supremum.NonStrict as NonStrict
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index 9bac46632e..d9bf470f5c 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -15,7 +15,8 @@ open import Function.Base using (id)
 open import Level
 open import Relation.Binary hiding (_⇔_)
 open import Relation.Binary.Construct.Closure.Reflexive
-open import Relation.Binary.PropositionalEquality as PropEq using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as PropEq using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as PropEq
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 open import Relation.Unary using (Pred)
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda
index 2ae6422870..26d04320b0 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda
@@ -10,7 +10,7 @@ module Relation.Binary.Construct.Closure.Reflexive.Properties.WithK where
 
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.Reflexive
-open import Relation.Binary.PropositionalEquality as PropEq using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Nullary.Negation using (contradiction)
 
 open import Relation.Binary.Construct.Closure.Reflexive.Properties public
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index 88e284c2bf..64c81b7151 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -11,8 +11,9 @@ module Relation.Binary.Construct.Closure.ReflexiveTransitive.Properties where
 open import Function
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
-open import Relation.Binary.PropositionalEquality as PropEq
+open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; refl; sym; cong; cong₂)
+import Relation.Binary.PropositionalEquality.Properties as PropEq
 import Relation.Binary.Reasoning.Preorder as PreR
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Construct/Closure/Transitive.agda b/src/Relation/Binary/Construct/Closure/Transitive.agda
index 1699dc0cb6..551b6f5cb5 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive.agda
@@ -13,7 +13,7 @@ open import Function.Bundles using (_⇔_; mk⇔)
 open import Induction.WellFounded
 open import Level
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 private
   variable
diff --git a/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda b/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
index 1eff43bbca..62816238d9 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
@@ -11,7 +11,7 @@ module Relation.Binary.Construct.Closure.Transitive.WithK where
 open import Function
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.Transitive
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 module _ {a ℓ} {A : Set a} {_∼_ : Rel A ℓ} where
 
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 0a4886531e..1ca855e122 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -22,7 +22,8 @@ open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
   using (_atₛ_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as P
 
 import Relation.Binary.HeterogeneousEquality.Core as Core
 
diff --git a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
index fb1db6b9fb..d2825a3548 100644
--- a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
+++ b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
@@ -12,7 +12,7 @@ open import Relation.Binary.HeterogeneousEquality.Quotients
 open import Level using (0ℓ)
 open import Relation.Binary
 open import Relation.Binary.HeterogeneousEquality hiding (isEquivalence)
-import Relation.Binary.PropositionalEquality as ≡
+import Relation.Binary.PropositionalEquality.Core as ≡
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Product
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
index fd83521783..5ed814cdc1 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
@@ -15,7 +15,7 @@ open import Data.Product using (_,_)
 open import Function.Base using (_⟨_⟩_)
 open import Level using (Level; _⊔_; suc)
 open import Relation.Binary as B using (_⇒_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Binary.Indexed.Homogeneous.Core
 open import Relation.Binary.Indexed.Homogeneous.Structures
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
index 1404bf7f52..38db83929a 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
@@ -21,7 +21,7 @@ open import Data.Product using (_,_)
 open import Function.Base using (_⟨_⟩_)
 open import Level using (Level; _⊔_; suc)
 open import Relation.Binary as B using (_⇒_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Indexed.Homogeneous.Definitions
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Reasoning/MultiSetoid.agda b/src/Relation/Binary/Reasoning/MultiSetoid.agda
index 994ac3c4b6..7737e6a779 100644
--- a/src/Relation/Binary/Reasoning/MultiSetoid.agda
+++ b/src/Relation/Binary/Reasoning/MultiSetoid.agda
@@ -29,7 +29,7 @@ module Relation.Binary.Reasoning.MultiSetoid where
 open import Function.Base using (flip)
 open import Level using (_⊔_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 import Relation.Binary.Reasoning.Setoid as EqR
 
diff --git a/src/Relation/Binary/Reflection.agda b/src/Relation/Binary/Reflection.agda
index 5a9ec838fb..20e2054cfc 100644
--- a/src/Relation/Binary/Reflection.agda
+++ b/src/Relation/Binary/Reflection.agda
@@ -14,7 +14,7 @@ open import Function.Base using (id; _⟨_⟩_)
 open import Function.Bundles using (module Equivalence)
 open import Level using (Level)
 open import Relation.Binary
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality.Core as P
 
 -- Think of the parameters as follows:
 --
diff --git a/src/Relation/Nary.agda b/src/Relation/Nary.agda
index 5e6ecb5b25..da804883f5 100644
--- a/src/Relation/Nary.agda
+++ b/src/Relation/Nary.agda
@@ -27,7 +27,7 @@ open import Function.Nary.NonDependent
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _because_; _×-dec_)
 import Relation.Unary as Unary
-open import Relation.Binary.PropositionalEquality using (_≡_; cong; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
 
 private
   variable
diff --git a/src/Relation/Nullary/Construct/Add/Extrema.agda b/src/Relation/Nullary/Construct/Add/Extrema.agda
index 4b88fb3b3e..3366d83c27 100644
--- a/src/Relation/Nullary/Construct/Add/Extrema.agda
+++ b/src/Relation/Nullary/Construct/Add/Extrema.agda
@@ -8,7 +8,7 @@
 
 module Relation.Nullary.Construct.Add.Extrema where
 
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Nullary.Construct.Add.Infimum  as Infimum  using (_₋)
 open import Relation.Nullary.Construct.Add.Supremum as Supremum using (_⁺)
 
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 4c0308343d..68394c051c 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -12,8 +12,8 @@ open import Relation.Nullary
 open import Relation.Nullary.Negation
 open import Relation.Binary hiding (_⇒_)
 import Relation.Binary.Construct.Always as Always
-open import Relation.Binary.PropositionalEquality as PropEq
-  using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as PropEq
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   as Trivial
 open import Data.Sum     as Sum  hiding (map)
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 25e21193be..e46392ee5d 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -36,7 +36,7 @@ open import Data.Unit.Base            using (⊤)
 open import Data.Word.Base   as Word  using (toℕ)
 open import Data.Product
 
-open import Relation.Binary.PropositionalEquality as Eq using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_; refl; cong)
 
 -- 'Data.String.Properties' defines this via 'Dec', so let's use the builtin
 -- for maximum speed.
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
index d04ecbc131..1eb2244f5d 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
@@ -22,7 +22,7 @@ open import Data.Vec                  using (Vec)
 open import Function
 open import Relation.Unary
 
-import Relation.Binary.PropositionalEquality as ≡
+import Relation.Binary.PropositionalEquality.Core as ≡
 
 open Homomorphism homo hiding (_^_)
 open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
index 6fda530012..ca622ab00a 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
@@ -21,7 +21,7 @@ open import Data.List.Kleene
 open import Data.Vec      using (Vec)
 
 import Data.Nat.Properties as ℕ-Prop
-import Relation.Binary.PropositionalEquality as ≡
+import Relation.Binary.PropositionalEquality.Core as ≡
 
 open Homomorphism homo
 open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index ce20b1e381..6f2eba31d7 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -13,20 +13,20 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Bool                                  using (Bool;true;false)
-open import Data.Nat.Base as ℕ                         using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
-open import Data.Nat.Properties as ℕₚ                  using (≤′-trans)
-open import Data.Vec.Base as Vec                       using (Vec; _∷_)
-open import Data.Fin                                   using (Fin; zero; suc)
-open import Data.List.Base                             using (_∷_; [])
+open import Data.Bool                                       using (Bool;true;false)
+open import Data.Nat.Base as ℕ                              using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
+open import Data.Nat.Properties as ℕₚ                       using (≤′-trans)
+open import Data.Vec.Base as Vec                            using (Vec; _∷_)
+open import Data.Fin                                        using (Fin; zero; suc)
+open import Data.List.Base                                  using (_∷_; [])
 open import Data.Unit using (tt)
 open import Data.List.Kleene
-open import Data.Product                               using (_,_; proj₁; proj₂; map₁; _×_)
-open import Data.Maybe                                 using (nothing; just)
+open import Data.Product                                    using (_,_; proj₁; proj₂; map₁; _×_)
+open import Data.Maybe                                      using (nothing; just)
 open import Function
-open import Level                                      using (lift)
-open import Relation.Nullary                           using (Dec; yes; no)
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
+open import Level                                           using (lift)
+open import Relation.Nullary                                using (Dec; yes; no)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 open Homomorphism homo hiding (_^_)
 open import Tactic.RingSolver.Core.Polynomial.Reasoning to
diff --git a/src/Text/Regex/Derivative/Brzozowski.agda b/src/Text/Regex/Derivative/Brzozowski.agda
index ca3768296b..d368bfc70d 100644
--- a/src/Text/Regex/Derivative/Brzozowski.agda
+++ b/src/Text/Regex/Derivative/Brzozowski.agda
@@ -19,7 +19,7 @@ open import Relation.Nullary.Decidable using (Dec; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Nullary.Decidable using (map′; ¬?)
 open import Relation.Binary using (Decidable)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 open DecPoset P? using (preorder) renaming (Carrier to A)
 open import Text.Regex.Base preorder as R hiding (_∣_; _∙_; _⋆)
diff --git a/src/Text/Regex/Properties.agda b/src/Text/Regex/Properties.agda
index 2652f9c8cf..cf807ca3f3 100644
--- a/src/Text/Regex/Properties.agda
+++ b/src/Text/Regex/Properties.agda
@@ -24,7 +24,6 @@ open import Relation.Nullary.Negation
 
 import Relation.Unary  as U
 open import Relation.Binary using (Decidable)
-import Relation.Binary.PropositionalEquality as P
 
 open DecPoset P? renaming (Carrier to A)
 open import Text.Regex.Base preorder
diff --git a/src/Text/Regex/Search.agda b/src/Text/Regex/Search.agda
index c802768e38..f5c8c780c3 100644
--- a/src/Text/Regex/Search.agda
+++ b/src/Text/Regex/Search.agda
@@ -30,7 +30,7 @@ open import Relation.Nullary using (Dec; ¬_; yes; no)
 open import Relation.Nullary.Decidable using (map′)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Binary using (Rel; Decidable; _⇒_)
-open import Relation.Binary.PropositionalEquality hiding (preorder)
+open import Relation.Binary.PropositionalEquality.Core
 
 open DecPoset P? using (preorder) renaming (Carrier to A)
 open import Text.Regex.Base preorder
diff --git a/src/Text/Regex/SmartConstructors.agda b/src/Text/Regex/SmartConstructors.agda
index eecda1284f..afe9d3f936 100644
--- a/src/Text/Regex/SmartConstructors.agda
+++ b/src/Text/Regex/SmartConstructors.agda
@@ -19,7 +19,7 @@ open import Data.Sum.Base using (inj₁; inj₂; fromInj₁; fromInj₂)
 
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary.PropositionalEquality using (refl)
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 open import Text.Regex.Base P as R hiding (_∣_; _∙_; _⋆)
 open import Text.Regex.Properties.Core P
diff --git a/src/Text/Regex/String/Unsafe.agda b/src/Text/Regex/String/Unsafe.agda
index bc6c9fc0d7..a46fcd5655 100644
--- a/src/Text/Regex/String/Unsafe.agda
+++ b/src/Text/Regex/String/Unsafe.agda
@@ -13,7 +13,7 @@ import Data.String.Unsafe as Stringₚ
 open import Function.Base using (_on_; id)
 open import Level using (0ℓ)
 open import Relation.Binary using (Rel; Decidable)
-open import Relation.Binary.PropositionalEquality using (_≡_; sym; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; sym; subst)
 open import Relation.Nullary.Decidable using (map′)
 
 ------------------------------------------------------------------------

From 20a346053a923877130ea22bd950dec667375d30 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Am=C3=A9lia=20Liao?= <me@amelia.how>
Date: Fri, 21 Jul 2023 13:04:52 -0300
Subject: [PATCH 068/316] Put indexed data types in the right universe

To fix agda/agda#6654, we've decided that large indices will no longer
be allowed by default. There is an infective flag `--large-indices` to
bring them back, but none of the uses of large indices in the standard
library were essential: to avoid complicated mutually-recursive PRs
across repos, I adjusted the levels to check with `--no-large-indices`
instead of adding the flag to the modules that used them.
---
 src/Data/Star/Decoration.agda    | 2 +-
 src/Data/Star/Pointer.agda       | 2 +-
 src/Reflection/AnnotatedAST.agda | 6 +++---
 3 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/Data/Star/Decoration.agda b/src/Data/Star/Decoration.agda
index 24428a5843..170b5abb07 100644
--- a/src/Data/Star/Decoration.agda
+++ b/src/Data/Star/Decoration.agda
@@ -28,7 +28,7 @@ data NonEmptyEdgePred {ℓ r p : Level} {I : Set ℓ} (T : Rel I r)
 -- Decorating an edge with more information.
 
 data DecoratedWith {ℓ r p : Level} {I : Set ℓ} {T : Rel I r} (P : EdgePred p T)
-       : Rel (NonEmpty (Star T)) p where
+       : Rel (NonEmpty (Star T)) (ℓ ⊔ r ⊔ p) where
   ↦ : ∀ {i j k} {x : T i j} {xs : Star T j k}
       (p : P x) → DecoratedWith P (nonEmpty (x ◅ xs)) (nonEmpty xs)
 
diff --git a/src/Data/Star/Pointer.agda b/src/Data/Star/Pointer.agda
index ed606807ce..bc86786f74 100644
--- a/src/Data/Star/Pointer.agda
+++ b/src/Data/Star/Pointer.agda
@@ -26,7 +26,7 @@ private
 
 data Pointer {T : Rel I r}
              (P : EdgePred p T) (Q : EdgePred q T)
-             : Rel (Maybe (NonEmpty (Star T))) (p ⊔ q) where
+             : Rel (Maybe (NonEmpty (Star T))) (ℓ ⊔ r ⊔ p ⊔ q) where
   step : ∀ {i j k} {x : T i j} {xs : Star T j k}
          (p : P x) → Pointer P Q (just (nonEmpty (x ◅ xs)))
                                  (just (nonEmpty xs))
diff --git a/src/Reflection/AnnotatedAST.agda b/src/Reflection/AnnotatedAST.agda
index 61d55def7d..8ebc54b919 100644
--- a/src/Reflection/AnnotatedAST.agda
+++ b/src/Reflection/AnnotatedAST.agda
@@ -35,8 +35,8 @@ Annotation : ∀ ℓ → Set (suc ℓ)
 Annotation ℓ = ∀ {u} → ⟦ u ⟧ → Set ℓ
 
 -- An annotated type is a family over an Annotation and a reflected term.
-Typeₐ : ∀ ℓ → Univ → Set (suc ℓ)
-Typeₐ ℓ u = Annotation ℓ → ⟦ u ⟧ → Set ℓ
+Typeₐ : ∀ ℓ → Univ → Set (suc (suc ℓ))
+Typeₐ ℓ u = Annotation ℓ → ⟦ u ⟧ → Set (suc ℓ)
 
 private
   variable
@@ -168,7 +168,7 @@ mutual
 
 -- An annotation function computes the top-level annotation given a
 -- term annotated at all sub-terms.
-AnnotationFun : Annotation ℓ → Set ℓ
+AnnotationFun : Annotation ℓ → Set (suc ℓ)
 AnnotationFun Ann = ∀ u {t : ⟦ u ⟧} → Annotated′ Ann t → Ann t
 
 

From c9d6f183155f3850d221f9edd8cfbe63d9317210 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Tue, 25 Jul 2023 15:40:35 +0200
Subject: [PATCH 069/316] CHANGELOG: fix whitespace violations: remove trailing
 whitespace

---
 CHANGELOG.md | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5891740538..f1448641fe 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2165,9 +2165,9 @@ Other minor changes
   take-suc-tabulate : (f : Fin n → A) (o : Fin n) → let m = toℕ o in take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f o
   drop-take-suc : (o : Fin (length xs)) → let m = toℕ o in drop m (take (suc m) xs) ≡ [ lookup xs o ]
   drop-take-suc-tabulate : (f : Fin n → A) (o : Fin n) → let m = toℕ o in drop m (take (suc m) (tabulate f)) ≡ [ f o ]
-  
-  take-all : n ≥ length xs → take n xs ≡ xs     
-  
+
+  take-all : n ≥ length xs → take n xs ≡ xs
+
   take-[] : ∀ m → take  m [] ≡ []
   drop-[] : ∀ m → drop  m [] ≡ []
   ```
@@ -2901,7 +2901,7 @@ Other minor changes
   foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
   foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
   ```
-  
+
 NonZero/Positive/Negative changes
 ---------------------------------
 

From 484c2448bfa883bbd1d85d2f759db16bb3e937d1 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Tue, 25 Jul 2023 15:43:48 +0200
Subject: [PATCH 070/316] CHANGELOG: fix whitespace violations: replace tabs by
 spaces

---
 CHANGELOG.md | 100 +++++++++++++++++++++++++--------------------------
 1 file changed, 50 insertions(+), 50 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index f1448641fe..092d73a881 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -371,42 +371,42 @@ Non-backwards compatible changes
   `m / n` without having to explicitly provide a proof, as instance search will fill it in
   for you. The full list of such operations changed is as follows:
     - In `Data.Nat.DivMod`: `_/_`, `_%_`, `_div_`, `_mod_`
-	- In `Data.Nat.Pseudorandom.LCG`: `Generator`
-	- In `Data.Integer.DivMod`: `_divℕ_`, `_div_`, `_modℕ_`, `_mod_`
-	- In `Data.Rational`: `mkℚ+`, `normalize`, `_/_`, `1/_`
-	- In `Data.Rational.Unnormalised`: `_/_`, `1/_`, `_÷_`
+    - In `Data.Nat.Pseudorandom.LCG`: `Generator`
+    - In `Data.Integer.DivMod`: `_divℕ_`, `_div_`, `_modℕ_`, `_mod_`
+    - In `Data.Rational`: `mkℚ+`, `normalize`, `_/_`, `1/_`
+    - In `Data.Rational.Unnormalised`: `_/_`, `1/_`, `_÷_`
 
 * At the moment, there are 4 different ways such instance arguments can be provided,
   listed in order of convenience and clarity:
     1. *Automatic basic instances* - the standard library provides instances based on the constructors of each
-	   numeric type in `Data.X.Base`. For example, `Data.Nat.Base` constains an instance of `NonZero (suc n)` for any `n`
-	   and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`. Consequently,
-	   if the argument is of the required form, these instances will always be filled in by instance search
-	   automatically, e.g.
-	   ```
-	   0/n≡0 : 0 / suc n ≡ 0
-	   ```
-	2. *Take the instance as an argument* - You can provide the instance argument as a parameter to your function
-	   and Agda's instance search will automatically use it in the correct place without you having to
-	   explicitly pass it, e.g.
-	   ```
-	   0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
-	   ```
-	3. *Define the instance locally* - You can define an instance argument in scope (e.g. in a `where` clause)
-	   and Agda's instance search will again find it automatically, e.g.
-	   ```
-	   instance
-	     n≢0 : NonZero n
-	     n≢0 = ...
-
-	   0/n≡0 : 0 / n ≡ 0
-	   ```
-	4. *Pass the instance argument explicitly* - Finally, if all else fails you can pass the
-	   instance argument explicitly into the function using `{{ }}`, e.g.
-	   ```
-	   0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
-	   ```
-	   Suitable constructors for `NonZero`/`Positive` etc. can be found in `Data.X.Base`.
+       numeric type in `Data.X.Base`. For example, `Data.Nat.Base` constains an instance of `NonZero (suc n)` for any `n`
+       and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`. Consequently,
+       if the argument is of the required form, these instances will always be filled in by instance search
+       automatically, e.g.
+       ```
+       0/n≡0 : 0 / suc n ≡ 0
+       ```
+    2. *Take the instance as an argument* - You can provide the instance argument as a parameter to your function
+       and Agda's instance search will automatically use it in the correct place without you having to
+       explicitly pass it, e.g.
+       ```
+       0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
+       ```
+    3. *Define the instance locally* - You can define an instance argument in scope (e.g. in a `where` clause)
+       and Agda's instance search will again find it automatically, e.g.
+       ```
+       instance
+         n≢0 : NonZero n
+         n≢0 = ...
+
+       0/n≡0 : 0 / n ≡ 0
+       ```
+    4. *Pass the instance argument explicitly* - Finally, if all else fails you can pass the
+       instance argument explicitly into the function using `{{ }}`, e.g.
+       ```
+       0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
+       ```
+       Suitable constructors for `NonZero`/`Positive` etc. can be found in `Data.X.Base`.
 
 * A full list of proofs that have changed to use instance arguments is available at the end of this file.
   Notable changes to proofs are now discussed below.
@@ -457,14 +457,14 @@ Non-backwards compatible changes
 * As a consequence of this, some proofs that relied on this reduction behaviour
   or on eta-equality may no longer go through. There are several ways to fix this:
   1. The principled way is to not rely on this reduction behaviour in the first place.
-	 The `Properties` files for rational numbers have been greatly expanded in `v1.7`
-	 and `v2.0`, and we believe most proofs should be able to be built up from existing
-	 proofs contained within these files.
+     The `Properties` files for rational numbers have been greatly expanded in `v1.7`
+     and `v2.0`, and we believe most proofs should be able to be built up from existing
+     proofs contained within these files.
   2. Alternatively, annotating any rational arguments to a proof with either
-	 `@record{}` or `@(mkℚ _ _ _)` should restore the old reduction behaviour for any
-	 terms involving those parameters.
+     `@record{}` or `@(mkℚ _ _ _)` should restore the old reduction behaviour for any
+     terms involving those parameters.
   3. Finally, if the above approaches are not viable then you may be forced to explicitly
-	 use `cong` combined with a lemma that proves the old reduction behaviour.
+     use `cong` combined with a lemma that proves the old reduction behaviour.
 
 ### Change to the definition of `LeftCancellative` and `RightCancellative`
 
@@ -478,20 +478,20 @@ Non-backwards compatible changes
 
 * Therefore the definitions have been changed as follows to make all their arguments explicit:
   - `LeftCancellative _•_`
-	- From: `∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z`
-	- To: `∀ x y z → (x • y) ≈ (x • z) → y ≈ z`
+    - From: `∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z`
+    - To: `∀ x y z → (x • y) ≈ (x • z) → y ≈ z`
 
   - `RightCancellative _•_`
     - From: `∀ {x} y z → (y • x) ≈ (z • x) → y ≈ z`
-	- To: `∀ x y z → (y • x) ≈ (z • x) → y ≈ z`
+    - To: `∀ x y z → (y • x) ≈ (z • x) → y ≈ z`
 
   - `AlmostLeftCancellative e _•_`
     - From: `∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
-	- To: `∀ x y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
 
   - `AlmostRightCancellative e _•_`
-	- From: `∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
-	- To: `∀ x y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
+    - From: `∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
 
 * Correspondingly some proofs of the above types will need additional arguments passed explicitly.
   Instances can easily be fixed by adding additional underscores, e.g.
@@ -643,7 +643,7 @@ Non-backwards compatible changes
   - `¬?` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Decidable.Core`
   - `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Reflects`.
   - `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved from `Relation.Nullary.Negation`
-	to `Relation.Nullary.Decidable`.
+    to `Relation.Nullary.Decidable`.
   - `fromDec` and `toDec` have been mvoed from `Data.Sum.Base` to `Data.Sum`.
 
 ### Refactoring of the unindexed Functor/Applicative/Monad hiearchy
@@ -883,13 +883,13 @@ Major improvements
 
 * The ring solver tactic has been greatly improved. In particular:
   1. When the solver is used for concrete ring types, e.g. ℤ, the equality can now use
-	 all the ring operations defined natively for that type, rather than having
-	 to use the operations defined in `AlmostCommutativeRing`. For example
-	 previously you could not use `Data.Integer.Base._*_` but instead had to
-	 use `AlmostCommutativeRing._*_`.
+     all the ring operations defined natively for that type, rather than having
+     to use the operations defined in `AlmostCommutativeRing`. For example
+     previously you could not use `Data.Integer.Base._*_` but instead had to
+     use `AlmostCommutativeRing._*_`.
   2. The solver now supports use of the subtraction operator `_-_` whenever
      it is defined immediately in terms of `_+_` and `-_`. This is the case for
-	 `Data.Integer` and `Data.Rational`.
+     `Data.Integer` and `Data.Rational`.
 
 ### Moved `_%_` and `_/_` operators to `Data.Nat.Base`
 

From f84aa763b37538e640ac6f92a55b561546dffabf Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Tue, 25 Jul 2023 15:49:38 +0200
Subject: [PATCH 071/316] CHANGELOG for #2030: fix large indices

---
 CHANGELOG.md | 7 +++++++
 1 file changed, 7 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 092d73a881..1da4157a62 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -733,6 +733,13 @@ Non-backwards compatible changes
 * In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
   the `--without-K` flag now use the `--cubical-compatible` flag instead.
 
+* To avoid _large indices_ that are by default no longer allowed in Agda 2.6.4,
+  universe levels have been increased in the following definitions:
+  - `Data.Star.Decoration.DecoratedWith`
+  - `Data.Star.Pointer.Pointer`
+  - `Reflection.AnnotatedAST.Typeₐ`
+  - `Reflection.AnnotatedAST.AnnotationFun`
+
 * The first two arguments of `m≡n⇒m-n≡0` (now `i≡j⇒i-j≡0`) in `Data.Integer.Base`
   have been made implicit.
 

From f64ce4d2fd135f530631e6f7bd5d1b5b582a7a24 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Fri, 28 Jul 2023 09:22:30 -0400
Subject: [PATCH 072/316] Simplify some `Relation.Binary` imports (#2009)

* Simplify some `Relation.Binary` imports

* Format

* Better naming scheme

* Fix tests

* Merge issues
---
 README/Data/Trie/NonDependent.agda                    |  2 +-
 README/Design/Hierarchies.agda                        |  2 +-
 src/Algebra/Consequences/Propositional.agda           |  4 +++-
 src/Algebra/Consequences/Setoid.agda                  |  5 ++++-
 src/Algebra/Construct/Initial.agda                    |  5 ++++-
 src/Algebra/FunctionProperties/Consequences.agda      |  5 ++++-
 src/Algebra/Lattice/Bundles/Raw.agda                  |  2 +-
 src/Algebra/Lattice/Structures.agda                   |  4 +++-
 src/Algebra/Lattice/Structures/Biased.agda            |  4 +++-
 src/Algebra/Module/Consequences.agda                  |  3 ++-
 src/Algebra/Module/Structures.agda                    |  4 +++-
 src/Algebra/Module/Structures/Biased.agda             |  4 +++-
 src/Algebra/Ordered/Bundles.agda                      |  2 +-
 .../Properties/CancellativeCommutativeSemiring.agda   |  2 +-
 src/Algebra/Properties/Semiring/Primality.agda        |  2 +-
 src/Algebra/Solver/Monoid.agda                        |  2 +-
 src/Algebra/Structures.agda                           |  4 +++-
 src/Algebra/Structures/Biased.agda                    |  4 +++-
 src/Data/String.agda                                  |  2 +-
 src/Effect/Monad/Partiality.agda                      |  8 +++++++-
 src/Function/Construct/Symmetry.agda                  |  4 +++-
 src/Function/Indexed/Bundles.agda                     |  2 +-
 src/Relation/Binary/Construct/Closure/Transitive.agda |  4 +++-
 .../Binary/Indexed/Heterogeneous/Bundles.agda         |  2 +-
 .../Binary/Indexed/Heterogeneous/Structures.agda      |  2 +-
 src/Relation/Binary/Indexed/Homogeneous/Bundles.agda  | 11 ++++++-----
 src/Relation/Binary/Indexed/Homogeneous/Core.agda     |  2 +-
 .../Binary/Indexed/Homogeneous/Structures.agda        |  6 ++++--
 .../Binary/Lattice/Properties/BoundedLattice.agda     |  2 +-
 src/Relation/Nullary/Universe.agda                    |  3 ++-
 src/Relation/Unary/PredicateTransformer.agda          |  2 +-
 src/Text/Regex.agda                                   |  2 +-
 src/Text/Regex/Base.agda                              |  2 +-
 src/Text/Regex/Derivative/Brzozowski.agda             |  4 ++--
 src/Text/Regex/Properties.agda                        |  4 ++--
 src/Text/Regex/Properties/Core.agda                   |  2 +-
 src/Text/Regex/Search.agda                            |  5 +++--
 src/Text/Regex/SmartConstructors.agda                 |  2 +-
 src/Text/Regex/String/Unsafe.agda                     |  3 ++-
 tests/data/trie/Main.agda                             |  2 +-
 40 files changed, 88 insertions(+), 48 deletions(-)

diff --git a/README/Data/Trie/NonDependent.agda b/README/Data/Trie/NonDependent.agda
index 2387a71242..974ce5c96d 100644
--- a/README/Data/Trie/NonDependent.agda
+++ b/README/Data/Trie/NonDependent.agda
@@ -62,7 +62,7 @@ open import Data.These       as These
 
 open import Function.Base using (case_of_; _$_; _∘′_; id; _on_)
 open import Relation.Nary
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary.Decidable using (¬?)
 
 open import Data.Trie Char.<-strictTotalOrder
diff --git a/README/Design/Hierarchies.agda b/README/Design/Hierarchies.agda
index 0a91586086..cc03919b85 100644
--- a/README/Design/Hierarchies.agda
+++ b/README/Design/Hierarchies.agda
@@ -10,7 +10,7 @@ module README.Design.Hierarchies where
 
 open import Data.Sum using (_⊎_)
 open import Level using (Level; _⊔_; suc)
-open import Relation.Binary using (_Preserves₂_⟶_⟶_)
+open import Relation.Binary.Core using (_Preserves₂_⟶_⟶_)
 
 private
   variable
diff --git a/src/Algebra/Consequences/Propositional.agda b/src/Algebra/Consequences/Propositional.agda
index 7c3030f4a6..6f2626889c 100644
--- a/src/Algebra/Consequences/Propositional.agda
+++ b/src/Algebra/Consequences/Propositional.agda
@@ -11,7 +11,9 @@ module Algebra.Consequences.Propositional
   {a} {A : Set a} where
 
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Relation.Binary using (Rel; Setoid; Symmetric; Total)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (Symmetric; Total)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary using (Pred)
 
diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index 07b75806c2..444a9296b8 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -7,7 +7,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid; Substitutive; Symmetric; Total)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions
+  using (Substitutive; Symmetric; Total)
 
 module Algebra.Consequences.Setoid {a ℓ} (S : Setoid a ℓ) where
 
diff --git a/src/Algebra/Construct/Initial.agda b/src/Algebra/Construct/Initial.agda
index e542446248..99d80b98a3 100644
--- a/src/Algebra/Construct/Initial.agda
+++ b/src/Algebra/Construct/Initial.agda
@@ -22,7 +22,10 @@ open import Algebra.Core using (Op₂)
 open import Algebra.Definitions using (Congruent₂)
 open import Algebra.Structures using (IsMagma; IsSemigroup; IsBand)
 open import Data.Empty.Polymorphic
-open import Relation.Binary using (Rel; Reflexive; Symmetric; Transitive; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/FunctionProperties/Consequences.agda b/src/Algebra/FunctionProperties/Consequences.agda
index 03c9113995..e77b0b6d4e 100644
--- a/src/Algebra/FunctionProperties/Consequences.agda
+++ b/src/Algebra/FunctionProperties/Consequences.agda
@@ -6,7 +6,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid; Substitutive; Symmetric; Total)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions
+  using (Substitutive; Symmetric; Total)
 
 module Algebra.FunctionProperties.Consequences
   {a ℓ} (S : Setoid a ℓ) where
diff --git a/src/Algebra/Lattice/Bundles/Raw.agda b/src/Algebra/Lattice/Bundles/Raw.agda
index 5ac79408a5..8b5501cb72 100644
--- a/src/Algebra/Lattice/Bundles/Raw.agda
+++ b/src/Algebra/Lattice/Bundles/Raw.agda
@@ -11,7 +11,7 @@ module Algebra.Lattice.Bundles.Raw where
 open import Algebra.Core
 open import Algebra.Bundles.Raw using (RawMagma)
 open import Level using (suc; _⊔_)
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 
 record RawLattice c ℓ : Set (suc (c ⊔ ℓ)) where
   infixr 7 _∧_
diff --git a/src/Algebra/Lattice/Structures.agda b/src/Algebra/Lattice/Structures.agda
index b02787eee0..fc40c0c300 100644
--- a/src/Algebra/Lattice/Structures.agda
+++ b/src/Algebra/Lattice/Structures.agda
@@ -16,7 +16,9 @@
 open import Algebra.Core
 open import Data.Product.Base using (proj₁; proj₂)
 open import Level using (_⊔_)
-open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Algebra.Lattice.Structures
   {a ℓ} {A : Set a}  -- The underlying set
diff --git a/src/Algebra/Lattice/Structures/Biased.agda b/src/Algebra/Lattice/Structures/Biased.agda
index fefe3bef67..b5e1e6c9fd 100644
--- a/src/Algebra/Lattice/Structures/Biased.agda
+++ b/src/Algebra/Lattice/Structures/Biased.agda
@@ -16,7 +16,9 @@ open import Algebra.Core
 open import Algebra.Consequences.Setoid
 open import Data.Product.Base using (proj₁; proj₂)
 open import Level using (_⊔_)
-open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Algebra.Lattice.Structures.Biased
   {a ℓ} {A : Set a}  -- The underlying set
diff --git a/src/Algebra/Module/Consequences.agda b/src/Algebra/Module/Consequences.agda
index 4741349edc..2e53f96d76 100644
--- a/src/Algebra/Module/Consequences.agda
+++ b/src/Algebra/Module/Consequences.agda
@@ -14,7 +14,8 @@ open import Algebra.Module.Core using (Opₗ; Opᵣ)
 open import Algebra.Module.Definitions
 open import Function.Base using (flip)
 open import Level using (Level)
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.Reasoning.Setoid as Rea
 
 private
diff --git a/src/Algebra/Module/Structures.agda b/src/Algebra/Module/Structures.agda
index 4a0e40c17c..5b071b1a63 100644
--- a/src/Algebra/Module/Structures.agda
+++ b/src/Algebra/Module/Structures.agda
@@ -6,7 +6,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Algebra.Module.Structures where
 
diff --git a/src/Algebra/Module/Structures/Biased.agda b/src/Algebra/Module/Structures/Biased.agda
index 74d825e335..99e48695ae 100644
--- a/src/Algebra/Module/Structures/Biased.agda
+++ b/src/Algebra/Module/Structures/Biased.agda
@@ -7,7 +7,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Algebra.Module.Structures.Biased where
 
diff --git a/src/Algebra/Ordered/Bundles.agda b/src/Algebra/Ordered/Bundles.agda
index fd69b7145d..896f9d4d1f 100644
--- a/src/Algebra/Ordered/Bundles.agda
+++ b/src/Algebra/Ordered/Bundles.agda
@@ -16,7 +16,7 @@ open import Algebra.Core
 open import Algebra.Bundles
 open import Algebra.Ordered.Structures
 open import Level using (suc; _⊔_)
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (Preorder; Poset)
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Properties/CancellativeCommutativeSemiring.agda b/src/Algebra/Properties/CancellativeCommutativeSemiring.agda
index f07f898eb2..e92fa5170e 100644
--- a/src/Algebra/Properties/CancellativeCommutativeSemiring.agda
+++ b/src/Algebra/Properties/CancellativeCommutativeSemiring.agda
@@ -9,7 +9,7 @@
 open import Algebra using (CancellativeCommutativeSemiring)
 open import Algebra.Definitions using (AlmostRightCancellative)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 
diff --git a/src/Algebra/Properties/Semiring/Primality.agda b/src/Algebra/Properties/Semiring/Primality.agda
index 2b0d226491..210d9fb1a0 100644
--- a/src/Algebra/Properties/Semiring/Primality.agda
+++ b/src/Algebra/Properties/Semiring/Primality.agda
@@ -9,7 +9,7 @@
 open import Algebra using (Semiring)
 open import Data.Sum.Base using (reduce)
 open import Function.Base using (flip)
-open import Relation.Binary using (Symmetric)
+open import Relation.Binary.Definitions using (Symmetric)
 
 module Algebra.Properties.Semiring.Primality
   {a ℓ} (R : Semiring a ℓ)
diff --git a/src/Algebra/Solver/Monoid.agda b/src/Algebra/Solver/Monoid.agda
index bf066f7745..23e994821c 100644
--- a/src/Algebra/Solver/Monoid.agda
+++ b/src/Algebra/Solver/Monoid.agda
@@ -20,7 +20,7 @@ open import Data.Nat.Base using (ℕ)
 open import Data.Product
 open import Data.Vec.Base using (Vec; lookup)
 open import Function.Base using (_∘_; _$_)
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.Reflection
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index eb9d49b2c9..4f9fc7fcb3 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -9,7 +9,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Algebra.Structures
   {a ℓ} {A : Set a}  -- The underlying set
diff --git a/src/Algebra/Structures/Biased.agda b/src/Algebra/Structures/Biased.agda
index 0aa549c66c..c0c6994e3f 100644
--- a/src/Algebra/Structures/Biased.agda
+++ b/src/Algebra/Structures/Biased.agda
@@ -11,7 +11,9 @@ open import Algebra.Core
 open import Algebra.Consequences.Setoid
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Level using (_⊔_)
-open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Algebra.Structures.Biased
   {a ℓ} {A : Set a}  -- The underlying set
diff --git a/src/Data/String.agda b/src/Data/String.agda
index fffe1ac300..7ff82dfac0 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -22,7 +22,7 @@ open import Data.Vec.Base as Vec using (Vec)
 open import Data.Char.Base as Char using (Char)
 import Data.Char.Properties as Char using (_≟_)
 open import Function
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Nullary.Decidable using (does)
 open import Relation.Unary using (Pred; Decidable)
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index 0751cb5b15..e6cfdd1c05 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -19,7 +19,13 @@ open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
-open import Relation.Binary as B hiding (Rel; _⇔_)
+open import Relation.Binary.Core as B hiding (Rel; _⇔_)
+open import Relation.Binary.Definitions
+  using (Decidable; Reflexive; Symmetric; Transitive)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsEquivalence)
+open import Relation.Binary.Bundles
+  using (Preorder; Setoid; Poset)
 import Relation.Binary.Properties.Setoid as SetoidProperties
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
diff --git a/src/Function/Construct/Symmetry.agda b/src/Function/Construct/Symmetry.agda
index 86fa273c57..1974dee391 100644
--- a/src/Function/Construct/Symmetry.agda
+++ b/src/Function/Construct/Symmetry.agda
@@ -11,7 +11,9 @@ module Function.Construct.Symmetry where
 open import Data.Product.Base using (_,_; swap; proj₁; proj₂)
 open import Function
 open import Level using (Level)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Symmetric; Transitive)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality
 
 private
diff --git a/src/Function/Indexed/Bundles.agda b/src/Function/Indexed/Bundles.agda
index 033cabd426..6cce0a3026 100644
--- a/src/Function/Indexed/Bundles.agda
+++ b/src/Function/Indexed/Bundles.agda
@@ -10,7 +10,7 @@ module Function.Indexed.Bundles where
 
 open import Relation.Unary using (Pred)
 open import Function.Bundles using (_⟶_; _↣_; _↠_; _⤖_; _⇔_; _↩_; _↪_; _↩↪_; _↔_)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core hiding (_⇔_)
 open import Level using (Level)
 
 private
diff --git a/src/Relation/Binary/Construct/Closure/Transitive.agda b/src/Relation/Binary/Construct/Closure/Transitive.agda
index 551b6f5cb5..ed20e48bb0 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive.agda
@@ -12,7 +12,9 @@ open import Function.Base
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Induction.WellFounded
 open import Level
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 private
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
index 089cad70ab..86dc3ccdaa 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
@@ -13,7 +13,7 @@ module Relation.Binary.Indexed.Heterogeneous.Bundles where
 
 open import Function.Base
 open import Level using (suc; _⊔_)
-open import Relation.Binary using (_⇒_)
+open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Indexed.Heterogeneous.Core
 open import Relation.Binary.Indexed.Heterogeneous.Structures
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda b/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
index c61022d362..7e27aab07f 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
@@ -17,7 +17,7 @@ module Relation.Binary.Indexed.Heterogeneous.Structures
 
 open import Function.Base
 open import Level using (suc; _⊔_)
-open import Relation.Binary using (_⇒_)
+open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Indexed.Heterogeneous.Definitions
 
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
index 5ed814cdc1..1f218cfc01 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
@@ -14,7 +14,8 @@ module Relation.Binary.Indexed.Homogeneous.Bundles where
 open import Data.Product using (_,_)
 open import Function.Base using (_⟨_⟩_)
 open import Level using (Level; _⊔_; suc)
-open import Relation.Binary as B using (_⇒_)
+open import Relation.Binary.Core using (_⇒_; Rel)
+open import Relation.Binary.Bundles as B
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Binary.Indexed.Homogeneous.Core
@@ -41,10 +42,10 @@ record IndexedSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where
 
   infix 4 _≉_
 
-  _≈_ : B.Rel Carrier _
+  _≈_ : Rel Carrier _
   _≈_ = Lift Carrierᵢ _≈ᵢ_
 
-  _≉_ : B.Rel Carrier _
+  _≉_ : Rel Carrier _
   x ≉ y = ¬ (x ≈ y)
 
   setoid : B.Setoid _ _
@@ -89,10 +90,10 @@ record IndexedPreorder {i} (I : Set i) c ℓ₁ ℓ₂ :
   Carrier : Set _
   Carrier = ∀ i → Carrierᵢ i
 
-  _≈_ : B.Rel Carrier _
+  _≈_ : Rel Carrier _
   x ≈ y = ∀ i → x i ≈ᵢ y i
 
-  _∼_ : B.Rel Carrier _
+  _∼_ : Rel Carrier _
   x ∼ y = ∀ i → x i ∼ᵢ y i
 
   preorder : B.Preorder _ _ _
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Core.agda b/src/Relation/Binary/Indexed/Homogeneous/Core.agda
index 6dc2e03f82..7281652597 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Core.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Core.agda
@@ -13,7 +13,7 @@ module Relation.Binary.Indexed.Homogeneous.Core where
 
 open import Level using (Level; _⊔_)
 open import Data.Product using (_×_)
-open import Relation.Binary as B using (REL; Rel)
+open import Relation.Binary.Core as B using (REL; Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.Indexed.Heterogeneous as I
 open import Relation.Unary.Indexed using (IPred)
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
index 38db83929a..0ccad66a6f 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
@@ -20,7 +20,9 @@ module Relation.Binary.Indexed.Homogeneous.Structures
 open import Data.Product using (_,_)
 open import Function.Base using (_⟨_⟩_)
 open import Level using (Level; _⊔_; suc)
-open import Relation.Binary as B using (_⇒_)
+open import Relation.Binary.Core using (_⇒_)
+import Relation.Binary.Definitions as B
+import Relation.Binary.Structures as B
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Indexed.Homogeneous.Definitions
 
@@ -97,7 +99,7 @@ record IsIndexedPreorder {ℓ₂} (_∼ᵢ_ : IRel A ℓ₂)
 
   -- Lifted properties
 
-  reflexive : Lift A _≈ᵢ_ B.⇒ Lift A _∼ᵢ_
+  reflexive : Lift A _≈ᵢ_ ⇒ Lift A _∼ᵢ_
   reflexive x≈y i = reflexiveᵢ (x≈y i)
 
   refl : B.Reflexive (Lift A _∼ᵢ_)
diff --git a/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda b/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda
index 95bf238945..1f3477b19d 100644
--- a/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda
@@ -15,7 +15,7 @@ open BoundedLattice L
 
 open import Algebra.Definitions _≈_
 open import Data.Product using (_,_)
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice
 open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
 
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 68394c051c..74b34607a6 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -10,7 +10,8 @@ module Relation.Nullary.Universe where
 
 open import Relation.Nullary
 open import Relation.Nullary.Negation
-open import Relation.Binary hiding (_⇒_)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.Construct.Always as Always
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.PropositionalEquality.Properties as PropEq
diff --git a/src/Relation/Unary/PredicateTransformer.agda b/src/Relation/Unary/PredicateTransformer.agda
index 518552d0d8..b6df4f2b8a 100644
--- a/src/Relation/Unary/PredicateTransformer.agda
+++ b/src/Relation/Unary/PredicateTransformer.agda
@@ -13,7 +13,7 @@ open import Function
 open import Level hiding (_⊔_)
 open import Relation.Nullary
 open import Relation.Unary
-open import Relation.Binary using (REL)
+open import Relation.Binary.Core using (REL)
 
 private
   variable
diff --git a/src/Text/Regex.agda b/src/Text/Regex.agda
index b6454f36ae..9b818d15b5 100644
--- a/src/Text/Regex.agda
+++ b/src/Text/Regex.agda
@@ -10,7 +10,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecPoset)
+open import Relation.Binary.Bundles using (DecPoset)
 
 module Text.Regex {a e r} (decPoset : DecPoset a e r) where
 
diff --git a/src/Text/Regex/Base.agda b/src/Text/Regex/Base.agda
index 839bad4181..e5cc2c24a4 100644
--- a/src/Text/Regex/Base.agda
+++ b/src/Text/Regex/Base.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Preorder)
+open import Relation.Binary.Bundles using (Preorder)
 
 module Text.Regex.Base {a e r} (P : Preorder a e r) where
 
diff --git a/src/Text/Regex/Derivative/Brzozowski.agda b/src/Text/Regex/Derivative/Brzozowski.agda
index d368bfc70d..9c66c831a4 100644
--- a/src/Text/Regex/Derivative/Brzozowski.agda
+++ b/src/Text/Regex/Derivative/Brzozowski.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecPoset)
+open import Relation.Binary.Bundles using (DecPoset)
 
 module Text.Regex.Derivative.Brzozowski {a e r} (P? : DecPoset a e r) where
 
@@ -18,7 +18,7 @@ open import Function.Base using (_$_; _∘′_; case_of_)
 open import Relation.Nullary.Decidable using (Dec; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Nullary.Decidable using (map′; ¬?)
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 open DecPoset P? using (preorder) renaming (Carrier to A)
diff --git a/src/Text/Regex/Properties.agda b/src/Text/Regex/Properties.agda
index cf807ca3f3..815bf04471 100644
--- a/src/Text/Regex/Properties.agda
+++ b/src/Text/Regex/Properties.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecPoset)
+open import Relation.Binary.Bundles using (DecPoset)
 
 module Text.Regex.Properties {a e r} (P? : DecPoset a e r) where
 
@@ -23,7 +23,7 @@ open import Relation.Nullary.Negation
   using (¬_; contradiction)
 
 import Relation.Unary  as U
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 
 open DecPoset P? renaming (Carrier to A)
 open import Text.Regex.Base preorder
diff --git a/src/Text/Regex/Properties/Core.agda b/src/Text/Regex/Properties/Core.agda
index 64faba7029..e1916fb5cf 100644
--- a/src/Text/Regex/Properties/Core.agda
+++ b/src/Text/Regex/Properties/Core.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Preorder)
+open import Relation.Binary.Bundles using (Preorder)
 
 module Text.Regex.Properties.Core {a e r} (P : Preorder a e r) where
 
diff --git a/src/Text/Regex/Search.agda b/src/Text/Regex/Search.agda
index f5c8c780c3..b96fb3847c 100644
--- a/src/Text/Regex/Search.agda
+++ b/src/Text/Regex/Search.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecPoset)
+open import Relation.Binary.Bundles using (DecPoset)
 
 module Text.Regex.Search {a e r} (P? : DecPoset a e r) where
 
@@ -29,7 +29,8 @@ open import Data.List.Relation.Binary.Suffix.Heterogeneous
 open import Relation.Nullary using (Dec; ¬_; yes; no)
 open import Relation.Nullary.Decidable using (map′)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary using (Rel; Decidable; _⇒_)
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality.Core
 
 open DecPoset P? using (preorder) renaming (Carrier to A)
diff --git a/src/Text/Regex/SmartConstructors.agda b/src/Text/Regex/SmartConstructors.agda
index afe9d3f936..3269d22da5 100644
--- a/src/Text/Regex/SmartConstructors.agda
+++ b/src/Text/Regex/SmartConstructors.agda
@@ -9,7 +9,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Preorder)
+open import Relation.Binary.Bundles using (Preorder)
 
 module Text.Regex.SmartConstructors {a e r} (P : Preorder a e r) where
 
diff --git a/src/Text/Regex/String/Unsafe.agda b/src/Text/Regex/String/Unsafe.agda
index a46fcd5655..118b48505a 100644
--- a/src/Text/Regex/String/Unsafe.agda
+++ b/src/Text/Regex/String/Unsafe.agda
@@ -12,7 +12,8 @@ open import Data.String.Base using (String; toList; fromList)
 import Data.String.Unsafe as Stringₚ
 open import Function.Base using (_on_; id)
 open import Level using (0ℓ)
-open import Relation.Binary using (Rel; Decidable)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; sym; subst)
 open import Relation.Nullary.Decidable using (map′)
 
diff --git a/tests/data/trie/Main.agda b/tests/data/trie/Main.agda
index 9dfac5cb26..3ce2ad3ecb 100644
--- a/tests/data/trie/Main.agda
+++ b/tests/data/trie/Main.agda
@@ -18,7 +18,7 @@ open import Data.These        as These
 
 open import Function.Base using (case_of_; _$_; _∘′_; id; _on_)
 open import Relation.Nary
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary.Decidable using (¬?)
 
 open import Data.Trie Char.<-strictTotalOrder

From 2c47650515f157c77dd597178731ae5bf31585b8 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Fri, 28 Jul 2023 11:01:15 -0400
Subject: [PATCH 073/316] Simplify Data.Sum and leftover Data.List imports
 (#2011)

---
 README/Case.agda                                              | 2 --
 README/Data/Container/FreeMonad.agda                          | 2 +-
 README/Design/Decidability.agda                               | 2 +-
 README/Design/Hierarchies.agda                                | 2 +-
 README/Nary.agda                                              | 2 +-
 src/Algebra/Construct/Flip/Op.agda                            | 2 +-
 src/Algebra/Construct/NaturalChoice/Min.agda                  | 2 +-
 src/Algebra/Construct/Subst/Equality.agda                     | 2 +-
 src/Algebra/Lattice/Construct/Subst/Equality.agda             | 1 -
 src/Algebra/Module/Definitions/Left.agda                      | 1 -
 src/Algebra/Module/Definitions/Right.agda                     | 1 -
 src/Data/Container/Indexed/Combinator.agda                    | 2 +-
 src/Data/List/Relation/Unary/Enumerates/Setoid.agda           | 2 +-
 .../List/Relation/Unary/Enumerates/Setoid/Properties.agda     | 2 +-
 src/Data/List/Relation/Unary/First.agda                       | 2 +-
 src/Data/List/Relation/Unary/Grouped.agda                     | 2 +-
 src/Data/List/Relation/Unary/Grouped/Properties.agda          | 2 +-
 src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda | 2 +-
 src/Data/Nat/Binary/Subtraction.agda                          | 2 +-
 src/Data/Nat/Combinatorics/Specification.agda                 | 2 +-
 src/Data/Nat/InfinitelyOften.agda                             | 2 +-
 src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda  | 2 +-
 src/Data/Tree/AVL/Map/Membership/Propositional.agda           | 2 +-
 .../Tree/AVL/Map/Membership/Propositional/Properties.agda     | 2 +-
 src/Data/Vec/Relation/Binary/Lex/Core.agda                    | 2 +-
 src/Effect/Monad/Error/Transformer.agda                       | 4 ++--
 src/Reflection/External.agda                                  | 2 +-
 src/Relation/Nullary/Universe.agda                            | 2 +-
 src/Relation/Unary/Algebra.agda                               | 2 +-
 29 files changed, 26 insertions(+), 31 deletions(-)

diff --git a/README/Case.agda b/README/Case.agda
index 358460a63b..450105aff9 100644
--- a/README/Case.agda
+++ b/README/Case.agda
@@ -12,8 +12,6 @@ module README.Case where
 open import Data.Fin   hiding (pred)
 open import Data.Maybe hiding (from-just)
 open import Data.Nat   hiding (pred)
-open import Data.List
-open import Data.Sum
 open import Data.Product
 open import Function.Base using (case_of_; case_return_of_)
 open import Relation.Nullary
diff --git a/README/Data/Container/FreeMonad.agda b/README/Data/Container/FreeMonad.agda
index 54f0935937..ab302c90f5 100644
--- a/README/Data/Container/FreeMonad.agda
+++ b/README/Data/Container/FreeMonad.agda
@@ -15,7 +15,7 @@ open import Data.Empty
 open import Data.Unit
 open import Data.Bool.Base using (Bool; true)
 open import Data.Nat
-open import Data.Sum using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Product.Base renaming (_×_ to _⟨×⟩_)
 open import Data.Container using (Container; _▷_)
 open import Data.Container.Combinator
diff --git a/README/Design/Decidability.agda b/README/Design/Decidability.agda
index e615af282e..38bb52d489 100644
--- a/README/Design/Decidability.agda
+++ b/README/Design/Decidability.agda
@@ -9,7 +9,7 @@
 module README.Design.Decidability where
 
 open import Data.Bool
-open import Data.List
+open import Data.List.Base using (List; []; _∷_)
 open import Data.List.Properties using (∷-injective)
 open import Data.Nat
 open import Data.Nat.Properties using (suc-injective)
diff --git a/README/Design/Hierarchies.agda b/README/Design/Hierarchies.agda
index cc03919b85..09d5504a91 100644
--- a/README/Design/Hierarchies.agda
+++ b/README/Design/Hierarchies.agda
@@ -8,7 +8,7 @@
 
 module README.Design.Hierarchies where
 
-open import Data.Sum using (_⊎_)
+open import Data.Sum.Base using (_⊎_)
 open import Level using (Level; _⊔_; suc)
 open import Relation.Binary.Core using (_Preserves₂_⟶_⟶_)
 
diff --git a/README/Nary.agda b/README/Nary.agda
index 826d7b74e6..da2c2fcca3 100644
--- a/README/Nary.agda
+++ b/README/Nary.agda
@@ -16,7 +16,7 @@ open import Data.Fin using (Fin; fromℕ; #_; inject₁)
 open import Data.List
 open import Data.List.Properties
 open import Data.Product using (_×_; _,_)
-open import Data.Sum using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂)
 open import Function
 open import Relation.Nullary
 open import Relation.Binary using (module Tri); open Tri
diff --git a/src/Algebra/Construct/Flip/Op.agda b/src/Algebra/Construct/Flip/Op.agda
index cc84ad59c9..4dd5849a6f 100644
--- a/src/Algebra/Construct/Flip/Op.agda
+++ b/src/Algebra/Construct/Flip/Op.agda
@@ -11,7 +11,7 @@ module Algebra.Construct.Flip.Op where
 
 open import Algebra
 import Data.Product.Base as Prod
-import Data.Sum as Sum
+import Data.Sum.Base as Sum
 open import Function.Base using (flip)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)
diff --git a/src/Algebra/Construct/NaturalChoice/Min.agda b/src/Algebra/Construct/NaturalChoice/Min.agda
index 100dac2193..41f895afe6 100644
--- a/src/Algebra/Construct/NaturalChoice/Min.agda
+++ b/src/Algebra/Construct/NaturalChoice/Min.agda
@@ -9,7 +9,7 @@
 open import Algebra.Core
 open import Algebra.Bundles
 open import Algebra.Construct.NaturalChoice.Base
-open import Data.Sum using (inj₁; inj₂; [_,_])
+open import Data.Sum.Base using (inj₁; inj₂; [_,_])
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 open import Relation.Binary
diff --git a/src/Algebra/Construct/Subst/Equality.agda b/src/Algebra/Construct/Subst/Equality.agda
index 636b1d8004..05d0277398 100644
--- a/src/Algebra/Construct/Subst/Equality.agda
+++ b/src/Algebra/Construct/Subst/Equality.agda
@@ -19,7 +19,7 @@ module Algebra.Construct.Subst.Equality
 
 open import Algebra.Definitions
 open import Algebra.Structures
-import Data.Sum as Sum
+import Data.Sum.Base as Sum
 open import Function.Base
 open import Relation.Binary.Construct.Subst.Equality equiv
 
diff --git a/src/Algebra/Lattice/Construct/Subst/Equality.agda b/src/Algebra/Lattice/Construct/Subst/Equality.agda
index 19b4884e8e..14a8f650cf 100644
--- a/src/Algebra/Lattice/Construct/Subst/Equality.agda
+++ b/src/Algebra/Lattice/Construct/Subst/Equality.agda
@@ -12,7 +12,6 @@
 open import Algebra.Core using (Op₂)
 open import Algebra.Definitions
 open import Algebra.Lattice.Structures
-import Data.Sum as Sum
 open import Data.Product as Prod
 open import Function.Base
 open import Relation.Binary.Core
diff --git a/src/Algebra/Module/Definitions/Left.agda b/src/Algebra/Module/Definitions/Left.agda
index 1175292fcb..e43f2ef38e 100644
--- a/src/Algebra/Module/Definitions/Left.agda
+++ b/src/Algebra/Module/Definitions/Left.agda
@@ -15,7 +15,6 @@ module Algebra.Module.Definitions.Left
   {a b ℓb} (A : Set a) {B : Set b} (_≈_ : Rel B ℓb)
   where
 
-open import Data.Sum
 open import Data.Product
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Module/Definitions/Right.agda b/src/Algebra/Module/Definitions/Right.agda
index 9f9dc67bf7..875dcfc594 100644
--- a/src/Algebra/Module/Definitions/Right.agda
+++ b/src/Algebra/Module/Definitions/Right.agda
@@ -16,7 +16,6 @@ module Algebra.Module.Definitions.Right
   where
 
 open import Data.Product
-open import Data.Sum
 
 ------------------------------------------------------------------------
 -- Binary operations
diff --git a/src/Data/Container/Indexed/Combinator.agda b/src/Data/Container/Indexed/Combinator.agda
index bc99e83f96..43721d4689 100644
--- a/src/Data/Container/Indexed/Combinator.agda
+++ b/src/Data/Container/Indexed/Combinator.agda
@@ -13,7 +13,7 @@ open import Data.Container.Indexed
 open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
 open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Product as Prod hiding (Σ) renaming (_×_ to _⟨×⟩_)
-open import Data.Sum renaming (_⊎_ to _⟨⊎⟩_)
+open import Data.Sum.Base renaming (_⊎_ to _⟨⊎⟩_)
 open import Data.Sum.Relation.Unary.All as All using (All)
 open import Function.Base as F hiding (id; const) renaming (_∘_ to _⟨∘⟩_)
 open import Function.Inverse using (_↔̇_; inverse)
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid.agda
index cdfd9c7ed5..4ac2f914d7 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.List
+open import Data.List.Base using (List)
 open import Level
 open import Relation.Binary
 
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index dc0114238a..f3feaf2f02 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -11,7 +11,7 @@ open import Data.List.Membership.Setoid.Properties as Membership
 open import Data.List.Relation.Unary.Any using (index)
 open import Data.List.Relation.Unary.Any.Properties using (lookup-index)
 open import Data.List.Relation.Unary.Enumerates.Setoid
-open import Data.Sum using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Sum.Relation.Binary.Pointwise
   using (_⊎ₛ_; inj₁; inj₂)
 open import Data.Product using (_,_; proj₁; proj₂)
diff --git a/src/Data/List/Relation/Unary/First.agda b/src/Data/List/Relation/Unary/First.agda
index b00f35e9d3..90b8bf26cb 100644
--- a/src/Data/List/Relation/Unary/First.agda
+++ b/src/Data/List/Relation/Unary/First.agda
@@ -16,7 +16,7 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.Product as Prod using (∃; -,_; _,_)
-open import Data.Sum as Sum using (_⊎_; inj₁; inj₂)
+open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function
 open import Relation.Unary
 open import Relation.Nullary
diff --git a/src/Data/List/Relation/Unary/Grouped.agda b/src/Data/List/Relation/Unary/Grouped.agda
index 51f3f28a17..b907069142 100644
--- a/src/Data/List/Relation/Unary/Grouped.agda
+++ b/src/Data/List/Relation/Unary/Grouped.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Unary.Grouped where
 
 open import Data.List.Base using (List; []; _∷_; map)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_; all?)
-open import Data.Sum using (_⊎_; inj₁; inj₂)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Product using (_×_; _,_)
 open import Relation.Binary as B using (REL; Rel)
 open import Relation.Unary as U using (Pred)
diff --git a/src/Data/List/Relation/Unary/Grouped/Properties.agda b/src/Data/List/Relation/Unary/Grouped/Properties.agda
index a8a72e6cec..65ac3d6ed6 100644
--- a/src/Data/List/Relation/Unary/Grouped/Properties.agda
+++ b/src/Data/List/Relation/Unary/Grouped/Properties.agda
@@ -9,7 +9,7 @@
 module Data.List.Relation.Unary.Grouped.Properties where
 
 open import Data.Bool using (true; false)
-open import Data.List
+open import Data.List.Base using ([]; [_]; _∷_; map; derun)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
diff --git a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
index 8945b2f220..2eda2ded13 100644
--- a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.List
+open import Data.List.Base using ([]; _∷_; deduplicate)
 import Data.List.Relation.Unary.Unique.DecSetoid as Unique
 open import Data.List.Relation.Unary.All.Properties using (all-filter)
 open import Data.List.Relation.Unary.Unique.Setoid.Properties
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index 8b095b137f..f1c0b64f56 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -17,7 +17,7 @@ open import Data.Nat.Binary.Base
 open import Data.Nat.Binary.Properties
 import Data.Nat.Properties as ℕₚ
 open import Data.Product using (_×_; _,_; proj₁; proj₂; ∃)
-open import Data.Sum using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Vec using ([]; _∷_)
 open import Function.Base using (_∘_; _$_)
 open import Level using (0ℓ)
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index 533a565d23..0b263eed54 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -16,7 +16,7 @@ open import Data.Nat.DivMod
 open import Data.Nat.Divisibility
 open import Data.Nat.Properties
 open import Data.Nat.Combinatorics.Base
-open import Data.Sum using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; trans; _≢_)
 open import Relation.Nullary.Decidable using (yes; no; does)
diff --git a/src/Data/Nat/InfinitelyOften.agda b/src/Data/Nat/InfinitelyOften.agda
index 0913f3fd5c..ba4dc24f1d 100644
--- a/src/Data/Nat/InfinitelyOften.agda
+++ b/src/Data/Nat/InfinitelyOften.agda
@@ -14,7 +14,7 @@ open import Data.Empty using (⊥-elim)
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Product as Prod hiding (map)
-open import Data.Sum hiding (map)
+open import Data.Sum.Base using (inj₁; inj₂; _⊎_)
 open import Function
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
index f2dec21b8c..0a803425f2 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
@@ -17,7 +17,7 @@ open import Data.Maybe.Properties using (just-injective)
 open import Data.Maybe.Relation.Unary.All as Maybe using (nothing; just)
 open import Data.Nat.Base using (ℕ)
 open import Data.Product as Prod using (∃; ∃-syntax; _×_; _,_; proj₁; proj₂)
-open import Data.Sum as Sum using (_⊎_; inj₁; inj₂)
+open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base as F
 open import Level using (Level)
 
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional.agda b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
index 7e5d297025..5e9498ea31 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
@@ -17,7 +17,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.Maybe.Base using (just; nothing; is-just)
 open import Data.Product as Prod using (_×_; ∃-syntax; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent renaming (Pointwise to _×ᴿ_)
-open import Data.Sum using (_⊎_; inj₁; inj₂)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_)
 open import Level using (Level)
 
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
index 91df2a1c0c..c05ed27177 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
@@ -17,7 +17,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.Maybe.Base using (just; nothing; is-just)
 open import Data.Product as Prod using (_×_; ∃-syntax; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent renaming (Pointwise to _×ᴿ_)
-open import Data.Sum using (_⊎_; inj₁; inj₂)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_)
 open import Level using (Level)
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index a3ff214a06..9b2bea47b7 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -12,7 +12,7 @@ open import Data.Empty
 open import Data.Nat using (ℕ; suc)
 import Data.Nat.Properties as ℕ
 open import Data.Product using (_×_; _,_; proj₁; proj₂; uncurry)
-open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_])
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Binary.Pointwise.Inductive using (Pointwise; []; _∷_)
 open import Function.Base using (flip)
diff --git a/src/Effect/Monad/Error/Transformer.agda b/src/Effect/Monad/Error/Transformer.agda
index d6d5f97626..308a20bbbd 100644
--- a/src/Effect/Monad/Error/Transformer.agda
+++ b/src/Effect/Monad/Error/Transformer.agda
@@ -41,7 +41,7 @@ record RawMonadError
 
 module Sumₗ where
 
-  open import Data.Sum using (inj₁; inj₂; [_,_]′)
+  open import Data.Sum.Base using (inj₁; inj₂; [_,_]′)
   open import Data.Sum.Effectful.Left.Transformer E a
 
   monadError : RawMonad M → RawMonadError (SumₗT M)
@@ -54,7 +54,7 @@ module Sumₗ where
 
 module Sumᵣ where
 
-  open import Data.Sum using (inj₁; inj₂; [_,_]′)
+  open import Data.Sum.Base using (inj₁; inj₂; [_,_]′)
   open import Data.Sum.Effectful.Right.Transformer a E
 
   monadError : RawMonad M → RawMonadError (SumᵣT M)
diff --git a/src/Reflection/External.agda b/src/Reflection/External.agda
index ba4e076426..e7e9ae989d 100644
--- a/src/Reflection/External.agda
+++ b/src/Reflection/External.agda
@@ -14,7 +14,7 @@ open import Data.Nat.Base using (ℕ; suc; zero; NonZero)
 open import Data.List.Base using (List; _∷_; [])
 open import Data.Product using (_×_; _,_)
 open import Data.String.Base as String using (String; _++_)
-open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_])
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Unit.Base using (⊤; tt)
 open import Function.Base using (case_of_; _$_; _∘_)
 open import Reflection hiding (name)
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 74b34607a6..6ab36647c3 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -17,7 +17,7 @@ open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.PropositionalEquality.Properties as PropEq
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   as Trivial
-open import Data.Sum     as Sum  hiding (map)
+open import Data.Sum.Base as Sum  hiding (map)
 open import Data.Sum.Relation.Binary.Pointwise
 open import Data.Product as Prod hiding (map)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
diff --git a/src/Relation/Unary/Algebra.agda b/src/Relation/Unary/Algebra.agda
index 2d121be0fa..600bab6c1f 100644
--- a/src/Relation/Unary/Algebra.agda
+++ b/src/Relation/Unary/Algebra.agda
@@ -15,7 +15,7 @@ import Algebra.Lattice.Structures as AlgebraicLatticeStructures
 import Algebra.Structures as AlgebraicStructures
 open import Data.Empty.Polymorphic using (⊥-elim)
 open import Data.Product as Product using (_,_; proj₁; proj₂; <_,_>; curry; uncurry)
-open import Data.Sum as Sum using (inj₁; inj₂; [_,_])
+open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])
 open import Data.Unit.Polymorphic using (tt)
 open import Function.Base using (id; const; _∘_)
 open import Level

From 77a24631f54c52970b7ab0e79009b376db49753b Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Fri, 28 Jul 2023 12:00:51 -0400
Subject: [PATCH 074/316] score didn t change when changing the  Import of
 Data.Nat.Show (#2025)

---
 src/Data/Fin/Show.agda       | 2 +-
 src/Data/Integer.agda        | 2 +-
 src/System/Clock.agda        | 2 +-
 src/System/Console/ANSI.agda | 2 +-
 src/Test/Golden.agda         | 2 +-
 src/Text/Printf.agda         | 2 +-
 6 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/src/Data/Fin/Show.agda b/src/Data/Fin/Show.agda
index 360cc8b1fc..8f9ae0583b 100644
--- a/src/Data/Fin/Show.agda
+++ b/src/Data/Fin/Show.agda
@@ -11,7 +11,7 @@ module Data.Fin.Show where
 open import Data.Fin.Base using (Fin; toℕ; fromℕ<)
 open import Data.Maybe.Base using (Maybe; nothing; just; _>>=_)
 open import Data.Nat as ℕ using (ℕ; _≤?_; _<?_)
-import Data.Nat.Show as ℕ
+import Data.Nat.Show as ℕ using (show; readMaybe)
 open import Data.String.Base using (String)
 open import Function.Base
 open import Relation.Nullary.Decidable using (yes; no)
diff --git a/src/Data/Integer.agda b/src/Data/Integer.agda
index ae8b8787ab..66c8100242 100644
--- a/src/Data/Integer.agda
+++ b/src/Data/Integer.agda
@@ -30,7 +30,7 @@ open import Data.Integer.Properties public
 -- Version 1.5
 -- Show
 
-import Data.Nat.Show as ℕ
+import Data.Nat.Show as ℕ using (show)
 open import Data.Sign as Sign using (Sign)
 open import Data.String.Base using (String; _++_)
 
diff --git a/src/System/Clock.agda b/src/System/Clock.agda
index ef0e91d1fa..d8b8745b17 100644
--- a/src/System/Clock.agda
+++ b/src/System/Clock.agda
@@ -12,7 +12,7 @@ open import Level using (Level; 0ℓ; Lift; lower)
 open import Data.Bool.Base using (if_then_else_)
 open import Data.Fin.Base using (Fin; toℕ)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _∸_; _^_; _<ᵇ_)
-import Data.Nat.Show as ℕ
+import Data.Nat.Show as ℕ using (show)
 open import Data.Nat.DivMod using (_/_)
 import Data.Nat.Properties as ℕₚ
 open import Data.String.Base using (String; _++_; padLeft)
diff --git a/src/System/Console/ANSI.agda b/src/System/Console/ANSI.agda
index e1a903a653..867a4cd856 100644
--- a/src/System/Console/ANSI.agda
+++ b/src/System/Console/ANSI.agda
@@ -10,7 +10,7 @@ module System.Console.ANSI where
 
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.Nat.Base using (ℕ; _+_)
-import Data.Nat.Show as ℕ
+import Data.Nat.Show as ℕ using (show)
 open import Data.String.Base using (String; concat; intersperse)
 open import Function.Base using (_$_; case_of_)
 
diff --git a/src/Test/Golden.agda b/src/Test/Golden.agda
index 81f9f2af1a..efbfc59ada 100644
--- a/src/Test/Golden.agda
+++ b/src/Test/Golden.agda
@@ -88,7 +88,7 @@ open import Data.List.Relation.Binary.Infix.Heterogeneous.Properties using (infi
 open import Data.List.Relation.Unary.Any using (any?)
 open import Data.Maybe.Base using (Maybe; just; nothing; fromMaybe)
 open import Data.Nat.Base using (ℕ; _≡ᵇ_; _<ᵇ_; _+_; _∸_)
-import Data.Nat.Show as ℕ
+import Data.Nat.Show as ℕ using (show)
 open import Data.Product using (_×_; _,_)
 open import Data.String as String using (String; lines; unlines; unwords; concat; _≟_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
diff --git a/src/Text/Printf.agda b/src/Text/Printf.agda
index 057b2718b8..dbb5a51d28 100644
--- a/src/Text/Printf.agda
+++ b/src/Text/Printf.agda
@@ -14,7 +14,7 @@ open import Function.Base using (id)
 import Data.Char.Base    as Cₛ
 import Data.Integer.Show as ℤₛ
 import Data.Float        as Fₛ
-import Data.Nat.Show     as ℕₛ
+import Data.Nat.Show     as ℕₛ  using (show)
 
 open import Text.Format as Format hiding (Error)
 open import Text.Printf.Generic

From 6301a5197d93ca3564c6a4ec670d9fab2b7cab77 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Fri, 28 Jul 2023 13:21:13 -0400
Subject: [PATCH 075/316] Simplify leftover `Function` imports (#2012)

* Simplify leftover `Function` imports

* `Function` -> `Function.Base`

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

---------

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
---
 README/Data/Container/Indexed.agda                        | 2 +-
 README/Design/Decidability.agda                           | 2 +-
 README/Function/Reasoning.agda                            | 2 +-
 README/Nary.agda                                          | 2 +-
 README/Tactic/RingSolver.agda                             | 1 -
 .../Lattice/Properties/BooleanAlgebra/Expression.agda     | 2 +-
 src/Algebra/Properties/AbelianGroup.agda                  | 2 +-
 src/Algebra/Solver/Ring.agda                              | 2 +-
 src/Algebra/Solver/Ring/Lemmas.agda                       | 2 +-
 src/Axiom/Extensionality/Heterogeneous.agda               | 2 +-
 src/Codata/Sized/Delay/Effectful.agda                     | 2 +-
 src/Data/Container/Relation/Binary/Pointwise.agda         | 2 +-
 src/Data/Container/Relation/Unary/Any.agda                | 2 +-
 src/Data/DifferenceList.agda                              | 2 +-
 src/Data/DifferenceNat.agda                               | 2 +-
 src/Data/DifferenceVec.agda                               | 2 +-
 src/Data/Graph/Acyclic.agda                               | 2 +-
 src/Data/Integer/Divisibility.agda                        | 2 +-
 src/Data/Integer/Divisibility/Signed.agda                 | 2 +-
 src/Data/List/NonEmpty/Effectful.agda                     | 2 +-
 src/Data/List/Properties.agda                             | 3 ++-
 .../List/Relation/Ternary/Interleaving/Properties.agda    | 2 +-
 .../Relation/Ternary/Interleaving/Setoid/Properties.agda  | 2 +-
 .../List/Relation/Unary/Enumerates/Setoid/Properties.agda | 4 +++-
 src/Data/List/Relation/Unary/First.agda                   | 2 +-
 src/Data/List/Relation/Unary/First/Properties.agda        | 2 +-
 src/Data/List/Reverse.agda                                | 2 +-
 src/Data/List/Zipper.agda                                 | 2 +-
 src/Data/List/Zipper/Properties.agda                      | 2 +-
 src/Data/Maybe/Properties.agda                            | 3 ++-
 src/Data/Maybe/Relation/Unary/All/Properties.agda         | 2 +-
 src/Data/Nat/Coprimality.agda                             | 2 +-
 src/Data/Nat/GCD.agda                                     | 2 +-
 src/Data/Nat/GCD/Lemmas.agda                              | 2 +-
 src/Data/Nat/InfinitelyOften.agda                         | 2 +-
 src/Data/Nat/LCM.agda                                     | 1 -
 src/Data/Product/Effectful/Examples.agda                  | 2 +-
 src/Data/Product/Properties/WithK.agda                    | 2 +-
 src/Data/Record.agda                                      | 2 +-
 src/Data/Star/BoundedVec.agda                             | 2 +-
 src/Data/Star/Decoration.agda                             | 2 +-
 src/Data/Star/Nat.agda                                    | 2 +-
 src/Data/String.agda                                      | 1 -
 src/Data/Sum/Effectful/Examples.agda                      | 2 +-
 src/Data/Sum/Properties.agda                              | 4 ++--
 src/Data/Sum/Relation/Binary/LeftOrder.agda               | 2 +-
 src/Data/These.agda                                       | 2 +-
 src/Data/These/Effectful/Left/Base.agda                   | 2 +-
 src/Data/These/Effectful/Right/Base.agda                  | 2 +-
 src/Data/Tree/AVL/NonEmpty.agda                           | 2 +-
 src/Data/Tree/Binary/Relation/Unary/All/Properties.agda   | 2 +-
 src/Data/Tree/Binary/Zipper.agda                          | 2 +-
 src/Data/Tree/Binary/Zipper/Properties.agda               | 2 +-
 src/Data/Trie.agda                                        | 2 +-
 src/Data/Universe/Indexed.agda                            | 2 +-
 src/Data/Vec/Bounded.agda                                 | 2 +-
 .../Vec/Functional/Relation/Unary/All/Properties.agda     | 2 +-
 src/Data/Vec/Functional/Relation/Unary/Any.agda           | 2 +-
 src/Data/Vec/Membership/Setoid.agda                       | 2 +-
 src/Data/Vec/Recursive/Effectful.agda                     | 2 +-
 src/Data/Word/Base.agda                                   | 2 +-
 src/Effect/Applicative/Predicate.agda                     | 2 +-
 src/Effect/Comonad.agda                                   | 2 +-
 src/Effect/Functor/Predicate.agda                         | 2 +-
 src/Effect/Monad/Continuation.agda                        | 2 +-
 src/Effect/Monad/Partiality/All.agda                      | 2 +-
 src/Effect/Monad/Predicate.agda                           | 2 +-
 src/Effect/Monad/Reader/Indexed.agda                      | 2 +-
 src/Effect/Monad/Writer/Indexed.agda                      | 2 +-
 src/Function/Construct/Composition.agda                   | 8 +++++++-
 src/Function/Construct/Symmetry.agda                      | 8 +++++++-
 src/Function/Endomorphism/Propositional.agda              | 2 +-
 src/Function/Properties.agda                              | 3 ++-
 .../Construct/Closure/ReflexiveTransitive/Properties.agda | 2 +-
 .../Closure/ReflexiveTransitive/Properties/WithK.agda     | 2 +-
 .../Binary/Construct/Closure/Transitive/WithK.agda        | 2 +-
 src/Relation/Binary/HeterogeneousEquality/Quotients.agda  | 2 +-
 src/Relation/Nullary/Indexed/Negation.agda                | 2 +-
 src/Relation/Nullary/Universe.agda                        | 2 +-
 src/Relation/Unary/PredicateTransformer.agda              | 2 +-
 src/System/Directory.agda                                 | 2 +-
 src/System/FilePath/Posix.agda                            | 1 -
 src/Tactic/MonoidSolver.agda                              | 2 +-
 src/Tactic/RingSolver/Core/NatSet.agda                    | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Addition.agda | 2 +-
 .../Core/Polynomial/Homomorphism/Exponentiation.agda      | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda   | 2 +-
 .../Core/Polynomial/Homomorphism/Multiplication.agda      | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Negation.agda | 2 +-
 src/Tactic/RingSolver/Core/Polynomial/Parameters.agda     | 1 -
 src/Tactic/RingSolver/NonReflective.agda                  | 2 +-
 91 files changed, 104 insertions(+), 92 deletions(-)

diff --git a/README/Data/Container/Indexed.agda b/README/Data/Container/Indexed.agda
index 0c94e182ec..7ace9736f1 100644
--- a/README/Data/Container/Indexed.agda
+++ b/README/Data/Container/Indexed.agda
@@ -12,7 +12,7 @@ open import Data.Unit
 open import Data.Empty
 open import Data.Nat.Base
 open import Data.Product
-open import Function
+open import Function.Base using (_∋_)
 open import Data.W.Indexed
 open import Data.Container.Indexed
 open import Data.Container.Indexed.WithK
diff --git a/README/Design/Decidability.agda b/README/Design/Decidability.agda
index 38bb52d489..cb41cf41ef 100644
--- a/README/Design/Decidability.agda
+++ b/README/Design/Decidability.agda
@@ -15,7 +15,7 @@ open import Data.Nat
 open import Data.Nat.Properties using (suc-injective)
 open import Data.Product
 open import Data.Unit
-open import Function
+open import Function.Base using (id;  _∘_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nary
 
diff --git a/README/Function/Reasoning.agda b/README/Function/Reasoning.agda
index 0c9afa4532..4b9835346e 100644
--- a/README/Function/Reasoning.agda
+++ b/README/Function/Reasoning.agda
@@ -36,7 +36,7 @@ open import Data.Nat
 open import Data.List.Base
 open import Data.Char.Base
 open import Data.String as String using (String; toList; fromList; _==_)
-open import Function
+open import Function.Base using (_∘_)
 open import Data.Bool hiding (_≤?_)
 open import Data.Product as P using (_×_; <_,_>; uncurry; proj₁)
 open import Agda.Builtin.Equality
diff --git a/README/Nary.agda b/README/Nary.agda
index da2c2fcca3..16e50392b3 100644
--- a/README/Nary.agda
+++ b/README/Nary.agda
@@ -17,7 +17,7 @@ open import Data.List
 open import Data.List.Properties
 open import Data.Product using (_×_; _,_)
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Function
+open import Function.Base using (id; flip; _∘′_)
 open import Relation.Nullary
 open import Relation.Binary using (module Tri); open Tri
 open import Relation.Binary.PropositionalEquality
diff --git a/README/Tactic/RingSolver.agda b/README/Tactic/RingSolver.agda
index fdbea87bbf..8071bee39c 100644
--- a/README/Tactic/RingSolver.agda
+++ b/README/Tactic/RingSolver.agda
@@ -29,7 +29,6 @@ instance
 -- Imports!
 
 open import Data.List.Base as List using (List; _∷_; [])
-open import Function
 open import Relation.Binary.PropositionalEquality
   using (subst; cong; _≡_; module ≡-Reasoning)
 open import Data.Bool as Bool using (Bool; true; false; if_then_else_)
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
index 9cdad57edc..88a5cc5518 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
@@ -25,7 +25,7 @@ import Function.Identity.Effectful as IdCat
 open import Data.Vec.Properties using (lookup-map)
 open import Data.Vec.Relation.Binary.Pointwise.Extensional as PW
   using (Pointwise; ext)
-open import Function
+open import Function.Base using (_∘_; _$_; flip)
 open import Relation.Binary.PropositionalEquality as P using (_≗_)
 import Relation.Binary.Reflection as Reflection
 
diff --git a/src/Algebra/Properties/AbelianGroup.agda b/src/Algebra/Properties/AbelianGroup.agda
index a1c2056ccb..6846f29d83 100644
--- a/src/Algebra/Properties/AbelianGroup.agda
+++ b/src/Algebra/Properties/AbelianGroup.agda
@@ -12,7 +12,7 @@ module Algebra.Properties.AbelianGroup
   {a ℓ} (G : AbelianGroup a ℓ) where
 
 open AbelianGroup G
-open import Function
+open import Function.Base using (_$_)
 open import Relation.Binary.Reasoning.Setoid setoid
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Solver/Ring.agda b/src/Algebra/Solver/Ring.agda
index e103b7a70e..5bc696da38 100644
--- a/src/Algebra/Solver/Ring.agda
+++ b/src/Algebra/Solver/Ring.agda
@@ -49,7 +49,7 @@ open import Data.Nat.Base using (ℕ; suc; zero)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Vec.Base using (Vec; []; _∷_; lookup)
 open import Data.Maybe.Base using (just; nothing)
-open import Function
+open import Function.Base using (_⟨_⟩_; _$_)
 open import Level using (_⊔_)
 
 infix  9 :-_ -H_ -N_
diff --git a/src/Algebra/Solver/Ring/Lemmas.agda b/src/Algebra/Solver/Ring/Lemmas.agda
index 0a9cd7ac73..944e30dfb7 100644
--- a/src/Algebra/Solver/Ring/Lemmas.agda
+++ b/src/Algebra/Solver/Ring/Lemmas.agda
@@ -24,7 +24,7 @@ open AlmostCommutativeRing r
 open import Algebra.Morphism
 open _-Raw-AlmostCommutative⟶_ morphism
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Function
+open import Function.Base using (_⟨_⟩_; _$_)
 
 lemma₀ : ∀ a b c x →
          (a + b) * x + c ≈ a * x + (b * x + c)
diff --git a/src/Axiom/Extensionality/Heterogeneous.agda b/src/Axiom/Extensionality/Heterogeneous.agda
index 0b10affb61..ed1be68791 100644
--- a/src/Axiom/Extensionality/Heterogeneous.agda
+++ b/src/Axiom/Extensionality/Heterogeneous.agda
@@ -9,7 +9,7 @@
 module Axiom.Extensionality.Heterogeneous where
 
 import Axiom.Extensionality.Propositional as P
-open import Function
+open import Function.Base using (_$_; _∘_)
 open import Level
 open import Relation.Binary.HeterogeneousEquality.Core
 open import Relation.Binary.PropositionalEquality.Core
diff --git a/src/Codata/Sized/Delay/Effectful.agda b/src/Codata/Sized/Delay/Effectful.agda
index 95254c3a13..2724c67d47 100644
--- a/src/Codata/Sized/Delay/Effectful.agda
+++ b/src/Codata/Sized/Delay/Effectful.agda
@@ -9,7 +9,7 @@
 module Codata.Sized.Delay.Effectful where
 
 open import Codata.Sized.Delay
-open import Function
+open import Function.Base using (id)
 open import Effect.Choice
 open import Effect.Empty
 open import Effect.Functor
diff --git a/src/Data/Container/Relation/Binary/Pointwise.agda b/src/Data/Container/Relation/Binary/Pointwise.agda
index 2b0b966b3d..7b20aa8dbb 100644
--- a/src/Data/Container/Relation/Binary/Pointwise.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise.agda
@@ -9,7 +9,7 @@
 module Data.Container.Relation.Binary.Pointwise where
 
 open import Data.Product using (_,_; Σ-syntax; -,_; proj₁; proj₂)
-open import Function
+open import Function.Base using (_∘_)
 open import Level using (_⊔_)
 open import Relation.Binary using (REL; _⇒_)
 open import Relation.Binary.PropositionalEquality.Core as P
diff --git a/src/Data/Container/Relation/Unary/Any.agda b/src/Data/Container/Relation/Unary/Any.agda
index 944734af3d..de3937f221 100644
--- a/src/Data/Container/Relation/Unary/Any.agda
+++ b/src/Data/Container/Relation/Unary/Any.agda
@@ -11,7 +11,7 @@ module Data.Container.Relation.Unary.Any where
 open import Level using (_⊔_)
 open import Relation.Unary using (Pred; _⊆_)
 open import Data.Product as Prod using (_,_; proj₂; ∃)
-open import Function
+open import Function.Base using (_∘′_; id)
 
 open import Data.Container.Core hiding (map)
 import Data.Container.Morphism as M
diff --git a/src/Data/DifferenceList.agda b/src/Data/DifferenceList.agda
index dc62baf000..3227e6ed8e 100644
--- a/src/Data/DifferenceList.agda
+++ b/src/Data/DifferenceList.agda
@@ -10,7 +10,7 @@ module Data.DifferenceList where
 
 open import Level using (Level)
 open import Data.List.Base as List using (List)
-open import Function
+open import Function.Base using (_⟨_⟩_)
 open import Data.Nat.Base
 
 private
diff --git a/src/Data/DifferenceNat.agda b/src/Data/DifferenceNat.agda
index 0d6c024e51..c0f33a1fea 100644
--- a/src/Data/DifferenceNat.agda
+++ b/src/Data/DifferenceNat.agda
@@ -10,7 +10,7 @@
 module Data.DifferenceNat where
 
 open import Data.Nat.Base as N using (ℕ)
-open import Function
+open import Function.Base using (_⟨_⟩_)
 
 infixl 6 _+_
 
diff --git a/src/Data/DifferenceVec.agda b/src/Data/DifferenceVec.agda
index acb50ffde0..c34382a051 100644
--- a/src/Data/DifferenceVec.agda
+++ b/src/Data/DifferenceVec.agda
@@ -10,7 +10,7 @@ module Data.DifferenceVec where
 
 open import Data.DifferenceNat
 open import Data.Vec.Base as V using (Vec)
-open import Function
+open import Function.Base using (_⟨_⟩_)
 import Data.Nat.Base as N
 
 infixr 5 _∷_ _++_
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index d7b298f0ef..586b28e278 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -26,7 +26,7 @@ open import Data.Empty
 open import Data.Unit.Base using (⊤; tt)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.List.Base as List using (List; []; _∷_)
-open import Function
+open import Function.Base using (_$_; _∘′_; _∘_; id)
 open import Relation.Nullary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
diff --git a/src/Data/Integer/Divisibility.agda b/src/Data/Integer/Divisibility.agda
index 44a97c6554..0f089b30fc 100644
--- a/src/Data/Integer/Divisibility.agda
+++ b/src/Data/Integer/Divisibility.agda
@@ -10,7 +10,7 @@
 
 module Data.Integer.Divisibility where
 
-open import Function
+open import Function.Base using (_on_; _$_)
 open import Data.Integer.Base
 open import Data.Integer.Properties
 import Data.Nat.Base as ℕ
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index d4a22e420a..db80bcb3a0 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -8,7 +8,7 @@
 
 module Data.Integer.Divisibility.Signed where
 
-open import Function
+open import Function.Base using (_⟨_⟩_; _$_; _$′_; _∘_; _∘′_)
 open import Data.Integer.Base
 open import Data.Integer.Properties
 open import Data.Integer.Divisibility as Unsigned
diff --git a/src/Data/List/NonEmpty/Effectful.agda b/src/Data/List/NonEmpty/Effectful.agda
index 654002c839..373ceceb26 100644
--- a/src/Data/List/NonEmpty/Effectful.agda
+++ b/src/Data/List/NonEmpty/Effectful.agda
@@ -16,7 +16,7 @@ open import Effect.Functor
 open import Effect.Applicative
 open import Effect.Monad
 open import Effect.Comonad
-open import Function
+open import Function.Base using (flip; _∘′_; _∘_)
 
 ------------------------------------------------------------------------
 -- List⁺ applicative functor
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index c4cf722ec6..c05d3251de 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -31,7 +31,8 @@ import Data.Product.Relation.Unary.All as Prod using (All)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Fin.Properties using (toℕ-cast)
-open import Function
+open import Function.Base using (id; _∘_; _∘′_; _∋_; _-⟨_∣; ∣_⟩-_; _$_; const; flip)
+open import Function.Definitions using (Injective)
 open import Level using (Level)
 open import Relation.Binary as B using (DecidableEquality)
 import Relation.Binary.Reasoning.Setoid as EqR
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
index 563adfc9d9..ba61947c85 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Nat.Properties using (+-suc)
 open import Data.List.Base hiding (_∷ʳ_)
 open import Data.List.Properties using (reverse-involutive)
 open import Data.List.Relation.Ternary.Interleaving hiding (map)
-open import Function
+open import Function.Base using (_$_)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong; module ≡-Reasoning)
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda
index f66628a628..cdd8f968aa 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda
@@ -16,7 +16,7 @@ open import Data.Bool.Base using (true; false)
 open import Relation.Unary using (Decidable)
 open import Relation.Nullary.Decidable using (does)
 open import Relation.Nullary.Decidable using (¬?)
-open import Function
+open import Function.Base using (_∘_)
 
 open import Data.List.Relation.Binary.Equality.Setoid S using (≋-refl)
 open import Data.List.Relation.Ternary.Interleaving.Setoid S
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index f3feaf2f02..78a365324b 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -17,7 +17,9 @@ open import Data.Sum.Relation.Binary.Pointwise
 open import Data.Product using (_,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
   using (_×ₛ_)
-open import Function
+open import Function.Base using (_∘_)
+open import Function.Bundles using (Surjection)
+open import Function.Definitions using (Surjective)
 open import Level
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
diff --git a/src/Data/List/Relation/Unary/First.agda b/src/Data/List/Relation/Unary/First.agda
index 90b8bf26cb..931195aea6 100644
--- a/src/Data/List/Relation/Unary/First.agda
+++ b/src/Data/List/Relation/Unary/First.agda
@@ -17,7 +17,7 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.Product as Prod using (∃; -,_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
-open import Function
+open import Function.Base using (id; _∘′_)
 open import Relation.Unary
 open import Relation.Nullary
 
diff --git a/src/Data/List/Relation/Unary/First/Properties.agda b/src/Data/List/Relation/Unary/First/Properties.agda
index 7885712d7e..f30e037022 100644
--- a/src/Data/List/Relation/Unary/First/Properties.agda
+++ b/src/Data/List/Relation/Unary/First/Properties.agda
@@ -15,7 +15,7 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (here; there)
 open import Data.List.Relation.Unary.First
 import Data.Sum as Sum
-open import Function
+open import Function.Base using (_∘′_; _$_; _∘_; id)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; refl; _≗_)
 open import Relation.Unary
 open import Relation.Nullary.Negation
diff --git a/src/Data/List/Reverse.agda b/src/Data/List/Reverse.agda
index 92b2aa5b0c..0464181932 100644
--- a/src/Data/List/Reverse.agda
+++ b/src/Data/List/Reverse.agda
@@ -10,7 +10,7 @@ module Data.List.Reverse where
 
 open import Data.List.Base as L hiding (reverse)
 open import Data.List.Properties
-open import Function
+open import Function.Base using (_$_)
 open import Relation.Binary.PropositionalEquality
 
 -- If you want to traverse a list from the end, then you can use the
diff --git a/src/Data/List/Zipper.agda b/src/Data/List/Zipper.agda
index 86f122e522..e2cc769ed2 100644
--- a/src/Data/List/Zipper.agda
+++ b/src/Data/List/Zipper.agda
@@ -11,7 +11,7 @@ module Data.List.Zipper where
 open import Data.Nat.Base
 open import Data.Maybe.Base as Maybe using (Maybe ; just ; nothing)
 open import Data.List.Base as List using (List ; [] ; _∷_)
-open import Function
+open import Function.Base using (_on_; flip)
 
 
 -- Definition
diff --git a/src/Data/List/Zipper/Properties.agda b/src/Data/List/Zipper/Properties.agda
index 3e3dea85d2..5c4ce7c7d8 100644
--- a/src/Data/List/Zipper/Properties.agda
+++ b/src/Data/List/Zipper/Properties.agda
@@ -15,7 +15,7 @@ open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
 open import Relation.Binary.PropositionalEquality
 open ≡-Reasoning
-open import Function
+open import Function.Base using (_on_; _$′_; _$_; flip)
 
 -- Invariant: Zipper represents a given list
 ------------------------------------------------------------------------
diff --git a/src/Data/Maybe/Properties.agda b/src/Data/Maybe/Properties.agda
index 6833f301e7..cb19c9c328 100644
--- a/src/Data/Maybe/Properties.agda
+++ b/src/Data/Maybe/Properties.agda
@@ -14,7 +14,8 @@ import Algebra.Definitions as Definitions
 open import Data.Maybe.Base
 open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
 open import Data.Product using (_,_)
-open import Function
+open import Function.Base using (_∋_; id; _∘_; _∘′_)
+open import Function.Definitions using (Injective)
 open import Level using (Level)
 open import Relation.Binary using (Decidable)
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/Maybe/Relation/Unary/All/Properties.agda b/src/Data/Maybe/Relation/Unary/All/Properties.agda
index 37d55adf0a..493f8679d0 100644
--- a/src/Data/Maybe/Relation/Unary/All/Properties.agda
+++ b/src/Data/Maybe/Relation/Unary/All/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Maybe.Relation.Unary.All as All
   using (All; nothing; just)
 open import Data.Maybe.Relation.Binary.Connected
 open import Data.Product using (_×_; _,_)
-open import Function
+open import Function.Base using (_∘_)
 open import Level
 open import Relation.Unary
 open import Relation.Binary.Core
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 13701b486c..04283143fd 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -19,7 +19,7 @@ open import Data.Nat.Primality
 open import Data.Nat.Properties
 open import Data.Nat.DivMod
 open import Data.Product as Prod
-open import Function
+open import Function.Base using (_∘_)
 open import Level using (0ℓ)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; trans; cong; subst; module ≡-Reasoning)
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index 9e21f6f812..c843ae215b 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -17,7 +17,7 @@ open import Data.Nat.Induction
   using (Acc; acc; <′-Rec; <′-recBuilder; <-wellFounded-fast)
 open import Data.Product
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
-open import Function
+open import Function.Base using (_$_; _∘_)
 open import Induction using (build)
 open import Induction.Lexicographic using (_⊗_; [_⊗_])
 open import Relation.Binary
diff --git a/src/Data/Nat/GCD/Lemmas.agda b/src/Data/Nat/GCD/Lemmas.agda
index 8d92924ddd..3f9a4b6d42 100644
--- a/src/Data/Nat/GCD/Lemmas.agda
+++ b/src/Data/Nat/GCD/Lemmas.agda
@@ -11,7 +11,7 @@ module Data.Nat.GCD.Lemmas where
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Nat.Solver
-open import Function
+open import Function.Base using (_$_)
 open import Relation.Binary.PropositionalEquality
 
 open +-*-Solver
diff --git a/src/Data/Nat/InfinitelyOften.agda b/src/Data/Nat/InfinitelyOften.agda
index ba4dc24f1d..9e89841633 100644
--- a/src/Data/Nat/InfinitelyOften.agda
+++ b/src/Data/Nat/InfinitelyOften.agda
@@ -15,7 +15,7 @@ open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Product as Prod hiding (map)
 open import Data.Sum.Base using (inj₁; inj₂; _⊎_)
-open import Function
+open import Function.Base using (_∘_; id)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Negation using (¬¬-Monad; call/cc)
diff --git a/src/Data/Nat/LCM.agda b/src/Data/Nat/LCM.agda
index 5ce1c7cd2a..af85ed64ad 100644
--- a/src/Data/Nat/LCM.agda
+++ b/src/Data/Nat/LCM.agda
@@ -17,7 +17,6 @@ open import Data.Nat.Properties
 open import Data.Nat.GCD
 open import Data.Product
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
-open import Function
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl; sym; trans; cong; cong₂; module ≡-Reasoning)
 open import Relation.Binary
diff --git a/src/Data/Product/Effectful/Examples.agda b/src/Data/Product/Effectful/Examples.agda
index daf05ce431..a393cb4f66 100644
--- a/src/Data/Product/Effectful/Examples.agda
+++ b/src/Data/Product/Effectful/Examples.agda
@@ -16,7 +16,7 @@ open import Effect.Functor using (RawFunctor)
 open import Effect.Monad using (RawMonad)
 open import Data.Product
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Function
+open import Function.Base using (id)
 import Function.Identity.Effectful as Id
 open import Relation.Binary using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
diff --git a/src/Data/Product/Properties/WithK.agda b/src/Data/Product/Properties/WithK.agda
index 2218ec4ca3..645ea6a6c4 100644
--- a/src/Data/Product/Properties/WithK.agda
+++ b/src/Data/Product/Properties/WithK.agda
@@ -9,7 +9,7 @@
 module Data.Product.Properties.WithK where
 
 open import Data.Product
-open import Function
+open import Function.Base using (_∋_)
 open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Record.agda b/src/Data/Record.agda
index b2328e89ad..edc19da8ff 100644
--- a/src/Data/Record.agda
+++ b/src/Data/Record.agda
@@ -14,7 +14,7 @@ open import Data.Empty
 open import Data.List.Base
 open import Data.Product hiding (proj₁; proj₂)
 open import Data.Unit.Polymorphic
-open import Function
+open import Function.Base using (id; _∘_)
 open import Level
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/Star/BoundedVec.agda b/src/Data/Star/BoundedVec.agda
index bae0c52f2d..2755554354 100644
--- a/src/Data/Star/BoundedVec.agda
+++ b/src/Data/Star/BoundedVec.agda
@@ -16,7 +16,7 @@ open import Data.Star.Decoration
 open import Data.Star.Pointer
 open import Data.Star.List using (List)
 open import Data.Unit
-open import Function
+open import Function.Base using (const)
 open import Relation.Binary
 open import Relation.Binary.Consequences
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
diff --git a/src/Data/Star/Decoration.agda b/src/Data/Star/Decoration.agda
index 170b5abb07..48c0f4f11a 100644
--- a/src/Data/Star/Decoration.agda
+++ b/src/Data/Star/Decoration.agda
@@ -9,7 +9,7 @@
 module Data.Star.Decoration where
 
 open import Data.Unit
-open import Function
+open import Function.Base using (flip)
 open import Level
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
diff --git a/src/Data/Star/Nat.agda b/src/Data/Star/Nat.agda
index 42cf1e6b9f..ef81051ff9 100644
--- a/src/Data/Star/Nat.agda
+++ b/src/Data/Star/Nat.agda
@@ -9,7 +9,7 @@
 module Data.Star.Nat where
 
 open import Data.Unit
-open import Function
+open import Function.Base using (const)
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Relation.Binary.Construct.Always using (Always)
diff --git a/src/Data/String.agda b/src/Data/String.agda
index 7ff82dfac0..8a6dd08045 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -21,7 +21,6 @@ open import Data.List.Relation.Binary.Lex.Strict using (Lex-<; Lex-≤)
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Char.Base as Char using (Char)
 import Data.Char.Properties as Char using (_≟_)
-open import Function
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Nullary.Decidable using (does)
diff --git a/src/Data/Sum/Effectful/Examples.agda b/src/Data/Sum/Effectful/Examples.agda
index f6850c7e19..773c6ca316 100644
--- a/src/Data/Sum/Effectful/Examples.agda
+++ b/src/Data/Sum/Effectful/Examples.agda
@@ -21,7 +21,7 @@ private
   module Examplesₗ {a b} {A : Set a} {B : Set b} where
 
     open import Agda.Builtin.Equality
-    open import Function
+    open import Function.Base using (id)
     module Sₗ = Sumₗ A b
 
     open RawFunctor Sₗ.functor
diff --git a/src/Data/Sum/Properties.agda b/src/Data/Sum/Properties.agda
index 3fede95b06..2c6e638731 100644
--- a/src/Data/Sum/Properties.agda
+++ b/src/Data/Sum/Properties.agda
@@ -10,8 +10,8 @@ module Data.Sum.Properties where
 
 open import Level
 open import Data.Sum.Base
-open import Function
-open import Function.Bundles using (mk↔′)
+open import Function.Base using (_∋_; _∘_; id)
+open import Function.Bundles using (mk↔′; _↔_)
 open import Relation.Binary using (Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index 34c676bcbb..2749640316 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -13,7 +13,7 @@ open import Data.Sum.Relation.Binary.Pointwise as PW
   using (Pointwise; inj₁; inj₂)
 open import Data.Product
 open import Data.Empty
-open import Function
+open import Function.Base using (_$_; _∘_)
 open import Level
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Data/These.agda b/src/Data/These.agda
index c8d042c8e9..45024ec599 100644
--- a/src/Data/These.agda
+++ b/src/Data/These.agda
@@ -11,7 +11,7 @@ module Data.These where
 open import Level
 open import Data.Maybe.Base using (Maybe; just; nothing; maybe′)
 open import Data.Sum.Base using (_⊎_; [_,_]′)
-open import Function
+open import Function.Base using (const; _∘′_; id; constᵣ)
 
 
 ------------------------------------------------------------------------
diff --git a/src/Data/These/Effectful/Left/Base.agda b/src/Data/These/Effectful/Left/Base.agda
index 04f569abdf..8f8f3c48bc 100644
--- a/src/Data/These/Effectful/Left/Base.agda
+++ b/src/Data/These/Effectful/Left/Base.agda
@@ -21,7 +21,7 @@ open import Data.These.Base
 open import Effect.Functor
 open import Effect.Applicative
 open import Effect.Monad
-open import Function
+open import Function.Base using (_∘_; flip)
 
 Theseₗ : Set (a ⊔ b) → Set (a ⊔ b)
 Theseₗ B = These A B
diff --git a/src/Data/These/Effectful/Right/Base.agda b/src/Data/These/Effectful/Right/Base.agda
index 91da52ab02..721426ae68 100644
--- a/src/Data/These/Effectful/Right/Base.agda
+++ b/src/Data/These/Effectful/Right/Base.agda
@@ -21,7 +21,7 @@ open import Data.These.Base
 open import Effect.Functor
 open import Effect.Applicative
 open import Effect.Monad
-open import Function
+open import Function.Base using (flip; _∘_)
 
 Theseᵣ : Set (a ⊔ b) → Set (a ⊔ b)
 Theseᵣ A = These A B
diff --git a/src/Data/Tree/AVL/NonEmpty.agda b/src/Data/Tree/AVL/NonEmpty.agda
index 622e2b4a58..d4af2edd58 100644
--- a/src/Data/Tree/AVL/NonEmpty.agda
+++ b/src/Data/Tree/AVL/NonEmpty.agda
@@ -23,7 +23,7 @@ open import Data.Maybe.Base hiding (map)
 open import Data.Nat.Base hiding (_<_; _⊔_; compare)
 open import Data.Product hiding (map)
 open import Data.Unit
-open import Function
+open import Function.Base using (_$_; _∘′_)
 open import Level using (_⊔_; Lift; lift)
 open import Relation.Unary
 
diff --git a/src/Data/Tree/Binary/Relation/Unary/All/Properties.agda b/src/Data/Tree/Binary/Relation/Unary/All/Properties.agda
index 198964ef4d..08361f8476 100644
--- a/src/Data/Tree/Binary/Relation/Unary/All/Properties.agda
+++ b/src/Data/Tree/Binary/Relation/Unary/All/Properties.agda
@@ -12,7 +12,7 @@ open import Level
 open import Data.Tree.Binary as Tree using (Tree; leaf; node)
 open import Data.Tree.Binary.Relation.Unary.All
 open import Relation.Unary
-open import Function
+open import Function.Base using (id)
 
 private
   variable
diff --git a/src/Data/Tree/Binary/Zipper.agda b/src/Data/Tree/Binary/Zipper.agda
index 0e5f4b8d05..5786ca7672 100644
--- a/src/Data/Tree/Binary/Zipper.agda
+++ b/src/Data/Tree/Binary/Zipper.agda
@@ -13,7 +13,7 @@ open import Data.Tree.Binary as BT using (Tree; node; leaf)
 open import Data.List.Base as List using (List; []; _∷_; sum; _++_; [_])
 open import Data.Maybe using (Maybe; nothing; just)
 open import Data.Nat.Base using (ℕ; suc; _+_)
-open import Function
+open import Function.Base using (_$_; _∘_)
 
 private
   variable
diff --git a/src/Data/Tree/Binary/Zipper/Properties.agda b/src/Data/Tree/Binary/Zipper/Properties.agda
index 20017ad162..107981bcbf 100644
--- a/src/Data/Tree/Binary/Zipper/Properties.agda
+++ b/src/Data/Tree/Binary/Zipper/Properties.agda
@@ -17,7 +17,7 @@ open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
 open import Relation.Binary.PropositionalEquality
 open ≡-Reasoning
-open import Function
+open import Function.Base using (_on_; _∘_; _$_)
 open import Level using (Level)
 
 private
diff --git a/src/Data/Trie.agda b/src/Data/Trie.agda
index 2ea6b2f963..7f06a6f0ed 100644
--- a/src/Data/Trie.agda
+++ b/src/Data/Trie.agda
@@ -20,7 +20,7 @@ import Data.List.NonEmpty as List⁺
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing; maybe′)
 open import Data.Product as Prod using (∃)
 open import Data.These.Base as These using (These)
-open import Function
+open import Function.Base using (_∘′_; const)
 open import Relation.Unary using (IUniversal; _⇒_)
 
 open StrictTotalOrder S
diff --git a/src/Data/Universe/Indexed.agda b/src/Data/Universe/Indexed.agda
index a11958e16f..cb6d9f4e56 100644
--- a/src/Data/Universe/Indexed.agda
+++ b/src/Data/Universe/Indexed.agda
@@ -10,7 +10,7 @@ module Data.Universe.Indexed where
 
 open import Data.Product
 open import Data.Universe
-open import Function
+open import Function.Base using (_∘_)
 open import Level
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Bounded.agda b/src/Data/Vec/Bounded.agda
index 2e1f286419..3a1aae623e 100644
--- a/src/Data/Vec/Bounded.agda
+++ b/src/Data/Vec/Bounded.agda
@@ -11,7 +11,7 @@ module Data.Vec.Bounded where
 open import Level using (Level)
 open import Data.Nat.Base
 open import Data.Vec as Vec using (Vec)
-open import Function
+open import Function.Base using (id)
 open import Relation.Binary using (_Preserves_⟶_)
 open import Relation.Unary using (Pred; Decidable)
 
diff --git a/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda b/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
index 3de5f61679..89dba8d259 100644
--- a/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
@@ -14,7 +14,7 @@ open import Data.Product as Σ using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Vec.Functional as VF hiding (map)
 open import Data.Vec.Functional.Relation.Unary.All
-open import Function
+open import Function.Base using (const; _∘_)
 open import Level using (Level)
 open import Relation.Unary
 
diff --git a/src/Data/Vec/Functional/Relation/Unary/Any.agda b/src/Data/Vec/Functional/Relation/Unary/Any.agda
index 83e0e6dfb3..de6c3570cb 100644
--- a/src/Data/Vec/Functional/Relation/Unary/Any.agda
+++ b/src/Data/Vec/Functional/Relation/Unary/Any.agda
@@ -13,7 +13,7 @@ open import Data.Fin.Properties using (any?)
 open import Data.Nat.Base
 open import Data.Product as Σ using (Σ; ∃; _×_; _,_; proj₁; proj₂)
 open import Data.Vec.Functional as VF hiding (map)
-open import Function
+open import Function.Base using (id)
 open import Level using (Level)
 open import Relation.Unary
 
diff --git a/src/Data/Vec/Membership/Setoid.agda b/src/Data/Vec/Membership/Setoid.agda
index 1151aecf6f..50ef16bb79 100644
--- a/src/Data/Vec/Membership/Setoid.agda
+++ b/src/Data/Vec/Membership/Setoid.agda
@@ -10,7 +10,7 @@ open import Relation.Binary using (Setoid; _Respects_)
 
 module Data.Vec.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where
 
-open import Function
+open import Function.Base using (_∘_; flip)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.Any as Any
   using (Any; here; there; index)
diff --git a/src/Data/Vec/Recursive/Effectful.agda b/src/Data/Vec/Recursive/Effectful.agda
index 567020352a..cc58d5af6d 100644
--- a/src/Data/Vec/Recursive/Effectful.agda
+++ b/src/Data/Vec/Recursive/Effectful.agda
@@ -14,7 +14,7 @@ open import Data.Vec.Recursive
 open import Effect.Functor
 open import Effect.Applicative
 open import Effect.Monad
-open import Function
+open import Function.Base using (_∘_; flip)
 
 ------------------------------------------------------------------------
 -- Functor and applicative
diff --git a/src/Data/Word/Base.agda b/src/Data/Word/Base.agda
index 34206371e1..1dcaea5731 100644
--- a/src/Data/Word/Base.agda
+++ b/src/Data/Word/Base.agda
@@ -10,7 +10,7 @@ module Data.Word.Base where
 
 open import Level using (zero)
 import Data.Nat.Base as ℕ
-open import Function
+open import Function.Base using (_on_)
 open import Relation.Binary using (Rel)
 open import Relation.Binary.PropositionalEquality
 
diff --git a/src/Effect/Applicative/Predicate.agda b/src/Effect/Applicative/Predicate.agda
index 0bfe6a9803..bcfec53170 100644
--- a/src/Effect/Applicative/Predicate.agda
+++ b/src/Effect/Applicative/Predicate.agda
@@ -13,7 +13,7 @@ module Effect.Applicative.Predicate where
 
 open import Effect.Functor.Predicate
 open import Data.Product
-open import Function
+open import Function.Base using (const; constᵣ)
 open import Level
 open import Relation.Unary
 open import Relation.Unary.PredicateTransformer using (Pt)
diff --git a/src/Effect/Comonad.agda b/src/Effect/Comonad.agda
index 67a4d7d43d..72cf07d732 100644
--- a/src/Effect/Comonad.agda
+++ b/src/Effect/Comonad.agda
@@ -11,7 +11,7 @@
 module Effect.Comonad where
 
 open import Level
-open import Function
+open import Function.Base using (id; _∘′_; flip)
 
 private
   variable
diff --git a/src/Effect/Functor/Predicate.agda b/src/Effect/Functor/Predicate.agda
index d24938b26a..70704e34c3 100644
--- a/src/Effect/Functor/Predicate.agda
+++ b/src/Effect/Functor/Predicate.agda
@@ -10,7 +10,7 @@
 
 module Effect.Functor.Predicate where
 
-open import Function
+open import Function.Base using (const)
 open import Level
 open import Relation.Unary
 open import Relation.Unary.PredicateTransformer using (PT)
diff --git a/src/Effect/Monad/Continuation.agda b/src/Effect/Monad/Continuation.agda
index bcc66ce265..2ff9dd4bb9 100644
--- a/src/Effect/Monad/Continuation.agda
+++ b/src/Effect/Monad/Continuation.agda
@@ -13,7 +13,7 @@ open import Effect.Applicative.Indexed
 open import Effect.Monad
 open import Function.Identity.Effectful as Id using (Identity)
 open import Effect.Monad.Indexed
-open import Function
+open import Function.Base using (flip)
 open import Level
 
 private
diff --git a/src/Effect/Monad/Partiality/All.agda b/src/Effect/Monad/Partiality/All.agda
index 7e90400a85..2954a5da2b 100644
--- a/src/Effect/Monad/Partiality/All.agda
+++ b/src/Effect/Monad/Partiality/All.agda
@@ -11,7 +11,7 @@ module Effect.Monad.Partiality.All where
 open import Effect.Monad
 open import Effect.Monad.Partiality as Partiality using (_⊥; ⇒≈)
 open import Codata.Musical.Notation
-open import Function
+open import Function.Base using (flip; _∘_)
 open import Level
 open import Relation.Binary using (_Respects_; IsEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
diff --git a/src/Effect/Monad/Predicate.agda b/src/Effect/Monad/Predicate.agda
index 5593b202d8..df3e58347c 100644
--- a/src/Effect/Monad/Predicate.agda
+++ b/src/Effect/Monad/Predicate.agda
@@ -15,7 +15,7 @@ open import Effect.Monad
 open import Effect.Monad.Indexed
 open import Data.Unit
 open import Data.Product
-open import Function
+open import Function.Base using (const; id; _∘_)
 open import Level
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary
diff --git a/src/Effect/Monad/Reader/Indexed.agda b/src/Effect/Monad/Reader/Indexed.agda
index 2260b07e09..28eab8d453 100644
--- a/src/Effect/Monad/Reader/Indexed.agda
+++ b/src/Effect/Monad/Reader/Indexed.agda
@@ -10,7 +10,7 @@ open import Level
 
 module Effect.Monad.Reader.Indexed {r} (R : Set r) (a : Level) where
 
-open import Function
+open import Function.Base using (const; flip; _∘_)
 open import Function.Identity.Effectful as Id using (Identity)
 open import Effect.Applicative.Indexed
 open import Effect.Monad.Indexed
diff --git a/src/Effect/Monad/Writer/Indexed.agda b/src/Effect/Monad/Writer/Indexed.agda
index 9720e72b3c..6d99dd9b43 100644
--- a/src/Effect/Monad/Writer/Indexed.agda
+++ b/src/Effect/Monad/Writer/Indexed.agda
@@ -16,7 +16,7 @@ open import Data.Unit.Polymorphic
 open import Effect.Applicative.Indexed
 open import Effect.Monad
 open import Effect.Monad.Indexed
-open import Function
+open import Function.Base using (_∘′_)
 open import Function.Identity.Effectful as Id using (Identity)
 
 private
diff --git a/src/Function/Construct/Composition.agda b/src/Function/Construct/Composition.agda
index a18b4b87b1..269448eaba 100644
--- a/src/Function/Construct/Composition.agda
+++ b/src/Function/Construct/Composition.agda
@@ -9,7 +9,13 @@
 module Function.Construct.Composition where
 
 open import Data.Product.Base using (_,_)
-open import Function
+open import Function.Base using (_∘_)
+open import Function.Bundles
+  using (Func; Injection; Surjection; Bijection; Equivalence; LeftInverse; RightInverse; Inverse; _⟶_; _↣_; _↠_; _⤖_; _⇔_; _↩_; _↪_; _↔_)
+open import Function.Definitions
+  using (Congruent; Injective; Surjective; Bijective; Inverseˡ; Inverseʳ; Inverseᵇ)
+open import Function.Structures
+  using (IsCongruent; IsInjection; IsSurjection; IsBijection; IsLeftInverse; IsRightInverse; IsInverse)
 open import Level using (Level)
 open import Relation.Binary hiding (_⇔_; IsEquivalence)
 
diff --git a/src/Function/Construct/Symmetry.agda b/src/Function/Construct/Symmetry.agda
index 1974dee391..14a7adc466 100644
--- a/src/Function/Construct/Symmetry.agda
+++ b/src/Function/Construct/Symmetry.agda
@@ -9,7 +9,13 @@
 module Function.Construct.Symmetry where
 
 open import Data.Product.Base using (_,_; swap; proj₁; proj₂)
-open import Function
+open import Function.Base using (_∘_)
+open import Function.Definitions
+  using (Bijective; Injective; Surjective; Inverseˡ; Inverseʳ; Inverseᵇ; Congruent)
+open import Function.Structures
+  using (IsBijection; IsCongruent; IsRightInverse; IsLeftInverse; IsInverse)
+open import Function.Bundles
+  using (Bijection; Equivalence; LeftInverse; RightInverse; Inverse; _⤖_; _⇔_; _↩_; _↪_; _↔_)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Symmetric; Transitive)
diff --git a/src/Function/Endomorphism/Propositional.agda b/src/Function/Endomorphism/Propositional.agda
index d14128edfe..9cb46af85f 100644
--- a/src/Function/Endomorphism/Propositional.agda
+++ b/src/Function/Endomorphism/Propositional.agda
@@ -18,7 +18,7 @@ open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Nat.Properties using (+-0-monoid; +-semigroup)
 open import Data.Product using (_,_)
 
-open import Function
+open import Function.Base using (id; _∘′_; _∋_)
 open import Function.Equality using (_⟨$⟩_)
 open import Relation.Binary using (_Preserves_⟶_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
diff --git a/src/Function/Properties.agda b/src/Function/Properties.agda
index b20807f64e..adfb3c73b6 100644
--- a/src/Function/Properties.agda
+++ b/src/Function/Properties.agda
@@ -9,7 +9,8 @@
 module Function.Properties where
 
 open import Axiom.Extensionality.Propositional using (Extensionality)
-open import Function
+open import Function.Base using (flip; _∘_)
+open import Function.Bundles using (_↔_; mk↔′; Inverse)
 open import Level
 open import Relation.Binary.PropositionalEquality.Core
   using (trans; cong; cong′)
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index 64c81b7151..963c5b6330 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -8,7 +8,7 @@
 
 module Relation.Binary.Construct.Closure.ReflexiveTransitive.Properties where
 
-open import Function
+open import Function.Base using (id; _∘_; _$_)
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Relation.Binary.PropositionalEquality.Core as PropEq
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
index c88cf76c8a..aa2fd6e0a5 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
@@ -11,7 +11,7 @@ module
   Relation.Binary.Construct.Closure.ReflexiveTransitive.Properties.WithK
   where
 
-open import Function
+open import Function.Base using (_∋_)
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda b/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
index 62816238d9..b432e3e1dd 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
@@ -8,7 +8,7 @@
 
 module Relation.Binary.Construct.Closure.Transitive.WithK where
 
-open import Function
+open import Function.Base using (_∋_)
 open import Relation.Binary
 open import Relation.Binary.Construct.Closure.Transitive
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
diff --git a/src/Relation/Binary/HeterogeneousEquality/Quotients.agda b/src/Relation/Binary/HeterogeneousEquality/Quotients.agda
index a429dafe8f..fcdc1e97fc 100644
--- a/src/Relation/Binary/HeterogeneousEquality/Quotients.agda
+++ b/src/Relation/Binary/HeterogeneousEquality/Quotients.agda
@@ -8,7 +8,7 @@
 
 module Relation.Binary.HeterogeneousEquality.Quotients where
 
-open import Function
+open import Function.Base using (_$_; const; _∘_)
 open import Level hiding (lift)
 open import Relation.Binary
 open import Relation.Binary.HeterogeneousEquality
diff --git a/src/Relation/Nullary/Indexed/Negation.agda b/src/Relation/Nullary/Indexed/Negation.agda
index 45c6a9e2a1..4a742baea3 100644
--- a/src/Relation/Nullary/Indexed/Negation.agda
+++ b/src/Relation/Nullary/Indexed/Negation.agda
@@ -10,7 +10,7 @@ module Relation.Nullary.Indexed.Negation where
 
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Empty.Polymorphic
-open import Function
+open import Function.Bundles using (_↔_)
 open import Function.Properties
 import Function.Construct.Identity as Identity
 open import Relation.Nullary.Indexed
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 6ab36647c3..5df3c1c423 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -21,7 +21,7 @@ open import Data.Sum.Base as Sum  hiding (map)
 open import Data.Sum.Relation.Binary.Pointwise
 open import Data.Product as Prod hiding (map)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Function
+open import Function.Base using (_∘_; id)
 import Function.Equality as FunS
 open import Data.Empty
 open import Effect.Applicative
diff --git a/src/Relation/Unary/PredicateTransformer.agda b/src/Relation/Unary/PredicateTransformer.agda
index b6df4f2b8a..0ae725588a 100644
--- a/src/Relation/Unary/PredicateTransformer.agda
+++ b/src/Relation/Unary/PredicateTransformer.agda
@@ -9,7 +9,7 @@
 module Relation.Unary.PredicateTransformer where
 
 open import Data.Product
-open import Function
+open import Function.Base using (_∘_)
 open import Level hiding (_⊔_)
 open import Relation.Nullary
 open import Relation.Unary
diff --git a/src/System/Directory.agda b/src/System/Directory.agda
index baee50bc2d..fc2a1578a9 100644
--- a/src/System/Directory.agda
+++ b/src/System/Directory.agda
@@ -17,7 +17,7 @@ open import Data.Maybe.Base using (Maybe)
 open import Data.Nat.Base using (ℕ)
 open import Data.String.Base using (String)
 open import Foreign.Haskell.Coerce using (coerce)
-open import Function
+open import Function.Base using (_∘′_)
 open import System.FilePath.Posix hiding (makeRelative)
 
 open import System.Directory.Primitive as Prim
diff --git a/src/System/FilePath/Posix.agda b/src/System/FilePath/Posix.agda
index 4e4a205d47..d486f1d342 100644
--- a/src/System/FilePath/Posix.agda
+++ b/src/System/FilePath/Posix.agda
@@ -15,7 +15,6 @@ open import IO.Base using (IO; lift)
 open import Data.Maybe.Base using (Maybe)
 open import Data.Product using (_×_)
 open import Data.Sum.Base using (_⊎_)
-open import Function
 
 open import Foreign.Haskell.Coerce
 
diff --git a/src/Tactic/MonoidSolver.agda b/src/Tactic/MonoidSolver.agda
index fcf193244b..991ff41556 100644
--- a/src/Tactic/MonoidSolver.agda
+++ b/src/Tactic/MonoidSolver.agda
@@ -73,7 +73,7 @@
 module Tactic.MonoidSolver where
 
 open import Algebra
-open import Function
+open import Function.Base using (_⟨_⟩_)
 
 open import Data.Bool         as Bool    using (Bool; _∨_; if_then_else_)
 open import Data.Maybe        as Maybe   using (Maybe; just; nothing; maybe)
diff --git a/src/Tactic/RingSolver/Core/NatSet.agda b/src/Tactic/RingSolver/Core/NatSet.agda
index 7ebbcebc22..d28b441d05 100644
--- a/src/Tactic/RingSolver/Core/NatSet.agda
+++ b/src/Tactic/RingSolver/Core/NatSet.agda
@@ -40,7 +40,7 @@ open import Data.Nat        as ℕ     using (ℕ; suc; zero)
 open import Data.List.Base  as List  using (List; _∷_; [])
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Bool       as Bool  using (Bool)
-open import Function
+open import Function.Base            using (const; _∘_)
 open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
index 1eb2244f5d..1f6de95f4f 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
@@ -19,7 +19,7 @@ open import Data.Product              using (_,_; _×_; proj₂)
 open import Data.List.Base            using (_∷_; [])
 open import Data.List.Kleene
 open import Data.Vec                  using (Vec)
-open import Function
+open import Function.Base using (_⟨_⟩_; flip)
 open import Relation.Unary
 
 import Relation.Binary.PropositionalEquality.Core as ≡
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
index ca622ab00a..9435251957 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
@@ -13,7 +13,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Exponentiation
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Function
+open import Function.Base using (_⟨_⟩_)
 
 open import Data.Nat.Base as ℕ using (ℕ; suc; zero; compare)
 open import Data.Product  using (_,_; _×_; proj₁; proj₂)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index 6f2eba31d7..3e1ce2bfe3 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -23,7 +23,7 @@ open import Data.Unit using (tt)
 open import Data.List.Kleene
 open import Data.Product                                    using (_,_; proj₁; proj₂; map₁; _×_)
 open import Data.Maybe                                      using (nothing; just)
-open import Function
+open import Function.Base                                   using (_⟨_⟩_)
 open import Level                                           using (lift)
 open import Relation.Nullary                                using (Dec; yes; no)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
index c322284a96..2baeac0f23 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
@@ -19,7 +19,7 @@ open import Data.Nat.Induction
 open import Data.Product           using (_×_; _,_; proj₁; proj₂; map₁)
 open import Data.List.Kleene
 open import Data.Vec               using (Vec)
-open import Function
+open import Function.Base using (_⟨_⟩_; flip)
 open import Induction.WellFounded
 open import Relation.Unary
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
index ebef6fc39e..dd84eafef2 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
@@ -18,7 +18,7 @@ open import Data.Vec              using (Vec)
 open import Data.Nat              using (_<′_)
 open import Data.Nat.Induction
 
-open import Function
+open import Function.Base using (_⟨_⟩_; flip)
 
 open Homomorphism homo
 open import Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas homo
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Parameters.agda b/src/Tactic/RingSolver/Core/Polynomial/Parameters.agda
index a622ffb49c..c7777160a8 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Parameters.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Parameters.agda
@@ -13,7 +13,6 @@ module Tactic.RingSolver.Core.Polynomial.Parameters where
 
 open import Algebra.Bundles using (RawRing)
 open import Data.Bool.Base using (Bool; T)
-open import Function
 open import Level
 open import Relation.Unary
 open import Tactic.RingSolver.Core.AlmostCommutativeRing
diff --git a/src/Tactic/RingSolver/NonReflective.agda b/src/Tactic/RingSolver/NonReflective.agda
index 1c65a6c1bb..ef99adc22e 100644
--- a/src/Tactic/RingSolver/NonReflective.agda
+++ b/src/Tactic/RingSolver/NonReflective.agda
@@ -18,7 +18,7 @@ module Tactic.RingSolver.NonReflective
   where
 
 open import Algebra.Morphism
-open import Function
+open import Function.Base using (id; _⟨_⟩_)
 open import Data.Bool.Base using (Bool; true; false; T; if_then_else_)
 open import Data.Maybe.Base
 open import Data.Empty using (⊥-elim)

From 2b1c341deaece45ba9df2096fc5c1094f208514c Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Fri, 28 Jul 2023 16:02:13 -0400
Subject: [PATCH 076/316] Simplify more `Data.product` imports (#2014)

* Simplify more `Data.Product` import to `Data.Product.Base`

* Missed an import
---
 src/Codata/Guarded/Stream/Properties.agda                      | 2 +-
 src/Codata/Guarded/Stream/Relation/Unary/All.agda              | 2 +-
 src/Codata/Musical/Covec.agda                                  | 2 +-
 src/Codata/Sized/Colist.agda                                   | 2 +-
 src/Codata/Sized/Colist/Properties.agda                        | 2 +-
 src/Codata/Sized/M/Bisimilarity.agda                           | 2 +-
 src/Codata/Sized/M/Properties.agda                             | 2 +-
 src/Codata/Sized/Stream/Effectful.agda                         | 2 +-
 src/Codata/Sized/Stream/Properties.agda                        | 2 +-
 src/Data/AVL/IndexedMap.agda                                   | 2 +-
 src/Data/Char/Properties.agda                                  | 2 +-
 src/Data/Digit/Properties.agda                                 | 2 +-
 src/Data/Fin/Induction.agda                                    | 2 +-
 src/Data/Fin/Permutation.agda                                  | 2 +-
 src/Data/Fin/Permutation/Components.agda                       | 2 +-
 src/Data/Fin/Permutation/Transposition/List.agda               | 2 +-
 src/Data/Integer/Properties.agda                               | 3 +--
 src/Data/List/Extrema/Core.agda                                | 2 +-
 src/Data/List/Extrema/Nat.agda                                 | 2 +-
 src/Data/List/Fresh/Properties.agda                            | 2 +-
 src/Data/List/Fresh/Relation/Unary/All.agda                    | 2 +-
 src/Data/List/Fresh/Relation/Unary/All/Properties.agda         | 2 +-
 src/Data/List/Fresh/Relation/Unary/Any/Properties.agda         | 2 +-
 src/Data/List/Nary/NonDependent.agda                           | 2 +-
 src/Data/List/NonEmpty/Effectful.agda                          | 2 +-
 src/Data/List/Relation/Binary/Disjoint/Setoid.agda             | 2 +-
 src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda  | 2 +-
 src/Data/List/Relation/Binary/Lex/Strict.agda                  | 2 +-
 .../Relation/Binary/Permutation/Propositional/Properties.agda  | 2 +-
 .../List/Relation/Binary/Prefix/Heterogeneous/Properties.agda  | 2 +-
 30 files changed, 30 insertions(+), 31 deletions(-)

diff --git a/src/Codata/Guarded/Stream/Properties.agda b/src/Codata/Guarded/Stream/Properties.agda
index 29ea54bb1d..df68a9c4b2 100644
--- a/src/Codata/Guarded/Stream/Properties.agda
+++ b/src/Codata/Guarded/Stream/Properties.agda
@@ -16,7 +16,7 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)
 open import Data.Nat.Base using (ℕ; zero; suc; _+_; _*_)
 import Data.Nat.GeneralisedArithmetic as ℕ
-open import Data.Product as Prod using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Prod using (_×_; _,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec; _∷_)
 open import Function.Base using (const; flip; id; _∘′_; _$′_; _⟨_⟩_; _∘₂′_)
 open import Level using (Level)
diff --git a/src/Codata/Guarded/Stream/Relation/Unary/All.agda b/src/Codata/Guarded/Stream/Relation/Unary/All.agda
index ad87a7e589..69171a8960 100644
--- a/src/Codata/Guarded/Stream/Relation/Unary/All.agda
+++ b/src/Codata/Guarded/Stream/Relation/Unary/All.agda
@@ -9,7 +9,7 @@
 module Codata.Guarded.Stream.Relation.Unary.All where
 
 open import Codata.Guarded.Stream using (Stream; head; tail)
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Level
 open import Relation.Unary
 
diff --git a/src/Codata/Musical/Covec.agda b/src/Codata/Musical/Covec.agda
index 88b15c070b..8993fd5aa8 100644
--- a/src/Codata/Musical/Covec.agda
+++ b/src/Codata/Musical/Covec.agda
@@ -14,7 +14,7 @@ open import Codata.Musical.Cofin using (Cofin; zero; suc)
 open import Codata.Musical.Colist as Colist using (Colist; []; _∷_)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.Vec.Base using (Vec; []; _∷_)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_)
 open import Level using (Level)
 open import Relation.Binary
diff --git a/src/Codata/Sized/Colist.agda b/src/Codata/Sized/Colist.agda
index e57a5b632e..0675721bab 100644
--- a/src/Codata/Sized/Colist.agda
+++ b/src/Codata/Sized/Colist.agda
@@ -12,7 +12,7 @@ open import Level using (Level)
 open import Size
 open import Data.Unit.Base
 open import Data.Nat.Base
-open import Data.Product using (_×_ ; _,_)
+open import Data.Product.Base using (_×_ ; _,_)
 open import Data.These.Base using (These; this; that; these)
 open import Data.Maybe.Base using (Maybe; nothing; just)
 open import Data.List.Base using (List; []; _∷_)
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index de5925e482..5b77551b7f 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -27,7 +27,7 @@ open import Data.Maybe.Base as Maybe using (Maybe; nothing; just)
 import Data.Maybe.Properties as Maybeₚ
 open import Data.Maybe.Relation.Unary.All using (All; nothing; just)
 open import Data.Nat.Base as ℕ using (zero; suc; z≤n; s≤s)
-open import Data.Product as Prod using (_×_; _,_; uncurry)
+open import Data.Product.Base as Prod using (_×_; _,_; uncurry)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Function.Base
diff --git a/src/Codata/Sized/M/Bisimilarity.agda b/src/Codata/Sized/M/Bisimilarity.agda
index 3a60a91d16..ef26860e66 100644
--- a/src/Codata/Sized/M/Bisimilarity.agda
+++ b/src/Codata/Sized/M/Bisimilarity.agda
@@ -14,7 +14,7 @@ open import Codata.Sized.Thunk
 open import Codata.Sized.M
 open import Data.Container.Core
 open import Data.Container.Relation.Binary.Pointwise using (Pointwise; _,_)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_)
 open import Relation.Binary
 import Relation.Binary.PropositionalEquality.Core as P
diff --git a/src/Codata/Sized/M/Properties.agda b/src/Codata/Sized/M/Properties.agda
index 4161f4d1f1..f9801448f5 100644
--- a/src/Codata/Sized/M/Properties.agda
+++ b/src/Codata/Sized/M/Properties.agda
@@ -15,7 +15,7 @@ open import Codata.Sized.M
 open import Codata.Sized.M.Bisimilarity
 open import Data.Container.Core as C hiding (map)
 import Data.Container.Morphism as Mp
-open import Data.Product as Prod using (_,_)
+open import Data.Product.Base as Prod using (_,_)
 open import Data.Product.Properties hiding (map-cong)
 open import Function.Base using (_$′_; _∘′_)
 import Relation.Binary.PropositionalEquality.Core as P
diff --git a/src/Codata/Sized/Stream/Effectful.agda b/src/Codata/Sized/Stream/Effectful.agda
index 00fe7eb2b1..a9636ad027 100644
--- a/src/Codata/Sized/Stream/Effectful.agda
+++ b/src/Codata/Sized/Stream/Effectful.agda
@@ -8,7 +8,7 @@
 
 module Codata.Sized.Stream.Effectful where
 
-open import Data.Product using (<_,_>)
+open import Data.Product.Base using (<_,_>)
 open import Codata.Sized.Stream
 open import Effect.Functor
 open import Effect.Applicative
diff --git a/src/Codata/Sized/Stream/Properties.agda b/src/Codata/Sized/Stream/Properties.agda
index 51b0719c7d..69685e12b2 100644
--- a/src/Codata/Sized/Stream/Properties.agda
+++ b/src/Codata/Sized/Stream/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Nat.GeneralisedArithmetic using (fold; fold-pull)
 open import Data.List.Base as List using ([]; _∷_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)
 import Data.List.Relation.Binary.Equality.Propositional as Eq
-open import Data.Product as Prod using (_,_)
+open import Data.Product.Base as Prod using (_,_)
 open import Data.Vec.Base as Vec using (_∷_)
 
 open import Function.Base using (id; _$_; _∘′_; const)
diff --git a/src/Data/AVL/IndexedMap.agda b/src/Data/AVL/IndexedMap.agda
index 25bccdee5f..84dbcceba4 100644
--- a/src/Data/AVL/IndexedMap.agda
+++ b/src/Data/AVL/IndexedMap.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product as Prod
+open import Data.Product.Base using (∃)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
 import Data.Tree.AVL.Value
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index 2bbf63ab17..52a997fd81 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -12,7 +12,7 @@ open import Data.Bool.Base using (Bool)
 open import Data.Char.Base
 import Data.Nat.Base as ℕ
 import Data.Nat.Properties as ℕₚ
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 
 open import Function.Base
 open import Relation.Nullary using (¬_; yes; no)
diff --git a/src/Data/Digit/Properties.agda b/src/Data/Digit/Properties.agda
index 37ead69a28..113bdde6ca 100644
--- a/src/Data/Digit/Properties.agda
+++ b/src/Data/Digit/Properties.agda
@@ -11,7 +11,7 @@ import Data.Char.Properties as Charₚ
 open import Data.Nat.Base using (ℕ)
 open import Data.Nat.Properties using (_≤?_)
 open import Data.Fin.Properties using (inject≤-injective)
-open import Data.Product using (_,_; proj₁)
+open import Data.Product.Base using (_,_; proj₁)
 open import Data.Vec.Relation.Unary.Unique.Propositional using (Unique)
 import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Uniqueₚ
 open import Data.Vec.Relation.Unary.AllPairs using (allPairs?)
diff --git a/src/Data/Fin/Induction.agda b/src/Data/Fin/Induction.agda
index 3c9bf54546..b0cf553d0d 100644
--- a/src/Data/Fin/Induction.agda
+++ b/src/Data/Fin/Induction.agda
@@ -13,7 +13,7 @@ open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _∸_; s≤s)
 open import Data.Nat.Properties using (n<1+n; ≤⇒≯)
 import Data.Nat.Induction as ℕ
 import Data.Nat.Properties as ℕ
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.Linked as Linked using (Linked; [-]; _∷_)
 import Data.Vec.Relation.Unary.Linked.Properties as Linkedₚ
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index f6a7820b52..852a609779 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -15,7 +15,7 @@ open import Data.Fin.Patterns
 open import Data.Fin.Properties
 import Data.Fin.Permutation.Components as PC
 open import Data.Nat.Base using (ℕ; suc; zero)
-open import Data.Product using (_,_; proj₂)
+open import Data.Product.Base using (_,_; proj₂)
 open import Function.Bundles using (_↔_; Injection; Inverse; mk↔′)
 open import Function.Construct.Composition using (_↔-∘_)
 open import Function.Construct.Identity using (↔-id)
diff --git a/src/Data/Fin/Permutation/Components.agda b/src/Data/Fin/Permutation/Components.agda
index b053cee86b..e75fa29eb2 100644
--- a/src/Data/Fin/Permutation/Components.agda
+++ b/src/Data/Fin/Permutation/Components.agda
@@ -13,7 +13,7 @@ open import Data.Fin.Base
 open import Data.Fin.Properties
 open import Data.Nat.Base as ℕ using (zero; suc; _∸_)
 import Data.Nat.Properties as ℕₚ
-open import Data.Product using (proj₂)
+open import Data.Product.Base using (proj₂)
 open import Function.Base using (_∘_)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (does; _because_; yes; no)
diff --git a/src/Data/Fin/Permutation/Transposition/List.agda b/src/Data/Fin/Permutation/Transposition/List.agda
index 07feee0020..4da8f08d37 100644
--- a/src/Data/Fin/Permutation/Transposition/List.agda
+++ b/src/Data/Fin/Permutation/Transposition/List.agda
@@ -14,7 +14,7 @@ open import Data.Fin.Permutation as P hiding (lift₀)
 import Data.Fin.Permutation.Components as PC
 open import Data.List.Base using (List; []; _∷_; map)
 open import Data.Nat.Base using (ℕ; suc; zero)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Function.Base using (_∘_)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; sym; cong; module ≡-Reasoning)
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 518c1456f4..78011e54b1 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -21,11 +21,10 @@ open import Data.Nat as ℕ
   hiding (module ℕ)
 import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
-open import Data.Product using (proj₁; proj₂; _,_)
+open import Data.Product.Base using (proj₁; proj₂; _,_; _×_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sign as Sign using (Sign) renaming (_*_ to _𝕊*_)
 import Data.Sign.Properties as 𝕊ₚ
-open import Data.Product using (_×_)
 open import Function.Base using (_∘_; _$_; id)
 open import Level using (0ℓ)
 open import Relation.Binary
diff --git a/src/Data/List/Extrema/Core.agda b/src/Data/List/Extrema/Core.agda
index 1fb59ae06d..0ed9dbe159 100644
--- a/src/Data/List/Extrema/Core.agda
+++ b/src/Data/List/Extrema/Core.agda
@@ -15,7 +15,7 @@ open import Algebra.Core
 open import Algebra.Definitions
 import Algebra.Construct.NaturalChoice.Min as Min
 import Algebra.Construct.NaturalChoice.Max as Max
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
diff --git a/src/Data/List/Extrema/Nat.agda b/src/Data/List/Extrema/Nat.agda
index 6e2899b8e6..89f513e2bf 100644
--- a/src/Data/List/Extrema/Nat.agda
+++ b/src/Data/List/Extrema/Nat.agda
@@ -18,7 +18,7 @@ open import Data.List.Base using (List)
 import Data.List.Extrema
 open import Data.List.Relation.Unary.Any as Any using (Any)
 open import Data.List.Relation.Unary.All as All using (All)
-open import Data.Product using (_×_; _,_; uncurry′)
+open import Data.Product.Base using (_×_; _,_; uncurry′)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 
diff --git a/src/Data/List/Fresh/Properties.agda b/src/Data/List/Fresh/Properties.agda
index cc8925eca6..b22347f92e 100644
--- a/src/Data/List/Fresh/Properties.agda
+++ b/src/Data/List/Fresh/Properties.agda
@@ -9,7 +9,7 @@
 module Data.List.Fresh.Properties where
 
 open import Level using (Level; _⊔_; Lift)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Relation.Nullary
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary as B using (Rel)
diff --git a/src/Data/List/Fresh/Relation/Unary/All.agda b/src/Data/List/Fresh/Relation/Unary/All.agda
index c013f934bb..b0e2115ce1 100644
--- a/src/Data/List/Fresh/Relation/Unary/All.agda
+++ b/src/Data/List/Fresh/Relation/Unary/All.agda
@@ -9,7 +9,7 @@
 module Data.List.Fresh.Relation.Unary.All where
 
 open import Level using (Level; _⊔_; Lift)
-open import Data.Product using (_×_; _,_; proj₁; uncurry)
+open import Data.Product.Base using (_×_; _,_; proj₁; uncurry)
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _×-dec_)
 open import Relation.Unary  as U
diff --git a/src/Data/List/Fresh/Relation/Unary/All/Properties.agda b/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
index f2c47657b5..8c2f0c80b4 100644
--- a/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
@@ -11,7 +11,7 @@ module Data.List.Fresh.Relation.Unary.All.Properties where
 open import Level using (Level; _⊔_; Lift)
 open import Data.Empty
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘′_)
 open import Relation.Nullary
 open import Relation.Unary  as U
diff --git a/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda b/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
index 09ec3d5333..8b47063949 100644
--- a/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
@@ -12,7 +12,7 @@ open import Level using (Level; _⊔_; Lift)
 open import Data.Bool.Base using (true; false)
 open import Data.Empty
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘′_)
 open import Relation.Nullary.Reflects using (invert)
diff --git a/src/Data/List/Nary/NonDependent.agda b/src/Data/List/Nary/NonDependent.agda
index 6fff2b0dac..f95da6345c 100644
--- a/src/Data/List/Nary/NonDependent.agda
+++ b/src/Data/List/Nary/NonDependent.agda
@@ -10,7 +10,7 @@ module Data.List.Nary.NonDependent where
 
 open import Data.Nat.Base using (zero; suc)
 open import Data.List.Base as List using (List; []; _∷_)
-open import Data.Product as Prod using (_,_)
+open import Data.Product.Base as Prod using (_,_)
 open import Data.Product.Nary.NonDependent using (Product)
 open import Function.Base using ()
 open import Function.Nary.NonDependent.Base
diff --git a/src/Data/List/NonEmpty/Effectful.agda b/src/Data/List/NonEmpty/Effectful.agda
index 373ceceb26..e2edccdf9f 100644
--- a/src/Data/List/NonEmpty/Effectful.agda
+++ b/src/Data/List/NonEmpty/Effectful.agda
@@ -11,7 +11,7 @@ module Data.List.NonEmpty.Effectful where
 open import Agda.Builtin.List
 import Data.List.Effectful as List
 open import Data.List.NonEmpty.Base
-open import Data.Product using (uncurry)
+open import Data.Product.Base using (uncurry)
 open import Effect.Functor
 open import Effect.Applicative
 open import Effect.Monad
diff --git a/src/Data/List/Relation/Binary/Disjoint/Setoid.agda b/src/Data/List/Relation/Binary/Disjoint/Setoid.agda
index d5003aab4e..6ea7e15cf7 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Setoid.agda
@@ -15,7 +15,7 @@ open import Relation.Nullary.Negation using (¬_)
 open import Function.Base using (_∘_)
 open import Data.List.Base using (List; []; [_]; _∷_)
 open import Data.List.Relation.Unary.Any using (here; there)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 
 open Setoid S renaming (Carrier to A)
 open import Data.List.Membership.Setoid S using (_∈_; _∉_)
diff --git a/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
index 3f0701a8ef..43a67d4d7d 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
@@ -14,7 +14,7 @@ import Data.List.Relation.Unary.Any as Any
 open import Data.List.Relation.Unary.All as All
 open import Data.List.Relation.Unary.All.Properties using (¬Any⇒All¬)
 open import Data.List.Relation.Unary.Any.Properties using (++⁻)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Relation.Binary
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/List/Relation/Binary/Lex/Strict.agda b/src/Data/List/Relation/Binary/Lex/Strict.agda
index fef11ad07e..c0428b347d 100644
--- a/src/Data/List/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/List/Relation/Binary/Lex/Strict.agda
@@ -14,7 +14,7 @@ module Data.List.Relation.Binary.Lex.Strict where
 open import Data.Empty using (⊥)
 open import Data.Unit.Base using (⊤; tt)
 open import Function.Base using (_∘_; id)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.List.Base using (List; []; _∷_)
 open import Level using (_⊔_)
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index 135d5a778d..dc7e34db86 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -13,7 +13,7 @@ open import Algebra.Definitions
 open import Algebra.Structures
 open import Data.Bool.Base using (Bool; true; false)
 open import Data.Nat using (suc)
-open import Data.Product using (-,_; proj₂)
+open import Data.Product.Base using (-,_; proj₂)
 open import Data.List.Base as List
 open import Data.List.Relation.Binary.Permutation.Propositional
 open import Data.List.Relation.Unary.Any using (Any; here; there)
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
index da5a26e6e0..21e61b4aa4 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
@@ -19,7 +19,7 @@ open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
 open import Data.List.Relation.Binary.Prefix.Heterogeneous as Prefix hiding (PrefixView; _++_)
 open import Data.Nat.Base using (ℕ; zero; suc; _≤_; z≤n; s≤s)
 open import Data.Nat.Properties using (suc-injective)
-open import Data.Product as Prod using (_×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Product.Base as Prod using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Function.Base
 
 open import Relation.Nullary.Negation using (¬_)

From f1c6e48051e9f0f94c36c8765ac7ad834eb7bddb Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Sat, 29 Jul 2023 01:09:33 -0400
Subject: [PATCH 077/316] Simplify import of
 `Data.List.Relation.Binary.Pointwise` in agda-stdl (#2019)

* Simplify import of `Data.List.Relation.Binary.Pointwise` in agda-stdl

* rectifications on Data.List.Relation.Binary.Pointwise import

* rectifications on Data.List.Relation.Binary.Pointwise import

* Delete weights-README.dot

* Delete weight.png

* Delete README.dot
---
 README/Case.agda                                              | 1 -
 README/Data/List/Relation/Binary/Equality.agda                | 2 +-
 README/Inspect.agda                                           | 3 ++-
 src/Codata/Sized/Colist/Bisimilarity.agda                     | 2 +-
 src/Codata/Sized/Stream/Bisimilarity.agda                     | 2 +-
 src/Data/Container/Indexed/FreeMonad.agda                     | 2 +-
 .../List/Relation/Binary/Infix/Heterogeneous/Properties.agda  | 3 +--
 src/Data/List/Relation/Binary/Lex/NonStrict.agda              | 2 +-
 src/Data/List/Relation/Binary/Permutation/Homogeneous.agda    | 4 +++-
 src/Data/List/Relation/Binary/Permutation/Setoid.agda         | 2 +-
 .../List/Relation/Binary/Prefix/Heterogeneous/Properties.agda | 2 +-
 src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda       | 2 +-
 src/Data/List/Relation/Ternary/Appending.agda                 | 2 +-
 src/Data/List/Relation/Ternary/Appending/Properties.agda      | 3 ++-
 src/Data/List/Relation/Ternary/Appending/Propositional.agda   | 3 +--
 .../Relation/Ternary/Appending/Propositional/Properties.agda  | 2 +-
 .../List/Relation/Ternary/Appending/Setoid/Properties.agda    | 2 +-
 src/Data/List/Relation/Ternary/Interleaving.agda              | 2 +-
 src/Data/List/Relation/Ternary/Interleaving/Properties.agda   | 4 +++-
 src/Data/List/Relation/Unary/All/Properties.agda              | 2 +-
 src/Data/String.agda                                          | 2 +-
 src/Text/Regex/Properties.agda                                | 3 +--
 22 files changed, 27 insertions(+), 25 deletions(-)

diff --git a/README/Case.agda b/README/Case.agda
index 450105aff9..15449391d4 100644
--- a/README/Case.agda
+++ b/README/Case.agda
@@ -16,7 +16,6 @@ open import Data.Product
 open import Function.Base using (case_of_; case_return_of_)
 open import Relation.Nullary
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
 -- Different types of pattern-matching lambdas
diff --git a/README/Data/List/Relation/Binary/Equality.agda b/README/Data/List/Relation/Binary/Equality.agda
index 6201dc5c5d..4ce5fe2478 100644
--- a/README/Data/List/Relation/Binary/Equality.agda
+++ b/README/Data/List/Relation/Binary/Equality.agda
@@ -54,7 +54,7 @@ open import Relation.Binary.PropositionalEquality using (_≗_)
 -- to `_≗_`. However instead of using the pointwise module directly
 -- to write:
 
-open import Data.List.Relation.Binary.Pointwise using (Pointwise)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise)
 
 lem₃ : Pointwise _≗_ ((λ x → x + 1) ∷ []) ((λ x → x + 2 ∸ 1) ∷ [])
 lem₃ = {!!}
diff --git a/README/Inspect.agda b/README/Inspect.agda
index cff492d008..002005abaa 100644
--- a/README/Inspect.agda
+++ b/README/Inspect.agda
@@ -15,7 +15,8 @@ module README.Inspect where
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Product
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality using (inspect)
 
 ------------------------------------------------------------------------
 -- Using inspect
diff --git a/src/Codata/Sized/Colist/Bisimilarity.agda b/src/Codata/Sized/Colist/Bisimilarity.agda
index 9efbcebbca..d66ade8b89 100644
--- a/src/Codata/Sized/Colist/Bisimilarity.agda
+++ b/src/Codata/Sized/Colist/Bisimilarity.agda
@@ -13,7 +13,7 @@ open import Size
 open import Codata.Sized.Thunk
 open import Codata.Sized.Colist
 open import Data.List.Base using (List; []; _∷_)
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.List.NonEmpty as List⁺  using (List⁺; _∷_)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
diff --git a/src/Codata/Sized/Stream/Bisimilarity.agda b/src/Codata/Sized/Stream/Bisimilarity.agda
index 74419c7809..c036d030b7 100644
--- a/src/Codata/Sized/Stream/Bisimilarity.agda
+++ b/src/Codata/Sized/Stream/Bisimilarity.agda
@@ -13,7 +13,7 @@ open import Codata.Sized.Thunk
 open import Codata.Sized.Stream
 open import Level
 open import Data.List.NonEmpty as List⁺ using (_∷_)
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as Eq
diff --git a/src/Data/Container/Indexed/FreeMonad.agda b/src/Data/Container/Indexed/FreeMonad.agda
index f2fe0d2aab..1c6ee81c1a 100644
--- a/src/Data/Container/Indexed/FreeMonad.agda
+++ b/src/Data/Container/Indexed/FreeMonad.agda
@@ -19,7 +19,7 @@ open import Data.Product
 open import Data.W.Indexed
 open import Relation.Unary
 open import Relation.Unary.PredicateTransformer
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 ------------------------------------------------------------------------
 
diff --git a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
index f9610d5422..cc067dc846 100644
--- a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
@@ -22,8 +22,7 @@ open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary using (REL; _⇒_; Decidable; Trans; Antisym)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_; refl; cong)
-
-open import Data.List.Relation.Binary.Pointwise as Pointwise using (Pointwise)
+open import Data.List.Relation.Binary.Pointwise.Base as Pointwise using (Pointwise)
 open import Data.List.Relation.Binary.Infix.Heterogeneous
 open import Data.List.Relation.Binary.Prefix.Heterogeneous
   as Prefix using (Prefix; []; _∷_)
diff --git a/src/Data/List/Relation/Binary/Lex/NonStrict.agda b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
index 83c8767cb3..4606c9cdbd 100644
--- a/src/Data/List/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
@@ -16,7 +16,7 @@ open import Function.Base
 open import Data.Unit.Base using (⊤; tt)
 open import Data.Product
 open import Data.List.Base
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; [])
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; [])
 import Data.List.Relation.Binary.Lex.Strict as Strict
 open import Level
 open import Relation.Nullary
diff --git a/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda b/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda
index 9e247e734b..6b9fd9c20d 100644
--- a/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda
@@ -9,8 +9,10 @@
 module Data.List.Relation.Binary.Permutation.Homogeneous where
 
 open import Data.List.Base using (List; _∷_)
-open import Data.List.Relation.Binary.Pointwise as Pointwise
+open import Data.List.Relation.Binary.Pointwise.Base as Pointwise
   using (Pointwise)
+open import Data.List.Relation.Binary.Pointwise.Properties as Pointwise
+  using (symmetric)
 open import Level using (Level; _⊔_)
 open import Relation.Binary
 
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid.agda b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
index 5a147d8008..a9a0cf6400 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
@@ -13,7 +13,7 @@ module Data.List.Relation.Binary.Permutation.Setoid
 
 open import Data.List.Base using (List; _∷_)
 import Data.List.Relation.Binary.Permutation.Homogeneous as Homogeneous
-import Data.List.Relation.Binary.Pointwise as Pointwise
+import Data.List.Relation.Binary.Pointwise.Properties as Pointwise using (refl)
 open import Data.List.Relation.Binary.Equality.Setoid S
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Level using (_⊔_)
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
index 21e61b4aa4..5706dbc36e 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
@@ -15,7 +15,7 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Base as List hiding (map; uncons)
 open import Data.List.Membership.Propositional.Properties using ([]∈inits)
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.List.Relation.Binary.Prefix.Heterogeneous as Prefix hiding (PrefixView; _++_)
 open import Data.Nat.Base using (ℕ; zero; suc; _≤_; z≤n; s≤s)
 open import Data.Nat.Properties using (suc-injective)
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
index 984b857c59..aeeb927d2d 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Binary.Suffix.Heterogeneous where
 open import Level
 open import Relation.Binary using (REL; _⇒_)
 open import Data.List.Base as List using (List; []; _∷_)
-open import Data.List.Relation.Binary.Pointwise as Pointwise
+open import Data.List.Relation.Binary.Pointwise.Base as Pointwise
   using (Pointwise; []; _∷_)
 
 module _ {a b r} {A : Set a} {B : Set b} (R : REL A B r) where
diff --git a/src/Data/List/Relation/Ternary/Appending.agda b/src/Data/List/Relation/Ternary/Appending.agda
index 0d1f935414..802f80aac3 100644
--- a/src/Data/List/Relation/Ternary/Appending.agda
+++ b/src/Data/List/Relation/Ternary/Appending.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Ternary.Appending {a b c} {A : Set a} {B : Set b} {C :
 
 open import Level using (Level; _⊔_)
 open import Data.List.Base as List using (List; []; _∷_)
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.Product as Prod using (∃₂; _×_; _,_; -,_)
 open import Relation.Binary using (REL)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
diff --git a/src/Data/List/Relation/Ternary/Appending/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Properties.agda
index 4508a7f719..6c02c49ffb 100644
--- a/src/Data/List/Relation/Ternary/Appending/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Properties.agda
@@ -10,7 +10,8 @@ module Data.List.Relation.Ternary.Appending.Properties where
 
 open import Data.List.Base using (List; [])
 open import Data.List.Relation.Ternary.Appending
-open import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Base as Pw using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Properties as Pw using (transitive)
 open import Level using (Level)
 open import Relation.Binary using (REL; Rel; Trans)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional.agda b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
index 6149e2a236..46567a01ed 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
@@ -14,8 +14,7 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.Product using (_,_)
 
 import Data.List.Properties as Listₚ
-import Data.List.Relation.Binary.Pointwise as Pw
-
+import Data.List.Relation.Binary.Pointwise as Pw using (≡⇒Pointwise-≡; Pointwise-≡⇒≡)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; setoid; refl; trans; cong₂; module ≡-Reasoning)
 
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
index 757d888012..54e19f59d2 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Ternary.Appending.Propositional.Properties {a} {A : Se
 
 open import Data.List.Base as List using (List; [])
 import Data.List.Properties as Listₚ
-import Data.List.Relation.Binary.Pointwise as Pw
+import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise-≡⇒≡)
 open import Data.List.Relation.Binary.Equality.Propositional using (_≋_)
 open import Data.List.Relation.Ternary.Appending.Propositional {A = A}
 open import Function.Base using (_∘′_)
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
index 0530b8b965..c205170359 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
@@ -12,7 +12,7 @@ module Data.List.Relation.Ternary.Appending.Setoid.Properties {c l} (S : Setoid
 
 open import Data.List.Base as List using (List; [])
 import Data.List.Properties as Listₚ
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; [])
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; [])
 import Data.List.Relation.Ternary.Appending.Properties as Appendingₚ
 open import Data.Product using (_,_)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
diff --git a/src/Data/List/Relation/Ternary/Interleaving.agda b/src/Data/List/Relation/Ternary/Interleaving.agda
index 62206daf2c..158701fc7e 100644
--- a/src/Data/List/Relation/Ternary/Interleaving.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Ternary.Interleaving where
 
 open import Level
 open import Data.List.Base as List using (List; []; _∷_; _++_)
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.Product as Prod using (∃; ∃₂; _×_; uncurry; _,_; -,_; proj₂)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
index ba61947c85..da3d687ba5 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
@@ -15,7 +15,9 @@ open import Data.List.Properties using (reverse-involutive)
 open import Data.List.Relation.Ternary.Interleaving hiding (map)
 open import Function.Base using (_$_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality
+--open import Relation.Binary.PropositionalEquality
+--  using (_≡_; refl; sym; cong; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; sym; cong; module ≡-Reasoning)
 open ≡-Reasoning
 
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index 004424df80..da3de23dd9 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -25,7 +25,7 @@ open import Data.List.Relation.Unary.All as All using
   )
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 import Data.List.Relation.Binary.Equality.Setoid as ListEq using (_≋_; []; _∷_)
-open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_)
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.All as Maybe using (just; nothing)
diff --git a/src/Data/String.agda b/src/Data/String.agda
index 8a6dd08045..6a63342ccd 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -16,7 +16,7 @@ import Data.Nat.Properties as ℕₚ
 open import Data.List.Base as List using (List; _∷_; []; [_])
 open import Data.List.NonEmpty as NE using (List⁺)
 open import Data.List.Extrema ℕₚ.≤-totalOrder
-open import Data.List.Relation.Binary.Pointwise using (Pointwise)
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise)
 open import Data.List.Relation.Binary.Lex.Strict using (Lex-<; Lex-≤)
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Char.Base as Char using (Char)
diff --git a/src/Text/Regex/Properties.agda b/src/Text/Regex/Properties.agda
index 815bf04471..02b6db2039 100644
--- a/src/Text/Regex/Properties.agda
+++ b/src/Text/Regex/Properties.agda
@@ -27,8 +27,7 @@ open import Relation.Binary.Definitions using (Decidable)
 
 open DecPoset P? renaming (Carrier to A)
 open import Text.Regex.Base preorder
-
-open import Data.List.Relation.Binary.Pointwise using ([])
+open import Data.List.Relation.Binary.Pointwise.Base using ([])
 open import Data.List.Relation.Ternary.Appending.Propositional {A = A}
 open import Data.List.Relation.Ternary.Appending.Propositional.Properties {A = A}
 

From ec52a7713553dc3a12ffc5b44dfa942b40ab3f0c Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sat, 29 Jul 2023 01:56:18 -0400
Subject: [PATCH 078/316] Bump stdlib and agda version in installation guide
 (#2027)

---
 notes/installation-guide.md | 12 ++++++------
 1 file changed, 6 insertions(+), 6 deletions(-)

diff --git a/notes/installation-guide.md b/notes/installation-guide.md
index f7f351b484..f05f21ed90 100644
--- a/notes/installation-guide.md
+++ b/notes/installation-guide.md
@@ -3,19 +3,19 @@ Installation instructions
 
 Note: the full story on installing Agda libraries can be found at [readthedocs](http://agda.readthedocs.io/en/latest/tools/package-system.html).
 
-Use version v1.7.1 of the standard library with Agda 2.6.2.2, 2.6.2.1, or 2.6.2.
+Use version v1.7.2 of the standard library with Agda 2.6.3.
 
 1. Navigate to a suitable directory `$HERE` (replace appropriately) where
    you would like to install the library.
 
-2. Download the tarball of v1.7.1 of the standard library. This can either be
+2. Download the tarball of v1.7.2 of the standard library. This can either be
    done manually by visiting the Github repository for the library, or via the
    command line as follows:
    ```
-   wget -O agda-stdlib.tar https://github.com/agda/agda-stdlib/archive/v1.7.1.tar.gz
+   wget -O agda-stdlib.tar https://github.com/agda/agda-stdlib/archive/v1.7.2.tar.gz
    ```
    Note that you can replace `wget` with other popular tools such as `curl` and that
-   you can replace `1.7.1` with any other version of the library you desire.
+   you can replace `1.7.2` with any other version of the library you desire.
 
 3. Extract the standard library from the tarball. Again this can either be
    done manually or via the command line as follows:
@@ -26,7 +26,7 @@ Use version v1.7.1 of the standard library with Agda 2.6.2.2, 2.6.2.1, or 2.6.2.
 4. [ OPTIONAL ] If using [cabal](https://www.haskell.org/cabal/) then run
    the commands to install via cabal:
    ```
-   cd agda-stdlib-1.7.1
+   cd agda-stdlib-1.7.2
    cabal install
    ```
 
@@ -40,7 +40,7 @@ Use version v1.7.1 of the standard library with Agda 2.6.2.2, 2.6.2.1, or 2.6.2.
 6. Register the standard library with Agda's package system by adding
    the following line to `$HOME/.agda/libraries`:
    ```
-   $HERE/agda-stdlib-1.7.1/standard-library.agda-lib
+   $HERE/agda-stdlib-1.7.2/standard-library.agda-lib
    ```
 
 Now, the standard library is ready to be used either:

From 55ec99dd03bc651abe7c63f58ffaf27a022ece65 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sat, 29 Jul 2023 04:40:59 -0400
Subject: [PATCH 079/316] Simplify more `Data.Product` imports to
 `Data.Product.Base` (#2036)

* Simplify more `Data.Product` import to `Data.Product.Base`

* Simplify more `Data.Product` import to `Data.Product.Base`

* Indent
---
 src/Function/Endomorphism/Propositional.agda                  | 2 +-
 src/Function/Nary/NonDependent.agda                           | 2 +-
 src/Function/Properties/Bijection.agda                        | 2 +-
 src/Function/Related.agda                                     | 2 +-
 src/Function/Related/TypeIsomorphisms/Solver.agda             | 2 +-
 src/Reflection/AST/Abstraction.agda                           | 2 +-
 src/Reflection/AST/Argument.agda                              | 2 +-
 src/Reflection/AST/Definition.agda                            | 2 +-
 src/Reflection/AST/Show.agda                                  | 2 +-
 src/Reflection/AST/Term.agda                                  | 4 ++--
 src/Reflection/AST/Traversal.agda                             | 2 +-
 src/Reflection/AST/Universe.agda                              | 2 +-
 src/Reflection/AnnotatedAST.agda                              | 2 +-
 src/Reflection/AnnotatedAST/Free.agda                         | 2 +-
 src/Reflection/External.agda                                  | 2 +-
 src/Relation/Binary/Construct/Subst/Equality.agda             | 2 +-
 src/Relation/Binary/Indexed/Homogeneous/Bundles.agda          | 2 +-
 src/Relation/Binary/Indexed/Homogeneous/Core.agda             | 2 +-
 src/Relation/Binary/Indexed/Homogeneous/Definitions.agda      | 2 +-
 src/Relation/Binary/Indexed/Homogeneous/Structures.agda       | 2 +-
 src/Relation/Binary/Lattice/Properties/BoundedLattice.agda    | 2 +-
 .../Binary/Lattice/Properties/DistributiveLattice.agda        | 2 +-
 src/Relation/Binary/Lattice/Properties/Lattice.agda           | 2 +-
 src/Relation/Binary/Morphism/Construct/Identity.agda          | 2 +-
 src/Relation/Binary/Morphism/OrderMonomorphism.agda           | 2 +-
 src/Relation/Binary/Properties/HeytingAlgebra.agda            | 2 +-
 src/Relation/Nary.agda                                        | 2 +-
 src/Relation/Nullary/Universe.agda                            | 2 +-
 src/Relation/Unary/Algebra.agda                               | 2 +-
 src/System/Environment.agda                                   | 2 +-
 src/System/FilePath/Posix.agda                                | 2 +-
 src/System/Process.agda                                       | 2 +-
 src/Tactic/MonoidSolver.agda                                  | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Addition.agda     | 2 +-
 .../Core/Polynomial/Homomorphism/Exponentiation.agda          | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda       | 2 +-
 .../Core/Polynomial/Homomorphism/Multiplication.agda          | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Negation.agda     | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Variables.agda    | 2 +-
 src/Tactic/RingSolver/Core/Polynomial/Semantics.agda          | 2 +-
 src/Test/Golden.agda                                          | 2 +-
 src/Text/Regex/Properties/Core.agda                           | 2 +-
 src/Text/Tabular/List.agda                                    | 2 +-
 src/Text/Tabular/Vec.agda                                     | 2 +-
 44 files changed, 45 insertions(+), 45 deletions(-)

diff --git a/src/Function/Endomorphism/Propositional.agda b/src/Function/Endomorphism/Propositional.agda
index 9cb46af85f..d53142c532 100644
--- a/src/Function/Endomorphism/Propositional.agda
+++ b/src/Function/Endomorphism/Propositional.agda
@@ -16,7 +16,7 @@ open import Algebra.Morphism; open Definitions
 
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Nat.Properties using (+-0-monoid; +-semigroup)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 
 open import Function.Base using (id; _∘′_; _∋_)
 open import Function.Equality using (_⟨$⟩_)
diff --git a/src/Function/Nary/NonDependent.agda b/src/Function/Nary/NonDependent.agda
index 3c29079b5e..aa9c35a0f5 100644
--- a/src/Function/Nary/NonDependent.agda
+++ b/src/Function/Nary/NonDependent.agda
@@ -17,7 +17,7 @@ module Function.Nary.NonDependent where
 open import Level using (Level; 0ℓ; _⊔_; Lift)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.Fin.Base using (Fin; zero; suc)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Nary.NonDependent
 open import Function.Base using (_∘′_; _$′_; const; flip)
 open import Relation.Unary using (IUniversal)
diff --git a/src/Function/Properties/Bijection.agda b/src/Function/Properties/Bijection.agda
index c8abd7db19..89aab1b58e 100644
--- a/src/Function/Properties/Bijection.agda
+++ b/src/Function/Properties/Bijection.agda
@@ -13,7 +13,7 @@ open import Level using (Level)
 open import Relation.Binary hiding (_⇔_)
 import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
 open import Function.Properties.Inverse using (Inverse⇒Equivalence)
 
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index ca06dceffb..12920055af 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -20,7 +20,7 @@ open import Function.Surjection  as Surj    using (Surjection)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
-open import Data.Product using (_,_; proj₁; proj₂; <_,_>)
+open import Data.Product.Base using (_,_; proj₁; proj₂; <_,_>)
 
 import Function.Related.Propositional as R
 import Function.Bundles as B
diff --git a/src/Function/Related/TypeIsomorphisms/Solver.agda b/src/Function/Related/TypeIsomorphisms/Solver.agda
index 29f19c8afd..52f27d15ed 100644
--- a/src/Function/Related/TypeIsomorphisms/Solver.agda
+++ b/src/Function/Related/TypeIsomorphisms/Solver.agda
@@ -13,7 +13,7 @@ module Function.Related.TypeIsomorphisms.Solver where
 open import Algebra using (CommutativeSemiring)
 import Algebra.Solver.Ring.NaturalCoefficients.Default
 open import Data.Empty.Polymorphic using (⊥)
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Data.Sum.Base using (_⊎_)
 open import Data.Unit.Polymorphic using (⊤)
 open import Level using (Level)
diff --git a/src/Reflection/AST/Abstraction.agda b/src/Reflection/AST/Abstraction.agda
index 793c729aec..55043355d7 100644
--- a/src/Reflection/AST/Abstraction.agda
+++ b/src/Reflection/AST/Abstraction.agda
@@ -8,7 +8,7 @@
 
 module Reflection.AST.Abstraction where
 
-open import Data.Product                               using (_×_; <_,_>; uncurry)
+open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Data.String as String                      using (String)
 open import Level
 open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
diff --git a/src/Reflection/AST/Argument.agda b/src/Reflection/AST/Argument.agda
index 0feb2deddd..449a45c4d8 100644
--- a/src/Reflection/AST/Argument.agda
+++ b/src/Reflection/AST/Argument.agda
@@ -9,7 +9,7 @@
 module Reflection.AST.Argument where
 
 open import Data.List.Base as List                     using (List; []; _∷_)
-open import Data.Product                               using (_×_; <_,_>; uncurry)
+open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
 open import Relation.Binary                            using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index 7ccabb6f51..ea005eebd3 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -10,7 +10,7 @@ module Reflection.AST.Definition where
 
 import Data.List.Properties as Listₚ                   using (≡-dec)
 import Data.Nat.Properties as ℕₚ                       using (_≟_)
-open import Data.Product                               using (_×_; <_,_>; uncurry)
+open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)
 open import Relation.Binary                            using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
diff --git a/src/Reflection/AST/Show.agda b/src/Reflection/AST/Show.agda
index f0291cf447..7650f6fc8e 100644
--- a/src/Reflection/AST/Show.agda
+++ b/src/Reflection/AST/Show.agda
@@ -15,7 +15,7 @@ import Data.Char as Char   using (show)
 import Data.Float as Float using (show)
 open import Data.List.Base hiding (_++_; intersperse)
 import Data.Nat.Show as ℕ using (show)
-open import Data.Product   using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.String as String
   using (String; _++_; intersperse; braces; parens; parensIfSpace; _<+>_)
 import Data.Word as Word   using (toℕ)
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index cfa7e70cb1..c850900464 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -10,8 +10,8 @@ module Reflection.AST.Term where
 
 open import Data.List.Base as List  hiding (_++_)
 open import Data.List.Properties    using (∷-dec)
-open import Data.Nat as ℕ          using (ℕ; zero; suc)
-open import Data.Product            using (_×_; _,_; <_,_>; uncurry; map₁)
+open import Data.Nat as ℕ           using (ℕ; zero; suc)
+open import Data.Product.Base       using (_×_; _,_; <_,_>; uncurry; map₁)
 open import Data.Product.Properties using (,-injective)
 open import Data.Maybe.Base         using (Maybe; just; nothing)
 open import Data.String as String   using (String)
diff --git a/src/Reflection/AST/Traversal.agda b/src/Reflection/AST/Traversal.agda
index c311567c7f..5437c52b73 100644
--- a/src/Reflection/AST/Traversal.agda
+++ b/src/Reflection/AST/Traversal.agda
@@ -13,7 +13,7 @@ module Reflection.AST.Traversal
 
 open import Data.Nat          using (ℕ; zero; suc; _+_)
 open import Data.List.Base    using (List; []; _∷_; _++_; reverse; length)
-open import Data.Product      using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.String       using (String)
 open import Function.Base     using (_∘_)
 open import Reflection        hiding (pure)
diff --git a/src/Reflection/AST/Universe.agda b/src/Reflection/AST/Universe.agda
index 334458ce2d..dac0af71ed 100644
--- a/src/Reflection/AST/Universe.agda
+++ b/src/Reflection/AST/Universe.agda
@@ -10,7 +10,7 @@ module Reflection.AST.Universe where
 
 open import Data.List.Base             using (List)
 open import Data.String.Base           using (String)
-open import Data.Product               using (_×_)
+open import Data.Product.Base          using (_×_)
 open import Reflection.AST.Argument    using (Arg)
 open import Reflection.AST.Abstraction using (Abs)
 open import Reflection.AST.Term        using (Term; Pattern; Sort; Clause)
diff --git a/src/Reflection/AnnotatedAST.agda b/src/Reflection/AnnotatedAST.agda
index 8ebc54b919..9b281bd4bd 100644
--- a/src/Reflection/AnnotatedAST.agda
+++ b/src/Reflection/AnnotatedAST.agda
@@ -17,7 +17,7 @@ open import Effect.Applicative           using (RawApplicative)
 open import Data.Bool.Base               using (Bool; false; true; if_then_else_)
 open import Data.List.Base               using (List; []; _∷_)
 open import Data.List.Relation.Unary.All using (All; _∷_; [])
-open import Data.Product                 using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base            using (_×_; _,_; proj₁; proj₂)
 open import Data.String.Base             using (String)
 
 open import Reflection                   hiding (pure)
diff --git a/src/Reflection/AnnotatedAST/Free.agda b/src/Reflection/AnnotatedAST/Free.agda
index 79ae7a96be..2017e2fd9f 100644
--- a/src/Reflection/AnnotatedAST/Free.agda
+++ b/src/Reflection/AnnotatedAST/Free.agda
@@ -12,7 +12,7 @@ open import Data.Bool.Base               using (if_then_else_)
 open import Data.Nat.Base                using (ℕ; _∸_; compare; _<ᵇ_; less; equal; greater)
 open import Data.List.Base               using (List; []; _∷_; [_]; concatMap; length)
 open import Data.List.Relation.Unary.All using (_∷_)
-open import Data.Product                 using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base            using (_×_; _,_; proj₁; proj₂)
 open import Data.String.Base             using (String)
 
 open import Reflection
diff --git a/src/Reflection/External.agda b/src/Reflection/External.agda
index e7e9ae989d..5a132d9362 100644
--- a/src/Reflection/External.agda
+++ b/src/Reflection/External.agda
@@ -12,7 +12,7 @@ import Agda.Builtin.Reflection.External as Builtin
 
 open import Data.Nat.Base using (ℕ; suc; zero; NonZero)
 open import Data.List.Base using (List; _∷_; [])
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.String.Base as String using (String; _++_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Unit.Base using (⊤; tt)
diff --git a/src/Relation/Binary/Construct/Subst/Equality.agda b/src/Relation/Binary/Construct/Subst/Equality.agda
index 4e0a3d0100..24e465f9f0 100644
--- a/src/Relation/Binary/Construct/Subst/Equality.agda
+++ b/src/Relation/Binary/Construct/Subst/Equality.agda
@@ -9,7 +9,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
 open import Relation.Binary
 
 module Relation.Binary.Construct.Subst.Equality
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
index 1f218cfc01..4bbd118c13 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
@@ -11,7 +11,7 @@
 
 module Relation.Binary.Indexed.Homogeneous.Bundles where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_⟨_⟩_)
 open import Level using (Level; _⊔_; suc)
 open import Relation.Binary.Core using (_⇒_; Rel)
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Core.agda b/src/Relation/Binary/Indexed/Homogeneous/Core.agda
index 7281652597..50f8b3a200 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Core.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Core.agda
@@ -12,7 +12,7 @@
 module Relation.Binary.Indexed.Homogeneous.Core where
 
 open import Level using (Level; _⊔_)
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Relation.Binary.Core as B using (REL; Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.Indexed.Heterogeneous as I
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda b/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda
index 0231def44f..6160c7e014 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda
@@ -11,7 +11,7 @@
 
 module Relation.Binary.Indexed.Homogeneous.Definitions where
 
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Level using (Level)
 import Relation.Binary as B
 open import Relation.Unary.Indexed using (IPred)
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
index 0ccad66a6f..77438dafdd 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
@@ -17,7 +17,7 @@ module Relation.Binary.Indexed.Homogeneous.Structures
   (_≈ᵢ_ : IRel A ℓ) -- The underlying indexed equality relation
   where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_⟨_⟩_)
 open import Level using (Level; _⊔_; suc)
 open import Relation.Binary.Core using (_⇒_)
diff --git a/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda b/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda
index 1f3477b19d..b665166517 100644
--- a/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/BoundedLattice.agda
@@ -14,7 +14,7 @@ module Relation.Binary.Lattice.Properties.BoundedLattice
 open BoundedLattice L
 
 open import Algebra.Definitions _≈_
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice
 open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
diff --git a/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda b/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda
index 8ee0462d39..e3bd0170eb 100644
--- a/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda
@@ -14,7 +14,7 @@ module Relation.Binary.Lattice.Properties.DistributiveLattice
 open DistributiveLattice L hiding (refl)
 
 open import Algebra.Definitions _≈_
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Relation.Binary
 open import Relation.Binary.Reasoning.Setoid setoid
 open import Relation.Binary.Lattice.Properties.Lattice lattice
diff --git a/src/Relation/Binary/Lattice/Properties/Lattice.agda b/src/Relation/Binary/Lattice/Properties/Lattice.agda
index a221c35b2e..633832a26c 100644
--- a/src/Relation/Binary/Lattice/Properties/Lattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/Lattice.agda
@@ -16,7 +16,7 @@ open Lattice L
 import Algebra.Lattice as Alg
 import Algebra.Structures as Alg
 open import Algebra.Definitions _≈_
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (flip)
 open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
diff --git a/src/Relation/Binary/Morphism/Construct/Identity.agda b/src/Relation/Binary/Morphism/Construct/Identity.agda
index c8ed9c6b03..db12c9793b 100644
--- a/src/Relation/Binary/Morphism/Construct/Identity.agda
+++ b/src/Relation/Binary/Morphism/Construct/Identity.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 open import Level using (Level)
 open import Relation.Binary
diff --git a/src/Relation/Binary/Morphism/OrderMonomorphism.agda b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
index dd2e0ef821..3d788184c2 100644
--- a/src/Relation/Binary/Morphism/OrderMonomorphism.agda
+++ b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
@@ -11,7 +11,7 @@
 
 open import Algebra.Morphism.Definitions
 open import Function.Base
-open import Data.Product using (_,_; map)
+open import Data.Product.Base using (_,_; map)
 open import Relation.Binary
 open import Relation.Binary.Morphism
 import Relation.Binary.Morphism.RelMonomorphism as RawRelation
diff --git a/src/Relation/Binary/Properties/HeytingAlgebra.agda b/src/Relation/Binary/Properties/HeytingAlgebra.agda
index f823a5b519..8786792e78 100644
--- a/src/Relation/Binary/Properties/HeytingAlgebra.agda
+++ b/src/Relation/Binary/Properties/HeytingAlgebra.agda
@@ -15,7 +15,7 @@ open HeytingAlgebra L
 
 open import Algebra.Core
 open import Algebra.Definitions _≈_
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_$_; flip; _∘_)
 open import Level using (_⊔_)
 open import Relation.Binary
diff --git a/src/Relation/Nary.agda b/src/Relation/Nary.agda
index da804883f5..36d1db0568 100644
--- a/src/Relation/Nary.agda
+++ b/src/Relation/Nary.agda
@@ -19,7 +19,7 @@ open import Data.Unit.Base
 open import Data.Bool.Base using (true; false)
 open import Data.Empty
 open import Data.Nat.Base using (zero; suc)
-open import Data.Product as Prod using (_×_; _,_)
+open import Data.Product.Base as Prod using (_×_; _,_)
 open import Data.Product.Nary.NonDependent
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_$_; _∘′_)
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 5df3c1c423..658f8c7bf7 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -19,7 +19,7 @@ import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   as Trivial
 open import Data.Sum.Base as Sum  hiding (map)
 open import Data.Sum.Relation.Binary.Pointwise
-open import Data.Product as Prod hiding (map)
+open import Data.Product.Base as Prod hiding (map)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function.Base using (_∘_; id)
 import Function.Equality as FunS
diff --git a/src/Relation/Unary/Algebra.agda b/src/Relation/Unary/Algebra.agda
index 600bab6c1f..514470de38 100644
--- a/src/Relation/Unary/Algebra.agda
+++ b/src/Relation/Unary/Algebra.agda
@@ -14,7 +14,7 @@ open import Algebra.Lattice.Bundles
 import Algebra.Lattice.Structures as AlgebraicLatticeStructures
 import Algebra.Structures as AlgebraicStructures
 open import Data.Empty.Polymorphic using (⊥-elim)
-open import Data.Product as Product using (_,_; proj₁; proj₂; <_,_>; curry; uncurry)
+open import Data.Product.Base as Product using (_,_; proj₁; proj₂; <_,_>; curry; uncurry)
 open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])
 open import Data.Unit.Polymorphic using (tt)
 open import Function.Base using (id; const; _∘_)
diff --git a/src/System/Environment.agda b/src/System/Environment.agda
index 228bf699ac..ec6833f64c 100644
--- a/src/System/Environment.agda
+++ b/src/System/Environment.agda
@@ -11,7 +11,7 @@ module System.Environment where
 open import IO using (IO; lift; run; ignore)
 open import Data.List.Base using (List)
 open import Data.Maybe.Base using (Maybe)
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Data.String.Base using (String)
 open import Data.Unit.Polymorphic using (⊤)
 open import Foreign.Haskell.Coerce
diff --git a/src/System/FilePath/Posix.agda b/src/System/FilePath/Posix.agda
index d486f1d342..f12cf8180e 100644
--- a/src/System/FilePath/Posix.agda
+++ b/src/System/FilePath/Posix.agda
@@ -13,7 +13,7 @@ open import Agda.Builtin.List using (List)
 open import Agda.Builtin.String using (String)
 open import IO.Base using (IO; lift)
 open import Data.Maybe.Base using (Maybe)
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Data.Sum.Base using (_⊎_)
 
 open import Foreign.Haskell.Coerce
diff --git a/src/System/Process.agda b/src/System/Process.agda
index 1473788365..4aff9ca4dd 100644
--- a/src/System/Process.agda
+++ b/src/System/Process.agda
@@ -10,7 +10,7 @@ module System.Process where
 
 open import Level using (Level)
 open import Data.List.Base using (List)
-open import Data.Product using (_×_; proj₁)
+open import Data.Product.Base using (_×_; proj₁)
 open import Data.String.Base using (String)
 open import Data.Unit.Polymorphic using (⊤)
 open import Foreign.Haskell.Coerce
diff --git a/src/Tactic/MonoidSolver.agda b/src/Tactic/MonoidSolver.agda
index 991ff41556..1448e41fe3 100644
--- a/src/Tactic/MonoidSolver.agda
+++ b/src/Tactic/MonoidSolver.agda
@@ -79,7 +79,7 @@ open import Data.Bool         as Bool    using (Bool; _∨_; if_then_else_)
 open import Data.Maybe        as Maybe   using (Maybe; just; nothing; maybe)
 open import Data.List.Base    as List    using (List; _∷_; [])
 open import Data.Nat          as ℕ       using (ℕ; suc; zero)
-open import Data.Product      as Product using (_×_; _,_)
+open import Data.Product.Base as Product using (_×_; _,_)
 
 open import Reflection.AST
 open import Reflection.AST.Term
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
index 1f6de95f4f..3ea01b2c17 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
@@ -15,7 +15,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Addition
 
 open import Data.Nat            as ℕ  using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
 open import Data.Nat.Properties as ℕₚ using (≤′-trans)
-open import Data.Product              using (_,_; _×_; proj₂)
+open import Data.Product.Base         using (_,_; _×_; proj₂)
 open import Data.List.Base            using (_∷_; [])
 open import Data.List.Kleene
 open import Data.Vec                  using (Vec)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
index 9435251957..eeacadb154 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
@@ -16,7 +16,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Exponentiation
 open import Function.Base using (_⟨_⟩_)
 
 open import Data.Nat.Base as ℕ using (ℕ; suc; zero; compare)
-open import Data.Product  using (_,_; _×_; proj₁; proj₂)
+open import Data.Product.Base  using (_,_; _×_; proj₁; proj₂)
 open import Data.List.Kleene
 open import Data.Vec      using (Vec)
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index 3e1ce2bfe3..90183bc4c9 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -21,7 +21,7 @@ open import Data.Fin                                        using (Fin; zero; su
 open import Data.List.Base                                  using (_∷_; [])
 open import Data.Unit using (tt)
 open import Data.List.Kleene
-open import Data.Product                                    using (_,_; proj₁; proj₂; map₁; _×_)
+open import Data.Product.Base                               using (_,_; proj₁; proj₂; map₁; _×_)
 open import Data.Maybe                                      using (nothing; just)
 open import Function.Base                                   using (_⟨_⟩_)
 open import Level                                           using (lift)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
index 2baeac0f23..c7ad6d6145 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
@@ -16,7 +16,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Multiplication
 open import Data.Nat.Base as ℕ     using (ℕ; suc; zero; _<′_; _≤′_; ≤′-step; ≤′-refl)
 open import Data.Nat.Properties    using (≤′-trans)
 open import Data.Nat.Induction
-open import Data.Product           using (_×_; _,_; proj₁; proj₂; map₁)
+open import Data.Product.Base      using (_×_; _,_; proj₁; proj₂; map₁)
 open import Data.List.Kleene
 open import Data.Vec               using (Vec)
 open import Function.Base using (_⟨_⟩_; flip)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
index dd84eafef2..59b99cf71b 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
@@ -13,7 +13,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Negation
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Product          using (_,_)
+open import Data.Product.Base     using (_,_)
 open import Data.Vec              using (Vec)
 open import Data.Nat              using (_<′_)
 open import Data.Nat.Induction
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda
index c3884e9e2a..9fcaca2f66 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda
@@ -13,7 +13,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Variables
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Product         using (_,_)
+open import Data.Product.Base    using (_,_)
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Fin             using (Fin)
 open import Data.List.Kleene
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
index 8cde662620..26e19d428a 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
@@ -16,7 +16,7 @@ module Tactic.RingSolver.Core.Polynomial.Semantics
 open import Data.Nat          using (ℕ; suc; zero; _≤′_; ≤′-step; ≤′-refl)
 open import Data.Vec          using (Vec; []; _∷_; uncons)
 open import Data.List.Base    using ([]; _∷_)
-open import Data.Product      using (_,_; _×_)
+open import Data.Product.Base using (_,_; _×_)
 open import Data.List.Kleene  using (_+; _*; ∹_; _&_; [])
 
 open Homomorphism homo hiding (_^_)
diff --git a/src/Test/Golden.agda b/src/Test/Golden.agda
index efbfc59ada..079e25720a 100644
--- a/src/Test/Golden.agda
+++ b/src/Test/Golden.agda
@@ -89,7 +89,7 @@ open import Data.List.Relation.Unary.Any using (any?)
 open import Data.Maybe.Base using (Maybe; just; nothing; fromMaybe)
 open import Data.Nat.Base using (ℕ; _≡ᵇ_; _<ᵇ_; _+_; _∸_)
 import Data.Nat.Show as ℕ using (show)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.String as String using (String; lines; unlines; unwords; concat; _≟_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Unit.Base using (⊤)
diff --git a/src/Text/Regex/Properties/Core.agda b/src/Text/Regex/Properties/Core.agda
index e1916fb5cf..51097a7656 100644
--- a/src/Text/Regex/Properties/Core.agda
+++ b/src/Text/Regex/Properties/Core.agda
@@ -17,7 +17,7 @@ open import Data.List.Base as List using (List; []; _∷_; _++_)
 
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 
 open import Relation.Nullary using (¬_; Dec; yes; no)
diff --git a/src/Text/Tabular/List.agda b/src/Text/Tabular/List.agda
index dd06ea809a..70fc09f3d5 100644
--- a/src/Text/Tabular/List.agda
+++ b/src/Text/Tabular/List.agda
@@ -11,7 +11,7 @@ module Text.Tabular.List where
 open import Data.String using (String)
 open import Data.List.Base
 import Data.Nat.Properties as ℕₚ
-open import Data.Product using (-,_; proj₂)
+open import Data.Product.Base using (-,_; proj₂)
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Vec.Bounded.Base as Vec≤ using (Vec≤)
 open import Function.Base
diff --git a/src/Text/Tabular/Vec.agda b/src/Text/Tabular/Vec.agda
index 74f899905c..1286ab1e14 100644
--- a/src/Text/Tabular/Vec.agda
+++ b/src/Text/Tabular/Vec.agda
@@ -9,7 +9,7 @@
 module Text.Tabular.Vec where
 
 open import Data.List.Base using (List)
-open import Data.Product as Prod using (uncurry)
+open import Data.Product.Base as Prod using (uncurry)
 open import Data.String using (String; rectangle; fromAlignment)
 open import Data.Vec.Base
 open import Function.Base

From 3352d4fcd0c5fc40c1b5492917da5ba125e6fc38 Mon Sep 17 00:00:00 2001
From: Maximilien Tirard <maximilien.tirard@icloud.com>
Date: Sat, 29 Jul 2023 12:34:49 +0100
Subject: [PATCH 080/316] Wrapping Comments & Fixing Code Delimiters (#2015)

---
 src/Algebra/Consequences/Setoid.agda          |  4 +-
 src/Algebra/Construct/LexProduct/Inner.agda   |  5 +-
 src/Algebra/Module/Structures.agda            | 10 ++--
 src/Algebra/Morphism/Consequences.agda        |  2 +-
 .../CommutativeMagma/Divisibility.agda        |  4 +-
 .../Properties/CommutativeSemigroup.agda      | 18 +++----
 .../CommutativeSemigroup/Divisibility.agda    |  6 +--
 .../Solver/Ring/NaturalCoefficients.agda      |  4 +-
 .../Ring/NaturalCoefficients/Default.agda     |  4 +-
 src/Codata/Sized/Delay/Properties.agda        | 12 ++---
 src/Codata/Sized/M/Bisimilarity.agda          |  4 +-
 src/Codata/Sized/Stream.agda                  |  8 ++--
 src/Data/AVL/Key.agda                         |  2 +-
 src/Data/AVL/Value.agda                       |  2 +-
 src/Data/Empty.agda                           |  8 ++--
 src/Data/Fin/Permutation/Components.agda      |  8 ++--
 src/Data/Fin/Properties.agda                  | 11 +++--
 src/Data/List/Extrema.agda                    | 12 ++---
 src/Data/List/Fresh.agda                      | 10 ++--
 src/Data/List/Kleene/AsList.agda              |  6 +--
 src/Data/List/Properties.agda                 |  5 +-
 src/Data/List/Relation/Binary/Lex/Core.agda   |  2 +-
 src/Data/List/Relation/Binary/Lex/Strict.agda |  6 +--
 .../Sublist/DecPropositional/Solver.agda      |  4 +-
 .../Sublist/Heterogeneous/Properties.agda     |  8 ++--
 .../Example/UniqueBoundVariables.agda         | 26 +++++-----
 .../Sublist/Propositional/Properties.agda     |  6 +--
 .../Binary/Sublist/Setoid/Properties.agda     |  2 +-
 src/Data/List/Relation/Unary/All.agda         |  3 +-
 src/Data/List/Relation/Unary/First.agda       |  2 +-
 .../Relation/Unary/Sorted/TotalOrder.agda     |  2 +-
 src/Data/Nat/Base.agda                        |  4 +-
 src/Data/Nat/Binary/Properties.agda           | 30 ++++++------
 src/Data/Nat/Binary/Subtraction.agda          |  2 +-
 src/Data/Nat/DivMod/Core.agda                 |  8 ++--
 src/Data/Nat/Induction.agda                   |  8 ++--
 src/Data/Nat/Logarithm.agda                   |  2 +-
 src/Data/Nat/Logarithm/Core.agda              |  2 +-
 src/Data/Nat/Primality.agda                   | 14 +++---
 src/Data/Nat/Properties.agda                  |  2 +-
 src/Data/Nat/PseudoRandom/LCG.agda            | 10 ++--
 src/Data/Product/Effectful/Examples.agda      | 10 ++--
 src/Data/Product/Nary/NonDependent.agda       | 24 +++++-----
 src/Data/Rational/Base.agda                   |  8 ++--
 src/Data/Rational/Unnormalised/Base.agda      | 31 ++++++------
 .../Rational/Unnormalised/Properties.agda     |  6 +--
 src/Data/String/Properties.agda               |  2 +-
 src/Data/Sum/Effectful/Examples.agda          | 10 ++--
 src/Data/Sum/Effectful/Left.agda              |  6 +--
 src/Data/Sum/Effectful/Left/Transformer.agda  |  6 +--
 src/Data/Sum/Effectful/Right/Transformer.agda |  6 +--
 src/Data/Sum/Relation/Binary/LeftOrder.agda   |  6 +--
 src/Data/Sum/Relation/Binary/Pointwise.agda   |  6 +--
 .../Relation/Unary/Any/Properties.agda        |  2 +-
 src/Data/Tree/AVL/Key.agda                    |  8 ++--
 .../Membership/Propositional/Properties.agda  |  4 +-
 src/Data/Tree/AVL/NonEmpty.agda               |  4 +-
 src/Data/Tree/AVL/Value.agda                  |  8 ++--
 src/Data/Trie/NonEmpty.agda                   |  8 ++--
 src/Data/Vec/Properties.agda                  |  6 +--
 src/Data/Vec/Recursive.agda                   | 14 +++---
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  | 28 +++++------
 src/Effect/Monad/Reader/Indexed.agda          |  4 +-
 src/Effect/Monad/Writer/Indexed.agda          |  4 +-
 src/Foreign/Haskell/Coerce.agda               | 48 ++++++++++---------
 src/Function/Metric/Definitions.agda          |  2 +-
 src/Function/Nary/NonDependent/Base.agda      | 18 +++----
 src/Function/Related/Propositional.agda       |  3 +-
 src/IO/Base.agda                              |  6 +--
 src/Reflection/AST/Abstraction.agda           |  4 +-
 src/Reflection/AST/Argument.agda              |  4 +-
 src/Reflection/AST/Name.agda                  | 10 ++--
 src/Reflection/AST/Term.agda                  |  8 ++--
 src/Reflection/AnnotatedAST.agda              |  7 +--
 src/Reflection/External.agda                  |  4 +-
 .../Construct/Add/Extrema/Equality.agda       | 10 ++--
 .../Construct/Add/Extrema/NonStrict.agda      | 14 +++---
 src/Relation/Binary/Morphism/Bundles.agda     |  4 +-
 src/Relation/Binary/Properties/Setoid.agda    | 14 +++---
 .../PropositionalEquality/Properties.agda     |  6 +--
 src/Relation/Binary/Reasoning/Setoid.agda     |  5 +-
 src/Relation/Nary.agda                        |  8 ++--
 src/Relation/Nullary/Decidable/Core.agda      |  7 +--
 src/Relation/Unary/Closure/Base.agda          | 10 ++--
 .../Unary/Polymorphic/Properties.agda         |  6 +--
 src/Relation/Unary/Properties.agda            | 12 ++---
 .../Unary/Relation/Binary/Subset.agda         |  4 +-
 src/System/Process.agda                       |  2 +-
 src/Tactic/Cong.agda                          | 16 +++----
 src/Tactic/MonoidSolver.agda                  | 24 +++++-----
 src/Tactic/RingSolver.agda                    |  6 +--
 src/Tactic/RingSolver/Core/NatSet.agda        | 18 +++----
 .../RingSolver/Core/Polynomial/Base.agda      | 34 ++++++-------
 .../Core/Polynomial/Homomorphism/Lemmas.agda  | 33 ++++++-------
 .../RingSolver/Core/Polynomial/Semantics.agda | 21 ++++----
 src/Test/Golden.agda                          |  2 +-
 src/Text/Format/Generic.agda                  | 10 ++--
 src/Text/Pretty/Core.agda                     |  9 ++--
 src/Text/Regex/Search.agda                    | 10 ++--
 99 files changed, 458 insertions(+), 440 deletions(-)

diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index 444a9296b8..f5cac63048 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -233,7 +233,7 @@ module _ {_•_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _•_) wh
     (x ⁻¹) • e       ≈⟨ idʳ (x ⁻¹) ⟩
     x ⁻¹             ∎
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Bisemigroup-like structures
 
 module _ {_•_ _◦_ : Op₂ A}
@@ -288,7 +288,7 @@ module _ {_•_ _◦_ : Op₂ A}
     (x ◦ (x • z)) • (y ◦ (x • z))  ≈˘⟨ ◦-distribʳ-• _ _ _ ⟩
     (x • y) ◦ (x • z)              ∎
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Ring-like structures
 
 module _ {_+_ _*_ : Op₂ A}
diff --git a/src/Algebra/Construct/LexProduct/Inner.agda b/src/Algebra/Construct/LexProduct/Inner.agda
index d235c1b8fd..9b6e06f0e6 100644
--- a/src/Algebra/Construct/LexProduct/Inner.agda
+++ b/src/Algebra/Construct/LexProduct/Inner.agda
@@ -165,8 +165,9 @@ cong₁ a≈b = cong₁₂ a≈b M.refl
 cong₂ : b ≈₁ c → lex a b x y ≈₂ lex a c x y
 cong₂ = cong₁₂ M.refl
 
--- It is possible to relax this. Instead of ∙ being selective and ◦ being associative it's also
--- possible for _◦_ to return a single idempotent element.
+-- It is possible to relax this. Instead of ∙ being selective and ◦
+-- being associative it's also possible for _◦_ to return a single
+-- idempotent element.
 assoc : Associative _≈₁_ _∙_ → Commutative _≈₁_ _∙_ →
         Selective _≈₁_ _∙_ → Associative _≈₂_ _◦_ →
         ∀ a b c x y z  → lex (a ∙ b) c (lex a b x y) z  ≈₂ lex a (b ∙ c) x (lex b c y z)
diff --git a/src/Algebra/Module/Structures.agda b/src/Algebra/Module/Structures.agda
index 5b071b1a63..d8a291bd13 100644
--- a/src/Algebra/Module/Structures.agda
+++ b/src/Algebra/Module/Structures.agda
@@ -166,8 +166,9 @@ module _ (commutativeSemiring : CommutativeSemiring r ℓr)
   open CommutativeSemiring commutativeSemiring renaming (Carrier to R)
 
   -- An R-semimodule is an R-R-bisemimodule where R is commutative.
-  -- This means that *ₗ and *ᵣ coincide up to mathematical equality, though it
-  -- may be that they do not coincide up to definitional equality.
+  -- This means that *ₗ and *ᵣ coincide up to mathematical equality,
+  -- though it may be that they do not coincide up to definitional
+  -- equality.
 
   record IsSemimodule (*ₗ : Opₗ R M) (*ᵣ : Opᵣ R M)
                       : Set (r ⊔ m ⊔ ℓr ⊔ ℓm) where
@@ -281,8 +282,9 @@ module _ (commutativeRing : CommutativeRing r ℓr)
   open CommutativeRing commutativeRing renaming (Carrier to R)
 
   -- An R-module is an R-R-bimodule where R is commutative.
-  -- This means that *ₗ and *ᵣ coincide up to mathematical equality, though it
-  -- may be that they do not coincide up to definitional equality.
+  -- This means that *ₗ and *ᵣ coincide up to mathematical equality,
+  -- though it may be that they do not coincide up to definitional
+  -- equality.
 
   record IsModule (*ₗ : Opₗ R M) (*ᵣ : Opᵣ R M) : Set (r ⊔ m ⊔ ℓr ⊔ ℓm) where
     field
diff --git a/src/Algebra/Morphism/Consequences.agda b/src/Algebra/Morphism/Consequences.agda
index ac7d6de2c5..c81e55a7bb 100644
--- a/src/Algebra/Morphism/Consequences.agda
+++ b/src/Algebra/Morphism/Consequences.agda
@@ -15,7 +15,7 @@ open import Function.Base using (id; _∘_)
 open import Function.Definitions
 import Relation.Binary.Reasoning.Setoid as EqR
 
----------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- If f and g are mutually inverse maps between A and B, g is congruent,
 -- f is a homomorphism, then g is a homomorphism.
 
diff --git a/src/Algebra/Properties/CommutativeMagma/Divisibility.agda b/src/Algebra/Properties/CommutativeMagma/Divisibility.agda
index 27f6ba56a5..3f6cae0f0c 100644
--- a/src/Algebra/Properties/CommutativeMagma/Divisibility.agda
+++ b/src/Algebra/Properties/CommutativeMagma/Divisibility.agda
@@ -17,12 +17,12 @@ open CommutativeMagma CM
 
 open import Relation.Binary.Reasoning.Setoid setoid
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Re-export the contents of magmas
 
 open import Algebra.Properties.Magma.Divisibility magma public
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Further properties
 
 x∣xy : ∀ x y → x ∣ x ∙ y
diff --git a/src/Algebra/Properties/CommutativeSemigroup.agda b/src/Algebra/Properties/CommutativeSemigroup.agda
index e09253ece7..0c27a5e96b 100644
--- a/src/Algebra/Properties/CommutativeSemigroup.agda
+++ b/src/Algebra/Properties/CommutativeSemigroup.agda
@@ -18,12 +18,12 @@ open import Algebra.Definitions _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
 open import Data.Product.Base using (_,_)
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Re-export the contents of semigroup
 
 open import Algebra.Properties.Semigroup semigroup public
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties
 
 interchange : Interchangable _∙_ _∙_
@@ -35,12 +35,12 @@ interchange a b c d = begin
   a ∙ (c ∙ (b ∙ d))  ≈˘⟨ assoc a c (b ∙ d) ⟩
   (a ∙ c) ∙ (b ∙ d)  ∎
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Permutation laws for _∙_ for three factors.
 
 -- There are five nontrivial permutations.
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Partitions (1,1).
 
 x∙yz≈y∙xz :  ∀ x y z → x ∙ (y ∙ z) ≈ y ∙ (x ∙ z)
@@ -72,7 +72,7 @@ x∙yz≈z∙xy x y z = begin
   (x ∙ y) ∙ z   ≈⟨ comm _ z ⟩
   z ∙ (x ∙ y)   ∎
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Partitions (1,2).
 
 -- These permutation laws are proved by composing the proofs for
@@ -93,7 +93,7 @@ x∙yz≈yz∙x x y z =  trans (x∙yz≈y∙zx _ _ _) (sym (assoc y z x))
 x∙yz≈zx∙y :  ∀ x y z → x ∙ (y ∙ z) ≈ (z ∙ x) ∙ y
 x∙yz≈zx∙y x y z =  trans (x∙yz≈z∙xy x y z) (sym (assoc z x y))
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Partitions (2,1).
 
 -- Their laws are proved by composing proofs for partitions (1,1) with
@@ -114,7 +114,7 @@ xy∙z≈y∙zx x y z =  trans (assoc x y z) (x∙yz≈y∙zx x y z)
 xy∙z≈z∙xy :  ∀ x y z → (x ∙ y) ∙ z ≈ z ∙ (x ∙ y)
 xy∙z≈z∙xy x y z =  trans (assoc x y z) (x∙yz≈z∙xy x y z)
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Partitions (2,2).
 
 -- These proofs are by composing with the proofs for (2,1).
@@ -134,13 +134,13 @@ xy∙z≈yz∙x x y z =  trans (xy∙z≈y∙zx x y z) (sym (assoc y z x))
 xy∙z≈zx∙y :  ∀ x y z → (x ∙ y) ∙ z ≈ (z ∙ x) ∙ y
 xy∙z≈zx∙y x y z =  trans (xy∙z≈z∙xy x y z) (sym (assoc z x y))
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- commutative semigroup has Jordan identity
 
 xy∙xx≈x∙yxx : ∀ x y → (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))
 xy∙xx≈x∙yxx x y = assoc x y ((x ∙ x))
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- commutative semigroup is left/right/middle semiMedial
 
 semimedialˡ : LeftSemimedial _∙_
diff --git a/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda b/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
index 66bec51115..da8d3abf45 100644
--- a/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
+++ b/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
@@ -18,18 +18,18 @@ open CommutativeSemigroup CS
 open import Algebra.Properties.CommutativeSemigroup CS using (x∙yz≈xz∙y; x∙yz≈y∙xz)
 open EqReasoning setoid
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Re-export the contents of divisibility over semigroups
 
 open import Algebra.Properties.Semigroup.Divisibility semigroup public
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Re-export the contents of divisibility over commutative magmas
 
 open import Algebra.Properties.CommutativeMagma.Divisibility commutativeMagma public
   using (x∣xy; xy≈z⇒x∣z; ∣-factors; ∣-factors-≈)
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- New properties
 
 x∣y∧z∣x/y⇒xz∣y : ∀ {x y z} → ((x/y , _) : x ∣ y) → z ∣ x/y → x ∙ z ∣ y
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients.agda b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
index fdd0a65e6b..6d3ab81b37 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
@@ -44,8 +44,8 @@ private
   -- There is a homomorphism from ℕ to R.
   --
   -- Note that the optimised _×_ is used rather than unoptimised _×ᵤ_.
-  -- If _×ᵤ_ were used, then Function.Related.TypeIsomorphisms.test would fail
-  -- to type-check.
+  -- If _×ᵤ_ were used, then Function.Related.TypeIsomorphisms.test
+  -- would fail to type-check.
 
   homomorphism : ℕ-ring -Raw-AlmostCommutative⟶ fromCommutativeSemiring R
   homomorphism = record
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
index 84003922a0..9b9671c527 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
@@ -5,8 +5,8 @@
 -- equality induced by ℕ.
 --
 -- This is sufficient for proving equalities that are independent of the
--- characteristic.  In particular, this is enough for equalities in rings of
--- characteristic 0.
+-- characteristic.  In particular, this is enough for equalities in
+-- rings of characteristic 0.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Codata/Sized/Delay/Properties.agda b/src/Codata/Sized/Delay/Properties.agda
index e491178c71..39a56d1ef9 100644
--- a/src/Codata/Sized/Delay/Properties.agda
+++ b/src/Codata/Sized/Delay/Properties.agda
@@ -80,16 +80,16 @@ module _ {a} {A B : Set a} where
   bind̅₂ (later s) {f} (later foo) =
     bind̅₂ (force s) foo
 
-  -- The extracted value of a bind is equivalent to the extracted value of its
-  -- second element
+  -- The extracted value of a bind is equivalent to the extracted value
+  -- of its second element
   extract-bind-⇓ : {d : Delay A Size.∞} → {f : A → Delay B Size.∞} →
                    (d⇓ : d ⇓) → (f⇓ : f (extract d⇓) ⇓) →
                    extract (bind-⇓ d⇓ {f} f⇓) ≡ extract f⇓
   extract-bind-⇓ (now a) f⇓ = Eq.refl
   extract-bind-⇓ (later t) f⇓ = extract-bind-⇓ t f⇓
 
-  -- If the right element of a bind returns a certain value so does the entire
-  -- bind
+  -- If the right element of a bind returns a certain value so does the
+  -- entire bind
   extract-bind̅₂-bind⇓ :
     (d : Delay A ∞) {f : A → Delay B ∞} →
     (bind⇓ : bind d f ⇓) →
@@ -98,8 +98,8 @@ module _ {a} {A B : Set a} where
   extract-bind̅₂-bind⇓ (later s) (later bind⇓) =
     extract-bind̅₂-bind⇓ (force s) bind⇓
 
-  -- Proof that the length of a bind-⇓ is equal to the sum of the length of its
-  -- components.
+  -- Proof that the length of a bind-⇓ is equal to the sum of the length
+  -- of its components.
   bind⇓-length :
       {d : Delay A ∞} {f : A → Delay B ∞} →
       (bind⇓ : bind d f ⇓) →
diff --git a/src/Codata/Sized/M/Bisimilarity.agda b/src/Codata/Sized/M/Bisimilarity.agda
index ef26860e66..8205ae92e3 100644
--- a/src/Codata/Sized/M/Bisimilarity.agda
+++ b/src/Codata/Sized/M/Bisimilarity.agda
@@ -24,8 +24,8 @@ data Bisim {s p} (C : Container s p) (i : Size) : Rel (M C ∞) (s ⊔ p) where
 
 module _ {s p} {C : Container s p} where
 
-  -- unfortunately the proofs are a lot nicer if we do not use the combinators
-  -- C.refl, C.sym and C.trans
+  -- unfortunately the proofs are a lot nicer if we do not use the
+  -- combinators C.refl, C.sym and C.trans
 
   refl : ∀ {i} → Reflexive (Bisim C i)
   refl {x = inf t} = inf (P.refl , λ where p .force → refl)
diff --git a/src/Codata/Sized/Stream.agda b/src/Codata/Sized/Stream.agda
index 4b7d6bd37d..34a9583448 100644
--- a/src/Codata/Sized/Stream.agda
+++ b/src/Codata/Sized/Stream.agda
@@ -127,8 +127,8 @@ chunksOf n = chunksOfAcc n id module ChunksOf where
 interleave : Stream A i → Thunk (Stream A) i → Stream A i
 interleave (x ∷ xs) ys = x ∷ λ where .force → interleave (ys .force) xs
 
--- This interleaving strategy can be generalised to an arbitrary non-empty
--- list of streams
+-- This interleaving strategy can be generalised to an arbitrary
+-- non-empty list of streams
 interleave⁺ : List⁺ (Stream A i) → Stream A i
 interleave⁺ xss =
   List⁺.map head xss
@@ -155,8 +155,8 @@ cantor (l ∷ ls) = zig (l ∷ []) (ls .force) module Cantor where
   zag (x ∷ []) zs (l ∷ ls) = x ∷ λ where .force → zig (l ∷⁺ zs) (ls .force)
   zag (x ∷ (y ∷ xs)) zs ls = x ∷ λ where .force → zag (y ∷ xs) zs ls
 
--- We can use the Cantor zig zag function to define a form of `bind' that
--- reaches any point of the plane in a finite amount of time.
+-- We can use the Cantor zig zag function to define a form of `bind'
+-- that reaches any point of the plane in a finite amount of time.
 plane : {B : A → Set b} → Stream A ∞ → ((a : A) → Stream (B a) ∞) →
         Stream (Σ A B) ∞
 plane as bs = cantor (map (λ a → map (a ,_) (bs a)) as)
diff --git a/src/Data/AVL/Key.agda b/src/Data/AVL/Key.agda
index 30eaf2f5aa..96023d342d 100644
--- a/src/Data/AVL/Key.agda
+++ b/src/Data/AVL/Key.agda
@@ -2,7 +2,7 @@
 -- The Agda standard library
 --
 -- This module is DEPRECATED.
------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
diff --git a/src/Data/AVL/Value.agda b/src/Data/AVL/Value.agda
index 4ddbc30003..d46bde80eb 100644
--- a/src/Data/AVL/Value.agda
+++ b/src/Data/AVL/Value.agda
@@ -2,7 +2,7 @@
 -- The Agda standard library
 --
 -- This module is DEPRECATED.
------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
diff --git a/src/Data/Empty.agda b/src/Data/Empty.agda
index ff98565038..990a9a1632 100644
--- a/src/Data/Empty.agda
+++ b/src/Data/Empty.agda
@@ -20,10 +20,10 @@ open import Data.Irrelevant using (Irrelevant)
 private
   data Empty : Set where
 
--- ⊥ is defined via Data.Irrelevant (a record with a single irrelevant field)
--- so that Agda can judgementally declare that all proofs of ⊥ are equal
--- to each other. In particular this means that all functions returning a
--- proof of ⊥ are equal.
+-- ⊥ is defined via Data.Irrelevant (a record with a single irrelevant
+-- field) so that Agda can judgementally declare that all proofs of ⊥
+-- are equal to each other. In particular this means that all functions
+-- returning a proof of ⊥ are equal.
 
 ⊥ : Set
 ⊥ = Irrelevant Empty
diff --git a/src/Data/Fin/Permutation/Components.agda b/src/Data/Fin/Permutation/Components.agda
index e75fa29eb2..5a2f27f7f8 100644
--- a/src/Data/Fin/Permutation/Components.agda
+++ b/src/Data/Fin/Permutation/Components.agda
@@ -22,9 +22,9 @@ open import Relation.Binary.PropositionalEquality
 open import Algebra.Definitions using (Involutive)
 open ≡-Reasoning
 
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 --  Functions
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 -- 'tranpose i j' swaps the places of 'i' and 'j'.
 
@@ -35,9 +35,9 @@ transpose i j k with does (k ≟ i)
 ...   | true  = i
 ...   | false = k
 
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 --  Properties
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 transpose-inverse : ∀ {n} (i j : Fin n) {k} →
                     transpose i j (transpose j i k) ≡ k
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index e181a291e9..615728e971 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -818,8 +818,9 @@ punchInᵢ≢i (suc i) (suc j) = punchInᵢ≢i i j ∘ suc-injective
 -- punchOut
 ------------------------------------------------------------------------
 
--- A version of 'cong' for 'punchOut' in which the inequality argument can be
--- changed out arbitrarily (reflecting the proof-irrelevance of that argument).
+-- A version of 'cong' for 'punchOut' in which the inequality argument
+-- can be changed out arbitrarily (reflecting the proof-irrelevance of
+-- that argument).
 
 punchOut-cong : ∀ (i : Fin (suc n)) {j k} {i≢j : i ≢ j} {i≢k : i ≢ k} →
                 j ≡ k → punchOut i≢j ≡ punchOut i≢k
@@ -829,9 +830,9 @@ punchOut-cong {_}     zero    {suc j} {suc k}            = suc-injective
 punchOut-cong {suc n} (suc i) {zero}  {zero}   _ = refl
 punchOut-cong {suc n} (suc i) {suc j} {suc k}    = cong suc ∘ punchOut-cong i ∘ suc-injective
 
--- An alternative to 'punchOut-cong' in the which the new inequality argument is
--- specific. Useful for enabling the omission of that argument during equational
--- reasoning.
+-- An alternative to 'punchOut-cong' in the which the new inequality
+-- argument is specific. Useful for enabling the omission of that
+-- argument during equational reasoning.
 
 punchOut-cong′ : ∀ (i : Fin (suc n)) {j k} {p : i ≢ j} (q : j ≡ k) →
                  punchOut p ≡ punchOut (p ∘ sym ∘ trans q ∘ sym)
diff --git a/src/Data/List/Extrema.agda b/src/Data/List/Extrema.agda
index 8d71aab6a7..5abc5edee9 100644
--- a/src/Data/List/Extrema.agda
+++ b/src/Data/List/Extrema.agda
@@ -30,7 +30,7 @@ open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; sym; subst) renaming (refl to ≡-refl)
 import Relation.Binary.Construct.On as On
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Setup
 
 open TotalOrder totalOrder renaming (Carrier to B)
@@ -43,7 +43,7 @@ private
     a p : Level
     A : Set a
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Functions
 
 argmin : (A → B) → A → List A → A
@@ -58,7 +58,7 @@ min = argmin id
 max : B → List B → B
 max = argmax id
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of argmin
 
 module _ {f : A → B} where
@@ -114,7 +114,7 @@ argmin-all f {⊤} {xs} {P = P}  p⊤ pxs with argmin-sel f ⊤ xs
 ... | inj₁ argmin≡⊤  = subst P (sym argmin≡⊤) p⊤
 ... | inj₂ argmin∈xs = lookup pxs argmin∈xs
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of argmax
 
 module _ {f : A → B} where
@@ -170,7 +170,7 @@ argmax-all f {P = P} {⊥} {xs} p⊥ pxs with argmax-sel f ⊥ xs
 ... | inj₁ argmax≡⊥  = subst P (sym argmax≡⊥) p⊥
 ... | inj₂ argmax∈xs = lookup pxs argmax∈xs
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of min
 
 min≤v⁺ : ∀ {v} ⊤ xs → ⊤ ≤ v ⊎ Any (_≤ v) xs → min ⊤ xs ≤ v
@@ -208,7 +208,7 @@ min-mono-⊆ : ∀ {⊥₁ ⊥₂ xs ys} → ⊥₁ ≤ ⊥₂ → xs ⊇ ys →
 min-mono-⊆ ⊥₁≤⊥₂ ys⊆xs = min[xs]≤min[ys]⁺ _ _ (inj₁ ⊥₁≤⊥₂)
   (tabulate (inj₂ ∘ Any.map (λ {≡-refl → refl}) ∘ ys⊆xs))
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of max
 
 max≤v⁺ : ∀ {v ⊥ xs} → ⊥ ≤ v → All (_≤ v) xs → max ⊥ xs ≤ v
diff --git a/src/Data/List/Fresh.agda b/src/Data/List/Fresh.agda
index 72c372003a..b5a5718765 100644
--- a/src/Data/List/Fresh.agda
+++ b/src/Data/List/Fresh.agda
@@ -1,9 +1,9 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Fresh lists, a proof relevant variant of Catarina Coquand's contexts in
--- "A Formalised Proof of the Soundness and Completeness of a Simply Typed
--- Lambda-Calculus with Explicit Substitutions"
+-- Fresh lists, a proof relevant variant of Catarina Coquand's contexts
+-- in "A Formalised Proof of the Soundness and Completeness of a Simply
+-- Typed Lambda-Calculus with Explicit Substitutions"
 ------------------------------------------------------------------------
 
 -- See README.Data.List.Fresh and README.Data.Trie.NonDependent for
@@ -52,8 +52,8 @@ module _ {a} (A : Set a) (R : Rel A r) where
   -- the erasure ⌊_⌋ of a decidable predicate, cf. Relation.Nary) or we
   -- do not care about the proof, it is convenient to get back list syntax.
 
-  -- We use a different symbol to avoid conflict when importing both Data.List
-  -- and Data.List.Fresh.
+  -- We use a different symbol to avoid conflict when importing both
+  -- Data.List and Data.List.Fresh.
   infixr 5 _∷#_
   pattern _∷#_ x xs = cons x xs _
 
diff --git a/src/Data/List/Kleene/AsList.agda b/src/Data/List/Kleene/AsList.agda
index bb2c52b603..0b6d4c033d 100644
--- a/src/Data/List/Kleene/AsList.agda
+++ b/src/Data/List/Kleene/AsList.agda
@@ -54,9 +54,9 @@ infixr 5 _∷_
 pattern _∷_ x xs = Kleene.∹ x Kleene.& xs
 
 ------------------------------------------------------------------------
--- The following functions change the type of the list (from ⁺ to * or vice
--- versa) in Data.KleeneList, so we reimplement them here to keep the
--- type the same.
+-- The following functions change the type of the list (from ⁺ to * or
+-- vice versa) in Data.KleeneList, so we reimplement them here to keep
+-- the type the same.
 
 scanr : (A → B → B) → B → List A → List B
 scanr f b xs = Kleene.∹ Kleene.scanr* f b xs
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index c05d3251de..454ee4b211 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -55,7 +55,7 @@ private
     D : Set d
     E : Set e
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- _∷_
 
 module _ {x y : A} {xs ys : List A} where
@@ -771,7 +771,8 @@ take-suc-tabulate : ∀ {n} (f : Fin n → A) (i : Fin n) → let m = toℕ i in
 take-suc-tabulate f i rewrite sym (toℕ-cast (sym (length-tabulate f)) i) | sym (lookup-tabulate f i)
   = take-suc (tabulate f) (cast _ i)
 
--- If you take at least as many elements from a list as it has, you get the whole list.
+-- If you take at least as many elements from a list as it has, you get
+-- the whole list.
 take-all :(n : ℕ) (xs : List A) → n ≥ length xs → take n xs ≡ xs
 take-all zero [] _ = refl
 take-all (suc _) [] _ = refl
diff --git a/src/Data/List/Relation/Binary/Lex/Core.agda b/src/Data/List/Relation/Binary/Lex/Core.agda
index 96549e013b..0b3e821194 100644
--- a/src/Data/List/Relation/Binary/Lex/Core.agda
+++ b/src/Data/List/Relation/Binary/Lex/Core.agda
@@ -35,7 +35,7 @@ data Lex {A : Set a} (P : Set)
   next : ∀ {x xs y ys} (x≈y : x ≈ y)
          (xs<ys : Lex P _≈_ _≺_ xs ys) → Lex P _≈_ _≺_ (x ∷ xs) (y ∷ ys)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Lexicographic orderings, using a strict ordering as the base
 
 Lex-< : {A : Set a} (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) →
diff --git a/src/Data/List/Relation/Binary/Lex/Strict.agda b/src/Data/List/Relation/Binary/Lex/Strict.agda
index c0428b347d..d79311ba14 100644
--- a/src/Data/List/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/List/Relation/Binary/Lex/Strict.agda
@@ -26,13 +26,13 @@ open import Data.List.Relation.Binary.Pointwise as Pointwise
 
 import Data.List.Relation.Binary.Lex as Core
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Re-exporting core definitions
 
 open Core public
   using (Lex-<; Lex-≤; base; halt; this; next; ¬≤-this; ¬≤-next)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Strict lexicographic ordering.
 
 module _ {a ℓ₁ ℓ₂} {A : Set a} where
@@ -132,7 +132,7 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
   { isStrictTotalOrder = <-isStrictTotalOrder isStrictTotalOrder
   } where open StrictTotalOrder sto
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Non-strict lexicographic ordering.
 
 module _ {a ℓ₁ ℓ₂} {A : Set a} where
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
index 6385ff0960..58d64ee738 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
@@ -1,8 +1,8 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- A solver for proving that one list is a sublist of the other for types
--- which enjoy decidable equalities.
+-- A solver for proving that one list is a sublist of the other for
+-- types which enjoy decidable equalities.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index f11e85258c..c34e1b8904 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -1,4 +1,4 @@
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Properties of the heterogeneous sublist relation
@@ -78,9 +78,9 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 -- Various functions' outputs are sublists
 
 -- These lemmas are generalisations of results of the form `f xs ⊆ xs`.
--- (where _⊆_ stands for Sublist R). If R is reflexive then we can indeed
--- obtain `f xs ⊆ xs` from `xs ⊆ ys → f xs ⊆ ys`. The other direction is
--- only true if R is both reflexive and transitive.
+-- (where _⊆_ stands for Sublist R). If R is reflexive then we can
+-- indeed obtain `f xs ⊆ xs` from `xs ⊆ ys → f xs ⊆ ys`. The other
+-- direction is only true if R is both reflexive and transitive.
 
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
index 9a681ff916..d2a252e0f0 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
@@ -141,9 +141,10 @@ variable
 
 -- Relating de Bruijn terms and uniquely named terms.
 --
--- The judgement δ ⊢ e ~ β ▷ t relates a de Bruijn term e with potentially free variables δ : Δ ⊆ Γ
--- to a named term t with exact bound variables β : B ⊆ Γ.  The intention is to relate exactly
--- the terms with the same meaning.
+-- The judgement δ ⊢ e ~ β ▷ t relates a de Bruijn term e with
+-- potentially free variables δ : Δ ⊆ Γ to a named term t with exact
+-- bound variables β : B ⊆ Γ.  The intention is to relate exactly the
+-- terms with the same meaning.
 --
 -- The judgement will imply the disjointness of Δ and B.
 
@@ -152,8 +153,8 @@ variable
 
 data _⊢_~_▷_ {Γ Δ : Cxt} (δ : Δ ⊆ Γ) : ∀{a} (e : Exp Δ a) {B} (β : B ⊆ Γ) (t : Tm β a) → Set where
 
-  -- Free de Bruijn index x : a ∈ Δ is related to free variable y : a ∈ Γ
-  -- if δ : Δ ⊆ Γ maps x to y.
+  -- Free de Bruijn index x : a ∈ Δ is related to free variable
+  -- y : a ∈ Γ if δ : Δ ⊆ Γ maps x to y.
 
   var : ∀{y} (δx≡y : lookup δ x ≡ y) (δ#β : Disjoint δ β)
       → δ ⊢ var x ~ β ▷ var! y
@@ -229,9 +230,10 @@ disjoint-fv-bv (app dₜ dᵤ βₜ⊎βᵤ) = disjoint⇒disjoint-to-union δ#
 
 -- Translating de Bruijn terms to uniquely named terms.
 --
--- Given a de Bruijn term Δ ⊢ e : a, we seek to produce a named term β ▷ t : a
--- that is related to the de Bruijn term.  On the way, we have to compute the
--- global context Γ that hosts all free and bound variables of t.
+-- Given a de Bruijn term Δ ⊢ e : a, we seek to produce a named term
+-- β ▷ t : a that is related to the de Bruijn term.  On the way, we have
+-- to compute the global context Γ that hosts all free and bound
+-- variables of t.
 
 -- Record (NamedOf e) collects all the outputs of the translation of e.
 
@@ -293,8 +295,8 @@ dB→Named {Δ = Δ} {a = a ⇒ b} (abs e) with dB→Named e
   z#β     = subst (λ □ → Disjoint □ β) eq (shrinkDisjointˡ [a]⊆aΔ zδ#β)
   z⊎β     = Disjoint→DisjointUnion z#β
 
--- For the translation of an application (f e) we have by induction hypothesis
--- two independent extensions δ₁ : Δ ⊆ Γ₁ and δ₂ : Δ ⊆ Γ₂
+-- For the translation of an application (f e) we have by induction
+-- hypothesis two independent extensions δ₁ : Δ ⊆ Γ₁ and δ₂ : Δ ⊆ Γ₂
 -- and two bound variable lists β₁ : B₁ ⊆ Γ₁ and β₂ : B₂ ⊆ Γ₂.
 -- We need to find a common global context Γ such that
 --
@@ -309,8 +311,8 @@ dB→Named {Δ = Δ} {a = a ⇒ b} (abs e) with dB→Named e
 --   β₁′ = β₁ϕ₁ : B₁ ⊆ Γ₁₂ and
 --   β₂′ = β₂ϕ₂ : B₂ ⊆ Γ₁₂ and
 --
--- may be overlapping, but we can separate them by enlarging the global context
--- to Γ with two embeddings
+-- may be overlapping, but we can separate them by enlarging the global
+-- context to Γ with two embeddings
 --
 --   γ₁ : Γ₁₂ ⊆ Γ
 --   γ₂ : Γ₁₂ ⊆ Γ
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
index 0c5c80e987..1378e745f5 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
@@ -149,8 +149,8 @@ lookup-⊆-trans = Any-resp-⊆-trans
 ------------------------------------------------------------------------
 -- The `lookup` function for `xs ⊆ ys` is injective.
 --
--- Note: `lookup` can be seen as a strictly increasing reindexing function
--- for indices into `xs`, producing indices into `ys`.
+-- Note: `lookup` can be seen as a strictly increasing reindexing
+-- function for indices into `xs`, producing indices into `ys`.
 
 lookup-injective : ∀ {P : Pred A ℓ} {xs ys} {τ : xs ⊆ ys} {i j : Any P xs} →
                    lookup τ i ≡ lookup τ j → i ≡ j
@@ -160,7 +160,7 @@ lookup-injective {τ = x≡y ∷ _} {here  _} {here  _} = cong here ∘′ subst
   -- However, this turns the following clause into a non-strict match.
 lookup-injective {τ = _   ∷ _} {there _} {there _} = cong there ∘′ lookup-injective ∘′ there-injective
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- from∈ ∘ to∈ turns a sublist morphism τ : x∷xs ⊆ ys into a morphism
 -- [x] ⊆ ys.  The same morphism is obtained by pre-composing τ with
 -- the canonial morphism [x] ⊆ x∷xs.
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
index b9c3d21968..abe9a20b73 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
@@ -1,4 +1,4 @@
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Properties of the setoid sublist relation
diff --git a/src/Data/List/Relation/Unary/All.agda b/src/Data/List/Relation/Unary/All.agda
index 52d2baa819..00d8e6f7b7 100644
--- a/src/Data/List/Relation/Unary/All.agda
+++ b/src/Data/List/Relation/Unary/All.agda
@@ -49,7 +49,8 @@ data All {A : Set a} (P : Pred A p) : Pred (List A) (a ⊔ p) where
   _∷_ : ∀ {x xs} (px : P x) (pxs : All P xs) → All P (x ∷ xs)
 
 -- All P xs is a finite map from indices x ∈ xs to content P x.
--- Relation pxs [ i ]= px states that, in map pxs, key i : x ∈ xs points to value px.
+-- Relation pxs [ i ]= px states that, in map pxs, key i : x ∈ xs points
+-- to value px.
 
 infix 4 _[_]=_
 
diff --git a/src/Data/List/Relation/Unary/First.agda b/src/Data/List/Relation/Unary/First.agda
index 931195aea6..bfe8c761b0 100644
--- a/src/Data/List/Relation/Unary/First.agda
+++ b/src/Data/List/Relation/Unary/First.agda
@@ -21,7 +21,7 @@ open import Function.Base using (id; _∘′_)
 open import Relation.Unary
 open import Relation.Nullary
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Basic type.
 
 module _ {p q} (P : Pred A p) (Q : Pred A q) where
diff --git a/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda b/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda
index e9295c61ff..0b6060bae9 100644
--- a/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda
+++ b/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda
@@ -19,7 +19,7 @@ open import Level using (_⊔_)
 open import Relation.Unary as U using (Pred; _⊆_)
 open import Relation.Binary as B
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Definition
 
 Sorted : Pred (List A) (a ⊔ ℓ₂)
diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index 4920e7751e..4ae0acc2dd 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -206,8 +206,8 @@ x ^ suc n = x * x ^ n
 ∣ suc x - suc y ∣ = ∣ x - y ∣
 
 -- Distance in terms of primitive operations.
--- This is much faster than `∣_-_∣` but harder to reason about. For proofs
--- involving this function, convert it to `∣_-_∣` with
+-- This is much faster than `∣_-_∣` but harder to reason about.
+-- For proofs involving this function, convert it to `∣_-_∣` with
 -- `Data.Nat.Properties.∣-∣≡∣-∣′`.
 ∣_-_∣′ : ℕ → ℕ → ℕ
 ∣ x - y ∣′ with x <ᵇ y
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 3ea094ec00..3c362222ed 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -273,7 +273,7 @@ x≢0⇒x>0 {1+[2 _ ]} _   =  0<odd
 <⇒≤ : _<_ ⇒ _≤_
 <⇒≤ = inj₁
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _<_ and toℕ & fromℕ.
 
 toℕ-mono-< : toℕ Preserves _<_ ⟶ ℕ._<_
@@ -334,7 +334,7 @@ toℕ-isMonomorphism-< = record
   ; cancel              = toℕ-cancel-<
   }
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Relational properties of _<_
 
 <-irrefl : Irreflexive _≡_ _<_
@@ -387,7 +387,7 @@ toℕ-isMonomorphism-< = record
 _<?_ : Decidable _<_
 _<?_ = tri⇒dec< <-cmp
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures for _<_
 
 <-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_
@@ -405,7 +405,7 @@ _<?_ = tri⇒dec< <-cmp
   ; compare       = <-cmp
   }
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Bundles for _<_
 
 <-strictPartialOrder : StrictPartialOrder _ _ _
@@ -418,7 +418,7 @@ _<?_ = tri⇒dec< <-cmp
   { isStrictTotalOrder =  <-isStrictTotalOrder
   }
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Other properties of _<_
 
 x<2[1+x] : ∀ x → x < 2[1+ x ]
@@ -432,9 +432,9 @@ x<1+[2x] zero     = 0<odd
 x<1+[2x] 2[1+ x ] = even<odd (x<2[1+x] x)
 x<1+[2x] 1+[2 x ] = odd<odd (x<1+[2x] x)
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _≤_
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Basic properties
 
 <⇒≱ : _<_ ⇒ _≱_
@@ -468,7 +468,7 @@ x<1+[2x] 1+[2 x ] = odd<odd (x<1+[2x] x)
 x≤0⇒x≡0 : x ≤ zero → x ≡ zero
 x≤0⇒x≡0 (inj₂ x≡0) = x≡0
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _<_ and toℕ & fromℕ.
 
 fromℕ-mono-≤ : fromℕ Preserves ℕ._≤_ ⟶ _≤_
@@ -499,7 +499,7 @@ toℕ-isMonomorphism-≤ = record
   ; cancel              = toℕ-cancel-≤
   }
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Relational properties of _≤_
 
 ≤-refl : Reflexive _≤_
@@ -537,7 +537,7 @@ x ≤? y with <-cmp x y
 ... | tri> _   _   y<x = no (<⇒≱ y<x)
 
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures
 
 ≤-isPreorder : IsPreorder _≡_ _≤_
@@ -566,7 +566,7 @@ x ≤? y with <-cmp x y
   ; _≤?_         = _≤?_
   }
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Bundles
 
 ≤-preorder : Preorder 0ℓ 0ℓ 0ℓ
@@ -589,7 +589,7 @@ x ≤? y with <-cmp x y
   { isDecTotalOrder = ≤-isDecTotalOrder
   }
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Equational reasoning for _≤_ and _<_
 
 module ≤-Reasoning = InequalityReasoning
@@ -599,9 +599,9 @@ module ≤-Reasoning = InequalityReasoning
   <-≤-trans ≤-<-trans
   hiding (step-≈; step-≈˘)
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _<ℕ_
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 <⇒<ℕ : ∀ {x y} → x < y → x <ℕ y
 <⇒<ℕ x<y = toℕ-mono-< x<y
@@ -799,7 +799,7 @@ private
 
 module Bin+CSemigroup = CommSemigProp +-commutativeSemigroup
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _+_ and _≤_
 
 +-mono-≤ : _+_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index f1c0b64f56..5ab5e243e0 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -134,7 +134,7 @@ x∸y≡0⇒x≤y {x} {y} = toℕ-cancel-≤ ∘ ℕₚ.m∸n≡0⇒m≤n ∘ tr
 x<y⇒y∸x>0 : x < y → y ∸ x > 0ᵇ
 x<y⇒y∸x>0 {x} {y} = toℕ-cancel-< ∘ subst (ℕ._> 0) (sym (toℕ-homo-∸ y x)) ∘ ℕₚ.m<n⇒0<n∸m ∘ toℕ-mono-<
 
----------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _∸_ and _+_
 
 [x∸y]+y≡x : x ≥ y → (x ∸ y) + y ≡ x
diff --git a/src/Data/Nat/DivMod/Core.agda b/src/Data/Nat/DivMod/Core.agda
index fed1a1ae0d..8ada6d77b2 100644
--- a/src/Data/Nat/DivMod/Core.agda
+++ b/src/Data/Nat/DivMod/Core.agda
@@ -21,7 +21,7 @@ open import Relation.Nullary.Negation using (contradiction)
 
 open ≤-Reasoning
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Helper lemmas that have no interpretation for _%_, only for modₕ
 
 private
@@ -38,7 +38,7 @@ private
     modₕ (a + suc acc) n b 0       ≡⟨ cong (λ v → modₕ v n b 0) (+-suc a acc) ⟩
     modₕ (suc a + acc) n b 0       ∎
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Lemmas for modₕ that also have an interpretation for _%_
 
 a[modₕ]1≡0 : ∀ a → modₕ 0 0 a 0 ≡ 0
@@ -120,7 +120,7 @@ a+n[modₕ]n≡a[modₕ]n acc (suc a) (suc n) rewrite +-suc acc n = begin-equali
   mod₁ = modₕ acc       (suc acc + n)
   mod₂ = modₕ (suc acc) (suc acc + n)
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Helper lemmas that have no interpretation for `_/_`, only for `divₕ`
 
 private
@@ -176,7 +176,7 @@ private
   divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₃  (eq , k<mod , mod≤1+j)) =
     divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₃ (eq , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d k<mod , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (<-transʳ z≤n k<mod) mod≤1+j))
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Lemmas for divₕ that also have an interpretation for _/_
 
 -- The quotient and remainder are related to the dividend and
diff --git a/src/Data/Nat/Induction.agda b/src/Data/Nat/Induction.agda
index 209f9f60bb..efdd57fe3d 100644
--- a/src/Data/Nat/Induction.agda
+++ b/src/Data/Nat/Induction.agda
@@ -90,10 +90,10 @@ module _ {ℓ : Level} where
 -- the first billion proofs. Use this when you require the function
 -- using the proof of well-foundedness to evaluate fast.
 --
--- IMPORTANT: You have to be a little bit careful when using this to always
--- make the function be strict in some other argument than the accessibility
--- proof, otherwise you will have neutral terms unfolding a billion times
--- before getting stuck.
+-- IMPORTANT: You have to be a little bit careful when using this to
+-- always make the function be strict in some other argument than the
+-- accessibility proof, otherwise you will have neutral terms unfolding
+-- a billion times before getting stuck.
 <-wellFounded-fast : WellFounded _<_
 <-wellFounded-fast = <-wellFounded-skip 1000000000
   where
diff --git a/src/Data/Nat/Logarithm.agda b/src/Data/Nat/Logarithm.agda
index 461fa540e6..49cdc0f362 100644
--- a/src/Data/Nat/Logarithm.agda
+++ b/src/Data/Nat/Logarithm.agda
@@ -1,4 +1,4 @@
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Logarithm base 2 and respective properties
diff --git a/src/Data/Nat/Logarithm/Core.agda b/src/Data/Nat/Logarithm/Core.agda
index 08321f5a0a..6a7cdb21e5 100644
--- a/src/Data/Nat/Logarithm/Core.agda
+++ b/src/Data/Nat/Logarithm/Core.agda
@@ -1,4 +1,4 @@
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Logarithm base 2 core definitions and properties
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index e634095eea..dda1122c3c 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -87,8 +87,8 @@ prime⇒¬composite {n@(suc (suc _))} n-prime (d , 2≤d , d<n , d∣n) =
 -- Euclid's lemma
 
 -- For p prime, if p ∣ m * n, then either p ∣ m or p ∣ n.
--- This demonstrates that the usual definition of prime numbers matches the
--- ring theoretic definition of a prime element of the semiring ℕ.
+-- This demonstrates that the usual definition of prime numbers matches
+-- the ring theoretic definition of a prime element of the semiring ℕ.
 -- This is useful for proving many other theorems involving prime numbers.
 euclidsLemma : ∀ m n {p} → Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n
 euclidsLemma m n {p@(suc (suc _))} p-prime p∣m*n = result
@@ -104,13 +104,13 @@ euclidsLemma m n {p@(suc (suc _))} p-prime p∣m*n = result
 
   result : p ∣ m ⊎ p ∣ n
   result with Bézout.lemma m p
-  -- if the GCD of m and p is zero then p must be zero, which is impossible as p
-  -- is a prime
+  -- if the GCD of m and p is zero then p must be zero, which is
+  -- impossible as p is a prime
   ... | Bézout.result 0 g _ = contradiction (0∣⇒≡0 (GCD.gcd∣n g)) λ()
 
-  -- if the GCD of m and p is one then m and p is coprime, and we know that for
-  -- some integers s and r, sm + rp = 1. We can use this fact to determine that p
-  -- divides n
+  -- if the GCD of m and p is one then m and p is coprime, and we know
+  -- that for some integers s and r, sm + rp = 1. We can use this fact
+  -- to determine that p divides n
   ... | Bézout.result 1 _ (Bézout.+- r s 1+sp≡rm) =
     inj₂ (flip ∣m+n∣m⇒∣n (n∣m*n*o s n) (begin
       p             ∣⟨ p∣rmn r ⟩
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index ac8a4dcaa5..2b9a2a4819 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -1551,7 +1551,7 @@ m≤n⇒n∸m≤n : ∀ {m n} → m ≤ n → n ∸ m ≤ n
 m≤n⇒n∸m≤n z≤n       = ≤-refl
 m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
 
----------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _∸_ and _+_
 
 +-∸-comm : ∀ {m} n {o} → o ≤ m → (m + n) ∸ o ≡ (m ∸ o) + n
diff --git a/src/Data/Nat/PseudoRandom/LCG.agda b/src/Data/Nat/PseudoRandom/LCG.agda
index c6c98843cf..32e55c7228 100644
--- a/src/Data/Nat/PseudoRandom/LCG.agda
+++ b/src/Data/Nat/PseudoRandom/LCG.agda
@@ -35,12 +35,12 @@ list (suc k) gen x = x ∷ list k gen (step gen x)
 -- Examples of parameters
 -- Taken from https://en.wikipedia.org/wiki/Linear_congruential_generator
 
--- /!\ As explained in that wikipedia entry, none of these are claimed to
--- be good parameters.
+-- /!\ As explained in that wikipedia entry, none of these are claimed
+-- to be good parameters.
 
--- Note also that if you need your output to have good properties you may
--- need to postprocess the stream of values to only use some of the bits of
--- the output!
+-- Note also that if you need your output to have good properties you
+-- may need to postprocess the stream of values to only use some of the
+-- bits of the output!
 
 glibc : Generator
 glibc = record
diff --git a/src/Data/Product/Effectful/Examples.agda b/src/Data/Product/Effectful/Examples.agda
index a393cb4f66..0ada48ad84 100644
--- a/src/Data/Product/Effectful/Examples.agda
+++ b/src/Data/Product/Effectful/Examples.agda
@@ -38,16 +38,16 @@ private
 
   open RawFunctor functor
 
-  -- This type to the right of × needs to be a "lifted" version of (B : Set b)
-  -- that lives in the universe (Set (a ⊔ b)).
+  -- This type to the right of × needs to be a "lifted" version of
+  -- (B : Set b) that lives in the universe (Set (a ⊔ b)).
   fmapIdₗ : (x : A.Carrier × Lift a B) → (id <$> x) ≈ x
   fmapIdₗ x = A.refl , refl
 
   open RawMonad monad
 
-  -- Now, let's show that "pure" is a unit for >>=. We use Lift in exactly
-  -- the same way as above. The data (x : B) then needs to be "lifted" to
-  -- this new type (Lift B).
+  -- Now, let's show that "pure" is a unit for >>=. We use Lift in
+  -- exactly the same way as above. The data (x : B) then needs to be
+  -- "lifted" to this new type (Lift B).
   pureUnitL : ∀ {x : B} {f : Lift a B → A.Carrier × Lift a B} →
                 (pure (lift x) >>= f) ≈ f (lift x)
   pureUnitL = A.identityˡ _ , refl
diff --git a/src/Data/Product/Nary/NonDependent.agda b/src/Data/Product/Nary/NonDependent.agda
index cb421f16b6..29bb483964 100644
--- a/src/Data/Product/Nary/NonDependent.agda
+++ b/src/Data/Product/Nary/NonDependent.agda
@@ -28,13 +28,14 @@ open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong
 
 open import Function.Nary.NonDependent.Base
 
--- Provided n Levels and a corresponding "vector" of `n` Sets, we can build a big
--- right-nested product type packing a value for each one of these Sets.
+-- Provided n Levels and a corresponding "vector" of `n` Sets, we can
+-- build a big right-nested product type packing a value for each one
+-- of these Sets.
 -- We have two distinct but equivalent definitions:
 -- the first which is always ⊤-terminated
--- the other which has a special case for n = 1 because we want our `(un)curryₙ`
--- functions to work for user-written functions and products and they rarely are
--- ⊤-terminated.
+-- the other which has a special case for n = 1 because we want our
+-- `(un)curryₙ` functions to work for user-written functions and
+-- products and they rarely are ⊤-terminated.
 
 Product⊤ : ∀ n {ls} → Sets n ls → Set (⨆ n ls)
 Product⊤ zero    as       = ⊤
@@ -143,8 +144,8 @@ fromEqualₙ (suc n@(suc _)) eq = uncurry (cong₂ _,_) (Prod.map₂ (fromEqual
 ------------------------------------------------------------------------
 -- projection of the k-th component
 
--- To know at which Set level the k-th projection out of an n-ary product
--- lives, we need to extract said level, by induction on k.
+-- To know at which Set level the k-th projection out of an n-ary
+-- product lives, we need to extract said level, by induction on k.
 
 Levelₙ : ∀ {n} → Levels n → Fin n → Level
 Levelₙ (l , _)  zero    = l
@@ -157,10 +158,11 @@ Projₙ : ∀ {n ls} → Sets n ls → ∀ k → Set (Levelₙ ls k)
 Projₙ (a , _)  zero    = a
 Projₙ (_ , as) (suc k) = Projₙ as k
 
--- Finally, provided a Product of these sets, we can extract the k-th value.
--- `projₙ` takes both `n` and `k` explicitly because we expect the user will
--- be using a concrete `k` (potentially manufactured using `Data.Fin`'s `#_`)
--- and it will not be possible to infer `n` from it.
+-- Finally, provided a Product of these sets, we can extract the k-th
+-- value. `projₙ` takes both `n` and `k` explicitly because we expect
+-- the user will be using a concrete `k` (potentially manufactured
+-- using `Data.Fin`'s `#_`) and it will not be possible to infer `n`
+-- from it.
 
 projₙ : ∀ n {ls} {as : Sets n ls} k → Product n as → Projₙ as k
 projₙ 1               zero    v        = v
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 666cac3dc9..53739b8571 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -143,7 +143,7 @@ toℚᵘ (mkℚ n d-1 _) = mkℚᵘ n d-1
 fromℚᵘ : ℚᵘ → ℚ
 fromℚᵘ (mkℚᵘ n d-1) = n / suc d-1
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Some constants
 
 0ℚ : ℚ
@@ -202,11 +202,11 @@ nonPositive {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonPositive {toℚᵘ p} (
 nonNegative : ∀ {p} → p ≥ 0ℚ → NonNegative p
 nonNegative {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonNegative {toℚᵘ p} (ℚᵘ.*≤* p≤q)
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Operations on rationals
 
--- For explanation of the `@record{}` annotations see notes in the equivalent
--- place in `Data.Rational.Unnormalised.Base`.
+-- For explanation of the `@record{}` annotations see notes in the
+-- equivalent place in `Data.Rational.Unnormalised.Base`.
 
 infix  8 -_ 1/_
 infixl 7 _*_ _÷_ _⊓_
diff --git a/src/Data/Rational/Unnormalised/Base.agda b/src/Data/Rational/Unnormalised/Base.agda
index c03041650c..ec104895be 100644
--- a/src/Data/Rational/Unnormalised/Base.agda
+++ b/src/Data/Rational/Unnormalised/Base.agda
@@ -117,7 +117,7 @@ infixl 7 _/_
 _/_ : (n : ℤ) (d : ℕ) .{{_ : ℕ.NonZero d}} → ℚᵘ
 n / suc d = mkℚᵘ n d
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Some constants
 
 0ℚᵘ : ℚᵘ
@@ -190,24 +190,25 @@ nonNegative : ∀ {p} → p ≥ 0ℚᵘ → NonNegative p
 nonNegative {mkℚᵘ +0       _} (*≤* _) = _
 nonNegative {mkℚᵘ +[1+ n ] _} (*≤* _) = _
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Operations on rationals
 
--- Explanation for `@record{}` everywhere: combined with no-eta-equality on
--- the record definition of ℚᵘ above, these annotations prevent the operations
--- from automatically expanding unless their arguments are explicitly pattern
--- matched on.
+-- Explanation for `@record{}` everywhere: combined with no-eta-equality
+-- on the record definition of ℚᵘ above, these annotations prevent the
+-- operations from automatically expanding unless their arguments are
+-- explicitly pattern matched on.
 --
--- For example prior to their addition, `p + q` would often be normalised by
--- Agda to `(↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)`. While in this
--- small example this isn't a big problem, it leads to an exponential blowup
--- when you have large arithmetic expressions which would often choke
--- both type-checking and the display code. For example, the normalised
--- form of `p + q + r + s + t + u` would be ~300 lines long.
+-- For example prior to their addition, `p + q` would often be
+-- normalised by Agda to `(↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)`.
+-- While in this small example this isn't a big problem, it leads to an
+-- exponential blowup when you have large arithmetic expressions which
+-- would often choke both type-checking and the display code. For
+-- example, the normalised form of `p + q + r + s + t + u` would be
+-- ~300 lines long.
 --
--- This is fundementally a problem with Agda, so if over-eager normalisation
--- is ever fixed in Agda (e.g. with glued representation of terms) these
--- annotations can be removed.
+-- This is fundementally a problem with Agda, so if over-eager
+-- normalisation is ever fixed in Agda (e.g. with glued representation
+-- of terms) these annotations can be removed.
 
 infix  8 -_ 1/_
 infixl 7 _*_ _÷_ _⊓_
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 6f0d490412..3c800a5a98 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -794,7 +794,7 @@ p≤p+q p q rewrite +-comm-≡ p q = p≤q+p p q
 +-mono-<-≤ : _+_ Preserves₂ _<_ ⟶ _≤_ ⟶ _<_
 +-mono-<-≤ {p} {q} {r} p<q q≤r = <-≤-trans (+-monoˡ-< r p<q) (+-monoʳ-≤ q q≤r)
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _+_ and predicates
 
 pos+pos⇒pos : ∀ p .{{_ : Positive p}} →
@@ -808,7 +808,7 @@ nonNeg+nonNeg⇒nonNeg : ∀ p .{{_ : NonNegative p}} →
 nonNeg+nonNeg⇒nonNeg p q = nonNegative
   (+-mono-≤ (nonNegative⁻¹ p) (nonNegative⁻¹ q))
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _-_
 
 +-minus-telescope : ∀ p q r → (p - q) + (q - r) ≃ p - r
@@ -1235,7 +1235,7 @@ private
 *-cancelʳ-<-nonPos : ∀ r .{{_ : NonPositive r}} → p * r < q * r → q < p
 *-cancelʳ-<-nonPos {p} {q} r rewrite *-comm-≡ p r | *-comm-≡ q r = *-cancelˡ-<-nonPos r
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _*_ and predicates
 
 pos*pos⇒pos : ∀ p .{{_ : Positive p}} →
diff --git a/src/Data/String/Properties.agda b/src/Data/String/Properties.agda
index b7ed9a93d6..beb0d869d8 100644
--- a/src/Data/String/Properties.agda
+++ b/src/Data/String/Properties.agda
@@ -78,7 +78,7 @@ x ≈? y = Pointwise.decidable Charₚ._≟_ (toList x) (toList y)
   { isDecEquivalence = ≈-isDecEquivalence
   }
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _≡_
 
 infix 4 _≟_
diff --git a/src/Data/Sum/Effectful/Examples.agda b/src/Data/Sum/Effectful/Examples.agda
index 773c6ca316..c05f334421 100644
--- a/src/Data/Sum/Effectful/Examples.agda
+++ b/src/Data/Sum/Effectful/Examples.agda
@@ -26,8 +26,8 @@ private
 
     open RawFunctor Sₗ.functor
 
-    -- This type to the right of ⊎ needs to be a "lifted" version of (B : Set b)
-    -- that lives in the universe (Set (a ⊔ b)).
+    -- This type to the right of ⊎ needs to be a "lifted" version of
+    -- (B : Set b) that lives in the universe (Set (a ⊔ b)).
     fmapId : (x : A ⊎ (Lift a B)) → (id <$> x) ≡ x
     fmapId (inj₁ x) = refl
     fmapId (inj₂ y) = refl
@@ -35,9 +35,9 @@ private
 
     open RawMonad   Sₗ.monad
 
-    -- Now, let's show that "pure" is a unit for >>=. We use Lift in exactly
-    -- the same way as above. The data (x : B) then needs to be "lifted" to
-    -- this new type (Lift B).
+    -- Now, let's show that "pure" is a unit for >>=. We use Lift in
+    -- exactly the same way as above. The data (x : B) then needs to be
+    -- "lifted" to this new type (Lift B).
     pureUnitL : ∀ {x : B} {f : Lift a B → A ⊎ (Lift a B)}
                   → (pure (lift x) >>= f) ≡ f (lift x)
     pureUnitL = refl
diff --git a/src/Data/Sum/Effectful/Left.agda b/src/Data/Sum/Effectful/Left.agda
index d86a7de26f..6d0f453b7b 100644
--- a/src/Data/Sum/Effectful/Left.agda
+++ b/src/Data/Sum/Effectful/Left.agda
@@ -18,9 +18,9 @@ open import Effect.Applicative
 open import Effect.Monad
 open import Function.Base
 
--- To minimize the universe level of the RawFunctor, we require that elements of
--- B are "lifted" to a copy of B at a higher universe level (a ⊔ b). See the
--- examples for how this is done.
+-- To minimize the universe level of the RawFunctor, we require that
+-- elements of B are "lifted" to a copy of B at a higher universe level
+-- (a ⊔ b). See the examples for how this is done.
 
 ------------------------------------------------------------------------
 -- Left-biased monad instance for _⊎_
diff --git a/src/Data/Sum/Effectful/Left/Transformer.agda b/src/Data/Sum/Effectful/Left/Transformer.agda
index cb45478f11..2d12a229fa 100644
--- a/src/Data/Sum/Effectful/Left/Transformer.agda
+++ b/src/Data/Sum/Effectful/Left/Transformer.agda
@@ -22,9 +22,9 @@ private
   variable
     M : Set (a ⊔ b) → Set (a ⊔ b)
 
--- To minimize the universe level of the RawFunctor, we require that elements of
--- B are "lifted" to a copy of B at a higher universe level (a ⊔ b). See the
--- examples for how this is done.
+-- To minimize the universe level of the RawFunctor, we require that
+-- elements of B are "lifted" to a copy of B at a higher universe level
+-- (a ⊔ b). See the examples for how this is done.
 
 open import Data.Sum.Effectful.Left A b using (Sumₗ)
 
diff --git a/src/Data/Sum/Effectful/Right/Transformer.agda b/src/Data/Sum/Effectful/Right/Transformer.agda
index 8fffb29c80..576d6fb379 100644
--- a/src/Data/Sum/Effectful/Right/Transformer.agda
+++ b/src/Data/Sum/Effectful/Right/Transformer.agda
@@ -22,9 +22,9 @@ private
   variable
     M : Set (a ⊔ b) → Set (a ⊔ b)
 
--- To minimize the universe level of the RawFunctor, we require that elements of
--- B are "lifted" to a copy of B at a higher universe level (a ⊔ b). See the
--- examples for how this is done.
+-- To minimize the universe level of the RawFunctor, we require that
+-- elements of B are "lifted" to a copy of B at a higher universe level
+-- (a ⊔ b). See the examples for how this is done.
 
 open import Data.Sum.Effectful.Right a B using (Sumᵣ)
 
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index 2749640316..2d23693e2d 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -20,7 +20,7 @@ import Relation.Nullary.Decidable as Dec
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Definition
 
 infixr 1 _⊎-<_
@@ -32,7 +32,7 @@ data _⊎-<_ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
   ₁∼₁ : ∀ {x y} (x∼₁y : x ∼₁ y) → (_∼₁_ ⊎-< _∼₂_) (inj₁ x) (inj₁ y)
   ₂∼₂ : ∀ {x y} (x∼₂y : x ∼₂ y) → (_∼₁_ ⊎-< _∼₂_) (inj₂ x) (inj₂ y)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Some properties which are preserved by _⊎-<_
 
 module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
@@ -127,7 +127,7 @@ module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
   ... | tri≈ x≮y x≈y x≯y = tri≈ (x≮y ∘ drop-inj₂) (inj₂ x≈y) (x≯y ∘ drop-inj₂)
   ... | tri> x≮y x≉y x>y = tri> (x≮y ∘ drop-inj₂) (x≉y ∘ PW.drop-inj₂) (₂∼₂ x>y)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Some collections of properties which are preserved
 
 module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index 23d10172c9..724b341c5e 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -20,7 +20,7 @@ open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Definition
 
 data Pointwise {a b c d r s}
@@ -30,7 +30,7 @@ data Pointwise {a b c d r s}
   inj₁ : ∀ {a c} → R a c → Pointwise R S (inj₁ a) (inj₁ c)
   inj₂ : ∀ {b d} → S b d → Pointwise R S (inj₂ b) (inj₂ d)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Relational properties
 
 module _ {a₁ a₂ ℓ₁ ℓ₂} {A₁ : Set a₁} {A₂ : Set a₂}
@@ -108,7 +108,7 @@ module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
                 (Pointwise ∼₁ ∼₂) Respects₂ (Pointwise ≈₁ ≈₂)
   ⊎-respects₂ (r₁ , l₁) (r₂ , l₂) = ⊎-respectsʳ r₁ r₂ , ⊎-respectsˡ l₁ l₂
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures
 
 module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
index 0a803425f2..ef06c5574a 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
@@ -244,7 +244,7 @@ module _ {V : Value v} where
                Any P (singleton k v l<k<u) → P (k , v)
   singleton⁻ k v l<k<u (here Pkv) = Pkv
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- insert
 
   module _ (k : Key) (f : Maybe (Val k) → Val k) where
diff --git a/src/Data/Tree/AVL/Key.agda b/src/Data/Tree/AVL/Key.agda
index 9a2b7e8521..dd41ffb9fd 100644
--- a/src/Data/Tree/AVL/Key.agda
+++ b/src/Data/Tree/AVL/Key.agda
@@ -3,7 +3,7 @@
 --
 -- Keys for AVL trees -- the original key type extended with a new
 -- minimum and maximum.
------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
@@ -29,7 +29,7 @@ import Relation.Binary.Construct.Add.Extrema.Strict
 open StrictTotalOrder sto renaming (Carrier to Key)
   using (_≈_; _<_; trans; irrefl; module Eq)
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Keys are augmented with new extrema (i.e. an artificial minimum and
 -- maximum)
 
@@ -43,7 +43,7 @@ open AddExtremaToSet public
   ; ⊤± to ⊤⁺
   )
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The equality is extended in a corresponding manner
 
 open AddExtremaToEquality _≈_ public
@@ -53,7 +53,7 @@ open AddExtremaToEquality _≈_ public
   ; [_]  to [_]ᴱ
   )
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The order is extended in a corresponding manner
 
 open AddExtremaToOrder _<_ public
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
index c05ed27177..050dd82f12 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
@@ -1,8 +1,8 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Properties of the membership relation for AVL Maps identifying values up to
--- propositional equality.
+-- Properties of the membership relation for AVL Maps identifying values
+-- up to propositional equality.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Data/Tree/AVL/NonEmpty.agda b/src/Data/Tree/AVL/NonEmpty.agda
index d4af2edd58..0e6b680d90 100644
--- a/src/Data/Tree/AVL/NonEmpty.agda
+++ b/src/Data/Tree/AVL/NonEmpty.agda
@@ -35,8 +35,8 @@ open Indexed using (⊥⁺; ⊤⁺; ⊥⁺<⊤⁺; ⊥⁺<[_]<⊤⁺; ⊥⁺<[_]
 ------------------------------------------------------------------------
 -- Types and functions with hidden indices
 
--- NB: the height is non-zero thus guaranteeing that the AVL tree contains
--- at least one value.
+-- NB: the height is non-zero thus guaranteeing that the AVL tree
+-- contains at least one value.
 
 data Tree⁺ {v} (V : Value v) : Set (a ⊔ v ⊔ ℓ₂) where
   tree : ∀ {h} → Indexed.Tree V ⊥⁺ ⊤⁺ (suc h) → Tree⁺ V
diff --git a/src/Data/Tree/AVL/Value.agda b/src/Data/Tree/AVL/Value.agda
index 4f8543b6ba..77af5cffdb 100644
--- a/src/Data/Tree/AVL/Value.agda
+++ b/src/Data/Tree/AVL/Value.agda
@@ -3,7 +3,7 @@
 --
 -- Values for AVL trees
 -- Values must respect the underlying equivalence on keys
------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
@@ -16,7 +16,7 @@ open import Level using (suc; _⊔_)
 import Function.Base as F
 open Setoid S renaming (Carrier to Key)
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- A Value
 
 record Value v : Set (a ⊔ ℓ ⊔ suc v) where
@@ -25,7 +25,7 @@ record Value v : Set (a ⊔ ℓ ⊔ suc v) where
     family   : Key → Set v
     respects : family Respects _≈_
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- A Key together with its value
 
 record K&_ {v} (V : Value v) : Set (a ⊔ v) where
@@ -43,7 +43,7 @@ module _ {v} {V : Value v} where
   fromPair : Σ Key (Value.family V) → K& V
   fromPair (k , v) = k , v
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The constant family of values
 
 -- The function `const` is defined using copatterns to prevent eager
diff --git a/src/Data/Trie/NonEmpty.agda b/src/Data/Trie/NonEmpty.agda
index 118a0db1bd..5fa0a458f0 100644
--- a/src/Data/Trie/NonEmpty.agda
+++ b/src/Data/Trie/NonEmpty.agda
@@ -35,10 +35,10 @@ open Value
 ------------------------------------------------------------------------
 -- Definition
 
--- A Trie⁺ is a tree branching over an alphabet of Keys. It stores values
--- indexed over the Word (i.e. List Key) that was read to reach them.
--- Each node in the Trie⁺ contains either a value, a non-empty Tree of
--- sub-Trie⁺ reached by reading an extra letter, or both.
+-- A Trie⁺ is a tree branching over an alphabet of Keys. It stores
+-- values indexed over the Word (i.e. List Key) that was read to reach
+-- them. Each node in the Trie⁺ contains either a value, a non-empty
+-- Tree of sub-Trie⁺ reached by reading an extra letter, or both.
 
 Word : Set k
 Word = List Key
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index ad356c78a8..c7719eedf1 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -151,7 +151,7 @@ take-drop-id (suc m) (x ∷ xs) = begin
     x ∷ xs
   ∎
 
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- truncate
 
 truncate-refl : (xs : Vec A n) → truncate ≤-refl xs ≡ xs
@@ -163,7 +163,7 @@ truncate-trans : ∀ {p} (m≤n : m ≤ n) (n≤p : n ≤ p) (xs : Vec A p) →
 truncate-trans z≤n       n≤p       xs = refl
 truncate-trans (s≤s m≤n) (s≤s n≤p) (x ∷ xs) = cong (x ∷_) (truncate-trans m≤n n≤p xs)
 
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- pad
 
 padRight-refl : (a : A) (xs : Vec A n) → padRight ≤-refl a xs ≡ xs
@@ -179,7 +179,7 @@ padRight-trans : ∀ {p} (m≤n : m ≤ n) (n≤p : n ≤ p) (a : A) (xs : Vec A
 padRight-trans z≤n       n≤p       a []       = padRight-replicate n≤p a
 padRight-trans (s≤s m≤n) (s≤s n≤p) a (x ∷ xs) = cong (x ∷_) (padRight-trans m≤n n≤p a xs)
 
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- truncate and padRight together
 
 truncate-padRight : (m≤n : m ≤ n) (a : A) (xs : Vec A m) →
diff --git a/src/Data/Vec/Recursive.agda b/src/Data/Vec/Recursive.agda
index 0ecc16f687..bd99c2064e 100644
--- a/src/Data/Vec/Recursive.agda
+++ b/src/Data/Vec/Recursive.agda
@@ -4,13 +4,13 @@
 -- Vectors defined by recursion
 ------------------------------------------------------------------------
 
--- What is the point of this module? The n-ary products below are intended
--- to be used with a fixed n, in which case the nil constructor can be
--- avoided: pairs are represented as pairs (x , y), not as triples
--- (x , y , unit).
--- Additionally, vectors defined by recursion enjoy η-rules. That is to say
--- that two vectors of known length are definitionally equal whenever their
--- elements are.
+-- What is the point of this module? The n-ary products below are
+-- intended to be used with a fixed n, in which case the nil constructor
+-- can be avoided: pairs are represented as pairs (x , y), not as
+-- triples (x , y , unit).
+-- Additionally, vectors defined by recursion enjoy η-rules. That is to
+-- say that two vectors of known length are definitionally equal
+-- whenever their elements are.
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index a1f11fb1b1..e33369a448 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -35,16 +35,16 @@ private
     a ℓ₁ ℓ₂ : Level
     A : Set a
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Re-exports
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 open import Data.Vec.Relation.Binary.Lex.Core as Core public
   using (base; this; next; ≰-this; ≰-next)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Definitions
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 module _ {A : Set a} (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) where
 
@@ -54,9 +54,9 @@ module _ {A : Set a} (_≈_ : Rel A ℓ₁) (_≺_ : Rel A ℓ₂) where
   Lex-≤ : ∀ {m n} → REL (Vec A m) (Vec A n) (a ⊔ ℓ₁ ⊔ ℓ₂)
   Lex-≤ = Core.Lex {A = A} ⊤ _≈_ _≺_
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of Lex-<
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
   private
@@ -138,7 +138,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
         uncons-Lex-wellFounded : WellFounded (×-Lex _≈_ _≺_ _<_ on uncons)
         uncons-Lex-wellFounded = On.wellFounded uncons (×-wellFounded' ≈-sym ≈-trans ≺-respʳ ≺-wf <-wellFounded)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures
 
   <-isStrictPartialOrder : IsStrictPartialOrder _≈_ _≺_ →
@@ -166,7 +166,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
     ; compare       = <-cmp O.Eq.sym O.compare
     } where module O = IsStrictTotalOrder ≺-isStrictTotalOrder
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Bundles for Lex-<
 
 <-strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ → ℕ → StrictPartialOrder _ _ _
@@ -184,9 +184,9 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
   { isStrictTotalOrder = <-isStrictTotalOrder isStrictTotalOrder {n = n}
   } where open StrictTotalOrder ≺-sto
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of Lex-≤
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
   private
@@ -245,7 +245,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
                  ∀ {m n} → Irrelevant (_≤_ {m} {n})
   ≤-irrelevant = Core.irrelevant ⊤-irrelevant
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures
 
   ≤-isPreorder : IsEquivalence _≈_ → Transitive _≺_ → _≺_ Respects₂ _≈_ →
@@ -286,7 +286,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
     ; _≤?_         = ≤-dec _≟_ _<?_
     } where open IsStrictTotalOrder ≺-isStrictTotalOrder
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Bundles
 
 ≤-preorder : Preorder a ℓ₁ ℓ₂ → ℕ → Preorder _ _ _
@@ -314,9 +314,9 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
   { isDecTotalOrder = ≤-isDecTotalOrder isStrictTotalOrder {n = n}
   } where open StrictTotalOrder ≺-sto
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Equational Reasoning
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 module ≤-Reasoning {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂}
                    (≈-isEquivalence : IsEquivalence _≈_)
diff --git a/src/Effect/Monad/Reader/Indexed.agda b/src/Effect/Monad/Reader/Indexed.agda
index 28eab8d453..6ece55aa68 100644
--- a/src/Effect/Monad/Reader/Indexed.agda
+++ b/src/Effect/Monad/Reader/Indexed.agda
@@ -30,7 +30,7 @@ IReaderT M i j A = R → M i j A
 
 module _ {M : IFun I (r ⊔ a)} where
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- Indexed reader applicative
 
   ReaderTIApplicative : RawIApplicative M → RawIApplicative (IReaderT M)
@@ -52,7 +52,7 @@ module _ {M : IFun I (r ⊔ a)} where
     ; _∣_ = λ m n r → m r ∣ n r
     } where open RawIAlternative Alt
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- Indexed reader monad
 
   ReaderTIMonad : RawIMonad M → RawIMonad (IReaderT M)
diff --git a/src/Effect/Monad/Writer/Indexed.agda b/src/Effect/Monad/Writer/Indexed.agda
index 6d99dd9b43..44eaf22196 100644
--- a/src/Effect/Monad/Writer/Indexed.agda
+++ b/src/Effect/Monad/Writer/Indexed.agda
@@ -34,7 +34,7 @@ module _ {M : IFun I (w ⊔ a)} {𝕎 : RawMonoid w ℓ} where
 
   open RawMonoid 𝕎 renaming (Carrier to W)
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- Indexed writer applicative
 
   WriterTIApplicative : RawIApplicative M → RawIApplicative (IWriterT 𝕎 M)
@@ -59,7 +59,7 @@ module _ {M : IFun I (w ⊔ a)} {𝕎 : RawMonoid w ℓ} where
     ; _∣_ = _∣_
     } where open RawIAlternative Alt
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- Indexed writer monad
 
   WriterTIMonad : RawIMonad M → RawIMonad (IWriterT 𝕎 M)
diff --git a/src/Foreign/Haskell/Coerce.agda b/src/Foreign/Haskell/Coerce.agda
index 6e98f5426c..f593d10af3 100644
--- a/src/Foreign/Haskell/Coerce.agda
+++ b/src/Foreign/Haskell/Coerce.agda
@@ -54,16 +54,17 @@ private
     C : Set c
     D : Set d
 
--- We define a simple indexed datatype `Coercible`. A value of `Coercible A B`
--- is a proof that ̀A` and `B` have the same underlying runtime representation.
--- The only possible proof is an incantation from the implementer begging to
--- be trusted.
+-- We define a simple indexed datatype `Coercible`. A value of
+-- `Coercible A B` is a proof that `A` and `B` have the same underlying
+-- runtime representation. The only possible proof is an incantation
+-- from the implementer begging to be trusted.
 
--- We need this type to be concrete so that overlapping instances can be checked
--- for equality: we do not care what proof we get as long as we get one.
+-- We need this type to be concrete so that overlapping instances can be
+-- checked for equality: we do not care what proof we get as long as we
+-- get one.
 
--- We need for it to be a data type rather than a record type so that Agda does
--- not mistakenly build arbitrary instances by η-expansion.
+-- We need for it to be a data type rather than a record type so that
+-- Agda does not mistakenly build arbitrary instances by η-expansion.
 
 data Coercible (A : Set a) (B : Set b) : Set where
   TrustMe : Coercible A B
@@ -71,9 +72,9 @@ data Coercible (A : Set a) (B : Set b) : Set where
 {-# FOREIGN GHC data AgdaCoercible l1 l2 a b = TrustMe #-}
 {-# COMPILE GHC Coercible = data AgdaCoercible (TrustMe) #-}
 
--- Once we get our hands on a proof that `Coercible A B` we postulate that it
--- is safe to convert an `A` into a `B`. This is done under the hood by using
--- `unsafeCoerce`.
+-- Once we get our hands on a proof that `Coercible A B` we postulate
+-- that it is safe to convert an `A` into a `B`. This is done under the
+-- hood by using `unsafeCoerce`.
 
 postulate coerce : {{_ : Coercible A B}} → A → B
 
@@ -95,9 +96,9 @@ Coercible₂ _ _ _ _ T U = ∀ {A B} → {{_ : Coercible A B}} → Coercible₁
 -- Nat
 
 -- Our first instance reveals one of Agda's secrets: natural numbers are
--- represented by (arbitrary precision) integers at runtime! Note that we
--- may only coerce in one direction: arbitrary integers may actually be
--- negative and will not do as mere natural numbers.
+-- represented by (arbitrary precision) integers at runtime! Note that
+-- we may only coerce in one direction: arbitrary integers may actually
+-- be negative and will not do as mere natural numbers.
 
 instance
 
@@ -132,8 +133,8 @@ instance
   nonEmpty-fromFFI : Coercible₁ a b FFI.NonEmpty STD.List⁺
   nonEmpty-fromFFI = TrustMe
 
--- We follow up with purely structural rules for builtin data types which
--- already have known low-level representations.
+-- We follow up with purely structural rules for builtin data types
+-- which already have known low-level representations.
 
 -- Maybe
 
@@ -156,14 +157,15 @@ instance
   coerce-fun : {{_ : Coercible A B}} → Coercible₁ c d (λ C → B → C) (λ D → A → D)
   coerce-fun = TrustMe
 
--- Finally we add a reflexivity proof to discharge all the dangling constraints
--- involving type variables and concrete builtin types such as `Bool`.
+-- Finally we add a reflexivity proof to discharge all the dangling
+-- constraints involving type variables and concrete builtin types such
+-- as `Bool`.
 
--- This rule overlaps with the purely structural ones: when attempting to prove
--- `Coercible (List A) (List A)`, should Agda use the proof obtained by `coerce-refl`
--- or the one obtained by `coerce-list coerce-refl`? Because we are using a
--- datatype with a single constructor these distinctions do not matter: both proofs
--- are definitionally equal.
+-- This rule overlaps with the purely structural ones: when attempting
+-- to prove `Coercible (List A) (List A)`, should Agda use the proof
+-- obtained by `coerce-refl` or the one obtained by `coerce-list coerce-refl`?
+-- Because we are using a datatype with a single constructor these
+-- distinctions do not matter: both proofs are definitionally equal.
 
   coerce-refl : Coercible A A
   coerce-refl = TrustMe
diff --git a/src/Function/Metric/Definitions.agda b/src/Function/Metric/Definitions.agda
index f0be9cb724..acecc24564 100644
--- a/src/Function/Metric/Definitions.agda
+++ b/src/Function/Metric/Definitions.agda
@@ -23,7 +23,7 @@ private
     A : Set a
     I : Set i
 
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties
 
 Congruent : Rel A ℓ₁ → Rel I ℓ₂ → DistanceFunction A I → Set _
diff --git a/src/Function/Nary/NonDependent/Base.agda b/src/Function/Nary/NonDependent/Base.agda
index efb24aec57..2f7347cdab 100644
--- a/src/Function/Nary/NonDependent/Base.agda
+++ b/src/Function/Nary/NonDependent/Base.agda
@@ -30,15 +30,15 @@ private
 ------------------------------------------------------------------------
 -- Type Definitions
 
--- We want to define n-ary function spaces and generic n-ary operations on
--- them such as (un)currying, zipWith, alignWith, etc.
+-- We want to define n-ary function spaces and generic n-ary operations
+-- on them such as (un)currying, zipWith, alignWith, etc.
 
 -- We want users to be able to use these seamlessly whenever n is concrete.
 
--- In other words, we want Agda to infer the sets `A₁, ⋯, Aₙ` when we write
--- `uncurryₙ n f` where `f` has type `A₁ → ⋯ → Aₙ → B`. For this to happen,
--- we need the structure in which these Sets are stored to effectively
--- η-expand to `A₁, ⋯, Aₙ` when the parameter `n` is known.
+-- In other words, we want Agda to infer the sets `A₁, ⋯, Aₙ` when we
+-- write `uncurryₙ n f` where `f` has type `A₁ → ⋯ → Aₙ → B`. For this
+-- to happen, we need the structure in which these Sets are stored to
+-- effectively η-expand to `A₁, ⋯, Aₙ` when the parameter `n` is known.
 
 -- Hence the following definitions:
 ------------------------------------------------------------------------
@@ -66,9 +66,9 @@ Sets : ∀ n (ls : Levels n) → Set (Level.suc (⨆ n ls))
 Sets zero    _        = ⊤
 Sets (suc n) (l , ls) = Set l × Sets n ls
 
--- Third, a function type whose domains are given by a "vector" of `n` Sets
--- `A₁, ⋯, Aₙ` and whose codomain is `B`. `Arrows` forms such a type of
--- shape `A₁ → ⋯ → Aₙ → B` by induction on `n`.
+-- Third, a function type whose domains are given by a "vector" of `n`
+-- Sets `A₁, ⋯, Aₙ` and whose codomain is `B`. `Arrows` forms such a
+-- type of shape `A₁ → ⋯ → Aₙ → B` by induction on `n`.
 
 Arrows : ∀ n {r ls} → Sets n ls → Set r → Set (r ⊔ (⨆ n ls))
 Arrows zero    _        b = b
diff --git a/src/Function/Related/Propositional.agda b/src/Function/Related/Propositional.agda
index 5d62c0718b..2c84095c12 100644
--- a/src/Function/Related/Propositional.agda
+++ b/src/Function/Related/Propositional.agda
@@ -305,7 +305,8 @@ module EquationalReasoning where
   _∼⟨_⟩_ : (A : Set a) → A ∼[ k ] B → B ∼[ k ] C → A ∼[ k ] C
   _ ∼⟨ A↝B ⟩ B↝C = K-trans A↝B B↝C
 
-  -- Isomorphisms and bijections can be combined with any other kind of relatedness.
+  -- Isomorphisms and bijections can be combined with any other kind of
+  -- relatedness.
 
   _⤖⟨_⟩_ : (A : Set a) → A ⤖ B → B ∼[ k ] C → A ∼[ k ] C
   A ⤖⟨ A⤖B ⟩ B⇔C = A ∼⟨ ⤖⇒ A⤖B ⟩ B⇔C
diff --git a/src/IO/Base.agda b/src/IO/Base.agda
index cce93d922b..8c44474935 100644
--- a/src/IO/Base.agda
+++ b/src/IO/Base.agda
@@ -78,9 +78,9 @@ module _ {A B : Set a} where
 -- Running programs
 
 -- A value of type `IO A` is a description of a computation that may
--- eventually produce an `A`. The `run` function converts this description
--- of a computation into calls to primitive functions that will actually
--- perform it.
+-- eventually produce an `A`. The `run` function converts this
+-- description of a computation into calls to primitive functions that
+-- will actually perform it.
 
 {-# NON_TERMINATING #-}
 run : IO A → Prim.IO A
diff --git a/src/Reflection/AST/Abstraction.agda b/src/Reflection/AST/Abstraction.agda
index 55043355d7..80dade0845 100644
--- a/src/Reflection/AST/Abstraction.agda
+++ b/src/Reflection/AST/Abstraction.agda
@@ -50,8 +50,8 @@ abs-injective = < abs-injective₁ , abs-injective₂ >
 
 -- We often need decidability of equality for Abs A when implementing it
 -- for A. Unfortunately ≡-dec makes the termination checker unhappy.
--- Instead, we can match on both Abs A and use unAbs-dec for an obviously
--- decreasing recursive call.
+-- Instead, we can match on both Abs A and use unAbs-dec for an
+-- obviously decreasing recursive call.
 
 unAbs : Abs A → A
 unAbs (abs s a) = a
diff --git a/src/Reflection/AST/Argument.agda b/src/Reflection/AST/Argument.agda
index 449a45c4d8..fe95ccedc1 100644
--- a/src/Reflection/AST/Argument.agda
+++ b/src/Reflection/AST/Argument.agda
@@ -79,8 +79,8 @@ arg-injective = < arg-injective₁ , arg-injective₂ >
 
 -- We often need decidability of equality for Arg A when implementing it
 -- for A. Unfortunately ≡-dec makes the termination checker unhappy.
--- Instead, we can match on both Arg A and use unArg-dec for an obviously
--- decreasing recursive call.
+-- Instead, we can match on both Arg A and use unArg-dec for an
+-- obviously decreasing recursive call.
 
 unArg : Arg A → A
 unArg (arg i a) = a
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index a46940e0a5..fb31bc777a 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -17,7 +17,7 @@ open import Relation.Binary                            using (Rel; Decidable; De
 open import Relation.Binary.Construct.On               using (decidable)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Re-export built-ins
 
 open import Agda.Builtin.Reflection public
@@ -26,16 +26,16 @@ open import Agda.Builtin.Reflection public
 open import Agda.Builtin.Reflection.Properties public
   renaming (primQNameToWord64sInjective to toWords-injective)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- More definitions
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 Names : Set
 Names = List Name
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Decidable equality for names
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 infix 4 _≈?_ _≟_ _≈_
 
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index c850900464..002ca3dd58 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -59,7 +59,7 @@ pattern vΠ[_∶_]_ s a ty    = Π[ s ∶ (vArg a) ] ty
 pattern hΠ[_∶_]_ s a ty    = Π[ s ∶ (hArg a) ] ty
 pattern iΠ[_∶_]_ s a ty    = Π[ s ∶ (iArg a) ] ty
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Utility functions
 
 getName : Term → Maybe Name
@@ -143,9 +143,9 @@ _≟-Patterns_ : Decidable (_≡_ {A = Args Pattern})
 _≟-Pattern_  : Decidable (_≡_ {A = Pattern})
 
 -- Decidable equality 'transformers'
--- We need to inline these because the terms are not sized so termination
--- would not obvious if we were to use higher-order functions such as
--- Data.List.Properties' ≡-dec
+-- We need to inline these because the terms are not sized so
+-- termination would not obvious if we were to use higher-order
+-- functions such as Data.List.Properties' ≡-dec
 
 abs s a ≟-AbsTerm abs s′ a′ = unAbs-dec (a ≟ a′)
 abs s a ≟-AbsType abs s′ a′ = unAbs-dec (a ≟ a′)
diff --git a/src/Reflection/AnnotatedAST.agda b/src/Reflection/AnnotatedAST.agda
index 9b281bd4bd..b839a2f2b6 100644
--- a/src/Reflection/AnnotatedAST.agda
+++ b/src/Reflection/AnnotatedAST.agda
@@ -3,9 +3,10 @@
 --
 -- Annotated reflected syntax.
 --
--- NOTE: This file does not check under --cubical-compatible due to restrictions
---       in the termination checker. In particular recursive functions
---       over a universe of types is not supported by --cubical-compatible.
+-- NOTE: This file does not check under --cubical-compatible due to
+--       restrictions in the termination checker. In particular
+--       recursive functions over a universe of types is not supported
+--       by --cubical-compatible.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --safe --with-K #-}
diff --git a/src/Reflection/External.agda b/src/Reflection/External.agda
index 5a132d9362..122eab9905 100644
--- a/src/Reflection/External.agda
+++ b/src/Reflection/External.agda
@@ -26,8 +26,8 @@ StdIn   = String
 StdErr  = String
 StdOut  = String
 
--- Representation for exit codes, assuming 0 is consistently used to indicate
--- success across platforms.
+-- Representation for exit codes, assuming 0 is consistently used to
+-- indicate success across platforms.
 data ExitCode : Set where
   exitSuccess : ExitCode
   exitFailure : (n : ℕ) {n≢0 : NonZero n} → ExitCode
diff --git a/src/Relation/Binary/Construct/Add/Extrema/Equality.agda b/src/Relation/Binary/Construct/Add/Extrema/Equality.agda
index 2829910dcc..b4992e112d 100644
--- a/src/Relation/Binary/Construct/Add/Extrema/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Extrema/Equality.agda
@@ -2,7 +2,7 @@
 -- The Agda standard library
 --
 -- A pointwise lifting of a relation to incorporate new extrema.
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
@@ -19,7 +19,7 @@ import Relation.Binary.Construct.Add.Infimum.Equality as AddInfimum
 import Relation.Binary.Construct.Add.Supremum.Equality as AddSupremum
 open import Relation.Nullary.Construct.Add.Extrema
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Definition
 
 private
@@ -28,14 +28,14 @@ private
 
 open Sup using () renaming (_≈⁺_ to _≈±_) public
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Useful pattern synonyms
 
 pattern ⊥±≈⊥± = Sup.[ Inf.⊥₋≈⊥₋ ]
 pattern [_] p = Sup.[ Inf.[ p ] ]
 pattern ⊤±≈⊤± = Sup.⊤⁺≈⊤⁺
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Relational properties
 
 [≈]-injective : ∀ {k l} → [ k ] ≈± [ l ] → k ≈ l
@@ -59,7 +59,7 @@ pattern ⊤±≈⊤± = Sup.⊤⁺≈⊤⁺
 ≈±-substitutive : ∀ {ℓ} → Substitutive _≈_ ℓ → Substitutive _≈±_ ℓ
 ≈±-substitutive = Sup.≈⁺-substitutive ∘′ Inf.≈₋-substitutive
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures
 
 ≈±-isEquivalence : IsEquivalence _≈_ → IsEquivalence _≈±_
diff --git a/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda b/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
index e160bb3b85..ac1880044d 100644
--- a/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
@@ -22,7 +22,7 @@ import Relation.Binary.Construct.Add.Infimum.NonStrict as AddInfimum
 import Relation.Binary.Construct.Add.Supremum.NonStrict as AddSupremum
 import Relation.Binary.Construct.Add.Extrema.Equality as Equality
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Definition
 
 private
@@ -31,7 +31,7 @@ private
 
 open Sup using () renaming (_≤⁺_ to _≤±_) public
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Useful pattern synonyms
 
 pattern ⊥±≤⊥±    = Sup.[ Inf.⊥₋≤ I.⊥₋ ]
@@ -51,7 +51,7 @@ _≤⊤± : ∀ k → k ≤± ⊤±
 [ k ] ≤⊤± = [ k ]≤⊤±
 ⊤±    ≤⊤± = ⊤±≤⊤±
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Relational properties
 
 [≤]-injective : ∀ {k l} → [ k ] ≤± [ l ] → k ≤ l
@@ -75,7 +75,7 @@ _≤⊤± : ∀ k → k ≤± ⊤±
 ≤±-irrelevant : Irrelevant _≤_ → Irrelevant _≤±_
 ≤±-irrelevant = Sup.≤⁺-irrelevant ∘′ Inf.≤₋-irrelevant
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Relational properties + propositional equality
 
 ≤±-reflexive-≡ : (_≡_ ⇒ _≤_) → (_≡_ ⇒ _≤±_)
@@ -84,7 +84,7 @@ _≤⊤± : ∀ k → k ≤± ⊤±
 ≤±-antisym-≡ : Antisymmetric _≡_ _≤_ → Antisymmetric _≡_ _≤±_
 ≤±-antisym-≡ = Sup.≤⁺-antisym-≡ ∘′ Inf.≤₋-antisym-≡
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Relational properties + setoid equality
 
 module _ {e} {_≈_ : Rel A e} where
@@ -97,7 +97,7 @@ module _ {e} {_≈_ : Rel A e} where
   ≤±-antisym : Antisymmetric _≈_ _≤_ → Antisymmetric _≈±_ _≤±_
   ≤±-antisym = Sup.≤⁺-antisym ∘′ Inf.≤₋-antisym
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures + propositional equality
 
 ≤±-isPreorder-≡ : IsPreorder _≡_ _≤_ → IsPreorder _≡_ _≤±_
@@ -115,7 +115,7 @@ module _ {e} {_≈_ : Rel A e} where
 ≤±-isDecTotalOrder-≡ : IsDecTotalOrder _≡_ _≤_ → IsDecTotalOrder _≡_ _≤±_
 ≤±-isDecTotalOrder-≡ = Sup.≤⁺-isDecTotalOrder-≡ ∘′ Inf.≤₋-isDecTotalOrder-≡
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures + setoid equality
 
 module _ {e} {_≈_ : Rel A e} where
diff --git a/src/Relation/Binary/Morphism/Bundles.agda b/src/Relation/Binary/Morphism/Bundles.agda
index 782c6af098..e8c1c84041 100644
--- a/src/Relation/Binary/Morphism/Bundles.agda
+++ b/src/Relation/Binary/Morphism/Bundles.agda
@@ -92,8 +92,8 @@ module _ (P : Poset ℓ₁ ℓ₂ ℓ₃) (Q : Poset ℓ₄ ℓ₅ ℓ₆) where
     open IsOrderHomomorphism isOrderHomomorphism public
 
 
-  -- Smart constructor that automatically constructs the congruence proof
-  -- from the monotonicity proof
+  -- Smart constructor that automatically constructs the congruence
+  -- proof from the monotonicity proof
   mkPosetHomo : ∀ f → f Preserves P._≤_ ⟶ Q._≤_ → PosetHomomorphism
   mkPosetHomo f mono = record
     { ⟦_⟧ = f
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index cca28fb5ad..050177c629 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -1,8 +1,8 @@
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Additional properties for setoids
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
@@ -16,9 +16,9 @@ module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where
 
 open Setoid S
 
-------------------------------------------------------------------------------
--- Every setoid is a preorder and partial order with respect to propositional
--- equality
+------------------------------------------------------------------------
+-- Every setoid is a preorder and partial order with respect to
+-- propositional equality
 
 isPreorder : IsPreorder _≡_ _≈_
 isPreorder = record
@@ -59,7 +59,7 @@ preorder = record
   { isPartialOrder = ≈-isPartialOrder
   }
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Properties of _≉_
 
 ≉-sym :  Symmetric _≉_
@@ -74,7 +74,7 @@ preorder = record
 ≉-resp₂ : _≉_ Respects₂ _≈_
 ≉-resp₂ = ≉-respʳ , ≉-respˡ
 
-------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Other properties
 
 respʳ-flip : _≈_ Respectsʳ (flip _≈_)
diff --git a/src/Relation/Binary/PropositionalEquality/Properties.agda b/src/Relation/Binary/PropositionalEquality/Properties.agda
index ad936254ff..48bb2a257e 100644
--- a/src/Relation/Binary/PropositionalEquality/Properties.agda
+++ b/src/Relation/Binary/PropositionalEquality/Properties.agda
@@ -3,9 +3,9 @@
 --
 -- Propositional equality
 --
--- This file contains some core properies of propositional equality which
--- are re-exported by Relation.Binary.PropositionalEquality. They are
--- ``equality rearrangement'' lemmas.
+-- This file contains some core properies of propositional equality
+-- which are re-exported by Relation.Binary.PropositionalEquality. They
+-- are ``equality rearrangement'' lemmas.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Relation/Binary/Reasoning/Setoid.agda b/src/Relation/Binary/Reasoning/Setoid.agda
index 0c43af7bc7..77b1879db1 100644
--- a/src/Relation/Binary/Reasoning/Setoid.agda
+++ b/src/Relation/Binary/Reasoning/Setoid.agda
@@ -15,8 +15,8 @@
 --   0         ∎
 
 -- Module `≡-Reasoning` in `Relation.Binary.PropositionalEquality`
--- is recommended for equational reasoning when the underlying equality is
--- `_≡_`.
+-- is recommended for equational reasoning when the underlying equality
+-- is `_≡_`.
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
@@ -29,7 +29,6 @@ open Setoid S
 ------------------------------------------------------------------------
 -- Reasoning combinators
 
--- open import Relation.Binary.Reasoning.PartialSetoid partialSetoid public
 open import Relation.Binary.Reasoning.Base.Single _≈_ refl trans as Base public
   hiding (step-∼)
 
diff --git a/src/Relation/Nary.agda b/src/Relation/Nary.agda
index 36d1db0568..6610b16439 100644
--- a/src/Relation/Nary.agda
+++ b/src/Relation/Nary.agda
@@ -105,10 +105,10 @@ module _ {n r ls} {as : Sets n ls} (P : as ⇉ Set r) where
 ------------------------------------------------------------------------
 -- Pointwise liftings of k-ary operators
 
--- Rather than having multiple ad-hoc lifting functions for various arities
--- we have a fully generic liftₙ functional which lifts a k-ary operator
--- to work with k n-ary functions whose respective codomains match the domains
--- of the operator.
+-- Rather than having multiple ad-hoc lifting functions for various
+-- arities we have a fully generic liftₙ functional which lifts a k-ary
+-- operator to work with k n-ary functions whose respective codomains
+-- match the domains of the operator.
 -- The type of liftₙ is fairly unreadable. Here it is written with ellipsis:
 
 -- liftₙ : ∀ k n. (B₁ → ⋯ → Bₖ → R) →
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index e06dc32239..a7c1009091 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -86,9 +86,10 @@ proof (p? →-dec q?) = proof p? →-reflects proof q?
 ------------------------------------------------------------------------
 -- Relationship with booleans
 
--- `isYes` is a stricter version of `does`. The lack of computation means that
--- we can recover the proposition `P` from `isYes P?` by unification. This is
--- useful when we are using the decision procedure for proof automation.
+-- `isYes` is a stricter version of `does`. The lack of computation
+-- means that we can recover the proposition `P` from `isYes P?` by
+-- unification. This is useful when we are using the decision procedure
+-- for proof automation.
 
 isYes : Dec P → Bool
 isYes (true  because _) = true
diff --git a/src/Relation/Unary/Closure/Base.agda b/src/Relation/Unary/Closure/Base.agda
index 992e87df65..4d80a5af1d 100644
--- a/src/Relation/Unary/Closure/Base.agda
+++ b/src/Relation/Unary/Closure/Base.agda
@@ -20,9 +20,9 @@ open import Relation.Unary using (Pred)
 
 -- Box
 
--- We start with the definition of □ ("box") which is named after the box
--- modality in modal logic. `□ T x` states that all the elements one step
--- away from `x` with respect to the relation R satisfy `T`.
+-- We start with the definition of □ ("box") which is named after the
+-- box modality in modal logic. `□ T x` states that all the elements one
+-- step away from `x` with respect to the relation R satisfy `T`.
 
 □ : ∀ {t} → Pred A t → Pred A (a ⊔ b ⊔ t)
 □ T x = ∀ {y} → R x y → T y
@@ -42,8 +42,8 @@ open import Relation.Unary using (Pred)
 -- diamond modality in modal logic. In modal logic, `◇ T x` states that
 -- there exists an element one step away from x with respect to the
 -- relation R that satisfies T. It is worth noting that the modal logic
--- metaphor breaks down here: this only is a closure operator if the step
--- we take is *backwards* with respect to R.
+-- metaphor breaks down here: this only is a closure operator if the
+-- step we take is *backwards* with respect to R.
 
 ◇ : ∀ {t} → Pred A t → Pred A (a ⊔ b ⊔ t)
 ◇ T x = Σ[ support ∈ A ] (R support x × T support)
diff --git a/src/Relation/Unary/Polymorphic/Properties.agda b/src/Relation/Unary/Polymorphic/Properties.agda
index 8d46f51807..3a59912e2b 100644
--- a/src/Relation/Unary/Polymorphic/Properties.agda
+++ b/src/Relation/Unary/Polymorphic/Properties.agda
@@ -21,7 +21,7 @@ private
     a ℓ ℓ₁ ℓ₂ : Level
     A : Set a
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The empty set
 
 ∅? : Decidable {A = A} {ℓ} ∅
@@ -33,7 +33,7 @@ private
 ∁∅-Universal : Universal {A = A} {ℓ} (∁ ∅)
 ∁∅-Universal _ ()
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The universe
 
 U? : Decidable {A = A} {ℓ} U
@@ -45,7 +45,7 @@ U-Universal _ = _
 ∁U-Empty : Empty {A = A} {ℓ} (∁ U)
 ∁U-Empty _ x∈∁U = x∈∁U _
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Subset properties
 
 ∅-⊆ : (P : Pred A ℓ₁) → ∅ {ℓ = ℓ₂} ⊆ P
diff --git a/src/Relation/Unary/Properties.agda b/src/Relation/Unary/Properties.agda
index f30a9e59b4..d090716c08 100644
--- a/src/Relation/Unary/Properties.agda
+++ b/src/Relation/Unary/Properties.agda
@@ -25,7 +25,7 @@ private
     A : Set a
     B : Set b
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The empty set
 
 ∅? : Decidable {A = A} ∅
@@ -37,7 +37,7 @@ private
 ∁∅-Universal : Universal {A = A} (∁ ∅)
 ∁∅-Universal = λ x x∈∅ → x∈∅
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The universe
 
 U? : Decidable {A = A} U
@@ -49,7 +49,7 @@ U-Universal = λ _ → _
 ∁U-Empty : Empty {A = A} (∁ U)
 ∁U-Empty = λ x x∈∁U → x∈∁U _
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Subset properties
 
 ∅-⊆ : (P : Pred A ℓ) → ∅ ⊆ P
@@ -169,7 +169,7 @@ U-Universal = λ _ → _
 ⊂′⇒⊂ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _⊂′_ _⊂_
 ⊂′⇒⊂ = Product.map ⊆′⇒⊆ (_∘ ⊆⇒⊆′)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Equality properties
 
 ≐-refl : Reflexive {A = Pred A ℓ} _≐_
@@ -196,7 +196,7 @@ U-Universal = λ _ → _
 ≐′⇒≐ : Binary._⇒_ {A = Pred A ℓ₁} {B = Pred A ℓ₂} _≐′_ _≐_
 ≐′⇒≐ = Product.map ⊆′⇒⊆ ⊆′⇒⊆
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Decidability properties
 
 ∁? : {P : Pred A ℓ} → Decidable P → Decidable (∁ P)
@@ -226,7 +226,7 @@ _⊎?_ P? Q? (inj₂ b) = Q? b
 _~? : {P : Pred (A × B) ℓ} → Decidable P → Decidable (P ~)
 _~? P? = P? ∘ swap
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Irrelevant properties
 
 U-irrelevant : Irrelevant {A = A} U
diff --git a/src/Relation/Unary/Relation/Binary/Subset.agda b/src/Relation/Unary/Relation/Binary/Subset.agda
index 77679f15bd..1069b99a73 100644
--- a/src/Relation/Unary/Relation/Binary/Subset.agda
+++ b/src/Relation/Unary/Relation/Binary/Subset.agda
@@ -15,7 +15,7 @@ open import Relation.Unary
 open import Relation.Unary.Properties
 open import Relation.Unary.Relation.Binary.Equality using (≐-isEquivalence; ≐′-isEquivalence)
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Structures
 
 module _ {a : Level} {A : Set a} {ℓ : Level} where
@@ -68,7 +68,7 @@ module _ {a : Level} {A : Set a} {ℓ : Level} where
     ; <-resp-≈ = ⊂′-resp-≐′
     }
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Bundles
 
 module _ {a : Level} (A : Set a) (ℓ : Level) where
diff --git a/src/System/Process.agda b/src/System/Process.agda
index 4aff9ca4dd..b2eacdacea 100644
--- a/src/System/Process.agda
+++ b/src/System/Process.agda
@@ -1,4 +1,4 @@
------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Calling external processes
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index e46392ee5d..515d904872 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -38,8 +38,8 @@ open import Data.Product
 
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_; refl; cong)
 
--- 'Data.String.Properties' defines this via 'Dec', so let's use the builtin
--- for maximum speed.
+-- 'Data.String.Properties' defines this via 'Dec', so let's use the
+-- builtin for maximum speed.
 import Agda.Builtin.String as String renaming (primStringEquality to _≡ᵇ_)
 
 open import Reflection
@@ -55,9 +55,9 @@ open import Reflection.AST.Term                 as Term
 
 open import Reflection.TCM.Syntax
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Utilities
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 private
   -- Descend past a variable.
@@ -116,7 +116,7 @@ private
     `y ← quoteTC y
     pure $ def (quote cong) $ `a ⟅∷⟆ `A ⟅∷⟆ level ⟅∷⟆ type ⟅∷⟆ vLam "ϕ" f ⟨∷⟩ `x ⟅∷⟆ `y ⟅∷⟆ eq ⟨∷⟩ []
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Anti-Unification
 --
 -- The core idea of the tactic is that we can compute the input
@@ -128,7 +128,7 @@ private
 -- For instance, the two terms 'suc (m + (m + 0)) + (m + 0)' and
 -- 'suc (m + m) + (m + 0)' would anti unify to 'suc (m + _) + (m + 0)'
 -- which we can then use to construct the lambda 'λ ϕ → suc (m + ϕ) + (m + 0)'.
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 private
   antiUnify        : ℕ → Term → Term → Term
@@ -202,9 +202,9 @@ private
     just []
 
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Rewriting
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 macro
   cong! : ∀ {a} {A : Set a} {x y : A} → x ≡ y → Term → TC ⊤
diff --git a/src/Tactic/MonoidSolver.agda b/src/Tactic/MonoidSolver.agda
index 1448e41fe3..c137e44990 100644
--- a/src/Tactic/MonoidSolver.agda
+++ b/src/Tactic/MonoidSolver.agda
@@ -90,9 +90,9 @@ open import Reflection.TCM.Syntax
 
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- The Expr type with homomorphism proofs
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 infixl 7 _∙′_
 data Expr {a} (A : Set a) : Set a where
@@ -147,9 +147,9 @@ module _ {m₁ m₂} (monoid : Monoid m₁ m₂) where
     [ x ⇓] ∙ [ y ⇓] ≈⟨ ∙-cong (homo x) (homo y) ⟩
     [ x ↓] ∙ [ y ↓] ∎
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Helpers for reflection
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 getArgs : Term → Maybe (Term × Term)
 getArgs (def _ xs) = go xs
@@ -160,9 +160,9 @@ getArgs (def _ xs) = go xs
   go _                      = nothing
 getArgs _ = nothing
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Getting monoid names
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 -- We try to be flexible here, by matching two kinds of names.
 -- The first is the field accessor for the monoid record itself.
@@ -191,9 +191,9 @@ findMonoidNames mon = do
     ; is-ε = buildMatcher (quote Monoid.ε)   (getName ε-altName)
     }
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Building Expr
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 -- We now define a function that takes an AST representing the LHS
 -- or RHS of the equation to solve and converts it into an AST
@@ -231,9 +231,9 @@ module _ (names : MonoidNames) where
     else [ t ↑]′
   buildExpr t = quote [_↑] ⟨ con ⟩ (t ⟨∷⟩ [])
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Constructing the solution
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 -- This function joins up the two homomorphism proofs. It constructs
 -- a proof of the following form:
@@ -252,9 +252,9 @@ constructSoln mon names lhs rhs =
     (quote homo ⟨ def ⟩ 2 ⋯⟅∷⟆ mon ⟨∷⟩ buildExpr names rhs ⟨∷⟩ []) ⟨∷⟩
     []
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Macro
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
 solve-macro : Term → Term → TC _
 solve-macro mon hole = do
diff --git a/src/Tactic/RingSolver.agda b/src/Tactic/RingSolver.agda
index 6441342e40..07e9f1a033 100644
--- a/src/Tactic/RingSolver.agda
+++ b/src/Tactic/RingSolver.agda
@@ -349,9 +349,9 @@ constructSolution `ring opTerms variables `lhs `rhs = do
     t' ← convertTerm `ring numVars opTerms varMap t
     pure $ `correct `ring numVars t' ρ
 
--- Use this macro when you want to solve something *under* a lambda. For example:
--- say you have a long proof, and you just want the solver to deal with an
--- intermediate step. Call it like so:
+-- Use this macro when you want to solve something *under* a lambda.
+-- For example: say you have a long proof, and you just want the solver
+-- to deal with an intermediate step. Call it like so:
 --
 --   lemma₃ : ∀ x y → x + y * 1 + 3 ≈ 2 + 1 + y + x
 --   lemma₃ x y = begin
diff --git a/src/Tactic/RingSolver/Core/NatSet.agda b/src/Tactic/RingSolver/Core/NatSet.agda
index d28b441d05..e297bedbc9 100644
--- a/src/Tactic/RingSolver/Core/NatSet.agda
+++ b/src/Tactic/RingSolver/Core/NatSet.agda
@@ -3,20 +3,20 @@
 --
 -- Simple implementation of sets of ℕ.
 --
--- Since ℕ is represented as unary numbers, simply having an ordered list of
--- numbers to represent a set is quite inefficient. For instance, to see if
--- 6 is in the set {1, 3, 4}, we have to do a comparison with 1, then 3, and
--- then 4. But 4 is equal to suc 3, so we should be able to share the work
--- accross those two comparisons.
+-- Since ℕ is represented as unary numbers, simply having an ordered
+-- list of numbers to represent a set is quite inefficient. For
+-- instance, to see if 6 is in the set {1, 3, 4}, we have to do a
+-- comparison with 1, then 3, and then 4. But 4 is equal to suc 3, so
+-- we should be able to share the work accross those two comparisons.
 --
 -- This module defines a type that represents {1, 3, 4} as:
 --
 --   1 ∷ 1 ∷ 0 ∷ []
 --
--- i.e. we only store the gaps. When checking if a number (the needle) is in the
--- set (the haystack), we subtract each successive member of the haystack from the
--- needle as we go. For example, to check if 6 is in the above set, we do the
--- following:
+-- i.e. we only store the gaps. When checking if a number (the needle)
+-- is in the set (the haystack), we subtract each successive member of
+-- the haystack from the needle as we go. For example, to check if 6 is
+-- in the above set, we do the following:
 --
 --   start:                  6 ∈? (1 ∷ 1 ∷ 0 ∷ [])
 --   test head:              6 ≟ 1
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Base.agda b/src/Tactic/RingSolver/Core/Polynomial/Base.agda
index 6cfece2d45..0a027d6d3f 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Base.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Base.agda
@@ -106,9 +106,9 @@ space≤′n : ∀ {n} (x : Fin n) → space x ≤′ n
 space≤′n zero    = ≤′-refl
 space≤′n (suc x) = ≤′-step (space≤′n x)
 
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Definition
--------------------------------------------------------------------------
+------------------------------------------------------------------------
 
 infixl 6 _Δ_
 record PowInd {c} (C : Set c) : Set c where
@@ -168,7 +168,7 @@ Normalised (_ Δ suc _ & _)   = ⊤
 open NonZero public
 open Poly public
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Special operations
 
 -- Decision procedure for Zero
@@ -212,17 +212,17 @@ _⊐↓_ : ∀ {i n} → Coeff i * → suc i ≤′ n → Poly n
 (∹ (x    Δ suc j & xs  )) ⊐↓ i≤n = ⅀ (x Δ suc j & xs)   ⊐ i≤n
 {-# INLINE _⊐↓_ #-}
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Standard operations
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Folds
 
--- These folds allow us to abstract over the proofs later: we try to avoid
--- using ∷↓ and ⊐↓ directly anywhere except here, so if we prove that this fold
--- acts the same on a normalised or non-normalised polynomial, we can prove th
--- same about any operation which uses it.
+-- These folds allow us to abstract over the proofs later: we try to
+-- avoid using ∷↓ and ⊐↓ directly anywhere except here, so if we prove
+-- that this fold acts the same on a normalised or non-normalised
+-- polynomial, we can prove th same about any operation which uses it.
 
 PolyF : ℕ → Set ℓ₁
 PolyF i = Poly i × Coeff i *
@@ -238,7 +238,7 @@ poly-map : ∀ {i} → (Poly i → Poly i) → Coeff i + → Coeff i *
 poly-map f = para (map₁ f)
 {-# INLINE poly-map #-}
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Addition
 
 -- The reason the following code is so verbose is termination
@@ -255,8 +255,8 @@ poly-map f = para (map₁ f)
 -- a list that was already pattern-matched on, the recursive call
 -- does not strictly decrease the size of its argument.
 --
--- Interestingly, if --cubical-compatible is turned off, we don't need the
--- helper function ⊞-coeffs; we could pattern match on _⊞_ directly.
+-- Interestingly, if --cubical-compatible is turned off, we don't need
+-- the helper function ⊞-coeffs; we could pattern match on _⊞_ directly.
 --
 -- _⊞_ {zero} (lift x) (lift y) = lift (x + y)
 -- _⊞_ {suc n} [] ys = ys
@@ -308,7 +308,7 @@ mutual
   ⊞-zip-r x i xs [] = ∹ x Δ i & xs
   ⊞-zip-r x i xs (∹ y Δ j & ys) = ⊞-zip (compare i j) x xs y ys
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Negation
 
 -- recurse on acc directly
@@ -322,7 +322,7 @@ mutual
 ⊟_ = ⊟-step (<′-wellFounded _)
 {-# INLINE ⊟_ #-}
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Multiplication
 
 mutual
@@ -385,7 +385,7 @@ _⊠_ : ∀ {n} → Poly n → Poly n → Poly n
 _⊠_ = ⊠-step′ (<′-wellFounded _)
 {-# INLINE _⊠_ #-}
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Constants and variables
 
 -- The constant polynomial
@@ -398,7 +398,7 @@ _⊠_ = ⊠-step′ (<′-wellFounded _)
 ι i = (κ 1# Δ 1 ∷↓ []) ⊐↓ space≤′n i
 {-# INLINE ι #-}
 
-----------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Exponentiation
 
 -- We try very hard to never do things like multiply by 1
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index 90183bc4c9..4bdfd74381 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -59,8 +59,8 @@ pow-add x y (suc i) j = go x y i j
     y ^ suc (suc i ℕ.+ suc j) * x          ∎
 
 -- Here we show a homomorphism on exponentiation, i.e. that using the
--- exponentiation function on polynomials and then evaluating is the same
--- as evaluating and then exponentiating.
+-- exponentiation function on polynomials and then evaluating is the
+-- same as evaluating and then exponentiating.
 pow-hom : ∀ {n} i
         → (xs : Coeff n +)
         → ∀ ρ ρs
@@ -93,10 +93,10 @@ pow-sucʳ : ∀ x i → x ^ suc i ≈ x ^ i * x
 pow-sucʳ x zero  = sym (*-identityˡ _)
 pow-sucʳ x (suc i) = refl
 
--- In the proper evaluation function, we avoid ever inserting an unnecessary 0#
--- like we do here. However, it is easier to prove with the form that does insert
--- 0#. So we write one here, and then prove that it's equivalent to the one that
--- adds a 0#.
+-- In the proper evaluation function, we avoid ever inserting an
+-- unnecessary 0# like we do here. However, it is easier to prove with
+-- the form that does insert 0#. So we write one here, and then prove
+-- that it's equivalent to the one that adds a 0#.
 ⅀?⟦_⟧ : ∀ {n} (xs : Coeff n *) → Carrier × Vec Carrier n → Carrier
 ⅀?⟦ []  ⟧ _ = 0#
 ⅀?⟦ ∹ x ⟧   = ⅀⟦ x ⟧
@@ -117,8 +117,8 @@ pow′-hom zero [] ρ ρs = refl
 pow′-hom (suc i) [] ρ ρs = zeroʳ _
 
 -- Here, we show that the normalising cons is correct.
--- This lets us prove with respect to the non-normalising form, especially
--- when we're using the folds.
+-- This lets us prove with respect to the non-normalising form,
+-- especially when we're using the folds.
 ∷↓-hom-0 : ∀ {n} (x : Poly n) → ∀ xs ρ ρs → ⅀?⟦ x Δ 0 ∷↓ xs ⟧ (ρ , ρs) ≈ (x , xs) ⟦∷⟧ (ρ , ρs)
 ∷↓-hom-0 x xs      ρ ρs with zero? x
 ∷↓-hom-0 x xs      ρ ρs | no ¬p = refl
@@ -161,9 +161,9 @@ pow′-hom (suc i) [] ρ ρs = zeroʳ _
 ⟦∷⟧-hom x []     ρ ρs = sym ((≪+ zeroʳ _) ⟨ trans ⟩ +-identityˡ _)
 ⟦∷⟧-hom x (∹ xs) ρ ρs = refl
 
--- This proves that injecting a polynomial into more variables is correct.
--- Basically, we show that if a polynomial doesn't care about the first few
--- variables, we can drop them from the input vector.
+-- This proves that injecting a polynomial into more variables is
+-- correct. Basically, we show that if a polynomial doesn't care about
+-- the first few variables, we can drop them from the input vector.
 ⅀-⊐↑-hom : ∀ {i n m}
          → (xs : Coeff i +)
          → (si≤n : suc i ≤′ n)
@@ -222,11 +222,12 @@ drop-1⇒lookup : ∀ {n} (i : Fin n) (ρs : Vec Carrier n) →
 drop-1⇒lookup zero    (ρ ∷ ρs) = ≡.refl
 drop-1⇒lookup (suc i) (ρ ∷ ρs) = drop-1⇒lookup i ρs
 
--- The fold: this function saves us hundreds of lines of proofs in the rest of the
--- homomorphism proof.
--- Many of the functions on polynomials are defined using para: this function allows
--- us to prove properties of those functions (in a foldr-fusion style) *ignoring*
--- optimisations we have made to the polynomial structure.
+-- The fold: this function saves us hundreds of lines of proofs in the
+-- rest of the homomorphism proof.
+-- Many of the functions on polynomials are defined using para: this
+-- function allows us to prove properties of those functions (in a
+-- foldr-fusion style) *ignoring* optimisations we have made to the
+-- polynomial structure.
 poly-foldR : ∀ {n} ρ ρs
         → ([f] : Fold n)
         → (f : Carrier → Carrier)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
index 26e19d428a..3d6e8c512f 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
@@ -36,13 +36,14 @@ x *⟨ ρ ⟩^ zero = x
 x *⟨ ρ ⟩^ suc i = ρ ^ (suc i) * x
 {-# INLINE _*⟨_⟩^_ #-}
 
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Evaluation
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Why do we have three functions here? Why are they so weird looking?
 --
--- These three functions are the main bottleneck for all of the proofs: as such,
--- slight changes can dramatically affect the length of proof code.
+-- These three functions are the main bottleneck for all of the proofs:
+-- as such, slight changes can dramatically affect the length of proof
+-- code.
 
 mutual
 
@@ -59,13 +60,13 @@ mutual
   ⟦ ⅀ xs ⊐ i≤n ⟧ Ρ = ⅀⟦ xs ⟧ (drop-1 i≤n Ρ)
   {-# INLINE ⟦_⟧ #-}
 
---------------------------------------------------------------------------------
+------------------------------------------------------------------------
 -- Performance
---------------------------------------------------------------------------------
--- As you might imagine, the implementation of the functions above seriously
--- affect performance. What you might not realise, though, is that the most
--- important component is the *order of the arguments*. For instance, if
--- we change:
+------------------------------------------------------------------------
+-- As you might imagine, the implementation of the functions above
+-- seriously affect performance. What you might not realise, though,
+-- is that the most important component is the *order of the arguments*.
+-- For instance, if we change:
 --
 --   (x , xs) ⟦∷⟧ (ρ , ρs) = ρ * ⅀⟦ xs ⟧ (ρ , ρs) + ⟦ x ⟧ ρs
 --
diff --git a/src/Test/Golden.agda b/src/Test/Golden.agda
index 079e25720a..4b032bd783 100644
--- a/src/Test/Golden.agda
+++ b/src/Test/Golden.agda
@@ -56,7 +56,7 @@
 --
 -- We provide an options parser (`options`) that takes the list of command line
 -- arguments and constructs this for you.
----
+--
 
 ------------------------------------------------------------------------
 -- Usage
diff --git a/src/Text/Format/Generic.agda b/src/Text/Format/Generic.agda
index d0fe551041..8a304f9c9a 100644
--- a/src/Text/Format/Generic.agda
+++ b/src/Text/Format/Generic.agda
@@ -49,7 +49,7 @@ module Format (spec : FormatSpec) where
 
   open FormatSpec spec
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- Basic types
 
   data Chunk : Set where
@@ -59,7 +59,7 @@ module Format (spec : FormatSpec) where
   Format : Set
   Format = List Chunk
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- Semantics
 
   size : Format → ℕ
@@ -72,11 +72,11 @@ module Format (spec : FormatSpec) where
   ⟦ Arg a  ∷ cs ⟧ = ArgType a , ⟦ cs ⟧
   ⟦ Raw _  ∷ cs ⟧ =             ⟦ cs ⟧
 
-  ------------------------------------------------------------------------
+  ----------------------------------------------------------------------
   -- Lexer: from Strings to Formats
 
-  -- Lexing may fail. To have a useful error message, we defined the following
-  -- enumerated type
+  -- Lexing may fail. To have a useful error message, we defined the
+  -- following enumerated type
 
   data Error : Set where
     UnexpectedEndOfString : String → Error
diff --git a/src/Text/Pretty/Core.agda b/src/Text/Pretty/Core.agda
index a97db1dda2..974b1af823 100644
--- a/src/Text/Pretty/Core.agda
+++ b/src/Text/Pretty/Core.agda
@@ -43,10 +43,11 @@ import Data.Refinement.Relation.Unary.All as Allᴿ
 ------------------------------------------------------------------------
 -- Block of text
 
--- Content is a representation of the first line and the middle of the block.
--- We use a tree rather than a list for the middle of the block so that we can
--- extend it with lines on the left and on the line for free. We will ultimately
--- render the block by traversing the tree left to right in a depth-first manner.
+-- Content is a representation of the first line and the middle of the
+-- block. We use a tree rather than a list for the middle of the block
+-- so that we can extend it with lines on the left and on the line for
+-- free. We will ultimately render the block by traversing the tree left
+-- to right in a depth-first manner.
 
 Content : Set
 Content = Maybe (String × Tree String ⊤)
diff --git a/src/Text/Regex/Search.agda b/src/Text/Regex/Search.agda
index b96fb3847c..efa1f74684 100644
--- a/src/Text/Regex/Search.agda
+++ b/src/Text/Regex/Search.agda
@@ -41,9 +41,9 @@ open import Text.Regex.Derivative.Brzozowski P?
 ------------------------------------------------------------------------
 -- Type corresponding to a match
 
--- Users have control over whether the match should start at the beginning
--- or stop at the end. So we have a precise type of spans ensuring their
--- demands are respected
+-- Users have control over whether the match should start at the
+-- beginning or stop at the end. So we have a precise type of spans
+-- ensuring their demands are respected
 Span : ∀ {r} → Regex → Rel A r → Rel (List A) (a ⊔ r)
 Span regex =
   if Regex.fromStart regex
@@ -144,8 +144,8 @@ module Infix where
   step⁻¹ {e} {acc} x []∉e []∉acc (inj₂ (mkMatch (.x ∷ ys) ys∈e (refl ∷ p)))
     = inj₂ (mkMatch ys (eat-complete x (acc ∣ e) (sum (inj₁ ys∈e))) p)
 
-  -- search non-deterministically: at each step, the `acc` regex is changed
-  -- to accomodate the fact the match may be starting just now
+  -- search non-deterministically: at each step, the `acc` regex is
+  -- changed to accomodate the fact the match may be starting just now
   searchND : ∀ xs e acc → [] ∉ e → Dec (Match (Infix _≡_) xs e ⊎ Match (Prefix _≡_) xs acc)
   searchND xs e acc []∉e with []∈? acc
   ... | yes []∈acc with Prefix.longest xs acc -- get the best match possible

From c4285333b6ec03dcc90727e7c64d5adec2ad099e Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 29 Jul 2023 13:38:56 +0200
Subject: [PATCH 081/316] Add new pattern synonym `divides-refl` to
 `Data.Nat.Divisibility.Core`

---
 CHANGELOG.md                        |  7 ++-
 src/Data/Nat/Coprimality.agda       | 10 ++---
 src/Data/Nat/DivMod.agda            | 22 +++++-----
 src/Data/Nat/Divisibility.agda      | 67 ++++++++++++++++-------------
 src/Data/Nat/Divisibility/Core.agda |  3 ++
 src/Data/Nat/GCD.agda               | 28 ++++++------
 src/Data/Nat/Primality.agda         |  2 +-
 7 files changed, 74 insertions(+), 65 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1da4157a62..8e5e356e52 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -908,7 +908,7 @@ Major improvements
   for them still live in `Data.Nat.DivMod` (which also publicly re-exports them
   to provide backwards compatability).
 
-* Beneficieries of this change include `Data.Rational.Unnormalised.Base` whose
+* Beneficiaries of this change include `Data.Rational.Unnormalised.Base` whose
   dependencies are now significantly smaller.
 
 ### Moved raw bundles from Data.X.Properties to Data.X.Base
@@ -2207,6 +2207,11 @@ Other minor changes
   suc-injective : Injective _≡_ _≡_ suc
   ```
 
+* Added a new pattern synonym to `Data.Nat.Divisibility.Core`:
+  ```agda
+  pattern divides-refl q = divides q refl
+  ```
+
 * Added new definitions and proofs to `Data.Nat.Primality`:
   ```agda
   Composite        : ℕ → Set
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 04283143fd..2836b4d1da 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -9,8 +9,6 @@
 module Data.Nat.Coprimality where
 
 open import Data.Empty
-open import Data.Fin.Base using (toℕ; fromℕ<)
-open import Data.Fin.Properties using (toℕ-fromℕ<)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.GCD
@@ -45,8 +43,8 @@ coprime⇒GCD≡1 : ∀ {m n} → Coprime m n → GCD m n 1
 coprime⇒GCD≡1 {m} {n} c = GCD.is (1∣ m , 1∣ n) (∣-reflexive ∘ c)
 
 GCD≡1⇒coprime : ∀ {m n} → GCD m n 1 → Coprime m n
-GCD≡1⇒coprime g cd with GCD.greatest g cd
-... | divides q eq = m*n≡1⇒n≡1 q _ (P.sym eq)
+GCD≡1⇒coprime g cd with divides q eq ← GCD.greatest g cd
+  = m*n≡1⇒n≡1 q _ (P.sym eq)
 
 coprime⇒gcd≡1 : ∀ {m n} → Coprime m n → gcd m n ≡ 1
 coprime⇒gcd≡1 coprime = GCD.unique (gcd-GCD _ _) (coprime⇒GCD≡1 coprime)
@@ -111,9 +109,9 @@ recompute {n} {d} c = Nullary.recompute (coprime? n d) c
 
 Bézout-coprime : ∀ {i j d} .{{_ : NonZero d}} →
                  Bézout.Identity d (i * d) (j * d) → Coprime i j
-Bézout-coprime {d = suc _} (Bézout.+- x y eq) (divides q₁ refl , divides q₂ refl) =
+Bézout-coprime {d = suc _} (Bézout.+- x y eq) (divides-refl q₁ , divides-refl q₂) =
   lem₁₀ y q₂ x q₁ eq
-Bézout-coprime {d = suc _} (Bézout.-+ x y eq) (divides q₁ refl , divides q₂ refl) =
+Bézout-coprime {d = suc _} (Bézout.-+ x y eq) (divides-refl q₁ , divides-refl q₂) =
   lem₁₀ x q₁ y q₂ eq
 
 -- Coprime numbers satisfy Bézout's identity.
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index b4c225d6a4..c879cec098 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -22,8 +22,6 @@ open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (False; toWitnessFalse)
 
-import Algebra.Properties.CommutativeSemigroup *-commutativeSemigroup as *-CS
-
 open ≤-Reasoning
 
 ------------------------------------------------------------------------
@@ -91,7 +89,8 @@ m*n≤o⇒[o∸m*n]%n≡o%n m {n} {o} m*n≤o = begin-equality
 
 m∣n⇒o%n%m≡o%m : ∀ m n o .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ∣ n →
                 o % n % m ≡ o % m
-m∣n⇒o%n%m≡o%m m n o (divides p refl) = begin-equality
+m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
+  o % n % m                ≡⟨⟩
   o % pm % m               ≡⟨ %-congˡ (m%n≡m∸m/n*n o pm) ⟩
   (o ∸ o / pm * pm) % m    ≡˘⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟩
   (o ∸ o / pm * p * m) % m ≡⟨ m*n≤o⇒[o∸m*n]%n≡o%n (o / pm * p) lem ⟩
@@ -171,7 +170,8 @@ m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-
     m′ * n′ + (m′ * j + (n′ + j * d) * k) * d         ∎
 
 %-remove-+ˡ : ∀ {m} n {d} .{{_ : NonZero d}} → d ∣ m → (m + n) % d ≡ n % d
-%-remove-+ˡ {m} n {d@(suc d-1)} (divides p refl) = begin-equality
+%-remove-+ˡ {m@.(p * d)} n {d@(suc _)} (divides-refl p) = begin-equality
+  (m + n)     % d ≡⟨⟩
   (p * d + n) % d ≡⟨ cong (_% d) (+-comm (p * d) n) ⟩
   (n + p * d) % d ≡⟨ [m+kn]%n≡m%n n p d ⟩
   n           % d ∎
@@ -191,7 +191,7 @@ m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-
 /-congʳ refl = refl
 
 0/n≡0 : ∀ n .{{_ : NonZero n}} → 0 / n ≡ 0
-0/n≡0 (suc n-1) = refl
+0/n≡0 (suc _) = refl
 
 n/1≡n : ∀ n → n / 1 ≡ n
 n/1≡n n = a[divₕ]1≡a 0 n
@@ -203,7 +203,7 @@ m*n/n≡m : ∀ m n .{{_ : NonZero n}} → m * n / n ≡ m
 m*n/n≡m m (suc n-1) = a*n[divₕ]n≡a 0 m n-1
 
 m/n*n≡m : ∀ {m n} .{{_ : NonZero n}} → n ∣ m → m / n * n ≡ m
-m/n*n≡m {_} {n@(suc n-1)} (divides q refl) = cong (_* n) (m*n/n≡m q n)
+m/n*n≡m {_} {n@(suc _)} (divides-refl q) = cong (_* n) (m*n/n≡m q n)
 
 m*[n/m]≡n : ∀ {m n} .{{_ : NonZero m}} → m ∣ n → m * (n / m) ≡ n
 m*[n/m]≡n {m} m∣n = trans (*-comm m (_ / m)) (m/n*n≡m m∣n)
@@ -261,14 +261,16 @@ m≥n⇒m/n>0 {m@(suc _)} {n@(suc _)} m≥n = begin
 
 +-distrib-/-∣ˡ : ∀ {m} n {d} .{{_ : NonZero d}} →
                  d ∣ m → (m + n) / d ≡ m / d + n / d
-+-distrib-/-∣ˡ {m} n {d} (divides p refl) = +-distrib-/ m n (begin-strict
++-distrib-/-∣ˡ {m@.(p * d)} n {d} (divides-refl p) = +-distrib-/ m n (begin-strict
+  m % d + n % d     ≡⟨⟩
   p * d % d + n % d ≡⟨ cong (_+ n % d) (m*n%n≡0 p d) ⟩
   n % d             <⟨ m%n<n n d ⟩
   d                 ∎)
 
 +-distrib-/-∣ʳ : ∀ m {n} {d} .{{_ : NonZero d}} →
                  d ∣ n → (m + n) / d ≡ m / d + n / d
-+-distrib-/-∣ʳ m {n} {d} (divides p refl) = +-distrib-/ m n (begin-strict
++-distrib-/-∣ʳ m {n@.(p * d)} {d} (divides-refl p) = +-distrib-/ m n (begin-strict
+  m % d + n % d     ≡⟨⟩
   m % d + p * d % d ≡⟨ cong (m % d +_) (m*n%n≡0 p d) ⟩
   m % d + 0         ≡⟨ +-identityʳ _ ⟩
   m % d             <⟨ m%n<n m d ⟩
@@ -284,7 +286,7 @@ m/n≡1+[m∸n]/n {m@(suc m-1)} {n@(suc n-1)} m≥n = begin-equality
 
 m*n/m*o≡n/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (m * o)}} →
               (m * n) / (m * o) ≡ n / o
-m*n/m*o≡n/o m@(suc m-1) n o = helper (<-wellFounded n)
+m*n/m*o≡n/o m@(suc _) n o = helper (<-wellFounded n)
   where
   helper : ∀ {n} → Acc _<_ n → (m * n) / (m * o) ≡ n / o
   helper {n} (acc rec) with n <? o
@@ -354,7 +356,7 @@ m/n/o≡m/[n*o] m n o = begin-equality
   m / n / o                             ≡⟨ /-congˡ {o = o} (/-congˡ (m≡m%n+[m/n]*n m n*o)) ⟩
   (m % n*o + m / n*o * n*o) / n / o     ≡⟨ /-congˡ (+-distrib-/-∣ʳ (m % n*o) lem₁) ⟩
   (m % n*o / n + m / n*o * n*o / n) / o ≡⟨ cong (λ # → (m % n*o / n + #) / o) lem₂ ⟩
-  (m % n*o / n + m / n*o * o) / o       ≡⟨ +-distrib-/-∣ʳ (m % n*o / n) (divides (m / n*o) refl) ⟩
+  (m % n*o / n + m / n*o * o) / o       ≡⟨ +-distrib-/-∣ʳ (m % n*o / n) (divides-refl (m / n*o)) ⟩
   m % n*o / n / o + m / n*o * o / o     ≡⟨ cong (m % n*o / n / o +_) (m*n/n≡m (m / n*o) o) ⟩
   m % n*o / n / o + m / n*o             ≡⟨ cong (_+ m / n*o) (m<n⇒m/n≡0 (m<n*o⇒m/o<n {n = o} lem₃)) ⟩
   m / n*o                               ∎
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index 26cd81bf27..1b1321906d 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -40,11 +40,7 @@ m%n≡0⇒n∣m m n eq = divides (m / n) (begin-equality
   where open ≤-Reasoning
 
 n∣m⇒m%n≡0 : ∀ m n .{{_ : NonZero n}} → n ∣ m → m % n ≡ 0
-n∣m⇒m%n≡0 m n (divides v eq) = begin-equality
-  m       % n  ≡⟨ cong (_% n) eq ⟩
-  (v * n) % n  ≡⟨ m*n%n≡0 v n ⟩
-  0            ∎
-  where open ≤-Reasoning
+n∣m⇒m%n≡0 .(q * n) n (divides-refl q) = m*n%n≡0 q n
 
 m%n≡0⇔n∣m : ∀ m n .{{_ : NonZero n}} → m % n ≡ 0 ⇔ n ∣ m
 m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)
@@ -65,6 +61,8 @@ m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)
 ------------------------------------------------------------------------
 -- _∣_ is a partial order
 
+-- these could/should inherit from Algebra.Properties.Monoid.Divisibility
+
 ∣-reflexive : _≡_ ⇒ _∣_
 ∣-reflexive {n} refl = divides 1 (sym (*-identityˡ n))
 
@@ -72,19 +70,19 @@ m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)
 ∣-refl = ∣-reflexive refl
 
 ∣-trans : Transitive _∣_
-∣-trans (divides p refl) (divides q refl) =
+∣-trans (divides-refl p) (divides-refl q) =
   divides (q * p) (sym (*-assoc q p _))
 
 ∣-antisym : Antisymmetric _≡_ _∣_
-∣-antisym {m}     {zero}  _ (divides q refl) = *-zeroʳ q
+∣-antisym {m}     {zero}  _ (divides-refl q) = *-zeroʳ q
 ∣-antisym {zero}  {n}     (divides p eq) _   = sym (trans eq (*-comm p 0))
 ∣-antisym {suc m} {suc n} p∣q           q∣p  = ≤-antisym (∣⇒≤ p∣q) (∣⇒≤ q∣p)
 
 infix 4 _∣?_
 
 _∣?_ : Decidable _∣_
-zero  ∣? zero   = yes (divides 0 refl)
-zero  ∣? suc m  = no ((λ()) ∘′ ∣-antisym (divides 0 refl))
+zero  ∣? zero   = yes (divides-refl 0)
+zero  ∣? suc m  = no ((λ()) ∘′ ∣-antisym (divides-refl 0))
 suc n ∣? m      = Dec.map (m%n≡0⇔n∣m m (suc n)) (m % suc n ≟ 0)
 
 ∣-isPreorder : IsPreorder _≡_ _∣_
@@ -133,7 +131,7 @@ infix 10 1∣_ _∣0
 1∣ n = divides n (sym (*-identityʳ n))
 
 _∣0 : ∀ n → n ∣ 0
-n ∣0 = divides 0 refl
+n ∣0 = divides-refl 0
 
 0∣⇒≡0 : ∀ {n} → 0 ∣ n → n ≡ 0
 0∣⇒≡0 {n} 0∣n = ∣-antisym (n ∣0) 0∣n
@@ -148,7 +146,7 @@ n∣n {n} = ∣-refl
 -- Properties of _∣_ and _+_
 
 ∣m∣n⇒∣m+n : ∀ {i m n} → i ∣ m → i ∣ n → i ∣ m + n
-∣m∣n⇒∣m+n (divides p refl) (divides q refl) =
+∣m∣n⇒∣m+n (divides-refl p) (divides-refl q) =
   divides (p + q) (sym (*-distribʳ-+ _ p q))
 
 ∣m+n∣m⇒∣n : ∀ {i m n} → i ∣ m + n → i ∣ m → i ∣ n
@@ -174,19 +172,20 @@ n∣m*n*o : ∀ m {n} o → n ∣ m * n * o
 n∣m*n*o m o = ∣-trans (n∣m*n m) (m∣m*n o)
 
 ∣m⇒∣m*n : ∀ {i m} n → i ∣ m → i ∣ m * n
-∣m⇒∣m*n {i} {m} n (divides q refl) = ∣-trans (n∣m*n q) (m∣m*n n)
+∣m⇒∣m*n {i} {m} n (divides-refl q) = ∣-trans (n∣m*n q) (m∣m*n n)
 
 ∣n⇒∣m*n : ∀ {i} m {n} → i ∣ n → i ∣ m * n
 ∣n⇒∣m*n m {n} rewrite *-comm m n = ∣m⇒∣m*n m
 
 m*n∣⇒m∣ : ∀ {i} m n → m * n ∣ i → m ∣ i
-m*n∣⇒m∣ m n (divides q refl) = ∣n⇒∣m*n q (m∣m*n n)
+m*n∣⇒m∣ m n (divides-refl q) = ∣n⇒∣m*n q (m∣m*n n)
 
 m*n∣⇒n∣ : ∀ {i} m n → m * n ∣ i → n ∣ i
 m*n∣⇒n∣ m n rewrite *-comm m n = m*n∣⇒m∣ n m
 
 *-monoʳ-∣ : ∀ {i j} k → i ∣ j → k * i ∣ k * j
-*-monoʳ-∣ {i} {j} k (divides q refl) = divides q $ begin-equality
+*-monoʳ-∣ {i} {j@.(q * i)} k (divides-refl q) = divides q $ begin-equality
+  k * j        ≡⟨⟩
   k * (q * i)  ≡⟨ sym (*-assoc k q i) ⟩
   (k * q) * i  ≡⟨ cong (_* i) (*-comm k q) ⟩
   (q * k) * i  ≡⟨ *-assoc q k i ⟩
@@ -226,27 +225,30 @@ m*n∣⇒n∣ m n rewrite *-comm m n = m*n∣⇒m∣ n m
 -- Properties of _∣_ and _/_
 
 m/n∣m : ∀ {m n} .{{_ : NonZero n}} → n ∣ m → m / n ∣ m
-m/n∣m {m} {n} (divides p refl) = begin
+m/n∣m {m@.(p * n)} {n} (divides-refl p) = begin
+  m / n     ≡⟨⟩
   p * n / n ≡⟨ m*n/n≡m p n ⟩
   p         ∣⟨ m∣m*n n ⟩
-  p * n     ∎
+  p * n     ≡⟨⟩
+  m         ∎
   where open ∣-Reasoning
 
 m*n∣o⇒m∣o/n : ∀ m n {o} .{{_ : NonZero n}} → m * n ∣ o → m ∣ o / n
-m*n∣o⇒m∣o/n m n {_} (divides p refl) = begin
+m*n∣o⇒m∣o/n m n {o@.(p * (m * n))} (divides-refl p) = begin
   m               ∣⟨ n∣m*n p ⟩
   p * m           ≡⟨ sym (*-identityʳ (p * m)) ⟩
   p * m * 1       ≡⟨ sym (cong (p * m *_) (n/n≡1 n)) ⟩
   p * m * (n / n) ≡⟨ sym (*-/-assoc (p * m) (n∣n {n})) ⟩
   p * m * n / n   ≡⟨ cong (_/ n) (*-assoc p m n) ⟩
-  p * (m * n) / n ∎
+  p * (m * n) / n ≡⟨⟩
+  o / n           ∎
   where open ∣-Reasoning
 
 m*n∣o⇒n∣o/m : ∀ m n {o} .{{_ : NonZero m}} → m * n ∣ o → n ∣ (o / m)
 m*n∣o⇒n∣o/m m n rewrite *-comm m n = m*n∣o⇒m∣o/n n m
 
 m∣n/o⇒m*o∣n : ∀ {m n o} .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → m * o ∣ n
-m∣n/o⇒m*o∣n {m} {n} {o} (divides p refl) m∣p*o/o = begin
+m∣n/o⇒m*o∣n {m} {n} {o} (divides-refl p) m∣p*o/o = begin
   m * o ∣⟨ *-monoˡ-∣ o (subst (m ∣_) (m*n/n≡m p o) m∣p*o/o) ⟩
   p * o ∎
   where open ∣-Reasoning
@@ -255,13 +257,14 @@ m∣n/o⇒o*m∣n : ∀ {m n o} .{{_ : NonZero o}} → o ∣ n → m ∣ n / o 
 m∣n/o⇒o*m∣n {m} {_} {o} rewrite *-comm o m = m∣n/o⇒m*o∣n
 
 m/n∣o⇒m∣o*n : ∀ {m n o} .{{_ : NonZero n}} → n ∣ m → m / n ∣ o → m ∣ o * n
-m/n∣o⇒m∣o*n {_} {n} {o} (divides p refl) p*n/n∣o = begin
+m/n∣o⇒m∣o*n {_} {n} {o} (divides-refl p) p*n/n∣o = begin
   p * n ∣⟨ *-monoˡ-∣ n (subst (_∣ o) (m*n/n≡m p n) p*n/n∣o) ⟩
   o * n ∎
   where open ∣-Reasoning
 
 m∣n*o⇒m/n∣o : ∀ {m n o} .{{_ : NonZero n}} → n ∣ m → m ∣ o * n → m / n ∣ o
-m∣n*o⇒m/n∣o {_} {n@(suc _)} {o} (divides p refl) pn∣on = begin
+m∣n*o⇒m/n∣o {m@.(p * n)} {n@(suc _)} {o} (divides-refl p) pn∣on = begin
+  m / n     ≡⟨⟩
   p * n / n ≡⟨ m*n/n≡m p n ⟩
   p         ∣⟨ *-cancelʳ-∣ n pn∣on ⟩
   o         ∎
@@ -271,24 +274,26 @@ m∣n*o⇒m/n∣o {_} {n@(suc _)} {o} (divides p refl) pn∣on = begin
 -- Properties of _∣_ and _%_
 
 ∣n∣m%n⇒∣m : ∀ {m n d} .{{_ : NonZero n}} → d ∣ n → d ∣ m % n → d ∣ m
-∣n∣m%n⇒∣m {m} {n} {d} (divides a n≡ad) (divides b m%n≡bd) =
+∣n∣m%n⇒∣m {m} {n@.(a * d)} {d} (divides-refl a) (divides b m%n≡bd) =
   divides (b + (m / n) * a) (begin-equality
     m                         ≡⟨ m≡m%n+[m/n]*n m n ⟩
-    m % n + (m / n) * n       ≡⟨ cong₂ _+_ m%n≡bd (cong (m / n *_) n≡ad) ⟩
+    m % n + (m / n) * n       ≡⟨ cong (_+ (m / n) * n) m%n≡bd ⟩
+    b * d + (m / n) * n       ≡⟨⟩
     b * d + (m / n) * (a * d) ≡⟨ sym (cong (b * d +_) (*-assoc (m / n) a d)) ⟩
     b * d + ((m / n) * a) * d ≡⟨ sym (*-distribʳ-+ d b _) ⟩
     (b + (m / n) * a) * d     ∎)
     where open ≤-Reasoning
 
 %-presˡ-∣ : ∀ {m n d} .{{_ : NonZero n}} → d ∣ m → d ∣ n → d ∣ m % n
-%-presˡ-∣ {m} {n} {d} (divides a refl) (divides b 1+n≡bd) =
-  divides (a ∸ ad/n * b) $ begin-equality
-    a * d % n              ≡⟨  m%n≡m∸m/n*n (a * d) n ⟩
-    a * d ∸ ad/n * n       ≡⟨  cong (λ v → a * d ∸ ad/n * v) 1+n≡bd ⟩
-    a * d ∸ ad/n * (b * d) ≡˘⟨ cong (a * d ∸_) (*-assoc ad/n b d) ⟩
-    a * d ∸ (ad/n * b) * d ≡˘⟨ *-distribʳ-∸ d a (ad/n * b) ⟩
-    (a ∸ ad/n * b) * d     ∎
-  where open ≤-Reasoning; ad/n = a * d / n
+%-presˡ-∣ {m@.(a * d)} {n} {d} (divides-refl a) (divides b 1+n≡bd) =
+  divides (a ∸ m / n * b) $ begin-equality
+    m % n                   ≡⟨  m%n≡m∸m/n*n m n ⟩
+    m ∸ m / n * n           ≡⟨  cong (λ v → m ∸ m / n * v) 1+n≡bd ⟩
+    m ∸ m / n * (b * d)     ≡˘⟨ cong (m ∸_) (*-assoc (m / n) b d) ⟩
+    m  ∸ (m / n * b) * d    ≡⟨⟩
+    a * d ∸ (m / n * b) * d ≡˘⟨ *-distribʳ-∸ d a (m / n * b) ⟩
+    (a ∸ m / n * b) * d     ∎
+  where open ≤-Reasoning
 
 ------------------------------------------------------------------------
 -- Properties of _∣_ and !_
diff --git a/src/Data/Nat/Divisibility/Core.agda b/src/Data/Nat/Divisibility/Core.agda
index 635a70c111..3b06bc2c96 100644
--- a/src/Data/Nat/Divisibility/Core.agda
+++ b/src/Data/Nat/Divisibility/Core.agda
@@ -40,6 +40,9 @@ open _∣_ using (quotient) public
 _∤_ : Rel ℕ 0ℓ
 m ∤ n = ¬ (m ∣ n)
 
+-- smart constructor
+
+pattern divides-refl q = divides q refl
 
 ------------------------------------------------------------------------
 -- Basic properties
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index c843ae215b..e702c512a6 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -56,8 +56,8 @@ gcd m n with <-cmp m n
 gcd′[m,n]∣m,n : ∀ {m n} rec n<m → gcd′ m n rec n<m ∣ m × gcd′ m n rec n<m ∣ n
 gcd′[m,n]∣m,n {m} {zero}  rec       n<m = ∣-refl , m ∣0
 gcd′[m,n]∣m,n {m} {suc n} (acc rec) n<m
-  with gcd′[m,n]∣m,n (rec _ n<m) (m%n<n m (suc n))
-... | gcd∣n , gcd∣m%n = ∣n∣m%n⇒∣m gcd∣n gcd∣m%n , gcd∣n
+  with gcd∣n , gcd∣m%n ← gcd′[m,n]∣m,n (rec _ n<m) (m%n<n m (suc n))
+  = ∣n∣m%n⇒∣m gcd∣n gcd∣m%n , gcd∣n
 
 gcd′-greatest : ∀ {m n c} rec n<m → c ∣ m → c ∣ n → c ∣ gcd′ m n rec n<m
 gcd′-greatest {m} {zero}  rec       n<m c∣m c∣n = c∣m
@@ -166,10 +166,8 @@ gcd-universality : ∀ {m n g} →
                    (∀ {d} → d ∣ m × d ∣ n → d ∣ g) →
                    (∀ {d} → d ∣ g → d ∣ m × d ∣ n) →
                    g ≡ gcd m n
-gcd-universality {m} {n} forwards backwards with backwards ∣-refl
-... | back₁ , back₂ = ∣-antisym
-  (gcd-greatest back₁ back₂)
-  (forwards (gcd[m,n]∣m m n , gcd[m,n]∣n m n))
+gcd-universality {m} {n} forwards backwards with back₁ , back₂ ← backwards ∣-refl
+  = ∣-antisym (gcd-greatest back₁ back₂) (forwards (gcd[m,n]∣m m n , gcd[m,n]∣n m n))
 
 -- This could be simplified with some nice backwards/forwards reasoning
 -- after the new function hierarchy is up and running.
@@ -180,10 +178,10 @@ gcd[cm,cn]/c≡gcd[m,n] c m n = gcd-universality forwards backwards
   forwards {d} (d∣m , d∣n) = m*n∣o⇒n∣o/m c d (gcd-greatest (*-monoʳ-∣ c d∣m) (*-monoʳ-∣ c d∣n))
 
   backwards : ∀ {d : ℕ} → d ∣ gcd (c * m) (c * n) / c → d ∣ m × d ∣ n
-  backwards {d} d∣gcd[cm,cn]/c with m∣n/o⇒o*m∣n (gcd-greatest (m∣m*n m) (m∣m*n n)) d∣gcd[cm,cn]/c
-  ... | cd∣gcd[cm,n] =
-    *-cancelˡ-∣ c (∣-trans cd∣gcd[cm,n] (gcd[m,n]∣m (c * m) _)) ,
-    *-cancelˡ-∣ c (∣-trans cd∣gcd[cm,n] (gcd[m,n]∣n (c * m) _))
+  backwards {d} d∣gcd[cm,cn]/c
+    with cd∣gcd[cm,n] ← m∣n/o⇒o*m∣n (gcd-greatest (m∣m*n m) (m∣m*n n)) d∣gcd[cm,cn]/c
+    = *-cancelˡ-∣ c (∣-trans cd∣gcd[cm,n] (gcd[m,n]∣m (c * m) _)) ,
+      *-cancelˡ-∣ c (∣-trans cd∣gcd[cm,n] (gcd[m,n]∣n (c * m) _))
 
 c*gcd[m,n]≡gcd[cm,cn] : ∀ c m n → c * gcd m n ≡ gcd (c * m) (c * n)
 c*gcd[m,n]≡gcd[cm,cn] zero      m n = P.sym gcd[0,0]≡0
@@ -256,8 +254,8 @@ module GCD where
   -- n + k.
 
   step : ∀ {n k d} → GCD n k d → GCD n (n + k) d
-  step g with GCD.commonDivisor g
-  step {n} {k} {d} g | (d₁ , d₂) = is (d₁ , ∣m∣n⇒∣m+n d₁ d₂) greatest′
+  step {n} {k} {d} g with d₁ , d₂ ← GCD.commonDivisor g
+    = is (d₁ , ∣m∣n⇒∣m+n d₁ d₂) greatest′
     where
     greatest′ : ∀ {d′} → d′ ∣ n × d′ ∣ n + k → d′ ∣ d
     greatest′ (d₁ , d₂) = GCD.greatest g (d₁ , ∣m+n∣m⇒∣n d₂ d₁)
@@ -292,7 +290,7 @@ GCD-* {c = suc _} (GCD.is (dc∣nc , dc∣mc) dc-greatest) =
 GCD-/ : ∀ {m n d c} .{{_ : NonZero c}} → c ∣ m → c ∣ n → c ∣ d →
         GCD m n d → GCD (m / c) (n / c) (d / c)
 GCD-/ {m} {n} {d} {c} {{x}}
-  (divides p P.refl) (divides q P.refl) (divides r P.refl) gcd
+  (divides-refl p) (divides-refl q) (divides-refl r) gcd
   rewrite m*n/n≡m p c {{x}} | m*n/n≡m q c {{x}} | m*n/n≡m r c {{x}} = GCD-* gcd
 
 GCD-/gcd : ∀ m n .{{_ : NonZero (gcd m n)}} → GCD (m / gcd m n) (n / gcd m n) 1
@@ -400,6 +398,4 @@ module Bézout where
   -- Bézout's identity can be recovered from the GCD.
 
   identity : ∀ {m n d} → GCD m n d → Identity d m n
-  identity {m} {n} g with lemma m n
-  ... | result d g′ b with GCD.unique g g′
-  ...   | P.refl = b
+  identity {m} {n} g with result d g′ b ← lemma m n rewrite GCD.unique g g′ = b
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index dda1122c3c..6ed6997c0b 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -21,7 +21,7 @@ open import Relation.Nullary.Decidable as Dec
 open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Unary using (Decidable)
 open import Relation.Binary.PropositionalEquality
-  using (refl; sym; cong; subst)
+  using (refl; cong)
 
 private
   variable

From 996795d9c8a41f00c676bbd6ee6ab9c8db473093 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sat, 29 Jul 2023 07:47:21 -0400
Subject: [PATCH 082/316] Simplify more `Relation.Binary` imports (#2034)

---
 src/Algebra/Properties/Monoid/Sum.agda                      | 2 +-
 src/Data/AVL/Map.agda                                       | 2 +-
 src/Data/AVL/NonEmpty.agda                                  | 2 +-
 src/Data/AVL/Sets.agda                                      | 2 +-
 src/Data/AVL/Value.agda                                     | 2 +-
 src/Data/Container/Morphism/Properties.agda                 | 2 +-
 src/Data/Container/Relation/Binary/Equality/Setoid.agda     | 2 +-
 src/Data/Container/Relation/Binary/Pointwise.agda           | 2 +-
 src/Data/Container/Relation/Unary/Any/Properties.agda       | 2 +-
 src/Data/Fin/Permutation.agda                               | 2 +-
 src/Data/Integer/Coprimality.agda                           | 3 ++-
 src/Data/List/Extrema/Core.agda                             | 3 ++-
 src/Data/List/Fresh/Membership/Setoid/Properties.agda       | 3 ++-
 src/Data/List/Fresh/Relation/Unary/All.agda                 | 2 +-
 src/Data/List/Fresh/Relation/Unary/All/Properties.agda      | 2 +-
 src/Data/List/Fresh/Relation/Unary/Any.agda                 | 2 +-
 src/Data/List/Fresh/Relation/Unary/Any/Properties.agda      | 2 +-
 src/Data/List/Membership/Propositional/Properties.agda      | 3 ++-
 .../List/Relation/Binary/Permutation/Setoid/Properties.agda | 5 ++++-
 src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda     | 2 +-
 .../Relation/Binary/Sublist/DecPropositional/Solver.agda    | 2 +-
 src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda | 2 +-
 .../List/Relation/Binary/Sublist/Heterogeneous/Core.agda    | 2 +-
 .../List/Relation/Binary/Sublist/Heterogeneous/Solver.agda  | 3 ++-
 src/Data/List/Relation/Binary/Sublist/Setoid.agda           | 3 ++-
 .../Relation/Binary/Subset/Propositional/Properties.agda    | 6 ++++--
 src/Data/List/Relation/Ternary/Appending.agda               | 2 +-
 src/Data/List/Relation/Ternary/Appending/Properties.agda    | 3 ++-
 src/Data/List/Relation/Ternary/Appending/Setoid.agda        | 2 +-
 .../List/Relation/Ternary/Appending/Setoid/Properties.agda  | 2 +-
 src/Data/List/Relation/Ternary/Interleaving/Setoid.agda     | 2 +-
 .../Relation/Ternary/Interleaving/Setoid/Properties.agda    | 2 +-
 src/Data/Tree/AVL.agda                                      | 2 +-
 src/Data/Trie/NonEmpty.agda                                 | 2 +-
 src/Data/Vec/Membership/DecPropositional.agda               | 2 +-
 src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda        | 2 +-
 src/Data/Vec/Relation/Unary/All/Properties.agda             | 2 +-
 src/Relation/Binary/Indexed/Homogeneous/Definitions.agda    | 5 +++--
 src/Relation/Nullary/Decidable.agda                         | 3 ++-
 39 files changed, 55 insertions(+), 41 deletions(-)

diff --git a/src/Algebra/Properties/Monoid/Sum.agda b/src/Algebra/Properties/Monoid/Sum.agda
index 1f522ca678..dacb936285 100644
--- a/src/Algebra/Properties/Monoid/Sum.agda
+++ b/src/Algebra/Properties/Monoid/Sum.agda
@@ -12,7 +12,7 @@ open import Data.Vec.Functional as Vector
 open import Data.Fin.Base using (zero; suc)
 open import Data.Unit using (tt)
 open import Function.Base using (_∘_)
-open import Relation.Binary as B using (_Preserves_⟶_)
+open import Relation.Binary.Core using (_Preserves_⟶_)
 open import Relation.Binary.PropositionalEquality as P using (_≗_; _≡_)
 
 module Algebra.Properties.Monoid.Sum {a ℓ} (M : Monoid a ℓ) where
diff --git a/src/Data/AVL/Map.agda b/src/Data/AVL/Map.agda
index 9950a1299b..d4768bb19e 100644
--- a/src/Data/AVL/Map.agda
+++ b/src/Data/AVL/Map.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.AVL.Map
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/AVL/NonEmpty.agda b/src/Data/AVL/NonEmpty.agda
index 57b0c83ac2..26398cb553 100644
--- a/src/Data/AVL/NonEmpty.agda
+++ b/src/Data/AVL/NonEmpty.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.AVL.NonEmpty
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂) where
diff --git a/src/Data/AVL/Sets.agda b/src/Data/AVL/Sets.agda
index 7a3d05a177..5d216695db 100644
--- a/src/Data/AVL/Sets.agda
+++ b/src/Data/AVL/Sets.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.AVL.Sets
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/AVL/Value.agda b/src/Data/AVL/Value.agda
index d46bde80eb..6338a876d8 100644
--- a/src/Data/AVL/Value.agda
+++ b/src/Data/AVL/Value.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.AVL.Value {a ℓ} (S : Setoid a ℓ) where
 
diff --git a/src/Data/Container/Morphism/Properties.agda b/src/Data/Container/Morphism/Properties.agda
index 0859cfe0be..fc71843263 100644
--- a/src/Data/Container/Morphism/Properties.agda
+++ b/src/Data/Container/Morphism/Properties.agda
@@ -11,7 +11,7 @@ module Data.Container.Morphism.Properties where
 open import Level using (_⊔_; suc)
 open import Function.Base as F using (_$_)
 open import Data.Product using (∃; proj₁; proj₂; _,_)
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
 
 open import Data.Container.Core
diff --git a/src/Data/Container/Relation/Binary/Equality/Setoid.agda b/src/Data/Container/Relation/Binary/Equality/Setoid.agda
index f970218f10..993aaef60b 100644
--- a/src/Data/Container/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Container/Relation/Binary/Equality/Setoid.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.Container.Relation.Binary.Equality.Setoid {c e} (S : Setoid c e) where
 
diff --git a/src/Data/Container/Relation/Binary/Pointwise.agda b/src/Data/Container/Relation/Binary/Pointwise.agda
index 7b20aa8dbb..6f9949dc4e 100644
--- a/src/Data/Container/Relation/Binary/Pointwise.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise.agda
@@ -11,7 +11,7 @@ module Data.Container.Relation.Binary.Pointwise where
 open import Data.Product using (_,_; Σ-syntax; -,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
 open import Level using (_⊔_)
-open import Relation.Binary using (REL; _⇒_)
+open import Relation.Binary.Core using (REL; _⇒_)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; subst; cong)
 
diff --git a/src/Data/Container/Relation/Unary/Any/Properties.agda b/src/Data/Container/Relation/Unary/Any/Properties.agda
index e0398eff8f..ceae1510c9 100644
--- a/src/Data/Container/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Container/Relation/Unary/Any/Properties.agda
@@ -22,7 +22,7 @@ open import Function.Inverse as Inv using (_↔_; inverse; module Inverse)
 open import Function.Related as Related using (Related; SK-sym)
 open import Function.Related.TypeIsomorphisms
 open import Relation.Unary using (Pred ; _∪_ ; _∩_)
-open import Relation.Binary using (REL)
+open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; _≗_; refl)
 
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 852a609779..9c7cbdf170 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -25,7 +25,7 @@ open import Function.Equality using (_⟨$⟩_)
 open import Function.Properties.Inverse using (↔⇒↣)
 open import Function.Base using (_∘_)
 open import Level using (0ℓ)
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary using (does; ¬_; yes; no)
 open import Relation.Nullary.Decidable using (dec-yes; dec-no)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/Integer/Coprimality.agda b/src/Data/Integer/Coprimality.agda
index c3cfafcbff..36eb45e19e 100644
--- a/src/Data/Integer/Coprimality.agda
+++ b/src/Data/Integer/Coprimality.agda
@@ -15,7 +15,8 @@ import Data.Nat.Coprimality as ℕ
 import Data.Nat.Divisibility as ℕ
 open import Function.Base using (_on_)
 open import Level using (0ℓ)
-open import Relation.Binary using (Rel; Decidable; Symmetric)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Decidable; Symmetric)
 open import Relation.Binary.PropositionalEquality.Core using (subst)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Extrema/Core.agda b/src/Data/List/Extrema/Core.agda
index 0ed9dbe159..674e359136 100644
--- a/src/Data/List/Extrema/Core.agda
+++ b/src/Data/List/Extrema/Core.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Trans; TotalOrder; Setoid)
+open import Relation.Binary.Definitions using (Trans)
+open import Relation.Binary.Bundles using (TotalOrder; Setoid)
 
 module Data.List.Extrema.Core
   {b ℓ₁ ℓ₂} (totalOrder : TotalOrder b ℓ₁ ℓ₂) where
diff --git a/src/Data/List/Fresh/Membership/Setoid/Properties.agda b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
index 0515b84efe..e287147516 100644
--- a/src/Data/List/Fresh/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Fresh.Membership.Setoid.Properties {c ℓ} (S : Setoid c ℓ) where
 
diff --git a/src/Data/List/Fresh/Relation/Unary/All.agda b/src/Data/List/Fresh/Relation/Unary/All.agda
index b0e2115ce1..2f560e82c2 100644
--- a/src/Data/List/Fresh/Relation/Unary/All.agda
+++ b/src/Data/List/Fresh/Relation/Unary/All.agda
@@ -13,7 +13,7 @@ open import Data.Product.Base using (_×_; _,_; proj₁; uncurry)
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _×-dec_)
 open import Relation.Unary  as U
-open import Relation.Binary as B using (Rel)
+open import Relation.Binary.Core using (Rel)
 
 open import Data.List.Fresh using (List#; []; cons; _∷#_; _#_)
 open import Data.List.Fresh.Relation.Unary.Any as Any using (Any; here; there)
diff --git a/src/Data/List/Fresh/Relation/Unary/All/Properties.agda b/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
index 8c2f0c80b4..b32f08ca25 100644
--- a/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Fresh/Relation/Unary/All/Properties.agda
@@ -15,7 +15,7 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘′_)
 open import Relation.Nullary
 open import Relation.Unary  as U
-open import Relation.Binary as B using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 open import Data.List.Fresh using (List#; []; cons; _∷#_; _#_)
diff --git a/src/Data/List/Fresh/Relation/Unary/Any.agda b/src/Data/List/Fresh/Relation/Unary/Any.agda
index 4c11dd7974..9c2050472c 100644
--- a/src/Data/List/Fresh/Relation/Unary/Any.agda
+++ b/src/Data/List/Fresh/Relation/Unary/Any.agda
@@ -16,7 +16,7 @@ open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _⊎-dec_)
 open import Relation.Unary  as U
-open import Relation.Binary as B using (Rel)
+open import Relation.Binary.Core using (Rel)
 
 open import Data.List.Fresh using (List#; []; cons; _∷#_; _#_)
 
diff --git a/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda b/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
index 8b47063949..70a2a6ee29 100644
--- a/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Fresh/Relation/Unary/Any/Properties.agda
@@ -18,7 +18,7 @@ open import Function.Base using (_∘′_)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary
 open import Relation.Unary  as U using (Pred)
-open import Relation.Binary as B using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Nary
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index 364c6d29b8..c7431c4949 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -34,7 +34,8 @@ open import Function.Inverse as Inv using (_↔_; module Inverse)
 import Function.Related as Related
 open import Function.Related.TypeIsomorphisms
 open import Level using (Level)
-open import Relation.Binary as B hiding (Decidable)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions as B hiding (Decidable)
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; _≢_; refl; sym; trans; cong; subst; →-to-⟶; _≗_)
 import Relation.Binary.Properties.DecTotalOrder as DTOProperties
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index 24b8bee2cf..e7d2b4039c 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -6,7 +6,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary as B hiding (Decidable)
+open import Relation.Binary.Core
+  using (Rel; _⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions as B hiding (Decidable)
 
 module Data.List.Relation.Binary.Permutation.Setoid.Properties
   {a ℓ} (S : Setoid a ℓ)
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
index bf107dfcc3..39d4439a86 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
@@ -13,7 +13,7 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise
   using (Pointwise; []; _∷_)
 open import Data.Product using (∃; _×_; _,_; uncurry)
-open import Relation.Binary using (REL; _⇒_)
+open import Relation.Binary.Core using (REL; _⇒_)
 
 module _ {a b r} {A : Set a} {B : Set b} (R : REL A B r) where
 
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
index 58d64ee738..a2532bc63b 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
@@ -7,7 +7,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 module Data.List.Relation.Binary.Sublist.DecPropositional.Solver
diff --git a/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda b/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda
index b007311b23..7564052444 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecSetoid/Solver.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecSetoid)
+open import Relation.Binary.Bundles using (DecSetoid)
 
 module Data.List.Relation.Binary.Sublist.DecSetoid.Solver {c ℓ} (S : DecSetoid c ℓ) where
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Core.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Core.agda
index f57ac4a876..4ef8ae7793 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Core.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Core.agda
@@ -13,7 +13,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (REL)
+open import Relation.Binary.Core using (REL)
 
 module Data.List.Relation.Binary.Sublist.Heterogeneous.Core
        {a b r} {A : Set a} {B : Set b} (R : REL A B r)
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index 2fe2447efb..a7afe038c7 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Reflexive; Decidable)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Reflexive; Decidable)
 
 module Data.List.Relation.Binary.Sublist.Heterogeneous.Solver
   {a r} {A : Set a} (R : Rel A r)
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid.agda b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
index 8915308cf2..eff26de0aa 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
@@ -9,7 +9,8 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 {-# OPTIONS --postfix-projections #-}
 
-open import Relation.Binary using (Setoid; Rel)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Binary.Sublist.Setoid
   {c ℓ} (S : Setoid c ℓ) where
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index 349586f59b..836c52fd70 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary hiding (Decidable)
+open import Relation.Binary.Definitions hiding (Decidable)
+open import Relation.Binary.Structures using (IsPreorder)
 
 module Data.List.Relation.Binary.Subset.Propositional.Properties
   where
@@ -35,7 +36,8 @@ open import Function.Equivalence using (module Equivalence)
 open import Level using (Level)
 open import Relation.Nullary using (¬_; yes; no)
 open import Relation.Unary using (Decidable; Pred) renaming (_⊆_ to _⋐_)
-open import Relation.Binary using (_⇒_; _Respects_)
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Preorder)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; _≗_; isEquivalence; subst; resp; refl; setoid; module ≡-Reasoning)
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
diff --git a/src/Data/List/Relation/Ternary/Appending.agda b/src/Data/List/Relation/Ternary/Appending.agda
index 802f80aac3..1ae20e905f 100644
--- a/src/Data/List/Relation/Ternary/Appending.agda
+++ b/src/Data/List/Relation/Ternary/Appending.agda
@@ -12,7 +12,7 @@ open import Level using (Level; _⊔_)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.Product as Prod using (∃₂; _×_; _,_; -,_)
-open import Relation.Binary using (REL)
+open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 private
diff --git a/src/Data/List/Relation/Ternary/Appending/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Properties.agda
index 6c02c49ffb..4063d504a7 100644
--- a/src/Data/List/Relation/Ternary/Appending/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Properties.agda
@@ -13,7 +13,8 @@ open import Data.List.Relation.Ternary.Appending
 open import Data.List.Relation.Binary.Pointwise.Base as Pw using (Pointwise; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise.Properties as Pw using (transitive)
 open import Level using (Level)
-open import Relation.Binary using (REL; Rel; Trans)
+open import Relation.Binary.Core using (REL; Rel)
+open import Relation.Binary.Definitions using (Trans)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid.agda b/src/Data/List/Relation/Ternary/Appending/Setoid.agda
index fc07b650af..0460f9ea21 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Ternary.Appending.Setoid
   {c ℓ} (S : Setoid c ℓ)
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
index c205170359..83d3add55e 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Ternary.Appending.Setoid.Properties {c l} (S : Setoid c l) where
 
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Setoid.agda b/src/Data/List/Relation/Ternary/Interleaving/Setoid.agda
index 006acdf84b..b245df5da7 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Setoid.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Setoid.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Ternary.Interleaving.Setoid {c ℓ} (S : Setoid c ℓ) where
 
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda
index cdd8f968aa..936bca38b4 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Setoid/Properties.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Ternary.Interleaving.Setoid.Properties
   {c ℓ} (S : Setoid c ℓ) where
diff --git a/src/Data/Tree/AVL.agda b/src/Data/Tree/AVL.agda
index d897161019..e8920b7a36 100644
--- a/src/Data/Tree/AVL.agda
+++ b/src/Data/Tree/AVL.agda
@@ -13,7 +13,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Trie/NonEmpty.agda b/src/Data/Trie/NonEmpty.agda
index 5fa0a458f0..96378d25e6 100644
--- a/src/Data/Trie/NonEmpty.agda
+++ b/src/Data/Trie/NonEmpty.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --sized-types #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Trie.NonEmpty {k e r} (S : StrictTotalOrder k e r) where
 
diff --git a/src/Data/Vec/Membership/DecPropositional.agda b/src/Data/Vec/Membership/DecPropositional.agda
index e1b100f0af..8004af2720 100644
--- a/src/Data/Vec/Membership/DecPropositional.agda
+++ b/src/Data/Vec/Membership/DecPropositional.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
diff --git a/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda b/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
index 5614602219..f3bd4cb1a9 100644
--- a/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
@@ -15,7 +15,7 @@ open import Data.Nat.Base using (ℕ)
 import Data.Vec.Relation.Binary.Equality.Setoid as Equality
 import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
 open import Level using (_⊔_)
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 
 open DecSetoid DS
 
diff --git a/src/Data/Vec/Relation/Unary/All/Properties.agda b/src/Data/Vec/Relation/Unary/All/Properties.agda
index 9e0f807632..d59627f783 100644
--- a/src/Data/Vec/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/All/Properties.agda
@@ -20,7 +20,7 @@ open import Level using (Level)
 open import Function.Base using (_∘_; id)
 open import Function.Inverse using (_↔_; inverse)
 open import Relation.Unary using (Pred) renaming (_⊆_ to _⋐_)
-open import Relation.Binary as B using (REL)
+open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; cong; cong₂; →-to-⟶)
 
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda b/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda
index 6160c7e014..cde714bc83 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Definitions.agda
@@ -13,7 +13,8 @@ module Relation.Binary.Indexed.Homogeneous.Definitions where
 
 open import Data.Product.Base using (_×_)
 open import Level using (Level)
-import Relation.Binary as B
+open import Relation.Binary.Core using (_⇒_)
+import Relation.Binary.Definitions as B
 open import Relation.Unary.Indexed using (IPred)
 
 open import Relation.Binary.Indexed.Homogeneous.Core
@@ -31,7 +32,7 @@ module _ (A : I → Set a) where
   syntax Implies A _∼₁_ _∼₂_ = _∼₁_ ⇒[ A ] _∼₂_
 
   Implies : IRel A ℓ₁ → IRel A ℓ₂ → Set _
-  Implies _∼₁_ _∼₂_ = ∀ {i} → _∼₁_ B.⇒ (_∼₂_ {i})
+  Implies _∼₁_ _∼₂_ = ∀ {i} → _∼₁_ ⇒ (_∼₂_ {i})
 
   Reflexive : IRel A ℓ → Set _
   Reflexive _∼_ = ∀ {i} → B.Reflexive (_∼_ {i})
diff --git a/src/Relation/Nullary/Decidable.agda b/src/Relation/Nullary/Decidable.agda
index b41a410391..5793725f0e 100644
--- a/src/Relation/Nullary/Decidable.agda
+++ b/src/Relation/Nullary/Decidable.agda
@@ -16,7 +16,8 @@ open import Data.Sum.Base as Sum hiding (map)
 open import Function.Base
 open import Function.Bundles using
   (Injection; module Injection; module Equivalence; _⇔_; _↔_; mk↔′)
-open import Relation.Binary using (Setoid; module Setoid; Decidable)
+open import Relation.Binary.Bundles using (Setoid; module Setoid)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary
 open import Relation.Nullary.Negation
 open import Relation.Nullary.Reflects using (invert)

From eed1b6d19b962d407ae611304451836684bf5f67 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sat, 29 Jul 2023 07:54:06 -0400
Subject: [PATCH 083/316] Rename and deprecate `excluded-middle` (#2026)

---
 CHANGELOG.md                                           |  3 ++-
 src/Axiom/DoubleNegationElimination.agda               |  2 +-
 src/Codata/Musical/Colist.agda                         |  4 ++--
 src/Data/List/Membership/Propositional/Properties.agda |  4 ++--
 src/Effect/Monad/Partiality.agda                       |  2 +-
 src/Relation/Nullary/Decidable/Core.agda               | 10 +++++++++-
 src/Relation/Nullary/Negation.agda                     |  6 +++---
 7 files changed, 20 insertions(+), 11 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8e5e356e52..8e74556b52 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -881,7 +881,8 @@ Non-backwards compatible changes
     lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
     lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
     ```
-
+  * `excluded-middle` in `Relation.Nullary.Decidable.Core` has been renamed to
+    `¬¬-excluded-middle`.
 
 Major improvements
 ------------------
diff --git a/src/Axiom/DoubleNegationElimination.agda b/src/Axiom/DoubleNegationElimination.agda
index 49e8b01eaa..777f08c017 100644
--- a/src/Axiom/DoubleNegationElimination.agda
+++ b/src/Axiom/DoubleNegationElimination.agda
@@ -32,4 +32,4 @@ em⇒dne : ∀ {ℓ} → ExcludedMiddle ℓ → DoubleNegationElimination ℓ
 em⇒dne em = decidable-stable em
 
 dne⇒em : ∀ {ℓ} → DoubleNegationElimination ℓ → ExcludedMiddle ℓ
-dne⇒em dne = dne excluded-middle
+dne⇒em dne = dne ¬¬-excluded-middle
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index 2ec01be70f..d70a101f6a 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -36,7 +36,7 @@ open import Relation.Binary.PropositionalEquality as P using (_≡_)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary
 open import Relation.Nullary.Negation
-open import Relation.Nullary.Decidable using (excluded-middle)
+open import Relation.Nullary.Decidable using (¬¬-excluded-middle)
 open import Relation.Unary using (Pred)
 
 private
@@ -244,7 +244,7 @@ not-finite-is-infinite (x ∷ xs) hyp =
 
 finite-or-infinite :
   (xs : Colist A) → ¬ ¬ (Finite xs ⊎ Infinite xs)
-finite-or-infinite xs = helper <$> excluded-middle
+finite-or-infinite xs = helper <$> ¬¬-excluded-middle
   where
   helper : Dec (Finite xs) → Finite xs ⊎ Infinite xs
   helper ( true because  [fin]) = inj₁ (invert [fin])
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index c7431c4949..f781e1678d 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -44,7 +44,7 @@ import Relation.Nullary.Reflects as Reflects
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (¬_; Dec; does; yes; no; _because_)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Nullary.Decidable using (excluded-middle)
+open import Relation.Nullary.Decidable using (¬¬-excluded-middle)
 
 private
   open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})
@@ -345,7 +345,7 @@ module _ {_•_ : Op₂ A} where
 
 finite : (f : ℕ ↣ A) → ∀ xs → ¬ (∀ i → Injection.to f ⟨$⟩ i ∈ xs)
 finite inj []       fᵢ∈[]   = ¬Any[] (fᵢ∈[] 0)
-finite inj (x ∷ xs) fᵢ∈x∷xs = excluded-middle helper
+finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
   where
   open Injection inj renaming (injective to f-inj)
 
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index e6cfdd1c05..194b079339 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -401,7 +401,7 @@ module _ {A : Set a} {_∼_ : A → A → Set ℓ} where
   now-or-never : Reflexive _∼_ →
                  ∀ {k} (x : A ⊥) →
                  ¬ ¬ ((∃ λ y → x ⇓[ other k ] y) ⊎ x ⇑[ other k ])
-  now-or-never refl x = helper <$> excluded-middle
+  now-or-never refl x = helper <$> ¬¬-excluded-middle
     where
     open RawMonad ¬¬-Monad
 
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index a7c1009091..2d827519d7 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -176,5 +176,13 @@ decidable-stable (no ¬p) ¬¬p = ⊥-elim (¬¬p ¬p)
 -- A double-negation-translated variant of excluded middle (or: every
 -- nullary relation is decidable in the double-negation monad).
 
+¬¬-excluded-middle : DoubleNegation (Dec P)
+¬¬-excluded-middle ¬h = ¬h (no (λ p → ¬h (yes p)))
+
 excluded-middle : DoubleNegation (Dec P)
-excluded-middle ¬h = ¬h (no (λ p → ¬h (yes p)))
+excluded-middle = ¬¬-excluded-middle
+
+{-# WARNING_ON_USAGE excluded-middle
+"Warning: excluded-middle was deprecated in v2.0.
+Please use ¬¬-excluded-middle instead."
+#-}
diff --git a/src/Relation/Nullary/Negation.agda b/src/Relation/Nullary/Negation.agda
index 5111db897f..59e96f3429 100644
--- a/src/Relation/Nullary/Negation.agda
+++ b/src/Relation/Nullary/Negation.agda
@@ -15,7 +15,7 @@ open import Data.Product.Base as Prod using (_,_; Σ; Σ-syntax; ∃; curry; unc
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
 open import Function.Base using (flip; _∘_; const; _∘′_)
 open import Level using (Level)
-open import Relation.Nullary.Decidable.Core using (Dec; yes; no; excluded-middle)
+open import Relation.Nullary.Decidable.Core using (Dec; yes; no; ¬¬-excluded-middle)
 open import Relation.Unary using (Universal)
 
 private
@@ -85,7 +85,7 @@ call/cc hyp ¬p = hyp (λ p → ⊥-elim (¬p p)) ¬p
 
 independence-of-premise : {R : Q → Set r} →
                           Q → (P → Σ Q R) → DoubleNegation (Σ[ x ∈ Q ] (P → R x))
-independence-of-premise {P = P} q f = ¬¬-map helper excluded-middle
+independence-of-premise {P = P} q f = ¬¬-map helper ¬¬-excluded-middle
   where
   helper : Dec P → _
   helper (yes p) = Prod.map₂ const (f p)
@@ -94,7 +94,7 @@ independence-of-premise {P = P} q f = ¬¬-map helper excluded-middle
 -- The independence of premise rule for binary sums.
 
 independence-of-premise-⊎ : (P → Q ⊎ R) → DoubleNegation ((P → Q) ⊎ (P → R))
-independence-of-premise-⊎ {P = P} f = ¬¬-map helper excluded-middle
+independence-of-premise-⊎ {P = P} f = ¬¬-map helper ¬¬-excluded-middle
   where
   helper : Dec P → _
   helper (yes p) = Sum.map const const (f p)

From c27b5cc4cebccd5bca800b66f1910400644748ea Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Sat, 29 Jul 2023 09:13:47 -0400
Subject: [PATCH 084/316] Simplified `String` imports (#2016)

---
 README/Data/List/Relation/Binary/Subset.agda | 3 ++-
 README/Data/Tree/AVL.agda                    | 2 +-
 README/Data/Trie/NonDependent.agda           | 3 ++-
 README/Function/Reasoning.agda               | 3 ++-
 README/IO.agda                               | 2 +-
 src/Data/Rational.agda                       | 2 +-
 src/Reflection/AST/Abstraction.agda          | 3 ++-
 src/Reflection/AST/Literal.agda              | 2 +-
 src/Reflection/AST/Show.agda                 | 4 ++--
 src/Reflection/AST/Term.agda                 | 3 ++-
 src/Reflection/AST/Traversal.agda            | 2 +-
 src/Test/Golden.agda                         | 3 ++-
 src/Text/Tabular/List.agda                   | 2 +-
 13 files changed, 20 insertions(+), 14 deletions(-)

diff --git a/README/Data/List/Relation/Binary/Subset.agda b/README/Data/List/Relation/Binary/Subset.agda
index f6ef7690a1..ccef725cca 100644
--- a/README/Data/List/Relation/Binary/Subset.agda
+++ b/README/Data/List/Relation/Binary/Subset.agda
@@ -21,7 +21,8 @@ module README.Data.List.Relation.Binary.Subset where
 -- tell Agda which equality relation to use.
 
 -- Decidable equality over Strings
-open import Data.String using (String; _≟_)
+open import Data.String.Base using (String)
+open import Data.String.Properties using (_≟_)
 
 -- Open the decidable membership module using Decidable ≡ over Strings
 open import Data.List.Membership.DecPropositional _≟_
diff --git a/README/Data/Tree/AVL.agda b/README/Data/Tree/AVL.agda
index 7994afed41..abba7bfd0b 100644
--- a/README/Data/Tree/AVL.agda
+++ b/README/Data/Tree/AVL.agda
@@ -21,7 +21,7 @@ import Data.Tree.AVL
 
 open import Data.Nat.Properties using (<-strictTotalOrder)
 open import Data.Product as Prod using (_,_; _,′_)
-open import Data.String using (String)
+open import Data.String.Base using (String)
 open import Data.Vec using (Vec; _∷_; [])
 open import Relation.Binary.PropositionalEquality
 
diff --git a/README/Data/Trie/NonDependent.agda b/README/Data/Trie/NonDependent.agda
index 974ce5c96d..9921942f6d 100644
--- a/README/Data/Trie/NonDependent.agda
+++ b/README/Data/Trie/NonDependent.agda
@@ -57,7 +57,8 @@ open import Data.List.Base   as List using (List; []; _∷_)
 open import Data.List.Fresh  as List# using (List#; []; _∷#_)
 open import Data.Maybe       as Maybe
 open import Data.Product     as Prod
-open import Data.String      as String using (String)
+open import Data.String.Base as String using (String)
+open import Data.String.Properties as String using (_≟_)
 open import Data.These       as These
 
 open import Function.Base using (case_of_; _$_; _∘′_; id; _on_)
diff --git a/README/Function/Reasoning.agda b/README/Function/Reasoning.agda
index 4b9835346e..c484fdea9f 100644
--- a/README/Function/Reasoning.agda
+++ b/README/Function/Reasoning.agda
@@ -35,7 +35,8 @@ module _ {A B C : Set} {A→B : A → B} {B→C : B → C} where
 open import Data.Nat
 open import Data.List.Base
 open import Data.Char.Base
-open import Data.String as String using (String; toList; fromList; _==_)
+open import Data.String.Base as String using (String; toList; fromList)
+open import Data.String.Properties as String using (_==_)
 open import Function.Base using (_∘_)
 open import Data.Bool hiding (_≤?_)
 open import Data.Product as P using (_×_; <_,_>; uncurry; proj₁)
diff --git a/README/IO.agda b/README/IO.agda
index 880ad1fa32..37553edd1d 100644
--- a/README/IO.agda
+++ b/README/IO.agda
@@ -11,7 +11,7 @@ module README.IO where
 open import Level
 open import Data.Nat.Base
 open import Data.Nat.Show using (show)
-open import Data.String using (String; _++_; lines)
+open import Data.String.Base using (String; _++_; lines)
 open import Data.Unit.Polymorphic
 open import IO
 
diff --git a/src/Data/Rational.agda b/src/Data/Rational.agda
index 260aa44951..2345466dcb 100644
--- a/src/Data/Rational.agda
+++ b/src/Data/Rational.agda
@@ -26,7 +26,7 @@ open import Data.Rational.Properties public
 -- Version 1.5
 
 import Data.Integer.Show as ℤ
-open import Data.String using (String; _++_)
+open import Data.String.Base using (String; _++_)
 
 show : ℚ → String
 show p = ℤ.show (↥ p) ++ "/" ++ ℤ.show (↧ p)
diff --git a/src/Reflection/AST/Abstraction.agda b/src/Reflection/AST/Abstraction.agda
index 80dade0845..3bf47f81a4 100644
--- a/src/Reflection/AST/Abstraction.agda
+++ b/src/Reflection/AST/Abstraction.agda
@@ -8,8 +8,9 @@
 
 module Reflection.AST.Abstraction where
 
+open import Data.String.Base as String                 using (String)
+open import Data.String.Properties as String           using (_≟_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
-open import Data.String as String                      using (String)
 open import Level
 open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
 open import Relation.Binary                            using (DecidableEquality)
diff --git a/src/Reflection/AST/Literal.agda b/src/Reflection/AST/Literal.agda
index ff8835236a..f400c9f1dc 100644
--- a/src/Reflection/AST/Literal.agda
+++ b/src/Reflection/AST/Literal.agda
@@ -12,7 +12,7 @@ open import Data.Bool.Base   using (Bool; true; false)
 import Data.Char as Char     using (_≟_)
 import Data.Float as Float   using (_≟_)
 import Data.Nat as ℕ        using (_≟_)
-import Data.String as String using (_≟_)
+import Data.String.Properties as String using (_≟_)
 import Data.Word as Word     using (_≟_)
 import Reflection.AST.Meta as Meta
 import Reflection.AST.Name as Name
diff --git a/src/Reflection/AST/Show.agda b/src/Reflection/AST/Show.agda
index 7650f6fc8e..efe77a4779 100644
--- a/src/Reflection/AST/Show.agda
+++ b/src/Reflection/AST/Show.agda
@@ -15,9 +15,9 @@ import Data.Char as Char   using (show)
 import Data.Float as Float using (show)
 open import Data.List.Base hiding (_++_; intersperse)
 import Data.Nat.Show as ℕ using (show)
+open import Data.String.Base as String using (String; _++_; intersperse; braces; parens; _<+>_)
+open import Data.String as String using (parensIfSpace)
 open import Data.Product.Base using (_×_; _,_)
-open import Data.String as String
-  using (String; _++_; intersperse; braces; parens; parensIfSpace; _<+>_)
 import Data.Word as Word   using (toℕ)
 open import Function.Base using (id; _∘′_; case_of_)
 open import Relation.Nullary.Decidable using (yes; no)
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index 002ca3dd58..b22641b2a9 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -14,7 +14,8 @@ open import Data.Nat as ℕ           using (ℕ; zero; suc)
 open import Data.Product.Base       using (_×_; _,_; <_,_>; uncurry; map₁)
 open import Data.Product.Properties using (,-injective)
 open import Data.Maybe.Base         using (Maybe; just; nothing)
-open import Data.String as String   using (String)
+open import Data.String.Base    using (String)
+open import Data.String.Properties as String hiding (_≟_)
 open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)
 open import Relation.Binary                            using (Decidable; DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
diff --git a/src/Reflection/AST/Traversal.agda b/src/Reflection/AST/Traversal.agda
index 5437c52b73..a71d93e191 100644
--- a/src/Reflection/AST/Traversal.agda
+++ b/src/Reflection/AST/Traversal.agda
@@ -13,8 +13,8 @@ module Reflection.AST.Traversal
 
 open import Data.Nat          using (ℕ; zero; suc; _+_)
 open import Data.List.Base    using (List; []; _∷_; _++_; reverse; length)
+open import Data.String.Base  using (String)
 open import Data.Product.Base using (_×_; _,_)
-open import Data.String       using (String)
 open import Function.Base     using (_∘_)
 open import Reflection        hiding (pure)
 
diff --git a/src/Test/Golden.agda b/src/Test/Golden.agda
index 4b032bd783..43bbd3037e 100644
--- a/src/Test/Golden.agda
+++ b/src/Test/Golden.agda
@@ -90,7 +90,8 @@ open import Data.Maybe.Base using (Maybe; just; nothing; fromMaybe)
 open import Data.Nat.Base using (ℕ; _≡ᵇ_; _<ᵇ_; _+_; _∸_)
 import Data.Nat.Show as ℕ using (show)
 open import Data.Product.Base using (_×_; _,_)
-open import Data.String as String using (String; lines; unlines; unwords; concat; _≟_)
+open import Data.String.Base as String using (String; lines; unlines; unwords; concat)
+open import Data.String.Properties as String using (_≟_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Unit.Base using (⊤)
 
diff --git a/src/Text/Tabular/List.agda b/src/Text/Tabular/List.agda
index 70fc09f3d5..1f1603424b 100644
--- a/src/Text/Tabular/List.agda
+++ b/src/Text/Tabular/List.agda
@@ -8,7 +8,7 @@
 
 module Text.Tabular.List where
 
-open import Data.String using (String)
+open import Data.String.Base using (String)
 open import Data.List.Base
 import Data.Nat.Properties as ℕₚ
 open import Data.Product.Base using (-,_; proj₂)

From de453c320c6db8a95a83498b235224bc2fba64f5 Mon Sep 17 00:00:00 2001
From: Alex Rice <alexrice999@hotmail.co.uk>
Date: Sun, 30 Jul 2023 11:03:07 +0100
Subject: [PATCH 085/316] Change `Function.Definitions` to use strict inverses
 (#1156)

---
 CHANGELOG.md                                  |  41 +++++-
 src/Algebra/Consequences/Setoid.agda          |  37 ++----
 .../Morphism/Construct/Composition.agda       |   2 +-
 .../Lattice/Morphism/Construct/Identity.agda  |   3 +-
 src/Algebra/Lattice/Morphism/Structures.agda  |   7 +-
 .../Morphism/Construct/Composition.agda       |  16 +--
 .../Module/Morphism/Construct/Identity.agda   |  17 +--
 src/Algebra/Module/Morphism/Structures.agda   |  42 +++---
 src/Algebra/Morphism/Consequences.agda        |  14 +-
 .../Morphism/Construct/Composition.agda       |  18 +--
 src/Algebra/Morphism/Construct/Identity.agda  |  19 +--
 src/Algebra/Morphism/Structures.agda          |  47 +++----
 src/Data/Fin/Permutation.agda                 |  20 +--
 src/Data/Fin/Properties.agda                  |  17 ++-
 .../Sublist/Heterogeneous/Properties.agda     |   4 +-
 .../Unary/Enumerates/Setoid/Properties.agda   |  12 +-
 src/Data/Nat/Binary/Properties.agda           |  17 ++-
 src/Data/Nat/Binary/Subtraction.agda          |   4 +-
 src/Data/Parity/Properties.agda               |  25 ++--
 src/Data/Product/Algebra.agda                 |  12 +-
 src/Data/Sum/Algebra.agda                     |  13 +-
 src/Data/Vec/N-ary.agda                       |   4 +-
 src/Data/Vec/Recursive.agda                   |  11 +-
 src/Function/Bundles.agda                     | 118 +++++++++--------
 src/Function/Consequences.agda                | 120 +++++++++++++-----
 src/Function/Consequences/Propositional.agda  |  91 +++++++++++++
 src/Function/Construct/Composition.agda       |  65 ++++------
 src/Function/Construct/Identity.agda          |  46 +++----
 src/Function/Construct/Symmetry.agda          |  44 +++----
 src/Function/Definitions.agda                 |  62 +++++----
 src/Function/Definitions/Core1.agda           |  25 ----
 src/Function/Definitions/Core2.agda           |  29 -----
 src/Function/Properties.agda                  |   4 +-
 src/Function/Properties/Bijection.agda        |  17 +--
 src/Function/Properties/Inverse.agda          |  12 +-
 src/Function/Properties/RightInverse.agda     |   4 +-
 src/Function/Properties/Surjection.agda       |  23 +++-
 src/Function/Related.agda                     |  20 +--
 src/Function/Structures.agda                  |  19 ++-
 .../Morphism/Construct/Composition.agda       |   8 +-
 .../Binary/Morphism/Construct/Identity.agda   |  11 +-
 41 files changed, 636 insertions(+), 484 deletions(-)
 create mode 100644 src/Function/Consequences/Propositional.agda
 delete mode 100644 src/Function/Definitions/Core1.agda
 delete mode 100644 src/Function/Definitions/Core2.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8e74556b52..b7cb6aa60b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -272,7 +272,7 @@ Non-backwards compatible changes
 * Added new aliases `Is(Meet/Join)(Bounded)Semilattice` for `Is(Bounded)Semilattice`
   which can be used to indicate meet/join-ness of the original structures.
 
-#### Function hierarchy
+#### Removal of the old function hierarchy
 
 * The switch to the new function hierarchy is complete and the following definitions
   now use the new definitions instead of the old ones:
@@ -345,9 +345,41 @@ Non-backwards compatible changes
   * `Relation.Nullary.Decidable`
     * `map`
 
-* The names of the fields in the records of the new hierarchy have been
-  changed from `f`, `g`, `cong₁`, `cong₂` to `to`, `from`, `to-cong`, `from-cong`.
 
+#### Changes to the new function hierarchy
+
+* The names of the fields in `Function.Bundles` have been
+  changed from `f`, `g`, `cong₁` and `cong₂` to `to`, `from`, `to-cong`, `from-cong`.
+
+* The module `Function.Definitions` no longer has two equalities as module arguments, as
+  they did not interact as intended with the re-exports from `Function.Definitions.(Core1/Core2)`.
+  The latter have been removed and their definitions folded into `Function.Definitions`.
+  
+* In `Function.Definitions` the types of `Surjective`, `Injective` and `Surjective` 
+  have been changed from:
+  ```
+  Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
+  Inverseˡ f g = ∀ y → f (g y) ≈₂ y
+  Inverseʳ f g = ∀ x → g (f x) ≈₁ x
+  ```
+  to:
+  ```
+  Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y
+  Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
+  Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
+  ```
+  This is for several reasons: i) the new definitions compose much more easily, ii) Agda
+  can better infer the equalities used.
+  
+  To ease backwards compatibility:
+   - the old definitions have been moved to the new names  `StrictlySurjective`, 
+	 `StrictlyInverseˡ` and `StrictlyInverseʳ`. 
+   - The records in  `Function.Structures` and `Function.Bundles` export proofs 
+	 of these under the names `strictlySurjective`, `strictlyInverseˡ` and 
+	 `strictlyInverseʳ`,
+   - Conversion functions have been added in both directions to
+	 `Function.Consequences(.Propositional)`. 
+  
 #### Proofs of non-zeroness/positivity/negativity as instance arguments
 
 * Many numeric operations in the library require their arguments to be non-zero,
@@ -2206,6 +2238,9 @@ Other minor changes
 * Added a new proof to `Data.Nat.Binary.Properties`:
   ```agda
   suc-injective : Injective _≡_ _≡_ suc
+  toℕ-inverseˡ  : Inverseˡ _≡_ _≡_ toℕ fromℕ
+  toℕ-inverseʳ  : Inverseʳ _≡_ _≡_ toℕ fromℕ
+  toℕ-inverseᵇ  : Inverseᵇ _≡_ _≡_ toℕ fromℕ
   ```
 
 * Added a new pattern synonym to `Data.Nat.Divisibility.Core`:
diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index f5cac63048..1aa2b8f4c8 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -19,8 +19,8 @@ open import Algebra.Core
 open import Algebra.Definitions _≈_
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Product.Base using (_,_)
-open import Function.Base using (_$_)
-import Function.Definitions as FunDefs
+open import Function.Base using (_$_; id; _∘_)
+open import Function.Definitions
 import Relation.Binary.Consequences as Bin
 open import Relation.Binary.Reasoning.Setoid S
 open import Relation.Unary using (Pred)
@@ -67,45 +67,32 @@ module _ {_•_ : Op₂ A} (cong : Congruent₂ _•_) where
     y • x             ∎
 
 ------------------------------------------------------------------------
--- Involutive/SelfInverse functions
-
-module _ {f : Op₁ A} (inv : Involutive f) where
-
-  open FunDefs _≈_ _≈_
-
-  involutive⇒surjective : Surjective f
-  involutive⇒surjective y = f y , inv y
+-- SelfInverse
 
 module _ {f : Op₁ A} (self : SelfInverse f) where
 
   selfInverse⇒involutive : Involutive f
   selfInverse⇒involutive = reflexive+selfInverse⇒involutive _≈_ refl self
 
-  private
-
-    inv = selfInverse⇒involutive
-
-  open FunDefs _≈_ _≈_
-
-  selfInverse⇒congruent : Congruent f
+  selfInverse⇒congruent : Congruent _≈_ _≈_ f
   selfInverse⇒congruent {x} {y} x≈y = sym (self (begin
-    f (f x) ≈⟨ inv x ⟩
+    f (f x) ≈⟨ selfInverse⇒involutive x ⟩
     x       ≈⟨ x≈y ⟩
     y       ∎))
 
-  selfInverse⇒inverseᵇ : Inverseᵇ f f
-  selfInverse⇒inverseᵇ = inv , inv
+  selfInverse⇒inverseᵇ : Inverseᵇ _≈_ _≈_ f f
+  selfInverse⇒inverseᵇ = self ∘ sym , self ∘ sym
 
-  selfInverse⇒surjective : Surjective f
-  selfInverse⇒surjective = involutive⇒surjective inv
+  selfInverse⇒surjective : Surjective _≈_ _≈_ f
+  selfInverse⇒surjective y = f y , self ∘ sym
 
-  selfInverse⇒injective : Injective f
+  selfInverse⇒injective : Injective _≈_ _≈_ f
   selfInverse⇒injective {x} {y} x≈y = begin
     x       ≈˘⟨ self x≈y ⟩
-    f (f y) ≈⟨ inv y ⟩
+    f (f y) ≈⟨ selfInverse⇒involutive y ⟩
     y       ∎
 
-  selfInverse⇒bijective : Bijective f
+  selfInverse⇒bijective : Bijective _≈_ _≈_ f
   selfInverse⇒bijective = selfInverse⇒injective , selfInverse⇒surjective
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Lattice/Morphism/Construct/Composition.agda b/src/Algebra/Lattice/Morphism/Construct/Composition.agda
index 42f609c8de..fe5266af3d 100644
--- a/src/Algebra/Lattice/Morphism/Construct/Composition.agda
+++ b/src/Algebra/Lattice/Morphism/Construct/Composition.agda
@@ -57,5 +57,5 @@ module _ {L₁ : RawLattice a ℓ₁}
     → IsLatticeIsomorphism L₁ L₃ (g ∘ f)
   isLatticeIsomorphism f-iso g-iso = record
     { isLatticeMonomorphism = isLatticeMonomorphism F.isLatticeMonomorphism G.isLatticeMonomorphism
-    ; surjective            = Func.surjective (_≈_ L₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective            = Func.surjective _ _ (_≈_ L₃) F.surjective G.surjective
     } where module F = IsLatticeIsomorphism f-iso; module G = IsLatticeIsomorphism g-iso
diff --git a/src/Algebra/Lattice/Morphism/Construct/Identity.agda b/src/Algebra/Lattice/Morphism/Construct/Identity.agda
index 8531643540..5fba1b976b 100644
--- a/src/Algebra/Lattice/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Lattice/Morphism/Construct/Identity.agda
@@ -13,6 +13,7 @@ open import Algebra.Lattice.Morphism.Structures
   using ( module LatticeMorphisms )
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
+import Function.Construct.Identity as Id
 open import Level using (Level)
 open import Relation.Binary.Morphism.Construct.Identity using (isRelHomomorphism)
 open import Relation.Binary.Definitions using (Reflexive)
@@ -40,5 +41,5 @@ module _ (L : RawLattice c ℓ) (open RawLattice L) (refl : Reflexive _≈_) whe
   isLatticeIsomorphism : IsLatticeIsomorphism id
   isLatticeIsomorphism = record
     { isLatticeMonomorphism = isLatticeMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
diff --git a/src/Algebra/Lattice/Morphism/Structures.agda b/src/Algebra/Lattice/Morphism/Structures.agda
index 7e7c34c163..87cf92e616 100644
--- a/src/Algebra/Lattice/Morphism/Structures.agda
+++ b/src/Algebra/Lattice/Morphism/Structures.agda
@@ -12,7 +12,7 @@ open import Algebra.Morphism
 open import Algebra.Lattice.Bundles
 import Algebra.Morphism.Definitions as MorphismDefinitions
 open import Level using (Level; _⊔_)
-import Function.Definitions as FunctionDefinitions
+open import Function.Definitions
 open import Relation.Binary.Morphism.Structures
 open import Relation.Binary.Core
 
@@ -44,7 +44,6 @@ module LatticeMorphisms (L₁ : RawLattice a ℓ₁) (L₂ : RawLattice b ℓ₂
   module ∧ = MagmaMorphisms ∧-rawMagma₁ ∧-rawMagma₂
 
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
 
 
   record IsLatticeHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
@@ -71,7 +70,7 @@ module LatticeMorphisms (L₁ : RawLattice a ℓ₁) (L₂ : RawLattice b ℓ₂
   record IsLatticeMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isLatticeHomomorphism : IsLatticeHomomorphism ⟦_⟧
-      injective             : Injective ⟦_⟧
+      injective             : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsLatticeHomomorphism isLatticeHomomorphism public
 
@@ -94,7 +93,7 @@ module LatticeMorphisms (L₁ : RawLattice a ℓ₁) (L₂ : RawLattice b ℓ₂
   record IsLatticeIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isLatticeMonomorphism : IsLatticeMonomorphism ⟦_⟧
-      surjective            : Surjective ⟦_⟧
+      surjective            : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsLatticeMonomorphism isLatticeMonomorphism public
 
diff --git a/src/Algebra/Module/Morphism/Construct/Composition.agda b/src/Algebra/Module/Morphism/Construct/Composition.agda
index 3aeb7a5f26..8c39496dd4 100644
--- a/src/Algebra/Module/Morphism/Construct/Composition.agda
+++ b/src/Algebra/Module/Morphism/Construct/Composition.agda
@@ -51,7 +51,7 @@ module _
                                 IsLeftSemimoduleIsomorphism M₁ M₃ (g ∘ f)
   isLeftSemimoduleIsomorphism f-iso g-iso = record
     { isLeftSemimoduleMonomorphism = isLeftSemimoduleMonomorphism F.isLeftSemimoduleMonomorphism G.isLeftSemimoduleMonomorphism
-    ; surjective                   = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective                   = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsLeftSemimoduleIsomorphism f-iso; module G = IsLeftSemimoduleIsomorphism g-iso
 
 module _
@@ -85,7 +85,7 @@ module _
                             IsLeftModuleIsomorphism M₁ M₃ (g ∘ f)
   isLeftModuleIsomorphism f-iso g-iso = record
     { isLeftModuleMonomorphism = isLeftModuleMonomorphism F.isLeftModuleMonomorphism G.isLeftModuleMonomorphism
-    ; surjective               = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective               = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsLeftModuleIsomorphism f-iso; module G = IsLeftModuleIsomorphism g-iso
 
 module _
@@ -119,7 +119,7 @@ module _
                                  IsRightSemimoduleIsomorphism M₁ M₃ (g ∘ f)
   isRightSemimoduleIsomorphism f-iso g-iso = record
     { isRightSemimoduleMonomorphism = isRightSemimoduleMonomorphism F.isRightSemimoduleMonomorphism G.isRightSemimoduleMonomorphism
-    ; surjective                    = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective                    = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsRightSemimoduleIsomorphism f-iso; module G = IsRightSemimoduleIsomorphism g-iso
 
 module _
@@ -153,7 +153,7 @@ module _
                              IsRightModuleIsomorphism M₁ M₃ (g ∘ f)
   isRightModuleIsomorphism f-iso g-iso = record
     { isRightModuleMonomorphism = isRightModuleMonomorphism F.isRightModuleMonomorphism G.isRightModuleMonomorphism
-    ; surjective                = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective                = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsRightModuleIsomorphism f-iso; module G = IsRightModuleIsomorphism g-iso
 
 module _
@@ -189,7 +189,7 @@ module _
                               IsBisemimoduleIsomorphism M₁ M₃ (g ∘ f)
   isBisemimoduleIsomorphism f-iso g-iso = record
     { isBisemimoduleMonomorphism = isBisemimoduleMonomorphism F.isBisemimoduleMonomorphism G.isBisemimoduleMonomorphism
-    ; surjective                 = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective                 = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsBisemimoduleIsomorphism f-iso; module G = IsBisemimoduleIsomorphism g-iso
 
 module _
@@ -225,7 +225,7 @@ module _
                           IsBimoduleIsomorphism M₁ M₃ (g ∘ f)
   isBimoduleIsomorphism f-iso g-iso = record
     { isBimoduleMonomorphism = isBimoduleMonomorphism F.isBimoduleMonomorphism G.isBimoduleMonomorphism
-    ; surjective             = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective             = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsBimoduleIsomorphism f-iso; module G = IsBimoduleIsomorphism g-iso
 
 module _
@@ -258,7 +258,7 @@ module _
                             IsSemimoduleIsomorphism M₁ M₃ (g ∘ f)
   isSemimoduleIsomorphism f-iso g-iso = record
     { isSemimoduleMonomorphism = isSemimoduleMonomorphism F.isSemimoduleMonomorphism G.isSemimoduleMonomorphism
-    ; surjective               = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective               = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsSemimoduleIsomorphism f-iso; module G = IsSemimoduleIsomorphism g-iso
 
 module _
@@ -291,5 +291,5 @@ module _
                         IsModuleIsomorphism M₁ M₃ (g ∘ f)
   isModuleIsomorphism f-iso g-iso = record
     { isModuleMonomorphism = isModuleMonomorphism F.isModuleMonomorphism G.isModuleMonomorphism
-    ; surjective           = Func.surjective (_≈ᴹ_ M₁) _ _ (≈ᴹ-trans M₃) G.⟦⟧-cong F.surjective G.surjective
+    ; surjective           = Func.surjective _ _ (_≈ᴹ_ M₃) F.surjective G.surjective
     } where module F = IsModuleIsomorphism f-iso; module G = IsModuleIsomorphism g-iso
diff --git a/src/Algebra/Module/Morphism/Construct/Identity.agda b/src/Algebra/Module/Morphism/Construct/Identity.agda
index 90f6940a16..55735c2b24 100644
--- a/src/Algebra/Module/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Module/Morphism/Construct/Identity.agda
@@ -23,6 +23,7 @@ open import Algebra.Module.Morphism.Structures
 open import Algebra.Morphism.Construct.Identity
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
+import Function.Construct.Identity as Id
 open import Level using (Level)
 
 private
@@ -48,7 +49,7 @@ module _ {semiring : Semiring r ℓr} (M : LeftSemimodule semiring m ℓm) where
   isLeftSemimoduleIsomorphism : IsLeftSemimoduleIsomorphism id
   isLeftSemimoduleIsomorphism = record
     { isLeftSemimoduleMonomorphism = isLeftSemimoduleMonomorphism
-    ; surjective                   = _, ≈ᴹ-refl
+    ; surjective                   = Id.surjective _
     }
 
 module _ {ring : Ring r ℓr} (M : LeftModule ring m ℓm) where
@@ -70,7 +71,7 @@ module _ {ring : Ring r ℓr} (M : LeftModule ring m ℓm) where
   isLeftModuleIsomorphism : IsLeftModuleIsomorphism id
   isLeftModuleIsomorphism = record
     { isLeftModuleMonomorphism = isLeftModuleMonomorphism
-    ; surjective               = _, ≈ᴹ-refl
+    ; surjective               = Id.surjective _
     }
 
 module _ {semiring : Semiring r ℓr} (M : RightSemimodule semiring m ℓm) where
@@ -92,7 +93,7 @@ module _ {semiring : Semiring r ℓr} (M : RightSemimodule semiring m ℓm) wher
   isRightSemimoduleIsomorphism : IsRightSemimoduleIsomorphism id
   isRightSemimoduleIsomorphism = record
     { isRightSemimoduleMonomorphism = isRightSemimoduleMonomorphism
-    ; surjective                    = _, ≈ᴹ-refl
+    ; surjective                    = Id.surjective _
     }
 
 module _ {ring : Ring r ℓr} (M : RightModule ring m ℓm) where
@@ -114,7 +115,7 @@ module _ {ring : Ring r ℓr} (M : RightModule ring m ℓm) where
   isRightModuleIsomorphism : IsRightModuleIsomorphism id
   isRightModuleIsomorphism = record
     { isRightModuleMonomorphism = isRightModuleMonomorphism
-    ; surjective                = _, ≈ᴹ-refl
+    ; surjective                = Id.surjective _
     }
 
 module _ {R-semiring : Semiring r ℓr} {S-semiring : Semiring s ℓs} (M : Bisemimodule R-semiring S-semiring m ℓm) where
@@ -137,7 +138,7 @@ module _ {R-semiring : Semiring r ℓr} {S-semiring : Semiring s ℓs} (M : Bise
   isBisemimoduleIsomorphism : IsBisemimoduleIsomorphism id
   isBisemimoduleIsomorphism = record
     { isBisemimoduleMonomorphism = isBisemimoduleMonomorphism
-    ; surjective                 = _, ≈ᴹ-refl
+    ; surjective                 = Id.surjective _
     }
 
 module _ {R-ring : Ring r ℓr} {S-ring : Ring r ℓr} (M : Bimodule R-ring S-ring m ℓm) where
@@ -160,7 +161,7 @@ module _ {R-ring : Ring r ℓr} {S-ring : Ring r ℓr} (M : Bimodule R-ring S-ri
   isBimoduleIsomorphism : IsBimoduleIsomorphism id
   isBimoduleIsomorphism = record
     { isBimoduleMonomorphism = isBimoduleMonomorphism
-    ; surjective             = _, ≈ᴹ-refl
+    ; surjective             = Id.surjective _
     }
 
 module _ {commutativeSemiring : CommutativeSemiring r ℓr} (M : Semimodule commutativeSemiring m ℓm) where
@@ -181,7 +182,7 @@ module _ {commutativeSemiring : CommutativeSemiring r ℓr} (M : Semimodule comm
   isSemimoduleIsomorphism : IsSemimoduleIsomorphism id
   isSemimoduleIsomorphism = record
     { isSemimoduleMonomorphism = isSemimoduleMonomorphism
-    ; surjective               = _, ≈ᴹ-refl
+    ; surjective               = Id.surjective _
     }
 
 module _ {commutativeRing : CommutativeRing r ℓr} (M : Module commutativeRing m ℓm) where
@@ -202,5 +203,5 @@ module _ {commutativeRing : CommutativeRing r ℓr} (M : Module commutativeRing
   isModuleIsomorphism : IsModuleIsomorphism id
   isModuleIsomorphism = record
     { isModuleMonomorphism = isModuleMonomorphism
-    ; surjective           = _, ≈ᴹ-refl
+    ; surjective           = Id.surjective _
     }
diff --git a/src/Algebra/Module/Morphism/Structures.agda b/src/Algebra/Module/Morphism/Structures.agda
index 3dc5fba3c5..96fb5ffa0a 100644
--- a/src/Algebra/Module/Morphism/Structures.agda
+++ b/src/Algebra/Module/Morphism/Structures.agda
@@ -11,7 +11,7 @@ open import Algebra.Bundles
 open import Algebra.Module.Bundles
 import Algebra.Module.Morphism.Definitions as MorphismDefinitions
 import Algebra.Morphism.Structures as MorphismStructures
-import Function.Definitions as FunctionDefinitions
+open import Function.Definitions
 open import Level
 
 private
@@ -28,7 +28,6 @@ module LeftSemimoduleMorphisms
   open LeftSemimodule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _≈ᴹ_ to _≈ᴹ₁_)
   open LeftSemimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open MorphismStructures.MonoidMorphisms (LeftSemimodule.+ᴹ-rawMonoid M₁) (LeftSemimodule.+ᴹ-rawMonoid M₂)
 
   record IsLeftSemimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
@@ -43,7 +42,7 @@ module LeftSemimoduleMorphisms
   record IsLeftSemimoduleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isLeftSemimoduleHomomorphism : IsLeftSemimoduleHomomorphism ⟦_⟧
-      injective                    : Injective ⟦_⟧
+      injective                    : Injective  _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsLeftSemimoduleHomomorphism isLeftSemimoduleHomomorphism public
 
@@ -60,7 +59,7 @@ module LeftSemimoduleMorphisms
   record IsLeftSemimoduleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isLeftSemimoduleMonomorphism : IsLeftSemimoduleMonomorphism ⟦_⟧
-      surjective                   : Surjective ⟦_⟧
+      surjective                   : Surjective  _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsLeftSemimoduleMonomorphism isLeftSemimoduleMonomorphism public
 
@@ -84,7 +83,6 @@ module LeftModuleMorphisms
   open LeftModule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _≈ᴹ_ to _≈ᴹ₁_)
   open LeftModule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open MorphismStructures.GroupMorphisms (LeftModule.+ᴹ-rawGroup M₁) (LeftModule.+ᴹ-rawGroup M₂)
   open LeftSemimoduleMorphisms (LeftModule.leftSemimodule M₁) (LeftModule.leftSemimodule M₂)
 
@@ -108,7 +106,7 @@ module LeftModuleMorphisms
   record IsLeftModuleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isLeftModuleHomomorphism : IsLeftModuleHomomorphism ⟦_⟧
-      injective                : Injective ⟦_⟧
+      injective                : Injective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsLeftModuleHomomorphism isLeftModuleHomomorphism public
 
@@ -130,7 +128,7 @@ module LeftModuleMorphisms
   record IsLeftModuleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isLeftModuleMonomorphism : IsLeftModuleMonomorphism ⟦_⟧
-      surjective               : Surjective ⟦_⟧
+      surjective               : Surjective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsLeftModuleMonomorphism isLeftModuleMonomorphism public
 
@@ -159,7 +157,6 @@ module RightSemimoduleMorphisms
   open RightSemimodule M₁ renaming (Carrierᴹ to A; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
   open RightSemimodule M₂ renaming (Carrierᴹ to B; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open MorphismStructures.MonoidMorphisms (RightSemimodule.+ᴹ-rawMonoid M₁) (RightSemimodule.+ᴹ-rawMonoid M₂)
 
   record IsRightSemimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
@@ -174,7 +171,7 @@ module RightSemimoduleMorphisms
   record IsRightSemimoduleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isRightSemimoduleHomomorphism : IsRightSemimoduleHomomorphism ⟦_⟧
-      injective                    : Injective ⟦_⟧
+      injective                    : Injective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsRightSemimoduleHomomorphism isRightSemimoduleHomomorphism public
 
@@ -191,7 +188,7 @@ module RightSemimoduleMorphisms
   record IsRightSemimoduleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isRightSemimoduleMonomorphism : IsRightSemimoduleMonomorphism ⟦_⟧
-      surjective                   : Surjective ⟦_⟧
+      surjective                   : Surjective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsRightSemimoduleMonomorphism isRightSemimoduleMonomorphism public
 
@@ -215,7 +212,6 @@ module RightModuleMorphisms
   open RightModule M₁ renaming (Carrierᴹ to A; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
   open RightModule M₂ renaming (Carrierᴹ to B; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open MorphismStructures.GroupMorphisms (RightModule.+ᴹ-rawGroup M₁) (RightModule.+ᴹ-rawGroup M₂)
   open RightSemimoduleMorphisms (RightModule.rightSemimodule M₁) (RightModule.rightSemimodule M₂)
 
@@ -238,7 +234,7 @@ module RightModuleMorphisms
   record IsRightModuleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isRightModuleHomomorphism : IsRightModuleHomomorphism ⟦_⟧
-      injective                : Injective ⟦_⟧
+      injective                : Injective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsRightModuleHomomorphism isRightModuleHomomorphism public
 
@@ -260,7 +256,7 @@ module RightModuleMorphisms
   record IsRightModuleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isRightModuleMonomorphism : IsRightModuleMonomorphism ⟦_⟧
-      surjective               : Surjective ⟦_⟧
+      surjective               : Surjective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsRightModuleMonomorphism isRightModuleMonomorphism public
 
@@ -292,7 +288,6 @@ module BisemimoduleMorphisms
   open Bisemimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_ using (Homomorphicₗ)
   open MorphismDefinitions S A B _≈ᴹ₂_ using (Homomorphicᵣ)
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open MorphismStructures.MonoidMorphisms (Bisemimodule.+ᴹ-rawMonoid M₁) (Bisemimodule.+ᴹ-rawMonoid M₂)
   open LeftSemimoduleMorphisms (Bisemimodule.leftSemimodule M₁) (Bisemimodule.leftSemimodule M₂)
   open RightSemimoduleMorphisms (Bisemimodule.rightSemimodule M₁) (Bisemimodule.rightSemimodule M₂)
@@ -321,7 +316,7 @@ module BisemimoduleMorphisms
   record IsBisemimoduleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ s ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isBisemimoduleHomomorphism : IsBisemimoduleHomomorphism ⟦_⟧
-      injective                  : Injective ⟦_⟧
+      injective                  : Injective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsBisemimoduleHomomorphism isBisemimoduleHomomorphism public
 
@@ -343,7 +338,7 @@ module BisemimoduleMorphisms
   record IsBisemimoduleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ s ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isBisemimoduleMonomorphism : IsBisemimoduleMonomorphism ⟦_⟧
-      surjective                 : Surjective ⟦_⟧
+      surjective                 : Surjective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsBisemimoduleMonomorphism isBisemimoduleMonomorphism public
 
@@ -375,7 +370,6 @@ module BimoduleMorphisms
   open Bimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_ using (Homomorphicₗ)
   open MorphismDefinitions S A B _≈ᴹ₂_ using (Homomorphicᵣ)
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open MorphismStructures.GroupMorphisms (Bimodule.+ᴹ-rawGroup M₁) (Bimodule.+ᴹ-rawGroup M₂)
   open LeftModuleMorphisms (Bimodule.leftModule M₁) (Bimodule.leftModule M₂)
   open RightModuleMorphisms (Bimodule.rightModule M₁) (Bimodule.rightModule M₂)
@@ -418,7 +412,7 @@ module BimoduleMorphisms
   record IsBimoduleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ s ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isBimoduleHomomorphism : IsBimoduleHomomorphism ⟦_⟧
-      injective              : Injective ⟦_⟧
+      injective              : Injective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsBimoduleHomomorphism isBimoduleHomomorphism public
 
@@ -459,7 +453,7 @@ module BimoduleMorphisms
   record IsBimoduleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ s ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isBimoduleMonomorphism : IsBimoduleMonomorphism ⟦_⟧
-      surjective             : Surjective ⟦_⟧
+      surjective             : Surjective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsBimoduleMonomorphism isBimoduleMonomorphism public
 
@@ -505,7 +499,6 @@ module SemimoduleMorphisms
 
   open Semimodule M₁ renaming (Carrierᴹ to A; _≈ᴹ_ to _≈ᴹ₁_)
   open Semimodule M₂ renaming (Carrierᴹ to B; _≈ᴹ_ to _≈ᴹ₂_)
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open BisemimoduleMorphisms (Semimodule.bisemimodule M₁) (Semimodule.bisemimodule M₂)
 
   record IsSemimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
@@ -517,7 +510,7 @@ module SemimoduleMorphisms
   record IsSemimoduleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isSemimoduleHomomorphism : IsSemimoduleHomomorphism ⟦_⟧
-      injective                : Injective ⟦_⟧
+      injective                : Injective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsSemimoduleHomomorphism isSemimoduleHomomorphism public
 
@@ -535,7 +528,7 @@ module SemimoduleMorphisms
   record IsSemimoduleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isSemimoduleMonomorphism : IsSemimoduleMonomorphism ⟦_⟧
-      surjective               : Surjective ⟦_⟧
+      surjective               : Surjective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsSemimoduleMonomorphism isSemimoduleMonomorphism public
 
@@ -558,7 +551,6 @@ module ModuleMorphisms
 
   open Module M₁ renaming (Carrierᴹ to A; _≈ᴹ_ to _≈ᴹ₁_)
   open Module M₂ renaming (Carrierᴹ to B; _≈ᴹ_ to _≈ᴹ₂_)
-  open FunctionDefinitions _≈ᴹ₁_ _≈ᴹ₂_
   open BimoduleMorphisms (Module.bimodule M₁) (Module.bimodule M₂)
   open SemimoduleMorphisms (Module.semimodule M₁) (Module.semimodule M₂)
 
@@ -576,7 +568,7 @@ module ModuleMorphisms
   record IsModuleMonomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isModuleHomomorphism : IsModuleHomomorphism ⟦_⟧
-      injective            : Injective ⟦_⟧
+      injective            : Injective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsModuleHomomorphism isModuleHomomorphism public
 
@@ -601,7 +593,7 @@ module ModuleMorphisms
   record IsModuleIsomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ m₂ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
       isModuleMonomorphism : IsModuleMonomorphism ⟦_⟧
-      surjective           : Surjective ⟦_⟧
+      surjective           : Surjective _≈ᴹ₁_ _≈ᴹ₂_ ⟦_⟧
 
     open IsModuleMonomorphism isModuleMonomorphism public
 
diff --git a/src/Algebra/Morphism/Consequences.agda b/src/Algebra/Morphism/Consequences.agda
index c81e55a7bb..96175740e6 100644
--- a/src/Algebra/Morphism/Consequences.agda
+++ b/src/Algebra/Morphism/Consequences.agda
@@ -29,10 +29,10 @@ module _ {α α= β β=} (M₁ : Magma α α=) (M₂ : Magma β β=) where
                       Inverseᵇ _≈₁_ _≈₂_ f g →
                       Homomorphic₂ _ _ _≈₂_ f _∙₁_ _∙₂_  →
                       Homomorphic₂ _ _ _≈₁_ g _∙₂_ _∙₁_
-  homomorphic₂-inv {f} {g} g-cong (f∘g=id , g∘f=id) homo x y = begin
-    g (x ∙₂ y)             ≈⟨ M₁.sym (g-cong (M₂.∙-cong (f∘g=id x) (f∘g=id y))) ⟩
-    g (f (g x) ∙₂ f (g y)) ≈⟨ M₁.sym (g-cong (homo (g x) (g y))) ⟩
-    g (f (g x ∙₁ g y))     ≈⟨ g∘f=id (g x ∙₁ g y) ⟩
+  homomorphic₂-inv {f} {g} g-cong (invˡ , invʳ) homo x y = begin
+    g (x ∙₂ y)             ≈˘⟨ g-cong (M₂.∙-cong (invˡ M₁.refl) (invˡ M₁.refl)) ⟩
+    g (f (g x) ∙₂ f (g y)) ≈˘⟨ g-cong (homo (g x) (g y)) ⟩
+    g (f (g x ∙₁ g y))     ≈⟨ invʳ M₂.refl ⟩
     g x ∙₁ g y             ∎
     where open EqR M₁.setoid
 
@@ -41,8 +41,8 @@ module _ {α α= β β=} (M₁ : Magma α α=) (M₂ : Magma β β=) where
                       Homomorphic₂ _ _ _≈₂_ f _∙₁_ _∙₂_  →
                       Homomorphic₂ _ _ _≈₁_ g _∙₂_ _∙₁_
   homomorphic₂-inj {f} {g} inj invˡ homo x y = inj (begin
-    f (g (x ∙₂ y))      ≈⟨ invˡ (x ∙₂ y) ⟩
-    x ∙₂ y              ≈⟨ M₂.sym (M₂.∙-cong (invˡ x) (invˡ y)) ⟩
-    f (g x) ∙₂ f (g y)  ≈⟨ M₂.sym (homo (g x) (g y)) ⟩
+    f (g (x ∙₂ y))      ≈⟨ invˡ M₁.refl ⟩
+    x ∙₂ y              ≈˘⟨ M₂.∙-cong (invˡ M₁.refl) (invˡ M₁.refl) ⟩
+    f (g x) ∙₂ f (g y)  ≈˘⟨ homo (g x) (g y) ⟩
     f (g x ∙₁ g y)      ∎)
     where open EqR M₂.setoid
diff --git a/src/Algebra/Morphism/Construct/Composition.agda b/src/Algebra/Morphism/Construct/Composition.agda
index 991fb93433..e3f3a3ad71 100644
--- a/src/Algebra/Morphism/Construct/Composition.agda
+++ b/src/Algebra/Morphism/Construct/Composition.agda
@@ -57,7 +57,7 @@ module _ {M₁ : RawMagma a ℓ₁}
     → IsMagmaIsomorphism M₁ M₃ (g ∘ f)
   isMagmaIsomorphism f-iso g-iso = record
     { isMagmaMonomorphism = isMagmaMonomorphism F.isMagmaMonomorphism G.isMagmaMonomorphism
-    ; surjective          = Func.surjective (_≈_ M₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective          = Func.surjective _ _ (_≈_ M₃) F.surjective G.surjective
     } where module F = IsMagmaIsomorphism f-iso; module G = IsMagmaIsomorphism g-iso
 
 
@@ -97,7 +97,7 @@ module _ {M₁ : RawMonoid a ℓ₁}
     → IsMonoidIsomorphism M₁ M₃ (g ∘ f)
   isMonoidIsomorphism f-iso g-iso = record
     { isMonoidMonomorphism = isMonoidMonomorphism F.isMonoidMonomorphism G.isMonoidMonomorphism
-    ; surjective           = Func.surjective (_≈_ M₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective           = Func.surjective _ _(_≈_ M₃)  F.surjective G.surjective
     } where module F = IsMonoidIsomorphism f-iso; module G = IsMonoidIsomorphism g-iso
 
 
@@ -138,7 +138,7 @@ module _ {G₁ : RawGroup a ℓ₁}
     → IsGroupIsomorphism G₁ G₃ (g ∘ f)
   isGroupIsomorphism f-iso g-iso = record
     { isGroupMonomorphism = isGroupMonomorphism F.isGroupMonomorphism G.isGroupMonomorphism
-    ; surjective          = Func.surjective (_≈_ G₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective          = Func.surjective  _ _ (_≈_ G₃) F.surjective G.surjective
     } where module F = IsGroupIsomorphism f-iso; module G = IsGroupIsomorphism g-iso
 
 
@@ -178,7 +178,7 @@ module _ {R₁ : RawNearSemiring a ℓ₁}
     → IsNearSemiringIsomorphism R₁ R₃ (g ∘ f)
   isNearSemiringIsomorphism f-iso g-iso = record
     { isNearSemiringMonomorphism = isNearSemiringMonomorphism F.isNearSemiringMonomorphism G.isNearSemiringMonomorphism
-    ; surjective                 = Func.surjective (_≈_ R₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective                 = Func.surjective _ _ (_≈_ R₃) F.surjective G.surjective
     } where module F = IsNearSemiringIsomorphism f-iso; module G = IsNearSemiringIsomorphism g-iso
 
 
@@ -220,7 +220,7 @@ module _
     → IsSemiringIsomorphism R₁ R₃ (g ∘ f)
   isSemiringIsomorphism f-iso g-iso = record
     { isSemiringMonomorphism = isSemiringMonomorphism F.isSemiringMonomorphism G.isSemiringMonomorphism
-    ; surjective             = Func.surjective (_≈_ R₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective             = Func.surjective _ _ (_≈_ R₃) F.surjective G.surjective
     } where module F = IsSemiringIsomorphism f-iso; module G = IsSemiringIsomorphism g-iso
 
 ------------------------------------------------------------------------
@@ -260,7 +260,7 @@ module _ {R₁ : RawRingWithoutOne a ℓ₁}
     → IsRingWithoutOneIsoMorphism R₁ R₃ (g ∘ f)
   isRingWithoutOneIsoMorphism f-iso g-iso = record
     { isRingWithoutOneMonomorphism = isRingWithoutOneMonomorphism F.isRingWithoutOneMonomorphism G.isRingWithoutOneMonomorphism
-    ; surjective         = Func.surjective (_≈_ R₁) (_≈_ R₂) (_≈_ R₃) ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective         = Func.surjective _ _ (_≈_ R₃) F.surjective G.surjective
     } where module F = IsRingWithoutOneIsoMorphism f-iso; module G = IsRingWithoutOneIsoMorphism g-iso
 
 ------------------------------------------------------------------------
@@ -300,7 +300,7 @@ module _ {R₁ : RawRing a ℓ₁}
     → IsRingIsomorphism R₁ R₃ (g ∘ f)
   isRingIsomorphism f-iso g-iso = record
     { isRingMonomorphism = isRingMonomorphism F.isRingMonomorphism G.isRingMonomorphism
-    ; surjective         = Func.surjective (_≈_ R₁) _ _ ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective         = Func.surjective _ _ (_≈_ R₃) F.surjective G.surjective
     } where module F = IsRingIsomorphism f-iso; module G = IsRingIsomorphism g-iso
 
 ------------------------------------------------------------------------
@@ -342,7 +342,7 @@ module _ {Q₁ : RawQuasigroup a ℓ₁}
     → IsQuasigroupIsomorphism Q₁ Q₃ (g ∘ f)
   isQuasigroupIsomorphism f-iso g-iso = record
     { isQuasigroupMonomorphism = isQuasigroupMonomorphism F.isQuasigroupMonomorphism G.isQuasigroupMonomorphism
-    ; surjective               = Func.surjective (_≈_ Q₁) (_≈_ Q₂) (_≈_ Q₃) ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective               = Func.surjective _ _ (_≈_ Q₃) F.surjective G.surjective
     } where module F = IsQuasigroupIsomorphism f-iso; module G = IsQuasigroupIsomorphism g-iso
 
 ------------------------------------------------------------------------
@@ -382,5 +382,5 @@ module _ {L₁ : RawLoop a ℓ₁}
     → IsLoopIsomorphism L₁ L₃ (g ∘ f)
   isLoopIsomorphism f-iso g-iso = record
     { isLoopMonomorphism = isLoopMonomorphism F.isLoopMonomorphism G.isLoopMonomorphism
-    ; surjective               = Func.surjective (_≈_ L₁) (_≈_ L₂) (_≈_ L₃) ≈₃-trans G.⟦⟧-cong F.surjective G.surjective
+    ; surjective         = Func.surjective _ _ (_≈_ L₃) F.surjective G.surjective
     } where module F = IsLoopIsomorphism f-iso; module G = IsLoopIsomorphism g-iso
diff --git a/src/Algebra/Morphism/Construct/Identity.agda b/src/Algebra/Morphism/Construct/Identity.agda
index 5529ec3c01..63e95e54e6 100644
--- a/src/Algebra/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Morphism/Construct/Identity.agda
@@ -22,6 +22,7 @@ open import Algebra.Morphism.Structures
         )
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
+import Function.Construct.Identity as Id
 open import Level using (Level)
 open import Relation.Binary.Morphism.Construct.Identity using (isRelHomomorphism)
 open import Relation.Binary.Definitions using (Reflexive)
@@ -51,7 +52,7 @@ module _ (M : RawMagma c ℓ) (open RawMagma M) (refl : Reflexive _≈_) where
   isMagmaIsomorphism : IsMagmaIsomorphism id
   isMagmaIsomorphism = record
     { isMagmaMonomorphism = isMagmaMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -75,7 +76,7 @@ module _ (M : RawMonoid c ℓ) (open RawMonoid M) (refl : Reflexive _≈_) where
   isMonoidIsomorphism : IsMonoidIsomorphism id
   isMonoidIsomorphism = record
     { isMonoidMonomorphism = isMonoidMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -99,7 +100,7 @@ module _ (G : RawGroup c ℓ) (open RawGroup G) (refl : Reflexive _≈_) where
   isGroupIsomorphism : IsGroupIsomorphism id
   isGroupIsomorphism = record
     { isGroupMonomorphism = isGroupMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -123,7 +124,7 @@ module _ (R : RawNearSemiring c ℓ) (open RawNearSemiring R) (refl : Reflexive
   isNearSemiringIsomorphism : IsNearSemiringIsomorphism id
   isNearSemiringIsomorphism = record
     { isNearSemiringMonomorphism = isNearSemiringMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -147,7 +148,7 @@ module _ (R : RawSemiring c ℓ) (open RawSemiring R) (refl : Reflexive _≈_) w
   isSemiringIsomorphism : IsSemiringIsomorphism id
   isSemiringIsomorphism = record
     { isSemiringMonomorphism = isSemiringMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -171,7 +172,7 @@ module _ (R : RawRingWithoutOne c ℓ) (open RawRingWithoutOne R) (refl : Reflex
   isRingWithoutOneIsoMorphism : IsRingWithoutOneIsoMorphism id
   isRingWithoutOneIsoMorphism = record
     { isRingWithoutOneMonomorphism = isRingWithoutOneMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -195,7 +196,7 @@ module _ (R : RawRing c ℓ) (open RawRing R) (refl : Reflexive _≈_) where
   isRingIsomorphism : IsRingIsomorphism id
   isRingIsomorphism = record
     { isRingMonomorphism = isRingMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -221,7 +222,7 @@ module _ (Q : RawQuasigroup c ℓ) (open RawQuasigroup Q) (refl : Reflexive _≈
   isQuasigroupIsomorphism : IsQuasigroupIsomorphism id
   isQuasigroupIsomorphism = record
     { isQuasigroupMonomorphism = isQuasigroupMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
 
 ------------------------------------------------------------------------
@@ -245,5 +246,5 @@ module _ (L : RawLoop c ℓ) (open RawLoop L) (refl : Reflexive _≈_) where
   isLoopIsomorphism : IsLoopIsomorphism id
   isLoopIsomorphism = record
     { isLoopMonomorphism = isLoopMonomorphism
-    ; surjective = _, refl
+    ; surjective = Id.surjective _
     }
diff --git a/src/Algebra/Morphism/Structures.agda b/src/Algebra/Morphism/Structures.agda
index 525cbf5da0..eb8970ba5c 100644
--- a/src/Algebra/Morphism/Structures.agda
+++ b/src/Algebra/Morphism/Structures.agda
@@ -14,7 +14,7 @@ open import Algebra.Core
 open import Algebra.Bundles
 import Algebra.Morphism.Definitions as MorphismDefinitions
 open import Level using (Level; _⊔_)
-import Function.Definitions as FunctionDefinitions
+open import Function.Definitions
 open import Relation.Binary.Morphism.Structures
 
 private
@@ -30,7 +30,6 @@ module MagmaMorphisms (M₁ : RawMagma a ℓ₁) (M₂ : RawMagma b ℓ₂) wher
   open RawMagma M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_)
   open RawMagma M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_)
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
 
 
   record IsMagmaHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
@@ -45,7 +44,7 @@ module MagmaMorphisms (M₁ : RawMagma a ℓ₁) (M₂ : RawMagma b ℓ₂) wher
   record IsMagmaMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isMagmaHomomorphism : IsMagmaHomomorphism ⟦_⟧
-      injective           : Injective ⟦_⟧
+      injective           : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsMagmaHomomorphism isMagmaHomomorphism public
 
@@ -59,7 +58,7 @@ module MagmaMorphisms (M₁ : RawMagma a ℓ₁) (M₂ : RawMagma b ℓ₂) wher
   record IsMagmaIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isMagmaMonomorphism : IsMagmaMonomorphism ⟦_⟧
-      surjective          : Surjective ⟦_⟧
+      surjective          : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsMagmaMonomorphism isMagmaMonomorphism public
 
@@ -79,7 +78,6 @@ module MonoidMorphisms (M₁ : RawMonoid a ℓ₁) (M₂ : RawMonoid b ℓ₂) w
   open RawMonoid M₁ renaming (Carrier to A; _≈_ to _≈₁_; _∙_ to _∙_; ε to ε₁)
   open RawMonoid M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; ε to ε₂)
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
   open MagmaMorphisms (RawMonoid.rawMagma M₁) (RawMonoid.rawMagma M₂)
 
 
@@ -94,7 +92,7 @@ module MonoidMorphisms (M₁ : RawMonoid a ℓ₁) (M₂ : RawMonoid b ℓ₂) w
   record IsMonoidMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isMonoidHomomorphism : IsMonoidHomomorphism ⟦_⟧
-      injective            : Injective ⟦_⟧
+      injective            : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsMonoidHomomorphism isMonoidHomomorphism public
 
@@ -111,7 +109,7 @@ module MonoidMorphisms (M₁ : RawMonoid a ℓ₁) (M₂ : RawMonoid b ℓ₂) w
   record IsMonoidIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isMonoidMonomorphism : IsMonoidMonomorphism ⟦_⟧
-      surjective           : Surjective ⟦_⟧
+      surjective           : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsMonoidMonomorphism isMonoidMonomorphism public
 
@@ -136,7 +134,6 @@ module GroupMorphisms (G₁ : RawGroup a ℓ₁) (G₂ : RawGroup b ℓ₂) wher
   open RawGroup G₂ renaming
     (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; _⁻¹ to _⁻¹₂; ε to ε₂)
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
   open MagmaMorphisms  (RawGroup.rawMagma  G₁) (RawGroup.rawMagma  G₂)
   open MonoidMorphisms (RawGroup.rawMonoid G₁) (RawGroup.rawMonoid G₂)
 
@@ -152,7 +149,7 @@ module GroupMorphisms (G₁ : RawGroup a ℓ₁) (G₂ : RawGroup b ℓ₂) wher
   record IsGroupMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isGroupHomomorphism : IsGroupHomomorphism ⟦_⟧
-      injective           : Injective ⟦_⟧
+      injective           : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsGroupHomomorphism isGroupHomomorphism public
       renaming (homo to ∙-homo)
@@ -170,7 +167,7 @@ module GroupMorphisms (G₁ : RawGroup a ℓ₁) (G₂ : RawGroup b ℓ₂) wher
   record IsGroupIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isGroupMonomorphism : IsGroupMonomorphism ⟦_⟧
-      surjective          : Surjective ⟦_⟧
+      surjective          : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsGroupMonomorphism isGroupMonomorphism public
 
@@ -207,7 +204,6 @@ module NearSemiringMorphisms (R₁ : RawNearSemiring a ℓ₁) (R₂ : RawNearSe
     module * = MagmaMorphisms *-rawMagma₁ *-rawMagma₂
 
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
 
 
   record IsNearSemiringHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
@@ -227,7 +223,7 @@ module NearSemiringMorphisms (R₁ : RawNearSemiring a ℓ₁) (R₂ : RawNearSe
   record IsNearSemiringMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isNearSemiringHomomorphism : IsNearSemiringHomomorphism ⟦_⟧
-      injective          : Injective ⟦_⟧
+      injective          : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsNearSemiringHomomorphism isNearSemiringHomomorphism public
 
@@ -250,7 +246,7 @@ module NearSemiringMorphisms (R₁ : RawNearSemiring a ℓ₁) (R₂ : RawNearSe
   record IsNearSemiringIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isNearSemiringMonomorphism : IsNearSemiringMonomorphism ⟦_⟧
-      surjective                 : Surjective ⟦_⟧
+      surjective                 : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsNearSemiringMonomorphism isNearSemiringMonomorphism public
 
@@ -292,7 +288,6 @@ module SemiringMorphisms (R₁ : RawSemiring a ℓ₁) (R₂ : RawSemiring b ℓ
     module * = MonoidMorphisms *-rawMonoid₁ *-rawMonoid₂
 
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
   open NearSemiringMorphisms rawNearSemiring₁ rawNearSemiring₂
 
   record IsSemiringHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
@@ -311,7 +306,7 @@ module SemiringMorphisms (R₁ : RawSemiring a ℓ₁) (R₂ : RawSemiring b ℓ
   record IsSemiringMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isSemiringHomomorphism : IsSemiringHomomorphism ⟦_⟧
-      injective              : Injective ⟦_⟧
+      injective              : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsSemiringHomomorphism isSemiringHomomorphism public
 
@@ -333,7 +328,7 @@ module SemiringMorphisms (R₁ : RawSemiring a ℓ₁) (R₂ : RawSemiring b ℓ
   record IsSemiringIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isSemiringMonomorphism : IsSemiringMonomorphism ⟦_⟧
-      surjective             : Surjective ⟦_⟧
+      surjective             : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsSemiringMonomorphism isSemiringMonomorphism public
 
@@ -375,7 +370,6 @@ module RingWithoutOneMorphisms (R₁ : RawRingWithoutOne a ℓ₁) (R₂ : RawRi
     module * = MagmaMorphisms *-rawMagma₁ *-rawMagma₂
 
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
 
   record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
@@ -394,7 +388,7 @@ module RingWithoutOneMorphisms (R₁ : RawRingWithoutOne a ℓ₁) (R₂ : RawRi
   record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isRingWithoutOneHomomorphism : IsRingWithoutOneHomomorphism ⟦_⟧
-      injective                    : Injective ⟦_⟧
+      injective                    : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsRingWithoutOneHomomorphism isRingWithoutOneHomomorphism public
 
@@ -417,7 +411,7 @@ module RingWithoutOneMorphisms (R₁ : RawRingWithoutOne a ℓ₁) (R₂ : RawRi
   record IsRingWithoutOneIsoMorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isRingWithoutOneMonomorphism : IsRingWithoutOneMonomorphism ⟦_⟧
-      surjective                   : Surjective ⟦_⟧
+      surjective                   : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsRingWithoutOneMonomorphism isRingWithoutOneMonomorphism public
 
@@ -462,7 +456,6 @@ module RingMorphisms (R₁ : RawRing a ℓ₁) (R₂ : RawRing b ℓ₂) where
   module * = MonoidMorphisms *-rawMonoid₁ *-rawMonoid₂
 
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
   open SemiringMorphisms rawSemiring₁ rawSemiring₂
 
 
@@ -482,7 +475,7 @@ module RingMorphisms (R₁ : RawRing a ℓ₁) (R₂ : RawRing b ℓ₂) where
   record IsRingMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isRingHomomorphism : IsRingHomomorphism ⟦_⟧
-      injective          : Injective ⟦_⟧
+      injective          : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsRingHomomorphism isRingHomomorphism public
 
@@ -518,7 +511,7 @@ module RingMorphisms (R₁ : RawRing a ℓ₁) (R₂ : RawRing b ℓ₂) where
   record IsRingIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isRingMonomorphism : IsRingMonomorphism ⟦_⟧
-      surjective         : Surjective ⟦_⟧
+      surjective         : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsRingMonomorphism isRingMonomorphism public
 
@@ -568,7 +561,6 @@ module QuasigroupMorphisms (Q₁ : RawQuasigroup a ℓ₁) (Q₂ : RawQuasigroup
   module // = MagmaMorphisms //-rawMagma₁ //-rawMagma₂
 
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
 
   record IsQuasigroupHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
@@ -601,7 +593,7 @@ module QuasigroupMorphisms (Q₁ : RawQuasigroup a ℓ₁) (Q₂ : RawQuasigroup
   record IsQuasigroupMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isQuasigroupHomomorphism : IsQuasigroupHomomorphism ⟦_⟧
-      injective                : Injective ⟦_⟧
+      injective                : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsQuasigroupHomomorphism isQuasigroupHomomorphism public
 
@@ -631,7 +623,7 @@ module QuasigroupMorphisms (Q₁ : RawQuasigroup a ℓ₁) (Q₂ : RawQuasigroup
   record IsQuasigroupIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isQuasigroupMonomorphism : IsQuasigroupMonomorphism ⟦_⟧
-      surjective               : Surjective ⟦_⟧
+      surjective               : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsQuasigroupMonomorphism isQuasigroupMonomorphism public
 
@@ -669,7 +661,6 @@ module LoopMorphisms (L₁ : RawLoop a ℓ₁) (L₂ : RawLoop b ℓ₂) where
                             \\-rawMagma to \\-rawMagma₂; //-rawMagma to //-rawMagma₂;
                             _≈_ to _≈₂_; _∙_ to _∙₂_; _\\_ to _\\₂_; _//_ to _//₂_ ; ε to ε₂)
   open MorphismDefinitions A B _≈₂_
-  open FunctionDefinitions _≈₁_ _≈₂_
 
   open QuasigroupMorphisms (RawLoop.rawQuasigroup L₁) (RawLoop.rawQuasigroup L₂)
 
@@ -683,14 +674,14 @@ module LoopMorphisms (L₁ : RawLoop a ℓ₁) (L₂ : RawLoop b ℓ₂) where
   record IsLoopMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isLoopHomomorphism   : IsLoopHomomorphism ⟦_⟧
-      injective            : Injective ⟦_⟧
+      injective            : Injective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsLoopHomomorphism isLoopHomomorphism public
 
   record IsLoopIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       isLoopMonomorphism   : IsLoopMonomorphism ⟦_⟧
-      surjective           : Surjective ⟦_⟧
+      surjective           : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
     open IsLoopMonomorphism isLoopMonomorphism public
 
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 9c7cbdf170..5fe0fa0b2c 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -20,7 +20,7 @@ open import Function.Bundles using (_↔_; Injection; Inverse; mk↔′)
 open import Function.Construct.Composition using (_↔-∘_)
 open import Function.Construct.Identity using (↔-id)
 open import Function.Construct.Symmetry using (↔-sym)
-open import Function.Definitions using (Inverseˡ; Inverseʳ)
+open import Function.Definitions
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Properties.Inverse using (↔⇒↣)
 open import Function.Base using (_∘_)
@@ -57,7 +57,7 @@ Permutation′ n = Permutation n n
 -- Helper functions
 
 permutation : ∀ (f : Fin m → Fin n) (g : Fin n → Fin m) →
-              Inverseˡ _≡_ _≡_ f g → Inverseʳ _≡_ _≡_ f g → Permutation m n
+              StrictlyInverseˡ _≡_ f g → StrictlyInverseʳ _≡_ f g → Permutation m n
 permutation = mk↔′
 
 infixl 5 _⟨$⟩ʳ_ _⟨$⟩ˡ_
@@ -68,10 +68,10 @@ _⟨$⟩ˡ_ : Permutation m n → Fin n → Fin m
 _⟨$⟩ˡ_ = Inverse.from
 
 inverseˡ : ∀ (π : Permutation m n) {i} → π ⟨$⟩ˡ (π ⟨$⟩ʳ i) ≡ i
-inverseˡ π = Inverse.inverseʳ π _
+inverseˡ π = Inverse.inverseʳ π P.refl
 
 inverseʳ : ∀ (π : Permutation m n) {i} → π ⟨$⟩ʳ (π ⟨$⟩ˡ i) ≡ i
-inverseʳ π = Inverse.inverseˡ π _
+inverseʳ π = Inverse.inverseˡ π P.refl
 
 ------------------------------------------------------------------------
 -- Equality
@@ -142,7 +142,7 @@ remove {m} {n} i π = permutation to from inverseˡ′ inverseʳ′
   from : Fin n → Fin m
   from j = punchOut {j = πˡ (punchIn (πʳ i) j)} from-punchOut
 
-  inverseʳ′ : Inverseʳ _≡_ _≡_ to from
+  inverseʳ′ : StrictlyInverseʳ _≡_ to from
   inverseʳ′ j = begin
     from (to j)                                                      ≡⟨⟩
     punchOut {i = i} {πˡ (punchIn (πʳ i) (punchOut to-punchOut))} _  ≡⟨ punchOut-cong′ i (cong πˡ (punchIn-punchOut _)) ⟩
@@ -150,7 +150,7 @@ remove {m} {n} i π = permutation to from inverseˡ′ inverseʳ′
     punchOut {i = i} {punchIn i j}                                _  ≡⟨ punchOut-punchIn i ⟩
     j                                                                ∎
 
-  inverseˡ′ : Inverseˡ _≡_ _≡_ to from
+  inverseˡ′ : StrictlyInverseˡ _≡_ to from
   inverseˡ′ j = begin
     to (from j)                                                       ≡⟨⟩
     punchOut {i = πʳ i} {πʳ (punchIn i (punchOut from-punchOut))}  _  ≡⟨ punchOut-cong′ (πʳ i) (cong πʳ (punchIn-punchOut _)) ⟩
@@ -171,11 +171,11 @@ lift₀ {m} {n} π = permutation to from inverseˡ′ inverseʳ′
   from 0F      = 0F
   from (suc i) = suc (π ⟨$⟩ˡ i)
 
-  inverseʳ′ : Inverseʳ _≡_ _≡_ to from
+  inverseʳ′ : StrictlyInverseʳ _≡_ to from
   inverseʳ′ 0F      = refl
   inverseʳ′ (suc j) = cong suc (inverseˡ π)
 
-  inverseˡ′ : Inverseˡ _≡_ _≡_ to from
+  inverseˡ′ : StrictlyInverseˡ _≡_ to from
   inverseˡ′ 0F      = refl
   inverseˡ′ (suc j) = cong suc (inverseʳ π)
 
@@ -194,7 +194,7 @@ insert {m} {n} i j π = permutation to from inverseˡ′ inverseʳ′
   ... | yes j≡k = i
   ... | no  j≢k = punchIn i (π ⟨$⟩ˡ punchOut j≢k)
 
-  inverseʳ′ : Inverseʳ _≡_ _≡_ to from
+  inverseʳ′ : StrictlyInverseʳ _≡_ to from
   inverseʳ′ k with i ≟ k
   ... | yes i≡k rewrite proj₂ (dec-yes (j ≟ j) refl) = i≡k
   ... | no  i≢k
@@ -207,7 +207,7 @@ insert {m} {n} i j π = permutation to from inverseˡ′ inverseʳ′
     punchIn i (punchOut i≢k)                                              ≡⟨ punchIn-punchOut i≢k ⟩
     k                                                                     ∎
 
-  inverseˡ′ : Inverseˡ _≡_ _≡_ to from
+  inverseˡ′ : StrictlyInverseˡ _≡_ to from
   inverseˡ′ k with j ≟ k
   ... | yes j≡k rewrite proj₂ (dec-yes (i ≟ i) refl) = j≡k
   ... | no  j≢k
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 615728e971..2bacfe58ca 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -31,9 +31,8 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)
 open import Data.Sum.Properties using ([,]-map; [,]-∘)
 open import Function.Base using (_∘_; id; _$_; flip)
 open import Function.Bundles using (Injection; _↣_; _⇔_; _↔_; mk⇔; mk↔′)
-open import Function.Definitions using (Injective)
-open import Function.Definitions.Core2 using (Surjective)
-open import Function.Consequences using (contraInjective)
+open import Function.Definitions using (Injective; Surjective)
+open import Function.Consequences.Propositional using (contraInjective)
 open import Function.Construct.Composition as Comp hiding (injective)
 open import Level using (Level)
 open import Relation.Binary as B hiding (Decidable; _⇔_)
@@ -870,10 +869,10 @@ punchOut-punchIn (suc i) {suc j} = cong suc (begin
 -- pinch
 ------------------------------------------------------------------------
 
-pinch-surjective : ∀ (i : Fin n) → Surjective _≡_ (pinch i)
-pinch-surjective _       zero    = zero , refl
-pinch-surjective zero    (suc j) = suc (suc j) , refl
-pinch-surjective (suc i) (suc j) = map suc (cong suc) (pinch-surjective i j)
+pinch-surjective : ∀ (i : Fin n) → Surjective _≡_ _≡_ (pinch i)
+pinch-surjective _       zero    = zero , λ { refl → refl }
+pinch-surjective zero    (suc j) = suc (suc j) , λ { refl → refl }
+pinch-surjective (suc i) (suc j) = map suc (λ {f refl → cong suc (f refl)}) (pinch-surjective i j)
 
 pinch-mono-≤ : ∀ (i : Fin n) → (pinch i) Preserves _≤_ ⟶ _≤_
 pinch-mono-≤ 0F      {0F}    {k}     0≤n       = z≤n
@@ -990,8 +989,8 @@ injective⇒≤ {zero}  {_}     {f} _   = z≤n
 injective⇒≤ {suc _} {zero}  {f} _   = contradiction (f zero) ¬Fin0
 injective⇒≤ {suc _} {suc _} {f} inj = s≤s (injective⇒≤ (λ eq →
   suc-injective (inj (punchOut-injective
-    (contraInjective _≡_ _≡_ inj 0≢1+n)
-    (contraInjective _≡_ _≡_ inj 0≢1+n) eq))))
+    (contraInjective inj 0≢1+n)
+    (contraInjective inj 0≢1+n) eq))))
 
 <⇒notInjective : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective _≡_ _≡_ f)
 <⇒notInjective n<m inj = ℕₚ.≤⇒≯ (injective⇒≤ inj) n<m
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index c34e1b8904..7e7a3a090d 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -28,7 +28,7 @@ open import Data.Product using (∃₂; _×_; _,_; <_,_>; proj₂; uncurry)
 
 open import Function.Base
 open import Function.Bundles using (_⤖_; _⇔_ ; mk⤖; mk⇔)
-
+open import Function.Consequences.Propositional using (strictlySurjective⇒surjective)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (Dec; does; _because_; yes; no; ¬_)
 open import Relation.Nullary.Decidable as Dec using (¬?)
@@ -345,7 +345,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   toAny∘fromAny≗id (there p) = P.cong there (toAny∘fromAny≗id p)
 
   Sublist-[x]-bijection : ∀ {x xs} → (Sublist R [ x ] xs) ⤖ (Any (R x) xs)
-  Sublist-[x]-bijection = mk⤖ (toAny-injective , < fromAny , toAny∘fromAny≗id >)
+  Sublist-[x]-bijection = mk⤖ (toAny-injective , strictlySurjective⇒surjective < fromAny , toAny∘fromAny≗id >)
 
 ------------------------------------------------------------------------
 -- Relational properties
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index 78a365324b..f01fd90a48 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -20,14 +20,14 @@ open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function.Base using (_∘_)
 open import Function.Bundles using (Surjection)
 open import Function.Definitions using (Surjective)
+open import Function.Consequences using (strictlySurjective⇒surjective)
 open import Level
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Properties.Setoid using (respʳ-flip)
 
 module Data.List.Relation.Unary.Enumerates.Setoid.Properties where
 
-open Setoid
 
 private
   variable
@@ -40,7 +40,7 @@ module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) (surj : Surjection S T) whe
   open Surjection surj
 
   map⁺ : ∀ {xs} → IsEnumeration S xs → IsEnumeration T (map to xs)
-  map⁺ _∈xs y with surjective y
+  map⁺ _∈xs y with strictlySurjective y
   ... | (x , fx≈y) = ∈-resp-≈ T fx≈y (∈-map⁺ S T cong (x ∈xs))
 
 ------------------------------------------------------------------------
@@ -84,7 +84,9 @@ module _ (S? : DecSetoid a ℓ₁) where
 -- lookup
 
 module _ (S : Setoid a ℓ₁) where
+  open Setoid S
 
   lookup-surjective : ∀ {xs} → IsEnumeration S xs →
-                      Surjective _≡_ (_≈_ S) (lookup xs)
-  lookup-surjective _∈xs y = index (y ∈xs) , sym S (lookup-index (y ∈xs))
+                      Surjective _≡_ _≈_ (lookup xs)
+  lookup-surjective _∈xs = strictlySurjective⇒surjective
+    trans (λ { P.refl → refl}) (λ y → index (y ∈xs) , sym (lookup-index (y ∈xs)))
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 3c362222ed..2294a08968 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -24,8 +24,8 @@ open import Data.Nat.Solver
 open import Data.Product using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _$_; id)
-open import Function.Definitions using (Injective)
-open import Function.Definitions.Core2 using (Surjective)
+open import Function.Definitions
+open import Function.Consequences.Propositional
 open import Level using (0ℓ)
 open import Relation.Binary
 open import Relation.Binary.Consequences
@@ -196,8 +196,8 @@ toℕ-injective {1+[2 x ]} {1+[2 y ]} 1+2xN≡1+2yN =  cong 1+[2_] x≡y
   xN≡yN   = ℕₚ.*-cancelˡ-≡ _ _ 2 2xN≡2yN
   x≡y     = toℕ-injective xN≡yN
 
-toℕ-surjective : Surjective _≡_ toℕ
-toℕ-surjective n = (fromℕ n , toℕ-fromℕ n)
+toℕ-surjective : Surjective _≡_ _≡_ toℕ
+toℕ-surjective = strictlySurjective⇒surjective (λ n → fromℕ n , toℕ-fromℕ n)
 
 toℕ-isRelHomomorphism : IsRelHomomorphism _≡_ _≡_ toℕ
 toℕ-isRelHomomorphism = record
@@ -215,6 +215,15 @@ fromℕ-injective {x} {y} f[x]≡f[y] = begin
 fromℕ-toℕ : fromℕ ∘ toℕ ≗ id
 fromℕ-toℕ = toℕ-injective ∘ toℕ-fromℕ ∘ toℕ
 
+toℕ-inverseˡ : Inverseˡ _≡_ _≡_ toℕ fromℕ
+toℕ-inverseˡ = strictlyInverseˡ⇒inverseˡ {f⁻¹ = fromℕ} toℕ toℕ-fromℕ
+
+toℕ-inverseʳ : Inverseʳ _≡_ _≡_ toℕ fromℕ
+toℕ-inverseʳ = strictlyInverseʳ⇒inverseʳ toℕ fromℕ-toℕ
+
+toℕ-inverseᵇ : Inverseᵇ _≡_ _≡_ toℕ fromℕ
+toℕ-inverseᵇ = toℕ-inverseˡ , toℕ-inverseʳ
+
 fromℕ-pred : ∀ n → fromℕ (ℕ.pred n) ≡ pred (fromℕ n)
 fromℕ-pred n = begin
   fromℕ (ℕ.pred n)        ≡⟨ cong (fromℕ ∘ ℕ.pred) (sym (toℕ-fromℕ n)) ⟩
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index 5ab5e243e0..a3e1de4a1b 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -109,8 +109,8 @@ toℕ-homo-∸ 1+[2 x ] 1+[2 y ] = begin
   where open ≡-Reasoning
 
 fromℕ-homo-∸ : ∀ m n → fromℕ (m ℕ.∸ n) ≡ (fromℕ m) ∸ (fromℕ n)
-fromℕ-homo-∸ = homomorphic₂-inv ∸-magma ℕₚ.∸-magma {toℕ}
-  (cong fromℕ) (toℕ-fromℕ , fromℕ-toℕ) toℕ-homo-∸
+fromℕ-homo-∸ = homomorphic₂-inv ∸-magma ℕₚ.∸-magma
+  (cong fromℕ) toℕ-inverseᵇ toℕ-homo-∸
 
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _≤_/_<_
diff --git a/src/Data/Parity/Properties.agda b/src/Data/Parity/Properties.agda
index b9fb8a5eb7..574c9c68c1 100644
--- a/src/Data/Parity/Properties.agda
+++ b/src/Data/Parity/Properties.agda
@@ -15,7 +15,8 @@ open import Data.Parity.Base as ℙ using (Parity; 0ℙ; 1ℙ; _⁻¹; toSign; f
 open import Data.Product using (_,_)
 open import Data.Sign.Base as 𝕊
 open import Function.Base using (_$_; id)
-open import Function.Definitions using (Injective; Surjective)
+open import Function.Definitions
+open import Function.Consequences.Propositional
 open import Level using (0ℓ)
 open import Relation.Binary
   using (Decidable; DecidableEquality; Setoid; DecSetoid; IsDecEquivalence)
@@ -402,23 +403,19 @@ p+p≡0ℙ 1ℙ = refl
 ⁻¹-homo-opposite 0ℙ = refl
 ⁻¹-homo-opposite 1ℙ = refl
 
-toSign-inverts-fromSign : ∀ {p s} → toSign p ≡ s → fromSign s ≡ p
-toSign-inverts-fromSign {0ℙ} refl = refl
-toSign-inverts-fromSign {1ℙ} refl = refl
+toSign-inverseʳ : Inverseʳ _≡_ _≡_ toSign fromSign
+toSign-inverseʳ {0ℙ} refl = refl
+toSign-inverseʳ {1ℙ} refl = refl
 
-fromSign-inverts-toSign : ∀ {s p} → fromSign s ≡ p → toSign p ≡ s
-fromSign-inverts-toSign { + }  refl = refl
-fromSign-inverts-toSign { - } refl = refl
+toSign-inverseˡ : Inverseˡ _≡_ _≡_ toSign fromSign
+toSign-inverseˡ { + }  refl = refl
+toSign-inverseˡ { - } refl = refl
 
 toSign-injective : Injective _≡_ _≡_ toSign
-toSign-injective {p} {q} eq = begin
-  p                   ≡⟨ sym (toSign-inverts-fromSign {p} refl) ⟩
-  fromSign (toSign p) ≡⟨ cong fromSign eq ⟩
-  fromSign (toSign q) ≡⟨ toSign-inverts-fromSign {q} refl ⟩
-  q ∎ where open ≡-Reasoning
+toSign-injective = inverseʳ⇒injective toSign toSign-inverseʳ
 
 toSign-surjective : Surjective _≡_ _≡_ toSign
-toSign-surjective s = (fromSign s) , fromSign-inverts-toSign {s} refl
+toSign-surjective = inverseˡ⇒surjective toSign-inverseˡ
 
 toSign-isMagmaHomomorphism : IsMagmaHomomorphism ℙ.+-rawMagma 𝕊.*-rawMagma toSign
 toSign-isMagmaHomomorphism = record
@@ -477,7 +474,7 @@ toSign-isGroupIsomorphism = record
 
 
 ------------------------------------------------------------------------
--- relating Nat and Parity
+-- Relating Nat and Parity
 
 -- successor and (_⁻¹)
 
diff --git a/src/Data/Product/Algebra.agda b/src/Data/Product/Algebra.agda
index c64057dcc0..0b268b6ffd 100644
--- a/src/Data/Product/Algebra.agda
+++ b/src/Data/Product/Algebra.agda
@@ -29,13 +29,7 @@ import Function.Definitions as FuncDef
 private
   variable
     a b c d p : Level
-    A : Set a
-    B : Set b
-    C : Set c
-    D : Set d
-
-  module _ {A : Set a} {B : Set b} where
-    open FuncDef {A = A} {B} _≡_ _≡_
+    A B C D : Set a
 
 ------------------------------------------------------------------------
 -- Properties of Σ
@@ -56,8 +50,8 @@ private
 -- × is a congruence
 ×-cong : A ↔ B → C ↔ D → (A × C) ↔ (B × D)
 ×-cong i j = mk↔′ (map I.to J.to) (map I.from J.from)
-  (λ {(a , b) → cong₂ _,_ (I.inverseˡ a) (J.inverseˡ b)})
-  (λ {(a , b) → cong₂ _,_ (I.inverseʳ a) (J.inverseʳ b)})
+  (λ {(a , b) → cong₂ _,_ (I.strictlyInverseˡ a) (J.strictlyInverseˡ b)})
+  (λ {(a , b) → cong₂ _,_ (I.strictlyInverseʳ a) (J.strictlyInverseʳ b)})
   where module I = Inverse i; module J = Inverse j
 
 -- × is commutative.
diff --git a/src/Data/Sum/Algebra.agda b/src/Data/Sum/Algebra.agda
index f69988942b..ca96ac9c1c 100644
--- a/src/Data/Sum/Algebra.agda
+++ b/src/Data/Sum/Algebra.agda
@@ -29,14 +29,7 @@ import Function.Definitions as FuncDef
 private
   variable
     a b c d : Level
-    A : Set a
-    B : Set b
-    C : Set c
-    D : Set d
-
-  -- The module is needed because we need to pass `A` and `B` to `FuncDef`
-  module _ {A : Set a} {B : Set b} where
-    open FuncDef {A = A} {B} _≡_ _≡_
+    A B C D : Set a
 
   ♯ : {B : ⊥ {a} → Set b} → (w : ⊥) → B w
   ♯ ()
@@ -46,8 +39,8 @@ private
 
 ⊎-cong : A ↔ B → C ↔ D → (A ⊎ C) ↔ (B ⊎ D)
 ⊎-cong i j = mk↔′ (map I.to J.to) (map I.from J.from)
-  [ cong inj₁ ∘ I.inverseˡ , cong inj₂ ∘ J.inverseˡ ]
-  [ cong inj₁ ∘ I.inverseʳ , cong inj₂ ∘ J.inverseʳ ]
+  [ cong inj₁ ∘ I.strictlyInverseˡ , cong inj₂ ∘ J.strictlyInverseˡ ]
+  [ cong inj₁ ∘ I.strictlyInverseʳ , cong inj₂ ∘ J.strictlyInverseʳ ]
   where module I = Inverse i; module J = Inverse j
 
 -- ⊎ is commutative.
diff --git a/src/Data/Vec/N-ary.agda b/src/Data/Vec/N-ary.agda
index 197552d753..9220f86547 100644
--- a/src/Data/Vec/N-ary.agda
+++ b/src/Data/Vec/N-ary.agda
@@ -184,5 +184,5 @@ module _ (ext : ∀ {a b} → Extensionality a b) where
   Vec↔N-ary (suc n) = let open Inverse (Vec↔N-ary n) in
     mk↔′ (λ vxs x → to λ xs → vxs (x ∷ xs))
     (λ any xs → from (any (head xs)) (tail xs))
-    (λ any → ext λ x → inverseˡ _)
-    (λ vxs → ext λ where (x ∷ xs) → cong (λ f → f xs) (inverseʳ (λ ys → vxs (x ∷ ys))))
+    (λ any → ext λ x → strictlyInverseˡ _)
+    (λ vxs → ext λ where (x ∷ xs) → cong (λ f → f xs) (strictlyInverseʳ (λ ys → vxs (x ∷ ys))))
diff --git a/src/Data/Vec/Recursive.agda b/src/Data/Vec/Recursive.agda
index bd99c2064e..4c8796d016 100644
--- a/src/Data/Vec/Recursive.agda
+++ b/src/Data/Vec/Recursive.agda
@@ -29,7 +29,7 @@ open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Unit.Polymorphic.Properties using (⊤↔⊤*)
 open import Data.Vec.Base as Vec using (Vec; _∷_)
 open import Data.Vec.N-ary using (N-ary)
-open import Function.Base using (_∘′_; _∘_; id)
+open import Function.Base using (_∘′_; _∘_; id; const)
 open import Function.Bundles using (_↔_; mk↔′; mk↔)
 open import Function.Properties.Inverse using (↔-isEquivalence; ↔-refl; ↔-sym; ↔-trans)
 open import Level using (Level; lift)
@@ -44,8 +44,8 @@ private
     B : Set b
     C : Set c
 
--- Types and patterns
 ------------------------------------------------------------------------
+-- Types and patterns
 
 infix 8 _^_
 _^_ : Set a → ℕ → Set a
@@ -61,8 +61,8 @@ a ∈[ 0    ] as               = ⊥
 a ∈[ 1    ] a′               = a ≡ a′
 a ∈[ suc n@(suc _) ] a′ , as = a ≡ a′ ⊎ a ∈[ n ] as
 
--- Basic operations
 ------------------------------------------------------------------------
+-- Basic operations
 
 cons : ∀ n → A → A ^ n → A ^ suc n
 cons 0       a _  = a
@@ -106,9 +106,8 @@ splitAt (suc m) n xs =
   let (ys , zs) = splitAt m n (tail (m Nat.+ n) xs) in
   cons m (head (m Nat.+ n) xs) ys , zs
 
-
--- Manipulating N-ary products
 ------------------------------------------------------------------------
+-- Manipulating N-ary products
 
 map : (A → B) → ∀ n → A ^ n → B ^ n
 map f 0      as       = lift tt
@@ -156,7 +155,7 @@ unzip : ∀ n → (A × B) ^ n → A ^ n × B ^ n
 unzip = unzipWith id
 
 lift↔ : ∀ n → A ↔ B → A ^ n ↔ B ^ n
-lift↔ 0               A↔B = mk↔ ((λ { [] → refl }) , (λ{ [] → refl }))
+lift↔ 0               A↔B = mk↔′ _ _ (const refl) (const refl)
 lift↔ 1               A↔B = A↔B
 lift↔ (suc n@(suc _)) A↔B = ×-cong A↔B (lift↔ n A↔B)
 
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index d113f14858..1b92bee909 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -20,7 +20,7 @@
 module Function.Bundles where
 
 open import Function.Base using (_∘_)
-import Function.Definitions as FunctionDefinitions
+open import Function.Definitions
 import Function.Structures as FunctionStructures
 open import Level using (Level; _⊔_; suc)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
@@ -28,6 +28,7 @@ open import Relation.Binary hiding (_⇔_)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as ≡
+open import Function.Consequences.Propositional
 open Setoid using (isEquivalence)
 
 private
@@ -42,8 +43,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
 
   open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)
   open Setoid To   using () renaming (Carrier to B; _≈_ to _≈₂_)
-  open FunctionDefinitions _≈₁_ _≈₂_
-  open FunctionStructures  _≈₁_ _≈₂_
+  open FunctionStructures _≈₁_ _≈₂_
 
 ------------------------------------------------------------------------
 -- Bundles with one element
@@ -53,7 +53,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
   record Func : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       to   : A → B
-      cong : to Preserves _≈₁_ ⟶ _≈₂_
+      cong : Congruent _≈₁_ _≈₂_ to
 
     isCongruent : IsCongruent to
     isCongruent = record
@@ -69,8 +69,8 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
   record Injection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       to          : A → B
-      cong        : to Preserves _≈₁_ ⟶ _≈₂_
-      injective   : Injective to
+      cong        : Congruent _≈₁_ _≈₂_ to
+      injective   : Injective _≈₁_ _≈₂_ to
 
     function : Func
     function = record
@@ -91,8 +91,8 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
   record Surjection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       to         : A → B
-      cong       : to Preserves _≈₁_ ⟶ _≈₂_
-      surjective : Surjective to
+      cong       : Congruent _≈₁_ _≈₂_ to
+      surjective : Surjective _≈₁_ _≈₂_ to
 
     to⁻ : B → A
     to⁻ = proj₁ ∘ surjective
@@ -104,25 +104,30 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       ; isEquivalence₂ = isEquivalence To
       }
 
-    open IsCongruent isCongruent public using (module Eq₁; module Eq₂)
-
     isSurjection : IsSurjection to
     isSurjection = record
       { isCongruent = isCongruent
       ; surjective  = surjective
       }
 
+    open IsSurjection isSurjection public
+      using
+      ( module Eq₁
+      ; module Eq₂
+      ; strictlySurjective
+      )
+
 
   record Bijection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       to        : A → B
-      cong      : to Preserves _≈₁_ ⟶ _≈₂_
-      bijective : Bijective to
+      cong      : Congruent _≈₁_ _≈₂_ to
+      bijective : Bijective _≈₁_ _≈₂_ to
 
-    injective : Injective to
+    injective : Injective _≈₁_ _≈₂_ to
     injective = proj₁ bijective
 
-    surjective : Surjective to
+    surjective : Surjective _≈₁_ _≈₂_ to
     surjective = proj₂ bijective
 
     injection : Injection
@@ -138,7 +143,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       }
 
     open Injection  injection  public using (isInjection)
-    open Surjection surjection public using (isSurjection; to⁻)
+    open Surjection surjection public using (isSurjection; to⁻;  strictlySurjective)
 
     isBijection : IsBijection to
     isBijection = record
@@ -156,7 +161,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
     field
       to        : A → B
       from      : B → A
-      to-cong   : to Preserves _≈₁_ ⟶ _≈₂_
+      to-cong   : Congruent _≈₁_ _≈₂_ to
       from-cong : from Preserves _≈₂_ ⟶ _≈₁_
 
 
@@ -164,9 +169,9 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
     field
       to        : A → B
       from      : B → A
-      to-cong   : to Preserves _≈₁_ ⟶ _≈₂_
+      to-cong   : Congruent _≈₁_ _≈₂_ to
       from-cong : from Preserves _≈₂_ ⟶ _≈₁_
-      inverseˡ  : Inverseˡ to from
+      inverseˡ  : Inverseˡ _≈₁_ _≈₂_ to from
 
     isCongruent : IsCongruent to
     isCongruent = record
@@ -175,8 +180,6 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       ; isEquivalence₂ = isEquivalence To
       }
 
-    open IsCongruent isCongruent public using (module Eq₁; module Eq₂)
-
     isLeftInverse : IsLeftInverse to from
     isLeftInverse = record
       { isCongruent = isCongruent
@@ -184,6 +187,9 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       ; inverseˡ    = inverseˡ
       }
 
+    open IsLeftInverse isLeftInverse public
+      using (module Eq₁; module Eq₂; strictlyInverseˡ)
+
     equivalence : Equivalence
     equivalence = record
       { to-cong   = to-cong
@@ -195,9 +201,9 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
     field
       to        : A → B
       from      : B → A
-      to-cong   : to Preserves _≈₁_ ⟶ _≈₂_
+      to-cong   : Congruent _≈₁_ _≈₂_ to
       from-cong : from Preserves _≈₂_ ⟶ _≈₁_
-      inverseʳ  : Inverseʳ to from
+      inverseʳ  : Inverseʳ _≈₁_ _≈₂_ to from
 
     isCongruent : IsCongruent to
     isCongruent = record
@@ -213,6 +219,9 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       ; inverseʳ    = inverseʳ
       }
 
+    open IsRightInverse isRightInverse public
+      using (module Eq₁; module Eq₂; strictlyInverseʳ)
+
     equivalence : Equivalence
     equivalence = record
       { to-cong   = to-cong
@@ -224,14 +233,14 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
     field
       to        : A → B
       from      : B → A
-      to-cong   : to Preserves _≈₁_ ⟶ _≈₂_
-      from-cong : from Preserves _≈₂_ ⟶ _≈₁_
-      inverse   : Inverseᵇ to from
+      to-cong   : Congruent _≈₁_ _≈₂_ to
+      from-cong : Congruent _≈₂_ _≈₁_ from
+      inverse   : Inverseᵇ _≈₁_ _≈₂_ to from
 
-    inverseˡ : Inverseˡ to from
+    inverseˡ : Inverseˡ _≈₁_ _≈₂_ to from
     inverseˡ = proj₁ inverse
 
-    inverseʳ : Inverseʳ to from
+    inverseʳ : Inverseʳ _≈₁_ _≈₂_ to from
     inverseʳ = proj₂ inverse
 
     leftInverse : LeftInverse
@@ -248,8 +257,8 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       ; inverseʳ  = inverseʳ
       }
 
-    open LeftInverse leftInverse   public using (isLeftInverse)
-    open RightInverse rightInverse public using (isRightInverse)
+    open LeftInverse leftInverse   public using (isLeftInverse; strictlyInverseˡ)
+    open RightInverse rightInverse public using (isRightInverse; strictlyInverseʳ)
 
     isInverse : IsInverse to from
     isInverse = record
@@ -268,21 +277,21 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       to         : A → B
       from₁      : B → A
       from₂      : B → A
-      to-cong    : to Preserves _≈₁_ ⟶ _≈₂_
-      from₁-cong : from₁ Preserves _≈₂_ ⟶ _≈₁_
-      from₂-cong : from₂ Preserves _≈₂_ ⟶ _≈₁_
+      to-cong    : Congruent _≈₁_ _≈₂_ to
+      from₁-cong : Congruent _≈₂_ _≈₁_ from₁
+      from₂-cong : Congruent _≈₂_ _≈₁_ from₂
 
 
   record BiInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
-      to         : A → B
-      from₁        : B → A
-      from₂        : B → A
-      to-cong     : to Preserves _≈₁_ ⟶ _≈₂_
-      from₁-cong     : from₁ Preserves _≈₂_ ⟶ _≈₁_
-      from₂-cong     : from₂ Preserves _≈₂_ ⟶ _≈₁_
-      inverseˡ  : Inverseˡ to from₁
-      inverseʳ  : Inverseʳ to from₂
+      to          : A → B
+      from₁       : B → A
+      from₂       : B → A
+      to-cong     : Congruent _≈₁_ _≈₂_ to
+      from₁-cong  : Congruent _≈₂_ _≈₁_ from₁
+      from₂-cong  : Congruent _≈₂_ _≈₁_ from₂
+      inverseˡ  : Inverseˡ _≈₁_ _≈₂_ to from₁
+      inverseʳ  : Inverseʳ _≈₁_ _≈₂_ to from₂
 
     to-isCongruent : IsCongruent to
     to-isCongruent = record
@@ -345,29 +354,27 @@ A ↔ B = Inverse (≡.setoid A) (≡.setoid B)
 
 module _ {A : Set a} {B : Set b} where
 
-  open FunctionDefinitions {A = A} {B} _≡_ _≡_
-
   mk⟶ : (A → B) → A ⟶ B
   mk⟶ to = record
     { to        = to
     ; cong      = ≡.cong to
     }
 
-  mk↣ : ∀ {to : A → B} → Injective to → A ↣ B
+  mk↣ : ∀ {to : A → B} → Injective _≡_ _≡_ to → A ↣ B
   mk↣ {to} inj = record
     { to         = to
     ; cong      = ≡.cong to
     ; injective = inj
     }
 
-  mk↠ : ∀ {to : A → B} → Surjective to → A ↠ B
+  mk↠ : ∀ {to : A → B} → Surjective _≡_ _≡_ to → A ↠ B
   mk↠ {to} surj = record
     { to         = to
     ; cong       = ≡.cong to
     ; surjective = surj
     }
 
-  mk⤖ : ∀ {to : A → B} → Bijective to → A ⤖ B
+  mk⤖ : ∀ {to : A → B} → Bijective _≡_ _≡_ to → A ⤖ B
   mk⤖ {to} bij = record
     { to        = to
     ; cong      = ≡.cong to
@@ -382,7 +389,7 @@ module _ {A : Set a} {B : Set b} where
     ; from-cong = ≡.cong from
     }
 
-  mk↩ : ∀ {to : A → B} {from : B → A} → Inverseˡ to from → A ↩ B
+  mk↩ : ∀ {to : A → B} {from : B → A} → Inverseˡ _≡_ _≡_ to from → A ↩ B
   mk↩ {to} {from} invˡ = record
     { to        = to
     ; from      = from
@@ -391,7 +398,7 @@ module _ {A : Set a} {B : Set b} where
     ; inverseˡ  = invˡ
     }
 
-  mk↪ : ∀ {to : A → B} {from : B → A} → Inverseʳ to from → A ↪ B
+  mk↪ : ∀ {to : A → B} {from : B → A} → Inverseʳ _≡_ _≡_ to from → A ↪ B
   mk↪ {to} {from} invʳ = record
     { to        = to
     ; from      = from
@@ -401,7 +408,7 @@ module _ {A : Set a} {B : Set b} where
     }
 
   mk↩↪ : ∀ {to : A → B} {from₁ : B → A} {from₂ : B → A} →
-         Inverseˡ to from₁ → Inverseʳ to from₂ → A ↩↪ B
+         Inverseˡ _≡_ _≡_ to from₁ → Inverseʳ _≡_ _≡_ to from₂ → A ↩↪ B
   mk↩↪ {to} {from₁} {from₂} invˡ invʳ = record
     { to         = to
     ; from₁      = from₁
@@ -413,7 +420,7 @@ module _ {A : Set a} {B : Set b} where
     ; inverseʳ   = invʳ
     }
 
-  mk↔ : ∀ {to : A → B} {from : B → A} → Inverseᵇ to from → A ↔ B
+  mk↔ : ∀ {to : A → B} {from : B → A} → Inverseᵇ _≡_ _≡_ to from → A ↔ B
   mk↔ {to} {from} inv = record
     { to        = to
     ; from      = from
@@ -422,6 +429,13 @@ module _ {A : Set a} {B : Set b} where
     ; inverse   = inv
     }
 
-  -- Sometimes the implicit arguments above cannot be inferred
-  mk↔′ : ∀ (to : A → B) (from : B → A) → Inverseˡ to from → Inverseʳ to from → A ↔ B
-  mk↔′ to from invˡ invʳ = mk↔ {to = to} {from = from} (invˡ , invʳ)
+  -- Strict variant of the above.
+  mk↔′ : ∀ (to : A → B) (from : B → A) →
+          StrictlyInverseˡ _≡_ to from →
+          StrictlyInverseʳ _≡_ to from →
+          A ↔ B
+  mk↔′ to from invˡ invʳ = mk↔ {to} {from}
+    ( strictlyInverseˡ⇒inverseˡ to invˡ
+    , strictlyInverseʳ⇒inverseʳ to invʳ
+    )
+
diff --git a/src/Function/Consequences.agda b/src/Function/Consequences.agda
index e8c6a0f498..242aa3d289 100644
--- a/src/Function/Consequences.agda
+++ b/src/Function/Consequences.agda
@@ -1,55 +1,113 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Relationships between properties of functions
+-- Relationships between properties of functions. See
+-- `Function.Consequences.Propositional` for specialisations to
+-- propositional equality.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module Function.Consequences where
 
-open import Data.Product.Base using (_,_)
+open import Data.Product.Base as Prod
 open import Function.Definitions
-open import Level using (Level)
-open import Relation.Binary
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+open import Level
+open import Relation.Binary.Core
+open import Relation.Binary.Definitions
 open import Relation.Nullary.Negation.Core using (¬_; contraposition)
 
 private
   variable
     a b ℓ₁ ℓ₂ : Level
-    A : Set a
-    B : Set b
+    A B : Set a
+    ≈₁ ≈₂ : Rel A ℓ₁
+    f f⁻¹ : A → B
 
-module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) {f f⁻¹} where
+------------------------------------------------------------------------
+-- Injective
+
+contraInjective : Injective ≈₁ ≈₂ f →
+                  ∀ {x y} → ¬ (≈₁ x y) → ¬ (≈₂ (f x) (f y))
+contraInjective inj p = contraposition inj p
+
+------------------------------------------------------------------------
+-- Inverseˡ
 
-  inverseˡ⇒surjective : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Surjective ≈₁ ≈₂ f
-  inverseˡ⇒surjective invˡ y = (f⁻¹ y , invˡ y)
+inverseˡ⇒surjective : ∀ (≈₂ : Rel B ℓ₂) →
+                      Inverseˡ ≈₁ ≈₂ f f⁻¹ →
+                      Surjective ≈₁ ≈₂ f
+inverseˡ⇒surjective ≈₂ invˡ y = (_ , invˡ)
 
-  inverseʳ⇒surjective : Inverseʳ ≈₁ ≈₂ f f⁻¹ → Surjective ≈₂ ≈₁ f⁻¹
-  inverseʳ⇒surjective invʳ y = (f y , invʳ y)
+------------------------------------------------------------------------
+-- Inverseʳ
+
+inverseʳ⇒injective : ∀ (≈₂ : Rel B ℓ₂) f →
+                     Reflexive ≈₂ →
+                     Symmetric ≈₁ →
+                     Transitive ≈₁ →
+                     Inverseʳ ≈₁ ≈₂ f f⁻¹ →
+                     Injective ≈₁ ≈₂ f
+inverseʳ⇒injective ≈₂ f refl sym trans invʳ {x} {y} fx≈fy =
+  trans (sym (invʳ refl)) (invʳ fx≈fy)
 
-module _ (From : Setoid a ℓ₁) {≈₂ : Rel B ℓ₂} where
+------------------------------------------------------------------------
+-- Inverseᵇ
 
-  open Setoid From using () renaming (Carrier to A; _≈_ to ≈₁)
+inverseᵇ⇒bijective : ∀ (≈₂ : Rel B ℓ₂) →
+                     Reflexive ≈₂ →
+                     Symmetric ≈₁ →
+                     Transitive ≈₁ →
+                     Inverseᵇ ≈₁ ≈₂ f f⁻¹ →
+                     Bijective ≈₁ ≈₂ f
+inverseᵇ⇒bijective {f = f} ≈₂ refl sym trans (invˡ , invʳ) =
+  (inverseʳ⇒injective ≈₂ f refl sym trans invʳ , inverseˡ⇒surjective ≈₂ invˡ)
 
-  inverseʳ⇒injective : ∀ {f f⁻¹} → Congruent ≈₂ ≈₁ f⁻¹ →
-                       Inverseʳ ≈₁ ≈₂ f f⁻¹ → Injective ≈₁ ≈₂ f
-  inverseʳ⇒injective {f} {f⁻¹} cong₂ invʳ {x} {y} x≈y = begin
-    x         ≈˘⟨ invʳ x ⟩
-    f⁻¹ (f x) ≈⟨  cong₂ x≈y ⟩
-    f⁻¹ (f y) ≈⟨  invʳ y ⟩
-    y         ∎
-    where open SetoidReasoning From
+------------------------------------------------------------------------
+-- StrictlySurjective
 
-  inverseᵇ⇒bijective : ∀ {f f⁻¹} → Congruent ≈₂ ≈₁ f⁻¹ → Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Bijective ≈₁ ≈₂ f
-  inverseᵇ⇒bijective cong₂ (invˡ , invʳ) =
-    (inverseʳ⇒injective cong₂ invʳ , inverseˡ⇒surjective ≈₁ ≈₂ invˡ)
+surjective⇒strictlySurjective : ∀ (≈₂ : Rel B ℓ₂) →
+                                 Reflexive ≈₁ →
+                                 Surjective ≈₁ ≈₂ f →
+                                 StrictlySurjective ≈₂ f
+surjective⇒strictlySurjective _ refl surj x =
+  Prod.map₂ (λ v → v refl) (surj x)
+
+strictlySurjective⇒surjective : Transitive ≈₂ →
+                                 Congruent ≈₁ ≈₂ f →
+                                 StrictlySurjective ≈₂ f →
+                                 Surjective ≈₁ ≈₂ f
+strictlySurjective⇒surjective trans cong surj x =
+  Prod.map₂ (λ fy≈x z≈y → trans (cong z≈y) fy≈x) (surj x)
+
+------------------------------------------------------------------------
+-- StrictlyInverseˡ
+
+inverseˡ⇒strictlyInverseˡ : ∀ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) →
+                            Reflexive ≈₁ →
+                            Inverseˡ ≈₁ ≈₂ f f⁻¹ →
+                            StrictlyInverseˡ ≈₂ f f⁻¹
+inverseˡ⇒strictlyInverseˡ _ _ refl sinv x = sinv refl
+
+strictlyInverseˡ⇒inverseˡ : Transitive ≈₂ →
+                            Congruent ≈₁ ≈₂ f →
+                            StrictlyInverseˡ ≈₂ f f⁻¹ →
+                            Inverseˡ ≈₁ ≈₂ f f⁻¹
+strictlyInverseˡ⇒inverseˡ trans cong sinv {x} y≈f⁻¹x =
+  trans (cong y≈f⁻¹x) (sinv x)
+
+------------------------------------------------------------------------
+-- StrictlyInverseʳ
 
-module _
-  {f : A → B} (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)
-  where
+inverseʳ⇒strictlyInverseʳ : ∀ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) →
+                            Reflexive ≈₂ →
+                            Inverseʳ ≈₁ ≈₂ f f⁻¹ →
+                            StrictlyInverseʳ ≈₁ f f⁻¹
+inverseʳ⇒strictlyInverseʳ _ _ refl sinv x = sinv refl
 
-  contraInjective : Injective _≈₁_ _≈₂_ f →
-                    ∀ {x y} → ¬ (x ≈₁ y) → ¬ (f x ≈₂ f y)
-  contraInjective inj p = contraposition inj p
+strictlyInverseʳ⇒inverseʳ : Transitive ≈₁ →
+                            Congruent ≈₂ ≈₁ f⁻¹ →
+                            StrictlyInverseʳ ≈₁ f f⁻¹ →
+                            Inverseʳ ≈₁ ≈₂ f f⁻¹
+strictlyInverseʳ⇒inverseʳ trans cong sinv {x} y≈f⁻¹x =
+  trans (cong y≈f⁻¹x) (sinv x)
diff --git a/src/Function/Consequences/Propositional.agda b/src/Function/Consequences/Propositional.agda
new file mode 100644
index 0000000000..d6067a7702
--- /dev/null
+++ b/src/Function/Consequences/Propositional.agda
@@ -0,0 +1,91 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Relationships between properties of functions where the equality
+-- over both the domain and codomain is assumed to be _≡_
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Consequences.Propositional where
+
+open import Function.Definitions
+open import Level
+open import Relation.Nullary.Negation.Core using (contraposition)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≢_; refl; sym; trans; cong)
+
+import Function.Consequences as C
+
+private
+  variable
+    a b ℓ₁ ℓ₂ : Level
+    A B : Set a
+    f f⁻¹ : A → B
+
+------------------------------------------------------------------------
+-- Injective
+
+contraInjective : Injective _≡_ _≡_ f →
+                  ∀ {x y} → x ≢ y → f x ≢ f y
+contraInjective inj p = contraposition inj p
+
+------------------------------------------------------------------------
+-- Inverseˡ
+
+inverseˡ⇒surjective : Inverseˡ _≡_ _≡_ f f⁻¹ → Surjective _≡_ _≡_ f
+inverseˡ⇒surjective = C.inverseˡ⇒surjective _≡_
+
+------------------------------------------------------------------------
+-- Inverseʳ
+
+inverseʳ⇒injective : ∀ f →
+                     Inverseʳ _≡_ _≡_ f f⁻¹ →
+                     Injective _≡_ _≡_ f
+inverseʳ⇒injective f = C.inverseʳ⇒injective _≡_ f refl sym trans
+
+------------------------------------------------------------------------
+-- Inverseᵇ
+
+inverseᵇ⇒bijective : Inverseᵇ _≡_ _≡_ f f⁻¹ → Bijective _≡_ _≡_ f
+inverseᵇ⇒bijective = C.inverseᵇ⇒bijective _≡_ refl sym trans
+
+------------------------------------------------------------------------
+-- StrictlySurjective
+
+surjective⇒strictlySurjective : Surjective _≡_ _≡_ f →
+                                 StrictlySurjective _≡_ f
+surjective⇒strictlySurjective =
+  C.surjective⇒strictlySurjective _≡_ refl
+
+strictlySurjective⇒surjective : StrictlySurjective _≡_ f →
+                                 Surjective _≡_ _≡_ f
+strictlySurjective⇒surjective =
+  C.strictlySurjective⇒surjective trans (cong _)
+
+------------------------------------------------------------------------
+-- StrictlyInverseˡ
+
+inverseˡ⇒strictlyInverseˡ : Inverseˡ _≡_ _≡_ f f⁻¹ →
+                            StrictlyInverseˡ _≡_ f f⁻¹
+inverseˡ⇒strictlyInverseˡ =
+  C.inverseˡ⇒strictlyInverseˡ _≡_ _≡_ refl
+
+strictlyInverseˡ⇒inverseˡ : ∀ f →
+                            StrictlyInverseˡ _≡_ f f⁻¹ →
+                            Inverseˡ _≡_ _≡_ f f⁻¹
+strictlyInverseˡ⇒inverseˡ f =
+  C.strictlyInverseˡ⇒inverseˡ trans (cong f)
+
+------------------------------------------------------------------------
+-- StrictlyInverseʳ
+
+inverseʳ⇒strictlyInverseʳ : Inverseʳ _≡_ _≡_ f f⁻¹ →
+                            StrictlyInverseʳ _≡_ f f⁻¹
+inverseʳ⇒strictlyInverseʳ = C.inverseʳ⇒strictlyInverseʳ _≡_ _≡_ refl
+
+strictlyInverseʳ⇒inverseʳ : ∀ f →
+                            StrictlyInverseʳ _≡_ f f⁻¹ →
+                            Inverseʳ _≡_ _≡_ f f⁻¹
+strictlyInverseʳ⇒inverseʳ {f⁻¹ = f⁻¹} _ =
+  C.strictlyInverseʳ⇒inverseʳ trans (cong f⁻¹)
diff --git a/src/Function/Construct/Composition.agda b/src/Function/Construct/Composition.agda
index 269448eaba..b745daa8e6 100644
--- a/src/Function/Construct/Composition.agda
+++ b/src/Function/Construct/Composition.agda
@@ -8,14 +8,11 @@
 
 module Function.Construct.Composition where
 
-open import Data.Product.Base using (_,_)
+open import Data.Product.Base as Product using (_,_)
 open import Function.Base using (_∘_)
 open import Function.Bundles
-  using (Func; Injection; Surjection; Bijection; Equivalence; LeftInverse; RightInverse; Inverse; _⟶_; _↣_; _↠_; _⤖_; _⇔_; _↩_; _↪_; _↔_)
 open import Function.Definitions
-  using (Congruent; Injective; Surjective; Bijective; Inverseˡ; Inverseʳ; Inverseᵇ)
 open import Function.Structures
-  using (IsCongruent; IsInjection; IsSurjection; IsBijection; IsLeftInverse; IsRightInverse; IsInverse)
 open import Level using (Level)
 open import Relation.Binary hiding (_⇔_; IsEquivalence)
 
@@ -39,39 +36,31 @@ module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)
               Injective ≈₁ ≈₃ (g ∘ f)
   injective f-inj g-inj = f-inj ∘ g-inj
 
-  surjective : Transitive ≈₃ → Congruent ≈₂ ≈₃ g →
-               Surjective ≈₁ ≈₂ f → Surjective ≈₂ ≈₃ g →
+  surjective : Surjective ≈₁ ≈₂ f → Surjective ≈₂ ≈₃ g →
                Surjective ≈₁ ≈₃ (g ∘ f)
-  surjective trans g-cong f-sur g-sur x with g-sur x
-  ... | y , fy≈x  with f-sur y
-  ...   | z , fz≈y = z , trans (g-cong fz≈y) fy≈x
+  surjective f-sur g-sur x with g-sur x
+  ... | y , gproof with f-sur y
+  ...   | z , fproof = z , gproof ∘ fproof
 
-  bijective : Transitive ≈₃ → Congruent ≈₂ ≈₃ g →
-              Bijective ≈₁ ≈₂ f → Bijective ≈₂ ≈₃ g →
+  bijective : Bijective ≈₁ ≈₂ f → Bijective ≈₂ ≈₃ g →
               Bijective ≈₁ ≈₃ (g ∘ f)
-  bijective trans g-cong (f-inj , f-sur) (g-inj , g-sur) =
-    injective f-inj g-inj , surjective trans g-cong f-sur g-sur
+  bijective = Product.zip′ injective surjective
 
 module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) (≈₃ : Rel C ℓ₃)
-         (f : A → B) {f⁻¹ : B → A} {g : B → C} (g⁻¹ : C → B)
+         {f : A → B} {f⁻¹ : B → A} {g : B → C} {g⁻¹ : C → B}
          where
 
-  inverseˡ : Transitive ≈₃ → Congruent ≈₂ ≈₃ g →
-             Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseˡ ≈₂ ≈₃ g g⁻¹ →
+  inverseˡ : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseˡ ≈₂ ≈₃ g g⁻¹ →
              Inverseˡ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
-  inverseˡ trn g-cong f-inv g-inv x = trn (g-cong (f-inv _)) (g-inv x)
+  inverseˡ f-inv g-inv = g-inv ∘ f-inv
 
-  inverseʳ : Transitive ≈₁ → Congruent ≈₂ ≈₁ f⁻¹ →
-             Inverseʳ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₃ g g⁻¹ →
+  inverseʳ : Inverseʳ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₃ g g⁻¹ →
              Inverseʳ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
-  inverseʳ trn f⁻¹-cong f-inv g-inv x = trn (f⁻¹-cong (g-inv _)) (f-inv x)
+  inverseʳ f-inv g-inv = f-inv ∘ g-inv
 
-  inverseᵇ : Transitive ≈₁ → Transitive ≈₃ →
-             Congruent ≈₂ ≈₃ g → Congruent ≈₂ ≈₁ f⁻¹ →
-             Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Inverseᵇ ≈₂ ≈₃ g g⁻¹ →
+  inverseᵇ : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Inverseᵇ ≈₂ ≈₃ g g⁻¹ →
              Inverseᵇ ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
-  inverseᵇ trn₁ trn₃ g-cong f⁻¹-cong (f-invˡ , f-invʳ) (g-invˡ , g-invʳ) =
-    inverseˡ trn₃ g-cong f-invˡ g-invˡ , inverseʳ trn₁ f⁻¹-cong f-invʳ g-invʳ
+  inverseᵇ = Product.zip′ inverseˡ inverseʳ
 
 ------------------------------------------------------------------------
 -- Structures
@@ -99,14 +88,14 @@ module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}
                  IsSurjection ≈₁ ≈₃ (g ∘ f)
   isSurjection f-surj g-surj = record
     { isCongruent = isCongruent F.isCongruent G.isCongruent
-    ; surjective  = surjective ≈₁ ≈₂ ≈₃ G.Eq₂.trans G.cong F.surjective G.surjective
+    ; surjective  = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective
     } where module F = IsSurjection f-surj; module G = IsSurjection g-surj
 
   isBijection : IsBijection ≈₁ ≈₂ f → IsBijection ≈₂ ≈₃ g →
                 IsBijection ≈₁ ≈₃ (g ∘ f)
   isBijection f-bij g-bij = record
     { isInjection = isInjection F.isInjection G.isInjection
-    ; surjective  = surjective ≈₁ ≈₂ ≈₃ G.Eq₂.trans G.cong F.surjective G.surjective
+    ; surjective  = surjective ≈₁ ≈₂ ≈₃ F.surjective G.surjective
     } where module F = IsBijection f-bij; module G = IsBijection g-bij
 
 module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}
@@ -117,23 +106,23 @@ module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {≈₃ : Rel C ℓ₃}
                   IsLeftInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
   isLeftInverse f-invˡ g-invˡ = record
     { isCongruent = isCongruent F.isCongruent G.isCongruent
-    ; from-cong       = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong
-    ; inverseˡ    = inverseˡ ≈₁ ≈₂ ≈₃ f _ G.Eq₂.trans G.to-cong F.inverseˡ G.inverseˡ
+    ; from-cong   = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong
+    ; inverseˡ    = inverseˡ ≈₁ ≈₂ ≈₃ F.inverseˡ G.inverseˡ
     } where module F = IsLeftInverse f-invˡ; module G = IsLeftInverse g-invˡ
 
   isRightInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsRightInverse ≈₂ ≈₃ g g⁻¹ →
                    IsRightInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
   isRightInverse f-invʳ g-invʳ = record
     { isCongruent = isCongruent F.isCongruent G.isCongruent
-    ; from-cong       = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong
-    ; inverseʳ    = inverseʳ ≈₁ ≈₂ ≈₃ _ g⁻¹ F.Eq₁.trans F.from-cong F.inverseʳ G.inverseʳ
+    ; from-cong   = congruent ≈₃ ≈₂ ≈₁ G.from-cong F.from-cong
+    ; inverseʳ    = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ
     } where module F = IsRightInverse f-invʳ; module G = IsRightInverse g-invʳ
 
   isInverse : IsInverse ≈₁ ≈₂ f f⁻¹ → IsInverse ≈₂ ≈₃ g g⁻¹ →
               IsInverse ≈₁ ≈₃ (g ∘ f) (f⁻¹ ∘ g⁻¹)
   isInverse f-inv g-inv = record
     { isLeftInverse = isLeftInverse F.isLeftInverse G.isLeftInverse
-    ; inverseʳ      = inverseʳ ≈₁ ≈₂ ≈₃ _ g⁻¹ F.Eq₁.trans F.from-cong F.inverseʳ G.inverseʳ
+    ; inverseʳ      = inverseʳ ≈₁ ≈₂ ≈₃ F.inverseʳ G.inverseʳ
     } where module F = IsInverse f-inv; module G = IsInverse g-inv
 
 ------------------------------------------------------------------------
@@ -159,15 +148,15 @@ module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where
   surjection : Surjection R S → Surjection S T → Surjection R T
   surjection surj₁ surj₂ = record
     { to         = G.to ∘ F.to
-    ; cong       = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong
-    ; surjective = surjective (≈ R) (≈ S) (≈ T) G.Eq₂.trans G.cong F.surjective G.surjective
+    ; cong       = congruent  (≈ R) (≈ S) (≈ T) F.cong G.cong
+    ; surjective = surjective (≈ R) (≈ S) (≈ T) F.surjective G.surjective
     } where module F = Surjection surj₁; module G = Surjection surj₂
 
   bijection : Bijection R S → Bijection S T → Bijection R T
   bijection bij₁ bij₂ = record
     { to        = G.to ∘ F.to
     ; cong      = congruent (≈ R) (≈ S) (≈ T) F.cong G.cong
-    ; bijective = bijective (≈ R) (≈ S) (≈ T) (trans T) G.cong F.bijective G.bijective
+    ; bijective = bijective (≈ R) (≈ S) (≈ T) F.bijective G.bijective
     } where module F = Bijection bij₁; module G = Bijection bij₂
 
   equivalence : Equivalence R S → Equivalence S T → Equivalence R T
@@ -184,7 +173,7 @@ module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where
     ; from      = F.from ∘ G.from
     ; to-cong   = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong
     ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong
-    ; inverseˡ  = inverseˡ  (≈ R) (≈ S) (≈ T) F.to _ (trans T) G.to-cong F.inverseˡ G.inverseˡ
+    ; inverseˡ  = inverseˡ  (≈ R) (≈ S) (≈ T) F.inverseˡ G.inverseˡ
     } where module F = LeftInverse invˡ₁; module G = LeftInverse invˡ₂
 
   rightInverse : RightInverse R S → RightInverse S T → RightInverse R T
@@ -193,7 +182,7 @@ module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where
     ; from      = F.from ∘ G.from
     ; to-cong   = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong
     ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong
-    ; inverseʳ  = inverseʳ  (≈ R) (≈ S) (≈ T) _ G.from (trans R) F.from-cong F.inverseʳ G.inverseʳ
+    ; inverseʳ  = inverseʳ  (≈ R) (≈ S) (≈ T) F.inverseʳ G.inverseʳ
     } where module F = RightInverse invʳ₁; module G = RightInverse invʳ₂
 
   inverse : Inverse R S → Inverse S T → Inverse R T
@@ -202,7 +191,7 @@ module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where
     ; from      = F.from ∘ G.from
     ; to-cong   = congruent (≈ R) (≈ S) (≈ T) F.to-cong G.to-cong
     ; from-cong = congruent (≈ T) (≈ S) (≈ R) G.from-cong F.from-cong
-    ; inverse   = inverseᵇ  (≈ R) (≈ S) (≈ T) _ G.from (trans R) (trans T) G.to-cong F.from-cong F.inverse G.inverse
+    ; inverse   = inverseᵇ  (≈ R) (≈ S) (≈ T) F.inverse G.inverse
     } where module F = Inverse inv₁; module G = Inverse inv₂
 
 ------------------------------------------------------------------------
diff --git a/src/Function/Construct/Identity.agda b/src/Function/Construct/Identity.agda
index bec0ecd3d6..e4ec7e3d3e 100644
--- a/src/Function/Construct/Identity.agda
+++ b/src/Function/Construct/Identity.agda
@@ -28,28 +28,28 @@ private
 
 module _ (_≈_ : Rel A ℓ) where
 
-  open Definitions _≈_ _≈_
+  open Definitions
 
-  congruent : Congruent id
+  congruent : Congruent _≈_ _≈_ id
   congruent = id
 
-  injective : Injective id
+  injective : Injective _≈_ _≈_ id
   injective = id
 
-  surjective : Reflexive _≈_ → Surjective id
-  surjective refl x = x , refl
+  surjective : Surjective _≈_ _≈_ id
+  surjective x = x , id
 
-  bijective : Reflexive _≈_ → Bijective id
-  bijective refl = injective , surjective refl
+  bijective : Bijective _≈_ _≈_ id
+  bijective = injective , surjective
 
-  inverseˡ : Reflexive _≈_ → Inverseˡ id id
-  inverseˡ refl x = refl
+  inverseˡ : Inverseˡ _≈_ _≈_ id id
+  inverseˡ = id
 
-  inverseʳ : Reflexive _≈_ → Inverseʳ id id
-  inverseʳ refl x = refl
+  inverseʳ : Inverseʳ _≈_ _≈_ id id
+  inverseʳ = id
 
-  inverseᵇ : Reflexive _≈_ → Inverseᵇ id id
-  inverseᵇ refl = inverseˡ refl , inverseʳ refl
+  inverseᵇ : Inverseᵇ _≈_ _≈_ id id
+  inverseᵇ = inverseˡ , inverseʳ
 
 ------------------------------------------------------------------------
 -- Structures
@@ -75,33 +75,33 @@ module _ {_≈_ : Rel A ℓ} (isEq : B.IsEquivalence _≈_) where
   isSurjection : IsSurjection id
   isSurjection = record
     { isCongruent = isCongruent
-    ; surjective  = surjective _≈_ refl
+    ; surjective  = surjective _≈_
     }
 
   isBijection : IsBijection id
   isBijection = record
     { isInjection = isInjection
-    ; surjective  = surjective _≈_ refl
+    ; surjective  = surjective _≈_
     }
 
   isLeftInverse : IsLeftInverse id id
   isLeftInverse = record
     { isCongruent = isCongruent
     ; from-cong   = id
-    ; inverseˡ    = inverseˡ _≈_ refl
+    ; inverseˡ    = inverseˡ _≈_
     }
 
   isRightInverse : IsRightInverse id id
   isRightInverse = record
     { isCongruent = isCongruent
     ; from-cong   = id
-    ; inverseʳ    = inverseʳ _≈_ refl
+    ; inverseʳ    = inverseʳ _≈_
     }
 
   isInverse : IsInverse id id
   isInverse = record
     { isLeftInverse = isLeftInverse
-    ; inverseʳ      = inverseʳ _≈_ refl
+    ; inverseʳ      = inverseʳ _≈_
     }
 
 ------------------------------------------------------------------------
@@ -128,14 +128,14 @@ module _ (S : Setoid a ℓ) where
   surjection = record
     { to         = id
     ; cong       = id
-    ; surjective = surjective _≈_ refl
+    ; surjective = surjective _≈_
     }
 
   bijection : Bijection S S
   bijection = record
     { to        = id
     ; cong      = id
-    ; bijective = bijective _≈_ refl
+    ; bijective = bijective _≈_
     }
 
   equivalence : Equivalence S S
@@ -152,7 +152,7 @@ module _ (S : Setoid a ℓ) where
     ; from      = id
     ; to-cong   = id
     ; from-cong = id
-    ; inverseˡ  = inverseˡ _≈_ refl
+    ; inverseˡ  = inverseˡ _≈_
     }
 
   rightInverse : RightInverse S S
@@ -161,7 +161,7 @@ module _ (S : Setoid a ℓ) where
     ; from      = id
     ; to-cong   = id
     ; from-cong = id
-    ; inverseʳ  = inverseʳ _≈_ refl
+    ; inverseʳ  = inverseʳ _≈_
     }
 
   inverse : Inverse S S
@@ -170,7 +170,7 @@ module _ (S : Setoid a ℓ) where
     ; from      = id
     ; to-cong   = id
     ; from-cong = id
-    ; inverse   = inverseᵇ _≈_ refl
+    ; inverse   = inverseᵇ _≈_
     }
 
 ------------------------------------------------------------------------
diff --git a/src/Function/Construct/Symmetry.agda b/src/Function/Construct/Symmetry.agda
index 14a7adc466..834ee0d4fa 100644
--- a/src/Function/Construct/Symmetry.agda
+++ b/src/Function/Construct/Symmetry.agda
@@ -18,9 +18,10 @@ open import Function.Bundles
   using (Bijection; Equivalence; LeftInverse; RightInverse; Inverse; _⤖_; _⇔_; _↩_; _↪_; _↔_)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.Definitions using (Symmetric; Transitive)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality
+  using (_≡_; cong; setoid)
 
 private
   variable
@@ -38,18 +39,19 @@ module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {f : A → B}
     f⁻¹      = proj₁ ∘ surj
     f∘f⁻¹≡id = proj₂ ∘ surj
 
-  injective : Symmetric ≈₂ → Transitive ≈₂ → Congruent ≈₁ ≈₂ f → Injective ≈₂ ≈₁ f⁻¹
-  injective sym trans cong gx≈gy = trans (trans (sym (f∘f⁻¹≡id _)) (cong gx≈gy)) (f∘f⁻¹≡id _)
+  injective : Reflexive ≈₁ → Symmetric ≈₂ → Transitive ≈₂ →
+              Congruent ≈₁ ≈₂ f → Injective ≈₂ ≈₁ f⁻¹
+  injective refl sym trans cong gx≈gy =
+    trans (trans (sym (f∘f⁻¹≡id _ refl)) (cong gx≈gy)) (f∘f⁻¹≡id _ refl)
 
-  surjective : Surjective ≈₂ ≈₁ f⁻¹
-  surjective x = f x , inj (proj₂ (surj (f x)))
+  surjective : Reflexive ≈₁ → Transitive ≈₂ → Surjective ≈₂ ≈₁ f⁻¹
+  surjective refl trans x = f x , inj ∘ trans (f∘f⁻¹≡id _ refl)
 
-  bijective : Symmetric ≈₂ → Transitive ≈₂ → Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹
-  bijective sym trans cong = injective sym trans cong , surjective
+  bijective : Reflexive ≈₁ → Symmetric ≈₂ → Transitive ≈₂ →
+              Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹
+  bijective refl sym trans cong = injective refl sym trans cong , surjective refl trans
 
-module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂)
-         (f : A → B) {f⁻¹ : B → A}
-         where
+module _ (≈₁ : Rel A ℓ₁) (≈₂ : Rel B ℓ₂) {f : A → B} {f⁻¹ : B → A} where
 
   inverseʳ : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Inverseʳ ≈₂ ≈₁ f⁻¹ f
   inverseʳ inv = inv
@@ -81,9 +83,9 @@ module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂}
         ; isEquivalence₁ = IB.Eq₂.isEquivalence
         ; isEquivalence₂ = IB.Eq₁.isEquivalence
         }
-      ; injective = injective IB.bijective IB.Eq₂.sym IB.Eq₂.trans IB.cong
+      ; injective = injective IB.bijective IB.Eq₁.refl IB.Eq₂.sym IB.Eq₂.trans IB.cong
       }
-    ; surjective = surjective {≈₂ = ≈₂} IB.bijective
+    ; surjective = surjective IB.bijective IB.Eq₁.refl IB.Eq₂.trans
     }
 
 module _ {≈₁ : Rel A ℓ₁} {f : A → B} (isBij : IsBijection ≈₁ _≡_ f) where
@@ -94,35 +96,33 @@ module _ {≈₁ : Rel A ℓ₁} {f : A → B} (isBij : IsBijection ≈₁ _≡_
   isBijection-≡ = isBijection isBij (IB.Eq₁.reflexive ∘ cong _)
     where module IB = IsBijection isBij
 
-module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂}
-         {f : A → B} {f⁻¹ : B → A}
-         where
+module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {f : A → B} {f⁻¹ : B → A} where
 
   isCongruent : IsCongruent ≈₁ ≈₂ f → Congruent ≈₂ ≈₁ f⁻¹ → IsCongruent ≈₂ ≈₁ f⁻¹
   isCongruent ic cg = record
     { cong           = cg
-    ; isEquivalence₁ = IC.isEquivalence₂
-    ; isEquivalence₂ = IC.isEquivalence₁
-    } where module IC = IsCongruent ic
+    ; isEquivalence₁ = F.isEquivalence₂
+    ; isEquivalence₂ = F.isEquivalence₁
+    } where module F = IsCongruent ic
 
   isLeftInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsLeftInverse ≈₂ ≈₁ f⁻¹ f
   isLeftInverse inv = record
     { isCongruent = isCongruent F.isCongruent F.from-cong
     ; from-cong   = F.cong₁
-    ; inverseˡ    = inverseˡ ≈₁ ≈₂ f {f⁻¹} F.inverseʳ
+    ; inverseˡ    = inverseˡ ≈₁ ≈₂ F.inverseʳ
     } where module F = IsRightInverse inv
 
   isRightInverse : IsLeftInverse ≈₁ ≈₂ f f⁻¹ → IsRightInverse ≈₂ ≈₁ f⁻¹ f
   isRightInverse inv = record
     { isCongruent = isCongruent F.isCongruent F.from-cong
     ; from-cong   = F.to-cong
-    ; inverseʳ    = inverseʳ ≈₁ ≈₂ f {f⁻¹} F.inverseˡ
+    ; inverseʳ    = inverseʳ ≈₁ ≈₂ F.inverseˡ
     } where module F = IsLeftInverse inv
 
   isInverse : IsInverse ≈₁ ≈₂ f f⁻¹ → IsInverse ≈₂ ≈₁ f⁻¹ f
   isInverse f-inv = record
     { isLeftInverse = isLeftInverse F.isRightInverse
-    ; inverseʳ      = inverseʳ ≈₁ ≈₂ f F.inverseˡ
+    ; inverseʳ      = inverseʳ ≈₁ ≈₂ F.inverseˡ
     } where module F = IsInverse f-inv
 
 ------------------------------------------------------------------------
@@ -140,7 +140,7 @@ module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} (bij : Bijection R S) where
   bijection cong = record
     { to        = from
     ; cong      = cong
-    ; bijective = bijective IB.bijective IB.Eq₂.sym IB.Eq₂.trans IB.cong
+    ; bijective = bijective IB.bijective IB.Eq₁.refl IB.Eq₂.sym IB.Eq₂.trans IB.cong
     }
 
 -- We can always flip a bijection if using the equality over the
diff --git a/src/Function/Definitions.agda b/src/Function/Definitions.agda
index bfeff3c05c..225dc53998 100644
--- a/src/Function/Definitions.agda
+++ b/src/Function/Definitions.agda
@@ -8,39 +8,57 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
+module Function.Definitions where
+
+open import Data.Product.Base using (∃; _×_)
+open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 
-module Function.Definitions
-  {a b ℓ₁ ℓ₂} {A : Set a} {B : Set b}
+private
+  variable
+    a ℓ₁ ℓ₂ : Level
+    A B : Set a
+
+------------------------------------------------------------------------
+-- Basic definitions
+
+module _
   (_≈₁_ : Rel A ℓ₁) -- Equality over the domain
   (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain
   where
 
-open import Data.Product.Base using (_×_)
-import Function.Definitions.Core1 as Core₁
-import Function.Definitions.Core2 as Core₂
-open import Level using (_⊔_)
+  Congruent : (A → B) → Set _
+  Congruent f = ∀ {x y} → x ≈₁ y → f x ≈₂ f y
 
-------------------------------------------------------------------------
--- Definitions
+  Injective : (A → B) → Set _
+  Injective f = ∀ {x y} → f x ≈₂ f y → x ≈₁ y
 
-Congruent : (A → B) → Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-Congruent f = ∀ {x y} → x ≈₁ y →  f x ≈₂ f y
+  Surjective : (A → B) → Set _
+  Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y
 
-Injective : (A → B) → Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-Injective f = ∀ {x y} → f x ≈₂ f y → x ≈₁ y
+  Bijective : (A → B) → Set _
+  Bijective f = Injective f × Surjective f
 
-open Core₂ {A = A} _≈₂_ public
-  using (Surjective)
+  Inverseˡ : (A → B) → (B → A) → Set _
+  Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
+
+  Inverseʳ : (A → B) → (B → A) → Set _
+  Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
+
+  Inverseᵇ : (A → B) → (B → A) → Set _
+  Inverseᵇ f g = Inverseˡ f g × Inverseʳ f g
+
+------------------------------------------------------------------------
+-- Strict definitions
 
-Bijective : (A → B) → Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-Bijective f = Injective f × Surjective f
+-- These are often easier to use once but much harder to compose and
+-- reason about.
 
-open Core₂ {A = A} _≈₂_ public
-  using (Inverseˡ)
+StrictlySurjective : Rel B ℓ₂ → (A → B) → Set _
+StrictlySurjective _≈₂_ f = ∀ y → ∃ λ x → f x ≈₂ y
 
-open Core₁ {B = B} _≈₁_ public
-  using (Inverseʳ)
+StrictlyInverseˡ : Rel B ℓ₂ → (A → B) → (B → A) → Set _
+StrictlyInverseˡ _≈₂_ f g = ∀ y → f (g y) ≈₂ y
 
-Inverseᵇ : (A → B) → (B → A) → Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-Inverseᵇ f g = Inverseˡ f g × Inverseʳ f g
+StrictlyInverseʳ : Rel A ℓ₁ → (A → B) → (B → A) → Set _
+StrictlyInverseʳ _≈₁_ f g = ∀ x → g (f x) ≈₁ x
diff --git a/src/Function/Definitions/Core1.agda b/src/Function/Definitions/Core1.agda
deleted file mode 100644
index b11ca26fea..0000000000
--- a/src/Function/Definitions/Core1.agda
+++ /dev/null
@@ -1,25 +0,0 @@
-------------------------------------------------------------------------
--- The Agda standard library
---
--- Definitions for types of functions that only require an equality
--- relation over the domain.
-------------------------------------------------------------------------
-
--- The contents of this file should usually be accessed from `Function`.
-
-{-# OPTIONS --cubical-compatible --safe #-}
-
-open import Relation.Binary.Core using (Rel)
-
-module Function.Definitions.Core1
-  {a b ℓ₁} {A : Set a} {B : Set b} (_≈₁_ : Rel A ℓ₁)
-  where
-
-open import Level using (_⊔_)
-
-------------------------------------------------------------------------
--- Definitions
-
--- (Note the name `RightInverse` is used for the bundle)
-Inverseʳ : (A → B) → (B → A) → Set (a ⊔ ℓ₁)
-Inverseʳ f g = ∀ x → g (f x) ≈₁ x
diff --git a/src/Function/Definitions/Core2.agda b/src/Function/Definitions/Core2.agda
deleted file mode 100644
index 74973d8433..0000000000
--- a/src/Function/Definitions/Core2.agda
+++ /dev/null
@@ -1,29 +0,0 @@
-------------------------------------------------------------------------
--- The Agda standard library
---
--- Definitions for types of functions that only require an equality
--- relation over the co-domain.
-------------------------------------------------------------------------
-
--- The contents of this file should usually be accessed from `Function`.
-
-{-# OPTIONS --cubical-compatible --safe #-}
-
-open import Relation.Binary.Core using (Rel)
-
-module Function.Definitions.Core2
-  {a b ℓ₂} {A : Set a} {B : Set b} (_≈₂_ : Rel B ℓ₂)
-  where
-
-open import Data.Product.Base using (∃)
-open import Level using (Level; _⊔_)
-
-------------------------------------------------------------------------
--- Definitions
-
-Surjective : (A → B) → Set (a ⊔ b ⊔ ℓ₂)
-Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
-
--- (Note the name `LeftInverse` is used for the bundle)
-Inverseˡ : (A → B) → (B → A) → Set (b ⊔ ℓ₂)
-Inverseˡ f g = ∀ x → f (g x) ≈₂ x
diff --git a/src/Function/Properties.agda b/src/Function/Properties.agda
index adfb3c73b6..69ae3506c1 100644
--- a/src/Function/Properties.agda
+++ b/src/Function/Properties.agda
@@ -43,6 +43,6 @@ private
 →-cong-↔ extAC extBD A↔B C↔D = mk↔′
   (λ h → C↔D.to   ∘ h ∘ A↔B.from)
   (λ g → C↔D.from ∘ g ∘ A↔B.to  )
-  (λ h → extBD λ x → trans (C↔D.inverseˡ _) (cong h (A↔B.inverseˡ x)))
-  (λ g → extAC λ x → trans (C↔D.inverseʳ _) (cong g (A↔B.inverseʳ x)))
+  (λ h → extBD λ x → trans (C↔D.strictlyInverseˡ _) (cong h (A↔B.strictlyInverseˡ x)))
+  (λ g → extAC λ x → trans (C↔D.strictlyInverseʳ _) (cong g (A↔B.strictlyInverseʳ x)))
   where module A↔B = Inverse A↔B; module C↔D = Inverse C↔D
diff --git a/src/Function/Properties/Bijection.agda b/src/Function/Properties/Bijection.agda
index 89aab1b58e..38d269195b 100644
--- a/src/Function/Properties/Bijection.agda
+++ b/src/Function/Properties/Bijection.agda
@@ -11,10 +11,10 @@ module Function.Properties.Bijection where
 open import Function.Bundles
 open import Level using (Level)
 open import Relation.Binary hiding (_⇔_)
-import Relation.Binary.PropositionalEquality.Properties as P
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+open import Relation.Binary.PropositionalEquality as P using (setoid)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
+open import Function.Properties.Surjection using (injective⇒to⁻-cong)
 open import Function.Properties.Inverse using (Inverse⇒Equivalence)
 
 import Function.Construct.Identity as Identity
@@ -55,18 +55,15 @@ trans = Composition.bijection
 -- Conversion functions
 
 Bijection⇒Inverse : Bijection S T → Inverse S T
-Bijection⇒Inverse {S = S} {T = T} b = record
+Bijection⇒Inverse bij = record
   { to        = to
   ; from      = to⁻
   ; to-cong   = cong
-  ; from-cong = λ {x} {y} x≈y → injective (begin
-      to (to⁻ x) ≈⟨ to∘to⁻ x ⟩
-      x          ≈⟨ x≈y ⟩
-      y          ≈˘⟨ to∘to⁻ y ⟩
-      to (to⁻ y) ∎)
-  ; inverse = to∘to⁻ , injective ∘ to∘to⁻ ∘ to
+  ; from-cong = injective⇒to⁻-cong surjection injective
+  ; inverse   = (λ y≈to⁻[x] → Eq₂.trans (cong y≈to⁻[x]) (to∘to⁻ _)) ,
+                (λ y≈to[x] → injective (Eq₂.trans (to∘to⁻ _) y≈to[x]))
   }
-  where open SetoidReasoning T; open Bijection b; to∘to⁻ = proj₂ ∘ surjective
+  where open Bijection bij; to∘to⁻ = proj₂ ∘ strictlySurjective
 
 Bijection⇒Equivalence : Bijection T S → Equivalence T S
 Bijection⇒Equivalence = Inverse⇒Equivalence ∘ Bijection⇒Inverse
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index 6ef0b45af0..f097945dec 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -57,17 +57,17 @@ open module ↔ {ℓ} = IsEquivalence (↔-isEquivalence {ℓ}) using ()
 -- Conversion functions
 
 Inverse⇒Injection : Inverse S T → Injection S T
-Inverse⇒Injection {S = S} I = record
+Inverse⇒Injection I = record
   { to = to
   ; cong = to-cong
-  ; injective = inverseʳ⇒injective S {f⁻¹ = from} from-cong inverseʳ
+  ; injective = inverseʳ⇒injective Eq₂._≈_ to Eq₂.refl Eq₁.sym Eq₁.trans inverseʳ
   } where open Inverse I
 
 Inverse⇒Bijection : Inverse S T → Bijection S T
-Inverse⇒Bijection {S = S} I = record
+Inverse⇒Bijection I = record
   { to        = to
   ; cong      = to-cong
-  ; bijective = inverseᵇ⇒bijective S from-cong inverse
+  ; bijective = inverseᵇ⇒bijective Eq₂._≈_ Eq₂.refl Eq₁.sym Eq₁.trans inverse
   } where open Inverse I
 
 Inverse⇒Equivalence : Inverse S T → Equivalence S T
@@ -93,6 +93,6 @@ module _ (ext : ∀ {a b} → Extensionality a b) where
   ↔-fun A↔B C↔D = mk↔′
     (λ a→c b → to C↔D (a→c (from A↔B b)))
     (λ b→d a → from C↔D (b→d (to A↔B a)))
-    (λ b→d → ext λ _ → P.trans (inverseˡ C↔D _ ) (P.cong b→d (inverseˡ A↔B _)))
-    (λ a→c → ext λ _ → P.trans (inverseʳ C↔D _ ) (P.cong a→c (inverseʳ A↔B _)))
+    (λ b→d → ext λ _ → P.trans (strictlyInverseˡ C↔D _ ) (P.cong b→d (strictlyInverseˡ A↔B _)))
+    (λ a→c → ext λ _ → P.trans (strictlyInverseʳ C↔D _ ) (P.cong a→c (strictlyInverseʳ A↔B _)))
     where open Inverse
diff --git a/src/Function/Properties/RightInverse.agda b/src/Function/Properties/RightInverse.agda
index f5aa05d9df..66b35e53f0 100644
--- a/src/Function/Properties/RightInverse.agda
+++ b/src/Function/Properties/RightInverse.agda
@@ -10,7 +10,7 @@ module Function.Properties.RightInverse where
 
 open import Function.Base
 open import Function.Bundles
-open import Function.Consequences using (inverseʳ⇒surjective)
+open import Function.Consequences using (inverseˡ⇒surjective)
 open import Level using (Level)
 open import Data.Product
 open import Relation.Binary using (Setoid; IsEquivalence)
@@ -27,7 +27,7 @@ RightInverse⇒Surjection : RightInverse S T → Surjection T S
 RightInverse⇒Surjection I = record
   { to         = from
   ; cong       = from-cong
-  ; surjective = λ a → to a , inverseʳ a
+  ; surjective = inverseˡ⇒surjective Eq₁._≈_ inverseʳ
   } where open RightInverse I
 
 ↪⇒↠ : B ↪ A → A ↠ B
diff --git a/src/Function/Properties/Surjection.agda b/src/Function/Properties/Surjection.agda
index ab164c43d0..fce294be76 100644
--- a/src/Function/Properties/Surjection.agda
+++ b/src/Function/Properties/Surjection.agda
@@ -9,20 +9,39 @@
 module Function.Properties.Surjection where
 
 open import Function.Base
+open import Function.Definitions
 open import Function.Bundles
 open import Level using (Level)
 open import Data.Product
+open import Relation.Binary.PropositionalEquality
+open import Relation.Binary using (Setoid)
+import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 private
   variable
-    a b : Level
+    a b ℓ : Level
     A B : Set a
+    T S : Setoid a ℓ
+
+injective⇒to⁻-cong : (surj : Surjection S T) →
+                      (open Surjection surj) →
+                      Injective Eq₁._≈_ Eq₂._≈_ to →
+                      Congruent Eq₂._≈_ Eq₁._≈_ to⁻
+injective⇒to⁻-cong {T = T} surj injective {x} {y} x≈y = injective $ begin
+  to (to⁻ x) ≈⟨ to∘to⁻ x ⟩
+  x          ≈⟨ x≈y ⟩
+  y          ≈˘⟨ to∘to⁻ y ⟩
+  to (to⁻ y) ∎
+  where
+  open SetoidReasoning T
+  open Surjection surj
+  to∘to⁻ = proj₂ ∘ strictlySurjective
 
 ------------------------------------------------------------------------
 -- Conversion functions
 
 ↠⇒↪ : A ↠ B → B ↪ A
-↠⇒↪ s = mk↪ {to = proj₁ ∘ surjective} {from = to} (proj₂ ∘ surjective)
+↠⇒↪ s = mk↪ {from = to} λ { refl → proj₂ (strictlySurjective _)}
   where open Surjection s
 
 ↠⇒⇔ : A ↠ B → A ⇔ B
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index 12920055af..cc80adff63 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -17,6 +17,7 @@ open import Function.Injection   as Inj     using (Injection; _↣_)
 open import Function.Inverse     as Inv     using (Inverse; _↔_)
 open import Function.LeftInverse as LeftInv using (LeftInverse)
 open import Function.Surjection  as Surj    using (Surjection)
+open import Function.Consequences.Propositional
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
@@ -94,12 +95,12 @@ toRelated {K = reverse-implication} rel = lam (B.Func.to rel)
 toRelated {K = equivalence}         rel = Eq.equivalence (B.Equivalence.to rel) (B.Equivalence.from rel)
 toRelated {K = injection}           rel = Inj.injection (B.Injection.to rel) (B.Injection.injective rel)
 toRelated {K = reverse-injection}   rel = lam (Inj.injection (B.Injection.to rel) (B.Injection.injective rel))
-toRelated {K = left-inverse}        rel =
-  LeftInv.leftInverse (B.RightInverse.to rel) (B.RightInverse.from rel) (B.RightInverse.inverseʳ rel)
-toRelated {K = surjection}          rel with B.Surjection.surjective rel
-... | surj = Surj.surjection (B.Surjection.to rel) (proj₁ ∘ surj) (proj₂ ∘ surj)
-toRelated {K = bijection}           rel with B.Bijection.bijective rel
-... | (inj , surj) = Inv.inverse (B.Bijection.to rel) (proj₁ ∘ surj) (inj ∘ proj₂ ∘ surj ∘ (B.Bijection.to rel)) (proj₂ ∘ surj)
+toRelated {K = left-inverse}        rel = LeftInv.leftInverse to from strictlyInverseʳ where open B.RightInverse rel
+toRelated {K = surjection}          rel = Surj.surjection to (proj₁ ∘ strictlySurjective) (proj₂ ∘ strictlySurjective)
+  where open B.Surjection rel
+toRelated {K = bijection}           rel =
+  Inv.inverse to (proj₁ ∘ strictlySurjective) (injective ∘ proj₂ ∘ strictlySurjective ∘ to) (proj₂ ∘ strictlySurjective)
+  where open B.Bijection rel
 
 fromRelated : {K : Kind} → A ∼[ K ] B → A R.∼[ K ] B
 fromRelated {K = implication}         rel = B.mk⟶ rel
@@ -108,13 +109,14 @@ fromRelated {K = equivalence}         record { to = to ; from = from } = B.mk⇔
 fromRelated {K = injection}           rel = B.mk↣ (Inj.Injection.injective rel)
 fromRelated {K = reverse-injection}   (lam app-↢) = B.mk↣ (Inj.Injection.injective app-↢)
 fromRelated {K = left-inverse}        record { to = to ; from = from ; left-inverse-of = left-inverse-of } =
-  B.mk↪ {to = to ⟨$⟩_} {from = from ⟨$⟩_} left-inverse-of
+  B.mk↪ {to = to ⟨$⟩_} {from = from ⟨$⟩_} (strictlyInverseʳ⇒inverseʳ (to ⟨$⟩_) left-inverse-of)
 fromRelated {K = surjection}          record { to = to ; surjective = surjective } with surjective
-... | record { from = from ; right-inverse-of = right-inverse-of } = B.mk↠ {to = to ⟨$⟩_} < from ⟨$⟩_ , right-inverse-of >
+... | record { from = from ; right-inverse-of = right-inverse-of } =
+  B.mk↠ {to = to ⟨$⟩_} < from ⟨$⟩_ , (λ { x P.refl → right-inverse-of x }) >
 fromRelated {K = bijection}           record { to = to ; from = from ; inverse-of = inverse-of } with inverse-of
 ... | record { left-inverse-of = left-inverse-of ; right-inverse-of = right-inverse-of } = B.mk⤖
   ((λ {x y} h → P.subst₂ P._≡_ (left-inverse-of x) (left-inverse-of y) (P.cong (from ⟨$⟩_) h)) ,
-  < from ⟨$⟩_ , right-inverse-of >)
+  < from ⟨$⟩_ , (λ { x P.refl → right-inverse-of x }) >)
 
 -- A non-infix synonym.
 
diff --git a/src/Function/Structures.agda b/src/Function/Structures.agda
index 770dab600e..5566e92c8b 100644
--- a/src/Function/Structures.agda
+++ b/src/Function/Structures.agda
@@ -15,8 +15,10 @@ module Function.Structures {a b ℓ₁ ℓ₂}
   {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain
   where
 
-open import Data.Product.Base using (_,_)
+open import Data.Product.Base as Product using (∃; _×_; _,_)
+open import Function.Base
 open import Function.Definitions
+open import Function.Consequences
 open import Level using (_⊔_)
 
 ------------------------------------------------------------------------
@@ -63,6 +65,9 @@ record IsSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
 
   open IsCongruent isCongruent public
 
+  strictlySurjective : StrictlySurjective _≈₂_ f
+  strictlySurjective x = Product.map₂ (λ v → v Eq₁.refl) (surjective x)
+
 
 record IsBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
   field
@@ -80,6 +85,9 @@ record IsBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     ; surjective  = surjective
     }
 
+  open IsSurjection isSurjection public
+    using (strictlySurjective)
+
 
 ------------------------------------------------------------------------
 -- Two element structures
@@ -94,6 +102,9 @@ record IsLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ 
   open IsCongruent isCongruent public
     renaming (cong to to-cong)
 
+  strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from
+  strictlyInverseˡ x = inverseˡ Eq₁.refl
+
 
 record IsRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
   field
@@ -104,6 +115,9 @@ record IsRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁
   open IsCongruent isCongruent public
     renaming (cong to cong₁)
 
+  strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from
+  strictlyInverseʳ x = inverseʳ Eq₂.refl
+
 
 record IsInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
   field
@@ -119,6 +133,9 @@ record IsInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ 
     ; inverseʳ    = inverseʳ
     }
 
+  open IsRightInverse isRightInverse public
+    using (strictlyInverseʳ)
+
   inverse : Inverseᵇ _≈₁_ _≈₂_ to from
   inverse = inverseˡ , inverseʳ
 
diff --git a/src/Relation/Binary/Morphism/Construct/Composition.agda b/src/Relation/Binary/Morphism/Construct/Composition.agda
index 202605b608..5f6fe9bf32 100644
--- a/src/Relation/Binary/Morphism/Construct/Composition.agda
+++ b/src/Relation/Binary/Morphism/Construct/Composition.agda
@@ -46,9 +46,9 @@ isRelIsomorphism : Transitive ≈₃ →
                    IsRelIsomorphism ≈₁ ≈₂ f →
                    IsRelIsomorphism ≈₂ ≈₃ g →
                    IsRelIsomorphism ≈₁ ≈₃ (g ∘ f)
-isRelIsomorphism {≈₁ = ≈₁} ≈₃-trans m₁ m₂ = record
+isRelIsomorphism  {≈₃ = ≈₃} ≈₃-trans m₁ m₂ = record
   { isMonomorphism = isRelMonomorphism F.isMonomorphism G.isMonomorphism
-  ; surjective     = surjective ≈₁ _ _ ≈₃-trans G.cong F.surjective G.surjective
+  ; surjective     = surjective _ _ ≈₃ F.surjective G.surjective
   } where module F = IsRelIsomorphism m₁; module G = IsRelIsomorphism m₂
 
 ------------------------------------------------------------------------
@@ -103,9 +103,9 @@ isOrderIsomorphism : Transitive ≈₃ →
                      IsOrderIsomorphism ≈₁ ≈₂ ∼₁ ∼₂ f →
                      IsOrderIsomorphism ≈₂ ≈₃ ∼₂ ∼₃ g →
                      IsOrderIsomorphism ≈₁ ≈₃ ∼₁ ∼₃ (g ∘ f)
-isOrderIsomorphism {≈₁ = ≈₁} ≈₃-trans m₁ m₂ = record
+isOrderIsomorphism {≈₃ = ≈₃}  ≈₃-trans m₁ m₂ = record
   { isOrderMonomorphism = isOrderMonomorphism F.isOrderMonomorphism G.isOrderMonomorphism
-  ; surjective          = surjective ≈₁ _ _ ≈₃-trans G.cong F.surjective G.surjective
+  ; surjective          = surjective _ _ ≈₃ F.surjective G.surjective
   } where module F = IsOrderIsomorphism m₁; module G = IsOrderIsomorphism m₂
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Morphism/Construct/Identity.agda b/src/Relation/Binary/Morphism/Construct/Identity.agda
index db12c9793b..302881b4aa 100644
--- a/src/Relation/Binary/Morphism/Construct/Identity.agda
+++ b/src/Relation/Binary/Morphism/Construct/Identity.agda
@@ -8,6 +8,7 @@
 
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
+import Function.Construct.Identity as Id
 open import Level using (Level)
 open import Relation.Binary
 open import Relation.Binary.Morphism.Structures
@@ -29,19 +30,19 @@ module _ (≈ : Rel A ℓ₁) where
 
   isRelHomomorphism : IsRelHomomorphism ≈ ≈ id
   isRelHomomorphism = record
-    { cong = id
+    { cong = Id.congruent ≈
     }
 
   isRelMonomorphism : IsRelMonomorphism ≈ ≈ id
   isRelMonomorphism = record
     { isHomomorphism = isRelHomomorphism
-    ; injective      = id
+    ; injective      = Id.injective ≈
     }
 
   isRelIsomorphism : Reflexive ≈ → IsRelIsomorphism ≈ ≈ id
   isRelIsomorphism refl = record
     { isMonomorphism = isRelMonomorphism
-    ; surjective     = λ y → y , refl
+    ; surjective     = Id.surjective ≈
     }
 
 ------------------------------------------------------------------------
@@ -76,14 +77,14 @@ module _ (≈ : Rel A ℓ₁) (∼ : Rel A ℓ₂) where
   isOrderMonomorphism : IsOrderMonomorphism ≈ ≈ ∼ ∼ id
   isOrderMonomorphism = record
     { isOrderHomomorphism = isOrderHomomorphism
-    ; injective           = id
+    ; injective           = Id.injective ≈
     ; cancel              = id
     }
 
   isOrderIsomorphism : Reflexive ≈ → IsOrderIsomorphism ≈ ≈ ∼ ∼ id
   isOrderIsomorphism refl = record
     { isOrderMonomorphism = isOrderMonomorphism
-    ; surjective          = λ y → y , refl
+    ; surjective          = Id.surjective ≈
     }
 
 ------------------------------------------------------------------------

From 3aa5e3a7fe5387eef717f856f648ce95bec32719 Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Mon, 31 Jul 2023 08:12:16 -0400
Subject: [PATCH 086/316] Proofs `take/drop/head -map-commute` added to
 Data.List.Properties (#1986)

---
 CHANGELOG.md                  |  4 ++++
 src/Data/List/Properties.agda | 27 +++++++++++++++++++++++++--
 2 files changed, 29 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b7cb6aa60b..3818f1a6f7 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2200,6 +2200,10 @@ Other minor changes
 
   length-isMagmaHomomorphism : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
   length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
+  
+  take-map : take n (map f xs) ≡ map f (take n xs)
+  drop-map : drop n (map f xs) ≡ map f (drop n xs)
+  head-map : head (map f xs) ≡ Maybe.map f (head xs)
 
   take-suc : (o : Fin (length xs)) → let m = toℕ o in take (suc m) xs ≡ take m xs ∷ʳ lookup xs o
   take-suc-tabulate : (f : Fin n → A) (o : Fin n) → let m = toℕ o in take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f o
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 454ee4b211..ca5ffe27f2 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -21,7 +21,7 @@ open import Data.List.Base as List
 open import Data.List.Membership.Propositional using (_∈_)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
-open import Data.Maybe.Base using (Maybe; just; nothing)
+open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.Properties
@@ -44,6 +44,7 @@ open import Relation.Nullary.Decidable as Decidable using (isYes; map′; ⌊_
 open import Relation.Unary using (Pred; Decidable; ∁)
 open import Relation.Unary.Properties using (∁?)
 
+
 open ≡-Reasoning
 
 private
@@ -761,6 +762,12 @@ length-take zero    xs       = refl
 length-take (suc n) []       = refl
 length-take (suc n) (x ∷ xs) = cong suc (length-take n xs)
 
+-- Take commutes with map.
+take-map : ∀ {f : A → B} (n : ℕ) xs → take n (map f xs) ≡ map f (take n xs)
+take-map zero xs = refl
+take-map (suc s) [] = refl
+take-map (suc s) (a ∷ xs) = cong (_ ∷_) (take-map s xs)
+
 take-suc : (xs : List A) (i : Fin (length xs)) → let m = toℕ i in
            take (suc m) xs ≡ take m xs ∷ʳ lookup xs i
 take-suc (x ∷ xs) zero    = refl
@@ -791,12 +798,17 @@ length-drop zero    xs       = refl
 length-drop (suc n) []       = refl
 length-drop (suc n) (x ∷ xs) = length-drop n xs
 
+-- Drop commutes with map.
+drop-map : ∀ {f : A → B} (n : ℕ) xs → drop n (map f xs) ≡ map f (drop n xs)
+drop-map zero xs = refl
+drop-map (suc n) [] = refl
+drop-map (suc n) (a ∷ xs) = drop-map n xs
+
 -- Dropping from an empty list does nothing.
 drop-[] : ∀ m → drop {A = A} m [] ≡ []
 drop-[] zero = refl
 drop-[] (suc m) = refl
 
-
 take++drop : ∀ n (xs : List A) → take n xs ++ drop n xs ≡ xs
 take++drop zero    xs       = refl
 take++drop (suc n) []       = refl
@@ -1118,6 +1130,17 @@ module _ {x y : A} where
 
 
 
+------------------------------------------------------------------------
+-- head
+
+-- 'commute' List.head and List.map to obtain a Maybe.map and List.head.
+head-map : ∀ {f : A → B} xs → head (map f xs) ≡ Maybe.map f (head xs)
+head-map [] = refl
+head-map (_ ∷ _) = refl
+
+
+
+
 ------------------------------------------------------------------------
 -- DEPRECATED
 ------------------------------------------------------------------------

From d4627c66ced4a0bb16b1c4c82d1f85f3b7086d13 Mon Sep 17 00:00:00 2001
From: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
Date: Mon, 31 Jul 2023 09:05:33 -0400
Subject: [PATCH 087/316] Simplified `Vec` import (#2018)

---
 README/Data/Tree/AVL.agda                                      | 2 +-
 src/Data/Nat/Binary/Subtraction.agda                           | 2 +-
 src/Data/Vec/Bounded.agda                                      | 3 ++-
 src/Data/Vec/Functional/Properties.agda                        | 2 +-
 src/Data/Vec/Membership/Propositional/Properties.agda          | 2 +-
 src/Data/Vec/Relation/Binary/Lex/Core.agda                     | 2 +-
 src/Data/Vec/Relation/Binary/Lex/NonStrict.agda                | 2 +-
 src/Data/Vec/Relation/Unary/Any/Properties.agda                | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Addition.agda      | 2 +-
 .../Core/Polynomial/Homomorphism/Exponentiation.agda           | 2 +-
 .../Core/Polynomial/Homomorphism/Multiplication.agda           | 2 +-
 .../RingSolver/Core/Polynomial/Homomorphism/Negation.agda      | 2 +-
 src/Tactic/RingSolver/Core/Polynomial/Semantics.agda           | 2 +-
 src/Tactic/RingSolver/NonReflective.agda                       | 2 +-
 14 files changed, 15 insertions(+), 14 deletions(-)

diff --git a/README/Data/Tree/AVL.agda b/README/Data/Tree/AVL.agda
index abba7bfd0b..1b20d9a3b0 100644
--- a/README/Data/Tree/AVL.agda
+++ b/README/Data/Tree/AVL.agda
@@ -22,7 +22,7 @@ import Data.Tree.AVL
 open import Data.Nat.Properties using (<-strictTotalOrder)
 open import Data.Product as Prod using (_,_; _,′_)
 open import Data.String.Base using (String)
-open import Data.Vec using (Vec; _∷_; [])
+open import Data.Vec.Base using (Vec; _∷_; [])
 open import Relation.Binary.PropositionalEquality
 
 open Data.Tree.AVL <-strictTotalOrder renaming (Tree to Tree′)
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index a3e1de4a1b..5dc7c8951f 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -18,7 +18,7 @@ open import Data.Nat.Binary.Properties
 import Data.Nat.Properties as ℕₚ
 open import Data.Product using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Data.Vec using ([]; _∷_)
+open import Data.Vec.Base using ([]; _∷_)
 open import Function.Base using (_∘_; _$_)
 open import Level using (0ℓ)
 open import Relation.Binary
diff --git a/src/Data/Vec/Bounded.agda b/src/Data/Vec/Bounded.agda
index 3a1aae623e..35ecc6a96d 100644
--- a/src/Data/Vec/Bounded.agda
+++ b/src/Data/Vec/Bounded.agda
@@ -10,7 +10,8 @@ module Data.Vec.Bounded where
 
 open import Level using (Level)
 open import Data.Nat.Base
-open import Data.Vec as Vec using (Vec)
+open import Data.Vec.Base using (Vec)
+import Data.Vec as Vec using (filter; takeWhile; dropWhile)
 open import Function.Base using (id)
 open import Relation.Binary using (_Preserves_⟶_)
 open import Relation.Unary using (Pred; Decidable)
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index 9c4486be47..e6cc006d89 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -16,7 +16,7 @@ open import Data.Product as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.List.Base as L using (List)
 import Data.List.Properties as Lₚ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
-open import Data.Vec as V using (Vec)
+open import Data.Vec.Base as V using (Vec)
 import Data.Vec.Properties as Vₚ
 open import Data.Vec.Functional
 open import Function.Base
diff --git a/src/Data/Vec/Membership/Propositional/Properties.agda b/src/Data/Vec/Membership/Propositional/Properties.agda
index 983e70ae63..f4722c9899 100644
--- a/src/Data/Vec/Membership/Propositional/Properties.agda
+++ b/src/Data/Vec/Membership/Propositional/Properties.agda
@@ -10,7 +10,7 @@ module Data.Vec.Membership.Propositional.Properties where
 
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Product as Prod using (_,_; ∃; _×_; -,_)
-open import Data.Vec hiding (here; there)
+open import Data.Vec.Base
 open import Data.Vec.Relation.Unary.Any using (here; there)
 open import Data.List.Base using ([]; _∷_)
 open import Data.List.Relation.Unary.Any using (here; there)
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index 9b2bea47b7..ddc4242496 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -12,8 +12,8 @@ open import Data.Empty
 open import Data.Nat using (ℕ; suc)
 import Data.Nat.Properties as ℕ
 open import Data.Product using (_×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
-open import Data.Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Binary.Pointwise.Inductive using (Pointwise; []; _∷_)
 open import Function.Base using (flip)
 open import Function.Bundles using (_⇔_; mk⇔)
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index d3c24ab42b..80a10442ef 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -15,7 +15,7 @@ open import Data.Empty
 open import Data.Unit using (⊤; tt)
 open import Data.Product using (proj₁; proj₂)
 open import Data.Nat using (ℕ)
-open import Data.Vec using (Vec; []; _∷_)
+open import Data.Vec.Base using (Vec; []; _∷_)
 import Data.Vec.Relation.Binary.Lex.Strict as Strict
 open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
   using (Pointwise; []; _∷_; head; tail)
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index 1c9f171d7b..fd6455da55 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -16,7 +16,7 @@ import Data.List.Relation.Unary.Any as List
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
 open import Data.Product as Prod using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
-open import Data.Vec hiding (here; there)
+open import Data.Vec.Base hiding (here; there)
 open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.Vec.Membership.Propositional
   using (_∈_; mapWith∈; find; lose)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
index 3ea01b2c17..bad78c4da2 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
@@ -18,7 +18,7 @@ open import Data.Nat.Properties as ℕₚ using (≤′-trans)
 open import Data.Product.Base         using (_,_; _×_; proj₂)
 open import Data.List.Base            using (_∷_; [])
 open import Data.List.Kleene
-open import Data.Vec                  using (Vec)
+open import Data.Vec.Base             using (Vec)
 open import Function.Base using (_⟨_⟩_; flip)
 open import Relation.Unary
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
index eeacadb154..4f6607d663 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
@@ -18,7 +18,7 @@ open import Function.Base using (_⟨_⟩_)
 open import Data.Nat.Base as ℕ using (ℕ; suc; zero; compare)
 open import Data.Product.Base  using (_,_; _×_; proj₁; proj₂)
 open import Data.List.Kleene
-open import Data.Vec      using (Vec)
+open import Data.Vec.Base      using (Vec)
 
 import Data.Nat.Properties as ℕ-Prop
 import Relation.Binary.PropositionalEquality.Core as ≡
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
index c7ad6d6145..c4e1ef756b 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
@@ -18,7 +18,7 @@ open import Data.Nat.Properties    using (≤′-trans)
 open import Data.Nat.Induction
 open import Data.Product.Base      using (_×_; _,_; proj₁; proj₂; map₁)
 open import Data.List.Kleene
-open import Data.Vec               using (Vec)
+open import Data.Vec.Base          using (Vec)
 open import Function.Base using (_⟨_⟩_; flip)
 open import Induction.WellFounded
 open import Relation.Unary
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
index 59b99cf71b..604f02c533 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
@@ -13,8 +13,8 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Negation
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
+open import Data.Vec.Base         using (Vec)
 open import Data.Product.Base     using (_,_)
-open import Data.Vec              using (Vec)
 open import Data.Nat              using (_<′_)
 open import Data.Nat.Induction
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
index 3d6e8c512f..4e3e672916 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
@@ -14,7 +14,7 @@ module Tactic.RingSolver.Core.Polynomial.Semantics
   where
 
 open import Data.Nat          using (ℕ; suc; zero; _≤′_; ≤′-step; ≤′-refl)
-open import Data.Vec          using (Vec; []; _∷_; uncons)
+open import Data.Vec.Base     using (Vec; []; _∷_; uncons)
 open import Data.List.Base    using ([]; _∷_)
 open import Data.Product.Base using (_,_; _×_)
 open import Data.List.Kleene  using (_+; _*; ∹_; _&_; [])
diff --git a/src/Tactic/RingSolver/NonReflective.agda b/src/Tactic/RingSolver/NonReflective.agda
index ef99adc22e..6aa8997bb0 100644
--- a/src/Tactic/RingSolver/NonReflective.agda
+++ b/src/Tactic/RingSolver/NonReflective.agda
@@ -24,7 +24,7 @@ open import Data.Maybe.Base
 open import Data.Empty using (⊥-elim)
 open import Data.Nat.Base using (ℕ)
 open import Data.Product
-open import Data.Vec hiding (_⊛_)
+open import Data.Vec.Base using (Vec)
 open import Data.Vec.N-ary
 
 open import Tactic.RingSolver.Core.Polynomial.Parameters

From a590f6ef55b04ed219ea79d6686af431c772df20 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 10 Aug 2023 03:33:32 -0400
Subject: [PATCH 088/316] More `Data.Product` simplifications (#2039)

---
 README/Data/Nat/Induction.agda                                  | 2 +-
 README/Data/Tree/AVL.agda                                       | 2 +-
 README/Function/Reasoning.agda                                  | 2 +-
 README/Nary.agda                                                | 2 +-
 src/Algebra/Lattice/Construct/Subst/Equality.agda               | 2 +-
 src/Codata/Guarded/M.agda                                       | 2 +-
 src/Codata/Guarded/Stream.agda                                  | 2 +-
 src/Codata/Musical/M.agda                                       | 2 +-
 src/Codata/Sized/Delay.agda                                     | 2 +-
 src/Codata/Sized/M.agda                                         | 2 +-
 src/Data/Container/Core.agda                                    | 2 +-
 src/Data/Container/FreeMonad.agda                               | 2 +-
 src/Data/Container/Indexed.agda                                 | 2 +-
 src/Data/Container/Relation/Binary/Pointwise.agda               | 2 +-
 src/Data/Container/Relation/Binary/Pointwise/Properties.agda    | 2 +-
 src/Data/List/Relation/Binary/Lex.agda                          | 2 +-
 src/Data/List/Relation/Binary/Pointwise.agda                    | 2 +-
 src/Data/List/Relation/Binary/Pointwise/Properties.agda         | 2 +-
 .../Sublist/Propositional/Example/UniqueBoundVariables.agda     | 2 +-
 .../List/Relation/Binary/Subset/Propositional/Properties.agda   | 2 +-
 src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda     | 2 +-
 src/Data/List/Relation/Ternary/Appending/Propositional.agda     | 2 +-
 src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda | 2 +-
 src/Data/List/Relation/Unary/All/Properties.agda                | 2 +-
 src/Data/List/Relation/Unary/AllPairs.agda                      | 2 +-
 src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda  | 2 +-
 src/Data/List/Relation/Unary/Grouped.agda                       | 2 +-
 src/Data/List/Relation/Unary/Linked.agda                        | 2 +-
 .../List/Relation/Unary/Unique/Propositional/Properties.agda    | 2 +-
 src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda      | 2 +-
 src/Data/List/Sort/MergeSort.agda                               | 2 +-
 src/Data/Maybe/Properties.agda                                  | 2 +-
 src/Data/Maybe/Relation/Unary/All.agda                          | 2 +-
 src/Data/Maybe/Relation/Unary/All/Properties.agda               | 2 +-
 src/Data/Nat/Coprimality.agda                                   | 2 +-
 src/Data/Nat/PseudoRandom/LCG/Unsafe.agda                       | 2 +-
 src/Data/Nat/Show.agda                                          | 2 +-
 src/Data/Parity/Properties.agda                                 | 2 +-
 src/Data/Product/Relation/Binary/Lex/NonStrict.agda             | 2 +-
 src/Data/Product/Relation/Binary/Pointwise/Dependent.agda       | 2 +-
 src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda    | 2 +-
 src/Data/Rational/Properties.agda                               | 2 +-
 src/Data/Rational/Unnormalised/Properties.agda                  | 2 +-
 src/Data/Record.agda                                            | 2 +-
 src/Data/Sign/Properties.agda                                   | 2 +-
 src/Data/Sum/Relation/Binary/Pointwise.agda                     | 2 +-
 src/Data/Tree/AVL.agda                                          | 2 +-
 src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda               | 2 +-
 src/Data/Tree/AVL/Map.agda                                      | 2 +-
 src/Data/Tree/AVL/NonEmpty.agda                                 | 2 +-
 src/Data/Tree/AVL/Sets.agda                                     | 2 +-
 src/Data/Tree/AVL/Sets/Membership.agda                          | 2 +-
 src/Data/Tree/AVL/Sets/Membership/Properties.agda               | 2 +-
 src/Data/Tree/AVL/Value.agda                                    | 2 +-
 src/Data/Tree/Rose/Show.agda                                    | 2 +-
 src/Data/Vec/Functional/Properties.agda                         | 2 +-
 .../Vec/Functional/Relation/Binary/Pointwise/Properties.agda    | 2 +-
 src/Data/Vec/Functional/Relation/Unary/All.agda                 | 2 +-
 src/Data/Vec/Functional/Relation/Unary/All/Properties.agda      | 2 +-
 src/Data/Vec/Properties.agda                                    | 2 +-
 src/Data/Vec/Recursive/Effectful.agda                           | 2 +-
 src/Data/Vec/Relation/Binary/Lex/Core.agda                      | 2 +-
 src/Data/Vec/Relation/Binary/Lex/NonStrict.agda                 | 2 +-
 src/Data/Vec/Relation/Binary/Lex/Strict.agda                    | 2 +-
 src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda           | 2 +-
 src/Data/Vec/Relation/Unary/All.agda                            | 2 +-
 src/Data/Vec/Relation/Unary/All/Properties.agda                 | 2 +-
 src/Data/Vec/Relation/Unary/AllPairs.agda                       | 2 +-
 src/Data/Vec/Relation/Unary/Linked.agda                         | 2 +-
 .../Vec/Relation/Unary/Unique/Propositional/Properties.agda     | 2 +-
 src/Data/W/Sized.agda                                           | 2 +-
 src/Effect/Applicative/Indexed.agda                             | 2 +-
 src/Effect/Monad/IO.agda                                        | 2 +-
 src/Effect/Monad/Reader/Transformer.agda                        | 2 +-
 src/Effect/Monad/State/Transformer.agda                         | 2 +-
 src/Effect/Monad/Writer.agda                                    | 2 +-
 src/Effect/Monad/Writer/Indexed.agda                            | 2 +-
 src/Effect/Monad/Writer/Transformer.agda                        | 2 +-
 src/Text/Printf/Generic.agda                                    | 2 +-
 tests/data/appending/TakeWhile.agda                             | 2 +-
 tests/monad/counting/Main.agda                                  | 2 +-
 tests/monad/fibonacci/Main.agda                                 | 2 +-
 tests/monad/pythagorean/Main.agda                               | 2 +-
 83 files changed, 83 insertions(+), 83 deletions(-)

diff --git a/README/Data/Nat/Induction.agda b/README/Data/Nat/Induction.agda
index d430d6d992..52b802b2b8 100644
--- a/README/Data/Nat/Induction.agda
+++ b/README/Data/Nat/Induction.agda
@@ -9,7 +9,7 @@ module README.Data.Nat.Induction where
 
 open import Data.Nat
 open import Data.Nat.Induction
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_)
 open import Induction.WellFounded
 open import Relation.Binary.PropositionalEquality
diff --git a/README/Data/Tree/AVL.agda b/README/Data/Tree/AVL.agda
index 1b20d9a3b0..0955887f23 100644
--- a/README/Data/Tree/AVL.agda
+++ b/README/Data/Tree/AVL.agda
@@ -20,7 +20,7 @@ import Data.Tree.AVL
 -- natural numbers as keys and vectors of strings as values.
 
 open import Data.Nat.Properties using (<-strictTotalOrder)
-open import Data.Product as Prod using (_,_; _,′_)
+open import Data.Product.Base as Prod using (_,_; _,′_)
 open import Data.String.Base using (String)
 open import Data.Vec.Base using (Vec; _∷_; [])
 open import Relation.Binary.PropositionalEquality
diff --git a/README/Function/Reasoning.agda b/README/Function/Reasoning.agda
index c484fdea9f..0125fa9d99 100644
--- a/README/Function/Reasoning.agda
+++ b/README/Function/Reasoning.agda
@@ -39,7 +39,7 @@ open import Data.String.Base as String using (String; toList; fromList)
 open import Data.String.Properties as String using (_==_)
 open import Function.Base using (_∘_)
 open import Data.Bool hiding (_≤?_)
-open import Data.Product as P using (_×_; <_,_>; uncurry; proj₁)
+open import Data.Product.Base as P using (_×_; <_,_>; uncurry; proj₁)
 open import Agda.Builtin.Equality
 
 -- This can give us for instance this decomposition of a function
diff --git a/README/Nary.agda b/README/Nary.agda
index 16e50392b3..5566bd48cf 100644
--- a/README/Nary.agda
+++ b/README/Nary.agda
@@ -15,7 +15,7 @@ open import Data.Nat.Properties
 open import Data.Fin using (Fin; fromℕ; #_; inject₁)
 open import Data.List
 open import Data.List.Properties
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (id; flip; _∘′_)
 open import Relation.Nullary
diff --git a/src/Algebra/Lattice/Construct/Subst/Equality.agda b/src/Algebra/Lattice/Construct/Subst/Equality.agda
index 14a8f650cf..e756319aaf 100644
--- a/src/Algebra/Lattice/Construct/Subst/Equality.agda
+++ b/src/Algebra/Lattice/Construct/Subst/Equality.agda
@@ -12,7 +12,7 @@
 open import Algebra.Core using (Op₂)
 open import Algebra.Definitions
 open import Algebra.Lattice.Structures
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
 open import Function.Base
 open import Relation.Binary.Core
 
diff --git a/src/Codata/Guarded/M.agda b/src/Codata/Guarded/M.agda
index eaf2e0eadd..876fde8031 100644
--- a/src/Codata/Guarded/M.agda
+++ b/src/Codata/Guarded/M.agda
@@ -11,7 +11,7 @@ module Codata.Guarded.M where
 open import Level
 open import Data.Container.Core hiding (map; Shape; Position)
 open import Function.Base
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 
 -- The family of M-types
 
diff --git a/src/Codata/Guarded/Stream.agda b/src/Codata/Guarded/Stream.agda
index 6024525cb4..6036158463 100644
--- a/src/Codata/Guarded/Stream.agda
+++ b/src/Codata/Guarded/Stream.agda
@@ -12,7 +12,7 @@ open import Level hiding (suc)
 open import Data.Nat.Base
 open import Function.Base
 open import Data.List.Base as List using (List; []; _∷_)
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.List.NonEmpty.Base as List⁺ using (List⁺; _∷_)
 open import Algebra.Core
diff --git a/src/Codata/Musical/M.agda b/src/Codata/Musical/M.agda
index 6c63de7ea2..5c724015bb 100644
--- a/src/Codata/Musical/M.agda
+++ b/src/Codata/Musical/M.agda
@@ -10,7 +10,7 @@ module Codata.Musical.M where
 
 open import Codata.Musical.Notation
 open import Level
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.Container.Core as C hiding (map)
 
 -- The family of M-types.
diff --git a/src/Codata/Sized/Delay.agda b/src/Codata/Sized/Delay.agda
index 0bb9c3470b..b4549089e7 100644
--- a/src/Codata/Sized/Delay.agda
+++ b/src/Codata/Sized/Delay.agda
@@ -16,7 +16,7 @@ open import Data.Empty
 open import Relation.Nullary
 open import Data.Nat.Base
 open import Data.Maybe.Base hiding (map ; fromMaybe ; zipWith ; alignWith ; zip ; align)
-open import Data.Product as P hiding (map ; zip)
+open import Data.Product.Base as P hiding (map ; zip)
 open import Data.Sum.Base as S hiding (map)
 open import Data.These.Base as T using (These; this; that; these)
 open import Function.Base using (id)
diff --git a/src/Codata/Sized/M.agda b/src/Codata/Sized/M.agda
index 063c276a95..844f2e3219 100644
--- a/src/Codata/Sized/M.agda
+++ b/src/Codata/Sized/M.agda
@@ -11,7 +11,7 @@ module Codata.Sized.M where
 open import Size
 open import Level
 open import Codata.Sized.Thunk using (Thunk; force)
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.Container.Core as C hiding (map)
 
 data M {s p} (C : Container s p) (i : Size) : Set (s ⊔ p) where
diff --git a/src/Data/Container/Core.agda b/src/Data/Container/Core.agda
index 22b26955d8..dc5134f585 100644
--- a/src/Data/Container/Core.agda
+++ b/src/Data/Container/Core.agda
@@ -9,7 +9,7 @@
 module Data.Container.Core where
 
 open import Level
-open import Data.Product as Prod using (Σ-syntax)
+open import Data.Product.Base as Prod using (Σ-syntax)
 open import Function.Base
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse using (_↔_; module Inverse)
diff --git a/src/Data/Container/FreeMonad.agda b/src/Data/Container/FreeMonad.agda
index 1d206a4b87..8a24ef1d36 100644
--- a/src/Data/Container/FreeMonad.agda
+++ b/src/Data/Container/FreeMonad.agda
@@ -10,7 +10,7 @@ module Data.Container.FreeMonad where
 
 open import Level using (Level; _⊔_)
 open import Data.Sum.Base using (inj₁; inj₂ ; [_,_]′)
-open import Data.Product using (_,_; -,_)
+open import Data.Product.Base using (_,_; -,_)
 open import Data.Container using (Container; ⟦_⟧; μ)
 open import Data.Container.Relation.Unary.All using (□; all)
 open import Data.Container.Combinator using (const; _⊎_)
diff --git a/src/Data/Container/Indexed.agda b/src/Data/Container/Indexed.agda
index a5b17d2aab..26aeaa25b0 100644
--- a/src/Data/Container/Indexed.agda
+++ b/src/Data/Container/Indexed.agda
@@ -12,7 +12,7 @@
 module Data.Container.Indexed where
 
 open import Level
-open import Data.Product as Prod hiding (map)
+open import Data.Product.Base as Prod hiding (map)
 open import Data.W.Indexed
 open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
 open import Function.Equality using (_⟨$⟩_)
diff --git a/src/Data/Container/Relation/Binary/Pointwise.agda b/src/Data/Container/Relation/Binary/Pointwise.agda
index 6f9949dc4e..4de4cd0939 100644
--- a/src/Data/Container/Relation/Binary/Pointwise.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise.agda
@@ -8,7 +8,7 @@
 
 module Data.Container.Relation.Binary.Pointwise where
 
-open import Data.Product using (_,_; Σ-syntax; -,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; Σ-syntax; -,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
 open import Level using (_⊔_)
 open import Relation.Binary.Core using (REL; _⇒_)
diff --git a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
index 3092037901..7ddc209860 100644
--- a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
@@ -11,7 +11,7 @@ module Data.Container.Relation.Binary.Pointwise.Properties where
 open import Axiom.Extensionality.Propositional
 open import Data.Container.Core
 open import Data.Container.Relation.Binary.Pointwise
-open import Data.Product using (_,_; Σ-syntax; -,_)
+open import Data.Product.Base using (_,_; Σ-syntax; -,_)
 open import Level using (_⊔_)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P
diff --git a/src/Data/List/Relation/Binary/Lex.agda b/src/Data/List/Relation/Binary/Lex.agda
index f0c8fa42df..c1a07e8b53 100644
--- a/src/Data/List/Relation/Binary/Lex.agda
+++ b/src/Data/List/Relation/Binary/Lex.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Binary.Lex where
 
 open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Unit.Base using (⊤; tt)
-open import Data.Product using (_×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Data.List.Base using (List; []; _∷_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Function.Base using (_∘_; flip; id)
diff --git a/src/Data/List/Relation/Binary/Pointwise.agda b/src/Data/List/Relation/Binary/Pointwise.agda
index 0852085a37..6a5cde5aec 100644
--- a/src/Data/List/Relation/Binary/Pointwise.agda
+++ b/src/Data/List/Relation/Binary/Pointwise.agda
@@ -12,7 +12,7 @@ open import Algebra.Core using (Op₂)
 open import Function.Base
 open import Function.Inverse using (Inverse)
 open import Data.Bool.Base using (true; false)
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.List.Base as List hiding (map; head; tail; uncons)
 open import Data.List.Properties using (≡-dec; length-++)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
diff --git a/src/Data/List/Relation/Binary/Pointwise/Properties.agda b/src/Data/List/Relation/Binary/Pointwise/Properties.agda
index 207f0fe70d..7dfba09296 100644
--- a/src/Data/List/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/List/Relation/Binary/Pointwise/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.List.Relation.Binary.Pointwise.Properties where
 
-open import Data.Product using (_,_; uncurry)
+open import Data.Product.Base using (_,_; uncurry)
 open import Data.List.Base using (List; []; _∷_)
 open import Level
 open import Relation.Binary.Core using (REL; _⇒_)
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
index d2a252e0f0..39a6fe65a1 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
@@ -36,7 +36,7 @@ open import Data.List.Relation.Binary.Sublist.Propositional.Properties using
   ; equalize-separators
   )
 
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 
 infixr 8 _⇒_
 infix 1 _⊢_~_▷_
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index 836c52fd70..222bcee92f 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -26,7 +26,7 @@ open import Data.List.Relation.Binary.Subset.Propositional
 open import Data.List.Relation.Binary.Permutation.Propositional
 import Data.List.Relation.Binary.Permutation.Propositional.Properties as Permutation
 open import Data.Nat using (ℕ; _≤_)
-import Data.Product as Prod
+import Data.Product.Base as Prod
 import Data.Sum.Base as Sum
 open import Effect.Monad
 open import Function.Base using (_∘_; _∘′_; id; _$_)
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index 2a700e468f..5229813a50 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -20,7 +20,7 @@ import Data.List.Relation.Binary.Sublist.Setoid as Sublist
 import Data.List.Relation.Binary.Equality.Setoid as Equality
 import Data.List.Relation.Binary.Permutation.Setoid as Permutation
 import Data.List.Relation.Binary.Permutation.Setoid.Properties as Permutationₚ
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; _∘₂_)
 open import Level using (Level)
 open import Relation.Nullary using (¬_; does; yes; no)
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional.agda b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
index 46567a01ed..f85ff2f480 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Ternary.Appending.Propositional
   where
 
 open import Data.List.Base as List using (List; []; _∷_)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 
 import Data.List.Properties as Listₚ
 import Data.List.Relation.Binary.Pointwise as Pw using (≡⇒Pointwise-≡; Pointwise-≡⇒≡)
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
index 83d3add55e..1e2ef9cd6a 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
@@ -14,7 +14,7 @@ open import Data.List.Base as List using (List; [])
 import Data.List.Properties as Listₚ
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; [])
 import Data.List.Relation.Ternary.Appending.Properties as Appendingₚ
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 open import Data.List.Relation.Ternary.Appending.Setoid S
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index da3de23dd9..29a645762e 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -31,7 +31,7 @@ open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.All as Maybe using (just; nothing)
 open import Data.Nat.Base using (zero; suc; s≤s; _<_; z<s; s<s)
 open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)
-open import Data.Product as Prod using (_×_; _,_; uncurry; uncurry′)
+open import Data.Product.Base as Prod using (_×_; _,_; uncurry; uncurry′)
 open import Function.Base
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
diff --git a/src/Data/List/Relation/Unary/AllPairs.agda b/src/Data/List/Relation/Unary/AllPairs.agda
index f2cbf29755..49bfac6ae0 100644
--- a/src/Data/List/Relation/Unary/AllPairs.agda
+++ b/src/Data/List/Relation/Unary/AllPairs.agda
@@ -13,7 +13,7 @@ module Data.List.Relation.Unary.AllPairs
 
 open import Data.List.Base using (List; []; _∷_)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
-open import Data.Product as Prod using (_,_; _×_; uncurry; <_,_>)
+open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (_⊔_)
 open import Relation.Binary as B using (Rel; _⇒_)
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index f01fd90a48..ba400055f3 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -14,7 +14,7 @@ open import Data.List.Relation.Unary.Enumerates.Setoid
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Sum.Relation.Binary.Pointwise
   using (_⊎ₛ_; inj₁; inj₂)
-open import Data.Product using (_,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
   using (_×ₛ_)
 open import Function.Base using (_∘_)
diff --git a/src/Data/List/Relation/Unary/Grouped.agda b/src/Data/List/Relation/Unary/Grouped.agda
index b907069142..87b11cabc3 100644
--- a/src/Data/List/Relation/Unary/Grouped.agda
+++ b/src/Data/List/Relation/Unary/Grouped.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Unary.Grouped where
 open import Data.List.Base using (List; []; _∷_; map)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_; all?)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Relation.Binary as B using (REL; Rel)
 open import Relation.Unary as U using (Pred)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/List/Relation/Unary/Linked.agda b/src/Data/List/Relation/Unary/Linked.agda
index cfafbc28b5..b5dbddecd7 100644
--- a/src/Data/List/Relation/Unary/Linked.agda
+++ b/src/Data/List/Relation/Unary/Linked.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Unary.Linked {a} {A : Set a} where
 
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
-open import Data.Product as Prod using (_,_; _×_; uncurry; <_,_>)
+open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
 open import Data.Maybe.Base using (just)
 open import Data.Maybe.Relation.Binary.Connected
   using (Connected; just; just-nothing)
diff --git a/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda b/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
index a223a2c9cc..7740dc83e3 100644
--- a/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
@@ -17,7 +17,7 @@ import Data.List.Relation.Unary.Unique.Setoid.Properties as Setoid
 open import Data.Fin.Base using (Fin)
 open import Data.Nat.Base using (_<_)
 open import Data.Nat.Properties using (<⇒≢)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡⇒≡×≡)
 open import Function.Base using (id; _∘_)
 open import Level using (Level)
diff --git a/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
index 33b66d2bd1..6ba7f803a1 100644
--- a/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
@@ -16,7 +16,7 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.All.Properties using (All¬⇒¬Any)
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs)
 open import Data.List.Relation.Unary.Unique.Setoid
-open import Data.Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 import Data.List.Relation.Unary.AllPairs.Properties as AllPairs
 open import Data.Fin.Base using (Fin)
diff --git a/src/Data/List/Sort/MergeSort.agda b/src/Data/List/Sort/MergeSort.agda
index 09ccdf2471..380665d1f1 100644
--- a/src/Data/List/Sort/MergeSort.agda
+++ b/src/Data/List/Sort/MergeSort.agda
@@ -24,7 +24,7 @@ open import Relation.Nullary.Decidable using (does)
 open import Data.Nat using (_<_; _>_; z<s; s<s)
 open import Data.Nat.Induction
 open import Data.Nat.Properties using (m<n⇒m<1+n)
-open import Data.Product as Prod using (_,_)
+open import Data.Product.Base as Prod using (_,_)
 open import Function.Base using (_∘_)
 open import Relation.Binary using (DecTotalOrder)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/Maybe/Properties.agda b/src/Data/Maybe/Properties.agda
index cb19c9c328..5a3b00f79f 100644
--- a/src/Data/Maybe/Properties.agda
+++ b/src/Data/Maybe/Properties.agda
@@ -13,7 +13,7 @@ import Algebra.Structures as Structures
 import Algebra.Definitions as Definitions
 open import Data.Maybe.Base
 open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_; id; _∘_; _∘′_)
 open import Function.Definitions using (Injective)
 open import Level using (Level)
diff --git a/src/Data/Maybe/Relation/Unary/All.agda b/src/Data/Maybe/Relation/Unary/All.agda
index a3818c38b2..c070bd279d 100644
--- a/src/Data/Maybe/Relation/Unary/All.agda
+++ b/src/Data/Maybe/Relation/Unary/All.agda
@@ -12,7 +12,7 @@ open import Effect.Applicative
 open import Effect.Monad
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.Any using (Any; just)
-open import Data.Product as Prod using (_,_)
+open import Data.Product.Base as Prod using (_,_)
 open import Function.Base using (id; _∘′_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
diff --git a/src/Data/Maybe/Relation/Unary/All/Properties.agda b/src/Data/Maybe/Relation/Unary/All/Properties.agda
index 493f8679d0..6c494632f1 100644
--- a/src/Data/Maybe/Relation/Unary/All/Properties.agda
+++ b/src/Data/Maybe/Relation/Unary/All/Properties.agda
@@ -12,7 +12,7 @@ open import Data.Maybe.Base
 open import Data.Maybe.Relation.Unary.All as All
   using (All; nothing; just)
 open import Data.Maybe.Relation.Binary.Connected
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Function.Base using (_∘_)
 open import Level
 open import Relation.Unary
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 2836b4d1da..94bfe8968c 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -16,7 +16,7 @@ open import Data.Nat.GCD.Lemmas
 open import Data.Nat.Primality
 open import Data.Nat.Properties
 open import Data.Nat.DivMod
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
 open import Function.Base using (_∘_)
 open import Level using (0ℓ)
 open import Relation.Binary.PropositionalEquality.Core as P
diff --git a/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda b/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda
index 17de9de089..de8427b6d4 100644
--- a/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda
+++ b/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda
@@ -11,7 +11,7 @@
 open import Codata.Sized.Stream using (Stream; unfold)
 open import Data.Nat.PseudoRandom.LCG
 open import Data.Nat using (ℕ)
-open import Data.Product using (<_,_>)
+open import Data.Product.Base using (<_,_>)
 open import Function.Base using (id)
 
 module Data.Nat.PseudoRandom.LCG.Unsafe where
diff --git a/src/Data/Nat/Show.agda b/src/Data/Nat/Show.agda
index 1777aca9d3..23ab91b0fd 100644
--- a/src/Data/Nat/Show.agda
+++ b/src/Data/Nat/Show.agda
@@ -17,7 +17,7 @@ open import Data.Maybe.Base as Maybe using (Maybe; nothing; _<∣>_; when)
 import Data.Maybe.Effectful as Maybe
 open import Data.Nat
 open import Data.Product.Base using (proj₁)
-open import Data.String.Base as String using (String; fromList; toList)
+open import Data.String.Base using (toList; fromList; String)
 open import Function.Base using (_∘′_; _∘_)
 open import Relation.Nullary.Decidable using (True)
 
diff --git a/src/Data/Parity/Properties.agda b/src/Data/Parity/Properties.agda
index 574c9c68c1..0d5d29607a 100644
--- a/src/Data/Parity/Properties.agda
+++ b/src/Data/Parity/Properties.agda
@@ -12,7 +12,7 @@ open import Algebra.Bundles
 open import Data.Empty
 open import Data.Nat.Base as ℕ using (zero; suc; parity)
 open import Data.Parity.Base as ℙ using (Parity; 0ℙ; 1ℙ; _⁻¹; toSign; fromSign)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sign.Base as 𝕊
 open import Function.Base using (_$_; id)
 open import Function.Definitions
diff --git a/src/Data/Product/Relation/Binary/Lex/NonStrict.agda b/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
index 259bc36789..e8752c1caf 100644
--- a/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
@@ -11,7 +11,7 @@
 
 module Data.Product.Relation.Binary.Lex.NonStrict where
 
-open import Data.Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Level using (Level)
 open import Relation.Binary
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
index 96f7211ce7..8a7dc43e83 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
@@ -8,7 +8,7 @@
 
 module Data.Product.Relation.Binary.Pointwise.Dependent where
 
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
 open import Level
 open import Function.Base
 open import Relation.Binary as B
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index fc77109043..16756bf375 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -8,7 +8,7 @@
 
 module Data.Product.Relation.Binary.Pointwise.NonDependent where
 
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
 open import Data.Product.Properties using (≡-dec)
 open import Data.Sum.Base
 open import Data.Unit.Base using (⊤)
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index f564ffe835..7c74cd2513 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -33,7 +33,7 @@ open import Data.Nat.Coprimality as C using (Coprime; coprime?)
 open import Data.Nat.Divisibility
 import Data.Nat.GCD as ℕ
 import Data.Nat.DivMod as ℕ
-open import Data.Product using (proj₁; proj₂; _×_; _,_; uncurry)
+open import Data.Product.Base using (proj₁; proj₂; _×_; _,_; uncurry)
 open import Data.Rational.Base
 open import Data.Rational.Unnormalised.Base as ℚᵘ
   using (ℚᵘ; mkℚᵘ; *≡*; *≤*; *<*)
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 3c800a5a98..5b7a295cb4 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -24,7 +24,7 @@ open import Data.Integer.Base as ℤ using (ℤ; +0; +[1+_]; -[1+_]; 0ℤ; 1ℤ;
 open import Data.Integer.Solver renaming (module +-*-Solver to ℤ-solver)
 import Data.Integer.Properties as ℤ
 open import Data.Rational.Unnormalised.Base
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (_⊎_; [_,_]′; inj₁; inj₂)
 import Data.Sign as Sign
 open import Function.Base using (_on_; _$_; _∘_; flip)
diff --git a/src/Data/Record.agda b/src/Data/Record.agda
index edc19da8ff..6d77f87622 100644
--- a/src/Data/Record.agda
+++ b/src/Data/Record.agda
@@ -12,7 +12,7 @@
 open import Data.Bool.Base using (true; false; if_then_else_)
 open import Data.Empty
 open import Data.List.Base
-open import Data.Product hiding (proj₁; proj₂)
+open import Data.Product.Base hiding (proj₁; proj₂)
 open import Data.Unit.Polymorphic
 open import Function.Base using (id; _∘_)
 open import Level
diff --git a/src/Data/Sign/Properties.agda b/src/Data/Sign/Properties.agda
index 5ca9a36e95..7ac07167e7 100644
--- a/src/Data/Sign/Properties.agda
+++ b/src/Data/Sign/Properties.agda
@@ -11,7 +11,7 @@ module Data.Sign.Properties where
 open import Algebra.Bundles
 open import Data.Empty
 open import Data.Sign.Base
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_$_; id)
 open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index 724b341c5e..2b9b4d763d 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -8,7 +8,7 @@
 
 module Data.Sum.Relation.Binary.Pointwise where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base as Sum
 open import Data.Sum.Properties
 open import Level using (_⊔_)
diff --git a/src/Data/Tree/AVL.agda b/src/Data/Tree/AVL.agda
index e8920b7a36..895a194085 100644
--- a/src/Data/Tree/AVL.agda
+++ b/src/Data/Tree/AVL.agda
@@ -24,7 +24,7 @@ import Data.DifferenceList as DiffList
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.Maybe.Base using (Maybe; nothing; just; is-just)
 open import Data.Nat.Base using (ℕ; suc)
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Function.Base as F
 open import Level using (Level; _⊔_)
 open import Relation.Unary using (IUniversal; _⇒_)
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
index 5add64a3cc..53c5caeeb0 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Indexed.Relation.Unary.All
   where
 
 open import Data.Nat.Base using (ℕ)
-open import Data.Product using (_,_; _×_)
+open import Data.Product.Base using (_,_; _×_)
 open import Data.Product.Nary.NonDependent
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
diff --git a/src/Data/Tree/AVL/Map.agda b/src/Data/Tree/AVL/Map.agda
index 65731a8f29..6869f3fe04 100644
--- a/src/Data/Tree/AVL/Map.agda
+++ b/src/Data/Tree/AVL/Map.agda
@@ -18,7 +18,7 @@ open import Data.Bool.Base using (Bool)
 open import Data.List.Base as List using (List)
 open import Data.Maybe.Base as Maybe using (Maybe)
 open import Data.Nat.Base using (ℕ)
-open import Data.Product as Prod using (_×_)
+open import Data.Product.Base as Prod using (_×_)
 open import Function.Base using (_∘′_)
 open import Level using (Level; _⊔_)
 
diff --git a/src/Data/Tree/AVL/NonEmpty.agda b/src/Data/Tree/AVL/NonEmpty.agda
index 0e6b680d90..51c6232077 100644
--- a/src/Data/Tree/AVL/NonEmpty.agda
+++ b/src/Data/Tree/AVL/NonEmpty.agda
@@ -21,7 +21,7 @@ open import Data.Empty
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_; _++⁺_)
 open import Data.Maybe.Base hiding (map)
 open import Data.Nat.Base hiding (_<_; _⊔_; compare)
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.Unit
 open import Function.Base using (_$_; _∘′_)
 open import Level using (_⊔_; Lift; lift)
diff --git a/src/Data/Tree/AVL/Sets.agda b/src/Data/Tree/AVL/Sets.agda
index b533605b95..976298933c 100644
--- a/src/Data/Tree/AVL/Sets.agda
+++ b/src/Data/Tree/AVL/Sets.agda
@@ -16,7 +16,7 @@ open import Data.Bool.Base using (Bool)
 open import Data.List.Base as List using (List)
 open import Data.Maybe.Base as Maybe
 open import Data.Nat.Base using (ℕ)
-open import Data.Product as Prod using (_×_; _,_; proj₁)
+open import Data.Product.Base as Prod using (_×_; _,_; proj₁)
 open import Data.Unit.Base
 open import Function.Base
 open import Level using (Level; _⊔_)
diff --git a/src/Data/Tree/AVL/Sets/Membership.agda b/src/Data/Tree/AVL/Sets/Membership.agda
index 53e0aea9f9..748aa60344 100644
--- a/src/Data/Tree/AVL/Sets/Membership.agda
+++ b/src/Data/Tree/AVL/Sets/Membership.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Sets.Membership
   where
 
 open import Data.Bool.Base using (true; false)
-open import Data.Product as Prod using (_,_; proj₁; proj₂)
+open import Data.Product.Base as Prod using (_,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
 open import Function.Base using (_∘_; _∘′_; const)
diff --git a/src/Data/Tree/AVL/Sets/Membership/Properties.agda b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
index 75fc1e9242..3c2c28ed71 100644
--- a/src/Data/Tree/AVL/Sets/Membership/Properties.agda
+++ b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Sets.Membership.Properties
   where
 
 open import Data.Bool.Base using (true; false)
-open import Data.Product as Prod using (_,_; proj₁; proj₂)
+open import Data.Product.Base as Prod using (_,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
 open import Function.Base using (_∘_; _∘′_; const)
diff --git a/src/Data/Tree/AVL/Value.agda b/src/Data/Tree/AVL/Value.agda
index 77af5cffdb..ff5e85dfa0 100644
--- a/src/Data/Tree/AVL/Value.agda
+++ b/src/Data/Tree/AVL/Value.agda
@@ -11,7 +11,7 @@ open import Relation.Binary using (Setoid; _Respects_)
 
 module Data.Tree.AVL.Value {a ℓ} (S : Setoid a ℓ) where
 
-open import Data.Product using (Σ; _,_)
+open import Data.Product.Base using (Σ; _,_)
 open import Level using (suc; _⊔_)
 import Function.Base as F
 open Setoid S renaming (Carrier to Key)
diff --git a/src/Data/Tree/Rose/Show.agda b/src/Data/Tree/Rose/Show.agda
index a0e9631622..5b293aeb70 100644
--- a/src/Data/Tree/Rose/Show.agda
+++ b/src/Data/Tree/Rose/Show.agda
@@ -14,7 +14,7 @@ open import Data.Bool.Base using (Bool; true; false; if_then_else_; _∧_)
 open import Data.DifferenceList as DList renaming (DiffList to DList) using ()
 open import Data.List.Base as List using (List; []; [_]; _∷_; _∷ʳ_)
 open import Data.Nat.Base using (ℕ; _∸_)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.String.Base hiding (show)
 open import Data.Tree.Rose using (Rose; node; map; fromBinary)
 open import Function.Base using (flip; _∘′_; id)
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index e6cc006d89..d0dbc16cda 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -12,7 +12,7 @@ open import Data.Empty using (⊥-elim)
 open import Data.Fin.Base
 open import Data.Nat as ℕ
 import Data.Nat.Properties as ℕₚ
-open import Data.Product as Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.List.Base as L using (List)
 import Data.List.Properties as Lₚ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
diff --git a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
index 0aa81b07a7..feff1ab55c 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
@@ -11,7 +11,7 @@ module Data.Vec.Functional.Relation.Binary.Pointwise.Properties where
 open import Data.Fin.Base using (zero; suc; _↑ˡ_; _↑ʳ_; splitAt)
 open import Data.Fin.Properties using (all?; splitAt-↑ˡ; splitAt-↑ʳ)
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Data.Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
   using () renaming (Pointwise to ×-Pointwise)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
diff --git a/src/Data/Vec/Functional/Relation/Unary/All.agda b/src/Data/Vec/Functional/Relation/Unary/All.agda
index 84c00eb687..56c385fae8 100644
--- a/src/Data/Vec/Functional/Relation/Unary/All.agda
+++ b/src/Data/Vec/Functional/Relation/Unary/All.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Functional.Relation.Unary.All where
 
 open import Data.Fin.Properties using (all?)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.Vec.Functional as VF hiding (map)
 open import Level using (Level)
 open import Relation.Unary
diff --git a/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda b/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
index 89dba8d259..85b5075142 100644
--- a/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
@@ -10,7 +10,7 @@ module Data.Vec.Functional.Relation.Unary.All.Properties where
 
 open import Data.Fin.Base using (zero; suc; _↑ˡ_; _↑ʳ_; splitAt)
 open import Data.Fin.Properties using (splitAt-↑ˡ; splitAt-↑ʳ)
-open import Data.Product as Σ using (_×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Product.Base as Σ using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Vec.Functional as VF hiding (map)
 open import Data.Vec.Functional.Relation.Unary.All
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index c7719eedf1..430cdf8d2b 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -15,7 +15,7 @@ open import Data.List.Base as List using (List)
 open import Data.Nat.Base
 open import Data.Nat.Properties
   using (+-assoc; m≤n⇒m≤1+n; ≤-refl; ≤-trans; suc-injective)
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
   using (_×_; _,_; proj₁; proj₂; <_,_>; uncurry)
 open import Data.Sum.Base using ([_,_]′)
 open import Data.Sum.Properties using ([,]-map)
diff --git a/src/Data/Vec/Recursive/Effectful.agda b/src/Data/Vec/Recursive/Effectful.agda
index cc58d5af6d..97aaa5b053 100644
--- a/src/Data/Vec/Recursive/Effectful.agda
+++ b/src/Data/Vec/Recursive/Effectful.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Recursive.Effectful where
 
 open import Agda.Builtin.Nat
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.Vec.Recursive
 open import Effect.Functor
 open import Effect.Applicative
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index ddc4242496..c091d0f98c 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -11,7 +11,7 @@ module Data.Vec.Relation.Binary.Lex.Core {a} {A : Set a} where
 open import Data.Empty
 open import Data.Nat using (ℕ; suc)
 import Data.Nat.Properties as ℕ
-open import Data.Product using (_×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Vec.Relation.Binary.Pointwise.Inductive using (Pointwise; []; _∷_)
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index 80a10442ef..82036a5352 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -13,7 +13,7 @@ module Data.Vec.Relation.Binary.Lex.NonStrict where
 
 open import Data.Empty
 open import Data.Unit using (⊤; tt)
-open import Data.Product using (proj₁; proj₂)
+open import Data.Product.Base using (proj₁; proj₂)
 open import Data.Nat using (ℕ)
 open import Data.Vec.Base using (Vec; []; _∷_)
 import Data.Vec.Relation.Binary.Lex.Strict as Strict
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index e33369a448..3bafcdca03 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -15,7 +15,7 @@ open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Unit.Base using (⊤; tt)
 open import Data.Unit.Properties using (⊤-irrelevant)
 open import Data.Nat.Base using (ℕ; suc)
-open import Data.Product using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Lex.Strict
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Vec.Base using (Vec; []; _∷_; uncons)
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
index 011c2b731b..85bd6df9da 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
@@ -10,7 +10,7 @@ module Data.Vec.Relation.Binary.Pointwise.Inductive where
 
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Data.Product using (_×_; _,_; uncurry; <_,_>)
+open import Data.Product.Base using (_×_; _,_; uncurry; <_,_>)
 open import Data.Vec.Base as Vec hiding ([_]; head; tail; map; lookup; uncons)
 open import Data.Vec.Relation.Unary.All using (All; []; _∷_)
 open import Level using (Level; _⊔_)
diff --git a/src/Data/Vec/Relation/Unary/All.agda b/src/Data/Vec/Relation/Unary/All.agda
index 2eda26c71e..f8fe5a4045 100644
--- a/src/Data/Vec/Relation/Unary/All.agda
+++ b/src/Data/Vec/Relation/Unary/All.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Relation.Unary.All where
 
 open import Data.Nat.Base using (ℕ; zero; suc; NonZero)
-open import Data.Product as Prod using (_×_; _,_; uncurry; <_,_>)
+open import Data.Product.Base as Prod using (_×_; _,_; uncurry; <_,_>)
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)
diff --git a/src/Data/Vec/Relation/Unary/All/Properties.agda b/src/Data/Vec/Relation/Unary/All/Properties.agda
index d59627f783..868620ff74 100644
--- a/src/Data/Vec/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/All/Properties.agda
@@ -12,7 +12,7 @@ open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base using ([]; _∷_)
 open import Data.List.Relation.Unary.All as List using ([]; _∷_)
-open import Data.Product as Prod using (_×_; _,_; uncurry; uncurry′)
+open import Data.Product.Base as Prod using (_×_; _,_; uncurry; uncurry′)
 open import Data.Vec.Base as Vec
 import Data.Vec.Properties as Vecₚ
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
diff --git a/src/Data/Vec/Relation/Unary/AllPairs.agda b/src/Data/Vec/Relation/Unary/AllPairs.agda
index 0fee880769..aeac84557b 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs.agda
@@ -14,7 +14,7 @@ module Data.Vec.Relation.Unary.AllPairs
 open import Data.Nat.Base using (suc)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
-open import Data.Product as Prod using (_,_; _×_; uncurry; <_,_>)
+open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (_⊔_)
 open import Relation.Binary as B using (Rel; _⇒_)
diff --git a/src/Data/Vec/Relation/Unary/Linked.agda b/src/Data/Vec/Relation/Unary/Linked.agda
index 2ca0182f03..1c90268d81 100644
--- a/src/Data/Vec/Relation/Unary/Linked.agda
+++ b/src/Data/Vec/Relation/Unary/Linked.agda
@@ -11,7 +11,7 @@ module Data.Vec.Relation.Unary.Linked {a} {A : Set a} where
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
-open import Data.Product as Prod using (_,_; _×_; uncurry; <_,_>)
+open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary as B using (Rel; _⇒_)
diff --git a/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda b/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
index 9ab4cd57f7..292283689c 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
@@ -16,7 +16,7 @@ import Data.Vec.Relation.Unary.Unique.Setoid.Properties as Setoid
 open import Data.Fin.Base using(Fin)
 open import Data.Nat.Base
 open import Data.Nat.Properties using (<⇒≢)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡⇒≡×≡)
 open import Function.Base using (id; _∘_)
 open import Level using (Level)
diff --git a/src/Data/W/Sized.agda b/src/Data/W/Sized.agda
index 72d742ff4b..0a0b2b6e90 100644
--- a/src/Data/W/Sized.agda
+++ b/src/Data/W/Sized.agda
@@ -11,7 +11,7 @@ module Data.W.Sized where
 open import Level
 open import Size
 open import Function.Base using (_$_; _∘_; const)
-open import Data.Product using (_,_; -,_; proj₂)
+open import Data.Product.Base using (_,_; -,_; proj₂)
 open import Data.Container.Core as Container using (Container; ⟦_⟧; Shape; Position; _⇒_; ⟪_⟫)
 open import Data.Container.Relation.Unary.All using (□; all)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Effect/Applicative/Indexed.agda b/src/Effect/Applicative/Indexed.agda
index de2b381934..30e9b632bb 100644
--- a/src/Effect/Applicative/Indexed.agda
+++ b/src/Effect/Applicative/Indexed.agda
@@ -12,7 +12,7 @@
 module Effect.Applicative.Indexed where
 
 open import Effect.Functor using (RawFunctor)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Function.Base
 open import Level
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
diff --git a/src/Effect/Monad/IO.agda b/src/Effect/Monad/IO.agda
index 8e9e820260..f52c0a3eff 100644
--- a/src/Effect/Monad/IO.agda
+++ b/src/Effect/Monad/IO.agda
@@ -10,7 +10,7 @@ open import Level
 
 module Effect.Monad.IO where
 
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Function.Base
 open import IO.Base using (IO)
 open import Effect.Functor
diff --git a/src/Effect/Monad/Reader/Transformer.agda b/src/Effect/Monad/Reader/Transformer.agda
index 96f95ecae4..04121c6938 100644
--- a/src/Effect/Monad/Reader/Transformer.agda
+++ b/src/Effect/Monad/Reader/Transformer.agda
@@ -116,7 +116,7 @@ liftReaderT MRead = record
   ; local = λ f mx → mkReaderT (λ r₂ → local f (runReaderT mx r₂))
   } where open RawMonadReader MRead
 
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Effect.Monad.Writer.Transformer.Base
 
 liftWriterT : (MR : RawMonoid r g) →
diff --git a/src/Effect/Monad/State/Transformer.agda b/src/Effect/Monad/State/Transformer.agda
index 7271117eec..7c6fc1156c 100644
--- a/src/Effect/Monad/State/Transformer.agda
+++ b/src/Effect/Monad/State/Transformer.agda
@@ -11,7 +11,7 @@ open import Level using (Level; suc; _⊔_)
 module Effect.Monad.State.Transformer where
 
 open import Algebra using (RawMonoid)
-open import Data.Product using (_×_; _,_; map₂; proj₁; proj₂)
+open import Data.Product.Base using (_×_; _,_; map₂; proj₁; proj₂)
 open import Data.Unit.Polymorphic.Base
 open import Effect.Choice
 open import Effect.Empty
diff --git a/src/Effect/Monad/Writer.agda b/src/Effect/Monad/Writer.agda
index 43f21deaa9..489c7f4347 100644
--- a/src/Effect/Monad/Writer.agda
+++ b/src/Effect/Monad/Writer.agda
@@ -9,7 +9,7 @@
 module Effect.Monad.Writer where
 
 open import Algebra using (RawMonoid)
-open import Data.Product using (_×_)
+open import Data.Product.Base using (_×_)
 open import Effect.Applicative
 open import Effect.Choice
 open import Effect.Empty
diff --git a/src/Effect/Monad/Writer/Indexed.agda b/src/Effect/Monad/Writer/Indexed.agda
index 44eaf22196..d0fb68a967 100644
--- a/src/Effect/Monad/Writer/Indexed.agda
+++ b/src/Effect/Monad/Writer/Indexed.agda
@@ -11,7 +11,7 @@ open import Level
 module Effect.Monad.Writer.Indexed (a : Level) where
 
 open import Algebra using (RawMonoid)
-open import Data.Product using (_×_; _,_; map₁)
+open import Data.Product.Base using (_×_; _,_; map₁)
 open import Data.Unit.Polymorphic
 open import Effect.Applicative.Indexed
 open import Effect.Monad
diff --git a/src/Effect/Monad/Writer/Transformer.agda b/src/Effect/Monad/Writer/Transformer.agda
index f239c08940..d0924cfe29 100644
--- a/src/Effect/Monad/Writer/Transformer.agda
+++ b/src/Effect/Monad/Writer/Transformer.agda
@@ -10,7 +10,7 @@
 module Effect.Monad.Writer.Transformer where
 
 open import Algebra using (RawMonoid)
-open import Data.Product using (_×_; _,_; proj₂; map₂)
+open import Data.Product.Base using (_×_; _,_; proj₂; map₂)
 open import Effect.Applicative
 open import Effect.Choice
 open import Effect.Empty
diff --git a/src/Text/Printf/Generic.agda b/src/Text/Printf/Generic.agda
index 6272351756..07c8856443 100644
--- a/src/Text/Printf/Generic.agda
+++ b/src/Text/Printf/Generic.agda
@@ -13,7 +13,7 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.Char.Base
 open import Data.Maybe.Base hiding (map)
 open import Data.Nat.Base using (ℕ)
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.Product.Nary.NonDependent
 open import Data.String.Base
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
diff --git a/tests/data/appending/TakeWhile.agda b/tests/data/appending/TakeWhile.agda
index 5a885c3fbe..50f582df27 100644
--- a/tests/data/appending/TakeWhile.agda
+++ b/tests/data/appending/TakeWhile.agda
@@ -7,7 +7,7 @@ open import Data.List.Base hiding (takeWhile)
 open import Data.List.Relation.Unary.All as List using ([]; _∷_)
 open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_; refl)
 open import Data.List.Relation.Ternary.Appending.Propositional
-open import Data.Product using (_×_; proj₁)
+open import Data.Product.Base using (_×_; proj₁)
 open import Data.Maybe.Relation.Unary.All as Maybe using (nothing; just)
 import Data.Nat
 open import Relation.Unary
diff --git a/tests/monad/counting/Main.agda b/tests/monad/counting/Main.agda
index e3613b4741..0660e0f224 100644
--- a/tests/monad/counting/Main.agda
+++ b/tests/monad/counting/Main.agda
@@ -4,7 +4,7 @@ module Main where
 
 open import Data.Nat.Base
 open import Data.Nat.Show using (show)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.String.Base using (String; _++_)
 open import IO.Base as IO using (IO; Main; run)
 open import IO.Finite using (putStrLn)
diff --git a/tests/monad/fibonacci/Main.agda b/tests/monad/fibonacci/Main.agda
index 93d7ab4e85..d1e46181bd 100644
--- a/tests/monad/fibonacci/Main.agda
+++ b/tests/monad/fibonacci/Main.agda
@@ -5,7 +5,7 @@ module Main where
 open import Data.Bool.Base using (Bool; true; false)
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Nat.Show using (show)
-open import Data.Product using (_,_)
+open import Data.Product.Base using (_,_)
 open import Data.String.Base using (String; _++_)
 
 open import Data.Default
diff --git a/tests/monad/pythagorean/Main.agda b/tests/monad/pythagorean/Main.agda
index d89de60e04..97bdea2064 100644
--- a/tests/monad/pythagorean/Main.agda
+++ b/tests/monad/pythagorean/Main.agda
@@ -10,7 +10,7 @@ open import Data.Maybe.Base using (just; nothing)
 open import Data.Maybe.Effectful.Transformer
 open import Data.Nat.Base using (ℕ; zero; suc; _+_; _^_; _≡ᵇ_)
 open import Data.Nat.Show using (show)
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 open import Data.String.Base using (_++_)
 open import Function.Base using (_$_)
 

From 17fbc46d84e40424d24a95c3101de0243be2ad69 Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Fri, 11 Aug 2023 02:14:20 +0000
Subject: [PATCH 089/316] Added Unnormalised Rational Field Structure (#1959)

---
 CHANGELOG.md                                  | 26 +++++++
 .../Rational/Unnormalised/Properties.agda     | 72 ++++++++++++++++++-
 2 files changed, 97 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 3818f1a6f7..e166023acc 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3243,6 +3243,32 @@ This is a full list of proofs that have changed form to use irrelevant instance
   <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
   ```
 
+* Added new module to `Data.Rational.Unnormalised.Properties`
+  ```agda
+  module ≃-Reasoning = SetoidReasoning ≃-setoid
+  ```
+
+* Added new functions to `Data.Rational.Unnormalised.Properties`
+  ```agda
+  0≠1 : 0ℚᵘ ≠ 1ℚᵘ
+  ≃-≠-irreflexive : Irreflexive _≃_ _≠_
+  ≠-symmetric : Symmetric _≠_
+  ≠-cotransitive : Cotransitive _≠_
+  ≠⇒invertible : p ≠ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
+  ```
+
+* Added new structures to `Data.Rational.Unnormalised.Properties`
+  ```agda
+  +-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
+  +-*-isHeytingField : IsHeytingField _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
+  ```
+
+* Added new bundles to `Data.Rational.Unnormalised.Properties`
+  ```agda
+  +-*-heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
+  +-*-heytingField : HeytingField 0ℓ 0ℓ 0ℓ
+  ```
+
 * Added new function to `Data.Vec.Relation.Binary.Pointwise.Inductive`
   ```agda
   cong-[_]≔ : Pointwise _∼_ xs ys → Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 5b7a295cb4..2945a8654a 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -10,12 +10,14 @@
 module Data.Rational.Unnormalised.Properties where
 
 open import Algebra
+open import Algebra.Apartness
 open import Algebra.Lattice
 import Algebra.Consequences.Setoid as Consequences
 open import Algebra.Consequences.Propositional
 open import Algebra.Construct.NaturalChoice.Base
 import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp
 import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
+open import Data.Empty using (⊥-elim)
 open import Data.Bool.Base using (T; true; false)
 open import Data.Nat.Base as ℕ using (suc; pred)
 import Data.Nat.Properties as ℕ
@@ -24,7 +26,7 @@ open import Data.Integer.Base as ℤ using (ℤ; +0; +[1+_]; -[1+_]; 0ℤ; 1ℤ;
 open import Data.Integer.Solver renaming (module +-*-Solver to ℤ-solver)
 import Data.Integer.Properties as ℤ
 open import Data.Rational.Unnormalised.Base
-open import Data.Product.Base using (_,_)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Data.Sum.Base using (_⊎_; [_,_]′; inj₁; inj₂)
 import Data.Sign as Sign
 open import Function.Base using (_on_; _$_; _∘_; flip)
@@ -36,6 +38,8 @@ open import Relation.Binary
 import Relation.Binary.Consequences as BC
 open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Properties.Poset as PosetProperties
+open import Relation.Nullary using (yes; no)
+import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 open import Algebra.Properties.CommutativeSemigroup ℤ.*-commutativeSemigroup
 
@@ -96,6 +100,21 @@ infix 4 _≃?_
 _≃?_ : Decidable _≃_
 p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* ↧ p)
 
+0≠1 : 0ℚᵘ ≠ 1ℚᵘ
+0≠1 = Dec.from-no (0ℚᵘ ≃? 1ℚᵘ)
+
+≃-≠-irreflexive : Irreflexive _≃_ _≠_
+≃-≠-irreflexive x≃y x≠y = x≠y x≃y
+
+≠-symmetric : Symmetric _≠_
+≠-symmetric x≠y y≃x = x≠y (≃-sym y≃x)
+
+≠-cotransitive : Cotransitive _≠_
+≠-cotransitive {x} {y} x≠y z with x ≃? z | z ≃? y
+... | no  x≠z | _       = inj₁ x≠z
+... | yes _   | no z≠y  = inj₂ z≠y
+... | yes x≃z | yes z≃y = ⊥-elim (x≠y (≃-trans x≃z z≃y))
+
 ≃-isEquivalence : IsEquivalence _≃_
 ≃-isEquivalence = record
   { refl  = ≃-refl
@@ -109,6 +128,17 @@ p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* 
   ; _≟_           = _≃?_
   }
 
+≠-isApartnessRelation : IsApartnessRelation _≃_ _≠_
+≠-isApartnessRelation = record
+  { irrefl  = ≃-≠-irreflexive
+  ; sym     = ≠-symmetric
+  ; cotrans = ≠-cotransitive
+  }
+
+≠-tight : Tight _≃_ _≠_
+proj₁ (≠-tight p q) ¬p≠q = Dec.decidable-stable (p ≃? q) ¬p≠q
+proj₂ (≠-tight p q) p≃q p≠q = p≠q p≃q
+
 ≃-setoid : Setoid 0ℓ 0ℓ
 ≃-setoid = record
   { isEquivalence = ≃-isEquivalence
@@ -119,6 +149,8 @@ p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* 
   { isDecEquivalence = ≃-isDecEquivalence
   }
 
+module ≃-Reasoning = SetoidReasoning ≃-setoid
+
 ------------------------------------------------------------------------
 -- Properties of -_
 ------------------------------------------------------------------------
@@ -1044,6 +1076,12 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
 *-inverseʳ : ∀ p .{{_ : NonZero p}} → p * 1/ p ≃ 1ℚᵘ
 *-inverseʳ p = ≃-trans (*-comm p (1/ p)) (*-inverseˡ p)
 
+≠⇒invertible : p ≠ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
+≠⇒invertible {p} {q} p≠q = _ , *-inverseˡ (p - q) , *-inverseʳ (p - q)
+  where instance
+  _ : NonZero (p - q)
+  _ = ≢-nonZero (p≠q ∘ p-q≃0⇒p≃q p q)
+
 *-zeroˡ : LeftZero _≃_ 0ℚᵘ _*_
 *-zeroˡ p@record{} = *≡* refl
 
@@ -1053,6 +1091,14 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
 *-zero : Zero _≃_ 0ℚᵘ _*_
 *-zero = *-zeroˡ , *-zeroʳ
 
+invertible⇒≠ : Invertible _≃_ 1ℚᵘ _*_ (p - q) → p ≠ q
+invertible⇒≠ {p} {q} (1/p-q , 1/x*x≃1 , x*1/x≃1) p≃q = 0≠1 (begin
+  0ℚᵘ             ≈˘⟨ *-zeroˡ 1/p-q ⟩
+  0ℚᵘ * 1/p-q     ≈˘⟨ *-congʳ (p≃q⇒p-q≃0 p q p≃q) ⟩
+  (p - q) * 1/p-q ≈⟨ x*1/x≃1 ⟩
+  1ℚᵘ             ∎)
+  where open ≃-Reasoning
+
 *-distribˡ-+ : _DistributesOverˡ_ _≃_ _*_ _+_
 *-distribˡ-+ p@record{} q@record{} r@record{} =
   let ↥p = ↥ p; ↧p = ↧ p
@@ -1293,6 +1339,20 @@ nonNeg*nonNeg⇒nonNeg p q = nonNegative
   ; *-comm = *-comm
   }
 
++-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
++-*-isHeytingCommutativeRing = record
+  { isCommutativeRing   = +-*-isCommutativeRing
+  ; isApartnessRelation = ≠-isApartnessRelation
+  ; #⇒invertible        = ≠⇒invertible
+  ; invertible⇒#        = invertible⇒≠
+  }
+
++-*-isHeytingField : IsHeytingField _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
++-*-isHeytingField = record
+  { isHeytingCommutativeRing = +-*-isHeytingCommutativeRing
+  ; tight                    = ≠-tight
+  }
+
 ------------------------------------------------------------------------
 -- Algebraic bundles
 
@@ -1326,6 +1386,16 @@ nonNeg*nonNeg⇒nonNeg p q = nonNegative
   { isCommutativeRing = +-*-isCommutativeRing
   }
 
++-*-heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
++-*-heytingCommutativeRing = record
+  { isHeytingCommutativeRing = +-*-isHeytingCommutativeRing
+  }
+
++-*-heytingField : HeytingField 0ℓ 0ℓ 0ℓ
++-*-heytingField = record
+  { isHeytingField = +-*-isHeytingField
+  }
+
 ------------------------------------------------------------------------
 -- Properties of 1/_
 ------------------------------------------------------------------------

From 10ed135b71703cff2e29aaa4e91c3bb55f295005 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 10 Aug 2023 22:25:07 -0400
Subject: [PATCH 090/316] More simplifications for `Relation.Binary` imports
 (#2038)

---
 README/Nary.agda                               |  2 +-
 src/Data/AVL.agda                              |  2 +-
 src/Data/AVL/Indexed.agda                      |  2 +-
 src/Data/AVL/NonEmpty/Propositional.agda       |  4 +++-
 src/Data/List/Extrema.agda                     |  2 +-
 src/Data/List/Fresh/Properties.agda            |  3 ++-
 src/Data/List/Membership/DecPropositional.agda |  2 +-
 src/Data/List/Membership/DecSetoid.agda        |  3 ++-
 src/Data/List/Properties.agda                  |  4 ++--
 .../Relation/Binary/Equality/DecSetoid.agda    |  2 +-
 .../List/Relation/Binary/Equality/Setoid.agda  |  6 ++++--
 .../Relation/Binary/Infix/Heterogeneous.agda   |  2 +-
 .../Binary/Infix/Heterogeneous/Properties.agda |  3 ++-
 src/Data/List/Relation/Binary/Pointwise.agda   |  5 ++++-
 .../Sublist/Propositional/Properties.agda      |  2 +-
 .../Binary/Sublist/Setoid/Properties.agda      |  5 +++--
 .../Relation/Binary/Suffix/Heterogeneous.agda  |  2 +-
 .../List/Relation/Unary/AllPairs/Core.agda     |  2 +-
 .../Relation/Unary/AllPairs/Properties.agda    |  3 ++-
 .../Relation/Unary/Grouped/Properties.agda     |  3 ++-
 src/Data/List/Relation/Unary/Linked.agda       |  3 ++-
 src/Data/List/Sort.agda                        |  2 +-
 src/Data/List/Sort/Base.agda                   |  2 +-
 src/Data/List/Sort/MergeSort.agda              |  2 +-
 src/Data/Maybe/Properties.agda                 |  2 +-
 src/Data/Nat/Divisibility/Core.agda            |  2 +-
 src/Data/Product/Effectful/Examples.agda       |  2 +-
 src/Data/Product/Properties.agda               |  2 +-
 .../Binary/Pointwise/Dependent/WithK.agda      |  2 +-
 src/Data/Sum/Properties.agda                   |  2 +-
 src/Data/These/Properties.agda                 |  2 +-
 src/Data/Tree/AVL/Indexed.agda                 |  4 ++--
 .../Tree/AVL/Indexed/Relation/Unary/All.agda   |  2 +-
 .../Tree/AVL/Indexed/Relation/Unary/Any.agda   |  2 +-
 .../Indexed/Relation/Unary/Any/Properties.agda |  4 ++--
 src/Data/Tree/AVL/Map.agda                     |  2 +-
 .../Tree/AVL/Map/Membership/Propositional.agda |  4 ++--
 .../Membership/Propositional/Properties.agda   |  4 ++--
 src/Data/Tree/AVL/Map/Relation/Unary/Any.agda  |  2 +-
 src/Data/Tree/AVL/NonEmpty.agda                |  2 +-
 src/Data/Tree/AVL/NonEmpty/Propositional.agda  |  4 +++-
 src/Data/Tree/AVL/Relation/Unary/Any.agda      |  2 +-
 src/Data/Tree/AVL/Sets.agda                    |  2 +-
 src/Data/Tree/AVL/Sets/Membership.agda         |  2 +-
 .../Tree/AVL/Sets/Membership/Properties.agda   |  2 +-
 src/Data/Tree/AVL/Value.agda                   |  3 ++-
 src/Data/Trie.agda                             |  3 ++-
 src/Data/Vec/Bounded.agda                      |  2 +-
 src/Data/Vec/Membership/DecSetoid.agda         |  2 +-
 src/Data/Vec/Membership/Setoid.agda            |  3 ++-
 src/Data/Vec/Relation/Unary/All.agda           |  3 ++-
 src/Data/Vec/Relation/Unary/AllPairs.agda      |  4 ++--
 src/Data/Vec/Relation/Unary/AllPairs/Core.agda |  2 +-
 .../Relation/Unary/AllPairs/Properties.agda    |  2 +-
 src/Data/Vec/Relation/Unary/Linked.agda        |  3 ++-
 .../Vec/Relation/Unary/Linked/Properties.agda  |  4 +++-
 .../Unary/Unique/Propositional/Properties.agda |  3 ++-
 src/Data/Vec/Relation/Unary/Unique/Setoid.agda |  3 ++-
 .../Unary/Unique/Setoid/Properties.agda        |  3 ++-
 src/Data/Word/Base.agda                        |  2 +-
 src/Effect/Monad/Partiality/All.agda           |  3 ++-
 src/Function/Endomorphism/Propositional.agda   |  2 +-
 src/Function/Properties/Equivalence.agda       |  3 ++-
 src/Function/Properties/Inverse.agda           |  3 ++-
 src/Function/Properties/RightInverse.agda      |  3 ++-
 src/Reflection/AST/Abstraction.agda            |  2 +-
 src/Reflection/AST/AlphaEquality.agda          | 12 ++++++------
 src/Reflection/AST/Argument.agda               |  2 +-
 src/Reflection/AST/Argument/Information.agda   |  2 +-
 src/Reflection/AST/Argument/Modality.agda      |  2 +-
 src/Reflection/AST/Argument/Quantity.agda      |  2 +-
 src/Reflection/AST/Argument/Relevance.agda     |  2 +-
 src/Reflection/AST/Argument/Visibility.agda    |  2 +-
 src/Reflection/AST/Definition.agda             |  2 +-
 src/Reflection/AST/Literal.agda                |  2 +-
 src/Reflection/AST/Meta.agda                   |  3 ++-
 src/Reflection/AST/Name.agda                   |  3 ++-
 src/Reflection/AST/Term.agda                   | 18 +++++++++---------
 78 files changed, 131 insertions(+), 99 deletions(-)

diff --git a/README/Nary.agda b/README/Nary.agda
index 5566bd48cf..b8b337832a 100644
--- a/README/Nary.agda
+++ b/README/Nary.agda
@@ -19,7 +19,7 @@ open import Data.Product.Base using (_×_; _,_)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (id; flip; _∘′_)
 open import Relation.Nullary
-open import Relation.Binary using (module Tri); open Tri
+open import Relation.Binary.Definitions using (module Tri); open Tri
 open import Relation.Binary.PropositionalEquality
 
 private
diff --git a/src/Data/AVL.agda b/src/Data/AVL.agda
index ca0015925b..14c695d0d4 100644
--- a/src/Data/AVL.agda
+++ b/src/Data/AVL.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.AVL
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/AVL/Indexed.agda b/src/Data/AVL/Indexed.agda
index a4ee4a1d96..9efe5e7500 100644
--- a/src/Data/AVL/Indexed.agda
+++ b/src/Data/AVL/Indexed.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.AVL.Indexed
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂) where
diff --git a/src/Data/AVL/NonEmpty/Propositional.agda b/src/Data/AVL/NonEmpty/Propositional.agda
index 70087c1f88..0318537f0e 100644
--- a/src/Data/AVL/NonEmpty/Propositional.agda
+++ b/src/Data/AVL/NonEmpty/Propositional.agda
@@ -6,7 +6,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; IsStrictTotalOrder; StrictTotalOrder)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsStrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 open import Relation.Binary.PropositionalEquality using (_≡_; refl; subst)
 
 module Data.AVL.NonEmpty.Propositional
diff --git a/src/Data/List/Extrema.agda b/src/Data/List/Extrema.agda
index 5abc5edee9..16e38db5bb 100644
--- a/src/Data/List/Extrema.agda
+++ b/src/Data/List/Extrema.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (TotalOrder; Setoid)
+open import Relation.Binary.Bundles using (TotalOrder; Setoid)
 
 module Data.List.Extrema
   {b ℓ₁ ℓ₂} (totalOrder : TotalOrder b ℓ₁ ℓ₂) where
diff --git a/src/Data/List/Fresh/Properties.agda b/src/Data/List/Fresh/Properties.agda
index b22347f92e..6619fbb18d 100644
--- a/src/Data/List/Fresh/Properties.agda
+++ b/src/Data/List/Fresh/Properties.agda
@@ -12,7 +12,8 @@ open import Level using (Level; _⊔_; Lift)
 open import Data.Product.Base using (_,_)
 open import Relation.Nullary
 open import Relation.Unary as U using (Pred)
-open import Relation.Binary as B using (Rel)
+import Relation.Binary.Definitions as B
+open import Relation.Binary.Core using (Rel)
 
 open import Data.List.Fresh
 
diff --git a/src/Data/List/Membership/DecPropositional.agda b/src/Data/List/Membership/DecPropositional.agda
index a59ce42c67..71cb595375 100644
--- a/src/Data/List/Membership/DecPropositional.agda
+++ b/src/Data/List/Membership/DecPropositional.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
diff --git a/src/Data/List/Membership/DecSetoid.agda b/src/Data/List/Membership/DecSetoid.agda
index 3d798abc5b..8f34aab9f6 100644
--- a/src/Data/List/Membership/DecSetoid.agda
+++ b/src/Data/List/Membership/DecSetoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Decidable; DecSetoid)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Bundles using (DecSetoid)
 open import Relation.Nullary.Decidable using (¬?)
 
 module Data.List.Membership.DecSetoid {a ℓ} (DS : DecSetoid a ℓ) where
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index ca5ffe27f2..dd4326a749 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -34,10 +34,10 @@ open import Data.Fin.Properties using (toℕ-cast)
 open import Function.Base using (id; _∘_; _∘′_; _∋_; _-⟨_∣; ∣_⟩-_; _$_; const; flip)
 open import Function.Definitions using (Injective)
 open import Level using (Level)
-open import Relation.Binary as B using (DecidableEquality)
+open import Relation.Binary.Definitions as B using (DecidableEquality)
 import Relation.Binary.Reasoning.Setoid as EqR
 open import Relation.Binary.PropositionalEquality as P hiding ([_])
-open import Relation.Binary as B using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (¬_; Dec; does; _because_; yes; no; contradiction)
 open import Relation.Nullary.Decidable as Decidable using (isYes; map′; ⌊_⌋; ¬?; _×-dec_)
diff --git a/src/Data/List/Relation/Binary/Equality/DecSetoid.agda b/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
index 45f73c0902..ec82d91dcf 100644
--- a/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
+++ b/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
@@ -14,7 +14,7 @@ module Data.List.Relation.Binary.Equality.DecSetoid
 import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
 import Data.List.Relation.Binary.Pointwise as PW
 open import Level
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open DecSetoid DS
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Equality/Setoid.agda b/src/Data/List/Relation/Binary/Equality/Setoid.agda
index f1ed5f457c..fc6842c38d 100644
--- a/src/Data/List/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Equality/Setoid.agda
@@ -7,7 +7,9 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Core using (Op₂)
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (Transitive; Symmetric; Reflexive; _Respects_)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Data.List.Relation.Binary.Equality.Setoid {a ℓ} (S : Setoid a ℓ) where
 
@@ -17,7 +19,7 @@ open import Data.List.Relation.Binary.Pointwise as PW using (Pointwise)
 open import Data.List.Relation.Unary.Unique.Setoid S using (Unique)
 open import Function.Base using (_∘_)
 open import Level
-open import Relation.Binary renaming (Rel to Rel₂)
+open import Relation.Binary.Core renaming (Rel to Rel₂)
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
 open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
 open import Relation.Unary as U using (Pred)
diff --git a/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda
index c9668fa1ec..b09af201e0 100644
--- a/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda
@@ -9,7 +9,7 @@
 module Data.List.Relation.Binary.Infix.Heterogeneous where
 
 open import Level
-open import Relation.Binary using (REL; _⇒_)
+open import Relation.Binary.Core using (REL; _⇒_)
 open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.Relation.Binary.Pointwise
   using (Pointwise)
diff --git a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
index cc067dc846..9af0162ded 100644
--- a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
@@ -20,7 +20,8 @@ open import Function.Base using (case_of_; _$′_)
 open import Relation.Nullary.Decidable using (yes; no; does; map′; _⊎-dec_)
 open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Unary as U using (Pred)
-open import Relation.Binary using (REL; _⇒_; Decidable; Trans; Antisym)
+open import Relation.Binary.Core using (REL; _⇒_)
+open import Relation.Binary.Definitions using (Decidable; Trans; Antisym)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_; refl; cong)
 open import Data.List.Relation.Binary.Pointwise.Base as Pointwise using (Pointwise)
 open import Data.List.Relation.Binary.Infix.Heterogeneous
diff --git a/src/Data/List/Relation/Binary/Pointwise.agda b/src/Data/List/Relation/Binary/Pointwise.agda
index 6a5cde5aec..b749a05bcc 100644
--- a/src/Data/List/Relation/Binary/Pointwise.agda
+++ b/src/Data/List/Relation/Binary/Pointwise.agda
@@ -25,7 +25,10 @@ open import Level
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec using (map′)
 open import Relation.Unary as U using (Pred)
-open import Relation.Binary renaming (Rel to Rel₂)
+open import Relation.Binary.Core renaming (Rel to Rel₂)
+open import Relation.Binary.Definitions using (_Respects_; _Respects₂_)
+open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder; Poset)
+open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence; IsPartialOrder; IsPreorder)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
index 1378e745f5..844dd1c996 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
@@ -22,7 +22,7 @@ import Data.List.Relation.Binary.Sublist.Setoid.Properties
 open import Data.Product using (∃; _,_; proj₂)
 open import Function.Base
 open import Level using (Level)
-open import Relation.Binary using (_Respects_)
+open import Relation.Binary.Definitions using (_Respects_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary using (Pred)
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
index abe9a20b73..df904a0452 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid; _⇒_; _Preserves_⟶_)
+open import Relation.Binary.Core using (Rel; _⇒_; _Preserves_⟶_)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Binary.Sublist.Setoid.Properties
   {c ℓ} (S : Setoid c ℓ) where
@@ -20,7 +21,7 @@ open import Data.Product using (∃; _,_; proj₂)
 open import Function.Base
 open import Function.Bundles using (_⇔_; _⤖_)
 open import Level
-open import Relation.Binary using () renaming (Decidable to Decidable₂)
+open import Relation.Binary.Definitions using () renaming (Decidable to Decidable₂)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Binary.Structures using (IsDecTotalOrder)
 open import Relation.Unary using (Pred; Decidable; Irrelevant)
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
index aeeb927d2d..441956829f 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
@@ -9,7 +9,7 @@
 module Data.List.Relation.Binary.Suffix.Heterogeneous where
 
 open import Level
-open import Relation.Binary using (REL; _⇒_)
+open import Relation.Binary.Core using (REL; _⇒_)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise.Base as Pointwise
   using (Pointwise; []; _∷_)
diff --git a/src/Data/List/Relation/Unary/AllPairs/Core.agda b/src/Data/List/Relation/Unary/AllPairs/Core.agda
index 04728171b0..899d2d76fd 100644
--- a/src/Data/List/Relation/Unary/AllPairs/Core.agda
+++ b/src/Data/List/Relation/Unary/AllPairs/Core.agda
@@ -12,7 +12,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 
 module Data.List.Relation.Unary.AllPairs.Core
   {a ℓ} {A : Set a} (R : Rel A ℓ) where
diff --git a/src/Data/List/Relation/Unary/AllPairs/Properties.agda b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
index 037279cfe1..c116fcb00f 100644
--- a/src/Data/List/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
@@ -19,7 +19,8 @@ open import Data.Nat.Base using (zero; suc; _<_; z≤n; s≤s)
 open import Data.Nat.Properties using (≤-refl; m<n⇒m<1+n)
 open import Function.Base using (_∘_; flip)
 open import Level using (Level)
-open import Relation.Binary using (Rel; DecSetoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (DecSetoid)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (does)
diff --git a/src/Data/List/Relation/Unary/Grouped/Properties.agda b/src/Data/List/Relation/Unary/Grouped/Properties.agda
index 65ac3d6ed6..f020cdd233 100644
--- a/src/Data/List/Relation/Unary/Grouped/Properties.agda
+++ b/src/Data/List/Relation/Unary/Grouped/Properties.agda
@@ -16,7 +16,8 @@ open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _
 open import Data.List.Relation.Unary.Grouped
 open import Function.Base using (_∘_)
 open import Function.Bundles using (module Equivalence; _⇔_)
-open import Relation.Binary as B using (REL; Rel)
+open import Relation.Binary.Definitions as B
+open import Relation.Binary.Core using (REL; Rel)
 open import Relation.Unary as U using (Pred)
 open import Relation.Nullary using (¬_; does; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/List/Relation/Unary/Linked.agda b/src/Data/List/Relation/Unary/Linked.agda
index b5dbddecd7..38fcdd1507 100644
--- a/src/Data/List/Relation/Unary/Linked.agda
+++ b/src/Data/List/Relation/Unary/Linked.agda
@@ -16,7 +16,8 @@ open import Data.Maybe.Relation.Binary.Connected
   using (Connected; just; just-nothing)
 open import Function.Base using (id; _∘_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary as B using (Rel; _⇒_)
+open import Relation.Binary.Definitions as B
+open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
diff --git a/src/Data/List/Sort.agda b/src/Data/List/Sort.agda
index d0939b4d49..acae2ce7f5 100644
--- a/src/Data/List/Sort.agda
+++ b/src/Data/List/Sort.agda
@@ -10,7 +10,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.List.Base using (List)
-open import Relation.Binary using (DecTotalOrder)
+open import Relation.Binary.Bundles using (DecTotalOrder)
 
 module Data.List.Sort
   {a ℓ₁ ℓ₂} (O : DecTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/List/Sort/Base.agda b/src/Data/List/Sort/Base.agda
index 0c20220e78..ae64c39bd3 100644
--- a/src/Data/List/Sort/Base.agda
+++ b/src/Data/List/Sort/Base.agda
@@ -9,7 +9,7 @@
 open import Data.List.Base using (List)
 open import Data.List.Relation.Binary.Permutation.Propositional
 open import Level using (_⊔_)
-open import Relation.Binary using (TotalOrder)
+open import Relation.Binary.Bundles using (TotalOrder)
 
 module Data.List.Sort.Base
   {a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/List/Sort/MergeSort.agda b/src/Data/List/Sort/MergeSort.agda
index 380665d1f1..9648718d0d 100644
--- a/src/Data/List/Sort/MergeSort.agda
+++ b/src/Data/List/Sort/MergeSort.agda
@@ -26,7 +26,7 @@ open import Data.Nat.Induction
 open import Data.Nat.Properties using (m<n⇒m<1+n)
 open import Data.Product.Base as Prod using (_,_)
 open import Function.Base using (_∘_)
-open import Relation.Binary using (DecTotalOrder)
+open import Relation.Binary.Bundles using (DecTotalOrder)
 open import Relation.Nullary.Negation using (¬_)
 
 module Data.List.Sort.MergeSort
diff --git a/src/Data/Maybe/Properties.agda b/src/Data/Maybe/Properties.agda
index 5a3b00f79f..4940da433f 100644
--- a/src/Data/Maybe/Properties.agda
+++ b/src/Data/Maybe/Properties.agda
@@ -17,7 +17,7 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_; id; _∘_; _∘′_)
 open import Function.Definitions using (Injective)
 open import Level using (Level)
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (map′)
diff --git a/src/Data/Nat/Divisibility/Core.agda b/src/Data/Nat/Divisibility/Core.agda
index 3b06bc2c96..2f8cdbe241 100644
--- a/src/Data/Nat/Divisibility/Core.agda
+++ b/src/Data/Nat/Divisibility/Core.agda
@@ -16,7 +16,7 @@ open import Data.Nat.Base using (ℕ; _*_)
 open import Data.Nat.Properties
 open import Level using (0ℓ)
 open import Relation.Nullary.Negation using (¬_)
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong₂; module ≡-Reasoning)
 
diff --git a/src/Data/Product/Effectful/Examples.agda b/src/Data/Product/Effectful/Examples.agda
index 0ada48ad84..f11631341e 100644
--- a/src/Data/Product/Effectful/Examples.agda
+++ b/src/Data/Product/Effectful/Examples.agda
@@ -18,7 +18,7 @@ open import Data.Product
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function.Base using (id)
 import Function.Identity.Effectful as Id
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Product/Properties.agda b/src/Data/Product/Properties.agda
index 1cc3c2383e..e741f8976a 100644
--- a/src/Data/Product/Properties.agda
+++ b/src/Data/Product/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Product.Base
 open import Function.Base using (_∋_; _∘_; id)
 open import Function.Bundles using (_↔_; mk↔′)
 open import Level using (Level)
-open import Relation.Binary using (DecidableEquality)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no)
 
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
index 13edd9aa66..7fce761f79 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
@@ -13,7 +13,7 @@ open import Data.Product
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Function.Base
 open import Function.Inverse using (Inverse)
-open import Relation.Binary using (_⇒_)
+open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.HeterogeneousEquality as H using (_≅_)
 open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
diff --git a/src/Data/Sum/Properties.agda b/src/Data/Sum/Properties.agda
index 2c6e638731..1ee329a2d5 100644
--- a/src/Data/Sum/Properties.agda
+++ b/src/Data/Sum/Properties.agda
@@ -12,7 +12,7 @@ open import Level
 open import Data.Sum.Base
 open import Function.Base using (_∋_; _∘_; id)
 open import Function.Bundles using (mk↔′; _↔_)
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (map′)
diff --git a/src/Data/These/Properties.agda b/src/Data/These/Properties.agda
index 62257ed1ad..a6c96a3208 100644
--- a/src/Data/These/Properties.agda
+++ b/src/Data/These/Properties.agda
@@ -11,7 +11,7 @@ module Data.These.Properties where
 open import Data.Product
 open import Data.These.Base
 open import Function.Base using (_∘_)
-open import Relation.Binary using (Decidable)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no; map′; _×-dec_)
 
diff --git a/src/Data/Tree/AVL/Indexed.agda b/src/Data/Tree/AVL/Indexed.agda
index 16ebdff70e..c322d680e3 100644
--- a/src/Data/Tree/AVL/Indexed.agda
+++ b/src/Data/Tree/AVL/Indexed.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Indexed
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂) where
@@ -19,7 +19,7 @@ open import Data.List.Base as List using (List)
 open import Data.DifferenceList using (DiffList; []; _∷_; _++_)
 open import Function.Base as F hiding (const)
 open import Relation.Unary
-open import Relation.Binary using (_Respects_; Tri; tri<; tri≈; tri>)
+open import Relation.Binary.Definitions using (_Respects_; Tri; tri<; tri≈; tri>)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 private
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
index 53c5caeeb0..545a925f00 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/All.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Indexed.Relation.Unary.All
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda
index 2b0d6f98a7..2710a373b5 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Indexed.Relation.Unary.Any
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
index ef06c5574a..9f4aa1aeca 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Indexed.Relation.Unary.Any.Properties
   {a ℓ₁ ℓ₂} (sto : StrictTotalOrder a ℓ₁ ℓ₂)
@@ -21,7 +21,7 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base as F
 open import Level using (Level)
 
-open import Relation.Binary using (_Respects_; tri<; tri≈; tri>)
+open import Relation.Binary.Definitions using (_Respects_; tri<; tri≈; tri>)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_) renaming (refl to ≡-refl)
 open import Relation.Nullary using (¬_; Dec; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/Tree/AVL/Map.agda b/src/Data/Tree/AVL/Map.agda
index 6869f3fe04..254c965241 100644
--- a/src/Data/Tree/AVL/Map.agda
+++ b/src/Data/Tree/AVL/Map.agda
@@ -8,7 +8,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Map
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional.agda b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
index 5e9498ea31..0873c66cd2 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
@@ -7,7 +7,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Map.Membership.Propositional
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
@@ -21,7 +21,7 @@ open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_)
 open import Level using (Level)
 
-open import Relation.Binary using (Transitive; Symmetric; _Respectsˡ_)
+open import Relation.Binary.Definitions using (Transitive; Symmetric; _Respectsˡ_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Intersection using (_∩_)
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
index 050dd82f12..aef88a23d7 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
@@ -7,7 +7,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Map.Membership.Propositional.Properties
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
@@ -21,7 +21,7 @@ open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_)
 open import Level using (Level)
 
-open import Relation.Binary using (Transitive; Symmetric; _Respectsˡ_)
+open import Relation.Binary.Definitions using (Transitive; Symmetric; _Respectsˡ_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Intersection using (_∩_)
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
index 4bd6961c3d..8a347bd5ff 100644
--- a/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Map.Relation.Unary.Any
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/NonEmpty.agda b/src/Data/Tree/AVL/NonEmpty.agda
index 51c6232077..79fd4eb842 100644
--- a/src/Data/Tree/AVL/NonEmpty.agda
+++ b/src/Data/Tree/AVL/NonEmpty.agda
@@ -11,7 +11,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.NonEmpty
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂) where
diff --git a/src/Data/Tree/AVL/NonEmpty/Propositional.agda b/src/Data/Tree/AVL/NonEmpty/Propositional.agda
index 82ea9125fd..632d4de428 100644
--- a/src/Data/Tree/AVL/NonEmpty/Propositional.agda
+++ b/src/Data/Tree/AVL/NonEmpty/Propositional.agda
@@ -6,7 +6,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; IsStrictTotalOrder; StrictTotalOrder)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsStrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; subst)
 
 module Data.Tree.AVL.NonEmpty.Propositional
diff --git a/src/Data/Tree/AVL/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Relation/Unary/Any.agda
index e0dbadfa83..a9429d4ed2 100644
--- a/src/Data/Tree/AVL/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Relation/Unary/Any.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Relation.Unary.Any
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/Sets.agda b/src/Data/Tree/AVL/Sets.agda
index 976298933c..f20fded531 100644
--- a/src/Data/Tree/AVL/Sets.agda
+++ b/src/Data/Tree/AVL/Sets.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Sets
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/Sets/Membership.agda b/src/Data/Tree/AVL/Sets/Membership.agda
index 748aa60344..f65022ac92 100644
--- a/src/Data/Tree/AVL/Sets/Membership.agda
+++ b/src/Data/Tree/AVL/Sets/Membership.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Sets.Membership
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/Sets/Membership/Properties.agda b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
index 3c2c28ed71..db3f0d38cc 100644
--- a/src/Data/Tree/AVL/Sets/Membership/Properties.agda
+++ b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (StrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Tree.AVL.Sets.Membership.Properties
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Tree/AVL/Value.agda b/src/Data/Tree/AVL/Value.agda
index ff5e85dfa0..ee51150477 100644
--- a/src/Data/Tree/AVL/Value.agda
+++ b/src/Data/Tree/AVL/Value.agda
@@ -7,7 +7,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid; _Respects_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (_Respects_)
 
 module Data.Tree.AVL.Value {a ℓ} (S : Setoid a ℓ) where
 
diff --git a/src/Data/Trie.agda b/src/Data/Trie.agda
index 7f06a6f0ed..694d7afe7b 100644
--- a/src/Data/Trie.agda
+++ b/src/Data/Trie.agda
@@ -9,7 +9,8 @@
 
 {-# OPTIONS --cubical-compatible --sized-types #-}
 
-open import Relation.Binary using (Rel; StrictTotalOrder)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.Trie {k e r} (S : StrictTotalOrder k e r) where
 
diff --git a/src/Data/Vec/Bounded.agda b/src/Data/Vec/Bounded.agda
index 35ecc6a96d..881e3b86e1 100644
--- a/src/Data/Vec/Bounded.agda
+++ b/src/Data/Vec/Bounded.agda
@@ -13,7 +13,7 @@ open import Data.Nat.Base
 open import Data.Vec.Base using (Vec)
 import Data.Vec as Vec using (filter; takeWhile; dropWhile)
 open import Function.Base using (id)
-open import Relation.Binary using (_Preserves_⟶_)
+open import Relation.Binary.Core using (_Preserves_⟶_)
 open import Relation.Unary using (Pred; Decidable)
 
 private
diff --git a/src/Data/Vec/Membership/DecSetoid.agda b/src/Data/Vec/Membership/DecSetoid.agda
index 10efd977d7..fac9138660 100644
--- a/src/Data/Vec/Membership/DecSetoid.agda
+++ b/src/Data/Vec/Membership/DecSetoid.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecSetoid)
+open import Relation.Binary.Bundles using (DecSetoid)
 
 module Data.Vec.Membership.DecSetoid {c ℓ} (DS : DecSetoid c ℓ) where
 
diff --git a/src/Data/Vec/Membership/Setoid.agda b/src/Data/Vec/Membership/Setoid.agda
index 50ef16bb79..57476d5f35 100644
--- a/src/Data/Vec/Membership/Setoid.agda
+++ b/src/Data/Vec/Membership/Setoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Setoid; _Respects_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (_Respects_)
 
 module Data.Vec.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where
 
diff --git a/src/Data/Vec/Relation/Unary/All.agda b/src/Data/Vec/Relation/Unary/All.agda
index f8fe5a4045..e8d24fbd65 100644
--- a/src/Data/Vec/Relation/Unary/All.agda
+++ b/src/Data/Vec/Relation/Unary/All.agda
@@ -19,7 +19,8 @@ open import Function.Base using (_∘_)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no)
 open import Relation.Unary hiding (_∈_)
-open import Relation.Binary using (Setoid; _Respects_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (_Respects_)
 open import Relation.Binary.PropositionalEquality.Core as P using (subst)
 
 private
diff --git a/src/Data/Vec/Relation/Unary/AllPairs.agda b/src/Data/Vec/Relation/Unary/AllPairs.agda
index aeac84557b..a702a1dbf9 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel; _⇒_)
 
 module Data.Vec.Relation.Unary.AllPairs
        {a ℓ} {A : Set a} {R : Rel A ℓ} where
@@ -17,7 +17,7 @@ open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (_⊔_)
-open import Relation.Binary as B using (Rel; _⇒_)
+open import Relation.Binary.Definitions as B
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
diff --git a/src/Data/Vec/Relation/Unary/AllPairs/Core.agda b/src/Data/Vec/Relation/Unary/AllPairs/Core.agda
index a101f14057..bb094cc5fb 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs/Core.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs/Core.agda
@@ -12,7 +12,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 
 module Data.Vec.Relation.Unary.AllPairs.Core
   {a ℓ} {A : Set a} (R : Rel A ℓ) where
diff --git a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
index 672c9fa485..2158246fbc 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
@@ -19,7 +19,7 @@ open import Data.Fin.Properties using (suc-injective)
 open import Data.Nat.Base using (zero; suc; _+_)
 open import Function.Base using (_∘_)
 open import Level using (Level)
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 
 private
diff --git a/src/Data/Vec/Relation/Unary/Linked.agda b/src/Data/Vec/Relation/Unary/Linked.agda
index 1c90268d81..368c22ddc1 100644
--- a/src/Data/Vec/Relation/Unary/Linked.agda
+++ b/src/Data/Vec/Relation/Unary/Linked.agda
@@ -14,7 +14,8 @@ open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary as B using (Rel; _⇒_)
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Definitions as B
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
diff --git a/src/Data/Vec/Relation/Unary/Linked/Properties.agda b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
index 340b6b81a2..c32e4c8df6 100644
--- a/src/Data/Vec/Relation/Unary/Linked/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
@@ -19,7 +19,9 @@ open import Data.Nat.Base using (ℕ; zero; suc; NonZero)
 open import Data.Nat.Properties using (<-pred)
 open import Level using (Level)
 open import Function.Base using (_∘_; flip; _on_)
-open import Relation.Binary using (Rel; Transitive; DecSetoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Transitive)
+open import Relation.Binary.Bundles using (DecSetoid)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (yes; no; does)
diff --git a/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda b/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
index 292283689c..8e1fe54b4f 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
@@ -20,7 +20,8 @@ open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡⇒≡×≡)
 open import Function.Base using (id; _∘_)
 open import Level using (Level)
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality
   using (refl; _≡_; _≢_; sym; setoid)
 open import Relation.Unary using (Pred; Decidable)
diff --git a/src/Data/Vec/Relation/Unary/Unique/Setoid.agda b/src/Data/Vec/Relation/Unary/Unique/Setoid.agda
index 3f0db9d300..76610e0485 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Setoid.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Setoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.Vec.Relation.Unary.Unique.Setoid {a ℓ} (S : Setoid a ℓ) where
 
diff --git a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
index 4c433c04f6..cb276cae9a 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
@@ -18,7 +18,8 @@ import Data.Vec.Relation.Unary.AllPairs.Properties as AllPairsₚ
 open import Data.Nat.Base using (ℕ; _+_)
 open import Function.Base using (_∘_; id)
 open import Level using (Level)
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Negation using (contradiction; contraposition)
 
diff --git a/src/Data/Word/Base.agda b/src/Data/Word/Base.agda
index 1dcaea5731..81a7ae6666 100644
--- a/src/Data/Word/Base.agda
+++ b/src/Data/Word/Base.agda
@@ -11,7 +11,7 @@ module Data.Word.Base where
 open import Level using (zero)
 import Data.Nat.Base as ℕ
 open import Function.Base using (_on_)
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
diff --git a/src/Effect/Monad/Partiality/All.agda b/src/Effect/Monad/Partiality/All.agda
index 2954a5da2b..c2db61c031 100644
--- a/src/Effect/Monad/Partiality/All.agda
+++ b/src/Effect/Monad/Partiality/All.agda
@@ -13,7 +13,8 @@ open import Effect.Monad.Partiality as Partiality using (_⊥; ⇒≈)
 open import Codata.Musical.Notation
 open import Function.Base using (flip; _∘_)
 open import Level
-open import Relation.Binary using (_Respects_; IsEquivalence)
+open import Relation.Binary.Definitions using (_Respects_)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 open Partiality._⊥
diff --git a/src/Function/Endomorphism/Propositional.agda b/src/Function/Endomorphism/Propositional.agda
index d53142c532..c41305f9e9 100644
--- a/src/Function/Endomorphism/Propositional.agda
+++ b/src/Function/Endomorphism/Propositional.agda
@@ -20,7 +20,7 @@ open import Data.Product.Base using (_,_)
 
 open import Function.Base using (id; _∘′_; _∋_)
 open import Function.Equality using (_⟨$⟩_)
-open import Relation.Binary using (_Preserves_⟶_)
+open import Relation.Binary.Core using (_Preserves_⟶_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
 import Relation.Binary.PropositionalEquality.Properties as P
 
diff --git a/src/Function/Properties/Equivalence.agda b/src/Function/Properties/Equivalence.agda
index 066316c01b..beaa7d8940 100644
--- a/src/Function/Properties/Equivalence.agda
+++ b/src/Function/Properties/Equivalence.agda
@@ -11,7 +11,8 @@ module Function.Properties.Equivalence where
 
 open import Function.Bundles using (Equivalence; _⇔_)
 open import Level
-open import Relation.Binary using (Setoid; IsEquivalence)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 import Relation.Binary.PropositionalEquality.Properties as Eq
 
 import Function.Construct.Identity as Identity
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index f097945dec..d430b36398 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -13,7 +13,8 @@ open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Bundles
 open import Level using (Level)
-open import Relation.Binary using (Setoid; IsEquivalence)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 import Relation.Binary.PropositionalEquality.Core as P
 import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
diff --git a/src/Function/Properties/RightInverse.agda b/src/Function/Properties/RightInverse.agda
index 66b35e53f0..531b7236e7 100644
--- a/src/Function/Properties/RightInverse.agda
+++ b/src/Function/Properties/RightInverse.agda
@@ -13,7 +13,8 @@ open import Function.Bundles
 open import Function.Consequences using (inverseˡ⇒surjective)
 open import Level using (Level)
 open import Data.Product
-open import Relation.Binary using (Setoid; IsEquivalence)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 private
   variable
diff --git a/src/Reflection/AST/Abstraction.agda b/src/Reflection/AST/Abstraction.agda
index 3bf47f81a4..43fb2f8302 100644
--- a/src/Reflection/AST/Abstraction.agda
+++ b/src/Reflection/AST/Abstraction.agda
@@ -13,7 +13,7 @@ open import Data.String.Properties as String           using (_≟_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Level
 open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 private
diff --git a/src/Reflection/AST/AlphaEquality.agda b/src/Reflection/AST/AlphaEquality.agda
index 2c8637e9dc..4849470d50 100644
--- a/src/Reflection/AST/AlphaEquality.agda
+++ b/src/Reflection/AST/AlphaEquality.agda
@@ -8,12 +8,12 @@
 
 module Reflection.AST.AlphaEquality where
 
-open import Data.Bool.Base             using (Bool; true; false; _∧_)
-open import Data.List.Base             using ([]; _∷_)
-open import Data.Nat.Base as ℕ        using (ℕ; zero; suc; _≡ᵇ_)
-open import Data.Product.Base          using (_,_)
-open import Relation.Nullary.Decidable using (⌊_⌋)
-open import Relation.Binary            using (DecidableEquality)
+open import Data.Bool.Base                using (Bool; true; false; _∧_)
+open import Data.List.Base                using ([]; _∷_)
+open import Data.Nat.Base as ℕ            using (ℕ; zero; suc; _≡ᵇ_)
+open import Data.Product.Base             using (_,_)
+open import Relation.Nullary.Decidable    using (⌊_⌋)
+open import Relation.Binary.Definitions   using (DecidableEquality)
 
 open import Reflection.AST.Abstraction
 open import Reflection.AST.Argument
diff --git a/src/Reflection/AST/Argument.agda b/src/Reflection/AST/Argument.agda
index fe95ccedc1..b97ec90f77 100644
--- a/src/Reflection/AST/Argument.agda
+++ b/src/Reflection/AST/Argument.agda
@@ -11,7 +11,7 @@ module Reflection.AST.Argument where
 open import Data.List.Base as List                     using (List; []; _∷_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 open import Level
 
diff --git a/src/Reflection/AST/Argument/Information.agda b/src/Reflection/AST/Argument/Information.agda
index e0f5be24a5..66fec3be77 100644
--- a/src/Reflection/AST/Argument/Information.agda
+++ b/src/Reflection/AST/Argument/Information.agda
@@ -10,7 +10,7 @@ module Reflection.AST.Argument.Information where
 
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable                 using (map′; _×-dec_)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 open import Reflection.AST.Argument.Modality as Modality using (Modality)
diff --git a/src/Reflection/AST/Argument/Modality.agda b/src/Reflection/AST/Argument/Modality.agda
index 02e481af58..489247031e 100644
--- a/src/Reflection/AST/Argument/Modality.agda
+++ b/src/Reflection/AST/Argument/Modality.agda
@@ -10,7 +10,7 @@ module Reflection.AST.Argument.Modality where
 
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable                 using (map′; _×-dec_)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 open import Reflection.AST.Argument.Relevance as Relevance using (Relevance)
diff --git a/src/Reflection/AST/Argument/Quantity.agda b/src/Reflection/AST/Argument/Quantity.agda
index 5b3b98e092..bc79d74b3c 100644
--- a/src/Reflection/AST/Argument/Quantity.agda
+++ b/src/Reflection/AST/Argument/Quantity.agda
@@ -9,7 +9,7 @@
 module Reflection.AST.Argument.Quantity where
 
 open import Relation.Nullary                           using (yes; no)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 ------------------------------------------------------------------------
diff --git a/src/Reflection/AST/Argument/Relevance.agda b/src/Reflection/AST/Argument/Relevance.agda
index 03ba0f4f0a..3ac78cf765 100644
--- a/src/Reflection/AST/Argument/Relevance.agda
+++ b/src/Reflection/AST/Argument/Relevance.agda
@@ -9,7 +9,7 @@
 module Reflection.AST.Argument.Relevance where
 
 open import Relation.Nullary                           using (yes; no)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 ------------------------------------------------------------------------
diff --git a/src/Reflection/AST/Argument/Visibility.agda b/src/Reflection/AST/Argument/Visibility.agda
index b5e9236043..b384609c93 100644
--- a/src/Reflection/AST/Argument/Visibility.agda
+++ b/src/Reflection/AST/Argument/Visibility.agda
@@ -9,7 +9,7 @@
 module Reflection.AST.Argument.Visibility where
 
 open import Relation.Nullary                           using (yes; no)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 ------------------------------------------------------------------------
diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index ea005eebd3..b5b5b32700 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -12,7 +12,7 @@ import Data.List.Properties as Listₚ                   using (≡-dec)
 import Data.Nat.Properties as ℕₚ                       using (_≟_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 
 import Reflection.AST.Argument as Arg
diff --git a/src/Reflection/AST/Literal.agda b/src/Reflection/AST/Literal.agda
index f400c9f1dc..00ac5cc652 100644
--- a/src/Reflection/AST/Literal.agda
+++ b/src/Reflection/AST/Literal.agda
@@ -18,7 +18,7 @@ import Reflection.AST.Meta as Meta
 import Reflection.AST.Name as Name
 open import Relation.Nullary                           using (yes; no)
 open import Relation.Nullary.Decidable                 using (map′; isYes)
-open import Relation.Binary                            using (DecidableEquality)
+open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 ------------------------------------------------------------------------
diff --git a/src/Reflection/AST/Meta.agda b/src/Reflection/AST/Meta.agda
index 56bda50d31..682a52c8ca 100644
--- a/src/Reflection/AST/Meta.agda
+++ b/src/Reflection/AST/Meta.agda
@@ -11,7 +11,8 @@ module Reflection.AST.Meta where
 import Data.Nat.Properties as ℕₚ                       using (_≟_)
 open import Function.Base                              using (_on_)
 open import Relation.Nullary.Decidable                 using (map′)
-open import Relation.Binary                            using (Rel; Decidable; DecidableEquality)
+open import Relation.Binary.Core                       using (Rel)
+open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)
 import Relation.Binary.Construct.On as On
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index fb31bc777a..ac478443fb 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -13,7 +13,8 @@ import Data.Product.Properties as Prodₚ using (≡-dec)
 import Data.Word.Properties as Wₚ       using (_≟_)
 open import Function.Base               using (_on_)
 open import Relation.Nullary.Decidable                 using (map′)
-open import Relation.Binary                            using (Rel; Decidable; DecidableEquality)
+open import Relation.Binary.Core                       using (Rel)
+open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)
 open import Relation.Binary.Construct.On               using (decidable)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index b22641b2a9..33ef68d6f0 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -8,16 +8,16 @@
 
 module Reflection.AST.Term where
 
-open import Data.List.Base as List  hiding (_++_)
-open import Data.List.Properties    using (∷-dec)
-open import Data.Nat as ℕ           using (ℕ; zero; suc)
-open import Data.Product.Base       using (_×_; _,_; <_,_>; uncurry; map₁)
-open import Data.Product.Properties using (,-injective)
-open import Data.Maybe.Base         using (Maybe; just; nothing)
-open import Data.String.Base    using (String)
-open import Data.String.Properties as String hiding (_≟_)
+open import Data.List.Base as List                     hiding (_++_)
+open import Data.List.Properties                       using (∷-dec)
+open import Data.Nat as ℕ                              using (ℕ; zero; suc)
+open import Data.Product.Base                          using (_×_; _,_; <_,_>; uncurry; map₁)
+open import Data.Product.Properties                    using (,-injective)
+open import Data.Maybe.Base                            using (Maybe; just; nothing)
+open import Data.String.Base                           using (String)
+open import Data.String.Properties as String           hiding (_≟_)
 open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)
-open import Relation.Binary                            using (Decidable; DecidableEquality)
+open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 
 open import Reflection.AST.Abstraction

From 35f151a3982b5853579904898e2958b60c335296 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Thu, 10 Aug 2023 22:30:25 -0400
Subject: [PATCH 091/316] Move just a few more things over to new Function
 hierarchy. (#2044)

---
 src/Data/List/Relation/Binary/Pointwise.agda     | 13 ++++++-------
 src/Data/List/Relation/Unary/All/Properties.agda |  6 +++---
 src/Data/Vec/Properties.agda                     | 10 +++++-----
 src/Data/Vec/Recursive/Properties.agda           |  4 ++--
 4 files changed, 16 insertions(+), 17 deletions(-)

diff --git a/src/Data/List/Relation/Binary/Pointwise.agda b/src/Data/List/Relation/Binary/Pointwise.agda
index b749a05bcc..2728418ddc 100644
--- a/src/Data/List/Relation/Binary/Pointwise.agda
+++ b/src/Data/List/Relation/Binary/Pointwise.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Binary.Pointwise where
 
 open import Algebra.Core using (Op₂)
 open import Function.Base
-open import Function.Inverse using (Inverse)
+open import Function.Bundles using (Inverse)
 open import Data.Bool.Base using (true; false)
 open import Data.Product.Base hiding (map)
 open import Data.List.Base as List hiding (map; head; tail; uncons)
@@ -269,10 +269,9 @@ Pointwise-≡⇒≡ (P.refl ∷ xs∼ys) with Pointwise-≡⇒≡ xs∼ys
 
 Pointwise-≡↔≡ : Inverse (setoid (P.setoid A)) (P.setoid (List A))
 Pointwise-≡↔≡ = record
-  { to         = record { _⟨$⟩_ = id; cong = Pointwise-≡⇒≡ }
-  ; from       = record { _⟨$⟩_ = id; cong = ≡⇒Pointwise-≡ }
-  ; inverse-of = record
-    { left-inverse-of  = λ _ → refl P.refl
-    ; right-inverse-of = λ _ → P.refl
-    }
+  { to = id
+  ; from = id
+  ; to-cong = Pointwise-≡⇒≡
+  ; from-cong = ≡⇒Pointwise-≡
+  ; inverse = Pointwise-≡⇒≡ , ≡⇒Pointwise-≡
   }
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index 29a645762e..af41a14041 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -33,9 +33,9 @@ open import Data.Nat.Base using (zero; suc; s≤s; _<_; z<s; s<s)
 open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)
 open import Data.Product.Base as Prod using (_×_; _,_; uncurry; uncurry′)
 open import Function.Base
+import Function.Bundles as B
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
-open import Function.Inverse using (_↔_; inverse)
 open import Function.Surjection using (_↠_; surjection)
 open import Level using (Level)
 open import Relation.Binary as B using (REL; Setoid; _Respects_)
@@ -405,8 +405,8 @@ mapMaybe⁺ {xs = x ∷ xs} {f = f} (px ∷ pxs) with f x
 ++⁻ []       p          = [] , p
 ++⁻ (x ∷ xs) (px ∷ pxs) = Prod.map (px ∷_) id (++⁻ _ pxs)
 
-++↔ : (All P xs × All P ys) ↔ All P (xs ++ ys)
-++↔ {xs = zs} = inverse (uncurry ++⁺) (++⁻ zs) ++⁻∘++⁺ (++⁺∘++⁻ zs)
+++↔ : (All P xs × All P ys) B.↔ All P (xs ++ ys)
+++↔ {xs = zs} = B.mk↔′ (uncurry ++⁺) (++⁻ zs) (++⁺∘++⁻ zs) ++⁻∘++⁺
   where
   ++⁺∘++⁻ : ∀ xs (p : All P (xs ++ ys)) → uncurry′ ++⁺ (++⁻ xs p) ≡ p
   ++⁺∘++⁻ []       p          = refl
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 430cdf8d2b..49924a7b31 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -21,7 +21,8 @@ open import Data.Sum.Base using ([_,_]′)
 open import Data.Sum.Properties using ([,]-map)
 open import Data.Vec.Base
 open import Function.Base
-open import Function.Inverse using (_↔_; inverse)
+-- open import Function.Inverse using (_↔_; inverse)
+open import Function.Bundles using (_↔_; mk↔′)
 open import Level using (Level)
 open import Relation.Binary hiding (Decidable)
 open import Relation.Binary.PropositionalEquality
@@ -199,9 +200,8 @@ lookup⇒[]= zero    (_ ∷ _)  refl = here
 lookup⇒[]= (suc i) (_ ∷ xs) p    = there (lookup⇒[]= i xs p)
 
 []=↔lookup : ∀ {i} → xs [ i ]= x ↔ lookup xs i ≡ x
-[]=↔lookup {i = i} =
-  inverse []=⇒lookup (lookup⇒[]= _ _)
-          lookup⇒[]=∘[]=⇒lookup ([]=⇒lookup∘lookup⇒[]= _ i)
+[]=↔lookup {xs = ys} {i = i} =
+  mk↔′ []=⇒lookup (lookup⇒[]= i ys) ([]=⇒lookup∘lookup⇒[]= _ i) lookup⇒[]=∘[]=⇒lookup
   where
   lookup⇒[]=∘[]=⇒lookup : ∀ {i} (p : xs [ i ]= x) →
                           lookup⇒[]= i xs ([]=⇒lookup p) ≡ p
@@ -706,7 +706,7 @@ zip∘unzip []         = refl
 zip∘unzip (xy ∷ xys) = cong (xy ∷_) (zip∘unzip xys)
 
 ×v↔v× : (Vec A n × Vec B n) ↔ Vec (A × B) n
-×v↔v× = inverse (uncurry zip) unzip (uncurry unzip∘zip) zip∘unzip
+×v↔v× = mk↔′ (uncurry zip) unzip zip∘unzip (uncurry unzip∘zip)
 
 ------------------------------------------------------------------------
 -- _⊛_
diff --git a/src/Data/Vec/Recursive/Properties.agda b/src/Data/Vec/Recursive/Properties.agda
index d557808cee..7a6a6e960b 100644
--- a/src/Data/Vec/Recursive/Properties.agda
+++ b/src/Data/Vec/Recursive/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Nat.Base hiding (_^_)
 open import Data.Product
 open import Data.Vec.Recursive
 open import Data.Vec.Base using (Vec; _∷_)
-open import Function.Inverse using (_↔_; inverse)
+open import Function.Bundles using (_↔_; mk↔′)
 open import Relation.Binary.PropositionalEquality.Core as P
 open ≡-Reasoning
 
@@ -82,7 +82,7 @@ toVec∘fromVec {n = suc n} (x Vec.∷ xs) = begin
   tl-prf = tail-cons-identity _ x (fromVec xs)
 
 ↔Vec : ∀ n → A ^ n ↔ Vec A n
-↔Vec n = inverse (toVec n) fromVec (fromVec∘toVec n) toVec∘fromVec
+↔Vec n = mk↔′ (toVec n) fromVec toVec∘fromVec (fromVec∘toVec n)
 
 ------------------------------------------------------------------------
 -- DEPRECATED NAMES

From 8a59faf48a9ac8bfaa9cbf1e3189203bb73d0222 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 10 Aug 2023 23:30:44 -0400
Subject: [PATCH 092/316] Final `Data.Product` import simplifications (#2052)

---
 README/Case.agda                                           | 1 -
 README/Data/Container/Indexed.agda                         | 2 +-
 README/Data/Integer.agda                                   | 1 -
 README/Data/List/Fresh.agda                                | 2 +-
 README/Data/Record.agda                                    | 2 +-
 README/Data/Trie/NonDependent.agda                         | 2 +-
 README/Data/Wrap.agda                                      | 2 +-
 README/Design/Decidability.agda                            | 2 +-
 README/Foreign/Haskell.agda                                | 2 +-
 README/Inspect.agda                                        | 2 +-
 README/Relation/Binary/TypeClasses.agda                    | 2 +-
 README/Text/Pretty.agda                                    | 2 +-
 src/Algebra/Apartness/Structures.agda                      | 2 +-
 src/Algebra/Construct/DirectProduct.agda                   | 2 +-
 src/Algebra/Construct/LexProduct.agda                      | 2 +-
 src/Algebra/Lattice/Construct/DirectProduct.agda           | 2 +-
 src/Algebra/Module/Construct/DirectProduct.agda            | 2 +-
 src/Algebra/Module/Definitions/Left.agda                   | 2 --
 src/Algebra/Module/Definitions/Right.agda                  | 2 --
 src/Algebra/Module/Morphism/ModuleHomomorphism.agda        | 2 +-
 src/Algebra/Morphism/Construct/Composition.agda            | 1 -
 src/Algebra/Morphism/GroupMonomorphism.agda                | 2 +-
 src/Algebra/Morphism/MagmaMonomorphism.agda                | 2 +-
 src/Algebra/Morphism/RingMonomorphism.agda                 | 2 +-
 src/Algebra/Properties/Loop.agda                           | 1 -
 src/Algebra/Properties/Magma/Divisibility.agda             | 2 +-
 src/Algebra/Properties/MiddleBolLoop.agda                  | 1 -
 src/Algebra/Properties/MoufangLoop.agda                    | 2 +-
 src/Algebra/Properties/Quasigroup.agda                     | 2 +-
 src/Algebra/Solver/Monoid.agda                             | 2 +-
 src/Codata/Musical/Colist.agda                             | 2 +-
 src/Codata/Musical/Colist/Infinite-merge.agda              | 2 +-
 src/Codata/Musical/Conversion.agda                         | 2 +-
 src/Codata/Musical/M/Indexed.agda                          | 2 +-
 src/Data/Container/Combinator.agda                         | 2 +-
 src/Data/Container/Combinator/Properties.agda              | 2 +-
 src/Data/Container/Indexed/Combinator.agda                 | 2 +-
 src/Data/Container/Indexed/Core.agda                       | 2 +-
 src/Data/Container/Indexed/FreeMonad.agda                  | 2 +-
 src/Data/Container/Indexed/WithK.agda                      | 3 ++-
 src/Data/Container/Morphism/Properties.agda                | 2 +-
 src/Data/Container/Relation/Unary/All.agda                 | 2 +-
 src/Data/Container/Relation/Unary/Any.agda                 | 2 +-
 src/Data/Container/Relation/Unary/Any/Properties.agda      | 2 +-
 src/Data/Fin/Subset.agda                                   | 2 +-
 src/Data/Graph/Acyclic.agda                                | 2 +-
 src/Data/Integer/GCD.agda                                  | 2 +-
 src/Data/List/Countdown.agda                               | 2 +-
 src/Data/List/Fresh/Membership/Setoid/Properties.agda      | 2 +-
 src/Data/List/Fresh/Relation/Unary/Any.agda                | 2 +-
 src/Data/List/Membership/Propositional/Properties.agda     | 2 +-
 .../List/Membership/Propositional/Properties/Core.agda     | 2 +-
 src/Data/List/Membership/Setoid/Properties.agda            | 2 +-
 src/Data/List/Relation/Binary/BagAndSetEquality.agda       | 2 +-
 src/Data/List/Relation/Binary/Lex/NonStrict.agda           | 1 -
 .../Binary/Permutation/Propositional/Properties.agda       | 2 +-
 .../Relation/Binary/Permutation/Setoid/Properties.agda     | 2 +-
 src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda    | 2 +-
 .../Relation/Binary/Sublist/Heterogeneous/Properties.agda  | 2 +-
 .../List/Relation/Binary/Sublist/Heterogeneous/Solver.agda | 2 +-
 .../Relation/Binary/Sublist/Propositional/Properties.agda  | 2 +-
 src/Data/List/Relation/Binary/Sublist/Setoid.agda          | 2 +-
 .../List/Relation/Binary/Sublist/Setoid/Properties.agda    | 2 +-
 src/Data/List/Relation/Ternary/Appending.agda              | 2 +-
 src/Data/List/Relation/Ternary/Interleaving.agda           | 2 +-
 src/Data/List/Relation/Unary/Any/Properties.agda           | 2 +-
 src/Data/List/Relation/Unary/First.agda                    | 2 +-
 src/Data/Maybe/Relation/Binary/Connected.agda              | 1 -
 src/Data/Maybe/Relation/Binary/Pointwise.agda              | 2 +-
 src/Data/Maybe/Relation/Unary/Any.agda                     | 2 +-
 src/Data/Nat/Binary/Properties.agda                        | 2 +-
 src/Data/Nat/Binary/Subtraction.agda                       | 2 +-
 src/Data/Nat/GCD.agda                                      | 3 ++-
 src/Data/Nat/InfinitelyOften.agda                          | 2 +-
 src/Data/Nat/LCM.agda                                      | 2 +-
 src/Data/Nat/Primality.agda                                | 2 +-
 src/Data/Product/Effectful/Examples.agda                   | 2 +-
 src/Data/Product/Function/Dependent/Propositional.agda     | 2 +-
 .../Product/Function/Dependent/Propositional/WithK.agda    | 2 +-
 src/Data/Product/Function/Dependent/Setoid.agda            | 2 +-
 src/Data/Product/Function/Dependent/Setoid/WithK.agda      | 2 +-
 src/Data/Product/Function/NonDependent/Propositional.agda  | 2 +-
 src/Data/Product/Function/NonDependent/Setoid.agda         | 2 +-
 src/Data/Product/Instances.agda                            | 2 +-
 src/Data/Product/Properties/WithK.agda                     | 2 +-
 .../Product/Relation/Binary/Pointwise/Dependent/WithK.agda | 2 +-
 src/Data/Sum/Relation/Binary/LeftOrder.agda                | 2 +-
 src/Data/These/Properties.agda                             | 2 +-
 src/Data/Tree/AVL/Indexed.agda                             | 2 +-
 src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda          | 2 +-
 .../Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda    | 2 +-
 src/Data/Tree/AVL/Indexed/WithK.agda                       | 1 -
 src/Data/Tree/AVL/IndexedMap.agda                          | 7 ++++---
 src/Data/Tree/AVL/Key.agda                                 | 2 +-
 src/Data/Tree/AVL/Map/Membership/Propositional.agda        | 2 +-
 .../Tree/AVL/Map/Membership/Propositional/Properties.agda  | 2 +-
 src/Data/Tree/AVL/Map/Relation/Unary/Any.agda              | 2 +-
 src/Data/Tree/AVL/Relation/Unary/Any.agda                  | 2 +-
 src/Data/Trie.agda                                         | 2 +-
 src/Data/Trie/NonEmpty.agda                                | 2 +-
 src/Data/Universe.agda                                     | 1 -
 src/Data/Universe/Indexed.agda                             | 2 +-
 src/Data/Vec/Functional/Relation/Unary/Any.agda            | 2 +-
 src/Data/Vec/Membership/Propositional/Properties.agda      | 4 ++--
 src/Data/Vec/Membership/Setoid.agda                        | 2 +-
 src/Data/Vec/Recursive/Properties.agda                     | 2 +-
 src/Data/Vec/Relation/Unary/Any.agda                       | 2 +-
 src/Data/Vec/Relation/Unary/Any/Properties.agda            | 2 +-
 src/Data/W/Indexed.agda                                    | 2 +-
 src/Data/W/WithK.agda                                      | 2 +-
 src/Effect/Applicative/Predicate.agda                      | 2 +-
 src/Effect/Monad/Predicate.agda                            | 2 +-
 src/Effect/Monad/State.agda                                | 2 +-
 src/Effect/Monad/State/Indexed.agda                        | 2 +-
 src/Foreign/Haskell/Coerce.agda                            | 2 +-
 src/Foreign/Haskell/Pair.agda                              | 2 +-
 src/Function/Bijection.agda                                | 1 -
 src/Function/Endomorphism/Setoid.agda                      | 2 +-
 src/Function/LeftInverse.agda                              | 1 -
 src/Function/Properties/RightInverse.agda                  | 2 +-
 src/Function/Properties/Surjection.agda                    | 2 +-
 src/Function/Related/TypeIsomorphisms.agda                 | 3 ++-
 src/Function/Surjection.agda                               | 1 -
 src/Induction/Lexicographic.agda                           | 2 +-
 src/Relation/Binary/Construct/Add/Infimum/Strict.agda      | 2 +-
 src/Relation/Binary/Construct/Add/Supremum/Strict.agda     | 2 +-
 src/Relation/Binary/Construct/Composition.agda             | 2 +-
 src/Relation/Binary/Construct/Flip/Ord.agda                | 2 +-
 src/Relation/Binary/Construct/FromPred.agda                | 1 -
 src/Relation/Binary/Construct/FromRel.agda                 | 1 -
 src/Relation/Binary/Construct/On.agda                      | 2 +-
 src/Relation/Binary/HeterogeneousEquality.agda             | 1 -
 .../Binary/HeterogeneousEquality/Quotients/Examples.agda   | 2 +-
 .../Binary/Lattice/Properties/BoundedJoinSemilattice.agda  | 2 +-
 .../Binary/Lattice/Properties/BoundedMeetSemilattice.agda  | 1 -
 .../Binary/Lattice/Properties/JoinSemilattice.agda         | 2 +-
 .../Binary/Lattice/Properties/MeetSemilattice.agda         | 1 -
 src/Relation/Binary/Rewriting.agda                         | 2 +-
 src/Relation/Unary/Indexed.agda                            | 2 +-
 src/Relation/Unary/PredicateTransformer.agda               | 2 +-
 src/Tactic/Cong.agda                                       | 2 +-
 src/Tactic/RingSolver/NonReflective.agda                   | 2 +-
 src/Text/Format/Generic.agda                               | 2 +-
 143 files changed, 131 insertions(+), 148 deletions(-)

diff --git a/README/Case.agda b/README/Case.agda
index 15449391d4..9de25c6e82 100644
--- a/README/Case.agda
+++ b/README/Case.agda
@@ -12,7 +12,6 @@ module README.Case where
 open import Data.Fin   hiding (pred)
 open import Data.Maybe hiding (from-just)
 open import Data.Nat   hiding (pred)
-open import Data.Product
 open import Function.Base using (case_of_; case_return_of_)
 open import Relation.Nullary
 open import Relation.Binary
diff --git a/README/Data/Container/Indexed.agda b/README/Data/Container/Indexed.agda
index 7ace9736f1..adf6cbc9e5 100644
--- a/README/Data/Container/Indexed.agda
+++ b/README/Data/Container/Indexed.agda
@@ -11,7 +11,7 @@ module README.Data.Container.Indexed where
 open import Data.Unit
 open import Data.Empty
 open import Data.Nat.Base
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_)
 open import Data.W.Indexed
 open import Data.Container.Indexed
diff --git a/README/Data/Integer.agda b/README/Data/Integer.agda
index e93333dcf5..23619757d3 100644
--- a/README/Data/Integer.agda
+++ b/README/Data/Integer.agda
@@ -51,7 +51,6 @@ ex₅ i j = ℤₚ.*-comm i j
 -- provides some combinators for equational reasoning.
 
 open P.≡-Reasoning
-open import Data.Product
 
 ex₆ : ∀ i j → i * (j + + 0) ≡ j * i
 ex₆ i j = begin
diff --git a/README/Data/List/Fresh.agda b/README/Data/List/Fresh.agda
index 27f7eb7ca0..4723c0631e 100644
--- a/README/Data/List/Fresh.agda
+++ b/README/Data/List/Fresh.agda
@@ -12,7 +12,7 @@ open import Data.Nat
 open import Data.List.Base
 open import Data.List.Fresh
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs)
-open import Data.Product
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Relation.Nary using (⌊_⌋; fromWitness)
 
 -- A sorted list of natural numbers can be seen as a fresh list
diff --git a/README/Data/Record.agda b/README/Data/Record.agda
index 5800d11e54..da0ae70dc4 100644
--- a/README/Data/Record.agda
+++ b/README/Data/Record.agda
@@ -11,7 +11,7 @@
 
 module README.Data.Record where
 
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Data.String
 open import Function.Base using (flip)
 open import Level
diff --git a/README/Data/Trie/NonDependent.agda b/README/Data/Trie/NonDependent.agda
index 9921942f6d..0484e6005a 100644
--- a/README/Data/Trie/NonDependent.agda
+++ b/README/Data/Trie/NonDependent.agda
@@ -56,7 +56,7 @@ import Data.Char.Properties  as Char
 open import Data.List.Base   as List using (List; []; _∷_)
 open import Data.List.Fresh  as List# using (List#; []; _∷#_)
 open import Data.Maybe       as Maybe
-open import Data.Product     as Prod
+open import Data.Product.Base as Prod using (_×_; ∃; proj₁; _,_)
 open import Data.String.Base as String using (String)
 open import Data.String.Properties as String using (_≟_)
 open import Data.These       as These
diff --git a/README/Data/Wrap.agda b/README/Data/Wrap.agda
index 35009e4ed4..b92a8a3c81 100644
--- a/README/Data/Wrap.agda
+++ b/README/Data/Wrap.agda
@@ -13,7 +13,7 @@ open import Data.Wrap
 open import Algebra
 open import Data.Nat
 open import Data.Nat.Properties
-open import Data.Product
+open import Data.Product.Base using (_,_; ∃; ∃₂; _×_)
 open import Level using (Level)
 open import Relation.Binary
 
diff --git a/README/Design/Decidability.agda b/README/Design/Decidability.agda
index cb41cf41ef..c6f086424f 100644
--- a/README/Design/Decidability.agda
+++ b/README/Design/Decidability.agda
@@ -13,7 +13,7 @@ open import Data.List.Base using (List; []; _∷_)
 open import Data.List.Properties using (∷-injective)
 open import Data.Nat
 open import Data.Nat.Properties using (suc-injective)
-open import Data.Product
+open import Data.Product.Base using (uncurry)
 open import Data.Unit
 open import Function.Base using (id;  _∘_)
 open import Relation.Binary.PropositionalEquality
diff --git a/README/Foreign/Haskell.agda b/README/Foreign/Haskell.agda
index 61e757ea8f..38c6c78c4e 100644
--- a/README/Foreign/Haskell.agda
+++ b/README/Foreign/Haskell.agda
@@ -22,7 +22,7 @@ open import Data.Bool.Base using (Bool; if_then_else_)
 open import Data.Char as Char
 open import Data.List.Base as List using (List; _∷_; []; takeWhile; dropWhile)
 open import Data.Maybe.Base using (Maybe; just; nothing)
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_)
 open import Function
 open import Relation.Nullary.Decidable
 
diff --git a/README/Inspect.agda b/README/Inspect.agda
index 002005abaa..9f40a8d65a 100644
--- a/README/Inspect.agda
+++ b/README/Inspect.agda
@@ -14,7 +14,7 @@ module README.Inspect where
 
 open import Data.Nat.Base
 open import Data.Nat.Properties
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Binary.PropositionalEquality using (inspect)
 
diff --git a/README/Relation/Binary/TypeClasses.agda b/README/Relation/Binary/TypeClasses.agda
index 023e98c3e4..a836389ee4 100644
--- a/README/Relation/Binary/TypeClasses.agda
+++ b/README/Relation/Binary/TypeClasses.agda
@@ -19,7 +19,7 @@ open import Data.List.Instances
 open import Data.List.Relation.Binary.Lex.NonStrict using (Lex-≤)
 open import Data.Nat.Base renaming (_≤_ to _≤ℕ_)
 open import Data.Nat.Instances
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_; Σ)
 open import Data.Product.Instances
 open import Data.Unit.Base renaming (_≤_ to _≤⊤_)
 open import Data.Unit.Instances
diff --git a/README/Text/Pretty.agda b/README/Text/Pretty.agda
index 2525ff5242..855c599f08 100644
--- a/README/Text/Pretty.agda
+++ b/README/Text/Pretty.agda
@@ -14,7 +14,7 @@ open import Data.Bool.Base
 open import Data.List.Base as List
 open import Data.List.NonEmpty as List⁺
 open import Data.Nat.Base
-open import Data.Product
+open import Data.Product.Base using (_×_; uncurry; _,_)
 open import Data.String.Base hiding (parens; _<+>_)
 open import Data.Vec.Base as Vec
 open import Function.Base
diff --git a/src/Algebra/Apartness/Structures.agda b/src/Algebra/Apartness/Structures.agda
index 3311234945..a8c5c88982 100644
--- a/src/Algebra/Apartness/Structures.agda
+++ b/src/Algebra/Apartness/Structures.agda
@@ -17,7 +17,7 @@ module Algebra.Apartness.Structures
   where
 
 open import Level using (_⊔_; suc)
-open import Data.Product using (∃-syntax; _×_; _,_; proj₂)
+open import Data.Product.Base using (∃-syntax; _×_; _,_; proj₂)
 open import Algebra.Definitions _≈_ using (Invertible)
 open import Algebra.Structures _≈_ using (IsCommutativeRing)
 open import Relation.Binary.Structures using (IsEquivalence; IsApartnessRelation)
diff --git a/src/Algebra/Construct/DirectProduct.agda b/src/Algebra/Construct/DirectProduct.agda
index 7722662114..5f7b042017 100644
--- a/src/Algebra/Construct/DirectProduct.agda
+++ b/src/Algebra/Construct/DirectProduct.agda
@@ -16,7 +16,7 @@
 module Algebra.Construct.DirectProduct where
 
 open import Algebra
-open import Data.Product
+open import Data.Product.Base using (_×_; zip; _,_; map; _<*>_; uncurry)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Level using (Level; _⊔_)
 
diff --git a/src/Algebra/Construct/LexProduct.agda b/src/Algebra/Construct/LexProduct.agda
index 1301ff7c04..984cdd57f2 100644
--- a/src/Algebra/Construct/LexProduct.agda
+++ b/src/Algebra/Construct/LexProduct.agda
@@ -8,7 +8,7 @@
 
 open import Algebra
 open import Data.Bool using (true; false)
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (Pointwise)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (_∘_)
diff --git a/src/Algebra/Lattice/Construct/DirectProduct.agda b/src/Algebra/Lattice/Construct/DirectProduct.agda
index 9e2a825c99..0c68071a51 100644
--- a/src/Algebra/Lattice/Construct/DirectProduct.agda
+++ b/src/Algebra/Lattice/Construct/DirectProduct.agda
@@ -16,7 +16,7 @@
 open import Algebra
 open import Algebra.Lattice
 import Algebra.Construct.DirectProduct as DirectProduct
-open import Data.Product
+open import Data.Product.Base using (_,_; _<*>_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Level using (Level; _⊔_)
 
diff --git a/src/Algebra/Module/Construct/DirectProduct.agda b/src/Algebra/Module/Construct/DirectProduct.agda
index 3c168564bd..840cc42cab 100644
--- a/src/Algebra/Module/Construct/DirectProduct.agda
+++ b/src/Algebra/Module/Construct/DirectProduct.agda
@@ -14,7 +14,7 @@ module Algebra.Module.Construct.DirectProduct where
 open import Algebra.Bundles
 open import Algebra.Construct.DirectProduct
 open import Algebra.Module.Bundles
-open import Data.Product
+open import Data.Product.Base using (map; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Level
 
diff --git a/src/Algebra/Module/Definitions/Left.agda b/src/Algebra/Module/Definitions/Left.agda
index e43f2ef38e..3c04e82cd0 100644
--- a/src/Algebra/Module/Definitions/Left.agda
+++ b/src/Algebra/Module/Definitions/Left.agda
@@ -15,8 +15,6 @@ module Algebra.Module.Definitions.Left
   {a b ℓb} (A : Set a) {B : Set b} (_≈_ : Rel B ℓb)
   where
 
-open import Data.Product
-
 ------------------------------------------------------------------------
 -- Binary operations
 
diff --git a/src/Algebra/Module/Definitions/Right.agda b/src/Algebra/Module/Definitions/Right.agda
index 875dcfc594..92b90a477c 100644
--- a/src/Algebra/Module/Definitions/Right.agda
+++ b/src/Algebra/Module/Definitions/Right.agda
@@ -15,8 +15,6 @@ module Algebra.Module.Definitions.Right
   {a b ℓb} (A : Set a) {B : Set b} (_≈_ : Rel B ℓb)
   where
 
-open import Data.Product
-
 ------------------------------------------------------------------------
 -- Binary operations
 
diff --git a/src/Algebra/Module/Morphism/ModuleHomomorphism.agda b/src/Algebra/Module/Morphism/ModuleHomomorphism.agda
index 47ae46de10..704ec3863c 100644
--- a/src/Algebra/Module/Morphism/ModuleHomomorphism.agda
+++ b/src/Algebra/Module/Morphism/ModuleHomomorphism.agda
@@ -25,7 +25,7 @@ module Algebra.Module.Morphism.ModuleHomomorphism
   where
 
 open import Axiom.DoubleNegationElimination
-open import Data.Product
+open import Data.Product.Base using (∃₂; _×_; _,_)
 open import Relation.Binary.Reasoning.MultiSetoid
 open import Relation.Nullary          using (¬_)
 open import Relation.Nullary.Negation using (contraposition)
diff --git a/src/Algebra/Morphism/Construct/Composition.agda b/src/Algebra/Morphism/Construct/Composition.agda
index e3f3a3ad71..5087de2ae3 100644
--- a/src/Algebra/Morphism/Construct/Composition.agda
+++ b/src/Algebra/Morphism/Construct/Composition.agda
@@ -10,7 +10,6 @@ module Algebra.Morphism.Construct.Composition where
 
 open import Algebra.Bundles
 open import Algebra.Morphism.Structures
-open import Data.Product
 open import Function.Base using (_∘_)
 import Function.Construct.Composition as Func
 open import Level using (Level)
diff --git a/src/Algebra/Morphism/GroupMonomorphism.agda b/src/Algebra/Morphism/GroupMonomorphism.agda
index b7a09ce92c..b862559938 100644
--- a/src/Algebra/Morphism/GroupMonomorphism.agda
+++ b/src/Algebra/Morphism/GroupMonomorphism.agda
@@ -26,7 +26,7 @@ open RawGroup G₂ renaming
 
 open import Algebra.Definitions
 open import Algebra.Structures
-open import Data.Product
+open import Data.Product.Base using (_,_)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Morphism/MagmaMonomorphism.agda b/src/Algebra/Morphism/MagmaMonomorphism.agda
index c952fe112e..30f19edb02 100644
--- a/src/Algebra/Morphism/MagmaMonomorphism.agda
+++ b/src/Algebra/Morphism/MagmaMonomorphism.agda
@@ -25,7 +25,7 @@ open RawMagma M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_)
 
 open import Algebra.Structures
 open import Algebra.Definitions
-open import Data.Product
+open import Data.Product.Base using (map)
 open import Data.Sum.Base using (inj₁; inj₂)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 import Relation.Binary.Morphism.RelMonomorphism isRelMonomorphism as RelMorphism
diff --git a/src/Algebra/Morphism/RingMonomorphism.agda b/src/Algebra/Morphism/RingMonomorphism.agda
index f9789e60b7..4b2c900946 100644
--- a/src/Algebra/Morphism/RingMonomorphism.agda
+++ b/src/Algebra/Morphism/RingMonomorphism.agda
@@ -28,7 +28,7 @@ open RawRing R₂ renaming
 
 open import Algebra.Definitions
 open import Algebra.Structures
-open import Data.Product
+open import Data.Product.Base using (proj₁; proj₂; _,_)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Properties/Loop.agda b/src/Algebra/Properties/Loop.agda
index 26bf6f4ce6..1ecb2c52f4 100644
--- a/src/Algebra/Properties/Loop.agda
+++ b/src/Algebra/Properties/Loop.agda
@@ -13,7 +13,6 @@ module Algebra.Properties.Loop {l₁ l₂} (L : Loop l₁ l₂) where
 open Loop L
 open import Algebra.Definitions _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Data.Product
 open import Algebra.Properties.Quasigroup
 
 x//x≈ε : ∀ x → x // x ≈ ε
diff --git a/src/Algebra/Properties/Magma/Divisibility.agda b/src/Algebra/Properties/Magma/Divisibility.agda
index 72907929df..d8ffc208f0 100644
--- a/src/Algebra/Properties/Magma/Divisibility.agda
+++ b/src/Algebra/Properties/Magma/Divisibility.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra using (Magma)
-open import Data.Product using (_×_; _,_; ∃; map; swap)
+open import Data.Product.Base using (_×_; _,_; ∃; map; swap)
 open import Relation.Binary.Definitions
 
 module Algebra.Properties.Magma.Divisibility {a ℓ} (M : Magma a ℓ) where
diff --git a/src/Algebra/Properties/MiddleBolLoop.agda b/src/Algebra/Properties/MiddleBolLoop.agda
index a50ce3b0d5..60f2898496 100644
--- a/src/Algebra/Properties/MiddleBolLoop.agda
+++ b/src/Algebra/Properties/MiddleBolLoop.agda
@@ -13,7 +13,6 @@ module Algebra.Properties.MiddleBolLoop {m₁ m₂} (M : MiddleBolLoop m₁ m₂
 open MiddleBolLoop M
 open import Algebra.Definitions _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Data.Product
 import Algebra.Properties.Loop as LoopProperties
 open LoopProperties loop public
 
diff --git a/src/Algebra/Properties/MoufangLoop.agda b/src/Algebra/Properties/MoufangLoop.agda
index 4740bc19b6..9b0e5ac94b 100644
--- a/src/Algebra/Properties/MoufangLoop.agda
+++ b/src/Algebra/Properties/MoufangLoop.agda
@@ -13,7 +13,7 @@ module Algebra.Properties.MoufangLoop {a ℓ} (M : MoufangLoop a ℓ) where
 open MoufangLoop M
 open import Algebra.Definitions _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Data.Product
+open import Data.Product.Base using (_,_)
 
 alternativeˡ : LeftAlternative _∙_
 alternativeˡ x y = begin
diff --git a/src/Algebra/Properties/Quasigroup.agda b/src/Algebra/Properties/Quasigroup.agda
index a583ae34f5..335a9ea7a3 100644
--- a/src/Algebra/Properties/Quasigroup.agda
+++ b/src/Algebra/Properties/Quasigroup.agda
@@ -13,7 +13,7 @@ module Algebra.Properties.Quasigroup {q₁ q₂} (Q : Quasigroup q₁ q₂) wher
 open Quasigroup Q
 open import Algebra.Definitions _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Data.Product
+open import Data.Product.Base using (_,_)
 
 cancelˡ : LeftCancellative _∙_
 cancelˡ x y z eq = begin
diff --git a/src/Algebra/Solver/Monoid.agda b/src/Algebra/Solver/Monoid.agda
index 23e994821c..f80a2847f6 100644
--- a/src/Algebra/Solver/Monoid.agda
+++ b/src/Algebra/Solver/Monoid.agda
@@ -17,7 +17,7 @@ import Data.List.Relation.Binary.Equality.DecPropositional as ListEq
 open import Data.Maybe.Base as Maybe
   using (Maybe; decToMaybe; From-just; from-just)
 open import Data.Nat.Base using (ℕ)
-open import Data.Product
+open import Data.Product.Base using (_×_; uncurry)
 open import Data.Vec.Base using (Vec; lookup)
 open import Function.Base using (_∘_; _$_)
 open import Relation.Binary.Definitions using (Decidable)
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index d70a101f6a..3ef672abc7 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -21,7 +21,7 @@ open import Data.Maybe.Relation.Unary.Any using (just)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.List.Base using (List; []; _∷_)
 open import Data.List.NonEmpty using (List⁺; _∷_)
-open import Data.Product as Prod using (∃; _×_; _,_)
+open import Data.Product.Base as Prod using (∃; _×_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Vec.Bounded as Vec≤ using (Vec≤)
 open import Function.Base
diff --git a/src/Codata/Musical/Colist/Infinite-merge.agda b/src/Codata/Musical/Colist/Infinite-merge.agda
index 677024808a..eb06c3b9cc 100644
--- a/src/Codata/Musical/Colist/Infinite-merge.agda
+++ b/src/Codata/Musical/Colist/Infinite-merge.agda
@@ -13,7 +13,7 @@ open import Codata.Musical.Colist as Colist hiding (_⋎_)
 open import Data.Nat.Base
 open import Data.Nat.Induction using (<′-wellFounded)
 open import Data.Nat.Properties
-open import Data.Product as Prod
+open import Data.Product.Base as Prod using (_×_; _,_; ∃; ∃₂; proj₁; proj₂)
 open import Data.Sum.Base
 open import Data.Sum.Properties
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
diff --git a/src/Codata/Musical/Conversion.agda b/src/Codata/Musical/Conversion.agda
index d7213daba9..4e5886e000 100644
--- a/src/Codata/Musical/Conversion.agda
+++ b/src/Codata/Musical/Conversion.agda
@@ -27,7 +27,7 @@ open import Codata.Musical.Covec hiding (module Covec)
 open import Codata.Musical.M hiding (module M)
 open import Codata.Musical.Notation
 open import Codata.Musical.Stream hiding (module Stream)
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Data.Container.Core as C using (Container)
 import Size
 
diff --git a/src/Codata/Musical/M/Indexed.agda b/src/Codata/Musical/M/Indexed.agda
index 964aca0695..3eefda55e6 100644
--- a/src/Codata/Musical/M/Indexed.agda
+++ b/src/Codata/Musical/M/Indexed.agda
@@ -11,7 +11,7 @@ module Codata.Musical.M.Indexed where
 
 open import Level
 open import Codata.Musical.Notation
-open import Data.Product
+open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Data.Container.Indexed.Core
 open import Function.Base using (_∘_)
 open import Relation.Unary
diff --git a/src/Data/Container/Combinator.agda b/src/Data/Container/Combinator.agda
index c68536427b..01e102cd5c 100644
--- a/src/Data/Container/Combinator.agda
+++ b/src/Data/Container/Combinator.agda
@@ -10,7 +10,7 @@ module Data.Container.Combinator where
 
 open import Level using (Level; _⊔_; lower)
 open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
-open import Data.Product as P using (_,_; <_,_>; proj₁; proj₂; ∃)
+open import Data.Product.Base as P using (_,_; <_,_>; proj₁; proj₂; ∃)
 open import Data.Sum.Base as S using ([_,_]′)
 open import Data.Unit.Polymorphic.Base using (⊤)
 import Function.Base as F
diff --git a/src/Data/Container/Combinator/Properties.agda b/src/Data/Container/Combinator/Properties.agda
index b93d6c7a7a..8d347abcae 100644
--- a/src/Data/Container/Combinator/Properties.agda
+++ b/src/Data/Container/Combinator/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Container.Core
 open import Data.Container.Combinator
 open import Data.Container.Relation.Unary.Any
 open import Data.Empty using (⊥-elim)
-open import Data.Product as Prod using (∃; _,_; proj₁; proj₂; <_,_>; uncurry; curry)
+open import Data.Product.Base as Prod using (∃; _,_; proj₁; proj₂; <_,_>; uncurry; curry)
 open import Data.Sum.Base as S using (inj₁; inj₂; [_,_]′; [_,_])
 open import Function.Base as F using (_∘′_)
 open import Function.Inverse as Inv using (_↔_; inverse; module Inverse)
diff --git a/src/Data/Container/Indexed/Combinator.agda b/src/Data/Container/Indexed/Combinator.agda
index 43721d4689..283a6a0314 100644
--- a/src/Data/Container/Indexed/Combinator.agda
+++ b/src/Data/Container/Indexed/Combinator.agda
@@ -12,7 +12,7 @@ open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Container.Indexed
 open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
 open import Data.Unit.Polymorphic.Base using (⊤)
-open import Data.Product as Prod hiding (Σ) renaming (_×_ to _⟨×⟩_)
+open import Data.Product.Base as Prod hiding (Σ) renaming (_×_ to _⟨×⟩_)
 open import Data.Sum.Base renaming (_⊎_ to _⟨⊎⟩_)
 open import Data.Sum.Relation.Unary.All as All using (All)
 open import Function.Base as F hiding (id; const) renaming (_∘_ to _⟨∘⟩_)
diff --git a/src/Data/Container/Indexed/Core.agda b/src/Data/Container/Indexed/Core.agda
index 91fb4b2e21..75d1a0d986 100644
--- a/src/Data/Container/Indexed/Core.agda
+++ b/src/Data/Container/Indexed/Core.agda
@@ -9,7 +9,7 @@
 module Data.Container.Indexed.Core where
 
 open import Level
-open import Data.Product
+open import Data.Product.Base using (Σ; Σ-syntax; _,_; ∃)
 open import Relation.Unary
 
 infix 5 _◃_/_
diff --git a/src/Data/Container/Indexed/FreeMonad.agda b/src/Data/Container/Indexed/FreeMonad.agda
index 1c6ee81c1a..294ab0bd34 100644
--- a/src/Data/Container/Indexed/FreeMonad.agda
+++ b/src/Data/Container/Indexed/FreeMonad.agda
@@ -15,7 +15,7 @@ open import Data.Container.Indexed
 open import Data.Container.Indexed.Combinator hiding (id; _∘_)
 open import Data.Empty
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Data.W.Indexed
 open import Relation.Unary
 open import Relation.Unary.PredicateTransformer
diff --git a/src/Data/Container/Indexed/WithK.agda b/src/Data/Container/Indexed/WithK.agda
index ca5f18205d..9897402264 100644
--- a/src/Data/Container/Indexed/WithK.agda
+++ b/src/Data/Container/Indexed/WithK.agda
@@ -15,7 +15,8 @@ module Data.Container.Indexed.WithK where
 
 open import Axiom.Extensionality.Heterogeneous using (Extensionality)
 open import Data.Container.Indexed hiding (module PlainMorphism)
-open import Data.Product as Prod hiding (map)
+open import Data.Product.Base
+  using (_,_; -,_; _×_; ∃; proj₁; proj₂; Σ-syntax)
 open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
 open import Level
 open import Relation.Unary using (Pred; _⊆_)
diff --git a/src/Data/Container/Morphism/Properties.agda b/src/Data/Container/Morphism/Properties.agda
index fc71843263..c0d84007b4 100644
--- a/src/Data/Container/Morphism/Properties.agda
+++ b/src/Data/Container/Morphism/Properties.agda
@@ -10,7 +10,7 @@ module Data.Container.Morphism.Properties where
 
 open import Level using (_⊔_; suc)
 open import Function.Base as F using (_$_)
-open import Data.Product using (∃; proj₁; proj₂; _,_)
+open import Data.Product.Base using (∃; proj₁; proj₂; _,_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
 
diff --git a/src/Data/Container/Relation/Unary/All.agda b/src/Data/Container/Relation/Unary/All.agda
index 65d593eea4..829bd3a0c9 100644
--- a/src/Data/Container/Relation/Unary/All.agda
+++ b/src/Data/Container/Relation/Unary/All.agda
@@ -10,7 +10,7 @@ module Data.Container.Relation.Unary.All where
 
 open import Level using (_⊔_)
 open import Relation.Unary using (Pred; _⊆_)
-open import Data.Product as Prod using (_,_; proj₁; proj₂; ∃)
+open import Data.Product.Base using (_,_; proj₁; proj₂; ∃)
 open import Function.Base using (_∘′_; id)
 
 open import Data.Container.Core hiding (map)
diff --git a/src/Data/Container/Relation/Unary/Any.agda b/src/Data/Container/Relation/Unary/Any.agda
index de3937f221..121391f4d6 100644
--- a/src/Data/Container/Relation/Unary/Any.agda
+++ b/src/Data/Container/Relation/Unary/Any.agda
@@ -10,7 +10,7 @@ module Data.Container.Relation.Unary.Any where
 
 open import Level using (_⊔_)
 open import Relation.Unary using (Pred; _⊆_)
-open import Data.Product as Prod using (_,_; proj₂; ∃)
+open import Data.Product.Base using (_,_; proj₂; ∃)
 open import Function.Base using (_∘′_; id)
 
 open import Data.Container.Core hiding (map)
diff --git a/src/Data/Container/Relation/Unary/Any/Properties.agda b/src/Data/Container/Relation/Unary/Any/Properties.agda
index ceae1510c9..d1d35c5a02 100644
--- a/src/Data/Container/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Container/Relation/Unary/Any/Properties.agda
@@ -10,7 +10,7 @@ module Data.Container.Relation.Unary.Any.Properties where
 
 open import Level
 open import Algebra
-open import Data.Product as Prod using (∃; _×_; ∃₂; _,_; proj₂)
+open import Data.Product.Base using (∃; _×_; ∃₂; _,_; proj₂)
 open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
diff --git a/src/Data/Fin/Subset.agda b/src/Data/Fin/Subset.agda
index 01926df438..b3d53106ff 100644
--- a/src/Data/Fin/Subset.agda
+++ b/src/Data/Fin/Subset.agda
@@ -13,7 +13,7 @@ open import Data.Bool using (not; _∧_; _∨_; _≟_)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base using (List; foldr; foldl)
 open import Data.Nat.Base using (ℕ)
-open import Data.Product using (∃; _×_)
+open import Data.Product.Base using (∃; _×_)
 open import Data.Vec.Base hiding (foldr; foldl)
 open import Relation.Nullary
 
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index 586b28e278..d40d078bd3 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -20,7 +20,7 @@ open import Data.Fin as Fin
   using (Fin; Fin′; zero; suc; #_; toℕ; _≟_; opposite) renaming (_ℕ-ℕ_ to _-_)
 import Data.Fin.Properties as FP
 import Data.Fin.Permutation.Components as PC
-open import Data.Product as Prod using (∃; _×_; _,_)
+open import Data.Product.Base as Prod using (∃; _×_; _,_)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; decToMaybe)
 open import Data.Empty
 open import Data.Unit.Base using (⊤; tt)
diff --git a/src/Data/Integer/GCD.agda b/src/Data/Integer/GCD.agda
index e087da4763..3f46877f45 100644
--- a/src/Data/Integer/GCD.agda
+++ b/src/Data/Integer/GCD.agda
@@ -13,7 +13,7 @@ open import Data.Integer.Divisibility
 open import Data.Integer.Properties
 open import Data.Nat.Base
 import Data.Nat.GCD as ℕ
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Relation.Binary.PropositionalEquality
 
 open import Algebra.Definitions {A = ℤ} _≡_ as Algebra
diff --git a/src/Data/List/Countdown.agda b/src/Data/List/Countdown.agda
index f1ef43ba8e..db8dc806ad 100644
--- a/src/Data/List/Countdown.agda
+++ b/src/Data/List/Countdown.agda
@@ -23,7 +23,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.List.Base hiding (lookup)
 open import Data.List.Relation.Unary.Any as Any using (here; there)
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Data.Product
+open import Data.Product.Base using (∃; _,_; _×_)
 open import Data.Sum.Base
 open import Data.Sum.Properties
 open import Relation.Nullary.Reflects using (invert)
diff --git a/src/Data/List/Fresh/Membership/Setoid/Properties.agda b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
index e287147516..d2f2634bf8 100644
--- a/src/Data/List/Fresh/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
@@ -15,7 +15,7 @@ open import Level using (Level; _⊔_)
 open import Data.Empty
 open import Data.Nat.Base
 open import Data.Nat.Properties
-open import Data.Product using (∃; _×_; _,_)
+open import Data.Product.Base using (∃; _×_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; fromInj₂)
 
 open import Function.Base using (id; _∘′_; _$_)
diff --git a/src/Data/List/Fresh/Relation/Unary/Any.agda b/src/Data/List/Fresh/Relation/Unary/Any.agda
index 9c2050472c..dbb9dd0dec 100644
--- a/src/Data/List/Fresh/Relation/Unary/Any.agda
+++ b/src/Data/List/Fresh/Relation/Unary/Any.agda
@@ -10,7 +10,7 @@ module Data.List.Fresh.Relation.Unary.Any where
 
 open import Level using (Level; _⊔_; Lift)
 open import Data.Empty
-open import Data.Product using (∃; _,_; -,_)
+open import Data.Product.Base using (∃; _,_; -,_)
 open import Data.Sum.Base using (_⊎_; [_,_]′; inj₁; inj₂)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index f781e1678d..dfef9fa160 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -22,7 +22,7 @@ open import Data.List.Relation.Binary.Equality.Propositional
 open import Data.List.Effectful using (monad)
 open import Data.Nat.Base using (ℕ; zero; suc; pred; s≤s; _≤_; _<_; _≤ᵇ_)
 open import Data.Nat.Properties
-open import Data.Product hiding (map)
+open import Data.Product.Base hiding (map)
 open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
diff --git a/src/Data/List/Membership/Propositional/Properties/Core.agda b/src/Data/List/Membership/Propositional/Properties/Core.agda
index 85ad85aab0..aec90ef8d1 100644
--- a/src/Data/List/Membership/Propositional/Properties/Core.agda
+++ b/src/Data/List/Membership/Propositional/Properties/Core.agda
@@ -17,7 +17,7 @@ open import Function.Inverse using (_↔_; inverse)
 open import Data.List.Base using (List)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
   using (_,_; proj₁; proj₂; uncurry′; ∃; _×_)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core as P
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 47768eed76..2fee07309f 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -20,7 +20,7 @@ import Data.List.Relation.Binary.Equality.Setoid as Equality
 import Data.List.Relation.Unary.Unique.Setoid as Unique
 open import Data.Nat.Base using (suc; z≤n; s≤s; _≤_; _<_)
 open import Data.Nat.Properties using (≤-trans; n≤1+n)
-open import Data.Product as Prod using (∃; _×_; _,_ ; ∃₂; proj₁; proj₂)
+open import Data.Product.Base as Prod using (∃; _×_; _,_ ; ∃₂; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (_$_; flip; _∘_; _∘′_; id)
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index ac88bd72ef..d48b3dae45 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -23,7 +23,7 @@ open import Data.List.Relation.Binary.Subset.Propositional.Properties
   using (⊆-preorder)
 open import Data.List.Relation.Binary.Permutation.Propositional
 open import Data.List.Relation.Binary.Permutation.Propositional.Properties
-open import Data.Product as Prod hiding (map)
+open import Data.Product.Base as Prod hiding (map)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base as Sum hiding (map)
 open import Data.Sum.Properties hiding (map-cong)
diff --git a/src/Data/List/Relation/Binary/Lex/NonStrict.agda b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
index 4606c9cdbd..e87e1e0b45 100644
--- a/src/Data/List/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
@@ -14,7 +14,6 @@ module Data.List.Relation.Binary.Lex.NonStrict where
 open import Data.Empty using (⊥)
 open import Function.Base
 open import Data.Unit.Base using (⊤; tt)
-open import Data.Product
 open import Data.List.Base
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; [])
 import Data.List.Relation.Binary.Lex.Strict as Strict
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index dc7e34db86..42430b90ea 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -21,7 +21,7 @@ open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Membership.Propositional
 open import Data.List.Membership.Propositional.Properties
 import Data.List.Properties as Lₚ
-open import Data.Product using (_,_; _×_; ∃; ∃₂)
+open import Data.Product.Base using (_,_; _×_; ∃; ∃₂)
 open import Function.Base using (_∘_; _⟨_⟩_)
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse as Inv using (inverse)
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index e7d2b4039c..57233a7b2a 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -32,7 +32,7 @@ import Data.List.Properties as Lₚ
 open import Data.Nat hiding (_⊔_)
 open import Data.Nat.Induction
 open import Data.Nat.Properties
-open import Data.Product using (_,_; _×_; ∃; ∃₂; proj₁; proj₂)
+open import Data.Product.Base using (_,_; _×_; ∃; ∃₂; proj₁; proj₂)
 open import Function.Base using (_∘_; _⟨_⟩_; flip)
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse as Inv using (inverse)
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
index 39d4439a86..75104bd39e 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
@@ -12,7 +12,7 @@ open import Level
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise
   using (Pointwise; []; _∷_)
-open import Data.Product using (∃; _×_; _,_; uncurry)
+open import Data.Product.Base using (∃; _×_; _,_; uncurry)
 open import Relation.Binary.Core using (REL; _⇒_)
 
 module _ {a b r} {A : Set a} {B : Set b} (R : REL A B r) where
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index 7e7a3a090d..46d5d7d2cc 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -24,7 +24,7 @@ open import Data.List.Relation.Binary.Sublist.Heterogeneous
 open import Data.Maybe.Relation.Unary.All as MAll using (nothing; just)
 open import Data.Nat.Base using (ℕ; _≤_; _≥_); open ℕ; open _≤_
 import Data.Nat.Properties as ℕₚ
-open import Data.Product using (∃₂; _×_; _,_; <_,_>; proj₂; uncurry)
+open import Data.Product.Base using (∃₂; _×_; _,_; <_,_>; proj₂; uncurry)
 
 open import Function.Base
 open import Function.Bundles using (_⤖_; _⇔_ ; mk⤖; mk⇔)
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index a7afe038c7..e7b95805be 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -23,7 +23,7 @@ open import Level using (_⊔_)
 open import Data.Fin as Fin
 open import Data.Maybe.Base as M
 open import Data.Nat.Base as Nat using (ℕ)
-open import Data.Product
+open import Data.Product.Base using (Σ-syntax; _,_)
 open import Data.Vec.Base as Vec using (Vec ; lookup)
 open import Data.List.Base hiding (lookup)
 open import Data.List.Properties
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
index 844dd1c996..c9d2ea6724 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
@@ -19,7 +19,7 @@ open import Data.List.Relation.Binary.Sublist.Propositional
   hiding (map)
 import Data.List.Relation.Binary.Sublist.Setoid.Properties
   as SetoidProperties
-open import Data.Product using (∃; _,_; proj₂)
+open import Data.Product.Base using (∃; _,_; proj₂)
 open import Function.Base
 open import Level using (Level)
 open import Relation.Binary.Definitions using (_Respects_)
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid.agda b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
index eff26de0aa..a07c6ffc25 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
@@ -24,7 +24,7 @@ import Data.List.Relation.Binary.Sublist.Heterogeneous.Core
   as HeterogeneousCore
 import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
   as HeterogeneousProperties
-open import Data.Product using (∃; ∃₂; _×_; _,_; proj₂)
+open import Data.Product.Base using (∃; ∃₂; _×_; _,_; proj₂)
 
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
index df904a0452..19fe765456 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
@@ -17,7 +17,7 @@ open import Data.List.Relation.Unary.Any using (Any)
 import Data.Maybe.Relation.Unary.All as Maybe
 open import Data.Nat.Base using (_≤_; _≥_)
 import Data.Nat.Properties as ℕₚ
-open import Data.Product using (∃; _,_; proj₂)
+open import Data.Product.Base using (∃; _,_; proj₂)
 open import Function.Base
 open import Function.Bundles using (_⇔_; _⤖_)
 open import Level
diff --git a/src/Data/List/Relation/Ternary/Appending.agda b/src/Data/List/Relation/Ternary/Appending.agda
index 1ae20e905f..1a2bab30e1 100644
--- a/src/Data/List/Relation/Ternary/Appending.agda
+++ b/src/Data/List/Relation/Ternary/Appending.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Ternary.Appending {a b c} {A : Set a} {B : Set b} {C :
 open import Level using (Level; _⊔_)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
-open import Data.Product as Prod using (∃₂; _×_; _,_; -,_)
+open import Data.Product.Base using (∃₂; _×_; _,_; -,_)
 open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
diff --git a/src/Data/List/Relation/Ternary/Interleaving.agda b/src/Data/List/Relation/Ternary/Interleaving.agda
index 158701fc7e..8818638518 100644
--- a/src/Data/List/Relation/Ternary/Interleaving.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving.agda
@@ -12,7 +12,7 @@ module Data.List.Relation.Ternary.Interleaving where
 open import Level
 open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
-open import Data.Product as Prod using (∃; ∃₂; _×_; uncurry; _,_; -,_; proj₂)
+open import Data.Product.Base as Prod using (∃; ∃₂; _×_; uncurry; _,_; -,_; proj₂)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
 open import Relation.Binary
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index 6a0020d269..8f3bcd640c 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -25,7 +25,7 @@ open import Data.Nat using (zero; suc; _<_; z<s; s<s; s≤s)
 open import Data.Nat.Properties using (_≟_; ≤∧≢⇒<; ≤-refl; m<n⇒m<1+n)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.Any as MAny using (just)
-open import Data.Product as Prod
+open import Data.Product.Base as Prod
   using (_×_; _,_; ∃; ∃₂; proj₁; proj₂; uncurry′)
 open import Data.Product.Properties
 open import Data.Product.Function.NonDependent.Propositional
diff --git a/src/Data/List/Relation/Unary/First.agda b/src/Data/List/Relation/Unary/First.agda
index bfe8c761b0..25eb9fc37a 100644
--- a/src/Data/List/Relation/Unary/First.agda
+++ b/src/Data/List/Relation/Unary/First.agda
@@ -15,7 +15,7 @@ open import Data.Fin.Base as Fin using (Fin; zero; suc)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
-open import Data.Product as Prod using (∃; -,_; _,_)
+open import Data.Product.Base using (∃; -,_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base using (id; _∘′_)
 open import Relation.Unary
diff --git a/src/Data/Maybe/Relation/Binary/Connected.agda b/src/Data/Maybe/Relation/Binary/Connected.agda
index 33bff58154..5c509e50fd 100644
--- a/src/Data/Maybe/Relation/Binary/Connected.agda
+++ b/src/Data/Maybe/Relation/Binary/Connected.agda
@@ -9,7 +9,6 @@
 module Data.Maybe.Relation.Binary.Connected where
 
 open import Level
-open import Data.Product
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Binary hiding (_⇔_)
diff --git a/src/Data/Maybe/Relation/Binary/Pointwise.agda b/src/Data/Maybe/Relation/Binary/Pointwise.agda
index 3ae5ad421e..a9946a6ef3 100644
--- a/src/Data/Maybe/Relation/Binary/Pointwise.agda
+++ b/src/Data/Maybe/Relation/Binary/Pointwise.agda
@@ -9,7 +9,7 @@
 module Data.Maybe.Relation.Binary.Pointwise where
 
 open import Level
-open import Data.Product
+open import Data.Product.Base using (∃; _×_; -,_; _,_)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Binary hiding (_⇔_)
diff --git a/src/Data/Maybe/Relation/Unary/Any.agda b/src/Data/Maybe/Relation/Unary/Any.agda
index ea81316c77..48cd402ce0 100644
--- a/src/Data/Maybe/Relation/Unary/Any.agda
+++ b/src/Data/Maybe/Relation/Unary/Any.agda
@@ -9,7 +9,7 @@
 module Data.Maybe.Relation.Unary.Any where
 
 open import Data.Maybe.Base using (Maybe; just; nothing)
-open import Data.Product as Prod using (∃; _,_; -,_)
+open import Data.Product.Base as Prod using (∃; _,_; -,_)
 open import Function.Base using (id)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 2294a08968..43496f3139 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -21,7 +21,7 @@ open import Data.Nat.Divisibility using (∣-refl)
 import Data.Nat.Base as ℕᵇ
 import Data.Nat.Properties as ℕₚ
 open import Data.Nat.Solver
-open import Data.Product using (_×_; _,_; proj₁; proj₂; ∃)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _$_; id)
 open import Function.Definitions
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index 5dc7c8951f..48eb4b4ae7 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -16,7 +16,7 @@ open import Data.Nat as ℕ using (ℕ)
 open import Data.Nat.Binary.Base
 open import Data.Nat.Binary.Properties
 import Data.Nat.Properties as ℕₚ
-open import Data.Product using (_×_; _,_; proj₁; proj₂; ∃)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Vec.Base using ([]; _∷_)
 open import Function.Base using (_∘_; _$_)
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index e702c512a6..f54b09e47f 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -15,7 +15,8 @@ open import Data.Nat.GCD.Lemmas
 open import Data.Nat.Properties
 open import Data.Nat.Induction
   using (Acc; acc; <′-Rec; <′-recBuilder; <-wellFounded-fast)
-open import Data.Product
+open import Data.Product.Base
+  using (_×_; _,_; proj₂; proj₁; swap; uncurry′; ∃; map)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_$_; _∘_)
 open import Induction using (build)
diff --git a/src/Data/Nat/InfinitelyOften.agda b/src/Data/Nat/InfinitelyOften.agda
index 9e89841633..f85ceb196a 100644
--- a/src/Data/Nat/InfinitelyOften.agda
+++ b/src/Data/Nat/InfinitelyOften.agda
@@ -13,7 +13,7 @@ open import Level using (Level; 0ℓ)
 open import Data.Empty using (⊥-elim)
 open import Data.Nat.Base
 open import Data.Nat.Properties
-open import Data.Product as Prod hiding (map)
+open import Data.Product.Base as Prod hiding (map)
 open import Data.Sum.Base using (inj₁; inj₂; _⊎_)
 open import Function.Base using (_∘_; id)
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/Nat/LCM.agda b/src/Data/Nat/LCM.agda
index af85ed64ad..ccb1fbbf5b 100644
--- a/src/Data/Nat/LCM.agda
+++ b/src/Data/Nat/LCM.agda
@@ -15,7 +15,7 @@ open import Data.Nat.Divisibility
 open import Data.Nat.DivMod
 open import Data.Nat.Properties
 open import Data.Nat.GCD
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_; uncurry′; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl; sym; trans; cong; cong₂; module ≡-Reasoning)
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index 6ed6997c0b..8e72b32be0 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -13,7 +13,7 @@ open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.GCD using (module GCD; module Bézout)
 open import Data.Nat.Properties
-open import Data.Product
+open import Data.Product.Base using (∃; _×_; map₂; _,_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (flip; _∘_; _∘′_)
 open import Relation.Nullary.Decidable as Dec
diff --git a/src/Data/Product/Effectful/Examples.agda b/src/Data/Product/Effectful/Examples.agda
index f11631341e..b4cc86618e 100644
--- a/src/Data/Product/Effectful/Examples.agda
+++ b/src/Data/Product/Effectful/Examples.agda
@@ -14,7 +14,7 @@ module Data.Product.Effectful.Examples
 open import Level using (Lift; lift; _⊔_)
 open import Effect.Functor using (RawFunctor)
 open import Effect.Monad using (RawMonad)
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function.Base using (id)
 import Function.Identity.Effectful as Id
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index d062e33d39..abae430992 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.Dependent.Propositional where
 
-open import Data.Product
+open import Data.Product.Base using (Σ; map; _,_)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Relation.Binary hiding (_⇔_)
diff --git a/src/Data/Product/Function/Dependent/Propositional/WithK.agda b/src/Data/Product/Function/Dependent/Propositional/WithK.agda
index 0b3838e12f..df3bafe613 100644
--- a/src/Data/Product/Function/Dependent/Propositional/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Propositional/WithK.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.Dependent.Propositional.WithK where
 
-open import Data.Product
+open import Data.Product.Base using (Σ)
 open import Data.Product.Function.Dependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Data.Product.Relation.Binary.Pointwise.Dependent.WithK
diff --git a/src/Data/Product/Function/Dependent/Setoid.agda b/src/Data/Product/Function/Dependent/Setoid.agda
index cc6d4a12fc..22f3f7937a 100644
--- a/src/Data/Product/Function/Dependent/Setoid.agda
+++ b/src/Data/Product/Function/Dependent/Setoid.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.Dependent.Setoid where
 
-open import Data.Product
+open import Data.Product.Base using (map; _,_)
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Function.Base
 open import Function.Equality as F using (_⟶_; _⟨$⟩_)
diff --git a/src/Data/Product/Function/Dependent/Setoid/WithK.agda b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
index c9468fb939..7362b3ade0 100644
--- a/src/Data/Product/Function/Dependent/Setoid/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.Dependent.Setoid.WithK where
 
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Data.Product.Function.Dependent.Setoid using (surjection)
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Relation.Binary
diff --git a/src/Data/Product/Function/NonDependent/Propositional.agda b/src/Data/Product/Function/NonDependent/Propositional.agda
index e686cbad6e..b4c9501304 100644
--- a/src/Data/Product/Function/NonDependent/Propositional.agda
+++ b/src/Data/Product/Function/NonDependent/Propositional.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.NonDependent.Propositional where
 
-open import Data.Product
+open import Data.Product.Base using (_×_; map)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Relation.Binary hiding (_⇔_)
diff --git a/src/Data/Product/Function/NonDependent/Setoid.agda b/src/Data/Product/Function/NonDependent/Setoid.agda
index eb1a369710..93e388da23 100644
--- a/src/Data/Product/Function/NonDependent/Setoid.agda
+++ b/src/Data/Product/Function/NonDependent/Setoid.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.NonDependent.Setoid where
 
-open import Data.Product
+open import Data.Product.Base using (map; _,_; <_,_>; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Relation.Binary
 open import Function.Equality as F using (_⟶_; _⟨$⟩_)
diff --git a/src/Data/Product/Instances.agda b/src/Data/Product/Instances.agda
index 222a254b0b..93794c3c58 100644
--- a/src/Data/Product/Instances.agda
+++ b/src/Data/Product/Instances.agda
@@ -8,7 +8,7 @@
 
 module Data.Product.Instances where
 
-open import Data.Product
+open import Data.Product.Base
   using (Σ)
 open import Data.Product.Properties
 open import Level
diff --git a/src/Data/Product/Properties/WithK.agda b/src/Data/Product/Properties/WithK.agda
index 645ea6a6c4..329cdaf675 100644
--- a/src/Data/Product/Properties/WithK.agda
+++ b/src/Data/Product/Properties/WithK.agda
@@ -8,7 +8,7 @@
 
 module Data.Product.Properties.WithK where
 
-open import Data.Product
+open import Data.Product.Base using (Σ; _,_)
 open import Function.Base using (_∋_)
 open import Relation.Binary.PropositionalEquality
 
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
index 7fce761f79..d388b14f91 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Relation.Binary.Pointwise.Dependent.WithK where
 
-open import Data.Product
+open import Data.Product.Base using (Σ; uncurry)
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Function.Base
 open import Function.Inverse using (Inverse)
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index 2d23693e2d..1ff3b18d05 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -11,7 +11,7 @@ module Data.Sum.Relation.Binary.LeftOrder where
 open import Data.Sum.Base as Sum
 open import Data.Sum.Relation.Binary.Pointwise as PW
   using (Pointwise; inj₁; inj₂)
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Data.Empty
 open import Function.Base using (_$_; _∘_)
 open import Level
diff --git a/src/Data/These/Properties.agda b/src/Data/These/Properties.agda
index a6c96a3208..e7111cfce1 100644
--- a/src/Data/These/Properties.agda
+++ b/src/Data/These/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.These.Properties where
 
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_; <_,_>; uncurry)
 open import Data.These.Base
 open import Function.Base using (_∘_)
 open import Relation.Binary.Definitions using (Decidable)
diff --git a/src/Data/Tree/AVL/Indexed.agda b/src/Data/Tree/AVL/Indexed.agda
index c322d680e3..94d1b0782b 100644
--- a/src/Data/Tree/AVL/Indexed.agda
+++ b/src/Data/Tree/AVL/Indexed.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Indexed
 
 open import Level using (Level; _⊔_)
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
-open import Data.Product using (Σ; ∃; _×_; _,_; proj₁)
+open import Data.Product.Base using (Σ; ∃; _×_; _,_; proj₁)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.List.Base as List using (List)
 open import Data.DifferenceList using (DiffList; []; _∷_; _++_)
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda
index 2710a373b5..f8c270f1a1 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Indexed.Relation.Unary.Any
   where
 
 open import Data.Nat.Base using (ℕ)
-open import Data.Product using (_,_; ∃; -,_; proj₁; proj₂)
+open import Data.Product.Base using (_,_; ∃; -,_; proj₁; proj₂)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (_∘′_; _∘_)
 open import Level using (Level; _⊔_)
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
index 9f4aa1aeca..02a09a098e 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
@@ -16,7 +16,7 @@ open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′)
 open import Data.Maybe.Properties using (just-injective)
 open import Data.Maybe.Relation.Unary.All as Maybe using (nothing; just)
 open import Data.Nat.Base using (ℕ)
-open import Data.Product as Prod using (∃; ∃-syntax; _×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Prod using (∃; ∃-syntax; _×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base as F
 open import Level using (Level)
diff --git a/src/Data/Tree/AVL/Indexed/WithK.agda b/src/Data/Tree/AVL/Indexed/WithK.agda
index 7467a3813f..169427e986 100644
--- a/src/Data/Tree/AVL/Indexed/WithK.agda
+++ b/src/Data/Tree/AVL/Indexed/WithK.agda
@@ -15,7 +15,6 @@ module Data.Tree.AVL.Indexed.WithK
 
 strictTotalOrder = record { isStrictTotalOrder = isStrictTotalOrder }
 
-open import Data.Product
 open import Data.Tree.AVL.Indexed strictTotalOrder as AVL hiding (Value)
 
 module _ {v} {V′ : Key → Set v} where
diff --git a/src/Data/Tree/AVL/IndexedMap.agda b/src/Data/Tree/AVL/IndexedMap.agda
index cb16cd6220..7b46f3391a 100644
--- a/src/Data/Tree/AVL/IndexedMap.agda
+++ b/src/Data/Tree/AVL/IndexedMap.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product as Prod
+open import Data.Product.Base
+  using (map₁; map₂; ∃; _×_; Σ-syntax; proj₁; _,_; -,_)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
 import Data.Tree.AVL.Value
@@ -76,10 +77,10 @@ member : ∀ {i} → Key i → Map → Bool
 member k = AVL.member (-, k)
 
 headTail : Map → Maybe (KV × Map)
-headTail m = Maybe.map (Prod.map₁ (toKV ∘′ AVL.toPair)) (AVL.headTail m)
+headTail m = Maybe.map (map₁ (toKV ∘′ AVL.toPair)) (AVL.headTail m)
 
 initLast : Map → Maybe (Map × KV)
-initLast m = Maybe.map (Prod.map₂ (toKV ∘′ AVL.toPair)) (AVL.initLast m)
+initLast m = Maybe.map (map₂ (toKV ∘′ AVL.toPair)) (AVL.initLast m)
 
 foldr : (∀ {k} → Value k → A → A) → A → Map → A
 foldr cons = AVL.foldr cons
diff --git a/src/Data/Tree/AVL/Key.agda b/src/Data/Tree/AVL/Key.agda
index dd41ffb9fd..5c47ccbacc 100644
--- a/src/Data/Tree/AVL/Key.agda
+++ b/src/Data/Tree/AVL/Key.agda
@@ -16,7 +16,7 @@ module Data.Tree.AVL.Key
 open import Level
 open import Data.Empty
 open import Data.Unit
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_ ; refl)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Construct.Add.Extrema
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional.agda b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
index 0873c66cd2..0a3ecd2337 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
@@ -15,7 +15,7 @@ module Data.Tree.AVL.Map.Membership.Propositional
 
 open import Data.Bool.Base using (true; false)
 open import Data.Maybe.Base using (just; nothing; is-just)
-open import Data.Product as Prod using (_×_; ∃-syntax; _,_; proj₁; proj₂)
+open import Data.Product.Base using (_×_; ∃-syntax; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent renaming (Pointwise to _×ᴿ_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_)
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
index aef88a23d7..64eb574b04 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
@@ -15,7 +15,7 @@ module Data.Tree.AVL.Map.Membership.Propositional.Properties
 
 open import Data.Bool.Base using (true; false)
 open import Data.Maybe.Base using (just; nothing; is-just)
-open import Data.Product as Prod using (_×_; ∃-syntax; _,_; proj₁; proj₂)
+open import Data.Product.Base as Prod using (_×_; ∃-syntax; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent renaming (Pointwise to _×ᴿ_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_)
diff --git a/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
index 8a347bd5ff..5ad457d54f 100644
--- a/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
@@ -12,7 +12,7 @@ module Data.Tree.AVL.Map.Relation.Unary.Any
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
   where
 
-open import Data.Product as Prod using (_×_; ∃; _,_)
+open import Data.Product.Base as Prod using (_×_; ∃; _,_)
 open import Function.Base using (_∘_; _∘′_; id)
 open import Level using (Level; _⊔_)
 
diff --git a/src/Data/Tree/AVL/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Relation/Unary/Any.agda
index a9429d4ed2..dfa96935ac 100644
--- a/src/Data/Tree/AVL/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Relation/Unary/Any.agda
@@ -12,7 +12,7 @@ module Data.Tree.AVL.Relation.Unary.Any
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
   where
 
-open import Data.Product as Prod using (∃)
+open import Data.Product.Base as Prod using (∃)
 open import Function.Base using (_∘_; _$_)
 open import Level using (Level; _⊔_)
 
diff --git a/src/Data/Trie.agda b/src/Data/Trie.agda
index 694d7afe7b..3c2de5daf1 100644
--- a/src/Data/Trie.agda
+++ b/src/Data/Trie.agda
@@ -19,7 +19,7 @@ open import Size
 open import Data.List.Base using (List; []; _∷_; _++_)
 import Data.List.NonEmpty as List⁺
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing; maybe′)
-open import Data.Product as Prod using (∃)
+open import Data.Product.Base using (∃)
 open import Data.These.Base as These using (These)
 open import Function.Base using (_∘′_; const)
 open import Relation.Unary using (IUniversal; _⇒_)
diff --git a/src/Data/Trie/NonEmpty.agda b/src/Data/Trie/NonEmpty.agda
index 96378d25e6..f32d20d92f 100644
--- a/src/Data/Trie/NonEmpty.agda
+++ b/src/Data/Trie/NonEmpty.agda
@@ -13,7 +13,7 @@ module Data.Trie.NonEmpty {k e r} (S : StrictTotalOrder k e r) where
 open import Level
 open import Size
 open import Effect.Monad
-open import Data.Product as Prod using (∃; uncurry; -,_)
+open import Data.Product.Base as Prod using (∃; uncurry; -,_)
 open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; [_]; concatMap)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′) hiding (module Maybe)
diff --git a/src/Data/Universe.agda b/src/Data/Universe.agda
index 516e0bd2a4..9bbfb4620b 100644
--- a/src/Data/Universe.agda
+++ b/src/Data/Universe.agda
@@ -8,7 +8,6 @@
 
 module Data.Universe where
 
-open import Data.Product
 open import Level
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Universe/Indexed.agda b/src/Data/Universe/Indexed.agda
index cb6d9f4e56..9807e38eb5 100644
--- a/src/Data/Universe/Indexed.agda
+++ b/src/Data/Universe/Indexed.agda
@@ -8,7 +8,7 @@
 
 module Data.Universe.Indexed where
 
-open import Data.Product
+open import Data.Product.Base using (∃; proj₂)
 open import Data.Universe
 open import Function.Base using (_∘_)
 open import Level
diff --git a/src/Data/Vec/Functional/Relation/Unary/Any.agda b/src/Data/Vec/Functional/Relation/Unary/Any.agda
index de6c3570cb..f41295bf04 100644
--- a/src/Data/Vec/Functional/Relation/Unary/Any.agda
+++ b/src/Data/Vec/Functional/Relation/Unary/Any.agda
@@ -11,7 +11,7 @@ module Data.Vec.Functional.Relation.Unary.Any where
 open import Data.Fin.Base using (zero; suc)
 open import Data.Fin.Properties using (any?)
 open import Data.Nat.Base
-open import Data.Product as Σ using (Σ; ∃; _×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Σ using (Σ; ∃; _×_; _,_; proj₁; proj₂)
 open import Data.Vec.Functional as VF hiding (map)
 open import Function.Base using (id)
 open import Level using (Level)
diff --git a/src/Data/Vec/Membership/Propositional/Properties.agda b/src/Data/Vec/Membership/Propositional/Properties.agda
index f4722c9899..205622c17a 100644
--- a/src/Data/Vec/Membership/Propositional/Properties.agda
+++ b/src/Data/Vec/Membership/Propositional/Properties.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Membership.Propositional.Properties where
 
 open import Data.Fin.Base using (Fin; zero; suc)
-open import Data.Product as Prod using (_,_; ∃; _×_; -,_)
+open import Data.Product.Base using (_,_; ∃; _×_; -,_; map₁; map₂)
 open import Data.Vec.Base
 open import Data.Vec.Relation.Unary.Any using (here; there)
 open import Data.List.Base using ([]; _∷_)
@@ -109,7 +109,7 @@ module _ {P : Pred A p} where
 
   fromAny : ∀ {n} {xs : Vec A n} → Any P xs → ∃ λ x → x ∈ xs × P x
   fromAny (here px) = -, here refl , px
-  fromAny (there v) = Prod.map₂ (Prod.map₁ there) (fromAny v)
+  fromAny (there v) = map₂ (map₁ there) (fromAny v)
 
   toAny : ∀ {n x} {xs : Vec A n} → x ∈ xs → P x → Any P xs
   toAny (here refl) px = here px
diff --git a/src/Data/Vec/Membership/Setoid.agda b/src/Data/Vec/Membership/Setoid.agda
index 57476d5f35..72e57e15f3 100644
--- a/src/Data/Vec/Membership/Setoid.agda
+++ b/src/Data/Vec/Membership/Setoid.agda
@@ -15,7 +15,7 @@ open import Function.Base using (_∘_; flip)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.Any as Any
   using (Any; here; there; index)
-open import Data.Product using (∃; _×_; _,_)
+open import Data.Product.Base using (∃; _×_; _,_)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Unary using (Pred)
 
diff --git a/src/Data/Vec/Recursive/Properties.agda b/src/Data/Vec/Recursive/Properties.agda
index 7a6a6e960b..85fa8e44e6 100644
--- a/src/Data/Vec/Recursive/Properties.agda
+++ b/src/Data/Vec/Recursive/Properties.agda
@@ -10,7 +10,7 @@ module Data.Vec.Recursive.Properties where
 
 open import Level using (Level)
 open import Data.Nat.Base hiding (_^_)
-open import Data.Product
+open import Data.Product.Base
 open import Data.Vec.Recursive
 open import Data.Vec.Base using (Vec; _∷_)
 open import Function.Bundles using (_↔_; mk↔′)
diff --git a/src/Data/Vec/Relation/Unary/Any.agda b/src/Data/Vec/Relation/Unary/Any.agda
index dcf142bcd6..de901fe571 100644
--- a/src/Data/Vec/Relation/Unary/Any.agda
+++ b/src/Data/Vec/Relation/Unary/Any.agda
@@ -13,7 +13,7 @@ open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Nat.Base using (ℕ; zero; suc; NonZero)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Vec.Base as Vec using (Vec; []; [_]; _∷_)
-open import Data.Product as Prod using (∃; _,_)
+open import Data.Product.Base as Prod using (∃; _,_)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Nullary.Decidable as Dec using (yes; no; _⊎-dec_)
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index fd6455da55..96f8874bd7 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -15,7 +15,7 @@ open import Data.List.Base using ([]; _∷_)
 import Data.List.Relation.Unary.Any as List
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
-open import Data.Product as Prod using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Prod using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
 open import Data.Vec.Base hiding (here; there)
 open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.Vec.Membership.Propositional
diff --git a/src/Data/W/Indexed.agda b/src/Data/W/Indexed.agda
index b356a22533..fbf3910427 100644
--- a/src/Data/W/Indexed.agda
+++ b/src/Data/W/Indexed.agda
@@ -10,7 +10,7 @@ module Data.W.Indexed where
 
 open import Level
 open import Data.Container.Indexed.Core
-open import Data.Product
+open import Data.Product.Base using (_,_; Σ)
 open import Relation.Unary
 
 -- The family of indexed W-types.
diff --git a/src/Data/W/WithK.agda b/src/Data/W/WithK.agda
index 0646bfb315..e88fb7cda5 100644
--- a/src/Data/W/WithK.agda
+++ b/src/Data/W/WithK.agda
@@ -8,7 +8,7 @@
 
 module Data.W.WithK where
 
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Data.Container.Core
 open import Data.W
 open import Agda.Builtin.Equality
diff --git a/src/Effect/Applicative/Predicate.agda b/src/Effect/Applicative/Predicate.agda
index bcfec53170..5d696b37b2 100644
--- a/src/Effect/Applicative/Predicate.agda
+++ b/src/Effect/Applicative/Predicate.agda
@@ -12,7 +12,7 @@
 module Effect.Applicative.Predicate where
 
 open import Effect.Functor.Predicate
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (const; constᵣ)
 open import Level
 open import Relation.Unary
diff --git a/src/Effect/Monad/Predicate.agda b/src/Effect/Monad/Predicate.agda
index df3e58347c..1210de7959 100644
--- a/src/Effect/Monad/Predicate.agda
+++ b/src/Effect/Monad/Predicate.agda
@@ -14,7 +14,7 @@ open import Effect.Applicative.Indexed
 open import Effect.Monad
 open import Effect.Monad.Indexed
 open import Data.Unit
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (const; id; _∘_)
 open import Level
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Effect/Monad/State.agda b/src/Effect/Monad/State.agda
index 222f6e7f48..41fc3336d8 100644
--- a/src/Effect/Monad/State.agda
+++ b/src/Effect/Monad/State.agda
@@ -9,7 +9,7 @@
 
 module Effect.Monad.State where
 
-open import Data.Product
+open import Data.Product.Base using (_×_)
 open import Effect.Choice
 open import Effect.Empty
 open import Effect.Functor
diff --git a/src/Effect/Monad/State/Indexed.agda b/src/Effect/Monad/State/Indexed.agda
index 1fc27655d8..4150554ac9 100644
--- a/src/Effect/Monad/State/Indexed.agda
+++ b/src/Effect/Monad/State/Indexed.agda
@@ -12,7 +12,7 @@ open import Effect.Applicative.Indexed
 open import Effect.Monad
 open import Function.Identity.Effectful as Id using (Identity)
 open import Effect.Monad.Indexed
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_; uncurry)
 open import Data.Unit
 open import Function.Base
 open import Level
diff --git a/src/Foreign/Haskell/Coerce.agda b/src/Foreign/Haskell/Coerce.agda
index f593d10af3..3eb411c325 100644
--- a/src/Foreign/Haskell/Coerce.agda
+++ b/src/Foreign/Haskell/Coerce.agda
@@ -39,7 +39,7 @@ import IO.Primitive            as STD
 import Data.List.Base          as STD
 import Data.List.NonEmpty.Base as STD
 import Data.Maybe.Base         as STD
-import Data.Product            as STD
+import Data.Product.Base       as STD
 import Data.Sum.Base           as STD
 
 import Foreign.Haskell.Pair          as FFI
diff --git a/src/Foreign/Haskell/Pair.agda b/src/Foreign/Haskell/Pair.agda
index 1f10b05169..84d4b8180b 100644
--- a/src/Foreign/Haskell/Pair.agda
+++ b/src/Foreign/Haskell/Pair.agda
@@ -9,7 +9,7 @@
 module Foreign.Haskell.Pair where
 
 open import Level
-open import Data.Product using (_×_; _,_)
+open import Data.Product.Base using (_×_; _,_)
 
 private
   variable
diff --git a/src/Function/Bijection.agda b/src/Function/Bijection.agda
index 1de607e957..8c7ce78ef3 100644
--- a/src/Function/Bijection.agda
+++ b/src/Function/Bijection.agda
@@ -13,7 +13,6 @@
 
 module Function.Bijection where
 
-open import Data.Product
 open import Level
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality as P
diff --git a/src/Function/Endomorphism/Setoid.agda b/src/Function/Endomorphism/Setoid.agda
index b02976db88..9d7974294c 100644
--- a/src/Function/Endomorphism/Setoid.agda
+++ b/src/Function/Endomorphism/Setoid.agda
@@ -21,7 +21,7 @@ open import Algebra.Morphism; open Definitions
 open import Function.Equality
 open import Data.Nat.Base using (ℕ; _+_); open ℕ
 open import Data.Nat.Properties
-open import Data.Product
+open import Data.Product.Base using (_,_)
 
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial as Trivial
 
diff --git a/src/Function/LeftInverse.agda b/src/Function/LeftInverse.agda
index f06fbc6c8a..23f9b4a900 100644
--- a/src/Function/LeftInverse.agda
+++ b/src/Function/LeftInverse.agda
@@ -13,7 +13,6 @@
 
 module Function.LeftInverse where
 
-open import Data.Product
 open import Level
 import Relation.Binary.Reasoning.Setoid as EqReasoning
 open import Relation.Binary
diff --git a/src/Function/Properties/RightInverse.agda b/src/Function/Properties/RightInverse.agda
index 531b7236e7..681e3e2b5a 100644
--- a/src/Function/Properties/RightInverse.agda
+++ b/src/Function/Properties/RightInverse.agda
@@ -12,7 +12,7 @@ open import Function.Base
 open import Function.Bundles
 open import Function.Consequences using (inverseˡ⇒surjective)
 open import Level using (Level)
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 
diff --git a/src/Function/Properties/Surjection.agda b/src/Function/Properties/Surjection.agda
index fce294be76..e0fccc598b 100644
--- a/src/Function/Properties/Surjection.agda
+++ b/src/Function/Properties/Surjection.agda
@@ -12,7 +12,7 @@ open import Function.Base
 open import Function.Definitions
 open import Function.Bundles
 open import Level using (Level)
-open import Data.Product
+open import Data.Product.Base using (proj₁; proj₂)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Binary using (Setoid)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index 58cf6bab88..9922c69fcf 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -15,7 +15,8 @@ open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Bool.Base using (true; false)
 open import Data.Empty using (⊥-elim)
 open import Data.Empty.Polymorphic using (⊥) renaming (⊥-elim to ⊥ₚ-elim)
-open import Data.Product as Prod hiding (swap)
+open import Data.Product.Base as Prod
+  using (_×_; Σ; curry; uncurry; _,_; -,_; <_,_>; proj₁; proj₂; ∃₂; ∃)
 open import Data.Product.Function.NonDependent.Propositional
 open import Data.Sum.Base as Sum
 open import Data.Sum.Properties using (swap-involutive)
diff --git a/src/Function/Surjection.agda b/src/Function/Surjection.agda
index 12f4490f3e..6861979551 100644
--- a/src/Function/Surjection.agda
+++ b/src/Function/Surjection.agda
@@ -19,7 +19,6 @@ open import Function.Equality as F
 open import Function.Equivalence using (Equivalence)
 open import Function.Injection           hiding (id; _∘_; injection)
 open import Function.LeftInverse as Left hiding (id; _∘_)
-open import Data.Product
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
 
diff --git a/src/Induction/Lexicographic.agda b/src/Induction/Lexicographic.agda
index 303f2c8dfa..6acd96b7f1 100644
--- a/src/Induction/Lexicographic.agda
+++ b/src/Induction/Lexicographic.agda
@@ -8,7 +8,7 @@
 
 module Induction.Lexicographic where
 
-open import Data.Product
+open import Data.Product.Base using (Σ; _,_; _×_)
 open import Induction
 open import Level
 
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
index 80a0727fc1..e4c8f3a974 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
@@ -15,7 +15,7 @@ module Relation.Binary.Construct.Add.Infimum.Strict
   {a ℓ} {A : Set a} (_<_ : Rel A ℓ) where
 
 open import Level using (_⊔_)
-open import Data.Product
+open import Data.Product.Base using (_,_; map)
 open import Function.Base
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
diff --git a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
index 0bbefc05fa..b09432961d 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
@@ -15,7 +15,7 @@ module Relation.Binary.Construct.Add.Supremum.Strict
   {a r} {A : Set a} (_<_ : Rel A r) where
 
 open import Level using (_⊔_)
-open import Data.Product
+open import Data.Product.Base using (_,_; map)
 open import Function.Base
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Relation/Binary/Construct/Composition.agda b/src/Relation/Binary/Construct/Composition.agda
index e34550fb58..7f8ba241b0 100644
--- a/src/Relation/Binary/Construct/Composition.agda
+++ b/src/Relation/Binary/Construct/Composition.agda
@@ -8,7 +8,7 @@
 
 module Relation.Binary.Construct.Composition where
 
-open import Data.Product
+open import Data.Product.Base using (∃; _×_; _,_)
 open import Function.Base
 open import Level
 open import Relation.Binary
diff --git a/src/Relation/Binary/Construct/Flip/Ord.agda b/src/Relation/Binary/Construct/Flip/Ord.agda
index ecfc4b6df4..4a2eea690f 100644
--- a/src/Relation/Binary/Construct/Flip/Ord.agda
+++ b/src/Relation/Binary/Construct/Flip/Ord.agda
@@ -12,7 +12,7 @@ open import Relation.Binary
 
 module Relation.Binary.Construct.Flip.Ord where
 
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (flip; _∘_)
 open import Level using (Level)
 
diff --git a/src/Relation/Binary/Construct/FromPred.agda b/src/Relation/Binary/Construct/FromPred.agda
index c95f95fb6f..e08997b9a7 100644
--- a/src/Relation/Binary/Construct/FromPred.agda
+++ b/src/Relation/Binary/Construct/FromPred.agda
@@ -16,7 +16,6 @@ module Relation.Binary.Construct.FromPred
   where
 
 open import Function.Base
-open import Data.Product
 
 open module Eq = Setoid S using (_≈_) renaming (Carrier to A)
 
diff --git a/src/Relation/Binary/Construct/FromRel.agda b/src/Relation/Binary/Construct/FromRel.agda
index fb842b7810..13fc1774c7 100644
--- a/src/Relation/Binary/Construct/FromRel.agda
+++ b/src/Relation/Binary/Construct/FromRel.agda
@@ -15,7 +15,6 @@ module Relation.Binary.Construct.FromRel
   {a r} {A : Set a} (_R_ : REL A (Carrier S) r) -- The relation
   where
 
-open import Data.Product
 open import Function.Base
 open import Level using (_⊔_)
 
diff --git a/src/Relation/Binary/Construct/On.agda b/src/Relation/Binary/Construct/On.agda
index 5206276f95..122b0d6584 100644
--- a/src/Relation/Binary/Construct/On.agda
+++ b/src/Relation/Binary/Construct/On.agda
@@ -8,7 +8,7 @@
 
 module Relation.Binary.Construct.On where
 
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_on_; _∘_)
 open import Induction.WellFounded using (WellFounded; Acc; acc)
 open import Level using (Level)
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 1ca855e122..3152bbf0cb 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -9,7 +9,6 @@
 module Relation.Binary.HeterogeneousEquality where
 
 import Axiom.Extensionality.Heterogeneous as Ext
-open import Data.Product
 open import Data.Unit.NonEta
 open import Function.Base
 open import Function.Inverse using (Inverse)
diff --git a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
index d2825a3548..2dbd39344a 100644
--- a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
+++ b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
@@ -15,7 +15,7 @@ open import Relation.Binary.HeterogeneousEquality hiding (isEquivalence)
 import Relation.Binary.PropositionalEquality.Core as ≡
 open import Data.Nat.Base
 open import Data.Nat.Properties
-open import Data.Product
+open import Data.Product.Base using (_×_; _,_)
 open import Function.Base
 
 open ≅-Reasoning
diff --git a/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
index b75cfa7093..a86918081f 100644
--- a/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
@@ -16,7 +16,7 @@ open BoundedJoinSemilattice J
 open import Algebra.Definitions _≈_
 open import Algebra.Ordered.Structures using (IsCommutativePomonoid)
 open import Algebra.Ordered.Bundles using (CommutativePomonoid)
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; flip)
 open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
diff --git a/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda b/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda
index b6c7be5a5c..622c8b6b0c 100644
--- a/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda
@@ -14,7 +14,6 @@ module Relation.Binary.Lattice.Properties.BoundedMeetSemilattice
 open BoundedMeetSemilattice M
 
 open import Algebra.Definitions _≈_
-open import Data.Product
 open import Function.Base using (_∘_; flip)
 open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
diff --git a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
index fb85a17fe0..faa51120cd 100644
--- a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
@@ -18,7 +18,7 @@ import Algebra.Structures as Alg
 open import Algebra.Definitions _≈_
 open import Algebra.Ordered.Structures using (IsPosemigroup)
 open import Algebra.Ordered.Bundles using (Posemigroup)
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; flip)
 open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
diff --git a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
index 34e4252f92..f072cb2199 100644
--- a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
@@ -14,7 +14,6 @@ module Relation.Binary.Lattice.Properties.MeetSemilattice
 open MeetSemilattice M
 
 open import Algebra.Definitions _≈_
-open import Data.Product
 open import Function.Base using (flip)
 open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
diff --git a/src/Relation/Binary/Rewriting.agda b/src/Relation/Binary/Rewriting.agda
index 9006765413..10d760e23b 100644
--- a/src/Relation/Binary/Rewriting.agda
+++ b/src/Relation/Binary/Rewriting.agda
@@ -10,7 +10,7 @@
 module Relation.Binary.Rewriting where
 
 open import Agda.Builtin.Equality using (_≡_ ; refl)
-open import Data.Product using (_×_ ; ∃ ; -,_; _,_ ; proj₁ ; proj₂)
+open import Data.Product.Base using (_×_ ; ∃ ; -,_; _,_ ; proj₁ ; proj₂)
 open import Data.Empty
 open import Function.Base using (flip)
 open import Induction.WellFounded
diff --git a/src/Relation/Unary/Indexed.agda b/src/Relation/Unary/Indexed.agda
index 1acfdc8cef..d1cdbb01f5 100644
--- a/src/Relation/Unary/Indexed.agda
+++ b/src/Relation/Unary/Indexed.agda
@@ -8,7 +8,7 @@
 
 module Relation.Unary.Indexed  where
 
-open import Data.Product using (∃; _×_)
+open import Data.Product.Base using (∃; _×_)
 open import Level
 open import Relation.Nullary.Negation using (¬_)
 
diff --git a/src/Relation/Unary/PredicateTransformer.agda b/src/Relation/Unary/PredicateTransformer.agda
index 0ae725588a..cd52a78985 100644
--- a/src/Relation/Unary/PredicateTransformer.agda
+++ b/src/Relation/Unary/PredicateTransformer.agda
@@ -8,7 +8,7 @@
 
 module Relation.Unary.PredicateTransformer where
 
-open import Data.Product
+open import Data.Product.Base using (∃)
 open import Function.Base using (_∘_)
 open import Level hiding (_⊔_)
 open import Relation.Nullary
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 515d904872..b7accfcc08 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -34,7 +34,7 @@ open import Data.Maybe.Base  as Maybe using (Maybe; just; nothing)
 open import Data.Nat.Base    as Nat   using (ℕ; zero; suc; _≡ᵇ_; _+_)
 open import Data.Unit.Base            using (⊤)
 open import Data.Word.Base   as Word  using (toℕ)
-open import Data.Product
+open import Data.Product.Base         using (_×_; map₁; _,_)
 
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_; refl; cong)
 
diff --git a/src/Tactic/RingSolver/NonReflective.agda b/src/Tactic/RingSolver/NonReflective.agda
index 6aa8997bb0..cc0b747037 100644
--- a/src/Tactic/RingSolver/NonReflective.agda
+++ b/src/Tactic/RingSolver/NonReflective.agda
@@ -23,7 +23,7 @@ open import Data.Bool.Base using (Bool; true; false; T; if_then_else_)
 open import Data.Maybe.Base
 open import Data.Empty using (⊥-elim)
 open import Data.Nat.Base using (ℕ)
-open import Data.Product
+open import Data.Product.Base using (_×_; proj₁; proj₂; _,_)
 open import Data.Vec.Base using (Vec)
 open import Data.Vec.N-ary
 
diff --git a/src/Text/Format/Generic.agda b/src/Text/Format/Generic.agda
index 8a304f9c9a..9969801832 100644
--- a/src/Text/Format/Generic.agda
+++ b/src/Text/Format/Generic.agda
@@ -14,7 +14,7 @@ open import Data.Char.Base using (Char)
 open import Data.List.Base as List
 open import Data.Maybe.Base as Maybe
 open import Data.Nat.Base
-open import Data.Product
+open import Data.Product.Base using (_,_)
 open import Data.Product.Nary.NonDependent
 open import Data.Sum.Base
 open import Data.String.Base

From 1d91f7567899ad4dc1df6c47d0fcf5626a7f474a Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Fri, 11 Aug 2023 03:31:05 +0000
Subject: [PATCH 093/316] Added properties of Heyting Commutative Ring (#1968)

---
 CHANGELOG.md                                  | 18 ++++
 .../Properties/HeytingCommutativeRing.agda    | 91 +++++++++++++++++--
 src/Algebra/Properties/CommutativeMonoid.agda | 17 ++++
 3 files changed, 118 insertions(+), 8 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e166023acc..b531a5d4a4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3284,6 +3284,24 @@ This is a full list of proofs that have changed form to use irrelevant instance
   ↭-reverse : (xs : List A) → reverse xs ↭ xs
   ```
 
+* Added new functions to `Algebra.Properties.CommutativeMonoid`
+  ```agda
+  invertibleˡ⇒invertibleʳ : LeftInvertible _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x
+  invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible _≈_ 0# _+_ x
+  invertibleˡ⇒invertible  : LeftInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x
+  invertibleʳ⇒invertible  : RightInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x
+  ```
+
+* Added new functions to `Algebra.Apartness.Bundles`
+  ```agda
+  invertibleˡ⇒# : LeftInvertible _≈_ 1# _*_ (x - y) → x # y
+  invertibleʳ⇒# : RightInvertible _≈_ 1# _*_ (x - y) → x # y
+  x#0y#0→xy#0   : x # 0# → y # 0# → x * y # 0#
+  #-sym         : Symmetric _#_
+  #-congʳ       : x ≈ y → x # z → y # z
+  #-congˡ       : y ≈ z → x # y → x # z
+  ```
+
 * Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`
   and `Data.List.Relation.Unary.Sorted.TotalOrder.Properties`.
   ```agda
diff --git a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
index 1f04cc2dd0..7f4f164f42 100644
--- a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
+++ b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
@@ -11,23 +11,98 @@ open import Algebra.Apartness.Bundles using (HeytingCommutativeRing)
 module Algebra.Apartness.Properties.HeytingCommutativeRing
   {c ℓ₁ ℓ₂} (HCR : HeytingCommutativeRing c ℓ₁ ℓ₂) where
 
-open import Data.Product.Base using (_,_; proj₂)
-open import Algebra using (CommutativeRing; RightIdentity)
+open import Function.Base using (_∘_)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
+open import Algebra using (CommutativeRing; RightIdentity; Invertible; LeftInvertible; RightInvertible)
 
 open HeytingCommutativeRing HCR
-open CommutativeRing commutativeRing using (ring)
+open CommutativeRing commutativeRing using (ring; *-commutativeMonoid)
 
-open import Algebra.Properties.Ring ring using (-0#≈0#)
+open import Algebra.Properties.Ring ring
+  using (-0#≈0#; -‿distribˡ-*; -‿distribʳ-*; -‿anti-homo-+; -‿involutive)
+open import Relation.Binary.Definitions using (Symmetric)
+import Relation.Binary.Reasoning.Setoid as ReasonSetoid
+open import Algebra.Properties.CommutativeMonoid *-commutativeMonoid
 
+private variable
+  x y z : Carrier
+
+invertibleˡ⇒# : LeftInvertible _≈_ 1# _*_ (x - y) → x # y
+invertibleˡ⇒# = invertible⇒# ∘ invertibleˡ⇒invertible
+
+invertibleʳ⇒# : RightInvertible _≈_ 1# _*_ (x - y) → x # y
+invertibleʳ⇒# = invertible⇒# ∘ invertibleʳ⇒invertible
 
 x-0≈x : RightIdentity _≈_ 0# _-_
 x-0≈x x = trans (+-congˡ -0#≈0#) (+-identityʳ x)
 
 1#0 : 1# # 0#
-1#0 = invertible⇒# (1# , 1*[x-0]≈x , [x-0]*1≈x)
+1#0 = invertibleˡ⇒# (1# , 1*[x-0]≈x)
   where
-  1*[x-0]≈x : ∀ {x} → 1# * (x - 0#) ≈ x
+  1*[x-0]≈x : 1# * (x - 0#) ≈ x
   1*[x-0]≈x {x} = trans (*-identityˡ (x - 0#)) (x-0≈x x)
 
-  [x-0]*1≈x : ∀ {x} → (x - 0#) * 1# ≈ x
-  [x-0]*1≈x {x} = trans (*-comm (x - 0#) 1#) 1*[x-0]≈x
+x#0y#0→xy#0 : x # 0# → y # 0# → x * y # 0#
+x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
+  where
+
+  helper : Invertible _≈_ 1# _*_ (x - 0#) → Invertible _≈_ 1# _*_ (y - 0#) → x * y # 0#
+  helper (x⁻¹ , x⁻¹*x≈1 , x*x⁻¹≈1) (y⁻¹ , y⁻¹*y≈1 , y*y⁻¹≈1)
+    = invertibleˡ⇒# (y⁻¹ * x⁻¹ , y⁻¹*x⁻¹*x*y≈1)
+    where
+    open ReasonSetoid setoid
+
+    y⁻¹*x⁻¹*x*y≈1 : y⁻¹ * x⁻¹ * (x * y - 0#) ≈ 1#
+    y⁻¹*x⁻¹*x*y≈1 = begin
+      y⁻¹ * x⁻¹ * (x * y - 0#)     ≈⟨ *-congˡ (x-0≈x (x * y)) ⟩
+      y⁻¹ * x⁻¹ * (x * y)          ≈⟨ *-assoc y⁻¹ x⁻¹ (x * y) ⟩
+      y⁻¹ * (x⁻¹ * (x * y))       ≈˘⟨ *-congˡ (*-assoc x⁻¹ x y) ⟩
+      y⁻¹ * ((x⁻¹ * x) * y)       ≈˘⟨ *-congˡ (*-congʳ (*-congˡ (x-0≈x x))) ⟩
+      y⁻¹ * ((x⁻¹ * (x - 0#)) * y) ≈⟨ *-congˡ (*-congʳ x⁻¹*x≈1) ⟩
+      y⁻¹ * (1# * y)               ≈⟨ *-congˡ (*-identityˡ y) ⟩
+      y⁻¹ * y                     ≈˘⟨ *-congˡ (x-0≈x y) ⟩
+      y⁻¹ * (y - 0#)               ≈⟨ y⁻¹*y≈1 ⟩
+      1# ∎
+
+#-sym : Symmetric _#_
+#-sym {x} {y} x#y = invertibleˡ⇒# (- x-y⁻¹ , x-y⁻¹*y-x≈1)
+  where
+  open ReasonSetoid setoid
+  InvX-Y : Invertible _≈_ 1# _*_ (x - y)
+  InvX-Y = #⇒invertible x#y
+
+  x-y⁻¹ = InvX-Y .proj₁
+
+  y-x≈-[x-y] : y - x ≈ - (x - y)
+  y-x≈-[x-y] = begin
+    y - x     ≈˘⟨ +-congʳ (-‿involutive y) ⟩
+    - - y - x ≈˘⟨ -‿anti-homo-+ x (- y) ⟩
+    - (x - y) ∎
+
+  x-y⁻¹*y-x≈1 : (- x-y⁻¹) * (y - x) ≈ 1#
+  x-y⁻¹*y-x≈1 = begin
+    (- x-y⁻¹) * (y - x)   ≈˘⟨ -‿distribˡ-* x-y⁻¹ (y - x) ⟩
+    - (x-y⁻¹ * (y - x))    ≈⟨ -‿cong (*-congˡ y-x≈-[x-y]) ⟩
+    - (x-y⁻¹ * - (x - y)) ≈˘⟨ -‿cong (-‿distribʳ-* x-y⁻¹ (x - y)) ⟩
+    - - (x-y⁻¹ * (x - y))  ≈⟨ -‿involutive (x-y⁻¹ * ((x - y))) ⟩
+    x-y⁻¹ * (x - y)        ≈⟨ InvX-Y .proj₂ .proj₁ ⟩
+    1# ∎
+
+#-congʳ : x ≈ y → x # z → y # z
+#-congʳ {x} {y} {z} x≈y x#z = helper (#⇒invertible x#z)
+  where
+
+  helper : Invertible _≈_ 1# _*_ (x - z) → y # z
+  helper (x-z⁻¹ , x-z⁻¹*x-z≈1# , x-z*x-z⁻¹≈1#)
+    = invertibleˡ⇒# (x-z⁻¹ , x-z⁻¹*y-z≈1)
+    where
+    open ReasonSetoid setoid
+
+    x-z⁻¹*y-z≈1 : x-z⁻¹ * (y - z) ≈ 1#
+    x-z⁻¹*y-z≈1 = begin
+      x-z⁻¹ * (y - z) ≈˘⟨ *-congˡ (+-congʳ x≈y) ⟩
+      x-z⁻¹ * (x - z)  ≈⟨ x-z⁻¹*x-z≈1# ⟩
+      1# ∎
+
+#-congˡ : y ≈ z → x # y → x # z
+#-congˡ y≈z x#y = #-sym (#-congʳ y≈z (#-sym x#y))
diff --git a/src/Algebra/Properties/CommutativeMonoid.agda b/src/Algebra/Properties/CommutativeMonoid.agda
index 091e88ec61..e83be61ac7 100644
--- a/src/Algebra/Properties/CommutativeMonoid.agda
+++ b/src/Algebra/Properties/CommutativeMonoid.agda
@@ -7,6 +7,8 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Bundles
+open import Algebra.Definitions
+open import Data.Product using (_,_; proj₂)
 
 module Algebra.Properties.CommutativeMonoid
   {g₁ g₂} (M : CommutativeMonoid g₁ g₂) where
@@ -23,3 +25,18 @@ open CommutativeMonoid M
   ; assoc     to +-assoc
   ; comm      to +-comm
   )
+
+private variable
+  x : Carrier
+
+invertibleˡ⇒invertibleʳ : LeftInvertible _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x
+invertibleˡ⇒invertibleʳ {x} (-x , -x+x≈1) = -x , trans (+-comm x -x) -x+x≈1
+
+invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible _≈_ 0# _+_ x
+invertibleʳ⇒invertibleˡ {x} (-x , x+-x≈1) = -x , trans (+-comm -x x) x+-x≈1
+
+invertibleˡ⇒invertible : LeftInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x
+invertibleˡ⇒invertible left@(-x , -x+x≈1) = -x , -x+x≈1 , invertibleˡ⇒invertibleʳ left .proj₂
+
+invertibleʳ⇒invertible : RightInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x
+invertibleʳ⇒invertible right@(-x , x+-x≈1) = -x , invertibleʳ⇒invertibleˡ right .proj₂ , x+-x≈1

From 61531e38b6cb1a62d1202e726e76f7f566afa603 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 10 Aug 2023 23:31:31 -0400
Subject: [PATCH 094/316] Add more properties for `xor` (#2035)

---
 CHANGELOG.md                  | 21 +++++++-
 src/Data/Bool/Properties.agda | 93 +++++++++++++++++++++++++++++------
 2 files changed, 98 insertions(+), 16 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b531a5d4a4..5394b704d0 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2018,10 +2018,27 @@ Other minor changes
   <-wellFounded : WellFounded _<_
   ∨-conicalˡ : LeftConical false _∨_
   ∨-conicalʳ : RightConical false _∨_
-  ∨-conical : Conical false _∨_
+  ∨-conical  : Conical false _∨_
   ∧-conicalˡ : LeftConical true _∧_
   ∧-conicalʳ : RightConical true _∧_
-  ∧-conical : Conical true _∧_
+  ∧-conical  : Conical true _∧_
+
+  true-xor            : true xor x ≡ not x
+  xor-same            : x xor x ≡ false
+  not-distribˡ-xor    : not (x xor y) ≡ (not x) xor y
+  not-distribʳ-xor    : not (x xor y) ≡ x xor (not y)
+  xor-assoc           : Associative _xor_
+  xor-comm            : Commutative _xor_
+  xor-identityˡ       : LeftIdentity false _xor_
+  xor-identityʳ       : RightIdentity false _xor_
+  xor-identity        : Identity false _xor_
+  xor-inverseˡ        : LeftInverse true not _xor_
+  xor-inverseʳ        : RightInverse true not _xor_
+  xor-inverse         : Inverse true not _xor_
+  ∧-distribˡ-xor      : _∧_ DistributesOverˡ _xor_
+  ∧-distribʳ-xor      : _∧_ DistributesOverʳ _xor_
+  ∧-distrib-xor       : _∧_ DistributesOver _xor_
+  xor-annihilates-not : (not x) xor (not y) ≡ x xor y
   ```
 
 * Added new functions in `Data.Fin.Base`:
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index 56448d8e2b..713ad3da34 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -627,20 +627,7 @@ true  <? _     = no  (λ())
   }
 
 ------------------------------------------------------------------------
--- Properties of _xor_
-
-xor-is-ok : ∀ x y → x xor y ≡ (x ∨ y) ∧ not (x ∧ y)
-xor-is-ok true  y = refl
-xor-is-ok false y = sym (∧-identityʳ _)
-
-xor-∧-commutativeRing : CommutativeRing 0ℓ 0ℓ
-xor-∧-commutativeRing = ⊕-∧-commutativeRing
-  where
-  open BooleanAlgebraProperties ∨-∧-booleanAlgebra
-  open XorRing _xor_ xor-is-ok
-
-------------------------------------------------------------------------
--- Miscellaneous other properties
+-- Properties of not
 
 not-involutive : Involutive not
 not-involutive true  = refl
@@ -660,6 +647,84 @@ not-¬ {false} refl ()
 ¬-not {false} {true}  _   = refl
 ¬-not {false} {false} x≢y = ⊥-elim (x≢y refl)
 
+------------------------------------------------------------------------
+-- Properties of _xor_
+
+xor-is-ok : ∀ x y → x xor y ≡ (x ∨ y) ∧ not (x ∧ y)
+xor-is-ok true  y = refl
+xor-is-ok false y = sym (∧-identityʳ _)
+
+true-xor : ∀ x → true xor x ≡ not x
+true-xor false = refl
+true-xor true  = refl
+
+xor-same : ∀ x → x xor x ≡ false
+xor-same false = refl
+xor-same true  = refl
+
+not-distribˡ-xor : ∀ x y → not (x xor y) ≡ (not x) xor y
+not-distribˡ-xor false y = refl
+not-distribˡ-xor true  y = not-involutive _
+
+not-distribʳ-xor : ∀ x y → not (x xor y) ≡ x xor (not y)
+not-distribʳ-xor false y = refl
+not-distribʳ-xor true  y = refl
+
+xor-assoc : Associative _xor_
+xor-assoc true  y z = sym (not-distribˡ-xor y z)
+xor-assoc false y z = refl
+
+xor-comm : Commutative _xor_
+xor-comm false false = refl
+xor-comm false true  = refl
+xor-comm true  false = refl
+xor-comm true  true  = refl
+
+xor-identityˡ : LeftIdentity false _xor_
+xor-identityˡ _ = refl
+
+xor-identityʳ : RightIdentity false _xor_
+xor-identityʳ false = refl
+xor-identityʳ true  = refl
+
+xor-identity : Identity false _xor_
+xor-identity = xor-identityˡ , xor-identityʳ
+
+xor-inverseˡ : LeftInverse true not _xor_
+xor-inverseˡ false = refl
+xor-inverseˡ true = refl
+
+xor-inverseʳ : RightInverse true not _xor_
+xor-inverseʳ x = xor-comm x (not x) ⟨ trans ⟩ xor-inverseˡ x
+
+xor-inverse : Inverse true not _xor_
+xor-inverse = xor-inverseˡ , xor-inverseʳ
+
+∧-distribˡ-xor : _∧_ DistributesOverˡ _xor_
+∧-distribˡ-xor false y z = refl
+∧-distribˡ-xor true  y z = refl
+
+∧-distribʳ-xor : _∧_ DistributesOverʳ _xor_
+∧-distribʳ-xor x false z    = refl
+∧-distribʳ-xor x true false = sym (xor-identityʳ x)
+∧-distribʳ-xor x true true  = sym (xor-same x)
+
+∧-distrib-xor : _∧_ DistributesOver _xor_
+∧-distrib-xor = ∧-distribˡ-xor , ∧-distribʳ-xor
+
+xor-annihilates-not : ∀ x y → (not x) xor (not y) ≡ x xor y
+xor-annihilates-not false y = not-involutive _
+xor-annihilates-not true  y = refl
+
+xor-∧-commutativeRing : CommutativeRing 0ℓ 0ℓ
+xor-∧-commutativeRing = ⊕-∧-commutativeRing
+  where
+  open BooleanAlgebraProperties ∨-∧-booleanAlgebra
+  open XorRing _xor_ xor-is-ok
+
+------------------------------------------------------------------------
+-- Miscellaneous other properties
+
 ⇔→≡ : {x y z : Bool} → x ≡ z ⇔ y ≡ z → x ≡ y
 ⇔→≡ {true } {true }         hyp = refl
 ⇔→≡ {true } {false} {true } hyp = sym (Equivalence.to hyp ⟨$⟩ refl)

From 4f692f82d390d7786cbe538021f6b0adb3f7a6d3 Mon Sep 17 00:00:00 2001
From: Ambroise <chaster_killer@hotmail.fr>
Date: Thu, 17 Aug 2023 02:00:46 +0100
Subject: [PATCH 095/316] Additional lemmas about lists and vectors (#2020)

---
 CHANGELOG.md                                  | 15 ++++++++++
 .../List/Membership/Setoid/Properties.agda    |  8 +++++
 src/Data/List/Properties.agda                 |  4 +++
 .../Membership/Propositional/Properties.agda  |  7 ++++-
 src/Data/Vec/Properties.agda                  | 30 +++++++++++++++++++
 5 files changed, 63 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5394b704d0..77b00a6ff3 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2203,6 +2203,7 @@ Other minor changes
   ```agda
   mapWith∈-id  : mapWith∈ xs (λ {x} _ → x) ≡ xs
   map-mapWith∈ : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
+  index-injective : index x₁∈xs ≡ index x₂∈xs → x₁ ≈ x₂
   ```
 
 * Add new proofs in `Data.List.Properties`:
@@ -2231,6 +2232,8 @@ Other minor changes
 
   take-[] : ∀ m → take  m [] ≡ []
   drop-[] : ∀ m → drop  m [] ≡ []
+
+  map-replicate : map f (replicate n x) ≡ replicate n (f x)
   ```
 
 * Added new patterns and definitions to `Data.Nat.Base`:
@@ -2599,6 +2602,8 @@ Other minor changes
   map-∷ʳ       : map f (xs ∷ʳ x) ≡ (map f xs) ∷ʳ (f x)
   map-reverse  : map f (reverse xs) ≡ reverse (map f xs)
   map-ʳ++      : map f (xs ʳ++ ys) ≡ map f xs ʳ++ map f ys
+  map-insert   : map f (insert xs i x) ≡ insert (map f xs) i (f x)
+  toList-map   : toList (map f xs) ≡ List.map f (toList xs)
 
   lookup-concat : lookup (concat xss) (combine i j) ≡ lookup (lookup xss i) j
 
@@ -2633,6 +2638,9 @@ Other minor changes
   reverse-injective  : reverse xs ≡ reverse ys → xs ≡ ys
 
   transpose-replicate : transpose (replicate xs) ≡ map replicate xs
+  toList-replicate    : toList (replicate {n = n} a) ≡ List.replicate n a
+
+  toList-++ : toList (xs ++ ys) ≡ toList xs List.++ toList ys
 
   toList-cast   : toList (cast eq xs) ≡ toList xs
   cast-is-id    : cast eq xs ≡ xs
@@ -2642,6 +2650,13 @@ Other minor changes
   lookup-cast   : lookup (cast eq xs) (Fin.cast eq i) ≡ lookup xs i
   lookup-cast₁  : lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)
   lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
+
+  zipwith-++ : zipWith f (xs ++ ys) (xs' ++ ys') ≡ zipWith f xs xs' ++ zipWith f ys ys'
+  ```
+
+* Added new proofs in `Data.Vec.Membership.Propositional.Properties`:
+  ```agda
+  index-∈-fromList⁺ : Any.index (∈-fromList⁺ v∈xs) ≡ indexₗ v∈xs
   ```
 
 * Added new proofs in `Data.Vec.Functional.Properties`:
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 2fee07309f..5488f9a037 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -11,6 +11,7 @@ module Data.List.Membership.Setoid.Properties where
 open import Algebra using (Op₂; Selective)
 open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base using (Fin; zero; suc)
+open import Data.Fin.Properties using (suc-injective)
 open import Data.List.Base
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Relation.Unary.All as All using (All)
@@ -62,6 +63,13 @@ module _ (S : Setoid c ℓ) where
   ∉-resp-≋ : ∀ {x} → (x ∉_) Respects _≋_
   ∉-resp-≋ xs≋ys v∉xs v∈ys = v∉xs (∈-resp-≋ (≋-sym xs≋ys) v∈ys)
 
+  -- index is injective in its first argument.
+
+  index-injective : ∀ {x₁ x₂ xs} (x₁∈xs : x₁ ∈ xs) (x₂∈xs : x₂ ∈ xs) →
+                    Any.index x₁∈xs ≡ Any.index x₂∈xs → x₁ ≈ x₂
+  index-injective (here x₁≈x)   (here x₂≈x)   _  = trans x₁≈x (sym x₂≈x)
+  index-injective (there x₁∈xs) (there x₂∈xs) eq = index-injective x₁∈xs x₂∈xs (suc-injective eq)
+
 ------------------------------------------------------------------------
 -- Irrelevance
 
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index dd4326a749..56c22c953a 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -118,6 +118,10 @@ map-injective finj {x ∷ xs} {y ∷ ys} eq =
   let fx≡fy , fxs≡fys = ∷-injective eq in
   cong₂ _∷_ (finj fx≡fy) (map-injective finj fxs≡fys)
 
+map-replicate : ∀ (f : A → B) n x → map f (replicate n x) ≡ replicate n (f x)
+map-replicate f zero    x = refl
+map-replicate f (suc n) x = cong (_ ∷_) (map-replicate f n x)
+
 ------------------------------------------------------------------------
 -- mapMaybe
 
diff --git a/src/Data/Vec/Membership/Propositional/Properties.agda b/src/Data/Vec/Membership/Propositional/Properties.agda
index 205622c17a..7ebca867ad 100644
--- a/src/Data/Vec/Membership/Propositional/Properties.agda
+++ b/src/Data/Vec/Membership/Propositional/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Product.Base using (_,_; ∃; _×_; -,_; map₁; map₂)
 open import Data.Vec.Base
 open import Data.Vec.Relation.Unary.Any using (here; there)
 open import Data.List.Base using ([]; _∷_)
-open import Data.List.Relation.Unary.Any using (here; there)
+open import Data.List.Relation.Unary.Any as ListAny using (here; there)
 open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.Vec.Membership.Propositional
 open import Data.List.Membership.Propositional
@@ -102,6 +102,11 @@ index-∈-lookup (suc i) (x ∷ xs) = cong suc (index-∈-lookup i xs)
 ∈-fromList⁻ {xs = _ ∷ _} (here refl)  = here refl
 ∈-fromList⁻ {xs = _ ∷ _} (there v∈xs) = there (∈-fromList⁻ v∈xs)
 
+index-∈-fromList⁺ : ∀ {v : A} {xs} → (v∈xs : v ∈ₗ xs) →
+                    Any.index (∈-fromList⁺ v∈xs) ≡ ListAny.index v∈xs
+index-∈-fromList⁺ (here refl) = refl
+index-∈-fromList⁺ (there v∈xs) = cong suc (index-∈-fromList⁺ v∈xs)
+
 ------------------------------------------------------------------------
 -- Relationship to Any
 
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 49924a7b31..57ffb79317 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -448,6 +448,12 @@ map-updateAt : ∀ {f : A → B} {g : A → A} {h : B → B}
 map-updateAt (x ∷ xs) zero    eq = cong (_∷ _) eq
 map-updateAt (x ∷ xs) (suc i) eq = cong (_ ∷_) (map-updateAt xs i eq)
 
+map-insert : ∀ (f : A → B) (x : A) (xs : Vec A n) (i : Fin (suc n)) →
+             map f (insert xs i x) ≡ insert (map f xs) i (f x)
+map-insert f _ []        Fin.zero = refl
+map-insert f _ (x' ∷ xs) Fin.zero = refl
+map-insert f x (x' ∷ xs) (Fin.suc i) = cong (_ ∷_) (map-insert f x xs i)
+
 map-[]≔ : ∀ (f : A → B) (xs : Vec A n) (i : Fin n) →
           map f (xs [ i ]≔ x) ≡ map f xs [ i ]≔ f x
 map-[]≔ f xs i = map-updateAt xs i refl
@@ -457,6 +463,11 @@ map-⊛ : ∀ (f : A → B → C) (g : A → B) (xs : Vec A n) →
 map-⊛ f g []       = refl
 map-⊛ f g (x ∷ xs) = cong (f x (g x) ∷_) (map-⊛ f g xs)
 
+toList-map : ∀ (f : A → B) (xs : Vec A n) →
+             toList (map f xs) ≡ List.map f (toList xs)
+toList-map f [] = refl
+toList-map f (x ∷ xs) = cong (f x List.∷_) (toList-map f xs)
+
 ------------------------------------------------------------------------
 -- _++_
 
@@ -509,6 +520,11 @@ lookup-splitAt (suc m) (x ∷ xs) ys (suc i) = trans
   (lookup-splitAt m xs ys i)
   (sym ([,]-map (Fin.splitAt m i)))
 
+toList-++ : (xs : Vec A n) (ys : Vec A m) →
+            toList (xs ++ ys) ≡ toList xs List.++ toList ys
+toList-++ []       ys = refl
+toList-++ (x ∷ xs) ys = cong (x List.∷_) (toList-++ xs ys)
+
 ------------------------------------------------------------------------
 -- concat
 
@@ -645,6 +661,15 @@ lookup-zipWith : ∀ (f : A → B → C) (i : Fin n) xs ys →
 lookup-zipWith _ zero    (x ∷ _)  (y ∷ _)   = refl
 lookup-zipWith _ (suc i) (_ ∷ xs) (_ ∷ ys)  = lookup-zipWith _ i xs ys
 
+zipWith-++ : ∀ (f : A → B → C)
+             (xs : Vec A n) (ys : Vec A m)
+             (xs' : Vec B n) (ys' : Vec B m) →
+             zipWith f (xs ++ ys) (xs' ++ ys') ≡
+             zipWith f xs xs' ++ zipWith f ys ys'
+zipWith-++ f []       ys []         ys' = refl
+zipWith-++ f (x ∷ xs) ys (x' ∷ xs') ys' =
+  cong (_ ∷_) (zipWith-++ f xs ys xs' ys')
+
 ------------------------------------------------------------------------
 -- zip
 
@@ -1002,6 +1027,11 @@ zipWith-replicate₂ _⊕_ []       y = refl
 zipWith-replicate₂ _⊕_ (x ∷ xs) y =
   cong (x ⊕ y ∷_) (zipWith-replicate₂ _⊕_ xs y)
 
+toList-replicate : ∀ (n : ℕ) (x : A) →
+                   toList (replicate {n = n} a) ≡ List.replicate n a
+toList-replicate zero    x = refl
+toList-replicate (suc n) x = cong (_ List.∷_) (toList-replicate n x)
+
 ------------------------------------------------------------------------
 -- tabulate
 

From b57c85bcd7358f4ad337a341601d4011523a4c51 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 17 Aug 2023 02:01:37 +0100
Subject: [PATCH 096/316] Removed redundant `import`s from `Data.Bool.Base`
 (#2062)

---
 src/Data/List/NonEmpty/Base.agda | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/src/Data/List/NonEmpty/Base.agda b/src/Data/List/NonEmpty/Base.agda
index 320cacd4b6..82f16c270a 100644
--- a/src/Data/List/NonEmpty/Base.agda
+++ b/src/Data/List/NonEmpty/Base.agda
@@ -9,8 +9,7 @@
 module Data.List.NonEmpty.Base where
 
 open import Level using (Level)
-open import Data.Bool.Base using (Bool; false; true; not; T)
-open import Data.Bool.Properties using (T?)
+open import Data.Bool.Base using (Bool; false; true)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.Maybe.Base using (Maybe ; nothing; just)
 open import Data.Nat.Base as ℕ

From 5c23f0baa2114c1723a191ce6a838be8d1f1e19b Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 20 Aug 2023 05:18:35 +0100
Subject: [PATCH 097/316] Tidying up `Data.String`  (#2061)

---
 src/Data/String.agda | 20 ++++++--------------
 1 file changed, 6 insertions(+), 14 deletions(-)

diff --git a/src/Data/String.agda b/src/Data/String.agda
index 6a63342ccd..0ad40779fa 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -8,23 +8,17 @@
 
 module Data.String where
 
-open import Data.Bool using (true; false; T?)
+open import Data.Bool.Base using (if_then_else_)
 open import Data.Char as Char using (Char)
-open import Function.Base
-open import Data.Nat.Base as ℕ using (ℕ; _∸_; ⌊_/2⌋; ⌈_/2⌉)
+open import Function.Base using (_∘_; _$_)
+open import Data.Nat.Base as ℕ using (ℕ)
 import Data.Nat.Properties as ℕₚ
-open import Data.List.Base as List using (List; _∷_; []; [_])
-open import Data.List.NonEmpty as NE using (List⁺)
+open import Data.List.Base as List using (List)
 open import Data.List.Extrema ℕₚ.≤-totalOrder
-open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise)
-open import Data.List.Relation.Binary.Lex.Strict using (Lex-<; Lex-≤)
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Char.Base as Char using (Char)
 import Data.Char.Properties as Char using (_≟_)
-open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
-open import Relation.Nullary.Decidable using (does)
-open import Relation.Unary using (Pred; Decidable)
+open import Relation.Nullary.Decidable.Core using (does)
 
 open import Data.List.Membership.DecPropositional Char._≟_
 
@@ -48,9 +42,7 @@ fromVec = fromList ∘ Vec.toList
 
 -- enclose string with parens if it contains a space character
 parensIfSpace : String → String
-parensIfSpace s with does (' ' ∈? toList s)
-... | true  = parens s
-... | false = s
+parensIfSpace s = if does (' ' ∈? toList s) then parens s else s
 
 
 ------------------------------------------------------------------------

From 476ca71e65972e72f1ae8ba8566cc1f5424ff836 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sun, 20 Aug 2023 00:19:20 -0400
Subject: [PATCH 098/316] More `Relation.Binary` simplifications (#2053)

---
 README/Case.agda                                     |  1 -
 README/Data/Record.agda                              |  2 +-
 README/Data/Wrap.agda                                |  1 -
 src/Algebra/Bundles.agda                             |  2 +-
 src/Algebra/Construct/LexProduct.agda                |  3 ++-
 src/Algebra/Construct/LiftedChoice.agda              |  3 ++-
 src/Algebra/Construct/NaturalChoice/Base.agda        |  2 +-
 src/Algebra/Construct/NaturalChoice/Max.agda         |  2 +-
 src/Algebra/Construct/NaturalChoice/MaxOp.agda       |  3 ++-
 src/Algebra/Construct/NaturalChoice/Min.agda         |  2 +-
 src/Algebra/Construct/NaturalChoice/MinMaxOp.agda    |  3 ++-
 src/Algebra/Construct/NaturalChoice/MinOp.agda       |  4 +++-
 src/Algebra/Lattice/Bundles.agda                     |  3 ++-
 src/Algebra/Lattice/Construct/LiftedChoice.agda      |  3 ++-
 .../Lattice/Construct/NaturalChoice/MaxOp.agda       |  2 +-
 .../Lattice/Construct/NaturalChoice/MinMaxOp.agda    |  2 +-
 .../Lattice/Construct/NaturalChoice/MinOp.agda       |  2 +-
 .../Lattice/Morphism/LatticeMonomorphism.agda        |  2 +-
 src/Algebra/Lattice/Properties/BooleanAlgebra.agda   |  1 -
 src/Algebra/Lattice/Properties/Lattice.agda          |  2 +-
 src/Algebra/Module/Bundles.agda                      |  2 +-
 src/Algebra/Module/Definitions.agda                  |  2 --
 src/Algebra/Module/Definitions/Bi.agda               |  2 +-
 src/Algebra/Module/Definitions/Left.agda             |  3 ++-
 src/Algebra/Module/Definitions/Right.agda            |  3 ++-
 src/Algebra/Morphism.agda                            |  2 +-
 .../CommutativeSemiring/Exp/TCOptimised.agda         |  1 -
 src/Algebra/Properties/Lattice.agda                  |  2 +-
 src/Algebra/Properties/Monoid/Divisibility.agda      |  5 ++++-
 src/Algebra/Properties/Monoid/Mult.agda              |  2 +-
 src/Algebra/Properties/Semiring/Exp.agda             |  2 +-
 src/Algebra/Properties/Semiring/Exp/TCOptimised.agda |  2 +-
 .../Semiring/Exp/TailRecursiveOptimised.agda         |  2 +-
 src/Algebra/Solver/Ring.agda                         |  1 -
 src/Algebra/Solver/Ring/AlmostCommutativeRing.agda   |  2 +-
 src/Algebra/Solver/Ring/Simple.agda                  |  2 +-
 .../Guarded/Stream/Relation/Binary/Pointwise.agda    |  6 +++++-
 src/Codata/Musical/Colist.agda                       |  4 +++-
 src/Codata/Musical/Colist/Bisimilarity.agda          |  5 ++++-
 .../Colist/Relation/Unary/Any/Properties.agda        |  2 +-
 src/Codata/Musical/Conat.agda                        |  4 +++-
 src/Codata/Musical/Covec.agda                        |  5 ++++-
 src/Codata/Musical/Stream.agda                       |  4 +++-
 src/Codata/Sized/Colist/Bisimilarity.agda            |  6 +++++-
 src/Codata/Sized/Conat/Bisimilarity.agda             |  3 ++-
 src/Codata/Sized/Covec/Bisimilarity.agda             |  3 ++-
 src/Codata/Sized/Cowriter/Bisimilarity.agda          |  6 +++++-
 src/Codata/Sized/Delay/Bisimilarity.agda             |  3 ++-
 src/Codata/Sized/M/Bisimilarity.agda                 |  6 +++++-
 src/Codata/Sized/Stream/Bisimilarity.agda            |  6 +++++-
 src/Data/Bool/Properties.agda                        |  8 +++++++-
 src/Data/Container/Indexed.agda                      | 10 +++++-----
 src/Data/Fin/Properties.agda                         |  7 ++++++-
 src/Data/Fin/Subset/Properties.agda                  |  7 ++++++-
 src/Data/List/Fresh.agda                             |  7 ++++---
 .../List/Fresh/Membership/Setoid/Properties.agda     |  2 +-
 src/Data/List/Membership/Setoid/Properties.agda      |  4 +++-
 src/Data/List/Relation/Binary/Lex.agda               |  5 ++++-
 .../Binary/Sublist/Heterogeneous/Properties.agda     |  7 ++++++-
 .../Relation/Binary/Subset/Setoid/Properties.agda    |  6 +++++-
 .../Binary/Suffix/Heterogeneous/Properties.agda      |  5 +++--
 src/Data/List/Relation/Unary/All.agda                |  3 ++-
 src/Data/List/Relation/Unary/All/Properties.agda     |  6 ++++--
 src/Data/List/Relation/Unary/AllPairs.agda           |  4 ++--
 src/Data/List/Relation/Unary/Any/Properties.agda     |  3 ++-
 src/Data/List/Relation/Unary/Grouped.agda            |  3 ++-
 src/Data/List/Relation/Unary/Linked/Properties.agda  |  4 +++-
 src/Data/List/Relation/Unary/Sorted/TotalOrder.agda  |  4 ++--
 .../Relation/Unary/Sorted/TotalOrder/Properties.agda |  3 ++-
 .../List/Relation/Unary/Unique/DecPropositional.agda |  3 ++-
 .../Unary/Unique/DecPropositional/Properties.agda    |  2 +-
 src/Data/List/Relation/Unary/Unique/DecSetoid.agda   |  2 +-
 .../Relation/Unary/Unique/DecSetoid/Properties.agda  |  2 +-
 .../Unary/Unique/Propositional/Properties.agda       |  3 ++-
 src/Data/List/Relation/Unary/Unique/Setoid.agda      |  3 ++-
 .../Relation/Unary/Unique/Setoid/Properties.agda     |  3 ++-
 src/Data/Maybe/Relation/Binary/Connected.agda        |  3 ++-
 src/Data/Maybe/Relation/Binary/Pointwise.agda        |  5 ++++-
 .../Product/Function/Dependent/Propositional.agda    |  1 -
 src/Data/Product/Function/Dependent/Setoid.agda      |  4 ++--
 .../Product/Function/Dependent/Setoid/WithK.agda     |  1 -
 .../Product/Function/NonDependent/Propositional.agda |  1 -
 .../Product/Relation/Binary/Pointwise/Dependent.agda | 12 +++++++-----
 src/Data/Unit/Properties.agda                        |  6 +++++-
 src/Data/Vec/Functional/Properties.agda              |  6 +++---
 src/Data/Vec/Properties.agda                         |  2 +-
 src/Data/Vec/Relation/Binary/Lex/Core.agda           |  5 ++++-
 .../Vec/Relation/Binary/Pointwise/Extensional.agda   |  5 ++++-
 src/Function/Bijection.agda                          |  2 +-
 src/Function/Bundles.agda                            |  3 ++-
 src/Function/Consequences.agda                       |  7 ++++---
 src/Function/Construct/Composition.agda              |  4 +++-
 src/Function/Construct/Identity.agda                 |  5 ++++-
 src/Function/Endomorphism/Setoid.agda                |  3 ++-
 src/Function/Equality.agda                           |  2 +-
 src/Function/Equivalence.agda                        |  3 ++-
 src/Function/Injection.agda                          |  2 +-
 src/Function/Inverse.agda                            |  3 ++-
 src/Function/LeftInverse.agda                        |  2 +-
 src/Function/Metric/Nat/Structures.agda              |  2 +-
 src/Function/Metric/Rational/Structures.agda         |  2 +-
 src/Function/Metric/Structures.agda                  |  3 ++-
 src/Function/Properties/Bijection.agda               |  4 +++-
 src/Function/Related.agda                            |  5 ++++-
 src/Function/Related/TypeIsomorphisms.agda           |  1 -
 src/Function/Structures.agda                         |  4 +++-
 src/Function/Surjection.agda                         |  2 +-
 src/Induction/WellFounded.agda                       |  4 +++-
 .../Construct/Closure/Reflexive/Properties.agda      |  6 +++++-
 .../RingSolver/Core/AlmostCommutativeRing.agda       |  2 +-
 110 files changed, 251 insertions(+), 130 deletions(-)

diff --git a/README/Case.agda b/README/Case.agda
index 9de25c6e82..30ce0221aa 100644
--- a/README/Case.agda
+++ b/README/Case.agda
@@ -14,7 +14,6 @@ open import Data.Maybe hiding (from-just)
 open import Data.Nat   hiding (pred)
 open import Function.Base using (case_of_; case_return_of_)
 open import Relation.Nullary
-open import Relation.Binary
 
 ------------------------------------------------------------------------
 -- Different types of pattern-matching lambdas
diff --git a/README/Data/Record.agda b/README/Data/Record.agda
index da0ae70dc4..63fea328ae 100644
--- a/README/Data/Record.agda
+++ b/README/Data/Record.agda
@@ -15,7 +15,7 @@ open import Data.Product.Base using (_,_)
 open import Data.String
 open import Function.Base using (flip)
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Definitions using (Symmetric; Transitive)
 
 import Data.Record as Record
 
diff --git a/README/Data/Wrap.agda b/README/Data/Wrap.agda
index b92a8a3c81..7273e787d1 100644
--- a/README/Data/Wrap.agda
+++ b/README/Data/Wrap.agda
@@ -15,7 +15,6 @@ open import Data.Nat
 open import Data.Nat.Properties
 open import Data.Product.Base using (_,_; ∃; ∃₂; _×_)
 open import Level using (Level)
-open import Relation.Binary
 
 private
   variable
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index 2598fde953..f20cf1ad03 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -14,7 +14,7 @@ module Algebra.Bundles where
 import Algebra.Bundles.Raw as Raw
 open import Algebra.Core
 open import Algebra.Structures
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 open import Function.Base
 import Relation.Nullary as N
 open import Level
diff --git a/src/Algebra/Construct/LexProduct.agda b/src/Algebra/Construct/LexProduct.agda
index 984cdd57f2..e4d51ff9ed 100644
--- a/src/Algebra/Construct/LexProduct.agda
+++ b/src/Algebra/Construct/LexProduct.agda
@@ -12,7 +12,8 @@ open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (Pointwise)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (_∘_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary using (¬_; does; yes; no)
 open import Relation.Nullary.Negation using (contradiction; contradiction₂)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
diff --git a/src/Algebra/Construct/LiftedChoice.agda b/src/Algebra/Construct/LiftedChoice.agda
index 91d3f1d546..2433c87b9d 100644
--- a/src/Algebra/Construct/LiftedChoice.agda
+++ b/src/Algebra/Construct/LiftedChoice.agda
@@ -11,7 +11,8 @@ open import Algebra
 module Algebra.Construct.LiftedChoice where
 
 open import Algebra.Consequences.Base
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_; _Preserves_⟶_)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Nullary using (¬_; yes; no)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Product.Base using (_×_; _,_)
diff --git a/src/Algebra/Construct/NaturalChoice/Base.agda b/src/Algebra/Construct/NaturalChoice/Base.agda
index 3ebd9b924f..6b4d4d13b7 100644
--- a/src/Algebra/Construct/NaturalChoice/Base.agda
+++ b/src/Algebra/Construct/NaturalChoice/Base.agda
@@ -10,7 +10,7 @@
 open import Algebra.Core
 open import Level as L hiding (_⊔_)
 open import Function.Base using (flip)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (TotalPreorder)
 open import Relation.Binary.Construct.Flip.EqAndOrd using ()
   renaming (totalPreorder to flipOrder)
 import Relation.Binary.Properties.TotalOrder as TotalOrderProperties
diff --git a/src/Algebra/Construct/NaturalChoice/Max.agda b/src/Algebra/Construct/NaturalChoice/Max.agda
index 212a52602f..3e4c8f7db8 100644
--- a/src/Algebra/Construct/NaturalChoice/Max.agda
+++ b/src/Algebra/Construct/NaturalChoice/Max.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (TotalOrder)
 
 module Algebra.Construct.NaturalChoice.Max
   {a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂) where
diff --git a/src/Algebra/Construct/NaturalChoice/MaxOp.agda b/src/Algebra/Construct/NaturalChoice/MaxOp.agda
index d6c7de61ba..8e4b032950 100644
--- a/src/Algebra/Construct/NaturalChoice/MaxOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MaxOp.agda
@@ -11,7 +11,8 @@ open import Algebra.Core
 open import Algebra.Construct.NaturalChoice.Base
 import Algebra.Construct.NaturalChoice.MinOp as MinOp
 open import Function.Base using (flip)
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_)
+open import Relation.Binary.Bundles using (TotalPreorder)
 open import Relation.Binary.Construct.Flip.EqAndOrd using ()
   renaming (totalPreorder to flipOrder)
 
diff --git a/src/Algebra/Construct/NaturalChoice/Min.agda b/src/Algebra/Construct/NaturalChoice/Min.agda
index 41f895afe6..6d2482a4a9 100644
--- a/src/Algebra/Construct/NaturalChoice/Min.agda
+++ b/src/Algebra/Construct/NaturalChoice/Min.agda
@@ -12,7 +12,7 @@ open import Algebra.Construct.NaturalChoice.Base
 open import Data.Sum.Base using (inj₁; inj₂; [_,_])
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (TotalOrder)
 import Algebra.Construct.NaturalChoice.MinOp as MinOp
 
 module Algebra.Construct.NaturalChoice.Min
diff --git a/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda b/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
index a0b7a945f2..04866f0ad0 100644
--- a/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
@@ -13,7 +13,8 @@ open import Algebra.Construct.NaturalChoice.Base
 open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id; _∘_; flip)
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_)
+open import Relation.Binary.Bundles using (TotalPreorder)
 open import Relation.Binary.Consequences
 
 module Algebra.Construct.NaturalChoice.MinMaxOp
diff --git a/src/Algebra/Construct/NaturalChoice/MinOp.agda b/src/Algebra/Construct/NaturalChoice/MinOp.agda
index 563e1662f1..744d32e1fe 100644
--- a/src/Algebra/Construct/NaturalChoice/MinOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MinOp.agda
@@ -13,7 +13,9 @@ open import Algebra.Construct.NaturalChoice.Base
 open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_])
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id; _∘_)
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Bundles using (TotalPreorder)
+open import Relation.Binary.Definitions using (Maximum; Minimum)
 open import Relation.Binary.Consequences
 
 module Algebra.Construct.NaturalChoice.MinOp
diff --git a/src/Algebra/Lattice/Bundles.agda b/src/Algebra/Lattice/Bundles.agda
index e278d0ffb4..28fd07d5b3 100644
--- a/src/Algebra/Lattice/Bundles.agda
+++ b/src/Algebra/Lattice/Bundles.agda
@@ -21,7 +21,8 @@ open import Algebra.Structures
 import Algebra.Lattice.Bundles.Raw as Raw
 open import Algebra.Lattice.Structures
 open import Level using (suc; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Core using (Rel)
 
 ------------------------------------------------------------------------
 -- Re-export definitions of 'raw' bundles
diff --git a/src/Algebra/Lattice/Construct/LiftedChoice.agda b/src/Algebra/Lattice/Construct/LiftedChoice.agda
index bcedcf2074..01cef531c6 100644
--- a/src/Algebra/Lattice/Construct/LiftedChoice.agda
+++ b/src/Algebra/Lattice/Construct/LiftedChoice.agda
@@ -9,7 +9,8 @@
 open import Algebra
 open import Algebra.Lattice
 open import Algebra.Construct.LiftedChoice
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Level using (Level)
 
 module Algebra.Lattice.Construct.LiftedChoice where
diff --git a/src/Algebra/Lattice/Construct/NaturalChoice/MaxOp.agda b/src/Algebra/Lattice/Construct/NaturalChoice/MaxOp.agda
index 445e685da0..4d00c28524 100644
--- a/src/Algebra/Lattice/Construct/NaturalChoice/MaxOp.agda
+++ b/src/Algebra/Lattice/Construct/NaturalChoice/MaxOp.agda
@@ -9,7 +9,7 @@
 
 open import Algebra.Construct.NaturalChoice.Base
 import Algebra.Lattice.Construct.NaturalChoice.MinOp as MinOp
-open import Relation.Binary
+open import Relation.Binary.Bundles using (TotalPreorder)
 
 module Algebra.Lattice.Construct.NaturalChoice.MaxOp
   {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂} (maxOp : MaxOperator O)
diff --git a/src/Algebra/Lattice/Construct/NaturalChoice/MinMaxOp.agda b/src/Algebra/Lattice/Construct/NaturalChoice/MinMaxOp.agda
index 058bd694cf..b574147807 100644
--- a/src/Algebra/Lattice/Construct/NaturalChoice/MinMaxOp.agda
+++ b/src/Algebra/Lattice/Construct/NaturalChoice/MinMaxOp.agda
@@ -8,7 +8,7 @@
 
 open import Algebra.Lattice.Bundles
 open import Algebra.Construct.NaturalChoice.Base
-open import Relation.Binary
+open import Relation.Binary.Bundles using (TotalPreorder)
 
 module Algebra.Lattice.Construct.NaturalChoice.MinMaxOp
   {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂}
diff --git a/src/Algebra/Lattice/Construct/NaturalChoice/MinOp.agda b/src/Algebra/Lattice/Construct/NaturalChoice/MinOp.agda
index 19468c0851..ac4328ded3 100644
--- a/src/Algebra/Lattice/Construct/NaturalChoice/MinOp.agda
+++ b/src/Algebra/Lattice/Construct/NaturalChoice/MinOp.agda
@@ -10,7 +10,7 @@
 open import Algebra.Bundles
 open import Algebra.Lattice.Bundles
 open import Algebra.Construct.NaturalChoice.Base
-open import Relation.Binary
+open import Relation.Binary.Bundles using (TotalPreorder)
 
 module Algebra.Lattice.Construct.NaturalChoice.MinOp
   {a ℓ₁ ℓ₂} {O : TotalPreorder a ℓ₁ ℓ₂} (minOp : MinOperator O) where
diff --git a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
index c2c8301409..fe3ceec5be 100644
--- a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
+++ b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
@@ -16,7 +16,7 @@ import Algebra.Consequences.Setoid as Consequences
 import Algebra.Morphism.MagmaMonomorphism as MagmaMonomorphisms
 import Algebra.Lattice.Properties.Lattice as LatticeProperties
 open import Data.Product.Base using (_,_; map)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.Morphism.RelMonomorphism as RelMonomorphisms
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
index cf90b022b9..85f22c39be 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
@@ -22,7 +22,6 @@ open import Algebra.Consequences.Setoid setoid
 open import Algebra.Bundles
 open import Algebra.Lattice.Structures _≈_
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Relation.Binary
 open import Function.Base using (id; _$_; _⟨_⟩_)
 open import Function.Bundles using (_⇔_; module Equivalence)
 open import Data.Product.Base using (_,_)
diff --git a/src/Algebra/Lattice/Properties/Lattice.agda b/src/Algebra/Lattice/Properties/Lattice.agda
index b5908f57b1..b83dcdea10 100644
--- a/src/Algebra/Lattice/Properties/Lattice.agda
+++ b/src/Algebra/Lattice/Properties/Lattice.agda
@@ -8,7 +8,7 @@
 
 open import Algebra.Lattice.Bundles
 import Algebra.Lattice.Properties.Semilattice as SemilatticeProperties
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Poset)
 import Relation.Binary.Lattice as R
 open import Function.Base
 open import Data.Product.Base using (_,_; swap)
diff --git a/src/Algebra/Module/Bundles.agda b/src/Algebra/Module/Bundles.agda
index e6b33ed065..f36e5bccd0 100644
--- a/src/Algebra/Module/Bundles.agda
+++ b/src/Algebra/Module/Bundles.agda
@@ -34,7 +34,7 @@ open import Algebra.Module.Definitions
 open import Algebra.Properties.Group
 open import Function.Base
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary    using (¬_)
 import Relation.Binary.Reasoning.Setoid as SetR
 
diff --git a/src/Algebra/Module/Definitions.agda b/src/Algebra/Module/Definitions.agda
index cb02f94469..19c1abdb46 100644
--- a/src/Algebra/Module/Definitions.agda
+++ b/src/Algebra/Module/Definitions.agda
@@ -7,8 +7,6 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
-
 module Algebra.Module.Definitions where
 
   import Algebra.Module.Definitions.Left as L
diff --git a/src/Algebra/Module/Definitions/Bi.agda b/src/Algebra/Module/Definitions/Bi.agda
index a6aa511af2..b85589b6fb 100644
--- a/src/Algebra/Module/Definitions/Bi.agda
+++ b/src/Algebra/Module/Definitions/Bi.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 
 -- The properties are parameterised by the three carriers and
 -- the result equality.
diff --git a/src/Algebra/Module/Definitions/Left.agda b/src/Algebra/Module/Definitions/Left.agda
index 3c04e82cd0..e8aa153507 100644
--- a/src/Algebra/Module/Definitions/Left.agda
+++ b/src/Algebra/Module/Definitions/Left.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core
+  using (Rel; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
 
 -- The properties are parameterised by the two carriers and
 -- the result equality.
diff --git a/src/Algebra/Module/Definitions/Right.agda b/src/Algebra/Module/Definitions/Right.agda
index 92b90a477c..d1a7a878ea 100644
--- a/src/Algebra/Module/Definitions/Right.agda
+++ b/src/Algebra/Module/Definitions/Right.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core
+  using (Rel; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
 
 -- The properties are parameterised by the two carriers and
 -- the result equality.
diff --git a/src/Algebra/Morphism.agda b/src/Algebra/Morphism.agda
index 59a6f22e23..1bba00bab2 100644
--- a/src/Algebra/Morphism.agda
+++ b/src/Algebra/Morphism.agda
@@ -13,7 +13,7 @@ open import Algebra
 import Algebra.Properties.Group as GroupP
 open import Function.Base
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
 import Relation.Binary.Reasoning.Setoid as EqR
 
 private
diff --git a/src/Algebra/Properties/CommutativeSemiring/Exp/TCOptimised.agda b/src/Algebra/Properties/CommutativeSemiring/Exp/TCOptimised.agda
index 6c06319032..67b37378bc 100644
--- a/src/Algebra/Properties/CommutativeSemiring/Exp/TCOptimised.agda
+++ b/src/Algebra/Properties/CommutativeSemiring/Exp/TCOptimised.agda
@@ -9,7 +9,6 @@
 open import Algebra
 open import Data.Nat.Base as ℕ using (zero; suc)
 import Data.Nat.Properties as ℕ
-open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 module Algebra.Properties.CommutativeSemiring.Exp.TCOptimised
diff --git a/src/Algebra/Properties/Lattice.agda b/src/Algebra/Properties/Lattice.agda
index ed5bfd00dc..57037296f6 100644
--- a/src/Algebra/Properties/Lattice.agda
+++ b/src/Algebra/Properties/Lattice.agda
@@ -8,7 +8,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Lattice.Bundles
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 open import Function.Base
 open import Function.Bundles using (module Equivalence; _⇔_)
 open import Data.Product.Base using (_,_; swap)
diff --git a/src/Algebra/Properties/Monoid/Divisibility.agda b/src/Algebra/Properties/Monoid/Divisibility.agda
index f2c8c1f20f..588f6ef375 100644
--- a/src/Algebra/Properties/Monoid/Divisibility.agda
+++ b/src/Algebra/Properties/Monoid/Divisibility.agda
@@ -8,7 +8,10 @@
 
 open import Algebra using (Monoid)
 open import Data.Product.Base using (_,_)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Preorder)
+open import Relation.Binary.Structures using (IsPreorder; IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive)
 
 module Algebra.Properties.Monoid.Divisibility
   {a ℓ} (M : Monoid a ℓ) where
diff --git a/src/Algebra/Properties/Monoid/Mult.agda b/src/Algebra/Properties/Monoid/Mult.agda
index 1a9e346585..7abdb583ca 100644
--- a/src/Algebra/Properties/Monoid/Mult.agda
+++ b/src/Algebra/Properties/Monoid/Mult.agda
@@ -8,7 +8,7 @@
 
 open import Algebra.Bundles using (Monoid)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; NonZero)
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 module Algebra.Properties.Monoid.Mult {a ℓ} (M : Monoid a ℓ) where
diff --git a/src/Algebra/Properties/Semiring/Exp.agda b/src/Algebra/Properties/Semiring/Exp.agda
index 1d50298ca5..5da80a99f1 100644
--- a/src/Algebra/Properties/Semiring/Exp.agda
+++ b/src/Algebra/Properties/Semiring/Exp.agda
@@ -8,7 +8,7 @@
 
 open import Algebra
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Data.Nat.Properties as ℕ
 
diff --git a/src/Algebra/Properties/Semiring/Exp/TCOptimised.agda b/src/Algebra/Properties/Semiring/Exp/TCOptimised.agda
index 23a11524cb..5642bde51c 100644
--- a/src/Algebra/Properties/Semiring/Exp/TCOptimised.agda
+++ b/src/Algebra/Properties/Semiring/Exp/TCOptimised.agda
@@ -9,7 +9,7 @@
 open import Algebra
 open import Data.Nat.Base as ℕ using (zero; suc)
 import Data.Nat.Properties as ℕ
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 module Algebra.Properties.Semiring.Exp.TCOptimised
diff --git a/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda b/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda
index 836c794e10..d661dde6b1 100644
--- a/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda
+++ b/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda
@@ -9,7 +9,7 @@
 open import Algebra
 open import Data.Nat.Base as ℕ using (zero; suc)
 import Data.Nat.Properties as ℕ
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
 
 module Algebra.Properties.Semiring.Exp.TailRecursiveOptimised
   {a ℓ} (S : Semiring a ℓ) where
diff --git a/src/Algebra/Solver/Ring.agda b/src/Algebra/Solver/Ring.agda
index 5bc696da38..50c191903b 100644
--- a/src/Algebra/Solver/Ring.agda
+++ b/src/Algebra/Solver/Ring.agda
@@ -39,7 +39,6 @@ open import Algebra.Morphism
 open _-Raw-AlmostCommutative⟶_ morphism renaming (⟦_⟧ to ⟦_⟧′)
 open import Algebra.Properties.Semiring.Exp semiring
 
-open import Relation.Binary
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Binary.Reasoning.Setoid setoid
 import Relation.Binary.PropositionalEquality.Core as PropEq
diff --git a/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda b/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
index 2bcfe92901..4c33c7b337 100644
--- a/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
+++ b/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
@@ -16,7 +16,7 @@ import Algebra.Morphism as Morphism
 import Algebra.Morphism.Definitions as MorphismDefinitions
 open import Function.Base using (id)
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 
 
 record IsAlmostCommutativeRing {a ℓ} {A : Set a} (_≈_ : Rel A ℓ)
diff --git a/src/Algebra/Solver/Ring/Simple.agda b/src/Algebra/Solver/Ring/Simple.agda
index 86a7d1ca04..93bcc1a7fa 100644
--- a/src/Algebra/Solver/Ring/Simple.agda
+++ b/src/Algebra/Solver/Ring/Simple.agda
@@ -8,7 +8,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Solver.Ring.AlmostCommutativeRing
-open import Relation.Binary
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.Consequences using (dec⇒weaklyDec)
 
 module Algebra.Solver.Ring.Simple
diff --git a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
index 3f610599a5..acc7153f52 100644
--- a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
+++ b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
@@ -12,7 +12,11 @@ open import Codata.Guarded.Stream as Stream using (Stream; head; tail)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Function.Base using (_∘_; _on_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; _⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions
+  using (Reflexive; Sym; Trans; Antisym; Symmetric; Transitive)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index 3ef672abc7..fdfbfc00b1 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -28,7 +28,9 @@ open import Function.Base
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse as Inv using (_↔_; _↔̇_; Inverse; inverse)
 open import Level using (_⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles using (Poset; Setoid; Preorder)
+open import Relation.Binary.Definitions using (Transitive; Antisymmetric)
 import Relation.Binary.Construct.FromRel as Ind
 import Relation.Binary.Reasoning.Preorder as PreR
 import Relation.Binary.Reasoning.PartialOrder as POR
diff --git a/src/Codata/Musical/Colist/Bisimilarity.agda b/src/Codata/Musical/Colist/Bisimilarity.agda
index 64a0c37d9c..304e20db4b 100644
--- a/src/Codata/Musical/Colist/Bisimilarity.agda
+++ b/src/Codata/Musical/Colist/Bisimilarity.agda
@@ -11,7 +11,10 @@ module Codata.Musical.Colist.Bisimilarity where
 open import Codata.Musical.Colist.Base
 open import Codata.Musical.Notation
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _=[_]⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 
 private
   variable
diff --git a/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda b/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
index 8ea859f4a7..874a338a85 100644
--- a/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
+++ b/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
@@ -21,7 +21,7 @@ open import Function.Base using (_∋_; _∘_)
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse as Inv using (_↔_; _↔̇_; Inverse; inverse)
 open import Level using (Level; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; refl; cong)
 open import Relation.Unary using (Pred)
diff --git a/src/Codata/Musical/Conat.agda b/src/Codata/Musical/Conat.agda
index d8e76d90be..f4dab55a6f 100644
--- a/src/Codata/Musical/Conat.agda
+++ b/src/Codata/Musical/Conat.agda
@@ -11,7 +11,9 @@ module Codata.Musical.Conat where
 open import Codata.Musical.Notation
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Function.Base using (_∋_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
diff --git a/src/Codata/Musical/Covec.agda b/src/Codata/Musical/Covec.agda
index 8993fd5aa8..2658fd0311 100644
--- a/src/Codata/Musical/Covec.agda
+++ b/src/Codata/Musical/Covec.agda
@@ -17,7 +17,10 @@ open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_; _=[_]⇒_)
+open import Relation.Binary.Bundles using (Setoid; Poset)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Antisymmetric)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 private
diff --git a/src/Codata/Musical/Stream.agda b/src/Codata/Musical/Stream.agda
index a51f8f41b5..d0d655af65 100644
--- a/src/Codata/Musical/Stream.agda
+++ b/src/Codata/Musical/Stream.agda
@@ -13,7 +13,9 @@ open import Codata.Musical.Notation
 open import Codata.Musical.Colist using (Colist; []; _∷_)
 open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Relation.Binary
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 private
diff --git a/src/Codata/Sized/Colist/Bisimilarity.agda b/src/Codata/Sized/Colist/Bisimilarity.agda
index d66ade8b89..4ea3a18047 100644
--- a/src/Codata/Sized/Colist/Bisimilarity.agda
+++ b/src/Codata/Sized/Colist/Bisimilarity.agda
@@ -15,7 +15,11 @@ open import Codata.Sized.Colist
 open import Data.List.Base using (List; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.List.NonEmpty as List⁺  using (List⁺; _∷_)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Sym; Trans)
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as Eq
 
diff --git a/src/Codata/Sized/Conat/Bisimilarity.agda b/src/Codata/Sized/Conat/Bisimilarity.agda
index ae3e306aca..1b5aaea922 100644
--- a/src/Codata/Sized/Conat/Bisimilarity.agda
+++ b/src/Codata/Sized/Conat/Bisimilarity.agda
@@ -12,7 +12,8 @@ open import Level using (0ℓ)
 open import Size
 open import Codata.Sized.Thunk
 open import Codata.Sized.Conat
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 infix 1 _⊢_≈_
 data _⊢_≈_ i : (m n : Conat ∞) → Set where
diff --git a/src/Codata/Sized/Covec/Bisimilarity.agda b/src/Codata/Sized/Covec/Bisimilarity.agda
index 7610c41946..685b7f180f 100644
--- a/src/Codata/Sized/Covec/Bisimilarity.agda
+++ b/src/Codata/Sized/Covec/Bisimilarity.agda
@@ -13,7 +13,8 @@ open import Size
 open import Codata.Sized.Thunk
 open import Codata.Sized.Conat hiding (_⊔_)
 open import Codata.Sized.Covec
-open import Relation.Binary
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Sym; Trans)
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 
 data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) (i : Size) :
diff --git a/src/Codata/Sized/Cowriter/Bisimilarity.agda b/src/Codata/Sized/Cowriter/Bisimilarity.agda
index 84ab0375c4..d9ce3e2d6f 100644
--- a/src/Codata/Sized/Cowriter/Bisimilarity.agda
+++ b/src/Codata/Sized/Cowriter/Bisimilarity.agda
@@ -12,7 +12,11 @@ open import Level using (Level; _⊔_)
 open import Size
 open import Codata.Sized.Thunk
 open import Codata.Sized.Cowriter
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Sym; Trans)
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as Eq
 
diff --git a/src/Codata/Sized/Delay/Bisimilarity.agda b/src/Codata/Sized/Delay/Bisimilarity.agda
index 4d9f702acc..f4845669ba 100644
--- a/src/Codata/Sized/Delay/Bisimilarity.agda
+++ b/src/Codata/Sized/Delay/Bisimilarity.agda
@@ -12,7 +12,8 @@ open import Size
 open import Codata.Sized.Thunk
 open import Codata.Sized.Delay
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Sym; Trans)
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 
 data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) i :
diff --git a/src/Codata/Sized/M/Bisimilarity.agda b/src/Codata/Sized/M/Bisimilarity.agda
index 8205ae92e3..9c641ec834 100644
--- a/src/Codata/Sized/M/Bisimilarity.agda
+++ b/src/Codata/Sized/M/Bisimilarity.agda
@@ -16,7 +16,11 @@ open import Data.Container.Core
 open import Data.Container.Relation.Binary.Pointwise using (Pointwise; _,_)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 import Relation.Binary.PropositionalEquality.Core as P
 
 data Bisim {s p} (C : Container s p) (i : Size) : Rel (M C ∞) (s ⊔ p) where
diff --git a/src/Codata/Sized/Stream/Bisimilarity.agda b/src/Codata/Sized/Stream/Bisimilarity.agda
index c036d030b7..f08d0daeb8 100644
--- a/src/Codata/Sized/Stream/Bisimilarity.agda
+++ b/src/Codata/Sized/Stream/Bisimilarity.agda
@@ -14,7 +14,11 @@ open import Codata.Sized.Stream
 open import Level
 open import Data.List.NonEmpty as List⁺ using (_∷_)
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Sym; Trans)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as Eq
 
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index 713ad3da34..3c073cd23a 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -21,7 +21,13 @@ open import Function.Equivalence
   using (_⇔_; equivalence; module Equivalence)
 open import Induction.WellFounded using (WellFounded; Acc; acc)
 open import Level using (Level; 0ℓ)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Poset; Preorder; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Decidable; Reflexive; Transitive; Antisymmetric; Minimum; Maximum; Total; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri<; tri>; _Respects₂_)
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Binary.PropositionalEquality.Properties
 open import Relation.Nullary.Reflects using (ofʸ; ofⁿ)
diff --git a/src/Data/Container/Indexed.agda b/src/Data/Container/Indexed.agda
index 26aeaa25b0..fa2e7fc33a 100644
--- a/src/Data/Container/Indexed.agda
+++ b/src/Data/Container/Indexed.agda
@@ -18,7 +18,7 @@ open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse using (_↔_; module Inverse)
 open import Relation.Unary using (Pred; _⊆_)
-import Relation.Binary as B
+open import Relation.Binary.Core using (Rel; REL)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_; refl)
 
 ------------------------------------------------------------------------
@@ -66,7 +66,7 @@ module _ {i₁ i₂ o₁ o₂}
   record ContainerMorphism {c₁ c₂ r₁ r₂ ℓ₁ ℓ₂}
          (C₁ : Container I₁ O₁ c₁ r₁) (C₂ : Container I₂ O₂ c₂ r₂)
          (f : I₁ → I₂) (g : O₁ → O₂)
-         (_∼_ : B.Rel I₂ ℓ₁) (_≈_ : B.REL (Set r₂) (Set r₁) ℓ₂)
+         (_∼_ : Rel I₂ ℓ₁) (_≈_ : REL (Set r₂) (Set r₁) ℓ₂)
          (_·_ : ∀ {A B} → A ≈ B → A → B) :
          Set (i₁ ⊔ i₂ ⊔ o₁ ⊔ o₂ ⊔ c₁ ⊔ c₂ ⊔ r₁ ⊔ r₂ ⊔ ℓ₁ ⊔ ℓ₂) where
     field
@@ -107,13 +107,13 @@ module _ {i o c r} {I : Set i} {O : Set o} where
 
   infixr 8 _⇒_ _⊸_ _⇒C_
 
-  _⇒_ : B.Rel (Container I O c r) _
+  _⇒_ : Rel (Container I O c r) _
   C₁ ⇒ C₂ = C₁ ⇒[ ⟨id⟩ / ⟨id⟩ ] C₂
 
-  _⊸_ : B.Rel (Container I O c r) _
+  _⊸_ : Rel (Container I O c r) _
   C₁ ⊸ C₂ = C₁ ⊸[ ⟨id⟩ / ⟨id⟩ ] C₂
 
-  _⇒C_ : B.Rel (Container I O c r) _
+  _⇒C_ : Rel (Container I O c r) _
   C₁ ⇒C C₂ = C₁ ⇒C[ ⟨id⟩ / ⟨id⟩ ] C₂
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 2bacfe58ca..e8a1cef5dc 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -35,7 +35,12 @@ open import Function.Definitions using (Injective; Surjective)
 open import Function.Consequences.Propositional using (contraInjective)
 open import Function.Construct.Composition as Comp hiding (injective)
 open import Level using (Level)
-open import Relation.Binary as B hiding (Decidable; _⇔_)
+open import Relation.Binary.Definitions as B hiding (Decidable)
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
+open import Relation.Binary.Bundles
+  using (Preorder; Setoid; DecSetoid; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_; module ≡-Reasoning)
 open import Relation.Nullary
diff --git a/src/Data/Fin/Subset/Properties.agda b/src/Data/Fin/Subset/Properties.agda
index 1cc8bd19f9..2f7246da6a 100644
--- a/src/Data/Fin/Subset/Properties.agda
+++ b/src/Data/Fin/Subset/Properties.agda
@@ -30,7 +30,12 @@ open import Data.Vec.Properties
 open import Function.Base using (_∘_; const; id; case_of_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level)
-open import Relation.Binary as B hiding (Decidable; _⇔_)
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsStrictPartialOrder; IsDecStrictPartialOrder)
+open import Relation.Binary.Bundles
+  using (Preorder; Poset; StrictPartialOrder; DecStrictPartialOrder)
+open import Relation.Binary.Definitions as B hiding (Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _⊎-dec_)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/List/Fresh.agda b/src/Data/List/Fresh.agda
index b5a5718765..3375b0d54a 100644
--- a/src/Data/List/Fresh.agda
+++ b/src/Data/List/Fresh.agda
@@ -22,9 +22,10 @@ open import Data.List.Relation.Unary.AllPairs using (AllPairs; []; _∷_)
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Function.Base using (_∘′_; flip; id; _on_)
-open import Relation.Nullary      using (does)
-open import Relation.Unary   as U using (Pred)
-open import Relation.Binary  as B using (Rel)
+open import Relation.Nullary using (does)
+open import Relation.Unary as U using (Pred)
+open import Relation.Binary.Core using (Rel)
+import Relation.Binary.Definitions as B
 open import Relation.Nary
 
 private
diff --git a/src/Data/List/Fresh/Membership/Setoid/Properties.agda b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
index d2f2634bf8..ce3f7df88b 100644
--- a/src/Data/List/Fresh/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
@@ -21,7 +21,7 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; fromInj₂)
 open import Function.Base using (id; _∘′_; _$_)
 open import Relation.Nullary
 open import Relation.Unary as U using (Pred)
-import Relation.Binary as B
+import Relation.Binary.Definitions as B
 import Relation.Binary.PropositionalEquality.Core as P
 open import Relation.Nary
 
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 5488f9a037..4072da5ed8 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -27,7 +27,9 @@ open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (_$_; flip; _∘_; _∘′_; id)
 open import Function.Inverse using (_↔_)
 open import Level using (Level)
-open import Relation.Binary as B hiding (Decidable)
+open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_; _Preserves_⟶_)
+open import Relation.Binary.Definitions as B hiding (Decidable)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Unary as U using (Decidable; Pred)
 open import Relation.Nullary using (¬_; does; _because_; yes; no)
diff --git a/src/Data/List/Relation/Binary/Lex.agda b/src/Data/List/Relation/Binary/Lex.agda
index c1a07e8b53..6c8fb4b323 100644
--- a/src/Data/List/Relation/Binary/Lex.agda
+++ b/src/Data/List/Relation/Binary/Lex.agda
@@ -19,7 +19,10 @@ open import Level using (_⊔_)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Decidable as Dec
   using (Dec; yes; no; _×-dec_; _⊎-dec_)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Symmetric; Transitive; Irreflexive; Asymmetric; Antisymmetric; Decidable; _Respects₂_; _Respects_)
 open import Data.List.Relation.Binary.Pointwise.Base
    using (Pointwise; []; _∷_; head; tail)
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index 46d5d7d2cc..b11e011d76 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -33,7 +33,12 @@ open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (Dec; does; _because_; yes; no; ¬_)
 open import Relation.Nullary.Decidable as Dec using (¬?)
 open import Relation.Unary as U using (Pred)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Bundles using (Preorder; Poset; DecPoset)
+open import Relation.Binary.Definitions
+  using (Reflexive; Trans; Antisym; Decidable; Irrelevant; Irreflexive)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsDecPartialOrder)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index 5229813a50..551b721d8d 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -26,7 +26,11 @@ open import Level using (Level)
 open import Relation.Nullary using (¬_; does; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Unary using (Pred; Decidable) renaming (_⊆_ to _⋐_)
-open import Relation.Binary hiding (Decidable)
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Definitions
+  using (Reflexive; Transitive; _Respectsʳ_; _Respectsˡ_; _Respects_)
+open import Relation.Binary.Bundles using (Setoid; Preorder)
+open import Relation.Binary.Structures using (IsPreorder)
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 
 open Setoid using (Carrier)
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
index 05b7725d31..3ea10495e6 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
@@ -24,8 +24,9 @@ open import Relation.Nullary using (Dec; does; ¬_)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Unary as U using (Pred)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary as B
-  using (REL; Rel; Trans; Antisym; Irrelevant; _⇒_)
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Definitions as B
+  using (Trans; Antisym; Irrelevant)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; sym; subst; subst₂)
 
diff --git a/src/Data/List/Relation/Unary/All.agda b/src/Data/List/Relation/Unary/All.agda
index 00d8e6f7b7..97da349699 100644
--- a/src/Data/List/Relation/Unary/All.agda
+++ b/src/Data/List/Relation/Unary/All.agda
@@ -23,7 +23,8 @@ open import Level using (Level; _⊔_)
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Unary hiding (_∈_)
-open import Relation.Binary using (Setoid; _Respects_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (_Respects_)
 open import Relation.Binary.PropositionalEquality.Core as P
 import Relation.Binary.PropositionalEquality.Properties as P
 
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index af41a14041..1a75b5eb35 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -38,7 +38,9 @@ open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
 open import Function.Surjection using (_↠_; surjection)
 open import Level using (Level)
-open import Relation.Binary as B using (REL; Setoid; _Respects_)
+open import Relation.Binary.Core using (REL)
+open import Relation.Binary.Bundles using (Setoid)
+import Relation.Binary.Definitions as B
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; cong; cong₂; _≗_)
 open import Relation.Nullary
@@ -731,7 +733,7 @@ module _ (S : Setoid c ℓ) where
   open Setoid S
   open ListEq S
 
-  respects : P Respects _≈_ → (All P) Respects _≋_
+  respects : P B.Respects _≈_ → (All P) B.Respects _≋_
   respects p≈ []            []         = []
   respects p≈ (x≈y ∷ xs≈ys) (px ∷ pxs) = p≈ x≈y px ∷ respects p≈ xs≈ys pxs
 
diff --git a/src/Data/List/Relation/Unary/AllPairs.agda b/src/Data/List/Relation/Unary/AllPairs.agda
index 49bfac6ae0..bed4a5972d 100644
--- a/src/Data/List/Relation/Unary/AllPairs.agda
+++ b/src/Data/List/Relation/Unary/AllPairs.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel)
+open import Relation.Binary.Core using (Rel; _⇒_)
 
 module Data.List.Relation.Unary.AllPairs
        {a ℓ} {A : Set a} {R : Rel A ℓ} where
@@ -16,7 +16,7 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (_⊔_)
-open import Relation.Binary as B using (Rel; _⇒_)
+open import Relation.Binary.Definitions as B
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index 8f3bcd640c..b50b962e26 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -40,7 +40,8 @@ open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
 open import Function.Inverse as Inv using (_↔_; inverse; Inverse)
 open import Function.Related as Related using (Kind; Related; SK-sym)
 open import Level using (Level)
-open import Relation.Binary as B hiding (_⇔_)
+open import Relation.Binary.Core using (Rel; REL)
+open import Relation.Binary.Definitions as B
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
 open import Relation.Unary as U
diff --git a/src/Data/List/Relation/Unary/Grouped.agda b/src/Data/List/Relation/Unary/Grouped.agda
index 87b11cabc3..f10bc1fd32 100644
--- a/src/Data/List/Relation/Unary/Grouped.agda
+++ b/src/Data/List/Relation/Unary/Grouped.agda
@@ -12,7 +12,8 @@ open import Data.List.Base using (List; []; _∷_; map)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_; all?)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Product.Base using (_×_; _,_)
-open import Relation.Binary as B using (REL; Rel)
+open import Relation.Binary.Core using (REL; Rel)
+open import Relation.Binary.Definitions as B
 open import Relation.Unary as U using (Pred)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Decidable as Dec using (yes; ¬?; _⊎-dec_; _×-dec_)
diff --git a/src/Data/List/Relation/Unary/Linked/Properties.agda b/src/Data/List/Relation/Unary/Linked/Properties.agda
index d969986a55..10ece2fdb3 100644
--- a/src/Data/List/Relation/Unary/Linked/Properties.agda
+++ b/src/Data/List/Relation/Unary/Linked/Properties.agda
@@ -22,7 +22,9 @@ open import Data.Maybe.Relation.Binary.Connected
   using (Connected; just; nothing; just-nothing; nothing-just)
 open import Level using (Level)
 open import Function.Base using (_∘_; flip; _on_)
-open import Relation.Binary using (Rel; Transitive; DecSetoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (DecSetoid)
+open import Relation.Binary.Definitions using (Transitive)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (yes; no; does)
diff --git a/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda b/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda
index 0b6060bae9..cfdf8bf275 100644
--- a/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda
+++ b/src/Data/List/Relation/Unary/Sorted/TotalOrder.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (TotalOrder)
+open import Relation.Binary.Bundles using (TotalOrder)
 
 module Data.List.Relation.Unary.Sorted.TotalOrder
   {a ℓ₁ ℓ₂} (totalOrder : TotalOrder a ℓ₁ ℓ₂) where
@@ -17,7 +17,7 @@ open import Data.List.Base using (List; []; _∷_)
 open import Data.List.Relation.Unary.Linked as Linked using (Linked)
 open import Level using (_⊔_)
 open import Relation.Unary as U using (Pred; _⊆_)
-open import Relation.Binary as B
+open import Relation.Binary.Definitions as B
 
 ------------------------------------------------------------------------
 -- Definition
diff --git a/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda b/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda
index 89582d2f88..c8c4210ed6 100644
--- a/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda
+++ b/src/Data/List/Relation/Unary/Sorted/TotalOrder/Properties.agda
@@ -23,7 +23,8 @@ open import Data.Maybe.Relation.Binary.Connected using (Connected; just)
 open import Data.Nat.Base using (ℕ; zero; suc; _<_)
 
 open import Level using (Level)
-open import Relation.Binary hiding (Decidable)
+open import Relation.Binary.Core using (_Preserves_⟶_)
+open import Relation.Binary.Bundles using (TotalOrder; DecTotalOrder; Preorder)
 import Relation.Binary.Properties.TotalOrder as TotalOrderProperties
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable using (yes; no)
diff --git a/src/Data/List/Relation/Unary/Unique/DecPropositional.agda b/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
index 3f552b6fa9..f2e3692103 100644
--- a/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecPropositional.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecSetoid; DecidableEquality)
+open import Relation.Binary.Bundles using (DecSetoid)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.List.Relation.Unary.Unique.DecPropositional
diff --git a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
index 6d28ed56d6..4184647ae2 100644
--- a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
@@ -10,7 +10,7 @@ open import Data.List
 import Data.List.Relation.Unary.Unique.DecSetoid.Properties as Setoid
 open import Data.List.Relation.Unary.All.Properties using (all-filter)
 open import Level
-open import Relation.Binary using (DecidableEquality)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.List.Relation.Unary.Unique.DecPropositional.Properties
diff --git a/src/Data/List/Relation/Unary/Unique/DecSetoid.agda b/src/Data/List/Relation/Unary/Unique/DecSetoid.agda
index 4b7133c8f1..bddc7b2727 100644
--- a/src/Data/List/Relation/Unary/Unique/DecSetoid.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecSetoid.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (DecSetoid)
+open import Relation.Binary.Bundles using (DecSetoid)
 import Data.List.Relation.Unary.AllPairs as AllPairs
 open import Relation.Unary using (Decidable)
 open import Relation.Nullary.Decidable using (¬?)
diff --git a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
index 2eda2ded13..9e97527ff8 100644
--- a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
@@ -11,7 +11,7 @@ import Data.List.Relation.Unary.Unique.DecSetoid as Unique
 open import Data.List.Relation.Unary.All.Properties using (all-filter)
 open import Data.List.Relation.Unary.Unique.Setoid.Properties
 open import Level
-open import Relation.Binary using (DecSetoid)
+open import Relation.Binary.Bundles using (DecSetoid)
 open import Relation.Nullary.Decidable using (¬?)
 
 module Data.List.Relation.Unary.Unique.DecSetoid.Properties where
diff --git a/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda b/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
index 7740dc83e3..e12b5cf47b 100644
--- a/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
@@ -21,7 +21,8 @@ open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡⇒≡×≡)
 open import Function.Base using (id; _∘_)
 open import Level using (Level)
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality
   using (refl; _≡_; _≢_; sym; setoid)
 open import Relation.Unary using (Pred; Decidable)
diff --git a/src/Data/List/Relation/Unary/Unique/Setoid.agda b/src/Data/List/Relation/Unary/Unique/Setoid.agda
index aad10762ac..041b68ec9b 100644
--- a/src/Data/List/Relation/Unary/Unique/Setoid.agda
+++ b/src/Data/List/Relation/Unary/Unique/Setoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Nullary.Negation using (¬_)
 
 module Data.List.Relation.Unary.Unique.Setoid {a ℓ} (S : Setoid a ℓ) where
diff --git a/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
index 6ba7f803a1..ffb587018a 100644
--- a/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/Setoid/Properties.agda
@@ -23,7 +23,8 @@ open import Data.Fin.Base using (Fin)
 open import Data.Nat.Base using (_<_)
 open import Function.Base using (_∘_; id)
 open import Level using (Level)
-open import Relation.Binary using (Rel; Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/Maybe/Relation/Binary/Connected.agda b/src/Data/Maybe/Relation/Binary/Connected.agda
index 5c509e50fd..8c174c9ea1 100644
--- a/src/Data/Maybe/Relation/Binary/Connected.agda
+++ b/src/Data/Maybe/Relation/Binary/Connected.agda
@@ -11,7 +11,8 @@ module Data.Maybe.Relation.Binary.Connected where
 open import Level
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Function.Bundles using (_⇔_; mk⇔)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (REL; _⇒_)
+open import Relation.Binary.Definitions using (Reflexive; Sym; Decidable)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Data/Maybe/Relation/Binary/Pointwise.agda b/src/Data/Maybe/Relation/Binary/Pointwise.agda
index a9946a6ef3..48fa144a2e 100644
--- a/src/Data/Maybe/Relation/Binary/Pointwise.agda
+++ b/src/Data/Maybe/Relation/Binary/Pointwise.agda
@@ -12,7 +12,10 @@ open import Level
 open import Data.Product.Base using (∃; _×_; -,_; _,_)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Function.Bundles using (_⇔_; mk⇔)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Bundles using (Setoid; DecSetoid)
+open import Relation.Binary.Definitions using (Reflexive; Sym; Trans; Decidable)
+open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index abae430992..b7333283ff 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -12,7 +12,6 @@ module Data.Product.Function.Dependent.Propositional where
 open import Data.Product.Base using (Σ; map; _,_)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Relation.Binary hiding (_⇔_)
 open import Function.Base
 open import Function.Equality using (_⟶_; _⟨$⟩_)
 open import Function.Equivalence as Equiv using (_⇔_; module Equivalence)
diff --git a/src/Data/Product/Function/Dependent/Setoid.agda b/src/Data/Product/Function/Dependent/Setoid.agda
index 22f3f7937a..71cd995b18 100644
--- a/src/Data/Product/Function/Dependent/Setoid.agda
+++ b/src/Data/Product/Function/Dependent/Setoid.agda
@@ -23,7 +23,8 @@ open import Function.LeftInverse as LeftInv
   using (LeftInverse; _↞_; _LeftInverseOf_; _RightInverseOf_; module LeftInverse)
 open import Function.Surjection as Surj
   using (Surjection; _↠_; module Surjection)
-open import Relation.Binary as B hiding (_⇔_)
+open import Relation.Binary.Core using (_=[_]⇒_)
+open import Relation.Binary.Bundles as B
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
@@ -57,7 +58,6 @@ private
     using () renaming (_≈_ to _≈₁_)
   open B.Setoid (setoid (P.setoid A₂) B₂)
     using () renaming (_≈_ to _≈₂_)
-  open B using (_=[_]⇒_)
 
   fg = map f (_⟨$⟩_ g)
 
diff --git a/src/Data/Product/Function/Dependent/Setoid/WithK.agda b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
index 7362b3ade0..3336ecb5e5 100644
--- a/src/Data/Product/Function/Dependent/Setoid/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
@@ -25,7 +25,6 @@ open import Function.LeftInverse as LeftInv
   using (LeftInverse; _↞_; _LeftInverseOf_; _RightInverseOf_; module LeftInverse)
 open import Function.Surjection as Surj
   using (Surjection; _↠_; module Surjection)
-open import Relation.Binary as B
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
diff --git a/src/Data/Product/Function/NonDependent/Propositional.agda b/src/Data/Product/Function/NonDependent/Propositional.agda
index b4c9501304..7dfb28207d 100644
--- a/src/Data/Product/Function/NonDependent/Propositional.agda
+++ b/src/Data/Product/Function/NonDependent/Propositional.agda
@@ -12,7 +12,6 @@ module Data.Product.Function.NonDependent.Propositional where
 open import Data.Product.Base using (_×_; map)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Relation.Binary hiding (_⇔_)
 open import Function.Equality using (_⟶_)
 open import Function.Equivalence as Eq using (_⇔_; module Equivalence)
 open import Function.Injection as Inj using (_↣_; module Injection)
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
index 8a7dc43e83..fd22ff6cce 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
@@ -11,8 +11,10 @@ module Data.Product.Relation.Binary.Pointwise.Dependent where
 open import Data.Product.Base as Prod
 open import Level
 open import Function.Base
-open import Relation.Binary as B
-  using (_⇒_; Setoid; IsEquivalence)
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions as B
 open import Relation.Binary.Indexed.Heterogeneous as I
   using (IREL; IRel; IndexedSetoid; IsIndexedEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
@@ -25,7 +27,7 @@ infixr 4 _,_
 record POINTWISE {a₁ a₂ b₁ b₂ ℓ₁ ℓ₂}
                  {A₁ : Set a₁} (B₁ : A₁ → Set b₁)
                  {A₂ : Set a₂} (B₂ : A₂ → Set b₂)
-                 (_R₁_ : B.REL A₁ A₂ ℓ₁) (_R₂_ : IREL B₁ B₂ ℓ₂)
+                 (_R₁_ : REL A₁ A₂ ℓ₁) (_R₂_ : IREL B₁ B₂ ℓ₂)
                  (xy₁ : Σ A₁ B₁) (xy₂ : Σ A₂ B₂)
                  : Set (a₁ ⊔ a₂ ⊔ b₁ ⊔ b₂ ⊔ ℓ₁ ⊔ ℓ₂) where
   constructor _,_
@@ -36,14 +38,14 @@ record POINTWISE {a₁ a₂ b₁ b₂ ℓ₁ ℓ₂}
 open POINTWISE public
 
 Pointwise : ∀ {a b ℓ₁ ℓ₂} {A : Set a} (B : A → Set b)
-            (_R₁_ : B.Rel A ℓ₁) (_R₂_ : IRel B ℓ₂) → B.Rel (Σ A B) _
+            (_R₁_ : Rel A ℓ₁) (_R₂_ : IRel B ℓ₂) → Rel (Σ A B) _
 Pointwise B = POINTWISE B B
 
 ------------------------------------------------------------------------
 -- Pointwise preserves many relational properties
 
 module _ {a b ℓ₁ ℓ₂} {A : Set a} {B : A → Set b}
-         {R : B.Rel A ℓ₁} {S : IRel B ℓ₂} where
+         {R : Rel A ℓ₁} {S : IRel B ℓ₂} where
 
   private
     R×S = Pointwise B R S
diff --git a/src/Data/Unit/Properties.agda b/src/Data/Unit/Properties.agda
index b31d8feb3d..5b4c0bee51 100644
--- a/src/Data/Unit/Properties.agda
+++ b/src/Data/Unit/Properties.agda
@@ -12,7 +12,11 @@ open import Data.Sum.Base
 open import Data.Unit.Base
 open import Level using (0ℓ)
 open import Relation.Nullary
-open import Relation.Binary hiding (Irrelevant)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Poset; DecTotalOrder)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions using (Decidable; Total; Antisymmetric)
 open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index d0dbc16cda..f3314754e0 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -21,7 +21,7 @@ import Data.Vec.Properties as Vₚ
 open import Data.Vec.Functional
 open import Function.Base
 open import Level using (Level)
-open import Relation.Binary as B
+open import Relation.Binary.Definitions using (DecidableEquality; Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable
   using (Dec; does; yes; no; map′; _×-dec_)
@@ -46,8 +46,8 @@ module _ {n} {xs ys : Vector A (suc n)} where
   ∷-injective : xs ≗ ys → head xs ≡ head ys × tail xs ≗ tail ys
   ∷-injective eq = eq zero , eq ∘ suc
 
-≗-dec : B.DecidableEquality A →
-        ∀ {n} → B.Decidable {A = Vector A n} _≗_
+≗-dec : DecidableEquality A →
+        ∀ {n} → Decidable {A = Vector A n} _≗_
 ≗-dec _≟_ {zero}  xs ys = yes λ ()
 ≗-dec _≟_ {suc n} xs ys =
   map′ (Product.uncurry ∷-cong) ∷-injective
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 57ffb79317..b370b155d4 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -24,7 +24,7 @@ open import Function.Base
 -- open import Function.Inverse using (_↔_; inverse)
 open import Function.Bundles using (_↔_; mk↔′)
 open import Level using (Level)
-open import Relation.Binary hiding (Decidable)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; _≢_; _≗_; refl; sym; trans; cong; cong₂; subst; module ≡-Reasoning)
 open import Relation.Unary using (Pred; Decidable)
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index c091d0f98c..7b35081911 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -18,7 +18,10 @@ open import Data.Vec.Relation.Binary.Pointwise.Inductive using (Pointwise; []; _
 open import Function.Base using (flip)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (Rel; REL)
+open import Relation.Binary.Definitions
+  using (Transitive; Symmetric; Asymmetric; Antisymmetric; Irreflexive; Trans; _Respects₂_; _Respectsˡ_; _Respectsʳ_; Decidable; Irrelevant)
+open import Relation.Binary.Structures using (IsPartialEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl; cong)
 import Relation.Nullary as Nullary
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
index c7b2b3d7ea..5dcaa6b900 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
@@ -18,7 +18,10 @@ open import Function.Base using (_∘_)
 open import Function.Bundles using (module Equivalence; _⇔_; mk⇔)
 open import Function.Properties.Equivalence using (⇔-setoid)
 open import Level using (Level; _⊔_; 0ℓ)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (Rel; REL; _⇒_; _=[_]⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Sym; Trans; Decidable)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Construct.Closure.Transitive as Plus
   hiding (equivalent; map)
diff --git a/src/Function/Bijection.agda b/src/Function/Bijection.agda
index 8c7ce78ef3..405d326b51 100644
--- a/src/Function/Bijection.agda
+++ b/src/Function/Bijection.agda
@@ -14,7 +14,7 @@
 module Function.Bijection where
 
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality as P
 open import Function.Equality as F
   using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index 1b92bee909..d100fc5343 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -24,7 +24,8 @@ open import Function.Definitions
 import Function.Structures as FunctionStructures
 open import Level using (Level; _⊔_; suc)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Core using (_Preserves_⟶_)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as ≡
diff --git a/src/Function/Consequences.agda b/src/Function/Consequences.agda
index 242aa3d289..60606cc673 100644
--- a/src/Function/Consequences.agda
+++ b/src/Function/Consequences.agda
@@ -12,9 +12,10 @@ module Function.Consequences where
 
 open import Data.Product.Base as Prod
 open import Function.Definitions
-open import Level
-open import Relation.Binary.Core
-open import Relation.Binary.Definitions
+open import Level using (Level)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
 open import Relation.Nullary.Negation.Core using (¬_; contraposition)
 
 private
diff --git a/src/Function/Construct/Composition.agda b/src/Function/Construct/Composition.agda
index b745daa8e6..75baef6256 100644
--- a/src/Function/Construct/Composition.agda
+++ b/src/Function/Construct/Composition.agda
@@ -14,7 +14,9 @@ open import Function.Bundles
 open import Function.Definitions
 open import Function.Structures
 open import Level using (Level)
-open import Relation.Binary hiding (_⇔_; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (Transitive)
 
 private
   variable
diff --git a/src/Function/Construct/Identity.agda b/src/Function/Construct/Identity.agda
index e4ec7e3d3e..0088445ba5 100644
--- a/src/Function/Construct/Identity.agda
+++ b/src/Function/Construct/Identity.agda
@@ -14,7 +14,10 @@ open import Function.Bundles
 import Function.Definitions as Definitions
 import Function.Structures as Structures
 open import Level using (Level)
-open import Relation.Binary as B hiding (_⇔_; IsEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures as B hiding (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
diff --git a/src/Function/Endomorphism/Setoid.agda b/src/Function/Endomorphism/Setoid.agda
index 9d7974294c..d9890c3a35 100644
--- a/src/Function/Endomorphism/Setoid.agda
+++ b/src/Function/Endomorphism/Setoid.agda
@@ -9,7 +9,8 @@
 -- Disabled to prevent warnings from deprecated names
 {-# OPTIONS --warn=noUserWarning #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Function.Endomorphism.Setoid {c e} (S : Setoid c e) where
 
diff --git a/src/Function/Equality.agda b/src/Function/Equality.agda
index bb32bca8cd..91eeb7ef73 100644
--- a/src/Function/Equality.agda
+++ b/src/Function/Equality.agda
@@ -15,7 +15,7 @@ module Function.Equality where
 
 import Function.Base as Fun
 open import Level using (Level; _⊔_)
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid; _=[_]⇒_)
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
diff --git a/src/Function/Equivalence.agda b/src/Function/Equivalence.agda
index 7bbbade93c..34ab390682 100644
--- a/src/Function/Equivalence.agda
+++ b/src/Function/Equivalence.agda
@@ -17,7 +17,8 @@ open import Function.Base using (flip)
 open import Function.Equality as F
   using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
 open import Level
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (Reflexive; TransFlip; Sym)
 import Relation.Binary.PropositionalEquality as P
 
 ------------------------------------------------------------------------
diff --git a/src/Function/Injection.agda b/src/Function/Injection.agda
index 4f746c2beb..9bd6277fd2 100644
--- a/src/Function/Injection.agda
+++ b/src/Function/Injection.agda
@@ -15,7 +15,7 @@ module Function.Injection where
 
 open import Function.Base as Fun using () renaming (_∘_ to _⟨∘⟩_)
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Function.Equality as F
   using (_⟶_; _⟨$⟩_ ; Π) renaming (_∘_ to _⟪∘⟫_)
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
diff --git a/src/Function/Inverse.agda b/src/Function/Inverse.agda
index 4d0185d4b6..87632836db 100644
--- a/src/Function/Inverse.agda
+++ b/src/Function/Inverse.agda
@@ -19,7 +19,8 @@ open import Function.Bijection hiding (id; _∘_; bijection)
 open import Function.Equality as F
   using (_⟶_) renaming (_∘_ to _⟪∘⟫_)
 open import Function.LeftInverse as Left hiding (id; _∘_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (Reflexive; TransFlip; Sym)
 open import Relation.Binary.PropositionalEquality as P using (_≗_; _≡_)
 open import Relation.Unary using (Pred)
 
diff --git a/src/Function/LeftInverse.agda b/src/Function/LeftInverse.agda
index 23f9b4a900..6f2c8d0db5 100644
--- a/src/Function/LeftInverse.agda
+++ b/src/Function/LeftInverse.agda
@@ -15,7 +15,7 @@ module Function.LeftInverse where
 
 open import Level
 import Relation.Binary.Reasoning.Setoid as EqReasoning
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Function.Equality as Eq
   using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
 open import Function.Equivalence using (Equivalence)
diff --git a/src/Function/Metric/Nat/Structures.agda b/src/Function/Metric/Nat/Structures.agda
index 839dc7b974..8eaa533faa 100644
--- a/src/Function/Metric/Nat/Structures.agda
+++ b/src/Function/Metric/Nat/Structures.agda
@@ -11,7 +11,7 @@ module Function.Metric.Nat.Structures where
 open import Data.Nat.Base hiding (suc)
 open import Function.Base using (const)
 open import Level using (Level; suc)
-open import Relation.Binary hiding (Symmetric)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 open import Function.Metric.Nat.Core
diff --git a/src/Function/Metric/Rational/Structures.agda b/src/Function/Metric/Rational/Structures.agda
index daa2b71a60..3a8acdaae6 100644
--- a/src/Function/Metric/Rational/Structures.agda
+++ b/src/Function/Metric/Rational/Structures.agda
@@ -9,7 +9,7 @@
 open import Data.Rational.Base
 open import Function.Base using (const)
 open import Level using (Level; suc)
-open import Relation.Binary hiding (Symmetric)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 open import Function.Metric.Rational.Core
diff --git a/src/Function/Metric/Structures.agda b/src/Function/Metric/Structures.agda
index 1523673557..68c85e48b4 100644
--- a/src/Function/Metric/Structures.agda
+++ b/src/Function/Metric/Structures.agda
@@ -9,7 +9,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary hiding (Symmetric)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsPartialOrder; IsEquivalence)
 
 module Function.Metric.Structures
   {a i ℓ₁ ℓ₂ ℓ₃} {A : Set a} {I : Set i}
diff --git a/src/Function/Properties/Bijection.agda b/src/Function/Properties/Bijection.agda
index 38d269195b..1028e694e8 100644
--- a/src/Function/Properties/Bijection.agda
+++ b/src/Function/Properties/Bijection.agda
@@ -10,7 +10,9 @@ module Function.Properties.Bijection where
 
 open import Function.Bundles
 open import Level using (Level)
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Trans)
 open import Relation.Binary.PropositionalEquality as P using (setoid)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index cc80adff63..aa661c9cb8 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -18,7 +18,10 @@ open import Function.Inverse     as Inv     using (Inverse; _↔_)
 open import Function.LeftInverse as LeftInv using (LeftInverse)
 open import Function.Surjection  as Surj    using (Surjection)
 open import Function.Consequences.Propositional
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Setoid; Preorder)
+open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)
+open import Relation.Binary.Definitions using (Reflexive; Trans; Sym)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 open import Data.Product.Base using (_,_; proj₁; proj₂; <_,_>)
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index 9922c69fcf..c2ba98dce4 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -28,7 +28,6 @@ open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence as Eq using (_⇔_; Equivalence)
 open import Function.Inverse as Inv using (_↔_; Inverse; inverse)
 open import Function.Related
-open import Relation.Binary hiding (_⇔_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ)
diff --git a/src/Function/Structures.agda b/src/Function/Structures.agda
index 5566e92c8b..a1cc0d6ceb 100644
--- a/src/Function/Structures.agda
+++ b/src/Function/Structures.agda
@@ -8,7 +8,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 
 module Function.Structures {a b ℓ₁ ℓ₂}
   {A : Set a} (_≈₁_ : Rel A ℓ₁) -- Equality over the domain
diff --git a/src/Function/Surjection.agda b/src/Function/Surjection.agda
index 6861979551..729b46a538 100644
--- a/src/Function/Surjection.agda
+++ b/src/Function/Surjection.agda
@@ -19,7 +19,7 @@ open import Function.Equality as F
 open import Function.Equivalence using (Equivalence)
 open import Function.Injection           hiding (id; _∘_; injection)
 open import Function.LeftInverse as Left hiding (id; _∘_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
 
 ------------------------------------------------------------------------
diff --git a/src/Induction/WellFounded.agda b/src/Induction/WellFounded.agda
index c1e206a0af..12b1719118 100644
--- a/src/Induction/WellFounded.agda
+++ b/src/Induction/WellFounded.agda
@@ -12,7 +12,9 @@ open import Data.Product.Base using (Σ; _,_; proj₁)
 open import Function.Base using (_on_)
 open import Induction
 open import Level using (Level; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions
+  using (Symmetric; _Respectsʳ_; _Respects_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Unary
 
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index d9bf470f5c..29defea059 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -13,7 +13,11 @@ open import Data.Sum.Base as Sum
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Function.Base using (id)
 open import Level
-open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.Core using (Rel; REL; _=[_]⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsStrictPartialOrder; IsPartialOrder; IsDecStrictPartialOrder; IsDecPartialOrder; IsStrictTotalOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Symmetric; Transitive; Reflexive; Asymmetric; Antisymmetric; Trichotomous; Total; Decidable; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects_; _Respects₂_)
 open import Relation.Binary.Construct.Closure.Reflexive
 open import Relation.Binary.PropositionalEquality.Core as PropEq using (_≡_; refl)
 import Relation.Binary.PropositionalEquality.Properties as PropEq
diff --git a/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda b/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda
index 2817b21cf4..5ed5b8f4ff 100644
--- a/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda
+++ b/src/Tactic/RingSolver/Core/AlmostCommutativeRing.agda
@@ -9,7 +9,7 @@
 module Tactic.RingSolver.Core.AlmostCommutativeRing where
 
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
 open import Algebra.Core using (Op₁; Op₂)
 open import Algebra.Structures using (IsCommutativeSemiring)
 open import Algebra.Definitions

From 6f839c8ad7c7d9b8391e5a263865d80a23840c4a Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Sun, 20 Aug 2023 00:38:17 -0400
Subject: [PATCH 099/316] Add `drop-drop` in `Data.List.Properties` (#2043)

---
 CHANGELOG.md                              | 22 +++++++++++++---------
 src/Codata/Guarded/Stream/Properties.agda | 12 +++++++++---
 src/Codata/Sized/Colist/Properties.agda   | 14 ++++++++++----
 src/Data/List/Properties.agda             |  8 +++++++-
 4 files changed, 39 insertions(+), 17 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 77b00a6ff3..fb718f81ed 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1080,12 +1080,14 @@ Deprecated names
   ```agda
   map-identity  ↦  map-id
   map-fusion    ↦  map-∘
+  drop-fusion   ↦  drop-drop
   ```
 
 * In `Codata.Sized.Colist.Properties`:
   ```agda
-  map-identity   ↦  map-id
-  map-map-fusion  ↦  map-∘
+  map-identity      ↦  map-id
+  map-map-fusion    ↦  map-∘
+  drop-drop-fusion  ↦  drop-drop
   ```
 
 * In `Codata.Sized.Covec.Properties`:
@@ -2216,24 +2218,26 @@ Other minor changes
   concatMap-map  : concatMap g (map f xs) ≡ concatMap (g ∘′ f) xs
   map-concatMap  : map f ∘′ concatMap g ≗ concatMap (map f ∘′ g)
 
-  length-isMagmaHomomorphism : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
+  length-isMagmaHomomorphism  : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
   length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
   
   take-map : take n (map f xs) ≡ map f (take n xs)
   drop-map : drop n (map f xs) ≡ map f (drop n xs)
   head-map : head (map f xs) ≡ Maybe.map f (head xs)
 
-  take-suc : (o : Fin (length xs)) → let m = toℕ o in take (suc m) xs ≡ take m xs ∷ʳ lookup xs o
-  take-suc-tabulate : (f : Fin n → A) (o : Fin n) → let m = toℕ o in take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f o
-  drop-take-suc : (o : Fin (length xs)) → let m = toℕ o in drop m (take (suc m) xs) ≡ [ lookup xs o ]
-  drop-take-suc-tabulate : (f : Fin n → A) (o : Fin n) → let m = toℕ o in drop m (take (suc m) (tabulate f)) ≡ [ f o ]
+  take-suc               : take (suc m) xs ≡ take m xs ∷ʳ lookup xs i
+  take-suc-tabulate      : take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f i
+  drop-take-suc          : drop m (take (suc m) xs) ≡ [ lookup xs i ]
+  drop-take-suc-tabulate : drop m (take (suc m) (tabulate f)) ≡ [ f i ]
 
   take-all : n ≥ length xs → take n xs ≡ xs
 
-  take-[] : ∀ m → take  m [] ≡ []
-  drop-[] : ∀ m → drop  m [] ≡ []
+  take-[] : take m [] ≡ []
+  drop-[] : drop m [] ≡ []
 
   map-replicate : map f (replicate n x) ≡ replicate n (f x)
+
+  drop-drop : drop n (drop m xs) ≡ drop (m + n) xs
   ```
 
 * Added new patterns and definitions to `Data.Nat.Base`:
diff --git a/src/Codata/Guarded/Stream/Properties.agda b/src/Codata/Guarded/Stream/Properties.agda
index df68a9c4b2..a07a614b59 100644
--- a/src/Codata/Guarded/Stream/Properties.agda
+++ b/src/Codata/Guarded/Stream/Properties.agda
@@ -268,9 +268,9 @@ take-zipWith (suc n) f as bs =
 ------------------------------------------------------------------------
 -- Properties of drop
 
-drop-fusion : ∀ m n (as : Stream A) → drop n (drop m as) ≡ drop (m + n) as
-drop-fusion zero    n as = P.refl
-drop-fusion (suc m) n as = drop-fusion m n (as .tail)
+drop-drop : ∀ m n (as : Stream A) → drop n (drop m as) ≡ drop (m + n) as
+drop-drop zero    n as = P.refl
+drop-drop (suc m) n as = drop-drop m n (as .tail)
 
 drop-zipWith : ∀ n (f : A → B → C) as bs →
                drop n (zipWith f as bs) ≡ zipWith f (drop n as) (drop n bs)
@@ -331,3 +331,9 @@ map-fusion = map-∘
 "Warning: map-fusion was deprecated in v2.0.
 Please use map-∘ instead."
 #-}
+
+drop-fusion = drop-drop
+{-# WARNING_ON_USAGE drop-fusion
+"Warning: drop-fusion was deprecated in v2.0.
+Please use drop-drop instead."
+#-}
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index 5b77551b7f..2ec1564627 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -197,11 +197,11 @@ drop-nil : ∀ m → i ⊢ drop {A = A} m [] ≈ []
 drop-nil zero    = []
 drop-nil (suc m) = []
 
-drop-drop-fusion : ∀ m n (as : Colist A ∞) →
+drop-drop : ∀ m n (as : Colist A ∞) →
                    i ⊢ drop n (drop m as) ≈ drop (m ℕ.+ n) as
-drop-drop-fusion zero    n as       = refl
-drop-drop-fusion (suc m) n []       = drop-nil n
-drop-drop-fusion (suc m) n (a ∷ as) = drop-drop-fusion m n (as .force)
+drop-drop zero    n as       = refl
+drop-drop (suc m) n []       = drop-nil n
+drop-drop (suc m) n (a ∷ as) = drop-drop m n (as .force)
 
 map-drop : ∀ (f : A → B) m as → i ⊢ map f (drop m as) ≈ drop m (map f as)
 map-drop f zero    as       = refl
@@ -351,3 +351,9 @@ map-map-fusion = map-∘
 "Warning: map-map-fusion was deprecated in v2.0.
 Please use map-∘ instead."
 #-}
+
+drop-drop-fusion = drop-drop
+{-# WARNING_ON_USAGE drop-drop-fusion
+"Warning: drop-drop-fusion was deprecated in v2.0.
+Please use drop-drop instead."
+#-}
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 56c22c953a..9bee6208b9 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -784,7 +784,7 @@ take-suc-tabulate f i rewrite sym (toℕ-cast (sym (length-tabulate f)) i) | sym
 
 -- If you take at least as many elements from a list as it has, you get
 -- the whole list.
-take-all :(n : ℕ) (xs : List A) → n ≥ length xs → take n xs ≡ xs
+take-all : (n : ℕ) (xs : List A) → n ≥ length xs → take n xs ≡ xs
 take-all zero [] _ = refl
 take-all (suc _) [] _ = refl
 take-all (suc n) (x ∷ xs) (s≤s pf) = cong (x ∷_) (take-all n xs pf)
@@ -828,6 +828,12 @@ drop-take-suc-tabulate : ∀ {n} (f : Fin n → A) (i : Fin n) → let m = toℕ
 drop-take-suc-tabulate f i rewrite sym (toℕ-cast (sym (length-tabulate f)) i) | sym (lookup-tabulate f i)
   = drop-take-suc (tabulate f) (cast _ i)
 
+-- Dropping m elements and then n elements is same as dropping m+n elements
+drop-drop : (m n : ℕ) → (xs : List A) → drop n (drop m xs) ≡ drop (m + n) xs
+drop-drop zero n xs = refl
+drop-drop (suc m) n [] = drop-[] n
+drop-drop (suc m) n (x ∷ xs) = drop-drop m n xs
+
 ------------------------------------------------------------------------
 -- splitAt
 

From da7bbde6cf9821a5db1b2e0f693d704d43f84a47 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 20 Aug 2023 05:57:25 +0100
Subject: [PATCH 100/316] Rename `push-function-into-if`

---
 CHANGELOG.md                  |  7 ++++++-
 src/Data/Bool/Properties.agda | 22 ++++++++++++++++++----
 2 files changed, 24 insertions(+), 5 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index fb718f81ed..5138c4017f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1114,6 +1114,11 @@ Deprecated names
   map-map-fusion  ↦  map-∘
   ```
 
+* In `Data.Bool.Properties` (Issue #2046):
+  ```
+  push-function-into-if ↦ if-float
+  ```
+
 * In `Data.Fin.Base`: two new, hopefully more memorable, names `↑ˡ` `↑ʳ`
   for the 'left', resp. 'right' injection of a Fin m into a 'larger' type,
   `Fin (m + n)`, resp. `Fin (n + m)`, with argument order to reflect the
@@ -1138,7 +1143,7 @@ Deprecated names
 
   As with Issue #1726 above: the deprecation of relation `_≺_` means that these definitions
   associated with wf-recursion are deprecated in favour of their `_<_` counterparts.
-  But it's not quite sensible to say that these definiton should be *renamed* to *anything*,
+  But it's not quite sensible to say that these definitions should be *renamed* to *anything*,
   least of all those counterparts... the type confusion would be intolerable.
 
 * In `Data.Fin.Properties`:
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index 3c073cd23a..ea22a37f68 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -768,7 +768,21 @@ proof (T? false) = ofⁿ λ()
 T?-diag : ∀ b → T b → True (T? b)
 T?-diag true  _ = _
 
-push-function-into-if : ∀ (f : A → B) x {y z} →
-                        f (if x then y else z) ≡ (if x then f y else f z)
-push-function-into-if _ true  = refl
-push-function-into-if _ false = refl
+if-float : ∀ (f : A → B) b {x y} →
+           f (if b then x else y) ≡ (if b then f x else f y)
+if-float _ true  = refl
+if-float _ false = refl
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+push-function-into-if = if-float
+{-# WARNING_ON_USAGE push-function-into-if
+"Warning: push-function-into-if was deprecated in v2.0.
+Please use if-float instead."
+#-}

From 6a544fcf91dc970c7079dfbc8bc43a2c50940ed1 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Sun, 27 Aug 2023 18:15:42 +0200
Subject: [PATCH 101/316] agda-stdlib-utils/AllNonAsciiChars: use List1.head
 instead of List.head

---
 AllNonAsciiChars.hs | 9 +++++----
 1 file changed, 5 insertions(+), 4 deletions(-)

diff --git a/AllNonAsciiChars.hs b/AllNonAsciiChars.hs
index fa9ac746d1..bc474296c4 100644
--- a/AllNonAsciiChars.hs
+++ b/AllNonAsciiChars.hs
@@ -4,7 +4,8 @@
 
 module Main where
 
-import qualified Data.List as L (sortBy, group, sort)
+import qualified Data.List as List (sortBy, sort)
+import qualified Data.List.NonEmpty as List1 (group, head)
 import Data.Char (isAscii, ord)
 import Data.Function (on)
 import Numeric (showHex)
@@ -27,9 +28,9 @@ main = do
   nonAsciiChars <-
     filter (not . isAscii) . T.unpack . T.concat <$> mapM readUTF8File agdaFiles
   let table :: [(Char, Int)]
-      table = L.sortBy (flip compare `on` snd) $
-              map (\cs -> (head cs, length cs)) $
-              L.group $ L.sort $ nonAsciiChars
+      table = List.sortBy (flip compare `on` snd) $
+              map (\cs -> (List1.head cs, length cs)) $
+              List1.group $ List.sort $ nonAsciiChars
 
   let codePoint :: Char -> T.Text
       codePoint c = T.pack $ showHex (ord c) ""

From 4103251088934dd5e86252c65ece726fe2dbcbf4 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Sun, 27 Aug 2023 18:19:49 +0200
Subject: [PATCH 102/316] Bump resolvers in stack-{9.4.5,9.6.2}.yaml

---
 stack-9.4.5.yaml | 2 +-
 stack-9.6.2.yaml | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/stack-9.4.5.yaml b/stack-9.4.5.yaml
index 67331480d1..790630b2e2 100644
--- a/stack-9.4.5.yaml
+++ b/stack-9.4.5.yaml
@@ -1,4 +1,4 @@
-resolver: lts-21.0
+resolver: lts-21.7
 compiler: ghc-9.4.5
 compiler-check: match-exact
 
diff --git a/stack-9.6.2.yaml b/stack-9.6.2.yaml
index 59f58a5299..dab6c80ebf 100644
--- a/stack-9.6.2.yaml
+++ b/stack-9.6.2.yaml
@@ -1,4 +1,4 @@
-resolver: nightly-2023-06-23
+resolver: nightly-2023-08-27
 compiler: ghc-9.6.2
 compiler-check: match-exact
 

From f60b970bec6d8f66bbd2a0123e88f6871b015fd5 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Sun, 27 Aug 2023 18:20:04 +0200
Subject: [PATCH 103/316] Bump Haskell CI to GHC 9.8.0 and 9.4.6; allow
 text-2.1

---
 .github/workflows/haskell-ci.yml | 44 ++++++++++++++++++++++++--------
 agda-stdlib-utils.cabal          |  7 ++---
 2 files changed, 38 insertions(+), 13 deletions(-)

diff --git a/.github/workflows/haskell-ci.yml b/.github/workflows/haskell-ci.yml
index f491e32dc6..b8c55c19be 100644
--- a/.github/workflows/haskell-ci.yml
+++ b/.github/workflows/haskell-ci.yml
@@ -8,9 +8,9 @@
 #
 # For more information, see https://github.com/haskell-CI/haskell-ci
 #
-# version: 0.16.4
+# version: 0.17.20230824
 #
-# REGENDATA ("0.16.4",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
+# REGENDATA ("0.17.20230824",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
 #
 name: Haskell-CI
 on:
@@ -44,14 +44,19 @@ jobs:
     strategy:
       matrix:
         include:
+          - compiler: ghc-9.8.0.20230822
+            compilerKind: ghc
+            compilerVersion: 9.8.0.20230822
+            setup-method: ghcup
+            allow-failure: true
           - compiler: ghc-9.6.2
             compilerKind: ghc
             compilerVersion: 9.6.2
             setup-method: ghcup
             allow-failure: false
-          - compiler: ghc-9.4.5
+          - compiler: ghc-9.4.6
             compilerKind: ghc
-            compilerVersion: 9.4.5
+            compilerVersion: 9.4.6
             setup-method: ghcup
             allow-failure: false
           - compiler: ghc-9.2.8
@@ -87,8 +92,9 @@ jobs:
           apt-get install -y --no-install-recommends gnupg ca-certificates dirmngr curl git software-properties-common libtinfo5
           if [ "${{ matrix.setup-method }}" = ghcup ]; then
             mkdir -p "$HOME/.ghcup/bin"
-            curl -sL https://downloads.haskell.org/ghcup/0.1.19.2/x86_64-linux-ghcup-0.1.19.2 > "$HOME/.ghcup/bin/ghcup"
+            curl -sL https://downloads.haskell.org/ghcup/0.1.19.5/x86_64-linux-ghcup-0.1.19.5 > "$HOME/.ghcup/bin/ghcup"
             chmod a+x "$HOME/.ghcup/bin/ghcup"
+            "$HOME/.ghcup/bin/ghcup" config add-release-channel https://raw.githubusercontent.com/haskell/ghcup-metadata/master/ghcup-prereleases-0.0.7.yaml;
             "$HOME/.ghcup/bin/ghcup" install ghc "$HCVER" || (cat "$HOME"/.ghcup/logs/*.* && false)
             "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
           else
@@ -96,8 +102,9 @@ jobs:
             apt-get update
             apt-get install -y "$HCNAME"
             mkdir -p "$HOME/.ghcup/bin"
-            curl -sL https://downloads.haskell.org/ghcup/0.1.19.2/x86_64-linux-ghcup-0.1.19.2 > "$HOME/.ghcup/bin/ghcup"
+            curl -sL https://downloads.haskell.org/ghcup/0.1.19.5/x86_64-linux-ghcup-0.1.19.5 > "$HOME/.ghcup/bin/ghcup"
             chmod a+x "$HOME/.ghcup/bin/ghcup"
+            "$HOME/.ghcup/bin/ghcup" config add-release-channel https://raw.githubusercontent.com/haskell/ghcup-metadata/master/ghcup-prereleases-0.0.7.yaml;
             "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
           fi
         env:
@@ -112,10 +119,12 @@ jobs:
           echo "CABAL_CONFIG=$HOME/.cabal/config" >> "$GITHUB_ENV"
           HCDIR=/opt/$HCKIND/$HCVER
           if [ "${{ matrix.setup-method }}" = ghcup ]; then
-            HC=$HOME/.ghcup/bin/$HCKIND-$HCVER
+            HC=$("$HOME/.ghcup/bin/ghcup" whereis ghc "$HCVER")
+            HCPKG=$(echo "$HC" | sed 's#ghc$#ghc-pkg#')
+            HADDOCK=$(echo "$HC" | sed 's#ghc$#haddock#')
             echo "HC=$HC" >> "$GITHUB_ENV"
-            echo "HCPKG=$HOME/.ghcup/bin/$HCKIND-pkg-$HCVER" >> "$GITHUB_ENV"
-            echo "HADDOCK=$HOME/.ghcup/bin/haddock-$HCVER" >> "$GITHUB_ENV"
+            echo "HCPKG=$HCPKG" >> "$GITHUB_ENV"
+            echo "HADDOCK=$HADDOCK" >> "$GITHUB_ENV"
             echo "CABAL=$HOME/.ghcup/bin/cabal-3.10.1.0 -vnormal+nowrap" >> "$GITHUB_ENV"
           else
             HC=$HCDIR/bin/$HCKIND
@@ -129,7 +138,7 @@ jobs:
           echo "HCNUMVER=$HCNUMVER" >> "$GITHUB_ENV"
           echo "ARG_TESTS=--enable-tests" >> "$GITHUB_ENV"
           echo "ARG_BENCH=--enable-benchmarks" >> "$GITHUB_ENV"
-          echo "HEADHACKAGE=false" >> "$GITHUB_ENV"
+          if [ $((HCNUMVER >= 90800)) -ne 0 ] ; then echo "HEADHACKAGE=true" >> "$GITHUB_ENV" ; else echo "HEADHACKAGE=false" >> "$GITHUB_ENV" ; fi
           echo "ARG_COMPILER=--$HCKIND --with-compiler=$HC" >> "$GITHUB_ENV"
           echo "GHCJSARITH=0" >> "$GITHUB_ENV"
         env:
@@ -158,6 +167,18 @@ jobs:
           repository hackage.haskell.org
             url: http://hackage.haskell.org/
           EOF
+          if $HEADHACKAGE; then
+          cat >> $CABAL_CONFIG <<EOF
+          repository head.hackage.ghc.haskell.org
+             url: https://ghc.gitlab.haskell.org/head.hackage/
+             secure: True
+             root-keys: 7541f32a4ccca4f97aea3b22f5e593ba2c0267546016b992dfadcd2fe944e55d
+                        26021a13b401500c8eb2761ca95c61f2d625bfef951b939a8124ed12ecf07329
+                        f76d08be13e9a61a377a85e2fb63f4c5435d40f8feb3e12eb05905edb8cdea89
+             key-threshold: 3
+          active-repositories: hackage.haskell.org, head.hackage.ghc.haskell.org:override
+          EOF
+          fi
           cat >> $CABAL_CONFIG <<EOF
           program-default-options
             ghc-options: $GHCJOBS +RTS -M3G -RTS
@@ -209,6 +230,9 @@ jobs:
           echo "    ghc-options: -Werror=missing-methods" >> cabal.project
           cat >> cabal.project <<EOF
           EOF
+          if $HEADHACKAGE; then
+          echo "allow-newer: $($HCPKG list --simple-output | sed -E 's/([a-zA-Z-]+)-[0-9.]+/*:\1,/g')" >> cabal.project
+          fi
           $HCPKG list --simple-output --names-only | perl -ne 'for (split /\s+/) { print "constraints: $_ installed\n" unless /^(agda-stdlib-utils)$/; }' >> cabal.project.local
           cat cabal.project
           cat cabal.project.local
diff --git a/agda-stdlib-utils.cabal b/agda-stdlib-utils.cabal
index 5ca3bd7055..a482bee069 100644
--- a/agda-stdlib-utils.cabal
+++ b/agda-stdlib-utils.cabal
@@ -6,8 +6,9 @@ description:     Helper programs for setting up the Agda standard library.
 license:         MIT
 
 tested-with:
+  GHC == 9.8.0
   GHC == 9.6.2
-  GHC == 9.4.5
+  GHC == 9.4.6
   GHC == 9.2.8
   GHC == 9.0.2
   GHC == 8.10.7
@@ -22,7 +23,7 @@ common common-build-parameters
     PatternSynonyms
 
   build-depends:
-      base          >= 4.12.0.0 && < 4.19
+      base          >= 4.12.0.0 && < 4.20
     , filemanip     >= 0.3.6.2  && < 0.4
 
 executable GenerateEverything
@@ -41,4 +42,4 @@ executable AllNonAsciiChars
   main-is:          AllNonAsciiChars.hs
 
   build-depends:
-      text          >= 1.2.3.1  && < 2.1
+      text          >= 1.2.3.1  && < 2.2

From e3571a345d105ea3c494214f638e006a247a15b6 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Wed, 30 Aug 2023 05:57:24 -0400
Subject: [PATCH 104/316] Rename `take-drop-id` and `take++drop` (#2069)

A useful improvement to consistency of names,
---
 CHANGELOG.md                  |  4 ++++
 README/Data/List.agda         |  2 +-
 src/Data/List/Properties.agda | 14 ++++++++++----
 src/Data/Vec/Properties.agda  | 14 ++++++++++----
 4 files changed, 25 insertions(+), 9 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5138c4017f..9e34dba19a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1246,6 +1246,8 @@ Deprecated names
 
   zipWith-identityˡ  ↦  zipWith-zeroˡ
   zipWith-identityʳ  ↦  zipWith-zeroʳ
+
+  take++drop ↦ take++drop≡id
   ```
 
 * In `Data.List.NonEmpty.Properties`:
@@ -1401,6 +1403,8 @@ Deprecated names
   []≔-++-raise    ↦ []≔-++-↑ʳ
   idIsFold        ↦ id-is-foldr
   sum-++-commute  ↦ sum-++
+
+  take-drop-id ↦ take++drop≡id
   ```
   and the type of the proof `zipWith-comm` has been generalised from:
   ```
diff --git a/README/Data/List.agda b/README/Data/List.agda
index 6cb8bba0b1..2c284daa78 100644
--- a/README/Data/List.agda
+++ b/README/Data/List.agda
@@ -48,7 +48,7 @@ lem₅ = refl
 open import Data.List.Properties
 
 lem₆ : ∀ n (xs : List ℕ) → take n xs ++ drop n xs ≡ xs
-lem₆ = take++drop
+lem₆ = take++drop≡id
 
 lem₇ : ∀ (xs : List ℕ) → reverse (reverse xs) ≡ xs
 lem₇ = reverse-involutive
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 9bee6208b9..9e86d12436 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -813,10 +813,10 @@ drop-[] : ∀ m → drop {A = A} m [] ≡ []
 drop-[] zero = refl
 drop-[] (suc m) = refl
 
-take++drop : ∀ n (xs : List A) → take n xs ++ drop n xs ≡ xs
-take++drop zero    xs       = refl
-take++drop (suc n) []       = refl
-take++drop (suc n) (x ∷ xs) = cong (x ∷_) (take++drop n xs)
+take++drop≡id : ∀ n (xs : List A) → take n xs ++ drop n xs ≡ xs
+take++drop≡id zero    xs       = refl
+take++drop≡id (suc n) []       = refl
+take++drop≡id (suc n) (x ∷ xs) = cong (x ∷_) (take++drop≡id n xs)
 
 drop-take-suc : (xs : List A) (i : Fin (length xs)) → let m = toℕ i in
            drop m (take (suc m) xs) ≡ [ lookup xs i ]
@@ -1212,3 +1212,9 @@ zipWith-identityʳ = zipWith-zeroʳ
 "Warning: zipWith-identityʳ was deprecated in v2.0.
 Please use zipWith-zeroʳ instead."
 #-}
+
+take++drop = take++drop≡id
+{-# WARNING_ON_USAGE take++drop
+"Warning: take++drop was deprecated in v2.0.
+Please use take++drop≡id instead."
+#-}
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index b370b155d4..56a2001f1e 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -140,15 +140,15 @@ drop-distr-map f (suc m) (x ∷ xs) = begin
 ------------------------------------------------------------------------
 -- take and drop together
 
-take-drop-id : ∀ (m : ℕ) (x : Vec A (m + n)) → take m x ++ drop m x ≡ x
-take-drop-id zero x = refl
-take-drop-id (suc m) (x ∷ xs) = begin
+take++drop≡id : ∀ (m : ℕ) (x : Vec A (m + n)) → take m x ++ drop m x ≡ x
+take++drop≡id zero x = refl
+take++drop≡id (suc m) (x ∷ xs) = begin
     take (suc m) (x ∷ xs) ++ drop (suc m) (x ∷ xs)
   ≡⟨ cong₂ _++_ (unfold-take m x xs) (unfold-drop m x xs) ⟩
     (x ∷ take m xs) ++ (drop m xs)
   ≡⟨⟩
     x ∷ (take m xs ++ drop m xs)
-  ≡⟨ cong (x ∷_) (take-drop-id m xs) ⟩
+  ≡⟨ cong (x ∷_) (take++drop≡id m xs) ⟩
     x ∷ xs
   ∎
 
@@ -1186,3 +1186,9 @@ sum-++-commute = sum-++
 "Warning: sum-++-commute was deprecated in v2.0.
 Please use sum-++ instead."
 #-}
+
+take-drop-id = take++drop≡id
+{-# WARNING_ON_USAGE take-drop-id
+"Warning: take-drop-id was deprecated in v2.0.
+Please use take++drop≡id instead."
+#-}

From 9311dd949802874d255b040f0ad3361626925c48 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 31 Aug 2023 20:08:08 +0530
Subject: [PATCH 105/316] Add `find`, `findIndex`, and `findIndices` for
 `Data.List` (#2042)

* `find*` functions for `Data.List`

both decidable predicate and boolean function variants

* Back to `Maybe.map`

* New type for findIndices

* Add comments

* Respect style guide

---------

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
---
 CHANGELOG.md                                  |  7 ++++
 src/Data/List/Base.agda                       | 32 ++++++++++++++++++-
 .../List/Membership/Setoid/Properties.agda    |  2 +-
 .../Binary/Subset/Setoid/Properties.agda      |  2 +-
 .../List/Relation/Unary/Any/Properties.agda   |  2 +-
 src/Function/Base.agda                        |  2 +-
 6 files changed, 42 insertions(+), 5 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9e34dba19a..812fe00b6c 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2173,6 +2173,13 @@ Other minor changes
   wordsByᵇ     : (A → Bool) → List A → List (List A)
   derunᵇ       : (A → A → Bool) → List A → List A
   deduplicateᵇ : (A → A → Bool) → List A → List A
+
+  findᵇ        : (A → Bool) → List A -> Maybe A
+  findIndexᵇ   : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)
+  findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)
+  find         : Decidable P → List A → Maybe A
+  findIndex    : Decidable P → (xs : List A) → Maybe $ Fin (length xs)
+  findIndices  : Decidable P → (xs : List A) → List $ Fin (length xs)
   ```
 
 * Added new functions and definitions to `Data.List.Base`:
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 2b25ece003..9817cf8648 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -20,7 +20,8 @@ open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _*_ ; _≤_ ; s≤s
 open import Data.Product.Base as Prod using (_×_; _,_; map₁; map₂′)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
-open import Function.Base using (id; _∘_ ; _∘′_; _∘₂_; const; flip)
+open import Function.Base
+  using (id; _∘_ ; _∘′_; _∘₂_; _$_; const; flip)
 open import Level using (Level)
 open import Relation.Nullary.Decidable.Core using (does; ¬?)
 open import Relation.Unary using (Pred; Decidable)
@@ -395,6 +396,26 @@ deduplicateᵇ : (A → A → Bool) → List A → List A
 deduplicateᵇ r [] = []
 deduplicateᵇ r (x ∷ xs) = x ∷ filterᵇ (not ∘ r x) (deduplicateᵇ r xs)
 
+-- Finds the first element satisfying the boolean predicate
+findᵇ : (A → Bool) → List A → Maybe A
+findᵇ p []       = nothing
+findᵇ p (x ∷ xs) = if p x then just x else findᵇ p xs
+
+-- Finds the index of the first element satisfying the boolean predicate
+findIndexᵇ : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)
+findIndexᵇ p []       = nothing
+findIndexᵇ p (x ∷ xs) = if p x
+  then just zero
+  else Maybe.map suc (findIndexᵇ p xs)
+
+-- Finds indices of all the elements satisfying the boolean predicate
+findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)
+findIndicesᵇ p []       = []
+findIndicesᵇ p (x ∷ xs) = if p x
+  then zero ∷ indices
+  else indices
+    where indices = map suc (findIndicesᵇ p xs)
+
 -- Equivalent functions that use a decidable predicate instead of a
 -- boolean function.
 
@@ -436,6 +457,15 @@ derun R? = derunᵇ (does ∘₂ R?)
 deduplicate : ∀ {R : Rel A p} → B.Decidable R → List A → List A
 deduplicate R? = deduplicateᵇ (does ∘₂ R?)
 
+find : ∀ {P : Pred A p} → Decidable P → List A → Maybe A
+find P? = findᵇ (does ∘ P?)
+
+findIndex : ∀ {P : Pred A p} → Decidable P → (xs : List A) → Maybe $ Fin (length xs)
+findIndex P? = findIndexᵇ (does ∘ P?)
+
+findIndices : ∀ {P : Pred A p} → Decidable P → (xs : List A) → List $ Fin (length xs)
+findIndices P? = findIndicesᵇ (does ∘ P?)
+
 ------------------------------------------------------------------------
 -- Actions on single elements
 
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 4072da5ed8..0dcca71055 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -12,7 +12,7 @@ open import Algebra using (Op₂; Selective)
 open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Fin.Properties using (suc-injective)
-open import Data.List.Base
+open import Data.List.Base hiding (find)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Relation.Unary.All as All using (All)
 import Data.List.Relation.Unary.Any.Properties as Any
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index 551b721d8d..abf5bdc86f 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -9,7 +9,7 @@
 module Data.List.Relation.Binary.Subset.Setoid.Properties where
 
 open import Data.Bool.Base using (Bool; true; false)
-open import Data.List.Base hiding (_∷ʳ_)
+open import Data.List.Base hiding (_∷ʳ_; find)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Relation.Unary.All as All using (All)
 import Data.List.Membership.Setoid as Membership
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index b50b962e26..d51a5cba64 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -12,7 +12,7 @@ open import Data.Bool.Base using (Bool; false; true; T)
 open import Data.Bool.Properties using (T-∨; T-≡)
 open import Data.Empty using (⊥)
 open import Data.Fin.Base using (Fin; zero; suc)
-open import Data.List.Base as List
+open import Data.List.Base as List hiding (find)
 open import Data.List.Properties using (ʳ++-defn)
 open import Data.List.Effectful as Listₑ using (monad)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
diff --git a/src/Function/Base.agda b/src/Function/Base.agda
index 69ab60bed0..9bdb996dc8 100644
--- a/src/Function/Base.agda
+++ b/src/Function/Base.agda
@@ -72,7 +72,7 @@ flip f = λ y x → f x y
 
 -- Application - note that _$_ is right associative, as in Haskell.
 -- If you want a left associative infix application operator, use
--- Category.Functor._<$>_ from Category.Monad.Identity.IdentityMonad.
+-- RawFunctor._<$>_ from Effect.Functor.
 
 _$_ : ∀ {A : Set a} {B : A → Set b} →
       ((x : A) → B x) → ((x : A) → B x)

From 9cdb0fb28a76e02aa866c6d35a540d839864fcc2 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Tue, 12 Sep 2023 05:52:24 +0530
Subject: [PATCH 106/316] Final `Relation.Binary` simplifications (#2068)

---
 src/Codata/Sized/Conat/Properties.agda                   | 2 +-
 src/Data/AVL/Indexed/WithK.agda                          | 3 ++-
 src/Data/AVL/IndexedMap.agda                             | 3 ++-
 src/Data/AVL/Key.agda                                    | 2 +-
 src/Data/Bool.agda                                       | 1 -
 src/Data/Char/Properties.agda                            | 8 +++++++-
 src/Data/Container/Properties.agda                       | 2 +-
 src/Data/Container/Related.agda                          | 2 +-
 src/Data/Container/Relation/Binary/Equality/Setoid.agda  | 5 ++++-
 .../Container/Relation/Binary/Pointwise/Properties.agda  | 5 ++++-
 src/Data/Fin/Induction.agda                              | 6 ++++--
 src/Data/Float/Properties.agda                           | 7 ++++++-
 src/Data/Integer/Divisibility.agda                       | 2 +-
 src/Data/Integer/Divisibility/Signed.agda                | 6 +++++-
 src/Data/Integer/Properties.agda                         | 8 +++++++-
 src/Data/List/Countdown.agda                             | 2 +-
 src/Data/List/Fresh/Membership/Setoid.agda               | 3 ++-
 src/Data/List/Membership/Setoid.agda                     | 3 ++-
 src/Data/List/Relation/Binary/BagAndSetEquality.agda     | 3 ++-
 .../List/Relation/Binary/Disjoint/Propositional.agda     | 2 --
 src/Data/List/Relation/Binary/Disjoint/Setoid.agda       | 3 ++-
 .../List/Relation/Binary/Disjoint/Setoid/Properties.agda | 3 ++-
 .../List/Relation/Binary/Equality/DecPropositional.agda  | 2 +-
 src/Data/List/Relation/Binary/Equality/DecSetoid.agda    | 4 +++-
 .../List/Relation/Binary/Equality/Propositional.agda     | 2 +-
 .../Relation/Binary/Infix/Homogeneous/Properties.agda    | 4 +++-
 src/Data/List/Relation/Binary/Lex/NonStrict.agda         | 8 +++++++-
 src/Data/List/Relation/Binary/Lex/Strict.agda            | 8 +++++++-
 .../List/Relation/Binary/Permutation/Homogeneous.agda    | 5 ++++-
 .../List/Relation/Binary/Permutation/Propositional.agda  | 5 ++++-
 .../Binary/Permutation/Propositional/Properties.agda     | 3 ++-
 src/Data/List/Relation/Binary/Permutation/Setoid.agda    | 6 +++++-
 .../Relation/Binary/Prefix/Heterogeneous/Properties.agda | 4 +++-
 .../Relation/Binary/Prefix/Homogeneous/Properties.agda   | 4 +++-
 .../List/Relation/Binary/Sublist/DecPropositional.agda   | 4 +++-
 src/Data/List/Relation/Binary/Sublist/DecSetoid.agda     | 5 ++++-
 src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda | 3 ++-
 src/Data/List/Relation/Binary/Sublist/Propositional.agda | 5 ++++-
 src/Data/List/Relation/Binary/Sublist/Setoid.agda        | 4 +++-
 src/Data/List/Relation/Binary/Subset/Setoid.agda         | 3 ++-
 .../Relation/Binary/Suffix/Homogeneous/Properties.agda   | 4 +++-
 src/Data/List/Relation/Ternary/Interleaving.agda         | 2 +-
 .../List/Relation/Ternary/Interleaving/Properties.agda   | 4 +---
 src/Data/List/Relation/Unary/Enumerates/Setoid.agda      | 2 +-
 .../Relation/Unary/Enumerates/Setoid/Properties.agda     | 2 +-
 src/Data/Nat/Binary/Properties.agda                      | 8 +++++++-
 src/Data/Nat/Coprimality.agda                            | 3 ++-
 src/Data/Nat/Divisibility.agda                           | 7 ++++++-
 src/Data/Nat/GCD.agda                                    | 2 +-
 src/Data/Nat/LCM.agda                                    | 1 -
 src/Data/Nat/Properties.agda                             | 9 ++++++++-
 src/Data/Product/Function/Dependent/Setoid/WithK.agda    | 1 -
 src/Data/Product/Function/NonDependent/Setoid.agda       | 3 ++-
 src/Data/Product/Relation/Binary/Lex/NonStrict.agda      | 8 +++++++-
 src/Data/Product/Relation/Binary/Lex/Strict.agda         | 8 +++++++-
 .../Product/Relation/Binary/Pointwise/NonDependent.agda  | 8 +++++++-
 src/Data/Rational/Properties.agda                        | 8 +++++++-
 src/Data/Rational/Unnormalised/Properties.agda           | 8 +++++++-
 src/Data/Record.agda                                     | 2 +-
 src/Data/Star/BoundedVec.agda                            | 2 +-
 src/Data/Star/Decoration.agda                            | 3 ++-
 src/Data/Star/Nat.agda                                   | 2 +-
 src/Data/Star/Vec.agda                                   | 1 -
 src/Data/String/Properties.agda                          | 8 +++++++-
 src/Data/Sum/Function/Setoid.agda                        | 3 ++-
 src/Data/Sum/Relation/Binary/LeftOrder.agda              | 8 +++++++-
 src/Data/Sum/Relation/Binary/Pointwise.agda              | 8 +++++++-
 src/Data/Tree/AVL/Indexed/WithK.agda                     | 4 +++-
 src/Data/Tree/AVL/IndexedMap.agda                        | 3 ++-
 src/Data/Tree/AVL/Key.agda                               | 4 +++-
 src/Data/Unit/Polymorphic/Properties.agda                | 7 ++++++-
 .../Vec/Functional/Relation/Binary/Equality/Setoid.agda  | 6 +++++-
 src/Data/Vec/Functional/Relation/Binary/Pointwise.agda   | 2 +-
 .../Functional/Relation/Binary/Pointwise/Properties.agda | 7 ++++++-
 .../Vec/Relation/Binary/Equality/DecPropositional.agda   | 2 +-
 src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda     | 3 ++-
 src/Data/Vec/Relation/Binary/Equality/Propositional.agda | 2 --
 src/Data/Vec/Relation/Binary/Equality/Setoid.agda        | 6 ++++--
 src/Data/Vec/Relation/Binary/Lex/NonStrict.agda          | 8 +++++++-
 src/Data/Vec/Relation/Binary/Lex/Strict.agda             | 8 +++++++-
 src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda    | 7 ++++++-
 src/Data/Vec/Relation/Unary/Any/Properties.agda          | 3 ++-
 src/Relation/Binary/Construct/Add/Extrema/Equality.agda  | 6 +++++-
 src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda | 6 +++++-
 src/Relation/Binary/Construct/Add/Extrema/Strict.agda    | 6 +++++-
 src/Relation/Binary/Construct/Add/Infimum/Equality.agda  | 2 +-
 src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda | 6 +++++-
 src/Relation/Binary/Construct/Add/Infimum/Strict.agda    | 6 +++++-
 src/Relation/Binary/Construct/Add/Point/Equality.agda    | 6 +++++-
 src/Relation/Binary/Construct/Add/Supremum/Equality.agda | 2 +-
 .../Binary/Construct/Add/Supremum/NonStrict.agda         | 6 +++++-
 src/Relation/Binary/Construct/Add/Supremum/Strict.agda   | 6 +++++-
 src/Relation/Binary/Construct/Always.agda                | 5 ++++-
 src/Relation/Binary/Construct/Closure/Equivalence.agda   | 6 +++++-
 .../Construct/Closure/Reflexive/Properties/WithK.agda    | 3 ++-
 .../Binary/Construct/Closure/ReflexiveTransitive.agda    | 4 +++-
 .../Closure/ReflexiveTransitive/Properties.agda          | 5 ++++-
 .../Closure/ReflexiveTransitive/Properties/WithK.agda    | 2 +-
 src/Relation/Binary/Construct/Closure/Symmetric.agda     | 3 ++-
 .../Binary/Construct/Closure/SymmetricTransitive.agda    | 7 ++++++-
 .../Binary/Construct/Closure/Transitive/WithK.agda       | 2 +-
 src/Relation/Binary/Construct/Composition.agda           | 5 ++++-
 src/Relation/Binary/Construct/Constant.agda              | 4 +++-
 src/Relation/Binary/Construct/Flip/EqAndOrd.agda         | 8 +++++++-
 src/Relation/Binary/Construct/Flip/Ord.agda              | 8 +++++++-
 src/Relation/Binary/Construct/FromPred.agda              | 5 ++++-
 src/Relation/Binary/Construct/FromRel.agda               | 5 ++++-
 src/Relation/Binary/Construct/Intersection.agda          | 6 +++++-
 src/Relation/Binary/Construct/NaturalOrder/Left.agda     | 8 +++++++-
 src/Relation/Binary/Construct/NaturalOrder/Right.agda    | 9 ++++++++-
 src/Relation/Binary/Construct/NonStrictToStrict.agda     | 6 +++++-
 src/Relation/Binary/Construct/On.agda                    | 8 +++++++-
 src/Relation/Binary/Construct/StrictToNonStrict.agda     | 6 +++++-
 src/Relation/Binary/Construct/Subst/Equality.agda        | 4 +++-
 src/Relation/Binary/Construct/Union.agda                 | 4 +++-
 src/Relation/Binary/HeterogeneousEquality.agda           | 5 ++++-
 src/Relation/Binary/HeterogeneousEquality/Quotients.agda | 2 +-
 .../Binary/HeterogeneousEquality/Quotients/Examples.agda | 3 ++-
 .../Binary/Indexed/Heterogeneous/Construct/At.agda       | 3 ++-
 .../Binary/Indexed/Heterogeneous/Construct/Trivial.agda  | 4 +++-
 .../Binary/Indexed/Homogeneous/Construct/At.agda         | 4 +++-
 src/Relation/Binary/Lattice/Bundles.agda                 | 3 ++-
 .../Lattice/Properties/BoundedJoinSemilattice.agda       | 1 -
 .../Lattice/Properties/BoundedMeetSemilattice.agda       | 1 -
 .../Binary/Lattice/Properties/DistributiveLattice.agda   | 2 +-
 .../Binary/Lattice/Properties/JoinSemilattice.agda       | 4 +++-
 src/Relation/Binary/Lattice/Properties/Lattice.agda      | 1 -
 .../Binary/Lattice/Properties/MeetSemilattice.agda       | 3 ++-
 src/Relation/Binary/Lattice/Structures.agda              | 4 +++-
 src/Relation/Binary/Morphism/Construct/Composition.agda  | 4 +++-
 src/Relation/Binary/Morphism/Construct/Constant.agda     | 4 +++-
 src/Relation/Binary/Morphism/Construct/Identity.agda     | 4 +++-
 src/Relation/Binary/Morphism/OrderMonomorphism.agda      | 6 +++++-
 src/Relation/Binary/Morphism/RelMonomorphism.agda        | 5 ++++-
 src/Relation/Binary/OrderMorphism.agda                   | 3 ++-
 src/Relation/Binary/Properties/DecTotalOrder.agda        | 5 ++++-
 src/Relation/Binary/Properties/HeytingAlgebra.agda       | 2 +-
 src/Relation/Binary/Properties/Poset.agda                | 7 ++++++-
 src/Relation/Binary/Properties/Preorder.agda             | 3 ++-
 src/Relation/Binary/Properties/Setoid.agda               | 5 ++++-
 src/Relation/Binary/Properties/StrictPartialOrder.agda   | 4 ++--
 src/Relation/Binary/Properties/StrictTotalOrder.agda     | 4 ++--
 src/Relation/Binary/Properties/TotalOrder.agda           | 4 +++-
 src/Relation/Binary/PropositionalEquality.agda           | 3 ++-
 .../Binary/PropositionalEquality/Properties.agda         | 7 ++++++-
 src/Relation/Binary/PropositionalEquality/WithK.agda     | 2 +-
 src/Relation/Binary/Reasoning/Base/Apartness.agda        | 4 +++-
 src/Relation/Binary/Reasoning/Base/Double.agda           | 3 ++-
 src/Relation/Binary/Reasoning/Base/Single.agda           | 3 ++-
 src/Relation/Binary/Reasoning/Base/Triple.agda           | 5 ++++-
 src/Relation/Binary/Reasoning/MultiSetoid.agda           | 2 +-
 src/Relation/Binary/Reasoning/PartialOrder.agda          | 2 +-
 src/Relation/Binary/Reasoning/PartialSetoid.agda         | 2 +-
 src/Relation/Binary/Reasoning/Preorder.agda              | 2 +-
 src/Relation/Binary/Reasoning/Setoid.agda                | 2 +-
 src/Relation/Binary/Reasoning/StrictPartialOrder.agda    | 2 +-
 src/Relation/Binary/Reflection.agda                      | 2 +-
 src/Relation/Unary/Closure/Base.agda                     | 3 ++-
 src/Relation/Unary/Closure/Preorder.agda                 | 2 +-
 src/Relation/Unary/Closure/StrictPartialOrder.agda       | 2 +-
 tests/system/directory/Main.agda                         | 2 +-
 161 files changed, 513 insertions(+), 169 deletions(-)

diff --git a/src/Codata/Sized/Conat/Properties.agda b/src/Codata/Sized/Conat/Properties.agda
index 43d5662a9a..54e211dc1f 100644
--- a/src/Codata/Sized/Conat/Properties.agda
+++ b/src/Codata/Sized/Conat/Properties.agda
@@ -16,7 +16,7 @@ open import Codata.Sized.Conat.Bisimilarity
 open import Function.Base using (_∋_)
 open import Relation.Nullary
 open import Relation.Nullary.Decidable using (map′)
-open import Relation.Binary
+open import Relation.Binary.Definitions using (Decidable)
 
 private
   variable
diff --git a/src/Data/AVL/Indexed/WithK.agda b/src/Data/AVL/Indexed/WithK.agda
index 15da148fa2..609fb7b45d 100644
--- a/src/Data/AVL/Indexed/WithK.agda
+++ b/src/Data/AVL/Indexed/WithK.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --with-K --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsStrictTotalOrder)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; subst)
 
 module Data.AVL.Indexed.WithK
diff --git a/src/Data/AVL/IndexedMap.agda b/src/Data/AVL/IndexedMap.agda
index 84dbcceba4..fad21c14c2 100644
--- a/src/Data/AVL/IndexedMap.agda
+++ b/src/Data/AVL/IndexedMap.agda
@@ -7,7 +7,8 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.Product.Base using (∃)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsStrictTotalOrder)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
 import Data.Tree.AVL.Value
 
diff --git a/src/Data/AVL/Key.agda b/src/Data/AVL/Key.agda
index 96023d342d..f1ca15bacd 100644
--- a/src/Data/AVL/Key.agda
+++ b/src/Data/AVL/Key.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 
 module Data.AVL.Key
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Bool.agda b/src/Data/Bool.agda
index 5fdbcc45e0..1270125254 100644
--- a/src/Data/Bool.agda
+++ b/src/Data/Bool.agda
@@ -9,7 +9,6 @@
 module Data.Bool where
 
 open import Relation.Nullary
-open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index 52a997fd81..970eb6249a 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -17,7 +17,13 @@ open import Data.Product.Base using (_,_)
 open import Function.Base
 open import Relation.Nullary using (¬_; yes; no)
 open import Relation.Nullary.Decidable using (map′; isYes)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; StrictPartialOrder; StrictTotalOrder; Preorder; Poset; DecPoset)
+open import Relation.Binary.Structures
+  using (IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Decidable; Trichotomous; Irreflexive; Transitive; Asymmetric; Antisymmetric; Symmetric; Substitutive; Reflexive; tri<; tri≈; tri>)
 import Relation.Binary.Construct.On as On
 import Relation.Binary.Construct.Subst.Equality as Subst
 import Relation.Binary.Construct.Closure.Reflexive as Refl
diff --git a/src/Data/Container/Properties.agda b/src/Data/Container/Properties.agda
index 9ee24ad878..720455fefa 100644
--- a/src/Data/Container/Properties.agda
+++ b/src/Data/Container/Properties.agda
@@ -9,7 +9,7 @@
 module Data.Container.Properties where
 
 import Function.Base as F
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 
 open import Data.Container.Core
 open import Data.Container.Relation.Binary.Equality.Setoid
diff --git a/src/Data/Container/Related.agda b/src/Data/Container/Related.agda
index a62603ac49..5f1715c765 100644
--- a/src/Data/Container/Related.agda
+++ b/src/Data/Container/Related.agda
@@ -12,7 +12,7 @@ module Data.Container.Related where
 open import Level using (_⊔_)
 open import Data.Container.Core
 import Function.Related as Related
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Preorder; Setoid)
 open import Data.Container.Membership
 
 open Related public
diff --git a/src/Data/Container/Relation/Binary/Equality/Setoid.agda b/src/Data/Container/Relation/Binary/Equality/Setoid.agda
index 993aaef60b..c530413939 100644
--- a/src/Data/Container/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Container/Relation/Binary/Equality/Setoid.agda
@@ -11,7 +11,10 @@ open import Relation.Binary.Bundles using (Setoid)
 module Data.Container.Relation.Binary.Equality.Setoid {c e} (S : Setoid c e) where
 
 open import Level using (_⊔_; suc)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 
 open import Data.Container.Core
 open import Data.Container.Relation.Binary.Pointwise
diff --git a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
index 7ddc209860..45b824cc56 100644
--- a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
@@ -13,7 +13,10 @@ open import Data.Container.Core
 open import Data.Container.Relation.Binary.Pointwise
 open import Data.Product.Base using (_,_; Σ-syntax; -,_)
 open import Level using (_⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; subst; cong)
 
diff --git a/src/Data/Fin/Induction.agda b/src/Data/Fin/Induction.agda
index b0cf553d0d..a15287a48f 100644
--- a/src/Data/Fin/Induction.agda
+++ b/src/Data/Fin/Induction.agda
@@ -21,8 +21,10 @@ open import Function.Base using (flip; _$_)
 open import Induction
 open import Induction.WellFounded as WF
 open import Level using (Level)
-open import Relation.Binary
-  using (Rel; Decidable; IsPartialOrder; IsStrictPartialOrder; StrictPartialOrder)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (StrictPartialOrder)
+open import Relation.Binary.Structures using (IsPartialOrder; IsStrictPartialOrder)
+open import Relation.Binary.Definitions using (Decidable)
 import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 import Relation.Binary.Construct.Flip.Ord as Ord
 import Relation.Binary.Construct.NonStrictToStrict as ToStrict
diff --git a/src/Data/Float/Properties.agda b/src/Data/Float/Properties.agda
index b2caebb616..a397dd53ec 100644
--- a/src/Data/Float/Properties.agda
+++ b/src/Data/Float/Properties.agda
@@ -17,7 +17,12 @@ import Data.Word.Base as Word
 import Data.Word.Properties as Wₚ
 open import Function.Base using (_∘_)
 open import Relation.Nullary.Decidable as RN using (map′)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Setoid; DecSetoid)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Substitutive; Decidable; DecidableEquality)
 import Relation.Binary.Construct.On as On
 open import Relation.Binary.PropositionalEquality
 
diff --git a/src/Data/Integer/Divisibility.agda b/src/Data/Integer/Divisibility.agda
index 0f089b30fc..a8c55a1e2a 100644
--- a/src/Data/Integer/Divisibility.agda
+++ b/src/Data/Integer/Divisibility.agda
@@ -18,7 +18,7 @@ import Data.Nat.Properties as ℕᵖ
 import Data.Nat.Divisibility as ℕᵈ
 import Data.Nat.Coprimality as ℕᶜ
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
 open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index db80bcb3a0..af06d57486 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -21,7 +21,11 @@ import Data.Nat.Properties as ℕ
 import Data.Sign as S
 import Data.Sign.Properties as SProp
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
+open import Relation.Binary.Bundles using (Preorder)
+open import Relation.Binary.Structures using (IsPreorder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Transitive; Decidable)
 open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Nullary.Decidable using (yes; no)
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 78011e54b1..ef7528495e 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -27,7 +27,13 @@ open import Data.Sign as Sign using (Sign) renaming (_*_ to _𝕊*_)
 import Data.Sign.Properties as 𝕊ₚ
 open import Function.Base using (_∘_; _$_; id)
 open import Level using (0ℓ)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Bundles using
+  (Setoid; DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsTotalPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri<; tri>)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary using (yes; no; ¬_)
 import Relation.Nullary.Reflects as Reflects
diff --git a/src/Data/List/Countdown.agda b/src/Data/List/Countdown.agda
index db8dc806ad..8d6761de27 100644
--- a/src/Data/List/Countdown.agda
+++ b/src/Data/List/Countdown.agda
@@ -8,7 +8,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Level using (0ℓ)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (DecSetoid)
 
 module Data.List.Countdown (D : DecSetoid 0ℓ 0ℓ) where
 
diff --git a/src/Data/List/Fresh/Membership/Setoid.agda b/src/Data/List/Fresh/Membership/Setoid.agda
index 6ef9ef4836..5e0b3fdc15 100644
--- a/src/Data/List/Fresh/Membership/Setoid.agda
+++ b/src/Data/List/Fresh/Membership/Setoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Fresh.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where
 
diff --git a/src/Data/List/Membership/Setoid.agda b/src/Data/List/Membership/Setoid.agda
index 047620041d..1b8a67fb7d 100644
--- a/src/Data/List/Membership/Setoid.agda
+++ b/src/Data/List/Membership/Setoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (_Respects_)
 
 module Data.List.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where
 
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index d48b3dae45..451ce28c10 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -36,7 +36,8 @@ open import Function.Inverse as Inv using (_↔_; Inverse; inverse)
 open import Function.Related as Related
   using (↔⇒; ⌊_⌋; ⌊_⌋→; ⇒→; K-refl; SK-sym)
 open import Function.Related.TypeIsomorphisms
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Preorder; Setoid)
 import Relation.Binary.Reasoning.Setoid as EqR
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Binary.PropositionalEquality as P
diff --git a/src/Data/List/Relation/Binary/Disjoint/Propositional.agda b/src/Data/List/Relation/Binary/Disjoint/Propositional.agda
index 6b78c64c85..f60f275b89 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Propositional.agda
@@ -6,8 +6,6 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
-
 module Data.List.Relation.Binary.Disjoint.Propositional
   {a} {A : Set a} where
 
diff --git a/src/Data/List/Relation/Binary/Disjoint/Setoid.agda b/src/Data/List/Relation/Binary/Disjoint/Setoid.agda
index 6ea7e15cf7..d912c28825 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Setoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Binary.Disjoint.Setoid {c ℓ} (S : Setoid c ℓ) where
 
diff --git a/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
index 43a67d4d7d..bc5a95fc77 100644
--- a/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Disjoint/Setoid/Properties.agda
@@ -16,7 +16,8 @@ open import Data.List.Relation.Unary.All.Properties using (¬Any⇒All¬)
 open import Data.List.Relation.Unary.Any.Properties using (++⁻)
 open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (Symmetric)
 open import Relation.Nullary.Negation using (¬_)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
index 1697020ed3..0be92107c7 100644
--- a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
@@ -10,7 +10,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality
 
 module Data.List.Relation.Binary.Equality.DecPropositional
diff --git a/src/Data/List/Relation/Binary/Equality/DecSetoid.agda b/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
index ec82d91dcf..670e3c981b 100644
--- a/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
+++ b/src/Data/List/Relation/Binary/Equality/DecSetoid.agda
@@ -6,7 +6,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (DecSetoid)
+open import Relation.Binary.Structures using (IsDecEquivalence)
+open import Relation.Binary.Definitions using (Decidable)
 
 module Data.List.Relation.Binary.Equality.DecSetoid
   {a ℓ} (DS : DecSetoid a ℓ) where
diff --git a/src/Data/List/Relation/Binary/Equality/Propositional.agda b/src/Data/List/Relation/Binary/Equality/Propositional.agda
index 9ecb2cfd6b..8b45be7d27 100644
--- a/src/Data/List/Relation/Binary/Equality/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Equality/Propositional.agda
@@ -10,7 +10,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
 
 module Data.List.Relation.Binary.Equality.Propositional {a} {A : Set a} where
 
diff --git a/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda b/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda
index cf7c2b7b27..0abb36a149 100644
--- a/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Infix/Homogeneous/Properties.agda
@@ -10,7 +10,9 @@ module Data.List.Relation.Binary.Infix.Homogeneous.Properties where
 
 open import Level
 open import Function.Base using (_∘′_)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsDecPartialOrder)
 
 open import Data.List.Relation.Binary.Pointwise as Pointwise using (Pointwise)
 open import Data.List.Relation.Binary.Infix.Heterogeneous
diff --git a/src/Data/List/Relation/Binary/Lex/NonStrict.agda b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
index e87e1e0b45..4afe0f8242 100644
--- a/src/Data/List/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/List/Relation/Binary/Lex/NonStrict.agda
@@ -19,7 +19,13 @@ open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; [])
 import Data.List.Relation.Binary.Lex.Strict as Strict
 open import Level
 open import Relation.Nullary
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Poset; StrictPartialOrder; DecTotalOrder; StrictTotalOrder; Preorder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsPartialOrder; IsStrictPartialOrder; IsTotalOrder; IsStrictTotalOrder; IsPreorder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Irreflexive; _Respects₂_; Antisymmetric; Asymmetric; Symmetric; Transitive; Decidable; Total; Trichotomous)
 import Relation.Binary.Construct.NonStrictToStrict as Conv
 
 import Data.List.Relation.Binary.Lex as Core
diff --git a/src/Data/List/Relation/Binary/Lex/Strict.agda b/src/Data/List/Relation/Binary/Lex/Strict.agda
index d79311ba14..9ea73457fe 100644
--- a/src/Data/List/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/List/Relation/Binary/Lex/Strict.agda
@@ -19,7 +19,13 @@ open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.List.Base using (List; []; _∷_)
 open import Level using (_⊔_)
 open import Relation.Nullary using (yes; no; ¬_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (StrictPartialOrder; StrictTotalOrder; Preorder; Poset; DecPoset; DecTotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Irreflexive; Symmetric; _Respects₂_; Total; Asymmetric; Antisymmetric; Transitive; Trichotomous; Decidable; tri≈; tri<; tri>)
 open import Relation.Binary.Consequences
 open import Data.List.Relation.Binary.Pointwise as Pointwise
    using (Pointwise; []; _∷_; head; tail)
diff --git a/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda b/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda
index 6b9fd9c20d..12015b3076 100644
--- a/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Homogeneous.agda
@@ -14,7 +14,10 @@ open import Data.List.Relation.Binary.Pointwise.Base as Pointwise
 open import Data.List.Relation.Binary.Pointwise.Properties as Pointwise
   using (symmetric)
 open import Level using (Level; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric)
 
 private
   variable
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional.agda b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
index 5d9068755b..aaea103279 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
@@ -10,7 +10,10 @@ module Data.List.Relation.Binary.Permutation.Propositional
   {a} {A : Set a} where
 
 open import Data.List.Base using (List; []; _∷_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Transitive)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.Reasoning.Setoid as EqReasoning
 
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index 42430b90ea..82475a505b 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -27,7 +27,8 @@ open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse as Inv using (inverse)
 open import Level using (Level)
 open import Relation.Unary using (Pred)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Definitions using (_Respects_; Decidable)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_ ; refl ; cong; cong₂; _≢_)
 open import Relation.Nullary
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid.agda b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
index a9a0cf6400..fe3f095ee5 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
@@ -6,7 +6,11 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 
 module Data.List.Relation.Binary.Permutation.Setoid
   {a ℓ} (S : Setoid a ℓ) where
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
index 5706dbc36e..202bff89b0 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
@@ -25,7 +25,9 @@ open import Function.Base
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no; _because_)
 open import Relation.Unary as U using (Pred)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Definitions
+  using (Trans; Antisym; Irrelevant; Decidable)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; _≢_)
 
 private
diff --git a/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda
index 97ee8cc41b..ad4291432e 100644
--- a/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Homogeneous/Properties.agda
@@ -10,7 +10,9 @@ module Data.List.Relation.Binary.Prefix.Homogeneous.Properties where
 
 open import Level
 open import Function.Base using (_∘′_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsDecPartialOrder)
 
 open import Data.List.Relation.Binary.Pointwise as Pointwise using (Pointwise)
 open import Data.List.Relation.Binary.Prefix.Heterogeneous
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
index 6604f4aeec..2e1878e906 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
@@ -8,7 +8,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (DecPoset)
+open import Relation.Binary.Structures using (IsDecPartialOrder)
+open import Relation.Binary.Definitions using (Decidable)
 open import Agda.Builtin.Equality using (_≡_)
 
 module Data.List.Relation.Binary.Sublist.DecPropositional
diff --git a/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda b/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda
index 654d6c6a0e..a184386034 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecSetoid.agda
@@ -8,7 +8,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (DecSetoid; DecPoset)
+open import Relation.Binary.Structures
+  using (IsDecPartialOrder)
+open import Relation.Binary.Definitions using (Decidable)
 
 module Data.List.Relation.Binary.Sublist.DecSetoid
   {c ℓ} (S : DecSetoid c ℓ) where
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
index 698dd789b0..7aa3f4dd5d 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
@@ -11,7 +11,8 @@
 open import Data.List.Base using (List; []; _∷_; [_])
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Level using (_⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; _⇒_)
+open import Relation.Binary.Definitions using (_⟶_Respects_; Min)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Unary using (Pred)
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional.agda b/src/Data/List/Relation/Binary/Sublist/Propositional.agda
index 7700e10e59..a0f816ad84 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional.agda
@@ -14,7 +14,10 @@ open import Data.List.Base using (List)
 open import Data.List.Relation.Binary.Equality.Propositional using (≋⇒≡)
 import Data.List.Relation.Binary.Sublist.Setoid as SetoidSublist
 open import Data.List.Relation.Unary.Any using (Any)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Preorder; Poset)
+open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
+open import Relation.Binary.Definitions using (Antisymmetric)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Unary using (Pred)
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid.agda b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
index a07c6ffc25..86e16ab80c 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
@@ -26,7 +26,9 @@ import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
   as HeterogeneousProperties
 open import Data.Product.Base using (∃; ∃₂; _×_; _,_; proj₂)
 
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Preorder; Poset)
+open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary using (¬_; Dec; yes; no)
 
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid.agda b/src/Data/List/Relation/Binary/Subset/Setoid.agda
index 3bb516fb5c..0968e06332 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Binary.Subset.Setoid
   {c ℓ} (S : Setoid c ℓ) where
diff --git a/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda b/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda
index 8f8e9c4a77..0a5298f431 100644
--- a/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Homogeneous/Properties.agda
@@ -10,7 +10,9 @@ module Data.List.Relation.Binary.Suffix.Homogeneous.Properties where
 
 open import Level
 open import Function.Base using (_∘′_)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsDecPartialOrder)
 
 open import Data.List.Relation.Binary.Pointwise as Pointwise using (Pointwise)
 open import Data.List.Relation.Binary.Suffix.Heterogeneous
diff --git a/src/Data/List/Relation/Ternary/Interleaving.agda b/src/Data/List/Relation/Ternary/Interleaving.agda
index 8818638518..08d9f728e7 100644
--- a/src/Data/List/Relation/Ternary/Interleaving.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving.agda
@@ -15,7 +15,7 @@ open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_
 open import Data.Product.Base as Prod using (∃; ∃₂; _×_; uncurry; _,_; -,_; proj₂)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; _⇒_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
index da3d687ba5..dff26bb6b2 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
@@ -14,9 +14,7 @@ open import Data.List.Base hiding (_∷ʳ_)
 open import Data.List.Properties using (reverse-involutive)
 open import Data.List.Relation.Ternary.Interleaving hiding (map)
 open import Function.Base using (_$_)
-open import Relation.Binary
---open import Relation.Binary.PropositionalEquality
---  using (_≡_; refl; sym; cong; module ≡-Reasoning)
+open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; sym; cong; module ≡-Reasoning)
 open ≡-Reasoning
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid.agda
index 4ac2f914d7..8da8806ce1 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid.agda
@@ -8,7 +8,7 @@
 
 open import Data.List.Base using (List)
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 
 module Data.List.Relation.Unary.Enumerates.Setoid
   {a ℓ} (S : Setoid a ℓ) where
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index ba400055f3..f5e0a53c5c 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -22,7 +22,7 @@ open import Function.Bundles using (Surjection)
 open import Function.Definitions using (Surjective)
 open import Function.Consequences using (strictlySurjective⇒surjective)
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid; DecSetoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Properties.Setoid using (respʳ-flip)
 
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 43496f3139..0072a2d325 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -27,7 +27,13 @@ open import Function.Base using (_∘_; _$_; id)
 open import Function.Definitions
 open import Function.Consequences.Propositional
 open import Level using (0ℓ)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; StrictPartialOrder; StrictTotalOrder; Preorder; Poset; TotalOrder; DecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Decidable; Irreflexive; Transitive; Reflexive; Antisymmetric; Total; Trichotomous; tri≈; tri<; tri>)
+open import Relation.Binary.Structures
+  using (IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
 open import Relation.Binary.Consequences
 open import Relation.Binary.Morphism
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphism
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 94bfe8968c..68237de411 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -23,7 +23,8 @@ open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; trans; cong; subst; module ≡-Reasoning)
 open import Relation.Nullary as Nullary hiding (recompute)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Symmetric; Decidable)
 
 open ≤-Reasoning
 
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index 1b1321906d..ea4814b964 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -18,7 +18,12 @@ open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (0ℓ)
 open import Relation.Nullary.Decidable as Dec using (False; yes; no)
 open import Relation.Nullary.Negation.Core using (contradiction)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Preorder; Poset)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Transitive; Antisymmetric; Decidable)
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst)
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index f54b09e47f..2aa87002ca 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -21,7 +21,7 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_$_; _∘_)
 open import Induction using (build)
 open import Induction.Lexicographic using (_⊗_; [_⊗_])
-open import Relation.Binary
+open import Relation.Binary.Definitions using (tri<; tri>; tri≈; Symmetric)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; subst; cong)
 open import Relation.Nullary.Decidable using (Dec)
diff --git a/src/Data/Nat/LCM.agda b/src/Data/Nat/LCM.agda
index ccb1fbbf5b..545870574f 100644
--- a/src/Data/Nat/LCM.agda
+++ b/src/Data/Nat/LCM.agda
@@ -19,7 +19,6 @@ open import Data.Product.Base using (_×_; _,_; uncurry′; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl; sym; trans; cong; cong₂; module ≡-Reasoning)
-open import Relation.Binary
 open import Relation.Nullary.Decidable using (False; fromWitnessFalse)
 
 private
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 2b9a2a4819..be41e093ac 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -31,7 +31,14 @@ open import Function.Bundles using (_↣_)
 open import Function.Metric.Nat
 open import Level using (0ℓ)
 open import Relation.Unary as U using (Pred)
-open import Relation.Binary
+open import Relation.Binary.Core
+  using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Bundles
+  using (DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsDecEquivalence; IsPreorder; IsTotalPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (DecidableEquality; Irrelevant; Reflexive; Antisymmetric; Transitive; Total; Decidable; Connex; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri>; tri<; _Respects₂_)
 open import Relation.Binary.Consequences using (flip-Connex)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary hiding (Irrelevant)
diff --git a/src/Data/Product/Function/Dependent/Setoid/WithK.agda b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
index 3336ecb5e5..b4c30a26aa 100644
--- a/src/Data/Product/Function/Dependent/Setoid/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
@@ -12,7 +12,6 @@ module Data.Product.Function.Dependent.Setoid.WithK where
 open import Data.Product.Base using (_,_)
 open import Data.Product.Function.Dependent.Setoid using (surjection)
 open import Data.Product.Relation.Binary.Pointwise.Dependent
-open import Relation.Binary
 open import Function.Base
 open import Function.Equality as F using (_⟶_; _⟨$⟩_)
 open import Function.Equivalence as Eq
diff --git a/src/Data/Product/Function/NonDependent/Setoid.agda b/src/Data/Product/Function/NonDependent/Setoid.agda
index 93e388da23..1d69d9c85b 100644
--- a/src/Data/Product/Function/NonDependent/Setoid.agda
+++ b/src/Data/Product/Function/NonDependent/Setoid.agda
@@ -11,7 +11,8 @@ module Data.Product.Function.NonDependent.Setoid where
 
 open import Data.Product.Base using (map; _,_; <_,_>; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Relation.Binary
+open import Relation.Binary.Core using (_=[_]⇒_)
+open import Relation.Binary.Bundles using (Setoid)
 open import Function.Equality as F using (_⟶_; _⟨$⟩_)
 open import Function.Equivalence as Eq
   using (Equivalence; _⇔_; module Equivalence)
diff --git a/src/Data/Product/Relation/Binary/Lex/NonStrict.agda b/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
index e8752c1caf..534ddd6bdc 100644
--- a/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/NonStrict.agda
@@ -14,7 +14,13 @@ module Data.Product.Relation.Binary.Lex.NonStrict where
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Poset; DecTotalOrder; TotalOrder)
+open import Relation.Binary.Structures
+  using (IsPartialOrder; IsEquivalence; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Transitive; Symmetric; Decidable; Antisymmetric; Total; Trichotomous; Irreflexive; Asymmetric; _Respects₂_; tri<; tri>; tri≈)
 open import Relation.Binary.Consequences
 import Relation.Binary.Construct.NonStrictToStrict as Conv
 open import Data.Product.Relation.Binary.Pointwise.NonDependent as Pointwise
diff --git a/src/Data/Product/Relation/Binary/Lex/Strict.agda b/src/Data/Product/Relation/Binary/Lex/Strict.agda
index d0a09fa4ac..90129350ea 100644
--- a/src/Data/Product/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/Strict.agda
@@ -20,7 +20,13 @@ open import Function.Base
 open import Induction.WellFounded
 open import Level
 open import Relation.Nullary.Decidable
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Preorder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsPreorder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Transitive; Symmetric; Irreflexive; Asymmetric; Total; Decidable; Antisymmetric; Trichotomous; _Respects₂_; _Respectsʳ_; _Respectsˡ_; tri<; tri>; tri≈)
 open import Relation.Binary.Consequences
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)
 
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index 16756bf375..ad40326ab4 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -26,7 +26,13 @@ open import Function.Related
 open import Function.Surjection as Surj
   using (Surjection; _↠_; module Surjection)
 open import Relation.Nullary.Decidable using (_×-dec_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Preorder; Setoid; DecSetoid; Poset; StrictPartialOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsStrictPartialOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Irreflexive; Symmetric; Transitive; Antisymmetric; Asymmetric; Total; Decidable; _Respects₂_; _Respects_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 7c74cd2513..8910cc5998 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -49,7 +49,13 @@ import Data.Sign as S
 open import Function.Base using (_∘_; _∘′_; _∘₂_; _$_; flip)
 open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; TotalPreorder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsTotalOrder; IsTotalPreorder; IsPartialOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri<; tri>; tri≈; _Respectsʳ_; _Respectsˡ_; _Respects₂_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Binary.Morphism.Structures
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 2945a8654a..169c3786ad 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -34,7 +34,13 @@ open import Level using (0ℓ)
 open import Relation.Nullary using (¬_; yes; no)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Nullary.Negation using (contradiction; contraposition)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsApartnessRelation; IsTotalPreorder; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Cotransitive; Tight; Decidable; Antisymmetric; Asymmetric; Total; Trans; Trichotomous; Irreflexive; Irrelevant; _Respectsˡ_; _Respectsʳ_; _Respects₂_; tri≈; tri<; tri>)
 import Relation.Binary.Consequences as BC
 open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Properties.Poset as PosetProperties
diff --git a/src/Data/Record.agda b/src/Data/Record.agda
index 6d77f87622..6992294a5e 100644
--- a/src/Data/Record.agda
+++ b/src/Data/Record.agda
@@ -16,7 +16,7 @@ open import Data.Product.Base hiding (proj₁; proj₂)
 open import Data.Unit.Polymorphic
 open import Function.Base using (id; _∘_)
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary
 open import Relation.Nullary.Decidable
diff --git a/src/Data/Star/BoundedVec.agda b/src/Data/Star/BoundedVec.agda
index 2755554354..5664cf847d 100644
--- a/src/Data/Star/BoundedVec.agda
+++ b/src/Data/Star/BoundedVec.agda
@@ -17,7 +17,7 @@ open import Data.Star.Pointer
 open import Data.Star.List using (List)
 open import Data.Unit
 open import Function.Base using (const)
-open import Relation.Binary
+open import Relation.Binary.Core using (_=[_]⇒_)
 open import Relation.Binary.Consequences
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 
diff --git a/src/Data/Star/Decoration.agda b/src/Data/Star/Decoration.agda
index 48c0f4f11a..d4e968388b 100644
--- a/src/Data/Star/Decoration.agda
+++ b/src/Data/Star/Decoration.agda
@@ -11,7 +11,8 @@ module Data.Star.Decoration where
 open import Data.Unit
 open import Function.Base using (flip)
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
+open import Relation.Binary.Definitions using (NonEmpty; nonEmpty)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 
 -- A predicate on relation "edges" (think of the relation as a graph).
diff --git a/src/Data/Star/Nat.agda b/src/Data/Star/Nat.agda
index ef81051ff9..aa1eb9a55b 100644
--- a/src/Data/Star/Nat.agda
+++ b/src/Data/Star/Nat.agda
@@ -10,7 +10,7 @@ module Data.Star.Nat where
 
 open import Data.Unit
 open import Function.Base using (const)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Relation.Binary.Construct.Always using (Always)
 
diff --git a/src/Data/Star/Vec.agda b/src/Data/Star/Vec.agda
index 2fd96ee75b..108fbe7399 100644
--- a/src/Data/Star/Vec.agda
+++ b/src/Data/Star/Vec.agda
@@ -14,7 +14,6 @@ open import Data.Star.Decoration
 open import Data.Star.Pointer as Pointer hiding (lookup)
 open import Data.Star.List using (List)
 open import Level using (Level)
-open import Relation.Binary
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Function.Base
 open import Data.Unit
diff --git a/src/Data/String/Properties.agda b/src/Data/String/Properties.agda
index beb0d869d8..9c5ae194e2 100644
--- a/src/Data/String/Properties.agda
+++ b/src/Data/String/Properties.agda
@@ -17,7 +17,13 @@ open import Data.String.Base
 open import Function.Base
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (map′; isYes)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; StrictPartialOrder; StrictTotalOrder; DecTotalOrder; DecPoset)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsDecPartialOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Substitutive; Decidable)
 open import Relation.Binary.PropositionalEquality.Core
 import Relation.Binary.Construct.On as On
 import Relation.Binary.PropositionalEquality.Properties as PropEq
diff --git a/src/Data/Sum/Function/Setoid.agda b/src/Data/Sum/Function/Setoid.agda
index 277d5feb98..f04798dd50 100644
--- a/src/Data/Sum/Function/Setoid.agda
+++ b/src/Data/Sum/Function/Setoid.agda
@@ -10,7 +10,8 @@ module Data.Sum.Function.Setoid where
 
 open import Data.Sum.Base
 open import Data.Sum.Relation.Binary.Pointwise
-open import Relation.Binary
+open import Relation.Binary.Core using (_=[_]⇒_)
+open import Relation.Binary.Bundles using (Setoid)
 open import Function.Equality as F using (_⟶_; _⟨$⟩_)
 open import Function.Equivalence as Eq
   using (Equivalence; _⇔_; module Equivalence)
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index 1ff3b18d05..397bc598dc 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -17,7 +17,13 @@ open import Function.Base using (_$_; _∘_)
 open import Level
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Preorder; Poset; StrictPartialOrder; TotalOrder; DecTotalOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsStrictPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Transitive; Asymmetric; Total; Decidable; Irreflexive; Antisymmetric; Trichotomous; _Respectsʳ_; _Respectsˡ_; _Respects₂_; tri<; tri>; tri≈)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index 2b9b4d763d..0c4bf6751c 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -16,7 +16,13 @@ open import Function.Base using (_∘_; id)
 open import Function.Inverse using (Inverse)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Preorder; Poset)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsStrictPartialOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Asymmetric; Substitutive; Decidable; Irreflexive; Antisymmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
diff --git a/src/Data/Tree/AVL/Indexed/WithK.agda b/src/Data/Tree/AVL/Indexed/WithK.agda
index 169427e986..4a8944757d 100644
--- a/src/Data/Tree/AVL/Indexed/WithK.agda
+++ b/src/Data/Tree/AVL/Indexed/WithK.agda
@@ -6,7 +6,9 @@
 
 {-# OPTIONS --with-K --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsStrictTotalOrder)
+open import Relation.Binary.Bundles using (StrictTotalOrder)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; subst)
 
 module Data.Tree.AVL.Indexed.WithK
diff --git a/src/Data/Tree/AVL/IndexedMap.agda b/src/Data/Tree/AVL/IndexedMap.agda
index 7b46f3391a..54e4d73bbb 100644
--- a/src/Data/Tree/AVL/IndexedMap.agda
+++ b/src/Data/Tree/AVL/IndexedMap.agda
@@ -8,7 +8,8 @@
 
 open import Data.Product.Base
   using (map₁; map₂; ∃; _×_; Σ-syntax; proj₁; _,_; -,_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsStrictTotalOrder)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong; subst)
 import Data.Tree.AVL.Value
 
diff --git a/src/Data/Tree/AVL/Key.agda b/src/Data/Tree/AVL/Key.agda
index 5c47ccbacc..6fd5a1b021 100644
--- a/src/Data/Tree/AVL/Key.agda
+++ b/src/Data/Tree/AVL/Key.agda
@@ -7,7 +7,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles
+  using (StrictTotalOrder; StrictPartialOrder)
+open import Relation.Binary.Definitions using (Reflexive)
 
 module Data.Tree.AVL.Key
   {a ℓ₁ ℓ₂} (sto : StrictTotalOrder a ℓ₁ ℓ₂)
diff --git a/src/Data/Unit/Polymorphic/Properties.agda b/src/Data/Unit/Polymorphic/Properties.agda
index dce385f19e..87519aa9ee 100644
--- a/src/Data/Unit/Polymorphic/Properties.agda
+++ b/src/Data/Unit/Polymorphic/Properties.agda
@@ -16,7 +16,12 @@ open import Data.Sum.Base using (inj₁)
 open import Data.Unit.Base renaming (⊤ to ⊤*)
 open import Data.Unit.Polymorphic.Base using (⊤; tt)
 open import Relation.Nullary
-open import Relation.Binary
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Decidable; Antisymmetric; Total)
 open import Relation.Binary.PropositionalEquality
 
 private
diff --git a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
index 603582bf02..5853bd1f68 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
@@ -12,7 +12,11 @@ open import Data.Vec.Functional.Relation.Binary.Pointwise
   using (Pointwise)
 import Data.Vec.Functional.Relation.Binary.Pointwise.Properties as PW
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 module Data.Vec.Functional.Relation.Binary.Equality.Setoid
diff --git a/src/Data/Vec/Functional/Relation/Binary/Pointwise.agda b/src/Data/Vec/Functional/Relation/Binary/Pointwise.agda
index a954853bc6..c238293aae 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Pointwise.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Pointwise.agda
@@ -10,7 +10,7 @@ module Data.Vec.Functional.Relation.Binary.Pointwise where
 
 open import Data.Vec.Functional as VF hiding (map)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; _⇒_)
 
 private
   variable
diff --git a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
index feff1ab55c..148fcf346f 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
@@ -19,7 +19,12 @@ open import Data.Vec.Functional as VF hiding (map)
 open import Data.Vec.Functional.Relation.Binary.Pointwise
 open import Function.Base
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL)
+open import Relation.Binary.Bundles using (Setoid; DecSetoid)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Transitive; Symmetric; Decidable)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 private
diff --git a/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda b/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
index da50443c31..06f8d50e7d 100644
--- a/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality
 
 module Data.Vec.Relation.Binary.Equality.DecPropositional
diff --git a/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda b/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
index f3bd4cb1a9..01fd39b051 100644
--- a/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/DecSetoid.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (DecSetoid)
+open import Relation.Binary.Structures using (IsDecEquivalence)
 
 module Data.Vec.Relation.Binary.Equality.DecSetoid
   {a ℓ} (DS : DecSetoid a ℓ) where
diff --git a/src/Data/Vec/Relation/Binary/Equality/Propositional.agda b/src/Data/Vec/Relation/Binary/Equality/Propositional.agda
index 15e85a6bcf..93b837fed6 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Propositional.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Propositional.agda
@@ -6,8 +6,6 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
-
 module Data.Vec.Relation.Binary.Equality.Propositional {a} {A : Set a} where
 
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
diff --git a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
index d40bd24998..7e2e743088 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
@@ -6,7 +6,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (REL)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Sym; Trans)
 
 module Data.Vec.Relation.Binary.Equality.Setoid
   {a ℓ} (S : Setoid a ℓ) where
@@ -18,7 +21,6 @@ open import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
   using (Pointwise)
 open import Function.Base
 open import Level using (_⊔_)
-open import Relation.Binary
 
 open Setoid S renaming (Carrier to A)
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index 82036a5352..487caa4c0e 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -20,7 +20,13 @@ import Data.Vec.Relation.Binary.Lex.Strict as Strict
 open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
   using (Pointwise; []; _∷_; head; tail)
 open import Function.Base using (id)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Poset; StrictPartialOrder; DecPoset; DecStrictPartialOrder; DecTotalOrder; StrictTotalOrder; Preorder; TotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsPartialOrder; IsStrictPartialOrder; IsDecPartialOrder; IsDecStrictPartialOrder; IsDecTotalOrder; IsStrictTotalOrder; IsPreorder; IsTotalOrder)
+open import Relation.Binary.Definitions
+  using (Irreflexive; _Respects₂_; Antisymmetric; Asymmetric; Symmetric; Trans; Decidable; Total; Trichotomous)
 import Relation.Binary.Construct.NonStrictToStrict as Conv
 open import Relation.Nullary hiding (Irrelevant)
 open import Level using (Level; _⊔_)
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 3bafcdca03..450c23809b 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -24,7 +24,13 @@ open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 open import Function.Base using (id; _on_; _∘_)
 open import Induction.WellFounded
 open import Relation.Nullary using (yes; no; ¬_)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Poset; StrictPartialOrder; DecPoset; DecStrictPartialOrder; DecTotalOrder; StrictTotalOrder; Preorder; TotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsPartialOrder; IsStrictPartialOrder; IsDecPartialOrder; IsDecStrictPartialOrder; IsDecTotalOrder; IsStrictTotalOrder; IsPreorder; IsTotalOrder; IsPartialEquivalence)
+open import Relation.Binary.Definitions
+  using (Irreflexive; _Respects₂_; _Respectsˡ_; _Respectsʳ_; Antisymmetric; Asymmetric; Symmetric; Trans; Decidable; Total; Trichotomous; Transitive; Irrelevant; tri≈; tri>; tri<)
 open import Relation.Binary.Consequences
 open import Relation.Binary.Construct.On as On using (wellFounded)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
index 85bd6df9da..c6835b73e1 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
@@ -16,7 +16,12 @@ open import Data.Vec.Relation.Unary.All using (All; []; _∷_)
 open import Level using (Level; _⊔_)
 open import Function.Base using (_∘_)
 open import Function.Bundles using (_⇔_; mk⇔)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Bundles using (Setoid; DecSetoid)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence)
+open import Relation.Binary.Definitions
+  using (Trans; Decidable; Reflexive; Sym)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Decidable using (yes; no; _×-dec_; map′)
 open import Relation.Unary using (Pred)
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index 96f8874bd7..6069a6b6db 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -28,7 +28,8 @@ open import Function.Inverse using (_↔_; inverse)
 open import Level using (Level)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Unary hiding (_∈_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (_Respects_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
 
 private
diff --git a/src/Relation/Binary/Construct/Add/Extrema/Equality.agda b/src/Relation/Binary/Construct/Add/Extrema/Equality.agda
index b4992e112d..d34e84602f 100644
--- a/src/Relation/Binary/Construct/Add/Extrema/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Extrema/Equality.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Extrema
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Decidable; Irrelevant; Substitutive)
 
 module Relation.Binary.Construct.Add.Extrema.Equality
   {a ℓ} {A : Set a} (_≈_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda b/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
index ac1880044d..35fdf49431 100644
--- a/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Extrema/NonStrict.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Extrema
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Decidable; Transitive; Minimum; Maximum; Total; Irrelevant; Antisymmetric)
 
 module Relation.Binary.Construct.Add.Extrema.NonStrict
   {a ℓ} {A : Set a} (_≤_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Extrema/Strict.agda b/src/Relation/Binary/Construct/Add/Extrema/Strict.agda
index 1195e265a9..4ed89565ae 100644
--- a/src/Relation/Binary/Construct/Add/Extrema/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Extrema/Strict.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Extrema
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures
+  using (IsStrictPartialOrder; IsDecStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Asymmetric; Transitive; Decidable; Irrelevant; Trichotomous; Irreflexive; Trans; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 
 module Relation.Binary.Construct.Add.Extrema.Strict
   {a ℓ} {A : Set a} (_<_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Equality.agda b/src/Relation/Binary/Construct/Add/Infimum/Equality.agda
index 4c1fd79449..6f5336fc44 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Equality.agda
@@ -9,7 +9,7 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Infimum
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 
 module Relation.Binary.Construct.Add.Infimum.Equality
   {a ℓ} {A : Set a} (_≈_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
index b6cd5e4bf6..5d5cb2a214 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Infimum
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Minimum; Transitive; Total; Decidable; Irrelevant; Antisymmetric)
 
 module Relation.Binary.Construct.Add.Infimum.NonStrict
   {a ℓ} {A : Set a} (_≤_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
index e4c8f3a974..ae96849340 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Infimum
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures
+  using (IsStrictPartialOrder; IsDecStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Asymmetric; Transitive; Decidable; Irrelevant; Irreflexive; Trans; Trichotomous; tri≈; tri<; tri>; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 
 module Relation.Binary.Construct.Add.Infimum.Strict
   {a ℓ} {A : Set a} (_<_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Point/Equality.agda b/src/Relation/Binary/Construct/Add/Point/Equality.agda
index 1715d6f2d1..704d963123 100644
--- a/src/Relation/Binary/Construct/Add/Point/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Point/Equality.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Point
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Decidable; Irrelevant; Substitutive)
 
 module Relation.Binary.Construct.Add.Point.Equality
   {a ℓ} {A : Set a} (_≈_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Supremum/Equality.agda b/src/Relation/Binary/Construct/Add/Supremum/Equality.agda
index 04f9376f8e..c0a18a68e5 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/Equality.agda
@@ -9,7 +9,7 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Supremum
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 
 module Relation.Binary.Construct.Add.Supremum.Equality
   {a ℓ} {A : Set a} (_≈_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
index a509871377..b550dd4924 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Supremum
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Maximum; Transitive; Total; Decidable; Irrelevant; Antisymmetric)
 
 module Relation.Binary.Construct.Add.Supremum.NonStrict
   {a ℓ} {A : Set a} (_≤_ : Rel A ℓ) where
diff --git a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
index b09432961d..d927cf45c4 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
@@ -9,7 +9,11 @@
 -- This module is designed to be used with
 -- Relation.Nullary.Construct.Add.Supremum
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures
+  using (IsStrictPartialOrder; IsDecStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Asymmetric; Transitive; Decidable; Irrelevant; Irreflexive; Trans; Trichotomous; tri≈; tri>; tri<; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 
 module Relation.Binary.Construct.Add.Supremum.Strict
   {a r} {A : Set a} (_<_ : Rel A r) where
diff --git a/src/Relation/Binary/Construct/Always.agda b/src/Relation/Binary/Construct/Always.agda
index 76eb5e5963..70c22e1034 100644
--- a/src/Relation/Binary/Construct/Always.agda
+++ b/src/Relation/Binary/Construct/Always.agda
@@ -8,7 +8,10 @@
 
 module Relation.Binary.Construct.Always where
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.Construct.Constant using (Const)
 open import Data.Unit.Polymorphic using (⊤; tt)
 
diff --git a/src/Relation/Binary/Construct/Closure/Equivalence.agda b/src/Relation/Binary/Construct/Closure/Equivalence.agda
index a4aea65266..622bf20e24 100644
--- a/src/Relation/Binary/Construct/Closure/Equivalence.agda
+++ b/src/Relation/Binary/Construct/Closure/Equivalence.agda
@@ -11,7 +11,11 @@ module Relation.Binary.Construct.Closure.Equivalence where
 
 open import Function.Base using (flip; id; _∘_; _on_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Transitive; Symmetric)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive as Star
   using (Star; ε; _◅◅_; reverse)
 open import Relation.Binary.Construct.Closure.Symmetric as SC
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda
index 26d04320b0..8684bf119a 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties/WithK.agda
@@ -8,7 +8,8 @@
 
 module Relation.Binary.Construct.Closure.Reflexive.Properties.WithK where
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Irrelevant; Irreflexive)
 open import Relation.Binary.Construct.Closure.Reflexive
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda
index ef9f4fe074..71ab8535d8 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda
@@ -8,7 +8,9 @@
 
 module Relation.Binary.Construct.Closure.ReflexiveTransitive where
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
+open import Relation.Binary.Definitions
+  using (Transitive; Trans; Sym; TransFlip; Reflexive)
 open import Function.Base
 open import Level using (_⊔_)
 
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index 963c5b6330..b2c173c190 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -9,7 +9,10 @@
 module Relation.Binary.Construct.Closure.ReflexiveTransitive.Properties where
 
 open import Function.Base using (id; _∘_; _$_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
+open import Relation.Binary.Bundles using (Preorder)
+open import Relation.Binary.Structures using (IsPreorder)
+open import Relation.Binary.Definitions using (Transitive; Reflexive)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; refl; sym; cong; cong₂)
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
index aa2fd6e0a5..ee4078a75c 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
@@ -12,7 +12,7 @@ module
   where
 
 open import Function.Base using (_∋_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Relation.Binary.PropositionalEquality
 
diff --git a/src/Relation/Binary/Construct/Closure/Symmetric.agda b/src/Relation/Binary/Construct/Closure/Symmetric.agda
index c1731b5ecd..369b368f20 100644
--- a/src/Relation/Binary/Construct/Closure/Symmetric.agda
+++ b/src/Relation/Binary/Construct/Closure/Symmetric.agda
@@ -10,7 +10,8 @@ module Relation.Binary.Construct.Closure.Symmetric where
 
 open import Function.Base using (id; _on_)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
+open import Relation.Binary.Definitions using (Symmetric)
 import Relation.Binary.Construct.On as On
 
 private
diff --git a/src/Relation/Binary/Construct/Closure/SymmetricTransitive.agda b/src/Relation/Binary/Construct/Closure/SymmetricTransitive.agda
index 0ef90ba9b4..5766ce704a 100644
--- a/src/Relation/Binary/Construct/Closure/SymmetricTransitive.agda
+++ b/src/Relation/Binary/Construct/Closure/SymmetricTransitive.agda
@@ -10,7 +10,12 @@ module Relation.Binary.Construct.Closure.SymmetricTransitive where
 
 open import Level
 open import Function.Base
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
+open import Relation.Binary.Bundles using (PartialSetoid; Setoid)
+open import Relation.Binary.Structures
+  using (IsPartialEquivalence; IsEquivalence)
+open import Relation.Binary.Definitions
+  using (Transitive; Symmetric; Reflexive)
 
 private
   variable
diff --git a/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda b/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
index b432e3e1dd..c05323903b 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive/WithK.agda
@@ -9,7 +9,7 @@
 module Relation.Binary.Construct.Closure.Transitive.WithK where
 
 open import Function.Base using (_∋_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Closure.Transitive
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
diff --git a/src/Relation/Binary/Construct/Composition.agda b/src/Relation/Binary/Construct/Composition.agda
index 7f8ba241b0..4d66ae5171 100644
--- a/src/Relation/Binary/Construct/Composition.agda
+++ b/src/Relation/Binary/Construct/Composition.agda
@@ -11,7 +11,10 @@ module Relation.Binary.Construct.Composition where
 open import Data.Product.Base using (∃; _×_; _,_)
 open import Function.Base
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Structures using (IsPreorder)
+open import Relation.Binary.Definitions
+  using (_Respects_; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Reflexive; Transitive)
 
 private
   variable
diff --git a/src/Relation/Binary/Construct/Constant.agda b/src/Relation/Binary/Construct/Constant.agda
index 1dad265ea7..8566dc6e30 100644
--- a/src/Relation/Binary/Construct/Constant.agda
+++ b/src/Relation/Binary/Construct/Constant.agda
@@ -9,7 +9,9 @@
 module Relation.Binary.Construct.Constant where
 
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
 
 private
   variable
diff --git a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
index c3c4dc2a18..78671dfbdc 100644
--- a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
+++ b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
@@ -8,7 +8,13 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; TotalPreorder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder; IsTotalPreorder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Asymmetric; Total; _Respects_; _Respects₂_; Minimum; Maximum; Irreflexive; Antisymmetric; Trichotomous; Decidable; tri<; tri>; tri≈)
 
 module Relation.Binary.Construct.Flip.EqAndOrd where
 
diff --git a/src/Relation/Binary/Construct/Flip/Ord.agda b/src/Relation/Binary/Construct/Flip/Ord.agda
index 4a2eea690f..742a315cc6 100644
--- a/src/Relation/Binary/Construct/Flip/Ord.agda
+++ b/src/Relation/Binary/Construct/Flip/Ord.agda
@@ -8,7 +8,13 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Asymmetric; Total; _Respects_; _Respects₂_; Minimum; Maximum; Irreflexive; Antisymmetric; Trichotomous; Decidable)
 
 module Relation.Binary.Construct.Flip.Ord where
 
diff --git a/src/Relation/Binary/Construct/FromPred.agda b/src/Relation/Binary/Construct/FromPred.agda
index e08997b9a7..eb39112719 100644
--- a/src/Relation/Binary/Construct/FromPred.agda
+++ b/src/Relation/Binary/Construct/FromPred.agda
@@ -7,7 +7,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles using (Setoid; Preorder)
+open import Relation.Binary.Structures using (IsPreorder)
+open import Relation.Binary.Definitions using (_Respects_; Reflexive; Transitive)
 open import Relation.Unary using (Pred)
 
 module Relation.Binary.Construct.FromPred
diff --git a/src/Relation/Binary/Construct/FromRel.agda b/src/Relation/Binary/Construct/FromRel.agda
index 13fc1774c7..3cac113912 100644
--- a/src/Relation/Binary/Construct/FromRel.agda
+++ b/src/Relation/Binary/Construct/FromRel.agda
@@ -7,7 +7,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Bundles using (Setoid; Preorder)
+open import Relation.Binary.Structures using (IsPreorder)
+open import Relation.Binary.Definitions using (_Respects_; Transitive)
 open Setoid using (Carrier)
 
 module Relation.Binary.Construct.FromRel
diff --git a/src/Relation/Binary/Construct/Intersection.agda b/src/Relation/Binary/Construct/Intersection.agda
index 17845536ca..0d26d675e6 100644
--- a/src/Relation/Binary/Construct/Intersection.agda
+++ b/src/Relation/Binary/Construct/Intersection.agda
@@ -12,7 +12,11 @@ open import Data.Product.Base
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Function.Base using (_∘_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsStrictPartialOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Antisymmetric; Decidable; _Respects_; _Respectsˡ_; _Respectsʳ_; _Respects₂_; Minimum; Maximum; Irreflexive)
 open import Relation.Nullary.Decidable using (yes; no; _×-dec_)
 
 private
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Left.agda b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
index e0ab59e9ca..22a5031d27 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Left.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
@@ -10,7 +10,13 @@
 open import Algebra.Core
 open import Data.Product.Base using (_,_; _×_)
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Preorder; Poset; DecPoset; TotalOrder; DecTotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Symmetric; Transitive; Reflexive; Antisymmetric; Total; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Decidable)
 open import Relation.Nullary.Negation using (¬_)
 import Relation.Binary.Reasoning.Setoid as EqReasoning
 open import Relation.Binary.Lattice using (Infimum)
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Right.agda b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
index 5278f0b116..80901f5939 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Right.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
@@ -10,7 +10,14 @@
 open import Algebra.Core using (Op₂)
 open import Data.Product.Base using (_,_; _×_)
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Preorder; Poset; DecPoset; TotalOrder; DecTotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Symmetric; Transitive; Reflexive; Antisymmetric; Total; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Decidable)
+
 open import Relation.Nullary.Negation using (¬_)
 import Relation.Binary.Reasoning.Setoid as EqReasoning
 
diff --git a/src/Relation/Binary/Construct/NonStrictToStrict.agda b/src/Relation/Binary/Construct/NonStrictToStrict.agda
index d9dd2c57ce..c2eb114004 100644
--- a/src/Relation/Binary/Construct/NonStrictToStrict.agda
+++ b/src/Relation/Binary/Construct/NonStrictToStrict.agda
@@ -6,7 +6,11 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Structures
+  using (IsPartialOrder; IsEquivalence; IsStrictPartialOrder; IsDecPartialOrder; IsDecStrictPartialOrder; IsTotalOrder; IsStrictTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Trichotomous; Antisymmetric; Symmetric; Total; Decidable; Irreflexive; Transitive; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Trans; Asymmetric; tri≈; tri<; tri>)
 
 module Relation.Binary.Construct.NonStrictToStrict
   {a ℓ₁ ℓ₂} {A : Set a} (_≈_ : Rel A ℓ₁) (_≤_ : Rel A ℓ₂) where
diff --git a/src/Relation/Binary/Construct/On.agda b/src/Relation/Binary/Construct/On.agda
index 122b0d6584..72959ac922 100644
--- a/src/Relation/Binary/Construct/On.agda
+++ b/src/Relation/Binary/Construct/On.agda
@@ -12,7 +12,13 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (_on_; _∘_)
 open import Induction.WellFounded using (WellFounded; Acc; acc)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Preorder; Setoid; DecSetoid; Poset; DecPoset; StrictPartialOrder; TotalOrder; DecTotalOrder; StrictTotalOrder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsStrictPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Reflexive; Irreflexive; Symmetric; Transitive; Antisymmetric; Asymmetric; Decidable; Total; Trichotomous; _Respects_; _Respects₂_)
 
 private
   variable
diff --git a/src/Relation/Binary/Construct/StrictToNonStrict.agda b/src/Relation/Binary/Construct/StrictToNonStrict.agda
index 42c76fc80f..d17b6017fc 100644
--- a/src/Relation/Binary/Construct/StrictToNonStrict.agda
+++ b/src/Relation/Binary/Construct/StrictToNonStrict.agda
@@ -10,7 +10,11 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsPreorder; IsStrictPartialOrder; IsPartialOrder; IsStrictTotalOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary.Definitions
+  using (Transitive; Symmetric; Irreflexive; Antisymmetric; Trichotomous; Decidable; Trans; Total; _Respects₂_; _Respectsʳ_; _Respectsˡ_; tri<; tri≈; tri>)
 
 module Relation.Binary.Construct.StrictToNonStrict
   {a ℓ₁ ℓ₂} {A : Set a}
diff --git a/src/Relation/Binary/Construct/Subst/Equality.agda b/src/Relation/Binary/Construct/Subst/Equality.agda
index 24e465f9f0..95172b57ee 100644
--- a/src/Relation/Binary/Construct/Subst/Equality.agda
+++ b/src/Relation/Binary/Construct/Subst/Equality.agda
@@ -10,7 +10,9 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.Product.Base as Prod
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇔_)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
 
 module Relation.Binary.Construct.Subst.Equality
   {a ℓ₁ ℓ₂} {A : Set a} {≈₁ : Rel A ℓ₁} {≈₂ : Rel A ℓ₂}
diff --git a/src/Relation/Binary/Construct/Union.agda b/src/Relation/Binary/Construct/Union.agda
index 2b70ce8500..45f72c8190 100644
--- a/src/Relation/Binary/Construct/Union.agda
+++ b/src/Relation/Binary/Construct/Union.agda
@@ -12,7 +12,9 @@ open import Data.Product.Base
 open import Data.Sum.Base as Sum
 open import Function.Base using (_∘_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
+open import Relation.Binary.Definitions
+  using (Reflexive; Total; Minimum; Maximum; Symmetric; Irreflexive; Decidable; _Respects_; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 open import Relation.Nullary.Decidable using (yes; no; _⊎-dec_)
 
 private
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 3152bbf0cb..866628a71c 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -15,7 +15,10 @@ open import Function.Inverse using (Inverse)
 open import Level
 open import Relation.Nullary hiding (Irrelevant)
 open import Relation.Unary using (Pred)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder)
+open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)
+open import Relation.Binary.Definitions using (Substitutive; Irrelevant; Decidable; _Respects₂_)
 open import Relation.Binary.Consequences
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
diff --git a/src/Relation/Binary/HeterogeneousEquality/Quotients.agda b/src/Relation/Binary/HeterogeneousEquality/Quotients.agda
index fcdc1e97fc..18bc88002a 100644
--- a/src/Relation/Binary/HeterogeneousEquality/Quotients.agda
+++ b/src/Relation/Binary/HeterogeneousEquality/Quotients.agda
@@ -10,7 +10,7 @@ module Relation.Binary.HeterogeneousEquality.Quotients where
 
 open import Function.Base using (_$_; const; _∘_)
 open import Level hiding (lift)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.HeterogeneousEquality
 open ≅-Reasoning
 
diff --git a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
index 2dbd39344a..5087b908fc 100644
--- a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
+++ b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
@@ -10,7 +10,8 @@ module Relation.Binary.HeterogeneousEquality.Quotients.Examples where
 
 open import Relation.Binary.HeterogeneousEquality.Quotients
 open import Level using (0ℓ)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.HeterogeneousEquality hiding (isEquivalence)
 import Relation.Binary.PropositionalEquality.Core as ≡
 open import Data.Nat.Base
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda b/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
index f9c6dd8712..a05a9aa139 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
@@ -9,7 +9,8 @@
 
 module Relation.Binary.Indexed.Heterogeneous.Construct.At where
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid; Preorder)
+open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)
 open import Relation.Binary.Indexed.Heterogeneous
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Construct/Trivial.agda b/src/Relation/Binary/Indexed/Heterogeneous/Construct/Trivial.agda
index 800d7e09f3..797e0587bc 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Construct/Trivial.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Construct/Trivial.agda
@@ -9,7 +9,9 @@
 module Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   {i} {I : Set i} where
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid; Preorder)
+open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)
 open import Relation.Binary.Indexed.Heterogeneous
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda b/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
index 9899c200b3..14668cfa49 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
@@ -7,7 +7,9 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Level
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder)
 open import Relation.Binary.Indexed.Homogeneous
 
 module Relation.Binary.Indexed.Homogeneous.Construct.At where
diff --git a/src/Relation/Binary/Lattice/Bundles.agda b/src/Relation/Binary/Lattice/Bundles.agda
index ffe5dbf50d..7752c79676 100644
--- a/src/Relation/Binary/Lattice/Bundles.agda
+++ b/src/Relation/Binary/Lattice/Bundles.agda
@@ -13,7 +13,8 @@ module Relation.Binary.Lattice.Bundles where
 
 open import Algebra.Core
 open import Level using (suc; _⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Poset; Setoid)
 open import Relation.Binary.Lattice.Structures
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
index a86918081f..276e3a9705 100644
--- a/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
@@ -18,7 +18,6 @@ open import Algebra.Ordered.Structures using (IsCommutativePomonoid)
 open import Algebra.Ordered.Bundles using (CommutativePomonoid)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; flip)
-open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
 open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
   using (isPosemigroup; ∨-comm)
diff --git a/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda b/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda
index 622c8b6b0c..b0c945c39f 100644
--- a/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/BoundedMeetSemilattice.agda
@@ -15,7 +15,6 @@ open BoundedMeetSemilattice M
 
 open import Algebra.Definitions _≈_
 open import Function.Base using (_∘_; flip)
-open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
 import Relation.Binary.Lattice.Properties.BoundedJoinSemilattice as J
 
diff --git a/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda b/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda
index e3bd0170eb..f287e0a05d 100644
--- a/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/DistributiveLattice.agda
@@ -15,7 +15,7 @@ open DistributiveLattice L hiding (refl)
 
 open import Algebra.Definitions _≈_
 open import Data.Product.Base using (_,_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Reasoning.Setoid setoid
 open import Relation.Binary.Lattice.Properties.Lattice lattice
 open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice
diff --git a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
index faa51120cd..c68467f577 100644
--- a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
@@ -20,7 +20,9 @@ open import Algebra.Ordered.Structures using (IsPosemigroup)
 open import Algebra.Ordered.Bundles using (Posemigroup)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; flip)
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves₂_⟶_⟶_)
+open import Relation.Binary.Structures using (IsDecPartialOrder)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.Properties.Poset poset
 open import Relation.Nullary using (¬_; yes; no)
 open import Relation.Nullary.Negation using (contraposition)
diff --git a/src/Relation/Binary/Lattice/Properties/Lattice.agda b/src/Relation/Binary/Lattice/Properties/Lattice.agda
index 633832a26c..fd95effaaf 100644
--- a/src/Relation/Binary/Lattice/Properties/Lattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/Lattice.agda
@@ -18,7 +18,6 @@ import Algebra.Structures as Alg
 open import Algebra.Definitions _≈_
 open import Data.Product.Base using (_,_)
 open import Function.Base using (flip)
-open import Relation.Binary
 open import Relation.Binary.Properties.Poset poset
 import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice as J
 import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice as M
diff --git a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
index f072cb2199..a78d4fea6a 100644
--- a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
@@ -15,7 +15,8 @@ open MeetSemilattice M
 
 open import Algebra.Definitions _≈_
 open import Function.Base using (flip)
-open import Relation.Binary
+open import Relation.Binary.Structures using (IsDecPartialOrder)
+open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.Properties.Poset poset
 import Relation.Binary.Lattice.Properties.JoinSemilattice as J
 
diff --git a/src/Relation/Binary/Lattice/Structures.agda b/src/Relation/Binary/Lattice/Structures.agda
index 07dbfad6a5..77d9604167 100644
--- a/src/Relation/Binary/Lattice/Structures.agda
+++ b/src/Relation/Binary/Lattice/Structures.agda
@@ -9,7 +9,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsPartialOrder)
+open import Relation.Binary.Definitions using (Minimum; Maximum)
 
 module Relation.Binary.Lattice.Structures
  {a ℓ₁ ℓ₂} {A : Set a}
diff --git a/src/Relation/Binary/Morphism/Construct/Composition.agda b/src/Relation/Binary/Morphism/Construct/Composition.agda
index 5f6fe9bf32..d86860b6bb 100644
--- a/src/Relation/Binary/Morphism/Construct/Composition.agda
+++ b/src/Relation/Binary/Morphism/Construct/Composition.agda
@@ -9,7 +9,9 @@
 open import Function.Base using (_∘_)
 open import Function.Construct.Composition using (surjective)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
+open import Relation.Binary.Definitions using (Transitive)
 open import Relation.Binary.Morphism.Bundles
 open import Relation.Binary.Morphism.Structures
 
diff --git a/src/Relation/Binary/Morphism/Construct/Constant.agda b/src/Relation/Binary/Morphism/Construct/Constant.agda
index d31c274bfa..f3a1bd2212 100644
--- a/src/Relation/Binary/Morphism/Construct/Constant.agda
+++ b/src/Relation/Binary/Morphism/Construct/Constant.agda
@@ -8,7 +8,9 @@
 
 open import Function.Base using (const)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid; Preorder)
+open import Relation.Binary.Definitions using (Reflexive)
 open import Relation.Binary.Morphism.Structures
 open import Relation.Binary.Morphism.Bundles
 
diff --git a/src/Relation/Binary/Morphism/Construct/Identity.agda b/src/Relation/Binary/Morphism/Construct/Identity.agda
index 302881b4aa..2a11d642ad 100644
--- a/src/Relation/Binary/Morphism/Construct/Identity.agda
+++ b/src/Relation/Binary/Morphism/Construct/Identity.agda
@@ -10,7 +10,9 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 import Function.Construct.Identity as Id
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
+open import Relation.Binary.Definitions using (Reflexive)
 open import Relation.Binary.Morphism.Structures
 open import Relation.Binary.Morphism.Bundles
 
diff --git a/src/Relation/Binary/Morphism/OrderMonomorphism.agda b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
index 3d788184c2..49d7c4e2e3 100644
--- a/src/Relation/Binary/Morphism/OrderMonomorphism.agda
+++ b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
@@ -12,7 +12,11 @@
 open import Algebra.Morphism.Definitions
 open import Function.Base
 open import Data.Product.Base using (_,_; map)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Structures
+  using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+open import Relation.Binary.Definitions
+  using (Irreflexive; Antisymmetric; Trichotomous; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 open import Relation.Binary.Morphism
 import Relation.Binary.Morphism.RelMonomorphism as RawRelation
 
diff --git a/src/Relation/Binary/Morphism/RelMonomorphism.agda b/src/Relation/Binary/Morphism/RelMonomorphism.agda
index e500fa4368..73a873fae5 100644
--- a/src/Relation/Binary/Morphism/RelMonomorphism.agda
+++ b/src/Relation/Binary/Morphism/RelMonomorphism.agda
@@ -10,7 +10,10 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Function.Base
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive; Total; Asymmetric; Decidable)
 open import Relation.Binary.Morphism
 
 module Relation.Binary.Morphism.RelMonomorphism
diff --git a/src/Relation/Binary/OrderMorphism.agda b/src/Relation/Binary/OrderMorphism.agda
index fbf3781c78..d516d5342e 100644
--- a/src/Relation/Binary/OrderMorphism.agda
+++ b/src/Relation/Binary/OrderMorphism.agda
@@ -14,7 +14,8 @@ module Relation.Binary.OrderMorphism where
 Use Relation.Binary.Reasoning.Morphism instead."
 #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (_=[_]⇒_)
+open import Relation.Binary.Bundles using (Poset; DecTotalOrder)
 open Poset
 import Function.Base as F
 open import Level
diff --git a/src/Relation/Binary/Properties/DecTotalOrder.agda b/src/Relation/Binary/Properties/DecTotalOrder.agda
index 386bb2189b..2923a0d047 100644
--- a/src/Relation/Binary/Properties/DecTotalOrder.agda
+++ b/src/Relation/Binary/Properties/DecTotalOrder.agda
@@ -6,7 +6,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Structures
+  using (IsDecTotalOrder; IsStrictTotalOrder)
+open import Relation.Binary.Bundles
+  using (DecTotalOrder; StrictTotalOrder)
 
 module Relation.Binary.Properties.DecTotalOrder
   {d₁ d₂ d₃} (DT : DecTotalOrder d₁ d₂ d₃) where
diff --git a/src/Relation/Binary/Properties/HeytingAlgebra.agda b/src/Relation/Binary/Properties/HeytingAlgebra.agda
index 8786792e78..87cb01899d 100644
--- a/src/Relation/Binary/Properties/HeytingAlgebra.agda
+++ b/src/Relation/Binary/Properties/HeytingAlgebra.agda
@@ -18,7 +18,7 @@ open import Algebra.Definitions _≈_
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_$_; flip; _∘_)
 open import Level using (_⊔_)
-open import Relation.Binary
+open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
 import Relation.Binary.Reasoning.PartialOrder as POR
 open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice
 open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
diff --git a/src/Relation/Binary/Properties/Poset.agda b/src/Relation/Binary/Properties/Poset.agda
index 5d1d6e9fe8..add0e20b28 100644
--- a/src/Relation/Binary/Properties/Poset.agda
+++ b/src/Relation/Binary/Properties/Poset.agda
@@ -7,7 +7,12 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Function.Base using (flip; _∘_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
+open import Relation.Binary.Bundles using (Poset; StrictPartialOrder)
+open import Relation.Binary.Structures
+  using (IsPartialOrder; IsStrictPartialOrder; IsDecPartialOrder)
+open import Relation.Binary.Definitions
+  using (_Respectsˡ_; _Respectsʳ_; Decidable)
 import Relation.Binary.Consequences as Consequences
 open import Relation.Nullary using (¬_; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Relation/Binary/Properties/Preorder.agda b/src/Relation/Binary/Properties/Preorder.agda
index e09bd0f3f1..dc9bd2913b 100644
--- a/src/Relation/Binary/Properties/Preorder.agda
+++ b/src/Relation/Binary/Properties/Preorder.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Preorder; Setoid)
+open import Relation.Binary.Structures using (IsPreorder)
 
 module Relation.Binary.Properties.Preorder
   {p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index 050177c629..968256077f 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -10,7 +10,10 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; id; _$_; flip)
 open import Relation.Nullary.Negation.Core using (¬_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
+open import Relation.Binary.Definitions
+  using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
+open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
 
 module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where
 
diff --git a/src/Relation/Binary/Properties/StrictPartialOrder.agda b/src/Relation/Binary/Properties/StrictPartialOrder.agda
index 7b6043af77..7420a2e24f 100644
--- a/src/Relation/Binary/Properties/StrictPartialOrder.agda
+++ b/src/Relation/Binary/Properties/StrictPartialOrder.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (StrictPartialOrder; Poset)
 
 module Relation.Binary.Properties.StrictPartialOrder
        {s₁ s₂ s₃} (SPO : StrictPartialOrder s₁ s₂ s₃) where
@@ -14,7 +14,7 @@ module Relation.Binary.Properties.StrictPartialOrder
 import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 import Relation.Binary.Construct.StrictToNonStrict
 
-open Relation.Binary.StrictPartialOrder SPO
+open StrictPartialOrder SPO
 
 ------------------------------------------------------------------------
 -- The inverse relation is also a strict partial order.
diff --git a/src/Relation/Binary/Properties/StrictTotalOrder.agda b/src/Relation/Binary/Properties/StrictTotalOrder.agda
index ab82e8ea54..f8b0062474 100644
--- a/src/Relation/Binary/Properties/StrictTotalOrder.agda
+++ b/src/Relation/Binary/Properties/StrictTotalOrder.agda
@@ -6,13 +6,13 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (StrictTotalOrder; DecTotalOrder)
 
 module Relation.Binary.Properties.StrictTotalOrder
        {s₁ s₂ s₃} (STO : StrictTotalOrder s₁ s₂ s₃)
        where
 
-open Relation.Binary.StrictTotalOrder STO
+open StrictTotalOrder STO
 open import Relation.Binary.Construct.StrictToNonStrict _≈_ _<_
 import Relation.Binary.Properties.StrictPartialOrder as SPO
 open import Relation.Binary.Consequences
diff --git a/src/Relation/Binary/Properties/TotalOrder.agda b/src/Relation/Binary/Properties/TotalOrder.agda
index 18e7095985..540de19905 100644
--- a/src/Relation/Binary/Properties/TotalOrder.agda
+++ b/src/Relation/Binary/Properties/TotalOrder.agda
@@ -6,7 +6,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (TotalOrder; DecTotalOrder)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Structures using (IsTotalOrder)
 
 module Relation.Binary.Properties.TotalOrder
   {t₁ t₂ t₃} (T : TotalOrder t₁ t₂ t₃) where
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 37c891312b..62bd7ca246 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -16,7 +16,8 @@ open import Level using (Level; _⊔_)
 open import Data.Product.Base using (∃)
 
 open import Relation.Nullary.Decidable using (yes; no; dec-yes-irr; dec-no)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
diff --git a/src/Relation/Binary/PropositionalEquality/Properties.agda b/src/Relation/Binary/PropositionalEquality/Properties.agda
index 48bb2a257e..9515b77700 100644
--- a/src/Relation/Binary/PropositionalEquality/Properties.agda
+++ b/src/Relation/Binary/PropositionalEquality/Properties.agda
@@ -14,7 +14,12 @@ module Relation.Binary.PropositionalEquality.Properties where
 
 open import Function.Base using (id; _∘_)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Preorder; Poset)
+open import Relation.Binary.Structures
+  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder)
+open import Relation.Binary.Definitions
+  using (Decidable; DecidableEquality)
 import Relation.Binary.Properties.Setoid as Setoid
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Unary using (Pred)
diff --git a/src/Relation/Binary/PropositionalEquality/WithK.agda b/src/Relation/Binary/PropositionalEquality/WithK.agda
index d14ec1d9f3..db49b97b2c 100644
--- a/src/Relation/Binary/PropositionalEquality/WithK.agda
+++ b/src/Relation/Binary/PropositionalEquality/WithK.agda
@@ -10,7 +10,7 @@
 module Relation.Binary.PropositionalEquality.WithK where
 
 open import Axiom.UniquenessOfIdentityProofs.WithK
-open import Relation.Binary
+open import Relation.Binary.Definitions using (Irrelevant)
 open import Relation.Binary.PropositionalEquality.Core
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Reasoning/Base/Apartness.agda b/src/Relation/Binary/Reasoning/Base/Apartness.agda
index 7a52568664..6e0e239a02 100644
--- a/src/Relation/Binary/Reasoning/Base/Apartness.agda
+++ b/src/Relation/Binary/Reasoning/Base/Apartness.agda
@@ -7,7 +7,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Transitive; Symmetric; Trans)
 
 module Relation.Binary.Reasoning.Base.Apartness {a ℓ₁ ℓ₂} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_#_ : Rel A ℓ₂}
diff --git a/src/Relation/Binary/Reasoning/Base/Double.agda b/src/Relation/Binary/Reasoning/Base/Double.agda
index 07761c5144..aa61052b0c 100644
--- a/src/Relation/Binary/Reasoning/Base/Double.agda
+++ b/src/Relation/Binary/Reasoning/Base/Double.agda
@@ -10,7 +10,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsPreorder)
 
 module Relation.Binary.Reasoning.Base.Double {a ℓ₁ ℓ₂} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} (isPreorder : IsPreorder _≈_ _∼_)
diff --git a/src/Relation/Binary/Reasoning/Base/Single.agda b/src/Relation/Binary/Reasoning/Base/Single.agda
index 97da6b06b8..43b553c7d1 100644
--- a/src/Relation/Binary/Reasoning/Base/Single.agda
+++ b/src/Relation/Binary/Reasoning/Base/Single.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Reflexive; Transitive)
 
 module Relation.Binary.Reasoning.Base.Single
   {a ℓ} {A : Set a} (_∼_ : Rel A ℓ)
diff --git a/src/Relation/Binary/Reasoning/Base/Triple.agda b/src/Relation/Binary/Reasoning/Base/Triple.agda
index c18aa9bf94..fe931722da 100644
--- a/src/Relation/Binary/Reasoning/Base/Triple.agda
+++ b/src/Relation/Binary/Reasoning/Base/Triple.agda
@@ -10,7 +10,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Structures using (IsPreorder)
+open import Relation.Binary.Definitions
+  using (Transitive; _Respects₂_; Trans)
 
 module Relation.Binary.Reasoning.Base.Triple {a ℓ₁ ℓ₂ ℓ₃} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} {_<_ : Rel A ℓ₃}
diff --git a/src/Relation/Binary/Reasoning/MultiSetoid.agda b/src/Relation/Binary/Reasoning/MultiSetoid.agda
index 7737e6a779..194e7ae410 100644
--- a/src/Relation/Binary/Reasoning/MultiSetoid.agda
+++ b/src/Relation/Binary/Reasoning/MultiSetoid.agda
@@ -28,7 +28,7 @@ module Relation.Binary.Reasoning.MultiSetoid where
 
 open import Function.Base using (flip)
 open import Level using (_⊔_)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 
 import Relation.Binary.Reasoning.Setoid as EqR
diff --git a/src/Relation/Binary/Reasoning/PartialOrder.agda b/src/Relation/Binary/Reasoning/PartialOrder.agda
index fbf1bef25f..80102d77f2 100644
--- a/src/Relation/Binary/Reasoning/PartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/PartialOrder.agda
@@ -39,7 +39,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Poset)
 
 module Relation.Binary.Reasoning.PartialOrder
   {p₁ p₂ p₃} (P : Poset p₁ p₂ p₃) where
diff --git a/src/Relation/Binary/Reasoning/PartialSetoid.agda b/src/Relation/Binary/Reasoning/PartialSetoid.agda
index 78f7c065d5..e679474016 100644
--- a/src/Relation/Binary/Reasoning/PartialSetoid.agda
+++ b/src/Relation/Binary/Reasoning/PartialSetoid.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (PartialSetoid)
 
 module Relation.Binary.Reasoning.PartialSetoid
   {s₁ s₂} (S : PartialSetoid s₁ s₂) where
diff --git a/src/Relation/Binary/Reasoning/Preorder.agda b/src/Relation/Binary/Reasoning/Preorder.agda
index 599be2f37b..933e8f0a45 100644
--- a/src/Relation/Binary/Reasoning/Preorder.agda
+++ b/src/Relation/Binary/Reasoning/Preorder.agda
@@ -23,7 +23,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Preorder)
 
 module Relation.Binary.Reasoning.Preorder
   {p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where
diff --git a/src/Relation/Binary/Reasoning/Setoid.agda b/src/Relation/Binary/Reasoning/Setoid.agda
index 77b1879db1..5b731767eb 100644
--- a/src/Relation/Binary/Reasoning/Setoid.agda
+++ b/src/Relation/Binary/Reasoning/Setoid.agda
@@ -20,7 +20,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 
 module Relation.Binary.Reasoning.Setoid {s₁ s₂} (S : Setoid s₁ s₂) where
 
diff --git a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
index 2e18d1fab7..8e2b7a3a3d 100644
--- a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
@@ -40,7 +40,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (StrictPartialOrder)
 
 module Relation.Binary.Reasoning.StrictPartialOrder
   {p₁ p₂ p₃} (S : StrictPartialOrder p₁ p₂ p₃) where
diff --git a/src/Relation/Binary/Reflection.agda b/src/Relation/Binary/Reflection.agda
index 20e2054cfc..a21288100a 100644
--- a/src/Relation/Binary/Reflection.agda
+++ b/src/Relation/Binary/Reflection.agda
@@ -13,7 +13,7 @@ open import Data.Vec.Base as Vec using (Vec; allFin)
 open import Function.Base using (id; _⟨_⟩_)
 open import Function.Bundles using (module Equivalence)
 open import Level using (Level)
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.PropositionalEquality.Core as P
 
 -- Think of the parameters as follows:
diff --git a/src/Relation/Unary/Closure/Base.agda b/src/Relation/Unary/Closure/Base.agda
index 4d80a5af1d..52ecac9ecf 100644
--- a/src/Relation/Unary/Closure/Base.agda
+++ b/src/Relation/Unary/Closure/Base.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Transitive; Reflexive)
 
 module Relation.Unary.Closure.Base {a b} {A : Set a} (R : Rel A b) where
 
diff --git a/src/Relation/Unary/Closure/Preorder.agda b/src/Relation/Unary/Closure/Preorder.agda
index 8c261b0283..374669aa30 100644
--- a/src/Relation/Unary/Closure/Preorder.agda
+++ b/src/Relation/Unary/Closure/Preorder.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (Preorder)
 
 module Relation.Unary.Closure.Preorder {a r e} (P : Preorder a e r) where
 
diff --git a/src/Relation/Unary/Closure/StrictPartialOrder.agda b/src/Relation/Unary/Closure/StrictPartialOrder.agda
index 889977fd02..81d5fb48f1 100644
--- a/src/Relation/Unary/Closure/StrictPartialOrder.agda
+++ b/src/Relation/Unary/Closure/StrictPartialOrder.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (StrictPartialOrder)
 
 module Relation.Unary.Closure.StrictPartialOrder
        {a r e} (P : StrictPartialOrder a e r) where
diff --git a/tests/system/directory/Main.agda b/tests/system/directory/Main.agda
index 0cde09f459..dabcdd433c 100644
--- a/tests/system/directory/Main.agda
+++ b/tests/system/directory/Main.agda
@@ -10,7 +10,7 @@ import Data.String.Properties as Stringₚ
 open import IO
 open import Function.Base using (_$_)
 
-open import Relation.Binary
+open import Relation.Binary.Bundles using (DecTotalOrder)
 import Relation.Binary.Construct.On as On
 
 open import System.Directory

From ff77df93edc1184af48efc99cb1dc81f1ed2b519 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 12 Sep 2023 01:22:49 +0100
Subject: [PATCH 107/316] Spellchecked `CHANGELOG` (#2078)

---
 CHANGELOG.md | 27 +++++++++++----------------
 1 file changed, 11 insertions(+), 16 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 812fe00b6c..0aa8b64abf 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -411,7 +411,7 @@ Non-backwards compatible changes
 * At the moment, there are 4 different ways such instance arguments can be provided,
   listed in order of convenience and clarity:
     1. *Automatic basic instances* - the standard library provides instances based on the constructors of each
-       numeric type in `Data.X.Base`. For example, `Data.Nat.Base` constains an instance of `NonZero (suc n)` for any `n`
+       numeric type in `Data.X.Base`. For example, `Data.Nat.Base` constrains an instance of `NonZero (suc n)` for any `n`
        and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`. Consequently,
        if the argument is of the required form, these instances will always be filled in by instance search
        automatically, e.g.
@@ -589,7 +589,7 @@ Non-backwards compatible changes
   ```
 
 * A new module `Reflection.AST` that re-exports the contents of the
-  submodules has been addeed.
+  submodules has been added.
 
 ### Implementation of division and modulus for `ℤ`
 
@@ -617,7 +617,7 @@ Non-backwards compatible changes
   * `IsSemiringWithoutAnnihilatingZero`
   * `IsRing`
 * To aid with migration, structures matching the old style ones have been added
-  to `Algebra.Structures.Biased`, with conversionFunctions:
+  to `Algebra.Structures.Biased`, with conversion functions:
   * `IsNearSemiring*` and `isNearSemiring*`
   * `IsSemiringWithoutOne*` and `isSemiringWithoutOne*`
   * `IsSemiringWithoutAnnihilatingZero*` and `isSemiringWithoutAnnihilatingZero*`
@@ -634,7 +634,7 @@ Non-backwards compatible changes
   ⊥ = Irrelevant Empty
   ```
   in order to make ⊥ proof irrelevant. Any two proofs of `⊥` or of a negated
-  statements are now *judgementally* equal to each other.
+  statements are now *judgmentally* equal to each other.
 
 * Consequently we have modified the following definitions:
   + In `Relation.Nullary.Decidable.Core`, the type of `dec-no` has changed
@@ -671,14 +671,14 @@ Non-backwards compatible changes
   however all their contents is re-exported by `Relation.Nullary` which is the easiest way to access
   it now.
 
-* In order to facilitate this reorganisation the following breaking moves have occured:
+* In order to facilitate this reorganisation the following breaking moves have occurred:
   - `¬?` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Decidable.Core`
   - `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Reflects`.
   - `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved from `Relation.Nullary.Negation`
     to `Relation.Nullary.Decidable`.
-  - `fromDec` and `toDec` have been mvoed from `Data.Sum.Base` to `Data.Sum`.
+  - `fromDec` and `toDec` have been moved from `Data.Sum.Base` to `Data.Sum`.
 
-### Refactoring of the unindexed Functor/Applicative/Monad hiearchy
+### Refactoring of the unindexed Functor/Applicative/Monad hierarchy
 
 * The unindexed versions are not defined in terms of the named versions anymore
 
@@ -698,7 +698,7 @@ Non-backwards compatible changes
 
 * `MonadT T` now returns a `MonadTd` record that packs both a proof that the
   `Monad M` transformed by `T` is a monad and that we can `lift` a computation
-  `M A` to a trasnformed computation `T M A`.
+  `M A` to a transformed computation `T M A`.
 
 * The monad transformer are not mere aliases anymore, they are record-wrapped
   which allows constraints such as `MonadIO (StateT S (ReaderT R IO))` to be
@@ -789,12 +789,12 @@ Non-backwards compatible changes
   previous implementation using the sum type `a ⟶ b ⊎ b ⟶ a`.
 
 * In `Algebra.Morphism.Structures`, `IsNearSemiringHomomorphism`,
-  `IsSemiringHomomorphism`, and `IsRingHomomorphism` have been redeisgned to
+  `IsSemiringHomomorphism`, and `IsRingHomomorphism` have been redesigned to
   build up from `IsMonoidHomomorphism`, `IsNearSemiringHomomorphism`, and
   `IsSemiringHomomorphism` respectively, adding a single property at each step.
   This means that they no longer need to have two separate proofs of
   `IsRelHomomorphism`. Similarly, `IsLatticeHomomorphism` is now built as
-  `IsRelHomomorphism` along with proofs that `_∧_` and `_∨_` are homorphic.
+  `IsRelHomomorphism` along with proofs that `_∧_` and `_∨_` are homomorphic.
 
   Also, `⁻¹-homo` in `IsRingHomomorphism` has been renamed to `-‿homo`.
 
@@ -939,7 +939,7 @@ Major improvements
 
 * To fix this, these operators have been moved to `Data.Nat.Base`. The properties
   for them still live in `Data.Nat.DivMod` (which also publicly re-exports them
-  to provide backwards compatability).
+  to provide backwards compatibility).
 
 * Beneficiaries of this change include `Data.Rational.Unnormalised.Base` whose
   dependencies are now significantly smaller.
@@ -1604,11 +1604,6 @@ New modules
   Data.List.NonEmpty.Relation.Unary.All
   ```
 
-* A small library for heterogenous equational reasoning on vectors:
-  ```
-  Data.Vec.Properties.Heterogeneous
-  ```
-
 * Show module for unnormalised rationals:
   ```
   Data.Rational.Unnormalised.Show

From 85d57fcd17ff5157a07d66148614075b11a5e4af Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 12 Sep 2023 03:46:00 +0100
Subject: [PATCH 108/316] #2075: Remove symmetry assumption from lexicographic
 well-foundedness (#2077)

---
 CHANGELOG.md                                  | 17 ++++++++++++++---
 .../Product/Relation/Binary/Lex/Strict.agda   | 19 ++++++++++---------
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  4 ++--
 src/Induction/WellFounded.agda                | 19 +++++++++++--------
 4 files changed, 37 insertions(+), 22 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0aa8b64abf..1e9e9dd82c 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -781,6 +781,12 @@ Non-backwards compatible changes
   properties about the orderings themselves the second index must be provided
   explicitly.
 
+* Issue #2075 (Johannes Waldmann): wellfoundedness of the lexicographic ordering
+  on products, defined in `Data.Product.Relation.Binary.Lex.Strict`, no longer
+  requires the assumption of symmetry for the first equality relation `_≈₁_`,
+  leading to a new lemma `Induction.WellFounded.Acc-resp-flip-≈`, and refactoring
+  of the previous proof `Induction.WellFounded.Acc-resp-≈`.
+
 * The operation `SymClosure` on relations in
   `Relation.Binary.Construct.Closure.Symmetric` has been reimplemented
   as a data type `SymClosure _⟶_ a b` that is parameterized by the
@@ -2515,13 +2521,13 @@ Other minor changes
   ×-≡,≡←≡ : p₁ ≡ p₂ → (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂)
   ```
 
-* Added new proof to `Data.Product.Relation.Binary.Lex.Strict`
+* Added new proofs to `Data.Product.Relation.Binary.Lex.Strict`
   ```agda
   ×-respectsʳ : Transitive _≈₁_ →
                 _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_
   ×-respectsˡ : Symmetric _≈₁_ → Transitive _≈₁_ →
                  _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_
-  ×-wellFounded' : Symmetric  _≈₁_ → Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ →
+  ×-wellFounded' : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ →
                    WellFounded _<₁_ → WellFounded _<₂_ → WellFounded _<ₗₑₓ_
   ```
 
@@ -2686,7 +2692,7 @@ Other minor changes
                 ∀ {m n} → _Respectsˡ_ (_<_ {m} {n}) _≋_
   <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ →
                 ∀ {m n} → _Respectsʳ_ (_<_ {m} {n}) _≋_
-  <-wellFounded : Symmetric _≈_ →  Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ →
+  <-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ →
                   ∀ {n} → WellFounded (_<_ {n})
 ```
 
@@ -3356,6 +3362,11 @@ This is a full list of proofs that have changed form to use irrelevant instance
   ⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys
   ```
 
+* Added new proof to `Induction.WellFounded`
+  ```agda
+  Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
+  ```
+
 * Added new file `Relation.Binary.Reasoning.Base.Apartness`
 
   This is how to use it:
diff --git a/src/Data/Product/Relation/Binary/Lex/Strict.agda b/src/Data/Product/Relation/Binary/Lex/Strict.agda
index 90129350ea..742c642d60 100644
--- a/src/Data/Product/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/Strict.agda
@@ -28,7 +28,7 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Definitions
   using (Transitive; Symmetric; Irreflexive; Asymmetric; Total; Decidable; Antisymmetric; Trichotomous; _Respects₂_; _Respectsʳ_; _Respectsˡ_; tri<; tri>; tri≈)
 open import Relation.Binary.Consequences
-open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 private
   variable
@@ -192,23 +192,24 @@ module _ {_≈₁_ : Rel A ℓ₁} {_<₁_ : Rel A ℓ₂} {_<₂_ : Rel B ℓ
   private
     _<ₗₑₓ_ = ×-Lex _≈₁_ _<₁_ _<₂_
 
-  ×-wellFounded' : Symmetric _≈₁_ → Transitive _≈₁_ →
+  ×-wellFounded' : Transitive _≈₁_ →
                    _<₁_ Respectsʳ _≈₁_ →
                    WellFounded _<₁_ →
                    WellFounded _<₂_ →
                    WellFounded _<ₗₑₓ_
-  ×-wellFounded' sym trans resp wf₁ wf₂ (x , y) = acc (×-acc (wf₁ x) (wf₂ y))
+  ×-wellFounded' trans resp wf₁ wf₂ (x , y) = acc (×-acc (wf₁ x) (wf₂ y))
     where
     ×-acc : ∀ {x y} →
             Acc _<₁_ x → Acc _<₂_ y →
             WfRec _<ₗₑₓ_ (Acc _<ₗₑₓ_) (x , y)
     ×-acc (acc rec₁) acc₂ (u , v) (inj₁ u<x)
       = acc (×-acc (rec₁ u u<x) (wf₂ v))
-    ×-acc {x₁} acc₁ (acc rec₂) (u , v) (inj₂ (u≈x , v<y))
-      = Acc-resp-≈ (Pointwise.×-symmetric {_∼₁_ = _≈₁_} {_∼₂_ = _≡_ } sym ≡.sym)
-                   (×-respectsʳ {_<₁_ = _<₁_} {_<₂_ = _<₂_} trans resp (≡.respʳ _<₂_))
-                   (sym u≈x , _≡_.refl)
-                   (acc (×-acc acc₁ (rec₂ v v<y)))
+    ×-acc acc₁ (acc rec₂) (u , v) (inj₂ (u≈x , v<y))
+      = Acc-resp-flip-≈
+        (×-respectsʳ {_<₁_ = _<₁_} {_<₂_ = _<₂_} trans resp (≡.respʳ _<₂_))
+        (u≈x , ≡.refl)
+        (acc (×-acc acc₁ (rec₂ v v<y)))
+
 
 module _ {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel B ℓ₂} where
 
@@ -218,7 +219,7 @@ module _ {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel B ℓ₂} where
   ×-wellFounded : WellFounded _<₁_ →
                   WellFounded _<₂_ →
                   WellFounded _<ₗₑₓ_
-  ×-wellFounded = ×-wellFounded' ≡.sym ≡.trans (≡.respʳ _<₁_)
+  ×-wellFounded = ×-wellFounded' ≡.trans (≡.respʳ _<₁_)
 
 ------------------------------------------------------------------------
 -- Collections of properties which are preserved by ×-Lex.
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 450c23809b..d488e8e9d2 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -130,7 +130,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
                  ∀ {m n} → Irrelevant (_<_ {m} {n})
   <-irrelevant = Core.irrelevant (λ ())
 
-  module _ (≈-sym : Symmetric _≈_) (≈-trans : Transitive _≈_) (≺-respʳ : _≺_ Respectsʳ _≈_ ) (≺-wf : WellFounded _≺_)
+  module _ (≈-trans : Transitive _≈_) (≺-respʳ : _≺_ Respectsʳ _≈_ ) (≺-wf : WellFounded _≺_)
     where
 
     <-wellFounded : ∀ {n} → WellFounded (_<_ {n})
@@ -142,7 +142,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
         <⇒uncons-Lex {x ∷ xs} {y ∷ ys} (next x≈y xs<ys) = inj₂ (x≈y , xs<ys)
 
         uncons-Lex-wellFounded : WellFounded (×-Lex _≈_ _≺_ _<_ on uncons)
-        uncons-Lex-wellFounded = On.wellFounded uncons (×-wellFounded' ≈-sym ≈-trans ≺-respʳ ≺-wf <-wellFounded)
+        uncons-Lex-wellFounded = On.wellFounded uncons (×-wellFounded' ≈-trans ≺-respʳ ≺-wf <-wellFounded)
 
 ------------------------------------------------------------------------
 -- Structures
diff --git a/src/Induction/WellFounded.agda b/src/Induction/WellFounded.agda
index 12b1719118..7615c99337 100644
--- a/src/Induction/WellFounded.agda
+++ b/src/Induction/WellFounded.agda
@@ -9,7 +9,7 @@
 module Induction.WellFounded where
 
 open import Data.Product.Base using (Σ; _,_; proj₁)
-open import Function.Base using (_on_)
+open import Function.Base using (_∘_; flip; _on_)
 open import Induction
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (Rel)
@@ -54,10 +54,13 @@ acc-inverse : ∀ {_<_ : Rel A ℓ} {x : A} (q : Acc _<_ x) →
               (y : A) → y < x → Acc _<_ y
 acc-inverse (acc rs) y y<x = rs y y<x
 
-Acc-resp-≈ : {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} → Symmetric _≈_ →
-             _<_ Respectsʳ _≈_ → (Acc _<_) Respects _≈_
-Acc-resp-≈ sym respʳ x≈y (acc rec) =
-  acc (λ z z<y → rec z (respʳ (sym x≈y) z<y))
+module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where
+
+  Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
+  Acc-resp-flip-≈ respʳ x≈y (acc rec) = acc λ z z<y → rec z (respʳ x≈y z<y)
+
+  Acc-resp-≈ : Symmetric _≈_ → _<_ Respectsʳ _≈_ → (Acc _<_) Respects _≈_
+  Acc-resp-≈ sym respʳ x≈y wf = Acc-resp-flip-≈ (respʳ ∘ sym) x≈y wf
 
 ------------------------------------------------------------------------
 -- Well-founded induction for the subset of accessible elements:
@@ -120,7 +123,7 @@ module Subrelation {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel A ℓ₂}
                    (<₁⇒<₂ : ∀ {x y} → x <₁ y → x <₂ y) where
 
   accessible : Acc _<₂_ ⊆ Acc _<₁_
-  accessible (acc rs) = acc (λ y y<x → accessible (rs y (<₁⇒<₂ y<x)))
+  accessible (acc rs) = acc λ y y<x → accessible (rs y (<₁⇒<₂ y<x))
 
   wellFounded : WellFounded _<₂_ → WellFounded _<₁_
   wellFounded wf = λ x → accessible (wf x)
@@ -131,7 +134,7 @@ module Subrelation {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel A ℓ₂}
 module InverseImage {_<_ : Rel B ℓ} (f : A → B) where
 
   accessible : ∀ {x} → Acc _<_ (f x) → Acc (_<_ on f) x
-  accessible (acc rs) = acc (λ y fy<fx → accessible (rs (f y) fy<fx))
+  accessible (acc rs) = acc λ y fy<fx → accessible (rs (f y) fy<fx)
 
   wellFounded : WellFounded _<_ → WellFounded (_<_ on f)
   wellFounded wf = λ x → accessible (wf (f x))
@@ -158,7 +161,7 @@ module TransitiveClosure {A : Set a} (_<_ : Rel A ℓ) where
     trans : ∀ {x y z} (x<y : x <⁺ y) (y<z : y <⁺ z) → x <⁺ z
 
   downwardsClosed : ∀ {x y} → Acc _<⁺_ y → x <⁺ y → Acc _<⁺_ x
-  downwardsClosed (acc rs) x<y = acc (λ z z<x → rs z (trans z<x x<y))
+  downwardsClosed (acc rs) x<y = acc λ z z<x → rs z (trans z<x x<y)
 
   mutual
 

From 28e0bd365be0fa0470f942dd182cf69a27b4e8ee Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Tue, 12 Sep 2023 04:46:55 +0200
Subject: [PATCH 109/316] Make sure RawRing defines rawRingWithoutOne (#1967)

---
 src/Algebra/Bundles/Raw.agda | 14 +++++++++-----
 1 file changed, 9 insertions(+), 5 deletions(-)

diff --git a/src/Algebra/Bundles/Raw.agda b/src/Algebra/Bundles/Raw.agda
index e7f9be1349..0b26aad1eb 100644
--- a/src/Algebra/Bundles/Raw.agda
+++ b/src/Algebra/Bundles/Raw.agda
@@ -213,14 +213,18 @@ record RawRing c ℓ : Set (suc (c ⊔ ℓ)) where
     ; *-rawMagma; *-rawMonoid
     )
 
-  +-rawGroup : RawGroup c ℓ
-  +-rawGroup = record
+  rawRingWithoutOne : RawRingWithoutOne c ℓ
+  rawRingWithoutOne = record
     { _≈_ = _≈_
-    ; _∙_ = _+_
-    ; ε   = 0#
-    ; _⁻¹ = -_
+    ; _+_ = _+_
+    ; _*_ = _*_
+    ; -_ = -_
+    ; 0# = 0#
     }
 
+  open RawRingWithoutOne rawRingWithoutOne public
+    using (+-rawGroup)
+
 ------------------------------------------------------------------------
 -- Raw bundles with 3 binary operations
 ------------------------------------------------------------------------

From 0bbd30c44337f9204fbab0428836f6cff416672d Mon Sep 17 00:00:00 2001
From: Akshobhya K M <akshobhyakm@gmail.com>
Date: Mon, 11 Sep 2023 22:47:44 -0400
Subject: [PATCH 110/316] Direct products and minor fixes (#1909)

---
 CHANGELOG.md                             |  5 ++
 src/Algebra/Bundles.agda                 |  7 +++
 src/Algebra/Construct/DirectProduct.agda | 63 ++++++++++++++++++++++++
 src/Algebra/Structures.agda              | 55 ++++++++++++++++++---
 4 files changed, 124 insertions(+), 6 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1e9e9dd82c..5121e27af9 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1888,6 +1888,11 @@ Other minor changes
   rightBolLoop : RightBolLoop a ℓ₁ → RightBolLoop b ℓ₂ → RightBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   middleBolLoop : MiddleBolLoop a ℓ₁ → MiddleBolLoop b ℓ₂ → MiddleBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   moufangLoop : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawRingWithoutOne : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  ringWithoutOne : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  nonAssociativeRing : NonAssociativeRing a ℓ₁ → NonAssociativeRing b ℓ₂ → NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  quasiring : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  nearring : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
  ```
 
 * Added new definition to `Algebra.Definitions`:
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index f20cf1ad03..5bc2d5a1d0 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -892,6 +892,12 @@ record NonAssociativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
   open AbelianGroup +-abelianGroup public
     using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)
 
+  *-unitalMagma : UnitalMagma _ _
+  *-unitalMagma = record { isUnitalMagma = *-isUnitalMagma}
+
+  open UnitalMagma *-unitalMagma public
+    using () renaming (magma to *-magma; identity to *-identity)
+
 record Nearring c ℓ : Set (suc (c ⊔ ℓ)) where
   infixl 7 _*_
   infixl 6 _+_
@@ -1158,3 +1164,4 @@ record MiddleBolLoop c ℓ : Set (suc (c ⊔ ℓ)) where
 
   open Loop loop public
     using (quasigroup)
+
diff --git a/src/Algebra/Construct/DirectProduct.agda b/src/Algebra/Construct/DirectProduct.agda
index 5f7b042017..ef9f377d29 100644
--- a/src/Algebra/Construct/DirectProduct.agda
+++ b/src/Algebra/Construct/DirectProduct.agda
@@ -61,6 +61,16 @@ rawSemiring R S = record
   ; 1#      = R.1# , S.1#
   } where module R = RawSemiring R; module S = RawSemiring S
 
+rawRingWithoutOne : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+rawRingWithoutOne R S = record
+  { Carrier = R.Carrier × S.Carrier
+  ; _≈_     = Pointwise R._≈_ S._≈_
+  ; _+_     = zip R._+_ S._+_
+  ; _*_     = zip R._*_ S._*_
+  ; -_      = map R.-_ S.-_
+  ; 0#      = R.0# , S.0#
+  } where module R = RawRingWithoutOne R; module S = RawRingWithoutOne S
+
 rawRing : RawRing a ℓ₁ → RawRing b ℓ₂ → RawRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
 rawRing R S = record
   { Carrier = R.Carrier × S.Carrier
@@ -317,6 +327,59 @@ kleeneAlgebra K L = record
       }
   } where module K = KleeneAlgebra K;  module L = KleeneAlgebra L
 
+ringWithoutOne : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+ringWithoutOne R S = record
+  { isRingWithoutOne = record
+      { +-isAbelianGroup = AbelianGroup.isAbelianGroup ((abelianGroup R.+-abelianGroup S.+-abelianGroup))
+      ; *-cong           = Semigroup.∙-cong (semigroup R.*-semigroup S.*-semigroup)
+      ; *-assoc          = Semigroup.assoc (semigroup R.*-semigroup S.*-semigroup)
+      ; distrib          = (λ x y z → (R.distribˡ , S.distribˡ) <*> x <*> y <*> z)
+                            , (λ x y z → (R.distribʳ , S.distribʳ) <*> x <*> y <*> z)
+      ; zero             = uncurry (λ x y → R.zeroˡ x , S.zeroˡ y)
+                            , uncurry (λ x y → R.zeroʳ x , S.zeroʳ y)
+      }
+
+  } where module R = RingWithoutOne R; module S = RingWithoutOne S
+
+nonAssociativeRing : NonAssociativeRing a ℓ₁ → NonAssociativeRing b ℓ₂ → NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+nonAssociativeRing R S = record
+  { isNonAssociativeRing = record
+      { +-isAbelianGroup = AbelianGroup.isAbelianGroup ((abelianGroup R.+-abelianGroup S.+-abelianGroup))
+      ; *-cong           = UnitalMagma.∙-cong (unitalMagma R.*-unitalMagma S.*-unitalMagma)
+      ; *-identity       = UnitalMagma.identity (unitalMagma R.*-unitalMagma S.*-unitalMagma)
+      ; distrib          = (λ x y z → (R.distribˡ , S.distribˡ) <*> x <*> y <*> z)
+                            , (λ x y z → (R.distribʳ , S.distribʳ) <*> x <*> y <*> z)
+      ; zero             = uncurry (λ x y → R.zeroˡ x , S.zeroˡ y)
+                            , uncurry (λ x y → R.zeroʳ x , S.zeroʳ y)
+      }
+
+  } where module R = NonAssociativeRing R; module S = NonAssociativeRing S
+
+quasiring : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+quasiring R S = record
+  { isQuasiring = record
+      { +-isMonoid = Monoid.isMonoid ((monoid R.+-monoid S.+-monoid))
+      ; *-cong           = Monoid.∙-cong (monoid R.*-monoid S.*-monoid)
+      ; *-assoc          = Monoid.assoc (monoid R.*-monoid S.*-monoid)
+      ; *-identity       = Monoid.identity ((monoid R.*-monoid S.*-monoid))
+      ; distrib          = (λ x y z → (R.distribˡ , S.distribˡ) <*> x <*> y <*> z)
+                            , (λ x y z → (R.distribʳ , S.distribʳ) <*> x <*> y <*> z)
+      ; zero             = uncurry (λ x y → R.zeroˡ x , S.zeroˡ y)
+                            , uncurry (λ x y → R.zeroʳ x , S.zeroʳ y)
+      }
+
+  } where module R = Quasiring R; module S = Quasiring S
+
+nearring : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+nearring R S =  record
+  { isNearring = record
+      { isQuasiring = Quasiring.isQuasiring (quasiring R.quasiring S.quasiring)
+      ; +-inverse   = (λ x → (R.+-inverseˡ , S.+-inverseˡ) <*> x)
+                      , (λ x → (R.+-inverseʳ , S.+-inverseʳ) <*> x)
+      ; ⁻¹-cong     = map R.⁻¹-cong S.⁻¹-cong
+      }
+  } where module R = Nearring R; module S = Nearring S
+
 ring : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
 ring R S = record
   { -_     = uncurry (λ x y → R.-_ x , S.-_ y)
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index 4f9fc7fcb3..e38bbe7ba9 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -594,6 +594,24 @@ record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
     ; isSemigroup   to +-isSemigroup
     )
 
+  distribˡ : * DistributesOverˡ +
+  distribˡ = proj₁ distrib
+
+  distribʳ : * DistributesOverʳ +
+  distribʳ = proj₂ distrib
+
+  zeroˡ : LeftZero 0# *
+  zeroˡ = proj₁ zero
+
+  zeroʳ : RightZero 0# *
+  zeroʳ = proj₂ zero
+
+  identityˡ : LeftIdentity 1# *
+  identityˡ = proj₁ *-identity
+
+  identityʳ : RightIdentity 1# *
+  identityʳ = proj₂ *-identity
+
   *-isMagma : IsMagma *
   *-isMagma = record
     { isEquivalence = isEquivalence
@@ -683,11 +701,11 @@ record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ
     ; assoc   = *-assoc
     }
 
-  open IsMagma *-isMagma public
+  open IsSemigroup *-isSemigroup public
     using ()
     renaming
-    ( ∙-congˡ  to *-congˡ
-    ; ∙-congʳ  to *-congʳ
+    ( ∙-congˡ   to *-congˡ
+    ; ∙-congʳ   to *-congʳ
     )
 
 ------------------------------------------------------------------------
@@ -698,7 +716,7 @@ record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a
   field
     +-isAbelianGroup : IsAbelianGroup + 0# -_
     *-cong           : Congruent₂ *
-    identity         : Identity 1# *
+    *-identity       : Identity 1# *
     distrib          : * DistributesOver +
     zero             : Zero 0# *
 
@@ -728,6 +746,18 @@ record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a
     ; isGroup                 to +-isGroup
     )
 
+  zeroˡ : LeftZero 0# *
+  zeroˡ = proj₁ zero
+
+  zeroʳ : RightZero 0# *
+  zeroʳ = proj₂ zero
+
+  distribˡ : * DistributesOverˡ +
+  distribˡ = proj₁ distrib
+
+  distribʳ : * DistributesOverʳ +
+  distribʳ = proj₂ distrib
+
   *-isMagma : IsMagma *
   *-isMagma = record
     { isEquivalence = isEquivalence
@@ -735,10 +765,23 @@ record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a
     }
 
   *-identityˡ : LeftIdentity 1# *
-  *-identityˡ = proj₁ identity
+  *-identityˡ = proj₁ *-identity
 
   *-identityʳ : RightIdentity 1# *
-  *-identityʳ = proj₂ identity
+  *-identityʳ = proj₂ *-identity
+
+  *-isUnitalMagma : IsUnitalMagma * 1#
+  *-isUnitalMagma = record
+    { isMagma = *-isMagma
+    ; identity = *-identity
+    }
+
+  open IsUnitalMagma *-isUnitalMagma public
+    using ()
+    renaming
+    ( ∙-congˡ   to *-congˡ
+    ; ∙-congʳ   to *-congʳ
+    )
 
 record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
   field

From d260bf02a5ba7a1cf43e5415784f2e3ea267091e Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Tue, 12 Sep 2023 12:19:21 +0530
Subject: [PATCH 111/316] Refine imports in `Reflection.AST` (#2072)

---
 src/Reflection/AST/Abstraction.agda          |  4 ++--
 src/Reflection/AST/AlphaEquality.agda        | 12 ++++++------
 src/Reflection/AST/Argument.agda             |  2 +-
 src/Reflection/AST/Argument/Information.agda |  2 +-
 src/Reflection/AST/Argument/Modality.agda    |  2 +-
 src/Reflection/AST/Argument/Quantity.agda    |  2 +-
 src/Reflection/AST/Argument/Relevance.agda   |  2 +-
 src/Reflection/AST/Argument/Visibility.agda  |  2 +-
 src/Reflection/AST/Definition.agda           |  2 +-
 src/Reflection/AST/Literal.agda              | 15 +++++++--------
 src/Reflection/AST/Meta.agda                 |  4 ++--
 src/Reflection/AST/Name.agda                 |  2 +-
 src/Reflection/AST/Show.agda                 | 10 +++++-----
 src/Reflection/AST/Term.agda                 |  3 ++-
 src/Reflection/AST/Traversal.agda            |  2 +-
 15 files changed, 33 insertions(+), 33 deletions(-)

diff --git a/src/Reflection/AST/Abstraction.agda b/src/Reflection/AST/Abstraction.agda
index 43fb2f8302..4f640bcb20 100644
--- a/src/Reflection/AST/Abstraction.agda
+++ b/src/Reflection/AST/Abstraction.agda
@@ -11,8 +11,8 @@ module Reflection.AST.Abstraction where
 open import Data.String.Base as String                 using (String)
 open import Data.String.Properties as String           using (_≟_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
-open import Level
-open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
+open import Level using (Level)
+open import Relation.Nullary.Decidable.Core            using (Dec; map′; _×-dec_)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
diff --git a/src/Reflection/AST/AlphaEquality.agda b/src/Reflection/AST/AlphaEquality.agda
index 4849470d50..ec0b556292 100644
--- a/src/Reflection/AST/AlphaEquality.agda
+++ b/src/Reflection/AST/AlphaEquality.agda
@@ -8,12 +8,12 @@
 
 module Reflection.AST.AlphaEquality where
 
-open import Data.Bool.Base                using (Bool; true; false; _∧_)
-open import Data.List.Base                using ([]; _∷_)
-open import Data.Nat.Base as ℕ            using (ℕ; zero; suc; _≡ᵇ_)
-open import Data.Product.Base             using (_,_)
-open import Relation.Nullary.Decidable    using (⌊_⌋)
-open import Relation.Binary.Definitions   using (DecidableEquality)
+open import Data.Bool.Base                   using (Bool; true; false; _∧_)
+open import Data.List.Base                   using ([]; _∷_)
+open import Data.Nat.Base as ℕ               using (ℕ; zero; suc; _≡ᵇ_)
+open import Data.Product.Base                using (_,_)
+open import Relation.Nullary.Decidable.Core  using (⌊_⌋)
+open import Relation.Binary.Definitions      using (DecidableEquality)
 
 open import Reflection.AST.Abstraction
 open import Reflection.AST.Argument
diff --git a/src/Reflection/AST/Argument.agda b/src/Reflection/AST/Argument.agda
index b97ec90f77..d21b3688cb 100644
--- a/src/Reflection/AST/Argument.agda
+++ b/src/Reflection/AST/Argument.agda
@@ -10,7 +10,7 @@ module Reflection.AST.Argument where
 
 open import Data.List.Base as List                     using (List; []; _∷_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable                 using (Dec; map′; _×-dec_)
+open import Relation.Nullary.Decidable.Core            using (Dec; map′; _×-dec_)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 open import Level
diff --git a/src/Reflection/AST/Argument/Information.agda b/src/Reflection/AST/Argument/Information.agda
index 66fec3be77..98c91ee818 100644
--- a/src/Reflection/AST/Argument/Information.agda
+++ b/src/Reflection/AST/Argument/Information.agda
@@ -9,7 +9,7 @@
 module Reflection.AST.Argument.Information where
 
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable                 using (map′; _×-dec_)
+open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
diff --git a/src/Reflection/AST/Argument/Modality.agda b/src/Reflection/AST/Argument/Modality.agda
index 489247031e..722c1cfa0e 100644
--- a/src/Reflection/AST/Argument/Modality.agda
+++ b/src/Reflection/AST/Argument/Modality.agda
@@ -9,7 +9,7 @@
 module Reflection.AST.Argument.Modality where
 
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable                 using (map′; _×-dec_)
+open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
diff --git a/src/Reflection/AST/Argument/Quantity.agda b/src/Reflection/AST/Argument/Quantity.agda
index bc79d74b3c..ff0a0d3c65 100644
--- a/src/Reflection/AST/Argument/Quantity.agda
+++ b/src/Reflection/AST/Argument/Quantity.agda
@@ -8,7 +8,7 @@
 
 module Reflection.AST.Argument.Quantity where
 
-open import Relation.Nullary                           using (yes; no)
+open import Relation.Nullary.Decidable.Core           using (yes; no)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 
diff --git a/src/Reflection/AST/Argument/Relevance.agda b/src/Reflection/AST/Argument/Relevance.agda
index 3ac78cf765..b2ef362d7d 100644
--- a/src/Reflection/AST/Argument/Relevance.agda
+++ b/src/Reflection/AST/Argument/Relevance.agda
@@ -8,7 +8,7 @@
 
 module Reflection.AST.Argument.Relevance where
 
-open import Relation.Nullary                           using (yes; no)
+open import Relation.Nullary.Decidable.Core            using (yes; no)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 
diff --git a/src/Reflection/AST/Argument/Visibility.agda b/src/Reflection/AST/Argument/Visibility.agda
index b384609c93..f8bd2afeee 100644
--- a/src/Reflection/AST/Argument/Visibility.agda
+++ b/src/Reflection/AST/Argument/Visibility.agda
@@ -8,7 +8,7 @@
 
 module Reflection.AST.Argument.Visibility where
 
-open import Relation.Nullary                           using (yes; no)
+open import Relation.Nullary.Decidable.Core            using (yes; no)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 
diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index b5b5b32700..b97fe47cae 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -11,7 +11,7 @@ module Reflection.AST.Definition where
 import Data.List.Properties as Listₚ                   using (≡-dec)
 import Data.Nat.Properties as ℕₚ                       using (_≟_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
-open import Relation.Nullary.Decidable                 using (map′; _×-dec_; yes; no)
+open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_; yes; no)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 
diff --git a/src/Reflection/AST/Literal.agda b/src/Reflection/AST/Literal.agda
index 00ac5cc652..abdd2b08a6 100644
--- a/src/Reflection/AST/Literal.agda
+++ b/src/Reflection/AST/Literal.agda
@@ -8,16 +8,15 @@
 
 module Reflection.AST.Literal where
 
-open import Data.Bool.Base   using (Bool; true; false)
-import Data.Char as Char     using (_≟_)
-import Data.Float as Float   using (_≟_)
-import Data.Nat as ℕ        using (_≟_)
-import Data.String.Properties as String using (_≟_)
-import Data.Word as Word     using (_≟_)
+open import Data.Bool.Base using (Bool; true; false)
+import Data.Char.Properties as Char
+import Data.Float.Properties as Float
+import Data.Nat.Properties as ℕ
+import Data.String.Properties as String
+import Data.Word.Properties as Word
 import Reflection.AST.Meta as Meta
 import Reflection.AST.Name as Name
-open import Relation.Nullary                           using (yes; no)
-open import Relation.Nullary.Decidable                 using (map′; isYes)
+open import Relation.Nullary.Decidable.Core            using (yes; no; map′; isYes)
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
diff --git a/src/Reflection/AST/Meta.agda b/src/Reflection/AST/Meta.agda
index 682a52c8ca..babcd23670 100644
--- a/src/Reflection/AST/Meta.agda
+++ b/src/Reflection/AST/Meta.agda
@@ -8,9 +8,9 @@
 
 module Reflection.AST.Meta where
 
-import Data.Nat.Properties as ℕₚ                       using (_≟_)
+import Data.Nat.Properties as ℕₚ
 open import Function.Base                              using (_on_)
-open import Relation.Nullary.Decidable                 using (map′)
+open import Relation.Nullary.Decidable.Core            using (map′)
 open import Relation.Binary.Core                       using (Rel)
 open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)
 import Relation.Binary.Construct.On as On
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index ac478443fb..c386912f08 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -12,7 +12,7 @@ open import Data.List.Base              using (List)
 import Data.Product.Properties as Prodₚ using (≡-dec)
 import Data.Word.Properties as Wₚ       using (_≟_)
 open import Function.Base               using (_on_)
-open import Relation.Nullary.Decidable                 using (map′)
+open import Relation.Nullary.Decidable.Core            using (map′)
 open import Relation.Binary.Core                       using (Rel)
 open import Relation.Binary.Definitions                using (Decidable; DecidableEquality)
 open import Relation.Binary.Construct.On               using (decidable)
diff --git a/src/Reflection/AST/Show.agda b/src/Reflection/AST/Show.agda
index efe77a4779..5d748e089b 100644
--- a/src/Reflection/AST/Show.agda
+++ b/src/Reflection/AST/Show.agda
@@ -11,16 +11,16 @@
 
 module Reflection.AST.Show where
 
-import Data.Char as Char   using (show)
-import Data.Float as Float using (show)
+import Data.Char.Base as Char
+import Data.Float.Base as Float
 open import Data.List.Base hiding (_++_; intersperse)
-import Data.Nat.Show as ℕ using (show)
+import Data.Nat.Show as ℕ
 open import Data.String.Base as String using (String; _++_; intersperse; braces; parens; _<+>_)
 open import Data.String as String using (parensIfSpace)
 open import Data.Product.Base using (_×_; _,_)
-import Data.Word as Word   using (toℕ)
+import Data.Word.Base as Word
 open import Function.Base using (id; _∘′_; case_of_)
-open import Relation.Nullary.Decidable using (yes; no)
+open import Relation.Nullary.Decidable.Core using (yes; no)
 
 open import Reflection.AST.Abstraction hiding (map)
 open import Reflection.AST.Argument hiding (map)
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index 33ef68d6f0..cdcefb55fc 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -10,7 +10,8 @@ module Reflection.AST.Term where
 
 open import Data.List.Base as List                     hiding (_++_)
 open import Data.List.Properties                       using (∷-dec)
-open import Data.Nat as ℕ                              using (ℕ; zero; suc)
+open import Data.Nat.Base                              using (ℕ; zero; suc)
+import Data.Nat.Properties as ℕ
 open import Data.Product.Base                          using (_×_; _,_; <_,_>; uncurry; map₁)
 open import Data.Product.Properties                    using (,-injective)
 open import Data.Maybe.Base                            using (Maybe; just; nothing)
diff --git a/src/Reflection/AST/Traversal.agda b/src/Reflection/AST/Traversal.agda
index a71d93e191..13f806d880 100644
--- a/src/Reflection/AST/Traversal.agda
+++ b/src/Reflection/AST/Traversal.agda
@@ -11,7 +11,7 @@ open import Effect.Applicative using (RawApplicative)
 module Reflection.AST.Traversal
   {F : Set → Set} (AppF : RawApplicative F) where
 
-open import Data.Nat          using (ℕ; zero; suc; _+_)
+open import Data.Nat.Base     using (ℕ; zero; suc; _+_)
 open import Data.List.Base    using (List; []; _∷_; _++_; reverse; length)
 open import Data.String.Base  using (String)
 open import Data.Product.Base using (_×_; _,_)

From 842a6596da9d7649701e7c806321428cdd546d12 Mon Sep 17 00:00:00 2001
From: shuhung <shhyou@users.noreply.github.com>
Date: Tue, 12 Sep 2023 02:04:19 -0500
Subject: [PATCH 112/316] =?UTF-8?q?Add=20some=20`=5F=E2=88=B7=CA=B3=5F`=20?=
 =?UTF-8?q?properties=20to=20`Data.Vec.Properties`=20(#2041)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                  | 11 +++++++++++
 src/Data/List/Properties.agda | 16 +++++++++++-----
 src/Data/Vec/Base.agda        |  8 ++++----
 src/Data/Vec/Properties.agda  | 33 ++++++++++++++++++++++++++++++++-
 4 files changed, 58 insertions(+), 10 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5121e27af9..599b64fcef 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -864,6 +864,9 @@ Non-backwards compatible changes
 
 * In `Data.Sum.Base` the definitions `fromDec` and `toDec` have been moved to `Data.Sum`.
 
+* In `Data.Vec.Base`: the definitions `init` and `last` have been changed from the `initLast`
+  view-derived implementation to direct recursive definitions.
+
 * In `Codata.Guarded.Stream` the following functions have been modified to have simpler definitions:
   * `cycle`
   * `interleave⁺`
@@ -1253,6 +1256,8 @@ Deprecated names
   zipWith-identityˡ  ↦  zipWith-zeroˡ
   zipWith-identityʳ  ↦  zipWith-zeroʳ
 
+  ʳ++-++  ↦  ++-ʳ++
+
   take++drop ↦ take++drop≡id
   ```
 
@@ -2658,6 +2663,12 @@ Other minor changes
   ∷ʳ-injectiveˡ : xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys
   ∷ʳ-injectiveʳ : xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y
 
+  unfold-∷ʳ : cast eq (xs ∷ʳ x) ≡ xs ++ [ x ]
+  init-∷ʳ   : init (xs ∷ʳ x) ≡ xs
+  last-∷ʳ   : last (xs ∷ʳ x) ≡ x
+  cast-∷ʳ   : cast eq (xs ∷ʳ x) ≡ (cast (cong pred eq) xs) ∷ʳ x
+  ++-∷ʳ     : cast eq ((xs ++ ys) ∷ʳ z) ≡ xs ++ (ys ∷ʳ z)
+
   reverse-∷          : reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
   reverse-involutive : Involutive _≡_ reverse
   reverse-reverse    : reverse xs ≡ ys → reverse ys ≡ xs
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 9e86d12436..eb403f8b91 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -993,11 +993,11 @@ module _ {P : Pred A p} (P? : Decidable P) where
 
 -- Reverse-append of append is reverse-append after reverse-append.
 
-ʳ++-++ : ∀ (xs {ys zs} : List A) → (xs ++ ys) ʳ++ zs ≡ ys ʳ++ xs ʳ++ zs
-ʳ++-++ [] = refl
-ʳ++-++ (x ∷ xs) {ys} {zs} = begin
+++-ʳ++ : ∀ (xs {ys zs} : List A) → (xs ++ ys) ʳ++ zs ≡ ys ʳ++ xs ʳ++ zs
+++-ʳ++ [] = refl
+++-ʳ++ (x ∷ xs) {ys} {zs} = begin
   (x ∷ xs ++ ys) ʳ++ zs   ≡⟨⟩
-  (xs ++ ys) ʳ++ x ∷ zs   ≡⟨ ʳ++-++ xs ⟩
+  (xs ++ ys) ʳ++ x ∷ zs   ≡⟨ ++-ʳ++ xs ⟩
   ys ʳ++ xs ʳ++ x ∷ zs    ≡⟨⟩
   ys ʳ++ (x ∷ xs) ʳ++ zs  ∎
 
@@ -1073,7 +1073,7 @@ reverse-++ : (xs ys : List A) →
                      reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
 reverse-++ xs ys = begin
   reverse (xs ++ ys)         ≡⟨⟩
-  (xs ++ ys) ʳ++ []          ≡⟨ ʳ++-++ xs ⟩
+  (xs ++ ys) ʳ++ []          ≡⟨ ++-ʳ++ xs ⟩
   ys ʳ++ xs ʳ++ []           ≡⟨⟩
   ys ʳ++ reverse xs          ≡⟨ ʳ++-defn ys ⟩
   reverse ys ++ reverse xs   ∎
@@ -1213,6 +1213,12 @@ zipWith-identityʳ = zipWith-zeroʳ
 Please use zipWith-zeroʳ instead."
 #-}
 
+ʳ++-++ = ++-ʳ++
+{-# WARNING_ON_USAGE ʳ++-++
+"Warning: ʳ++-++ was deprecated in v2.0.
+Please use ++-ʳ++ instead."
+#-}
+
 take++drop = take++drop≡id
 {-# WARNING_ON_USAGE take++drop
 "Warning: take++drop was deprecated in v2.0.
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 9f6d86b792..0bf095eaa1 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -345,12 +345,12 @@ initLast {n = suc n} (x ∷ xs) with initLast xs
 ... | (ys , y , refl) = (x ∷ ys , y , refl)
 
 init : Vec A (1 + n) → Vec A n
-init xs with initLast xs
-... | (ys , y , refl) = ys
+init {n = zero}  (x ∷ _)  = []
+init {n = suc n} (x ∷ xs) = x ∷ init xs
 
 last : Vec A (1 + n) → A
-last xs with initLast xs
-... | (ys , y , refl) = y
+last {n = zero}  (x ∷ _)  = x
+last {n = suc n} (_ ∷ xs) = last xs
 
 ------------------------------------------------------------------------
 -- Other operations
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 56a2001f1e..b6c1f165be 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -399,7 +399,7 @@ cast-is-id eq (x ∷ xs) = cong (x ∷_) (cast-is-id (suc-injective eq) xs)
 subst-is-cast : (eq : m ≡ n) (xs : Vec A m) → subst (Vec A) eq xs ≡ cast eq xs
 subst-is-cast refl xs = sym (cast-is-id refl xs)
 
-cast-trans : .(eq₁ : m ≡ n) (eq₂ : n ≡ o) (xs : Vec A m) →
+cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (xs : Vec A m) →
              cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
 cast-trans {m = zero}  {n = zero}  {o = zero}  eq₁ eq₂ [] = refl
 cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (x ∷ xs) =
@@ -864,6 +864,12 @@ map-is-foldr f = foldr-universal (Vec _) (λ x ys → f x ∷ ys) (map f) refl (
 ------------------------------------------------------------------------
 -- _∷ʳ_
 
+-- snoc is snoc
+
+unfold-∷ʳ : ∀ .(eq : suc n ≡ n + 1) x (xs : Vec A n) → cast eq (xs ∷ʳ x) ≡ xs ++ [ x ]
+unfold-∷ʳ eq x []       = refl
+unfold-∷ʳ eq x (y ∷ xs) = cong (y ∷_) (unfold-∷ʳ (cong pred eq) x xs)
+
 ∷ʳ-injective : ∀ (xs ys : Vec A n) → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
 ∷ʳ-injective []       []        refl = (refl , refl)
 ∷ʳ-injective (x ∷ xs) (y  ∷ ys) eq   with ∷-injective eq
@@ -885,12 +891,37 @@ foldr-∷ʳ : ∀ (B : ℕ → Set b) (f : FoldrOp A B) {e} y (ys : Vec A n) →
 foldr-∷ʳ B f y []       = refl
 foldr-∷ʳ B f y (x ∷ xs) = cong (f x) (foldr-∷ʳ B f y xs)
 
+-- init, last and _∷ʳ_
+
+init-∷ʳ : ∀ x (xs : Vec A n) → init (xs ∷ʳ x) ≡ xs
+init-∷ʳ x []       = refl
+init-∷ʳ x (y ∷ xs) = cong (y ∷_) (init-∷ʳ x xs)
+
+last-∷ʳ : ∀ x (xs : Vec A n) → last (xs ∷ʳ x) ≡ x
+last-∷ʳ x []       = refl
+last-∷ʳ x (y ∷ xs) = last-∷ʳ x xs
+
 -- map and _∷ʳ_
 
 map-∷ʳ : ∀ (f : A → B) x (xs : Vec A n) → map f (xs ∷ʳ x) ≡ map f xs ∷ʳ f x
 map-∷ʳ f x []       = refl
 map-∷ʳ f x (y ∷ xs) = cong (f y ∷_) (map-∷ʳ f x xs)
 
+-- cast and _∷ʳ_
+
+cast-∷ʳ : ∀ .(eq : suc n ≡ suc m) x (xs : Vec A n) →
+          cast eq (xs ∷ʳ x) ≡ (cast (cong pred eq) xs) ∷ʳ x
+cast-∷ʳ {m = zero}  eq x []       = refl
+cast-∷ʳ {m = suc m} eq x (y ∷ xs) = cong (y ∷_) (cast-∷ʳ (cong pred eq) x xs)
+
+-- _++_ and _∷ʳ_
+
+++-∷ʳ : ∀ .(eq : suc (m + n) ≡ m + suc n) z (xs : Vec A m) (ys : Vec A n) →
+        cast eq ((xs ++ ys) ∷ʳ z) ≡ xs ++ (ys ∷ʳ z)
+++-∷ʳ {m = zero}  eq z []       []       = refl
+++-∷ʳ {m = zero}  eq z []       (y ∷ ys) = cong (y ∷_) (++-∷ʳ refl z [] ys)
+++-∷ʳ {m = suc m} eq z (x ∷ xs) ys       = cong (x ∷_) (++-∷ʳ (cong pred eq) z xs ys)
+
 ------------------------------------------------------------------------
 -- reverse
 

From 9698a4a6d6b1ca6c922adc127598544a878b36c6 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 13 Sep 2023 01:51:18 +0100
Subject: [PATCH 113/316] Monadic join (#2079)

---
 CHANGELOG.md                            |  6 ++++--
 src/Data/List/Effectful.agda            |  4 ++++
 src/Data/Maybe/Effectful.agda           |  3 +++
 src/Data/Sum/Effectful/Left.agda        |  3 +++
 src/Data/Sum/Effectful/Right.agda       |  3 +++
 src/Data/Vec.agda                       |  2 +-
 src/Data/Vec/Base.agda                  |  3 +--
 src/Data/Vec/Effectful.agda             |  5 ++++-
 src/Data/Vec/Effectful/Transformer.agda |  2 +-
 src/Effect/Monad.agda                   | 11 ++++++++++-
 src/Effect/Monad/Identity.agda          |  4 ++++
 src/Effect/Monad/Partiality.agda        |  3 +++
 src/Effect/Monad/Reader.agda            |  3 +++
 src/Effect/Monad/State.agda             |  3 +++
 src/Effect/Monad/Writer.agda            |  3 +++
 15 files changed, 50 insertions(+), 8 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 599b64fcef..20f4c153b8 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -692,6 +692,9 @@ Non-backwards compatible changes
   This reorganisation means in particular that the functor/applicative of a monad
   are not computed using `_>>=_`. This may break proofs.
 
+* When `F : Set f → Set f` we moreover have a definable join/μ operator
+  `join : (M : RawMonad F) → F (F A) → F A`.
+
 * We now have `RawEmpty` and `RawChoice` respectively packing `empty : M A` and
   `(<|>) : M A → M A → M A`. `RawApplicativeZero`, `RawAlternative`, `RawMonadZero`,
   `RawMonadPlus` are all defined in terms of these.
@@ -2604,14 +2607,13 @@ Other minor changes
 
   diagonal           : Vec (Vec A n) n → Vec A n
   DiagonalBind._>>=_ : Vec A n → (A → Vec B n) → Vec B n
-  join               : Vec (Vec A n) n → Vec A n
 
   _ʳ++_              : Vec A m → Vec A n → Vec A (m + n)
 
   cast : .(eq : m ≡ n) → Vec A m → Vec A n
   ```
 
-* Added new instance in `Data.Vec.Categorical`:
+* Added new instance in `Data.Vec.Effectful`:
   ```agda
   monad : RawMonad (λ (A : Set a) → Vec A n)
   ```
diff --git a/src/Data/List/Effectful.agda b/src/Data/List/Effectful.agda
index fa7280f68f..3d4e557468 100644
--- a/src/Data/List/Effectful.agda
+++ b/src/Data/List/Effectful.agda
@@ -25,6 +25,7 @@ open P.≡-Reasoning
 private
   variable
     ℓ : Level
+    A : Set ℓ
 
 ------------------------------------------------------------------------
 -- List applicative functor
@@ -66,6 +67,9 @@ monad = record
   ; _>>=_  = flip concatMap
   }
 
+join : List (List A) → List A
+join = Join.join monad
+
 monadZero : ∀ {ℓ} → RawMonadZero {ℓ} List
 monadZero = record
   { rawMonad = monad
diff --git a/src/Data/Maybe/Effectful.agda b/src/Data/Maybe/Effectful.agda
index 405b4eab42..7e0bf0216b 100644
--- a/src/Data/Maybe/Effectful.agda
+++ b/src/Data/Maybe/Effectful.agda
@@ -67,6 +67,9 @@ monad = record
   ; _>>=_ = _>>=_
   }
 
+join : Maybe (Maybe A) → Maybe A
+join = Join.join monad
+
 monadZero : RawMonadZero {f} Maybe
 monadZero = record
   { rawMonad = monad
diff --git a/src/Data/Sum/Effectful/Left.agda b/src/Data/Sum/Effectful/Left.agda
index 6d0f453b7b..68c632afe4 100644
--- a/src/Data/Sum/Effectful/Left.agda
+++ b/src/Data/Sum/Effectful/Left.agda
@@ -62,6 +62,9 @@ monad = record
   ; _>>=_ = [ const ∘′ inj₁ , _|>′_ ]′
   }
 
+join : {B : Set (a ⊔ b)} → Sumₗ (Sumₗ B) → Sumₗ B
+join = Join.join monad
+
 ------------------------------------------------------------------------
 -- Get access to other monadic functions
 
diff --git a/src/Data/Sum/Effectful/Right.agda b/src/Data/Sum/Effectful/Right.agda
index 19b4995b1e..7658967128 100644
--- a/src/Data/Sum/Effectful/Right.agda
+++ b/src/Data/Sum/Effectful/Right.agda
@@ -56,6 +56,9 @@ monad = record
   ; _>>=_ = [ _|>′_ , const ∘′ inj₂ ]′
   }
 
+join : {A : Set (a ⊔ b)} → Sumᵣ (Sumᵣ A) → Sumᵣ A
+join = Join.join monad
+
 monadZero : B → RawMonadZero Sumᵣ
 monadZero b = record
   { rawMonad = monad
diff --git a/src/Data/Vec.agda b/src/Data/Vec.agda
index d9f1b89664..62b16e0cc6 100644
--- a/src/Data/Vec.agda
+++ b/src/Data/Vec.agda
@@ -9,7 +9,7 @@
 
 -- See `Data.Vec.Functional` for an alternative implementation as
 -- functions from finite sets, which is more suitable for reasoning
--- about fixed sized vectors and for when ease of retrevial is
+-- about fixed sized vectors and for when ease of retrieval is
 -- important.
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 0bf095eaa1..82177e63b4 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -189,8 +189,6 @@ module DiagonalBind where
   _>>=_ : Vec A n → (A → Vec B n) → Vec B n
   xs >>= f = diagonal (map f xs)
 
-  join : Vec (Vec A n) n → Vec A n
-  join = _>>= id
 
 ------------------------------------------------------------------------
 -- Operations for reducing vectors
@@ -358,3 +356,4 @@ last {n = suc n} (_ ∷ xs) = last xs
 transpose : Vec (Vec A n) m → Vec (Vec A m) n
 transpose []         = replicate []
 transpose (as ∷ ass) = replicate _∷_ ⊛ as ⊛ transpose ass
+
diff --git a/src/Data/Vec/Effectful.agda b/src/Data/Vec/Effectful.agda
index 5d8e438537..3d4c1a75e7 100644
--- a/src/Data/Vec/Effectful.agda
+++ b/src/Data/Vec/Effectful.agda
@@ -14,7 +14,7 @@ open import Data.Vec.Base as Vec hiding (_⊛_)
 open import Data.Vec.Properties
 open import Effect.Applicative as App using (RawApplicative)
 open import Effect.Functor as Fun using (RawFunctor)
-open import Effect.Monad using (RawMonad; RawMonadT; mkRawMonad)
+open import Effect.Monad using (RawMonad; module Join; RawMonadT; mkRawMonad)
 import Function.Identity.Effectful as Id
 open import Function.Base
 open import Level using (Level)
@@ -46,6 +46,9 @@ monad = record
   ; _>>=_ = DiagonalBind._>>=_
   }
 
+join : Vec (Vec A n) n → Vec A n
+join = Join.join monad
+
 ------------------------------------------------------------------------
 -- Get access to other monadic functions
 
diff --git a/src/Data/Vec/Effectful/Transformer.agda b/src/Data/Vec/Effectful/Transformer.agda
index 9c3847f40e..cf491fdb84 100644
--- a/src/Data/Vec/Effectful/Transformer.agda
+++ b/src/Data/Vec/Effectful/Transformer.agda
@@ -50,7 +50,7 @@ monad {f} {g} M = record
   ; _>>=_ = λ ma k → mkVecT $ do
                       a ← runVecT ma
                       bs ← mapM {a = f} (runVecT ∘′ k) a
-                      pure (Vec.DiagonalBind.join bs)
+                      pure (Vec.diagonal bs)
   } where open RawMonad M; open Vec.TraversableM {m = f} {n = g} M
 
 monadT : RawMonadT {f} {g} (VecT n)
diff --git a/src/Effect/Monad.agda b/src/Effect/Monad.agda
index 57e410bf3d..0448f0d6cd 100644
--- a/src/Effect/Monad.agda
+++ b/src/Effect/Monad.agda
@@ -16,7 +16,7 @@ open import Data.Unit.Polymorphic.Base using (⊤)
 open import Effect.Choice
 open import Effect.Empty
 open import Effect.Applicative
-open import Function.Base using (flip; _$′_; _∘′_)
+open import Function.Base using (id; flip; _$′_; _∘′_)
 open import Level using (Level; suc; _⊔_)
 
 private
@@ -58,7 +58,16 @@ record RawMonad (F : Set f → Set g) : Set (suc f ⊔ g) where
   unless : Bool → F ⊤ → F ⊤
   unless = when ∘′ not
 
+-- When level g=f, a join/μ operator is definable
+
+module Join {F : Set f → Set f} (M : RawMonad F) where
+  open RawMonad M
+
+  join : F (F A) → F A
+  join = _>>= id
+
 -- Smart constructor
+
 module _ where
 
   open RawMonad
diff --git a/src/Effect/Monad/Identity.agda b/src/Effect/Monad/Identity.agda
index 3cb7bf9f21..b828d00fcc 100644
--- a/src/Effect/Monad/Identity.agda
+++ b/src/Effect/Monad/Identity.agda
@@ -18,6 +18,7 @@ open import Level
 private
   variable
     a : Level
+    A : Set a
 
 record Identity (A : Set a) : Set a where
   constructor mkIdentity
@@ -47,3 +48,6 @@ comonad = record
   { extract = runIdentity
   ; extend  = λ f a → mkIdentity (f a)
   }
+
+join : Identity (Identity A) → Identity A
+join = Join.join monad
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index 194b079339..5fb6ec3455 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -72,6 +72,9 @@ monad = record
   ; _>>=_  = bind
   }
 
+join : (A ⊥) ⊥ → A ⊥
+join = Join.join monad
+
 private module M {f} = RawMonad (monad {f})
 
 -- Non-termination.
diff --git a/src/Effect/Monad/Reader.agda b/src/Effect/Monad/Reader.agda
index 79267a5ab2..42e6091d00 100644
--- a/src/Effect/Monad/Reader.agda
+++ b/src/Effect/Monad/Reader.agda
@@ -52,6 +52,9 @@ applicative = Trans.applicative Id.applicative
 monad : RawMonad (Reader R)
 monad = Trans.monad Id.monad
 
+join : Reader R (Reader R A) → Reader R A
+join = Join.join monad
+
 ------------------------------------------------------------------------
 -- Reader monad specifics
 
diff --git a/src/Effect/Monad/State.agda b/src/Effect/Monad/State.agda
index 41fc3336d8..2b06c8b087 100644
--- a/src/Effect/Monad/State.agda
+++ b/src/Effect/Monad/State.agda
@@ -59,6 +59,9 @@ applicative = Trans.applicative Id.monad
 monad : RawMonad (State S)
 monad = Trans.monad Id.monad
 
+join : State S (State S A) → State S A
+join = Join.join monad
+
 ------------------------------------------------------------------------
 -- State monad specifics
 
diff --git a/src/Effect/Monad/Writer.agda b/src/Effect/Monad/Writer.agda
index 489c7f4347..60f4dc2b72 100644
--- a/src/Effect/Monad/Writer.agda
+++ b/src/Effect/Monad/Writer.agda
@@ -55,6 +55,9 @@ module _ {𝕎 : RawMonoid w ℓ} where
   monad : RawMonad (Writer 𝕎)
   monad = Trans.monad Id.monad
 
+  join : Writer 𝕎 (Writer 𝕎 A) → Writer 𝕎 A
+  join = Join.join monad
+
   ----------------------------------------------------------------------
   -- Writer monad specifics
 

From ad13b40473f55305cce7700cc737c685db196dfc Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Wed, 13 Sep 2023 07:16:25 +0530
Subject: [PATCH 114/316] =?UTF-8?q?Move=20`=E2=89=A1-setoid`=20to=20`Funct?=
 =?UTF-8?q?ion.Indexed.Relation.Binary.Equality`=20(#2047)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 .../Permutation/Propositional/Properties.agda |  1 -
 .../Binary/Permutation/Setoid/Properties.agda |  2 --
 .../Function/Dependent/Propositional.agda     |  2 +-
 .../Function/Dependent/Setoid/WithK.agda      |  2 +-
 .../Function/NonDependent/Propositional.agda  |  1 -
 .../Binary/Pointwise/NonDependent.agda        |  2 +-
 src/Function/Endomorphism/Setoid.agda         |  2 +-
 .../Indexed/Relation/Binary/Equality.agda     | 27 +++++++++++++++++++
 .../Binary/PropositionalEquality.agda         |  3 ++-
 src/Relation/Nullary/Universe.agda            |  4 +--
 10 files changed, 35 insertions(+), 11 deletions(-)
 create mode 100644 src/Function/Indexed/Relation/Binary/Equality.agda

diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index 82475a505b..90196f8744 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -23,7 +23,6 @@ open import Data.List.Membership.Propositional.Properties
 import Data.List.Properties as Lₚ
 open import Data.Product.Base using (_,_; _×_; ∃; ∃₂)
 open import Function.Base using (_∘_; _⟨_⟩_)
-open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse as Inv using (inverse)
 open import Level using (Level)
 open import Relation.Unary using (Pred)
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index 57233a7b2a..bc7af2193f 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -34,8 +34,6 @@ open import Data.Nat.Induction
 open import Data.Nat.Properties
 open import Data.Product.Base using (_,_; _×_; ∃; ∃₂; proj₁; proj₂)
 open import Function.Base using (_∘_; _⟨_⟩_; flip)
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Inverse as Inv using (inverse)
 open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index b7333283ff..bde9e92c44 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -13,7 +13,7 @@ open import Data.Product.Base using (Σ; map; _,_)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function.Base
-open import Function.Equality using (_⟶_; _⟨$⟩_)
+open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence as Equiv using (_⇔_; module Equivalence)
 open import Function.HalfAdjointEquivalence using (↔→≃; _≃_)
 open import Function.Injection as Inj
diff --git a/src/Data/Product/Function/Dependent/Setoid/WithK.agda b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
index b4c30a26aa..bb89c1b95b 100644
--- a/src/Data/Product/Function/Dependent/Setoid/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
@@ -13,7 +13,7 @@ open import Data.Product.Base using (_,_)
 open import Data.Product.Function.Dependent.Setoid using (surjection)
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Function.Base
-open import Function.Equality as F using (_⟶_; _⟨$⟩_)
+open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence as Eq
   using (Equivalence; _⇔_; module Equivalence)
 open import Function.Injection as Inj
diff --git a/src/Data/Product/Function/NonDependent/Propositional.agda b/src/Data/Product/Function/NonDependent/Propositional.agda
index 7dfb28207d..a67490b8a5 100644
--- a/src/Data/Product/Function/NonDependent/Propositional.agda
+++ b/src/Data/Product/Function/NonDependent/Propositional.agda
@@ -12,7 +12,6 @@ module Data.Product.Function.NonDependent.Propositional where
 open import Data.Product.Base using (_×_; map)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Function.Equality using (_⟶_)
 open import Function.Equivalence as Eq using (_⇔_; module Equivalence)
 open import Function.Injection as Inj using (_↣_; module Injection)
 open import Function.Inverse as Inv using (_↔_; module Inverse)
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index ad40326ab4..00629a2b5d 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -13,7 +13,7 @@ open import Data.Product.Properties using (≡-dec)
 open import Data.Sum.Base
 open import Data.Unit.Base using (⊤)
 open import Function.Base
-open import Function.Equality as F using (_⟶_; _⟨$⟩_)
+open import Function.Equality using (_⟨$⟩_)
 open import Function.Equivalence as Eq
   using (Equivalence; _⇔_; module Equivalence)
 open import Function.Injection as Inj
diff --git a/src/Function/Endomorphism/Setoid.agda b/src/Function/Endomorphism/Setoid.agda
index d9890c3a35..a715ef5262 100644
--- a/src/Function/Endomorphism/Setoid.agda
+++ b/src/Function/Endomorphism/Setoid.agda
@@ -19,7 +19,7 @@ open import Agda.Builtin.Equality
 open import Algebra
 open import Algebra.Structures
 open import Algebra.Morphism; open Definitions
-open import Function.Equality
+open import Function.Equality using (setoid; _⟶_; id; _∘_; cong)
 open import Data.Nat.Base using (ℕ; _+_); open ℕ
 open import Data.Nat.Properties
 open import Data.Product.Base using (_,_)
diff --git a/src/Function/Indexed/Relation/Binary/Equality.agda b/src/Function/Indexed/Relation/Binary/Equality.agda
new file mode 100644
index 0000000000..ef9d6b3657
--- /dev/null
+++ b/src/Function/Indexed/Relation/Binary/Equality.agda
@@ -0,0 +1,27 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Function setoids and related constructions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Indexed.Relation.Binary.Equality where
+
+open import Relation.Binary using (Setoid)
+open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)
+
+-- A variant of setoid which uses the propositional equality setoid
+-- for the domain, and a more convenient definition of _≈_.
+
+≡-setoid : ∀ {f t₁ t₂} (From : Set f) → IndexedSetoid From t₁ t₂ → Setoid _ _
+≡-setoid From To = record
+  { Carrier       = (x : From) → Carrier x
+  ; _≈_           = λ f g → ∀ x → f x ≈ g x
+  ; isEquivalence = record
+    { refl  = λ {f} x → refl
+    ; sym   = λ f∼g x → sym (f∼g x)
+    ; trans = λ f∼g g∼h x → trans (f∼g x) (g∼h x)
+    }
+  } where open IndexedSetoid To
+
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 62bd7ca246..ca3ea60e4d 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -11,7 +11,8 @@ module Relation.Binary.PropositionalEquality where
 import Axiom.Extensionality.Propositional as Ext
 open import Axiom.UniquenessOfIdentityProofs
 open import Function.Base using (id; _∘_)
-open import Function.Equality using (Π; _⟶_; ≡-setoid)
+open import Function.Equality using (Π; _⟶_)
+open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)
 open import Level using (Level; _⊔_)
 open import Data.Product.Base using (∃)
 
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 658f8c7bf7..3057b16eb6 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -22,7 +22,7 @@ open import Data.Sum.Relation.Binary.Pointwise
 open import Data.Product.Base as Prod hiding (map)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Function.Base using (_∘_; id)
-import Function.Equality as FunS
+open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)
 open import Data.Empty
 open import Effect.Applicative
 open import Effect.Monad
@@ -53,7 +53,7 @@ mutual
   setoid (K P)     _ = PropEq.setoid P
   setoid (F₁ ∨ F₂) P = (setoid F₁ P) ⊎ₛ (setoid F₂ P)
   setoid (F₁ ∧ F₂) P = (setoid F₁ P) ×ₛ (setoid F₂ P)
-  setoid (P₁ ⇒ F₂) P = FunS.≡-setoid P₁
+  setoid (P₁ ⇒ F₂) P = ≡-setoid P₁
                          (Trivial.indexedSetoid (setoid F₂ P))
   setoid (¬¬ F)    P = Always.setoid (¬ ¬ ⟦ F ⟧ P) _
 

From 3cf21ecd39d85051f048dcba52b4922e40482221 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 13 Sep 2023 02:55:10 +0100
Subject: [PATCH 115/316] Making (more) arguments implicit in lexicographic
 ordering lemmas

---
 CHANGELOG.md                                  | 5 ++++-
 src/Data/List/Relation/Binary/Lex/Strict.agda | 6 +++---
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  | 8 ++++----
 3 files changed, 11 insertions(+), 8 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 20f4c153b8..8215de843f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -784,6 +784,9 @@ Non-backwards compatible changes
   properties about the orderings themselves the second index must be provided
   explicitly.
 
+* The argument `xs` in `xs≮[]` in `Data.{List|Vec}.Relation.Binary.Lex.Strict`
+  introduced in PRs #1648 and #1672 has now been made implicit.
+
 * Issue #2075 (Johannes Waldmann): wellfoundedness of the lexicographic ordering
   on products, defined in `Data.Product.Relation.Binary.Lex.Strict`, no longer
   requires the assumption of symmetry for the first equality relation `_≈₁_`,
@@ -2705,7 +2708,7 @@ Other minor changes
 
 * Added new proofs in `Data.Vec.Relation.Binary.Lex.Strict`:
   ```agda
-  xs≮[] : ∀ {n} (xs : Vec A n) → ¬ xs < []
+  xs≮[] : {xs : Vec A n} → ¬ xs < []
   <-respectsˡ : IsPartialEquivalence _≈_ → _≺_ Respectsˡ _≈_ →
                 ∀ {m n} → _Respectsˡ_ (_<_ {m} {n}) _≋_
   <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ →
diff --git a/src/Data/List/Relation/Binary/Lex/Strict.agda b/src/Data/List/Relation/Binary/Lex/Strict.agda
index 9ea73457fe..7b7f5d2baf 100644
--- a/src/Data/List/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/List/Relation/Binary/Lex/Strict.agda
@@ -51,11 +51,11 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
       _≋_ = Pointwise _≈_
       _<_ = Lex-< _≈_ _≺_
 
-    xs≮[] : ∀ xs → ¬ xs < []
-    xs≮[] _ (base ())
+    xs≮[] : ∀ {xs} → ¬ xs < []
+    xs≮[] (base ())
 
     ¬[]<[] : ¬ [] < []
-    ¬[]<[] = xs≮[] []
+    ¬[]<[] = xs≮[]
 
     <-irreflexive : Irreflexive _≈_ _≺_ → Irreflexive _≋_ _<_
     <-irreflexive irr (x≈y ∷ xs≋ys) (this x<y)     = irr x≈y x<y
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index d488e8e9d2..54ab06bd97 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -69,11 +69,11 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
     _≋_ = Pointwise _≈_
     _<_ = Lex-< _≈_ _≺_
 
-  xs≮[] : ∀ {n} (xs : Vec A n) → ¬ xs < []
-  xs≮[] _ (base ())
+  xs≮[] : ∀ {n} {xs : Vec A n} → ¬ xs < []
+  xs≮[] (base ())
 
   ¬[]<[] : ¬ [] < []
-  ¬[]<[] = xs≮[] []
+  ¬[]<[] = xs≮[]
 
   module _ (≺-irrefl : Irreflexive _≈_ _≺_) where
 
@@ -134,7 +134,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
     where
 
     <-wellFounded : ∀ {n} → WellFounded (_<_ {n})
-    <-wellFounded {0}     [] = acc λ ys ys<[] → ⊥-elim (xs≮[] ys ys<[])
+    <-wellFounded {0}     [] = acc λ ys ys<[] → ⊥-elim (xs≮[] ys<[])
     <-wellFounded {suc n} xs = Subrelation.wellFounded <⇒uncons-Lex uncons-Lex-wellFounded xs
       where
         <⇒uncons-Lex : {xs ys : Vec A (suc n)} → xs < ys → (×-Lex _≈_ _≺_ _<_ on uncons) xs ys

From 1b07c4a88da4dd6a46e66b92d0c22471c1cd0915 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 13 Sep 2023 13:50:49 +0100
Subject: [PATCH 116/316] `WellFounded` proofs for transitive closure (#2082)

---
 CHANGELOG.md                                  |  7 ++++++
 .../Binary/Construct/Closure/Transitive.agda  | 23 +++++++++++--------
 2 files changed, 21 insertions(+), 9 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8215de843f..202b5f4a41 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2951,6 +2951,13 @@ Other minor changes
                     f Preserves₂ ≈₁ ⟶ ≈₁ ⟶ ≈₂
   ```
 
+* Added new proofs to `Relation.Binary.Construct.Closure.Transitive`:
+  ```
+  accessible⁻ : ∀ {x} → Acc _∼⁺_ x → Acc _∼_ x
+  wellFounded⁻ : WellFounded _∼⁺_ → WellFounded _∼_
+  accessible : ∀ {x} → Acc _∼_ x → Acc _∼⁺_ x
+  ```
+
 * Added new operations in `Relation.Binary.PropositionalEquality.Properties`:
   ```
   J       : (B : (y : A) → x ≡ y → Set b) (p : x ≡ y) → B x refl → B y p
diff --git a/src/Relation/Binary/Construct/Closure/Transitive.agda b/src/Relation/Binary/Construct/Closure/Transitive.agda
index ed20e48bb0..7c12fa7bcd 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive.agda
@@ -58,6 +58,7 @@ module _ {_∼_ : Rel A ℓ} where
 module _ (_∼_ : Rel A ℓ) where
   private
     _∼⁺_ = TransClosure _∼_
+    module ∼⊆∼⁺ = Subrelation {_<₂_ = _∼⁺_} [_]
 
   reflexive : Reflexive _∼_ → Reflexive _∼⁺_
   reflexive refl = [ refl ]
@@ -69,17 +70,21 @@ module _ (_∼_ : Rel A ℓ) where
   transitive : Transitive _∼⁺_
   transitive = _++_
 
-  wellFounded : WellFounded _∼_ → WellFounded _∼⁺_
-  wellFounded wf = λ x → acc (accessible′ (wf x))
-    where
-    downwardsClosed : ∀ {x y} → Acc _∼⁺_ y → x ∼ y → Acc _∼⁺_ x
-    downwardsClosed (acc rec) x∼y = acc (λ z z∼x → rec z (z∼x ∷ʳ x∼y))
+  accessible⁻ : ∀ {x} → Acc _∼⁺_ x → Acc _∼_ x
+  accessible⁻ = ∼⊆∼⁺.accessible
 
-    accessible′ : ∀ {x} → Acc _∼_ x → WfRec _∼⁺_ (Acc _∼⁺_) x
-    accessible′ (acc rec) y [ y∼x ]      = acc (accessible′ (rec y y∼x))
-    accessible′ acc[x]    y (y∼z ∷ z∼⁺x) =
-      downwardsClosed (accessible′ acc[x] _ z∼⁺x) y∼z
+  wellFounded⁻ : WellFounded _∼⁺_ → WellFounded _∼_
+  wellFounded⁻ = ∼⊆∼⁺.wellFounded
 
+  accessible : ∀ {x} → Acc _∼_ x → Acc _∼⁺_ x
+  accessible acc[x] = acc (wf-acc acc[x])
+    where
+    wf-acc : ∀ {x} → Acc _∼_ x → WfRec _∼⁺_ (Acc _∼⁺_) x
+    wf-acc (acc rec) _ [ y∼x ]   = acc (wf-acc (rec _ y∼x))
+    wf-acc acc[x] _ (y∼z ∷ z∼⁺x) = acc-inverse (wf-acc acc[x] _ z∼⁺x) _ [ y∼z ]
+
+  wellFounded : WellFounded _∼_ → WellFounded _∼⁺_
+  wellFounded wf x = accessible (wf x)
 
 
 ------------------------------------------------------------------------

From fc3624c4f0b681179c058165fceb53dfab4a7a90 Mon Sep 17 00:00:00 2001
From: shuhung <shhyou@users.noreply.github.com>
Date: Thu, 14 Sep 2023 20:57:35 -0500
Subject: [PATCH 117/316] Add properties of Vector operations `reverse`, `_++_`
 and `fromList` (#2045)

---
 CHANGELOG.md                 | 13 +++++-
 src/Data/Fin/Properties.agda |  2 +-
 src/Data/Vec/Properties.agda | 87 +++++++++++++++++++++++++++++++++---
 3 files changed, 94 insertions(+), 8 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 202b5f4a41..0485996a5c 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2684,7 +2684,6 @@ Other minor changes
 
   toList-++ : toList (xs ++ ys) ≡ toList xs List.++ toList ys
 
-  toList-cast   : toList (cast eq xs) ≡ toList xs
   cast-is-id    : cast eq xs ≡ xs
   subst-is-cast : subst (Vec A) eq xs ≡ cast eq xs
   cast-trans    : cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
@@ -2692,8 +2691,20 @@ Other minor changes
   lookup-cast   : lookup (cast eq xs) (Fin.cast eq i) ≡ lookup xs i
   lookup-cast₁  : lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)
   lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
+  cast-reverse : cast eq ∘ reverse ≗ reverse ∘ cast eq
 
   zipwith-++ : zipWith f (xs ++ ys) (xs' ++ ys') ≡ zipWith f xs xs' ++ zipWith f ys ys'
+
+  ++-assoc     : cast eq ((xs ++ ys) ++ zs) ≡ xs ++ (ys ++ zs)
+  ++-identityʳ : cast eq (xs ++ []) ≡ xs
+  init-reverse : init ∘ reverse ≗ reverse ∘ tail
+  last-reverse : last ∘ reverse ≗ head
+  reverse-++   : cast eq (reverse (xs ++ ys)) ≡ reverse ys ++ reverse xs
+
+  toList-cast   : toList (cast eq xs) ≡ toList xs
+  cast-fromList : cast _ (fromList xs) ≡ fromList ys
+  fromList-map  : cast _ (fromList (List.map f xs)) ≡ map f (fromList xs)
+  fromList-++   : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
   ```
 
 * Added new proofs in `Data.Vec.Membership.Propositional.Properties`:
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index e8a1cef5dc..0f520afda5 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -267,7 +267,7 @@ cast-is-id eq (suc k) = cong suc (cast-is-id (ℕₚ.suc-injective eq) k)
 subst-is-cast : (eq : m ≡ n) (k : Fin m) → subst Fin eq k ≡ cast eq k
 subst-is-cast refl k = sym (cast-is-id refl k)
 
-cast-trans : .(eq₁ : m ≡ n) (eq₂ : n ≡ o) (k : Fin m) →
+cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (k : Fin m) →
              cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k
 cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ zero = refl
 cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (suc k) =
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index b6c1f165be..47ff393d0f 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -12,9 +12,10 @@ open import Algebra.Definitions
 open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base as Fin using (Fin; zero; suc; toℕ; fromℕ<; _↑ˡ_; _↑ʳ_)
 open import Data.List.Base as List using (List)
+import Data.List.Properties as Listₚ
 open import Data.Nat.Base
 open import Data.Nat.Properties
-  using (+-assoc; m≤n⇒m≤1+n; ≤-refl; ≤-trans; suc-injective)
+  using (+-assoc; m≤n⇒m≤1+n; ≤-refl; ≤-trans; suc-injective; +-comm; +-suc)
 open import Data.Product.Base as Prod
   using (_×_; _,_; proj₁; proj₂; <_,_>; uncurry)
 open import Data.Sum.Base using ([_,_]′)
@@ -387,11 +388,6 @@ lookup∘update′ {i = i} {j} i≢j xs y = lookup∘updateAt′ i j i≢j xs
 ------------------------------------------------------------------------
 -- cast
 
-toList-cast : ∀ .(eq : m ≡ n) (xs : Vec A m) → toList (cast eq xs) ≡ toList xs
-toList-cast {n = zero}  eq []       = refl
-toList-cast {n = suc _} eq (x ∷ xs) =
-  cong (x List.∷_) (toList-cast (cong pred eq) xs)
-
 cast-is-id : .(eq : m ≡ m) (xs : Vec A m) → cast eq xs ≡ xs
 cast-is-id eq []       = refl
 cast-is-id eq (x ∷ xs) = cong (x ∷_) (cast-is-id (suc-injective eq) xs)
@@ -488,6 +484,15 @@ toList-map f (x ∷ xs) = cong (f x List.∷_) (toList-map f xs)
 ++-injective ws xs eq =
   (++-injectiveˡ ws xs eq , ++-injectiveʳ ws xs eq)
 
+++-assoc : ∀ .(eq : (m + n) + o ≡ m + (n + o)) (xs : Vec A m) (ys : Vec A n) (zs : Vec A o) →
+           cast eq ((xs ++ ys) ++ zs) ≡ xs ++ (ys ++ zs)
+++-assoc eq []       ys zs = cast-is-id eq (ys ++ zs)
+++-assoc eq (x ∷ xs) ys zs = cong (x ∷_) (++-assoc (cong pred eq) xs ys zs)
+
+++-identityʳ : ∀ .(eq : n + zero ≡ n) (xs : Vec A n) → cast eq (xs ++ []) ≡ xs
+++-identityʳ eq []       = refl
+++-identityʳ eq (x ∷ xs) = cong (x ∷_) (++-identityʳ (cong pred eq) xs)
+
 lookup-++-< : ∀ (xs : Vec A m) (ys : Vec A n) →
               ∀ i (i<m : toℕ i < m) →
               lookup (xs ++ ys) i  ≡ lookup xs (Fin.fromℕ< i<m)
@@ -970,6 +975,20 @@ reverse-injective {xs = xs} {ys} eq = begin
   reverse (reverse ys) ≡⟨  reverse-involutive ys ⟩
   ys                   ∎
 
+-- init and last of reverse
+
+init-reverse : init ∘ reverse ≗ reverse ∘ tail {A = A} {n = n}
+init-reverse (x ∷ xs) = begin
+  init (reverse (x ∷ xs))   ≡⟨ cong init (reverse-∷ x xs) ⟩
+  init (reverse xs ∷ʳ x)    ≡⟨ init-∷ʳ x (reverse xs) ⟩
+  reverse xs                ∎
+
+last-reverse : last ∘ reverse ≗ head {A = A} {n = n}
+last-reverse (x ∷ xs) = begin
+  last (reverse (x ∷ xs))   ≡⟨ cong last (reverse-∷ x xs) ⟩
+  last (reverse xs ∷ʳ x)    ≡⟨ last-∷ʳ x (reverse xs) ⟩
+  x                         ∎
+
 -- map and reverse
 
 map-reverse : ∀ (f : A → B) (xs : Vec A n) →
@@ -983,6 +1002,39 @@ map-reverse f (x ∷ xs) = begin
   reverse (f x ∷ map f xs)  ≡⟨⟩
   reverse (map f (x ∷ xs))  ∎
 
+-- append and reverse
+
+reverse-++ : ∀ .(eq : m + n ≡ n + m) (xs : Vec A m) (ys : Vec A n) →
+             cast eq (reverse (xs ++ ys)) ≡ reverse ys ++ reverse xs
+reverse-++ {m = zero}  {n = n} eq []       ys = begin
+  cast _ (reverse ys)                ≡˘⟨ cong (cast eq) (++-identityʳ (sym eq) (reverse ys)) ⟩
+  cast _ (cast _ (reverse ys ++ [])) ≡⟨ cast-trans (sym eq) eq (reverse ys ++ []) ⟩
+  cast _ (reverse ys ++ [])          ≡⟨ cast-is-id (trans (sym eq) eq) (reverse ys ++ []) ⟩
+  reverse ys ++ []                   ≡⟨⟩
+  reverse ys ++ reverse []           ∎
+reverse-++ {m = suc m} {n = n} eq (x ∷ xs) ys = begin
+  cast eq (reverse (x ∷ xs ++ ys))                ≡⟨ cong (cast eq) (reverse-∷ x (xs ++ ys)) ⟩
+  cast eq (reverse (xs ++ ys) ∷ʳ x)               ≡˘⟨ cast-trans eq₂ eq₁ (reverse (xs ++ ys) ∷ʳ x) ⟩
+  (cast eq₁ ∘ cast eq₂) (reverse (xs ++ ys) ∷ʳ x) ≡⟨ cong (cast eq₁) (cast-∷ʳ _ x (reverse (xs ++ ys))) ⟩
+  cast eq₁ ((cast eq₃ (reverse (xs ++ ys))) ∷ʳ x) ≡⟨ cong (cast eq₁) (cong (_∷ʳ x) (reverse-++ _ xs ys)) ⟩
+  cast eq₁ ((reverse ys ++ reverse xs)      ∷ʳ x) ≡⟨ ++-∷ʳ _ x (reverse ys) (reverse xs) ⟩
+  reverse ys ++ (reverse xs ∷ʳ x)                 ≡˘⟨ cong (reverse ys ++_) (reverse-∷ x xs) ⟩
+  reverse ys ++ (reverse (x ∷ xs))                ∎
+  where
+  eq₁ = sym (+-suc n m)
+  eq₂ = cong suc (+-comm m n)
+  eq₃ = cong pred eq₂
+
+cast-reverse : ∀ .(eq : m ≡ n) → cast eq ∘ reverse {A = A} {n = m} ≗ reverse ∘ cast eq
+cast-reverse {n = zero}  eq []       = refl
+cast-reverse {n = suc n} eq (x ∷ xs) = begin
+  cast eq (reverse (x ∷ xs))              ≡⟨ cong (cast eq) (reverse-∷ x xs) ⟩
+  cast eq (reverse xs ∷ʳ x)               ≡⟨ cast-∷ʳ eq x (reverse xs) ⟩
+  (cast (cong pred eq) (reverse xs)) ∷ʳ x ≡⟨ cong (_∷ʳ x) (cast-reverse (cong pred eq) xs) ⟩
+  (reverse (cast (cong pred eq) xs)) ∷ʳ x ≡˘⟨ reverse-∷ x (cast (cong pred eq) xs) ⟩
+  reverse (x ∷ cast (cong pred eq) xs)    ≡⟨⟩
+  reverse (cast eq (x ∷ xs))              ∎
+
 ------------------------------------------------------------------------
 -- _ʳ++_
 
@@ -1161,6 +1213,29 @@ toList∘fromList : (xs : List A) → toList (fromList xs) ≡ xs
 toList∘fromList List.[]       = refl
 toList∘fromList (x List.∷ xs) = cong (x List.∷_) (toList∘fromList xs)
 
+toList-cast : ∀ .(eq : m ≡ n) (xs : Vec A m) → toList (cast eq xs) ≡ toList xs
+toList-cast {n = zero}  eq []       = refl
+toList-cast {n = suc _} eq (x ∷ xs) =
+  cong (x List.∷_) (toList-cast (cong pred eq) xs)
+
+cast-fromList : ∀ {xs ys : List A} (eq : xs ≡ ys) →
+                cast (cong List.length eq) (fromList xs) ≡ fromList ys
+cast-fromList {xs = List.[]}     {ys = List.[]}     eq = refl
+cast-fromList {xs = x List.∷ xs} {ys = y List.∷ ys} eq =
+  let x≡y , xs≡ys = Listₚ.∷-injective eq in begin
+  x ∷ cast (cong (pred ∘ List.length) eq) (fromList xs) ≡⟨ cong (_ ∷_) (cast-fromList xs≡ys) ⟩
+  x ∷ fromList ys                                       ≡⟨ cong (_∷ _) x≡y ⟩
+  y ∷ fromList ys                                       ∎
+
+fromList-map : ∀ (f : A → B) (xs : List A) →
+               cast (Listₚ.length-map f xs) (fromList (List.map f xs)) ≡ map f (fromList xs)
+fromList-map f List.[]       = refl
+fromList-map f (x List.∷ xs) = cong (f x ∷_) (fromList-map f xs)
+
+fromList-++ : ∀ (xs : List A) {ys : List A} →
+              cast (Listₚ.length-++ xs) (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
+fromList-++ List.[]       {ys} = cast-is-id refl (fromList ys)
+fromList-++ (x List.∷ xs) {ys} = cong (x ∷_) (fromList-++ xs)
 
 ------------------------------------------------------------------------
 -- DEPRECATED NAMES

From d95bb96142b491c0297823d9f2f9fa9014b62b98 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 17 Sep 2023 03:40:16 +0100
Subject: [PATCH 118/316] Rename `minor changes` section in CHANGELOG

---
 CHANGELOG.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0485996a5c..0e447d1d5b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1765,8 +1765,8 @@ New modules
   Function.Indexed.Bundles
   ```
 
-Other minor changes
--------------------
+Additions to existing modules
+-----------------------------
 
 * Added new proof to `Data.Maybe.Properties`
   ```agda

From e417e4afd2ff5f4a8b9f41cabb71095111f341cf Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Sat, 16 Sep 2023 22:56:15 -0400
Subject: [PATCH 119/316] Improve implementation of `splitAt`, `take` and
 `drop` in `Data.List`. (#2056)

---
 CHANGELOG.md                                  |   5 +
 src/Data/Vec/Base.agda                        |  38 +++---
 src/Data/Vec/Properties.agda                  | 126 +++++++-----------
 .../Vec/Relation/Unary/All/Properties.agda    |   6 +-
 .../Relation/Unary/AllPairs/Properties.agda   |   6 +-
 5 files changed, 76 insertions(+), 105 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0e447d1d5b..420d85fddc 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1409,6 +1409,11 @@ Deprecated names
 
 * In `Data.Vec.Properties`:
   ```
+  take-distr-zipWith ↦  take-zipWith
+  take-distr-map     ↦  take-map
+  drop-distr-zipWith ↦  drop-zipWith
+  drop-distr-map     ↦  drop-map
+  
   updateAt-id-relative      ↦  updateAt-id-local
   updateAt-compose-relative ↦  updateAt-∘-local
   updateAt-compose          ↦  updateAt-∘
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 82177e63b4..6752ec1b06 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -12,11 +12,11 @@ open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Nat.Base
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List)
-open import Data.Product.Base as Prod using (∃; ∃₂; _×_; _,_)
+open import Data.Product.Base as Prod using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Function.Base using (const; _∘′_; id; _∘_)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; trans; cong)
 open import Relation.Nullary.Decidable.Core using (does)
 open import Relation.Unary using (Pred; Decidable)
 
@@ -259,26 +259,23 @@ allFin _ = tabulate id
 
 splitAt : ∀ m {n} (xs : Vec A (m + n)) →
           ∃₂ λ (ys : Vec A m) (zs : Vec A n) → xs ≡ ys ++ zs
-splitAt zero    xs                = ([] , xs , refl)
-splitAt (suc m) (x ∷ xs)          with splitAt m xs
-splitAt (suc m) (x ∷ .(ys ++ zs)) | (ys , zs , refl) =
-  ((x ∷ ys) , zs , refl)
+splitAt zero    xs                = [] , xs , refl
+splitAt (suc m) (x ∷ xs) =
+  let ys , zs , eq = splitAt m xs in x ∷ ys , zs , cong (x ∷_) eq
 
 take : ∀ m {n} → Vec A (m + n) → Vec A m
-take m xs          with splitAt m xs
-take m .(ys ++ zs) | (ys , zs , refl) = ys
+take m xs = proj₁ (splitAt m xs)
 
 drop : ∀ m {n} → Vec A (m + n) → Vec A n
-drop m xs          with splitAt m xs
-drop m .(ys ++ zs) | (ys , zs , refl) = zs
+drop m xs = proj₁ (proj₂ (splitAt m xs))
 
 group : ∀ n k (xs : Vec A (n * k)) →
         ∃ λ (xss : Vec (Vec A k) n) → xs ≡ concat xss
 group zero    k []                  = ([] , refl)
-group (suc n) k xs                  with splitAt k xs
-group (suc n) k .(ys ++ zs)         | (ys , zs , refl) with group n k zs
-group (suc n) k .(ys ++ concat zss) | (ys , ._ , refl) | (zss , refl) =
-  ((ys ∷ zss) , refl)
+group (suc n) k xs  = 
+  let ys , zs , eq-split = splitAt k xs in
+  let zss , eq-group     = group n k zs in
+   (ys ∷ zss) , trans eq-split (cong (ys ++_) eq-group)
 
 split : Vec A n → Vec A ⌈ n /2⌉ × Vec A ⌊ n /2⌋
 split []           = ([]     , [])
@@ -338,17 +335,16 @@ xs ʳ++ ys = foldl (Vec _ ∘ (_+ _)) (λ rev x → x ∷ rev) ys xs
 -- init and last
 
 initLast : ∀ (xs : Vec A (1 + n)) → ∃₂ λ ys y → xs ≡ ys ∷ʳ y
-initLast {n = zero}  (x ∷ []) = ([] , x , refl)
-initLast {n = suc n} (x ∷ xs) with initLast xs
-... | (ys , y , refl) = (x ∷ ys , y , refl)
+initLast {n = zero}  (x ∷ []) = [] , x , refl
+initLast {n = suc n} (x ∷ xs) =
+  let ys , y , eq = initLast xs in
+  x ∷ ys , y , cong (x ∷_) eq
 
 init : Vec A (1 + n) → Vec A n
-init {n = zero}  (x ∷ _)  = []
-init {n = suc n} (x ∷ xs) = x ∷ init xs
+init xs = proj₁ (initLast xs)
 
 last : Vec A (1 + n) → A
-last {n = zero}  (x ∷ _)  = x
-last {n = suc n} (_ ∷ xs) = last xs
+last xs = proj₁ (proj₂ (initLast xs))
 
 ------------------------------------------------------------------------
 -- Other operations
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 47ff393d0f..8113d0a932 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -72,86 +72,43 @@ private
 -- take
 
 unfold-take : ∀ n x (xs : Vec A (n + m)) → take (suc n) (x ∷ xs) ≡ x ∷ take n xs
-unfold-take n x xs with splitAt n xs
-... | xs , ys , refl = refl
-
-take-distr-zipWith : ∀ (f : A → B → C) →
-                     (xs : Vec A (m + n)) (ys : Vec B (m + n)) →
-                     take m (zipWith f xs ys) ≡ zipWith f (take m xs) (take m ys)
-take-distr-zipWith {m = zero}  f xs       ys       = refl
-take-distr-zipWith {m = suc m} f (x ∷ xs) (y ∷ ys) = begin
-    take (suc m) (zipWith f (x ∷ xs) (y ∷ ys))
-  ≡⟨⟩
-    take (suc m) (f x y ∷ (zipWith f xs ys))
-  ≡⟨ unfold-take m (f x y) (zipWith f xs ys) ⟩
-    f x y ∷ take m (zipWith f xs ys)
-  ≡⟨ cong (f x y ∷_) (take-distr-zipWith f xs ys) ⟩
-    f x y ∷ (zipWith f (take m xs) (take m ys))
-  ≡⟨⟩
-    zipWith f (x ∷ (take m xs)) (y ∷ (take m ys))
-  ≡˘⟨ cong₂ (zipWith f) (unfold-take m x xs) (unfold-take m y ys) ⟩
-    zipWith f (take (suc m) (x ∷ xs)) (take (suc m) (y ∷ ys))
-  ∎
-
-take-distr-map : ∀ (f : A → B) (m : ℕ) (xs : Vec A (m + n)) →
-                 take m (map f xs) ≡ map f (take m xs)
-take-distr-map f zero xs = refl
-take-distr-map f (suc m) (x ∷ xs) = begin
-  take (suc m) (map f (x ∷ xs)) ≡⟨⟩
-  take (suc m) (f x ∷ map f xs) ≡⟨ unfold-take m (f x) (map f xs) ⟩
-  f x ∷ (take m (map f xs))     ≡⟨ cong (f x ∷_) (take-distr-map f m xs) ⟩
-  f x ∷ (map f (take m xs))     ≡⟨⟩
-  map f (x ∷ take m xs)         ≡˘⟨ cong (map f) (unfold-take m x xs) ⟩
-  map f (take (suc m) (x ∷ xs)) ∎
+unfold-take n x xs = refl
+
+take-zipWith : ∀ (f : A → B → C) →
+               (xs : Vec A (m + n)) (ys : Vec B (m + n)) →
+               take m (zipWith f xs ys) ≡ zipWith f (take m xs) (take m ys)
+take-zipWith {m = zero}  f xs       ys       = refl
+take-zipWith {m = suc m} f (x ∷ xs) (y ∷ ys) = cong (f x y ∷_) (take-zipWith f xs ys)
+
+take-map : ∀ (f : A → B) (m : ℕ) (xs : Vec A (m + n)) →
+           take m (map f xs) ≡ map f (take m xs)
+take-map f zero    xs       = refl
+take-map f (suc m) (x ∷ xs) = cong (f x ∷_) (take-map f m xs)
 
 ------------------------------------------------------------------------
 -- drop
 
 unfold-drop : ∀ n x (xs : Vec A (n + m)) →
               drop (suc n) (x ∷ xs) ≡ drop n xs
-unfold-drop n x xs with splitAt n xs
-... | xs , ys , refl = refl
-
-drop-distr-zipWith : (f : A → B → C) →
-                     (x : Vec A (m + n)) (y : Vec B (m + n)) →
-                     drop m (zipWith f x y) ≡ zipWith f (drop m x) (drop m y)
-drop-distr-zipWith {m = zero} f   xs       ys = refl
-drop-distr-zipWith {m = suc m} f (x ∷ xs) (y ∷ ys) = begin
-    drop (suc m) (zipWith f (x ∷ xs) (y ∷ ys))
-  ≡⟨⟩
-    drop (suc m) (f x y ∷ (zipWith f xs ys))
-  ≡⟨ unfold-drop m (f x y) (zipWith f xs ys) ⟩
-    drop m (zipWith f xs ys)
-  ≡⟨ drop-distr-zipWith f xs ys ⟩
-    zipWith f (drop m xs) (drop m ys)
-  ≡˘⟨ cong₂ (zipWith f) (unfold-drop m x xs) (unfold-drop m y ys) ⟩
-    zipWith f (drop (suc m) (x ∷ xs)) (drop (suc m) (y ∷ ys))
-  ∎
-
-drop-distr-map : ∀ (f : A → B) (m : ℕ) (x : Vec A (m + n)) →
-                 drop m (map f x) ≡ map f (drop m x)
-drop-distr-map f zero x = refl
-drop-distr-map f (suc m) (x ∷ xs) = begin
-  drop (suc m) (map f (x ∷ xs)) ≡⟨⟩
-  drop (suc m) (f x ∷ map f xs) ≡⟨  unfold-drop m (f x) (map f xs) ⟩
-  drop m (map f xs)             ≡⟨  drop-distr-map f m xs ⟩
-  map f (drop m xs)             ≡˘⟨ cong (map f) (unfold-drop m x xs) ⟩
-  map f (drop (suc m) (x ∷ xs)) ∎
+unfold-drop n x xs = refl
+
+drop-zipWith : (f : A → B → C) →
+               (xs : Vec A (m + n)) (ys : Vec B (m + n)) →
+               drop m (zipWith f xs ys) ≡ zipWith f (drop m xs) (drop m ys)
+drop-zipWith {m = zero}  f   xs       ys     = refl
+drop-zipWith {m = suc m} f (x ∷ xs) (y ∷ ys) = drop-zipWith f xs ys
+
+drop-map : ∀ (f : A → B) (m : ℕ) (xs : Vec A (m + n)) →
+           drop m (map f xs) ≡ map f (drop m xs)
+drop-map f zero    xs       = refl
+drop-map f (suc m) (x ∷ xs) = drop-map f m xs
 
 ------------------------------------------------------------------------
 -- take and drop together
 
-take++drop≡id : ∀ (m : ℕ) (x : Vec A (m + n)) → take m x ++ drop m x ≡ x
-take++drop≡id zero x = refl
-take++drop≡id (suc m) (x ∷ xs) = begin
-    take (suc m) (x ∷ xs) ++ drop (suc m) (x ∷ xs)
-  ≡⟨ cong₂ _++_ (unfold-take m x xs) (unfold-drop m x xs) ⟩
-    (x ∷ take m xs) ++ (drop m xs)
-  ≡⟨⟩
-    x ∷ (take m xs ++ drop m xs)
-  ≡⟨ cong (x ∷_) (take++drop≡id m xs) ⟩
-    x ∷ xs
-  ∎
+take++drop≡id : ∀ (m : ℕ) (xs : Vec A (m + n)) → take m xs ++ drop m xs ≡ xs
+take++drop≡id zero    xs       = refl
+take++drop≡id (suc m) (x ∷ xs) = cong (x ∷_) (take++drop≡id m xs)
 
 ------------------------------------------------------------------------
 -- truncate
@@ -216,12 +173,8 @@ lookup⇒[]= (suc i) (_ ∷ xs) p    = there (lookup⇒[]= i xs p)
 
 lookup-inject≤-take : ∀ m (m≤m+n : m ≤ m + n) (i : Fin m) (xs : Vec A (m + n)) →
                       lookup xs (Fin.inject≤ i m≤m+n) ≡ lookup (take m xs) i
-lookup-inject≤-take (suc m) m≤m+n zero (x ∷ xs)
-  rewrite unfold-take m x xs = refl
-lookup-inject≤-take (suc (suc m)) (s≤s m≤m+n) (suc zero) (x ∷ y ∷ xs)
-  rewrite unfold-take (suc m) x (y ∷ xs) | unfold-take m y xs = refl
-lookup-inject≤-take (suc (suc m)) (s≤s (s≤s m≤m+n)) (suc (suc i)) (x ∷ y ∷ xs)
-  rewrite unfold-take (suc m) x (y ∷ xs) | unfold-take m y xs = lookup-inject≤-take m m≤m+n i xs
+lookup-inject≤-take (suc m) m≤m+n zero (x ∷ xs) = refl
+lookup-inject≤-take (suc m) (s≤s m≤m+n) (suc i) (x ∷ xs) = lookup-inject≤-take m m≤m+n i xs
 
 ------------------------------------------------------------------------
 -- updateAt (_[_]%=_)
@@ -1298,3 +1251,24 @@ take-drop-id = take++drop≡id
 "Warning: take-drop-id was deprecated in v2.0.
 Please use take++drop≡id instead."
 #-}
+
+take-distr-zipWith = take-zipWith
+{-# WARNING_ON_USAGE take-distr-zipWith
+"Warning: take-distr-zipWith was deprecated in v2.0.
+Please use take-zipWith instead."
+#-}
+take-distr-map = take-map
+{-# WARNING_ON_USAGE take-distr-map
+"Warning: take-distr-map was deprecated in v2.0.
+Please use take-map instead."
+#-}
+drop-distr-zipWith = drop-zipWith
+{-# WARNING_ON_USAGE drop-distr-zipWith
+"Warning: drop-distr-zipWith was deprecated in v2.0.
+Please use tdrop-zipWith instead."
+#-}
+drop-distr-map = drop-map
+{-# WARNING_ON_USAGE drop-distr-map
+"Warning: drop-distr-map was deprecated in v2.0.
+Please use drop-map instead."
+#-}
diff --git a/src/Data/Vec/Relation/Unary/All/Properties.agda b/src/Data/Vec/Relation/Unary/All/Properties.agda
index 868620ff74..9cfc066635 100644
--- a/src/Data/Vec/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/All/Properties.agda
@@ -148,13 +148,11 @@ module _ {P : A → Set p} where
 
 drop⁺ : ∀ m {xs} → All P {m + n} xs → All P {n} (drop m xs)
 drop⁺ zero pxs = pxs
-drop⁺ (suc m) {x ∷ xs} (px ∷ pxs)
-  rewrite Vecₚ.unfold-drop m x xs = drop⁺ m pxs
+drop⁺ (suc m) {x ∷ xs} (px ∷ pxs) = drop⁺ m pxs
 
 take⁺ : ∀ m {xs} → All P {m + n} xs → All P {m} (take m xs)
 take⁺ zero pxs = []
-take⁺ (suc m) {x ∷ xs} (px ∷ pxs)
-  rewrite Vecₚ.unfold-take m x xs = px ∷ take⁺ m pxs
+take⁺ (suc m) {x ∷ xs} (px ∷ pxs) = px ∷ take⁺ m pxs
 
 ------------------------------------------------------------------------
 -- toList
diff --git a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
index 2158246fbc..ffc0ec6c4c 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
@@ -70,13 +70,11 @@ module _ {R : Rel A ℓ} where
 
   take⁺ : ∀ {n} m {xs} → AllPairs R {m + n} xs → AllPairs R {m} (take m xs)
   take⁺ zero pxs = []
-  take⁺ (suc m) {x ∷ xs} (px ∷ pxs)
-    rewrite Vecₚ.unfold-take m x xs = Allₚ.take⁺ m px ∷ take⁺ m pxs
+  take⁺ (suc m) {x ∷ xs} (px ∷ pxs) = Allₚ.take⁺ m px ∷ take⁺ m pxs
 
   drop⁺ : ∀ {n} m {xs} → AllPairs R {m + n} xs → AllPairs R {n} (drop m xs)
   drop⁺ zero pxs = pxs
-  drop⁺ (suc m) {x ∷ xs} (_ ∷ pxs)
-    rewrite Vecₚ.unfold-drop m x xs = drop⁺ m pxs
+  drop⁺ (suc m) (_ ∷ pxs) = drop⁺ m pxs
 
 ------------------------------------------------------------------------
 -- tabulate

From 1baeb09a00e0b4593a6cda7b738945dd79d3761a Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 18 Sep 2023 01:58:38 +0100
Subject: [PATCH 120/316] Add a recursive view of `Fin n` (#1923)

---
 CHANGELOG.md                            |   7 ++
 README/Data/Fin/Relation/Unary/Top.agda | 100 ++++++++++++++++++++++++
 src/Data/Fin/Properties.agda            |   3 +
 src/Data/Fin/Relation/Unary/Top.agda    |  79 +++++++++++++++++++
 4 files changed, 189 insertions(+)
 create mode 100644 README/Data/Fin/Relation/Unary/Top.agda
 create mode 100644 src/Data/Fin/Relation/Unary/Top.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 420d85fddc..b1e8ab2fa6 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1585,6 +1585,11 @@ New modules
   Data.Default
   ```
 
+* A small library defining a structurally recursive view of `Fin n`:
+  ```
+  Data.Fin.Relation.Unary.Top
+  ```
+
 * A small library for a non-empty fresh list:
   ```
   Data.List.Fresh.NonEmpty
@@ -2146,6 +2151,8 @@ Additions to existing modules
   cast-is-id    : cast eq k ≡ k
   subst-is-cast : subst Fin eq k ≡ cast eq k
   cast-trans    : cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k
+
+  fromℕ≢inject₁      : {i : Fin n} → fromℕ n ≢ inject₁ i
   ```
 
 * Added new functions in `Data.Integer.Base`:
diff --git a/README/Data/Fin/Relation/Unary/Top.agda b/README/Data/Fin/Relation/Unary/Top.agda
new file mode 100644
index 0000000000..e602932827
--- /dev/null
+++ b/README/Data/Fin/Relation/Unary/Top.agda
@@ -0,0 +1,100 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Example use of the 'top' view of Fin
+--
+-- This is an example of a view of (elements of) a datatype,
+-- here i : Fin (suc n), which exhibits every such i as
+-- * either, i = fromℕ n
+-- * or, i = inject₁ j for a unique j : Fin n
+--
+-- Using this view, we can redefine certain operations in `Data.Fin.Base`,
+-- together with their corresponding properties in `Data.Fin.Properties`.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module README.Data.Fin.Relation.Unary.Top where
+
+open import Data.Nat.Base using (ℕ; zero; suc; _∸_; _≤_)
+open import Data.Nat.Properties using (n∸n≡0; +-∸-assoc; ≤-reflexive)
+open import Data.Fin.Base using (Fin; zero; suc; toℕ; fromℕ; inject₁; _>_)
+open import Data.Fin.Properties using (toℕ-fromℕ; toℕ<n; toℕ-inject₁)
+open import Data.Fin.Induction hiding (>-weakInduction)
+open import Data.Fin.Relation.Unary.Top
+import Induction.WellFounded as WF
+open import Level using (Level)
+open import Relation.Binary.PropositionalEquality
+open import Relation.Unary using (Pred)
+
+private
+  variable
+    ℓ : Level
+    n : ℕ
+
+------------------------------------------------------------------------
+-- Reimplementation of `Data.Fin.Base.opposite`, and its properties
+
+-- Definition
+
+opposite : Fin n → Fin n
+opposite {suc n} i with view i
+... | ‵fromℕ     = zero
+... | ‵inject₁ j = suc (opposite {n} j)
+
+-- Properties
+
+opposite-zero≡fromℕ : ∀ n → opposite {suc n} zero ≡ fromℕ n
+opposite-zero≡fromℕ zero    = refl
+opposite-zero≡fromℕ (suc n) = cong suc (opposite-zero≡fromℕ n)
+
+opposite-fromℕ≡zero : ∀ n → opposite {suc n} (fromℕ n) ≡ zero
+opposite-fromℕ≡zero n rewrite view-fromℕ n = refl
+
+opposite-suc≡inject₁-opposite : (j : Fin n) →
+                                opposite (suc j) ≡ inject₁ (opposite j)
+opposite-suc≡inject₁-opposite {suc n} i with view i
+... | ‵fromℕ     = refl
+... | ‵inject₁ j = cong suc (opposite-suc≡inject₁-opposite {n} j)
+
+opposite-involutive : (j : Fin n) → opposite (opposite j) ≡ j
+opposite-involutive {suc n} zero
+  rewrite opposite-zero≡fromℕ n
+        | view-fromℕ n              = refl
+opposite-involutive {suc n} (suc i)
+  rewrite opposite-suc≡inject₁-opposite i
+        | view-inject₁ (opposite i) = cong suc (opposite-involutive i)
+
+opposite-suc : (j : Fin n) → toℕ (opposite (suc j)) ≡ toℕ (opposite j)
+opposite-suc j = begin
+  toℕ (opposite (suc j))      ≡⟨ cong toℕ (opposite-suc≡inject₁-opposite j) ⟩
+  toℕ (inject₁ (opposite j))  ≡⟨ toℕ-inject₁ (opposite j) ⟩
+  toℕ (opposite j)            ∎ where open ≡-Reasoning
+
+opposite-prop : (j : Fin n) → toℕ (opposite j) ≡ n ∸ suc (toℕ j)
+opposite-prop {suc n} i with view i
+... | ‵fromℕ  rewrite toℕ-fromℕ n | n∸n≡0 n = refl
+... | ‵inject₁ j = begin
+  suc (toℕ (opposite j)) ≡⟨ cong suc (opposite-prop j) ⟩
+  suc (n ∸ suc (toℕ j))  ≡˘⟨ +-∸-assoc 1 (toℕ<n j) ⟩
+  n ∸ toℕ j              ≡˘⟨ cong (n ∸_) (toℕ-inject₁ j) ⟩
+  n ∸ toℕ (inject₁ j)    ∎ where open ≡-Reasoning
+
+------------------------------------------------------------------------
+-- Reimplementation of `Data.Fin.Induction.>-weakInduction`
+
+open WF using (Acc; acc)
+
+>-weakInduction : (P : Pred (Fin (suc n)) ℓ) →
+                  P (fromℕ n) →
+                  (∀ i → P (suc i) → P (inject₁ i)) →
+                  ∀ i → P i
+>-weakInduction P Pₙ Pᵢ₊₁⇒Pᵢ i = induct (>-wellFounded i)
+  where
+  induct : ∀ {i} → Acc _>_ i → P i
+  induct {i} (acc rec) with view i
+  ... | ‵fromℕ = Pₙ
+  ... | ‵inject₁ j = Pᵢ₊₁⇒Pᵢ j (induct (rec _ inject₁[j]+1≤[j+1]))
+    where
+    inject₁[j]+1≤[j+1] : suc (toℕ (inject₁ j)) ≤ toℕ (suc j)
+    inject₁[j]+1≤[j+1] = ≤-reflexive (toℕ-inject₁ (suc j))
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 0f520afda5..bcb1cd4745 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -449,6 +449,9 @@ toℕ-inject {i = suc i} (suc j) = cong suc (toℕ-inject j)
 -- inject₁
 ------------------------------------------------------------------------
 
+fromℕ≢inject₁ : fromℕ n ≢ inject₁ i
+fromℕ≢inject₁ {i = suc i} eq = fromℕ≢inject₁ {i = i} (suc-injective eq)
+
 inject₁-injective : inject₁ i ≡ inject₁ j → i ≡ j
 inject₁-injective {i = zero}  {zero}  i≡j = refl
 inject₁-injective {i = suc i} {suc j} i≡j =
diff --git a/src/Data/Fin/Relation/Unary/Top.agda b/src/Data/Fin/Relation/Unary/Top.agda
new file mode 100644
index 0000000000..b6a567a9a8
--- /dev/null
+++ b/src/Data/Fin/Relation/Unary/Top.agda
@@ -0,0 +1,79 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The 'top' view of Fin.
+--
+-- This is an example of a view of (elements of) a datatype,
+-- here i : Fin (suc n), which exhibits every such i as
+-- * either, i = fromℕ n
+-- * or, i = inject₁ j for a unique j : Fin n
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Fin.Relation.Unary.Top where
+
+open import Data.Nat.Base using (ℕ; zero; suc)
+open import Data.Fin.Base using (Fin; zero; suc; fromℕ; inject₁)
+open import Relation.Binary.PropositionalEquality.Core
+
+private
+  variable
+    n : ℕ
+    i : Fin n
+
+------------------------------------------------------------------------
+-- The View, considered as a unary relation on Fin n
+
+-- NB `Data.Fin.Properties.fromℕ≢inject₁` establishes that the following
+-- inductively defined family on `Fin n` has constructors which target
+-- *disjoint* instances of the family (moreover at indices `n = suc _`);
+-- hence the interpretations of the View constructors will also be disjoint.
+
+data View : (i : Fin n) → Set where
+  ‵fromℕ : View (fromℕ n)
+  ‵inj₁ : View i → View (inject₁ i)
+
+pattern ‵inject₁ i = ‵inj₁ {i = i} _
+
+-- The view covering function, witnessing soundness of the view
+
+view : (i : Fin n) → View i
+view zero = view-zero where
+  view-zero : View (zero {n})
+  view-zero {n = zero}  = ‵fromℕ
+  view-zero {n = suc _} = ‵inj₁ view-zero
+view (suc i) with view i
+... | ‵fromℕ     = ‵fromℕ
+... | ‵inject₁ i = ‵inj₁ (view (suc i))
+
+-- Interpretation of the view constructors
+
+⟦_⟧ : {i : Fin n} → View i → Fin n
+⟦ ‵fromℕ ⟧     = fromℕ _
+⟦ ‵inject₁ i ⟧ = inject₁ i
+
+-- Completeness of the view
+
+view-complete : (v : View i) → ⟦ v ⟧ ≡ i
+view-complete ‵fromℕ    = refl
+view-complete (‵inj₁ _) = refl
+
+-- 'Computational' behaviour of the covering function
+
+view-fromℕ : ∀ n → view (fromℕ n) ≡ ‵fromℕ
+view-fromℕ zero                         = refl
+view-fromℕ (suc n) rewrite view-fromℕ n = refl
+
+view-inject₁ : (i : Fin n) → view (inject₁ i) ≡ ‵inj₁ (view i)
+view-inject₁ zero                           = refl
+view-inject₁ (suc i) rewrite view-inject₁ i = refl
+
+-- Uniqueness of the view
+
+view-unique : (v : View i) → view i ≡ v
+view-unique ‵fromℕ            = view-fromℕ _
+view-unique (‵inj₁ {i = i} v) = begin
+  view (inject₁ i) ≡⟨ view-inject₁ i ⟩
+  ‵inj₁ (view i)   ≡⟨ cong ‵inj₁ (view-unique v) ⟩
+  ‵inj₁ v          ∎ where open ≡-Reasoning

From 85431162d429065e836d64320c8ef418da53d7f2 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Tue, 26 Sep 2023 09:20:17 +0900
Subject: [PATCH 121/316] Use new style `Function` hierarchy everywhere.
 (#1895)

---
 CHANGELOG.md                                  |  14 +-
 src/Codata/Musical/Colist.agda                |  16 +-
 src/Codata/Musical/Colist/Infinite-merge.agda | 197 ++++---
 .../Colist/Relation/Unary/Any/Properties.agda |  40 +-
 src/Data/Bool/Properties.agda                 |  32 +-
 src/Data/Container/Combinator/Properties.agda |  16 +-
 src/Data/Container/Core.agda                  |   5 +-
 src/Data/Container/Indexed.agda               |   7 +-
 src/Data/Container/Indexed/Combinator.agda    |  48 +-
 src/Data/Container/Related.agda               |  10 +-
 .../Relation/Unary/Any/Properties.agda        |  50 +-
 src/Data/Fin/Permutation.agda                 |   5 +-
 src/Data/Fin/Properties.agda                  |  14 +-
 .../Membership/Propositional/Properties.agda  |  28 +-
 .../Propositional/Properties/Core.agda        |   4 +-
 .../List/Membership/Setoid/Properties.agda    |   2 +-
 .../Relation/Binary/BagAndSetEquality.agda    | 309 ++++++-----
 .../Permutation/Propositional/Properties.agda |   1 -
 .../Subset/Propositional/Properties.agda      |  32 +-
 .../List/Relation/Unary/All/Properties.agda   |  17 +-
 .../List/Relation/Unary/Any/Properties.agda   | 115 ++--
 src/Data/Product/Algebra.agda                 |  24 +-
 .../Function/Dependent/Propositional.agda     | 504 +++++++++---------
 .../Dependent/Propositional/WithK.agda        |  51 +-
 .../Product/Function/Dependent/Setoid.agda    | 371 +++++++------
 .../Function/Dependent/Setoid/WithK.agda      |  67 +--
 .../Function/NonDependent/Propositional.agda  | 121 +++--
 .../Product/Function/NonDependent/Setoid.agda | 232 ++++----
 src/Data/Product/Properties.agda              |   8 +-
 .../Binary/Pointwise/Dependent/WithK.agda     |  39 +-
 .../Binary/Pointwise/NonDependent.agda        | 387 ++++++--------
 src/Data/Sum/Algebra.agda                     |  12 +-
 src/Data/Sum/Function/Propositional.agda      | 125 ++---
 src/Data/Sum/Function/Setoid.agda             | 282 +++++-----
 src/Data/Sum/Properties.agda                  |   4 +-
 src/Data/Sum/Relation/Binary/Pointwise.agda   | 366 ++++++-------
 src/Data/Vec/N-ary.agda                       |   6 +-
 src/Data/Vec/Properties.agda                  |   6 +-
 src/Data/Vec/Recursive.agda                   |   4 +-
 src/Data/Vec/Recursive/Properties.agda        |   4 +-
 .../Vec/Relation/Unary/All/Properties.agda    |   4 +-
 .../Vec/Relation/Unary/Any/Properties.agda    |  30 +-
 src/Function/Bijection.agda                   |  17 +-
 src/Function/Bundles.agda                     |  64 ++-
 src/Function/Dependent/Bundles.agda           |  50 ++
 src/Function/Equality.agda                    |  21 +-
 src/Function/Equivalence.agda                 |  17 +-
 src/Function/HalfAdjointEquivalence.agda      |   8 +-
 src/Function/Indexed/Bundles.agda             |  22 +-
 src/Function/Injection.agda                   |  13 +-
 src/Function/Inverse.agda                     |  17 +-
 src/Function/LeftInverse.agda                 |  17 +-
 src/Function/Properties.agda                  |   8 +-
 src/Function/Properties/Equivalence.agda      |  42 +-
 src/Function/Properties/Injection.agda        |  52 ++
 src/Function/Properties/Inverse.agda          |  88 ++-
 .../Inverse/HalfAdjointEquivalence.agda       | 121 +++++
 src/Function/Properties/RightInverse.agda     |  56 +-
 src/Function/Properties/Surjection.agda       |  41 +-
 src/Function/Related.agda                     |  29 +-
 src/Function/Related/Propositional.agda       |  57 +-
 src/Function/Related/TypeIsomorphisms.agda    | 236 ++++----
 .../Related/TypeIsomorphisms/Solver.agda      |   9 +-
 src/Function/Surjection.agda                  |  17 +-
 .../Binary/HeterogeneousEquality.agda         |  14 +-
 .../Binary/PropositionalEquality.agda         |  17 +-
 src/Relation/Nullary/Decidable.agda           |   6 +-
 src/Relation/Nullary/Universe.agda            |   4 +-
 68 files changed, 2466 insertions(+), 2186 deletions(-)
 create mode 100644 src/Function/Dependent/Bundles.agda
 create mode 100644 src/Function/Properties/Injection.agda
 create mode 100644 src/Function/Properties/Inverse/HalfAdjointEquivalence.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b1e8ab2fa6..cdbdb88ef1 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -274,8 +274,14 @@ Non-backwards compatible changes
 
 #### Removal of the old function hierarchy
 
-* The switch to the new function hierarchy is complete and the following definitions
-  now use the new definitions instead of the old ones:
+* The switch to the new function hierarchy is complete and the following modules
+  have been completely switched over to use the new definitions:
+  ```
+  Data.Sum.Function.Setoid
+  Data.Sum.Function.Propositional
+  ```
+  
+* Additionally the following proofs now use the new definitions instead of the old ones:
   * `Algebra.Lattice.Properties.BooleanAlgebra`
   * `Algebra.Properties.CommutativeMonoid.Sum`
   * `Algebra.Properties.Lattice`
@@ -1080,6 +1086,10 @@ Deprecated modules
 
 * The module `Data.Nat.Properties.Core` has been deprecated, and its one entry moved to `Data.Nat.Properties`
 
+### Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`
+
+* This module has been deprecated, as none of its contents actually depended on axiom K. The contents has been moved to `Data.Product.Function.Dependent.Setoid`.
+
 Deprecated names
 ----------------
 
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index fdfbfc00b1..21dde992bc 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -25,8 +25,7 @@ open import Data.Product.Base as Prod using (∃; _×_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Vec.Bounded as Vec≤ using (Vec≤)
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Inverse as Inv using (_↔_; _↔̇_; Inverse; inverse)
+open import Function.Bundles
 open import Level using (_⊔_)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Bundles using (Poset; Setoid; Preorder)
@@ -100,14 +99,11 @@ data _⊑_ {A : Set a} : Rel (Colist A) a where
 -- Any can be expressed using _∈_ (and vice versa).
 
 Any-∈ : ∀ {xs} → Any P xs ↔ ∃ λ x → x ∈ xs × P x
-Any-∈ {P = P} = record
-  { to         = P.→-to-⟶ to
-  ; from       = P.→-to-⟶ (λ { (x , x∈xs , p) → from x∈xs p })
-  ; inverse-of = record
-    { left-inverse-of  = from∘to
-    ; right-inverse-of = λ { (x , x∈xs , p) → to∘from x∈xs p }
-    }
-  }
+Any-∈ {P = P} = mk↔ₛ′
+  to
+  (λ { (x , x∈xs , p) → from x∈xs p })
+  (λ { (x , x∈xs , p) → to∘from x∈xs p })
+  from∘to
   where
   to : ∀ {xs} → Any P xs → ∃ λ x → x ∈ xs × P x
   to (here  p) = _ , here P.refl , p
diff --git a/src/Codata/Musical/Colist/Infinite-merge.agda b/src/Codata/Musical/Colist/Infinite-merge.agda
index eb06c3b9cc..769c0ccc94 100644
--- a/src/Codata/Musical/Colist/Infinite-merge.agda
+++ b/src/Codata/Musical/Colist/Infinite-merge.agda
@@ -18,14 +18,22 @@ open import Data.Sum.Base
 open import Data.Sum.Properties
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Inverse as Inv using (_↔_; Inverse; inverse)
-import Function.Related as Related
+open import Function.Bundles
+open import Function.Properties.Inverse using (↔-refl; ↔-sym; ↔⇒↣)
+import Function.Related.Propositional as Related
 open import Function.Related.TypeIsomorphisms
+open import Level
+open import Relation.Unary using (Pred)
 import Induction.WellFounded as WF
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.Construct.On as On
 
+private
+  variable
+    a p : Level
+    A : Set a
+    P : Pred A p
+
 ------------------------------------------------------------------------
 -- Some code that is used to work around Agda's syntactic guardedness
 -- checker.
@@ -34,74 +42,70 @@ private
 
   infixr 5 _∷_ _⋎_
 
-  data ColistP {a} (A : Set a) : Set a where
+  data ColistP (A : Set a) : Set a where
     []  : ColistP A
     _∷_ : A → ∞ (ColistP A) → ColistP A
     _⋎_ : ColistP A → ColistP A → ColistP A
 
-  data ColistW {a} (A : Set a) : Set a where
+  data ColistW (A : Set a) : Set a where
     []  : ColistW A
     _∷_ : A → ColistP A → ColistW A
 
-  program : ∀ {a} {A : Set a} → Colist A → ColistP A
+  program : Colist A → ColistP A
   program []       = []
   program (x ∷ xs) = x ∷ ♯ program (♭ xs)
 
   mutual
 
-    _⋎W_ : ∀ {a} {A : Set a} → ColistW A → ColistP A → ColistW A
+    _⋎W_ : ColistW A → ColistP A → ColistW A
     []       ⋎W ys = whnf ys
     (x ∷ xs) ⋎W ys = x ∷ (ys ⋎ xs)
 
-    whnf : ∀ {a} {A : Set a} → ColistP A → ColistW A
+    whnf : ColistP A → ColistW A
     whnf []        = []
     whnf (x ∷ xs)  = x ∷ ♭ xs
     whnf (xs ⋎ ys) = whnf xs ⋎W ys
 
   mutual
 
-    ⟦_⟧P : ∀ {a} {A : Set a} → ColistP A → Colist A
+    ⟦_⟧P : ColistP A → Colist A
     ⟦ xs ⟧P = ⟦ whnf xs ⟧W
 
-    ⟦_⟧W : ∀ {a} {A : Set a} → ColistW A → Colist A
+    ⟦_⟧W : ColistW A → Colist A
     ⟦ [] ⟧W     = []
     ⟦ x ∷ xs ⟧W = x ∷ ♯ ⟦ xs ⟧P
 
   mutual
 
-    ⋎-homP : ∀ {a} {A : Set a} (xs : ColistP A) {ys} →
-             ⟦ xs ⋎ ys ⟧P ≈ ⟦ xs ⟧P Colist.⋎ ⟦ ys ⟧P
+    ⋎-homP : ∀ (xs : ColistP A) {ys} → ⟦ xs ⋎ ys ⟧P ≈ ⟦ xs ⟧P Colist.⋎ ⟦ ys ⟧P
     ⋎-homP xs = ⋎-homW (whnf xs) _
 
-    ⋎-homW : ∀ {a} {A : Set a} (xs : ColistW A) ys →
-             ⟦ xs ⋎W ys ⟧W ≈ ⟦ xs ⟧W Colist.⋎ ⟦ ys ⟧P
+    ⋎-homW : ∀ (xs : ColistW A) ys → ⟦ xs ⋎W ys ⟧W ≈ ⟦ xs ⟧W Colist.⋎ ⟦ ys ⟧P
     ⋎-homW (x ∷ xs) ys = x ∷ ♯ ⋎-homP ys
     ⋎-homW []       ys = begin ⟦ ys ⟧P ∎
       where open ≈-Reasoning
 
-  ⟦program⟧P : ∀ {a} {A : Set a} (xs : Colist A) →
-               ⟦ program xs ⟧P ≈ xs
+  ⟦program⟧P : ∀ (xs : Colist A) → ⟦ program xs ⟧P ≈ xs
   ⟦program⟧P []       = []
   ⟦program⟧P (x ∷ xs) = x ∷ ♯ ⟦program⟧P (♭ xs)
 
-  Any-⋎P : ∀ {a p} {A : Set a} {P : A → Set p} xs {ys} →
+  Any-⋎P : ∀ xs {ys} →
            Any P ⟦ program xs ⋎ ys ⟧P ↔ (Any P xs ⊎ Any P ⟦ ys ⟧P)
   Any-⋎P {P = P} xs {ys} =
-    Any P ⟦ program xs ⋎ ys ⟧P                ↔⟨ Any-cong Inv.id (⋎-homP (program xs)) ⟩
+    Any P ⟦ program xs ⋎ ys ⟧P                ↔⟨ Any-cong ↔-refl (⋎-homP (program xs)) ⟩
     Any P (⟦ program xs ⟧P Colist.⋎ ⟦ ys ⟧P)  ↔⟨ Any-⋎ _ ⟩
-    (Any P ⟦ program xs ⟧P ⊎ Any P ⟦ ys ⟧P)   ↔⟨ Any-cong Inv.id (⟦program⟧P _) ⊎-cong (_ ∎) ⟩
+    (Any P ⟦ program xs ⟧P ⊎ Any P ⟦ ys ⟧P)   ↔⟨ Any-cong ↔-refl (⟦program⟧P _) ⊎-cong (_ ∎) ⟩
     (Any P xs ⊎ Any P ⟦ ys ⟧P)                ∎
     where open Related.EquationalReasoning
 
   index-Any-⋎P :
-    ∀ {a p} {A : Set a} {P : A → Set p} xs {ys}
-    (p : Any P ⟦ program xs ⋎ ys ⟧P) →
-    index p ≥′ [ index , index ]′ (Inverse.to (Any-⋎P xs) ⟨$⟩ p)
+    ∀ xs {ys} (p : Any P ⟦ program xs ⋎ ys ⟧P) →
+    index p ≥′ [ index , index ]′ (Inverse.to (Any-⋎P xs) p)
   index-Any-⋎P xs p
     with       Any-resp      id  (⋎-homW (whnf (program xs)) _) p
        | index-Any-resp {f = id} (⋎-homW (whnf (program xs)) _) p
   index-Any-⋎P xs p | q | q≡p
-    with Inverse.to (Any-⋎ ⟦ program xs ⟧P) ⟨$⟩ q
+    with Inverse.to (Any-⋎ ⟦ program xs ⟧P)  q
        |       index-Any-⋎ ⟦ program xs ⟧P      q
   index-Any-⋎P xs p | q | q≡p | inj₂ r | r≤q rewrite q≡p = r≤q
   index-Any-⋎P xs p | q | q≡p | inj₁ r | r≤q
@@ -115,95 +119,90 @@ private
 
 private
 
-  merge′ : ∀ {a} {A : Set a} → Colist (A × Colist A) → ColistP A
+  merge′ : Colist (A × Colist A) → ColistP A
   merge′ []               = []
   merge′ ((x , xs) ∷ xss) = x ∷ ♯ (program xs ⋎ merge′ (♭ xss))
 
-merge : ∀ {a} {A : Set a} → Colist (A × Colist A) → Colist A
+merge : Colist (A × Colist A) → Colist A
 merge xss = ⟦ merge′ xss ⟧P
 
 ------------------------------------------------------------------------
 -- Any lemma for merge.
 
-module _ {a p} {A : Set a} {P : A → Set p} where
-
-  Any-merge : ∀ xss → Any P (merge xss) ↔ Any (λ { (x , xs) → P x ⊎ Any P xs }) xss
-  Any-merge xss = inverse (proj₁ ∘ to xss) from (proj₂ ∘ to xss) to∘from
+Any-merge : ∀ xss → Any P (merge xss) ↔ Any (λ { (x , xs) → P x ⊎ Any P xs }) xss
+Any-merge {P = P} xss = mk↔ₛ′ (proj₁ ∘ to xss) from to∘from (proj₂ ∘ to xss)
+  where
+  open P.≡-Reasoning
+
+  -- The from function.
+
+  Q = λ { (x , xs) → P x ⊎ Any P xs }
+
+  from : ∀ {xss} → Any Q xss → Any P (merge xss)
+  from (here (inj₁ p))        = here p
+  from (here (inj₂ p))        = there (Inverse.from (Any-⋎P _)  (inj₁ p))
+  from (there {x = _ , xs} p) = there (Inverse.from (Any-⋎P xs) (inj₂ (from p)))
+
+  -- The from function is injective.
+
+  from-injective : ∀ {xss} (p₁ p₂ : Any Q xss) →
+                   from p₁ ≡ from p₂ → p₁ ≡ p₂
+  from-injective (here (inj₁ p))  (here (inj₁ .p)) P.refl = P.refl
+  from-injective (here (inj₂ p₁)) (here (inj₂ p₂)) eq     =
+    P.cong (here ∘ inj₂) $
+    inj₁-injective $
+    Injection.injective (↔⇒↣ (↔-sym (Any-⋎P _))) $
+    there-injective eq
+  from-injective (here (inj₂ p₁)) (there p₂) eq with
+    Injection.injective (↔⇒↣ (↔-sym (Any-⋎P _)))
+                        {x = inj₁ p₁} {y = inj₂ (from p₂)}
+                        (there-injective eq)
+  ... | ()
+  from-injective (there p₁) (here (inj₂ p₂)) eq with
+    Injection.injective (↔⇒↣ (↔-sym (Any-⋎P _)))
+                           {x = inj₂ (from p₁)} {y = inj₁ p₂}
+                           (there-injective eq)
+  ... | ()
+  from-injective (there {x = _ , xs} p₁) (there p₂) eq =
+    P.cong there $
+    from-injective p₁ p₂ $
+    inj₂-injective $
+    Injection.injective (↔⇒↣ (↔-sym (Any-⋎P xs))) $
+    there-injective eq
+  -- The to function (defined as a right inverse of from).
+
+  Input = ∃ λ xss → Any P (merge xss)
+
+  InputPred : Input → Set _
+  InputPred (xss , p) = ∃ λ (q : Any Q xss) → from q ≡ p
+
+  to : ∀ xss p → InputPred (xss , p)
+  to xss p =
+    WF.All.wfRec (On.wellFounded size <′-wellFounded) _
+                 InputPred step (xss , p)
     where
-    open P.≡-Reasoning
-
-    -- The from function.
-
-    Q = λ { (x , xs) → P x ⊎ Any P xs }
-
-    from : ∀ {xss} → Any Q xss → Any P (merge xss)
-    from (here (inj₁ p))        = here p
-    from (here (inj₂ p))        = there (Inverse.from (Any-⋎P _)  ⟨$⟩ inj₁ p)
-    from (there {x = _ , xs} p) = there (Inverse.from (Any-⋎P xs) ⟨$⟩ inj₂ (from p))
-
-    -- The from function is injective.
-
-    from-injective : ∀ {xss} (p₁ p₂ : Any Q xss) →
-                     from p₁ ≡ from p₂ → p₁ ≡ p₂
-    from-injective (here (inj₁ p))  (here (inj₁ .p)) P.refl = P.refl
-    from-injective (here (inj₂ p₁)) (here (inj₂ p₂)) eq     =
-      P.cong (here ∘ inj₂) $
-      inj₁-injective $
-      Inverse.injective (Inv.sym (Any-⋎P _)) {x = inj₁ p₁} {y = inj₁ p₂} $
-      there-injective eq
-    from-injective (here (inj₂ p₁)) (there p₂) eq
-      with Inverse.injective (Inv.sym (Any-⋎P _))
-                             {x = inj₁ p₁} {y = inj₂ (from p₂)}
-                             (there-injective eq)
-    ... | ()
-    from-injective (there p₁) (here (inj₂ p₂)) eq
-      with Inverse.injective (Inv.sym (Any-⋎P _))
-                             {x = inj₂ (from p₁)} {y = inj₁ p₂}
-                             (there-injective eq)
-    ... | ()
-    from-injective (there {x = _ , xs} p₁) (there p₂) eq =
-      P.cong there $
-      from-injective p₁ p₂ $
-      inj₂-injective $
-      Inverse.injective (Inv.sym (Any-⋎P xs))
-                        {x = inj₂ (from p₁)} {y = inj₂ (from p₂)} $
-      there-injective eq
-
-    -- The to function (defined as a right inverse of from).
-
-    Input = ∃ λ xss → Any P (merge xss)
-
-    Pred : Input → Set _
-    Pred (xss , p) = ∃ λ (q : Any Q xss) → from q ≡ p
-
-    to : ∀ xss p → Pred (xss , p)
-    to = λ xss p →
-      WF.All.wfRec (On.wellFounded size <′-wellFounded) _
-                   Pred step (xss , p)
-      where
-      size : Input → ℕ
-      size (_ , p) = index p
-
-      step : ∀ p → WF.WfRec (_<′_ on size) Pred p → Pred p
-      step ([]             , ())      rec
-      step ((x , xs) ∷ xss , here  p) rec = here (inj₁ p) , P.refl
-      step ((x , xs) ∷ xss , there p) rec
-        with Inverse.to (Any-⋎P xs) ⟨$⟩ p
-           | Inverse.left-inverse-of (Any-⋎P xs) p
-           | index-Any-⋎P xs p
-      ... | inj₁ q | P.refl | _   = here (inj₂ q) , P.refl
-      ... | inj₂ q | P.refl | q≤p =
-        Prod.map there
-                 (P.cong (there ∘ _⟨$⟩_ (Inverse.from (Any-⋎P xs)) ∘ inj₂))
-                 (rec (♭ xss , q) (s≤′s q≤p))
-
-    to∘from = λ p → from-injective _ _ (proj₂ (to xss (from p)))
+    size : Input → ℕ
+    size (_ , p) = index p
+
+    step : ∀ p → WF.WfRec (_<′_ on size) InputPred p → InputPred p
+    step ([]             , ())      rec
+    step ((x , xs) ∷ xss , here  p) rec = here (inj₁ p) , P.refl
+    step ((x , xs) ∷ xss , there p) rec
+      with Inverse.to (Any-⋎P xs) p
+         | Inverse.strictlyInverseʳ (Any-⋎P xs) p
+         | index-Any-⋎P xs p
+    ... | inj₁ q | P.refl | _   = here (inj₂ q) , P.refl
+    ... | inj₂ q | P.refl | q≤p =
+      Prod.map there
+               (P.cong (there ∘ (Inverse.from (Any-⋎P xs)) ∘ inj₂))
+               (rec (♭ xss , q) (s≤′s q≤p))
+
+  to∘from = λ p → from-injective _ _ (proj₂ (to xss (from p)))
 
 -- Every member of xss is a member of merge xss, and vice versa (with
 -- equal multiplicities).
 
-∈-merge : ∀ {a} {A : Set a} {y : A} xss →
-          y ∈ merge xss ↔ ∃₂ λ x xs → (x , xs) ∈ xss × (y ≡ x ⊎ y ∈ xs)
+∈-merge : ∀ {y : A} xss → y ∈ merge xss ↔ ∃₂ λ x xs → (x , xs) ∈ xss × (y ≡ x ⊎ y ∈ xs)
 ∈-merge {y = y} xss =
   y ∈ merge xss                                           ↔⟨ Any-merge _ ⟩
   Any (λ { (x , xs) → y ≡ x ⊎ y ∈ xs }) xss               ↔⟨ Any-∈ ⟩
diff --git a/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda b/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
index 874a338a85..f7173c7e91 100644
--- a/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
+++ b/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
@@ -18,8 +18,7 @@ open import Data.Nat.Base using (suc; _≥′_; ≤′-refl; ≤′-step)
 open import Data.Nat.Properties using (s≤′s)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′; [_,_])
 open import Function.Base using (_∋_; _∘_)
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Inverse as Inv using (_↔_; _↔̇_; Inverse; inverse)
+open import Function.Bundles
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality as P
@@ -53,35 +52,29 @@ Any-resp f (x ∷ xs≈) (there p) = there (Any-resp f (♭ xs≈) p)
 -- isomorphic types.
 
 Any-cong : ∀ {a p q} {A : Set a} {P : A → Set p} {Q : A → Set q}
-           {xs ys} → P ↔̇ Q → xs ≈ ys → Any P xs ↔ Any Q ys
-Any-cong {A = A} {P} {Q} {xs} {ys} P↔Q = λ xs≈ys → record
-  { to         = P.→-to-⟶ (to xs≈ys)
-  ; from       = P.→-to-⟶ (from xs≈ys)
-  ; inverse-of = record
-    { left-inverse-of  = from∘to _
-    ; right-inverse-of = to∘from _
-    }
-  }
+           {xs ys} → (∀ {i} → P i ↔ Q i) → xs ≈ ys → Any P xs ↔ Any Q ys
+Any-cong {A = A} {P} {Q} {xs} {ys} P↔Q xs≈ys =
+  mk↔ₛ′ (to xs≈ys) (from xs≈ys) (to∘from _) (from∘to _)
   where
   open Setoid (setoid _) using (sym)
 
   to : ∀ {xs ys} → xs ≈ ys → Any P xs → Any Q ys
-  to xs≈ys = Any-resp (Inverse.to P↔Q ⟨$⟩_) xs≈ys
+  to xs≈ys = Any-resp (Inverse.to P↔Q) xs≈ys
 
   from : ∀ {xs ys} → xs ≈ ys → Any Q ys → Any P xs
-  from xs≈ys = Any-resp (Inverse.from P↔Q ⟨$⟩_) (sym xs≈ys)
+  from xs≈ys = Any-resp (Inverse.from P↔Q) (sym xs≈ys)
 
   to∘from : ∀ {xs ys} (xs≈ys : xs ≈ ys) (q : Any Q ys) →
             to xs≈ys (from xs≈ys q) ≡ q
   to∘from (x ∷ xs≈) (there q) = P.cong there (to∘from (♭ xs≈) q)
   to∘from (x ∷ xs≈) (here qx) =
-    P.cong here (Inverse.right-inverse-of P↔Q qx)
+    P.cong here (Inverse.strictlyInverseˡ P↔Q qx)
 
   from∘to : ∀ {xs ys} (xs≈ys : xs ≈ ys) (p : Any P xs) →
             from xs≈ys (to xs≈ys p) ≡ p
   from∘to (x ∷ xs≈) (there p) = P.cong there (from∘to (♭ xs≈) p)
   from∘to (x ∷ xs≈) (here px) =
-    P.cong here (Inverse.left-inverse-of P↔Q px)
+    P.cong here (Inverse.strictlyInverseʳ P↔Q px)
 
 ------------------------------------------------------------------------
 -- map
@@ -97,8 +90,8 @@ module _ {f : A → B} where
   map⁺ (there p) = there (map⁺ p)
 
   Any-map : ∀ {xs} → Any P (map f xs) ↔ Any (P ∘ f) xs
-  Any-map {xs = xs} = inverse map⁻ map⁺ from∘to to∘from where
-
+  Any-map {xs = xs} = mk↔ₛ′ map⁻ map⁺ to∘from from∘to
+    where
     from∘to : ∀ {xs} (p : Any P (map f xs)) → map⁺ (map⁻ p) ≡ p
     from∘to {xs = x ∷ xs} (here px) = refl
     from∘to {xs = x ∷ xs} (there p) = cong there (from∘to p)
@@ -130,14 +123,7 @@ mutual
 
 Any-⋎ : ∀ {a p} {A : Set a} {P : A → Set p} xs {ys} →
         Any P (xs ⋎ ys) ↔ (Any P xs ⊎ Any P ys)
-Any-⋎ {P = P} = λ xs → record
-  { to         = P.→-to-⟶ (⋎⁻ xs)
-  ; from       = P.→-to-⟶ (⋎⁺ xs)
-  ; inverse-of = record
-    { left-inverse-of  = from∘to xs
-    ; right-inverse-of = to∘from xs
-    }
-  }
+Any-⋎ {P = P} xs = mk↔ₛ′ (⋎⁻ xs) (⋎⁺ xs) (to∘from xs) (from∘to xs)
   where
 
   from∘to : ∀ xs {ys} (p : Any P (xs ⋎ ys)) → ⋎⁺ xs (⋎⁻ xs p) ≡ p
@@ -186,10 +172,10 @@ index-Any-resp (x ∷ xs≈) (there p) =
 -- no larger than that of the input proof.
 
 index-Any-⋎ : ∀ xs {ys} (p : Any P (xs ⋎ ys)) →
-              index p ≥′ [ index , index ]′ (Inverse.to (Any-⋎ xs) ⟨$⟩ p)
+              index p ≥′ [ index , index ]′ (Inverse.to (Any-⋎ xs) p)
 index-Any-⋎ []                 p         = ≤′-refl
 index-Any-⋎ (x ∷ xs)           (here px) = ≤′-refl
 index-Any-⋎ (x ∷ xs) {ys = ys} (there p)
-  with Inverse.to (Any-⋎ ys) ⟨$⟩ p | index-Any-⋎ ys p
+  with Inverse.to (Any-⋎ ys) p | index-Any-⋎ ys p
 ... | inj₁ q | q≤p = ≤′-step q≤p
 ... | inj₂ q | q≤p = s≤′s    q≤p
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index ea22a37f68..0449279a7d 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -16,9 +16,7 @@ open import Data.Empty
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Function.Base using (_⟨_⟩_; const; id)
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence
-  using (_⇔_; equivalence; module Equivalence)
+open import Function.Bundles hiding (LeftInverse; RightInverse; Inverse)
 open import Induction.WellFounded using (WellFounded; Acc; acc)
 open import Level using (Level; 0ℓ)
 open import Relation.Binary.Core using (_⇒_)
@@ -733,29 +731,29 @@ xor-∧-commutativeRing = ⊕-∧-commutativeRing
 
 ⇔→≡ : {x y z : Bool} → x ≡ z ⇔ y ≡ z → x ≡ y
 ⇔→≡ {true } {true }         hyp = refl
-⇔→≡ {true } {false} {true } hyp = sym (Equivalence.to hyp ⟨$⟩ refl)
-⇔→≡ {true } {false} {false} hyp = Equivalence.from hyp ⟨$⟩ refl
-⇔→≡ {false} {true } {true } hyp = Equivalence.from hyp ⟨$⟩ refl
-⇔→≡ {false} {true } {false} hyp = sym (Equivalence.to hyp ⟨$⟩ refl)
+⇔→≡ {true } {false} {true } hyp = sym (Equivalence.to hyp refl)
+⇔→≡ {true } {false} {false} hyp = Equivalence.from hyp refl
+⇔→≡ {false} {true } {true } hyp = Equivalence.from hyp refl
+⇔→≡ {false} {true } {false} hyp = sym (Equivalence.to hyp refl)
 ⇔→≡ {false} {false}         hyp = refl
 
 T-≡ : ∀ {x} → T x ⇔ x ≡ true
-T-≡ {false} = equivalence (λ ())       (λ ())
-T-≡ {true}  = equivalence (const refl) (const _)
+T-≡ {false} = mk⇔ (λ ())       (λ ())
+T-≡ {true}  = mk⇔ (const refl) (const _)
 
 T-not-≡ : ∀ {x} → T (not x) ⇔ x ≡ false
-T-not-≡ {false} = equivalence (const refl) (const _)
-T-not-≡ {true}  = equivalence (λ ())       (λ ())
+T-not-≡ {false} = mk⇔ (const refl) (const _)
+T-not-≡ {true}  = mk⇔ (λ ())       (λ ())
 
 T-∧ : ∀ {x y} → T (x ∧ y) ⇔ (T x × T y)
-T-∧ {true}  {true}  = equivalence (const (_ , _)) (const _)
-T-∧ {true}  {false} = equivalence (λ ())          proj₂
-T-∧ {false} {_}     = equivalence (λ ())          proj₁
+T-∧ {true}  {true}  = mk⇔ (const (_ , _)) (const _)
+T-∧ {true}  {false} = mk⇔ (λ ())          proj₂
+T-∧ {false} {_}     = mk⇔ (λ ())          proj₁
 
 T-∨ : ∀ {x y} → T (x ∨ y) ⇔ (T x ⊎ T y)
-T-∨ {true}  {_}     = equivalence inj₁ (const _)
-T-∨ {false} {true}  = equivalence inj₂ (const _)
-T-∨ {false} {false} = equivalence inj₁ [ id , id ]
+T-∨ {true}  {_}     = mk⇔ inj₁ (const _)
+T-∨ {false} {true}  = mk⇔ inj₂ (const _)
+T-∨ {false} {false} = mk⇔ inj₁ [ id , id ]
 
 T-irrelevant : U.Irrelevant T
 T-irrelevant {true}  _  _  = refl
diff --git a/src/Data/Container/Combinator/Properties.agda b/src/Data/Container/Combinator/Properties.agda
index 8d347abcae..6f7c83e993 100644
--- a/src/Data/Container/Combinator/Properties.agda
+++ b/src/Data/Container/Combinator/Properties.agda
@@ -16,7 +16,7 @@ open import Data.Empty using (⊥-elim)
 open import Data.Product.Base as Prod using (∃; _,_; proj₁; proj₂; <_,_>; uncurry; curry)
 open import Data.Sum.Base as S using (inj₁; inj₂; [_,_]′; [_,_])
 open import Function.Base as F using (_∘′_)
-open import Function.Inverse as Inv using (_↔_; inverse; module Inverse)
+open import Function.Bundles
 open import Level using (_⊔_; lower)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
 
@@ -27,12 +27,12 @@ open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
 module Identity where
 
   correct : ∀ {s p x} {X : Set x} → ⟦ id {s} {p} ⟧ X ↔ F.id X
-  correct {X = X} = inverse from-id to-id (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ from-id to-id (λ _ → P.refl) (λ _ → P.refl)
 
 module Constant (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′) where
 
   correct : ∀ {x p y} (X : Set x) {Y : Set y} → ⟦ const {x} {p ⊔ y} X ⟧ Y ↔ F.const X Y
-  correct {x} {y} X {Y} = inverse (from-const X) (to-const X) from∘to λ _ → P.refl
+  correct {x} {y} X {Y} = mk↔ₛ′ (from-const X) (to-const X) (λ _ → P.refl) from∘to
     where
     from∘to : (x : ⟦ const X ⟧ Y) → to-const X (proj₁ x) ≡ x
     from∘to xs = P.cong (proj₁ xs ,_) (ext (λ x → ⊥-elim (lower x)))
@@ -40,13 +40,13 @@ module Constant (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′) where
 module Composition {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) where
 
   correct : ∀ {x} {X : Set x} → ⟦ C₁ ∘ C₂ ⟧ X ↔ (⟦ C₁ ⟧ F.∘ ⟦ C₂ ⟧) X
-  correct {X = X} = inverse (from-∘ C₁ C₂) (to-∘ C₁ C₂) (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ (from-∘ C₁ C₂) (to-∘ C₁ C₂) (λ _ → P.refl) (λ _ → P.refl)
 
 module Product (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′)
        {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) where
 
   correct : ∀ {x} {X : Set x} →  ⟦ C₁ × C₂ ⟧ X ↔ (⟦ C₁ ⟧ X Prod.× ⟦ C₂ ⟧ X)
-  correct {X = X} = inverse (from-× C₁ C₂) (to-× C₁ C₂) from∘to (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ (from-× C₁ C₂) (to-× C₁ C₂) (λ _ → P.refl) from∘to
     where
     from∘to : (to-× C₁ C₂) F.∘ (from-× C₁ C₂) ≗ F.id
     from∘to (s , f) =
@@ -55,12 +55,12 @@ module Product (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′)
 module IndexedProduct {i s p} {I : Set i} (Cᵢ : I → Container s p) where
 
   correct : ∀ {x} {X : Set x} → ⟦ Π I Cᵢ ⟧ X ↔ (∀ i → ⟦ Cᵢ i ⟧ X)
-  correct {X = X} = inverse (from-Π I Cᵢ) (to-Π I Cᵢ) (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ (from-Π I Cᵢ) (to-Π I Cᵢ) (λ _ → P.refl) (λ _ → P.refl)
 
 module Sum {s₁ s₂ p} (C₁ : Container s₁ p) (C₂ : Container s₂ p) where
 
   correct : ∀ {x} {X : Set x} → ⟦ C₁ ⊎ C₂ ⟧ X ↔ (⟦ C₁ ⟧ X S.⊎ ⟦ C₂ ⟧ X)
-  correct {X = X} = inverse (from-⊎ C₁ C₂) (to-⊎ C₁ C₂) from∘to to∘from
+  correct {X = X} = mk↔ₛ′ (from-⊎ C₁ C₂) (to-⊎ C₁ C₂) to∘from from∘to
     where
     from∘to : (to-⊎ C₁ C₂) F.∘ (from-⊎ C₁ C₂) ≗ F.id
     from∘to (inj₁ s₁ , f) = P.refl
@@ -72,7 +72,7 @@ module Sum {s₁ s₂ p} (C₁ : Container s₁ p) (C₂ : Container s₂ p) whe
 module IndexedSum {i s p} {I : Set i} (C : I → Container s p) where
 
   correct : ∀ {x} {X : Set x} → ⟦ Σ I C ⟧ X ↔ (∃ λ i → ⟦ C i ⟧ X)
-  correct {X = X} = inverse (from-Σ I C) (to-Σ I C) (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ (from-Σ I C) (to-Σ I C) (λ _ → P.refl) (λ _ → P.refl)
 
 module ConstantExponentiation {i s p} {I : Set i} (C : Container s p) where
 
diff --git a/src/Data/Container/Core.agda b/src/Data/Container/Core.agda
index dc5134f585..f03f757c7e 100644
--- a/src/Data/Container/Core.agda
+++ b/src/Data/Container/Core.agda
@@ -11,8 +11,7 @@ module Data.Container.Core where
 open import Level
 open import Data.Product.Base as Prod using (Σ-syntax)
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Inverse using (_↔_; module Inverse)
+open import Function using (Inverse; _↔_)
 open import Relation.Unary using (Pred; _⊆_)
 
 -- Definition of Containers
@@ -63,7 +62,7 @@ record _⊸_ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Containe
   morphism : C₁ ⇒ C₂
   morphism = record
     { shape    = shape⊸
-    ; position = _⟨$⟩_ (Inverse.to position⊸)
+    ; position = Inverse.to position⊸
     }
 
   ⟪_⟫⊸ : ∀ {x} {X : Set x} → ⟦ C₁ ⟧ X → ⟦ C₂ ⟧ X
diff --git a/src/Data/Container/Indexed.agda b/src/Data/Container/Indexed.agda
index fa2e7fc33a..65b36f0527 100644
--- a/src/Data/Container/Indexed.agda
+++ b/src/Data/Container/Indexed.agda
@@ -15,8 +15,7 @@ open import Level
 open import Data.Product.Base as Prod hiding (map)
 open import Data.W.Indexed
 open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Inverse using (_↔_; module Inverse)
+open import Function using (_↔_; Inverse)
 open import Relation.Unary using (Pred; _⊆_)
 open import Relation.Binary.Core using (Rel; REL)
 open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_; refl)
@@ -91,7 +90,7 @@ module _ {i₁ i₂ o₁ o₂}
              Container I₁ O₁ c₁ r₁ → (I₁ → I₂) → (O₁ → O₂) →
              Container I₂ O₂ c₂ r₂ → Set _
   C₁ ⊸[ f / g ] C₂ = ContainerMorphism C₁ C₂ f g _≡_ _↔_
-                                       (λ r₂↔r₁ r₂ → Inverse.to r₂↔r₁ ⟨$⟩ r₂)
+                                       (λ r₂↔r₁ r₂ → Inverse.to r₂↔r₁ r₂)
 
   -- Cartesian container morphism.
 
@@ -168,7 +167,7 @@ module LinearMorphism
   morphism : C₁ ⇒ C₂
   morphism = record
     { command  = command m
-    ; response = _⟨$⟩_ (Inverse.to (response m))
+    ; response = Inverse.to (response m)
     ; coherent = coherent m
     }
 
diff --git a/src/Data/Container/Indexed/Combinator.agda b/src/Data/Container/Indexed/Combinator.agda
index 283a6a0314..b35a1ec887 100644
--- a/src/Data/Container/Indexed/Combinator.agda
+++ b/src/Data/Container/Indexed/Combinator.agda
@@ -16,7 +16,8 @@ open import Data.Product.Base as Prod hiding (Σ) renaming (_×_ to _⟨×⟩_)
 open import Data.Sum.Base renaming (_⊎_ to _⟨⊎⟩_)
 open import Data.Sum.Relation.Unary.All as All using (All)
 open import Function.Base as F hiding (id; const) renaming (_∘_ to _⟨∘⟩_)
-open import Function.Inverse using (_↔̇_; inverse)
+open import Function.Bundles using (mk↔ₛ′)
+open import Function.Indexed.Bundles using (_↔ᵢ_)
 open import Level
 open import Relation.Unary using (Pred; _⊆_; _∪_; _∩_; ⋃; ⋂)
   renaming (_⟨×⟩_ to _⟪×⟫_; _⟨⊙⟩_ to _⟪⊙⟫_; _⟨⊎⟩_ to _⟪⊎⟫_)
@@ -145,8 +146,8 @@ const[ X ]⟶ C = Π {X = X} (F.const C)
 
 module Identity where
 
-  correct : {X : Pred O ℓ} → ⟦ id {c = c}{r} ⟧ X ↔̇ F.id X
-  correct {X = X} = inverse to from (λ _ → refl) (λ _ → refl)
+  correct : {X : Pred O ℓ} → ⟦ id {c = c}{r} ⟧ X ↔ᵢ F.id X
+  correct {X = X} = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)
     where
     to : ∀ {x} → ⟦ id ⟧ X x → F.id X x
     to xs = proj₂ xs _
@@ -156,15 +157,8 @@ module Identity where
 
 module Constant (ext : ∀ {ℓ} → Extensionality ℓ ℓ) where
 
-  correct : (X : Pred O ℓ₁) {Y : Pred O ℓ₂} → ⟦ const X ⟧ Y ↔̇ F.const X Y
-  correct X {Y} = record
-    { to         = P.→-to-⟶ to
-    ; from       = P.→-to-⟶ from
-    ; inverse-of = record
-      { right-inverse-of = λ _ → refl
-      ; left-inverse-of  = to∘from
-      }
-    }
+  correct : (X : Pred O ℓ₁) {Y : Pred O ℓ₂} → ⟦ const X ⟧ Y ↔ᵢ F.const X Y
+  correct X {Y} = mk↔ₛ′ to from (λ _ → refl) to∘from
     where
     to : ⟦ const X ⟧ Y ⊆ X
     to = proj₁
@@ -178,15 +172,15 @@ module Constant (ext : ∀ {ℓ} → Extensionality ℓ ℓ) where
 module Duality where
 
   correct : (C : Container I O c r) (X : Pred I ℓ) →
-            ⟦ C ^⊥ ⟧ X ↔̇ (λ o → (c : Command C o) → ∃ λ r → X (next C c r))
-  correct C X = inverse (λ { (f , g) → < f , g > }) (λ f → proj₁ ⟨∘⟩ f , proj₂ ⟨∘⟩ f)
+            ⟦ C ^⊥ ⟧ X ↔ᵢ (λ o → (c : Command C o) → ∃ λ r → X (next C c r))
+  correct C X = mk↔ₛ′ (λ { (f , g) → < f , g > }) (λ f → proj₁ ⟨∘⟩ f , proj₂ ⟨∘⟩ f)
                         (λ _ → refl) (λ _ → refl)
 
 module Composition where
 
   correct : (C₁ : Container J K c r) (C₂ : Container I J c r) →
-            {X : Pred I ℓ} → ⟦ C₁ ∘ C₂ ⟧ X ↔̇ (⟦ C₁ ⟧ ⟨∘⟩ ⟦ C₂ ⟧) X
-  correct C₁ C₂ {X} = inverse to from (λ _ → refl) (λ _ → refl)
+            {X : Pred I ℓ} → ⟦ C₁ ∘ C₂ ⟧ X ↔ᵢ (⟦ C₁ ⟧ ⟨∘⟩ ⟦ C₂ ⟧) X
+  correct C₁ C₂ {X} = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)
     where
     to : ⟦ C₁ ∘ C₂ ⟧ X ⊆ ⟦ C₁ ⟧ (⟦ C₂ ⟧ X)
     to ((c , f) , g) = (c , < f , curry g >)
@@ -197,8 +191,8 @@ module Composition where
 module Product (ext : ∀ {ℓ} → Extensionality ℓ ℓ) where
 
   correct : (C₁ C₂ : Container I O c r) {X : Pred I _} →
-            ⟦ C₁ × C₂ ⟧ X ↔̇ (⟦ C₁ ⟧ X ∩ ⟦ C₂ ⟧ X)
-  correct C₁ C₂ {X} = inverse to from from∘to (λ _ → refl)
+            ⟦ C₁ × C₂ ⟧ X ↔ᵢ (⟦ C₁ ⟧ X ∩ ⟦ C₂ ⟧ X)
+  correct C₁ C₂ {X} = mk↔ₛ′ to from (λ _ → refl) from∘to
     where
     to : ⟦ C₁ × C₂ ⟧ X ⊆ ⟦ C₁ ⟧ X ∩ ⟦ C₂ ⟧ X
     to ((c₁ , c₂) , k) = ((c₁ , k ⟨∘⟩ inj₁) , (c₂ , k ⟨∘⟩ inj₂))
@@ -213,8 +207,8 @@ module Product (ext : ∀ {ℓ} → Extensionality ℓ ℓ) where
 module IndexedProduct where
 
   correct : (C : X → Container I O c r) {Y : Pred I ℓ} →
-            ⟦ Π C ⟧ Y ↔̇ ⋂[ x ∶ X ] ⟦ C x ⟧ Y
-  correct {X = X} C {Y} = inverse to from (λ _ → refl) (λ _ → refl)
+            ⟦ Π C ⟧ Y ↔ᵢ ⋂[ x ∶ X ] ⟦ C x ⟧ Y
+  correct {X = X} C {Y} = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)
     where
     to : ⟦ Π C ⟧ Y ⊆ ⋂[ x ∶ X ] ⟦ C x ⟧ Y
     to (c , k) = λ x → (c x , λ r → k (x , r))
@@ -225,8 +219,8 @@ module IndexedProduct where
 module Sum (ext : ∀ {ℓ₁ ℓ₂} → Extensionality ℓ₁ ℓ₂) where
 
   correct : (C₁ C₂ : Container I O c r) {X : Pred I ℓ} →
-            ⟦ C₁ ⊎ C₂ ⟧ X ↔̇ (⟦ C₁ ⟧ X ∪ ⟦ C₂ ⟧ X)
-  correct C₁ C₂ {X} = inverse to from from∘to to∘from
+            ⟦ C₁ ⊎ C₂ ⟧ X ↔ᵢ (⟦ C₁ ⟧ X ∪ ⟦ C₂ ⟧ X)
+  correct C₁ C₂ {X} = mk↔ₛ′ to from to∘from from∘to
     where
     to : ⟦ C₁ ⊎ C₂ ⟧ X ⊆ ⟦ C₁ ⟧ X ∪ ⟦ C₂ ⟧ X
     to (inj₁ c₁ , k) = inj₁ (c₁ , λ r → k (All.inj₁ r))
@@ -246,8 +240,8 @@ module Sum (ext : ∀ {ℓ₁ ℓ₂} → Extensionality ℓ₁ ℓ₂) where
 module Sum′ where
 
   correct : (C₁ C₂ : Container I O c r) {X : Pred I ℓ} →
-            ⟦ C₁ ⊎′ C₂ ⟧ X ↔̇ (⟦ C₁ ⟧ X ∪ ⟦ C₂ ⟧ X)
-  correct C₁ C₂ {X} = inverse to from from∘to to∘from
+            ⟦ C₁ ⊎′ C₂ ⟧ X ↔ᵢ (⟦ C₁ ⟧ X ∪ ⟦ C₂ ⟧ X)
+  correct C₁ C₂ {X} = mk↔ₛ′ to from to∘from from∘to
     where
     to : ⟦ C₁ ⊎′ C₂ ⟧ X ⊆ ⟦ C₁ ⟧ X ∪ ⟦ C₂ ⟧ X
     to (inj₁ c₁ , k) = inj₁ (c₁ , k)
@@ -266,8 +260,8 @@ module Sum′ where
 module IndexedSum where
 
   correct : (C : X → Container I O c r) {Y : Pred I ℓ} →
-            ⟦ Σ C ⟧ Y ↔̇ ⋃[ x ∶ X ] ⟦ C x ⟧ Y
-  correct {X = X} C {Y} = inverse to from (λ _ → refl) (λ _ → refl)
+            ⟦ Σ C ⟧ Y ↔ᵢ ⋃[ x ∶ X ] ⟦ C x ⟧ Y
+  correct {X = X} C {Y} = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)
     where
     to : ⟦ Σ C ⟧ Y ⊆ ⋃[ x ∶ X ] ⟦ C x ⟧ Y
     to ((x , c) , k) = (x , (c , k))
@@ -278,5 +272,5 @@ module IndexedSum where
 module ConstantExponentiation where
 
   correct : (C : Container I O c r) {Y : Pred I ℓ} →
-            ⟦ const[ X ]⟶ C ⟧ Y ↔̇ (⋂ X (F.const (⟦ C ⟧ Y)))
+            ⟦ const[ X ]⟶ C ⟧ Y ↔ᵢ (⋂ X (F.const (⟦ C ⟧ Y)))
   correct C {Y} = IndexedProduct.correct (F.const C) {Y}
diff --git a/src/Data/Container/Related.agda b/src/Data/Container/Related.agda
index 5f1715c765..a5af9e334f 100644
--- a/src/Data/Container/Related.agda
+++ b/src/Data/Container/Related.agda
@@ -11,17 +11,17 @@ module Data.Container.Related where
 
 open import Level using (_⊔_)
 open import Data.Container.Core
-import Function.Related as Related
+import Function.Related.Propositional as Related
 open import Relation.Binary.Bundles using (Preorder; Setoid)
 open import Data.Container.Membership
 
 open Related public
-  using (Kind; Symmetric-kind)
+  using (Kind; SymmetricKind)
   renaming ( implication         to subset
-           ; reverse-implication to superset
+           ; reverseImplication  to superset
            ; equivalence         to set
            ; injection           to subbag
-           ; reverse-injection   to superbag
+           ; reverseInjection    to superbag
            ; bijection           to bag
            )
 
@@ -29,7 +29,7 @@ open Related public
             Preorder (s ⊔ p ⊔ ℓ) (s ⊔ p ⊔ ℓ) (p ⊔ ℓ)
 [ k ]-Order C X = Related.InducedPreorder₂ k (_∈_ {C = C} {X = X})
 
-[_]-Equality : ∀ {s p ℓ} → Symmetric-kind → Container s p → Set ℓ →
+[_]-Equality : ∀ {s p ℓ} → SymmetricKind → Container s p → Set ℓ →
                Setoid (s ⊔ p ⊔ ℓ) (p ⊔ ℓ)
 [ k ]-Equality C X = Related.InducedEquivalence₂ k (_∈_ {C = C} {X = X})
 
diff --git a/src/Data/Container/Relation/Unary/Any/Properties.agda b/src/Data/Container/Relation/Unary/Any/Properties.agda
index d1d35c5a02..7bf97642a4 100644
--- a/src/Data/Container/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Container/Relation/Unary/Any/Properties.agda
@@ -11,15 +11,15 @@ module Data.Container.Relation.Unary.Any.Properties where
 open import Level
 open import Algebra
 open import Data.Product.Base using (∃; _×_; ∃₂; _,_; proj₂)
+open import Data.Product.Properties using (Σ-≡,≡→≡)
 open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence using (equivalence)
-open import Function.HalfAdjointEquivalence using (_≃_; ↔→≃)
-open import Function.Inverse as Inv using (_↔_; inverse; module Inverse)
-open import Function.Related as Related using (Related; SK-sym)
+open import Function.Bundles
+open import Function.Properties.Inverse using (↔-refl)
+open import Function.Properties.Inverse.HalfAdjointEquivalence using (_≃_; ↔⇒≃)
+open import Function.Related.Propositional as Related using (Related; SK-sym)
 open import Function.Related.TypeIsomorphisms
 open import Relation.Unary using (Pred ; _∪_ ; _∩_)
 open import Relation.Binary.Core using (REL)
@@ -43,12 +43,12 @@ module _ {s p} (C : Container s p) {x} {X : Set x} {ℓ} {P : Pred X ℓ} where
 -- ◇ can be unwrapped to reveal the Σ type
 
   ↔Σ : ∀ {xs : ⟦ C ⟧ X} → ◇ C P xs ↔ ∃ λ p → P (proj₂ xs p)
-  ↔Σ {xs} = inverse ◇.proof any (λ _ → P.refl) (λ _ → P.refl)
+  ↔Σ {xs} = mk↔ₛ′ ◇.proof any (λ _ → P.refl) (λ _ → P.refl)
 
 -- ◇ can be expressed using _∈_.
 
   ↔∈ : ∀ {xs : ⟦ C ⟧ X} → ◇ C P xs ↔ (∃ λ x → x ∈ xs × P x)
-  ↔∈ {xs} = inverse to from (λ _ → P.refl) (to∘from) where
+  ↔∈ {xs} = mk↔ₛ′ to from to∘from (λ _ → P.refl) where
 
     to : ◇ C P xs → ∃ λ x → x ∈ xs × P x
     to (any (p , Px)) = (proj₂ xs p , (any (p , P.refl)) , Px)
@@ -68,7 +68,7 @@ module _ {s p} {C : Container s p} {x} {X : Set x}
          Related k (◇ C P₁ xs₁) (◇ C P₂ xs₂)
   cong {k} {xs₁} {xs₂} P₁↔P₂ xs₁≈xs₂ =
     ◇ C P₁ xs₁               ↔⟨ ↔∈ C ⟩
-    (∃ λ x → x ∈ xs₁ × P₁ x) ∼⟨ Σ.cong Inv.id (xs₁≈xs₂ ×-cong P₁↔P₂ _) ⟩
+    (∃ λ x → x ∈ xs₁ × P₁ x) ∼⟨ Σ.cong ↔-refl (xs₁≈xs₂ ×-cong P₁↔P₂ _) ⟩
     (∃ λ x → x ∈ xs₂ × P₂ x) ↔⟨ SK-sym (↔∈ C) ⟩
     ◇ C P₂ xs₂               ∎
 
@@ -83,16 +83,16 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
          ◈ xs (◈ ys ∘ P) ↔ ◈ ys (◈ xs ∘ flip P)
   swap {xs} {ys} =
     ◇ _ (λ x → ◇ _ (P x) ys) xs                ↔⟨ ↔∈ C₁ ⟩
-    (∃ λ x → x ∈ xs × ◇ _ (P x) ys)            ↔⟨ Σ.cong Inv.id $ Σ.cong Inv.id $ ↔∈ C₂ ⟩
-    (∃ λ x → x ∈ xs × ∃ λ y → y ∈ ys × P x y)  ↔⟨ Σ.cong Inv.id (λ {x} → ∃∃↔∃∃ (λ _ y → y ∈ ys × P x y)) ⟩
+    (∃ λ x → x ∈ xs × ◇ _ (P x) ys)            ↔⟨ Σ.cong ↔-refl $ Σ.cong ↔-refl $ ↔∈ C₂ ⟩
+    (∃ λ x → x ∈ xs × ∃ λ y → y ∈ ys × P x y)  ↔⟨ Σ.cong ↔-refl (λ {x} → ∃∃↔∃∃ (λ _ y → y ∈ ys × P x y)) ⟩
     (∃₂ λ x y → x ∈ xs × y ∈ ys × P x y)       ↔⟨ ∃∃↔∃∃ (λ x y → x ∈ xs × y ∈ ys × P x y) ⟩
-    (∃₂ λ y x → x ∈ xs × y ∈ ys × P x y)       ↔⟨ Σ.cong Inv.id (λ {y} → Σ.cong Inv.id (λ {x} →
+    (∃₂ λ y x → x ∈ xs × y ∈ ys × P x y)       ↔⟨ Σ.cong ↔-refl (λ {y} → Σ.cong ↔-refl (λ {x} →
       (x ∈ xs × y ∈ ys × P x y)                     ↔⟨ SK-sym Σ-assoc ⟩
-      ((x ∈ xs × y ∈ ys) × P x y)                   ↔⟨ Σ.cong (×-comm _ _) Inv.id ⟩
+      ((x ∈ xs × y ∈ ys) × P x y)                   ↔⟨ Σ.cong (×-comm _ _) ↔-refl ⟩
       ((y ∈ ys × x ∈ xs) × P x y)                   ↔⟨ Σ-assoc ⟩
       (y ∈ ys × x ∈ xs × P x y)                     ∎)) ⟩
-    (∃₂ λ y x → y ∈ ys × x ∈ xs × P x y)       ↔⟨ Σ.cong Inv.id (λ {y} → ∃∃↔∃∃ {B = y ∈ ys} (λ x _ → x ∈ xs × P x y)) ⟩
-    (∃ λ y → y ∈ ys × ∃ λ x → x ∈ xs × P x y)  ↔⟨ Σ.cong Inv.id (Σ.cong Inv.id (SK-sym (↔∈ C₁))) ⟩
+    (∃₂ λ y x → y ∈ ys × x ∈ xs × P x y)       ↔⟨ Σ.cong ↔-refl (λ {y} → ∃∃↔∃∃ {B = y ∈ ys} (λ x _ → x ∈ xs × P x y)) ⟩
+    (∃ λ y → y ∈ ys × ∃ λ x → x ∈ xs × P x y)  ↔⟨ Σ.cong ↔-refl (Σ.cong ↔-refl (SK-sym (↔∈ C₁))) ⟩
     (∃ λ y → y ∈ ys × ◇ _ (flip P y) xs)       ↔⟨ SK-sym (↔∈ C₂) ⟩
     ◇ _ (λ y → ◇ _ (flip P y) xs) ys           ∎
 
@@ -103,16 +103,16 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
 
   flatten : ∀ (xss : ⟦ C₁ ⟧ (⟦ C₂ ⟧ X)) →
             ◇ C₁ (◇ C₂ P) xss ↔
-            ◇ (C₁ C.∘ C₂) P (Inverse.from (Composition.correct C₁ C₂) ⟨$⟩ xss)
-  flatten xss = inverse t f (λ _ → P.refl) (λ _ → P.refl) where
+            ◇ (C₁ C.∘ C₂) P (Inverse.from (Composition.correct C₁ C₂) xss)
+  flatten xss = mk↔ₛ′ t f (λ _ → P.refl) (λ _ → P.refl) where
 
     ◇₁ = ◇ C₁; ◇₂ = ◇ C₂; ◇₁₂ = ◇ (C₁ C.∘ C₂)
     open Inverse
 
-    t : ◇₁ (◇₂ P) xss → ◇₁₂ P (from (Composition.correct C₁ C₂) ⟨$⟩ xss)
+    t : ◇₁ (◇₂ P) xss → ◇₁₂ P (from (Composition.correct C₁ C₂) xss)
     t (any (p₁ , (any (p₂ , p)))) = any (any (p₁ , p₂) , p)
 
-    f : ◇₁₂ P (from (Composition.correct C₁ C₂) ⟨$⟩ xss) → ◇₁ (◇₂ P) xss
+    f : ◇₁₂ P (from (Composition.correct C₁ C₂) xss) → ◇₁ (◇₂ P) xss
     f (any (any (p₁ , p₂) , p)) = any (p₁ , any (p₂ , p))
 
 -- Sums commute with ◇ (for a fixed instance of a given container).
@@ -121,7 +121,7 @@ module _ {s p} {C : Container s p} {x} {X : Set x}
          {ℓ ℓ′} {P : Pred X ℓ} {Q : Pred X ℓ′} where
 
   ◇⊎↔⊎◇ : ∀ {xs : ⟦ C ⟧ X} → ◇ C (P ∪ Q) xs ↔ (◇ C P xs ⊎ ◇ C Q xs)
-  ◇⊎↔⊎◇ {xs} = inverse to from from∘to to∘from
+  ◇⊎↔⊎◇ {xs} = mk↔ₛ′ to from to∘from from∘to
     where
     to : ◇ C (λ x → P x ⊎ Q x) xs → ◇ C P xs ⊎ ◇ C Q xs
     to (any (pos , inj₁ p)) = inj₁ (any (pos , p))
@@ -144,7 +144,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
 
   ×◇↔◇◇× : ∀ {xs : ⟦ C₁ ⟧ X} {ys : ⟦ C₂ ⟧ Y} →
            ◇ C₁ (λ x → ◇ C₂ (λ y → P x × Q y) ys) xs ↔ (◇ C₁ P xs × ◇ C₂ Q ys)
-  ×◇↔◇◇× {xs} {ys} = inverse to from (λ _ → P.refl) (λ _ → P.refl)
+  ×◇↔◇◇× {xs} {ys} = mk↔ₛ′ to from (λ _ → P.refl) (λ _ → P.refl)
     where
     ◇₁ = ◇ C₁; ◇₂ = ◇ C₂
 
@@ -194,7 +194,7 @@ module _ {s p} (C : Container s p) {x y} {X : Set x} {Y : Set y}
     x ∈ map f₂ xs₂           ∎
     where
     helper : ∀ y → (x ≡ f₁ y) ↔ (x ≡ f₂ y)
-    helper y rewrite f₁≗f₂ y = Inv.id
+    helper y rewrite f₁≗f₂ y = ↔-refl
 
 -- Uses of linear morphisms can be removed.
 
@@ -202,13 +202,13 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
          {x} {X : Set x} {ℓ} (P : Pred X ℓ) where
 
   remove-linear : ∀ {xs : ⟦ C₁ ⟧ X} (m : C₁ ⊸ C₂) → ◇ C₂ P (⟪ m ⟫⊸ xs) ↔ ◇ C₁ P xs
-  remove-linear {xs} m = Inv.inverse t f f∘t t∘f
+  remove-linear {xs} m = mk↔ₛ′ t f t∘f f∘t
     where
     open _≃_
     open P.≡-Reasoning renaming (_∎ to _∎′)
 
     position⊸m : ∀ {s} → Position C₂ (shape⊸ m s) ≃ Position C₁ s
-    position⊸m = ↔→≃ (position⊸ m)
+    position⊸m = ↔⇒≃ (position⊸ m)
 
     ◇₁ = ◇ C₁; ◇₂ = ◇ C₂
 
@@ -301,10 +301,10 @@ module _ {s₁ s₂ s₃ p₁ p₂ p₃}
   join↔◇ : (join′ : (C₁ C.∘ C₂) ⊸ C₃) (xss : ⟦ C₁ ⟧ (⟦ C₂ ⟧ X)) →
            let join : ∀ {X} → ⟦ C₁ ⟧ (⟦ C₂ ⟧ X) → ⟦ C₃ ⟧ X
                join = λ {_} → ⟪ join′ ⟫⊸ ∘
-                      _⟨$⟩_ (Inverse.from (Composition.correct C₁ C₂)) in
+                      (Inverse.from (Composition.correct C₁ C₂)) in
            ◇ C₃ P (join xss) ↔ ◇ C₁ (◇ C₂ P) xss
   join↔◇ join xss =
     ◇ C₃ P (⟪ join ⟫⊸ xss′) ↔⟨ remove-linear P join ⟩
     ◇ (C₁ C.∘ C₂) P xss′    ↔⟨ SK-sym $ flatten P xss ⟩
     ◇ C₁ (◇ C₂ P) xss       ∎
-    where xss′ = Inverse.from (Composition.correct C₁ C₂) ⟨$⟩ xss
+    where xss′ = Inverse.from (Composition.correct C₁ C₂) xss
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 5fe0fa0b2c..00b65247fe 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -16,12 +16,11 @@ open import Data.Fin.Properties
 import Data.Fin.Permutation.Components as PC
 open import Data.Nat.Base using (ℕ; suc; zero)
 open import Data.Product.Base using (_,_; proj₂)
-open import Function.Bundles using (_↔_; Injection; Inverse; mk↔′)
+open import Function.Bundles using (_↔_; Injection; Inverse; mk↔ₛ′)
 open import Function.Construct.Composition using (_↔-∘_)
 open import Function.Construct.Identity using (↔-id)
 open import Function.Construct.Symmetry using (↔-sym)
 open import Function.Definitions
-open import Function.Equality using (_⟨$⟩_)
 open import Function.Properties.Inverse using (↔⇒↣)
 open import Function.Base using (_∘_)
 open import Level using (0ℓ)
@@ -58,7 +57,7 @@ Permutation′ n = Permutation n n
 
 permutation : ∀ (f : Fin m → Fin n) (g : Fin n → Fin m) →
               StrictlyInverseˡ _≡_ f g → StrictlyInverseʳ _≡_ f g → Permutation m n
-permutation = mk↔′
+permutation = mk↔ₛ′
 
 infixl 5 _⟨$⟩ʳ_ _⟨$⟩ˡ_
 _⟨$⟩ʳ_ : Permutation m n → Fin m → Fin n
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index bcb1cd4745..99121e1b42 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -30,7 +30,7 @@ open import Data.Product.Algebra using (×-cong)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)
 open import Data.Sum.Properties using ([,]-map; [,]-∘)
 open import Function.Base using (_∘_; id; _$_; flip)
-open import Function.Bundles using (Injection; _↣_; _⇔_; _↔_; mk⇔; mk↔′)
+open import Function.Bundles using (Injection; _↣_; _⇔_; _↔_; mk⇔; mk↔ₛ′)
 open import Function.Definitions using (Injective; Surjective)
 open import Function.Consequences.Propositional using (contraInjective)
 open import Function.Construct.Composition as Comp hiding (injective)
@@ -69,13 +69,13 @@ private
 -- Bundles
 
 0↔⊥ : Fin 0 ↔ ⊥
-0↔⊥ = mk↔′ ¬Fin0 (λ ()) (λ ()) (λ ())
+0↔⊥ = mk↔ₛ′ ¬Fin0 (λ ()) (λ ()) (λ ())
 
 1↔⊤ : Fin 1 ↔ ⊤
-1↔⊤ = mk↔′ (λ { 0F → tt }) (λ { tt → 0F }) (λ { tt → refl }) λ { 0F → refl }
+1↔⊤ = mk↔ₛ′ (λ { 0F → tt }) (λ { tt → 0F }) (λ { tt → refl }) λ { 0F → refl }
 
 2↔Bool : Fin 2 ↔ Bool
-2↔Bool = mk↔′ (λ { 0F → false; 1F → true }) (λ { false → 0F ; true → 1F })
+2↔Bool = mk↔ₛ′ (λ { 0F → false; 1F → true }) (λ { false → 0F ; true → 1F })
   (λ { false → refl ; true → refl }) (λ { 0F → refl ; 1F → refl })
 
 ------------------------------------------------------------------------
@@ -617,7 +617,7 @@ splitAt-≥ (suc m) (suc i) (s≤s i≥m) = cong (Sum.map suc id) (splitAt-≥ m
 -- Bundles
 
 +↔⊎ : Fin (m ℕ.+ n) ↔ (Fin m ⊎ Fin n)
-+↔⊎ {m} {n} = mk↔′ (splitAt m {n}) (join m n) (splitAt-join m n) (join-splitAt m n)
++↔⊎ {m} {n} = mk↔ₛ′ (splitAt m {n}) (join m n) (splitAt-join m n) (join-splitAt m n)
 
 ------------------------------------------------------------------------
 -- remQuot
@@ -709,7 +709,7 @@ combine-surjective {m} {n} i with remQuot {m} n i in eq
 -- Bundles
 
 *↔× : Fin (m ℕ.* n) ↔ (Fin m × Fin n)
-*↔× {m} {n} = mk↔′ (remQuot {m} n) (uncurry combine)
+*↔× {m} {n} = mk↔ₛ′ (remQuot {m} n) (uncurry combine)
   (uncurry remQuot-combine)
   (combine-remQuot {m} n)
 
@@ -760,7 +760,7 @@ finToFun-funToFin {suc m} {n} f (suc i) =
 -- Bundles
 
 ^↔→ : Extensionality _ _ → Fin (m ^ n) ↔ (Fin n → Fin m)
-^↔→ {m} {n} ext = mk↔′ finToFun funToFin
+^↔→ {m} {n} ext = mk↔ₛ′ finToFun funToFin
   (ext ∘ finToFun-funToFin)
   (funToFin-finToFin {n} {m})
 
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index dfef9fa160..06240c6be5 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -23,16 +23,16 @@ open import Data.List.Effectful using (monad)
 open import Data.Nat.Base using (ℕ; zero; suc; pred; s≤s; _≤_; _<_; _≤ᵇ_)
 open import Data.Nat.Properties
 open import Data.Product.Base hiding (map)
+open import Data.Product.Properties using (×-≡,≡↔≡)
 open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence using (module Equivalence)
-open import Function.Injection using (Injection; Injective; _↣_)
-open import Function.Inverse as Inv using (_↔_; module Inverse)
-import Function.Related as Related
+open import Function.Definitions
+import Function.Related.Propositional as Related
+open import Function.Bundles
 open import Function.Related.TypeIsomorphisms
+open import Function.Construct.Identity using (↔-id)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions as B hiding (Decidable)
@@ -149,7 +149,7 @@ module _ {v : A} where
   concat-∈↔ : ∀ {xss : List (List A)} →
               (∃ λ xs → v ∈ xs × xs ∈ xss) ↔ v ∈ concat xss
   concat-∈↔ {xss} =
-    (∃ λ xs → v ∈ xs × xs ∈ xss)  ↔⟨ Σ.cong Inv.id $ ×-comm _ _ ⟩
+    (∃ λ xs → v ∈ xs × xs ∈ xss)  ↔⟨ Σ.cong (↔-id _) $ ×-comm _ _ ⟩
     (∃ λ xs → xs ∈ xss × v ∈ xs)  ↔⟨ Any↔ ⟩
     Any (Any (v ≡_)) xss          ↔⟨ concat↔ ⟩
     v ∈ concat xss                ∎
@@ -284,8 +284,8 @@ module _ (_≈?_ : B.Decidable {A = A} _≡_) where
 ⊛-∈↔ : ∀ (fs : List (A → B)) {xs y} →
        (∃₂ λ f x → f ∈ fs × x ∈ xs × y ≡ f x) ↔ y ∈ (fs ⊛ xs)
 ⊛-∈↔ fs {xs} {y} =
-  (∃₂ λ f x → f ∈ fs × x ∈ xs × y ≡ f x)       ↔⟨ Σ.cong Inv.id (∃∃↔∃∃ _) ⟩
-  (∃ λ f → f ∈ fs × ∃ λ x → x ∈ xs × y ≡ f x)  ↔⟨ Σ.cong Inv.id ((_ ∎) ⟨ _×-cong_ ⟩ Any↔) ⟩
+  (∃₂ λ f x → f ∈ fs × x ∈ xs × y ≡ f x)       ↔⟨ Σ.cong (↔-id _) (∃∃↔∃∃ _) ⟩
+  (∃ λ f → f ∈ fs × ∃ λ x → x ∈ xs × y ≡ f x)  ↔⟨ Σ.cong (↔-id _) (↔-id _ ⟨ _×-cong_ ⟩ Any↔) ⟩
   (∃ λ f → f ∈ fs × Any (_≡_ y ∘ f) xs)        ↔⟨ Any↔ ⟩
   Any (λ f → Any (_≡_ y ∘ f) xs) fs            ↔⟨ ⊛↔ ⟩
   y ∈ (fs ⊛ xs)                                ∎
@@ -298,7 +298,7 @@ module _ (_≈?_ : B.Decidable {A = A} _≡_) where
        (x ∈ xs × y ∈ ys) ↔ (x , y) ∈ (xs ⊗ ys)
 ⊗-∈↔ {xs = xs} {ys} {x} {y} =
   (x ∈ xs × y ∈ ys)             ↔⟨ ⊗↔′ ⟩
-  Any (x ≡_ ⟨×⟩ y ≡_) (xs ⊗ ys) ↔⟨ Any-cong ×-≡×≡↔≡,≡ (_ ∎) ⟩
+  Any (x ≡_ ⟨×⟩ y ≡_) (xs ⊗ ys) ↔⟨ Any-cong (λ _ → ×-≡,≡↔≡) (↔-id _) ⟩
   (x , y) ∈ (xs ⊗ ys)           ∎
   where
   open Related.EquationalReasoning
@@ -343,14 +343,14 @@ module _ {_•_ : Op₂ A} where
 -- Only a finite number of distinct elements can be members of a
 -- given list.
 
-finite : (f : ℕ ↣ A) → ∀ xs → ¬ (∀ i → Injection.to f ⟨$⟩ i ∈ xs)
+finite : (inj : ℕ ↣ A) → ∀ xs → ¬ (∀ i → Injection.to inj i ∈ xs)
 finite inj []       fᵢ∈[]   = ¬Any[] (fᵢ∈[] 0)
 finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
   where
   open Injection inj renaming (injective to f-inj)
 
   f : ℕ → _
-  f = to ⟨$⟩_
+  f = to
 
   not-x : ∀ {i} → f i ≢ x → f i ∈ xs
   not-x {i} fᵢ≢x with fᵢ∈x∷xs i
@@ -377,7 +377,7 @@ finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
     ... | true  | p = ∈-if-not-i (<⇒≢ (s≤s p))
     ... | false | p = ∈-if-not-i (<⇒≢ (≰⇒> p) ∘ sym)
 
-    f′-injective′ : Injective {B = P.setoid _} (→-to-⟶ f′)
+    f′-injective′ : Injective _≡_ _≡_ f′
     f′-injective′ {j} {k} eq with i ≤ᵇ j | Reflects.invert (≤ᵇ-reflects-≤ i j)
                                 | i ≤ᵇ k | Reflects.invert (≤ᵇ-reflects-≤ i k)
     ... | true  | p | true  | q = P.cong pred (f-inj eq)
@@ -385,8 +385,10 @@ finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
     ... | false | p | true  | q = contradiction (f-inj eq) (lemma q p ∘ sym)
     ... | false | p | false | q = f-inj eq
 
+    f′-inj : ℕ ↣ _
     f′-inj = record
-      { to        = →-to-⟶ f′
+      { to        = f′
+      ; cong      = P.cong f′
       ; injective = f′-injective′
       }
 
diff --git a/src/Data/List/Membership/Propositional/Properties/Core.agda b/src/Data/List/Membership/Propositional/Properties/Core.agda
index aec90ef8d1..fadd49bb4b 100644
--- a/src/Data/List/Membership/Propositional/Properties/Core.agda
+++ b/src/Data/List/Membership/Propositional/Properties/Core.agda
@@ -13,7 +13,7 @@
 module Data.List.Membership.Propositional.Properties.Core where
 
 open import Function.Base using (flip; id; _∘_)
-open import Function.Inverse using (_↔_; inverse)
+open import Function.Bundles
 open import Data.List.Base using (List)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional
@@ -80,7 +80,7 @@ module _ {P : Pred A p} where
   ∃∈-Any = uncurry′ lose ∘ proj₂
 
   Any↔ : ∀ {xs} → (∃ λ x → x ∈ xs × P x) ↔ Any P xs
-  Any↔ = inverse ∃∈-Any find from∘to lose∘find
+  Any↔ = mk↔ₛ′ ∃∈-Any find lose∘find from∘to
     where
     from∘to : ∀ v → find (∃∈-Any v) ≡ v
     from∘to p = find∘lose _ (proj₁ (proj₂ p)) (proj₂ (proj₂ p))
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 0dcca71055..1a74ace45f 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -25,7 +25,7 @@ open import Data.Product.Base as Prod using (∃; _×_; _,_ ; ∃₂; proj₁; p
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (_$_; flip; _∘_; _∘′_; id)
-open import Function.Inverse using (_↔_)
+open import Function.Bundles using (_↔_)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_; _Preserves_⟶_)
 open import Relation.Binary.Definitions as B hiding (Decidable)
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 451ce28c10..982f018367 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -30,12 +30,12 @@ open import Data.Sum.Properties hiding (map-cong)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
 open import Data.Unit.Polymorphic.Base
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-import Function.Equivalence as FE
-open import Function.Inverse as Inv using (_↔_; Inverse; inverse)
-open import Function.Related as Related
+open import Function.Bundles using (_↔_; Inverse; Equivalence; mk↔ₛ′; mk⇔)
+open import Function.Related.Propositional as Related
   using (↔⇒; ⌊_⌋; ⌊_⌋→; ⇒→; K-refl; SK-sym)
 open import Function.Related.TypeIsomorphisms
+open import Function.Properties.Inverse using (↔-sym; ↔-trans; to-from)
+open import Level using (Level)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Bundles using (Preorder; Setoid)
 import Relation.Binary.Reasoning.Setoid as EqR
@@ -45,27 +45,34 @@ open import Relation.Binary.PropositionalEquality as P
 open import Relation.Nullary
 open import Data.List.Membership.Propositional.Properties
 
+private
+  variable
+    a b : Level
+    A B : Set a
+    x y : A
+    ws xs ys zs : List A
+
 ------------------------------------------------------------------------
 -- Definitions
 
-open Related public using (Kind; Symmetric-kind) renaming
+open Related public using (Kind; SymmetricKind) renaming
   ( implication         to subset
-  ; reverse-implication to superset
+  ; reverseImplication  to superset
   ; equivalence         to set
   ; injection           to subbag
-  ; reverse-injection   to superbag
+  ; reverseInjection    to superbag
   ; bijection           to bag
   )
 
-[_]-Order : Kind → ∀ {a} → Set a → Preorder _ _ _
-[ k ]-Order A = Related.InducedPreorder₂ k {A = A} _∈_
+[_]-Order : Kind → Set a → Preorder _ _ _
+[ k ]-Order A = Related.InducedPreorder₂ {A = A} k _∈_
 
-[_]-Equality : Symmetric-kind → ∀ {a} → Set a → Setoid _ _
-[ k ]-Equality A = Related.InducedEquivalence₂ k {A = A} _∈_
+[_]-Equality : SymmetricKind → Set a → Setoid _ _
+[ k ]-Equality A = Related.InducedEquivalence₂ {A = A} k _∈_
 
 infix 4 _∼[_]_
 
-_∼[_]_ : ∀ {a} {A : Set a} → List A → Kind → List A → Set _
+_∼[_]_ : List A → Kind → List A → Set _
 _∼[_]_ {A = A} xs k ys = Preorder._∼_ ([ k ]-Order A) xs ys
 
 private
@@ -77,8 +84,7 @@ private
 ------------------------------------------------------------------------
 -- Bag equality implies the other relations.
 
-bag-=⇒ : ∀ {k a} {A : Set a} {xs ys : List A} →
-         xs ∼[ bag ] ys → xs ∼[ k ] ys
+bag-=⇒ : ∀ {k} → xs ∼[ bag ] ys → xs ∼[ k ] ys
 bag-=⇒ xs≈ys = ↔⇒ xs≈ys
 
 ------------------------------------------------------------------------
@@ -96,11 +102,11 @@ module ⊆-Reasoning where
 
   step-⊆ = PreOrder.step-∼
 
-  step-∈ : ∀ {a} {A : Set a} x {xs ys : List A} →
+  step-∈ : ∀ x {xs ys : List A} →
            xs IsRelatedTo ys → x ∈ xs → x ∈ ys
   step-∈ x xs⊆ys x∈xs = (begin xs⊆ys) x∈xs
 
-  step-∼ : ∀ {k a} {A : Set a} xs {ys zs : List A} →
+  step-∼ : ∀ {k} xs {ys zs : List A} →
            ys IsRelatedTo zs → xs ∼[ ⌊ k ⌋→ ] ys → xs IsRelatedTo zs
   step-∼ xs ys⊆zs xs≈ys = step-⊆ xs ys⊆zs (⇒→ xs≈ys)
 
@@ -113,12 +119,12 @@ module ⊆-Reasoning where
 ------------------------------------------------------------------------
 -- _∷_
 
-module _ {a k} {A : Set a} {x y : A} {xs ys} where
+module _ {k} {x y : A} {xs ys} where
 
   ∷-cong : x ≡ y → xs ∼[ k ] ys → x ∷ xs ∼[ k ] y ∷ ys
   ∷-cong refl xs≈ys {y} = begin
     y ∈ x ∷ xs        ↔⟨ SK-sym $ ∷↔ (y ≡_) ⟩
-    (y ≡ x ⊎ y ∈ xs)  ∼⟨ (y ≡ x ∎) ⊎-cong xs≈ys ⟩
+    (y ≡ x ⊎ y ∈ xs)  ∼⟨ K-refl ⊎-cong xs≈ys ⟩
     (y ≡ x ⊎ y ∈ ys)  ↔⟨ ∷↔ (y ≡_) ⟩
     y ∈ x ∷ ys        ∎
     where open Related.EquationalReasoning
@@ -126,7 +132,7 @@ module _ {a k} {A : Set a} {x y : A} {xs ys} where
 ------------------------------------------------------------------------
 -- map
 
-module _ {ℓ k} {A B : Set ℓ} {f g : A → B} {xs ys} where
+module _ {k} {f g : A → B} {xs ys} where
 
   map-cong : f ≗ g → xs ∼[ k ] ys → map f xs ∼[ k ] map g ys
   map-cong f≗g xs≈ys {x} = begin
@@ -138,19 +144,16 @@ module _ {ℓ k} {A B : Set ℓ} {f g : A → B} {xs ys} where
     open Related.EquationalReasoning
 
     helper : ∀ y → x ≡ f y ↔ x ≡ g y
-    helper y = record
-      { to         = P.→-to-⟶ (λ x≡fy → P.trans x≡fy (        f≗g y))
-      ; from       = P.→-to-⟶ (λ x≡gy → P.trans x≡gy (P.sym $ f≗g y))
-      ; inverse-of = record
-        { left-inverse-of  = λ { P.refl → P.trans-symʳ (f≗g y) }
-        ; right-inverse-of = λ { P.refl → P.trans-symˡ (f≗g y) }
-        }
-      }
+    helper y = mk↔ₛ′
+      (λ x≡fy → P.trans x≡fy (        f≗g y))
+      (λ x≡gy → P.trans x≡gy (P.sym $ f≗g y))
+      (λ { P.refl → P.trans-symˡ (f≗g y) })
+      λ { P.refl → P.trans-symʳ (f≗g y) }
 
 ------------------------------------------------------------------------
 -- _++_
 
-module _ {a k} {A : Set a} {xs₁ xs₂ ys₁ ys₂ : List A} where
+module _ {k} {xs₁ xs₂ ys₁ ys₂ : List A} where
 
   ++-cong : xs₁ ∼[ k ] xs₂ → ys₁ ∼[ k ] ys₂ →
             xs₁ ++ ys₁ ∼[ k ] xs₂ ++ ys₂
@@ -164,7 +167,7 @@ module _ {a k} {A : Set a} {xs₁ xs₂ ys₁ ys₂ : List A} where
 ------------------------------------------------------------------------
 -- concat
 
-module _ {a k} {A : Set a} {xss yss : List (List A)} where
+module _ {k} {xss yss : List (List A)} where
 
   concat-cong : xss ∼[ k ] yss → concat xss ∼[ k ] concat yss
   concat-cong xss≈yss {x} = begin
@@ -177,7 +180,7 @@ module _ {a k} {A : Set a} {xss yss : List (List A)} where
 ------------------------------------------------------------------------
 -- _>>=_
 
-module _ {ℓ k} {A B : Set ℓ} {xs ys} {f g : A → List B} where
+module _ {k} {A B : Set a} {xs ys} {f g : A → List B} where
 
   >>=-cong : xs ∼[ k ] ys → (∀ x → f x ∼[ k ] g x) →
              (xs >>= f) ∼[ k ] (ys >>= g)
@@ -192,7 +195,7 @@ module _ {ℓ k} {A B : Set ℓ} {xs ys} {f g : A → List B} where
 ------------------------------------------------------------------------
 -- _⊛_
 
-module _ {ℓ k} {A B : Set ℓ} {fs gs : List (A → B)} {xs ys} where
+module _ {k} {A B : Set a} {fs gs : List (A → B)} {xs ys} where
 
   ⊛-cong : fs ∼[ k ] gs → xs ∼[ k ] ys → (fs ⊛ xs) ∼[ k ] (gs ⊛ ys)
   ⊛-cong fs≈gs xs≈ys {x} = begin
@@ -221,8 +224,7 @@ module _ {ℓ k} {A B : Set ℓ} {xs₁ xs₂ : List A} {ys₁ ys₂ : List B} w
 -- _++_ and [] form a commutative monoid, with either bag or set
 -- equality as the underlying equality.
 
-commutativeMonoid : ∀ {a} → Symmetric-kind → Set a →
-                    CommutativeMonoid _ _
+commutativeMonoid : SymmetricKind → Set a → CommutativeMonoid _ _
 commutativeMonoid {a} k A = record
   { Carrier             = List A
   ; _≈_                 = _∼[ ⌊ k ⌋ ]_
@@ -237,10 +239,8 @@ commutativeMonoid {a} k A = record
       ; assoc         = λ xs ys zs →
                           Eq.reflexive (LP.++-assoc xs ys zs)
       }
-    ; identityˡ = λ xs {x} → x ∈ xs ∎
-    ; comm      = λ xs ys {x} →
-                    x ∈ xs ++ ys  ↔⟨ ++↔++ xs ys ⟩
-                    x ∈ ys ++ xs  ∎
+    ; identityˡ = λ xs → K-refl
+    ; comm      = λ xs ys → ↔⇒ (++↔++ xs ys)
     }
   }
   where open Related.EquationalReasoning
@@ -248,25 +248,24 @@ commutativeMonoid {a} k A = record
 -- The only list which is bag or set equal to the empty list (or a
 -- subset or subbag of the list) is the empty list itself.
 
-empty-unique : ∀ {k a} {A : Set a} {xs : List A} →
-               xs ∼[ ⌊ k ⌋→ ] [] → xs ≡ []
+empty-unique : ∀ {k} → xs ∼[ ⌊ k ⌋→ ] [] → xs ≡ []
 empty-unique {xs = []}    _    = refl
 empty-unique {xs = _ ∷ _} ∷∼[] with ⇒→ ∷∼[] (here refl)
 ... | ()
 
 -- _++_ is idempotent (under set equality).
 
-++-idempotent : ∀ {a} {A : Set a} → Idempotent {A = List A} _∼[ set ]_ _++_
-++-idempotent {a} xs {x} =
-  x ∈ xs ++ xs  ∼⟨ FE.equivalence ([ id , id ]′ ∘ _⟨$⟩_ (Inverse.from $ ++↔))
-                                  (_⟨$⟩_ (Inverse.to $ ++↔) ∘ inj₁) ⟩
+++-idempotent : Idempotent {A = List A} _∼[ set ]_ _++_
+++-idempotent xs {x} =
+  x ∈ xs ++ xs  ∼⟨ mk⇔ ([ id , id ]′ ∘ (Inverse.from $ ++↔))
+                                  (Inverse.to ++↔ ∘ inj₁) ⟩
   x ∈ xs        ∎
   where open Related.EquationalReasoning
 
 -- The list monad's bind distributes from the left over _++_.
 
 >>=-left-distributive :
-  ∀ {ℓ} {A B : Set ℓ} (xs : List A) {f g : A → List B} →
+  ∀ (xs : List A) {f g : A → List B} →
   (xs >>= λ x → f x ++ g x) ∼[ bag ] (xs >>= f) ++ (xs >>= g)
 >>=-left-distributive {ℓ} xs {f} {g} {y} =
   y ∈ (xs >>= λ x → f x ++ g x)                      ↔⟨ SK-sym $ >>=↔ ⟩
@@ -280,7 +279,7 @@ empty-unique {xs = _ ∷ _} ∷∼[] with ⇒→ ∷∼[] (here refl)
 -- The same applies to _⊛_.
 
 ⊛-left-distributive :
-  ∀ {ℓ} {A B : Set ℓ} (fs : List (A → B)) xs₁ xs₂ →
+  ∀ (fs : List (A → B)) xs₁ xs₂ →
   (fs ⊛ (xs₁ ++ xs₂)) ∼[ bag ] (fs ⊛ xs₁) ++ (fs ⊛ xs₂)
 ⊛-left-distributive {B = B} fs xs₁ xs₂ = begin
   fs ⊛ (xs₁ ++ xs₂)                       ≡⟨ Applicative.unfold-⊛ fs (xs₁ ++ xs₂) ⟩
@@ -300,10 +299,9 @@ private
   -- If x ∷ xs is set equal to x ∷ ys, then xs and ys are not
   -- necessarily set equal.
 
-  ¬-drop-cons : ∀ {a} {A : Set a} {x : A} →
+  ¬-drop-cons : ∀ {x : A} →
     ¬ (∀ {xs ys} → x ∷ xs ∼[ set ] x ∷ ys → xs ∼[ set ] ys)
-  ¬-drop-cons {x = x} drop-cons
-    with FE.Equivalence.to x∼[] ⟨$⟩ here refl
+  ¬-drop-cons {x = x} drop-cons with Equivalence.to x∼[] (here refl)
     where
     x,x≈x :  (x ∷ x ∷ []) ∼[ set ] [ x ]
     x,x≈x = ++-idempotent [ x ]
@@ -314,9 +312,8 @@ private
 
 -- However, the corresponding property does hold for bag equality.
 
-drop-cons : ∀ {a} {A : Set a} {x : A} {xs ys} →
-            x ∷ xs ∼[ bag ] x ∷ ys → xs ∼[ bag ] ys
-drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
+drop-cons : ∀ {x : A} {xs ys} → x ∷ xs ∼[ bag ] x ∷ ys → xs ∼[ bag ] ys
+drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
   ⊎-left-cancellative
     (∼→⊎↔⊎ x∷xs≈x∷ys)
     (lemma x∷xs≈x∷ys)
@@ -329,21 +326,20 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
   -- List membership can be expressed as "there is an index which
   -- points to the element".
 
-  ∈-index : ∀ {a} {A : Set a} {z}
-            (xs : List A) → z ∈ xs ↔ ∃ λ i → z ≡ lookup xs i
+  ∈-index : ∀ {z} (xs : List A) → z ∈ xs ↔ ∃ λ i → z ≡ lookup xs i
   ∈-index {z = z} [] =
     z ∈ []                              ↔⟨ SK-sym ⊥↔Any[] ⟩
-    ⊥                                   ↔⟨ SK-sym $ inverse (λ { (() , _) }) (λ ()) (λ { (() , _) }) (λ ()) ⟩
+    ⊥                                   ↔⟨ mk↔ₛ′ (λ ()) (λ { (() , _) }) (λ { (() , _) }) (λ ()) ⟩
     (∃ λ (i : Fin 0) → z ≡ lookup [] i)  ∎
     where
     open Related.EquationalReasoning
   ∈-index {z = z} (x ∷ xs) =
     z ∈ x ∷ xs                         ↔⟨ SK-sym (∷↔ _) ⟩
     (z ≡ x ⊎ z ∈ xs)                   ↔⟨ K-refl ⊎-cong ∈-index xs ⟩
-    (z ≡ x ⊎ ∃ λ i → z ≡ lookup xs i)  ↔⟨ SK-sym $ inverse (λ { (zero , p) → inj₁ p; (suc i , p) → inj₂ (i , p) })
-                                                          (λ { (inj₁ p) → zero , p; (inj₂ (i , p)) → suc i , p })
-                                                          (λ { (zero , _) → refl; (suc _ , _) → refl })
-                                                          (λ { (inj₁ _) → refl; (inj₂ _) → refl }) ⟩
+    (z ≡ x ⊎ ∃ λ i → z ≡ lookup xs i)  ↔⟨ mk↔ₛ′ (λ { (inj₁ p) → zero , p; (inj₂ (i , p)) → suc i , p })
+                                                  (λ { (zero , p) → inj₁ p; (suc i , p) → inj₂ (i , p) })
+                                                  (λ { (zero , _) → refl; (suc _ , _) → refl })
+                                                  (λ { (inj₁ _) → refl; (inj₂ _) → refl }) ⟩
     (∃ λ i → z ≡ lookup (x ∷ xs) i)    ∎
     where
     open Related.EquationalReasoning
@@ -352,7 +348,7 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
 
   index-of : ∀ {a} {A : Set a} {z} {xs : List A} →
              z ∈ xs → Fin (length xs)
-  index-of = proj₁ ∘ (Inverse.to (∈-index _) ⟨$⟩_)
+  index-of = proj₁ ∘ (Inverse.to (∈-index _))
 
   -- The type ∃ λ z → z ∈ xs is isomorphic to Fin n, where n is the
   -- length of xs.
@@ -365,7 +361,7 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
   Fin-length xs =
     (∃ λ z → z ∈ xs)                   ↔⟨ Σ.cong K-refl (∈-index xs) ⟩
     (∃ λ z → ∃ λ i → z ≡ lookup xs i)  ↔⟨ ∃∃↔∃∃ _ ⟩
-    (∃ λ i → ∃ λ z → z ≡ lookup xs i)  ↔⟨ Σ.cong K-refl (inverse _ (λ _ → _ , refl) (λ { (_ , refl) → refl }) (λ _ → refl)) ⟩
+    (∃ λ i → ∃ λ z → z ≡ lookup xs i)  ↔⟨ Σ.cong K-refl (mk↔ₛ′ _ (λ _ → _ , refl) (λ _ → refl) (λ { (_ , refl) → refl })) ⟩
     (Fin (length xs) × ⊤)              ↔⟨ ×-identityʳ _ _ ⟩
     Fin (length xs)                    ∎
     where
@@ -390,33 +386,33 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
   index-of-commutes :
     ∀ {a} {A : Set a} {z : A} {xs ys} →
     (xs≈ys : xs ∼[ bag ] ys) (p : z ∈ xs) →
-    index-of (Inverse.to xs≈ys ⟨$⟩ p) ≡
-    Inverse.to (Fin-length-cong xs≈ys) ⟨$⟩ index-of p
+    index-of (Inverse.to xs≈ys p) ≡
+    Inverse.to (Fin-length-cong xs≈ys) (index-of p)
   index-of-commutes {z = z} {xs} {ys} xs≈ys p =
-    index-of (to xs≈ys ⟨$⟩ p)                                        ≡⟨ lemma z p ⟩
+    index-of (to xs≈ys p)                                        ≡⟨ lemma z p ⟩
 
-    index-of (to xs≈ys ⟨$⟩ proj₂
-      (from (Fin-length xs) ⟨$⟩ (to (Fin-length xs) ⟨$⟩ (z , p))))   ≡⟨⟩
+    index-of (to xs≈ys (proj₂
+      (from (Fin-length xs) (to (Fin-length xs) (z , p)))))   ≡⟨⟩
 
-    index-of (proj₂ (Prod.map id (to xs≈ys ⟨$⟩_)
-      (from (Fin-length xs) ⟨$⟩ (to (Fin-length xs) ⟨$⟩ (z , p)))))  ≡⟨⟩
+    index-of (proj₂ (Prod.map id (to xs≈ys)
+      (from (Fin-length xs) (to (Fin-length xs) (z , p)))))  ≡⟨⟩
 
-    to (Fin-length ys) ⟨$⟩ Prod.map id (to xs≈ys ⟨$⟩_)
-      (from (Fin-length xs) ⟨$⟩ index-of p)                          ≡⟨⟩
+    to (Fin-length ys) (Prod.map id (to xs≈ys)
+      (from (Fin-length xs) (index-of p)))                          ≡⟨⟩
 
-    to (Fin-length-cong xs≈ys) ⟨$⟩ index-of p                        ∎
+    to (Fin-length-cong xs≈ys) (index-of p)                        ∎
     where
     open P.≡-Reasoning
     open Inverse
 
     lemma :
       ∀ z p →
-      index-of (to xs≈ys ⟨$⟩ p) ≡
-      index-of (to xs≈ys ⟨$⟩ proj₂
-        (from (Fin-length xs) ⟨$⟩ (to (Fin-length xs) ⟨$⟩ (z , p))))
-    lemma z p with to (Fin-length xs) ⟨$⟩ (z , p)
-                 | left-inverse-of (Fin-length xs) (z , p)
-    lemma .(lookup xs i) .(from (∈-index xs) ⟨$⟩ (i , refl)) | i | refl =
+      index-of (to xs≈ys p) ≡
+      index-of (to xs≈ys (proj₂
+        (from (Fin-length xs) (to (Fin-length xs) (z , p)))))
+    lemma z p with to (Fin-length xs) (z , p)
+                 | strictlyInverseʳ (Fin-length xs) (z , p)
+    lemma .(lookup xs i) .(from (∈-index xs) (i , refl)) | i | refl =
       refl
 
   -- Bag equivalence isomorphisms preserve index equality. Note that
@@ -428,13 +424,13 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
     ∀ {a} {A : Set a} {z : A} {xs ys} {p q : z ∈ xs}
     (xs≈ys : xs ∼[ bag ] ys) →
     index-of p ≡ index-of q →
-    index-of (Inverse.to xs≈ys ⟨$⟩ p) ≡
-    index-of (Inverse.to xs≈ys ⟨$⟩ q)
+    index-of (Inverse.to xs≈ys p) ≡
+    index-of (Inverse.to xs≈ys q)
   index-equality-preserved {p = p} {q} xs≈ys eq =
-    index-of (Inverse.to xs≈ys ⟨$⟩ p)                  ≡⟨ index-of-commutes xs≈ys p ⟩
-    Inverse.to (Fin-length-cong xs≈ys) ⟨$⟩ index-of p  ≡⟨ P.cong (Inverse.to (Fin-length-cong xs≈ys) ⟨$⟩_) eq ⟩
-    Inverse.to (Fin-length-cong xs≈ys) ⟨$⟩ index-of q  ≡⟨ P.sym $ index-of-commutes xs≈ys q ⟩
-    index-of (Inverse.to xs≈ys ⟨$⟩ q)                  ∎
+    index-of (Inverse.to xs≈ys p)                  ≡⟨ index-of-commutes xs≈ys p ⟩
+    Inverse.to (Fin-length-cong xs≈ys) (index-of p)  ≡⟨ P.cong (Inverse.to (Fin-length-cong xs≈ys)) eq ⟩
+    Inverse.to (Fin-length-cong xs≈ys) (index-of q)  ≡⟨ P.sym $ index-of-commutes xs≈ys q ⟩
+    index-of (Inverse.to xs≈ys q)                  ∎
     where
     open P.≡-Reasoning
 
@@ -458,14 +454,14 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
   ⊎-left-cancellative :
     ∀ {a b c} {A : Set a} {B : Set b} {C : Set c}
     (f : (A ⊎ B) ↔ (A ⊎ C)) →
-    Well-behaved (Inverse.to   f ⟨$⟩_) →
-    Well-behaved (Inverse.from f ⟨$⟩_) →
+    Well-behaved (Inverse.to   f) →
+    Well-behaved (Inverse.from f) →
     B ↔ C
-  ⊎-left-cancellative {A = A} = λ inv to-hyp from-hyp → inverse
-    (g (to   inv ⟨$⟩_) to-hyp)
-    (g (from inv ⟨$⟩_) from-hyp)
-    (g∘g         inv  to-hyp from-hyp)
+  ⊎-left-cancellative {A = A} = λ inv to-hyp from-hyp → mk↔ₛ′
+    (g (to   inv) to-hyp)
+    (g (from inv) from-hyp)
     (g∘g (SK-sym inv) from-hyp to-hyp)
+    (g∘g         inv  to-hyp from-hyp)
     where
     open Inverse
 
@@ -491,44 +487,44 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
 
     g∘g : ∀ {b c} {B : Set b} {C : Set c}
           (f : (A ⊎ B) ↔ (A ⊎ C)) →
-          (to-hyp   : Well-behaved (to   f ⟨$⟩_)) →
-          (from-hyp : Well-behaved (from f ⟨$⟩_)) →
-          ∀ b → g (from f ⟨$⟩_) from-hyp (g (to f ⟨$⟩_) to-hyp b) ≡ b
+          (to-hyp   : Well-behaved (to   f)) →
+          (from-hyp : Well-behaved (from f)) →
+          ∀ b → g (from f) from-hyp (g (to f) to-hyp b) ≡ b
     g∘g f to-hyp from-hyp b = g∘g′
       where
       open P.≡-Reasoning
 
-      g∘g′ : g (from f ⟨$⟩_) from-hyp (g (to f ⟨$⟩_) to-hyp b) ≡ b
-      g∘g′ with inspect (to f ⟨$⟩ inj₂ b)
-      g∘g′ | inj₂ c , eq₁ with inspect (from f ⟨$⟩ inj₂ c)
-      g∘g′ | inj₂ c , eq₁ | inj₂ b′ , eq₂ = inj₂-injective (
-                                              inj₂ b′            ≡⟨ P.sym eq₂ ⟩
-                                              from f ⟨$⟩ inj₂ c  ≡⟨ to-from f eq₁ ⟩
-                                              inj₂ b             ∎)
-      g∘g′ | inj₂ c , eq₁ | inj₁ a  , eq₂ with
+      g∘g′ : g (from f) from-hyp (g (to f) to-hyp b) ≡ b
+      g∘g′ with inspect (to f (inj₂ b))
+      g∘g′ | inj₂ c , eq₁ with inspect (from f (inj₂ c))
+      ...   | inj₂ b′ , eq₂ = inj₂-injective (
+        inj₂ b′            ≡⟨ P.sym eq₂ ⟩
+        from f (inj₂ c)   ≡⟨ to-from f eq₁ ⟩
+        inj₂ b            ∎)
+      ...   | inj₁ a  , eq₂ with
         inj₁ a             ≡⟨ P.sym eq₂ ⟩
-        from f ⟨$⟩ inj₂ c  ≡⟨ to-from f eq₁ ⟩
+        from f (inj₂ c)    ≡⟨ to-from f eq₁ ⟩
         inj₂ b             ∎
       ... | ()
-      g∘g′ | inj₁ a , eq₁ with inspect (to f ⟨$⟩ inj₁ a)
+      g∘g′ | inj₁ a , eq₁ with inspect (to f (inj₁ a))
       g∘g′ | inj₁ a , eq₁ | inj₁ a′ , eq₂ = ⊥-elim $ to-hyp eq₁ eq₂
-      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ with inspect (from f ⟨$⟩ inj₂ c)
+      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ with inspect (from f (inj₂ c))
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₂ b′ , eq₃ with
-        inj₁ a             ≡⟨ P.sym $ to-from f eq₂ ⟩
-        from f ⟨$⟩ inj₂ c  ≡⟨ eq₃ ⟩
+        inj₁ a             ≡⟨ P.sym (to-from f eq₂) ⟩
+        from f (inj₂ c)    ≡⟨ eq₃ ⟩
         inj₂ b′            ∎
       ... | ()
-      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ with inspect (from f ⟨$⟩ inj₁ a′)
+      g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ with inspect (from f $ inj₁ a′)
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₁ a″ , eq₄ = ⊥-elim $ from-hyp eq₃ eq₄
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₂ b′ , eq₄ = inj₂-injective (
         let lemma =
-              inj₁ a′            ≡⟨ P.sym eq₃ ⟩
-              from f ⟨$⟩ inj₂ c  ≡⟨ to-from f eq₂ ⟩
+              inj₁ a′             ≡⟨ P.sym eq₃ ⟩
+              from f (inj₂ c)    ≡⟨ to-from f eq₂ ⟩
               inj₁ a             ∎
         in
         inj₂ b′             ≡⟨ P.sym eq₄ ⟩
-        from f ⟨$⟩ inj₁ a′  ≡⟨ P.cong ((from f ⟨$⟩_) ∘ inj₁) $ inj₁-injective lemma ⟩
-        from f ⟨$⟩ inj₁ a   ≡⟨ to-from f eq₁ ⟩
+        from f (inj₁ a′)    ≡⟨ P.cong (from f ∘ inj₁) $ inj₁-injective lemma ⟩
+        from f (inj₁ a)    ≡⟨ to-from f eq₁ ⟩
         inj₂ b              ∎)
 
   -- Some final lemmas.
@@ -537,7 +533,7 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
     ∀ {x : A} {xs ys} →
     x ∷ xs ∼[ bag ] x ∷ ys →
     ∀ {z} → (z ≡ x ⊎ z ∈ xs) ↔ (z ≡ x ⊎ z ∈ ys)
-  ∼→⊎↔⊎ {x} {xs} {ys} x∷xs≈x∷ys {z} =
+  ∼→⊎↔⊎ {x = x} {xs} {ys} x∷xs≈x∷ys {z} =
     (z ≡ x ⊎ z ∈ xs)  ↔⟨ ∷↔ _ ⟩
     z ∈ x ∷ xs        ↔⟨ x∷xs≈x∷ys ⟩
     z ∈ x ∷ ys        ↔⟨ SK-sym (∷↔ _) ⟩
@@ -546,20 +542,20 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
     open Related.EquationalReasoning
 
   lemma : ∀ {xs ys} (inv : x ∷ xs ∼[ bag ] x ∷ ys) {z} →
-          Well-behaved (Inverse.to (∼→⊎↔⊎ inv {z}) ⟨$⟩_)
+          Well-behaved (Inverse.to (∼→⊎↔⊎ inv {z}))
   lemma {xs} inv {b = z∈xs} {a = p} {a′ = q} hyp₁ hyp₂ with
     zero                                                                  ≡⟨⟩
     index-of {xs = x ∷ xs} (here p)                                       ≡⟨⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) ⟨$⟩ inj₁ p)                         ≡⟨ P.cong (index-of ∘ (to (∷↔ (_ ≡_)) ⟨$⟩_)) $ P.sym $
+    index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₁ p)                         ≡⟨ P.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ P.sym $
                                                                                to-from (∼→⊎↔⊎ inv) {x = inj₁ p} hyp₂ ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) ⟨$⟩ (from (∼→⊎↔⊎ inv) ⟨$⟩ inj₁ q))  ≡⟨ P.cong index-of $
-                                                                               right-inverse-of (∷↔ _) (from inv ⟨$⟩ here q) ⟩
-    index-of {xs = x ∷ xs} (to (SK-sym inv) ⟨$⟩ here q)                   ≡⟨ index-equality-preserved (SK-sym inv) refl ⟩
-    index-of {xs = x ∷ xs} (to (SK-sym inv) ⟨$⟩ here p)                   ≡⟨ P.cong index-of $ P.sym $
-                                                                               right-inverse-of (∷↔ _) (from inv ⟨$⟩ here p) ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) ⟨$⟩ (from (∼→⊎↔⊎ inv) ⟨$⟩ inj₁ p))  ≡⟨ P.cong (index-of ∘ (to (∷↔ (_ ≡_)) ⟨$⟩_)) $
+    index-of {xs = x ∷ xs} (to (∷↔ _) $ (from (∼→⊎↔⊎ inv) $ inj₁ q))  ≡⟨ P.cong index-of $
+                                                                               strictlyInverseˡ (∷↔ _) (from inv (here q)) ⟩
+    index-of {xs = x ∷ xs} (to (SK-sym inv) $ here q)                   ≡⟨ index-equality-preserved (SK-sym inv) refl ⟩
+    index-of {xs = x ∷ xs} (to (SK-sym inv) $ here p)                   ≡⟨ P.cong index-of $ P.sym $
+                                                                               strictlyInverseˡ (∷↔ _) (from inv (here p)) ⟩
+    index-of {xs = x ∷ xs} (to (∷↔ _) (from (∼→⊎↔⊎ inv) $ inj₁ p))  ≡⟨ P.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $
                                                                                to-from (∼→⊎↔⊎ inv) {x = inj₂ z∈xs} hyp₁ ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) ⟨$⟩ inj₂ z∈xs)                      ≡⟨⟩
+    index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₂ z∈xs)                      ≡⟨⟩
     index-of {xs = x ∷ xs} (there z∈xs)                                   ≡⟨⟩
     suc (index-of {xs = xs} z∈xs)                                         ∎
     where
@@ -570,42 +566,39 @@ drop-cons {A = A} {x} {xs} {ys} x∷xs≈x∷ys =
 ------------------------------------------------------------------------
 -- Relationships to other relations
 
-module _ {a} {A : Set a} where
-
-  ↭⇒∼bag : _↭_ ⇒ _∼[ bag ]_
-  ↭⇒∼bag xs↭ys {v} = inverse (to xs↭ys) (from xs↭ys) (from∘to xs↭ys) (to∘from xs↭ys)
-    where
-    to : ∀ {xs ys} → xs ↭ ys → v ∈ xs → v ∈ ys
-    to xs↭ys = Any-resp-↭ {A = A} xs↭ys
-
-    from : ∀ {xs ys} → xs ↭ ys → v ∈ ys → v ∈ xs
-    from xs↭ys = Any-resp-↭ (↭-sym xs↭ys)
-
-    from∘to : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ xs) → from p (to p q) ≡ q
-    from∘to refl          v∈xs                 = refl
-    from∘to (prep _ _)    (here refl)          = refl
-    from∘to (prep _ p)    (there v∈xs)         = P.cong there (from∘to p v∈xs)
-    from∘to (swap x y p)  (here refl)          = refl
-    from∘to (swap x y p)  (there (here refl))  = refl
-    from∘to (swap x y p)  (there (there v∈xs)) = P.cong (there ∘ there) (from∘to p v∈xs)
-    from∘to (trans p₁ p₂) v∈xs
-      rewrite from∘to p₂ (Any-resp-↭ p₁ v∈xs)
-            | from∘to p₁ v∈xs                  = refl
-
-    to∘from : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ ys) → to p (from p q) ≡ q
-    to∘from p with from∘to (↭-sym p)
-    ... | res rewrite ↭-sym-involutive p = res
-
-  ∼bag⇒↭ : _∼[ bag ]_ ⇒ _↭_
-  ∼bag⇒↭ {[]} eq with empty-unique {A = A} (Inv.sym eq)
-  ... | refl = refl
-  ∼bag⇒↭ {x ∷ xs} eq with ∈-∃++ (to ⟨$⟩ (here P.refl))
-    where open Inv.Inverse (eq {x})
-  ... | zs₁ , zs₂ , p rewrite p = begin
-    x ∷ xs           <⟨ ∼bag⇒↭ (drop-cons (Inv._∘_ (comm zs₁ (x ∷ zs₂)) eq)) ⟩
-    x ∷ (zs₂ ++ zs₁) <⟨ ++-comm zs₂ zs₁ ⟩
-    x ∷ (zs₁ ++ zs₂) ↭˘⟨ shift x zs₁ zs₂ ⟩
-    zs₁ ++ x ∷ zs₂   ∎
-    where
-    open CommutativeMonoid (commutativeMonoid bag A)
-    open PermutationReasoning
+↭⇒∼bag : _↭_ {A = A} ⇒ _∼[ bag ]_
+↭⇒∼bag {A = A} xs↭ys {v} = mk↔ₛ′ (to xs↭ys) (from xs↭ys) (to∘from xs↭ys) (from∘to xs↭ys)
+  where
+  to : ∀ {xs ys} → xs ↭ ys → v ∈ xs → v ∈ ys
+  to xs↭ys = Any-resp-↭ {A = A} xs↭ys
+
+  from : ∀ {xs ys} → xs ↭ ys → v ∈ ys → v ∈ xs
+  from xs↭ys = Any-resp-↭ (↭-sym xs↭ys)
+
+  from∘to : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ xs) → from p (to p q) ≡ q
+  from∘to refl          v∈xs                 = refl
+  from∘to (prep _ _)    (here refl)          = refl
+  from∘to (prep _ p)    (there v∈xs)         = P.cong there (from∘to p v∈xs)
+  from∘to (swap x y p)  (here refl)          = refl
+  from∘to (swap x y p)  (there (here refl))  = refl
+  from∘to (swap x y p)  (there (there v∈xs)) = P.cong (there ∘ there) (from∘to p v∈xs)
+  from∘to (trans p₁ p₂) v∈xs
+    rewrite from∘to p₂ (Any-resp-↭ p₁ v∈xs)
+          | from∘to p₁ v∈xs                  = refl
+
+  to∘from : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ ys) → to p (from p q) ≡ q
+  to∘from p with from∘to (↭-sym p)
+  ... | res rewrite ↭-sym-involutive p = res
+
+∼bag⇒↭ : _∼[ bag ]_ ⇒ _↭_ {A = A}
+∼bag⇒↭ {A = A} {[]} eq with empty-unique (↔-sym eq)
+... | refl = refl
+∼bag⇒↭ {A = A} {x ∷ xs} eq with ∈-∃++ (Inverse.to (eq {x}) (here P.refl))
+... | zs₁ , zs₂ , p rewrite p = begin
+  x ∷ xs           <⟨ ∼bag⇒↭ (drop-cons (↔-trans eq (comm zs₁ (x ∷ zs₂)))) ⟩
+  x ∷ (zs₂ ++ zs₁) <⟨ ++-comm zs₂ zs₁ ⟩
+  x ∷ (zs₁ ++ zs₂) ↭˘⟨ shift x zs₁ zs₂ ⟩
+  zs₁ ++ x ∷ zs₂   ∎
+  where
+  open CommutativeMonoid (commutativeMonoid bag A)
+  open PermutationReasoning
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index 90196f8744..1b33a771a0 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -23,7 +23,6 @@ open import Data.List.Membership.Propositional.Properties
 import Data.List.Properties as Lₚ
 open import Data.Product.Base using (_,_; _×_; ∃; ∃₂)
 open import Function.Base using (_∘_; _⟨_⟩_)
-open import Function.Inverse as Inv using (inverse)
 open import Level using (Level)
 open import Relation.Unary using (Pred)
 open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index 222bcee92f..71313d087e 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -30,9 +30,7 @@ import Data.Product.Base as Prod
 import Data.Sum.Base as Sum
 open import Effect.Monad
 open import Function.Base using (_∘_; _∘′_; id; _$_)
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Inverse as Inv using (_↔_; module Inverse)
-open import Function.Equivalence using (module Equivalence)
+open import Function.Bundles using (_↔_; Inverse; Equivalence)
 open import Level using (Level)
 open import Relation.Nullary using (¬_; yes; no)
 open import Relation.Unary using (Decidable; Pred) renaming (_⊆_ to _⋐_)
@@ -131,9 +129,9 @@ All-resp-⊇ = Setoidₚ.All-resp-⊇ (setoid _) (subst _)
 
 map⁺ : ∀ (f : A → B) → xs ⊆ ys → map f xs ⊆ map f ys
 map⁺ f xs⊆ys =
-  _⟨$⟩_ (Inverse.to (map-∈↔ f)) ∘
+  Inverse.to (map-∈↔ f) ∘
   Prod.map₂ (Prod.map₁ xs⊆ys) ∘
-  _⟨$⟩_ (Inverse.from (map-∈↔ f))
+  Inverse.from (map-∈↔ f)
 
 ------------------------------------------------------------------------
 -- ∷
@@ -172,9 +170,9 @@ module _ {xss yss : List (List A)} where
 
   concat⁺ : xss ⊆ yss → concat xss ⊆ concat yss
   concat⁺ xss⊆yss =
-    _⟨$⟩_ (Inverse.to concat-∈↔) ∘
+    Inverse.to concat-∈↔ ∘
     Prod.map₂ (Prod.map₂ xss⊆yss) ∘
-    _⟨$⟩_ (Inverse.from concat-∈↔)
+    Inverse.from concat-∈↔
 
 ------------------------------------------------------------------------
 -- applyUpTo
@@ -189,9 +187,9 @@ module _ {A B : Set a} (f g : A → List B) where
 
   >>=⁺ : xs ⊆ ys → (∀ {x} → f x ⊆ g x) → (xs >>= f) ⊆ (ys >>= g)
   >>=⁺ xs⊆ys f⊆g =
-    _⟨$⟩_ (Inverse.to >>=-∈↔) ∘
+    Inverse.to >>=-∈↔ ∘
     Prod.map₂ (Prod.map xs⊆ys f⊆g) ∘
-    _⟨$⟩_ (Inverse.from >>=-∈↔)
+    Inverse.from >>=-∈↔
 
 ------------------------------------------------------------------------
 -- _⊛_
@@ -200,9 +198,9 @@ module _ {A B : Set a} {fs gs : List (A → B)} where
 
   ⊛⁺ : fs ⊆ gs → xs ⊆ ys → (fs ⊛ xs) ⊆ (gs ⊛ ys)
   ⊛⁺ fs⊆gs xs⊆ys =
-    _⟨$⟩_ (Inverse.to $ ⊛-∈↔ gs) ∘
+    (Inverse.to $ ⊛-∈↔ gs) ∘
     Prod.map₂ (Prod.map₂ (Prod.map fs⊆gs (Prod.map₁ xs⊆ys))) ∘
-    _⟨$⟩_ (Inverse.from $ ⊛-∈↔ fs)
+    (Inverse.from $ ⊛-∈↔ fs)
 
 ------------------------------------------------------------------------
 -- _⊗_
@@ -211,9 +209,9 @@ module _ {A B : Set a} {ws xs : List A} {ys zs : List B} where
 
   ⊗⁺ : ws ⊆ xs → ys ⊆ zs → (ws ⊗ ys) ⊆ (xs ⊗ zs)
   ⊗⁺ ws⊆xs ys⊆zs =
-    _⟨$⟩_ (Inverse.to ⊗-∈↔) ∘
+    Inverse.to ⊗-∈↔ ∘
     Prod.map ws⊆xs ys⊆zs ∘
-    _⟨$⟩_ (Inverse.from ⊗-∈↔)
+    Inverse.from ⊗-∈↔
 
 ------------------------------------------------------------------------
 -- any
@@ -222,9 +220,9 @@ module _ (p : A → Bool) {xs ys} where
 
   any⁺ : xs ⊆ ys → T (any p xs) → T (any p ys)
   any⁺ xs⊆ys =
-    _⟨$⟩_ (Equivalence.to Any.any⇔) ∘
+    Equivalence.to Any.any⇔ ∘
     Any-resp-⊆ xs⊆ys ∘
-    _⟨$⟩_ (Equivalence.from Any.any⇔)
+    Equivalence.from Any.any⇔
 
 ------------------------------------------------------------------------
 -- mapWith∈
@@ -236,12 +234,12 @@ module _ {xs : List A} {f : ∀ {x} → x ∈ xs → B}
   mapWith∈⁺ : (xs⊆ys : xs ⊆ ys) → (∀ {x} → f {x} ≗ g ∘ xs⊆ys) →
                 mapWith∈ xs f ⊆ mapWith∈ ys g
   mapWith∈⁺ xs⊆ys f≈g {x} =
-    _⟨$⟩_ (Inverse.to Any.mapWith∈↔) ∘
+    Inverse.to Any.mapWith∈↔ ∘
     Prod.map₂ (Prod.map xs⊆ys (λ {x∈xs} x≡fx∈xs → begin
       x               ≡⟨ x≡fx∈xs ⟩
       f x∈xs          ≡⟨ f≈g x∈xs ⟩
       g (xs⊆ys x∈xs)  ∎)) ∘
-    _⟨$⟩_ (Inverse.from Any.mapWith∈↔)
+    Inverse.from Any.mapWith∈↔
     where open ≡-Reasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index 1a75b5eb35..6af4fd2a8c 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -33,10 +33,7 @@ open import Data.Nat.Base using (zero; suc; s≤s; _<_; z<s; s<s)
 open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)
 open import Data.Product.Base as Prod using (_×_; _,_; uncurry; uncurry′)
 open import Function.Base
-import Function.Bundles as B
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
-open import Function.Surjection using (_↠_; surjection)
+open import Function.Bundles
 open import Level using (Level)
 open import Relation.Binary.Core using (REL)
 open import Relation.Binary.Bundles using (Setoid)
@@ -109,7 +106,7 @@ Any¬⇒¬All (here  ¬p) = ¬p           ∘ All.head
 Any¬⇒¬All (there ¬p) = Any¬⇒¬All ¬p ∘ All.tail
 
 ¬Any↠All¬ : ∀ {xs} → (¬ Any P xs) ↠ All (¬_ ∘ P) xs
-¬Any↠All¬ = surjection (¬Any⇒All¬ _) All¬⇒¬Any to∘from
+¬Any↠All¬ = mk↠ₛ {to = ¬Any⇒All¬ _} (λ y → All¬⇒¬Any y , to∘from y)
   where
   to∘from : ∀ {xs} (¬p : All (¬_ ∘ P) xs) → ¬Any⇒All¬ xs (All¬⇒¬Any ¬p) ≡ ¬p
   to∘from []         = refl
@@ -127,7 +124,7 @@ Any¬⇒¬All (there ¬p) = Any¬⇒¬All ¬p ∘ All.tail
     }
 
 Any¬⇔¬All : ∀ {xs} → Decidable P → Any (¬_ ∘ P) xs ⇔ (¬ All P xs)
-Any¬⇔¬All dec = equivalence Any¬⇒¬All (¬All⇒Any¬ dec _)
+Any¬⇔¬All dec = mk⇔ Any¬⇒¬All (¬All⇒Any¬ dec _)
 
 private
   -- If equality of functions were extensional, then the logical
@@ -407,8 +404,8 @@ mapMaybe⁺ {xs = x ∷ xs} {f = f} (px ∷ pxs) with f x
 ++⁻ []       p          = [] , p
 ++⁻ (x ∷ xs) (px ∷ pxs) = Prod.map (px ∷_) id (++⁻ _ pxs)
 
-++↔ : (All P xs × All P ys) B.↔ All P (xs ++ ys)
-++↔ {xs = zs} = B.mk↔′ (uncurry ++⁺) (++⁻ zs) (++⁺∘++⁻ zs) ++⁻∘++⁺
+++↔ : (All P xs × All P ys) ↔ All P (xs ++ ys)
+++↔ {xs = zs} = mk↔ₛ′ (uncurry ++⁺) (++⁻ zs) (++⁺∘++⁻ zs) ++⁻∘++⁺
   where
   ++⁺∘++⁻ : ∀ xs (p : All P (xs ++ ys)) → uncurry′ ++⁺ (++⁻ xs p) ≡ p
   ++⁺∘++⁻ []       p          = refl
@@ -708,12 +705,12 @@ module _ (p : A → Bool) where
 
   all⁺ : ∀ xs → T (all p xs) → All (T ∘ p) xs
   all⁺ []       _     = []
-  all⁺ (x ∷ xs) px∷xs with Equivalence.to (T-∧ {p x}) ⟨$⟩ px∷xs
+  all⁺ (x ∷ xs) px∷xs with Equivalence.to (T-∧ {p x}) px∷xs
   ... | (px , pxs) = px ∷ all⁺ xs pxs
 
   all⁻ : All (T ∘ p) xs → T (all p xs)
   all⁻ []         = _
-  all⁻ (px ∷ pxs) = Equivalence.from T-∧ ⟨$⟩ (px , all⁻ pxs)
+  all⁻ (px ∷ pxs) = Equivalence.from T-∧ (px , all⁻ pxs)
 
 ------------------------------------------------------------------------
 -- All is anti-monotone.
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index d51a5cba64..c72bd0707f 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -35,10 +35,9 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
 open import Effect.Monad
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence using (_⇔_; equivalence; Equivalence)
-open import Function.Inverse as Inv using (_↔_; inverse; Inverse)
-open import Function.Related as Related using (Kind; Related; SK-sym)
+open import Function.Bundles
+import Function.Properties.Inverse as Inverse
+open import Function.Related.Propositional as Related using (Kind; Related)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel; REL)
 open import Relation.Binary.Definitions as B
@@ -56,9 +55,7 @@ private
 private
   variable
     a b c p q r ℓ : Level
-    A : Set a
-    B : Set b
-    C : Set c
+    A B C : Set a
     P Q R : Pred A p
     x y : A
     xs ys : List A
@@ -90,8 +87,8 @@ Any-cong : ∀ {k : Kind} → (∀ x → Related k (P x) (Q x)) →
            (∀ {z} → Related k (z ∈ xs) (z ∈ ys)) →
            Related k (Any P xs) (Any Q ys)
 Any-cong {P = P} {Q = Q} {xs = xs} {ys} P↔Q xs≈ys =
-  Any P xs                ↔⟨ SK-sym Any↔ ⟩
-  (∃ λ x → x ∈ xs × P x)  ∼⟨ Σ.cong Inv.id (xs≈ys ×-cong P↔Q _) ⟩
+  Any P xs                ↔⟨ Related.SK-sym Any↔ ⟩
+  (∃ λ x → x ∈ xs × P x)  ∼⟨ Σ.cong Inverse.↔-refl (xs≈ys ×-cong P↔Q _) ⟩
   (∃ λ x → x ∈ ys × Q x)  ↔⟨ Any↔ ⟩
   Any Q ys                ∎
   where open Related.EquationalReasoning
@@ -147,19 +144,19 @@ swap-invol (there pxys)       =
 
 swap↔ : ∀ {P : A → B → Set ℓ} →
        Any (λ x → Any (P x) ys) xs ↔ Any (λ y → Any (flip P y) xs) ys
-swap↔ = inverse swap swap swap-invol swap-invol
+swap↔ = mk↔ₛ′ swap swap swap-invol swap-invol
 
 ------------------------------------------------------------------------
 -- Lemmas relating Any to ⊥
 
 ⊥↔Any⊥ : ⊥ ↔ Any (const ⊥) xs
-⊥↔Any⊥ = inverse (λ()) (λ p → from p) (λ()) (λ p → from p)
+⊥↔Any⊥ = mk↔ₛ′ (λ()) (λ p → from p) (λ p → from p) (λ())
   where
   from : Any (const ⊥) xs → B
   from (there p) = from p
 
 ⊥↔Any[] : ⊥ ↔ Any P []
-⊥↔Any[] = inverse (λ()) (λ()) (λ()) (λ())
+⊥↔Any[] = mk↔ₛ′ (λ()) (λ()) (λ()) (λ())
 
 ------------------------------------------------------------------------
 -- Lemmas relating Any to ⊤
@@ -167,18 +164,18 @@ swap↔ = inverse swap swap swap-invol swap-invol
 -- These introduction and elimination rules are not inverses, though.
 
 any⁺ : ∀ (p : A → Bool) → Any (T ∘ p) xs → T (any p xs)
-any⁺ p (here  px)          = Equivalence.from T-∨ ⟨$⟩ inj₁ px
+any⁺ p (here  px)          = Equivalence.from T-∨ (inj₁ px)
 any⁺ p (there {x = x} pxs) with p x
 ... | true  = _
 ... | false = any⁺ p pxs
 
 any⁻ : ∀ (p : A → Bool) xs → T (any p xs) → Any (T ∘ p) xs
 any⁻ p (x ∷ xs) px∷xs with p x in eq
-... | true  = here (Equivalence.from T-≡ ⟨$⟩ eq)
+... | true  = here (Equivalence.from T-≡ eq)
 ... | false = there (any⁻ p xs px∷xs)
 
 any⇔ : ∀ {p : A → Bool} → Any (T ∘ p) xs ⇔ T (any p xs)
-any⇔ = equivalence (any⁺ _) (any⁻ _ _)
+any⇔ = mk⇔ (any⁺ _) (any⁻ _ _)
 
 ------------------------------------------------------------------------
 -- Sums commute with Any
@@ -192,7 +189,7 @@ Any-⊎⁻ (here (inj₂ q)) = inj₂ (here q)
 Any-⊎⁻ (there p)       = Sum.map there there (Any-⊎⁻ p)
 
 ⊎↔ : (Any P xs ⊎ Any Q xs) ↔ Any (λ x → P x ⊎ Q x) xs
-⊎↔ {P = P} {Q = Q} = inverse Any-⊎⁺ Any-⊎⁻ from∘to to∘from
+⊎↔ {P = P} {Q = Q} = mk↔ₛ′ Any-⊎⁺ Any-⊎⁻ to∘from from∘to
   where
   from∘to : (p : Any P xs ⊎ Any Q xs) → Any-⊎⁻ (Any-⊎⁺ p) ≡ p
   from∘to (inj₁ (here  p)) = refl
@@ -204,8 +201,8 @@ Any-⊎⁻ (there p)       = Sum.map there there (Any-⊎⁻ p)
   to∘from (here (inj₁ p)) = refl
   to∘from (here (inj₂ q)) = refl
   to∘from (there p) with Any-⊎⁻ p | to∘from p
-  to∘from (there .(Any.map inj₁ p)) | inj₁ p | refl = refl
-  to∘from (there .(Any.map inj₂ q)) | inj₂ q | refl = refl
+  ... | inj₁ p | refl = refl
+  ... | inj₂ q | refl = refl
 
 ------------------------------------------------------------------------
 -- Products "commute" with Any.
@@ -220,7 +217,7 @@ Any-×⁻ pq with Prod.map₂ (Prod.map₂ find) (find pq)
 
 ×↔ : ∀ {xs ys} →
      (Any P xs × Any Q ys) ↔ Any (λ x → Any (λ y → P x × Q y) ys) xs
-×↔ {P = P} {Q = Q} {xs} {ys} = inverse Any-×⁺ Any-×⁻ from∘to to∘from
+×↔ {P = P} {Q = Q} {xs} {ys} = mk↔ₛ′ Any-×⁺ Any-×⁻ to∘from from∘to
   where
   open P.≡-Reasoning
 
@@ -259,8 +256,7 @@ Any-×⁻ pq with Prod.map₂ (Prod.map₂ find) (find pq)
 
 
   to∘from : ∀ pq → Any-×⁺ {xs = xs} (Any-×⁻ pq) ≡ pq
-  to∘from pq
-    with find pq
+  to∘from pq with find pq
       | (λ (f : (proj₁ (find pq) ≡_) ⋐ _) → map∘find pq {f})
   ... | (x , x∈xs , pq′) | lem₁
     with find pq′
@@ -328,7 +324,7 @@ module _ {f : A → B} where
   map⁻∘map⁺ P (there p) = P.cong there (map⁻∘map⁺ P p)
 
   map↔ : Any (P ∘ f) xs ↔ Any P (List.map f xs)
-  map↔ = inverse map⁺ map⁻ (map⁻∘map⁺ _) map⁺∘map⁻
+  map↔ = mk↔ₛ′ map⁺ map⁻ map⁺∘map⁻ (map⁻∘map⁺ _)
 
   gmap : P ⋐ Q ∘ f → Any P ⋐ Any Q ∘ map f
   gmap g = map⁺ ∘ Any.map g
@@ -377,7 +373,7 @@ module _ {P : A → Set p} where
   ++⁻∘++⁺ (x ∷ xs)      (inj₂ p)         rewrite ++⁻∘++⁺ xs      (inj₂ p) = refl
 
   ++↔ : ∀ {xs ys} → (Any P xs ⊎ Any P ys) ↔ Any P (xs ++ ys)
-  ++↔ {xs = xs} = inverse [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs) (++⁻∘++⁺ xs) (++⁺∘++⁻ xs)
+  ++↔ {xs = xs} = mk↔ₛ′ [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs) (++⁺∘++⁻ xs) (++⁻∘++⁺ xs)
 
   ++-comm : ∀ xs ys → Any P (xs ++ ys) → Any P (ys ++ xs)
   ++-comm xs ys = [ ++⁺ʳ ys , ++⁺ˡ ]′ ∘ ++⁻ xs
@@ -399,8 +395,8 @@ module _ {P : A → Set p} where
           | ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₁ p) = refl
 
   ++↔++ : ∀ xs ys → Any P (xs ++ ys) ↔ Any P (ys ++ xs)
-  ++↔++ xs ys = inverse (++-comm xs ys) (++-comm ys xs)
-                        (++-comm∘++-comm xs) (++-comm∘++-comm ys)
+  ++↔++ xs ys = mk↔ₛ′ (++-comm xs ys) (++-comm ys xs)
+                        (++-comm∘++-comm ys) (++-comm∘++-comm xs)
 
   ++-insert : ∀ xs {ys} → P x → Any P (xs ++ [ x ] ++ ys)
   ++-insert xs Px = ++⁺ʳ xs (++⁺ˡ (singleton⁺ Px))
@@ -447,7 +443,7 @@ module _ {P : A → Set p} where
       P.cong there $ concat⁻∘concat⁺ p
 
   concat↔ : ∀ {xss} → Any (Any P) xss ↔ Any P (concat xss)
-  concat↔ {xss} = inverse concat⁺ (concat⁻ xss) concat⁻∘concat⁺ (concat⁺∘concat⁻ xss)
+  concat↔ {xss} = mk↔ₛ′ concat⁺ (concat⁻ xss) (concat⁺∘concat⁻ xss) concat⁻∘concat⁺
 
 ------------------------------------------------------------------------
 -- cartesianProductWith
@@ -497,18 +493,16 @@ applyUpTo⁻ f {suc n} (there p) with applyUpTo⁻ (f ∘ suc) p
 ------------------------------------------------------------------------
 -- applyDownFrom
 
-module _ {P : A → Set p} where
-
-  applyDownFrom⁺ : ∀ f {i n} → P (f i) → i < n → Any P (applyDownFrom f n)
-  applyDownFrom⁺ f {i} {suc n} p (s≤s i≤n) with i ≟ n
-  ... | yes P.refl = here p
-  ... | no  i≢n    = there (applyDownFrom⁺ f p (≤∧≢⇒< i≤n i≢n))
+applyDownFrom⁺ : ∀ f {i n} → P (f i) → i < n → Any P (applyDownFrom f n)
+applyDownFrom⁺ f {i} {suc n} p (s≤s i≤n) with i ≟ n
+... | yes P.refl = here p
+... | no  i≢n    = there (applyDownFrom⁺ f p (≤∧≢⇒< i≤n i≢n))
 
-  applyDownFrom⁻ : ∀ f {n} → Any P (applyDownFrom f n) →
-                   ∃ λ i → i < n × P (f i)
-  applyDownFrom⁻ f {suc n} (here p)  = n , ≤-refl , p
-  applyDownFrom⁻ f {suc n} (there p) with applyDownFrom⁻ f p
-  ... | i , i<n , pf = i , m<n⇒m<1+n i<n , pf
+applyDownFrom⁻ : ∀ f {n} → Any P (applyDownFrom f n) →
+                 ∃ λ i → i < n × P (f i)
+applyDownFrom⁻ f {suc n} (here p)  = n , ≤-refl , p
+applyDownFrom⁻ f {suc n} (there p) with applyDownFrom⁻ f p
+... | i , i<n , pf = i , m<n⇒m<1+n i<n , pf
 
 ------------------------------------------------------------------------
 -- tabulate
@@ -602,7 +596,7 @@ module _ {P : B → Set p} where
 
   mapWith∈↔ : ∀ {xs : List A} {f : ∀ {x} → x ∈ xs → B} →
                 (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) ↔ Any P (mapWith∈ xs f)
-  mapWith∈↔ = inverse (mapWith∈⁺ _) (mapWith∈⁻ _ _) (from∘to _) (to∘from _ _)
+  mapWith∈↔ = mk↔ₛ′ (mapWith∈⁺ _) (mapWith∈⁻ _ _) (to∘from _ _) (from∘to _)
     where
     from∘to : ∀ {xs : List A} (f : ∀ {x} → x ∈ xs → B)
               (p : ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) →
@@ -645,34 +639,30 @@ reverse⁻ ps with reverseAcc⁻ [] _ ps
 ------------------------------------------------------------------------
 -- pure
 
-module _ {P : A → Set p} where
-
-  pure⁺ : P x → Any P (pure x)
-  pure⁺ = here
+pure⁺ : P x → Any P (pure x)
+pure⁺ = here
 
-  pure⁻ : Any P (pure x) → P x
-  pure⁻ (here p) = p
+pure⁻ : Any P (pure x) → P x
+pure⁻ (here p) = p
 
-  pure⁺∘pure⁻ : (p : Any P (pure x)) → pure⁺ (pure⁻ p) ≡ p
-  pure⁺∘pure⁻ (here p) = refl
+pure⁺∘pure⁻ : (p : Any P (pure x)) → pure⁺ (pure⁻ p) ≡ p
+pure⁺∘pure⁻ (here p) = refl
 
-  pure⁻∘pure⁺ : (p : P x) → pure⁻ (pure⁺ p) ≡ p
-  pure⁻∘pure⁺ p = refl
+pure⁻∘pure⁺ : (p : P x) → pure⁻ {P = P} (pure⁺ p) ≡ p
+pure⁻∘pure⁺ p = refl
 
-  pure↔ : P x ↔ Any P (pure x)
-  pure↔ = inverse pure⁺ pure⁻ pure⁻∘pure⁺ pure⁺∘pure⁻
+pure↔ : P x ↔ Any P (pure x)
+pure↔ {P = P} = mk↔ₛ′ pure⁺ pure⁻ pure⁺∘pure⁻ (pure⁻∘pure⁺ {P = P})
 
 ------------------------------------------------------------------------
 -- _∷_
 
-module _ (P : Pred A p) where
-
-  ∷↔ : (P x ⊎ Any P xs) ↔ Any P (x ∷ xs)
-  ∷↔ {x = x} {xs} =
-    (P x         ⊎ Any P xs)  ↔⟨ pure↔ {P = P} ⊎-cong (Any P xs ∎) ⟩
-    (Any P [ x ] ⊎ Any P xs)  ↔⟨ ++↔ {P = P} {xs = [ x ]} ⟩
-    Any P (x ∷ xs)            ∎
-    where open Related.EquationalReasoning
+∷↔ : (P : Pred A p) → (P x ⊎ Any P xs) ↔ Any P (x ∷ xs)
+∷↔ {x = x} {xs} P =
+  (P x         ⊎ Any P xs)  ↔⟨ pure↔ ⊎-cong (Any P xs ∎) ⟩
+  (Any P [ x ] ⊎ Any P xs)  ↔⟨ ++↔ ⟩
+  Any P (x ∷ xs)            ∎
+  where open Related.EquationalReasoning
 
 ------------------------------------------------------------------------
 -- _>>=_
@@ -696,16 +686,15 @@ module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where
   Any (λ f → Any (Any P ∘ pure ∘ f) xs) fs     ↔⟨ Any-cong (λ _ → >>=↔ ) (_ ∎) ⟩
   Any (λ f → Any P (xs >>= pure ∘ f)) fs       ↔⟨ >>=↔ ⟩
   Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡˘⟨ P.cong (Any P) (Listₑ.Applicative.unfold-⊛ fs xs) ⟩
-  Any P (fs ⊛ xs)                              ∎
+  Any P (fs ⊛ xs)                               ∎
   where open Related.EquationalReasoning
 
+
 -- An alternative introduction rule for _⊛_
 
-⊛⁺′ : ∀ {P : A → Set ℓ} {Q : B → Set ℓ} {fs : List (A → B)} {xs} →
+⊛⁺′ : ∀ {P : Pred A ℓ} {Q : Pred B ℓ} {fs : List (A → B)} {xs} →
       Any (P ⟨→⟩ Q) fs → Any P xs → Any Q (fs ⊛ xs)
-⊛⁺′ pq p =
-  Inverse.to ⊛↔ ⟨$⟩
-    Any.map (λ pq → Any.map (λ {x} → pq {x}) p) pq
+⊛⁺′ pq p = Inverse.to ⊛↔ (Any.map (λ pq → Any.map (λ {x} → pq {x}) p) pq)
 
 ------------------------------------------------------------------------
 -- _⊗_
diff --git a/src/Data/Product/Algebra.agda b/src/Data/Product/Algebra.agda
index 0b268b6ffd..aed57d4ed3 100644
--- a/src/Data/Product/Algebra.agda
+++ b/src/Data/Product/Algebra.agda
@@ -17,7 +17,7 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sum.Algebra
 open import Data.Unit.Polymorphic using (⊤; tt)
 open import Function.Base using (_∘′_)
-open import Function.Bundles using (_↔_; Inverse; mk↔′)
+open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)
 open import Function.Properties.Inverse using (↔-isEquivalence)
 open import Level using (Level; suc)
 open import Relation.Binary.PropositionalEquality.Core
@@ -37,19 +37,19 @@ private
 -- Σ is associative
 Σ-assoc : {B : A → Set b} {C : (a : A) → B a → Set c} →
           Σ (Σ A B) (uncurry C) ↔ Σ A (λ a → Σ (B a) (C a))
-Σ-assoc = mk↔′ assocʳ assocˡ cong′ cong′
+Σ-assoc = mk↔ₛ′ assocʳ assocˡ cong′ cong′
 
 -- Σ is associative, alternate formulation
 Σ-assoc-alt : {B : A → Set b} {C : Σ A B → Set c} →
           Σ (Σ A B) C ↔ Σ A (λ a → Σ (B a) (curry C a))
-Σ-assoc-alt = mk↔′ assocʳ-curried assocˡ-curried cong′ cong′
+Σ-assoc-alt = mk↔ₛ′ assocʳ-curried assocˡ-curried cong′ cong′
 
 ------------------------------------------------------------------------
 -- Algebraic properties
 
 -- × is a congruence
 ×-cong : A ↔ B → C ↔ D → (A × C) ↔ (B × D)
-×-cong i j = mk↔′ (map I.to J.to) (map I.from J.from)
+×-cong i j = mk↔ₛ′ (map I.to J.to) (map I.from J.from)
   (λ {(a , b) → cong₂ _,_ (I.strictlyInverseˡ a) (J.strictlyInverseˡ b)})
   (λ {(a , b) → cong₂ _,_ (I.strictlyInverseʳ a) (J.strictlyInverseʳ b)})
   where module I = Inverse i; module J = Inverse j
@@ -57,44 +57,44 @@ private
 -- × is commutative.
 -- (we don't use Commutative because it isn't polymorphic enough)
 ×-comm : (A : Set a) (B : Set b) → (A × B) ↔ (B × A)
-×-comm _ _ = mk↔′ swap swap swap-involutive swap-involutive
+×-comm _ _ = mk↔ₛ′ swap swap swap-involutive swap-involutive
 
 module _ (ℓ : Level) where
 
   -- × is associative
   ×-assoc : Associative {ℓ = ℓ} _↔_ _×_
-  ×-assoc _ _ _ = mk↔′ assocʳ′ assocˡ′ cong′ cong′
+  ×-assoc _ _ _ = mk↔ₛ′ assocʳ′ assocˡ′ cong′ cong′
 
   -- ⊤ is the identity for ×
   ×-identityˡ : LeftIdentity {ℓ = ℓ} _↔_ ⊤ _×_
-  ×-identityˡ _ = mk↔′ proj₂ (tt ,_) cong′ cong′
+  ×-identityˡ _ = mk↔ₛ′ proj₂ (tt ,_) cong′ cong′
 
   ×-identityʳ : RightIdentity {ℓ = ℓ} _↔_ ⊤ _×_
-  ×-identityʳ _ = mk↔′ proj₁ (_, tt) cong′ cong′
+  ×-identityʳ _ = mk↔ₛ′ proj₁ (_, tt) cong′ cong′
 
   ×-identity : Identity _↔_ ⊤ _×_
   ×-identity = ×-identityˡ , ×-identityʳ
 
   -- ⊥ is the zero for ×
   ×-zeroˡ : LeftZero {ℓ = ℓ} _↔_ ⊥ _×_
-  ×-zeroˡ A = mk↔′ proj₁ ⊥-elim ⊥-elim λ ()
+  ×-zeroˡ A = mk↔ₛ′ proj₁ ⊥-elim ⊥-elim λ ()
 
   ×-zeroʳ : RightZero {ℓ = ℓ} _↔_ ⊥ _×_
-  ×-zeroʳ A = mk↔′ proj₂ ⊥-elim ⊥-elim λ ()
+  ×-zeroʳ A = mk↔ₛ′ proj₂ ⊥-elim ⊥-elim λ ()
 
   ×-zero : Zero _↔_ ⊥ _×_
   ×-zero = ×-zeroˡ , ×-zeroʳ
 
   -- × distributes over ⊎
   ×-distribˡ-⊎ : _DistributesOverˡ_ {ℓ = ℓ} _↔_ _×_ _⊎_
-  ×-distribˡ-⊎ _ _ _ = mk↔′
+  ×-distribˡ-⊎ _ _ _ = mk↔ₛ′
     (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)
     [ map₂ inj₁ , map₂ inj₂ ]′
     Sum.[ cong′ , cong′ ]
     (uncurry λ _ → Sum.[ cong′ , cong′ ])
 
   ×-distribʳ-⊎ : _DistributesOverʳ_ {ℓ = ℓ} _↔_ _×_ _⊎_
-  ×-distribʳ-⊎ _ _ _ = mk↔′
+  ×-distribʳ-⊎ _ _ _ = mk↔ₛ′
     (uncurry [ curry inj₁ , curry inj₂ ]′)
     [ map₁ inj₁ , map₁ inj₂ ]′
     Sum.[ cong′ , cong′ ]
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index bde9e92c44..04cb6a7097 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -9,325 +9,307 @@
 
 module Data.Product.Function.Dependent.Propositional where
 
-open import Data.Product.Base using (Σ; map; _,_)
-open import Data.Product.Function.NonDependent.Setoid
-open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence as Equiv using (_⇔_; module Equivalence)
-open import Function.HalfAdjointEquivalence using (↔→≃; _≃_)
-open import Function.Injection as Inj
-  using (Injective; _↣_; module Injection)
-open import Function.Inverse as Inv using (_↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (_↞_; _LeftInverseOf_; module LeftInverse)
-open import Function.Related
+open import Data.Product.Base as Prod
+open import Data.Product.Function.NonDependent.Setoid using ()
+open import Data.Product.Relation.Binary.Pointwise.NonDependent using ()
+open import Data.Product.Properties using (Σ-≡,≡→≡; Σ-≡,≡↔≡; Σ-≡,≡←≡)
+open import Level using (Level; 0ℓ)
 open import Function.Related.TypeIsomorphisms
-open import Function.Surjection as Surj using (_↠_; module Surjection)
+open import Function.Related.Propositional
+open import Function.Base
+open import Function.Properties.Inverse
+open import Function.Properties.RightInverse
+open import Function.Properties.Inverse.HalfAdjointEquivalence
+open import Function.Consequences.Propositional
+  using (inverseʳ⇒injective; strictlySurjective⇒surjective)
+open import Function.Definitions using (Inverseˡ; Inverseʳ; Injective; StrictlySurjective)
+open import Function.Bundles
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
 
+private
+  variable
+    i a b c d : Level
+    I J : Set i
+    A B : I → Set a
+
+------------------------------------------------------------------------
+-- Functions
+
+module _ where
+  open Func
+
+  Σ-⟶ : (I⟶J : I ⟶ J) →
+         (∀ {i} → A i ⟶ B (to I⟶J i)) →
+         Σ I A ⟶ Σ J B
+  Σ-⟶ I⟶J A⟶B = mk⟶ $ Prod.map (to I⟶J) (to A⟶B)
+
 ------------------------------------------------------------------------
--- Combinators for various function types
-
-module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
-         {b₁ b₂} {B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
-         where
-
-  ⇔ : (A₁⇔A₂ : A₁ ⇔ A₂) →
-      (∀ {x} → B₁ x → B₂ (Equivalence.to   A₁⇔A₂ ⟨$⟩ x)) →
-      (∀ {y} → B₂ y → B₁ (Equivalence.from A₁⇔A₂ ⟨$⟩ y)) →
-      Σ A₁ B₁ ⇔ Σ A₂ B₂
-  ⇔ A₁⇔A₂ B-to B-from = Equiv.equivalence
-    (map (Equivalence.to   A₁⇔A₂ ⟨$⟩_) B-to)
-    (map (Equivalence.from A₁⇔A₂ ⟨$⟩_) B-from)
-
-  ⇔-↠ : ∀ (A₁↠A₂ : A₁ ↠ A₂) →
-        (∀ {x} → _⇔_ (B₁ x) (B₂ (Surjection.to A₁↠A₂ ⟨$⟩ x))) →
-        _⇔_ (Σ A₁ B₁) (Σ A₂ B₂)
-  ⇔-↠ A₁↠A₂ B₁⇔B₂ = Equiv.equivalence
-    (map (Surjection.to   A₁↠A₂ ⟨$⟩_) (Equivalence.to B₁⇔B₂ ⟨$⟩_))
-    (map (Surjection.from A₁↠A₂ ⟨$⟩_)
-       ((Equivalence.from B₁⇔B₂ ⟨$⟩_) ∘
-        P.subst B₂ (P.sym $ Surjection.right-inverse-of A₁↠A₂ _)))
+-- Equivalences
+
+module _ where
+  open Surjection
+
+  Σ-⇔ : (I↠J : I ↠ J) →
+         (∀ {i} → A i ⇔ B (to I↠J i)) →
+         Σ I A ⇔ Σ J B
+  Σ-⇔ {B = B} I↠J A⇔B = mk⇔
+    (map (to  I↠J) (Equivalence.to A⇔B))
+    (map (to⁻ I↠J) (Equivalence.from A⇔B ∘ P.subst B (P.sym (proj₂ (surjective I↠J _) P.refl))))
 
   -- See also Data.Product.Relation.Binary.Pointwise.Dependent.WithK.↣.
 
-  ↣ : ∀ (A₁↔A₂ : A₁ ↔ A₂) →
-      (∀ {x} → B₁ x ↣ B₂ (Inverse.to A₁↔A₂ ⟨$⟩ x)) →
-      Σ A₁ B₁ ↣ Σ A₂ B₂
-  ↣ A₁↔A₂ B₁↣B₂ = Inj.injection to to-injective
+------------------------------------------------------------------------
+-- Injections
+
+module _ where
+
+  Σ-↣ : (I↔J : I ↔ J) →
+         (∀ {i} → A i ↣ B (Inverse.to I↔J i)) →
+         Σ I A ↣ Σ J B
+  Σ-↣ {I = I} {J = J} {A = A} {B = B} I↔J A↣B = mk↣ to-injective
     where
     open P.≡-Reasoning
 
-    A₁≃A₂ = ↔→≃ A₁↔A₂
+    I≃J = ↔⇒≃ I↔J
 
     subst-application′ :
-      let open _≃_ A₁≃A₂ in
-      {x₁ x₂ : A₁} {y : B₁ (from (to x₁))}
-      (g : ∀ x → B₁ (from (to x)) → B₂ (to x)) (eq : to x₁ ≡ to x₂) →
-      P.subst B₂ eq (g x₁ y) ≡ g x₂ (P.subst B₁ (P.cong from eq) y)
+      let open _≃_ I≃J in
+      {x₁ x₂ : I} {y : A (from (to x₁))}
+      (g : ∀ x → A (from (to x)) → B (to x))
+      (eq : to x₁ ≡ to x₂) →
+      P.subst B eq (g x₁ y) ≡ g x₂ (P.subst A (P.cong from eq) y)
     subst-application′ {x₁} {x₂} {y} g eq =
-      P.subst B₂ eq (g x₁ y)                      ≡⟨ P.cong (P.subst B₂ eq) (P.sym (g′-lemma _ _)) ⟩
-      P.subst B₂ eq (g′ (to x₁) y)                ≡⟨ P.subst-application B₁ g′ eq ⟩
-      g′ (to x₂) (P.subst B₁ (P.cong from eq) y)  ≡⟨ g′-lemma _ _ ⟩
-      g x₂ (P.subst B₁ (P.cong from eq) y)        ∎
+      P.subst B eq (g x₁ y)                      ≡⟨ P.cong (P.subst B eq) (P.sym (g′-lemma _ _)) ⟩
+      P.subst B eq (g′ (to x₁) y)                ≡⟨ P.subst-application A g′ eq ⟩
+      g′ (to x₂) (P.subst A (P.cong from eq) y)  ≡⟨ g′-lemma _ _ ⟩
+      g x₂ (P.subst A (P.cong from eq) y)        ∎
       where
-      open _≃_ A₁≃A₂
+      open _≃_ I≃J
 
-      g′ : ∀ x → B₁ (from x) → B₂ x
+      g′ : ∀ x → A (from x) → B x
       g′ x =
-        P.subst B₂ (right-inverse-of x) ∘
+        P.subst B (right-inverse-of x) ∘
         g (from x) ∘
-        P.subst B₁ (P.sym (P.cong from (right-inverse-of x)))
+        P.subst A (P.sym (P.cong from (right-inverse-of x)))
 
       g′-lemma : ∀ x y → g′ (to x) y ≡ g x y
       g′-lemma x y =
-        P.subst B₂ (right-inverse-of (to x))
+        P.subst B (right-inverse-of (to x))
           (g (from (to x)) $
-           P.subst B₁ (P.sym (P.cong from (right-inverse-of (to x)))) y)  ≡⟨ P.cong (λ p → P.subst B₂ p (g (from (to x))
-                                                                                                           (P.subst B₁ (P.sym (P.cong from p)) y)))
+           P.subst A (P.sym (P.cong from (right-inverse-of (to x)))) y)  ≡⟨ P.cong (λ p → P.subst B p (g (from (to x))
+                                                                                                           (P.subst A (P.sym (P.cong from p)) y)))
                                                                                (P.sym (left-right x)) ⟩
-        P.subst B₂ (P.cong to (left-inverse-of x))
+        P.subst B (P.cong to (left-inverse-of x))
           (g (from (to x)) $
-           P.subst B₁
+           P.subst A
              (P.sym (P.cong from (P.cong to (left-inverse-of x))))
              y)                                                           ≡⟨ lemma _ ⟩
 
         g x y                                                             ∎
         where
-        lemma :
-          ∀ {x′} eq {y : B₁ (from (to x′))} →
-          P.subst B₂ (P.cong to eq)
-            (g (from (to x))
-               (P.subst B₁ (P.sym (P.cong from (P.cong to eq))) y)) ≡
-          g x′ y
+        lemma : ∀ {x′} eq {y : A (from (to x′))} →
+                  P.subst B (P.cong to eq)
+                    (g (from (to x))
+                      (P.subst A (P.sym (P.cong from (P.cong to eq))) y)) ≡
+                  g x′ y
         lemma P.refl = P.refl
 
-    to = map (_≃_.to A₁≃A₂) (Injection.to B₁↣B₂ ⟨$⟩_)
+    open Injection
 
-    to-injective : Injective (P.→-to-⟶ {B = P.setoid _} to)
+    to′ : Σ I A → Σ J B
+    to′ = Prod.map (_≃_.to I≃J) (to A↣B)
+
+    to-injective : Injective _≡_ _≡_ to′
     to-injective {(x₁ , x₂)} {(y₁ , y₂)} =
+
       Σ-≡,≡→≡ ∘′
 
-      map (_≃_.injective A₁≃A₂) (λ {eq₁} eq₂ →
+      map (_≃_.injective I≃J) (λ {eq₁} eq₂ → injective A↣B (
+              to A↣B (P.subst A (_≃_.injective I≃J eq₁) x₂)             ≡⟨⟩
+
+              (let eq =
+                      P.trans (P.sym (_≃_.left-inverse-of I≃J x₁))
+                        (P.trans (P.cong (_≃_.from I≃J) eq₁)
+                          (P.trans (_≃_.left-inverse-of I≃J y₁)
+                            P.refl)) in
+              to A↣B (P.subst A eq x₂))                                   ≡⟨ P.cong (λ p → to A↣B
+                                                                                             (P.subst A
+                                                                                               (P.trans (P.sym (_≃_.left-inverse-of I≃J _))
+                                                                                                  (P.trans (P.cong (_≃_.from I≃J) eq₁) p))
+                                                                                               x₂))
+                                                                               (P.trans-reflʳ _) ⟩
 
-        let lemma =
+              (let eq = P.trans (P.sym (_≃_.left-inverse-of I≃J x₁))
+                          (P.trans (P.cong (_≃_.from I≃J) eq₁)
+                            (_≃_.left-inverse-of I≃J y₁)) in
+              to A↣B (P.subst A eq x₂))                                  ≡⟨ P.cong (to A↣B)
+                                                                               (P.sym (P.subst-subst (P.sym (_≃_.left-inverse-of I≃J _)))) ⟩
+
+              to A↣B ((P.subst A (P.trans (P.cong (_≃_.from I≃J) eq₁)
+                             (_≃_.left-inverse-of I≃J y₁)) $
+               P.subst A (P.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ P.cong (to A↣B)
+                                                                               (P.sym (P.subst-subst (P.cong (_≃_.from I≃J) eq₁))) ⟩
+              to A↣B (
+              (P.subst A (_≃_.left-inverse-of I≃J y₁) $
+               P.subst A (P.cong (_≃_.from I≃J) eq₁) $
+               P.subst A (P.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ P.sym (subst-application′
+                                                                                      (λ x y → to A↣B
+                                                                                                 (P.subst A (_≃_.left-inverse-of I≃J x) y))
+                                                                                      eq₁) ⟩
+              P.subst B eq₁ (to A↣B $
+                 (P.subst A (_≃_.left-inverse-of I≃J x₁) $
+                  P.subst A (P.sym (_≃_.left-inverse-of I≃J x₁)) x₂))  ≡⟨ P.cong (P.subst B eq₁ ∘ to A↣B)
+                                                                               (P.subst-subst (P.sym (_≃_.left-inverse-of I≃J _))) ⟩
 
-              Injection.to B₁↣B₂ ⟨$⟩
-              P.subst B₁ (_≃_.injective A₁≃A₂ eq₁) x₂                     ≡⟨⟩
+              (let eq = P.trans (P.sym (_≃_.left-inverse-of I≃J x₁))
+                          (_≃_.left-inverse-of I≃J x₁) in
+              P.subst B eq₁ (to A↣B (P.subst A eq x₂)))                  ≡⟨ P.cong (λ p → P.subst B eq₁ (to A↣B (P.subst A p x₂)))
+                                                                              (P.trans-symˡ (_≃_.left-inverse-of I≃J _)) ⟩
+              P.subst B eq₁ (to A↣B (P.subst A P.refl x₂))               ≡⟨⟩
 
-              Injection.to B₁↣B₂ ⟨$⟩
-              P.subst B₁
-                (P.trans (P.sym (_≃_.left-inverse-of A₁≃A₂ x₁))
-                   (P.trans (P.cong (_≃_.from A₁≃A₂) eq₁)
-                      (P.trans (_≃_.left-inverse-of A₁≃A₂ y₁)
-                         P.refl)))
-                x₂                                                        ≡⟨ P.cong (λ p → Injection.to B₁↣B₂ ⟨$⟩
-                                                                                             P.subst B₁
-                                                                                               (P.trans (P.sym (_≃_.left-inverse-of A₁≃A₂ _))
-                                                                                                  (P.trans (P.cong (_≃_.from A₁≃A₂) eq₁) p))
-                                                                                               x₂)
-                                                                               (P.trans-reflʳ _) ⟩
-              Injection.to B₁↣B₂ ⟨$⟩
-              P.subst B₁
-                (P.trans (P.sym (_≃_.left-inverse-of A₁≃A₂ x₁))
-                   (P.trans (P.cong (_≃_.from A₁≃A₂) eq₁)
-                      (_≃_.left-inverse-of A₁≃A₂ y₁)))
-                x₂                                                        ≡⟨ P.cong (Injection.to B₁↣B₂ ⟨$⟩_)
-                                                                               (P.sym (P.subst-subst (P.sym (_≃_.left-inverse-of A₁≃A₂ _)))) ⟩
-              Injection.to B₁↣B₂ ⟨$⟩
-              (P.subst B₁ (P.trans (P.cong (_≃_.from A₁≃A₂) eq₁)
-                             (_≃_.left-inverse-of A₁≃A₂ y₁)) $
-               P.subst B₁ (P.sym (_≃_.left-inverse-of A₁≃A₂ x₁)) x₂)      ≡⟨ P.cong (Injection.to B₁↣B₂ ⟨$⟩_)
-                                                                               (P.sym (P.subst-subst (P.cong (_≃_.from A₁≃A₂) eq₁))) ⟩
-              Injection.to B₁↣B₂ ⟨$⟩
-              (P.subst B₁ (_≃_.left-inverse-of A₁≃A₂ y₁) $
-               P.subst B₁ (P.cong (_≃_.from A₁≃A₂) eq₁) $
-               P.subst B₁ (P.sym (_≃_.left-inverse-of A₁≃A₂ x₁)) x₂)      ≡⟨ P.sym (subst-application′
-                                                                                      (λ x y → Injection.to B₁↣B₂ ⟨$⟩
-                                                                                                 P.subst B₁ (_≃_.left-inverse-of A₁≃A₂ x) y)
-                                                                                      eq₁) ⟩
-              P.subst B₂ eq₁
-                (Injection.to B₁↣B₂ ⟨$⟩
-                 (P.subst B₁ (_≃_.left-inverse-of A₁≃A₂ x₁) $
-                  P.subst B₁ (P.sym (_≃_.left-inverse-of A₁≃A₂ x₁)) x₂))  ≡⟨ P.cong (P.subst B₂ eq₁ ∘ (Injection.to B₁↣B₂ ⟨$⟩_))
-                                                                               (P.subst-subst (P.sym (_≃_.left-inverse-of A₁≃A₂ _))) ⟩
-              P.subst B₂ eq₁
-                (Injection.to B₁↣B₂ ⟨$⟩
-                 P.subst B₁
-                   (P.trans (P.sym (_≃_.left-inverse-of A₁≃A₂ x₁))
-                      (_≃_.left-inverse-of A₁≃A₂ x₁))
-                   x₂)                                                    ≡⟨ P.cong (λ p → P.subst B₂ eq₁
-                                                                                             (Injection.to B₁↣B₂ ⟨$⟩ P.subst B₁ p x₂))
-                                                                               (P.trans-symˡ (_≃_.left-inverse-of A₁≃A₂ _)) ⟩
-              P.subst B₂ eq₁
-                (Injection.to B₁↣B₂ ⟨$⟩ P.subst B₁ P.refl x₂)             ≡⟨⟩
-
-              P.subst B₂ eq₁ (Injection.to B₁↣B₂ ⟨$⟩ x₂)                  ≡⟨ eq₂ ⟩
-
-              Injection.to B₁↣B₂ ⟨$⟩ y₂                                   ∎
-
-        in
-
-        P.subst B₁ (_≃_.injective A₁≃A₂ eq₁) x₂  ≡⟨ Injection.injective B₁↣B₂ lemma ⟩
-        y₂                                       ∎) ∘
+              P.subst B eq₁ (to A↣B x₂)                                  ≡⟨ eq₂ ⟩
+
+              to A↣B y₂                                                  ∎
+
+        )) ∘
 
       Σ-≡,≡←≡
 
-  ↞ : (A₁↞A₂ : A₁ ↞ A₂) →
-      (∀ {x} → B₁ (LeftInverse.from A₁↞A₂ ⟨$⟩ x) ↞ B₂ x) →
-      Σ A₁ B₁ ↞ Σ A₂ B₂
-  ↞ A₁↞A₂ B₁↞B₂ = record
-    { to              = P.→-to-⟶ to
-    ; from            = P.→-to-⟶ from
-    ; left-inverse-of = left-inverse-of
-    }
+------------------------------------------------------------------------
+-- Surjections
+
+module _ where
+  open Surjection
+
+  Σ-↠ : (I↠J : I ↠ J) →
+       (∀ {x} → A x ↠ B (to I↠J x)) →
+       Σ I A ↠ Σ J B
+  Σ-↠ {I = I} {J = J} {A = A} {B = B} I↠J A↠B =
+    mk↠ (strictlySurjective⇒surjective strictlySurjective′)
     where
-    open P.≡-Reasoning
+    to′ : Σ I A → Σ J B
+    to′ = map (to I↠J) (to A↠B)
 
-    from = map (LeftInverse.from A₁↞A₂ ⟨$⟩_) (LeftInverse.from B₁↞B₂ ⟨$⟩_)
+    backcast : ∀ {i} → B i → B (to I↠J (to⁻ I↠J i))
+    backcast = P.subst B (P.sym (to∘to⁻ I↠J _))
 
-    to   = map
-      (LeftInverse.to A₁↞A₂ ⟨$⟩_)
-      (λ {x} y →
-         LeftInverse.to B₁↞B₂ ⟨$⟩
-           P.subst B₁ (P.sym (LeftInverse.left-inverse-of A₁↞A₂ x)) y)
+    to⁻′ : Σ J B → Σ I A
+    to⁻′ = map (to⁻ I↠J) (Surjection.to⁻ A↠B ∘ backcast)
 
-    left-inverse-of : ∀ p → from (to p) ≡ p
-    left-inverse-of (x , y) = Σ-≡,≡→≡
-      ( LeftInverse.left-inverse-of A₁↞A₂ x
-      , (P.subst B₁ (LeftInverse.left-inverse-of A₁↞A₂ x)
-           (LeftInverse.from B₁↞B₂ ⟨$⟩ (LeftInverse.to B₁↞B₂ ⟨$⟩
-              (P.subst B₁ (P.sym (LeftInverse.left-inverse-of A₁↞A₂ x))
-                 y)))                                                    ≡⟨ P.cong (P.subst B₁ _) (LeftInverse.left-inverse-of B₁↞B₂ _) ⟩
+    strictlySurjective′ : StrictlySurjective _≡_ to′
+    strictlySurjective′ (x , y) = to⁻′ (x , y) , Σ-≡,≡→≡
+      ( to∘to⁻ I↠J x
+      , (P.subst B (to∘to⁻ I↠J x) (to A↠B (to⁻ A↠B (backcast y))) ≡⟨ P.cong (P.subst B _) (to∘to⁻ A↠B _) ⟩
+         P.subst B (to∘to⁻ I↠J x) (backcast y)                      ≡⟨ P.subst-subst-sym (to∘to⁻ I↠J x) ⟩
+         y                                                          ∎)
+      ) where open P.≡-Reasoning
 
-         P.subst B₁ (LeftInverse.left-inverse-of A₁↞A₂ x)
-           (P.subst B₁ (P.sym (LeftInverse.left-inverse-of A₁↞A₂ x))
-              y)                                                         ≡⟨ P.subst-subst-sym (LeftInverse.left-inverse-of A₁↞A₂ x) ⟩
 
-         y                                                               ∎)
-      )
+------------------------------------------------------------------------
+-- Left inverses
 
-  ↠ : (A₁↠A₂ : A₁ ↠ A₂) →
-      (∀ {x} → B₁ x ↠ B₂ (Surjection.to A₁↠A₂ ⟨$⟩ x)) →
-      Σ A₁ B₁ ↠ Σ A₂ B₂
-  ↠ A₁↠A₂ B₁↠B₂ = record
-    { to         = P.→-to-⟶ to
-    ; surjective = record
-      { from             = P.→-to-⟶ from
-      ; right-inverse-of = right-inverse-of
-      }
-    }
+module _ where
+  open LeftInverse
+
+  Σ-↩ : (I↩J : I ↩ J) →
+         (∀ {i} → A i ↩ B (to I↩J i)) →
+         Σ I A ↩ Σ J B
+  Σ-↩ {I = I} {J = J} {A = A} {B = B} I↩J A↩B = mk↩ {to = to′ } {from = from′} inv
     where
-    open P.≡-Reasoning
+    to′ : Σ I A → Σ J B
+    to′ = map (to I↩J) (to A↩B)
 
-    to   = map (Surjection.to A₁↠A₂ ⟨$⟩_)
-                    (Surjection.to B₁↠B₂ ⟨$⟩_)
-    from = map
-      (Surjection.from A₁↠A₂ ⟨$⟩_)
-      (λ {x} y →
-         Surjection.from B₁↠B₂ ⟨$⟩
-           P.subst B₂ (P.sym (Surjection.right-inverse-of A₁↠A₂ x)) y)
-
-    right-inverse-of : ∀ p → to (from p) ≡ p
-    right-inverse-of (x , y) = Σ-≡,≡→≡
-      ( Surjection.right-inverse-of A₁↠A₂ x
-      , (P.subst B₂ (Surjection.right-inverse-of A₁↠A₂ x)
-           (Surjection.to B₁↠B₂ ⟨$⟩ (Surjection.from B₁↠B₂ ⟨$⟩
-              (P.subst B₂ (P.sym (Surjection.right-inverse-of A₁↠A₂ x))
-                 y)))                                                    ≡⟨ P.cong (P.subst B₂ _) (Surjection.right-inverse-of B₁↠B₂ _) ⟩
-
-         P.subst B₂ (Surjection.right-inverse-of A₁↠A₂ x)
-           (P.subst B₂ (P.sym (Surjection.right-inverse-of A₁↠A₂ x))
-              y)                                                         ≡⟨ P.subst-subst-sym (Surjection.right-inverse-of A₁↠A₂ x) ⟩
-
-         y                                                               ∎)
-      )
+    backcast : ∀ {j} → B j → B (to I↩J (from I↩J j))
+    backcast = P.subst B (P.sym (inverseˡ I↩J P.refl))
+
+    from′ : Σ J B → Σ I A
+    from′ = map (from I↩J) (from A↩B ∘ backcast)
+
+    inv : Inverseˡ _≡_ _≡_ to′ from′
+    inv {j , b} P.refl = Σ-≡,≡→≡ (strictlyInverseˡ I↩J j  , (
+      begin
+        P.subst B (inverseˡ I↩J P.refl) (to A↩B (from A↩B (backcast b))) ≡⟨ P.cong (P.subst B _) (inverseˡ A↩B P.refl) ⟩
+        P.subst B (inverseˡ I↩J P.refl) (backcast b)                       ≡⟨ P.subst-subst-sym (inverseˡ I↩J _) ⟩
+        b                                                                  ∎)) where open P.≡-Reasoning
+
+------------------------------------------------------------------------
+-- Right inverses
 
-  ↔ : (A₁↔A₂ : A₁ ↔ A₂) →
-      (∀ {x} → B₁ x ↔ B₂ (Inverse.to A₁↔A₂ ⟨$⟩ x)) →
-      Σ A₁ B₁ ↔ Σ A₂ B₂
-  ↔ A₁↔A₂ B₁↔B₂ = Inv.inverse
-    (Surjection.to   surjection′ ⟨$⟩_)
-    (Surjection.from surjection′ ⟨$⟩_)
+------------------------------------------------------------------------
+-- Inverses
+
+module _ where
+  open Inverse
+
+  Σ-↔ : (I↔J : I ↔ J) →
+      (∀ {x} → A x ↔ B (to I↔J x)) →
+      Σ I A ↔ Σ J B
+  Σ-↔ {I = I} {J = J} {A = A} {B = B} I↔J A↔B = mk↔ₛ′
+    (Surjection.to surjection′)
+    (Surjection.to⁻ surjection′)
+    (Surjection.to∘to⁻ surjection′)
     left-inverse-of
-    (Surjection.right-inverse-of surjection′)
     where
     open P.≡-Reasoning
 
-    A₁≃A₂ = ↔→≃ A₁↔A₂
-
-    surjection′ : _↠_ (Σ A₁ B₁) (Σ A₂ B₂)
-    surjection′ =
-      ↠ (Inverse.surjection (_≃_.inverse A₁≃A₂))
-        (Inverse.surjection B₁↔B₂)
-
-    left-inverse-of :
-      ∀ p → Surjection.from surjection′ ⟨$⟩
-              (Surjection.to surjection′ ⟨$⟩ p) ≡ p
-    left-inverse-of (x , y) = Σ-≡,≡→≡
-      ( _≃_.left-inverse-of A₁≃A₂ x
-      , (P.subst B₁ (_≃_.left-inverse-of A₁≃A₂ x)
-           (Inverse.from B₁↔B₂ ⟨$⟩
-              (P.subst B₂ (P.sym (_≃_.right-inverse-of A₁≃A₂
-                                    (_≃_.to A₁≃A₂ x)))
-                 (Inverse.to B₁↔B₂ ⟨$⟩ y)))                   ≡⟨ P.subst-application B₂ (λ _ → Inverse.from B₁↔B₂ ⟨$⟩_) _ ⟩
-
-         Inverse.from B₁↔B₂ ⟨$⟩
-           (P.subst B₂ (P.cong (_≃_.to A₁≃A₂)
-                          (_≃_.left-inverse-of A₁≃A₂ x))
-              (P.subst B₂ (P.sym (_≃_.right-inverse-of A₁≃A₂
-                                    (_≃_.to A₁≃A₂ x)))
-                 (Inverse.to B₁↔B₂ ⟨$⟩ y)))                   ≡⟨ P.cong (λ eq → Inverse.from B₁↔B₂ ⟨$⟩ P.subst B₂ eq
-                                                                                  (P.subst B₂ (P.sym (_≃_.right-inverse-of A₁≃A₂ _)) _))
-                                                                   (_≃_.left-right A₁≃A₂ _) ⟩
-         Inverse.from B₁↔B₂ ⟨$⟩
-           (P.subst B₂ (_≃_.right-inverse-of A₁≃A₂
-                          (_≃_.to A₁≃A₂ x))
-              (P.subst B₂ (P.sym (_≃_.right-inverse-of A₁≃A₂
-                                    (_≃_.to A₁≃A₂ x)))
-                 (Inverse.to B₁↔B₂ ⟨$⟩ y)))                   ≡⟨ P.cong (Inverse.from B₁↔B₂ ⟨$⟩_)
-                                                                   (P.subst-subst-sym (_≃_.right-inverse-of A₁≃A₂ _)) ⟩
-
-         Inverse.from B₁↔B₂ ⟨$⟩ (Inverse.to B₁↔B₂ ⟨$⟩ y)      ≡⟨ Inverse.left-inverse-of B₁↔B₂ _ ⟩
+    I≃J = ↔⇒≃ I↔J
+
+    surjection′ : Σ I A ↠ Σ J B
+    surjection′ = Σ-↠ (↔⇒↠ (≃⇒↔ I≃J)) (↔⇒↠ A↔B)
+
+    left-inverse-of : ∀ p → Surjection.to⁻ surjection′ (Surjection.to surjection′ p) ≡ p
+    left-inverse-of (x , y) = to Σ-≡,≡↔≡
+      ( _≃_.left-inverse-of I≃J x
+      , (P.subst A (_≃_.left-inverse-of I≃J x)
+           (from A↔B
+              (P.subst B (P.sym (_≃_.right-inverse-of I≃J
+                                    (_≃_.to I≃J x)))
+                 (to A↔B y)))                   ≡⟨ P.subst-application B (λ _ → from A↔B) _ ⟩
+
+         from A↔B
+           (P.subst B (P.cong (_≃_.to I≃J)
+                          (_≃_.left-inverse-of I≃J x))
+              (P.subst B (P.sym (_≃_.right-inverse-of I≃J
+                                    (_≃_.to I≃J x)))
+                 (to A↔B y)))                   ≡⟨ P.cong (λ eq → from A↔B (P.subst B eq
+                                                                                  (P.subst B (P.sym (_≃_.right-inverse-of I≃J _)) _)))
+                                                                   (_≃_.left-right I≃J _) ⟩
+         from A↔B
+           (P.subst B (_≃_.right-inverse-of I≃J
+                          (_≃_.to I≃J x))
+              (P.subst B (P.sym (_≃_.right-inverse-of I≃J
+                                    (_≃_.to I≃J x)))
+                 (to A↔B y)))                   ≡⟨ P.cong (from A↔B)
+                                                                   (P.subst-subst-sym (_≃_.right-inverse-of I≃J _)) ⟩
+
+         from A↔B (to A↔B y)      ≡⟨ Inverse.strictlyInverseʳ A↔B _ ⟩
 
          y                                                    ∎)
       )
 
-private
 
-  swap-coercions : ∀ {k a₁ a₂ b₁ b₂} {A₁ : Set a₁} {A₂ : Set a₂}
-    {B₁ : A₁ → Set b₁} (B₂ : A₂ → Set b₂)
-    (A₁↔A₂ : _↔_ A₁ A₂) →
-    (∀ {x} → B₁ x ∼[ k ] B₂ (Inverse.to A₁↔A₂ ⟨$⟩ x)) →
-    ∀ {x} → B₁ (Inverse.from A₁↔A₂ ⟨$⟩ x) ∼[ k ] B₂ x
-  swap-coercions {k} {B₁ = B₁} B₂ A₁↔A₂ eq {x} =
-    B₁ (Inverse.from A₁↔A₂ ⟨$⟩ x)
-      ∼⟨ eq ⟩
-    B₂ (Inverse.to A₁↔A₂ ⟨$⟩ (Inverse.from A₁↔A₂ ⟨$⟩ x))
-      ↔⟨ K-reflexive
-         (P.cong B₂ $ Inverse.right-inverse-of A₁↔A₂ x) ⟩
-    B₂ x
-      ∎
+private module _ where
+  open Inverse
+
+  swap-coercions : ∀ {k} (B : J → Set b)
+    (I↔J : _↔_ I J) →
+    (∀ {x} → A x ∼[ k ] B (to I↔J x)) →
+    ∀ {x} → A (from I↔J x) ∼[ k ] B x
+  swap-coercions {A = A} B I↔J eq {x} =
+    A (from I↔J x)           ∼⟨ eq ⟩
+    B (to I↔J (from I↔J x)) ↔⟨ K-reflexive (P.cong B $ strictlyInverseˡ I↔J x) ⟩
+    B x                       ∎
     where open EquationalReasoning
 
-cong : ∀ {k a₁ a₂ b₁ b₂}
-         {A₁ : Set a₁} {A₂ : Set a₂}
-         {B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
-       (A₁↔A₂ : _↔_ A₁ A₂) →
-       (∀ {x} → B₁ x ∼[ k ] B₂ (Inverse.to A₁↔A₂ ⟨$⟩ x)) →
-       Σ A₁ B₁ ∼[ k ] Σ A₂ B₂
-cong {implication}                   =
-  λ A₁↔A₂ → map (_⟨$⟩_ (Inverse.to A₁↔A₂))
-cong {reverse-implication} {B₂ = B₂} =
-  λ A₁↔A₂ B₁←B₂ → lam (map (_⟨$⟩_ (Inverse.from A₁↔A₂))
-    (app-← (swap-coercions B₂ A₁↔A₂ B₁←B₂)))
-cong {equivalence}                   = ⇔-↠ ∘ Inverse.surjection
-cong {injection}                     = ↣
-cong {reverse-injection}   {B₂ = B₂} =
-  λ A₁↔A₂ B₁↢B₂ → lam (↣ (Inv.sym A₁↔A₂)
-    (app-↢ (swap-coercions B₂ A₁↔A₂ B₁↢B₂)))
-cong {left-inverse}                  =
-  λ A₁↔A₂ → ↞ (Inverse.left-inverse A₁↔A₂) ∘ swap-coercions _ A₁↔A₂
-cong {surjection}                    = ↠ ∘ Inverse.surjection
-cong {bijection}                     = ↔
+
+cong : ∀ {k} (I↔J : I ↔ J) →
+       (∀ {x} → A x ∼[ k ] B (Inverse.to I↔J x)) →
+       Σ I A ∼[ k ] Σ J B
+cong {k = implication}                I↔J A⟶B = Σ-⟶ (↔⇒⟶ I↔J) A⟶B
+cong {B = B} {k = reverseImplication} I↔J A⟵B = Σ-⟶ (↔⇒⟵ I↔J) (swap-coercions {k = reverseImplication} B I↔J A⟵B)
+cong {k = equivalence}                I↔J A⇔B = Σ-⇔ (↔⇒↠ I↔J) A⇔B
+cong {k = injection}                  I↔J A↣B = Σ-↣ I↔J A↣B
+cong {B = B} {k = reverseInjection}   I↔J A↢B = Σ-↣ (↔-sym I↔J) (swap-coercions {k = reverseInjection} B I↔J A↢B)
+cong {B = B} {k = leftInverse}        I↔J A↩B = ↩⇒↪ (Σ-↩ (↔⇒↩ (↔-sym I↔J)) (↪⇒↩ (swap-coercions {k = leftInverse} B I↔J A↩B)))
+cong {k = surjection}                 I↔J A↠B = Σ-↠ (↔⇒↠ I↔J) A↠B
+cong {k = bijection}                  I↔J A↔B = Σ-↔ I↔J A↔B
diff --git a/src/Data/Product/Function/Dependent/Propositional/WithK.agda b/src/Data/Product/Function/Dependent/Propositional/WithK.agda
index df3bafe613..83aa4a9baa 100644
--- a/src/Data/Product/Function/Dependent/Propositional/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Propositional/WithK.agda
@@ -9,30 +9,41 @@
 
 module Data.Product.Function.Dependent.Propositional.WithK where
 
-open import Data.Product.Base using (Σ)
-open import Data.Product.Function.Dependent.Setoid
+open import Data.Product.Base
+open import Data.Product.Properties
+open import Data.Product.Function.Dependent.Setoid using (injection)
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Data.Product.Relation.Binary.Pointwise.Dependent.WithK
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Injection as Inj using (_↣_; module Injection)
-open import Function.Inverse as Inv using (_↔_; module Inverse)
-import Relation.Binary.HeterogeneousEquality as H
+open import Relation.Binary.Indexed.Heterogeneous.Construct.At using (_atₛ_)
+open import Relation.Binary.HeterogeneousEquality as H
+open import Level using (Level)
+open import Function
+open import Function.Properties.Injection
+open import Function.Properties.Inverse as Inverse
+open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
+
+private
+  variable
+    i a : Level
+    I J : Set i
+    A B : I → Set a
 
 ------------------------------------------------------------------------
 -- Combinator for Injection
 
-module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
-         {b₁ b₂} {B₁ : A₁ → Set b₁} {B₂ : A₂ → Set b₂}
-         where
+module _ where
+  open Injection
 
-  ↣ : ∀ (A₁↣A₂ : A₁ ↣ A₂) →
-      (∀ {x} → B₁ x ↣ B₂ (Injection.to A₁↣A₂ ⟨$⟩ x)) →
-      Σ A₁ B₁ ↣ Σ A₂ B₂
-  ↣ A₁↣A₂ B₁↣B₂ =
-    Inverse.injection Pointwise-≡↔≡ ⟨∘⟩
-    injection (H.indexedSetoid B₂) A₁↣A₂
-      (Inverse.injection (H.≡↔≅ B₂) ⟨∘⟩
-       B₁↣B₂ ⟨∘⟩
-       Inverse.injection (Inv.sym (H.≡↔≅ B₁))) ⟨∘⟩
-    Inverse.injection (Inv.sym Pointwise-≡↔≡)
-    where open Inj using () renaming (_∘_ to _⟨∘⟩_)
+  Σ-↣ : (I↣J : I ↣ J) →
+         (∀ {i} → A i ↣ B (to I↣J i)) →
+         Σ I A ↣ Σ J B
+  Σ-↣ {A = A} {B = B} I↣J A↣B =
+    ↣-trans (Inverse⇒Injection (Inverse.sym Pointwise-≡↔≡)) $
+      ↣-trans (injection I↣J Aᵢ↣Bᵢ) $
+        Inverse⇒Injection Pointwise-≡↔≡
+    where
+    Aᵢ↣Bᵢ : (∀ {i} → Injection (H.indexedSetoid A atₛ i) (H.indexedSetoid B atₛ (Injection.to I↣J i)))
+    Aᵢ↣Bᵢ =
+      ↣-trans (Inverse⇒Injection (Inverse.sym (H.≡↔≅ A))) $
+        ↣-trans A↣B $
+          Inverse⇒Injection (H.≡↔≅ B)
diff --git a/src/Data/Product/Function/Dependent/Setoid.agda b/src/Data/Product/Function/Dependent/Setoid.agda
index 71cd995b18..9866971920 100644
--- a/src/Data/Product/Function/Dependent/Setoid.agda
+++ b/src/Data/Product/Function/Dependent/Setoid.agda
@@ -2,27 +2,25 @@
 -- The Agda standard library
 --
 -- Dependent product combinators for setoid equality preserving
--- functions
+-- functions.
+--
+-- NOTE: the first component of the equality is propositional equality.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module Data.Product.Function.Dependent.Setoid where
 
-open import Data.Product.Base using (map; _,_)
-open import Data.Product.Relation.Binary.Pointwise.Dependent
-open import Function.Base
-open import Function.Equality as F using (_⟶_; _⟨$⟩_)
-open import Function.Equivalence as Eq
-  using (Equivalence; _⇔_; module Equivalence)
-open import Function.Injection as Inj
-  using (Injection; Injective; _↣_; module Injection)
-open import Function.Inverse as Inv
-  using (Inverse; _↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (LeftInverse; _↞_; _LeftInverseOf_; _RightInverseOf_; module LeftInverse)
-open import Function.Surjection as Surj
-  using (Surjection; _↠_; module Surjection)
+open import Data.Product.Base using (map; _,_; proj₁; proj₂)
+open import Data.Product.Relation.Binary.Pointwise.Dependent as Σ
+open import Level using (Level)
+open import Function
+open import Function.Consequences
+open import Function.Properties.Injection
+open import Function.Properties.Surjection
+open import Function.Properties.Equivalence
+open import Function.Properties.RightInverse
+import Function.Properties.Inverse as InverseProperties
 open import Relation.Binary.Core using (_=[_]⇒_)
 open import Relation.Binary.Bundles as B
 open import Relation.Binary.Indexed.Heterogeneous
@@ -32,167 +30,226 @@ open import Relation.Binary.Indexed.Heterogeneous.Construct.At
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
+private
+  variable
+    i a b ℓ₁ ℓ₂ : Level
+    I J : Set i
+    A B : IndexedSetoid I a ℓ₁
+
 ------------------------------------------------------------------------
 -- Properties related to "relatedness"
 ------------------------------------------------------------------------
 
+private module _ (A : IndexedSetoid I a ℓ₁) where
+  open IndexedSetoid A
+
+  cast : ∀ {i j} → j ≡ i → Carrier i → Carrier j
+  cast j≡i = P.subst Carrier (P.sym $ j≡i)
+
+  cast-cong : ∀ {i j} {x y : Carrier i}
+               (j≡i : j ≡ i) →
+               x ≈ y →
+               cast j≡i x ≈ cast j≡i y
+  cast-cong P.refl p = p
+
+  cast-eq : ∀ {i j x} (eq : i ≡ j) → cast eq x ≈ x
+  cast-eq P.refl = IndexedSetoid.refl A
+
 private
+  _×ₛ_ : (I : Set i) → IndexedSetoid I a ℓ₁ → Setoid _ _
+  I ×ₛ A = Σ.setoid (P.setoid I) A
 
-  subst-cong : ∀ {i a p} {I : Set i} {A : I → Set a}
-               (P : ∀ {i} → A i → A i → Set p) {i i′} {x y : A i}
-               (i≡i′ : i ≡ i′) →
-               P x y → P (P.subst A i≡i′ x) (P.subst A i≡i′ y)
-  subst-cong P P.refl p = p
-
-⟶ : ∀ {a₁ a₂ b₁ b₁′ b₂ b₂′}
-      {A₁ : Set a₁} {A₂ : Set a₂}
-      {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′)
-    (f : A₁ → A₂) → (∀ {x} → (B₁ atₛ x) ⟶ (B₂ atₛ (f x))) →
-    setoid (P.setoid A₁) B₁ ⟶ setoid (P.setoid A₂) B₂
-⟶ {A₁ = A₁} {A₂} {B₁} B₂ f g = record
-  { _⟨$⟩_ = fg
-  ; cong  = fg-cong
-  }
-  where
-  open B.Setoid (setoid (P.setoid A₁) B₁)
-    using () renaming (_≈_ to _≈₁_)
-  open B.Setoid (setoid (P.setoid A₂) B₂)
-    using () renaming (_≈_ to _≈₂_)
-
-  fg = map f (_⟨$⟩_ g)
-
-  fg-cong : _≈₁_ =[ fg ]⇒ _≈₂_
-  fg-cong (P.refl , ∼) = (P.refl , F.cong g ∼)
-
-
-module _ {a₁ a₂ b₁ b₁′ b₂ b₂′} {A₁ : Set a₁} {A₂ : Set a₂} where
-
-  equivalence : {B₁ : IndexedSetoid A₁ b₁ b₁′} {B₂ : IndexedSetoid A₂ b₂ b₂′}
-    (A₁⇔A₂ : A₁ ⇔ A₂) →
-    (∀ {x} → _⟶_ (B₁ atₛ x) (B₂ atₛ (Equivalence.to   A₁⇔A₂ ⟨$⟩ x))) →
-    (∀ {y} → _⟶_ (B₂ atₛ y) (B₁ atₛ (Equivalence.from A₁⇔A₂ ⟨$⟩ y))) →
-    Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
-  equivalence {B₁} {B₂} A₁⇔A₂ B-to B-from = record
-    { to   = ⟶ B₂ (_⟨$⟩_ (to   A₁⇔A₂)) B-to
-    ; from = ⟶ B₁ (_⟨$⟩_ (from A₁⇔A₂)) B-from
-    } where open Equivalence
-
-  equivalence-↞ : (B₁ : IndexedSetoid A₁ b₁ b₁′) {B₂ : IndexedSetoid A₂ b₂ b₂′}
-    (A₁↞A₂ : A₁ ↞ A₂) →
-    (∀ {x} → Equivalence (B₁ atₛ (LeftInverse.from A₁↞A₂ ⟨$⟩ x))
-                         (B₂ atₛ x)) →
-    Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
-  equivalence-↞ B₁ {B₂} A₁↞A₂ B₁⇔B₂ =
-    equivalence (LeftInverse.equivalence A₁↞A₂) B-to B-from
+------------------------------------------------------------------------
+-- Functions
+
+module _ where
+  open Func
+  open Setoid
+
+  function :
+    (f : I ⟶ J) →
+    (∀ {i} → Func (A atₛ i) (B atₛ (to f i))) →
+    Func (I ×ₛ A) (J ×ₛ B)
+  function {I = I} {J = J} {A = A} {B = B} I⟶J A⟶B = record
+    { to    = to′
+    ; cong  = cong′
+    }
     where
-    B-to : ∀ {x} → _⟶_ (B₁ atₛ x) (B₂ atₛ (LeftInverse.to A₁↞A₂ ⟨$⟩ x))
-    B-to = record
-      { _⟨$⟩_ = λ x → Equivalence.to B₁⇔B₂ ⟨$⟩
-                      P.subst (IndexedSetoid.Carrier B₁)
-                         (P.sym $ LeftInverse.left-inverse-of A₁↞A₂ _)
-                         x
-      ; cong  = F.cong (Equivalence.to B₁⇔B₂) ∘
-              subst-cong (λ {x} → IndexedSetoid._≈_ B₁ {x} {x})
-                         (P.sym (LeftInverse.left-inverse-of A₁↞A₂ _))
-      }
+    to′ = map (to I⟶J) (to A⟶B)
+
+    cong′ : Congruent (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′
+    cong′ (P.refl , ∼) = (P.refl , cong A⟶B ∼)
 
-    B-from : ∀ {y} → _⟶_ (B₂ atₛ y) (B₁ atₛ (LeftInverse.from A₁↞A₂ ⟨$⟩ y))
-    B-from = Equivalence.from B₁⇔B₂
+------------------------------------------------------------------------
+-- Equivalences
+
+module _ where
+  open Equivalence
+
+  equivalence :
+    (I⇔J : I ⇔ J) →
+    (∀ {i} → Func (A atₛ i) (B atₛ (to   I⇔J i))) →
+    (∀ {j} → Func (B atₛ j) (A atₛ (from I⇔J j))) →
+    Equivalence (I ×ₛ A) (J ×ₛ B)
+  equivalence I⇔J A⟶B B⟶A = mkEquivalence
+    (function (⇔⇒⟶ I⇔J) A⟶B)
+    (function (⇔⇒⟵ I⇔J) B⟶A)
+
+  equivalence-↪ :
+    (I↪J : I ↪ J) →
+    (∀ {i} → Equivalence (A atₛ (RightInverse.from I↪J i)) (B atₛ i)) →
+    Equivalence (I ×ₛ A) (J ×ₛ B)
+  equivalence-↪ {A = A} {B = B} I↪J A⇔B =
+    equivalence (RightInverse.equivalence I↪J) A→B (fromFunction A⇔B)
+    where
+    A→B : ∀ {i} → Func (A atₛ i) (B atₛ (RightInverse.to I↪J i))
+    A→B = record
+      { to   = to      A⇔B ∘ cast      A (RightInverse.strictlyInverseʳ I↪J _)
+      ; cong = to-cong A⇔B ∘ cast-cong A (RightInverse.strictlyInverseʳ I↪J _)
+      }
 
-  equivalence-↠ : {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′)
-    (A₁↠A₂ : A₁ ↠ A₂) →
-    (∀ {x} → Equivalence (B₁ atₛ x) (B₂ atₛ (Surjection.to A₁↠A₂ ⟨$⟩ x))) →
-    Equivalence (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
-  equivalence-↠ {B₁ = B₁} B₂ A₁↠A₂ B₁⇔B₂ =
-    equivalence (Surjection.equivalence A₁↠A₂) B-to B-from
+  equivalence-↠ :
+    (I↠J : I ↠ J) →
+    (∀ {x} → Equivalence (A atₛ x) (B atₛ (Surjection.to I↠J x))) →
+    Equivalence (I ×ₛ A) (J ×ₛ B)
+  equivalence-↠ {A = A} {B = B} I↠J A⇔B =
+    equivalence (↠⇒⇔ I↠J) B-to B-from
     where
-    B-to : ∀ {x} → _⟶_ (B₁ atₛ x) (B₂ atₛ (Surjection.to A₁↠A₂ ⟨$⟩ x))
-    B-to = Equivalence.to B₁⇔B₂
+    B-to : ∀ {x} → Func (A atₛ x) (B atₛ (Surjection.to I↠J x))
+    B-to = toFunction A⇔B
 
-    B-from : ∀ {y} → _⟶_ (B₂ atₛ y) (B₁ atₛ (Surjection.from A₁↠A₂ ⟨$⟩ y))
+    B-from : ∀ {y} → Func (B atₛ y) (A atₛ (Surjection.to⁻ I↠J y))
     B-from = record
-      { _⟨$⟩_ = λ x → Equivalence.from B₁⇔B₂ ⟨$⟩
-                      P.subst (IndexedSetoid.Carrier B₂)
-                         (P.sym $ Surjection.right-inverse-of A₁↠A₂ _)
-                         x
-      ; cong  = F.cong (Equivalence.from B₁⇔B₂) ∘
-              subst-cong (λ {x} → IndexedSetoid._≈_ B₂ {x} {x})
-                         (P.sym (Surjection.right-inverse-of A₁↠A₂ _))
+      { to   = from      A⇔B ∘ cast      B (Surjection.to∘to⁻ I↠J _)
+      ; cong = from-cong A⇔B ∘ cast-cong B (Surjection.to∘to⁻ I↠J _)
       }
 
-  injection : {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′) →
-    (A₁↣A₂ : A₁ ↣ A₂) →
-    (∀ {x} → Injection (B₁ atₛ x) (B₂ atₛ (Injection.to A₁↣A₂ ⟨$⟩ x))) →
-    Injection (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
-  injection {B₁ = B₁} B₂ A₁↣A₂ B₁↣B₂ = record
-    { to        = to
-    ; injective = inj
-    }
+------------------------------------------------------------------------
+-- Injections
+
+module _ where
+  open Injection hiding (function)
+  open IndexedSetoid
+
+  injection :
+    (I↣J : I ↣ J) →
+    (∀ {i} → Injection (A atₛ i) (B atₛ (Injection.to I↣J i))) →
+    Injection (I ×ₛ A) (J ×ₛ B)
+  injection {I = I} {J = J} {A = A} {B = B} I↣J A↣B = mkInjection func inj
     where
-    to = ⟶ B₂ (Injection.to A₁↣A₂ ⟨$⟩_) (Injection.to B₁↣B₂)
+    func : Func (I ×ₛ A) (J ×ₛ B)
+    func = function (Injection.function I↣J) (Injection.function A↣B)
 
-    inj : Injective to
-    inj (x , y) =
-      Injection.injective A₁↣A₂ x ,
-      lemma (Injection.injective A₁↣A₂ x) y
+    inj : Injective (Func.Eq₁._≈_ func) (Func.Eq₂._≈_ func) (Func.to func)
+    inj (to[i]≡to[j] , y) =
+      injective I↣J to[i]≡to[j] ,
+      lemma (injective I↣J to[i]≡to[j]) y
       where
       lemma :
-        ∀ {x x′}
-          {y : IndexedSetoid.Carrier B₁ x} {y′ : IndexedSetoid.Carrier B₁ x′} →
-        x ≡ x′ →
-        (eq : IndexedSetoid._≈_ B₂ (Injection.to B₁↣B₂ ⟨$⟩ y)
-                              (Injection.to B₁↣B₂ ⟨$⟩ y′)) →
-        IndexedSetoid._≈_ B₁ y y′
-      lemma P.refl = Injection.injective B₁↣B₂
-
-  left-inverse : (B₁ : IndexedSetoid A₁ b₁ b₁′) {B₂ : IndexedSetoid A₂ b₂ b₂′} →
-    (A₁↞A₂ : A₁ ↞ A₂) →
-    (∀ {x} → LeftInverse (B₁ atₛ (LeftInverse.from A₁↞A₂ ⟨$⟩ x))
-                         (B₂ atₛ x)) →
-    LeftInverse (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
-  left-inverse B₁ {B₂} A₁↞A₂ B₁↞B₂ = record
-    { to              = Equivalence.to   eq
-    ; from            = Equivalence.from eq
-    ; left-inverse-of = left
-    }
+        ∀ {i j} {x : Carrier A i} {y : Carrier A j} →
+          i ≡ j →
+          (_≈_ B (to A↣B x) (to A↣B y)) →
+          _≈_ A x y
+      lemma P.refl = Injection.injective A↣B
+
+------------------------------------------------------------------------
+-- Surjections
+
+module _ where
+  open Surjection hiding (function)
+  open Setoid
+
+  surjection :
+    (I↠J : I ↠ J) →
+    (∀ {x} → Surjection (A atₛ x) (B atₛ (to I↠J x))) →
+    Surjection (I ×ₛ A) (J ×ₛ B)
+  surjection {I = I} {J = J} {A = A} {B = B} I↠J A↠B =
+    mkSurjection func surj
     where
-    eq = equivalence-↞ B₁ A₁↞A₂ (LeftInverse.equivalence B₁↞B₂)
-
-    left : Equivalence.from eq LeftInverseOf Equivalence.to eq
-    left (x , y) =
-      LeftInverse.left-inverse-of A₁↞A₂ x ,
-      IndexedSetoid.trans B₁
-        (LeftInverse.left-inverse-of B₁↞B₂ _)
-        (lemma (P.sym (LeftInverse.left-inverse-of A₁↞A₂ x)))
-      where
-      lemma :
-        ∀ {x x′ y} (eq : x ≡ x′) →
-        IndexedSetoid._≈_ B₁ (P.subst (IndexedSetoid.Carrier B₁) eq y) y
-      lemma P.refl = IndexedSetoid.refl B₁
-
-  surjection : {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′) →
-    (A₁↠A₂ : A₁ ↠ A₂) →
-    (∀ {x} → Surjection (B₁ atₛ x) (B₂ atₛ (Surjection.to A₁↠A₂ ⟨$⟩ x))) →
-    Surjection (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
-  surjection B₂ A₁↠A₂ B₁↠B₂ = record
-    { to         = Equivalence.to eq
-    ; surjective = record
-      { from             = Equivalence.from eq
-      ; right-inverse-of = right
-      }
+    func : Func (I ×ₛ A) (J ×ₛ B)
+    func = function (Surjection.function I↠J) (Surjection.function A↠B)
+
+    to⁻′ : Carrier (J ×ₛ B) → Carrier (I ×ₛ A)
+    to⁻′ (j , y) = to⁻ I↠J j , to⁻ A↠B (cast B (Surjection.to∘to⁻ I↠J _) y)
+
+    strictlySurj : StrictlySurjective (Func.Eq₂._≈_ func) (Func.to func)
+    strictlySurj (j , y) = to⁻′ (j , y) ,
+      to∘to⁻ I↠J j , IndexedSetoid.trans B (to∘to⁻ A↠B _) (cast-eq B (to∘to⁻ I↠J j))
+
+    surj : Surjective (Func.Eq₁._≈_ func) (Func.Eq₂._≈_ func) (Func.to func)
+    surj = strictlySurjective⇒surjective (trans (J ×ₛ B)) (Func.cong func) strictlySurj
+
+------------------------------------------------------------------------
+-- LeftInverse
+
+module _ where
+  open RightInverse
+  open Setoid
+
+  left-inverse :
+    (I↪J : I ↪ J) →
+    (∀ {j} → RightInverse (A atₛ (from I↪J j)) (B atₛ j)) →
+    RightInverse (I ×ₛ A) (J ×ₛ B)
+  left-inverse {I = I} {J = J} {A = A} {B = B} I↪J A↪B =
+    mkRightInverse equiv invʳ
+    where
+    equiv : Equivalence (I ×ₛ A) (J ×ₛ B)
+    equiv = equivalence-↪ I↪J (RightInverse.equivalence A↪B)
+
+    strictlyInvʳ : StrictlyInverseʳ (_≈_ (I ×ₛ A)) (Equivalence.to equiv) (Equivalence.from equiv)
+    strictlyInvʳ (i , x) = strictlyInverseʳ I↪J i , IndexedSetoid.trans A (strictlyInverseʳ A↪B _) (cast-eq A (strictlyInverseʳ I↪J i))
+
+    invʳ : Inverseʳ (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) (Equivalence.to equiv) (Equivalence.from equiv)
+    invʳ = strictlyInverseʳ⇒inverseʳ {f⁻¹ = Equivalence.from equiv} (trans (I ×ₛ A)) (Equivalence.from-cong equiv) strictlyInvʳ
+
+
+------------------------------------------------------------------------
+-- Inverses
+
+module _ where
+  open Inverse hiding (inverse)
+  open Setoid
+
+  inverse : (I↔J : I ↔ J) →
+            (∀ {i} → Inverse (A atₛ i) (B atₛ (to I↔J i))) →
+            Inverse (I ×ₛ A) (J ×ₛ B)
+  inverse {I = I} {J = J} {A = A} {B = B} I↔J A↔B = record
+    { to = to′
+    ; from = from′
+    ; to-cong = to′-cong
+    ; from-cong = from′-cong
+    ; inverse = invˡ , invʳ
     }
     where
-    eq = equivalence-↠ B₂ A₁↠A₂ (Surjection.equivalence B₁↠B₂)
-
-    right : Equivalence.from eq RightInverseOf Equivalence.to eq
-    right (x , y) =
-      Surjection.right-inverse-of A₁↠A₂ x ,
-      IndexedSetoid.trans B₂
-        (Surjection.right-inverse-of B₁↠B₂ _)
-        (lemma (P.sym $ Surjection.right-inverse-of A₁↠A₂ x))
-      where
-      lemma : ∀ {x x′ y} (eq : x ≡ x′) →
-              IndexedSetoid._≈_ B₂ (P.subst (IndexedSetoid.Carrier B₂) eq y) y
-      lemma P.refl = IndexedSetoid.refl B₂
+    to′ : Carrier (I ×ₛ A) → Carrier (J ×ₛ B)
+    to′ (i , x) = to I↔J i , to A↔B x
+
+    to′-cong : Congruent (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′
+    to′-cong (P.refl , x≈y) = to-cong I↔J P.refl , to-cong A↔B x≈y
+
+    from′ : Carrier (J ×ₛ B) → Carrier (I ×ₛ A)
+    from′ (j , y) = from I↔J j , from A↔B (cast B (strictlyInverseˡ I↔J _) y)
+
+    from′-cong : Congruent (_≈_ (J ×ₛ B)) (_≈_ (I ×ₛ A)) from′
+    from′-cong (P.refl , x≈y) = from-cong I↔J P.refl , from-cong A↔B (cast-cong B (strictlyInverseˡ I↔J _) x≈y)
+
+    strictlyInvˡ : StrictlyInverseˡ (_≈_ (J ×ₛ B)) to′ from′
+    strictlyInvˡ (i , x) = strictlyInverseˡ I↔J i ,
+        IndexedSetoid.trans B (strictlyInverseˡ A↔B _)
+          (cast-eq B (strictlyInverseˡ I↔J i))
+
+    invˡ : Inverseˡ (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′ from′
+    invˡ = strictlyInverseˡ⇒inverseˡ {≈₁ = _≈_ (I ×ₛ A)} {f⁻¹ = from′} (trans (J ×ₛ B)) to′-cong strictlyInvˡ
+
+    lem : ∀ {i j} → i ≡ j → ∀ {x : IndexedSetoid.Carrier B (to I↔J i)} {y : IndexedSetoid.Carrier B (to I↔J j)} →
+          IndexedSetoid._≈_ B x y →
+          IndexedSetoid._≈_ A (from A↔B x) (from A↔B y)
+    lem P.refl x≈y = from-cong A↔B x≈y
+
+    strictlyInvʳ : StrictlyInverseʳ (_≈_ (I ×ₛ A)) to′ from′
+    strictlyInvʳ (i , x) = strictlyInverseʳ I↔J i ,
+      IndexedSetoid.trans A (lem (strictlyInverseʳ I↔J _) (cast-eq B (strictlyInverseˡ I↔J _))) (strictlyInverseʳ A↔B _)
+
+    invʳ : Inverseʳ (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′ from′
+    invʳ = strictlyInverseʳ⇒inverseʳ {f⁻¹ = from′} (trans (I ×ₛ A)) from′-cong strictlyInvʳ
 
-  -- See also Data.Product.Function.Dependent.Setoid.WithK.inverse.
diff --git a/src/Data/Product/Function/Dependent/Setoid/WithK.agda b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
index bb89c1b95b..75a3ff4ed1 100644
--- a/src/Data/Product/Function/Dependent/Setoid/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Setoid/WithK.agda
@@ -1,70 +1,17 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Dependent product combinators for setoid equality preserving
--- functions
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --with-K --safe #-}
 
 module Data.Product.Function.Dependent.Setoid.WithK where
 
-open import Data.Product.Base using (_,_)
-open import Data.Product.Function.Dependent.Setoid using (surjection)
-open import Data.Product.Relation.Binary.Pointwise.Dependent
-open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence as Eq
-  using (Equivalence; _⇔_; module Equivalence)
-open import Function.Injection as Inj
-  using (Injection; Injective; _↣_; module Injection)
-open import Function.Inverse as Inv
-  using (Inverse; _↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (LeftInverse; _↞_; _LeftInverseOf_; _RightInverseOf_; module LeftInverse)
-open import Function.Surjection as Surj
-  using (Surjection; _↠_; module Surjection)
-open import Relation.Binary.Indexed.Heterogeneous
-  using (IndexedSetoid)
-open import Relation.Binary.Indexed.Heterogeneous.Construct.At
-  using (_atₛ_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Data.Product.Function.Dependent.Setoid public
+  using (inverse)
 
-------------------------------------------------------------------------
--- Combinator for Inverse
-
-module _ {a₁ a₂ b₁ b₁′ b₂ b₂′} {A₁ : Set a₁} {A₂ : Set a₂} where
-
-  inverse : {B₁ : IndexedSetoid A₁ b₁ b₁′} (B₂ : IndexedSetoid A₂ b₂ b₂′) →
-    (A₁↔A₂ : A₁ ↔ A₂) →
-    (∀ {x} → Inverse (B₁ atₛ x) (B₂ atₛ (Inverse.to A₁↔A₂ ⟨$⟩ x))) →
-    Inverse (setoid (P.setoid A₁) B₁) (setoid (P.setoid A₂) B₂)
-  inverse {B₁} B₂ A₁↔A₂ B₁↔B₂ = record
-    { to         = Surjection.to   surj
-    ; from       = Surjection.from surj
-    ; inverse-of = record
-      { left-inverse-of  = left
-      ; right-inverse-of = Surjection.right-inverse-of surj
-      }
-    }
-    where
-    surj = surjection B₂ (Inverse.surjection A₁↔A₂)
-                         (Inverse.surjection B₁↔B₂)
-
-    left : Surjection.from surj LeftInverseOf Surjection.to surj
-    left (x , y) =
-      Inverse.left-inverse-of A₁↔A₂ x ,
-      IndexedSetoid.trans B₁
-        (lemma (P.sym (Inverse.left-inverse-of A₁↔A₂ x))
-               (P.sym (Inverse.right-inverse-of A₁↔A₂
-                       (Inverse.to A₁↔A₂ ⟨$⟩ x))))
-        (Inverse.left-inverse-of B₁↔B₂ y)
-      where
-      lemma :
-        ∀ {x x′ y} → x ≡ x′ →
-        (eq : (Inverse.to A₁↔A₂ ⟨$⟩ x) ≡ (Inverse.to A₁↔A₂ ⟨$⟩ x′)) →
-        IndexedSetoid._≈_ B₁
-          (Inverse.from B₁↔B₂ ⟨$⟩ P.subst (IndexedSetoid.Carrier B₂) eq y)
-          (Inverse.from B₁↔B₂ ⟨$⟩ y)
-      lemma P.refl P.refl = IndexedSetoid.refl B₁
+{-# WARNING_ON_IMPORT
+"Data.Product.Function.Dependent.Setoid.WithK was deprecated in v2.0.
+Use Data.Product.Function.Dependent.Setoid instead."
+#-}
diff --git a/src/Data/Product/Function/NonDependent/Propositional.agda b/src/Data/Product/Function/NonDependent/Propositional.agda
index a67490b8a5..5e7f3d1496 100644
--- a/src/Data/Product/Function/NonDependent/Propositional.agda
+++ b/src/Data/Product/Function/NonDependent/Propositional.agda
@@ -12,66 +12,69 @@ module Data.Product.Function.NonDependent.Propositional where
 open import Data.Product.Base using (_×_; map)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Function.Equivalence as Eq using (_⇔_; module Equivalence)
-open import Function.Injection as Inj using (_↣_; module Injection)
-open import Function.Inverse as Inv using (_↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (_↞_; _LeftInverseOf_; module LeftInverse)
-open import Function.Related
-open import Function.Surjection as Surj using (_↠_; module Surjection)
+open import Function
+open import Function.Properties.Inverse as Inv
+open import Function.Related.Propositional
+open import Function.Construct.Composition as Compose
+open import Level using (Level; _⊔_)
+open import Relation.Binary hiding (_⇔_)
+open import Relation.Binary.PropositionalEquality using (setoid)
+
+private
+  variable
+    a b c d : Level
+    A B C D : Set a
+
+------------------------------------------------------------------------
+-- Helper lemma
+
+private
+  liftViaInverse : {R : ∀ {a b ℓ₁ ℓ₂} → REL (Setoid a ℓ₁) (Setoid b ℓ₂) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)} →
+                   (∀ {a b c ℓ₁ ℓ₂ ℓ₃} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} → R S T → R T U → R S U) →
+                   (∀ {a b ℓ₁ ℓ₂} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} → Inverse S T → R S T) →
+                   (R (setoid A) (setoid C) → R (setoid B) (setoid D) → R (setoid A ×ₛ setoid B) (setoid C ×ₛ setoid D)) →
+                   R (setoid A) (setoid C) → R (setoid B) (setoid D) →
+                   R (setoid (A × B)) (setoid (C × D))
+  liftViaInverse trans inv⇒R lift RAC RBD =
+    Inv.transportVia trans inv⇒R (Inv.sym Pointwise-≡↔≡) (lift RAC RBD) Pointwise-≡↔≡
 
 ------------------------------------------------------------------------
 -- Combinators for various function types
 
-module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
-
-  infixr 2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_
-
-  _×-⇔_ : A ⇔ B → C ⇔ D → (A × C) ⇔ (B × D)
-  _×-⇔_ A⇔B C⇔D =
-    Inverse.equivalence Pointwise-≡↔≡           ⟨∘⟩
-    (A⇔B ×-equivalence C⇔D)                     ⟨∘⟩
-    Eq.sym (Inverse.equivalence Pointwise-≡↔≡)
-    where open Eq using () renaming (_∘_ to _⟨∘⟩_)
-
-  _×-↣_ : A ↣ B → C ↣ D → (A × C) ↣ (B × D)
-  _×-↣_ A↣B C↣D =
-    Inverse.injection Pointwise-≡↔≡           ⟨∘⟩
-    (A↣B ×-injection C↣D)                     ⟨∘⟩
-    Inverse.injection (Inv.sym Pointwise-≡↔≡)
-    where open Inj using () renaming (_∘_ to _⟨∘⟩_)
-
-  _×-↞_ : A ↞ B → C ↞ D → (A × C) ↞ (B × D)
-  _×-↞_ A↞B C↞D =
-    Inverse.left-inverse Pointwise-≡↔≡           ⟨∘⟩
-    (A↞B ×-left-inverse C↞D)                     ⟨∘⟩
-    Inverse.left-inverse (Inv.sym Pointwise-≡↔≡)
-    where open LeftInv using () renaming (_∘_ to _⟨∘⟩_)
-
-  _×-↠_ : A ↠ B → C ↠ D → (A × C) ↠ (B × D)
-  _×-↠_ A↠B C↠D =
-    Inverse.surjection Pointwise-≡↔≡           ⟨∘⟩
-    (A↠B ×-surjection C↠D)                     ⟨∘⟩
-    Inverse.surjection (Inv.sym Pointwise-≡↔≡)
-    where open Surj using () renaming (_∘_ to _⟨∘⟩_)
-
-  _×-↔_ : A ↔ B → C ↔ D → (A × C) ↔ (B × D)
-  _×-↔_ A↔B C↔D =
-    Pointwise-≡↔≡          ⟨∘⟩
-    (A↔B ×-inverse C↔D)    ⟨∘⟩
-    Inv.sym Pointwise-≡↔≡
-    where open Inv using () renaming (_∘_ to _⟨∘⟩_)
-
-module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
-
-  infixr 2 _×-cong_
-
-  _×-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A × C) ∼[ k ] (B × D)
-  _×-cong_ {implication}         = λ f g →      map        f         g
-  _×-cong_ {reverse-implication} = λ f g → lam (map (app-← f) (app-← g))
-  _×-cong_ {equivalence}         = _×-⇔_
-  _×-cong_ {injection}           = _×-↣_
-  _×-cong_ {reverse-injection}   = λ f g → lam (app-↢ f ×-↣ app-↢ g)
-  _×-cong_ {left-inverse}        = _×-↞_
-  _×-cong_ {surjection}          = _×-↠_
-  _×-cong_ {bijection}           = _×-↔_
+infixr 2 _×-⇔_ _×-↣_ _×-↠_ _×-⤖_ _×-↩_ _×-↪_ _×-↔_
+
+_×-⟶_ : A ⟶ B → C ⟶ D → (A × C) ⟶ (B × D)
+_×-⟶_ = liftViaInverse Compose.function Inv.toFunction _×-function_
+
+_×-⇔_ : A ⇔ B → C ⇔ D → (A × C) ⇔ (B × D)
+_×-⇔_ = liftViaInverse Compose.equivalence Inverse⇒Equivalence _×-equivalence_
+
+_×-↣_ : A ↣ B → C ↣ D → (A × C) ↣ (B × D)
+_×-↣_ = liftViaInverse Compose.injection Inverse⇒Injection _×-injection_
+
+_×-↠_ : A ↠ B → C ↠ D → (A × C) ↠ (B × D)
+_×-↠_ = liftViaInverse Compose.surjection Inverse⇒Surjection _×-surjection_
+
+_×-⤖_ : A ⤖ B → C ⤖ D → (A × C) ⤖ (B × D)
+_×-⤖_ = liftViaInverse Compose.bijection Inverse⇒Bijection _×-bijection_
+
+_×-↩_ : A ↩ B → C ↩ D → (A × C) ↩ (B × D)
+_×-↩_ = liftViaInverse Compose.leftInverse Inverse.leftInverse _×-leftInverse_
+
+_×-↪_ : A ↪ B → C ↪ D → (A × C) ↪ (B × D)
+_×-↪_ = liftViaInverse Compose.rightInverse Inverse.rightInverse _×-rightInverse_
+
+_×-↔_ : A ↔ B → C ↔ D → (A × C) ↔ (B × D)
+_×-↔_ = liftViaInverse Compose.inverse id _×-inverse_
+
+infixr 2 _×-cong_
+
+_×-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A × C) ∼[ k ] (B × D)
+_×-cong_ {k = implication}         = _×-⟶_
+_×-cong_ {k = reverseImplication}  = _×-⟶_
+_×-cong_ {k = equivalence}         = _×-⇔_
+_×-cong_ {k = injection}           = _×-↣_
+_×-cong_ {k = reverseInjection}    = _×-↣_
+_×-cong_ {k = leftInverse}         = _×-↪_
+_×-cong_ {k = surjection}          = _×-↠_
+_×-cong_ {k = bijection}           = _×-↔_
diff --git a/src/Data/Product/Function/NonDependent/Setoid.agda b/src/Data/Product/Function/NonDependent/Setoid.agda
index 1d69d9c85b..653f122bb5 100644
--- a/src/Data/Product/Function/NonDependent/Setoid.agda
+++ b/src/Data/Product/Function/NonDependent/Setoid.agda
@@ -9,140 +9,126 @@
 
 module Data.Product.Function.NonDependent.Setoid where
 
-open import Data.Product.Base using (map; _,_; <_,_>; proj₁; proj₂)
+open import Data.Product.Base as Prod
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Relation.Binary.Core using (_=[_]⇒_)
-open import Relation.Binary.Bundles using (Setoid)
-open import Function.Equality as F using (_⟶_; _⟨$⟩_)
-open import Function.Equivalence as Eq
-  using (Equivalence; _⇔_; module Equivalence)
-open import Function.Injection as Inj
-  using (Injection; _↣_; module Injection)
-open import Function.Inverse as Inv
-  using (Inverse; _↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (LeftInverse; _↞_; _LeftInverseOf_; module LeftInverse)
-open import Function.Related
-open import Function.Surjection as Surj
-  using (Surjection; _↠_; module Surjection)
+open import Level using (Level)
+open import Relation.Binary
+open import Function
+
+private
+  variable
+    a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂ : Level
+    a ℓ : Level
+    A B C D : Setoid a ℓ
 
 ------------------------------------------------------------------------
 -- Combinators for equality preserving functions
 
-module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
-          {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
-          {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
-          where
-
-  infixr 2 _×-⟶_
+proj₁ₛ : Func (A ×ₛ B) A
+proj₁ₛ = record { to = proj₁ ; cong = proj₁ }
 
-  _×-⟶_ : (A ⟶ B) → (C ⟶ D) → (A ×ₛ C) ⟶ (B ×ₛ D)
-  _×-⟶_ f g = record
-    { _⟨$⟩_ = fg
-    ; cong  = fg-cong
-    }
-    where
-    open Setoid (A ×ₛ C) using () renaming (_≈_ to _≈AC_)
-    open Setoid (B ×ₛ D) using () renaming (_≈_ to _≈BD_)
+proj₂ₛ : Func (A ×ₛ B) B
+proj₂ₛ = record { to = proj₂ ; cong = proj₂ }
 
-    fg = map (f ⟨$⟩_) (g ⟨$⟩_)
+<_,_>ₛ : Func A B → Func A C → Func A (B ×ₛ C)
+< f , g >ₛ = record
+  { to   = < to   f , to   g >
+  ; cong = < cong f , cong g >
+  } where open Func
 
-    fg-cong : _≈AC_ =[ fg ]⇒ _≈BD_
-    fg-cong (_∼₁_ , _∼₂_) = (F.cong f _∼₁_ , F.cong g _∼₂_)
+swapₛ : Func (A ×ₛ B) (B ×ₛ A)
+swapₛ = < proj₂ₛ , proj₁ₛ >ₛ
 
-module _ {a₁ a₂ b₁ b₂ c₁ c₂}
-         {A : Setoid a₁ a₂} {B : Setoid b₁ b₂} {C : Setoid c₁ c₂}
-         where
-
-  <_,_>ₛ : (A ⟶ B) → (A ⟶ C) → A ⟶ (B ×ₛ C)
-  < f , g >ₛ = record
-    { _⟨$⟩_ = < f ⟨$⟩_ , g ⟨$⟩_ >
-    ; cong = < F.cong f , F.cong g >
-    }
+------------------------------------------------------------------------
+-- Function bundles
+
+_×-function_ : Func A B → Func C D → Func (A ×ₛ C) (B ×ₛ D)
+f ×-function g = record
+  { to    = Prod.map (to f) (to g)
+  ; cong  = Prod.map (cong f) (cong g)
+  } where open Func
+
+infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_
+
+_×-equivalence_ : Equivalence A B → Equivalence C D →
+                  Equivalence (A ×ₛ C) (B ×ₛ D)
+_×-equivalence_ f g = record
+  { to        = Prod.map (to f) (to g)
+  ; from      = Prod.map (from f) (from g)
+  ; to-cong   = Prod.map (to-cong f) (to-cong g)
+  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  } where open Equivalence
+
+_×-injection_ : Injection A B → Injection C D →
+                Injection (A ×ₛ C) (B ×ₛ D)
+f ×-injection g = record
+  { to        = Prod.map (to f) (to g)
+  ; cong      = Prod.map (cong f) (cong g)
+  ; injective = Prod.map (injective f) (injective g)
+  } where open Injection
+
+_×-surjection_ : Surjection A B → Surjection C D →
+                 Surjection (A ×ₛ C) (B ×ₛ D)
+f ×-surjection g = record
+  { to         = Prod.map (to f) (to g)
+  ; cong       = Prod.map (cong f) (cong g)
+  ; surjective = λ y → Prod.zip _,_ (λ ff gg x₂ → (ff (proj₁ x₂)) , (gg (proj₂ x₂))) (surjective f (proj₁ y)) (surjective g (proj₂ y))
+  } where open Surjection
+
+_×-bijection_ : Bijection A B → Bijection C D →
+                Bijection (A ×ₛ C) (B ×ₛ D)
+f ×-bijection g = record
+  { to         = Prod.map (to f) (to g)
+  ; cong       = Prod.map (cong f) (cong g)
+  ; bijective  = Prod.map (injective f) (injective g) ,
+                 λ { (y₀ , y₁) → Prod.zip _,_ (λ {ff gg (x₀ , x₁) → ff x₀ , gg x₁}) (surjective f y₀) (surjective g y₁)}
+  } where open Bijection
+
+_×-leftInverse_ : LeftInverse A B → LeftInverse C D →
+                  LeftInverse (A ×ₛ C) (B ×ₛ D)
+f ×-leftInverse g = record
+  { to        = Prod.map (to f) (to g)
+  ; from      = Prod.map (from f) (from g)
+  ; to-cong   = Prod.map (to-cong f) (to-cong g)
+  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  ; inverseˡ   = λ x → inverseˡ f (proj₁ x) , inverseˡ g (proj₂ x)
+  } where open LeftInverse
+
+_×-rightInverse_ : RightInverse A B → RightInverse C D →
+                   RightInverse (A ×ₛ C) (B ×ₛ D)
+f ×-rightInverse g = record
+  { to        = Prod.map (to f) (to g)
+  ; from      = Prod.map (from f) (from g)
+  ; to-cong   = Prod.map (to-cong f) (to-cong g)
+  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  ; inverseʳ   = λ x → inverseʳ f (proj₁ x) , inverseʳ g (proj₂ x)
+  } where open RightInverse
+
+infixr 2 _×-surjection_ _×-inverse_
+
+_×-inverse_ : Inverse A B → Inverse C D →
+              Inverse (A ×ₛ C) (B ×ₛ D)
+f ×-inverse g = record
+  { to        = Prod.map (to f) (to g)
+  ; from      = Prod.map (from f) (from g)
+  ; to-cong   = Prod.map (to-cong f) (to-cong g)
+  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  ; inverse   = (λ x → inverseˡ f (proj₁ x) , inverseˡ g (proj₂ x)) ,
+                (λ x → inverseʳ f (proj₁ x) , inverseʳ g (proj₂ x))
+  } where open Inverse
 
-module _ {a₁ a₂ b₁ b₂} {A : Setoid a₁ a₂} {B : Setoid b₁ b₂} where
 
-  proj₁ₛ : (A ×ₛ B) ⟶ A
-  proj₁ₛ = record { _⟨$⟩_ = proj₁ ; cong = proj₁ }
 
-  proj₂ₛ : (A ×ₛ B) ⟶ B
-  proj₂ₛ = record { _⟨$⟩_ = proj₂ ; cong = proj₂ }
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
 
-  swapₛ : (A ×ₛ B) ⟶ (B ×ₛ A)
-  swapₛ = < proj₂ₛ , proj₁ₛ >ₛ
+-- Version 2.0
 
-------------------------------------------------------------------------
--- Combinators for more complex function types
-
-module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
-          {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
-          {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
-          where
-
-  infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_
-
-  _×-equivalence_ : Equivalence A B → Equivalence C D →
-                    Equivalence (A ×ₛ C) (B ×ₛ D)
-  _×-equivalence_ A⇔B C⇔D = record
-    { to   = to   A⇔B ×-⟶ to   C⇔D
-    ; from = from A⇔B ×-⟶ from C⇔D
-    } where open Equivalence
-
-  _×-injection_ : Injection A B → Injection C D →
-                  Injection (A ×ₛ C) (B ×ₛ D)
-  A↣B ×-injection C↣D = record
-    { to        = to A↣B ×-⟶ to C↣D
-    ; injective = map (injective A↣B) (injective C↣D)
-    } where open Injection
-
-  _×-left-inverse_ : LeftInverse A B → LeftInverse C D →
-                     LeftInverse (A ×ₛ C) (B ×ₛ D)
-  A↞B ×-left-inverse C↞D = record
-    { to              = Equivalence.to eq
-    ; from            = Equivalence.from eq
-    ; left-inverse-of = left
-    }
-    where
-    open LeftInverse
-    eq = LeftInverse.equivalence A↞B ×-equivalence
-         LeftInverse.equivalence C↞D
-
-    left : Equivalence.from eq LeftInverseOf Equivalence.to eq
-    left (x , y) = (left-inverse-of A↞B x , left-inverse-of C↞D y)
-
-module _ {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
-  {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
-  {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
-  where
-
-  infixr 2 _×-surjection_ _×-inverse_
-
-  _×-surjection_ : Surjection A B → Surjection C D →
-                   Surjection (A ×ₛ C) (B ×ₛ D)
-  A↠B ×-surjection C↠D = record
-    { to         = LeftInverse.from inv
-    ; surjective = record
-      { from             = LeftInverse.to inv
-      ; right-inverse-of = LeftInverse.left-inverse-of inv
-      }
-    }
-    where
-    open Surjection
-    inv = right-inverse A↠B ×-left-inverse right-inverse C↠D
-
-  _×-inverse_ : Inverse A B → Inverse C D →
-                Inverse (A ×ₛ C) (B ×ₛ D)
-  A↔B ×-inverse C↔D = record
-    { to         = Surjection.to   surj
-    ; from       = Surjection.from surj
-    ; inverse-of = record
-      { left-inverse-of  = LeftInverse.left-inverse-of inv
-      ; right-inverse-of = Surjection.right-inverse-of surj
-      }
-    }
-    where
-    open Inverse
-    surj = Inverse.surjection   A↔B ×-surjection
-           Inverse.surjection   C↔D
-    inv  = Inverse.left-inverse A↔B ×-left-inverse
-           Inverse.left-inverse C↔D
+_×-left-inverse_ = _×-leftInverse_
+{-# WARNING_ON_USAGE _×-left-inverse_
+"Warning: _×-left-inverse_ was deprecated in v2.0.
+Please use _×-leftInverse_ instead."
+#-}
diff --git a/src/Data/Product/Properties.agda b/src/Data/Product/Properties.agda
index e741f8976a..5e8509b76e 100644
--- a/src/Data/Product/Properties.agda
+++ b/src/Data/Product/Properties.agda
@@ -11,7 +11,7 @@ module Data.Product.Properties where
 open import Axiom.UniquenessOfIdentityProofs
 open import Data.Product.Base
 open import Function.Base using (_∋_; _∘_; id)
-open import Function.Bundles using (_↔_; mk↔′)
+open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
@@ -82,7 +82,7 @@ module _ {A : Set a} {B : A → Set b} {p₁@(a₁ , b₁) p₂@(a₂ , b₂) :
     right-inverse-of refl = refl
 
   Σ-≡,≡↔≡ : (∃ λ (p : a₁ ≡ a₂) → subst B p b₁ ≡ b₂) ↔ p₁ ≡ p₂
-  Σ-≡,≡↔≡ = mk↔′ Σ-≡,≡→≡ Σ-≡,≡←≡ right-inverse-of left-inverse-of
+  Σ-≡,≡↔≡ = mk↔ₛ′ Σ-≡,≡→≡ Σ-≡,≡←≡ right-inverse-of left-inverse-of
 
 -- the non-dependent case. Proofs are exactly as above, and straightforward.
 module _ {p₁@(a₁ , b₁) p₂@(a₂ , b₂) : A × B} where
@@ -93,7 +93,7 @@ module _ {p₁@(a₁ , b₁) p₂@(a₂ , b₂) : A × B} where
   ×-≡,≡←≡ refl = refl , refl
 
   ×-≡,≡↔≡ : (a₁ ≡ a₂ × b₁ ≡ b₂) ↔ p₁ ≡ p₂
-  ×-≡,≡↔≡ = mk↔′
+  ×-≡,≡↔≡ = mk↔ₛ′
     ×-≡,≡→≡
     ×-≡,≡←≡
     (λ { refl          → refl        })
@@ -103,7 +103,7 @@ module _ {p₁@(a₁ , b₁) p₂@(a₂ , b₂) : A × B} where
 -- The order of ∃₂ can be swapped
 
 ∃∃↔∃∃ : (R : A → B → Set ℓ) → (∃₂ λ x y → R x y) ↔ (∃₂ λ y x → R x y)
-∃∃↔∃∃ R = mk↔′ to from cong′ cong′
+∃∃↔∃∃ R = mk↔ₛ′ to from cong′ cong′
   where
   to : (∃₂ λ x y → R x y) → (∃₂ λ y x → R x y)
   to (x , y , Rxy) = (y , x , Rxy)
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
index d388b14f91..decb0c779f 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
@@ -9,35 +9,38 @@
 
 module Data.Product.Relation.Binary.Pointwise.Dependent.WithK where
 
-open import Data.Product.Base using (Σ; uncurry)
+open import Data.Product.Base using (Σ; uncurry; _,_)
 open import Data.Product.Relation.Binary.Pointwise.Dependent
 open import Function.Base
-open import Function.Inverse using (Inverse)
+open import Function.Bundles using (Inverse)
+open import Level using (Level)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.HeterogeneousEquality as H using (_≅_)
 open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
+private
+  variable
+    a b : Level
+    I : Set a
+    A : I → Set a
+
 ------------------------------------------------------------------------
 -- The propositional equality setoid over sigma types can be
 -- decomposed using Pointwise
 
-module _ {a b} {A : Set a} {B : A → Set b} where
-
-  Pointwise-≡⇒≡ : Pointwise B _≡_ (λ x y → x ≅ y) ⇒ _≡_
-  Pointwise-≡⇒≡ (P.refl , H.refl) = P.refl
+Pointwise-≡⇒≡ : Pointwise A _≡_ (λ x y → x ≅ y) ⇒ _≡_
+Pointwise-≡⇒≡ (P.refl , H.refl) = P.refl
 
-  ≡⇒Pointwise-≡ : _≡_ ⇒ Pointwise B _≡_ (λ x y → x ≅ y)
-  ≡⇒Pointwise-≡ P.refl = (P.refl , H.refl)
+≡⇒Pointwise-≡ : _≡_ ⇒ Pointwise A _≡_ (λ x y → x ≅ y)
+≡⇒Pointwise-≡ P.refl = (P.refl , H.refl)
 
-  Pointwise-≡↔≡ : Inverse (setoid (P.setoid A) (H.indexedSetoid B))
-                  (P.setoid (Σ A B))
-  Pointwise-≡↔≡ = record
-    { to         = record { _⟨$⟩_ = id; cong = Pointwise-≡⇒≡ }
-    ; from       = record { _⟨$⟩_ = id; cong = ≡⇒Pointwise-≡ }
-    ; inverse-of = record
-      { left-inverse-of  = uncurry (λ _ _ → (P.refl , H.refl))
-      ; right-inverse-of = λ _ → P.refl
-      }
-    }
+Pointwise-≡↔≡ : Inverse (setoid (P.setoid I) (H.indexedSetoid A)) (P.setoid (Σ I A))
+Pointwise-≡↔≡ = record
+  { to         = id
+  ; to-cong    = Pointwise-≡⇒≡
+  ; from       = id
+  ; from-cong  = ≡⇒Pointwise-≡
+  ; inverse    = (λ {(P.refl , H.refl) → P.refl}) , λ {P.refl → (P.refl , H.refl)}
+  }
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index 00629a2b5d..efa64a8605 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -12,241 +12,192 @@ open import Data.Product.Base as Prod
 open import Data.Product.Properties using (≡-dec)
 open import Data.Sum.Base
 open import Data.Unit.Base using (⊤)
-open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence as Eq
-  using (Equivalence; _⇔_; module Equivalence)
-open import Function.Injection as Inj
-  using (Injection; _↣_; module Injection)
-open import Function.Inverse as Inv
-  using (Inverse; _↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (LeftInverse; _↞_; _LeftInverseOf_; module LeftInverse)
-open import Function.Related
-open import Function.Surjection as Surj
-  using (Surjection; _↠_; module Surjection)
+open import Level using (Level; _⊔_; 0ℓ)
+open import Function
 open import Relation.Nullary.Decidable using (_×-dec_)
-open import Relation.Binary.Core using (Rel; _⇒_)
-open import Relation.Binary.Bundles
-  using (Preorder; Setoid; DecSetoid; Poset; StrictPartialOrder)
-open import Relation.Binary.Structures
-  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsStrictPartialOrder)
-open import Relation.Binary.Definitions
-  using (Reflexive; Irreflexive; Symmetric; Transitive; Antisymmetric; Asymmetric; Total; Decidable; _Respects₂_; _Respects_)
+open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
-module _ {a₁ a₂ ℓ₁ ℓ₂} {A₁ : Set a₁} {A₂ : Set a₂} where
+private
+  variable
+    a b ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level
+    A B : Set a
+    R S ≈₁ ≈₂ : Rel A ℓ₁
 
 ------------------------------------------------------------------------
--- Pointwise lifting
+-- Definition
 
-  Pointwise : Rel A₁ ℓ₁ → Rel A₂ ℓ₂ → Rel (A₁ × A₂) _
-  Pointwise _∼₁_ _∼₂_ = (_∼₁_ on proj₁) -×- (_∼₂_ on proj₂)
+Pointwise : Rel A ℓ₁ → Rel B ℓ₂ → Rel (A × B) (ℓ₁ ⊔ ℓ₂)
+Pointwise R S = (R on proj₁) -×- (S on proj₂)
 
 ------------------------------------------------------------------------
 -- Pointwise preserves many relational properties
 
-  ×-reflexive : ∀ {_≈₁_ _∼₁_ _≈₂_ _∼₂_} →
-    _≈₁_ ⇒ _∼₁_ → _≈₂_ ⇒ _∼₂_ →
-    (Pointwise _≈₁_ _≈₂_) ⇒ (Pointwise _∼₁_ _∼₂_)
-  ×-reflexive refl₁ refl₂ (x∼y₁ , x∼y₂) = refl₁ x∼y₁ , refl₂ x∼y₂
-
-  ×-refl : ∀ {_∼₁_ _∼₂_} →
-    Reflexive _∼₁_ → Reflexive _∼₂_ →
-    Reflexive (Pointwise _∼₁_ _∼₂_)
-  ×-refl refl₁ refl₂ = refl₁ , refl₂
-
-  ×-irreflexive₁ : ∀ {_≈₁_ _<₁_ _≈₂_ _<₂_} →
-    Irreflexive _≈₁_ _<₁_ →
-    Irreflexive (Pointwise _≈₁_ _≈₂_) (Pointwise _<₁_ _<₂_)
-  ×-irreflexive₁ ir x≈y x<y = ir (proj₁ x≈y) (proj₁ x<y)
-
-  ×-irreflexive₂ : ∀ {_≈₁_ _<₁_ _≈₂_ _<₂_} →
-    Irreflexive _≈₂_ _<₂_ →
-    Irreflexive (Pointwise _≈₁_ _≈₂_) (Pointwise _<₁_ _<₂_)
-  ×-irreflexive₂ ir x≈y x<y = ir (proj₂ x≈y) (proj₂ x<y)
-
-  ×-symmetric : ∀ {_∼₁_ _∼₂_} → Symmetric _∼₁_ → Symmetric _∼₂_ →
-    Symmetric (Pointwise _∼₁_ _∼₂_)
-  ×-symmetric sym₁ sym₂ (x∼y₁ , x∼y₂) = sym₁ x∼y₁ , sym₂ x∼y₂
-
-  ×-transitive : ∀ {_∼₁_ _∼₂_} → Transitive _∼₁_ → Transitive _∼₂_ →
-    Transitive (Pointwise _∼₁_ _∼₂_)
-  ×-transitive trans₁ trans₂ x∼y y∼z =
-    trans₁ (proj₁ x∼y) (proj₁ y∼z) ,
-    trans₂ (proj₂ x∼y) (proj₂ y∼z)
-
-  ×-antisymmetric : ∀ {_≈₁_ _≤₁_ _≈₂_ _≤₂_} →
-    Antisymmetric _≈₁_ _≤₁_ → Antisymmetric _≈₂_ _≤₂_ →
-    Antisymmetric (Pointwise _≈₁_ _≈₂_) (Pointwise _≤₁_ _≤₂_)
-  ×-antisymmetric antisym₁ antisym₂ (x≤y₁ , x≤y₂) (y≤x₁ , y≤x₂) =
-    (antisym₁ x≤y₁ y≤x₁ , antisym₂ x≤y₂ y≤x₂)
-
-  ×-asymmetric₁ : ∀ {_<₁_ _∼₂_} → Asymmetric _<₁_ →
-    Asymmetric (Pointwise _<₁_ _∼₂_)
-  ×-asymmetric₁ asym₁ x<y y<x = asym₁ (proj₁ x<y) (proj₁ y<x)
-
-  ×-asymmetric₂ : ∀ {_∼₁_ _<₂_} → Asymmetric _<₂_ →
-    Asymmetric (Pointwise _∼₁_ _<₂_)
-  ×-asymmetric₂ asym₂ x<y y<x = asym₂ (proj₂ x<y) (proj₂ y<x)
-
-  ×-respects₂ : ∀ {_≈₁_ _∼₁_ _≈₂_ _∼₂_} →
-    _∼₁_ Respects₂ _≈₁_ → _∼₂_ Respects₂ _≈₂_ →
-    (Pointwise _∼₁_ _∼₂_) Respects₂ (Pointwise _≈₁_ _≈₂_)
-  ×-respects₂ {_≈₁_} {_∼₁_} {_≈₂_} {_∼₂_} resp₁ resp₂ = resp¹ , resp²
-    where
-    _∼_ = Pointwise _∼₁_ _∼₂_
-    _≈_ = Pointwise _≈₁_ _≈₂_
-
-    resp¹ : ∀ {x} → (x ∼_) Respects _≈_
-    resp¹ y≈y′ x∼y = proj₁ resp₁ (proj₁ y≈y′) (proj₁ x∼y) ,
-                     proj₁ resp₂ (proj₂ y≈y′) (proj₂ x∼y)
-
-    resp² : ∀ {y} → (_∼ y) Respects _≈_
-    resp² x≈x′ x∼y = proj₂ resp₁ (proj₁ x≈x′) (proj₁ x∼y) ,
-                     proj₂ resp₂ (proj₂ x≈x′) (proj₂ x∼y)
-
-  ×-total : ∀ {_∼₁_ _∼₂_} → Symmetric _∼₁_ →
-    Total _∼₁_ → Total _∼₂_ →
-    Total (Pointwise _∼₁_ _∼₂_)
-  ×-total sym₁ total₁ total₂ (x₁ , x₂) (y₁ , y₂)
-    with total₁ x₁ y₁ | total₂ x₂ y₂
-  ... | inj₁ x₁∼y₁ | inj₁ x₂∼y₂ = inj₁ (     x₁∼y₁ , x₂∼y₂)
-  ... | inj₁ x₁∼y₁ | inj₂ y₂∼x₂ = inj₂ (sym₁ x₁∼y₁ , y₂∼x₂)
-  ... | inj₂ y₁∼x₁ | inj₂ y₂∼x₂ = inj₂ (     y₁∼x₁ , y₂∼x₂)
-  ... | inj₂ y₁∼x₁ | inj₁ x₂∼y₂ = inj₁ (sym₁ y₁∼x₁ , x₂∼y₂)
-
-  ×-decidable : ∀ {_∼₁_ _∼₂_} →
-    Decidable _∼₁_ → Decidable _∼₂_ →
-    Decidable (Pointwise _∼₁_ _∼₂_)
-  ×-decidable _≟₁_ _≟₂_ (x₁ , x₂) (y₁ , y₂) =
-    (x₁ ≟₁ y₁) ×-dec (x₂ ≟₂ y₂)
-
-  -- Some collections of properties which are preserved by ×-Rel.
-
-  ×-isEquivalence : ∀ {_≈₁_ _≈₂_} →
-    IsEquivalence _≈₁_ → IsEquivalence _≈₂_ →
-    IsEquivalence (Pointwise _≈₁_ _≈₂_)
-  ×-isEquivalence {_≈₁_ = _≈₁_} {_≈₂_ = _≈₂_} eq₁ eq₂ = record
-    { refl  = ×-refl        {_∼₁_ = _≈₁_} {_∼₂_ = _≈₂_}
-                            (refl  eq₁) (refl  eq₂)
-    ; sym   = ×-symmetric   {_∼₁_ = _≈₁_} {_∼₂_ = _≈₂_}
-                            (sym   eq₁) (sym   eq₂)
-    ; trans = ×-transitive  {_∼₁_ = _≈₁_} {_∼₂_ = _≈₂_}
-                            (trans eq₁) (trans eq₂)
-    }
-    where open IsEquivalence
-
-  ×-isDecEquivalence : ∀ {_≈₁_ _≈₂_} →
-    IsDecEquivalence _≈₁_ → IsDecEquivalence _≈₂_ →
-    IsDecEquivalence (Pointwise _≈₁_ _≈₂_)
-  ×-isDecEquivalence eq₁ eq₂ = record
-    { isEquivalence = ×-isEquivalence
-                        (isEquivalence eq₁) (isEquivalence eq₂)
-    ; _≟_           = ×-decidable (_≟_ eq₁) (_≟_ eq₂)
-    }
-    where open IsDecEquivalence
-
-  ×-isPreorder : ∀ {_≈₁_ _∼₁_ _≈₂_ _∼₂_} →
-    IsPreorder _≈₁_ _∼₁_ → IsPreorder _≈₂_ _∼₂_ →
-    IsPreorder (Pointwise _≈₁_ _≈₂_) (Pointwise _∼₁_ _∼₂_)
-  ×-isPreorder {_∼₁_ = _∼₁_} {_∼₂_ = _∼₂_} pre₁ pre₂ = record
-    { isEquivalence = ×-isEquivalence
-                        (isEquivalence pre₁) (isEquivalence pre₂)
-    ; reflexive     = ×-reflexive {_∼₁_ = _∼₁_} {_∼₂_ = _∼₂_}
-                        (reflexive pre₁) (reflexive pre₂)
-    ; trans         = ×-transitive {_∼₁_ = _∼₁_} {_∼₂_ = _∼₂_}
-                        (trans pre₁) (trans pre₂)
-    }
-    where open IsPreorder
-
-  ×-isPartialOrder : ∀ {_≈₁_ _≤₁_ _≈₂_ _≤₂_} →
-    IsPartialOrder _≈₁_ _≤₁_ → IsPartialOrder _≈₂_ _≤₂_ →
-    IsPartialOrder (Pointwise _≈₁_ _≈₂_) (Pointwise _≤₁_ _≤₂_)
-  ×-isPartialOrder {_≤₁_ = _≤₁_} {_≤₂_ = _≤₂_} po₁ po₂ = record
-    { isPreorder = ×-isPreorder (isPreorder po₁) (isPreorder po₂)
-    ; antisym    = ×-antisymmetric {_≤₁_ = _≤₁_} {_≤₂_ = _≤₂_}
-                     (antisym po₁) (antisym po₂)
-    }
-    where open IsPartialOrder
-
-  ×-isStrictPartialOrder : ∀ {_≈₁_ _<₁_ _≈₂_ _<₂_} →
-    IsStrictPartialOrder _≈₁_ _<₁_ → IsStrictPartialOrder _≈₂_ _<₂_ →
-    IsStrictPartialOrder (Pointwise _≈₁_ _≈₂_) (Pointwise _<₁_ _<₂_)
-  ×-isStrictPartialOrder {_<₁_ = _<₁_} {_≈₂_ = _≈₂_} {_<₂_ = _<₂_}
-                           spo₁ spo₂ =
-    record
-      { isEquivalence = ×-isEquivalence
-                          (isEquivalence spo₁) (isEquivalence spo₂)
-      ; irrefl        = ×-irreflexive₁ {_<₁_ = _<₁_} {_≈₂_} {_<₂_}
-                          (irrefl spo₁)
-      ; trans         = ×-transitive {_∼₁_ = _<₁_} {_<₂_}
-                          (trans spo₁) (trans spo₂)
-      ; <-resp-≈      = ×-respects₂ (<-resp-≈ spo₁) (<-resp-≈ spo₂)
-      }
-    where open IsStrictPartialOrder
+×-reflexive : ≈₁ ⇒ R → ≈₂ ⇒ S → Pointwise ≈₁ ≈₂ ⇒ Pointwise R S
+×-reflexive refl₁ refl₂ = Prod.map refl₁ refl₂
+
+×-refl : Reflexive R → Reflexive S → Reflexive (Pointwise R S)
+×-refl refl₁ refl₂ = refl₁ , refl₂
+
+×-irreflexive₁ : Irreflexive ≈₁ R →
+                 Irreflexive (Pointwise ≈₁ ≈₂) (Pointwise R S)
+×-irreflexive₁ ir x≈y x<y = ir (proj₁ x≈y) (proj₁ x<y)
+
+×-irreflexive₂ : Irreflexive ≈₂ S →
+                 Irreflexive (Pointwise ≈₁ ≈₂) (Pointwise R S)
+×-irreflexive₂ ir x≈y x<y = ir (proj₂ x≈y) (proj₂ x<y)
+
+×-symmetric : Symmetric R → Symmetric S → Symmetric (Pointwise R S)
+×-symmetric sym₁ sym₂ = Prod.map sym₁ sym₂
+
+×-transitive : Transitive R → Transitive S → Transitive (Pointwise R S)
+×-transitive trans₁ trans₂ = Prod.zip trans₁ trans₂
+
+×-antisymmetric : Antisymmetric ≈₁ R → Antisymmetric ≈₂ S →
+                  Antisymmetric (Pointwise ≈₁ ≈₂) (Pointwise R S)
+×-antisymmetric antisym₁ antisym₂ = Prod.zip antisym₁ antisym₂
+
+×-asymmetric₁ : Asymmetric R → Asymmetric (Pointwise R S)
+×-asymmetric₁ asym₁ x<y y<x = asym₁ (proj₁ x<y) (proj₁ y<x)
+
+×-asymmetric₂ : Asymmetric S → Asymmetric (Pointwise R S)
+×-asymmetric₂ asym₂ x<y y<x = asym₂ (proj₂ x<y) (proj₂ y<x)
+
+×-respectsʳ : R Respectsʳ ≈₁ → S Respectsʳ ≈₂ →
+             (Pointwise R S) Respectsʳ (Pointwise ≈₁ ≈₂)
+×-respectsʳ resp₁ resp₂ = Prod.zip resp₁ resp₂
+
+×-respectsˡ : R Respectsˡ ≈₁ → S Respectsˡ ≈₂ →
+             (Pointwise R S) Respectsˡ (Pointwise ≈₁ ≈₂)
+×-respectsˡ resp₁ resp₂ = Prod.zip resp₁ resp₂
+
+×-respects₂ : R Respects₂ ≈₁ → S Respects₂ ≈₂ →
+              (Pointwise R S) Respects₂ (Pointwise ≈₁ ≈₂)
+×-respects₂ = Prod.zip ×-respectsʳ ×-respectsˡ
+
+×-total : Symmetric R → Total R → Total S → Total (Pointwise R S)
+×-total sym₁ total₁ total₂ (x₁ , x₂) (y₁ , y₂)
+  with total₁ x₁ y₁ | total₂ x₂ y₂
+... | inj₁ x₁∼y₁ | inj₁ x₂∼y₂ = inj₁ (     x₁∼y₁ , x₂∼y₂)
+... | inj₁ x₁∼y₁ | inj₂ y₂∼x₂ = inj₂ (sym₁ x₁∼y₁ , y₂∼x₂)
+... | inj₂ y₁∼x₁ | inj₂ y₂∼x₂ = inj₂ (     y₁∼x₁ , y₂∼x₂)
+... | inj₂ y₁∼x₁ | inj₁ x₂∼y₂ = inj₁ (sym₁ y₁∼x₁ , x₂∼y₂)
+
+×-decidable : Decidable R → Decidable S → Decidable (Pointwise R S)
+×-decidable _≟₁_ _≟₂_ (x₁ , x₂) (y₁ , y₂) = (x₁ ≟₁ y₁) ×-dec (x₂ ≟₂ y₂)
+
+------------------------------------------------------------------------
+-- Structures can also be combined.
+
+-- Some collections of properties which are preserved by ×-Rel.
+
+×-isEquivalence : IsEquivalence R → IsEquivalence S →
+                  IsEquivalence (Pointwise R S)
+×-isEquivalence {R = R} {S = S} eq₁ eq₂ = record
+  { refl  = ×-refl {R = R} {S = S} (refl eq₁) (refl eq₂)
+  ; sym   = ×-symmetric {R = R} {S = S} (sym eq₁) (sym eq₂)
+  ; trans = ×-transitive {R = R} {S = S} (trans eq₁) (trans eq₂)
+  } where open IsEquivalence
+
+×-isDecEquivalence : IsDecEquivalence R → IsDecEquivalence S →
+                     IsDecEquivalence (Pointwise R S)
+×-isDecEquivalence eq₁ eq₂ = record
+  { isEquivalence = ×-isEquivalence
+                      (isEquivalence eq₁) (isEquivalence eq₂)
+  ; _≟_           = ×-decidable (_≟_ eq₁) (_≟_ eq₂)
+  } where open IsDecEquivalence
+
+×-isPreorder : IsPreorder ≈₁ R → IsPreorder ≈₂ S →
+               IsPreorder (Pointwise ≈₁ ≈₂) (Pointwise R S)
+×-isPreorder {R = R} {S = S} pre₁ pre₂ = record
+  { isEquivalence = ×-isEquivalence
+                      (isEquivalence pre₁) (isEquivalence pre₂)
+  ; reflexive     = ×-reflexive {R = R} {S = S}
+                      (reflexive pre₁) (reflexive pre₂)
+  ; trans         = ×-transitive {R = R} {S = S}
+                      (trans pre₁) (trans pre₂)
+  } where open IsPreorder
+
+×-isPartialOrder : IsPartialOrder ≈₁ R → IsPartialOrder ≈₂ S →
+                   IsPartialOrder (Pointwise ≈₁ ≈₂) (Pointwise R S)
+×-isPartialOrder {R = R} {S = S} po₁ po₂ = record
+  { isPreorder = ×-isPreorder (isPreorder po₁) (isPreorder po₂)
+  ; antisym    = ×-antisymmetric {R = R} {S = S}
+                   (antisym po₁) (antisym po₂)
+  } where open IsPartialOrder
+
+×-isStrictPartialOrder : IsStrictPartialOrder ≈₁ R →
+                         IsStrictPartialOrder ≈₂ S →
+                         IsStrictPartialOrder (Pointwise ≈₁ ≈₂) (Pointwise R S)
+×-isStrictPartialOrder {R = R} {≈₂ = ≈₂} {S = S} spo₁ spo₂ = record
+  { isEquivalence = ×-isEquivalence
+                      (isEquivalence spo₁) (isEquivalence spo₂)
+  ; irrefl        = ×-irreflexive₁ {R = R} {≈₂ = ≈₂} {S = S}
+                      (irrefl spo₁)
+  ; trans         = ×-transitive {R = R} {S = S}
+                      (trans spo₁) (trans spo₂)
+  ; <-resp-≈      = ×-respects₂ (<-resp-≈ spo₁) (<-resp-≈ spo₂)
+  } where open IsStrictPartialOrder
 
 ------------------------------------------------------------------------
--- "Bundles" can also be combined.
-
-module _ {ℓ₁ ℓ₂ ℓ₃ ℓ₄} where
-
-  ×-preorder : Preorder ℓ₁ ℓ₂ _ → Preorder ℓ₃ ℓ₄ _ → Preorder _ _ _
-  ×-preorder p₁ p₂ = record
-    { isPreorder = ×-isPreorder (isPreorder p₁) (isPreorder p₂)
-    } where open Preorder
-
-  ×-setoid : Setoid ℓ₁ ℓ₂ → Setoid ℓ₃ ℓ₄ → Setoid _ _
-  ×-setoid s₁ s₂ = record
-    { isEquivalence =
-        ×-isEquivalence (isEquivalence s₁) (isEquivalence s₂)
-    } where open Setoid
-
-  ×-decSetoid : DecSetoid ℓ₁ ℓ₂ → DecSetoid ℓ₃ ℓ₄ → DecSetoid _ _
-  ×-decSetoid s₁ s₂ = record
-    { isDecEquivalence =
-        ×-isDecEquivalence (isDecEquivalence s₁) (isDecEquivalence s₂)
-    } where open DecSetoid
-
-  ×-poset : Poset ℓ₁ ℓ₂ _ → Poset ℓ₃ ℓ₄ _ → Poset _ _ _
-  ×-poset s₁ s₂ = record
-    { isPartialOrder = ×-isPartialOrder (isPartialOrder s₁)
-                       (isPartialOrder s₂)
-    } where open Poset
-
-  ×-strictPartialOrder :
-    StrictPartialOrder ℓ₁ ℓ₂ _ → StrictPartialOrder ℓ₃ ℓ₄ _ →
-    StrictPartialOrder _ _ _
-  ×-strictPartialOrder s₁ s₂ = record
-    { isStrictPartialOrder = ×-isStrictPartialOrder
-                               (isStrictPartialOrder s₁)
-                               (isStrictPartialOrder s₂)
-    } where open StrictPartialOrder
-
-  -- A piece of infix notation for combining setoids
-  infix 4 _×ₛ_
-  _×ₛ_ : Setoid ℓ₁ ℓ₂ → Setoid ℓ₃ ℓ₄ → Setoid _ _
-  _×ₛ_ = ×-setoid
+-- Bundles
+
+×-setoid : Setoid a ℓ₁ → Setoid b ℓ₂ → Setoid _ _
+×-setoid s₁ s₂ = record
+  { isEquivalence =
+      ×-isEquivalence (isEquivalence s₁) (isEquivalence s₂)
+  } where open Setoid
+
+×-decSetoid : DecSetoid a ℓ₁ → DecSetoid b ℓ₂ → DecSetoid _ _
+×-decSetoid s₁ s₂ = record
+  { isDecEquivalence =
+      ×-isDecEquivalence (isDecEquivalence s₁) (isDecEquivalence s₂)
+  } where open DecSetoid
+
+×-preorder : Preorder a ℓ₁ ℓ₂ → Preorder b ℓ₃ ℓ₄ → Preorder _ _ _
+×-preorder p₁ p₂ = record
+  { isPreorder = ×-isPreorder (isPreorder p₁) (isPreorder p₂)
+  } where open Preorder
+
+×-poset : Poset a ℓ₁ ℓ₂ → Poset b ℓ₃ ℓ₄ → Poset _ _ _
+×-poset s₁ s₂ = record
+  { isPartialOrder = ×-isPartialOrder (isPartialOrder s₁)
+                     (isPartialOrder s₂)
+  } where open Poset
+
+×-strictPartialOrder : StrictPartialOrder a ℓ₁ ℓ₂ →
+                       StrictPartialOrder b ℓ₃ ℓ₄ →
+                       StrictPartialOrder _ _ _
+×-strictPartialOrder s₁ s₂ = record
+  { isStrictPartialOrder = ×-isStrictPartialOrder
+                             (isStrictPartialOrder s₁)
+                             (isStrictPartialOrder s₂)
+  } where open StrictPartialOrder
+
+------------------------------------------------------------------------
+-- Additional notation
+
+-- Infix combining setoids
+infix 4 _×ₛ_
+_×ₛ_ : Setoid a ℓ₁ → Setoid b ℓ₂ → Setoid _ _
+_×ₛ_ = ×-setoid
 
 ------------------------------------------------------------------------
 -- The propositional equality setoid over products can be
 -- decomposed using ×-Rel
 
-module _ {a b} {A : Set a} {B : Set b} where
-
-  ≡×≡⇒≡ : Pointwise _≡_ _≡_ ⇒ _≡_ {A = A × B}
-  ≡×≡⇒≡ (P.refl , P.refl) = P.refl
+≡×≡⇒≡ : Pointwise _≡_ _≡_ ⇒ _≡_ {A = A × B}
+≡×≡⇒≡ (P.refl , P.refl) = P.refl
 
-  ≡⇒≡×≡ : _≡_ {A = A × B} ⇒ Pointwise _≡_ _≡_
-  ≡⇒≡×≡ P.refl = (P.refl , P.refl)
+≡⇒≡×≡ : _≡_ {A = A × B} ⇒ Pointwise _≡_ _≡_
+≡⇒≡×≡ P.refl = (P.refl , P.refl)
 
-  Pointwise-≡↔≡ : Inverse (P.setoid A ×ₛ P.setoid B) (P.setoid (A × B))
-  Pointwise-≡↔≡ = record
-    { to         = record { _⟨$⟩_ = id; cong = ≡×≡⇒≡ }
-    ; from       = record { _⟨$⟩_ = id; cong = ≡⇒≡×≡ }
-    ; inverse-of = record
-      { left-inverse-of  = λ _ → (P.refl , P.refl)
-      ; right-inverse-of = λ _ → P.refl
-      }
-    }
+Pointwise-≡↔≡ : Inverse (P.setoid A ×ₛ P.setoid B) (P.setoid (A × B))
+Pointwise-≡↔≡ = record
+  { to         = id
+  ; from       = id
+  ; to-cong    = ≡×≡⇒≡
+  ; from-cong  = ≡⇒≡×≡
+  ; inverse    = ≡×≡⇒≡ , ≡⇒≡×≡
+  }
diff --git a/src/Data/Sum/Algebra.agda b/src/Data/Sum/Algebra.agda
index ca96ac9c1c..f128af8075 100644
--- a/src/Data/Sum/Algebra.agda
+++ b/src/Data/Sum/Algebra.agda
@@ -16,7 +16,7 @@ open import Data.Sum.Properties
 open import Data.Unit.Polymorphic using (⊤; tt)
 open import Function.Base using (id; _∘_)
 open import Function.Properties.Inverse using (↔-isEquivalence)
-open import Function.Bundles using (_↔_; Inverse; mk↔′)
+open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)
 open import Level using (Level; suc)
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; cong; cong′)
@@ -38,7 +38,7 @@ private
 -- Algebraic properties
 
 ⊎-cong : A ↔ B → C ↔ D → (A ⊎ C) ↔ (B ⊎ D)
-⊎-cong i j = mk↔′ (map I.to J.to) (map I.from J.from)
+⊎-cong i j = mk↔ₛ′ (map I.to J.to) (map I.from J.from)
   [ cong inj₁ ∘ I.strictlyInverseˡ , cong inj₂ ∘ J.strictlyInverseˡ ]
   [ cong inj₁ ∘ I.strictlyInverseʳ , cong inj₂ ∘ J.strictlyInverseʳ ]
   where module I = Inverse i; module J = Inverse j
@@ -46,21 +46,21 @@ private
 -- ⊎ is commutative.
 -- We don't use Commutative because it isn't polymorphic enough.
 ⊎-comm : (A : Set a) (B : Set b) → (A ⊎ B) ↔ (B ⊎ A)
-⊎-comm _ _ = mk↔′ swap swap swap-involutive swap-involutive
+⊎-comm _ _ = mk↔ₛ′ swap swap swap-involutive swap-involutive
 
 module _ (ℓ : Level) where
 
   -- ⊎ is associative
   ⊎-assoc : Associative {ℓ = ℓ} _↔_ _⊎_
-  ⊎-assoc _ _ _ = mk↔′ assocʳ assocˡ
+  ⊎-assoc _ _ _ = mk↔ₛ′ assocʳ assocˡ
     [ cong′ , [ cong′ , cong′ ] ] [ [ cong′ , cong′ ] , cong′ ]
 
   -- ⊥ is an identity for ⊎
   ⊎-identityˡ : LeftIdentity {ℓ = ℓ} _↔_ ⊥ _⊎_
-  ⊎-identityˡ A = mk↔′ [ ♯ , id ] inj₂ cong′ [ ♯ , cong′ ]
+  ⊎-identityˡ A = mk↔ₛ′ [ ♯ , id ] inj₂ cong′ [ ♯ , cong′ ]
 
   ⊎-identityʳ : RightIdentity {ℓ = ℓ} _↔_ ⊥ _⊎_
-  ⊎-identityʳ _ = mk↔′ [ id , ♯ ] inj₁ cong′ [ cong′ , ♯ ]
+  ⊎-identityʳ _ = mk↔ₛ′ [ id , ♯ ] inj₁ cong′ [ cong′ , ♯ ]
 
   ⊎-identity : Identity _↔_ ⊥ _⊎_
   ⊎-identity = ⊎-identityˡ , ⊎-identityʳ
diff --git a/src/Data/Sum/Function/Propositional.agda b/src/Data/Sum/Function/Propositional.agda
index a22aa64c4a..b7830bc895 100644
--- a/src/Data/Sum/Function/Propositional.agda
+++ b/src/Data/Sum/Function/Propositional.agda
@@ -10,67 +10,72 @@ module Data.Sum.Function.Propositional where
 
 open import Data.Sum.Base
 open import Data.Sum.Function.Setoid
-open import Data.Sum.Relation.Binary.Pointwise using (Pointwise-≡↔≡)
-open import Function.Equivalence as Eq using (_⇔_; module Equivalence)
-open import Function.Injection as Inj using (_↣_; module Injection)
-open import Function.Inverse as Inv using (_↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (_↞_; _LeftInverseOf_; module LeftInverse)
-open import Function.Related
-open import Function.Surjection as Surj using (_↠_; module Surjection)
+open import Data.Sum.Relation.Binary.Pointwise using (Pointwise-≡↔≡; _⊎ₛ_)
+open import Function.Construct.Composition as Compose
+open import Function.Related.Propositional
+open import Function
+open import Function.Properties.Inverse as Inv
+open import Level using (Level; _⊔_)
+open import Relation.Binary using (REL; Setoid)
+open import Relation.Binary.PropositionalEquality using (setoid)
+
+private
+  variable
+    a b c d : Level
+    A B C D : Set a
+
+
+------------------------------------------------------------------------
+-- Helper lemma
+
+private
+  liftViaInverse : {R : ∀ {a b ℓ₁ ℓ₂} → REL (Setoid a ℓ₁) (Setoid b ℓ₂) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)} →
+                   (∀ {a b c ℓ₁ ℓ₂ ℓ₃} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} → R S T → R T U → R S U) →
+                   (∀ {a b ℓ₁ ℓ₂} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} → Inverse S T → R S T) →
+                   (R (setoid A) (setoid C) → R (setoid B) (setoid D) → R (setoid A ⊎ₛ setoid B) (setoid C ⊎ₛ setoid D)) →
+                   R (setoid A) (setoid C) → R (setoid B) (setoid D) →
+                   R (setoid (A ⊎ B)) (setoid (C ⊎ D))
+  liftViaInverse trans inv⇒R lift RAC RBD =
+    Inv.transportVia trans inv⇒R (Inv.sym (Pointwise-≡↔≡ _ _)) (lift RAC RBD) (Pointwise-≡↔≡ _ _)
 
 ------------------------------------------------------------------------
 -- Combinators for various function types
 
-module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
-
-  infixr 1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_
-
-  _⊎-⇔_ : A ⇔ B → C ⇔ D → (A ⊎ C) ⇔ (B ⊎ D)
-  _⊎-⇔_ A⇔B C⇔D =
-    Inverse.equivalence (Pointwise-≡↔≡ B D) ⟨∘⟩
-    (A⇔B ⊎-equivalence C⇔D) ⟨∘⟩
-    Eq.sym (Inverse.equivalence (Pointwise-≡↔≡ A C))
-    where open Eq using () renaming (_∘_ to _⟨∘⟩_)
-
-  _⊎-↣_ : A ↣ B → C ↣ D → (A ⊎ C) ↣ (B ⊎ D)
-  _⊎-↣_ A↣B C↣D =
-    Inverse.injection (Pointwise-≡↔≡ B D) ⟨∘⟩
-    (A↣B ⊎-injection C↣D) ⟨∘⟩
-    Inverse.injection (Inv.sym (Pointwise-≡↔≡ A C))
-    where open Inj using () renaming (_∘_ to _⟨∘⟩_)
-
-  _⊎-↞_ : A ↞ B → C ↞ D → (A ⊎ C) ↞ (B ⊎ D)
-  _⊎-↞_ A↞B C↞D =
-    Inverse.left-inverse (Pointwise-≡↔≡ B D) ⟨∘⟩
-    (A↞B ⊎-left-inverse C↞D) ⟨∘⟩
-    Inverse.left-inverse (Inv.sym (Pointwise-≡↔≡ A C))
-    where open LeftInv using () renaming (_∘_ to _⟨∘⟩_)
-
-  _⊎-↠_ : A ↠ B → C ↠ D → (A ⊎ C) ↠ (B ⊎ D)
-  _⊎-↠_ A↠B C↠D =
-    Inverse.surjection (Pointwise-≡↔≡ B D) ⟨∘⟩
-    (A↠B ⊎-surjection C↠D) ⟨∘⟩
-    Inverse.surjection (Inv.sym (Pointwise-≡↔≡ A C))
-    where open Surj using () renaming (_∘_ to _⟨∘⟩_)
-
-  _⊎-↔_ : A ↔ B → C ↔ D → (A ⊎ C) ↔ (B ⊎ D)
-  _⊎-↔_ A↔B C↔D =
-    Pointwise-≡↔≡ B D ⟨∘⟩
-    (A↔B ⊎-inverse C↔D) ⟨∘⟩
-    Inv.sym (Pointwise-≡↔≡ A C)
-    where open Inv using () renaming (_∘_ to _⟨∘⟩_)
-
-module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} where
-
-  infixr 1 _⊎-cong_
-
-  _⊎-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A ⊎ C) ∼[ k ] (B ⊎ D)
-  _⊎-cong_ {implication}         = map
-  _⊎-cong_ {reverse-implication} = λ f g → lam (map (app-← f) (app-← g))
-  _⊎-cong_ {equivalence}         = _⊎-⇔_
-  _⊎-cong_ {injection}           = _⊎-↣_
-  _⊎-cong_ {reverse-injection}   = λ f g → lam (app-↢ f ⊎-↣ app-↢ g)
-  _⊎-cong_ {left-inverse}        = _⊎-↞_
-  _⊎-cong_ {surjection}          = _⊎-↠_
-  _⊎-cong_ {bijection}           = _⊎-↔_
+infixr 1 _⊎-⟶_ _⊎-⇔_ _⊎-↣_ _⊎-↩_ _⊎-↪_ _⊎-↔_
+
+_⊎-⟶_ : A ⟶ B → C ⟶ D → (A ⊎ C) ⟶ (B ⊎ D)
+_⊎-⟶_ = liftViaInverse Compose.function Inv.toFunction _⊎-function_
+
+
+_⊎-⇔_ : A ⇔ B → C ⇔ D → (A ⊎ C) ⇔ (B ⊎ D)
+_⊎-⇔_ = liftViaInverse Compose.equivalence Inverse⇒Equivalence _⊎-equivalence_
+
+_⊎-↣_ : A ↣ B → C ↣ D → (A ⊎ C) ↣ (B ⊎ D)
+_⊎-↣_ = liftViaInverse Compose.injection Inverse⇒Injection _⊎-injection_
+
+_⊎-↠_ : A ↠ B → C ↠ D → (A ⊎ C) ↠ (B ⊎ D)
+_⊎-↠_ = liftViaInverse Compose.surjection Inverse⇒Surjection _⊎-surjection_
+
+_⊎-↩_ : A ↩ B → C ↩ D → (A ⊎ C) ↩ (B ⊎ D)
+_⊎-↩_ = liftViaInverse Compose.leftInverse Inverse.leftInverse _⊎-leftInverse_
+
+_⊎-↪_ : A ↪ B → C ↪ D → (A ⊎ C) ↪ (B ⊎ D)
+_⊎-↪_ = liftViaInverse Compose.rightInverse Inverse.rightInverse _⊎-rightInverse_
+
+_⊎-⤖_ : A ⤖ B → C ⤖ D → (A ⊎ C) ⤖ (B ⊎ D)
+_⊎-⤖_ = liftViaInverse Compose.bijection Inverse⇒Bijection _⊎-bijection_
+
+_⊎-↔_ : A ↔ B → C ↔ D → (A ⊎ C) ↔ (B ⊎ D)
+_⊎-↔_ = liftViaInverse Compose.inverse id _⊎-inverse_
+
+infixr 1 _⊎-cong_
+
+_⊎-cong_ : ∀ {k} → A ∼[ k ] B → C ∼[ k ] D → (A ⊎ C) ∼[ k ] (B ⊎ D)
+_⊎-cong_ {k = implication}         = _⊎-⟶_
+_⊎-cong_ {k = reverseImplication}  = _⊎-⟶_
+_⊎-cong_ {k = equivalence}         = _⊎-⇔_
+_⊎-cong_ {k = injection}           = _⊎-↣_
+_⊎-cong_ {k = reverseInjection}    = _⊎-↣_
+_⊎-cong_ {k = leftInverse}         = _⊎-↪_
+_⊎-cong_ {k = surjection}          = _⊎-↠_
+_⊎-cong_ {k = bijection}           = _⊎-↔_
diff --git a/src/Data/Sum/Function/Setoid.agda b/src/Data/Sum/Function/Setoid.agda
index f04798dd50..d0b28634fa 100644
--- a/src/Data/Sum/Function/Setoid.agda
+++ b/src/Data/Sum/Function/Setoid.agda
@@ -8,149 +8,167 @@
 
 module Data.Sum.Function.Setoid where
 
-open import Data.Sum.Base
-open import Data.Sum.Relation.Binary.Pointwise
-open import Relation.Binary.Core using (_=[_]⇒_)
-open import Relation.Binary.Bundles using (Setoid)
-open import Function.Equality as F using (_⟶_; _⟨$⟩_)
-open import Function.Equivalence as Eq
-  using (Equivalence; _⇔_; module Equivalence)
-open import Function.Injection as Inj
-  using (Injection; _↣_; module Injection)
-open import Function.Inverse as Inv
-  using (Inverse; _↔_; module Inverse)
-open import Function.LeftInverse as LeftInv
-  using (LeftInverse; _↞_; _LeftInverseOf_; module LeftInverse)
-open import Function.Related
-open import Function.Surjection as Surj
-  using (Surjection; _↠_; module Surjection)
+open import Data.Product.Base as Prod using (_,_)
+open import Data.Sum.Base as Sum
+open import Data.Sum.Relation.Binary.Pointwise as Pointwise
+open import Relation.Binary
+open import Function.Base
+open import Function.Bundles
+open import Function.Definitions
+open import Level
+
+private
+  variable
+    a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂ : Level
+    a ℓ : Level
+    A B C D : Set a
+    ≈₁ ≈₂ ≈₃ ≈₄ : Rel A ℓ
+    S T U V : Setoid a ℓ
 
 ------------------------------------------------------------------------
 -- Combinators for equality preserving functions
 
-module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
-          {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
-          {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
-          where
+inj₁ₛ : Func S (S ⊎ₛ T)
+inj₁ₛ = record { to = inj₁ ; cong = inj₁ }
 
-  infixr 1 _⊎-⟶_
+inj₂ₛ : Func T (S ⊎ₛ T)
+inj₂ₛ = record { to = inj₂ ; cong = inj₂ }
 
-  _⊎-⟶_ : (A ⟶ B) → (C ⟶ D) → (A ⊎ₛ C) ⟶ (B ⊎ₛ D)
-  _⊎-⟶_ f g = record
-    { _⟨$⟩_ = fg
-    ; cong  = fg-cong
-    }
-    where
-    open Setoid (A ⊎ₛ C) using () renaming (_≈_ to _≈AC_)
-    open Setoid (B ⊎ₛ D) using () renaming (_≈_ to _≈BD_)
+[_,_]ₛ : Func S U → Func T U → Func (S ⊎ₛ T) U
+[ f , g ]ₛ = record
+  { to   = [ to f , to g ]
+  ; cong = λ where
+    (inj₁ x∼₁y) → cong f x∼₁y
+    (inj₂ x∼₂y) → cong g x∼₂y
+  } where open Func
 
-    fg = map (_⟨$⟩_ f) (_⟨$⟩_ g)
+swapₛ : Func (S ⊎ₛ T) (T ⊎ₛ S)
+swapₛ = [ inj₂ₛ , inj₁ₛ ]ₛ
 
-    fg-cong : _≈AC_ =[ fg ]⇒ _≈BD_
-    fg-cong (inj₁ x∼₁y) = inj₁ (F.cong f x∼₁y)
-    fg-cong (inj₂ x∼₂y) = inj₂ (F.cong g x∼₂y)
-
-module _ {a₁ a₂ b₁ b₂} {A : Setoid a₁ a₂} {B : Setoid b₁ b₂} where
+------------------------------------------------------------------------
+-- Definitions
+
+⊎-injective : ∀ {f g} →
+              Injective ≈₁ ≈₂ f →
+              Injective ≈₃ ≈₄ g →
+              Injective (Pointwise ≈₁ ≈₃) (Pointwise ≈₂ ≈₄) (Sum.map f g)
+⊎-injective f-inj g-inj {inj₁ x} {inj₁ y} (inj₁ x∼₁y) = inj₁ (f-inj x∼₁y)
+⊎-injective f-inj g-inj {inj₂ x} {inj₂ y} (inj₂ x∼₂y) = inj₂ (g-inj x∼₂y)
+
+⊎-strictlySurjective : ∀ {f : A → B} {g : C → D} →
+              StrictlySurjective ≈₁ f →
+              StrictlySurjective ≈₂ g →
+              StrictlySurjective (Pointwise ≈₁ ≈₂) (Sum.map f g)
+⊎-strictlySurjective f-sur g-sur =
+  [ Prod.map inj₁ inj₁ ∘ f-sur
+  , Prod.map inj₂ inj₂ ∘ g-sur
+  ]
+
+⊎-surjective : ∀ {f : A → B} {g : C → D} →
+              Surjective ≈₁ ≈₂ f →
+              Surjective ≈₃ ≈₄ g →
+              Surjective (Pointwise ≈₁ ≈₃) (Pointwise ≈₂ ≈₄) (Sum.map f g)
+⊎-surjective f-sur g-sur =
+  [ Prod.map inj₁ (λ { fwd (inj₁ x) → inj₁ (fwd x)}) ∘ f-sur
+  , Prod.map inj₂ (λ { fwd (inj₂ y) → inj₂ (fwd y)}) ∘ g-sur
+  ]
+
+
+infixr 1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_
 
-  inj₁ₛ : A ⟶ (A ⊎ₛ B)
-  inj₁ₛ = record { _⟨$⟩_ = inj₁ ; cong = inj₁ }
+------------------------------------------------------------------------
+-- Function bundles
+
+_⊎-function_ : Func S T → Func U V → Func (S ⊎ₛ U) (T ⊎ₛ V)
+S→T ⊎-function U→V = record
+  { to    = Sum.map (to S→T) (to U→V)
+  ; cong  = Pointwise.map (cong S→T) (cong U→V)
+  } where open Func
+
+_⊎-equivalence_ : Equivalence S T → Equivalence U V →
+                  Equivalence (S ⊎ₛ U) (T ⊎ₛ V)
+S⇔T ⊎-equivalence U⇔V = record
+  { to        = Sum.map (to S⇔T) (to U⇔V)
+  ; from      = Sum.map (from S⇔T) (from U⇔V)
+  ; to-cong   = Pointwise.map (to-cong S⇔T) (to-cong U⇔V)
+  ; from-cong = Pointwise.map (from-cong S⇔T) (from-cong U⇔V)
+  } where open Equivalence
+
+_⊎-injection_ : Injection S T → Injection U V →
+                Injection (S ⊎ₛ U) (T ⊎ₛ V)
+S↣T ⊎-injection U↣V = record
+  { to        = Sum.map (to S↣T) (to U↣V)
+  ; cong      = Pointwise.map (cong S↣T) (cong U↣V)
+  ; injective = ⊎-injective (injective S↣T) (injective U↣V)
+  } where open Injection
+
+infixr 1 _⊎-surjection_ _⊎-inverse_
+_⊎-surjection_ : Surjection S T → Surjection U V →
+                 Surjection (S ⊎ₛ U) (T ⊎ₛ V)
+S↠T ⊎-surjection U↠V = record
+  { to              = Sum.map (to S↠T) (to U↠V)
+  ; cong            = Pointwise.map (cong S↠T) (cong U↠V)
+  ; surjective      = ⊎-surjective (surjective S↠T) (surjective U↠V)
+  } where open Surjection
+
+_⊎-bijection_ : Bijection S T → Bijection U V →
+                 Bijection (S ⊎ₛ U) (T ⊎ₛ V)
+S⤖T ⊎-bijection U⤖V = record
+  { to        = Sum.map (to S⤖T) (to U⤖V)
+  ; cong      = Pointwise.map (cong S⤖T) (cong U⤖V)
+  ; bijective = ⊎-injective (injective S⤖T) (injective U⤖V) ,
+                ⊎-surjective (surjective S⤖T) (surjective U⤖V)
+  } where open Bijection
+
+_⊎-leftInverse_ : LeftInverse S T → LeftInverse U V →
+                  LeftInverse (S ⊎ₛ U) (T ⊎ₛ V)
+S↩T ⊎-leftInverse U↩V = record
+  { to              = Sum.map (to S↩T) (to U↩V)
+  ; from            = Sum.map (from S↩T) (from U↩V)
+  ; to-cong         = Pointwise.map (to-cong S↩T) (to-cong U↩V)
+  ; from-cong       = Pointwise.map (from-cong S↩T) (from-cong U↩V)
+  ; inverseˡ        = λ { {inj₁ _} {.(inj₁ _)} (inj₁ x) → inj₁ (inverseˡ S↩T x)
+                        ; {inj₂ _} {.(inj₂ _)} (inj₂ x) → inj₂ (inverseˡ U↩V x)}
+  } where open LeftInverse
+
+_⊎-rightInverse_ : RightInverse S T → RightInverse U V →
+                   RightInverse (S ⊎ₛ U) (T ⊎ₛ V)
+S↪T ⊎-rightInverse U↪V = record
+  { to              = Sum.map (to S↪T) (to U↪V)
+  ; from            = Sum.map (from S↪T) (from U↪V)
+  ; to-cong         = Pointwise.map (to-cong S↪T) (to-cong U↪V)
+  ; from-cong       = Pointwise.map (from-cong S↪T) (from-cong U↪V)
+  ; inverseʳ        = λ { {inj₁ _} (inj₁ x) → inj₁ (inverseʳ S↪T x)
+                        ; {inj₂ _} (inj₂ x) → inj₂ (inverseʳ U↪V x)
+                        }
+  } where open RightInverse
+
+_⊎-inverse_ : Inverse S T → Inverse U V →
+              Inverse (S ⊎ₛ U) (T ⊎ₛ V)
+S↔T ⊎-inverse U↔V = record
+  { to        = Sum.map (to S↔T) (to U↔V)
+  ; from      = Sum.map (from S↔T) (from U↔V)
+  ; to-cong   = Pointwise.map (to-cong S↔T) (to-cong U↔V)
+  ; from-cong = Pointwise.map (from-cong S↔T) (from-cong U↔V)
+  ; inverse   = (λ { {inj₁ _} (inj₁ x) → inj₁ (inverseˡ S↔T x)
+                   ; {inj₂ _} (inj₂ x) → inj₂ (inverseˡ U↔V x)}) ,
+                 λ { {inj₁ _} (inj₁ x) → inj₁ (inverseʳ S↔T x)
+                   ; {inj₂ _} (inj₂ x) → inj₂ (inverseʳ U↔V x)
+                   }
+  } where open Inverse
 
-  inj₂ₛ : B ⟶ (A ⊎ₛ B)
-  inj₂ₛ = record { _⟨$⟩_ = inj₂ ; cong = inj₂ }
 
-  [_,_]ₛ : ∀ {c₁ c₂} {C : Setoid c₁ c₂} →
-           (A ⟶ C) → (B ⟶ C) → (A ⊎ₛ B) ⟶ C
-  [ f , g ]ₛ = record
-    { _⟨$⟩_ = [ f ⟨$⟩_ , g ⟨$⟩_ ]
-    ; cong = λ where
-      (inj₁ x∼₁y) → F.cong f x∼₁y
-      (inj₂ x∼₂y) → F.cong g x∼₂y
-    }
 
-module _ {a₁ a₂ b₁ b₂} {A : Setoid a₁ a₂} {B : Setoid b₁ b₂} where
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
 
-  swapₛ : (A ⊎ₛ B) ⟶ (B ⊎ₛ A)
-  swapₛ = [ inj₂ₛ , inj₁ₛ ]ₛ
+-- Version 2.0
 
-------------------------------------------------------------------------
--- Combinators for more complex function types
-
-module _  {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
-          {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
-          {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
-          where
-
-  infixr 1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_
-
-  _⊎-equivalence_ : Equivalence A B → Equivalence C D →
-                    Equivalence (A ⊎ₛ C) (B ⊎ₛ D)
-  A⇔B ⊎-equivalence C⇔D = record
-    { to   = to   A⇔B ⊎-⟶ to   C⇔D
-    ; from = from A⇔B ⊎-⟶ from C⇔D
-    } where open Equivalence
-
-  _⊎-injection_ : Injection A B → Injection C D →
-                  Injection (A ⊎ₛ C) (B ⊎ₛ D)
-  _⊎-injection_ A↣B C↣D = record
-    { to        = to A↣B ⊎-⟶ to C↣D
-    ; injective = inj _ _
-    }
-    where
-    open Injection
-    open Setoid (A ⊎ₛ C) using () renaming (_≈_ to _≈AC_)
-    open Setoid (B ⊎ₛ D) using () renaming (_≈_ to _≈BD_)
-
-    inj : ∀ x y →
-          (to A↣B ⊎-⟶ to C↣D) ⟨$⟩ x ≈BD (to A↣B ⊎-⟶ to C↣D) ⟨$⟩ y →
-          x ≈AC y
-    inj (inj₁ x) (inj₁ y) (inj₁ x∼₁y) = inj₁ (injective A↣B x∼₁y)
-    inj (inj₂ x) (inj₂ y) (inj₂ x∼₂y) = inj₂ (injective C↣D x∼₂y)
-
-  _⊎-left-inverse_ : LeftInverse A B → LeftInverse C D →
-                     LeftInverse (A ⊎ₛ C) (B ⊎ₛ D)
-  A↞B ⊎-left-inverse C↞D = record
-    { to              = Equivalence.to   eq
-    ; from            = Equivalence.from eq
-    ; left-inverse-of = [ (λ x → inj₁ (left-inverse-of A↞B x))
-                        , (λ x → inj₂ (left-inverse-of C↞D x)) ]
-    }
-    where
-    open LeftInverse
-    eq = LeftInverse.equivalence A↞B ⊎-equivalence
-         LeftInverse.equivalence C↞D
-
-module _ {a₁ a₂ b₁ b₂ c₁ c₂ d₁ d₂}
-         {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
-         {C : Setoid c₁ c₂} {D : Setoid d₁ d₂}
-         where
-
-  infixr 1 _⊎-surjection_ _⊎-inverse_
-  _⊎-surjection_ : Surjection A B → Surjection C D →
-                   Surjection (A ⊎ₛ C) (B ⊎ₛ D)
-  A↠B ⊎-surjection C↠D = record
-    { to         = LeftInverse.from inv
-    ; surjective = record
-      { from             = LeftInverse.to inv
-      ; right-inverse-of = LeftInverse.left-inverse-of inv
-      }
-    }
-    where
-    open Surjection
-    inv = right-inverse A↠B ⊎-left-inverse right-inverse C↠D
-
-  _⊎-inverse_ : Inverse A B → Inverse C D →
-                Inverse (A ⊎ₛ C) (B ⊎ₛ D)
-  A↔B ⊎-inverse C↔D = record
-    { to         = Surjection.to   surj
-    ; from       = Surjection.from surj
-    ; inverse-of = record
-      { left-inverse-of  = LeftInverse.left-inverse-of inv
-      ; right-inverse-of = Surjection.right-inverse-of surj
-      }
-    }
-    where
-    open Inverse
-    surj = Inverse.surjection   A↔B ⊎-surjection
-           Inverse.surjection   C↔D
-    inv  = Inverse.left-inverse A↔B ⊎-left-inverse
-           Inverse.left-inverse C↔D
+_⊎-left-inverse_ = _⊎-leftInverse_
+{-# WARNING_ON_USAGE _⊎-left-inverse_
+"Warning: _⊎-left-inverse_ was deprecated in v2.0.
+Please use _⊎-leftInverse_ instead."
+#-}
diff --git a/src/Data/Sum/Properties.agda b/src/Data/Sum/Properties.agda
index 1ee329a2d5..d0ba12c2fd 100644
--- a/src/Data/Sum/Properties.agda
+++ b/src/Data/Sum/Properties.agda
@@ -11,7 +11,7 @@ module Data.Sum.Properties where
 open import Level
 open import Data.Sum.Base
 open import Function.Base using (_∋_; _∘_; id)
-open import Function.Bundles using (mk↔′; _↔_)
+open import Function.Bundles using (mk↔ₛ′; _↔_)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
@@ -47,7 +47,7 @@ swap-involutive : swap {A = A} {B = B} ∘ swap ≗ id
 swap-involutive = [ (λ _ → refl) , (λ _ → refl) ]
 
 swap-↔ : (A ⊎ B) ↔ (B ⊎ A)
-swap-↔ = mk↔′ swap swap swap-involutive swap-involutive
+swap-↔ = mk↔ₛ′ swap swap swap-involutive swap-involutive
 
 map-id : map {A = A} {B = B} id id ≗ id
 map-id (inj₁ _) = refl
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index 0c4bf6751c..943b6f5e49 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -9,230 +9,210 @@
 module Data.Sum.Relation.Binary.Pointwise where
 
 open import Data.Product.Base using (_,_)
-open import Data.Sum.Base as Sum
+open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.Sum.Properties
-open import Level using (_⊔_)
-open import Function.Base using (_∘_; id)
-open import Function.Inverse using (Inverse)
+open import Level using (Level; _⊔_)
+open import Function.Base using (const; _∘_; id)
+open import Function.Bundles using (Inverse; mk↔)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
-open import Relation.Binary.Core using (REL; Rel; _⇒_)
-open import Relation.Binary.Bundles
-  using (Setoid; DecSetoid; Preorder; Poset)
-open import Relation.Binary.Structures
-  using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder; IsStrictPartialOrder)
-open import Relation.Binary.Definitions
-  using (Reflexive; Symmetric; Transitive; Asymmetric; Substitutive; Decidable; Irreflexive; Antisymmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
+open import Relation.Binary
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 
+private
+  variable
+    a b c d ℓ₁ ℓ₂ ℓ₃ ℓ : Level
+    A B C D : Set ℓ
+    R S T U : REL A B ℓ
+    ≈₁ ≈₂ : Rel A ℓ
+
 ------------------------------------------------------------------------
 -- Definition
 
-data Pointwise {a b c d r s}
-               {A : Set a} {B : Set b} {C : Set c} {D : Set d}
-               (R : REL A C r) (S : REL B D s)
-               : REL (A ⊎ B) (C ⊎ D) (a ⊔ b ⊔ c ⊔ d ⊔ r ⊔ s) where
+data Pointwise {A : Set a} {B : Set b} {C : Set c} {D : Set d}
+               (R : REL A C ℓ₁) (S : REL B D ℓ₂)
+               : REL (A ⊎ B) (C ⊎ D) (a ⊔ b ⊔ c ⊔ d ⊔ ℓ₁ ⊔ ℓ₂) where
   inj₁ : ∀ {a c} → R a c → Pointwise R S (inj₁ a) (inj₁ c)
   inj₂ : ∀ {b d} → S b d → Pointwise R S (inj₂ b) (inj₂ d)
 
+----------------------------------------------------------------------
+-- Functions
+
+map : ∀ {f : A → C} {g : B → D} →
+      R =[ f ]⇒ T → S =[ g ]⇒ U →
+      Pointwise R S =[ Sum.map f g ]⇒ Pointwise T U
+map R⇒T _ (inj₁ x) = inj₁ (R⇒T x)
+map _ S⇒U (inj₂ x) = inj₂ (S⇒U x)
+
 ------------------------------------------------------------------------
 -- Relational properties
 
-module _ {a₁ a₂ ℓ₁ ℓ₂} {A₁ : Set a₁} {A₂ : Set a₂}
-         {∼₁ : Rel A₁ ℓ₁} {∼₂ : Rel A₂ ℓ₂}
-         where
-
-  drop-inj₁ : ∀ {x y} → Pointwise ∼₁ ∼₂ (inj₁ x) (inj₁ y) → ∼₁ x y
-  drop-inj₁ (inj₁ x) = x
-
-  drop-inj₂ : ∀ {x y} → Pointwise ∼₁ ∼₂ (inj₂ x) (inj₂ y) → ∼₂ x y
-  drop-inj₂ (inj₂ x) = x
-
-  ⊎-refl : Reflexive ∼₁ → Reflexive ∼₂ →
-           Reflexive (Pointwise ∼₁ ∼₂)
-  ⊎-refl refl₁ refl₂ {inj₁ x} = inj₁ refl₁
-  ⊎-refl refl₁ refl₂ {inj₂ y} = inj₂ refl₂
-
-  ⊎-symmetric : Symmetric ∼₁ → Symmetric ∼₂ →
-                Symmetric (Pointwise ∼₁ ∼₂)
-  ⊎-symmetric sym₁ sym₂ (inj₁ x) = inj₁ (sym₁ x)
-  ⊎-symmetric sym₁ sym₂ (inj₂ x) = inj₂ (sym₂ x)
-
-  ⊎-transitive : Transitive ∼₁ → Transitive ∼₂ →
-                 Transitive (Pointwise ∼₁ ∼₂)
-  ⊎-transitive trans₁ trans₂ (inj₁ x) (inj₁ y) = inj₁ (trans₁ x y)
-  ⊎-transitive trans₁ trans₂ (inj₂ x) (inj₂ y) = inj₂ (trans₂ x y)
-
-  ⊎-asymmetric : Asymmetric ∼₁ → Asymmetric ∼₂ →
-                 Asymmetric (Pointwise ∼₁ ∼₂)
-  ⊎-asymmetric asym₁ asym₂ (inj₁ x) = λ { (inj₁ y) → asym₁ x y }
-  ⊎-asymmetric asym₁ asym₂ (inj₂ x) = λ { (inj₂ y) → asym₂ x y }
-
-  ⊎-substitutive : ∀ {ℓ₃} → Substitutive ∼₁ ℓ₃ → Substitutive ∼₂ ℓ₃ →
-                   Substitutive (Pointwise ∼₁ ∼₂) ℓ₃
-  ⊎-substitutive subst₁ subst₂ P (inj₁ x) = subst₁ (P ∘ inj₁) x
-  ⊎-substitutive subst₁ subst₂ P (inj₂ x) = subst₂ (P ∘ inj₂) x
-
-  ⊎-decidable : Decidable ∼₁ → Decidable ∼₂ →
-                Decidable (Pointwise ∼₁ ∼₂)
-  ⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₁ y) = Dec.map′ inj₁ drop-inj₁ (x ≟₁ y)
-  ⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₂ y) = no λ()
-  ⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₁ y) = no λ()
-  ⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₂ y) = Dec.map′ inj₂ drop-inj₂ (x ≟₂ y)
-
-module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
-         {ℓ₁ ℓ₂} {∼₁ : Rel A₁ ℓ₁} {≈₁ : Rel A₁ ℓ₂}
-         {ℓ₃ ℓ₄} {∼₂ : Rel A₂ ℓ₃} {≈₂ : Rel A₂ ℓ₄} where
-
-  ⊎-reflexive : ≈₁ ⇒ ∼₁ → ≈₂ ⇒ ∼₂ →
-                (Pointwise ≈₁ ≈₂) ⇒ (Pointwise ∼₁ ∼₂)
-  ⊎-reflexive refl₁ refl₂ (inj₁ x) = inj₁ (refl₁ x)
-  ⊎-reflexive refl₁ refl₂ (inj₂ x) = inj₂ (refl₂ x)
-
-  ⊎-irreflexive : Irreflexive ≈₁ ∼₁ → Irreflexive ≈₂ ∼₂ →
-                  Irreflexive (Pointwise ≈₁ ≈₂) (Pointwise ∼₁ ∼₂)
-  ⊎-irreflexive irrefl₁ irrefl₂ (inj₁ x) (inj₁ y) = irrefl₁ x y
-  ⊎-irreflexive irrefl₁ irrefl₂ (inj₂ x) (inj₂ y) = irrefl₂ x y
-
-  ⊎-antisymmetric : Antisymmetric ≈₁ ∼₁ → Antisymmetric ≈₂ ∼₂ →
-                    Antisymmetric (Pointwise ≈₁ ≈₂) (Pointwise ∼₁ ∼₂)
-  ⊎-antisymmetric antisym₁ antisym₂ (inj₁ x) (inj₁ y) = inj₁ (antisym₁ x y)
-  ⊎-antisymmetric antisym₁ antisym₂ (inj₂ x) (inj₂ y) = inj₂ (antisym₂ x y)
-
-  ⊎-respectsˡ : ∼₁ Respectsˡ ≈₁ → ∼₂ Respectsˡ ≈₂ →
-                (Pointwise ∼₁ ∼₂) Respectsˡ (Pointwise ≈₁ ≈₂)
-  ⊎-respectsˡ resp₁ resp₂ (inj₁ x) (inj₁ y) = inj₁ (resp₁ x y)
-  ⊎-respectsˡ resp₁ resp₂ (inj₂ x) (inj₂ y) = inj₂ (resp₂ x y)
-
-  ⊎-respectsʳ : ∼₁ Respectsʳ ≈₁ → ∼₂ Respectsʳ ≈₂ →
-                (Pointwise ∼₁ ∼₂) Respectsʳ (Pointwise ≈₁ ≈₂)
-  ⊎-respectsʳ resp₁ resp₂ (inj₁ x) (inj₁ y) = inj₁ (resp₁ x y)
-  ⊎-respectsʳ resp₁ resp₂ (inj₂ x) (inj₂ y) = inj₂ (resp₂ x y)
-
-  ⊎-respects₂ : ∼₁ Respects₂ ≈₁ → ∼₂ Respects₂ ≈₂ →
-                (Pointwise ∼₁ ∼₂) Respects₂ (Pointwise ≈₁ ≈₂)
-  ⊎-respects₂ (r₁ , l₁) (r₂ , l₂) = ⊎-respectsʳ r₁ r₂ , ⊎-respectsˡ l₁ l₂
+drop-inj₁ : ∀ {x y} → Pointwise R S (inj₁ x) (inj₁ y) → R x y
+drop-inj₁ (inj₁ x) = x
+
+drop-inj₂ : ∀ {x y} → Pointwise R S (inj₂ x) (inj₂ y) → S x y
+drop-inj₂ (inj₂ x) = x
+
+⊎-refl : Reflexive R → Reflexive S → Reflexive (Pointwise R S)
+⊎-refl refl₁ refl₂ {inj₁ x} = inj₁ refl₁
+⊎-refl refl₁ refl₂ {inj₂ y} = inj₂ refl₂
+
+⊎-symmetric : Symmetric R → Symmetric S →
+              Symmetric (Pointwise R S)
+⊎-symmetric sym₁ sym₂ (inj₁ x) = inj₁ (sym₁ x)
+⊎-symmetric sym₁ sym₂ (inj₂ x) = inj₂ (sym₂ x)
+
+⊎-transitive : Transitive R → Transitive S →
+               Transitive (Pointwise R S)
+⊎-transitive trans₁ trans₂ (inj₁ x) (inj₁ y) = inj₁ (trans₁ x y)
+⊎-transitive trans₁ trans₂ (inj₂ x) (inj₂ y) = inj₂ (trans₂ x y)
+
+⊎-asymmetric : Asymmetric R → Asymmetric S →
+               Asymmetric (Pointwise R S)
+⊎-asymmetric asym₁ asym₂ (inj₁ x) = λ { (inj₁ y) → asym₁ x y }
+⊎-asymmetric asym₁ asym₂ (inj₂ x) = λ { (inj₂ y) → asym₂ x y }
+
+⊎-substitutive : Substitutive R ℓ₃ → Substitutive S ℓ₃ →
+                 Substitutive (Pointwise R S) ℓ₃
+⊎-substitutive subst₁ subst₂ P (inj₁ x) = subst₁ (P ∘ inj₁) x
+⊎-substitutive subst₁ subst₂ P (inj₂ x) = subst₂ (P ∘ inj₂) x
+
+⊎-decidable : Decidable R → Decidable S → Decidable (Pointwise R S)
+⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₁ y) = Dec.map′ inj₁ drop-inj₁ (x ≟₁ y)
+⊎-decidable _≟₁_ _≟₂_ (inj₁ x) (inj₂ y) = no λ()
+⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₁ y) = no λ()
+⊎-decidable _≟₁_ _≟₂_ (inj₂ x) (inj₂ y) = Dec.map′ inj₂ drop-inj₂ (x ≟₂ y)
+
+⊎-reflexive : ≈₁ ⇒ R → ≈₂ ⇒ S →
+              (Pointwise ≈₁ ≈₂) ⇒ (Pointwise R S)
+⊎-reflexive refl₁ refl₂ (inj₁ x) = inj₁ (refl₁ x)
+⊎-reflexive refl₁ refl₂ (inj₂ x) = inj₂ (refl₂ x)
+
+⊎-irreflexive : Irreflexive ≈₁ R → Irreflexive ≈₂ S →
+                Irreflexive (Pointwise ≈₁ ≈₂) (Pointwise R S)
+⊎-irreflexive irrefl₁ irrefl₂ (inj₁ x) (inj₁ y) = irrefl₁ x y
+⊎-irreflexive irrefl₁ irrefl₂ (inj₂ x) (inj₂ y) = irrefl₂ x y
+
+⊎-antisymmetric : Antisymmetric ≈₁ R → Antisymmetric ≈₂ S →
+                  Antisymmetric (Pointwise ≈₁ ≈₂) (Pointwise R S)
+⊎-antisymmetric antisym₁ antisym₂ (inj₁ x) (inj₁ y) = inj₁ (antisym₁ x y)
+⊎-antisymmetric antisym₁ antisym₂ (inj₂ x) (inj₂ y) = inj₂ (antisym₂ x y)
+
+⊎-respectsˡ : R Respectsˡ ≈₁ → S Respectsˡ ≈₂ →
+              (Pointwise R S) Respectsˡ (Pointwise ≈₁ ≈₂)
+⊎-respectsˡ resp₁ resp₂ (inj₁ x) (inj₁ y) = inj₁ (resp₁ x y)
+⊎-respectsˡ resp₁ resp₂ (inj₂ x) (inj₂ y) = inj₂ (resp₂ x y)
+
+⊎-respectsʳ : R Respectsʳ ≈₁ → S Respectsʳ ≈₂ →
+              (Pointwise R S) Respectsʳ (Pointwise ≈₁ ≈₂)
+⊎-respectsʳ resp₁ resp₂ (inj₁ x) (inj₁ y) = inj₁ (resp₁ x y)
+⊎-respectsʳ resp₁ resp₂ (inj₂ x) (inj₂ y) = inj₂ (resp₂ x y)
+
+⊎-respects₂ : R Respects₂ ≈₁ → S Respects₂ ≈₂ →
+              (Pointwise R S) Respects₂ (Pointwise ≈₁ ≈₂)
+⊎-respects₂ (r₁ , l₁) (r₂ , l₂) = ⊎-respectsʳ r₁ r₂ , ⊎-respectsˡ l₁ l₂
 
 ------------------------------------------------------------------------
 -- Structures
 
-module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
-         {ℓ₁ ℓ₂} {≈₁ : Rel A₁ ℓ₁} {≈₂ : Rel A₂ ℓ₂}
-         where
-
-  ⊎-isEquivalence : IsEquivalence ≈₁ → IsEquivalence ≈₂ →
-                    IsEquivalence (Pointwise ≈₁ ≈₂)
-  ⊎-isEquivalence eq₁ eq₂ = record
-    { refl  = ⊎-refl       (refl  eq₁) (refl  eq₂)
-    ; sym   = ⊎-symmetric  (sym   eq₁) (sym   eq₂)
-    ; trans = ⊎-transitive (trans eq₁) (trans eq₂)
-    }
-    where open IsEquivalence
-
-  ⊎-isDecEquivalence : IsDecEquivalence ≈₁ → IsDecEquivalence ≈₂ →
-                       IsDecEquivalence (Pointwise ≈₁ ≈₂)
-  ⊎-isDecEquivalence eq₁ eq₂ = record
-    { isEquivalence =
-        ⊎-isEquivalence (isEquivalence eq₁) (isEquivalence eq₂)
-    ; _≟_           = ⊎-decidable (_≟_ eq₁) (_≟_ eq₂)
-    }
-    where open IsDecEquivalence
-
-module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
-         {ℓ₁ ℓ₂} {≈₁ : Rel A₁ ℓ₁} {∼₁ : Rel A₁ ℓ₂}
-         {ℓ₃ ℓ₄} {≈₂ : Rel A₂ ℓ₃} {∼₂ : Rel A₂ ℓ₄} where
-
-  ⊎-isPreorder : IsPreorder ≈₁ ∼₁ → IsPreorder ≈₂ ∼₂ →
-                 IsPreorder (Pointwise ≈₁ ≈₂) (Pointwise ∼₁ ∼₂)
-  ⊎-isPreorder pre₁ pre₂ = record
-    { isEquivalence =
-        ⊎-isEquivalence (isEquivalence pre₁) (isEquivalence pre₂)
-    ; reflexive     = ⊎-reflexive (reflexive pre₁) (reflexive pre₂)
-    ; trans         = ⊎-transitive (trans pre₁) (trans pre₂)
-    }
-    where open IsPreorder
-
-  ⊎-isPartialOrder : IsPartialOrder ≈₁ ∼₁ →
-                     IsPartialOrder ≈₂ ∼₂ →
-                     IsPartialOrder
-                       (Pointwise ≈₁ ≈₂) (Pointwise ∼₁ ∼₂)
-  ⊎-isPartialOrder po₁ po₂ = record
-    { isPreorder = ⊎-isPreorder (isPreorder po₁) (isPreorder po₂)
-    ; antisym    = ⊎-antisymmetric (antisym po₁) (antisym    po₂)
-    }
-    where open IsPartialOrder
-
-  ⊎-isStrictPartialOrder : IsStrictPartialOrder ≈₁ ∼₁ →
-                           IsStrictPartialOrder ≈₂ ∼₂ →
-                           IsStrictPartialOrder
-                             (Pointwise ≈₁ ≈₂) (Pointwise ∼₁ ∼₂)
-  ⊎-isStrictPartialOrder spo₁ spo₂ = record
-    { isEquivalence =
-        ⊎-isEquivalence (isEquivalence spo₁) (isEquivalence spo₂)
-    ; irrefl        = ⊎-irreflexive (irrefl   spo₁) (irrefl   spo₂)
-    ; trans         = ⊎-transitive  (trans    spo₁) (trans    spo₂)
-    ; <-resp-≈      = ⊎-respects₂   (<-resp-≈ spo₁) (<-resp-≈ spo₂)
-    }
-    where open IsStrictPartialOrder
+⊎-isEquivalence : IsEquivalence ≈₁ → IsEquivalence ≈₂ →
+                  IsEquivalence (Pointwise ≈₁ ≈₂)
+⊎-isEquivalence eq₁ eq₂ = record
+  { refl  = ⊎-refl       (refl  eq₁) (refl  eq₂)
+  ; sym   = ⊎-symmetric  (sym   eq₁) (sym   eq₂)
+  ; trans = ⊎-transitive (trans eq₁) (trans eq₂)
+  } where open IsEquivalence
+
+⊎-isDecEquivalence : IsDecEquivalence ≈₁ → IsDecEquivalence ≈₂ →
+                     IsDecEquivalence (Pointwise ≈₁ ≈₂)
+⊎-isDecEquivalence eq₁ eq₂ = record
+  { isEquivalence =
+      ⊎-isEquivalence (isEquivalence eq₁) (isEquivalence eq₂)
+  ; _≟_           = ⊎-decidable (_≟_ eq₁) (_≟_ eq₂)
+  } where open IsDecEquivalence
+
+⊎-isPreorder : IsPreorder ≈₁ R → IsPreorder ≈₂ S →
+               IsPreorder (Pointwise ≈₁ ≈₂) (Pointwise R S)
+⊎-isPreorder pre₁ pre₂ = record
+  { isEquivalence =
+      ⊎-isEquivalence (isEquivalence pre₁) (isEquivalence pre₂)
+  ; reflexive     = ⊎-reflexive (reflexive pre₁) (reflexive pre₂)
+  ; trans         = ⊎-transitive (trans pre₁) (trans pre₂)
+  } where open IsPreorder
+
+⊎-isPartialOrder : IsPartialOrder ≈₁ R → IsPartialOrder ≈₂ S →
+                   IsPartialOrder
+                     (Pointwise ≈₁ ≈₂) (Pointwise R S)
+⊎-isPartialOrder po₁ po₂ = record
+  { isPreorder = ⊎-isPreorder (isPreorder po₁) (isPreorder po₂)
+  ; antisym    = ⊎-antisymmetric (antisym po₁) (antisym    po₂)
+  } where open IsPartialOrder
+
+⊎-isStrictPartialOrder : IsStrictPartialOrder ≈₁ R →
+                         IsStrictPartialOrder ≈₂ S →
+                         IsStrictPartialOrder
+                           (Pointwise ≈₁ ≈₂) (Pointwise R S)
+⊎-isStrictPartialOrder spo₁ spo₂ = record
+  { isEquivalence =
+      ⊎-isEquivalence (isEquivalence spo₁) (isEquivalence spo₂)
+  ; irrefl        = ⊎-irreflexive (irrefl   spo₁) (irrefl   spo₂)
+  ; trans         = ⊎-transitive  (trans    spo₁) (trans    spo₂)
+  ; <-resp-≈      = ⊎-respects₂   (<-resp-≈ spo₁) (<-resp-≈ spo₂)
+  } where open IsStrictPartialOrder
 
 ------------------------------------------------------------------------
 -- Bundles
 
-module _ {a b c d} where
-
-  ⊎-setoid : Setoid a b → Setoid c d → Setoid _ _
-  ⊎-setoid s₁ s₂ = record
-    { isEquivalence =
-        ⊎-isEquivalence (isEquivalence s₁) (isEquivalence s₂)
-    } where open Setoid
-
-  ⊎-decSetoid : DecSetoid a b → DecSetoid c d → DecSetoid _ _
-  ⊎-decSetoid ds₁ ds₂ = record
-    { isDecEquivalence =
-        ⊎-isDecEquivalence (isDecEquivalence ds₁) (isDecEquivalence ds₂)
-    } where open DecSetoid
+⊎-setoid : Setoid a b → Setoid c d → Setoid _ _
+⊎-setoid s₁ s₂ = record
+  { isEquivalence =
+      ⊎-isEquivalence (isEquivalence s₁) (isEquivalence s₂)
+  } where open Setoid
+
+⊎-decSetoid : DecSetoid a b → DecSetoid c d → DecSetoid _ _
+⊎-decSetoid ds₁ ds₂ = record
+  { isDecEquivalence =
+      ⊎-isDecEquivalence (isDecEquivalence ds₁) (isDecEquivalence ds₂)
+  } where open DecSetoid
+
+⊎-preorder : Preorder a b ℓ₁ → Preorder c d ℓ₂ → Preorder _ _ _
+⊎-preorder p₁ p₂ = record
+  { isPreorder   =
+      ⊎-isPreorder (isPreorder p₁) (isPreorder p₂)
+  } where open Preorder
+
+⊎-poset : Poset a b c → Poset a b c → Poset _ _ _
+⊎-poset po₁ po₂ = record
+  { isPartialOrder =
+      ⊎-isPartialOrder (isPartialOrder po₁) (isPartialOrder po₂)
+  } where open Poset
 
-  -- Some additional notation for combining setoids
-  infix 4 _⊎ₛ_
-  _⊎ₛ_ : Setoid a b → Setoid c d → Setoid _ _
-  _⊎ₛ_ = ⊎-setoid
-
-module _ {a b c d e f} where
-
-  ⊎-preorder : Preorder a b c → Preorder d e f → Preorder _ _ _
-  ⊎-preorder p₁ p₂ = record
-    { isPreorder   =
-        ⊎-isPreorder (isPreorder p₁) (isPreorder p₂)
-    } where open Preorder
+------------------------------------------------------------------------
+-- Additional notation
 
-  ⊎-poset : Poset a b c → Poset a b c → Poset _ _ _
-  ⊎-poset po₁ po₂ = record
-    { isPartialOrder =
-        ⊎-isPartialOrder (isPartialOrder po₁) (isPartialOrder po₂)
-    } where open Poset
+-- Infix combining setoids
+infix 4 _⊎ₛ_
+_⊎ₛ_ : Setoid a b → Setoid c d → Setoid _ _
+_⊎ₛ_ = ⊎-setoid
 
 ------------------------------------------------------------------------
 -- The propositional equality setoid over products can be
 -- decomposed using Pointwise
 
-module _ {a b} {A : Set a} {B : Set b} where
-
-  Pointwise-≡⇒≡ : (Pointwise _≡_ _≡_) ⇒ _≡_ {A = A ⊎ B}
-  Pointwise-≡⇒≡ (inj₁ x) = P.cong inj₁ x
-  Pointwise-≡⇒≡ (inj₂ x) = P.cong inj₂ x
+Pointwise-≡⇒≡ : (Pointwise _≡_ _≡_) ⇒ _≡_ {A = A ⊎ B}
+Pointwise-≡⇒≡ (inj₁ x) = P.cong inj₁ x
+Pointwise-≡⇒≡ (inj₂ x) = P.cong inj₂ x
 
-  ≡⇒Pointwise-≡ : _≡_ {A = A ⊎ B} ⇒ (Pointwise _≡_ _≡_)
-  ≡⇒Pointwise-≡ P.refl = ⊎-refl P.refl P.refl
+≡⇒Pointwise-≡ : _≡_ {A = A ⊎ B} ⇒ (Pointwise _≡_ _≡_)
+≡⇒Pointwise-≡ P.refl = ⊎-refl P.refl P.refl
 
-Pointwise-≡↔≡ : ∀ {a b} (A : Set a) (B : Set b) →
-                Inverse (P.setoid A ⊎ₛ P.setoid B)
-                        (P.setoid (A ⊎ B))
+Pointwise-≡↔≡ : (A : Set a) (B : Set b) →
+                 Inverse (P.setoid A ⊎ₛ P.setoid B) (P.setoid (A ⊎ B))
 Pointwise-≡↔≡ _ _ = record
-  { to         = record { _⟨$⟩_ = id; cong = Pointwise-≡⇒≡ }
-  ; from       = record { _⟨$⟩_ = id; cong = ≡⇒Pointwise-≡ }
-  ; inverse-of = record
-    { left-inverse-of  = λ _ → ⊎-refl P.refl P.refl
-    ; right-inverse-of = λ _ → P.refl
-    }
+  { to        = id
+  ; from      = id
+  ; to-cong   = Pointwise-≡⇒≡
+  ; from-cong = ≡⇒Pointwise-≡
+  ; inverse   = Pointwise-≡⇒≡ , ≡⇒Pointwise-≡
   }
diff --git a/src/Data/Vec/N-ary.agda b/src/Data/Vec/N-ary.agda
index 9220f86547..13c7cb29ae 100644
--- a/src/Data/Vec/N-ary.agda
+++ b/src/Data/Vec/N-ary.agda
@@ -9,7 +9,7 @@
 module Data.Vec.N-ary where
 
 open import Axiom.Extensionality.Propositional using (Extensionality)
-open import Function.Bundles using (_↔_; Inverse; mk↔′)
+open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)
 open import Data.Nat.Base hiding (_⊔_)
 open import Data.Product.Base as Prod using (∃; _,_)
 open import Data.Vec.Base using (Vec; []; _∷_; head; tail)
@@ -179,10 +179,10 @@ Eqʰ-to-Eq (suc n) _∼_ eq = λ _ → Eqʰ-to-Eq n _∼_ eq
 module _ (ext : ∀ {a b} → Extensionality a b) where
 
   Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B
-  Vec↔N-ary zero = mk↔′ (λ vxs → vxs []) (flip constᵣ) (λ _ → refl)
+  Vec↔N-ary zero = mk↔ₛ′ (λ vxs → vxs []) (flip constᵣ) (λ _ → refl)
     (λ vxs → ext λ where [] → refl)
   Vec↔N-ary (suc n) = let open Inverse (Vec↔N-ary n) in
-    mk↔′ (λ vxs x → to λ xs → vxs (x ∷ xs))
+    mk↔ₛ′ (λ vxs x → to λ xs → vxs (x ∷ xs))
     (λ any xs → from (any (head xs)) (tail xs))
     (λ any → ext λ x → strictlyInverseˡ _)
     (λ vxs → ext λ where (x ∷ xs) → cong (λ f → f xs) (strictlyInverseʳ (λ ys → vxs (x ∷ ys))))
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 8113d0a932..38ba95ae7c 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -23,7 +23,7 @@ open import Data.Sum.Properties using ([,]-map)
 open import Data.Vec.Base
 open import Function.Base
 -- open import Function.Inverse using (_↔_; inverse)
-open import Function.Bundles using (_↔_; mk↔′)
+open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
@@ -159,7 +159,7 @@ lookup⇒[]= (suc i) (_ ∷ xs) p    = there (lookup⇒[]= i xs p)
 
 []=↔lookup : ∀ {i} → xs [ i ]= x ↔ lookup xs i ≡ x
 []=↔lookup {xs = ys} {i = i} =
-  mk↔′ []=⇒lookup (lookup⇒[]= i ys) ([]=⇒lookup∘lookup⇒[]= _ i) lookup⇒[]=∘[]=⇒lookup
+  mk↔ₛ′ []=⇒lookup (lookup⇒[]= i ys) ([]=⇒lookup∘lookup⇒[]= _ i) lookup⇒[]=∘[]=⇒lookup
   where
   lookup⇒[]=∘[]=⇒lookup : ∀ {i} (p : xs [ i ]= x) →
                           lookup⇒[]= i xs ([]=⇒lookup p) ≡ p
@@ -689,7 +689,7 @@ zip∘unzip []         = refl
 zip∘unzip (xy ∷ xys) = cong (xy ∷_) (zip∘unzip xys)
 
 ×v↔v× : (Vec A n × Vec B n) ↔ Vec (A × B) n
-×v↔v× = mk↔′ (uncurry zip) unzip zip∘unzip (uncurry unzip∘zip)
+×v↔v× = mk↔ₛ′ (uncurry zip) unzip zip∘unzip (uncurry unzip∘zip)
 
 ------------------------------------------------------------------------
 -- _⊛_
diff --git a/src/Data/Vec/Recursive.agda b/src/Data/Vec/Recursive.agda
index 4c8796d016..baaf4eca37 100644
--- a/src/Data/Vec/Recursive.agda
+++ b/src/Data/Vec/Recursive.agda
@@ -30,7 +30,7 @@ open import Data.Unit.Polymorphic.Properties using (⊤↔⊤*)
 open import Data.Vec.Base as Vec using (Vec; _∷_)
 open import Data.Vec.N-ary using (N-ary)
 open import Function.Base using (_∘′_; _∘_; id; const)
-open import Function.Bundles using (_↔_; mk↔′; mk↔)
+open import Function.Bundles using (_↔_; mk↔ₛ′; mk↔)
 open import Function.Properties.Inverse using (↔-isEquivalence; ↔-refl; ↔-sym; ↔-trans)
 open import Level using (Level; lift)
 open import Relation.Unary using (IUniversal; Universal; _⇒_)
@@ -155,7 +155,7 @@ unzip : ∀ n → (A × B) ^ n → A ^ n × B ^ n
 unzip = unzipWith id
 
 lift↔ : ∀ n → A ↔ B → A ^ n ↔ B ^ n
-lift↔ 0               A↔B = mk↔′ _ _ (const refl) (const refl)
+lift↔ 0               A↔B = mk↔ₛ′ _ _ (const refl) (const refl)
 lift↔ 1               A↔B = A↔B
 lift↔ (suc n@(suc _)) A↔B = ×-cong A↔B (lift↔ n A↔B)
 
diff --git a/src/Data/Vec/Recursive/Properties.agda b/src/Data/Vec/Recursive/Properties.agda
index 85fa8e44e6..b012e59a80 100644
--- a/src/Data/Vec/Recursive/Properties.agda
+++ b/src/Data/Vec/Recursive/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Nat.Base hiding (_^_)
 open import Data.Product.Base
 open import Data.Vec.Recursive
 open import Data.Vec.Base using (Vec; _∷_)
-open import Function.Bundles using (_↔_; mk↔′)
+open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Relation.Binary.PropositionalEquality.Core as P
 open ≡-Reasoning
 
@@ -82,7 +82,7 @@ toVec∘fromVec {n = suc n} (x Vec.∷ xs) = begin
   tl-prf = tail-cons-identity _ x (fromVec xs)
 
 ↔Vec : ∀ n → A ^ n ↔ Vec A n
-↔Vec n = mk↔′ (toVec n) fromVec toVec∘fromVec (fromVec∘toVec n)
+↔Vec n = mk↔ₛ′ (toVec n) fromVec toVec∘fromVec (fromVec∘toVec n)
 
 ------------------------------------------------------------------------
 -- DEPRECATED NAMES
diff --git a/src/Data/Vec/Relation/Unary/All/Properties.agda b/src/Data/Vec/Relation/Unary/All/Properties.agda
index 9cfc066635..4b8951ff9d 100644
--- a/src/Data/Vec/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/All/Properties.agda
@@ -18,7 +18,7 @@ import Data.Vec.Properties as Vecₚ
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 open import Level using (Level)
 open import Function.Base using (_∘_; id)
-open import Function.Inverse using (_↔_; inverse)
+open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Relation.Unary using (Pred) renaming (_⊆_ to _⋐_)
 open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality
@@ -98,7 +98,7 @@ gmap g = map⁺ ∘ All.map g
 
 ++↔ : {xs : Vec A m} {ys : Vec A n} →
       (All P xs × All P ys) ↔ All P (xs ++ ys)
-++↔ {xs = xs} = inverse (uncurry ++⁺) (++⁻ xs) ++⁻∘++⁺ (++⁺∘++⁻ xs)
+++↔ {xs = xs} = mk↔ₛ′ (uncurry ++⁺) (++⁻ xs) (++⁺∘++⁻ xs) ++⁻∘++⁺
 
 ------------------------------------------------------------------------
 -- concat
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index 6069a6b6db..437e7933e7 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -23,8 +23,8 @@ open import Data.Vec.Membership.Propositional
 open import Data.Vec.Relation.Binary.Pointwise.Inductive
   using (Pointwise; []; _∷_)
 open import Function.Base
-open import Function.Inverse using (_↔_; inverse)
-  renaming (_∘_ to _∘↔_; id to id↔)
+open import Function.Bundles using (_↔_; mk↔ₛ′)
+open import Function.Properties.Inverse using (↔-refl; ↔-trans)
 open import Level using (Level)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Unary hiding (_∈_)
@@ -133,19 +133,19 @@ module _ {P : A → B → Set ℓ} where
 
   swap↔ : ∀ {n m} {xs : Vec A n} {ys : Vec B m} →
           Any (λ x → Any (P x) ys) xs ↔ Any (λ y → Any (flip P y) xs) ys
-  swap↔ = inverse swap swap swap-invol swap-invol
+  swap↔ = mk↔ₛ′ swap swap swap-invol swap-invol
 
 ------------------------------------------------------------------------
 -- Lemmas relating Any to ⊥
 
 ⊥↔Any⊥ : ∀ {n} {xs : Vec A n} → ⊥ ↔ Any (const ⊥) xs
-⊥↔Any⊥ = inverse (λ ()) (λ p → from p) (λ ()) (λ p → from p)
+⊥↔Any⊥ = mk↔ₛ′ (λ ()) (λ p → from p) (λ p → from p) (λ ())
   where
   from : ∀ {n xs} → Any (const ⊥) {n} xs → ∀ {b} {B : Set b} → B
   from (there p) = from p
 
 ⊥↔Any[] : ∀ {P : Pred A p} → ⊥ ↔ Any P []
-⊥↔Any[] = inverse (λ()) (λ()) (λ()) (λ())
+⊥↔Any[] = mk↔ₛ′ (λ()) (λ()) (λ()) (λ())
 
 ------------------------------------------------------------------------
 -- Sums commute with Any
@@ -161,7 +161,7 @@ module _ {P : Pred A p} {Q : A → Set q} where
   Any-⊎⁻ (there p)       = Sum.map there there (Any-⊎⁻ p)
 
   ⊎↔ : ∀ {n} {xs : Vec A n} → (Any P xs ⊎ Any Q xs) ↔ Any (λ x → P x ⊎ Q x) xs
-  ⊎↔ = inverse Any-⊎⁺ Any-⊎⁻ from∘to to∘from
+  ⊎↔ = mk↔ₛ′ Any-⊎⁺ Any-⊎⁻ to∘from from∘to
     where
     from∘to : ∀ {n} {xs : Vec A n} (p : Any P xs ⊎ Any Q xs) → Any-⊎⁻ (Any-⊎⁺ p) ≡ p
     from∘to (inj₁ (here  p)) = refl
@@ -217,7 +217,7 @@ module _ {P : Pred A p} where
   singleton⁻∘singleton⁺ p = refl
 
   singleton↔ : ∀ {x} → P x ↔ Any P [ x ]
-  singleton↔ = inverse singleton⁺ singleton⁻ singleton⁻∘singleton⁺ singleton⁺∘singleton⁻
+  singleton↔ = mk↔ₛ′ singleton⁺ singleton⁻ singleton⁺∘singleton⁻ singleton⁻∘singleton⁺
 
 ------------------------------------------------------------------------
 -- map
@@ -246,7 +246,7 @@ module _ {f : A → B} where
 
   map↔ : ∀ {P : Pred B p} {n} {xs : Vec A n} →
          Any (P ∘ f) xs ↔ Any P (map f xs)
-  map↔ = inverse map⁺ map⁻ (map⁻∘map⁺ _) map⁺∘map⁻
+  map↔ = mk↔ₛ′ map⁺ map⁻ map⁺∘map⁻ (map⁻∘map⁺ _)
 
 ------------------------------------------------------------------------
 -- _++_
@@ -283,7 +283,7 @@ module _ {P : Pred A p} where
 
   ++↔ : ∀ {n m} {xs : Vec A n} {ys : Vec A m} →
         (Any P xs ⊎ Any P ys) ↔ Any P (xs ++ ys)
-  ++↔ {xs = xs} = inverse [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs) (++⁻∘++⁺ xs) (++⁺∘++⁻ xs)
+  ++↔ {xs = xs} = mk↔ₛ′ [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs) (++⁺∘++⁻ xs) (++⁻∘++⁺ xs)
 
   ++-comm : ∀ {n m} (xs : Vec A n) (ys : Vec A m) →
             Any P (xs ++ ys) → Any P (ys ++ xs)
@@ -306,8 +306,8 @@ module _ {P : Pred A p} where
             | ++⁻∘++⁺ ys {ys = x ∷ xs} (inj₁ p) = refl
 
   ++↔++ : ∀ {n m} (xs : Vec A n) (ys : Vec A m) → Any P (xs ++ ys) ↔ Any P (ys ++ xs)
-  ++↔++ xs ys = inverse (++-comm xs ys) (++-comm ys xs)
-                        (++-comm∘++-comm xs) (++-comm∘++-comm ys)
+  ++↔++ xs ys = mk↔ₛ′ (++-comm xs ys) (++-comm ys xs)
+                        (++-comm∘++-comm ys) (++-comm∘++-comm xs)
 
   ++-insert : ∀ {n m x} (xs : Vec A n) {ys : Vec A m} → P x → Any P (xs ++ [ x ] ++ ys)
   ++-insert xs Px = ++⁺ʳ xs (++⁺ˡ (singleton⁺ Px))
@@ -351,7 +351,7 @@ module _ {P : Pred A p} where
             | concat⁻∘concat⁺ p = refl
 
   concat↔ : ∀ {n m} {xss : Vec (Vec A n) m} → Any (Any P) xss ↔ Any P (concat xss)
-  concat↔ {xss = xss} = inverse concat⁺ (concat⁻ xss) concat⁻∘concat⁺ (concat⁺∘concat⁻ xss)
+  concat↔ {xss = xss} = mk↔ₛ′ concat⁺ (concat⁻ xss) (concat⁺∘concat⁻ xss) concat⁻∘concat⁺
 
 ------------------------------------------------------------------------
 -- tabulate
@@ -388,7 +388,7 @@ module _ {P : Pred B p} where
 
   mapWith∈↔ : ∀ {n} {xs : Vec A n} {f : ∀ {x} → x ∈ xs → B} →
     (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) ↔ Any P (mapWith∈ xs f)
-  mapWith∈↔ = inverse (mapWith∈⁺ _) (mapWith∈⁻ _ _) (from∘to _) (to∘from _ _)
+  mapWith∈↔ = mk↔ₛ′ (mapWith∈⁺ _) (mapWith∈⁻ _ _) (to∘from _ _) (from∘to _)
     where
     from∘to : ∀ {n} {xs : Vec A n} (f : ∀ {x} → x ∈ xs → B)
               (p : ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) →
@@ -409,7 +409,7 @@ module _ {P : Pred B p} where
 
 ∷↔ : ∀ {n} (P : Pred A p) {x} {xs : Vec A n} →
      (P x ⊎ Any P xs) ↔ Any P (x ∷ xs)
-∷↔ P {x} {xs} = ++↔ ∘↔ (singleton↔ ⊎-cong id↔)
+∷↔ P {x} {xs} = ↔-trans (singleton↔ ⊎-cong ↔-refl) ++↔
 
 ------------------------------------------------------------------------
 -- _>>=_
@@ -419,4 +419,4 @@ module _ {A B : Set a} {P : Pred B p} {m} {f : A → Vec B m} where
   open CartesianBind
 
   >>=↔ : ∀ {n} {xs : Vec A n} → Any (Any P ∘ f) xs ↔ Any P (xs >>= f)
-  >>=↔ = concat↔ ∘↔ map↔
+  >>=↔ = ↔-trans map↔ concat↔
diff --git a/src/Function/Bijection.agda b/src/Function/Bijection.agda
index 405d326b51..bb7f2d2cce 100644
--- a/src/Function/Bijection.agda
+++ b/src/Function/Bijection.agda
@@ -1,18 +1,19 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Bijections
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
-
--- Note: use of the standard function hierarchy is encouraged. The
--- module `Function` re-exports `Bijective`, `IsBijection` and
--- `Bijection`. The alternative definitions found in this file will
--- eventually be deprecated.
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.Bijection where
 
+{-# WARNING_ON_IMPORT
+"Function.Bijection was deprecated in v2.0.
+Use the standard function hierarchy in Function/Function.Bundles instead."
+#-}
+
 open import Level
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality as P
@@ -89,11 +90,11 @@ bijection : ∀ {f t} {From : Set f} {To : Set t} →
             (∀ x → to (from x) ≡ x) →
             From ⤖ To
 bijection to from inj invʳ = record
-  { to        = P.→-to-⟶ to
+  { to        = F.→-to-⟶ to
   ; bijective = record
     { injective  = inj
     ; surjective = record
-      { from             = P.→-to-⟶ from
+      { from             = F.→-to-⟶ from
       ; right-inverse-of = invʳ
       }
     }
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index d100fc5343..1e479cf46a 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -66,7 +66,6 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
     open IsCongruent isCongruent public
       using (module Eq₁; module Eq₂)
 
-
   record Injection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       to          : A → B
@@ -95,16 +94,15 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       cong       : Congruent _≈₁_ _≈₂_ to
       surjective : Surjective _≈₁_ _≈₂_ to
 
-    to⁻ : B → A
-    to⁻ = proj₁ ∘ surjective
-
-    isCongruent : IsCongruent to
-    isCongruent = record
-      { cong           = cong
-      ; isEquivalence₁ = isEquivalence From
-      ; isEquivalence₂ = isEquivalence To
+    function : Func
+    function = record
+      { to   = to
+      ; cong = cong
       }
 
+    open Func function public
+      hiding (to; cong)
+
     isSurjection : IsSurjection to
     isSurjection = record
       { isCongruent = isCongruent
@@ -113,11 +111,15 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
 
     open IsSurjection isSurjection public
       using
-      ( module Eq₁
-      ; module Eq₂
-      ; strictlySurjective
+      ( strictlySurjective
       )
 
+    to⁻ : B → A
+    to⁻ = proj₁ ∘ surjective
+
+    to∘to⁻ : ∀ x → to (to⁻ x) ≈₂ x
+    to∘to⁻ = proj₂ ∘ strictlySurjective
+
 
   record Bijection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
@@ -158,12 +160,38 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
 ------------------------------------------------------------------------
 -- Bundles with two elements
 
+module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
+
+  open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)
+  open Setoid To   using () renaming (Carrier to B; _≈_ to _≈₂_)
+  open FunctionStructures _≈₁_ _≈₂_
+
   record Equivalence : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       to        : A → B
       from      : B → A
       to-cong   : Congruent _≈₁_ _≈₂_ to
-      from-cong : from Preserves _≈₂_ ⟶ _≈₁_
+      from-cong : Congruent _≈₂_ _≈₁_ from
+
+    toFunction : Func From To
+    toFunction = record
+      { to = to
+      ; cong = to-cong
+      }
+
+    open Func toFunction public
+      using (module Eq₁; module Eq₂)
+      renaming (isCongruent to to-isCongrunet)
+
+    fromFunction : Func To From
+    fromFunction = record
+      { to = from
+      ; cong = from-cong
+      }
+
+    open Func fromFunction public
+      using ()
+      renaming (isCongruent to from-isCongrunet)
 
 
   record LeftInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
@@ -171,7 +199,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       to        : A → B
       from      : B → A
       to-cong   : Congruent _≈₁_ _≈₂_ to
-      from-cong : from Preserves _≈₂_ ⟶ _≈₁_
+      from-cong : Congruent _≈₂_ _≈₁_ from
       inverseˡ  : Inverseˡ _≈₁_ _≈₂_ to from
 
     isCongruent : IsCongruent to
@@ -430,12 +458,16 @@ module _ {A : Set a} {B : Set b} where
     ; inverse   = inv
     }
 
+
   -- Strict variant of the above.
-  mk↔′ : ∀ (to : A → B) (from : B → A) →
+  mk↠ₛ : ∀ {to : A → B} → StrictlySurjective _≡_ to → A ↠ B
+  mk↠ₛ = mk↠ ∘ strictlySurjective⇒surjective
+
+  mk↔ₛ′ : ∀ (to : A → B) (from : B → A) →
           StrictlyInverseˡ _≡_ to from →
           StrictlyInverseʳ _≡_ to from →
           A ↔ B
-  mk↔′ to from invˡ invʳ = mk↔ {to} {from}
+  mk↔ₛ′ to from invˡ invʳ = mk↔ {to} {from}
     ( strictlyInverseˡ⇒inverseˡ to invˡ
     , strictlyInverseʳ⇒inverseʳ to invʳ
     )
diff --git a/src/Function/Dependent/Bundles.agda b/src/Function/Dependent/Bundles.agda
new file mode 100644
index 0000000000..177e2f9b9f
--- /dev/null
+++ b/src/Function/Dependent/Bundles.agda
@@ -0,0 +1,50 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bundles for types of functions
+------------------------------------------------------------------------
+
+-- The contents of this file should usually be accessed from `Function`.
+
+-- Note that these bundles differ from those found elsewhere in other
+-- library hierarchies as they take Setoids as parameters. This is
+-- because a function is of no use without knowing what its domain and
+-- codomain is, as well which equalities are being considered over them.
+-- One consequence of this is that they are not built from the
+-- definitions found in `Function.Structures` as is usually the case in
+-- other library hierarchies, as this would duplicate the equality
+-- axioms.
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Dependent.Bundles where
+
+open import Level using (Level; _⊔_)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)
+
+private
+  variable
+    a b ℓ₁ ℓ₂ : Level
+
+------------------------------------------------------------------------
+-- Setoid bundles
+------------------------------------------------------------------------
+
+module _
+  (From : Setoid a ℓ₁)
+  (To : IndexedSetoid (Setoid.Carrier From) b ℓ₂)
+  where
+
+  open Setoid From using () renaming (Carrier to A; _≈_ to _≈₁_)
+  open IndexedSetoid To using () renaming (Carrier to B; _≈_ to _≈₂_)
+
+------------------------------------------------------------------------
+-- Bundles with one element
+
+  -- Called `Func` rather than `Function` in order to avoid clashing
+  -- with the top-level module.
+  record Func : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+    field
+      to   : (x : A) → B x
+      cong : ∀ {x y} → x ≈₁ y → to x ≈₂ to y
diff --git a/src/Function/Equality.agda b/src/Function/Equality.agda
index 91eeb7ef73..4ce068b45c 100644
--- a/src/Function/Equality.agda
+++ b/src/Function/Equality.agda
@@ -1,18 +1,18 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Function setoids and related constructions
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
--- Note: use of the standard function hierarchy is encouraged. The
--- module `Function` re-exports `Congruent`, `IsBijection` and
--- `Bijection`. The alternative definitions found in this file will
--- eventually be deprecated.
-
 module Function.Equality where
 
+{-# WARNING_ON_IMPORT
+"Function.Equality was deprecated in v2.0.
+Use the standard function hierarchy in Function/Function.Bundles instead."
+#-}
+
 import Function.Base as Fun
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Bundles using (Setoid)
@@ -20,6 +20,8 @@ open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid; _=[_]⇒_)
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   as Trivial
+import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Properties as P
 
 ------------------------------------------------------------------------
 -- Functions which preserve equality
@@ -122,3 +124,10 @@ flip {B = B} f = record
     ; cong  = λ a₁≈a₂ → cong f a₁≈a₂ (Setoid.refl B) }
   ; cong  = λ b₁≈b₂ a₁≈a₂ → cong f a₁≈a₂ b₁≈b₂
   }
+
+→-to-⟶ : ∀ {a b ℓ} {A : Set a} {B : Setoid b ℓ} →
+         (A → Setoid.Carrier B) → P.setoid A ⟶ B
+→-to-⟶ {B = B} to = record
+  { _⟨$⟩_ = to
+  ; cong = λ { P.refl → Setoid.refl B }
+  }
diff --git a/src/Function/Equivalence.agda b/src/Function/Equivalence.agda
index 34ab390682..b25f2a062f 100644
--- a/src/Function/Equivalence.agda
+++ b/src/Function/Equivalence.agda
@@ -1,21 +1,22 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Equivalence (coinhabitance)
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.Equivalence where
 
--- Note: use of the standard function hierarchy is encouraged. The
--- module `Function` re-exports `Congruent` and `IsCongruent`.
--- The alternative definitions found in this file will eventually be
--- deprecated.
+{-# WARNING_ON_IMPORT
+"Function.Equivalence was deprecated in v2.0.
+Use the standard function hierarchy in Function/Function.Bundles instead."
+#-}
 
 open import Function.Base using (flip)
 open import Function.Equality as F
-  using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
+  using (_⟶_; _⟨$⟩_; →-to-⟶) renaming (_∘_ to _⟪∘⟫_)
 open import Level
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (Reflexive; TransFlip; Sym)
@@ -43,8 +44,8 @@ From ⇔ To = Equivalence (P.setoid From) (P.setoid To)
 equivalence : ∀ {f t} {From : Set f} {To : Set t} →
               (From → To) → (To → From) → From ⇔ To
 equivalence to from = record
-  { to   = P.→-to-⟶ to
-  ; from = P.→-to-⟶ from
+  { to   = →-to-⟶ to
+  ; from = →-to-⟶ from
   }
 
 ------------------------------------------------------------------------
diff --git a/src/Function/HalfAdjointEquivalence.agda b/src/Function/HalfAdjointEquivalence.agda
index 59b11d5f69..b78c1bc139 100644
--- a/src/Function/HalfAdjointEquivalence.agda
+++ b/src/Function/HalfAdjointEquivalence.agda
@@ -1,13 +1,19 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Half adjoint equivalences
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.HalfAdjointEquivalence where
 
+{-# WARNING_ON_IMPORT
+"Function.HalfAdjointEquivalence was deprecated in v2.0.
+Use Function.Properties.Inverse.HalfAdjointEquivalence instead."
+#-}
+
 open import Function.Base
 open import Function.Equality using (_⟨$⟩_)
 open import Function.Inverse as Inv using (_↔_; module Inverse)
diff --git a/src/Function/Indexed/Bundles.agda b/src/Function/Indexed/Bundles.agda
index 6cce0a3026..682f19e634 100644
--- a/src/Function/Indexed/Bundles.agda
+++ b/src/Function/Indexed/Bundles.agda
@@ -15,36 +15,38 @@ open import Level using (Level)
 
 private
   variable
-    a b ℓ₁ ℓ₂ : Level
+    a b i : Level
+    I : Set i
 
 ------------------------------------------------------------------------
 -- Bundles specialised for lifting relations to indexed sets
 ------------------------------------------------------------------------
 
 infix 3 _⟶ᵢ_ _↣ᵢ_ _↠ᵢ_ _⤖ᵢ_ _⇔ᵢ_ _↩ᵢ_ _↪ᵢ_ _↩↪ᵢ_ _↔ᵢ_
-_⟶ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+
+_⟶ᵢ_ : Pred I a → Pred I b → Set _
 A ⟶ᵢ B = ∀ {i} → A i ⟶ B i
 
-_↣ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_↣ᵢ_ : Pred I a → Pred I b → Set _
 A ↣ᵢ B = ∀ {i} → A i ↣ B i
 
-_↠ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_↠ᵢ_ : Pred I a → Pred I b → Set _
 A ↠ᵢ B = ∀ {i} → A i ↠ B i
 
-_⤖ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_⤖ᵢ_ : Pred I a → Pred I b → Set _
 A ⤖ᵢ B = ∀ {i} → A i ⤖ B i
 
-_⇔ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_⇔ᵢ_ : Pred I a → Pred I b → Set _
 A ⇔ᵢ B = ∀ {i} → A i ⇔ B i
 
-_↩ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_↩ᵢ_ : Pred I a → Pred I b → Set _
 A ↩ᵢ B = ∀ {i} → A i ↩ B i
 
-_↪ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_↪ᵢ_ : Pred I a → Pred I b → Set _
 A ↪ᵢ B = ∀ {i} → A i ↪ B i
 
-_↩↪ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_↩↪ᵢ_ : Pred I a → Pred I b → Set _
 A ↩↪ᵢ B = ∀ {i} → A i ↩↪ B i
 
-_↔ᵢ_ : ∀ {i} {I : Set i} → Pred I a → Pred I b → Set _
+_↔ᵢ_ : Pred I a → Pred I b → Set _
 A ↔ᵢ B = ∀ {i} → A i ↔ B i
diff --git a/src/Function/Injection.agda b/src/Function/Injection.agda
index 9bd6277fd2..910c91aa35 100644
--- a/src/Function/Injection.agda
+++ b/src/Function/Injection.agda
@@ -1,10 +1,11 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Injections
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
+{-# OPTIONS --warn=noUserWarning #-}
 
 -- Note: use of the standard function hierarchy is encouraged. The
 -- module `Function` re-exports `Injective`, `IsInjection` and
@@ -13,6 +14,11 @@
 
 module Function.Injection where
 
+{-# WARNING_ON_IMPORT
+"Function.Injection was deprecated in v2.0.
+Use the standard function hierarchy in Function/Function.Bundles instead."
+#-}
+
 open import Function.Base as Fun using () renaming (_∘_ to _⟨∘⟩_)
 open import Level
 open import Relation.Binary.Bundles using (Setoid)
@@ -54,7 +60,10 @@ From ↣ To = Injection (P.setoid From) (P.setoid To)
 injection : ∀ {f t} {From : Set f} {To : Set t} → (to : From → To) →
             (∀ {x y} → to x ≡ to y → x ≡ y) → From ↣ To
 injection to injective = record
-  { to        = P.→-to-⟶ to
+  { to        = record
+    { _⟨$⟩_ = to
+    ; cong = P.cong to
+    }
   ; injective = injective
   }
 
diff --git a/src/Function/Inverse.agda b/src/Function/Inverse.agda
index 87632836db..e2cb395ea5 100644
--- a/src/Function/Inverse.agda
+++ b/src/Function/Inverse.agda
@@ -1,18 +1,19 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Inverses
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
-
--- Note: use of the standard function hierarchy is encouraged. The
--- module `Function` re-exports `Inverseᵇ`, `IsInverse` and
--- `Inverse`. The alternative definitions found in this file will
--- eventually be deprecated.
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.Inverse where
 
+{-# WARNING_ON_IMPORT
+"Function.Inverse was deprecated in v2.0.
+Use the standard function hierarchy in Function/Function.Bundles instead."
+#-}
+
 open import Level
 open import Function.Base using (flip)
 open import Function.Bijection hiding (id; _∘_; bijection)
@@ -92,8 +93,8 @@ inverse : ∀ {f t} {From : Set f} {To : Set t} →
           (∀ x → to (from x) ≡ x) →
           From ↔ To
 inverse to from from∘to to∘from = record
-  { to   = P.→-to-⟶ to
-  ; from = P.→-to-⟶ from
+  { to   = F.→-to-⟶ to
+  ; from = F.→-to-⟶ from
   ; inverse-of = record
     { left-inverse-of  = from∘to
     ; right-inverse-of = to∘from
diff --git a/src/Function/LeftInverse.agda b/src/Function/LeftInverse.agda
index 6f2c8d0db5..7ff0fda218 100644
--- a/src/Function/LeftInverse.agda
+++ b/src/Function/LeftInverse.agda
@@ -1,18 +1,19 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Left inverses
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
-
--- Note: use of the standard function hierarchy is encouraged. The
--- module `Function` re-exports `Inverseˡ`, `IsLeftInverse` and
--- `LeftInverse`. The alternative definitions found in this file will
--- eventually be deprecated.
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.LeftInverse where
 
+{-# WARNING_ON_IMPORT
+"Function.LeftInverse was deprecated in v2.0.
+Use the standard function hierarchy in Function/Function.Bundles instead."
+#-}
+
 open import Level
 import Relation.Binary.Reasoning.Setoid as EqReasoning
 open import Relation.Binary.Bundles using (Setoid)
@@ -96,8 +97,8 @@ leftInverse : ∀ {f t} {From : Set f} {To : Set t} →
               (∀ x → from (to x) ≡ x) →
               From ↞ To
 leftInverse to from invˡ = record
-  { to              = P.→-to-⟶ to
-  ; from            = P.→-to-⟶ from
+  { to              = Eq.→-to-⟶ to
+  ; from            = Eq.→-to-⟶ from
   ; left-inverse-of = invˡ
   }
 
diff --git a/src/Function/Properties.agda b/src/Function/Properties.agda
index 69ae3506c1..08692108bc 100644
--- a/src/Function/Properties.agda
+++ b/src/Function/Properties.agda
@@ -10,7 +10,7 @@ module Function.Properties where
 
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Function.Base using (flip; _∘_)
-open import Function.Bundles using (_↔_; mk↔′; Inverse)
+open import Function.Bundles using (_↔_; mk↔ₛ′; Inverse)
 open import Level
 open import Relation.Binary.PropositionalEquality.Core
   using (trans; cong; cong′)
@@ -25,14 +25,14 @@ private
 -- Implicit and explicit function spaces are isomorphic
 
 Π↔Π : {B : A → Set b} → ((x : A) → B x) ↔ ({x : A} → B x)
-Π↔Π = mk↔′ (λ f {x} → f x) (λ f _ → f) cong′ cong′
+Π↔Π = mk↔ₛ′ (λ f {x} → f x) (λ f _ → f) cong′ cong′
 
 ------------------------------------------------------------------------
 -- Function spaces can be reordered
 
 ΠΠ↔ΠΠ : (R : A → B → Set p) →
         ((x : A) (y : B) → R x y) ↔ ((y : B) (x : A) → R x y)
-ΠΠ↔ΠΠ _ = mk↔′ flip flip cong′ cong′
+ΠΠ↔ΠΠ _ = mk↔ₛ′ flip flip cong′ cong′
 
 ------------------------------------------------------------------------
 -- Assuming extensionality then → preserves ↔
@@ -40,7 +40,7 @@ private
 →-cong-↔ : {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
          Extensionality a c → Extensionality b d →
          A ↔ B → C ↔ D → (A → C) ↔ (B → D)
-→-cong-↔ extAC extBD A↔B C↔D = mk↔′
+→-cong-↔ extAC extBD A↔B C↔D = mk↔ₛ′
   (λ h → C↔D.to   ∘ h ∘ A↔B.from)
   (λ g → C↔D.from ∘ g ∘ A↔B.to  )
   (λ h → extBD λ x → trans (C↔D.strictlyInverseˡ _) (cong h (A↔B.strictlyInverseˡ x)))
diff --git a/src/Function/Properties/Equivalence.agda b/src/Function/Properties/Equivalence.agda
index beaa7d8940..308a29525d 100644
--- a/src/Function/Properties/Equivalence.agda
+++ b/src/Function/Properties/Equivalence.agda
@@ -9,7 +9,7 @@
 
 module Function.Properties.Equivalence where
 
-open import Function.Bundles using (Equivalence; _⇔_)
+open import Function.Bundles
 open import Level
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
@@ -22,6 +22,19 @@ import Function.Construct.Composition as Composition
 private
   variable
     a ℓ : Level
+    A B : Set a
+    S T : Setoid a ℓ
+
+------------------------------------------------------------------------
+-- Constructors
+
+mkEquivalence : Func S T → Func T S → Equivalence S T
+mkEquivalence f g = record
+  { to = to f
+  ; from = to g
+  ; to-cong = cong f
+  ; from-cong = cong g
+  } where open Func
 
 ------------------------------------------------------------------------
 -- Setoid bundles
@@ -33,6 +46,13 @@ isEquivalence = record
   ; trans = Composition.equivalence
   }
 
+setoid : (s₁ s₂ : Level) → Setoid (suc (s₁ ⊔ s₂)) (s₁ ⊔ s₂)
+setoid s₁ s₂ = record
+  { Carrier       = Setoid s₁ s₂
+  ; _≈_           = Equivalence
+  ; isEquivalence = isEquivalence
+  }
+
 ------------------------------------------------------------------------
 -- Propositional bundles
 
@@ -43,20 +63,18 @@ isEquivalence = record
   ; trans = Composition.equivalence
   }
 
-
-------------------------------------------------------------------------
--- Setoids
-
-setoid : (s₁ s₂ : Level) → Setoid (suc (s₁ ⊔ s₂)) (s₁ ⊔ s₂)
-setoid s₁ s₂ = record
-  { Carrier       = Setoid s₁ s₂
-  ; _≈_           = Equivalence
-  ; isEquivalence = isEquivalence
-  }
-
 ⇔-setoid : (ℓ : Level) → Setoid (suc ℓ) ℓ
 ⇔-setoid ℓ = record
   { Carrier       = Set ℓ
   ; _≈_           = _⇔_
   ; isEquivalence = ⇔-isEquivalence
   }
+
+⟶×⟵⇒⇔ : A ⟶ B → B ⟶ A → A ⇔ B
+⟶×⟵⇒⇔ = mkEquivalence
+
+⇔⇒⟶ : A ⇔ B → A ⟶ B
+⇔⇒⟶ = Equivalence.toFunction
+
+⇔⇒⟵ : A ⇔ B → B ⟶ A
+⇔⇒⟵ = Equivalence.fromFunction
diff --git a/src/Function/Properties/Injection.agda b/src/Function/Properties/Injection.agda
new file mode 100644
index 0000000000..bca8ac4dc5
--- /dev/null
+++ b/src/Function/Properties/Injection.agda
@@ -0,0 +1,52 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties for injections
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Properties.Injection where
+
+open import Function.Base
+open import Function.Definitions
+open import Function.Bundles
+import Function.Construct.Identity as Identity
+import Function.Construct.Composition as Compose
+open import Level using (Level)
+open import Data.Product.Base using (proj₁; proj₂)
+open import Relation.Binary.PropositionalEquality
+open import Relation.Binary using (Setoid)
+import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+
+private
+  variable
+    a b ℓ : Level
+    A B : Set a
+    T S U : Setoid a ℓ
+
+------------------------------------------------------------------------
+-- Constructors
+
+mkInjection : (f : Func S T) (open Func f) →
+              Injective Eq₁._≈_ Eq₂._≈_ to  →
+              Injection S T
+mkInjection f injective = record
+  { Func f
+  ; injective = injective
+  }
+
+------------------------------------------------------------------------
+-- Conversion functions
+
+↣⇒⟶ : A ↣ B → A ⟶ B
+↣⇒⟶ = Injection.function
+
+------------------------------------------------------------------------
+-- Properties
+
+↣-refl : Injection S S
+↣-refl = Identity.injection _
+
+↣-trans : Injection S T → Injection T U → Injection S U
+↣-trans = Compose.injection
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index d430b36398..e25f010a3c 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -1,8 +1,7 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Some functional properties are Equivalence Relations
---   This file is meant to be imported qualified.
+-- Properties of inverses.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -12,10 +11,12 @@ module Function.Properties.Inverse where
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Bundles
-open import Level using (Level)
+import Function.Properties.RightInverse as RightInverse
+open import Level using (Level; _⊔_)
+open import Relation.Binary.Core using (REL)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
-import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 open import Function.Consequences
@@ -28,35 +29,53 @@ private
   variable
     a b ℓ ℓ₁ ℓ₂ : Level
     A B C D : Set a
-    S T : Setoid a ℓ
+    S T U V : Setoid a ℓ
 
 ------------------------------------------------------------------------
 -- Setoid bundles
 
+open Identity    public using () renaming (inverse to refl)
+open Symmetry    public using () renaming (inverse to sym)
+open Composition public using () renaming (inverse to trans)
+
 isEquivalence : IsEquivalence (Inverse {a} {b})
 isEquivalence = record
   { refl  = λ {x} → Identity.inverse x
-  ; sym   = Symmetry.inverse
-  ; trans = Composition.inverse
+  ; sym   = sym
+  ; trans = trans
   }
 
 ------------------------------------------------------------------------
 -- Propositional bundles
 
+↔-refl : A ↔ A
+↔-refl = Identity.↔-id _
+
+↔-sym : A ↔ B → B ↔ A
+↔-sym = Symmetry.↔-sym
+
+↔-trans : A ↔ B → B ↔ C → A ↔ C
+↔-trans = Composition._↔-∘_
+
 -- need to η-expand for everything to line up properly
 ↔-isEquivalence : IsEquivalence {ℓ = ℓ} _↔_
 ↔-isEquivalence = record
-  { refl  = λ {x} → Identity.inverse (P.setoid x)
-  ; sym   = Symmetry.inverse
-  ; trans = Composition.inverse
+  { refl  = ↔-refl
+  ; sym   = ↔-sym
+  ; trans = ↔-trans
   }
 
-open module ↔ {ℓ} = IsEquivalence (↔-isEquivalence {ℓ}) using ()
-  renaming (refl to ↔-refl; sym to ↔-sym; trans to ↔-trans) public
-
 ------------------------------------------------------------------------
 -- Conversion functions
 
+toFunction : Inverse S T → Func S T
+toFunction I = record { to = to ; cong = to-cong }
+  where open Inverse I
+
+fromFunction : Inverse S T → Func T S
+fromFunction I = record { to = from ; cong = from-cong }
+  where open Inverse I
+
 Inverse⇒Injection : Inverse S T → Injection S T
 Inverse⇒Injection I = record
   { to = to
@@ -64,6 +83,13 @@ Inverse⇒Injection I = record
   ; injective = inverseʳ⇒injective Eq₂._≈_ to Eq₂.refl Eq₁.sym Eq₁.trans inverseʳ
   } where open Inverse I
 
+Inverse⇒Surjection : Inverse S T → Surjection S T
+Inverse⇒Surjection {S = S} {T = T} I = record
+  { to = to
+  ; cong = to-cong
+  ; surjective = inverseˡ⇒surjective (_≈_ T) inverseˡ
+  } where open Inverse I; open Setoid
+
 Inverse⇒Bijection : Inverse S T → Bijection S T
 Inverse⇒Bijection I = record
   { to        = to
@@ -79,21 +105,55 @@ Inverse⇒Equivalence I = record
   ; from-cong = from-cong
   } where open Inverse I
 
+↔⇒⟶ : A ↔ B → A ⟶ B
+↔⇒⟶ = toFunction
+
+↔⇒⟵ : A ↔ B → B ⟶ A
+↔⇒⟵ = fromFunction
+
 ↔⇒↣ : A ↔ B → A ↣ B
 ↔⇒↣ = Inverse⇒Injection
 
+↔⇒↠ : A ↔ B → A ↠ B
+↔⇒↠ = Inverse⇒Surjection
+
 ↔⇒⤖ : A ↔ B → A ⤖ B
 ↔⇒⤖ = Inverse⇒Bijection
 
 ↔⇒⇔ : A ↔ B → A ⇔ B
 ↔⇒⇔ = Inverse⇒Equivalence
 
+↔⇒↩ : A ↔ B → A ↩ B
+↔⇒↩ = Inverse.leftInverse
+
+↔⇒↪ : A ↔ B → A ↪ B
+↔⇒↪ = Inverse.rightInverse
+
+-- The functions above can be combined with the following lemma to
+-- transport an arbitrary relation R (e.g. Injection) across
+-- inverses.
+transportVia : {R : ∀ {a b ℓ₁ ℓ₂} → REL (Setoid a ℓ₁) (Setoid b ℓ₂) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)} →
+               (∀ {a b c ℓ₁ ℓ₂ ℓ₃} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} {U : Setoid c ℓ₃} → R S T → R T U → R S U) →
+               (∀ {a b ℓ₁ ℓ₂} {S : Setoid a ℓ₁} {T : Setoid b ℓ₂} → Inverse S T → R S T) →
+               Inverse S T → R T U → Inverse U V → R S V
+transportVia R-trans inv⇒R IBA RBC ICD =
+  R-trans (inv⇒R IBA) (R-trans RBC (inv⇒R ICD))
+
+------------------------------------------------------------------------
+-- Other
+
 module _ (ext : ∀ {a b} → Extensionality a b) where
 
   ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)
-  ↔-fun A↔B C↔D = mk↔′
+  ↔-fun A↔B C↔D = mk↔ₛ′
     (λ a→c b → to C↔D (a→c (from A↔B b)))
     (λ b→d a → from C↔D (b→d (to A↔B a)))
     (λ b→d → ext λ _ → P.trans (strictlyInverseˡ C↔D _ ) (P.cong b→d (strictlyInverseˡ A↔B _)))
     (λ a→c → ext λ _ → P.trans (strictlyInverseʳ C↔D _ ) (P.cong a→c (strictlyInverseʳ A↔B _)))
     where open Inverse
+
+module _ (I : Inverse S T) where
+  open Inverse I
+
+  to-from : ∀ {x y} → to x Eq₂.≈ y → from y Eq₁.≈ x
+  to-from = RightInverse.to-from rightInverse
diff --git a/src/Function/Properties/Inverse/HalfAdjointEquivalence.agda b/src/Function/Properties/Inverse/HalfAdjointEquivalence.agda
new file mode 100644
index 0000000000..4ea2965f99
--- /dev/null
+++ b/src/Function/Properties/Inverse/HalfAdjointEquivalence.agda
@@ -0,0 +1,121 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Half adjoint equivalences
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Properties.Inverse.HalfAdjointEquivalence where
+
+open import Function.Base
+open import Function
+open import Level
+open import Relation.Binary.PropositionalEquality
+
+private
+  variable
+    a b : Level
+    A B : Set a
+
+-- Half adjoint equivalences (see the HoTT book).
+--
+-- They are inverses with an extra coherence condition that the left
+-- and right inversion proofs interact the right way with `cong`.
+
+infix 4 _≃_
+
+record _≃_ (A : Set a) (B : Set b) : Set (a ⊔ b) where
+  field
+    to               : A → B
+    from             : B → A
+    left-inverse-of  : ∀ x → from (to x) ≡ x
+    right-inverse-of : ∀ x → to (from x) ≡ x
+    left-right       : ∀ x → cong to (left-inverse-of x) ≡ right-inverse-of (to x)
+
+  -- The forward direction of a half adjoint equivalence is injective.
+
+  injective : ∀ {x y} → to x ≡ to y → x ≡ y
+  injective {x} {y} to-x≡to-y =
+    x            ≡⟨ sym (left-inverse-of _) ⟩
+    from (to x)  ≡⟨ cong from to-x≡to-y ⟩
+    from (to y)  ≡⟨ left-inverse-of _ ⟩
+    y            ∎
+    where open ≡-Reasoning
+
+-- Half adjoint equivalences can be turned into inverses.
+
+≃⇒↔ : A ≃ B → A ↔ B
+≃⇒↔ A≃B = mk↔ₛ′ to from right-inverse-of left-inverse-of
+  where open _≃_ A≃B
+
+-- Inverses can be turned into half adjoint equivalences.
+--
+-- (This proof is based on one in the HoTT book.)
+
+↔⇒≃ : A ↔ B → A ≃ B
+↔⇒≃ A↔B = record
+  { to               = to
+  ; from             = from
+  ; left-inverse-of  = strictlyInverseʳ
+  ; right-inverse-of = right-inverse-of
+  ; left-right       = left-right
+  }
+  where
+  open ≡-Reasoning
+  open module A↔B = Inverse A↔B
+
+  right-inverse-of : ∀ x → to (from x) ≡ x
+  right-inverse-of x =
+    to (from x)               ≡⟨ sym (A↔B.strictlyInverseˡ _) ⟩
+    to (from (to (from x)))   ≡⟨ cong to (strictlyInverseʳ  _) ⟩
+    to (from x)               ≡⟨ A↔B.strictlyInverseˡ _ ⟩
+    x                         ∎
+
+  left-right :
+    ∀ x →
+    cong to (strictlyInverseʳ x) ≡ right-inverse-of (to x)
+  left-right x =
+    cong to (strictlyInverseʳ x)               ≡⟨⟩
+
+    trans refl (cong to (strictlyInverseʳ _))  ≡⟨ cong (λ p → trans p (cong to (strictlyInverseʳ _)))
+                                                          (sym (trans-symˡ (A↔B.strictlyInverseˡ _))) ⟩
+    trans (trans (sym (A↔B.strictlyInverseˡ _))
+               (A↔B.strictlyInverseˡ _))
+      (cong to (strictlyInverseʳ _))           ≡⟨ trans-assoc (sym (A↔B.strictlyInverseˡ _)) ⟩
+
+    trans (sym (A↔B.strictlyInverseˡ _))
+      (trans (A↔B.strictlyInverseˡ _)
+         (cong to (strictlyInverseʳ _)))       ≡⟨ cong (trans (sym (A↔B.strictlyInverseˡ _))) lemma ⟩
+
+    trans (sym (A↔B.strictlyInverseˡ _))
+      (trans (cong to (strictlyInverseʳ _))
+         (trans (A↔B.strictlyInverseˡ _) refl))      ≡⟨⟩
+
+    right-inverse-of (to x)                      ∎
+    where
+    lemma =
+      trans (A↔B.strictlyInverseˡ _)
+        (cong to (strictlyInverseʳ _))             ≡⟨ cong (trans (A↔B.strictlyInverseˡ _)) (sym (cong-id _)) ⟩
+
+      trans (A↔B.strictlyInverseˡ _)
+        (cong id (cong to (strictlyInverseʳ _)))   ≡⟨ sym (naturality A↔B.strictlyInverseˡ) ⟩
+
+      trans (cong (to ∘ from)
+                 (cong to (strictlyInverseʳ _)))
+        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (λ p → trans p (A↔B.strictlyInverseˡ _))
+                                                              (sym (cong-∘ _)) ⟩
+      trans (cong (to ∘ from ∘ to)
+                      (strictlyInverseʳ _))
+        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (λ p → trans p (A↔B.strictlyInverseˡ _))
+                                                              (cong-∘ _) ⟩
+      trans (cong to
+                 (cong (from ∘ to)
+                    (strictlyInverseʳ _)))
+        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (λ p → trans (cong to p) (strictlyInverseˡ (to x)))
+                                                              (cong-≡id strictlyInverseʳ) ⟩
+      trans (cong to (strictlyInverseʳ _))
+        (A↔B.strictlyInverseˡ _)                         ≡⟨ cong (trans (cong to (strictlyInverseʳ _)))
+                                                              (sym (trans-reflʳ _)) ⟩
+      trans (cong to (strictlyInverseʳ _))
+        (trans (A↔B.strictlyInverseˡ _) refl)            ∎
diff --git a/src/Function/Properties/RightInverse.agda b/src/Function/Properties/RightInverse.agda
index 681e3e2b5a..ae8bc266de 100644
--- a/src/Function/Properties/RightInverse.agda
+++ b/src/Function/Properties/RightInverse.agda
@@ -1,7 +1,7 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Conversions for right inverses
+-- Properties of right inverses
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -9,6 +9,7 @@
 module Function.Properties.RightInverse where
 
 open import Function.Base
+open import Function.Definitions
 open import Function.Bundles
 open import Function.Consequences using (inverseˡ⇒surjective)
 open import Level using (Level)
@@ -19,10 +20,40 @@ open import Relation.Binary.Structures using (IsEquivalence)
 private
   variable
     ℓ₁ ℓ₂ a b : Level
-    A : Set a
-    B : Set b
-    S : Setoid a ℓ₁
-    T : Setoid b ℓ₂
+    A B : Set a
+    S T : Setoid a ℓ₁
+
+------------------------------------------------------------------------
+-- Constructors
+
+mkRightInverse : (e : Equivalence S T) (open Equivalence e) →
+                 Inverseʳ Eq₁._≈_ Eq₂._≈_ to from →
+                 RightInverse S T
+mkRightInverse eq invʳ = record
+  { Equivalence eq
+  ; inverseʳ = invʳ
+  }
+
+------------------------------------------------------------------------
+-- Conversion
+
+RightInverse⇒LeftInverse : RightInverse S T → LeftInverse T S
+RightInverse⇒LeftInverse I = record
+  { to         = from
+  ; from       = to
+  ; to-cong    = from-cong
+  ; from-cong  = to-cong
+  ; inverseˡ   = inverseʳ
+  } where open RightInverse I
+
+LeftInverse⇒RightInverse : LeftInverse S T → RightInverse T S
+LeftInverse⇒RightInverse I = record
+  { to         = from
+  ; from       = to
+  ; to-cong    = from-cong
+  ; from-cong  = to-cong
+  ; inverseʳ    = inverseˡ
+  } where open LeftInverse I
 
 RightInverse⇒Surjection : RightInverse S T → Surjection T S
 RightInverse⇒Surjection I = record
@@ -33,3 +64,18 @@ RightInverse⇒Surjection I = record
 
 ↪⇒↠ : B ↪ A → A ↠ B
 ↪⇒↠ = RightInverse⇒Surjection
+
+↪⇒↩ : B ↪ A → A ↩ B
+↪⇒↩ = RightInverse⇒LeftInverse
+
+↩⇒↪ : B ↩ A → A ↪ B
+↩⇒↪ = LeftInverse⇒RightInverse
+
+------------------------------------------------------------------------
+-- Other
+
+module _ (R : RightInverse S T) where
+  open RightInverse R
+
+  to-from : ∀ {x y} → to x Eq₂.≈ y → from y Eq₁.≈ x
+  to-from eq = Eq₁.trans (from-cong (Eq₂.sym eq)) (strictlyInverseʳ _)
diff --git a/src/Function/Properties/Surjection.agda b/src/Function/Properties/Surjection.agda
index e0fccc598b..d16670639c 100644
--- a/src/Function/Properties/Surjection.agda
+++ b/src/Function/Properties/Surjection.agda
@@ -1,7 +1,7 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Conversions for surjections
+-- Properties of surjections
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -23,6 +23,34 @@ private
     A B : Set a
     T S : Setoid a ℓ
 
+------------------------------------------------------------------------
+-- Constructors
+
+mkSurjection : (f : Func S T) (open Func f) →
+              Surjective Eq₁._≈_ Eq₂._≈_ to  →
+              Surjection S T
+mkSurjection f surjective = record
+  { Func f
+  ; surjective = surjective
+  }
+
+------------------------------------------------------------------------
+-- Conversion functions
+
+↠⇒⟶ : A ↠ B → A ⟶ B
+↠⇒⟶ = Surjection.function
+
+↠⇒↪ : A ↠ B → B ↪ A
+↠⇒↪ s = mk↪ {from = to} λ { refl → proj₂ (strictlySurjective _)}
+  where open Surjection s
+
+↠⇒⇔ : A ↠ B → A ⇔ B
+↠⇒⇔ s = mk⇔ to (proj₁ ∘ surjective)
+  where open Surjection s
+
+------------------------------------------------------------------------
+-- Other
+
 injective⇒to⁻-cong : (surj : Surjection S T) →
                       (open Surjection surj) →
                       Injective Eq₁._≈_ Eq₂._≈_ to →
@@ -35,15 +63,4 @@ injective⇒to⁻-cong {T = T} surj injective {x} {y} x≈y = injective $ begin
   where
   open SetoidReasoning T
   open Surjection surj
-  to∘to⁻ = proj₂ ∘ strictlySurjective
 
-------------------------------------------------------------------------
--- Conversion functions
-
-↠⇒↪ : A ↠ B → B ↪ A
-↠⇒↪ s = mk↪ {from = to} λ { refl → proj₂ (strictlySurjective _)}
-  where open Surjection s
-
-↠⇒⇔ : A ↠ B → A ⇔ B
-↠⇒⇔ s = mk⇔ to (proj₁ ∘ surjective)
-  where open Surjection s
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index aa661c9cb8..ce67d9580b 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -1,14 +1,19 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- A universe which includes several kinds of "relatedness" for sets,
--- such as equivalences, surjections and bijections
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.Related where
 
+{-# WARNING_ON_IMPORT
+"Function.Related was deprecated in v2.0.
+Use Function.Related.Propositional instead."
+#-}
+
 open import Level
 open import Function.Base
 open import Function.Equality using (_⟨$⟩_)
@@ -92,6 +97,11 @@ A ∼[ left-inverse        ] B = LeftInverse (P.setoid A) (P.setoid B)
 A ∼[ surjection          ] B = Surjection  (P.setoid A) (P.setoid B)
 A ∼[ bijection           ] B = Inverse     (P.setoid A) (P.setoid B)
 
+-- A non-infix synonym.
+
+Related : Kind → ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Set ℓ₂ → Set _
+Related k A B = A ∼[ k ] B
+
 toRelated : {K : Kind} → A R.∼[ K ] B → A ∼[ K ] B
 toRelated {K = implication}         rel = B.Func.to rel
 toRelated {K = reverse-implication} rel = lam (B.Func.to rel)
@@ -102,8 +112,8 @@ toRelated {K = left-inverse}        rel = LeftInv.leftInverse to from strictlyIn
 toRelated {K = surjection}          rel = Surj.surjection to (proj₁ ∘ strictlySurjective) (proj₂ ∘ strictlySurjective)
   where open B.Surjection rel
 toRelated {K = bijection}           rel =
-  Inv.inverse to (proj₁ ∘ strictlySurjective) (injective ∘ proj₂ ∘ strictlySurjective ∘ to) (proj₂ ∘ strictlySurjective)
-  where open B.Bijection rel
+             Inv.inverse to from strictlyInverseʳ strictlyInverseˡ
+  where open B.Inverse rel
 
 fromRelated : {K : Kind} → A ∼[ K ] B → A R.∼[ K ] B
 fromRelated {K = implication}         rel = B.mk⟶ rel
@@ -116,15 +126,8 @@ fromRelated {K = left-inverse}        record { to = to ; from = from ; left-inve
 fromRelated {K = surjection}          record { to = to ; surjective = surjective } with surjective
 ... | record { from = from ; right-inverse-of = right-inverse-of } =
   B.mk↠ {to = to ⟨$⟩_} < from ⟨$⟩_ , (λ { x P.refl → right-inverse-of x }) >
-fromRelated {K = bijection}           record { to = to ; from = from ; inverse-of = inverse-of } with inverse-of
-... | record { left-inverse-of = left-inverse-of ; right-inverse-of = right-inverse-of } = B.mk⤖
-  ((λ {x y} h → P.subst₂ P._≡_ (left-inverse-of x) (left-inverse-of y) (P.cong (from ⟨$⟩_) h)) ,
-  < from ⟨$⟩_ , (λ { x P.refl → right-inverse-of x }) >)
-
--- A non-infix synonym.
-
-Related : Kind → ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Set ℓ₂ → Set _
-Related k A B = A ∼[ k ] B
+fromRelated {K = bijection}           rel = B.mk↔ₛ′ (to ⟨$⟩_) (from ⟨$⟩_) right-inverse-of left-inverse-of
+  where open Inverse rel
 
 -- The bijective equality implies any kind of relatedness.
 
diff --git a/src/Function/Related/Propositional.agda b/src/Function/Related/Propositional.agda
index 2c84095c12..2eb6aa91c9 100644
--- a/src/Function/Related/Propositional.agda
+++ b/src/Function/Related/Propositional.agda
@@ -19,7 +19,7 @@ import Relation.Binary.PropositionalEquality.Properties as P
 open import Function.Properties.Surjection   using (↠⇒↪; ↠⇒⇔)
 open import Function.Properties.RightInverse using (↪⇒↠)
 open import Function.Properties.Bijection    using (⤖⇒↔; ⤖⇒⇔)
-open import Function.Properties.Inverse      using (↔⇒⤖)
+open import Function.Properties.Inverse      using (↔⇒⤖; ↔⇒⇔; ↔⇒↣; ↔⇒↠)
 
 import Function.Construct.Symmetry    as Symmetry
 import Function.Construct.Identity    as Identity
@@ -46,9 +46,7 @@ data Kind : Set where
 private
   variable
     a b c p : Level
-    A : Set a
-    B : Set b
-    C : Set c
+    A B C : Set a
     k : Kind
 
 -- Interpretation of the codes above. The code "bijection" is
@@ -65,7 +63,7 @@ A ∼[ injection          ] B = A ↣ B
 A ∼[ reverseInjection   ] B = B ↣ A
 A ∼[ leftInverse        ] B = A ↪ B
 A ∼[ surjection         ] B = A ↠ B
-A ∼[ bijection          ] B = A ⤖ B
+A ∼[ bijection          ] B = A ↔ B
 
 -- A non-infix synonym.
 
@@ -74,20 +72,20 @@ Related k A B = A ∼[ k ] B
 
 -- The bijective equality implies any kind of relatedness.
 
-⤖⇒ : A ∼[ bijection ] B → A ∼[ k ] B
-⤖⇒ {k = implication}        = mk⟶ ∘ Bijection.to
-⤖⇒ {k = reverseImplication} = mk⟶ ∘ Inverse.from ∘ ⤖⇒↔
-⤖⇒ {k = equivalence}        = ⤖⇒⇔
-⤖⇒ {k = injection}          = Bijection.injection
-⤖⇒ {k = reverseInjection}   = Bijection.injection ∘ ↔⇒⤖ ∘ Symmetry.inverse ∘ ⤖⇒↔
-⤖⇒ {k = leftInverse}        = Inverse.rightInverse ∘ ⤖⇒↔
-⤖⇒ {k = surjection}         = Bijection.surjection
-⤖⇒ {k = bijection}          = id
+↔⇒ : A ∼[ bijection ] B → A ∼[ k ] B
+↔⇒ {k = implication}        = mk⟶ ∘ Inverse.to
+↔⇒ {k = reverseImplication} = mk⟶ ∘ Inverse.from
+↔⇒ {k = equivalence}        = ↔⇒⇔
+↔⇒ {k = injection}          = ↔⇒↣
+↔⇒ {k = reverseInjection}   = ↔⇒↣ ∘ Symmetry.inverse
+↔⇒ {k = leftInverse}        = Inverse.rightInverse
+↔⇒ {k = surjection}         = ↔⇒↠
+↔⇒ {k = bijection}          = id
 
 -- Propositional equality also implies any kind of relatedness.
 
 ≡⇒ : A ≡ B → A ∼[ k ] B
-≡⇒ P.refl = ⤖⇒ (Identity.⤖-id _)
+≡⇒ P.refl = ↔⇒ (Identity.↔-id _)
 
 ------------------------------------------------------------------------
 -- Special kinds of kinds
@@ -135,7 +133,7 @@ data ForwardKind : Set where
 ⇒→ {k = injection}   = Injection.to
 ⇒→ {k = leftInverse} = RightInverse.to
 ⇒→ {k = surjection}  = Surjection.to
-⇒→ {k = bijection}   = Bijection.to
+⇒→ {k = bijection}   = Inverse.to
 
 -- Kinds whose interpretation include a function which "goes backwards".
 
@@ -165,7 +163,7 @@ data BackwardKind : Set where
 ⇒← {k = reverseInjection}   = Injection.to
 ⇒← {k = leftInverse}        = RightInverse.from
 ⇒← {k = surjection}         = RightInverse.to ∘ ↠⇒↪
-⇒← {k = bijection}          = Inverse.from ∘ ⤖⇒↔
+⇒← {k = bijection}          = Inverse.from
 
 -- Kinds whose interpretation include functions going in both
 -- directions.
@@ -190,7 +188,7 @@ data EquivalenceKind : Set where
 ⇒⇔ {k = equivalence} = id
 ⇒⇔ {k = leftInverse} = RightInverse.equivalence
 ⇒⇔ {k = surjection}  = ↠⇒⇔
-⇒⇔ {k = bijection}   = ⤖⇒⇔
+⇒⇔ {k = bijection}   = ↔⇒⇔
 
 -- Conversions between special kinds.
 
@@ -236,7 +234,7 @@ reverse {k = injection}          = id
 reverse {k = reverseInjection}   = id
 reverse {k = leftInverse}        = ↪⇒↠
 reverse {k = surjection}         = ↠⇒↪
-reverse {k = bijection}          = ↔⇒⤖ ∘ Symmetry.↔-sym ∘ ⤖⇒↔
+reverse {k = bijection}          = Symmetry.↔-sym
 
 ------------------------------------------------------------------------
 -- For a fixed universe level every kind is a preorder and each
@@ -250,7 +248,7 @@ K-refl {k = injection}          = Identity.↣-id _
 K-refl {k = reverseInjection}   = Identity.↣-id _
 K-refl {k = leftInverse}        = Identity.↪-id _
 K-refl {k = surjection}         = Identity.↠-id _
-K-refl {k = bijection}          = Identity.⤖-id _
+K-refl {k = bijection}          = Identity.↔-id _
 
 K-reflexive : _≡_ Relation.Binary.⇒ Related {a} k
 K-reflexive P.refl = K-refl
@@ -265,7 +263,7 @@ K-trans {k = injection}          = Composition._↣-∘_
 K-trans {k = reverseInjection}   = flip Composition._↣-∘_
 K-trans {k = leftInverse}        = Composition._↪-∘_
 K-trans {k = surjection}         = Composition._↠-∘_
-K-trans {k = bijection}          = Composition._⤖-∘_
+K-trans {k = bijection}          = Composition._↔-∘_
 
 SK-sym : ∀ {k} → Sym (Related {a} {b} ⌊ k ⌋)
                      (Related {b} {a} ⌊ k ⌋)
@@ -282,10 +280,10 @@ SK-isEquivalence k = record
 SK-setoid : SymmetricKind → (ℓ : Level) → Setoid _ _
 SK-setoid k ℓ = record { isEquivalence = SK-isEquivalence {ℓ} k }
 
-K-isPreorder : ∀ k → IsPreorder {ℓ = a} _⤖_ (Related k)
+K-isPreorder : ∀ k → IsPreorder {ℓ = a} _↔_ (Related k)
 K-isPreorder k = record
   { isEquivalence = SK-isEquivalence bijection
-  ; reflexive     = ⤖⇒
+  ; reflexive     = ↔⇒
   ; trans         = K-trans
   }
 
@@ -300,7 +298,11 @@ K-preorder k ℓ = record { isPreorder = K-isPreorder k }
 module EquationalReasoning where
 
   infix  3 _∎
-  infixr 2 _∼⟨_⟩_ _⤖⟨_⟩_ _↔⟨_⟩_ _↔⟨⟩_ _≡⟨_⟩_
+  infixr 2 _∼⟨_⟩_ _⤖⟨_⟩_ _↔⟨_⟩_ _↔⟨⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_
+  infix  1 begin_
+
+  begin_ : ∀ {k} → A ∼[ k ] B → A ∼[ k ] B
+  begin_ x∼y = x∼y
 
   _∼⟨_⟩_ : (A : Set a) → A ∼[ k ] B → B ∼[ k ] C → A ∼[ k ] C
   _ ∼⟨ A↝B ⟩ B↝C = K-trans A↝B B↝C
@@ -309,10 +311,10 @@ module EquationalReasoning where
   -- relatedness.
 
   _⤖⟨_⟩_ : (A : Set a) → A ⤖ B → B ∼[ k ] C → A ∼[ k ] C
-  A ⤖⟨ A⤖B ⟩ B⇔C = A ∼⟨ ⤖⇒ A⤖B ⟩ B⇔C
+  A ⤖⟨ A⤖B ⟩ B⇔C = A ∼⟨ ↔⇒ (⤖⇒↔ A⤖B) ⟩ B⇔C
 
   _↔⟨_⟩_ : (A : Set a) → A ↔ B → B ∼[ k ] C → A ∼[ k ] C
-  A ↔⟨ A↔B ⟩ B⇔C = A ∼⟨ ⤖⇒ (↔⇒⤖ A↔B) ⟩ B⇔C
+  A ↔⟨ A↔B ⟩ B⇔C = A ∼⟨ ↔⇒ A↔B ⟩ B⇔C
 
   _↔⟨⟩_ : (A : Set a) → A ∼[ k ] B → A ∼[ k ] B
   A ↔⟨⟩ A⇔B = A⇔B
@@ -320,6 +322,9 @@ module EquationalReasoning where
   _≡⟨_⟩_ : (A : Set a) → A ≡ B → B ∼[ k ] C → A ∼[ k ] C
   A ≡⟨ A≡B ⟩ B⇔C = A ∼⟨ ≡⇒ A≡B ⟩ B⇔C
 
+  _≡˘⟨_⟩_ : (A : Set a) → B ≡ A → B ∼[ k ] C → A ∼[ k ] C
+  A ≡˘⟨ B≡A ⟩ B⇔C = A ∼⟨ ≡⇒ (P.sym B≡A) ⟩ B⇔C
+
   _∎ : (A : Set a) → A ∼[ k ] A
   A ∎ = K-refl
 
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index c2ba98dce4..830bb6f8bb 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -24,50 +24,52 @@ open import Data.Sum.Function.Propositional using (_⊎-cong_)
 open import Data.Unit.Polymorphic using (⊤)
 open import Level using (Level; Lift; 0ℓ; suc)
 open import Function.Base
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence as Eq using (_⇔_; Equivalence)
-open import Function.Inverse as Inv using (_↔_; Inverse; inverse)
-open import Function.Related
+open import Function.Bundles
+open import Function.Related.Propositional
+import Function.Construct.Identity as Identity
+open import Relation.Binary hiding (_⇔_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ)
 import Relation.Nullary.Indexed as I
 open import Relation.Nullary.Decidable using (True)
 
+private
+  variable
+    a b c d : Level
+    A B C D : Set a
+
 ------------------------------------------------------------------------
 -- Properties of Σ and _×_
 
 -- Σ is associative
-Σ-assoc : ∀ {a b c}
-            {A : Set a} {B : A → Set b} {C : (a : A) → B a → Set c} →
+Σ-assoc : ∀ {A : Set a} {B : A → Set b} {C : (a : A) → B a → Set c} →
           Σ (Σ A B) (uncurry C) ↔ Σ A (λ a → Σ (B a) (C a))
-Σ-assoc = inverse (λ where ((a , b) , c) → (a , b , c))
-                  (λ where (a , b , c) → ((a , b) , c))
-                  (λ _ → P.refl) (λ _ → P.refl)
+Σ-assoc = mk↔ₛ′ Prod.assocʳ Prod.assocˡ (λ _ → P.refl) (λ _ → P.refl)
 
 -- × is commutative
 
-×-comm : ∀ {a b} (A : Set a) (B : Set b) → (A × B) ↔ (B × A)
-×-comm _ _ = inverse Prod.swap Prod.swap (λ _ → P.refl) λ _ → P.refl
+×-comm : ∀ (A : Set a) (B : Set b) → (A × B) ↔ (B × A)
+×-comm _ _ = mk↔ₛ′ Prod.swap Prod.swap (λ _ → P.refl) λ _ → P.refl
 
 -- × has ⊤ as its identity
 
-×-identityˡ : ∀ ℓ → LeftIdentity _↔_ (⊤ {ℓ}) _×_
-×-identityˡ _ _ = inverse proj₂ -,_ (λ _ → P.refl) (λ _ → P.refl)
+×-identityˡ : ∀ ℓ → LeftIdentity  {ℓ = ℓ} _↔_ ⊤ _×_
+×-identityˡ _ _ = mk↔ₛ′ proj₂ -,_ (λ _ → P.refl) (λ _ → P.refl)
 
-×-identityʳ : ∀ ℓ → RightIdentity _↔_ (⊤ {ℓ}) _×_
-×-identityʳ _ _ = inverse proj₁ (_, _) (λ _ → P.refl) (λ _ → P.refl)
+×-identityʳ : ∀ ℓ → RightIdentity  {ℓ = ℓ} _↔_ ⊤ _×_
+×-identityʳ _ _ = mk↔ₛ′ proj₁ (_, _) (λ _ → P.refl) (λ _ → P.refl)
 
 ×-identity : ∀ ℓ → Identity _↔_ ⊤ _×_
 ×-identity ℓ = ×-identityˡ ℓ , ×-identityʳ ℓ
 
 -- × has ⊥ has its zero
 
-×-zeroˡ : ∀ ℓ → LeftZero _↔_ (⊥ {ℓ}) _×_
-×-zeroˡ ℓ A = inverse proj₁ < id , ⊥ₚ-elim > (λ { () }) (λ _ → P.refl)
+×-zeroˡ : ∀ ℓ → LeftZero {ℓ = ℓ} _↔_ ⊥ _×_
+×-zeroˡ ℓ A = mk↔ₛ′ proj₁ < id , ⊥ₚ-elim > (λ _ → P.refl) (λ { () })
 
-×-zeroʳ : ∀ ℓ → RightZero _↔_ (⊥ {ℓ}) _×_
-×-zeroʳ ℓ A = inverse proj₂ < ⊥ₚ-elim , id > (λ { () }) λ _ → P.refl
+×-zeroʳ : ∀ ℓ → RightZero {ℓ = ℓ} _↔_ ⊥ _×_
+×-zeroʳ ℓ A = mk↔ₛ′ proj₂ < ⊥ₚ-elim , id > (λ _ → P.refl) (λ { () })
 
 ×-zero : ∀ ℓ → Zero _↔_ ⊥ _×_
 ×-zero ℓ  = ×-zeroˡ ℓ , ×-zeroʳ ℓ
@@ -78,26 +80,26 @@ open import Relation.Nullary.Decidable using (True)
 -- ⊎ is associative
 
 ⊎-assoc : ∀ ℓ → Associative {ℓ = ℓ} _↔_ _⊎_
-⊎-assoc ℓ _ _ _ = inverse
+⊎-assoc ℓ _ _ _ = mk↔ₛ′
   [ [ inj₁ , inj₂ ∘′ inj₁ ]′ , inj₂ ∘′ inj₂ ]′
   [ inj₁ ∘′ inj₁ , [ inj₁ ∘′ inj₂ , inj₂ ]′ ]′
-  [ [ (λ _ → P.refl) , (λ _ → P.refl) ] , (λ _ → P.refl) ]
   [ (λ _ → P.refl) , [ (λ _ → P.refl) , (λ _ → P.refl) ] ]
+  [ [ (λ _ → P.refl) , (λ _ → P.refl) ] , (λ _ → P.refl) ]
 
 -- ⊎ is commutative
 
-⊎-comm : ∀ {a b} (A : Set a) (B : Set b) → (A ⊎ B) ↔ (B ⊎ A)
-⊎-comm _ _ = inverse swap swap swap-involutive swap-involutive
+⊎-comm : ∀ (A : Set a) (B : Set b) → (A ⊎ B) ↔ (B ⊎ A)
+⊎-comm _ _ = mk↔ₛ′ swap swap swap-involutive swap-involutive
 
 -- ⊎ has ⊥ as its identity
 
 ⊎-identityˡ : ∀ ℓ → LeftIdentity _↔_ (⊥ {ℓ}) _⊎_
-⊎-identityˡ _ _ = inverse [ (λ ()) , id ]′ inj₂
-                          [ (λ ()) , (λ _ → P.refl) ] (λ _ → P.refl)
+⊎-identityˡ _ _ = mk↔ₛ′ [ (λ ()) , id ]′ inj₂ (λ _ → P.refl)
+                          [ (λ ()) , (λ _ → P.refl) ]
 
 ⊎-identityʳ : ∀ ℓ → RightIdentity _↔_ (⊥ {ℓ}) _⊎_
-⊎-identityʳ _ _ = inverse [ id , (λ ()) ]′ inj₁
-                          [ (λ _ → P.refl) , (λ ()) ] (λ _ → P.refl)
+⊎-identityʳ _ _ = mk↔ₛ′ [ id , (λ ()) ]′ inj₁ (λ _ → P.refl)
+                          [ (λ _ → P.refl) , (λ ()) ]
 
 ⊎-identity : ∀ ℓ → Identity _↔_ ⊥ _⊎_
 ⊎-identity ℓ = ⊎-identityˡ ℓ , ⊎-identityʳ ℓ
@@ -108,18 +110,18 @@ open import Relation.Nullary.Decidable using (True)
 -- × distributes over ⊎
 
 ×-distribˡ-⊎ : ∀ ℓ → _DistributesOverˡ_ {ℓ = ℓ} _↔_ _×_ _⊎_
-×-distribˡ-⊎ ℓ _ _ _ = inverse
+×-distribˡ-⊎ ℓ _ _ _ = mk↔ₛ′
   (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)
   [ Prod.map₂ inj₁ , Prod.map₂ inj₂ ]′
-  (uncurry λ _ → [ (λ _ → P.refl) , (λ _ → P.refl) ])
   [ (λ _ → P.refl) , (λ _ → P.refl) ]
+  (uncurry λ _ → [ (λ _ → P.refl) , (λ _ → P.refl) ])
 
 ×-distribʳ-⊎ : ∀ ℓ → _DistributesOverʳ_ {ℓ = ℓ} _↔_ _×_ _⊎_
-×-distribʳ-⊎ ℓ _ _ _ = inverse
+×-distribʳ-⊎ ℓ _ _ _ = mk↔ₛ′
   (uncurry [ curry inj₁ , curry inj₂ ]′)
   [ Prod.map₁ inj₁ , Prod.map₁ inj₂ ]′
-  (uncurry [ (λ _ _ → P.refl) , (λ _ _ → P.refl) ])
   [ (λ _ → P.refl) , (λ _ → P.refl) ]
+  (uncurry [ (λ _ _ → P.refl) , (λ _ _ → P.refl) ])
 
 ×-distrib-⊎ : ∀ ℓ → _DistributesOver_ {ℓ = ℓ} _↔_ _×_ _⊎_
 ×-distrib-⊎ ℓ = ×-distribˡ-⊎ ℓ , ×-distribʳ-⊎ ℓ
@@ -131,11 +133,11 @@ open import Relation.Nullary.Decidable using (True)
 
 ×-isMagma : ∀ k ℓ → IsMagma {Level.suc ℓ} (Related ⌊ k ⌋) _×_
 ×-isMagma k ℓ = record
-  { isEquivalence = SK-isEquivalence k ℓ
+  { isEquivalence = SK-isEquivalence k
   ; ∙-cong        = _×-cong_
   }
 
-×-magma : Symmetric-kind → (ℓ : Level) → Magma _ _
+×-magma : SymmetricKind → (ℓ : Level) → Magma _ _
 ×-magma k ℓ = record
   { isMagma = ×-isMagma k ℓ
   }
@@ -146,7 +148,7 @@ open import Relation.Nullary.Decidable using (True)
   ; assoc   = λ _ _ _ → ↔⇒ Σ-assoc
   }
 
-×-semigroup : Symmetric-kind → (ℓ : Level) → Semigroup _ _
+×-semigroup : SymmetricKind → (ℓ : Level) → Semigroup _ _
 ×-semigroup k ℓ = record
   { isSemigroup = ×-isSemigroup k ℓ
   }
@@ -157,7 +159,7 @@ open import Relation.Nullary.Decidable using (True)
   ; identity    = (↔⇒ ∘ ×-identityˡ ℓ) , (↔⇒ ∘ ×-identityʳ ℓ)
   }
 
-×-monoid : Symmetric-kind → (ℓ : Level) → Monoid _ _
+×-monoid : SymmetricKind → (ℓ : Level) → Monoid _ _
 ×-monoid k ℓ = record
   { isMonoid = ×-isMonoid k ℓ
   }
@@ -168,7 +170,7 @@ open import Relation.Nullary.Decidable using (True)
   ; comm     = λ _ _ → ↔⇒ (×-comm _ _)
   }
 
-×-commutativeMonoid : Symmetric-kind → (ℓ : Level) → CommutativeMonoid _ _
+×-commutativeMonoid : SymmetricKind → (ℓ : Level) → CommutativeMonoid _ _
 ×-commutativeMonoid k ℓ = record
   { isCommutativeMonoid = ×-isCommutativeMonoid k ℓ
   }
@@ -177,11 +179,11 @@ open import Relation.Nullary.Decidable using (True)
 
 ⊎-isMagma : ∀ k ℓ → IsMagma {Level.suc ℓ} (Related ⌊ k ⌋) _⊎_
 ⊎-isMagma k ℓ = record
-  { isEquivalence = SK-isEquivalence k ℓ
+  { isEquivalence = SK-isEquivalence k
   ; ∙-cong        = _⊎-cong_
   }
 
-⊎-magma : Symmetric-kind → (ℓ : Level) → Magma _ _
+⊎-magma : SymmetricKind → (ℓ : Level) → Magma _ _
 ⊎-magma k ℓ = record
   { isMagma = ⊎-isMagma k ℓ
   }
@@ -192,7 +194,7 @@ open import Relation.Nullary.Decidable using (True)
   ; assoc   = λ A B C → ↔⇒ (⊎-assoc ℓ A B C)
   }
 
-⊎-semigroup : Symmetric-kind → (ℓ : Level) → Semigroup _ _
+⊎-semigroup : SymmetricKind → (ℓ : Level) → Semigroup _ _
 ⊎-semigroup k ℓ = record
   { isSemigroup = ⊎-isSemigroup k ℓ
   }
@@ -203,7 +205,7 @@ open import Relation.Nullary.Decidable using (True)
   ; identity    = (↔⇒ ∘ ⊎-identityˡ ℓ) , (↔⇒ ∘ ⊎-identityʳ ℓ)
   }
 
-⊎-monoid : Symmetric-kind → (ℓ : Level) → Monoid _ _
+⊎-monoid : SymmetricKind → (ℓ : Level) → Monoid _ _
 ⊎-monoid k ℓ = record
   { isMonoid = ⊎-isMonoid k ℓ
   }
@@ -214,7 +216,7 @@ open import Relation.Nullary.Decidable using (True)
   ; comm     = λ _ _ → ↔⇒ (⊎-comm _ _)
   }
 
-⊎-commutativeMonoid : Symmetric-kind → (ℓ : Level) →
+⊎-commutativeMonoid : SymmetricKind → (ℓ : Level) →
                       CommutativeMonoid _ _
 ⊎-commutativeMonoid k ℓ = record
   { isCommutativeMonoid = ⊎-isCommutativeMonoid k ℓ
@@ -229,7 +231,7 @@ open import Relation.Nullary.Decidable using (True)
   ; zeroˡ                 = ↔⇒ ∘ ×-zeroˡ ℓ
   }
 
-×-⊎-commutativeSemiring : Symmetric-kind → (ℓ : Level) →
+×-⊎-commutativeSemiring : SymmetricKind → (ℓ : Level) →
                           CommutativeSemiring (Level.suc ℓ) ℓ
 ×-⊎-commutativeSemiring k ℓ = record
   { isCommutativeSemiring = ×-⊎-isCommutativeSemiring k ℓ
@@ -240,11 +242,11 @@ open import Relation.Nullary.Decidable using (True)
 
 ΠΠ↔ΠΠ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →
         ((x : A) (y : B) → P x y) ↔ ((y : B) (x : A) → P x y)
-ΠΠ↔ΠΠ _ = inverse flip flip (λ _ → P.refl) (λ _ → P.refl)
+ΠΠ↔ΠΠ _ = mk↔ₛ′ flip flip (λ _ → P.refl) (λ _ → P.refl)
 
 ∃∃↔∃∃ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →
         (∃₂ λ x y → P x y) ↔ (∃₂ λ y x → P x y)
-∃∃↔∃∃ P = inverse to from (λ _ → P.refl) (λ _ → P.refl)
+∃∃↔∃∃ P = mk↔ₛ′ to from (λ _ → P.refl) (λ _ → P.refl)
   where
   to : (∃₂ λ x y → P x y) → (∃₂ λ y x → P x y)
   to (x , y , Pxy) = (y , x , Pxy)
@@ -255,54 +257,48 @@ open import Relation.Nullary.Decidable using (True)
 ------------------------------------------------------------------------
 -- Implicit and explicit function spaces are isomorphic
 
-Π↔Π : ∀ {a b} {A : Set a} {B : A → Set b} →
+Π↔Π : ∀ {A : Set a} {B : A → Set b} →
       ((x : A) → B x) ↔ ({x : A} → B x)
-Π↔Π = inverse (λ f {x} → f x) (λ f x → f) (λ _ → P.refl) (λ _ → P.refl)
+Π↔Π = mk↔ₛ′ _$- λ- (λ _ → P.refl) (λ _ → P.refl)
 
 ------------------------------------------------------------------------
 -- _→_ preserves the symmetric relations
 
-_→-cong-⇔_ :
-  ∀ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
-  A ⇔ B → C ⇔ D → (A → C) ⇔ (B → D)
-A⇔B →-cong-⇔ C⇔D = Eq.equivalence
-  (λ f x → Equivalence.to   C⇔D ⟨$⟩ f (Equivalence.from A⇔B ⟨$⟩ x))
-  (λ f x → Equivalence.from C⇔D ⟨$⟩ f (Equivalence.to   A⇔B ⟨$⟩ x))
-
-→-cong :
-  ∀ {a b c d} →
-  Extensionality a c → Extensionality b d →
-  ∀ {k} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
-  A ∼[ ⌊ k ⌋ ] B → C ∼[ ⌊ k ⌋ ] D → (A → C) ∼[ ⌊ k ⌋ ] (B → D)
-→-cong extAC extBD {equivalence} A⇔B C⇔D = A⇔B →-cong-⇔ C⇔D
-→-cong extAC extBD {bijection}   A↔B C↔D = record
-  { to         = Equivalence.to   A→C⇔B→D
-  ; from       = Equivalence.from A→C⇔B→D
-  ; inverse-of = record
-    { left-inverse-of  = λ f → extAC λ x → begin
-        Inverse.from C↔D ⟨$⟩ (Inverse.to C↔D ⟨$⟩
-          f (Inverse.from A↔B ⟨$⟩ (Inverse.to A↔B ⟨$⟩ x)))  ≡⟨ Inverse.left-inverse-of C↔D _ ⟩
-        f (Inverse.from A↔B ⟨$⟩ (Inverse.to A↔B ⟨$⟩ x))     ≡⟨ P.cong f $ Inverse.left-inverse-of A↔B x ⟩
-        f x                                                 ∎
-    ; right-inverse-of = λ f → extBD λ x → begin
-        Inverse.to C↔D ⟨$⟩ (Inverse.from C↔D ⟨$⟩
-          f (Inverse.to A↔B ⟨$⟩ (Inverse.from A↔B ⟨$⟩ x)))  ≡⟨ Inverse.right-inverse-of C↔D _ ⟩
-        f (Inverse.to A↔B ⟨$⟩ (Inverse.from A↔B ⟨$⟩ x))     ≡⟨ P.cong f $ Inverse.right-inverse-of A↔B x ⟩
-        f x                                                 ∎
-    }
-  }
-  where
-  open P.≡-Reasoning
-  A→C⇔B→D = ↔⇒ A↔B →-cong-⇔ ↔⇒ C↔D
+→-cong-⇔ : A ⇔ B → C ⇔ D → (A → C) ⇔ (B → D)
+→-cong-⇔ A⇔B C⇔D = mk⇔
+  (λ f → to C⇔D ∘ f ∘ from A⇔B)
+  (λ f → from C⇔D ∘ f ∘ to A⇔B)
+  where open Equivalence
+
+→-cong-↔ : Extensionality a c → Extensionality b d →
+             {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
+             A ↔ B → C ↔ D → (A → C) ↔ (B → D)
+→-cong-↔ ext₁ ext₂ A↔B C↔D = mk↔ₛ′
+  (λ f → to C↔D ∘ f ∘ from A↔B)
+  (λ f → from C↔D ∘ f ∘ to A↔B)
+  (λ f → ext₂ λ x → begin
+    to C↔D (from C↔D (f (to A↔B (from A↔B x)))) ≡⟨ strictlyInverseˡ C↔D _ ⟩
+    f (to A↔B (from A↔B x))                     ≡⟨ P.cong f $ strictlyInverseˡ A↔B x ⟩
+    f x                                         ∎)
+  (λ f → ext₁ λ x → begin
+    from C↔D (to C↔D (f (from A↔B (to A↔B x))))  ≡⟨ strictlyInverseʳ C↔D _ ⟩
+    f (from A↔B (to A↔B x))                        ≡⟨ P.cong f $ strictlyInverseʳ A↔B x ⟩
+    f x                                              ∎)
+  where open Inverse; open P.≡-Reasoning
+
+→-cong :  Extensionality a c → Extensionality b d →
+          ∀ {k} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
+          A ∼[ ⌊ k ⌋ ] B → C ∼[ ⌊ k ⌋ ] D → (A → C) ∼[ ⌊ k ⌋ ] (B → D)
+→-cong extAC extBD {equivalence} = →-cong-⇔
+→-cong extAC extBD {bijection}   = →-cong-↔ extAC extBD
 
 ------------------------------------------------------------------------
 -- ¬_ (at Level 0) preserves the symmetric relations
 
-¬-cong-⇔ : ∀ {a b} {A : Set a} {B : Set b} →
-           A ⇔ B → (¬ A) ⇔ (¬ B)
-¬-cong-⇔ A⇔B =  A⇔B →-cong-⇔ Eq.id
+¬-cong-⇔ : A ⇔ B → (¬ A) ⇔ (¬ B)
+¬-cong-⇔ A⇔B = →-cong-⇔ A⇔B (Identity.⇔-id _)
 
-¬-cong : ∀ {a b} → Extensionality a 0ℓ → Extensionality b 0ℓ →
+¬-cong : Extensionality a 0ℓ → Extensionality b 0ℓ →
          ∀ {k} {A : Set a} {B : Set b} →
          A ∼[ ⌊ k ⌋ ] B → (¬ A) ∼[ ⌊ k ⌋ ] (¬ B)
 ¬-cong extA extB A≈B = →-cong extA extB A≈B (K-reflexive P.refl)
@@ -313,17 +309,17 @@ A⇔B →-cong-⇔ C⇔D = Eq.equivalence
 -- The type of the following proof is a bit more general.
 
 Related-cong :
-  ∀ {k a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
+  ∀ {k} →
   A ∼[ ⌊ k ⌋ ] B → C ∼[ ⌊ k ⌋ ] D → (A ∼[ ⌊ k ⌋ ] C) ⇔ (B ∼[ ⌊ k ⌋ ] D)
-Related-cong {A = A} {B} {C} {D} A≈B C≈D =
-  Eq.equivalence (λ A≈C → B  ∼⟨ SK-sym A≈B ⟩
-                          A  ∼⟨ A≈C ⟩
-                          C  ∼⟨ C≈D ⟩
-                          D  ∎)
-                 (λ B≈D → A  ∼⟨ A≈B ⟩
-                          B  ∼⟨ B≈D ⟩
-                          D  ∼⟨ SK-sym C≈D ⟩
-                          C  ∎)
+Related-cong {A = A} {B = B} {C = C} {D = D} A≈B C≈D = mk⇔
+  (λ A≈C → B  ∼⟨ SK-sym A≈B ⟩
+            A  ∼⟨ A≈C ⟩
+            C  ∼⟨ C≈D ⟩
+            D  ∎)
+  (λ B≈D → A  ∼⟨ A≈B ⟩
+            B  ∼⟨ B≈D ⟩
+            D  ∼⟨ SK-sym C≈D ⟩
+            C  ∎)
   where open EquationalReasoning
 
 ------------------------------------------------------------------------
@@ -332,58 +328,6 @@ Related-cong {A = A} {B} {C} {D} A≈B C≈D =
 True↔ : ∀ {p} {P : Set p}
         (dec : Dec P) → ((p₁ p₂ : P) → p₁ ≡ p₂) → True dec ↔ P
 True↔ ( true because  [p]) irr =
-  inverse (λ _ → invert [p]) (λ _ → _) (λ _ → P.refl) (irr _)
+  mk↔ₛ′ (λ _ → invert [p]) (λ _ → _) (irr _) (λ _ → P.refl)
 True↔ (false because ofⁿ ¬p) _ =
-  inverse (λ()) (invert (ofⁿ ¬p)) (λ ()) (⊥-elim ∘ ¬p)
-
-------------------------------------------------------------------------
--- Equality between pairs can be expressed as a pair of equalities
-
-module _ {a b} {A : Set a} {B : A → Set b} {p₁ p₂ : Σ A B} where
-  Σ-≡,≡→≡ : Σ (proj₁ p₁ ≡ proj₁ p₂)
-              (λ p → P.subst B p (proj₂ p₁) ≡ proj₂ p₂) →
-            p₁ ≡ p₂
-  Σ-≡,≡→≡ (P.refl , P.refl) = P.refl
-
-  Σ-≡,≡←≡ : p₁ ≡ p₂ →
-            Σ (proj₁ p₁ ≡ proj₁ p₂)
-              (λ p → P.subst B p (proj₂ p₁) ≡ proj₂ p₂)
-  Σ-≡,≡←≡ P.refl = P.refl , P.refl
-
-  private
-
-    left-inverse-of : (p : Σ (proj₁ p₁ ≡ proj₁ p₂)
-                             (λ x → P.subst B x (proj₂ p₁) ≡ proj₂ p₂)) →
-                      Σ-≡,≡←≡ (Σ-≡,≡→≡ p) ≡ p
-    left-inverse-of (P.refl , P.refl) = P.refl
-
-    right-inverse-of : (p : p₁ ≡ p₂) → Σ-≡,≡→≡ (Σ-≡,≡←≡ p) ≡ p
-    right-inverse-of P.refl = P.refl
-
-  Σ-≡,≡↔≡ : (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) →
-               P.subst B p (proj₂ p₁) ≡ proj₂ p₂) ↔
-            p₁ ≡ p₂
-  Σ-≡,≡↔≡ = inverse Σ-≡,≡→≡ Σ-≡,≡←≡ left-inverse-of right-inverse-of
-
-module _ {a b} {A : Set a} {B : Set b} {p₁ p₂ : A × B} where
-  ×-≡,≡→≡ : (proj₁ p₁ ≡ proj₁ p₂) × (proj₂ p₁ ≡ proj₂ p₂) → p₁ ≡ p₂
-  ×-≡,≡→≡ (P.refl , P.refl) = P.refl
-
-  ×-≡,≡←≡ : p₁ ≡ p₂ →
-            (proj₁ p₁ ≡ proj₁ p₂) × (proj₂ p₁ ≡ proj₂ p₂)
-  ×-≡,≡←≡ P.refl = P.refl , P.refl
-
-  private
-    left-inverse-of : (p : (proj₁ p₁ ≡ proj₁ p₂) × (proj₂ p₁ ≡ proj₂ p₂)) →
-                      ×-≡,≡←≡ (×-≡,≡→≡ p) ≡ p
-    left-inverse-of (P.refl , P.refl) = P.refl
-
-    right-inverse-of : (p : p₁ ≡ p₂) → ×-≡,≡→≡ (×-≡,≡←≡ p) ≡ p
-    right-inverse-of P.refl = P.refl
-
-  ×-≡,≡↔≡ : (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂) ↔ p₁ ≡ p₂
-  ×-≡,≡↔≡ = inverse ×-≡,≡→≡ ×-≡,≡←≡ left-inverse-of right-inverse-of
-
-×-≡×≡↔≡,≡ : ∀ {a b} {A : Set a} {B : Set b} {x y} (p : A × B) →
-            (x ≡ proj₁ p × y ≡ proj₂ p) ↔ (x , y) ≡ p
-×-≡×≡↔≡,≡ _ = ×-≡,≡↔≡
+  mk↔ₛ′ (λ()) (invert (ofⁿ ¬p)) (⊥-elim ∘ ¬p) (λ ())
diff --git a/src/Function/Related/TypeIsomorphisms/Solver.agda b/src/Function/Related/TypeIsomorphisms/Solver.agda
index 52f27d15ed..2d952ba82a 100644
--- a/src/Function/Related/TypeIsomorphisms/Solver.agda
+++ b/src/Function/Related/TypeIsomorphisms/Solver.agda
@@ -17,14 +17,15 @@ open import Data.Product.Base using (_×_)
 open import Data.Sum.Base using (_⊎_)
 open import Data.Unit.Polymorphic using (⊤)
 open import Level using (Level)
-open import Function.Inverse as Inv using (_↔_)
-open import Function.Related as Related
+open import Function.Bundles using (_↔_)
+open import Function.Properties.Inverse using (↔-refl)
+open import Function.Related.Propositional as Related
 open import Function.Related.TypeIsomorphisms
 
 ------------------------------------------------------------------------
 -- The solver
 
-module ×-⊎-Solver (k : Symmetric-kind) {ℓ} =
+module ×-⊎-Solver (k : SymmetricKind) {ℓ} =
   Algebra.Solver.Ring.NaturalCoefficients.Default
     (×-⊎-commutativeSemiring k ℓ)
 
@@ -39,5 +40,5 @@ private
          (⊤ × A × (B ⊎ C)) ↔ (A × B ⊎ C × (⊥ ⊎ A))
   test = solve 3 (λ A B C → con 1 :* (A :* (B :+ C)) :=
                             A :* B :+ C :* (con 0 :+ A))
-                 Inv.id
+                 ↔-refl
     where open ×-⊎-Solver bijection
diff --git a/src/Function/Surjection.agda b/src/Function/Surjection.agda
index 729b46a538..bd96c50efb 100644
--- a/src/Function/Surjection.agda
+++ b/src/Function/Surjection.agda
@@ -1,18 +1,19 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Surjections
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
-
--- Note: use of the standard function hierarchy is encouraged. The
--- module `Function` re-exports `Surjective`, `IsSurjection` and
--- `Surjection`. The alternative definitions found in this file will
--- eventually be deprecated.
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.Surjection where
 
+{-# WARNING_ON_IMPORT
+"Function.Surjection was deprecated in v2.0.
+Use the standard function hierarchy in Function/Function.Bundles instead."
+#-}
+
 open import Level
 open import Function.Equality as F
   using (_⟶_) renaming (_∘_ to _⟪∘⟫_)
@@ -94,9 +95,9 @@ surjection : ∀ {f t} {From : Set f} {To : Set t} →
              (∀ x → to (from x) ≡ x) →
              From ↠ To
 surjection to from surjective = record
-  { to         = P.→-to-⟶ to
+  { to         = F.→-to-⟶ to
   ; surjective = record
-    { from             = P.→-to-⟶ from
+    { from             = F.→-to-⟶ from
     ; right-inverse-of = surjective
     }
   }
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 866628a71c..8c7d26cb2a 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -10,8 +10,9 @@ module Relation.Binary.HeterogeneousEquality where
 
 import Axiom.Extensionality.Heterogeneous as Ext
 open import Data.Unit.NonEta
+open import Data.Product.Base using (_,_)
 open import Function.Base
-open import Function.Inverse using (Inverse)
+open import Function.Bundles using (Inverse)
 open import Level
 open import Relation.Nullary hiding (Irrelevant)
 open import Relation.Unary using (Pred)
@@ -183,12 +184,11 @@ indexedSetoid B = record
 ≡↔≅ : ∀ {A : Set a} (B : A → Set b) {x : A} →
       Inverse (P.setoid (B x)) ((indexedSetoid B) atₛ x)
 ≡↔≅ B = record
-  { to         = record { _⟨$⟩_ = id; cong = ≡-to-≅ }
-  ; from       = record { _⟨$⟩_ = id; cong = ≅-to-≡ }
-  ; inverse-of = record
-    { left-inverse-of  = λ _ → refl
-    ; right-inverse-of = λ _ → refl
-    }
+  { to         = id
+  ; to-cong    = ≡-to-≅
+  ; from       = id
+  ; from-cong  = ≅-to-≡
+  ; inverse    = (λ { P.refl → refl }) , λ { refl → P.refl }
   }
 
 decSetoid : Decidable {A = A} {B = A} (λ x y → x ≅ y) →
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index ca3ea60e4d..418e9b2ade 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -11,7 +11,7 @@ module Relation.Binary.PropositionalEquality where
 import Axiom.Extensionality.Propositional as Ext
 open import Axiom.UniquenessOfIdentityProofs
 open import Function.Base using (id; _∘_)
-open import Function.Equality using (Π; _⟶_)
+import Function.Dependent.Bundles as Dependent
 open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)
 open import Level using (Level; _⊔_)
 open import Data.Product.Base using (∃)
@@ -27,9 +27,7 @@ import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
 private
   variable
     a b c ℓ p : Level
-    A : Set a
-    B : Set b
-    C : Set c
+    A B C : Set a
 
 ------------------------------------------------------------------------
 -- Re-export contents modules that make up the parts
@@ -50,15 +48,16 @@ _≗_ : (f g : A → B) → Set _
 _≗_ {A = A} {B = B} = Setoid._≈_ (A →-setoid B)
 
 :→-to-Π : ∀ {A : Set a} {B : IndexedSetoid A b ℓ} →
-          ((x : A) → IndexedSetoid.Carrier B x) → Π (setoid A) B
+          ((x : A) → IndexedSetoid.Carrier B x) →
+          Dependent.Func (setoid A) B
 :→-to-Π {B = B} f = record
-  { _⟨$⟩_ = f
-  ; cong  = λ { refl → IndexedSetoid.refl B }
+  { to = f
+  ; cong = λ { refl → IndexedSetoid.refl B }
   }
-  where open IndexedSetoid B using (_≈_)
 
 →-to-⟶ : ∀ {A : Set a} {B : Setoid b ℓ} →
-         (A → Setoid.Carrier B) → setoid A ⟶ B
+         (A → Setoid.Carrier B) →
+         Dependent.Func (setoid A) (Trivial.indexedSetoid B)
 →-to-⟶ = :→-to-Π
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Nullary/Decidable.agda b/src/Relation/Nullary/Decidable.agda
index 5793725f0e..37d3e0904a 100644
--- a/src/Relation/Nullary/Decidable.agda
+++ b/src/Relation/Nullary/Decidable.agda
@@ -15,7 +15,7 @@ open import Data.Product.Base as Prod hiding (map)
 open import Data.Sum.Base as Sum hiding (map)
 open import Function.Base
 open import Function.Bundles using
-  (Injection; module Injection; module Equivalence; _⇔_; _↔_; mk↔′)
+  (Injection; module Injection; module Equivalence; _⇔_; _↔_; mk↔ₛ′)
 open import Relation.Binary.Bundles using (Setoid; module Setoid)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary
@@ -60,8 +60,8 @@ module _ {a₁ a₂ b₁ b₂} {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
 -- A lemma relating True and Dec
 
 True-↔ : (dec : Dec P) → Irrelevant P → True dec ↔ P
-True-↔ (true  because  [p]) irr = mk↔′ (λ _ → invert [p]) _ (irr (invert [p])) cong′
-True-↔ (false because ofⁿ ¬p) _ = mk↔′ (λ ()) (invert (ofⁿ ¬p)) (⊥-elim ∘ ¬p) λ ()
+True-↔ (true  because  [p]) irr = mk↔ₛ′ (λ _ → invert [p]) _ (irr (invert [p])) cong′
+True-↔ (false because ofⁿ ¬p) _ = mk↔ₛ′ (λ ()) (invert (ofⁿ ¬p)) (⊥-elim ∘ ¬p) λ ()
 
 ------------------------------------------------------------------------
 -- Result of decidability
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 3057b16eb6..8d72845f07 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -18,9 +18,9 @@ import Relation.Binary.PropositionalEquality.Properties as PropEq
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   as Trivial
 open import Data.Sum.Base as Sum  hiding (map)
-open import Data.Sum.Relation.Binary.Pointwise
+open import Data.Sum.Relation.Binary.Pointwise using (_⊎ₛ_; inj₁; inj₂)
 open import Data.Product.Base as Prod hiding (map)
-open import Data.Product.Relation.Binary.Pointwise.NonDependent
+open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 open import Function.Base using (_∘_; id)
 open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)
 open import Data.Empty

From 4ef86f2d6b17264732f7c8b2ee6ddc0ff4a09cf1 Mon Sep 17 00:00:00 2001
From: fredins <fredin.martin@gmail.com>
Date: Tue, 26 Sep 2023 02:21:02 +0200
Subject: [PATCH 122/316] Fix typo and whitespace violation (#2104)

---
 src/Data/Container/FreeMonad.agda | 2 +-
 src/Data/Vec/Base.agda            | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/Data/Container/FreeMonad.agda b/src/Data/Container/FreeMonad.agda
index 8a24ef1d36..d848915983 100644
--- a/src/Data/Container/FreeMonad.agda
+++ b/src/Data/Container/FreeMonad.agda
@@ -47,7 +47,7 @@ infix  1 _⋆_
 ------------------------------------------------------------------------
 -- Type definition
 
--- The free moand can be defined as the least fixpoint `μ (C ⋆C X)`
+-- The free monad can be defined as the least fixpoint `μ (C ⋆C X)`
 
 _⋆C_ : ∀ {x s p} → Container s p → Set x → Container (s ⊔ x) p
 C ⋆C X = const X ⊎ C
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 6752ec1b06..ed8817c66e 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -272,7 +272,7 @@ drop m xs = proj₁ (proj₂ (splitAt m xs))
 group : ∀ n k (xs : Vec A (n * k)) →
         ∃ λ (xss : Vec (Vec A k) n) → xs ≡ concat xss
 group zero    k []                  = ([] , refl)
-group (suc n) k xs  = 
+group (suc n) k xs  =
   let ys , zs , eq-split = splitAt k xs in
   let zss , eq-group     = group n k zs in
    (ys ∷ zss) , trans eq-split (cong (ys ++_) eq-group)

From 3f41852c6969fe90ed32d7737625aac188b1184b Mon Sep 17 00:00:00 2001
From: Akshobhya K M <akshobhyakm@gmail.com>
Date: Mon, 25 Sep 2023 23:12:36 -0400
Subject: [PATCH 123/316] Add Kleene Algebra morphism with composition and
 identity construct (#1936)

---
 CHANGELOG.md                                  |  4 ++
 src/Algebra/Bundles.agda                      |  2 +-
 src/Algebra/Bundles/Raw.agda                  | 42 ++++++++++++++++---
 .../Morphism/Construct/Composition.agda       | 40 ++++++++++++++++++
 src/Algebra/Morphism/Construct/Identity.agda  | 26 ++++++++++++
 src/Algebra/Morphism/Structures.agda          | 42 +++++++++++++++++++
 6 files changed, 149 insertions(+), 7 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index cdbdb88ef1..d8b06495e9 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -988,6 +988,7 @@ Major improvements
   * `RawRing`
   * `RawQuasigroup`
   * `RawLoop`
+  * `RawKleeneAlgebra`
 * A new module `Algebra.Lattice.Bundles.Raw` is also introduced.
   * `RawLattice` has been moved from `Algebra.Lattice.Bundles` to this new module.
 
@@ -2059,6 +2060,9 @@ Additions to existing modules
   record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
   record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
   record IsRingWithoutOneIsoMorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
   ```
 
 * Added new proofs in `Data.Bool.Properties`:
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index 5bc2d5a1d0..7fd88f5b42 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -26,7 +26,7 @@ open Raw public
   using (RawMagma; RawMonoid; RawGroup
         ; RawNearSemiring; RawSemiring
         ; RawRingWithoutOne; RawRing
-        ; RawQuasigroup; RawLoop)
+        ; RawQuasigroup; RawLoop; RawKleeneAlgebra)
 
 ------------------------------------------------------------------------
 -- Bundles with 1 binary operation
diff --git a/src/Algebra/Bundles/Raw.agda b/src/Algebra/Bundles/Raw.agda
index 0b26aad1eb..727440d85e 100644
--- a/src/Algebra/Bundles/Raw.agda
+++ b/src/Algebra/Bundles/Raw.agda
@@ -153,12 +153,12 @@ record RawRingWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where
   infixl 6 _+_
   infix  4 _≈_
   field
-    Carrier           : Set c
-    _≈_               : Rel Carrier ℓ
-    _+_               : Op₂ Carrier
-    _*_               : Op₂ Carrier
-    -_                : Op₁ Carrier
-    0#                : Carrier
+    Carrier : Set c
+    _≈_     : Rel Carrier ℓ
+    _+_     : Op₂ Carrier
+    _*_     : Op₂ Carrier
+    -_      : Op₁ Carrier
+    0#      : Carrier
 
   +-rawGroup : RawGroup c ℓ
   +-rawGroup = record
@@ -289,3 +289,33 @@ record RawLoop  c ℓ : Set (suc (c ⊔ ℓ)) where
 
   open RawQuasigroup rawQuasigroup public
     using (_≉_ ; ∙-rawMagma; \\-rawMagma; //-rawMagma)
+
+record RawKleeneAlgebra c ℓ : Set (suc (c ⊔ ℓ)) where
+  infix  8 _⋆
+  infixl 7 _*_
+  infixl 6 _+_
+  infix  4 _≈_
+  field
+    Carrier : Set c
+    _≈_     : Rel Carrier ℓ
+    _+_     : Op₂ Carrier
+    _*_     : Op₂ Carrier
+    _⋆      : Op₁ Carrier
+    0#      : Carrier
+    1#      : Carrier
+
+  rawSemiring : RawSemiring c ℓ
+  rawSemiring = record
+    { _≈_ = _≈_
+    ; _+_ = _+_
+    ; _*_ = _*_
+    ; 0#  = 0#
+    ; 1#  = 1#
+    }
+
+  open RawSemiring rawSemiring public
+    using
+    ( _≉_
+    ; +-rawMagma; +-rawMonoid
+    ; *-rawMagma; *-rawMonoid
+    )
diff --git a/src/Algebra/Morphism/Construct/Composition.agda b/src/Algebra/Morphism/Construct/Composition.agda
index 5087de2ae3..17d5f2798b 100644
--- a/src/Algebra/Morphism/Construct/Composition.agda
+++ b/src/Algebra/Morphism/Construct/Composition.agda
@@ -383,3 +383,43 @@ module _ {L₁ : RawLoop a ℓ₁}
     { isLoopMonomorphism = isLoopMonomorphism F.isLoopMonomorphism G.isLoopMonomorphism
     ; surjective         = Func.surjective _ _ (_≈_ L₃) F.surjective G.surjective
     } where module F = IsLoopIsomorphism f-iso; module G = IsLoopIsomorphism g-iso
+
+------------------------------------------------------------------------
+-- KleeneAlgebra
+
+module _ {K₁ : RawKleeneAlgebra a ℓ₁}
+         {K₂ : RawKleeneAlgebra b ℓ₂}
+         {K₃ : RawKleeneAlgebra c ℓ₃}
+         (open RawKleeneAlgebra)
+         (≈₃-trans : Transitive (_≈_ K₃))
+         {f : Carrier K₁ → Carrier K₂}
+         {g : Carrier K₂ → Carrier K₃}
+         where
+
+  isKleeneAlgebraHomomorphism
+    : IsKleeneAlgebraHomomorphism K₁ K₂ f
+    → IsKleeneAlgebraHomomorphism K₂ K₃ g
+    → IsKleeneAlgebraHomomorphism K₁ K₃ (g ∘ f)
+  isKleeneAlgebraHomomorphism f-homo g-homo = record
+    { isSemiringHomomorphism = isSemiringHomomorphism ≈₃-trans F.isSemiringHomomorphism G.isSemiringHomomorphism
+    ; ⋆-homo              = λ x → ≈₃-trans (G.⟦⟧-cong (F.⋆-homo x)) (G.⋆-homo (f x))
+    } where module F = IsKleeneAlgebraHomomorphism f-homo; module G = IsKleeneAlgebraHomomorphism g-homo
+
+  isKleeneAlgebraMonomorphism
+    : IsKleeneAlgebraMonomorphism K₁ K₂ f
+    → IsKleeneAlgebraMonomorphism K₂ K₃ g
+    → IsKleeneAlgebraMonomorphism K₁ K₃ (g ∘ f)
+  isKleeneAlgebraMonomorphism f-mono g-mono = record
+    { isKleeneAlgebraHomomorphism = isKleeneAlgebraHomomorphism F.isKleeneAlgebraHomomorphism G.isKleeneAlgebraHomomorphism
+    ; injective = F.injective ∘ G.injective
+    } where module F = IsKleeneAlgebraMonomorphism f-mono;  module G = IsKleeneAlgebraMonomorphism g-mono
+
+  isKleeneAlgebraIsomorphism
+    : IsKleeneAlgebraIsomorphism K₁ K₂ f
+    → IsKleeneAlgebraIsomorphism K₂ K₃ g
+    → IsKleeneAlgebraIsomorphism K₁ K₃ (g ∘ f)
+  isKleeneAlgebraIsomorphism f-iso g-iso = record
+    { isKleeneAlgebraMonomorphism = isKleeneAlgebraMonomorphism F.isKleeneAlgebraMonomorphism G.isKleeneAlgebraMonomorphism
+    ; surjective                  = Func.surjective (_≈_ K₁) (_≈_ K₂) (_≈_ K₃) F.surjective G.surjective
+    } where module F = IsKleeneAlgebraIsomorphism f-iso; module G = IsKleeneAlgebraIsomorphism g-iso
+
diff --git a/src/Algebra/Morphism/Construct/Identity.agda b/src/Algebra/Morphism/Construct/Identity.agda
index 63e95e54e6..881a8cf15c 100644
--- a/src/Algebra/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Morphism/Construct/Identity.agda
@@ -19,6 +19,7 @@ open import Algebra.Morphism.Structures
         ; module RingMorphisms
         ; module QuasigroupMorphisms
         ; module LoopMorphisms
+        ; module KleeneAlgebraMorphisms
         )
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
@@ -248,3 +249,28 @@ module _ (L : RawLoop c ℓ) (open RawLoop L) (refl : Reflexive _≈_) where
     { isLoopMonomorphism = isLoopMonomorphism
     ; surjective = Id.surjective _
     }
+
+------------------------------------------------------------------------
+-- KleeneAlgebra
+
+module _ (K : RawKleeneAlgebra c ℓ) (open RawKleeneAlgebra K) (refl : Reflexive _≈_) where
+  open KleeneAlgebraMorphisms K K
+
+  isKleeneAlgebraHomomorphism : IsKleeneAlgebraHomomorphism id
+  isKleeneAlgebraHomomorphism = record
+    { isSemiringHomomorphism = isSemiringHomomorphism _ refl
+    ; ⋆-homo = λ _ → refl
+    }
+
+  isKleeneAlgebraMonomorphism : IsKleeneAlgebraMonomorphism id
+  isKleeneAlgebraMonomorphism = record
+    { isKleeneAlgebraHomomorphism = isKleeneAlgebraHomomorphism
+    ; injective = id
+    }
+
+  isKleeneAlgebraIsomorphism : IsKleeneAlgebraIsomorphism id
+  isKleeneAlgebraIsomorphism = record
+    { isKleeneAlgebraMonomorphism = isKleeneAlgebraMonomorphism
+    ; surjective = Id.surjective _
+    }
+
diff --git a/src/Algebra/Morphism/Structures.agda b/src/Algebra/Morphism/Structures.agda
index eb8970ba5c..e73fa7e9e6 100644
--- a/src/Algebra/Morphism/Structures.agda
+++ b/src/Algebra/Morphism/Structures.agda
@@ -685,6 +685,47 @@ module LoopMorphisms (L₁ : RawLoop a ℓ₁) (L₂ : RawLoop b ℓ₂) where
 
     open IsLoopMonomorphism isLoopMonomorphism public
 
+------------------------------------------------------------------------
+-- Morphisms over Kleene algebra structures
+------------------------------------------------------------------------
+module KleeneAlgebraMorphisms (R₁ : RawKleeneAlgebra a ℓ₁) (R₂ : RawKleeneAlgebra b ℓ₂) where
+
+  open RawKleeneAlgebra R₁ renaming
+    ( Carrier to A; _≈_ to _≈₁_
+    ; _⋆ to _⋆₁
+    ; rawSemiring to rawSemiring₁
+    )
+
+  open RawKleeneAlgebra R₂ renaming
+    ( Carrier to B; _≈_ to _≈₂_
+    ; _⋆ to _⋆₂
+    ; rawSemiring to rawSemiring₂
+    )
+
+  open MorphismDefinitions A B _≈₂_
+  open SemiringMorphisms rawSemiring₁ rawSemiring₂
+
+  record IsKleeneAlgebraHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
+    field
+      isSemiringHomomorphism : IsSemiringHomomorphism ⟦_⟧
+      ⋆-homo :  Homomorphic₁ ⟦_⟧ _⋆₁ _⋆₂
+
+    open IsSemiringHomomorphism isSemiringHomomorphism public
+
+  record IsKleeneAlgebraMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
+    field
+      isKleeneAlgebraHomomorphism   : IsKleeneAlgebraHomomorphism ⟦_⟧
+      injective                     : Injective  _≈₁_ _≈₂_ ⟦_⟧
+
+    open IsKleeneAlgebraHomomorphism isKleeneAlgebraHomomorphism public
+
+  record IsKleeneAlgebraIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+    field
+      isKleeneAlgebraMonomorphism   : IsKleeneAlgebraMonomorphism ⟦_⟧
+      surjective                    : Surjective  _≈₁_ _≈₂_ ⟦_⟧
+
+    open IsKleeneAlgebraMonomorphism isKleeneAlgebraMonomorphism public
+
 ------------------------------------------------------------------------
 -- Re-export contents of modules publicly
 
@@ -697,3 +738,4 @@ open RingWithoutOneMorphisms public
 open RingMorphisms public
 open QuasigroupMorphisms public
 open LoopMorphisms public
+open KleeneAlgebraMorphisms public

From 437f6baa1d21f834d62de9b102563706e4582ee6 Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Tue, 26 Sep 2023 00:58:32 -0300
Subject: [PATCH 124/316] Added foldr of permutation of Commutative Monoid
 (#1944)

---
 CHANGELOG.md                                  |  7 ++++-
 .../Binary/Permutation/Setoid/Properties.agda | 29 +++++++++++++++++++
 2 files changed, 35 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d8b06495e9..b7b96f4fdb 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3361,6 +3361,11 @@ This is a full list of proofs that have changed form to use irrelevant instance
   <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
   ```
 
+* In `Data.List.Relation.Binary.Permutation.Setoid.Properties`:
+  ```agda
+  foldr-commMonoid : xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys
+  ```
+
 * Added new module to `Data.Rational.Unnormalised.Properties`
   ```agda
   module ≃-Reasoning = SetoidReasoning ≃-setoid
@@ -3442,4 +3447,4 @@ This is a full list of proofs that have changed form to use irrelevant instance
       b #⟨ b#c ⟩
       c ≈⟨ c≈d ⟩
       d ∎
-    ```
+    ```
\ No newline at end of file
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index bc7af2193f..ce16a7528a 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -16,6 +16,7 @@ module Data.List.Relation.Binary.Permutation.Setoid.Properties
   where
 
 open import Algebra
+import Algebra.Properties.CommutativeMonoid as ACM
 open import Data.Bool.Base using (true; false)
 open import Data.List.Base as List hiding (head; tail)
 open import Data.List.Relation.Binary.Pointwise as Pointwise
@@ -36,6 +37,7 @@ open import Data.Product.Base using (_,_; _×_; ∃; ∃₂; proj₁; proj₂)
 open import Function.Base using (_∘_; _⟨_⟩_; flip)
 open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred; Decidable)
+import Relation.Binary.Reasoning.Setoid as RelSetoid
 open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_ ; refl; sym; cong; cong₂; subst; _≢_)
@@ -474,3 +476,30 @@ module _ {ℓ} {R : Rel A ℓ} (R? : B.Decidable R) where
 ++↭ʳ++ : ∀ (xs ys : List A) → xs ++ ys ↭ xs ʳ++ ys
 ++↭ʳ++ []       ys = ↭-refl
 ++↭ʳ++ (x ∷ xs) ys = ↭-trans (↭-sym (↭-shift xs ys)) (++↭ʳ++ xs (x ∷ ys))
+
+------------------------------------------------------------------------
+-- foldr of Commutative Monoid
+
+module _ {_∙_ : Op₂ A} {ε : A} (isCmonoid : IsCommutativeMonoid _≈_ _∙_ ε) where
+  open module CM = IsCommutativeMonoid isCmonoid
+
+  private
+    module S = RelSetoid setoid
+
+    cmonoid : CommutativeMonoid _ _
+    cmonoid = record { isCommutativeMonoid = isCmonoid }
+
+  open ACM cmonoid
+
+  foldr-commMonoid : ∀ {xs ys} → xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys
+  foldr-commMonoid (refl []) = CM.refl
+  foldr-commMonoid (refl (x≈y ∷ xs≈ys)) = ∙-cong x≈y (foldr-commMonoid (Permutation.refl xs≈ys))
+  foldr-commMonoid (prep x≈y xs↭ys) = ∙-cong x≈y (foldr-commMonoid xs↭ys)
+  foldr-commMonoid (swap {xs} {ys} {x} {y} {x′} {y′} x≈x′ y≈y′ xs↭ys) = S.begin
+    x ∙ (y ∙ foldr _∙_ ε xs)   S.≈⟨ ∙-congˡ (∙-congˡ (foldr-commMonoid xs↭ys)) ⟩
+    x ∙ (y ∙ foldr _∙_ ε ys)   S.≈˘⟨ assoc x y (foldr _∙_ ε ys) ⟩
+    (x ∙ y) ∙ foldr _∙_ ε ys   S.≈⟨ ∙-congʳ (comm x y) ⟩
+    (y ∙ x) ∙ foldr _∙_ ε ys   S.≈⟨ ∙-congʳ (∙-cong y≈y′ x≈x′) ⟩
+    (y′ ∙ x′) ∙ foldr _∙_ ε ys S.≈⟨ assoc y′ x′ (foldr _∙_ ε ys) ⟩
+    y′ ∙ (x′ ∙ foldr _∙_ ε ys) S.∎
+  foldr-commMonoid (trans xs↭ys ys↭zs) = CM.trans (foldr-commMonoid xs↭ys) (foldr-commMonoid ys↭zs)

From 670d1e89d5f0a2a05d6790ee7bac0f78777f0afa Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Tue, 26 Sep 2023 12:58:42 +0900
Subject: [PATCH 125/316] Add `begin-irrefl` reasoning combinator (#1470)

---
 CHANGELOG.md                                  | 26 +++++++++++++++++++
 src/Data/Integer/Properties.agda              |  1 +
 .../List/Relation/Unary/Any/Properties.agda   |  1 -
 src/Data/Nat/Binary/Properties.agda           | 19 +++++++++-----
 src/Data/Nat/Properties.agda                  |  1 +
 src/Data/Rational/Properties.agda             |  6 +++--
 .../Rational/Unnormalised/Properties.agda     |  6 +++--
 .../Relation/Unary/Any/Properties.agda        | 13 +++++-----
 .../Vec/Relation/Binary/Lex/NonStrict.agda    |  1 +
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  2 ++
 .../Binary/Reasoning/Base/Triple.agda         | 16 +++++++++---
 .../Binary/Reasoning/PartialOrder.agda        |  5 +++-
 .../Binary/Reasoning/StrictPartialOrder.agda  |  2 ++
 13 files changed, 76 insertions(+), 23 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b7b96f4fdb..2fc788c86d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -769,6 +769,30 @@ Non-backwards compatible changes
   IO.Instances
   ```
 
+### Changes to triple reasoning interface
+
+* The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof that the strict
+  relation is irreflexive. 
+  
+* This allows the following new proof combinator:
+  ```agda
+  begin-contradiction : (r : x IsRelatedTo x) → {s : True (IsStrict? r)} → A
+  ```
+  that takes a proof that a value is strictly less than itself and then applies the principle of explosion.
+  
+* Specialised versions of this combinator are available in the following derived modules:
+  ```
+  Data.Nat.Properties
+  Data.Nat.Binary.Properties
+  Data.Integer.Properties
+  Data.Rational.Unnormalised.Properties
+  Data.Rational.Properties
+  Data.Vec.Relation.Binary.Lex.Strict
+  Data.Vec.Relation.Binary.Lex.NonStrict
+  Relation.Binary.Reasoning.StrictPartialOrder
+  Relation.Binary.Reasoning.PartialOrder
+  ```
+
 ### Other
 
 * In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
@@ -992,6 +1016,8 @@ Major improvements
 * A new module `Algebra.Lattice.Bundles.Raw` is also introduced.
   * `RawLattice` has been moved from `Algebra.Lattice.Bundles` to this new module.
 
+* In `Relation.Binary.Reasoning.Base.Triple`, added a new parameter `<-irrefl : Irreflexive _≈_ _<_`
+
 Deprecated modules
 ------------------
 
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index ef7528495e..5760b050ec 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -365,6 +365,7 @@ i≮i = <-irrefl refl
 module ≤-Reasoning where
   open import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
+    <-irrefl
     <-trans
     (resp₂ _<_)
     <⇒≤
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index c72bd0707f..aba1a777ee 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -254,7 +254,6 @@ Any-×⁻ pq with Prod.map₂ (Prod.map₂ find) (find pq)
 
     (p , q) ∎
 
-
   to∘from : ∀ pq → Any-×⁺ {xs = xs} (Any-×⁻ pq) ≡ pq
   to∘from pq with find pq
       | (λ (f : (proj₁ (find pq) ≡_) ⋐ _) → map∘find pq {f})
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 0072a2d325..638482131f 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -38,7 +38,6 @@ open import Relation.Binary.Consequences
 open import Relation.Binary.Morphism
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphism
 open import Relation.Binary.PropositionalEquality
-import Relation.Binary.Reasoning.Base.Triple as InequalityReasoning
 open import Relation.Nullary using (¬_; yes; no)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Nullary.Negation using (contradiction)
@@ -607,12 +606,18 @@ x ≤? y with <-cmp x y
 ------------------------------------------------------------------------
 -- Equational reasoning for _≤_ and _<_
 
-module ≤-Reasoning = InequalityReasoning
-  ≤-isPreorder
-  <-trans
-  (resp₂ _<_) <⇒≤
-  <-≤-trans ≤-<-trans
-  hiding (step-≈; step-≈˘)
+module ≤-Reasoning where
+
+  open import Relation.Binary.Reasoning.Base.Triple
+    ≤-isPreorder
+    <-irrefl
+    <-trans
+    (resp₂ _<_)
+    <⇒≤
+    <-≤-trans
+    ≤-<-trans
+    public
+    hiding (step-≈; step-≈˘)
 
 ------------------------------------------------------------------------
 -- Properties of _<ℕ_
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index be41e093ac..5ed9b64e43 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -495,6 +495,7 @@ m<1+n⇒m≤n (s≤s m≤n) = m≤n
 module ≤-Reasoning where
   open import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
+    <-irrefl
     <-trans
     (resp₂ _<_)
     <⇒≤
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 8910cc5998..959573218a 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -713,13 +713,16 @@ _>?_ = flip _<?_
 module ≤-Reasoning where
   import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
+    <-irrefl
     <-trans
     (resp₂ _<_)
     <⇒≤
     <-≤-trans
     ≤-<-trans
     as Triple
-  open Triple public hiding (step-≈; step-≈˘)
+
+  open Triple public
+    hiding (step-≈; step-≈˘)
 
   infixr 2 step-≃ step-≃˘
 
@@ -729,7 +732,6 @@ module ≤-Reasoning where
   syntax step-≃  x y∼z x≃y = x ≃⟨  x≃y ⟩ y∼z
   syntax step-≃˘ x y∼z y≃x = x ≃˘⟨ y≃x ⟩ y∼z
 
-
 ------------------------------------------------------------------------
 -- Properties of Positive/NonPositive/Negative/NonNegative and _≤_/_<_
 
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 169c3786ad..2ede5bb44c 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -542,13 +542,16 @@ _>?_ = flip _<?_
 module ≤-Reasoning where
   import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
+    <-irrefl
     <-trans
     <-resp-≃
     <⇒≤
     <-≤-trans
     ≤-<-trans
     as Triple
-  open Triple public hiding (step-≈; step-≈˘)
+
+  open Triple public
+    hiding (step-≈; step-≈˘)
 
   infixr 2 step-≃ step-≃˘
 
@@ -558,7 +561,6 @@ module ≤-Reasoning where
   syntax step-≃  x y∼z x≃y = x ≃⟨  x≃y ⟩ y∼z
   syntax step-≃˘ x y∼z y≃x = x ≃˘⟨ y≃x ⟩ y∼z
 
-
 ------------------------------------------------------------------------
 -- Properties of ↥_/↧_
 
diff --git a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
index 02a09a098e..49da62b10b 100644
--- a/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Tree/AVL/Indexed/Relation/Unary/Any/Properties.agda
@@ -35,7 +35,6 @@ open import Relation.Binary.Construct.Add.Extrema.Strict _<_ using ([<]-injectiv
 
 import Relation.Binary.Reasoning.StrictPartialOrder as <-Reasoning
 
-
 private
   variable
     v p q : Level
@@ -278,33 +277,33 @@ module _ {V : Value v} where
                                      joinʳ⁺-right⁺ kv lk ku′ bal (Any-insertWith-just ku seg′ pr rp)
 
     -- impossible cases
-    ... | here eq  | tri< k<k′ _ _ = flip contradiction (irrefl⁺ [ k ]) $ begin-strict
+    ... | here eq  | tri< k<k′ _ _ = begin-contradiction
       [ k  ] <⟨ [ k<k′ ]ᴿ ⟩
       [ k′ ] ≈⟨ [ sym eq ]ᴱ ⟩
       [ k  ] ∎
-    ... | here eq  | tri> _ _ k>k′ = flip contradiction (irrefl⁺ [ k ]) $ begin-strict
+    ... | here eq  | tri> _ _ k>k′ = begin-contradiction
       [ k  ] ≈⟨ [ eq ]ᴱ ⟩
       [ k′ ] <⟨ [ k>k′ ]ᴿ ⟩
       [ k  ] ∎
-    ... | left lp  | tri≈ _ k≈k′ _ = flip contradiction (irrefl⁺ [ k ]) $ begin-strict
+    ... | left lp  | tri≈ _ k≈k′ _ = begin-contradiction
       let k″ = Any.lookup lp .key; k≈k″ = lookup-result lp; (_ , k″<k′) = lookup-bounded lp in
       [ k  ] ≈⟨ [ k≈k″ ]ᴱ ⟩
       [ k″ ] <⟨ k″<k′ ⟩
       [ k′ ] ≈⟨ [ sym k≈k′ ]ᴱ ⟩
       [ k  ] ∎
-    ... | left lp  | tri> _ _ k>k′ = flip contradiction (irrefl⁺ [ k ]) $ begin-strict
+    ... | left lp  | tri> _ _ k>k′ = begin-contradiction
       let k″ = Any.lookup lp .key; k≈k″ = lookup-result lp; (_ , k″<k′) = lookup-bounded lp in
       [ k  ] ≈⟨ [ k≈k″ ]ᴱ ⟩
       [ k″ ] <⟨ k″<k′ ⟩
       [ k′ ] <⟨ [ k>k′ ]ᴿ ⟩
       [ k  ] ∎
-    ... | right rp | tri< k<k′ _ _ = flip contradiction (irrefl⁺ [ k ]) $ begin-strict
+    ... | right rp | tri< k<k′ _ _ = begin-contradiction
       let k″ = Any.lookup rp .key; k≈k″ = lookup-result rp; (k′<k″ , _) = lookup-bounded rp in
       [ k  ] <⟨ [ k<k′ ]ᴿ ⟩
       [ k′ ] <⟨ k′<k″ ⟩
       [ k″ ] ≈⟨ [ sym k≈k″ ]ᴱ ⟩
       [ k  ] ∎
-    ... | right rp | tri≈ _ k≈k′ _ = flip contradiction (irrefl⁺ [ k ]) $ begin-strict
+    ... | right rp | tri≈ _ k≈k′ _ = begin-contradiction
       let k″ = Any.lookup rp .key; k≈k″ = lookup-result rp; (k′<k″ , _) = lookup-bounded rp in
       [ k  ] ≈⟨ [ k≈k′ ]ᴱ ⟩
       [ k′ ] <⟨ k′<k″ ⟩
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index 487caa4c0e..ff1837b85d 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -261,6 +261,7 @@ module ≤-Reasoning  {_≈_ : Rel A ℓ₁} {_≼_ : Rel A ℓ₂}
 
   open import Relation.Binary.Reasoning.Base.Triple
     (≤-isPreorder ≼-po {n})
+    <-irrefl
     (<-trans ≼-po)
     (<-resp₂ isEquivalence ≤-resp-≈)
     <⇒≤
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 54ab06bd97..8c41fdfbd4 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -326,6 +326,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
 
 module ≤-Reasoning {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂}
                    (≈-isEquivalence : IsEquivalence _≈_)
+                   (≺-irrefl : Irreflexive _≈_ _≺_)
                    (≺-trans : Transitive _≺_)
                    (≺-resp-≈ : _≺_ Respects₂ _≈_)
                    (n : ℕ)
@@ -336,6 +337,7 @@ module ≤-Reasoning {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂}
 
   open import Relation.Binary.Reasoning.Base.Triple
     (≤-isPreorder ≈-isEquivalence ≺-trans ≺-resp-≈)
+    (<-irrefl ≺-irrefl)
     (<-trans ≈-isPartialEquivalence ≺-resp-≈ ≺-trans)
     (<-respects₂ ≈-isPartialEquivalence ≺-resp-≈)
     (<⇒≤ {m = n})
diff --git a/src/Relation/Binary/Reasoning/Base/Triple.agda b/src/Relation/Binary/Reasoning/Base/Triple.agda
index fe931722da..75aeaaef17 100644
--- a/src/Relation/Binary/Reasoning/Base/Triple.agda
+++ b/src/Relation/Binary/Reasoning/Base/Triple.agda
@@ -13,12 +13,13 @@
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Structures using (IsPreorder)
 open import Relation.Binary.Definitions
-  using (Transitive; _Respects₂_; Trans)
+  using (Transitive; _Respects₂_; Trans; Irreflexive)
 
 module Relation.Binary.Reasoning.Base.Triple {a ℓ₁ ℓ₂ ℓ₃} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} {_<_ : Rel A ℓ₃}
   (isPreorder : IsPreorder _≈_ _≤_)
-  (<-trans : Transitive _<_) (<-resp-≈ : _<_ Respects₂ _≈_) (<⇒≤ : _<_ ⇒ _≤_)
+  (<-irrefl : Irreflexive _≈_ _<_) (<-trans : Transitive _<_) (<-resp-≈ : _<_ Respects₂ _≈_)
+  (<⇒≤ : _<_ ⇒ _≤_)
   (<-≤-trans : Trans _<_ _≤_ _<_) (≤-<-trans : Trans _≤_ _<_ _<_)
   where
 
@@ -27,6 +28,8 @@ open import Function.Base using (case_of_; id)
 open import Level using (Level; _⊔_; Lift; lift)
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; sym)
+open import Relation.Nullary using (¬_)
+open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Nullary.Decidable.Core
   using (Dec; yes; no; True; toWitness)
 
@@ -79,7 +82,7 @@ extractEquality (isEquality x≈y) = x≈y
 -- See `Relation.Binary.Reasoning.Base.Partial` for the design decisions
 -- behind these combinators.
 
-infix  1 begin_ begin-strict_ begin-equality_
+infix  1 begin_ begin-strict_ begin-equality_ begin-contradiction_
 infixr 2 step-< step-≤ step-≈ step-≈˘ step-≡ step-≡˘ _≡⟨⟩_
 infix  3 _∎
 
@@ -96,6 +99,13 @@ begin-strict_ r {s} = extractStrict (toWitness s)
 begin-equality_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsEquality? r)} → x ≈ y
 begin-equality_ r {s} = extractEquality (toWitness s)
 
+begin-contradiction_ : ∀ {x} (r : x IsRelatedTo x) {s : True (IsStrict? r)} →
+                      ∀ {a} {A : Set a} → A
+begin-contradiction_ {x} r {s} = contradiction x<x (<-irrefl Eq.refl)
+  where
+  x<x : x < x
+  x<x = extractStrict (toWitness s)
+
 -- Step with the strict relation
 
 step-< : ∀ (x : A) {y z} → y IsRelatedTo z → x < y → x IsRelatedTo z
diff --git a/src/Relation/Binary/Reasoning/PartialOrder.agda b/src/Relation/Binary/Reasoning/PartialOrder.agda
index 80102d77f2..a6cb213e1a 100644
--- a/src/Relation/Binary/Reasoning/PartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/PartialOrder.agda
@@ -45,13 +45,16 @@ module Relation.Binary.Reasoning.PartialOrder
   {p₁ p₂ p₃} (P : Poset p₁ p₂ p₃) where
 
 open Poset P
-import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as Strict
+open import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_
+  as Strict
+  using (_<_)
 
 ------------------------------------------------------------------------
 -- Re-export contents of base module
 
 open import Relation.Binary.Reasoning.Base.Triple
   isPreorder
+  Strict.<-irrefl
   (Strict.<-trans isPartialOrder)
   (Strict.<-resp-≈ isEquivalence ≤-resp-≈)
   Strict.<⇒≤
diff --git a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
index 8e2b7a3a3d..71511d5192 100644
--- a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
@@ -53,9 +53,11 @@ import Relation.Binary.Construct.StrictToNonStrict _≈_ _<_ as NonStrict
 
 open import Relation.Binary.Reasoning.Base.Triple
   (NonStrict.isPreorder₂ isStrictPartialOrder)
+  irrefl
   trans
   <-resp-≈
   NonStrict.<⇒≤
   (NonStrict.<-≤-trans trans <-respʳ-≈)
   (NonStrict.≤-<-trans Eq.sym trans <-respˡ-≈)
   public
+

From 52c1702208d113dfa78c741abfe49871570a9f5a Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 27 Sep 2023 02:29:36 +0100
Subject: [PATCH 126/316] Refactor some lookup and truncation lemmas (#2101)

---
 CHANGELOG.md                 | 13 ++++++-
 src/Data/Fin/Base.agda       |  7 ++--
 src/Data/Fin/Properties.agda |  8 +++++
 src/Data/Vec/Base.agda       |  2 +-
 src/Data/Vec/Properties.agda | 70 ++++++++++++++++++++++++++++--------
 5 files changed, 80 insertions(+), 20 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2fc788c86d..5a42872be0 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1464,6 +1464,8 @@ Deprecated names
   sum-++-commute  ↦ sum-++
 
   take-drop-id ↦ take++drop≡id
+
+  lookup-inject≤-take ↦ lookup-take-inject≤
   ```
   and the type of the proof `zipWith-comm` has been generalised from:
   ```
@@ -2193,6 +2195,9 @@ Additions to existing modules
   cast-trans    : cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k
 
   fromℕ≢inject₁      : {i : Fin n} → fromℕ n ≢ inject₁ i
+
+  inject≤-trans      : inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (≤-trans m≤n n≤o)
+  inject≤-irrelevant : inject≤ i m≤n ≡ inject≤ i m≤n′
   ```
 
 * Added new functions in `Data.Integer.Base`:
@@ -2738,12 +2743,13 @@ Additions to existing modules
 
   cast-is-id    : cast eq xs ≡ xs
   subst-is-cast : subst (Vec A) eq xs ≡ cast eq xs
+  cast-sym      : cast eq xs ≡ ys → cast (sym eq) ys ≡ xs
   cast-trans    : cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
   map-cast      : map f (cast eq xs) ≡ cast eq (map f xs)
   lookup-cast   : lookup (cast eq xs) (Fin.cast eq i) ≡ lookup xs i
   lookup-cast₁  : lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)
   lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
-  cast-reverse : cast eq ∘ reverse ≗ reverse ∘ cast eq
+  cast-reverse  : cast eq ∘ reverse ≗ reverse ∘ cast eq
 
   zipwith-++ : zipWith f (xs ++ ys) (xs' ++ ys') ≡ zipWith f xs xs' ++ zipWith f ys ys'
 
@@ -2757,6 +2763,11 @@ Additions to existing modules
   cast-fromList : cast _ (fromList xs) ≡ fromList ys
   fromList-map  : cast _ (fromList (List.map f xs)) ≡ map f (fromList xs)
   fromList-++   : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
+
+  truncate≡take       : .(eq : n ≡ m + o) → truncate m≤n xs ≡ take m (cast eq xs)
+  take≡truncate       : take m xs ≡ truncate (m≤m+n m n) xs
+  lookup-truncate     : lookup (truncate m≤n xs) i ≡ lookup xs (Fin.inject≤ i m≤n)
+  lookup-take-inject≤ : lookup (take m xs) i ≡ lookup xs (Fin.inject≤ i (m≤m+n m n))
   ```
 
 * Added new proofs in `Data.Vec.Membership.Propositional.Properties`:
diff --git a/src/Data/Fin/Base.agda b/src/Data/Fin/Base.agda
index b591301ac0..2d5b0f638e 100644
--- a/src/Data/Fin/Base.agda
+++ b/src/Data/Fin/Base.agda
@@ -12,7 +12,6 @@
 module Data.Fin.Base where
 
 open import Data.Bool.Base using (Bool; true; false; T; not)
-open import Data.Empty using (⊥-elim)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; z≤n; s≤s; z<s; s<s; _^_)
 open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
@@ -115,13 +114,13 @@ inject₁ zero    = zero
 inject₁ (suc i) = suc (inject₁ i)
 
 inject≤ : Fin m → m ℕ.≤ n → Fin n
-inject≤ {_} {suc n} zero    _        = zero
+inject≤ {_} {suc n} zero    _         =  zero
 inject≤ {_} {suc n} (suc i) (s≤s m≤n) = suc (inject≤ i m≤n)
 
 -- lower₁ "i" _ = "i".
 
 lower₁ : ∀ (i : Fin (suc n)) → n ≢ toℕ i → Fin n
-lower₁ {zero}  zero    ne = ⊥-elim (ne refl)
+lower₁ {zero}  zero    ne = contradiction refl ne
 lower₁ {suc n} zero    _  = zero
 lower₁ {suc n} (suc i) ne = suc (lower₁ i (ne ∘ cong suc))
 
@@ -252,7 +251,7 @@ opposite {suc n} (suc i) = inject₁ (opposite i)
 -- McBride's "First-order unification by structural recursion".
 
 punchOut : ∀ {i j : Fin (suc n)} → i ≢ j → Fin n
-punchOut {_}     {zero}   {zero}  i≢j = ⊥-elim (i≢j refl)
+punchOut {_}     {zero}   {zero}  i≢j = contradiction refl i≢j
 punchOut {_}     {zero}   {suc j} _   = j
 punchOut {suc _} {suc i}  {zero}  _   = zero
 punchOut {suc _} {suc i}  {suc j} i≢j = suc (punchOut (i≢j ∘ cong suc))
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 99121e1b42..ef2073269f 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -547,12 +547,20 @@ inject≤-idempotent {_} {suc n} {suc o} zero    _   _   _ = refl
 inject≤-idempotent {_} {suc n} {suc o} (suc i) (s≤s m≤n) (s≤s n≤o) (s≤s m≤o) =
   cong suc (inject≤-idempotent i m≤n n≤o m≤o)
 
+inject≤-trans : ∀ (i : Fin m) (m≤n : m ℕ.≤ n) (n≤o : n ℕ.≤ o) →
+                inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (ℕₚ.≤-trans m≤n n≤o)
+inject≤-trans i m≤n n≤o = inject≤-idempotent i m≤n n≤o _
+
 inject≤-injective : ∀ (m≤n m≤n′ : m ℕ.≤ n) i j →
                     inject≤ i m≤n ≡ inject≤ j m≤n′ → i ≡ j
 inject≤-injective (s≤s p) (s≤s q) zero    zero    eq = refl
 inject≤-injective (s≤s p) (s≤s q) (suc i) (suc j) eq =
   cong suc (inject≤-injective p q i j (suc-injective eq))
 
+inject≤-irrelevant : ∀ (m≤n m≤n′ : m ℕ.≤ n) i →
+                    inject≤ i m≤n ≡ inject≤ i m≤n′
+inject≤-irrelevant m≤n m≤n′ i =  cong (inject≤ i) (ℕₚ.≤-irrelevant m≤n m≤n′)
+
 ------------------------------------------------------------------------
 -- pred
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index ed8817c66e..507ac05463 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -290,7 +290,7 @@ uncons (x ∷ xs) = x , xs
 
 -- Take the first 'm' elements of a vector.
 truncate : ∀ {m n} → m ≤ n → Vec A n → Vec A m
-truncate z≤n      _        = []
+truncate {m = zero} _ _    = []
 truncate (s≤s le) (x ∷ xs) = x ∷ (truncate le xs)
 
 -- Pad out a vector with extra elements.
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 38ba95ae7c..c5b92ec2a1 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -10,27 +10,27 @@ module Data.Vec.Properties where
 
 open import Algebra.Definitions
 open import Data.Bool.Base using (true; false)
-open import Data.Fin.Base as Fin using (Fin; zero; suc; toℕ; fromℕ<; _↑ˡ_; _↑ʳ_)
+open import Data.Fin.Base as Fin
+  using (Fin; zero; suc; toℕ; fromℕ<; _↑ˡ_; _↑ʳ_)
 open import Data.List.Base as List using (List)
 import Data.List.Properties as Listₚ
 open import Data.Nat.Base
 open import Data.Nat.Properties
-  using (+-assoc; m≤n⇒m≤1+n; ≤-refl; ≤-trans; suc-injective; +-comm; +-suc)
+  using (+-assoc; m≤n⇒m≤1+n; m≤m+n; ≤-refl; ≤-trans; ≤-irrelevant; ≤⇒≤″; suc-injective; +-comm; +-suc)
 open import Data.Product.Base as Prod
   using (_×_; _,_; proj₁; proj₂; <_,_>; uncurry)
 open import Data.Sum.Base using ([_,_]′)
 open import Data.Sum.Properties using ([,]-map)
 open import Data.Vec.Base
 open import Function.Base
--- open import Function.Inverse using (_↔_; inverse)
 open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; _≢_; _≗_; refl; sym; trans; cong; cong₂; subst; module ≡-Reasoning)
 open import Relation.Unary using (Pred; Decidable)
-open import Relation.Nullary.Decidable using (Dec; does; yes; no; _×-dec_; map′)
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Decidable.Core using (Dec; does; yes; no; _×-dec_; map′)
+open import Relation.Nullary.Negation.Core using (contradiction)
 
 open ≡-Reasoning
 
@@ -122,6 +122,19 @@ truncate-trans : ∀ {p} (m≤n : m ≤ n) (n≤p : n ≤ p) (xs : Vec A p) →
 truncate-trans z≤n       n≤p       xs = refl
 truncate-trans (s≤s m≤n) (s≤s n≤p) (x ∷ xs) = cong (x ∷_) (truncate-trans m≤n n≤p xs)
 
+truncate-irrelevant : (m≤n₁ m≤n₂ : m ≤ n) → truncate {A = A} m≤n₁ ≗ truncate m≤n₂
+truncate-irrelevant m≤n₁ m≤n₂ xs = cong (λ m≤n → truncate m≤n xs) (≤-irrelevant m≤n₁ m≤n₂)
+
+truncate≡take : (m≤n : m ≤ n) (xs : Vec A n) .(eq : n ≡ m + o) →
+                truncate m≤n xs ≡ take m (cast eq xs)
+truncate≡take z≤n       _        eq = refl
+truncate≡take (s≤s m≤n) (x ∷ xs) eq = cong (x ∷_) (truncate≡take m≤n xs (suc-injective eq))
+
+take≡truncate : ∀ m (xs : Vec A (m + n)) →
+                take m xs ≡ truncate (m≤m+n m n) xs
+take≡truncate zero    _        = refl
+take≡truncate (suc m) (x ∷ xs) = cong (x ∷_) (take≡truncate m xs)
+
 ------------------------------------------------------------------------
 -- pad
 
@@ -171,10 +184,20 @@ lookup⇒[]= (suc i) (_ ∷ xs) p    = there (lookup⇒[]= i xs p)
   []=⇒lookup∘lookup⇒[]= (x ∷ xs) zero    refl = refl
   []=⇒lookup∘lookup⇒[]= (x ∷ xs) (suc i) p    = []=⇒lookup∘lookup⇒[]= xs i p
 
-lookup-inject≤-take : ∀ m (m≤m+n : m ≤ m + n) (i : Fin m) (xs : Vec A (m + n)) →
-                      lookup xs (Fin.inject≤ i m≤m+n) ≡ lookup (take m xs) i
-lookup-inject≤-take (suc m) m≤m+n zero (x ∷ xs) = refl
-lookup-inject≤-take (suc m) (s≤s m≤m+n) (suc i) (x ∷ xs) = lookup-inject≤-take m m≤m+n i xs
+lookup-truncate : (m≤n : m ≤ n) (xs : Vec A n) (i : Fin m) →
+                  lookup (truncate m≤n xs) i ≡ lookup xs (Fin.inject≤ i m≤n)
+lookup-truncate (s≤s m≤m+n) (_ ∷ _)  zero    = refl
+lookup-truncate (s≤s m≤m+n) (_ ∷ xs) (suc i) = lookup-truncate m≤m+n xs i
+
+lookup-take-inject≤ : (xs : Vec A (m + n)) (i : Fin m) →
+                      lookup (take m xs) i ≡ lookup xs (Fin.inject≤ i (m≤m+n m n))
+lookup-take-inject≤ {m = m} {n = n} xs i = begin
+  lookup (take _ xs) i
+    ≡⟨ cong (λ ys → lookup ys i) (take≡truncate m xs) ⟩
+  lookup (truncate _ xs) i
+    ≡⟨ lookup-truncate (m≤m+n m n) xs i ⟩
+  lookup xs (Fin.inject≤ i (m≤m+n m n))
+    ∎ where open ≡-Reasoning
 
 ------------------------------------------------------------------------
 -- updateAt (_[_]%=_)
@@ -348,6 +371,13 @@ cast-is-id eq (x ∷ xs) = cong (x ∷_) (cast-is-id (suc-injective eq) xs)
 subst-is-cast : (eq : m ≡ n) (xs : Vec A m) → subst (Vec A) eq xs ≡ cast eq xs
 subst-is-cast refl xs = sym (cast-is-id refl xs)
 
+cast-sym : .(eq : m ≡ n) {xs : Vec A m} {ys : Vec A n} →
+           cast eq xs ≡ ys → cast (sym eq) ys ≡ xs
+cast-sym eq {xs = []}     {ys = []}     _ = refl
+cast-sym eq {xs = x ∷ xs} {ys = y ∷ ys} xxs[eq]≡yys =
+  let x≡y , xs[eq]≡ys = ∷-injective xxs[eq]≡yys
+  in cong₂ _∷_ (sym x≡y) (cast-sym (suc-injective eq) xs[eq]≡ys)
+
 cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (xs : Vec A m) →
              cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
 cast-trans {m = zero}  {n = zero}  {o = zero}  eq₁ eq₂ [] = refl
@@ -399,9 +429,9 @@ map-updateAt (x ∷ xs) (suc i) eq = cong (_ ∷_) (map-updateAt xs i eq)
 
 map-insert : ∀ (f : A → B) (x : A) (xs : Vec A n) (i : Fin (suc n)) →
              map f (insert xs i x) ≡ insert (map f xs) i (f x)
-map-insert f _ []        Fin.zero = refl
-map-insert f _ (x' ∷ xs) Fin.zero = refl
-map-insert f x (x' ∷ xs) (Fin.suc i) = cong (_ ∷_) (map-insert f x xs i)
+map-insert f _ []        zero    = refl
+map-insert f _ (x' ∷ xs) zero    = refl
+map-insert f x (x' ∷ xs) (suc i) = cong (_ ∷_) (map-insert f x xs i)
 
 map-[]≔ : ∀ (f : A → B) (xs : Vec A n) (i : Fin n) →
           map f (xs [ i ]≔ x) ≡ map f xs [ i ]≔ f x
@@ -1245,13 +1275,11 @@ sum-++-commute = sum-++
 "Warning: sum-++-commute was deprecated in v2.0.
 Please use sum-++ instead."
 #-}
-
 take-drop-id = take++drop≡id
 {-# WARNING_ON_USAGE take-drop-id
 "Warning: take-drop-id was deprecated in v2.0.
 Please use take++drop≡id instead."
 #-}
-
 take-distr-zipWith = take-zipWith
 {-# WARNING_ON_USAGE take-distr-zipWith
 "Warning: take-distr-zipWith was deprecated in v2.0.
@@ -1272,3 +1300,17 @@ drop-distr-map = drop-map
 "Warning: drop-distr-map was deprecated in v2.0.
 Please use drop-map instead."
 #-}
+lookup-inject≤-take : ∀ m (m≤m+n : m ≤ m + n) (i : Fin m) (xs : Vec A (m + n)) →
+                      lookup xs (Fin.inject≤ i m≤m+n) ≡ lookup (take m xs) i
+lookup-inject≤-take m m≤m+n i xs = sym (begin
+  lookup (take m xs) i
+    ≡⟨ lookup-take-inject≤ xs i ⟩
+  lookup xs (Fin.inject≤ i _)
+    ≡⟨ cong ((lookup xs) ∘ (Fin.inject≤ i)) (≤-irrelevant _ _) ⟩
+  lookup xs (Fin.inject≤ i m≤m+n)
+    ∎) where open ≡-Reasoning
+{-# WARNING_ON_USAGE lookup-inject≤-take
+"Warning: lookup-inject≤-take was deprecated in v2.0.
+Please use lookup-take-inject≤ or lookup-truncate, take≡truncate instead."
+#-}
+

From 2589f899bbf939dd1236484a7494a8a84d1a4ed2 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 27 Sep 2023 19:43:08 +0900
Subject: [PATCH 127/316] Add style-guide note about local suffix (#2109)

---
 notes/style-guide.md | 10 ++++++++++
 1 file changed, 10 insertions(+)

diff --git a/notes/style-guide.md b/notes/style-guide.md
index 2f9fa18553..d6201abe88 100644
--- a/notes/style-guide.md
+++ b/notes/style-guide.md
@@ -461,6 +461,16 @@ word within a compound word is capitalized except for the first word.
   map⁻ : Pointwise R (map f as) (map g bs) → Pointwise (λ a b → R (f a) (g b)) as bs
   ```
 
+* When specifying a property over a container, there are usually two choices. Either
+  assume the property holds for generally (e.g. `map id xs ≡ xs`) or a assume that
+  it only holds for the elements within the container (e.g. `All (λ x → f x ≡ x) xs → map f xs ≡ xs`).
+  The naming convention is to add a `-local` suffix on to the name of the latter variety.
+  e.g.
+  ```agda
+  map-id       :  map id xs ≡ xs
+  map-id-local :  All (λ x → f x ≡ x) xs → map f xs ≡ xs
+  ```
+
 #### Keywords
 
 * If the name of something clashes with a keyword in Agda, then convention

From 824cca076f8bb89c11aea27df11e4b46657c8758 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 28 Sep 2023 07:10:37 +0900
Subject: [PATCH 128/316] Weakened pre-conditions of grouped map lemmas (#2108)

---
 CHANGELOG.md                                  | 16 ++++++++-
 .../List/Relation/Unary/All/Properties.agda   | 15 ++++++--
 .../Relation/Unary/Grouped/Properties.agda    | 34 +++++++------------
 3 files changed, 40 insertions(+), 25 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5a42872be0..4e95417e85 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -851,6 +851,10 @@ Non-backwards compatible changes
   found in `Data.List.NonEmpty.Properties` under the names `groupSeqs-groups`
   and `ungroupSeqs` and `groupSeqs`.
 
+* In `Data.List.Relation.Unary.Grouped.Properties` the proofs `map⁺` and `map⁻`
+  have had their preconditions weakened so the equivalences no longer require congruence
+  proofs.
+
 * The constructors `+0` and `+[1+_]` from `Data.Integer.Base` are no longer
   exported by `Data.Rational.Base`. You will have to open `Data.Integer(.Base)`
   directly to use them.
@@ -1349,6 +1353,11 @@ Deprecated names
   map-with-∈↔  ↦  mapWith∈↔
   ```
 
+* In `Data.List.Relation.Unary.All.Properties`:
+  ```
+  gmap  ↦  gmap⁺
+  ```
+
 * In `Data.Nat.Properties`:
   ```
   suc[pred[n]]≡n  ↦  suc-pred
@@ -3462,6 +3471,11 @@ This is a full list of proofs that have changed form to use irrelevant instance
   #-congˡ       : y ≈ z → x # y → x # z
   ```
 
+* Added new proof to `Data.List.Relation.Unary.All.Properties`:
+  ```agda
+  gmap⁻ : Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P
+  ```
+
 * Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`
   and `Data.List.Relation.Unary.Sorted.TotalOrder.Properties`.
   ```agda
@@ -3484,4 +3498,4 @@ This is a full list of proofs that have changed form to use irrelevant instance
       b #⟨ b#c ⟩
       c ≈⟨ c≈d ⟩
       d ∎
-    ```
\ No newline at end of file
+    ```
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index 6af4fd2a8c..da617af44b 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -371,8 +371,11 @@ map⁻ {xs = _ ∷ _} (p ∷ ps) = p ∷ map⁻ ps
 
 -- A variant of All.map.
 
-gmap : ∀ {f : A → B} → P ⋐ Q ∘ f → All P ⋐ All Q ∘ map f
-gmap g = map⁺ ∘ All.map g
+gmap⁺ : ∀ {f : A → B} → P ⋐ Q ∘ f → All P ⋐ All Q ∘ map f
+gmap⁺ g = map⁺ ∘ All.map g
+
+gmap⁻ : ∀ {f : A → B} → Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P
+gmap⁻ g = All.map g ∘ map⁻
 
 ------------------------------------------------------------------------
 -- mapMaybe
@@ -679,7 +682,7 @@ replicate⁻ (px ∷ _) = px
 
 inits⁺ : All P xs → All (All P) (inits xs)
 inits⁺ []         = [] ∷ []
-inits⁺ (px ∷ pxs) = [] ∷ gmap (px ∷_) (inits⁺ pxs)
+inits⁺ (px ∷ pxs) = [] ∷ gmap⁺ (px ∷_) (inits⁺ pxs)
 
 inits⁻ : ∀ xs → All (All P) (inits xs) → All P xs
 inits⁻ []               pxs                   = []
@@ -773,3 +776,9 @@ updateAt-cong-relative = updateAt-cong-local
 "Warning: updateAt-cong-relative was deprecated in v2.0.
 Please use updateAt-cong-local instead."
 #-}
+
+gmap = gmap⁺
+{-# WARNING_ON_USAGE gmap
+"Warning: gmap was deprecated in v2.0.
+Please use gmap⁺ instead."
+#-}
diff --git a/src/Data/List/Relation/Unary/Grouped/Properties.agda b/src/Data/List/Relation/Unary/Grouped/Properties.agda
index f020cdd233..2907b9a665 100644
--- a/src/Data/List/Relation/Unary/Grouped/Properties.agda
+++ b/src/Data/List/Relation/Unary/Grouped/Properties.agda
@@ -14,10 +14,10 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
 open import Data.List.Relation.Unary.Grouped
-open import Function.Base using (_∘_)
-open import Function.Bundles using (module Equivalence; _⇔_)
+open import Data.Product.Base using (_,_)
+open import Function.Base using (_∘_; _on_)
 open import Relation.Binary.Definitions as B
-open import Relation.Binary.Core using (REL; Rel)
+open import Relation.Binary.Core using (_⇔_; REL; Rel)
 open import Relation.Unary as U using (Pred)
 open import Relation.Nullary using (¬_; does; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
@@ -26,31 +26,23 @@ open import Level
 private
   variable
     a b c p q : Level
-    A : Set a
-    B : Set b
-    C : Set c
+    A B C : Set a
 
 ------------------------------------------------------------------------
 -- map
 
 module _ (P : Rel A p) (Q : Rel B q) where
 
-  map⁺ : ∀ {f xs} → (∀ {x y} → P x y ⇔ Q (f x) (f y)) → Grouped P xs → Grouped Q (map f xs)
-  map⁺ {f} {[]} P⇔Q [] = []
-  map⁺ {f} {x ∷ xs} P⇔Q (all[¬Px,xs] ∷≉ g) = aux all[¬Px,xs] ∷≉ map⁺ P⇔Q g where
-    aux : ∀ {ys} → All (λ y → ¬ P x y) ys → All (λ y → ¬ Q (f x) y) (map f ys)
-    aux [] = []
-    aux (py ∷ pys) = py ∘ Equivalence.from P⇔Q ∷ aux pys
-  map⁺ {f} {x₁ ∷ x₂ ∷ xs} P⇔Q (Px₁x₂ ∷≈ g) = Equivalence.to P⇔Q Px₁x₂ ∷≈ map⁺ P⇔Q g
+  map⁺ : ∀ {f xs} → P ⇔ (Q on f) → Grouped P xs → Grouped Q (map f xs)
+  map⁺ P⇔Q            []                      = []
+  map⁺ P⇔Q@(_ , Q⇒P) (all[¬Px,xs] ∷≉ g[xs]) = All.gmap⁺ (_∘ Q⇒P) all[¬Px,xs] ∷≉ map⁺ P⇔Q g[xs]
+  map⁺ P⇔Q@(P⇒Q , _) (Px₁x₂ ∷≈ g[xs])       = P⇒Q Px₁x₂ ∷≈ map⁺ P⇔Q g[xs]
 
-  map⁻ : ∀ {f xs} → (∀ {x y} → P x y ⇔ Q (f x) (f y)) → Grouped Q (map f xs) → Grouped P xs
-  map⁻ {f} {[]} P⇔Q [] = []
-  map⁻ {f} {x ∷ []} P⇔Q ([] ∷≉ []) = [] ∷≉ []
-  map⁻ {f} {x₁ ∷ x₂ ∷ xs} P⇔Q (all[¬Qx,xs] ∷≉ g) = aux all[¬Qx,xs] ∷≉ map⁻ P⇔Q g where
-    aux : ∀ {ys} → All (λ y → ¬ Q (f x₁) y) (map f ys) → All (λ y → ¬ P x₁ y) ys
-    aux {[]} [] = []
-    aux {y ∷ ys} (py ∷ pys) = py ∘ Equivalence.to P⇔Q ∷ aux pys
-  map⁻ {f} {x₁ ∷ x₂ ∷ xs} P⇔Q (Qx₁x₂ ∷≈ g) = Equivalence.from P⇔Q Qx₁x₂ ∷≈ map⁻ P⇔Q g
+  map⁻ : ∀ {f xs} → P ⇔ (Q on f) → Grouped Q (map f xs) → Grouped P xs
+  map⁻ {xs = []}          P⇔Q            []                  = []
+  map⁻ {xs = _ ∷ []}     P⇔Q            ([] ∷≉ [])         = [] ∷≉ []
+  map⁻ {xs = _ ∷ _ ∷ _} P⇔Q@(P⇒Q , _) (all[¬Qx,xs] ∷≉ g) = All.gmap⁻ (_∘ P⇒Q) all[¬Qx,xs] ∷≉ map⁻ P⇔Q g
+  map⁻ {xs = _ ∷ _ ∷ _} P⇔Q@(_ , Q⇒P) (Qx₁x₂ ∷≈ g)       = Q⇒P Qx₁x₂ ∷≈ map⁻ P⇔Q g
 
 ------------------------------------------------------------------------
 -- [_]

From 29e22dfc3045fc84635340073931cc7bd13fef7e Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 28 Sep 2023 07:11:43 +0900
Subject: [PATCH 129/316] Undeprecate inspect idiom (#2107)

---
 CHANGELOG.md                                  |  8 -------
 .../Binary/PropositionalEquality.agda         | 23 +++++++------------
 2 files changed, 8 insertions(+), 23 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 4e95417e85..f88667b87b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1556,14 +1556,6 @@ Deprecated names
 * This fixes the fact we had picked the wrong name originally. The erased modality
   corresponds to @0 whereas the irrelevance one corresponds to `.`.
 
-### Deprecated `Relation.Binary.PropositionalEquality.inspect`
-    in favour of `with ... in ...` syntax (issue #1580; PRs #1630, #1930)
-
-* In `Relation.Binary.PropositionalEquality`
-  both the record type `Reveal_·_is_`
-  and its principal mode of use, `inspect`,
-  have been deprecated in favour of the new `with ... in ...` syntax.
-  See the documentation of [with-abstraction equality](https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality)
 
 New modules
 -----------
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 418e9b2ade..6ff59d8c28 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -104,15 +104,17 @@ module _ (_≟_ : DecidableEquality A) {x y : A} where
   ≢-≟-identity = dec-no (x ≟ y)
 
 
+------------------------------------------------------------------------
+-- Inspect
 
+-- Inspect can be used when you want to pattern match on the result r
+-- of some expression e, and you also need to "remember" that r ≡ e.
 
-------------------------------------------------------------------------
--- DEPRECATED NAMES
-------------------------------------------------------------------------
--- Please use the new names as continuing support for the old names is
--- not guaranteed.
+-- See README.Inspect for an explanation of how/why to use this.
 
--- Version 2.0
+-- Normally (but not always) the new `with ... in` syntax described at
+-- https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality
+-- can be used instead."
 
 record Reveal_·_is_ {A : Set a} {B : A → Set b}
                     (f : (x : A) → B x) (x : A) (y : B x) :
@@ -123,12 +125,3 @@ record Reveal_·_is_ {A : Set a} {B : A → Set b}
 inspect : ∀ {A : Set a} {B : A → Set b}
           (f : (x : A) → B x) (x : A) → Reveal f · x is f x
 inspect f x = [ refl ]
-
-{-# WARNING_ON_USAGE Reveal_·_is_
-"Warning: Reveal_·_is_ was deprecated in v2.0.
-Please use new `with ... in` syntax described at https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality instead."
-#-}
-{-# WARNING_ON_USAGE inspect
-"Warning: inspect was deprecated in v2.0.
-Please use new `with ... in` syntax described at https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality instead."
-#-}

From 8a2550f07fe5e014c9e7e50d5d40a9767fd99fd7 Mon Sep 17 00:00:00 2001
From: Nils Anders Danielsson <nad@cse.gu.se>
Date: Thu, 28 Sep 2023 01:00:57 +0200
Subject: [PATCH 130/316] Add some lemmas related to renamings and
 substitutions (#1750)

---
 CHANGELOG.md                                  |  25 +++
 .../Data/Fin/Substitution/UntypedLambda.agda  | 103 +++++++++++
 src/Data/Fin/Substitution.agda                |  56 +++---
 src/Data/Fin/Substitution/Example.agda        |  10 +-
 src/Data/Fin/Substitution/Lemmas.agda         | 170 ++++++++++++------
 src/Data/Fin/Substitution/List.agda           |  16 +-
 6 files changed, 294 insertions(+), 86 deletions(-)
 create mode 100644 README/Data/Fin/Substitution/UntypedLambda.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index f88667b87b..35a0991cac 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1117,6 +1117,10 @@ Deprecated modules
 
 * The module `Data.Nat.Properties.Core` has been deprecated, and its one entry moved to `Data.Nat.Properties`
 
+### Deprecation of `Data.Fin.Substitution.Example`
+
+* The module `Data.Fin.Substitution.Example` has been deprecated, and moved to `README.Data.Fin.Substitution.UntypedLambda`
+
 ### Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`
 
 * This module has been deprecated, as none of its contents actually depended on axiom K. The contents has been moved to `Data.Product.Function.Dependent.Setoid`.
@@ -2201,6 +2205,27 @@ Additions to existing modules
   inject≤-irrelevant : inject≤ i m≤n ≡ inject≤ i m≤n′
   ```
 
+* Changed the fixity of `Data.Fin.Substitution.TermSubst._/Var_`.
+  ```agda
+  infix 8 ↦ infixl 8
+  ```
+
+* Added new lemmas in `Data.Fin.Substitution.Lemmas.TermLemmas`:
+  ```
+  map-var≡ : {ρ₁ : Sub Fin m n} {ρ₂ : Sub T m n} {f : Fin m → Fin n} →
+             (∀ x → lookup ρ₁ x ≡ f x) →
+             (∀ x → lookup ρ₂ x ≡ T.var (f x)) →
+             map T.var ρ₁ ≡ ρ₂
+  wk≡wk : map T.var VarSubst.wk ≡ T.wk {n = n}
+  id≡id : map T.var VarSubst.id ≡ T.id {n = n}
+  sub≡sub : {x : Fin n} → map T.var (VarSubst.sub x) ≡ T.sub (T.var x)
+  ↑≡↑ : {ρ : Sub Fin m n} → map T.var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
+  /Var≡/ : {ρ : Sub Fin m n} {t} → t /Var ρ ≡ t T./ map T.var ρ
+  sub-renaming-commutes : {ρ : Sub T m n} →
+    t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
+  sub-commutes-with-renaming : {ρ : Sub Fin m n} →
+    t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)
+
 * Added new functions in `Data.Integer.Base`:
   ```
   _^_ : ℤ → ℕ → ℤ
diff --git a/README/Data/Fin/Substitution/UntypedLambda.agda b/README/Data/Fin/Substitution/UntypedLambda.agda
new file mode 100644
index 0000000000..126959958f
--- /dev/null
+++ b/README/Data/Fin/Substitution/UntypedLambda.agda
@@ -0,0 +1,103 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- An example of how Data.Fin.Substitution can be used: a definition
+-- of substitution for the untyped λ-calculus, along with some lemmas
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module README.Data.Fin.Substitution.UntypedLambda where
+
+open import Data.Fin.Substitution
+open import Data.Fin.Substitution.Lemmas
+open import Data.Nat.Base hiding (_/_)
+open import Data.Fin.Base using (Fin)
+open import Data.Vec.Base
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; cong; cong₂; module ≡-Reasoning)
+open import Relation.Binary.Construct.Closure.ReflexiveTransitive
+  using (Star; ε; _◅_)
+
+open ≡-Reasoning
+
+private
+  variable
+    m n : ℕ
+
+------------------------------------------------------------------------
+-- A representation of the untyped λ-calculus. Uses de Bruijn indices.
+
+infixl 9 _·_
+
+data Lam (n : ℕ) : Set where
+  var : (x : Fin n) → Lam n
+  ƛ   : (t : Lam (suc n)) → Lam n
+  _·_ : (t₁ t₂ : Lam n) → Lam n
+
+------------------------------------------------------------------------
+-- Code for applying substitutions.
+
+module LamApp {ℓ} {T : ℕ → Set ℓ} (l : Lift T Lam) where
+  open Lift l hiding (var)
+
+  -- Applies a substitution to a term.
+
+  infixl 8 _/_
+
+  _/_ : Lam m → Sub T m n → Lam n
+  var x   / ρ = lift (lookup ρ x)
+  ƛ t     / ρ = ƛ (t / ρ ↑)
+  t₁ · t₂ / ρ = (t₁ / ρ) · (t₂ / ρ)
+
+  open Application (record { _/_ = _/_ }) using (_/✶_)
+
+  -- Some lemmas about _/_.
+
+  ƛ-/✶-↑✶ : ∀ k {t} (ρs : Subs T m n) →
+            ƛ t /✶ ρs ↑✶ k ≡ ƛ (t /✶ ρs ↑✶ suc k)
+  ƛ-/✶-↑✶ k ε        = refl
+  ƛ-/✶-↑✶ k (ρ ◅ ρs) = cong (_/ _) (ƛ-/✶-↑✶ k ρs)
+
+  ·-/✶-↑✶ : ∀ k {t₁ t₂} (ρs : Subs T m n) →
+            t₁ · t₂ /✶ ρs ↑✶ k ≡ (t₁ /✶ ρs ↑✶ k) · (t₂ /✶ ρs ↑✶ k)
+  ·-/✶-↑✶ k ε        = refl
+  ·-/✶-↑✶ k (ρ ◅ ρs) = cong (_/ _) (·-/✶-↑✶ k ρs)
+
+lamSubst : TermSubst Lam
+lamSubst = record { var = var; app = LamApp._/_ }
+
+open TermSubst lamSubst hiding (var)
+
+------------------------------------------------------------------------
+-- Substitution lemmas.
+
+lamLemmas : TermLemmas Lam
+lamLemmas = record
+  { termSubst = lamSubst
+  ; app-var   = refl
+  ; /✶-↑✶     = Lemma./✶-↑✶
+  }
+  where
+  module Lemma {T₁ T₂} {lift₁ : Lift T₁ Lam} {lift₂ : Lift T₂ Lam} where
+
+    open Lifted lift₁ using () renaming (_↑✶_ to _↑✶₁_; _/✶_ to _/✶₁_)
+    open Lifted lift₂ using () renaming (_↑✶_ to _↑✶₂_; _/✶_ to _/✶₂_)
+
+    /✶-↑✶ : (ρs₁ : Subs T₁ m n) (ρs₂ : Subs T₂ m n) →
+            (∀ k x → var x /✶₁ ρs₁ ↑✶₁ k ≡ var x /✶₂ ρs₂ ↑✶₂ k) →
+             ∀ k t → t     /✶₁ ρs₁ ↑✶₁ k ≡ t     /✶₂ ρs₂ ↑✶₂ k
+    /✶-↑✶ ρs₁ ρs₂ hyp k (var x) = hyp k x
+    /✶-↑✶ ρs₁ ρs₂ hyp k (ƛ t)   = begin
+      ƛ t /✶₁ ρs₁ ↑✶₁ k        ≡⟨ LamApp.ƛ-/✶-↑✶ _ k ρs₁ ⟩
+      ƛ (t /✶₁ ρs₁ ↑✶₁ suc k)  ≡⟨ cong ƛ (/✶-↑✶ ρs₁ ρs₂ hyp (suc k) t) ⟩
+      ƛ (t /✶₂ ρs₂ ↑✶₂ suc k)  ≡⟨ sym (LamApp.ƛ-/✶-↑✶ _ k ρs₂) ⟩
+      ƛ t /✶₂ ρs₂ ↑✶₂ k        ∎
+    /✶-↑✶ ρs₁ ρs₂ hyp k (t₁ · t₂) = begin
+      t₁ · t₂ /✶₁ ρs₁ ↑✶₁ k                    ≡⟨ LamApp.·-/✶-↑✶ _ k ρs₁ ⟩
+      (t₁ /✶₁ ρs₁ ↑✶₁ k) · (t₂ /✶₁ ρs₁ ↑✶₁ k)  ≡⟨ cong₂ _·_ (/✶-↑✶ ρs₁ ρs₂ hyp k t₁)
+                                                            (/✶-↑✶ ρs₁ ρs₂ hyp k t₂) ⟩
+      (t₁ /✶₂ ρs₂ ↑✶₂ k) · (t₂ /✶₂ ρs₂ ↑✶₂ k)  ≡⟨ sym (LamApp.·-/✶-↑✶ _ k ρs₂) ⟩
+      t₁ · t₂ /✶₂ ρs₂ ↑✶₂ k                    ∎
+
+open TermLemmas lamLemmas public hiding (var)
diff --git a/src/Data/Fin/Substitution.agda b/src/Data/Fin/Substitution.agda
index 18769dde37..84a4484715 100644
--- a/src/Data/Fin/Substitution.agda
+++ b/src/Data/Fin/Substitution.agda
@@ -4,12 +4,12 @@
 -- Substitutions
 ------------------------------------------------------------------------
 
--- Uses a variant of Conor McBride's technique from "Type-Preserving
--- Renaming and Substitution".
+-- Uses a variant of Conor McBride's technique from his paper
+-- "Type-Preserving Renaming and Substitution".
 
--- See Data.Fin.Substitution.Example for an example of how this module
--- can be used: a definition of substitution for the untyped
--- λ-calculus.
+-- See README.Data.Fin.Substitution.UntypedLambda for an example
+-- of how this module can be used: a definition of substitution for
+-- the untyped λ-calculus.
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
@@ -24,23 +24,29 @@ open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Level using (Level; _⊔_; 0ℓ)
 open import Relation.Unary using (Pred)
 
+private
+  variable
+    ℓ ℓ₁ ℓ₂ : Level
+    k m n o : ℕ
+
+
 ------------------------------------------------------------------------
 -- General functionality
 
 -- A Sub T m n is a substitution which, when applied to something with
 -- at most m variables, yields something with at most n variables.
 
-Sub : ∀ {ℓ} → Pred ℕ ℓ → ℕ → ℕ → Set ℓ
+Sub : Pred ℕ ℓ → ℕ → ℕ → Set ℓ
 Sub T m n = Vec (T n) m
 
 -- A /reversed/ sequence of matching substitutions.
 
-Subs : ∀ {ℓ} → Pred ℕ ℓ → ℕ → ℕ → Set ℓ
+Subs : Pred ℕ ℓ → ℕ → ℕ → Set ℓ
 Subs T = flip (Star (flip (Sub T)))
 
 -- Some simple substitutions.
 
-record Simple {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
+record Simple (T : Pred ℕ ℓ) : Set ℓ where
   infix  10 _↑
   infixl 10 _↑⋆_ _↑✶_
 
@@ -50,40 +56,39 @@ record Simple {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
 
   -- Lifting.
 
-  _↑ : ∀ {m n} → Sub T m n → Sub T (suc m) (suc n)
+  _↑ : Sub T m n → Sub T (suc m) (suc n)
   ρ ↑ = var zero ∷ map weaken ρ
 
-  _↑⋆_ : ∀ {m n} → Sub T m n → ∀ k → Sub T (k + m) (k + n)
+  _↑⋆_ : Sub T m n → ∀ k → Sub T (k + m) (k + n)
   ρ ↑⋆ zero  = ρ
   ρ ↑⋆ suc k = (ρ ↑⋆ k) ↑
 
-  _↑✶_ : ∀ {m n} → Subs T m n → ∀ k → Subs T (k + m) (k + n)
+  _↑✶_ : Subs T m n → ∀ k → Subs T (k + m) (k + n)
   ρs ↑✶ k = Star.gmap (_+_ k) (λ ρ → ρ ↑⋆ k) ρs
 
   -- The identity substitution.
 
-  id : ∀ {n} → Sub T n n
+  id : Sub T n n
   id {zero}  = []
   id {suc n} = id ↑
 
   -- Weakening.
 
-  wk⋆ : ∀ k {n} → Sub T n (k + n)
+  wk⋆ : ∀ k → Sub T n (k + n)
   wk⋆ zero    = id
   wk⋆ (suc k) = map weaken (wk⋆ k)
 
-  wk : ∀ {n} → Sub T n (suc n)
+  wk : Sub T n (suc n)
   wk = wk⋆ 1
 
   -- A substitution which only replaces the first variable.
 
-  sub : ∀ {n} → T n → Sub T (suc n) n
+  sub : T n → Sub T (suc n) n
   sub t = t ∷ id
 
 -- Application of substitutions.
 
-record Application {ℓ₁ ℓ₂ : Level} (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ₂) :
-    Set (ℓ₁ ⊔ ℓ₂) where
+record Application (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ₂) : Set (ℓ₁ ⊔ ℓ₂) where
   infixl 8 _/_ _/✶_
   infixl 9 _⊙_
 
@@ -93,17 +98,17 @@ record Application {ℓ₁ ℓ₂ : Level} (T₁ : Pred ℕ ℓ₁) (T₂ : Pred
 
   -- Reverse composition. (Fits well with post-application.)
 
-  _⊙_ : ∀ {m n k} → Sub T₁ m n → Sub T₂ n k → Sub T₁ m k
-  ρ₁ ⊙ ρ₂ = map (λ t → t / ρ₂) ρ₁
+  _⊙_ : Sub T₁ m n → Sub T₂ n o → Sub T₁ m o
+  ρ₁ ⊙ ρ₂ = map (_/ ρ₂) ρ₁
 
   -- Application of multiple substitutions.
 
-  _/✶_ : ∀ {m n} → T₁ m → Subs T₂ m n → T₁ n
+  _/✶_ : T₁ m → Subs T₂ m n → T₁ n
   _/✶_ = Star.gfold Fun.id _ (flip _/_) {k = zero}
 
 -- A combination of the two records above.
 
-record Subst {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
+record Subst (T : Pred ℕ ℓ) : Set ℓ where
   field
     simple      : Simple      T
     application : Application T T
@@ -113,7 +118,7 @@ record Subst {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
 
   -- Composition of multiple substitutions.
 
-  ⨀ : ∀ {m n} → Subs T m n → Sub T m n
+  ⨀ : Subs T m n → Sub T m n
   ⨀ ε        = id
   ⨀ (ρ ◅ ε)  = ρ  -- Convenient optimisation; simplifies some proofs.
   ⨀ (ρ ◅ ρs) = ⨀ ρs ⊙ ρ
@@ -123,8 +128,7 @@ record Subst {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
 
 -- Liftings from T₁ to T₂.
 
-record Lift {ℓ₁ ℓ₂ : Level} (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ₂) :
-    Set (ℓ₁ ⊔ ℓ₂) where
+record Lift (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ₂) : Set (ℓ₁ ⊔ ℓ₂) where
   field
     simple : Simple T₁
     lift   : ∀ {n} → T₁ n → T₂ n
@@ -160,9 +164,9 @@ record TermSubst (T : Pred ℕ 0ℓ) : Set₁ where
   varLift : Lift Fin T
   varLift = record { simple = VarSubst.simple; lift = var }
 
-  infix 8 _/Var_
+  infixl 8 _/Var_
 
-  _/Var_ : ∀ {m n} → T m → Sub Fin m n → T n
+  _/Var_ : T m → Sub Fin m n → T n
   _/Var_ = app varLift
 
   simple : Simple T
diff --git a/src/Data/Fin/Substitution/Example.agda b/src/Data/Fin/Substitution/Example.agda
index 8b3c903a7c..02c98c539f 100644
--- a/src/Data/Fin/Substitution/Example.agda
+++ b/src/Data/Fin/Substitution/Example.agda
@@ -1,14 +1,20 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- An example of how Data.Fin.Substitution can be used: a definition
--- of substitution for the untyped λ-calculus, along with some lemmas
+-- This module is DEPRECATED. Please see
+-- `README.Data.Nat.Fin.Substitution.UntypedLambda` instead.
+--
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module Data.Fin.Substitution.Example where
 
+{-# WARNING_ON_IMPORT
+"Data.Fin.Substitution.Example was deprecated in v2.0.
+Please see README.Data.Nat.Fin.Substitution.UntypedLambda instead."
+#-}
+
 open import Data.Fin.Substitution
 open import Data.Fin.Substitution.Lemmas
 open import Data.Nat.Base hiding (_/_)
diff --git a/src/Data/Fin/Substitution/Lemmas.agda b/src/Data/Fin/Substitution/Lemmas.agda
index 6109f28726..ffa26e64fb 100644
--- a/src/Data/Fin/Substitution/Lemmas.agda
+++ b/src/Data/Fin/Substitution/Lemmas.agda
@@ -8,7 +8,6 @@
 
 module Data.Fin.Substitution.Lemmas where
 
-import Effect.Applicative.Indexed as Applicative
 open import Data.Fin.Substitution
 open import Data.Nat hiding (_⊔_; _/_)
 open import Data.Fin.Base using (Fin; zero; suc; lift)
@@ -23,25 +22,32 @@ open PropEq.≡-Reasoning
 open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred)
 
+private
+  variable
+    ℓ ℓ₁ ℓ₂ : Level
+    m n o p : ℕ
+
+------------------------------------------------------------------------
 -- A lemma which does not refer to any substitutions.
 
-lift-commutes : ∀ {n} k j (x : Fin (j + (k + n))) →
+lift-commutes : ∀ k j (x : Fin (j + (k + n))) →
                 lift j suc (lift j (lift k suc) x) ≡
                 lift j (lift (suc k) suc) (lift j suc x)
 lift-commutes k zero    x       = refl
 lift-commutes k (suc j) zero    = refl
 lift-commutes k (suc j) (suc x) = cong suc (lift-commutes k j x)
 
+------------------------------------------------------------------------
 -- The modules below prove a number of substitution lemmas, on the
 -- assumption that the underlying substitution machinery satisfies
 -- certain properties.
 
-record Lemmas₀ {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
+record Lemmas₀ (T : Pred ℕ ℓ) : Set ℓ where
   field simple : Simple T
 
   open Simple simple
 
-  extensionality : ∀ {m n} {ρ₁ ρ₂ : Sub T m n} →
+  extensionality : {ρ₁ ρ₂ : Sub T m n} →
                    (∀ x → lookup ρ₁ x ≡ lookup ρ₂ x) → ρ₁ ≡ ρ₂
   extensionality {ρ₁ = []}      {[]}       hyp = refl
   extensionality {ρ₁ = t₁ ∷ ρ₁} { t₂ ∷ ρ₂} hyp with hyp zero
@@ -54,7 +60,7 @@ record Lemmas₀ {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
     (id ↑⋆ k) ↑ ≡⟨ cong _↑ (id-↑⋆ k) ⟩
     id        ↑ ∎
 
-  lookup-map-weaken-↑⋆ : ∀ {m n} k x {ρ : Sub T m n} →
+  lookup-map-weaken-↑⋆ : ∀ k x {ρ : Sub T m n} →
                          lookup (map weaken ρ ↑⋆ k) x ≡
                          lookup ((ρ ↑) ↑⋆ k) (lift k suc x)
   lookup-map-weaken-↑⋆ zero    x           = refl
@@ -65,7 +71,7 @@ record Lemmas₀ {ℓ : Level} (T : Pred ℕ ℓ) : Set ℓ where
     weaken (lookup ((ρ ↑) ↑⋆ k) (lift k suc x))      ≡⟨ sym $ VecProp.lookup-map (lift k suc x) weaken ((ρ ↑) ↑⋆ k) ⟩
     lookup (map weaken ((ρ ↑) ↑⋆ k)) (lift k suc x)  ∎
 
-record Lemmas₁ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
+record Lemmas₁ (T : Pred ℕ ℓ) : Set ℓ where
   field lemmas₀ : Lemmas₀ T
 
   open Lemmas₀ lemmas₀
@@ -73,7 +79,7 @@ record Lemmas₁ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
 
   field weaken-var : ∀ {n} {x : Fin n} → weaken (var x) ≡ var (suc x)
 
-  lookup-map-weaken : ∀ {m n} x {y} {ρ : Sub T m n} →
+  lookup-map-weaken : ∀ x {y} {ρ : Sub T m n} →
                       lookup             ρ  x ≡ var      y →
                       lookup (map weaken ρ) x ≡ var (suc y)
   lookup-map-weaken x {y} {ρ} hyp = begin
@@ -84,14 +90,14 @@ record Lemmas₁ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
 
   mutual
 
-    lookup-id : ∀ {n} (x : Fin n) → lookup id x ≡ var x
+    lookup-id : (x : Fin n) → lookup id x ≡ var x
     lookup-id zero    = refl
     lookup-id (suc x) = lookup-wk x
 
-    lookup-wk : ∀ {n} (x : Fin n) → lookup wk x ≡ var (suc x)
+    lookup-wk : (x : Fin n) → lookup wk x ≡ var (suc x)
     lookup-wk x = lookup-map-weaken x {ρ = id} (lookup-id x)
 
-  lookup-↑⋆ : ∀ {m n} (f : Fin m → Fin n) {ρ : Sub T m n} →
+  lookup-↑⋆ : (f : Fin m → Fin n) {ρ : Sub T m n} →
               (∀ x → lookup ρ x ≡ var (f x)) →
               ∀ k x → lookup (ρ ↑⋆ k) x ≡ var (lift k f x)
   lookup-↑⋆ f         hyp zero    x       = hyp x
@@ -99,7 +105,7 @@ record Lemmas₁ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
   lookup-↑⋆ f {ρ = ρ} hyp (suc k) (suc x) =
     lookup-map-weaken x {ρ = ρ ↑⋆ k} (lookup-↑⋆ f hyp k x)
 
-  lookup-lift-↑⋆ : ∀ {m n} (f : Fin n → Fin m) {ρ : Sub T m n} →
+  lookup-lift-↑⋆ : (f : Fin n → Fin m) {ρ : Sub T m n} →
                    (∀ x → lookup ρ (f x) ≡ var x) →
                    ∀ k x → lookup (ρ ↑⋆ k) (lift k f x) ≡ var x
   lookup-lift-↑⋆ f         hyp zero    x       = hyp x
@@ -107,22 +113,22 @@ record Lemmas₁ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
   lookup-lift-↑⋆ f {ρ = ρ} hyp (suc k) (suc x) =
     lookup-map-weaken (lift k f x) {ρ = ρ ↑⋆ k} (lookup-lift-↑⋆ f hyp k x)
 
-  lookup-wk-↑⋆ : ∀ {n} k (x : Fin (k + n)) →
+  lookup-wk-↑⋆ : ∀ k (x : Fin (k + n)) →
                  lookup (wk ↑⋆ k) x ≡ var (lift k suc x)
   lookup-wk-↑⋆ = lookup-↑⋆ suc lookup-wk
 
-  lookup-wk-↑⋆-↑⋆ : ∀ {n} k j (x : Fin (j + (k + n))) →
+  lookup-wk-↑⋆-↑⋆ : ∀ k j (x : Fin (j + (k + n))) →
                     lookup (wk ↑⋆ k ↑⋆ j) x ≡
                     var (lift j (lift k suc) x)
   lookup-wk-↑⋆-↑⋆ k = lookup-↑⋆ (lift k suc) (lookup-wk-↑⋆ k)
 
-  lookup-sub-↑⋆ : ∀ {n t} k (x : Fin (k + n)) →
+  lookup-sub-↑⋆ : ∀ {t} k (x : Fin (k + n)) →
                   lookup (sub t ↑⋆ k) (lift k suc x) ≡ var x
   lookup-sub-↑⋆ = lookup-lift-↑⋆ suc lookup-id
 
   open Lemmas₀ lemmas₀ public
 
-record Lemmas₂ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
+record Lemmas₂ (T : Pred ℕ ℓ) : Set ℓ where
   field
     lemmas₁     : Lemmas₁ T
     application : Application T T
@@ -136,18 +142,18 @@ record Lemmas₂ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
 
   field var-/ : ∀ {m n x} {ρ : Sub T m n} → var x / ρ ≡ lookup ρ x
 
-  suc-/-sub : ∀ {n x} {t : T n} → var (suc x) / sub t ≡ var x
+  suc-/-sub : ∀ {x} {t : T n} → var (suc x) / sub t ≡ var x
   suc-/-sub {x = x} {t} = begin
     var (suc x) / sub t     ≡⟨ var-/ ⟩
     lookup (sub t) (suc x)  ≡⟨ refl ⟩
     lookup id x             ≡⟨ lookup-id x ⟩
     var x                   ∎
 
-  lookup-⊙ : ∀ {m n k} x {ρ₁ : Sub T m n} {ρ₂ : Sub T n k} →
+  lookup-⊙ : ∀ x {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
              lookup (ρ₁ ⊙ ρ₂) x ≡ lookup ρ₁ x / ρ₂
   lookup-⊙ x {ρ₁} {ρ₂} = VecProp.lookup-map x (λ t → t / ρ₂) ρ₁
 
-  lookup-⨀ : ∀ {m n} x (ρs : Subs T m n) →
+  lookup-⨀ : ∀ x (ρs : Subs T m n) →
              lookup (⨀ ρs) x ≡ var x /✶ ρs
   lookup-⨀ x ε                = lookup-id x
   lookup-⨀ x (ρ ◅ ε)          = sym var-/
@@ -158,14 +164,14 @@ record Lemmas₂ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     var x /✶ ρs / ρ        ∎
     where ρs = ρ′ ◅ ρs′
 
-  id-⊙ : ∀ {m n} {ρ : Sub T m n} → id ⊙ ρ ≡ ρ
+  id-⊙ : {ρ : Sub T m n} → id ⊙ ρ ≡ ρ
   id-⊙ {ρ = ρ} = extensionality λ x → begin
     lookup (id ⊙ ρ) x  ≡⟨ lookup-⊙ x {ρ₁ = id} ⟩
     lookup  id x / ρ   ≡⟨ cong₂ _/_ (lookup-id x) refl ⟩
     var x        / ρ   ≡⟨ var-/ ⟩
     lookup ρ x         ∎
 
-  lookup-wk-↑⋆-⊙ : ∀ {m n} k {x} {ρ : Sub T (k + suc m) n} →
+  lookup-wk-↑⋆-⊙ : ∀ k {x} {ρ : Sub T (k + suc m) n} →
                    lookup (wk ↑⋆ k ⊙ ρ) x ≡ lookup ρ (lift k suc x)
   lookup-wk-↑⋆-⊙ k {x} {ρ} = begin
     lookup (wk ↑⋆ k ⊙ ρ) x   ≡⟨ lookup-⊙ x {ρ₁ = wk ↑⋆ k} ⟩
@@ -173,14 +179,14 @@ record Lemmas₂ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     var (lift k suc x) / ρ   ≡⟨ var-/ ⟩
     lookup ρ (lift k suc x)  ∎
 
-  wk-⊙-sub′ : ∀ {n} {t : T n} k → wk ↑⋆ k ⊙ sub t ↑⋆ k ≡ id
+  wk-⊙-sub′ : ∀ {t : T n} k → wk ↑⋆ k ⊙ sub t ↑⋆ k ≡ id
   wk-⊙-sub′ {t = t} k = extensionality λ x → begin
     lookup (wk ↑⋆ k ⊙ sub t ↑⋆ k) x     ≡⟨ lookup-wk-↑⋆-⊙ k ⟩
     lookup (sub t ↑⋆ k) (lift k suc x)  ≡⟨ lookup-sub-↑⋆ k x ⟩
     var x                               ≡⟨ sym (lookup-id x) ⟩
     lookup id x                         ∎
 
-  wk-⊙-sub : ∀ {n} {t : T n} → wk ⊙ sub t ≡ id
+  wk-⊙-sub : {t : T n} → wk ⊙ sub t ≡ id
   wk-⊙-sub = wk-⊙-sub′ zero
 
   var-/-wk-↑⋆ : ∀ {n} k (x : Fin (k + n)) →
@@ -190,7 +196,7 @@ record Lemmas₂ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     lookup (wk ↑⋆ k) x  ≡⟨ lookup-wk-↑⋆ k x ⟩
     var (lift k suc x)  ∎
 
-  wk-↑⋆-⊙-wk : ∀ {n} k j →
+  wk-↑⋆-⊙-wk : ∀ k j →
                wk {n} ↑⋆ k ↑⋆ j ⊙ wk ↑⋆ j ≡
                wk ↑⋆ j ⊙ wk ↑⋆ suc k ↑⋆ j
   wk-↑⋆-⊙-wk k j = extensionality λ x → begin
@@ -207,7 +213,7 @@ record Lemmas₂ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
   open Subst   subst   public hiding (simple; application)
   open Lemmas₁ lemmas₁ public
 
-record Lemmas₃ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
+record Lemmas₃ (T : Pred ℕ ℓ) : Set ℓ where
   field lemmas₂ : Lemmas₂ T
 
   open Lemmas₂ lemmas₂
@@ -217,7 +223,7 @@ record Lemmas₃ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
             (∀ k x → var x /✶ ρs₁ ↑✶ k ≡ var x /✶ ρs₂ ↑✶ k) →
             ∀ k t → t /✶ ρs₁ ↑✶ k ≡ t /✶ ρs₂ ↑✶ k
 
-  /✶-↑✶′ : ∀ {m n} (ρs₁ ρs₂ : Subs T m n) →
+  /✶-↑✶′ : (ρs₁ ρs₂ : Subs T m n) →
            (∀ k → ⨀ (ρs₁ ↑✶ k) ≡ ⨀ (ρs₂ ↑✶ k)) →
            ∀ k t → t /✶ ρs₁ ↑✶ k ≡ t /✶ ρs₂ ↑✶ k
   /✶-↑✶′ ρs₁ ρs₂ hyp = /✶-↑✶ ρs₁ ρs₂ (λ k x → begin
@@ -226,10 +232,10 @@ record Lemmas₃ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     lookup (⨀ (ρs₂ ↑✶ k)) x  ≡⟨ lookup-⨀ x (ρs₂ ↑✶ k) ⟩
     var x /✶ ρs₂ ↑✶ k        ∎)
 
-  id-vanishes : ∀ {n} (t : T n) → t / id ≡ t
+  id-vanishes : (t : T n) → t / id ≡ t
   id-vanishes = /✶-↑✶′ (ε ▻ id) ε id-↑⋆ zero
 
-  ⊙-id : ∀ {m n} {ρ : Sub T m n} → ρ ⊙ id ≡ ρ
+  ⊙-id : {ρ : Sub T m n} → ρ ⊙ id ≡ ρ
   ⊙-id {ρ = ρ} = begin
     map (λ t → t / id) ρ  ≡⟨ VecProp.map-cong id-vanishes ρ ⟩
     map Fun.id         ρ  ≡⟨ VecProp.map-id ρ ⟩
@@ -237,7 +243,7 @@ record Lemmas₃ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
 
   open Lemmas₂ lemmas₂ public hiding (wk-⊙-sub′)
 
-record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
+record Lemmas₄ (T : Pred ℕ ℓ) : Set ℓ where
   field lemmas₃ : Lemmas₃ T
 
   open Lemmas₃ lemmas₃
@@ -246,7 +252,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
 
   private
 
-    ↑-distrib′ : ∀ {m n k} {ρ₁ : Sub T m n} {ρ₂ : Sub T n k} →
+    ↑-distrib′ : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
                  (∀ t → t / ρ₂ / wk ≡ t / wk / ρ₂ ↑) →
                  (ρ₁ ⊙ ρ₂) ↑ ≡ ρ₁ ↑ ⊙ ρ₂ ↑
     ↑-distrib′ {ρ₁ = ρ₁} {ρ₂} hyp = begin
@@ -265,7 +271,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
         map (λ t → weaken t / ρ₂ ↑) ρ₁        ≡⟨ VecProp.map-∘ _ _ _ ⟩
         map (λ t → t / ρ₂ ↑) (map weaken ρ₁)  ∎
 
-    ↑⋆-distrib′ : ∀ {m n o} {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
+    ↑⋆-distrib′ : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
                   (∀ k t → t / ρ₂ ↑⋆ k / wk ≡ t / wk / ρ₂ ↑⋆ suc k) →
                   ∀ k → (ρ₁ ⊙ ρ₂) ↑⋆ k ≡ ρ₁ ↑⋆ k ⊙ ρ₂ ↑⋆ k
     ↑⋆-distrib′                hyp zero    = refl
@@ -274,7 +280,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
       (ρ₁ ↑⋆ k ⊙ ρ₂ ↑⋆ k) ↑      ≡⟨ ↑-distrib′ (hyp k) ⟩
       ρ₁ ↑⋆ suc k ⊙ ρ₂ ↑⋆ suc k  ∎
 
-  map-weaken : ∀ {m n} {ρ : Sub T m n} → map weaken ρ ≡ ρ ⊙ wk
+  map-weaken : {ρ : Sub T m n} → map weaken ρ ≡ ρ ⊙ wk
   map-weaken {ρ = ρ} = begin
     map weaken ρ          ≡⟨ VecProp.map-cong (λ _ → sym /-wk) ρ ⟩
     map (λ t → t / wk) ρ  ≡⟨ refl ⟩
@@ -282,7 +288,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
 
   private
 
-    ⊙-wk′ : ∀ {m n} {ρ : Sub T m n} k →
+    ⊙-wk′ : ∀ {ρ : Sub T m n} k →
             ρ ↑⋆ k ⊙ wk ↑⋆ k ≡ wk ↑⋆ k ⊙ ρ ↑ ↑⋆ k
     ⊙-wk′ {ρ = ρ} k = sym (begin
       wk ↑⋆ k ⊙ ρ ↑ ↑⋆ k  ≡⟨ lemma ⟩
@@ -297,31 +303,30 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
         lookup (ρ ↑ ↑⋆ k) (lift k suc x)  ≡⟨ sym (lookup-map-weaken-↑⋆ k x) ⟩
         lookup (map weaken ρ ↑⋆ k) x      ∎
 
-  ⊙-wk : ∀ {m n} {ρ : Sub T m n} → ρ ⊙ wk ≡ wk ⊙ ρ ↑
+  ⊙-wk : {ρ : Sub T m n} → ρ ⊙ wk ≡ wk ⊙ ρ ↑
   ⊙-wk = ⊙-wk′ zero
 
-  wk-commutes : ∀ {m n} {ρ : Sub T m n} t →
+  wk-commutes : ∀ {ρ : Sub T m n} t →
                 t / ρ / wk ≡ t / wk / ρ ↑
   wk-commutes {ρ = ρ} = /✶-↑✶′ (ε ▻ ρ ▻ wk) (ε ▻ wk ▻ ρ ↑) ⊙-wk′ zero
 
-  ↑⋆-distrib : ∀ {m n o} {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
+  ↑⋆-distrib : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
                ∀ k → (ρ₁ ⊙ ρ₂) ↑⋆ k ≡ ρ₁ ↑⋆ k ⊙ ρ₂ ↑⋆ k
   ↑⋆-distrib = ↑⋆-distrib′ (λ _ → wk-commutes)
 
-  /-⊙ : ∀ {m n k} {ρ₁ : Sub T m n} {ρ₂ : Sub T n k} t →
+  /-⊙ : ∀ {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} t →
         t / ρ₁ ⊙ ρ₂ ≡ t / ρ₁ / ρ₂
   /-⊙ {ρ₁ = ρ₁} {ρ₂} t =
     /✶-↑✶′ (ε ▻ ρ₁ ⊙ ρ₂) (ε ▻ ρ₁ ▻ ρ₂) ↑⋆-distrib zero t
 
-  ⊙-assoc : ∀ {m n k o}
-              {ρ₁ : Sub T m n} {ρ₂ : Sub T n k} {ρ₃ : Sub T k o} →
+  ⊙-assoc : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} {ρ₃ : Sub T o p} →
             ρ₁ ⊙ (ρ₂ ⊙ ρ₃) ≡ (ρ₁ ⊙ ρ₂) ⊙ ρ₃
   ⊙-assoc {ρ₁ = ρ₁} {ρ₂} {ρ₃} = begin
     map (λ t → t / ρ₂ ⊙ ρ₃) ρ₁                  ≡⟨ VecProp.map-cong /-⊙ ρ₁ ⟩
     map (λ t → t / ρ₂ / ρ₃) ρ₁                  ≡⟨ VecProp.map-∘ _ _ _ ⟩
     map (λ t → t / ρ₃) (map (λ t → t / ρ₂) ρ₁)  ∎
 
-  map-weaken-⊙-sub : ∀ {m n} {ρ : Sub T m n} {t} → map weaken ρ ⊙ sub t ≡ ρ
+  map-weaken-⊙-sub : ∀ {ρ : Sub T m n} {t} → map weaken ρ ⊙ sub t ≡ ρ
   map-weaken-⊙-sub {ρ = ρ} {t} = begin
     map weaken ρ ⊙ sub t  ≡⟨ cong₂ _⊙_ map-weaken refl ⟩
     ρ ⊙ wk ⊙ sub t        ≡⟨ sym ⊙-assoc ⟩
@@ -329,7 +334,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     ρ ⊙ id                ≡⟨ ⊙-id ⟩
     ρ                     ∎
 
-  sub-⊙ : ∀ {m n} {ρ : Sub T m n} t → sub t ⊙ ρ ≡ ρ ↑ ⊙ sub (t / ρ)
+  sub-⊙ : ∀ {ρ : Sub T m n} t → sub t ⊙ ρ ≡ ρ ↑ ⊙ sub (t / ρ)
   sub-⊙ {ρ = ρ} t = begin
     sub t ⊙ ρ                           ≡⟨ refl ⟩
     t / ρ ∷ id ⊙ ρ                      ≡⟨ cong (_∷_ (t / ρ)) id-⊙ ⟩
@@ -337,7 +342,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     t / ρ ∷ map weaken ρ ⊙ sub (t / ρ)  ≡⟨ cong₂ _∷_ (sym var-/) refl ⟩
     ρ ↑ ⊙ sub (t / ρ)                   ∎
 
-  suc-/-↑ : ∀ {m n} {ρ : Sub T m n} x →
+  suc-/-↑ : ∀ {ρ : Sub T m n} x →
             var (suc x) / ρ ↑ ≡ var x / ρ / wk
   suc-/-↑ {ρ = ρ} x = begin
     var (suc x) / ρ ↑        ≡⟨ var-/ ⟩
@@ -346,7 +351,7 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     lookup ρ x / wk          ≡⟨ cong₂ _/_ (sym var-/) refl ⟩
     var x / ρ / wk           ∎
 
-  weaken-↑ : ∀ {k n} t {ρ : Sub T k n} → weaken t / (ρ ↑) ≡ weaken (t / ρ)
+  weaken-↑ : ∀ t {ρ : Sub T m n} → weaken t / (ρ ↑) ≡ weaken (t / ρ)
   weaken-↑ t {ρ} = begin
     weaken t / (ρ ↑) ≡⟨ cong (_/ ρ ↑) (sym /-wk) ⟩
     t / wk / ρ ↑     ≡⟨ sym (wk-commutes t) ⟩
@@ -357,10 +362,11 @@ record Lemmas₄ {ℓ} (T : Pred ℕ ℓ) : Set ℓ where
     hiding (/✶-↑✶; /✶-↑✶′; wk-↑⋆-⊙-wk;
             lookup-wk-↑⋆-⊙; lookup-map-weaken-↑⋆)
 
+------------------------------------------------------------------------
 -- For an example of how AppLemmas can be used, see
 -- Data.Fin.Substitution.List.
 
-record AppLemmas {ℓ₁ ℓ₂} (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ₂) : Set (ℓ₁ ⊔ ℓ₂) where
+record AppLemmas (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ₂) : Set (ℓ₁ ⊔ ℓ₂) where
   field
     application : Application T₁ T₂
     lemmas₄     : Lemmas₄ T₂
@@ -376,7 +382,7 @@ record AppLemmas {ℓ₁ ℓ₂} (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ
 
   private module L₄ = Lemmas₄ lemmas₄
 
-  /-⨀ : ∀ {m n} t (ρs : Subs T₂ m n) → t / ⨀ ρs ≡ t /✶ ρs
+  /-⨀ : ∀ t (ρs : Subs T₂ m n) → t / ⨀ ρs ≡ t /✶ ρs
   /-⨀ t ε                = id-vanishes t
   /-⨀ t (ρ ◅ ε)          = refl
   /-⨀ t (ρ ◅ (ρ′ ◅ ρs′)) = begin
@@ -385,7 +391,7 @@ record AppLemmas {ℓ₁ ℓ₂} (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ
     t /✶ ρs / ρ   ∎
     where ρs = ρ′ ◅ ρs′
 
-  ⨀→/✶ : ∀ {m n} (ρs₁ ρs₂ : Subs T₂ m n) →
+  ⨀→/✶ : (ρs₁ ρs₂ : Subs T₂ m n) →
          ⨀ ρs₁ ≡ ⨀ ρs₂ → ∀ t → t /✶ ρs₁ ≡ t /✶ ρs₂
   ⨀→/✶ ρs₁ ρs₂ hyp t = begin
     t /✶ ρs₁   ≡⟨ sym (/-⨀ t ρs₁) ⟩
@@ -393,19 +399,19 @@ record AppLemmas {ℓ₁ ℓ₂} (T₁ : Pred ℕ ℓ₁) (T₂ : Pred ℕ ℓ
     t / ⨀ ρs₂  ≡⟨ /-⨀ t ρs₂ ⟩
     t /✶ ρs₂   ∎
 
-  wk-commutes : ∀ {m n} {ρ : Sub T₂ m n} t →
+  wk-commutes : ∀ {ρ : Sub T₂ m n} t →
                 t / ρ / wk ≡ t / wk / ρ ↑
   wk-commutes {ρ = ρ} = ⨀→/✶ (ε ▻ ρ ▻ wk) (ε ▻ wk ▻ ρ ↑) L₄.⊙-wk
 
-  sub-commutes : ∀ {m n} {t′} {ρ : Sub T₂ m n} t →
+  sub-commutes : ∀ {t′} {ρ : Sub T₂ m n} t →
                  t / sub t′ / ρ ≡ t / ρ ↑ / sub (t′ ⊘ ρ)
   sub-commutes {t′ = t′} {ρ} =
     ⨀→/✶ (ε ▻ sub t′ ▻ ρ) (ε ▻ ρ ↑ ▻ sub (t′ ⊘ ρ)) (L₄.sub-⊙ t′)
 
-  wk-sub-vanishes : ∀ {n t′} (t : T₁ n) → t / wk / sub t′ ≡ t
+  wk-sub-vanishes : ∀ {t′} (t : T₁ n) → t / wk / sub t′ ≡ t
   wk-sub-vanishes {t′ = t′} = ⨀→/✶ (ε ▻ wk ▻ sub t′) ε L₄.wk-⊙-sub
 
-  /-weaken : ∀ {m n} {ρ : Sub T₂ m n} t → t / map weaken ρ ≡ t / ρ / wk
+  /-weaken : ∀ {ρ : Sub T₂ m n} t → t / map weaken ρ ≡ t / ρ / wk
   /-weaken {ρ = ρ} = ⨀→/✶ (ε ▻ map weaken ρ) (ε ▻ ρ ▻ wk) L₄.map-weaken
 
   open Application application public
@@ -527,19 +533,75 @@ record TermLemmas (T : ℕ → Set) : Set₁ where
 
   open Lemmas₅ lemmas₅ public hiding (lemmas₃)
 
-  wk-⊙-∷ : ∀ {m n} (t : T n) (ρ : Sub T m n) → (T.wk T.⊙ (t ∷ ρ)) ≡ ρ
-  wk-⊙-∷ t ρ = extensionality (λ x → begin
+  wk-⊙-∷ : (t : T n) (ρ : Sub T m n) → (T.wk T.⊙ (t ∷ ρ)) ≡ ρ
+  wk-⊙-∷ t ρ = extensionality λ x → begin
     lookup (T.wk T.⊙ (t ∷ ρ)) x ≡⟨ L₃.lookup-wk-↑⋆-⊙ 0 {ρ = t ∷ ρ} ⟩
-    lookup ρ x                  ∎)
+    lookup ρ x                  ∎
 
-  weaken-∷ : ∀ {k n} (t₁ : T k) {t₂ : T n} {ρ : Sub T k n} → T.weaken t₁ T./ (t₂ ∷ ρ) ≡ t₁ T./ ρ
+  weaken-∷ : (t₁ : T m) {t₂ : T n} {ρ : Sub T m n} →
+    T.weaken t₁ T./ (t₂ ∷ ρ) ≡ t₁ T./ ρ
   weaken-∷ t₁ {t₂} {ρ} = begin
     T.weaken t₁ T./ (t₂ ∷ ρ)   ≡⟨ cong (T._/ (t₂ ∷ ρ)) (sym /-wk) ⟩
     (t₁ T./ T.wk) T./ (t₂ ∷ ρ) ≡⟨ ⨀→/✶ ((t₂ ∷ ρ) ◅ T.wk ◅ ε) (ρ ◅ ε) (wk-⊙-∷ t₂ ρ) t₁ ⟩
     t₁ T./ ρ                   ∎
 
-  weaken-sub : ∀ {n} (t₁ : T n) {t₂ : T n} → T.weaken t₁ T./ (T.sub t₂) ≡ t₁
+  weaken-sub : (t₁ : T n) {t₂ : T n} → T.weaken t₁ T./ (T.sub t₂) ≡ t₁
   weaken-sub t₁ {t₂} = begin
     T.weaken t₁ T./ (T.sub t₂) ≡⟨ weaken-∷ t₁ ⟩
     t₁ T./ T.id                ≡⟨ id-vanishes t₁ ⟩
     t₁                         ∎
+
+  -- Lemmas relating renamings to substitutions.
+
+  map-var≡ : {ρ₁ : Sub Fin m n} {ρ₂ : Sub T m n} {f : Fin m → Fin n} →
+             (∀ x → lookup ρ₁ x ≡ f x) →
+             (∀ x → lookup ρ₂ x ≡ T.var (f x)) →
+             map T.var ρ₁ ≡ ρ₂
+  map-var≡ {ρ₁ = ρ₁} {ρ₂ = ρ₂} {f = f} hyp₁ hyp₂ = extensionality λ x →
+    lookup (map T.var ρ₁) x  ≡⟨ VecProp.lookup-map x _ ρ₁ ⟩
+    T.var (lookup ρ₁ x)      ≡⟨ cong T.var $ hyp₁ x ⟩
+    T.var (f x)              ≡⟨ sym $ hyp₂ x ⟩
+    lookup ρ₂ x              ∎
+
+  wk≡wk : map T.var VarSubst.wk ≡ T.wk {n = n}
+  wk≡wk = map-var≡ VarLemmas.lookup-wk lookup-wk
+
+  id≡id : map T.var VarSubst.id ≡ T.id {n = n}
+  id≡id = map-var≡ VarLemmas.lookup-id lookup-id
+
+  sub≡sub : {x : Fin n} → map T.var (VarSubst.sub x) ≡ T.sub (T.var x)
+  sub≡sub = cong (_ ∷_) id≡id
+
+  ↑≡↑ : {ρ : Sub Fin m n} → map T.var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
+  ↑≡↑ {ρ = ρ} = map-var≡
+    (VarLemmas.lookup-↑⋆ (lookup ρ) (λ _ → refl) 1)
+    (lookup-↑⋆ (lookup ρ) (λ _ → VecProp.lookup-map _ _ ρ) 1)
+
+  /Var≡/ : ∀ {ρ : Sub Fin m n} {t} → t /Var ρ ≡ t T./ map T.var ρ
+  /Var≡/ {ρ = ρ} {t = t} =
+    /✶-↑✶ (ε ▻ ρ) (ε ▻ map T.var ρ)
+      (λ k x →
+         T.var x /Var ρ VarSubst.↑⋆ k        ≡⟨ app-var ⟩
+         T.var (lookup (ρ VarSubst.↑⋆ k) x)  ≡⟨ cong T.var $ VarLemmas.lookup-↑⋆ _ (λ _ → refl) k _ ⟩
+         T.var (lift k (VarSubst._/ ρ) x)    ≡⟨ sym $ lookup-↑⋆ _ (λ _ → VecProp.lookup-map _ _ ρ) k _ ⟩
+         lookup (map T.var ρ T.↑⋆ k) x       ≡⟨ sym app-var ⟩
+         T.var x T./ map T.var ρ T.↑⋆ k      ∎)
+      zero t
+
+  sub-renaming-commutes : ∀ {t x} {ρ : Sub T m n} →
+    t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
+  sub-renaming-commutes {t = t} {x = x} {ρ = ρ} =
+    t /Var VarSubst.sub x T./ ρ             ≡⟨ cong (T._/ ρ) /Var≡/ ⟩
+    t T./ map T.var (VarSubst.sub x) T./ ρ  ≡⟨ cong (λ ρ′ → t T./ ρ′ T./ ρ) sub≡sub ⟩
+    t T./ T.sub (T.var x) T./ ρ             ≡⟨ sub-commutes _ ⟩
+    t T./ ρ T.↑ T./ T.sub (T.var x T./ ρ)   ≡⟨ cong (λ t′ → t T./ ρ T.↑ T./ T.sub t′) app-var ⟩
+    t T./ ρ T.↑ T./ T.sub (lookup ρ x)      ∎
+
+  sub-commutes-with-renaming : ∀ {t t′} {ρ : Sub Fin m n} →
+    t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)
+  sub-commutes-with-renaming {t = t} {t′ = t′} {ρ = ρ} =
+    t T./ T.sub t′ /Var ρ                                          ≡⟨ /Var≡/ ⟩
+    t T./ T.sub t′ T./ map T.var ρ                                 ≡⟨ sub-commutes _ ⟩
+    t T./ map T.var ρ T.↑ T./ T.sub (t′ T./ map T.var ρ)           ≡⟨ sym $ cong (λ ρ′ → t T./ ρ′ T./ T.sub (t′ T./ map T.var ρ)) ↑≡↑ ⟩
+    t T./ map T.var (ρ VarSubst.↑) T./ T.sub (t′ T./ map T.var ρ)  ≡⟨ sym $ cong₂ (λ t ρ → t T./ T.sub ρ) /Var≡/ /Var≡/ ⟩
+    t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)                      ∎
diff --git a/src/Data/Fin/Substitution/List.agda b/src/Data/Fin/Substitution/List.agda
index 63a7e39279..ce5ff3e27c 100644
--- a/src/Data/Fin/Substitution/List.agda
+++ b/src/Data/Fin/Substitution/List.agda
@@ -19,17 +19,25 @@ open import Data.List.Properties
 open import Data.Fin.Substitution
 import Function.Base as Fun
 open import Relation.Binary.PropositionalEquality
-open ≡-Reasoning
 
 private
+  variable
+    m n : ℕ
+
+  ListT : ℕ → Set ℓ
+  ListT = List Fun.∘ T
+
   open module L = Lemmas₄ lemmas₄ using (_/_; id; _⊙_)
 
+------------------------------------------------------------------------
+-- Listwise application of a substitution, plus lemmas about it
+
 infixl 8 _//_
 
-_//_ : ∀ {m n} → List (T m) → Sub T m n → List (T n)
+_//_ : ListT m → Sub T m n → ListT n
 ts // ρ = map (λ σ → σ / ρ) ts
 
-appLemmas : AppLemmas (λ n → List (T n)) T
+appLemmas : AppLemmas ListT T
 appLemmas = record
   { application = record { _/_ = _//_ }
   ; lemmas₄     = lemmas₄
@@ -41,7 +49,7 @@ appLemmas = record
       ts // ρ₁ ⊙ ρ₂               ≡⟨ map-cong L./-⊙ ts ⟩
       map (λ σ → σ / ρ₁ / ρ₂) ts  ≡⟨ map-∘ ts ⟩
       ts // ρ₁ // ρ₂              ∎
-  }
+  } where open ≡-Reasoning
 
 open AppLemmas appLemmas public
   hiding (_/_) renaming (_/✶_ to _//✶_)

From 6c068956c926f995e80c713b176ddf6e3e985274 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Fri, 29 Sep 2023 00:45:46 +0100
Subject: [PATCH 131/316] Proof of the Binomial Theorem for
 `(Commutative)Semiring` [supersedes #1287] (#1928)

---
 CHANGELOG.md                                  |   8 +-
 .../CommutativeSemiring/Binomial.agda         |  24 ++
 src/Algebra/Properties/Monoid/Sum.agda        |   6 +-
 src/Algebra/Properties/Semiring/Binomial.agda | 235 ++++++++++++++++++
 4 files changed, 271 insertions(+), 2 deletions(-)
 create mode 100644 src/Algebra/Properties/CommutativeSemiring/Binomial.agda
 create mode 100644 src/Algebra/Properties/Semiring/Binomial.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 35a0991cac..4d6a543f67 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1804,6 +1804,12 @@ New modules
   Algebra.Properties.Loop
   ```
 
+* Properties of (Commutative)Semiring: the Binomial Theorem
+  ```
+  Algebra.Properties.CommutativeSemiring.Binomial
+  Algebra.Properties.Semiring.Binomial
+  ```
+
 * Some n-ary functions manipulating lists
   ```
   Data.List.Nary.NonDependent
@@ -2020,7 +2026,7 @@ Additions to existing modules
 
 * Added new proof to `Algebra.Properties.Monoid.Sum`:
   ```agda
-  sum-init-last : ∀ {n} (t : Vector _ (suc n)) → sum t ≈ sum (init t) + last t
+  sum-init-last : (t : Vector _ (suc n)) → sum t ≈ sum (init t) + last t
   ```
 
 * Added new proofs to `Algebra.Properties.Semigroup`:
diff --git a/src/Algebra/Properties/CommutativeSemiring/Binomial.agda b/src/Algebra/Properties/CommutativeSemiring/Binomial.agda
new file mode 100644
index 0000000000..7a5524f3e8
--- /dev/null
+++ b/src/Algebra/Properties/CommutativeSemiring/Binomial.agda
@@ -0,0 +1,24 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The Binomial Theorem for Commutative Semirings
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Algebra.Bundles
+  using (CommutativeSemiring)
+
+module Algebra.Properties.CommutativeSemiring.Binomial {a ℓ} (S : CommutativeSemiring a ℓ) where
+
+open CommutativeSemiring S
+open import Algebra.Properties.Semiring.Exp semiring using (_^_)
+import Algebra.Properties.Semiring.Binomial semiring as Binomial
+open Binomial public hiding (theorem)
+
+------------------------------------------------------------------------
+-- Here it is
+
+theorem : ∀ n x y → (x + y) ^ n ≈ binomialExpansion x y n
+theorem n x y = Binomial.theorem x y (*-comm x y) n
+
diff --git a/src/Algebra/Properties/Monoid/Sum.agda b/src/Algebra/Properties/Monoid/Sum.agda
index dacb936285..2d8605a2b0 100644
--- a/src/Algebra/Properties/Monoid/Sum.agda
+++ b/src/Algebra/Properties/Monoid/Sum.agda
@@ -42,13 +42,17 @@ open import Algebra.Definitions.RawMonoid rawMonoid public
 ------------------------------------------------------------------------
 -- An alternative mathematical-style syntax for sumₜ
 
-infixl 10 sum-syntax
+infixl 10 sum-syntax sum⁺-syntax
 
 sum-syntax : ∀ n → Vector Carrier n → Carrier
 sum-syntax _ = sum
 
 syntax sum-syntax n (λ i → x) = ∑[ i < n ] x
 
+sum⁺-syntax = sum-syntax ∘ suc
+
+syntax sum⁺-syntax n (λ i → x) = ∑[ i ≤ n ] x
+
 ------------------------------------------------------------------------
 -- Properties
 
diff --git a/src/Algebra/Properties/Semiring/Binomial.agda b/src/Algebra/Properties/Semiring/Binomial.agda
new file mode 100644
index 0000000000..b4d5ef81db
--- /dev/null
+++ b/src/Algebra/Properties/Semiring/Binomial.agda
@@ -0,0 +1,235 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The Binomial Theorem for *-commuting elements in a Semiring
+--
+-- Freely adapted from PR #1287 by Maciej Piechotka (@uzytkownik)
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Algebra.Bundles using (Semiring)
+open import Data.Bool.Base using (true)
+open import Data.Nat.Base as Nat hiding (_+_; _*_; _^_)
+open import Data.Nat.Combinatorics
+  using (_C_; nCn≡1; nC1≡n; nCk+nC[k+1]≡[n+1]C[k+1])
+open import Data.Nat.Properties as Nat
+  using (<⇒<ᵇ; n<1+n; n∸n≡0; +-∸-assoc)
+open import Data.Fin.Base as Fin
+  using (Fin; zero; suc; toℕ; fromℕ; inject₁)
+open import Data.Fin.Patterns using (0F)
+open import Data.Fin.Properties as Fin
+  using (toℕ<n; toℕ-fromℕ; toℕ-inject₁)
+open import Data.Fin.Relation.Unary.Top
+  using (view; ‵fromℕ; ‵inject₁; view-fromℕ; view-inject₁)
+open import Function.Base using (_∘_)
+open import Relation.Binary.PropositionalEquality.Core as ≡
+  using (_≡_; _≢_; cong)
+
+module Algebra.Properties.Semiring.Binomial
+  {a ℓ} (S : Semiring a ℓ)
+  (open Semiring S)
+  (x y : Carrier)
+  where
+
+open import Algebra.Definitions _≈_
+open import Algebra.Properties.Semiring.Sum S
+open import Algebra.Properties.Semiring.Mult S
+open import Algebra.Properties.Semiring.Exp S
+open import Relation.Binary.Reasoning.Setoid setoid
+
+------------------------------------------------------------------------
+-- Definitions
+
+-- Note - `n` could be implicit in many of these definitions, but the
+-- code is more readable if left explicit.
+
+binomial : (n : ℕ) → Fin (suc n) → Carrier
+binomial n k = (x ^ toℕ k) * (y ^ (n ∸ toℕ k))
+
+binomialTerm : (n : ℕ) → Fin (suc n) → Carrier
+binomialTerm n k = (n C toℕ k) × binomial n k
+
+binomialExpansion : ℕ → Carrier
+binomialExpansion n = ∑[ k ≤ n ] (binomialTerm n k)
+
+term₁ : (n : ℕ) → Fin (suc (suc n)) → Carrier
+term₁ n zero    = 0#
+term₁ n (suc k) = x * (binomialTerm n k)
+
+term₂ : (n : ℕ) → Fin (suc (suc n)) → Carrier
+term₂ n k with view k
+... | ‵fromℕ     = 0#
+... | ‵inject₁ j = y * binomialTerm n j
+
+sum₁ : ℕ → Carrier
+sum₁ n = ∑[ k ≤ suc n ] term₁ n k
+
+sum₂ : ℕ → Carrier
+sum₂ n = ∑[ k ≤ suc n ] term₂ n k
+
+------------------------------------------------------------------------
+-- Properties
+
+term₂[n,n+1]≈0# : ∀ n → term₂ n (fromℕ (suc n)) ≈ 0# 
+term₂[n,n+1]≈0# n rewrite view-fromℕ (suc n) = refl
+
+lemma₁ : ∀ n → x * binomialExpansion n ≈ sum₁ n
+lemma₁ n = begin
+  x * binomialExpansion n                ≈⟨ *-distribˡ-sum x (binomialTerm n) ⟩
+  ∑[ i ≤ n ] (x * binomialTerm n i)      ≈˘⟨ +-identityˡ _ ⟩
+  0# + ∑[ i ≤ n ] (x * binomialTerm n i) ≡⟨⟩
+  0# + ∑[ i ≤ n ] term₁ n (suc i)        ≡⟨⟩
+  sum₁ n                                 ∎
+
+lemma₂ : ∀ n → y * binomialExpansion n ≈ sum₂ n
+lemma₂ n = begin
+  y * binomialExpansion n                       ≈⟨ *-distribˡ-sum y (binomialTerm n) ⟩
+  ∑[ i ≤ n ] (y * binomialTerm n i)             ≈˘⟨ +-identityʳ _ ⟩
+  ∑[ i ≤ n ] (y * binomialTerm n i) + 0#        ≈⟨ +-cong (sum-cong-≋ lemma₂-inject₁) (sym (term₂[n,n+1]≈0# n)) ⟩
+  (∑[ i ≤ n ] term₂-inject₁ i) + term₂ n [n+1]  ≈˘⟨ sum-init-last (term₂ n) ⟩
+  sum (term₂ n)                                 ≡⟨⟩
+  sum₂ n                                        ∎
+  where
+    [n+1] = fromℕ (suc n)
+
+    term₂-inject₁ : (i : Fin (suc n)) → Carrier
+    term₂-inject₁ i = term₂ n (inject₁ i)
+
+    lemma₂-inject₁ : ∀ i → y * binomialTerm n i ≈ term₂-inject₁ i
+    lemma₂-inject₁ i rewrite view-inject₁ i = refl
+
+------------------------------------------------------------------------
+-- Next, a lemma which is independent of commutativity
+
+x*lemma : ∀ {n} (i : Fin (suc n)) →
+          x * binomialTerm n i ≈ (n C toℕ i) × binomial (suc n) (suc i)
+x*lemma {n} i = begin
+  x * binomialTerm n i                        ≡⟨⟩
+  x * ((n C k) × (x ^ k * y ^ (n ∸ k)))       ≈˘⟨ *-congˡ (×-assoc-* (n C k) _ _) ⟩
+  x * ((n C k) × x ^ k * y ^ (n ∸ k))         ≈˘⟨ *-assoc x ((n C k) × x ^ k) _ ⟩
+  (x * (n C k) × x ^ k) * y ^ (n ∸ k)         ≈⟨ *-congʳ (×-comm-* (n C k) _ _) ⟩
+  (n C k) × x ^ (suc k) * y ^ (n ∸ k)         ≡⟨⟩
+  (n C k) × x ^ (suc k) * y ^ (suc n ∸ suc k) ≈⟨ ×-assoc-* (n C k) _ _ ⟩
+  (n C k) × binomial (suc n) (suc i)          ∎
+  where k = toℕ i
+
+------------------------------------------------------------------------
+-- Next, a lemma which does require commutativity
+
+y*lemma : x * y ≈ y * x → ∀ {n : ℕ} (j : Fin n) → 
+          y * binomialTerm n (suc j) ≈ (n C toℕ (suc j)) × binomial (suc n) (suc (inject₁ j))
+y*lemma x*y≈y*x {n} j = begin
+  y * binomialTerm n (suc j)
+    ≈⟨ ×-comm-* nC[j+1] y (binomial n (suc j)) ⟩
+  nC[j+1] × (y * binomial n (suc j))
+    ≈⟨ ×-congʳ nC[j+1] (y*x^m*y^n≈x^m*y^[n+1] x*y≈y*x [j+1] [n-j-1]) ⟩
+  nC[j+1] × (x ^ [j+1] * y ^ [n-j])
+    ≈˘⟨ ×-congʳ nC[j+1] (*-congʳ (^-congʳ x (cong suc k≡j))) ⟩
+  nC[j+1] × (x ^ [k+1] * y ^ [n-j])
+    ≈˘⟨ ×-congʳ nC[j+1] (*-congˡ (^-congʳ y [n-k]≡[n-j])) ⟩
+  nC[j+1] × (x ^ [k+1] * y ^ [n-k])
+    ≡⟨⟩
+  nC[j+1] × (x ^ [k+1] * y ^ [n+1]-[k+1])
+    ≡⟨⟩
+  (n C toℕ (suc j)) × binomial (suc n) (suc i) ∎
+  where
+    i           = inject₁ j
+    k           = toℕ i
+    [k+1]       = ℕ.suc k
+    [j+1]       = toℕ (suc j)
+    [n-k]       = n ∸ k
+    [n+1]-[k+1] = [n-k]
+    [n-j-1]     = n ∸ [j+1]
+    [n-j]       = ℕ.suc [n-j-1]
+    nC[j+1]     = n C [j+1]
+
+    k≡j : k ≡ toℕ j
+    k≡j = toℕ-inject₁ j
+
+    [n-k]≡[n-j] : [n-k] ≡ [n-j]
+    [n-k]≡[n-j] = ≡.trans (cong (n ∸_) k≡j) (+-∸-assoc 1 (toℕ<n j))
+
+------------------------------------------------------------------------
+-- Now, a lemma characterising the sum of the term₁ and term₂ expressions
+
+private
+  n<ᵇ1+n : ∀ n → (n Nat.<ᵇ suc n) ≡ true
+  n<ᵇ1+n n with true ← n Nat.<ᵇ suc n | _ ← <⇒<ᵇ (n<1+n n) = ≡.refl
+
+term₁+term₂≈term : x * y ≈ y * x → ∀ n i → term₁ n i + term₂ n i ≈ binomialTerm (suc n) i
+
+term₁+term₂≈term x*y≈y*x n 0F = begin
+  term₁ n 0F + term₂ n 0F      ≡⟨⟩
+  0# + y * (1 × (1# * y ^ n))  ≈⟨ +-identityˡ _ ⟩
+  y * (1 × (1# * y ^ n))       ≈⟨ *-congˡ (+-identityʳ (1# * y ^ n)) ⟩
+  y * (1# * y ^ n)             ≈⟨ *-congˡ (*-identityˡ (y ^ n)) ⟩
+  y * y ^ n                    ≡⟨⟩
+  y ^ suc n                    ≈˘⟨ *-identityˡ (y ^ suc n) ⟩
+  1# * y ^ suc n               ≈˘⟨ +-identityʳ (1# * y ^ suc n) ⟩
+  1 × (1# * y ^ suc n)         ≡⟨⟩
+  binomialTerm (suc n) 0F      ∎
+
+term₁+term₂≈term x*y≈y*x n (suc i) with view i
+... | ‵fromℕ {n}
+{- remembering that i = fromℕ n, definitionally -}
+  rewrite toℕ-fromℕ n
+    | nCn≡1 n
+    | n∸n≡0 n
+    | n<ᵇ1+n n
+    = begin
+  x * ((x ^ n * 1#) + 0#) + 0# ≈⟨ +-identityʳ _ ⟩
+  x * ((x ^ n * 1#) + 0#)      ≈⟨ *-congˡ (+-identityʳ _) ⟩
+  x * (x ^ n * 1#)             ≈˘⟨ *-assoc _ _ _ ⟩
+  x * x ^ n * 1#               ≈˘⟨ +-identityʳ _ ⟩
+  1 × (x * x ^ n * 1#)         ∎
+
+... | ‵inject₁ j
+{- remembering that i = inject₁ j, definitionally -}
+    = begin
+  (x * binomialTerm n i) + (y * binomialTerm n (suc j))
+    ≈⟨ +-cong (x*lemma i) (y*lemma x*y≈y*x j) ⟩
+  (nCk × [x^k+1]*[y^n-k]) + (nC[j+1] × [x^k+1]*[y^n-k])
+    ≈˘⟨ +-congˡ (×-congˡ nC[k+1]≡nC[j+1]) ⟩
+  (nCk × [x^k+1]*[y^n-k]) + (nC[k+1] × [x^k+1]*[y^n-k])
+    ≈˘⟨ ×-homo-+ [x^k+1]*[y^n-k] nCk nC[k+1] ⟩
+  (nCk Nat.+ nC[k+1]) × [x^k+1]*[y^n-k]
+    ≡⟨ cong (_× [x^k+1]*[y^n-k]) (nCk+nC[k+1]≡[n+1]C[k+1] n k) ⟩
+  ((suc n) C (suc k)) × [x^k+1]*[y^n-k]
+    ≡⟨⟩
+  binomialTerm (suc n) (suc i) ∎
+  where
+    k               = toℕ i
+    [k+1]           = ℕ.suc k
+    [j+1]           = toℕ (suc j)
+    nCk             = n C k
+    nC[k+1]         = n C [k+1]
+    nC[j+1]         = n C [j+1]
+    [x^k+1]*[y^n-k] = binomial (suc n) (suc i)
+
+    nC[k+1]≡nC[j+1] : nC[k+1] ≡ nC[j+1]
+    nC[k+1]≡nC[j+1] = cong ((n C_) ∘ suc) (toℕ-inject₁ j)
+
+------------------------------------------------------------------------
+-- Finally, the main result
+
+theorem : x * y ≈ y * x → ∀ n → (x + y) ^ n ≈ binomialExpansion n
+theorem x*y≈y*x zero    = begin
+  (x + y) ^ 0                     ≡⟨⟩
+  1#                              ≈˘⟨ 1×-identityʳ 1# ⟩
+  1 × 1#                          ≈˘⟨ *-identityʳ (1 × 1#) ⟩
+  (1 × 1#) * 1#                   ≈⟨ ×-assoc-* 1 1# 1# ⟩
+  1 × (1# * 1#)                   ≡⟨⟩
+  1 × (x ^ 0 * y ^ 0)             ≈˘⟨ +-identityʳ _ ⟩
+  1 × (x ^ 0 * y ^ 0) + 0#        ≡⟨⟩
+  (0 C 0) × (x ^ 0 * y ^ 0) + 0#  ≡⟨⟩
+  binomialExpansion 0             ∎
+theorem x*y≈y*x n+1@(suc n) = begin
+  (x + y) ^ n+1                                     ≡⟨⟩
+  (x + y) * (x + y) ^ n                             ≈⟨ *-congˡ (theorem x*y≈y*x n) ⟩
+  (x + y) * binomialExpansion n                     ≈⟨ distribʳ _ _ _ ⟩
+  x * binomialExpansion n + y * binomialExpansion n ≈⟨ +-cong (lemma₁ n) (lemma₂ n) ⟩
+  sum₁ n + sum₂ n                                   ≈˘⟨ ∑-distrib-+ (term₁ n) (term₂ n) ⟩
+  ∑[ i ≤ n+1 ] (term₁ n i + term₂ n i)              ≈⟨ sum-cong-≋ (term₁+term₂≈term x*y≈y*x n) ⟩
+  ∑[ i ≤ n+1 ] binomialTerm n+1 i                   ≡⟨⟩
+  binomialExpansion n+1                             ∎

From 64686b6ee1362ab606b44ea989fb58b7c8461f94 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 29 Sep 2023 13:26:37 +0900
Subject: [PATCH 132/316] Modernise `Relation.Nullary` code (#2110)

---
 CHANGELOG.md                             |   5 +
 src/Codata/Musical/Colist.agda           |   4 +-
 src/Data/Nat/InfinitelyOften.agda        |   3 +-
 src/Relation/Nullary/Decidable.agda      |  73 +++++------
 src/Relation/Nullary/Decidable/Core.agda | 151 ++++++++++++-----------
 src/Relation/Nullary/Negation.agda       |  68 +++++-----
 src/Relation/Nullary/Negation/Core.agda  |  59 +++++----
 src/Relation/Nullary/Reflects.agda       |  80 ++++++------
 src/Relation/Nullary/Universe.agda       |   2 +-
 9 files changed, 220 insertions(+), 225 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 4d6a543f67..3ba0274349 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1555,6 +1555,11 @@ Deprecated names
   invPreorder   ↦ converse-preorder
   ```
 
+* In `Relation.Nullary.Decidable.Core`:
+  ```
+  excluded-middle  ↦  ¬¬-excluded-middle
+  ```
+
 ### Renamed Data.Erased to Data.Irrelevant
 
 * This fixes the fact we had picked the wrong name originally. The erased modality
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index 21dde992bc..2f59082e22 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -67,8 +67,8 @@ take (suc n) []       = Vec≤.[]
 take (suc n) (x ∷ xs) = x Vec≤.∷ take n (♭ xs)
 
 
-module ¬¬Monad {p} where
-  open RawMonad (¬¬-Monad {p}) public
+module ¬¬Monad {a} where
+  open RawMonad (¬¬-Monad {a}) public
 open ¬¬Monad  -- we don't want the RawMonad content to be opened publicly
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Nat/InfinitelyOften.agda b/src/Data/Nat/InfinitelyOften.agda
index f85ceb196a..6d86336193 100644
--- a/src/Data/Nat/InfinitelyOften.agda
+++ b/src/Data/Nat/InfinitelyOften.agda
@@ -20,7 +20,8 @@ open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Negation using (¬¬-Monad; call/cc)
 open import Relation.Unary using (Pred; _∪_; _⊆_)
-open RawMonad (¬¬-Monad {p = 0ℓ})
+
+open RawMonad (¬¬-Monad {a = 0ℓ})
 
 private
   variable
diff --git a/src/Relation/Nullary/Decidable.agda b/src/Relation/Nullary/Decidable.agda
index 37d3e0904a..cb5f72b6e0 100644
--- a/src/Relation/Nullary/Decidable.agda
+++ b/src/Relation/Nullary/Decidable.agda
@@ -11,22 +11,20 @@ module Relation.Nullary.Decidable where
 open import Level using (Level)
 open import Data.Bool.Base using (true; false; if_then_else_)
 open import Data.Empty using (⊥-elim)
-open import Data.Product.Base as Prod hiding (map)
-open import Data.Sum.Base as Sum hiding (map)
+open import Data.Product.Base using (∃; _,_)
 open import Function.Base
 open import Function.Bundles using
   (Injection; module Injection; module Equivalence; _⇔_; _↔_; mk↔ₛ′)
 open import Relation.Binary.Bundles using (Setoid; module Setoid)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary
-open import Relation.Nullary.Negation
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong′)
 
 private
   variable
-    p q r : Level
-    P Q R : Set p
+    a b ℓ₁ ℓ₂ : Level
+    A B : Set a
 
 ------------------------------------------------------------------------
 -- Re-exporting the core definitions
@@ -36,56 +34,49 @@ open import Relation.Nullary.Decidable.Core public
 ------------------------------------------------------------------------
 -- Maps
 
-map : P ⇔ Q → Dec P → Dec Q
-map P⇔Q = map′ to from
-  where open Equivalence P⇔Q
+map : A ⇔ B → Dec A → Dec B
+map A⇔B = map′ to from
+  where open Equivalence A⇔B
 
-module _ {a₁ a₂ b₁ b₂} {A : Setoid a₁ a₂} {B : Setoid b₁ b₂}
-         (inj : Injection A B)
-  where
-
-  open Injection inj
-  open Setoid A using () renaming (_≈_ to _≈A_)
-  open Setoid B using () renaming (_≈_ to _≈B_)
-
-  -- If there is an injection from one setoid to another, and the
-  -- latter's equivalence relation is decidable, then the former's
-  -- equivalence relation is also decidable.
-
-  via-injection : Decidable _≈B_ → Decidable _≈A_
-  via-injection dec x y =
-    map′ injective cong (dec (to x) (to y))
+-- If there is an injection from one setoid to another, and the
+-- latter's equivalence relation is decidable, then the former's
+-- equivalence relation is also decidable.
+via-injection : {S : Setoid a ℓ₁} {T : Setoid b ℓ₂}
+                (inj : Injection S T) (open Injection inj) →
+                Decidable Eq₂._≈_ → Decidable Eq₁._≈_
+via-injection inj _≟_ x y = map′ injective cong (to x ≟ to y)
+  where open Injection inj
 
 ------------------------------------------------------------------------
 -- A lemma relating True and Dec
 
-True-↔ : (dec : Dec P) → Irrelevant P → True dec ↔ P
-True-↔ (true  because  [p]) irr = mk↔ₛ′ (λ _ → invert [p]) _ (irr (invert [p])) cong′
-True-↔ (false because ofⁿ ¬p) _ = mk↔ₛ′ (λ ()) (invert (ofⁿ ¬p)) (⊥-elim ∘ ¬p) λ ()
+True-↔ : (a? : Dec A) → Irrelevant A → True a? ↔ A
+True-↔ (true  because [a]) irr = mk↔ₛ′ (λ _ → invert [a]) _ (irr (invert [a])) cong′
+True-↔ (false because ofⁿ ¬a) _ = mk↔ₛ′ (λ ()) (invert (ofⁿ ¬a)) (⊥-elim ∘ ¬a) λ ()
 
 ------------------------------------------------------------------------
 -- Result of decidability
 
-isYes≗does : (P? : Dec P) → isYes P? ≡ does P?
+isYes≗does : (a? : Dec A) → isYes a? ≡ does a?
 isYes≗does (true  because _) = refl
 isYes≗does (false because _) = refl
 
-dec-true : (p? : Dec P) → P → does p? ≡ true
-dec-true (true  because   _ ) p = refl
-dec-true (false because [¬p]) p = ⊥-elim (invert [¬p] p)
+dec-true : (a? : Dec A) → A → does a? ≡ true
+dec-true (true  because   _ ) a = refl
+dec-true (false because [¬a]) a = ⊥-elim (invert [¬a] a)
 
-dec-false : (p? : Dec P) → ¬ P → does p? ≡ false
-dec-false (false because  _ ) ¬p = refl
-dec-false (true  because [p]) ¬p = ⊥-elim (¬p (invert [p]))
+dec-false : (a? : Dec A) → ¬ A → does a? ≡ false
+dec-false (false because  _ ) ¬a = refl
+dec-false (true  because [a]) ¬a = ⊥-elim (¬a (invert [a]))
 
-dec-yes : (p? : Dec P) → P → ∃ λ p′ → p? ≡ yes p′
-dec-yes p? p with dec-true p? p
-dec-yes (yes p′) p | refl = p′ , refl
+dec-yes : (a? : Dec A) → A → ∃ λ a → a? ≡ yes a
+dec-yes a? a with dec-true a? a
+dec-yes (yes a′) a | refl = a′ , refl
 
-dec-no : (p? : Dec P) (¬p : ¬ P) → p? ≡ no ¬p
-dec-no p? ¬p with dec-false p? ¬p
+dec-no : (a? : Dec A) (¬a : ¬ A) → a? ≡ no ¬a
+dec-no a? ¬a with dec-false a? ¬a
 dec-no (no _) _ | refl = refl
 
-dec-yes-irr : (p? : Dec P) → Irrelevant P → (p : P) → p? ≡ yes p
-dec-yes-irr p? irr p with dec-yes p? p
-... | p′ , eq rewrite irr p p′ = eq
+dec-yes-irr : (a? : Dec A) → Irrelevant A → (a : A) → a? ≡ yes a
+dec-yes-irr a? irr a with dec-yes a? a
+... | a′ , eq rewrite irr a a′ = eq
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index 2d827519d7..166b1f264e 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -24,9 +24,8 @@ open import Relation.Nullary.Negation.Core
 
 private
   variable
-    p q : Level
-    P : Set p
-    Q : Set q
+    a b : Level
+    A B : Set a
 
 ------------------------------------------------------------------------
 -- Definition.
@@ -41,25 +40,38 @@ private
 
 infix 2 _because_
 
-record Dec {p} (P : Set p) : Set p where
+record Dec (A : Set a) : Set a where
   constructor _because_
   field
     does  : Bool
-    proof : Reflects P does
+    proof : Reflects A does
 
 open Dec public
 
-pattern yes p =  true because ofʸ  p
-pattern no ¬p = false because ofⁿ ¬p
+pattern yes a =  true because ofʸ  a
+pattern no ¬a = false because ofⁿ ¬a
+
+------------------------------------------------------------------------
+-- Flattening
+
+module _ {A : Set a} where
+
+  From-yes : Dec A → Set a
+  From-yes (true  because _) = A
+  From-yes (false because _) = Lift a ⊤
+
+  From-no : Dec A → Set a
+  From-no (false because _) = ¬ A
+  From-no (true  because _) = Lift a ⊤
 
 ------------------------------------------------------------------------
 -- Recompute
 
 -- Given an irrelevant proof of a decidable type, a proof can
 -- be recomputed and subsequently used in relevant contexts.
-recompute : ∀ {a} {A : Set a} → Dec A → .A → A
-recompute (yes x) _ = x
-recompute (no ¬p) x = ⊥-elim (¬p x)
+recompute : Dec A → .A → A
+recompute (yes a) _ = a
+recompute (no ¬a) a = ⊥-elim (¬a a)
 
 ------------------------------------------------------------------------
 -- Interaction with negation, sum, product etc.
@@ -67,42 +79,42 @@ recompute (no ¬p) x = ⊥-elim (¬p x)
 infixr 1 _⊎-dec_
 infixr 2 _×-dec_ _→-dec_
 
-¬? : Dec P → Dec (¬ P)
-does  (¬? p?) = not (does p?)
-proof (¬? p?) = ¬-reflects (proof p?)
+¬? : Dec A → Dec (¬ A)
+does  (¬? a?) = not (does a?)
+proof (¬? a?) = ¬-reflects (proof a?)
 
-_×-dec_ : Dec P → Dec Q → Dec (P × Q)
-does  (p? ×-dec q?) = does p? ∧ does q?
-proof (p? ×-dec q?) = proof p? ×-reflects proof q?
+_×-dec_ : Dec A → Dec B → Dec (A × B)
+does  (a? ×-dec b?) = does a? ∧ does b?
+proof (a? ×-dec b?) = proof a? ×-reflects proof b?
 
-_⊎-dec_ : Dec P → Dec Q → Dec (P ⊎ Q)
-does  (p? ⊎-dec q?) = does p? ∨ does q?
-proof (p? ⊎-dec q?) = proof p? ⊎-reflects proof q?
+_⊎-dec_ : Dec A → Dec B → Dec (A ⊎ B)
+does  (a? ⊎-dec b?) = does a? ∨ does b?
+proof (a? ⊎-dec b?) = proof a? ⊎-reflects proof b?
 
-_→-dec_ : Dec P → Dec Q → Dec (P → Q)
-does  (p? →-dec q?) = not (does p?) ∨ does q?
-proof (p? →-dec q?) = proof p? →-reflects proof q?
+_→-dec_ : Dec A → Dec B → Dec (A → B)
+does  (a? →-dec b?) = not (does a?) ∨ does b?
+proof (a? →-dec b?) = proof a? →-reflects proof b?
 
 ------------------------------------------------------------------------
 -- Relationship with booleans
 
 -- `isYes` is a stricter version of `does`. The lack of computation
--- means that we can recover the proposition `P` from `isYes P?` by
+-- means that we can recover the proposition `P` from `isYes a?` by
 -- unification. This is useful when we are using the decision procedure
 -- for proof automation.
 
-isYes : Dec P → Bool
+isYes : Dec A → Bool
 isYes (true  because _) = true
 isYes (false because _) = false
 
-isNo : Dec P → Bool
+isNo : Dec A → Bool
 isNo = not ∘ isYes
 
-True : Dec P → Set
-True Q = T (isYes Q)
+True : Dec A → Set
+True = T ∘ isYes
 
-False : Dec P → Set
-False Q = T (isNo Q)
+False : Dec A → Set
+False = T ∘ isNo
 
 -- The traditional name for isYes is ⌊_⌋, indicating the stripping of evidence.
 ⌊_⌋ = isYes
@@ -111,77 +123,72 @@ False Q = T (isNo Q)
 -- Witnesses
 
 -- Gives a witness to the "truth".
-toWitness : {Q : Dec P} → True Q → P
-toWitness {Q = true  because [p]} _  = invert [p]
-toWitness {Q = false because  _ } ()
+toWitness : {a? : Dec A} → True a? → A
+toWitness {a? = true  because [a]} _  = invert [a]
+toWitness {a? = false because  _ } ()
 
 -- Establishes a "truth", given a witness.
-fromWitness : {Q : Dec P} → P → True Q
-fromWitness {Q = true  because   _ } = const _
-fromWitness {Q = false because [¬p]} = invert [¬p]
+fromWitness : {a? : Dec A} → A → True a?
+fromWitness {a? = true  because   _ } = const _
+fromWitness {a? = false because [¬a]} = invert [¬a]
 
 -- Variants for False.
-toWitnessFalse : {Q : Dec P} → False Q → ¬ P
-toWitnessFalse {Q = true  because   _ } ()
-toWitnessFalse {Q = false because [¬p]} _  = invert [¬p]
-
-fromWitnessFalse : {Q : Dec P} → ¬ P → False Q
-fromWitnessFalse {Q = true  because [p]} = flip _$_ (invert [p])
-fromWitnessFalse {Q = false because  _ } = const _
+toWitnessFalse : {a? : Dec A} → False a? → ¬ A
+toWitnessFalse {a? = true  because   _ } ()
+toWitnessFalse {a? = false because [¬a]} _  = invert [¬a]
 
-module _ {p} {P : Set p} where
+fromWitnessFalse : {a? : Dec A} → ¬ A → False a?
+fromWitnessFalse {a? = true  because [a]} = flip _$_ (invert [a])
+fromWitnessFalse {a? = false because  _ } = const _
 
 -- If a decision procedure returns "yes", then we can extract the
 -- proof using from-yes.
-
-  From-yes : Dec P → Set p
-  From-yes (true  because _) = P
-  From-yes (false because _) = Lift p ⊤
-
-  from-yes : (p : Dec P) → From-yes p
-  from-yes (true  because [p]) = invert [p]
-  from-yes (false because _ ) = _
+from-yes : (a? : Dec A) → From-yes a?
+from-yes (true  because [a]) = invert [a]
+from-yes (false because _ ) = _
 
 -- If a decision procedure returns "no", then we can extract the proof
 -- using from-no.
-
-  From-no : Dec P → Set p
-  From-no (false because _) = ¬ P
-  From-no (true  because _) = Lift p ⊤
-
-  from-no : (p : Dec P) → From-no p
-  from-no (false because [¬p]) = invert [¬p]
-  from-no (true  because   _ ) = _
+from-no : (a? : Dec A) → From-no a?
+from-no (false because [¬a]) = invert [¬a]
+from-no (true  because   _ ) = _
 
 ------------------------------------------------------------------------
 -- Maps
 
-map′ : (P → Q) → (Q → P) → Dec P → Dec Q
-does  (map′ P→Q Q→P p?)                   = does p?
-proof (map′ P→Q Q→P (true  because  [p])) = ofʸ (P→Q (invert [p]))
-proof (map′ P→Q Q→P (false because [¬p])) = ofⁿ (invert [¬p] ∘ Q→P)
+map′ : (A → B) → (B → A) → Dec A → Dec B
+does  (map′ A→B B→A a?)                   = does a?
+proof (map′ A→B B→A (true  because  [a])) = ofʸ (A→B (invert [a]))
+proof (map′ A→B B→A (false because [¬a])) = ofⁿ (invert [¬a] ∘ B→A)
 
 ------------------------------------------------------------------------
 -- Relationship with double-negation
 
 -- Decidable predicates are stable.
 
-decidable-stable : Dec P → Stable P
-decidable-stable (yes p) ¬¬p = p
-decidable-stable (no ¬p) ¬¬p = ⊥-elim (¬¬p ¬p)
+decidable-stable : Dec A → Stable A
+decidable-stable (yes a) ¬¬a = a
+decidable-stable (no ¬a) ¬¬a = ⊥-elim (¬¬a ¬a)
 
-¬-drop-Dec : Dec (¬ ¬ P) → Dec (¬ P)
-¬-drop-Dec ¬¬p? = map′ negated-stable contradiction (¬? ¬¬p?)
+¬-drop-Dec : Dec (¬ ¬ A) → Dec (¬ A)
+¬-drop-Dec ¬¬a? = map′ negated-stable contradiction (¬? ¬¬a?)
 
 -- A double-negation-translated variant of excluded middle (or: every
 -- nullary relation is decidable in the double-negation monad).
 
-¬¬-excluded-middle : DoubleNegation (Dec P)
-¬¬-excluded-middle ¬h = ¬h (no (λ p → ¬h (yes p)))
+¬¬-excluded-middle : DoubleNegation (Dec A)
+¬¬-excluded-middle ¬?a = ¬?a (no (λ a → ¬?a (yes a)))
 
-excluded-middle : DoubleNegation (Dec P)
-excluded-middle = ¬¬-excluded-middle
 
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+excluded-middle = ¬¬-excluded-middle
 {-# WARNING_ON_USAGE excluded-middle
 "Warning: excluded-middle was deprecated in v2.0.
 Please use ¬¬-excluded-middle instead."
diff --git a/src/Relation/Nullary/Negation.agda b/src/Relation/Nullary/Negation.agda
index 59e96f3429..4d9ee54208 100644
--- a/src/Relation/Nullary/Negation.agda
+++ b/src/Relation/Nullary/Negation.agda
@@ -16,15 +16,13 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
 open import Function.Base using (flip; _∘_; const; _∘′_)
 open import Level using (Level)
 open import Relation.Nullary.Decidable.Core using (Dec; yes; no; ¬¬-excluded-middle)
-open import Relation.Unary using (Universal)
+open import Relation.Unary using (Universal; Pred)
 
 private
   variable
-    a p q r w : Level
-    A : Set a
-    P : Set p
-    Q : Set q
-    R : Set r
+    a b c d p w : Level
+    A B C D : Set a
+    P : Pred A p
     Whatever : Set w
 
 ------------------------------------------------------------------------
@@ -35,22 +33,20 @@ open import Relation.Nullary.Negation.Core public
 ------------------------------------------------------------------------
 -- Quantifier juggling
 
-module _ {P : A → Set p} where
+∃⟶¬∀¬ : ∃ P → ¬ (∀ x → ¬ P x)
+∃⟶¬∀¬ = flip uncurry
 
-  ∃⟶¬∀¬ : ∃ P → ¬ (∀ x → ¬ P x)
-  ∃⟶¬∀¬ = flip uncurry
+∀⟶¬∃¬ : (∀ x → P x) → ¬ ∃ λ x → ¬ P x
+∀⟶¬∃¬ ∀xPx (x , ¬Px) = ¬Px (∀xPx x)
 
-  ∀⟶¬∃¬ : (∀ x → P x) → ¬ ∃ λ x → ¬ P x
-  ∀⟶¬∃¬ ∀xPx (x , ¬Px) = ¬Px (∀xPx x)
+¬∃⟶∀¬ : ¬ ∃ (λ x → P x) → ∀ x → ¬ P x
+¬∃⟶∀¬ = curry
 
-  ¬∃⟶∀¬ : ¬ ∃ (λ x → P x) → ∀ x → ¬ P x
-  ¬∃⟶∀¬ = curry
+∀¬⟶¬∃ : (∀ x → ¬ P x) → ¬ ∃ (λ x → P x)
+∀¬⟶¬∃ = uncurry
 
-  ∀¬⟶¬∃ : (∀ x → ¬ P x) → ¬ ∃ (λ x → P x)
-  ∀¬⟶¬∃ = uncurry
-
-  ∃¬⟶¬∀ : ∃ (λ x → ¬ P x) → ¬ (∀ x → P x)
-  ∃¬⟶¬∀ = flip ∀⟶¬∃¬
+∃¬⟶¬∀ : ∃ (λ x → ¬ P x) → ¬ (∀ x → P x)
+∃¬⟶¬∀ = flip ∀⟶¬∃¬
 
 ------------------------------------------------------------------------
 -- Double Negation
@@ -58,15 +54,14 @@ module _ {P : A → Set p} where
 -- Double-negation is a monad (if we assume that all elements of ¬ ¬ P
 -- are equal).
 
-¬¬-Monad : RawMonad {p} DoubleNegation
+¬¬-Monad : RawMonad {a} DoubleNegation
 ¬¬-Monad = mkRawMonad
   DoubleNegation
   contradiction
   (λ x f → negated-stable (¬¬-map f x))
 
-¬¬-push : {Q : P → Set q} →
-          DoubleNegation Π[ Q ] → Π[ DoubleNegation ∘ Q ]
-¬¬-push ¬¬P⟶Q P ¬Q = ¬¬P⟶Q (λ P⟶Q → ¬Q (P⟶Q P))
+¬¬-push : DoubleNegation Π[ P ] → Π[ DoubleNegation ∘ P ]
+¬¬-push ¬¬∀P a ¬Pa = ¬¬∀P (λ ∀P → ¬Pa (∀P a))
 
 -- If Whatever is instantiated with ¬ ¬ something, then this function
 -- is call with current continuation in the double-negation monad, or,
@@ -77,26 +72,25 @@ module _ {P : A → Set p} where
 -- that case this function can be used (with Whatever instantiated to
 -- ⊥).
 
-call/cc : ((P → Whatever) → DoubleNegation P) → DoubleNegation P
-call/cc hyp ¬p = hyp (λ p → ⊥-elim (¬p p)) ¬p
+call/cc : ((A → Whatever) → DoubleNegation A) → DoubleNegation A
+call/cc hyp ¬a = hyp (λ a → ⊥-elim (¬a a)) ¬a
 
 -- The "independence of premise" rule, in the double-negation monad.
--- It is assumed that the index set (Q) is inhabited.
+-- It is assumed that the index set (A) is inhabited.
 
-independence-of-premise : {R : Q → Set r} →
-                          Q → (P → Σ Q R) → DoubleNegation (Σ[ x ∈ Q ] (P → R x))
-independence-of-premise {P = P} q f = ¬¬-map helper ¬¬-excluded-middle
+independence-of-premise : A → (B → Σ A P) → DoubleNegation (Σ[ x ∈ A ] (B → P x))
+independence-of-premise {A = A} {B = B} {P = P} q f = ¬¬-map helper ¬¬-excluded-middle
   where
-  helper : Dec P → _
+  helper : Dec B → Σ[ x ∈ A ] (B → P x)
   helper (yes p) = Prod.map₂ const (f p)
   helper (no ¬p) = (q , ⊥-elim ∘′ ¬p)
 
 -- The independence of premise rule for binary sums.
 
-independence-of-premise-⊎ : (P → Q ⊎ R) → DoubleNegation ((P → Q) ⊎ (P → R))
-independence-of-premise-⊎ {P = P} f = ¬¬-map helper ¬¬-excluded-middle
+independence-of-premise-⊎ : (A → B ⊎ C) → DoubleNegation ((A → B) ⊎ (A → C))
+independence-of-premise-⊎ {A = A} {B = B} {C = C} f = ¬¬-map helper ¬¬-excluded-middle
   where
-  helper : Dec P → _
+  helper : Dec A → (A → B) ⊎ (A → C)
   helper (yes p) = Sum.map const const (f p)
   helper (no ¬p) = inj₁ (⊥-elim ∘′ ¬p)
 
@@ -106,12 +100,10 @@ private
   -- independence-of-premise (for simplicity it is assumed that Q and
   -- R have the same type here):
 
-  corollary : {Q R : Set q} →
-              (P → Q ⊎ R) → DoubleNegation ((P → Q) ⊎ (P → R))
-  corollary {P = P} {Q} {R} f =
-    ¬¬-map helper (independence-of-premise
-                     true ([ _,_ true , _,_ false ] ∘′ f))
+  corollary : {B C : Set b} → (A → B ⊎ C) → DoubleNegation ((A → B) ⊎ (A → C))
+  corollary {A = A} {B = B} {C = C} f =
+    ¬¬-map helper (independence-of-premise true ([ _,_ true , _,_ false ] ∘′ f))
     where
-    helper : ∃ (λ b → P → if b then Q else R) → (P → Q) ⊎ (P → R)
+    helper : ∃ (λ b → A → if b then B else C) → (A → B) ⊎ (A → C)
     helper (true  , f) = inj₁ f
     helper (false , f) = inj₂ f
diff --git a/src/Relation/Nullary/Negation/Core.agda b/src/Relation/Nullary/Negation/Core.agda
index 5c4efaf4fb..cbe9ac6da9 100644
--- a/src/Relation/Nullary/Negation/Core.agda
+++ b/src/Relation/Nullary/Negation/Core.agda
@@ -9,9 +9,7 @@
 module Relation.Nullary.Negation.Core where
 
 open import Data.Bool.Base using (not)
-open import Data.Empty using (⊥)
-open import Data.Empty.Irrelevant using (⊥-elim)
-open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
+open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Sum.Base using (_⊎_; [_,_]; inj₁; inj₂)
 open import Function.Base using (flip; _$_; _∘_; const)
 open import Level
@@ -19,9 +17,7 @@ open import Level
 private
   variable
     a p q w : Level
-    A : Set a
-    P : Set p
-    Q : Set q
+    A B C : Set a
     Whatever : Set w
 
 ------------------------------------------------------------------------
@@ -29,48 +25,51 @@ private
 
 infix 3 ¬_
 ¬_ : Set a → Set a
-¬ P = P → ⊥
+¬ A = A → ⊥
+
+------------------------------------------------------------------------
+-- Stability.
 
 -- Double-negation
-DoubleNegation : Set p → Set p
-DoubleNegation P = ¬ ¬ P
+DoubleNegation : Set a → Set a
+DoubleNegation A = ¬ ¬ A
 
 -- Stability under double-negation.
-Stable : Set p → Set p
-Stable P = ¬ ¬ P → P
+Stable : Set a → Set a
+Stable A = ¬ ¬ A → A
 
 ------------------------------------------------------------------------
--- Relationship to product and sum
+-- Relationship to sum
 
 infixr 1 _¬-⊎_
-_¬-⊎_ : ¬ P → ¬ Q → ¬ (P ⊎ Q)
+_¬-⊎_ : ¬ A → ¬ B → ¬ (A ⊎ B)
 _¬-⊎_ = [_,_]
 
 ------------------------------------------------------------------------
 -- Uses of negation
 
-contradiction : P → ¬ P → Whatever
-contradiction p ¬p = ⊥-elim (¬p p)
-
-contradiction₂ : P ⊎ Q → ¬ P → ¬ Q → Whatever
-contradiction₂ (inj₁ p) ¬p ¬q = contradiction p ¬p
-contradiction₂ (inj₂ q) ¬p ¬q = contradiction q ¬q
+contradiction : A → ¬ A → Whatever
+contradiction a ¬a = ⊥-elim (¬a a)
 
-contraposition : (P → Q) → ¬ Q → ¬ P
-contraposition f ¬q p = contradiction (f p) ¬q
+contradiction₂ : A ⊎ B → ¬ A → ¬ B → Whatever
+contradiction₂ (inj₁ a) ¬a ¬b = contradiction a ¬a
+contradiction₂ (inj₂ b) ¬a ¬b = contradiction b ¬b
 
--- Note also the following use of flip:
-private
-  note : (P → ¬ Q) → Q → ¬ P
-  note = flip
+contraposition : (A → B) → ¬ B → ¬ A
+contraposition f ¬b a = contradiction (f a) ¬b
 
 -- Everything is stable in the double-negation monad.
-stable : ¬ ¬ Stable P
-stable ¬[¬¬p→p] = ¬[¬¬p→p] (λ ¬¬p → ⊥-elim (¬¬p (¬[¬¬p→p] ∘ const)))
+stable : ¬ ¬ Stable A
+stable ¬[¬¬a→a] = ¬[¬¬a→a] (λ ¬¬a → ⊥-elim (¬¬a (¬[¬¬a→a] ∘ const)))
 
 -- Negated predicates are stable.
-negated-stable : Stable (¬ P)
-negated-stable ¬¬¬P P = ¬¬¬P (λ ¬P → ¬P P)
+negated-stable : Stable (¬ A)
+negated-stable ¬¬¬a a = ¬¬¬a (_$ a)
 
-¬¬-map : (P → Q) → ¬ ¬ P → ¬ ¬ Q
+¬¬-map : (A → B) → ¬ ¬ A → ¬ ¬ B
 ¬¬-map f = contraposition (contraposition f)
+
+-- Note also the following use of flip:
+private
+  note : (A → ¬ B) → B → ¬ A
+  note = flip
diff --git a/src/Relation/Nullary/Reflects.agda b/src/Relation/Nullary/Reflects.agda
index 66556e8ae6..4f97bd8709 100644
--- a/src/Relation/Nullary/Reflects.agda
+++ b/src/Relation/Nullary/Reflects.agda
@@ -21,18 +21,18 @@ open import Relation.Nullary.Negation.Core
 
 private
   variable
-    p q : Level
-    P Q : Set p
+    a : Level
+    A B : Set a
 
 ------------------------------------------------------------------------
 -- `Reflects` idiom.
 
--- The truth value of P is reflected by a boolean value.
--- `Reflects P b` is equivalent to `if b then P else ¬ P`.
+-- The truth value of A is reflected by a boolean value.
+-- `Reflects A b` is equivalent to `if b then A else ¬ A`.
 
-data Reflects {p} (P : Set p) : Bool → Set p where
-  ofʸ : ( p :   P) → Reflects P true
-  ofⁿ : (¬p : ¬ P) → Reflects P false
+data Reflects (A : Set a) : Bool → Set a where
+  ofʸ : ( a :   A) → Reflects A true
+  ofⁿ : (¬a : ¬ A) → Reflects A false
 
 ------------------------------------------------------------------------
 -- Constructors and destructors
@@ -41,57 +41,57 @@ data Reflects {p} (P : Set p) : Bool → Set p where
 -- that the `if` expressions have already been evaluated away.
 -- In this case, `of` works like the relevant constructor (`ofⁿ` or
 -- `ofʸ`), and `invert` strips off the constructor to just give either
--- the proof of `P` or the proof of `¬ P`.
+-- the proof of `A` or the proof of `¬ A`.
 
-of : ∀ {b} → if b then P else ¬ P → Reflects P b
-of {b = false} ¬p = ofⁿ ¬p
-of {b = true }  p = ofʸ p
+of : ∀ {b} → if b then A else ¬ A → Reflects A b
+of {b = false} ¬a = ofⁿ ¬a
+of {b = true }  a = ofʸ a
 
-invert : ∀ {b} → Reflects P b → if b then P else ¬ P
-invert (ofʸ  p) = p
-invert (ofⁿ ¬p) = ¬p
+invert : ∀ {b} → Reflects A b → if b then A else ¬ A
+invert (ofʸ  a) = a
+invert (ofⁿ ¬a) = ¬a
 
 ------------------------------------------------------------------------
 -- Interaction with negation, product, sums etc.
 
--- If we can decide P, then we can decide its negation.
-¬-reflects : ∀ {b} → Reflects P b → Reflects (¬ P) (not b)
-¬-reflects (ofʸ  p) = ofⁿ (_$ p)
-¬-reflects (ofⁿ ¬p) = ofʸ ¬p
+-- If we can decide A, then we can decide its negation.
+¬-reflects : ∀ {b} → Reflects A b → Reflects (¬ A) (not b)
+¬-reflects (ofʸ  a) = ofⁿ (_$ a)
+¬-reflects (ofⁿ ¬a) = ofʸ ¬a
 
--- If we can decide P and Q then we can decide their product
+-- If we can decide A and Q then we can decide their product
 infixr 2 _×-reflects_
-_×-reflects_ : ∀ {a b} → Reflects P a → Reflects Q b →
-               Reflects (P × Q) (a ∧ b)
-ofʸ  p ×-reflects ofʸ  q = ofʸ (p , q)
-ofʸ  p ×-reflects ofⁿ ¬q = ofⁿ (¬q ∘ proj₂)
-ofⁿ ¬p ×-reflects _      = ofⁿ (¬p ∘ proj₁)
+_×-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
+               Reflects (A × B) (a ∧ b)
+ofʸ  a ×-reflects ofʸ  b = ofʸ (a , b)
+ofʸ  a ×-reflects ofⁿ ¬b = ofⁿ (¬b ∘ proj₂)
+ofⁿ ¬a ×-reflects _      = ofⁿ (¬a ∘ proj₁)
 
 
 infixr 1 _⊎-reflects_
-_⊎-reflects_ : ∀ {a b} → Reflects P a → Reflects Q b →
-               Reflects (P ⊎ Q) (a ∨ b)
-ofʸ  p ⊎-reflects      _ = ofʸ (inj₁ p)
-ofⁿ ¬p ⊎-reflects ofʸ  q = ofʸ (inj₂ q)
-ofⁿ ¬p ⊎-reflects ofⁿ ¬q = ofⁿ (¬p ¬-⊎ ¬q)
+_⊎-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
+               Reflects (A ⊎ B) (a ∨ b)
+ofʸ  a ⊎-reflects      _ = ofʸ (inj₁ a)
+ofⁿ ¬a ⊎-reflects ofʸ  b = ofʸ (inj₂ b)
+ofⁿ ¬a ⊎-reflects ofⁿ ¬b = ofⁿ (¬a ¬-⊎ ¬b)
 
 infixr 2 _→-reflects_
-_→-reflects_ : ∀ {a b} → Reflects P a → Reflects Q b →
-                Reflects (P → Q) (not a ∨ b)
-ofʸ  p →-reflects ofʸ  q = ofʸ (const q)
-ofʸ  p →-reflects ofⁿ ¬q = ofⁿ (¬q ∘ (_$ p))
-ofⁿ ¬p →-reflects _      = ofʸ (⊥-elim ∘ ¬p)
+_→-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
+                Reflects (A → B) (not a ∨ b)
+ofʸ  a →-reflects ofʸ  b = ofʸ (const b)
+ofʸ  a →-reflects ofⁿ ¬b = ofⁿ (¬b ∘ (_$ a))
+ofⁿ ¬a →-reflects _      = ofʸ (⊥-elim ∘ ¬a)
 
 ------------------------------------------------------------------------
 -- Other lemmas
 
-fromEquivalence : ∀ {b} → (T b → P) → (P → T b) → Reflects P b
+fromEquivalence : ∀ {b} → (T b → A) → (A → T b) → Reflects A b
 fromEquivalence {b = true}  sound complete = ofʸ (sound _)
 fromEquivalence {b = false} sound complete = ofⁿ complete
 
 -- `Reflects` is deterministic.
-det : ∀ {b b′} → Reflects P b → Reflects P b′ → b ≡ b′
-det (ofʸ  p) (ofʸ  p′) = refl
-det (ofʸ  p) (ofⁿ ¬p′) = ⊥-elim (¬p′ p)
-det (ofⁿ ¬p) (ofʸ  p′) = ⊥-elim (¬p p′)
-det (ofⁿ ¬p) (ofⁿ ¬p′) = refl
+det : ∀ {b b′} → Reflects A b → Reflects A b′ → b ≡ b′
+det (ofʸ  a) (ofʸ  _) = refl
+det (ofʸ  a) (ofⁿ ¬a) = contradiction a ¬a
+det (ofⁿ ¬a) (ofʸ  a) = contradiction a ¬a
+det (ofⁿ ¬a) (ofⁿ  _) = refl
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 8d72845f07..34e087ec69 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -116,7 +116,7 @@ sequence {AF = AF} A extract-⊥ sequence-⇒ = helper
 -- Some lemmas about double negation.
 
 private
-  open module M {p} = RawMonad (¬¬-Monad {p = p})
+  open module M {a} = RawMonad (¬¬-Monad {a = a})
 
 ¬¬-pull : ∀ {p} (F : PropF p) {P} →
           ⟦ F ⟧ (¬ ¬ P) → ¬ ¬ ⟦ F ⟧ P

From 45cb6b4010899834e8b06cafd3ee5cc7ac76a5fe Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Mon, 2 Oct 2023 07:59:09 +0530
Subject: [PATCH 133/316] Add new fixities (#2116)

Co-authored-by: Sofia El Boufi--Crouzet <136095087+Sofia-Insa@users.noreply.github.com>
---
 CHANGELOG.md                                  | 28 +++++++++++++++++++
 README/Design/Fixity.agda                     |  2 ++
 .../Container/Relation/Binary/Pointwise.agda  |  1 +
 src/Data/Nat/Combinatorics.agda               |  2 +-
 src/Data/Nat/Combinatorics/Base.agda          |  4 +++
 src/Data/Nat/Combinatorics/Specification.agda | 18 ++++++------
 src/Data/Refinement.agda                      |  1 +
 src/Data/Tree/AVL/Value.agda                  |  1 +
 src/Effect/Monad/Partiality.agda              |  4 +++
 src/Effect/Monad/Partiality/All.agda          |  4 +++
 src/Effect/Monad/Predicate.agda               |  2 +-
 src/Foreign/Haskell/Pair.agda                 |  1 +
 src/Function/Construct/Composition.agda       |  2 ++
 src/Reflection/AST/DeBruijn.agda              |  2 ++
 src/Reflection/AST/Traversal.agda             |  1 +
 src/Reflection/AnnotatedAST.agda              |  1 +
 .../Binary/Construct/Composition.agda         |  2 ++
 .../Quotients/Examples.agda                   |  6 ++--
 .../Indexed/Heterogeneous/Construct/At.agda   |  2 ++
 .../Indexed/Homogeneous/Construct/At.agda     |  2 ++
 src/Relation/Unary/Indexed.agda               |  2 ++
 src/Relation/Unary/PredicateTransformer.agda  |  4 +++
 src/Relation/Unary/Properties.agda            |  6 ++++
 src/Tactic/RingSolver.agda                    |  2 ++
 src/Tactic/RingSolver/Core/Expression.agda    |  1 +
 src/Text/Regex/Derivative/Brzozowski.agda     |  2 ++
 src/Text/Regex/Properties.agda                |  2 ++
 src/Text/Regex/String/Unsafe.agda             |  2 ++
 28 files changed, 94 insertions(+), 13 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 3ba0274349..0b18a95448 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -120,6 +120,34 @@ Bug-fixes
   infixl  6 _+ℤ_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
   infix   4 _≉_ _≈ᵢ_ _≤ᵢ_                                    (Relation.Binary.Indexed.Homogeneous.Bundles)
   infixr  5 _∷ᴹ_ _∷⁻¹ᴹ_                                      (Text.Regex.Search)
+  infixr  4 _,_                                              (Data.Refinement)
+  infixr  4 _,_                                              (Data.Container.Relation.Binary.Pointwise)
+  infixr  4 _,_                                              (Data.Tree.AVL.Value)
+  infixr  4 _,_                                              (Foreign.Haskell.Pair)
+  infixr  4 _,_                                              (Reflection.AnnotatedAST)
+  infixr  4 _,_                                              (Reflection.AST.Traversal)
+  infixl  6.5 _P′_ _P_ _C′_ _C_                              (Data.Nat.Combinatorics.Base)
+  infixl  1 _>>=-cong_ _≡->>=-cong_                          (Effect.Monad.Partiality)
+  infixl  1 _?>=′_                                           (Effect.Monad.Predicate)
+  infixl  1 _>>=-cong_ _>>=-congP_                           (Effect.Monad.Partiality.All)
+  infix   4 _∈FV_                                            (Reflection.AST.DeBruijn)
+  infixr  9 _;_                                              (Relation.Binary.Construct.Composition)
+  infixl  6 _+²_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
+  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Heterogeneous.Construct.At)
+  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Homogeneous.Construct.At)
+  infix   4 _∈_ _∉_                                          (Relation.Unary.Indexed)
+  infixr  9 _⍮_                                              (Relation.Unary.PredicateTransformer)
+  infix   8 ∼_                                               (Relation.Unary.PredicateTransformer)
+  infix   2 _×?_ _⊙?_                                        (Relation.Unary.Properties)
+  infix   10 _~?                                             (Relation.Unary.Properties)
+  infixr  1 _⊎?_                                             (Relation.Unary.Properties)
+  infixr  7 _∩?_                                             (Relation.Unary.Properties)
+  infixr  6 _∪?_                                             (Relation.Unary.Properties)
+  infixl  6 _`⊜_                                             (Tactic.RingSolver)
+  infix   8 ⊝_                                               (Tactic.RingSolver.Core.Expression)
+  infix   4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]                  (Text.Regex.Properties)
+  infix   4 _∈?_ _∉?_                                        (Text.Regex.Derivative.Brzozowski)
+  infix   4 _∈_ _∉_ _∈?_ _∉?_                                (Text.Regex.String.Unsafe)
   ```
 
 * In `System.Exit`, the `ExitFailure` constructor is now carrying an integer
diff --git a/README/Design/Fixity.agda b/README/Design/Fixity.agda
index 87907e49b6..85153c1ec4 100644
--- a/README/Design/Fixity.agda
+++ b/README/Design/Fixity.agda
@@ -23,6 +23,8 @@ module README.Design.Fixity where
 -- ternary reasoning infix 1 _⊢_≈_
 -- composition infixr 9 _∘_
 -- application infixr -1 _$_ _$!_
+-- combinatorics infixl 6.5 _P_ _P′_ _C_ _C′_
+-- pair infixr 4 _,_
 
 -- Reasoning:
 -- QED  infix  3 _∎
diff --git a/src/Data/Container/Relation/Binary/Pointwise.agda b/src/Data/Container/Relation/Binary/Pointwise.agda
index 4de4cd0939..2bd20bc1c5 100644
--- a/src/Data/Container/Relation/Binary/Pointwise.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise.agda
@@ -26,6 +26,7 @@ module _ {s p} (C : Container s p) where
     constructor _,_
     field shape    : proj₁ cx ≡ proj₁ cy
           position : ∀ p → R (proj₂ cx p) (proj₂ cy (subst _ shape p))
+  infixr 4 _,_
 
 module _ {s p} {C : Container s p} {x y} {X : Set x} {Y : Set y}
          {ℓ ℓ′} {R : REL X Y ℓ} {R′ : REL X Y ℓ′}
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index 850cd01857..fc6ef422b4 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -75,7 +75,7 @@ nCk≡nC[n∸k] {k} {n} k≤n = begin-equality
     _ = k !* (n ∸ k) !≢0
     _ = (n ∸ k) !* (n ∸ (n ∸ k)) !≢0
 
-nCk≡nPk/k! : k ≤ n → n C k ≡ (n P k / k !) {{k !≢0}}
+nCk≡nPk/k! : k ≤ n → n C k ≡ ((n P k) / k !) {{k !≢0}}
 nCk≡nPk/k! {k} {n} k≤n = begin-equality
   n C k                   ≡⟨ nCk≡n!/k![n-k]! k≤n ⟩
   n ! / (k ! * (n ∸ k) !) ≡˘⟨ /-congʳ (*-comm ((n ∸ k)!) (k !)) ⟩
diff --git a/src/Data/Nat/Combinatorics/Base.agda b/src/Data/Nat/Combinatorics/Base.agda
index 3ee7066441..149665c801 100644
--- a/src/Data/Nat/Combinatorics/Base.agda
+++ b/src/Data/Nat/Combinatorics/Base.agda
@@ -25,6 +25,8 @@ open import Data.Nat.Properties using (_!≢0)
 
 -- n P k = n ! / (n ∸ k) !
 
+infixl 6.5 _P′_ _P_
+
 -- Base definition. Only valid for k ≤ n.
 
 _P′_ : ℕ → ℕ → ℕ
@@ -44,6 +46,8 @@ n P k = if k ≤ᵇ n then n P′ k else 0
 
 -- n C k = n ! / (k ! * (n ∸ k) !)
 
+infixl 6.5 _C′_ _C_
+
 -- Base definition. Only valid for k ≤ n.
 
 _C′_ : ℕ → ℕ → ℕ
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index 0b263eed54..522e3104ed 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -57,20 +57,20 @@ nP′k≡n[n∸1P′k∸1] : ∀ n k → .{{NonZero n}} → .{{NonZero k}} →
 nP′k≡n[n∸1P′k∸1] n           (suc zero)            = refl
 nP′k≡n[n∸1P′k∸1] n@(suc n-1) k@(suc k-1@(suc k-2)) = begin-equality
   n P′ k                        ≡⟨⟩
-  (n ∸ k-1) * n P′ k-1          ≡⟨ cong ((n ∸ k-1) *_) (nP′k≡n[n∸1P′k∸1] n k-1) ⟩
-  (n ∸ k-1) * (n * n-1 P′ k-2)  ≡⟨ x∙yz≈y∙xz (n ∸ k-1) n (n-1 P′ k-2) ⟩
-  n * ((n ∸ k-1) * n-1 P′ k-2)  ≡⟨⟩
-  n * n-1 P′ k-1                ∎
+  (n ∸ k-1) * (n P′ k-1)        ≡⟨ cong ((n ∸ k-1) *_) (nP′k≡n[n∸1P′k∸1] n k-1) ⟩
+  (n ∸ k-1) * (n * (n-1 P′ k-2))  ≡⟨ x∙yz≈y∙xz (n ∸ k-1) n (n-1 P′ k-2) ⟩
+  n * ((n ∸ k-1) * (n-1 P′ k-2))  ≡⟨⟩
+  n * (n-1 P′ k-1)              ∎
   where open ≤-Reasoning; open *-CS
 
 P′-rec : ∀ {n k} → k ≤ n → .{{NonZero k}} →
         n P′ k ≡ k * (pred n P′ pred k) + pred n P′ k
 P′-rec n@{suc n-1} k@{suc k-1} k≤n = begin-equality
-  n P′ k                                      ≡⟨ nP′k≡n[n∸1P′k∸1] n k ⟩
-  n * n-1 P′ k-1                              ≡˘⟨ cong (_* n-1 P′ k-1) (m+[n∸m]≡n {k} {n} k≤n) ⟩
-  (k + (n ∸ k)) * n-1 P′ k-1                  ≡⟨ *-distribʳ-+ (n-1 P′ k-1) k (n ∸ k) ⟩
-  k * (n-1 P′ k-1) + (n-1 ∸ k-1) * n-1 P′ k-1 ≡⟨⟩
-  k * (n-1 P′ k-1) + n-1 P′ k                 ∎
+  n P′ k                                        ≡⟨ nP′k≡n[n∸1P′k∸1] n k ⟩
+  n * (n-1 P′ k-1)                              ≡˘⟨ cong (_* (n-1 P′ k-1)) (m+[n∸m]≡n {k} {n} k≤n) ⟩
+  (k + (n ∸ k)) * (n-1 P′ k-1)                  ≡⟨ *-distribʳ-+ (n-1 P′ k-1) k (n ∸ k) ⟩
+  k * (n-1 P′ k-1) + (n-1 ∸ k-1) * (n-1 P′ k-1) ≡⟨⟩
+  k * (n-1 P′ k-1) + (n-1 P′ k)                 ∎
   where open ≤-Reasoning
 
 nP′n≡n! : ∀ n → n P′ n ≡ n !
diff --git a/src/Data/Refinement.agda b/src/Data/Refinement.agda
index 8671e8593d..dd63e0918a 100644
--- a/src/Data/Refinement.agda
+++ b/src/Data/Refinement.agda
@@ -23,6 +23,7 @@ record Refinement {a p} (A : Set a) (P : A → Set p) : Set (a ⊔ p) where
   constructor _,_
   field value : A
         proof : Irrelevant (P value)
+infixr 4 _,_
 open Refinement public
 
 -- The syntax declaration below is meant to mimick set comprehension.
diff --git a/src/Data/Tree/AVL/Value.agda b/src/Data/Tree/AVL/Value.agda
index ee51150477..6305bd5c2c 100644
--- a/src/Data/Tree/AVL/Value.agda
+++ b/src/Data/Tree/AVL/Value.agda
@@ -34,6 +34,7 @@ record K&_ {v} (V : Value v) : Set (a ⊔ v) where
   field
     key   : Key
     value : Value.family V key
+infixr 4 _,_
 open K&_ public
 
 module _ {v} {V : Value v} where
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index 5fb6ec3455..ed0e3efbc1 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -517,6 +517,8 @@ module _ {A B : Set s}
 
   -- Bind preserves all the relations.
 
+  infixl 1 _>>=-cong_
+
   _>>=-cong_ :
     ∀ {k} {x₁ x₂ : A ⊥} {f₁ f₂ : A → B ⊥} → let open M in
     Rel _∼A_ k x₁ x₂ →
@@ -578,6 +580,8 @@ module _ {A B : Set ℓ} {_∼_ : B → B → Set ℓ} where
 
   -- A variant of _>>=-cong_.
 
+  infixl 1 _≡->>=-cong_
+
   _≡->>=-cong_ :
     ∀ {k} {x₁ x₂ : A ⊥} {f₁ f₂ : A → B ⊥} → let open M in
     Rel _≡_ k x₁ x₂ →
diff --git a/src/Effect/Monad/Partiality/All.agda b/src/Effect/Monad/Partiality/All.agda
index c2db61c031..c5bf2da9ef 100644
--- a/src/Effect/Monad/Partiality/All.agda
+++ b/src/Effect/Monad/Partiality/All.agda
@@ -44,6 +44,8 @@ data All {A : Set a} (P : A → Set p) : A ⊥ → Set (a ⊔ p) where
 
 -- Bind preserves All in the following way:
 
+infixl 1 _>>=-cong_
+
 _>>=-cong_ : ∀ {p q} {P : A → Set p} {Q : B → Set q}
                {x : A ⊥} {f : A → B ⊥} →
              All P x → (∀ {x} → P x → All Q (f x)) →
@@ -118,6 +120,8 @@ module Alternative {a p : Level} where
     _≳⟨_⟩P_     : ∀ x {y} (x≳y : x ≳ y) (p : AllP P y) → AllP P x
     _⟨_⟩P       : ∀ x (p : AllP P x) → AllP P x
 
+  infixl 1 _>>=-congP_
+
   private
 
     -- WHNFs.
diff --git a/src/Effect/Monad/Predicate.agda b/src/Effect/Monad/Predicate.agda
index 1210de7959..7a46fb38a8 100644
--- a/src/Effect/Monad/Predicate.agda
+++ b/src/Effect/Monad/Predicate.agda
@@ -29,7 +29,7 @@ private
 
 record RawPMonad {I : Set i} (M : Pt I (i ⊔ ℓ)) : Set (suc i ⊔ suc ℓ) where
 
-  infixl 1 _?>=_ _?>_ _>?>_
+  infixl 1 _?>=_ _?>_ _>?>_ _?>=′_
   infixr 1 _=<?_ _<?<_
 
   -- ``Demonic'' operations (the opponent chooses the state).
diff --git a/src/Foreign/Haskell/Pair.agda b/src/Foreign/Haskell/Pair.agda
index 84d4b8180b..3487497204 100644
--- a/src/Foreign/Haskell/Pair.agda
+++ b/src/Foreign/Haskell/Pair.agda
@@ -24,6 +24,7 @@ record Pair (A : Set a) (B : Set b) : Set (a ⊔ b) where
   constructor _,_
   field  fst : A
          snd : B
+infixr 4 _,_
 open Pair public
 
 {-# FOREIGN GHC type AgdaPair l1 l2 a b = (a , b) #-}
diff --git a/src/Function/Construct/Composition.agda b/src/Function/Construct/Composition.agda
index 75baef6256..1d25753a9c 100644
--- a/src/Function/Construct/Composition.agda
+++ b/src/Function/Construct/Composition.agda
@@ -234,6 +234,8 @@ _↔-∘_ = inverse
 
 -- Version v2.0
 
+infix 8 _∘-⟶_ _∘-↣_ _∘-↠_ _∘-⤖_ _∘-⇔_ _∘-↩_ _∘-↪_ _∘-↔_
+
 _∘-⟶_ = _⟶-∘_
 {-# WARNING_ON_USAGE _∘-⟶_
 "Warning: _∘-⟶_ was deprecated in v2.0.
diff --git a/src/Reflection/AST/DeBruijn.agda b/src/Reflection/AST/DeBruijn.agda
index 8d05f19952..2236138550 100644
--- a/src/Reflection/AST/DeBruijn.agda
+++ b/src/Reflection/AST/DeBruijn.agda
@@ -128,5 +128,7 @@ module _ where
     actions : ℕ → Actions
     actions i = record defaultActions { onVar = fvVar i }
 
+  infix 4 _∈FV_
+
   _∈FV_ : ℕ → Term → Bool
   i ∈FV t = traverseTerm (actions i) (0 , []) t
diff --git a/src/Reflection/AST/Traversal.agda b/src/Reflection/AST/Traversal.agda
index 13f806d880..815642bf05 100644
--- a/src/Reflection/AST/Traversal.agda
+++ b/src/Reflection/AST/Traversal.agda
@@ -31,6 +31,7 @@ record Cxt : Set where
   field
     len     : ℕ
     context : List (String × Arg Term)
+infixr 4 _,_
 
 private
   _∷cxt_ : String × Arg Term → Cxt → Cxt
diff --git a/src/Reflection/AnnotatedAST.agda b/src/Reflection/AnnotatedAST.agda
index b839a2f2b6..aac574dea9 100644
--- a/src/Reflection/AnnotatedAST.agda
+++ b/src/Reflection/AnnotatedAST.agda
@@ -83,6 +83,7 @@ Argₐ Tyₐ = ⟪ Argₐ′ Tyₐ ⟫
 
 data Namedₐ′ (Tyₐ : Typeₐ ℓ u) : Typeₐ ℓ (⟨named⟩ u) where
   _,_ : ∀ {t} x → Tyₐ Ann t → Namedₐ′ Tyₐ Ann (x , t)
+infixr 4 _,_
 
 Namedₐ : Typeₐ ℓ u → Typeₐ ℓ (⟨named⟩ u)
 Namedₐ Tyₐ = ⟪ Namedₐ′ Tyₐ ⟫
diff --git a/src/Relation/Binary/Construct/Composition.agda b/src/Relation/Binary/Construct/Composition.agda
index 4d66ae5171..0f7a3cb02c 100644
--- a/src/Relation/Binary/Construct/Composition.agda
+++ b/src/Relation/Binary/Construct/Composition.agda
@@ -26,6 +26,8 @@ private
 ------------------------------------------------------------------------
 -- Definition
 
+infixr 9 _;_
+
 _;_ : {A : Set a} {B : Set b} {C : Set c} →
       REL A B ℓ₁ → REL B C ℓ₂ → REL A C (b ⊔ ℓ₁ ⊔ ℓ₂)
 L ; R = λ i j → ∃ λ k → L i k × R k j
diff --git a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
index 5087b908fc..12092a07ac 100644
--- a/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
+++ b/src/Relation/Binary/HeterogeneousEquality/Quotients/Examples.agda
@@ -59,10 +59,12 @@ module Integers (quot : Quotients 0ℓ 0ℓ) where
 
   open Quotient Int renaming (Q to ℤ)
 
+  infixl 6 _+²_
+
   _+²_ : ℕ² → ℕ² → ℕ²
   (x₁ , y₁) +² (x₂ , y₂) = x₁ + x₂ , y₁ + y₂
 
-  +²-cong : ∀{a b a′ b′} → a ∼ a′ → b ∼ b′ → a +² b ∼ a′ +² b′
+  +²-cong : ∀{a b a′ b′} → a ∼ a′ → b ∼ b′ → (a +² b) ∼ (a′ +² b′)
   +²-cong {a₁ , b₁} {c₁ , d₁} {a₂ , b₂} {c₂ , d₂} ab∼cd₁ ab∼cd₂ = begin
     (a₁ + c₁) + (b₂ + d₂) ≡⟨ ≡.cong (_+ (b₂ + d₂)) (+-comm a₁ c₁) ⟩
     (c₁ + a₁) + (b₂ + d₂) ≡⟨ +-assoc c₁ a₁ (b₂ + d₂) ⟩
@@ -128,7 +130,7 @@ module Integers (quot : Quotients 0ℓ 0ℓ) where
         abs (zero² +² a)   ≅⟨ compat-abs (+²-identityˡ a) ⟩
         abs a              ∎
 
-    +²-assoc : (i j k : ℕ²) → (i +² j) +² k ∼ i +² (j +² k)
+    +²-assoc : (i j k : ℕ²) → ((i +² j) +² k) ∼ (i +² (j +² k))
     +²-assoc (a , b) (c , d) (e , f) = begin
       ((a + c) + e) + (b + (d + f)) ≡⟨ ≡.cong (_+ (b + (d + f))) (+-assoc a c e) ⟩
       (a + (c + e)) + (b + (d + f)) ≡⟨ ≡.cong ((a + (c + e)) +_) (≡.sym (+-assoc b d f)) ⟩
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda b/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
index a05a9aa139..2bd79000e8 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
@@ -64,5 +64,7 @@ module _ {a i} {I : Set i} where
 
 module _ {a i} {I : Set i} where
 
+  infixr -1 _atₛ_
+
   _atₛ_ : ∀ {ℓ} → IndexedSetoid I a ℓ → I → Setoid a ℓ
   _atₛ_ = setoid
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda b/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
index 14668cfa49..f8911c770b 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Construct/At.agda
@@ -68,5 +68,7 @@ preorder O index = record
 ------------------------------------------------------------------------
 -- Some useful shorthand infix notation
 
+infixr -1 _atₛ_
+
 _atₛ_ : ∀ {ℓ} → IndexedSetoid I a ℓ → I → Setoid a ℓ
 _atₛ_ = setoid
diff --git a/src/Relation/Unary/Indexed.agda b/src/Relation/Unary/Indexed.agda
index d1cdbb01f5..d0d1a62979 100644
--- a/src/Relation/Unary/Indexed.agda
+++ b/src/Relation/Unary/Indexed.agda
@@ -17,6 +17,8 @@ IPred A ℓ = ∀ {i} → A i → Set ℓ
 
 module _ {i a} {I : Set i} {A : I → Set a} where
 
+  infix 4 _∈_ _∉_
+
   _∈_ : ∀ {ℓ} → (∀ i → A i) → IPred A ℓ → Set _
   x ∈ P = ∀ i → P (x i)
 
diff --git a/src/Relation/Unary/PredicateTransformer.agda b/src/Relation/Unary/PredicateTransformer.agda
index cd52a78985..2134605502 100644
--- a/src/Relation/Unary/PredicateTransformer.agda
+++ b/src/Relation/Unary/PredicateTransformer.agda
@@ -34,6 +34,8 @@ Pt A ℓ = PT A A ℓ ℓ
 ------------------------------------------------------------------------
 -- Composition and identity
 
+infixr 9 _⍮_
+
 _⍮_ : PT B C ℓ₂ ℓ₃ → PT A B ℓ₁ ℓ₂ → PT A C ℓ₁ _
 S ⍮ T = S ∘ T
 
@@ -53,6 +55,8 @@ magic = λ _ → U
 
 -- Negation.
 
+infix 8 ∼_
+
 ∼_ : PT A B ℓ₁ ℓ₂ → PT A B ℓ₁ ℓ₂
 ∼ T = ∁ ∘ T
 
diff --git a/src/Relation/Unary/Properties.agda b/src/Relation/Unary/Properties.agda
index d090716c08..520f5e93c8 100644
--- a/src/Relation/Unary/Properties.agda
+++ b/src/Relation/Unary/Properties.agda
@@ -202,6 +202,12 @@ U-Universal = λ _ → _
 ∁? : {P : Pred A ℓ} → Decidable P → Decidable (∁ P)
 ∁? P? x = ¬? (P? x)
 
+infix 2 _×?_ _⊙?_
+infix 10 _~?
+infixr 1 _⊎?_
+infixr 7 _∩?_
+infixr 6 _∪?_
+
 _∪?_ : {P : Pred A ℓ₁} {Q : Pred A ℓ₂} →
        Decidable P → Decidable Q → Decidable (P ∪ Q)
 _∪?_ P? Q? x = (P? x) ⊎-dec (Q? x)
diff --git a/src/Tactic/RingSolver.agda b/src/Tactic/RingSolver.agda
index 07e9f1a033..59475a3129 100644
--- a/src/Tactic/RingSolver.agda
+++ b/src/Tactic/RingSolver.agda
@@ -141,6 +141,8 @@ module RingSolverReflection (ring : Term) (numberOfVariables : ℕ) where
   `I : Term → Term
   `I x = quote Ι $ᵉ (x ⟨∷⟩ [])
 
+  infixl 6 _`⊜_
+
   _`⊜_ : Term → Term → Term
   x `⊜ y = quote _⊜_  $ʳ (`numberOfVariables ⟅∷⟆ x ⟨∷⟩ y ⟨∷⟩ [])
 
diff --git a/src/Tactic/RingSolver/Core/Expression.agda b/src/Tactic/RingSolver/Core/Expression.agda
index cfea6fdf52..566e71ac96 100644
--- a/src/Tactic/RingSolver/Core/Expression.agda
+++ b/src/Tactic/RingSolver/Core/Expression.agda
@@ -17,6 +17,7 @@ open import Algebra
 infixl 6 _⊕_
 infixl 7 _⊗_
 infixr 8 _⊛_
+infix 8 ⊝_
 
 data Expr {a} (A : Set a) (n : ℕ) : Set a where
   Κ   : A → Expr A n                   -- Constant
diff --git a/src/Text/Regex/Derivative/Brzozowski.agda b/src/Text/Regex/Derivative/Brzozowski.agda
index 9c66c831a4..178986eb65 100644
--- a/src/Text/Regex/Derivative/Brzozowski.agda
+++ b/src/Text/Regex/Derivative/Brzozowski.agda
@@ -107,6 +107,8 @@ eat-complete′ x (e R.⋆) (refl ∷ eq) (star (sum (inj₂ (prod (refl ∷ app
 ------------------------------------------------------------------------
 -- Consequence: matching is decidable
 
+infix 4 _∈?_ _∉?_
+
 _∈?_ : Decidable _∈_
 []       ∈? e = []∈? e
 (x ∷ xs) ∈? e = map′ (eat-sound x e) (eat-complete x e) (xs ∈? eat x e)
diff --git a/src/Text/Regex/Properties.agda b/src/Text/Regex/Properties.agda
index 02b6db2039..6b24e0aa16 100644
--- a/src/Text/Regex/Properties.agda
+++ b/src/Text/Regex/Properties.agda
@@ -49,6 +49,8 @@ open import Text.Regex.Properties.Core preorder public
              $ ([]∈? e) ×-dec ([]∈? f)
 []∈? (e ⋆)   = yes (star (sum (inj₁ ε)))
 
+infix 4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]
+
 _∈ᴿ?_ : Decidable _∈ᴿ_
 c ∈ᴿ? [ a ]     = map′ [_] (λ where [ eq ] → eq) (c ≟ a)
 c ∈ᴿ? (lb ─ ub) = map′ (uncurry _─_) (λ where (ge ─ le) → ge , le)
diff --git a/src/Text/Regex/String/Unsafe.agda b/src/Text/Regex/String/Unsafe.agda
index 118b48505a..c8c5eb7781 100644
--- a/src/Text/Regex/String/Unsafe.agda
+++ b/src/Text/Regex/String/Unsafe.agda
@@ -26,6 +26,8 @@ open import Text.Regex.String as Regex public
 ------------------------------------------------------------------------
 -- Specialised definitions
 
+infix 4 _∈_ _∉_ _∈?_ _∉?_
+
 _∈_ : String → Exp → Set
 str ∈ e = toList str Regex.∈ e
 

From e29233421d696717990cda61f5b38d0c7adc5642 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 2 Oct 2023 03:34:20 +0100
Subject: [PATCH 134/316] Adds setoid export to
 `Algebra.Bundles.SemiringWithoutOne`

---
 src/Algebra/Bundles.agda    | 4 ++--
 src/Algebra/Structures.agda | 4 +++-
 2 files changed, 5 insertions(+), 3 deletions(-)

diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index 7fd88f5b42..9c1c14fec0 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -504,7 +504,7 @@ record SemiringWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where
 
   open NearSemiring nearSemiring public
     using
-    ( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-semigroup
+    ( +-rawMagma; +-magma; +-unitalMagma; +-semigroup
     ; +-rawMonoid; +-monoid
     ; *-rawMagma; *-magma; *-semigroup
     ; rawNearSemiring
@@ -542,7 +542,7 @@ record CommutativeSemiringWithoutOne c ℓ : Set (suc (c ⊔ ℓ)) where
 
   open SemiringWithoutOne semiringWithoutOne public
     using
-    ( _≉_; +-rawMagma; +-magma; +-unitalMagma; +-semigroup; +-commutativeSemigroup
+    ( +-rawMagma; +-magma; +-unitalMagma; +-semigroup; +-commutativeSemigroup
     ; *-rawMagma; *-magma; *-semigroup
     ; +-rawMonoid; +-monoid; +-commutativeMonoid
     ; nearSemiring; rawNearSemiring
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index e38bbe7ba9..ed55226d67 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -358,7 +358,7 @@ record IsSemiringWithoutOne (+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where
     zero                  : Zero 0# *
 
   open IsCommutativeMonoid +-isCommutativeMonoid public
-    using (isEquivalence)
+    using (setoid)
     renaming
     ( comm                   to +-comm
     ; isMonoid               to +-isMonoid
@@ -366,6 +366,8 @@ record IsSemiringWithoutOne (+ * : Op₂ A) (0# : A) : Set (a ⊔ ℓ) where
     ; isCommutativeSemigroup to +-isCommutativeSemigroup
     )
 
+  open Setoid setoid public
+
   *-isMagma : IsMagma *
   *-isMagma = record
     { isEquivalence = isEquivalence

From a8916ba24f819f506dd189a0d801679d4453569e Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 2 Oct 2023 09:48:55 +0100
Subject: [PATCH 135/316] #453: added `Dense` relations and `DenseLinearOrder`
 (#2111)

---
 CHANGELOG.md                                  | 52 ++++++++++++------
 src/Data/Rational/Properties.agda             | 31 ++++++++++-
 .../Rational/Unnormalised/Properties.agda     | 55 +++++++++++++++++--
 src/Data/Tree/AVL/Indexed/WithK.agda          |  1 +
 src/Data/Tree/AVL/NonEmpty/Propositional.agda |  4 +-
 src/Relation/Binary/Bundles.agda              | 18 ++++++
 src/Relation/Binary/Definitions.agda          |  7 ++-
 src/Relation/Binary/Structures.agda           |  8 +++
 8 files changed, 148 insertions(+), 28 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0b18a95448..54ac43951a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -308,7 +308,7 @@ Non-backwards compatible changes
   Data.Sum.Function.Setoid
   Data.Sum.Function.Propositional
   ```
-  
+
 * Additionally the following proofs now use the new definitions instead of the old ones:
   * `Algebra.Lattice.Properties.BooleanAlgebra`
   * `Algebra.Properties.CommutativeMonoid.Sum`
@@ -388,8 +388,8 @@ Non-backwards compatible changes
 * The module `Function.Definitions` no longer has two equalities as module arguments, as
   they did not interact as intended with the re-exports from `Function.Definitions.(Core1/Core2)`.
   The latter have been removed and their definitions folded into `Function.Definitions`.
-  
-* In `Function.Definitions` the types of `Surjective`, `Injective` and `Surjective` 
+
+* In `Function.Definitions` the types of `Surjective`, `Injective` and `Surjective`
   have been changed from:
   ```
   Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
@@ -404,16 +404,16 @@ Non-backwards compatible changes
   ```
   This is for several reasons: i) the new definitions compose much more easily, ii) Agda
   can better infer the equalities used.
-  
+
   To ease backwards compatibility:
-   - the old definitions have been moved to the new names  `StrictlySurjective`, 
-	 `StrictlyInverseˡ` and `StrictlyInverseʳ`. 
-   - The records in  `Function.Structures` and `Function.Bundles` export proofs 
-	 of these under the names `strictlySurjective`, `strictlyInverseˡ` and 
-	 `strictlyInverseʳ`,
+   - the old definitions have been moved to the new names  `StrictlySurjective`,
+         `StrictlyInverseˡ` and `StrictlyInverseʳ`.
+   - The records in  `Function.Structures` and `Function.Bundles` export proofs
+         of these under the names `strictlySurjective`, `strictlyInverseˡ` and
+         `strictlyInverseʳ`,
    - Conversion functions have been added in both directions to
-	 `Function.Consequences(.Propositional)`. 
-  
+         `Function.Consequences(.Propositional)`.
+
 #### Proofs of non-zeroness/positivity/negativity as instance arguments
 
 * Many numeric operations in the library require their arguments to be non-zero,
@@ -800,14 +800,14 @@ Non-backwards compatible changes
 ### Changes to triple reasoning interface
 
 * The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof that the strict
-  relation is irreflexive. 
-  
+  relation is irreflexive.
+
 * This allows the following new proof combinator:
   ```agda
   begin-contradiction : (r : x IsRelatedTo x) → {s : True (IsStrict? r)} → A
   ```
   that takes a proof that a value is strictly less than itself and then applies the principle of explosion.
-  
+
 * Specialised versions of this combinator are available in the following derived modules:
   ```
   Data.Nat.Properties
@@ -1442,6 +1442,11 @@ Deprecated names
   *-rawMonoid ↦ *-1-rawMonoid
   ```
 
+* In `Data.Rational.Unnormalised.Properties`:
+  ```
+  ≤-steps  ↦  p≤q⇒p≤r+q
+  ```
+
 * In `Data.Sum.Properties`:
   ```agda
   [,]-∘-distr      ↦  [,]-∘
@@ -1491,7 +1496,7 @@ Deprecated names
   take-distr-map     ↦  take-map
   drop-distr-zipWith ↦  drop-zipWith
   drop-distr-map     ↦  drop-map
-  
+
   updateAt-id-relative      ↦  updateAt-id-local
   updateAt-compose-relative ↦  updateAt-∘-local
   updateAt-compose          ↦  updateAt-∘
@@ -2375,7 +2380,7 @@ Additions to existing modules
 
   length-isMagmaHomomorphism  : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
   length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
-  
+
   take-map : take n (map f xs) ≡ map f (take n xs)
   drop-map : drop n (map f xs) ≡ map f (drop n xs)
   head-map : head (map f xs) ≡ Maybe.map f (head xs)
@@ -3058,6 +3063,7 @@ Additions to existing modules
 
 * Added new definitions in `Relation.Binary.Definitions`:
   ```
+  Dense        _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y
   Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)
   Tight    _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)
 
@@ -3072,11 +3078,13 @@ Additions to existing modules
 
 * Added new definitions in `Relation.Binary.Bundles`:
   ```
+  record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   record ApartnessRelation c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   ```
 
 * Added new definitions in `Relation.Binary.Structures`:
   ```
+  record IsDenseLinearOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   record IsApartnessRelation (_#_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   ```
 
@@ -3310,7 +3318,6 @@ This is a full list of proofs that have changed form to use irrelevant instance
   negative<positive    : ∀ {p q} → .(Negative p) → .(Positive q) → p < q
   nonNeg∧nonPos⇒0      : ∀ {p} → .(NonNegative p) → .(NonPositive p) → p ≃ 0ℚᵘ
 
-  ≤-steps : ∀ {p q r} → NonNegative r → p ≤ q → p ≤ r + q
   p≤p+q   : ∀ {p q} → NonNegative q → p ≤ p + q
   p≤q+p   : ∀ {p} → NonNegative p → ∀ {q} → q ≤ p + q
 
@@ -3468,6 +3475,13 @@ This is a full list of proofs that have changed form to use irrelevant instance
   foldr-commMonoid : xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys
   ```
 
+* Added new proof, structure, and bundle to `Data.Rational.Properties`
+  ```agda
+  <-dense              : Dense _<_
+  <-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
+  <-denseLinearOrder   : DenseLinearOrder 0ℓ 0ℓ 0ℓ
+  ```
+
 * Added new module to `Data.Rational.Unnormalised.Properties`
   ```agda
   module ≃-Reasoning = SetoidReasoning ≃-setoid
@@ -3480,16 +3494,20 @@ This is a full list of proofs that have changed form to use irrelevant instance
   ≠-symmetric : Symmetric _≠_
   ≠-cotransitive : Cotransitive _≠_
   ≠⇒invertible : p ≠ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
+
+  <-dense : Dense _<_
   ```
 
 * Added new structures to `Data.Rational.Unnormalised.Properties`
   ```agda
+  <-isDenseLinearOrder : IsDenseLinearOrder _≃_ _<_
   +-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
   +-*-isHeytingField : IsHeytingField _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
   ```
 
 * Added new bundles to `Data.Rational.Unnormalised.Properties`
   ```agda
+  <-denseLinearOrder : DenseLinearOrder 0ℓ 0ℓ 0ℓ
   +-*-heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
   +-*-heytingField : HeytingField 0ℓ 0ℓ 0ℓ
   ```
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 959573218a..8dfbb57f13 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -51,11 +51,11 @@ open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
 open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
 open import Relation.Binary.Bundles
-  using (Setoid; DecSetoid; TotalPreorder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+  using (Setoid; DecSetoid; TotalPreorder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; DenseLinearOrder)
 open import Relation.Binary.Structures
-  using (IsPreorder; IsTotalOrder; IsTotalPreorder; IsPartialOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+  using (IsPreorder; IsTotalOrder; IsTotalPreorder; IsPartialOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder; IsDenseLinearOrder)
 open import Relation.Binary.Definitions
-  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri<; tri>; tri≈; _Respectsʳ_; _Respectsˡ_; _Respects₂_)
+  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; Dense; Trans; Trichotomous; tri<; tri>; tri≈; _Respectsʳ_; _Respectsˡ_; _Respects₂_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Binary.Morphism.Structures
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms
@@ -614,6 +614,20 @@ toℚᵘ-isOrderMonomorphism-< = record
 <-asym : Asymmetric _<_
 <-asym (*<* p<q) (*<* q<p) = ℤ.<-asym p<q q<p
 
+<-dense : Dense _<_
+<-dense {p} {q} p<q = let
+    m , p<ᵘm , m<ᵘq = ℚᵘ.<-dense (toℚᵘ-mono-< p<q)
+
+    m≃m : m ≃ᵘ toℚᵘ (fromℚᵘ m)
+    m≃m = ℚᵘ.≃-sym (toℚᵘ-fromℚᵘ m)
+
+    p<m : p < fromℚᵘ m
+    p<m = toℚᵘ-cancel-< (ℚᵘ.<-respʳ-≃ m≃m p<ᵘm)
+
+    m<q : fromℚᵘ m < q
+    m<q = toℚᵘ-cancel-< (ℚᵘ.<-respˡ-≃ m≃m m<ᵘq)
+  in fromℚᵘ m , p<m , m<q
+
 <-≤-trans : Trans _<_ _≤_ _<_
 <-≤-trans {p} {q} {r} (*<* p<q) (*≤* q≤r) = *<*
   (ℤ.*-cancelʳ-<-nonNeg _ (begin-strict
@@ -693,6 +707,12 @@ _>?_ = flip _<?_
   ; compare       = <-cmp
   }
 
+<-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
+<-isDenseLinearOrder = record
+  { isStrictTotalOrder = <-isStrictTotalOrder
+  ; dense              = <-dense
+  }
+
 ------------------------------------------------------------------------
 -- Bundles
 
@@ -706,6 +726,11 @@ _>?_ = flip _<?_
   { isStrictTotalOrder = <-isStrictTotalOrder
   }
 
+<-denseLinearOrder : DenseLinearOrder 0ℓ 0ℓ 0ℓ
+<-denseLinearOrder = record
+  { isDenseLinearOrder = <-isDenseLinearOrder
+  }
+
 ------------------------------------------------------------------------
 -- A specialised module for reasoning about the _≤_ and _<_ relations
 ------------------------------------------------------------------------
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 2ede5bb44c..c04b05d2f3 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -1,4 +1,4 @@
-------------------------------------------------------------------------
+-----------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Properties of unnormalized Rational numbers
@@ -17,7 +17,6 @@ open import Algebra.Consequences.Propositional
 open import Algebra.Construct.NaturalChoice.Base
 import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp
 import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
-open import Data.Empty using (⊥-elim)
 open import Data.Bool.Base using (T; true; false)
 open import Data.Nat.Base as ℕ using (suc; pred)
 import Data.Nat.Properties as ℕ
@@ -36,11 +35,11 @@ import Relation.Nullary.Decidable as Dec
 open import Relation.Nullary.Negation using (contradiction; contraposition)
 open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
 open import Relation.Binary.Bundles
-  using (Setoid; DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
+  using (Setoid; DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; DenseLinearOrder)
 open import Relation.Binary.Structures
-  using (IsEquivalence; IsDecEquivalence; IsApartnessRelation; IsTotalPreorder; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
+  using (IsEquivalence; IsDecEquivalence; IsApartnessRelation; IsTotalPreorder; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder; IsDenseLinearOrder)
 open import Relation.Binary.Definitions
-  using (Reflexive; Symmetric; Transitive; Cotransitive; Tight; Decidable; Antisymmetric; Asymmetric; Total; Trans; Trichotomous; Irreflexive; Irrelevant; _Respectsˡ_; _Respectsʳ_; _Respects₂_; tri≈; tri<; tri>)
+  using (Reflexive; Symmetric; Transitive; Cotransitive; Tight; Decidable; Antisymmetric; Asymmetric; Dense; Total; Trans; Trichotomous; Irreflexive; Irrelevant; _Respectsˡ_; _Respectsʳ_; _Respects₂_; tri≈; tri<; tri>)
 import Relation.Binary.Consequences as BC
 open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Properties.Poset as PosetProperties
@@ -119,7 +118,7 @@ p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* 
 ≠-cotransitive {x} {y} x≠y z with x ≃? z | z ≃? y
 ... | no  x≠z | _       = inj₁ x≠z
 ... | yes _   | no z≠y  = inj₂ z≠y
-... | yes x≃z | yes z≃y = ⊥-elim (x≠y (≃-trans x≃z z≃y))
+... | yes x≃z | yes z≃y = contradiction (≃-trans x≃z z≃y) x≠y
 
 ≃-isEquivalence : IsEquivalence _≃_
 ≃-isEquivalence = record
@@ -412,6 +411,39 @@ drop-*<* (*<* pq<qp) = pq<qp
 <-asym : Asymmetric _<_
 <-asym (*<* x<y) = ℤ.<-asym x<y ∘ drop-*<*
 
+<-dense : Dense _<_
+<-dense {p} {q} (*<* p<q) = m , p<m , m<q
+  where
+  open ℤ.≤-Reasoning
+  m : ℚᵘ
+  m = mkℚᵘ (↥ p ℤ.+ ↥ q) (pred (↧ₙ p ℕ.+ ↧ₙ q))
+
+  p<m : p < m
+  p<m = *<* (begin-strict
+    ↥ p ℤ.* ↧ m
+      ≡⟨⟩
+    ↥ p ℤ.* (↧ p ℤ.+ ↧ q)
+      ≡⟨ ℤ.*-distribˡ-+ (↥ p) (↧ p) (↧ q) ⟩
+    ↥ p ℤ.* ↧ p ℤ.+ ↥ p ℤ.* ↧ q
+      <⟨ ℤ.+-monoʳ-< (↥ p ℤ.* ↧ p) p<q ⟩
+    ↥ p ℤ.* ↧ p ℤ.+ ↥ q ℤ.* ↧ p
+      ≡˘⟨ ℤ.*-distribʳ-+ (↧ p) (↥ p) (↥ q) ⟩
+    (↥ p ℤ.+ ↥ q) ℤ.* ↧ p
+      ≡⟨⟩
+    ↥ m ℤ.* ↧ p ∎)
+
+  m<q : m < q
+  m<q = *<* (begin-strict
+    ↥ m ℤ.* ↧ q
+      ≡⟨ ℤ.*-distribʳ-+ (↧ q) (↥ p) (↥ q) ⟩
+    ↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ q
+      <⟨ ℤ.+-monoˡ-< (↥ q ℤ.* ↧ q) p<q ⟩
+    ↥ q ℤ.* ↧ p ℤ.+ ↥ q ℤ.* ↧ q
+      ≡˘⟨ ℤ.*-distribˡ-+ (↥ q) (↧ p) (↧ q) ⟩
+    ↥ q ℤ.* (↧ p ℤ.+ ↧ q)
+      ≡⟨⟩
+    ↥ q ℤ.* ↧ m ∎)
+
 ≤-<-trans : Trans _≤_ _<_ _<_
 ≤-<-trans {p} {q} {r} (*≤* p≤q) (*<* q<r) = *<* $
   ℤ.*-cancelʳ-<-nonNeg _ $ begin-strict
@@ -517,6 +549,12 @@ _>?_ = flip _<?_
   ; compare       = <-cmp
   }
 
+<-isDenseLinearOrder : IsDenseLinearOrder _≃_ _<_
+<-isDenseLinearOrder = record
+  { isStrictTotalOrder = <-isStrictTotalOrder
+  ; dense              = <-dense
+  }
+
 ------------------------------------------------------------------------
 -- Bundles
 
@@ -535,6 +573,11 @@ _>?_ = flip _<?_
   { isStrictTotalOrder = <-isStrictTotalOrder
   }
 
+<-denseLinearOrder : DenseLinearOrder 0ℓ 0ℓ 0ℓ
+<-denseLinearOrder = record
+  { isDenseLinearOrder = <-isDenseLinearOrder
+  }
+
 ------------------------------------------------------------------------
 -- A specialised module for reasoning about the _≤_ and _<_ relations
 ------------------------------------------------------------------------
diff --git a/src/Data/Tree/AVL/Indexed/WithK.agda b/src/Data/Tree/AVL/Indexed/WithK.agda
index 4a8944757d..607d63a5a8 100644
--- a/src/Data/Tree/AVL/Indexed/WithK.agda
+++ b/src/Data/Tree/AVL/Indexed/WithK.agda
@@ -15,6 +15,7 @@ module Data.Tree.AVL.Indexed.WithK
        {k r} (Key : Set k) {_<_ : Rel Key r}
        (isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_) where
 
+strictTotalOrder : StrictTotalOrder _ _ _
 strictTotalOrder = record { isStrictTotalOrder = isStrictTotalOrder }
 
 open import Data.Tree.AVL.Indexed strictTotalOrder as AVL hiding (Value)
diff --git a/src/Data/Tree/AVL/NonEmpty/Propositional.agda b/src/Data/Tree/AVL/NonEmpty/Propositional.agda
index 632d4de428..c449e4fb70 100644
--- a/src/Data/Tree/AVL/NonEmpty/Propositional.agda
+++ b/src/Data/Tree/AVL/NonEmpty/Propositional.agda
@@ -17,7 +17,9 @@ module Data.Tree.AVL.NonEmpty.Propositional
 
 open import Level
 
-private strictTotalOrder = record { isStrictTotalOrder = isStrictTotalOrder}
+private
+  strictTotalOrder : StrictTotalOrder _ _ _
+  strictTotalOrder = record { isStrictTotalOrder = isStrictTotalOrder}
 open import Data.Tree.AVL.Value (StrictTotalOrder.Eq.setoid strictTotalOrder)
 import Data.Tree.AVL.NonEmpty strictTotalOrder as AVL⁺
 
diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index 5a0621629f..a774d45d69 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -299,6 +299,24 @@ record StrictTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) wh
   Please use Eq.decSetoid instead."
   #-}
 
+record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
+  infix 4 _≈_ _<_
+  field
+    Carrier            : Set c
+    _≈_                : Rel Carrier ℓ₁
+    _<_                : Rel Carrier ℓ₂
+    isDenseLinearOrder : IsDenseLinearOrder _≈_ _<_
+
+  open IsDenseLinearOrder isDenseLinearOrder public
+    using (isStrictTotalOrder; dense)
+
+  strictTotalOrder : StrictTotalOrder c ℓ₁ ℓ₂
+  strictTotalOrder = record
+    { isStrictTotalOrder = isStrictTotalOrder
+    }
+
+  open StrictTotalOrder strictTotalOrder public
+
 
 ------------------------------------------------------------------------
 -- Apartness relations
diff --git a/src/Relation/Binary/Definitions.agda b/src/Relation/Binary/Definitions.agda
index 2ed315191b..6854074986 100644
--- a/src/Relation/Binary/Definitions.agda
+++ b/src/Relation/Binary/Definitions.agda
@@ -13,7 +13,7 @@ module Relation.Binary.Definitions where
 open import Agda.Builtin.Equality using (_≡_)
 
 open import Data.Maybe.Base using (Maybe)
-open import Data.Product.Base using (_×_)
+open import Data.Product.Base using (_×_; ∃-syntax)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_on_; flip)
 open import Level
@@ -88,6 +88,11 @@ Irreflexive _≈_ _<_ = ∀ {x y} → x ≈ y → ¬ (x < y)
 Asymmetric : Rel A ℓ → Set _
 Asymmetric _<_ = ∀ {x y} → x < y → ¬ (y < x)
 
+-- Density
+
+Dense : Rel A ℓ → Set _
+Dense _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y
+
 -- Generalised connex - exactly one of the two relations holds.
 
 Connex : REL A B ℓ₁ → REL B A ℓ₂ → Set _
diff --git a/src/Relation/Binary/Structures.agda b/src/Relation/Binary/Structures.agda
index 965e41691b..5b59e900be 100644
--- a/src/Relation/Binary/Structures.agda
+++ b/src/Relation/Binary/Structures.agda
@@ -280,6 +280,14 @@ record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) wher
     using (irrefl; asym; <-respʳ-≈; <-respˡ-≈; <-resp-≈)
 
 
+record IsDenseLinearOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
+  field
+    isStrictTotalOrder : IsStrictTotalOrder _<_
+    dense              : Dense _<_
+
+  open IsStrictTotalOrder isStrictTotalOrder public
+
+
 ------------------------------------------------------------------------
 -- Apartness relations
 ------------------------------------------------------------------------

From 33cb598a97e714ce25017c3d3c63116c27da2f12 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 3 Oct 2023 01:46:37 +0100
Subject: [PATCH 136/316] Rectifies the negated equality symbol in
 `Data.Rational.Unnormalised.*` (#2118)

---
 CHANGELOG.md                                  |  9 ++-
 src/Data/Rational/Base.agda                   |  4 +-
 src/Data/Rational/Properties.agda             |  6 +-
 src/Data/Rational/Unnormalised/Base.agda      | 14 ++--
 .../Rational/Unnormalised/Properties.agda     | 66 +++++++++----------
 5 files changed, 54 insertions(+), 45 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 54ac43951a..d9db593cef 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -892,7 +892,7 @@ Non-backwards compatible changes
   (issue #1437) to conform with the defined setoid equality `_≃_` on `Rational`s:
   ```agda
   step-≈  ↦  step-≃
-  step-≃˘ ↦  step-≃˘
+  step-≈˘ ↦  step-≃˘
   ```
   with corresponding associated syntax:
   ```agda
@@ -1399,6 +1399,13 @@ Deprecated names
   <-step          ↦  m<n⇒m<1+n
   ```
 
+* In `Data.Rational.Unnormalised.Base`:
+  ```
+  _≠_  ↦  _≄_
+  +-rawMonoid ↦ +-0-rawMonoid
+  *-rawMonoid ↦ *-1-rawMonoid
+  ```
+
 * In `Data.Rational.Unnormalised.Properties`:
   ```
   ↥[p/q]≡p         ↦  ↥[n/d]≡n
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 53739b8571..5f5a35e6f9 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -19,8 +19,8 @@ open import Data.Rational.Unnormalised.Base as ℚᵘ using (ℚᵘ; mkℚᵘ)
 open import Data.Sum.Base using (inj₂)
 open import Function.Base using (id)
 open import Level using (0ℓ)
-open import Relation.Nullary using (¬_; recompute)
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Decidable.Core using (recompute)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Unary using (Pred)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 8dfbb57f13..eab0c0cf2c 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -59,9 +59,9 @@ open import Relation.Binary.Definitions
 open import Relation.Binary.PropositionalEquality
 open import Relation.Binary.Morphism.Structures
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms
-open import Relation.Nullary.Decidable as Dec
-  using (True; False; fromWitness; fromWitnessFalse; toWitnessFalse; yes; no; recompute; map′; _×-dec_)
-open import Relation.Nullary.Negation using (¬_; contradiction; contraposition)
+open import Relation.Nullary.Decidable.Core as Dec
+  using (yes; no; recompute; map′; _×-dec_)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 
 open import Algebra.Definitions {A = ℚ} _≡_
 open import Algebra.Structures  {A = ℚ} _≡_
diff --git a/src/Data/Rational/Unnormalised/Base.agda b/src/Data/Rational/Unnormalised/Base.agda
index ec104895be..dc5bd144dc 100644
--- a/src/Data/Rational/Unnormalised/Base.agda
+++ b/src/Data/Rational/Unnormalised/Base.agda
@@ -14,8 +14,7 @@ open import Data.Integer.Base as ℤ
   using (ℤ; +_; +0; +[1+_]; -[1+_]; +<+; +≤+)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Level using (0ℓ)
-open import Relation.Nullary.Negation using (¬_)
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Unary using (Pred)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core
@@ -66,8 +65,8 @@ infix 4 _≃_ _≠_
 data _≃_ : Rel ℚᵘ 0ℓ where
   *≡* : ∀ {p q} → (↥ p ℤ.* ↧ q) ≡ (↥ q ℤ.* ↧ p) → p ≃ q
 
-_≠_ : Rel ℚᵘ 0ℓ
-p ≠ q = ¬ (p ≃ q)
+_≄_ : Rel ℚᵘ 0ℓ
+p ≄ q = ¬ (p ≃ q)
 
 ------------------------------------------------------------------------
 -- Ordering of rationals
@@ -156,7 +155,7 @@ NonNegative p = ℤ.NonNegative (↥ p)
 -- from ℤ but it requires importing `Data.Integer.Properties` which
 -- we would like to avoid doing.
 
-≢-nonZero : ∀ {p} → p ≠ 0ℚᵘ → NonZero p
+≢-nonZero : ∀ {p} → p ≄ 0ℚᵘ → NonZero p
 ≢-nonZero {mkℚᵘ -[1+ _ ] _      } _   = _
 ≢-nonZero {mkℚᵘ +[1+ _ ] _      } _   = _
 ≢-nonZero {mkℚᵘ +0       zero   } p≢0 = contradiction (*≡* refl) p≢0
@@ -378,3 +377,8 @@ Please use +-0-rawMonoid instead."
 "Warning: *-rawMonoid was deprecated in v2.0
 Please use *-1-rawMonoid instead."
 #-}
+_≠_ = _≄_
+{-# WARNING_ON_USAGE _≠_
+"Warning: _≠_ was deprecated in v2.0
+Please use _≄_ instead."
+#-}
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index c04b05d2f3..4e86729725 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -30,9 +30,8 @@ open import Data.Sum.Base using (_⊎_; [_,_]′; inj₁; inj₂)
 import Data.Sign as Sign
 open import Function.Base using (_on_; _$_; _∘_; flip)
 open import Level using (0ℓ)
-open import Relation.Nullary using (¬_; yes; no)
-import Relation.Nullary.Decidable as Dec
-open import Relation.Nullary.Negation using (contradiction; contraposition)
+open import Relation.Nullary.Decidable.Core as Dec using (yes; no)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
 open import Relation.Binary.Bundles
   using (Setoid; DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; DenseLinearOrder)
@@ -43,7 +42,6 @@ open import Relation.Binary.Definitions
 import Relation.Binary.Consequences as BC
 open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Properties.Poset as PosetProperties
-open import Relation.Nullary using (yes; no)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 open import Algebra.Properties.CommutativeSemigroup ℤ.*-commutativeSemigroup
@@ -105,20 +103,20 @@ infix 4 _≃?_
 _≃?_ : Decidable _≃_
 p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* ↧ p)
 
-0≠1 : 0ℚᵘ ≠ 1ℚᵘ
-0≠1 = Dec.from-no (0ℚᵘ ≃? 1ℚᵘ)
+0≄1 : 0ℚᵘ ≄ 1ℚᵘ
+0≄1 = Dec.from-no (0ℚᵘ ≃? 1ℚᵘ)
 
-≃-≠-irreflexive : Irreflexive _≃_ _≠_
-≃-≠-irreflexive x≃y x≠y = x≠y x≃y
+≃-≄-irreflexive : Irreflexive _≃_ _≄_
+≃-≄-irreflexive x≃y x≄y = x≄y x≃y
 
-≠-symmetric : Symmetric _≠_
-≠-symmetric x≠y y≃x = x≠y (≃-sym y≃x)
+≄-symmetric : Symmetric _≄_
+≄-symmetric x≄y y≃x = x≄y (≃-sym y≃x)
 
-≠-cotransitive : Cotransitive _≠_
-≠-cotransitive {x} {y} x≠y z with x ≃? z | z ≃? y
-... | no  x≠z | _       = inj₁ x≠z
-... | yes _   | no z≠y  = inj₂ z≠y
-... | yes x≃z | yes z≃y = contradiction (≃-trans x≃z z≃y) x≠y
+≄-cotransitive : Cotransitive _≄_
+≄-cotransitive {x} {y} x≄y z with x ≃? z | z ≃? y
+... | no  x≄z | _       = inj₁ x≄z
+... | yes _   | no z≄y  = inj₂ z≄y
+... | yes x≃z | yes z≃y = contradiction (≃-trans x≃z z≃y) x≄y
 
 ≃-isEquivalence : IsEquivalence _≃_
 ≃-isEquivalence = record
@@ -133,16 +131,16 @@ p ≃? q = Dec.map′ *≡* drop-*≡* (↥ p ℤ.* ↧ q ℤ.≟ ↥ q ℤ.* 
   ; _≟_           = _≃?_
   }
 
-≠-isApartnessRelation : IsApartnessRelation _≃_ _≠_
-≠-isApartnessRelation = record
-  { irrefl  = ≃-≠-irreflexive
-  ; sym     = ≠-symmetric
-  ; cotrans = ≠-cotransitive
+≄-isApartnessRelation : IsApartnessRelation _≃_ _≄_
+≄-isApartnessRelation = record
+  { irrefl  = ≃-≄-irreflexive
+  ; sym     = ≄-symmetric
+  ; cotrans = ≄-cotransitive
   }
 
-≠-tight : Tight _≃_ _≠_
-proj₁ (≠-tight p q) ¬p≠q = Dec.decidable-stable (p ≃? q) ¬p≠q
-proj₂ (≠-tight p q) p≃q p≠q = p≠q p≃q
+≄-tight : Tight _≃_ _≄_
+proj₁ (≄-tight p q) ¬p≄q = Dec.decidable-stable (p ≃? q) ¬p≄q
+proj₂ (≄-tight p q) p≃q p≄q = p≄q p≃q
 
 ≃-setoid : Setoid 0ℓ 0ℓ
 ≃-setoid = record
@@ -1127,11 +1125,11 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
 *-inverseʳ : ∀ p .{{_ : NonZero p}} → p * 1/ p ≃ 1ℚᵘ
 *-inverseʳ p = ≃-trans (*-comm p (1/ p)) (*-inverseˡ p)
 
-≠⇒invertible : p ≠ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
-≠⇒invertible {p} {q} p≠q = _ , *-inverseˡ (p - q) , *-inverseʳ (p - q)
+≄⇒invertible : p ≄ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
+≄⇒invertible {p} {q} p≄q = _ , *-inverseˡ (p - q) , *-inverseʳ (p - q)
   where instance
   _ : NonZero (p - q)
-  _ = ≢-nonZero (p≠q ∘ p-q≃0⇒p≃q p q)
+  _ = ≢-nonZero (p≄q ∘ p-q≃0⇒p≃q p q)
 
 *-zeroˡ : LeftZero _≃_ 0ℚᵘ _*_
 *-zeroˡ p@record{} = *≡* refl
@@ -1142,8 +1140,8 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
 *-zero : Zero _≃_ 0ℚᵘ _*_
 *-zero = *-zeroˡ , *-zeroʳ
 
-invertible⇒≠ : Invertible _≃_ 1ℚᵘ _*_ (p - q) → p ≠ q
-invertible⇒≠ {p} {q} (1/p-q , 1/x*x≃1 , x*1/x≃1) p≃q = 0≠1 (begin
+invertible⇒≄ : Invertible _≃_ 1ℚᵘ _*_ (p - q) → p ≄ q
+invertible⇒≄ {p} {q} (1/p-q , 1/x*x≃1 , x*1/x≃1) p≃q = 0≄1 (begin
   0ℚᵘ             ≈˘⟨ *-zeroˡ 1/p-q ⟩
   0ℚᵘ * 1/p-q     ≈˘⟨ *-congʳ (p≃q⇒p-q≃0 p q p≃q) ⟩
   (p - q) * 1/p-q ≈⟨ x*1/x≃1 ⟩
@@ -1390,18 +1388,18 @@ nonNeg*nonNeg⇒nonNeg p q = nonNegative
   ; *-comm = *-comm
   }
 
-+-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
++-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
 +-*-isHeytingCommutativeRing = record
   { isCommutativeRing   = +-*-isCommutativeRing
-  ; isApartnessRelation = ≠-isApartnessRelation
-  ; #⇒invertible        = ≠⇒invertible
-  ; invertible⇒#        = invertible⇒≠
+  ; isApartnessRelation = ≄-isApartnessRelation
+  ; #⇒invertible        = ≄⇒invertible
+  ; invertible⇒#        = invertible⇒≄
   }
 
-+-*-isHeytingField : IsHeytingField _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
++-*-isHeytingField : IsHeytingField _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
 +-*-isHeytingField = record
   { isHeytingCommutativeRing = +-*-isHeytingCommutativeRing
-  ; tight                    = ≠-tight
+  ; tight                    = ≄-tight
   }
 
 ------------------------------------------------------------------------

From 698dfeea386a0a29c02d49816189ca9612f7f829 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Wed, 4 Oct 2023 12:58:28 +0530
Subject: [PATCH 137/316] Sync insert, remove, and update functionalities for
 `List` and `Vec` (#2049)

---
 CHANGELOG.md                                  | 110 ++++++++--
 .../Properties/CommutativeMonoid/Sum.agda     |   8 +-
 src/Algebra/Properties/Semiring/Binomial.agda |   4 +-
 src/Data/List/Base.agda                       |  34 ++-
 src/Data/List/Properties.agda                 |  54 ++++-
 .../List/Relation/Unary/All/Properties.agda   |   2 +-
 src/Data/List/Relation/Unary/Any.agda         |   4 +-
 src/Data/Vec/Base.agda                        |  42 ++--
 src/Data/Vec/Functional.agda                  |  92 ++++----
 src/Data/Vec/Functional/Properties.agda       | 202 ++++++++++--------
 src/Data/Vec/Properties.agda                  | 188 ++++++++++------
 11 files changed, 489 insertions(+), 251 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d9db593cef..f787065436 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -797,6 +797,50 @@ Non-backwards compatible changes
   IO.Instances
   ```
 
+### Standardisation of `insertAt`/`updateAt`/`removeAt`
+
+* Previously, the names and argument order of index-based insertion, update and removal functions for
+  various types of lists and vectors were inconsistent.
+
+* To fix this the names have all been standardised to `insertAt`/`updateAt`/`removeAt`.
+
+* Correspondingly the following changes have occurred:
+
+* In `Data.List.Base` the following have been added:
+  ```agda
+  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
+  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
+  removeAt : (xs : List A) → Fin (length xs) → List A
+  ```
+  and the following has been deprecated
+  ```
+  _─_ ↦ removeAt
+  ```
+  
+* In `Data.Vec.Base`:
+  ```agda
+  insert ↦ insertAt
+  remove ↦ removeAt
+  
+  updateAt : Fin n → (A → A) → Vec A n → Vec A n
+    ↦
+  updateAt : Vec A n → Fin n → (A → A) → Vec A n
+  ```
+
+* In `Data.Vec.Functional`:
+  ```agda
+  remove : Fin (suc n) → Vector A (suc n) → Vector A n
+    ↦
+  removeAt : Vector A (suc n) → Fin (suc n) → Vector A n
+ 
+  updateAt : Fin n → (A → A) → Vector A n → Vector A n
+    ↦
+  updateAt : Vector A n → Fin n → (A → A) → Vector A n
+  ```
+  
+* The old names (and the names of all proofs about these functions) have been deprecated appropriately.
+
+
 ### Changes to triple reasoning interface
 
 * The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof that the strict
@@ -1321,6 +1365,11 @@ Deprecated names
   +-isAbelianGroup ↦ +-0-isAbelianGroup
   ```
 
+* In `Data.List.Base`:
+  ```
+  _─_  ↦  removeAt
+  ```
+
 * In `Data.List.Properties`:
   ```agda
   map-id₂         ↦  map-id-local
@@ -1337,7 +1386,10 @@ Deprecated names
 
   ʳ++-++  ↦  ++-ʳ++
 
-  take++drop ↦ take++drop≡id
+  take++drop  ↦  take++drop≡id
+
+  length-─  ↦  length-removeAt
+  map-─     ↦  map-removeAt
   ```
 
 * In `Data.List.NonEmpty.Properties`:
@@ -1350,8 +1402,8 @@ Deprecated names
 * In `Data.List.Relation.Unary.All.Properties`:
   ```agda
   updateAt-id-relative      ↦  updateAt-id-local
-  updateAt-compose-relative ↦  updateAt-∘-local
-  updateAt-compose          ↦  updateAt-∘
+  updateAt-compose-relative ↦  updateAt-updateAt-local
+  updateAt-compose          ↦  updateAt-updateAt
   updateAt-cong-relative    ↦  updateAt-cong-local
   ```
 
@@ -1497,6 +1549,12 @@ Deprecated names
   map-compose     ↦  map-∘
   ```
 
+* In `Data.Vec.Base`:
+  ```
+  remove  ↦  removeAt
+  insert  ↦  insertAt
+  ```
+
 * In `Data.Vec.Properties`:
   ```
   take-distr-zipWith ↦  take-zipWith
@@ -1505,8 +1563,8 @@ Deprecated names
   drop-distr-map     ↦  drop-map
 
   updateAt-id-relative      ↦  updateAt-id-local
-  updateAt-compose-relative ↦  updateAt-∘-local
-  updateAt-compose          ↦  updateAt-∘
+  updateAt-compose-relative ↦  updateAt-updateAt-local
+  updateAt-compose          ↦  updateAt-updateAt
   updateAt-cong-relative    ↦  updateAt-cong-local
 
   []%=-compose    ↦  []%=-∘
@@ -1516,7 +1574,15 @@ Deprecated names
   idIsFold        ↦ id-is-foldr
   sum-++-commute  ↦ sum-++
 
-  take-drop-id ↦ take++drop≡id
+  take-drop-id  ↦  take++drop≡id
+
+  map-insert       ↦  map-insertAt
+  
+  insert-lookup    ↦  insertAt-lookup
+  insert-punchIn   ↦  insertAt-punchIn
+  remove-PunchOut  ↦  removeAt-punchOut
+  remove-insert    ↦  removeAt-insertAt
+  insert-remove    ↦  insertAt-removeAt
 
   lookup-inject≤-take ↦ lookup-take-inject≤
   ```
@@ -1532,11 +1598,17 @@ Deprecated names
 * In `Data.Vec.Functional.Properties`:
   ```
   updateAt-id-relative      ↦  updateAt-id-local
-  updateAt-compose-relative ↦  updateAt-∘-local
-  updateAt-compose          ↦  updateAt-∘
+  updateAt-compose-relative ↦  updateAt-updateAt-local
+  updateAt-compose          ↦  updateAt-updateAt
   updateAt-cong-relative    ↦  updateAt-cong-local
 
   map-updateAt              ↦  map-updateAt-local
+  
+  insert-lookup             ↦  insertAt-lookup
+  insert-punchIn            ↦  insertAt-punchIn
+  remove-punchOut           ↦  removeAt-punchOut
+  remove-insert             ↦  removeAt-insertAt
+  insert-remove             ↦  insertAt-removeAt
   ```
   NB. This last one is complicated by the *addition* of a 'global' property `map-updateAt`
 
@@ -2312,7 +2384,7 @@ Additions to existing modules
   gcd-zero  : Zero 1ℤ gcd
   ```
 
-* Added new functions in `Data.List`:
+* Added new functions in `Data.List.Base`:
   ```agda
   takeWhileᵇ   : (A → Bool) → List A → List A
   dropWhileᵇ   : (A → Bool) → List A → List A
@@ -2331,14 +2403,14 @@ Additions to existing modules
   find         : Decidable P → List A → Maybe A
   findIndex    : Decidable P → (xs : List A) → Maybe $ Fin (length xs)
   findIndices  : Decidable P → (xs : List A) → List $ Fin (length xs)
-  ```
 
-* Added new functions and definitions to `Data.List.Base`:
-  ```agda
-  catMaybes : List (Maybe A) → List A
-  ap : List (A → B) → List A → List B
-  ++-rawMagma : Set a → RawMagma a _
+  catMaybes       : List (Maybe A) → List A
+  ap              : List (A → B) → List A → List B
+  ++-rawMagma     : Set a → RawMagma a _
   ++-[]-rawMonoid : Set a → RawMonoid a _
+
+  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
+  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
   ```
 
 * Added new proofs in `Data.List.Relation.Binary.Lex.Strict`:
@@ -2405,6 +2477,11 @@ Additions to existing modules
   map-replicate : map f (replicate n x) ≡ replicate n (f x)
 
   drop-drop : drop n (drop m xs) ≡ drop (m + n) xs
+
+  length-insertAt   : length (insertAt xs i v) ≡ suc (length xs)
+  length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
+  removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
+  insertAt-removeAt : insertAt (removeAt xs i) (cast (sym (lengthAt-removeAt xs i)) i) (lookup xs i) ≡ xs
   ```
 
 * Added new patterns and definitions to `Data.Nat.Base`:
@@ -2841,6 +2918,9 @@ Additions to existing modules
   fromList-map  : cast _ (fromList (List.map f xs)) ≡ map f (fromList xs)
   fromList-++   : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
 
+  length-toList   : List.length (toList xs) ≡ length xs
+  toList-insertAt : toList (insertAt xs i v) ≡ List.insertAt (toList xs) (Fin.cast (cong suc (sym (length-toList xs))) i) v
+
   truncate≡take       : .(eq : n ≡ m + o) → truncate m≤n xs ≡ take m (cast eq xs)
   take≡truncate       : take m xs ≡ truncate (m≤m+n m n) xs
   lookup-truncate     : lookup (truncate m≤n xs) i ≡ lookup xs (Fin.inject≤ i m≤n)
diff --git a/src/Algebra/Properties/CommutativeMonoid/Sum.agda b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
index 6cb417bc69..5873d4eb61 100644
--- a/src/Algebra/Properties/CommutativeMonoid/Sum.agda
+++ b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
@@ -46,7 +46,7 @@ open import Algebra.Properties.Monoid.Sum monoid public
 
 -- When summing over a function from a finite set, we can pull out any
 -- value and move it to the front.
-sum-remove : ∀ {n} {i : Fin (suc n)} t → sum t ≈ t i + sum (remove i t)
+sum-remove : ∀ {n} {i : Fin (suc n)} t → sum t ≈ t i + sum (removeAt t i)
 sum-remove {_}     {zero}   xs = refl
 sum-remove {suc n} {suc i}  xs = begin
   t₀ + ∑t           ≈⟨ +-congˡ (sum-remove t) ⟩
@@ -57,7 +57,7 @@ sum-remove {suc n} {suc i}  xs = begin
   t₀ = head xs
   tᵢ = t i
   ∑t = sum t
-  ∑t′ = sum (remove i t)
+  ∑t′ = sum (removeAt t i)
 
 -- The '∑' operator distributes over addition.
 ∑-distrib-+ : ∀ {n} (f g : Vector Carrier n) → ∑[ i < n ] (f i + g i) ≈ ∑[ i < n ] f i + ∑[ i < n ] g i
@@ -93,10 +93,10 @@ sum-permute {suc m} {suc n} f π = begin
   f 0F  + sum f/0                          ≡˘⟨ P.cong (_+ sum f/0) (P.cong f (Perm.inverseʳ π)) ⟩
   πf π₀ + sum f/0                          ≈⟨ +-congˡ (sum-permute f/0 (Perm.remove π₀ π)) ⟩
   πf π₀ + sum (rearrange (π/0 ⟨$⟩ʳ_) f/0)  ≡˘⟨ P.cong (πf π₀ +_) (sum-cong-≗ (P.cong f ∘ Perm.punchIn-permute′ π 0F)) ⟩
-  πf π₀ + sum (remove π₀ πf)               ≈⟨ sym (sum-remove πf) ⟩
+  πf π₀ + sum (removeAt πf π₀)             ≈⟨ sym (sum-remove πf) ⟩
   sum πf                                   ∎
   where
-  f/0 = remove 0F f
+  f/0 = removeAt f 0F
   π₀ = π ⟨$⟩ˡ 0F
   π/0 = Perm.remove π₀ π
   πf = rearrange (π ⟨$⟩ʳ_) f
diff --git a/src/Algebra/Properties/Semiring/Binomial.agda b/src/Algebra/Properties/Semiring/Binomial.agda
index b4d5ef81db..122da9fcb9 100644
--- a/src/Algebra/Properties/Semiring/Binomial.agda
+++ b/src/Algebra/Properties/Semiring/Binomial.agda
@@ -71,7 +71,7 @@ sum₂ n = ∑[ k ≤ suc n ] term₂ n k
 ------------------------------------------------------------------------
 -- Properties
 
-term₂[n,n+1]≈0# : ∀ n → term₂ n (fromℕ (suc n)) ≈ 0# 
+term₂[n,n+1]≈0# : ∀ n → term₂ n (fromℕ (suc n)) ≈ 0#
 term₂[n,n+1]≈0# n rewrite view-fromℕ (suc n) = refl
 
 lemma₁ : ∀ n → x * binomialExpansion n ≈ sum₁ n
@@ -117,7 +117,7 @@ x*lemma {n} i = begin
 ------------------------------------------------------------------------
 -- Next, a lemma which does require commutativity
 
-y*lemma : x * y ≈ y * x → ∀ {n : ℕ} (j : Fin n) → 
+y*lemma : x * y ≈ y * x → ∀ {n : ℕ} (j : Fin n) →
           y * binomialTerm n (suc j) ≈ (n C toℕ (suc j)) × binomial (suc n) (suc (inject₁ j))
 y*lemma x*y≈y*x {n} j = begin
   y * binomialTerm n (suc j)
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 9817cf8648..b96a0451cc 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -196,6 +196,14 @@ tails : List A → List (List A)
 tails []       = [] ∷ []
 tails (x ∷ xs) = (x ∷ xs) ∷ tails xs
 
+insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
+insertAt xs       zero    v = v ∷ xs
+insertAt (x ∷ xs) (suc i) v = x ∷ insertAt xs i v
+
+updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
+updateAt (x ∷ xs) zero    f = f x ∷ xs
+updateAt (x ∷ xs) (suc i) f = x ∷ updateAt xs i f
+
 -- Scans
 
 scanr : (A → B → B) → B → List A → List B
@@ -330,6 +338,10 @@ splitAt zero    xs       = ([] , xs)
 splitAt (suc n) []       = ([] , [])
 splitAt (suc n) (x ∷ xs) = Prod.map₁ (x ∷_) (splitAt n xs)
 
+removeAt : (xs : List A) → Fin (length xs) → List A
+removeAt (x ∷ xs) zero     = xs
+removeAt (x ∷ xs) (suc i)  = x ∷ removeAt xs i
+
 -- The following are functions which split a list up using boolean
 -- predicates. However, in practice they are difficult to use and
 -- prove properties about, and are mainly provided for advanced use
@@ -469,19 +481,18 @@ findIndices P? = findIndicesᵇ (does ∘ P?)
 ------------------------------------------------------------------------
 -- Actions on single elements
 
-infixl 5 _[_]%=_ _[_]∷=_ _─_
+infixl 5 _[_]%=_ _[_]∷=_
+
+-- xs [ i ]%= f  modifies the i-th element of xs according to f
 
 _[_]%=_ : (xs : List A) → Fin (length xs) → (A → A) → List A
-(x ∷ xs) [ zero  ]%= f = f x ∷ xs
-(x ∷ xs) [ suc k ]%= f = x ∷ (xs [ k ]%= f)
+xs [ i ]%= f = updateAt xs i f
+
+-- xs [ i ]≔ y  overwrites the i-th element of xs with y
 
 _[_]∷=_ : (xs : List A) → Fin (length xs) → A → List A
 xs [ k ]∷= v = xs [ k ]%= const v
 
-_─_ : (xs : List A) → Fin (length xs) → List A
-(x ∷ xs) ─ zero  = xs
-(x ∷ xs) ─ suc k = x ∷ (xs ─ k)
-
 ------------------------------------------------------------------------
 -- Conditional versions of cons and snoc
 
@@ -527,3 +538,12 @@ _∷ʳ'_ = InitLast._∷ʳ′_
 "Warning: _∷ʳ'_ (ending in an apostrophe) was deprecated in v1.4.
 Please use _∷ʳ′_ (ending in a prime) instead."
 #-}
+
+-- Version 2.0
+
+infixl 5 _─_
+_─_ = removeAt
+{-# WARNING_ON_USAGE _─_
+"Warning: _─_ was deprecated in v2.0.
+Please use removeAt instead."
+#-}
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index eb403f8b91..48f533f08d 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -746,17 +746,41 @@ map-∷= (x ∷ xs) zero    v f = refl
 map-∷= (x ∷ xs) (suc k) v f = cong (f x ∷_) (map-∷= xs k v f)
 
 ------------------------------------------------------------------------
--- _─_
+-- insertAt
 
-length-─ : ∀ (xs : List A) k → length (xs ─ k) ≡ pred (length xs)
-length-─ (x ∷ xs) zero        = refl
-length-─ (x ∷ y ∷ xs) (suc k) = cong suc (length-─ (y ∷ xs) k)
+length-insertAt : ∀ (xs : List A) (i : Fin (suc (length xs))) v →
+                  length (insertAt xs i v) ≡ suc (length xs)
+length-insertAt xs       zero    v = refl
+length-insertAt (x ∷ xs) (suc i) v = cong suc (length-insertAt xs i v)
 
-map-─ : ∀ xs k (f : A → B) →
-        let eq = sym (length-map f xs) in
-        map f (xs ─ k) ≡ map f xs ─ cast eq k
-map-─ (x ∷ xs) zero    f = refl
-map-─ (x ∷ xs) (suc k) f = cong (f x ∷_) (map-─ xs k f)
+------------------------------------------------------------------------
+-- removeAt
+
+length-removeAt : ∀ (xs : List A) k → length (removeAt xs k) ≡ pred (length xs)
+length-removeAt (x ∷ xs) zero            = refl
+length-removeAt (x ∷ xs@(_ ∷ _)) (suc k) = cong suc (length-removeAt xs k)
+
+length-removeAt′ : ∀ (xs : List A) k → length xs ≡ suc (length (removeAt xs k))
+length-removeAt′ xs@(_ ∷ _) k rewrite length-removeAt xs k = refl
+
+map-removeAt : ∀ xs k (f : A → B) →
+            let eq = sym (length-map f xs) in
+            map f (removeAt xs k) ≡ removeAt (map f xs) (cast eq k)
+map-removeAt (x ∷ xs) zero    f = refl
+map-removeAt (x ∷ xs) (suc k) f = cong (f x ∷_) (map-removeAt xs k f)
+
+------------------------------------------------------------------------
+ -- insertAt and removeAt
+
+removeAt-insertAt : ∀ (xs : List A) (i : Fin (suc (length xs))) v →
+  removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
+removeAt-insertAt xs       zero    v = refl
+removeAt-insertAt (x ∷ xs) (suc i) v = cong (_ ∷_) (removeAt-insertAt xs i v)
+
+insertAt-removeAt : (xs : List A) (i : Fin (length xs)) →
+  insertAt (removeAt xs i) (cast (length-removeAt′ xs i) i) (lookup xs i) ≡ xs
+insertAt-removeAt (x ∷ xs) zero    = refl
+insertAt-removeAt (x ∷ xs) (suc i) = cong (x ∷_) (insertAt-removeAt xs i)
 
 ------------------------------------------------------------------------
 -- take
@@ -1224,3 +1248,15 @@ take++drop = take++drop≡id
 "Warning: take++drop was deprecated in v2.0.
 Please use take++drop≡id instead."
 #-}
+
+length-─ = length-removeAt
+{-# WARNING_ON_USAGE length-─
+"Warning: length-─ was deprecated in v2.0.
+Please use length-removeAt instead."
+#-}
+
+map-─ = map-removeAt
+{-# WARNING_ON_USAGE map-─
+"Warning: map-─ was deprecated in v2.0.
+Please use map-removeAt instead."
+#-}
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index da617af44b..1b55bcd2b8 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Bool.Base using (Bool; T; true; false)
 open import Data.Bool.Properties using (T-∧)
 open import Data.Empty
 open import Data.Fin.Base using (Fin; zero; suc)
-open import Data.List.Base as List hiding (lookup)
+open import Data.List.Base as List hiding (lookup; updateAt)
 open import Data.List.Properties as Listₚ using (partition-defn)
 open import Data.List.Membership.Propositional
 open import Data.List.Membership.Propositional.Properties
diff --git a/src/Data/List/Relation/Unary/Any.agda b/src/Data/List/Relation/Unary/Any.agda
index eb010e2a39..169f6020aa 100644
--- a/src/Data/List/Relation/Unary/Any.agda
+++ b/src/Data/List/Relation/Unary/Any.agda
@@ -10,7 +10,7 @@ module Data.List.Relation.Unary.Any where
 
 open import Data.Empty
 open import Data.Fin.Base using (Fin; zero; suc)
-open import Data.List.Base as List using (List; []; [_]; _∷_)
+open import Data.List.Base as List using (List; []; [_]; _∷_; removeAt)
 open import Data.Product.Base as Prod using (∃; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Level using (Level; _⊔_)
@@ -69,7 +69,7 @@ _∷=_ {xs = xs} x∈xs v = xs List.[ index x∈xs ]∷= v
 
 infixl 4 _─_
 _─_ : {P : Pred A p} → ∀ xs → Any P xs → List A
-xs ─ x∈xs = xs List.─ index x∈xs
+xs ─ x∈xs = removeAt xs (index x∈xs)
 
 -- If any element satisfies P, then P is satisfied.
 
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 507ac05463..469549e1ed 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -64,29 +64,27 @@ iterate : (A → A) → A → ∀ {n} → Vec A n
 iterate s z {zero}  = []
 iterate s z {suc n} = z ∷ iterate s (s z)
 
-insert : Vec A n → Fin (suc n) → A → Vec A (suc n)
-insert xs       zero     v = v ∷ xs
-insert (x ∷ xs) (suc i)  v = x ∷ insert xs i v
+insertAt : Vec A n → Fin (suc n) → A → Vec A (suc n)
+insertAt xs       zero     v = v ∷ xs
+insertAt (x ∷ xs) (suc i)  v = x ∷ insertAt xs i v
 
-remove : Vec A (suc n) → Fin (suc n) → Vec A n
-remove (_ ∷ xs)     zero     = xs
-remove (x ∷ y ∷ xs) (suc i)  = x ∷ remove (y ∷ xs) i
+removeAt : Vec A (suc n) → Fin (suc n) → Vec A n
+removeAt (x ∷ xs)         zero    = xs
+removeAt (x ∷ xs@(_ ∷ _)) (suc i) = x ∷ removeAt xs i
 
-updateAt : Fin n → (A → A) → Vec A n → Vec A n
-updateAt zero    f (x ∷ xs) = f x ∷ xs
-updateAt (suc i) f (x ∷ xs) = x   ∷ updateAt i f xs
+updateAt : Vec A n → Fin n → (A → A) → Vec A n
+updateAt (x ∷ xs) zero    f = f x ∷ xs
+updateAt (x ∷ xs) (suc i) f = x   ∷ updateAt xs i f
 
 -- xs [ i ]%= f  modifies the i-th element of xs according to f
 
-infixl 6 _[_]%=_
+infixl 6 _[_]%=_ _[_]≔_
 
 _[_]%=_ : Vec A n → Fin n → (A → A) → Vec A n
-xs [ i ]%= f = updateAt i f xs
+xs [ i ]%= f = updateAt xs i f
 
 -- xs [ i ]≔ y  overwrites the i-th element of xs with y
 
-infixl 6 _[_]≔_
-
 _[_]≔_ : Vec A n → Fin n → A → Vec A n
 xs [ i ]≔ y = xs [ i ]%= const y
 
@@ -353,3 +351,21 @@ transpose : Vec (Vec A n) m → Vec (Vec A m) n
 transpose []         = replicate []
 transpose (as ∷ ass) = replicate _∷_ ⊛ as ⊛ transpose ass
 
+------------------------------------------------------------------------
+-- DEPRECATED
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+remove = removeAt
+{-# WARNING_ON_USAGE remove
+"Warning: remove was deprecated in v2.0.
+Please use removeAt instead."
+#-}
+insert = insertAt
+{-# WARNING_ON_USAGE insert
+"Warning: insert was deprecated in v2.0.
+Please use insertAt instead."
+#-}
diff --git a/src/Data/Vec/Functional.agda b/src/Data/Vec/Functional.agda
index 091886bef5..7b217a415b 100644
--- a/src/Data/Vec/Functional.agda
+++ b/src/Data/Vec/Functional.agda
@@ -31,9 +31,8 @@ infixl 1 _>>=_
 private
   variable
     a b c : Level
-    A : Set a
-    B : Set b
-    C : Set c
+    A B C : Set a
+    m n : ℕ
 
 ------------------------------------------------------------------------
 -- Definition
@@ -44,13 +43,13 @@ Vector A n = Fin n → A
 ------------------------------------------------------------------------
 -- Conversion
 
-toVec : ∀ {n} → Vector A n → Vec A n
+toVec : Vector A n → Vec A n
 toVec = V.tabulate
 
-fromVec : ∀ {n} → Vec A n → Vector A n
+fromVec : Vec A n → Vector A n
 fromVec = V.lookup
 
-toList : ∀ {n} → Vector A n → List A
+toList : Vector A n → List A
 toList = L.tabulate
 
 fromList : ∀ (xs : List A) → Vector A (L.length xs)
@@ -62,39 +61,39 @@ fromList = L.lookup
 [] : Vector A zero
 [] ()
 
-_∷_ : ∀ {n} → A → Vector A n → Vector A (suc n)
+_∷_ : A → Vector A n → Vector A (suc n)
 (x ∷ xs) zero    = x
 (x ∷ xs) (suc i) = xs i
 
-length : ∀ {n} → Vector A n → ℕ
+length : Vector A n → ℕ
 length {n = n} _ = n
 
-head : ∀ {n} → Vector A (suc n) → A
+head : Vector A (suc n) → A
 head xs = xs zero
 
-tail : ∀ {n} → Vector A (suc n) → Vector A n
+tail : Vector A (suc n) → Vector A n
 tail xs = xs ∘ suc
 
-uncons : ∀ {n} → Vector A (suc n) → A × Vector A n
+uncons : Vector A (suc n) → A × Vector A n
 uncons xs = head xs , tail xs
 
-replicate : ∀ {n} → A → Vector A n
+replicate : A → Vector A n
 replicate = const
 
-insert : ∀ {n} → Vector A n → Fin (suc n) → A → Vector A (suc n)
-insert {n = n}     xs zero    v zero    = v
-insert {n = n}     xs zero    v (suc j) = xs j
-insert {n = suc n} xs (suc i) v zero    = head xs
-insert {n = suc n} xs (suc i) v (suc j) = insert (tail xs) i v j
+insertAt : Vector A n → Fin (suc n) → A → Vector A (suc n)
+insertAt {n = n}     xs zero    v zero    = v
+insertAt {n = n}     xs zero    v (suc j) = xs j
+insertAt {n = suc n} xs (suc i) v zero    = head xs
+insertAt {n = suc n} xs (suc i) v (suc j) = insertAt (tail xs) i v j
 
-remove : ∀ {n} → Fin (suc n) → Vector A (suc n) → Vector A n
-remove i t = t ∘ punchIn i
+removeAt : Vector A (suc n) → Fin (suc n) → Vector A n
+removeAt t i = t ∘ punchIn i
 
-updateAt : ∀ {n} → Fin n → (A → A) → Vector A n → Vector A n
-updateAt {n = suc n} zero    f xs zero    = f (head xs)
-updateAt {n = suc n} zero    f xs (suc j) = xs (suc j)
-updateAt {n = suc n} (suc i) f xs zero    = head xs
-updateAt {n = suc n} (suc i) f xs (suc j) = updateAt i f (tail xs) j
+updateAt : Vector A n → Fin n → (A → A) → Vector A n
+updateAt {n = suc n} xs zero    f zero    = f (head xs)
+updateAt {n = suc n} xs zero    f (suc j) = xs (suc j)
+updateAt {n = suc n} xs (suc i) f zero    = head xs
+updateAt {n = suc n} xs (suc i) f (suc j) = updateAt (tail xs) i f j
 
 ------------------------------------------------------------------------
 -- Transformations
@@ -102,10 +101,10 @@ updateAt {n = suc n} (suc i) f xs (suc j) = updateAt i f (tail xs) j
 map : (A → B) → ∀ {n} → Vector A n → Vector B n
 map f xs = f ∘ xs
 
-_++_ : ∀ {m n} → Vector A m → Vector A n → Vector A (m ℕ.+ n)
+_++_ : Vector A m → Vector A n → Vector A (m ℕ.+ n)
 _++_ {m = m} xs ys i = [ xs , ys ] (splitAt m i)
 
-concat : ∀ {m n} → Vector (Vector A m) n → Vector A (n ℕ.* m)
+concat : Vector (Vector A m) n → Vector A (n ℕ.* m)
 concat {m = m} xss i = uncurry (flip xss) (quotRem m i)
 
 foldr : (A → B → B) → B → ∀ {n} → Vector A n → B
@@ -116,25 +115,25 @@ foldl : (B → A → B) → B → ∀ {n} → Vector A n → B
 foldl f z {n = zero}  xs = z
 foldl f z {n = suc n} xs = foldl f (f z (head xs)) (tail xs)
 
-rearrange : ∀ {m n} → (Fin m → Fin n) → Vector A n → Vector A m
+rearrange : (Fin m → Fin n) → Vector A n → Vector A m
 rearrange r xs = xs ∘ r
 
-_⊛_ : ∀ {n} → Vector (A → B) n → Vector A n → Vector B n
+_⊛_ : Vector (A → B) n → Vector A n → Vector B n
 _⊛_ = _ˢ_
 
-_>>=_ : ∀ {m n} → Vector A m → (A → Vector B n) → Vector B (m ℕ.* n)
+_>>=_ : Vector A m → (A → Vector B n) → Vector B (m ℕ.* n)
 xs >>= f = concat (map f xs)
 
 zipWith : (A → B → C) → ∀ {n} → Vector A n → Vector B n → Vector C n
 zipWith f xs ys i = f (xs i) (ys i)
 
-unzipWith : ∀ {n} → (A → B × C) → Vector A n → Vector B n × Vector C n
+unzipWith : (A → B × C) → Vector A n → Vector B n × Vector C n
 unzipWith f xs = proj₁ ∘ f ∘ xs , proj₂ ∘ f ∘ xs
 
-zip : ∀ {n} → Vector A n → Vector B n → Vector (A × B) n
+zip : Vector A n → Vector B n → Vector (A × B) n
 zip = zipWith _,_
 
-unzip : ∀ {n} → Vector (A × B) n → Vector A n × Vector B n
+unzip : Vector (A × B) n → Vector A n × Vector B n
 unzip = unzipWith id
 
 take : ∀ m {n} → Vector A (m ℕ.+ n) → Vector A m
@@ -143,14 +142,35 @@ take _ {n = n} xs = xs ∘ (_↑ˡ n)
 drop : ∀ m {n} → Vector A (m ℕ.+ n) → Vector A n
 drop m xs = xs ∘ (m ↑ʳ_)
 
-reverse : ∀ {n} → Vector A n → Vector A n
+reverse : Vector A n → Vector A n
 reverse xs = xs ∘ opposite
 
-init : ∀ {n} → Vector A (suc n) → Vector A n
+init : Vector A (suc n) → Vector A n
 init xs = xs ∘ inject₁
 
-last : ∀ {n} → Vector A (suc n) → A
+last : Vector A (suc n) → A
 last {n = n} xs = xs (fromℕ n)
 
-transpose : ∀ {m n} → Vector (Vector A n) m → Vector (Vector A m) n
+transpose : Vector (Vector A n) m → Vector (Vector A m) n
 transpose = flip
+
+------------------------------------------------------------------------
+-- DEPRECATED
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+remove : Fin (suc n) → Vector A (suc n) → Vector A n
+remove = flip removeAt
+{-# WARNING_ON_USAGE remove
+"Warning: remove was deprecated in v2.0.
+Please use removeAt instead.
+NOTE: argument order has been flipped."
+#-}
+insert = insertAt
+{-# WARNING_ON_USAGE insert
+"Warning: insert was deprecated in v2.0.
+Please use insertAt instead."
+#-}
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index f3314754e0..93f95b423b 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -31,13 +31,12 @@ import Data.Fin.Properties as Finₚ
 private
   variable
     a b c : Level
-    A : Set a
-    B : Set b
-    C : Set c
+    A B C : Set a
+    m n : ℕ
 
 ------------------------------------------------------------------------
 
-module _ {n} {xs ys : Vector A (suc n)} where
+module _ {xs ys : Vector A (suc n)} where
 
   ∷-cong : head xs ≡ head ys → tail xs ≗ tail ys → xs ≗ ys
   ∷-cong eq _ zero    = eq
@@ -46,10 +45,9 @@ module _ {n} {xs ys : Vector A (suc n)} where
   ∷-injective : xs ≗ ys → head xs ≡ head ys × tail xs ≗ tail ys
   ∷-injective eq = eq zero , eq ∘ suc
 
-≗-dec : DecidableEquality A →
-        ∀ {n} → Decidable {A = Vector A n} _≗_
-≗-dec _≟_ {zero}  xs ys = yes λ ()
-≗-dec _≟_ {suc n} xs ys =
+≗-dec : DecidableEquality A → Decidable {A = Vector A n} _≗_
+≗-dec {n = zero}  _≟_ xs ys = yes λ ()
+≗-dec {n = suc n} _≟_ xs ys =
   map′ (Product.uncurry ∷-cong) ∷-injective
        (head xs ≟ head ys ×-dec ≗-dec _≟_ (tail xs) (tail ys))
 
@@ -58,15 +56,15 @@ module _ {n} {xs ys : Vector A (suc n)} where
 
 -- (+) updateAt i actually updates the element at index i.
 
-updateAt-updates : ∀ {n} (i : Fin n) {f : A → A} (xs : Vector A n) →
-                   updateAt i f xs i ≡ f (xs i)
+updateAt-updates : ∀ (i : Fin n) {f : A → A} (xs : Vector A n) →
+                   updateAt xs i f i ≡ f (xs i)
 updateAt-updates zero    xs = refl
 updateAt-updates (suc i) xs = updateAt-updates i (tail xs)
 
 -- (-) updateAt i does not touch the elements at other indices.
 
-updateAt-minimal : ∀ {n} (i j : Fin n) {f : A → A} (xs : Vector A n) →
-                   i ≢ j → updateAt j f xs i ≡ xs i
+updateAt-minimal : ∀ (i j : Fin n) {f : A → A} (xs : Vector A n) →
+                   i ≢ j → updateAt xs j f i ≡ xs i
 updateAt-minimal zero    zero    xs 0≢0 = ⊥-elim (0≢0 refl)
 updateAt-minimal zero    (suc j) xs _   = refl
 updateAt-minimal (suc i) zero    xs _   = refl
@@ -74,46 +72,46 @@ updateAt-minimal (suc i) (suc j) xs i≢j = updateAt-minimal i j (tail xs) (i≢
 
 -- updateAt i is a monoid morphism from A → A to Vector A n → Vector A n.
 
-updateAt-id-local : ∀ {n} (i : Fin n) {f : A → A} (xs : Vector A n) →
+updateAt-id-local : ∀ (i : Fin n) {f : A → A} (xs : Vector A n) →
                     f (xs i) ≡ xs i →
-                    updateAt i f xs ≗ xs
+                    updateAt xs i f ≗ xs
 updateAt-id-local zero    xs eq zero    = eq
 updateAt-id-local zero    xs eq (suc j) = refl
 updateAt-id-local (suc i) xs eq zero    = refl
 updateAt-id-local (suc i) xs eq (suc j) = updateAt-id-local i (tail xs) eq j
 
-updateAt-id : ∀ {n} (i : Fin n) (xs : Vector A n) →
-              updateAt i id xs ≗ xs
+updateAt-id : ∀ (i : Fin n) (xs : Vector A n) →
+              updateAt xs i id ≗ xs
 updateAt-id i xs = updateAt-id-local i xs refl
 
-updateAt-∘-local : ∀ {n} (i : Fin n) {f g h : A → A} (xs : Vector A n) →
-                   f (g (xs i)) ≡ h (xs i) →
-                   updateAt i f (updateAt i g xs) ≗ updateAt i h xs
-updateAt-∘-local zero    xs eq zero    = eq
-updateAt-∘-local zero    xs eq (suc j) = refl
-updateAt-∘-local (suc i) xs eq zero    = refl
-updateAt-∘-local (suc i) xs eq (suc j) = updateAt-∘-local i (tail xs) eq j
+updateAt-updateAt-local : ∀ (i : Fin n) {f g h : A → A} (xs : Vector A n) →
+                          f (g (xs i)) ≡ h (xs i) →
+                          updateAt (updateAt xs i g) i f ≗ updateAt xs i h
+updateAt-updateAt-local zero    xs eq zero    = eq
+updateAt-updateAt-local zero    xs eq (suc j) = refl
+updateAt-updateAt-local (suc i) xs eq zero    = refl
+updateAt-updateAt-local (suc i) xs eq (suc j) = updateAt-updateAt-local i (tail xs) eq j
 
-updateAt-∘ : ∀ {n} (i : Fin n) {f g : A → A} (xs : Vector A n) →
-             updateAt i f (updateAt i g xs) ≗ updateAt i (f ∘ g) xs
-updateAt-∘ i xs = updateAt-∘-local i xs refl
+updateAt-updateAt : ∀ (i : Fin n) {f g : A → A} (xs : Vector A n) →
+                    updateAt (updateAt xs i g) i f ≗ updateAt xs i (f ∘ g)
+updateAt-updateAt i xs = updateAt-updateAt-local i xs refl
 
-updateAt-cong-local : ∀ {n} (i : Fin n) {f g : A → A} (xs : Vector A n) →
+updateAt-cong-local : ∀ (i : Fin n) {f g : A → A} (xs : Vector A n) →
                       f (xs i) ≡ g (xs i) →
-                      updateAt i f xs ≗ updateAt i g xs
+                      updateAt xs i f ≗ updateAt xs i g
 updateAt-cong-local zero    xs eq zero    = eq
 updateAt-cong-local zero    xs eq (suc j) = refl
 updateAt-cong-local (suc i) xs eq zero    = refl
 updateAt-cong-local (suc i) xs eq (suc j) = updateAt-cong-local i (tail xs) eq j
 
-updateAt-cong : ∀ {n} (i : Fin n) {f g : A → A} →
-                f ≗ g → (xs : Vector A n) → updateAt i f xs ≗ updateAt i g xs
+updateAt-cong : ∀ (i : Fin n) {f g : A → A} → f ≗ g → (xs : Vector A n) →
+                updateAt xs i f ≗ updateAt xs i g
 updateAt-cong i eq xs = updateAt-cong-local i xs (eq (xs i))
 
 -- The order of updates at different indices i ≢ j does not matter.
 
-updateAt-commutes : ∀ {n} (i j : Fin n) {f g : A → A} → i ≢ j → (xs : Vector A n) →
-                    updateAt i f (updateAt j g xs) ≗ updateAt j g (updateAt i f xs)
+updateAt-commutes : ∀ (i j : Fin n) {f g : A → A} → i ≢ j → (xs : Vector A n) →
+                    updateAt (updateAt xs j g) i f ≗ updateAt (updateAt xs i f) j g
 updateAt-commutes zero    zero    0≢0 xs k       = ⊥-elim (0≢0 refl)
 updateAt-commutes zero    (suc j) _   xs zero    = refl
 updateAt-commutes zero    (suc j) _   xs (suc k) = refl
@@ -125,59 +123,59 @@ updateAt-commutes (suc i) (suc j) i≢j xs (suc k) = updateAt-commutes i j (i≢
 ------------------------------------------------------------------------
 -- map
 
-map-id : ∀ {n} → (xs : Vector A n) → map id xs ≗ xs
+map-id : (xs : Vector A n) → map id xs ≗ xs
 map-id xs = λ _ → refl
 
-map-cong : ∀ {n} {f g : A → B} → f ≗ g → (xs : Vector A n) → map f xs ≗ map g xs
+map-cong : ∀ {f g : A → B} → f ≗ g → (xs : Vector A n) → map f xs ≗ map g xs
 map-cong f≗g xs = f≗g ∘ xs
 
-map-∘ : ∀ {n} {f : B → C} {g : A → B} →
-        (xs : Vector A n) → map (f ∘ g) xs ≗ map f (map g xs)
+map-∘ : ∀ {f : B → C} {g : A → B} (xs : Vector A n) →
+        map (f ∘ g) xs ≗ map f (map g xs)
 map-∘ xs = λ _ → refl
 
-lookup-map : ∀ {n} (i : Fin n) (f : A → B) (xs : Vector A n) →
+lookup-map : ∀ (i : Fin n) (f : A → B) (xs : Vector A n) →
              map f xs i ≡ f (xs i)
 lookup-map i f xs = refl
 
-map-updateAt-local : ∀ {n} {f : A → B} {g : A → A} {h : B → B}
+map-updateAt-local : ∀ {f : A → B} {g : A → A} {h : B → B}
                      (xs : Vector A n) (i : Fin n) →
                      f (g (xs i)) ≡ h (f (xs i)) →
-                     map f (updateAt i g xs) ≗ updateAt i h (map f xs)
+                     map f (updateAt xs i g) ≗ updateAt (map f xs) i h
 map-updateAt-local {n = suc n}       {f = f} xs zero    eq zero    = eq
 map-updateAt-local {n = suc n}       {f = f} xs zero    eq (suc j) = refl
 map-updateAt-local {n = suc (suc n)} {f = f} xs (suc i) eq zero    = refl
 map-updateAt-local {n = suc (suc n)} {f = f} xs (suc i) eq (suc j) = map-updateAt-local {f = f} (tail xs) i eq j
 
-map-updateAt : ∀ {n} {f : A → B} {g : A → A} {h : B → B} →
+map-updateAt : ∀ {f : A → B} {g : A → A} {h : B → B} →
                f ∘ g ≗ h ∘ f →
                (xs : Vector A n) (i : Fin n) →
-               map f (updateAt i g xs) ≗ updateAt i h (map f xs)
+               map f (updateAt xs i g) ≗ updateAt (map f xs) i h
 map-updateAt {f = f} {g = g} f∘g≗h∘f xs i = map-updateAt-local {f = f} {g = g} xs i (f∘g≗h∘f (xs i))
 
 ------------------------------------------------------------------------
 -- _++_
 
-lookup-++-< : ∀ {m n} (xs : Vector A m) (ys : Vector A n) →
+lookup-++-< : ∀ (xs : Vector A m) (ys : Vector A n) →
               ∀ i (i<m : toℕ i ℕ.< m) →
               (xs ++ ys) i ≡ xs (fromℕ< i<m)
 lookup-++-< {m = m} xs ys i i<m = cong Sum.[ xs , ys ] (Finₚ.splitAt-< m i i<m)
 
-lookup-++-≥ : ∀ {m n} (xs : Vector A m) (ys : Vector A n) →
+lookup-++-≥ : ∀ (xs : Vector A m) (ys : Vector A n) →
               ∀ i (i≥m : toℕ i ℕ.≥ m) →
               (xs ++ ys) i ≡ ys (reduce≥ i i≥m)
 lookup-++-≥ {m = m} xs ys i i≥m = cong Sum.[ xs , ys ] (Finₚ.splitAt-≥ m i i≥m)
 
-lookup-++ˡ : ∀ {m n} (xs : Vector A m) (ys : Vector A n) i →
+lookup-++ˡ : ∀ (xs : Vector A m) (ys : Vector A n) i →
              (xs ++ ys) (i ↑ˡ n) ≡ xs i
 lookup-++ˡ {m = m} {n = n} xs ys i = cong Sum.[ xs , ys ] (Finₚ.splitAt-↑ˡ m i n)
 
-lookup-++ʳ : ∀ {m n} (xs : Vector A m) (ys : Vector A n) i →
+lookup-++ʳ : ∀ (xs : Vector A m) (ys : Vector A n) i →
              (xs ++ ys) (m ↑ʳ i) ≡ ys i
 lookup-++ʳ {m = m} {n = n} xs ys i = cong Sum.[ xs , ys ] (Finₚ.splitAt-↑ʳ m n i)
 
-module _ {m} {ys ys′ : Vector A m} where
+module _ {ys ys′ : Vector A m} where
 
-  ++-cong : ∀ {n} (xs xs′ : Vector A n) →
+  ++-cong : ∀ (xs xs′ : Vector A n) →
             xs ≗ xs′ → ys ≗ ys′ → xs ++ ys ≗ xs′ ++ ys′
   ++-cong {n} xs xs′ eq₁ eq₂ i with toℕ i ℕₚ.<? n
   ... | yes i<n = begin
@@ -193,7 +191,7 @@ module _ {m} {ys ys′ : Vector A m} where
     (xs′ ++ ys′) i             ∎
     where open ≡-Reasoning
 
-  ++-injectiveˡ : ∀ {n} (xs xs′ : Vector A n) →
+  ++-injectiveˡ : ∀ (xs xs′ : Vector A n) →
                   xs ++ ys ≗ xs′ ++ ys′ → xs ≗ xs′
   ++-injectiveˡ xs xs′ eq i = begin
     xs i                   ≡˘⟨ lookup-++ˡ xs ys i ⟩
@@ -202,8 +200,7 @@ module _ {m} {ys ys′ : Vector A m} where
     xs′ i                  ∎
     where open ≡-Reasoning
 
-  ++-injectiveʳ : ∀ {n} (xs xs′ : Vector A n) →
-                  xs ++ ys ≗ xs′ ++ ys′ → ys ≗ ys′
+  ++-injectiveʳ : ∀ (xs xs′ : Vector A n) → xs ++ ys ≗ xs′ ++ ys′ → ys ≗ ys′
   ++-injectiveʳ {n} xs xs′ eq i = begin
     ys i                   ≡˘⟨ lookup-++ʳ xs ys i ⟩
     (xs ++ ys) (n ↑ʳ i)    ≡⟨ eq (n ↑ʳ i)   ⟩
@@ -211,56 +208,56 @@ module _ {m} {ys ys′ : Vector A m} where
     ys′ i                  ∎
     where open ≡-Reasoning
 
-  ++-injective : ∀ {n} (xs xs′ : Vector A n) →
+  ++-injective : ∀ (xs xs′ : Vector A n) →
                  xs ++ ys ≗ xs′ ++ ys′ → xs ≗ xs′ × ys ≗ ys′
   ++-injective xs xs′ eq = ++-injectiveˡ xs xs′ eq , ++-injectiveʳ xs xs′ eq
 
 ------------------------------------------------------------------------
--- insert
+-- insertAt
 
-insert-lookup : ∀ {n} (xs : Vector A n) (i : Fin (suc n)) (v : A) →
-                insert xs i v i ≡ v
-insert-lookup {n = n}     xs zero    v = refl
-insert-lookup {n = suc n} xs (suc i) v = insert-lookup (tail xs) i v
+insertAt-lookup : ∀ (xs : Vector A n) (i : Fin (suc n)) (v : A) →
+                  insertAt xs i v i ≡ v
+insertAt-lookup {n = n}     xs zero    v = refl
+insertAt-lookup {n = suc n} xs (suc i) v = insertAt-lookup (tail xs) i v
 
-insert-punchIn : ∀ {n} (xs : Vector A n) (i : Fin (suc n)) (v : A)
-                 (j : Fin n) →
-                 insert xs i v (punchIn i j) ≡ xs j
-insert-punchIn {n = suc n} xs zero    v j       = refl
-insert-punchIn {n = suc n} xs (suc i) v zero    = refl
-insert-punchIn {n = suc n} xs (suc i) v (suc j) = insert-punchIn (tail xs) i v j
+insertAt-punchIn : ∀ (xs : Vector A n) (i : Fin (suc n)) (v : A)
+                   (j : Fin n) →
+                   insertAt xs i v (punchIn i j) ≡ xs j
+insertAt-punchIn {n = suc n} xs zero    v j       = refl
+insertAt-punchIn {n = suc n} xs (suc i) v zero    = refl
+insertAt-punchIn {n = suc n} xs (suc i) v (suc j) = insertAt-punchIn (tail xs) i v j
 
 ------------------------------------------------------------------------
--- remove
+-- removeAt
 
-remove-punchOut : ∀ {n} (xs : Vector A (suc n))
+removeAt-punchOut : ∀ (xs : Vector A (suc n))
                   {i : Fin (suc n)} {j : Fin (suc n)} (i≢j : i ≢ j) →
-                  remove i xs (punchOut i≢j) ≡ xs j
-remove-punchOut {n = n}     xs {zero}  {zero}  i≢j = ⊥-elim (i≢j refl)
-remove-punchOut {n = suc n} xs {zero}  {suc j} i≢j = refl
-remove-punchOut {n = suc n} xs {suc i} {zero}  i≢j = refl
-remove-punchOut {n = suc n} xs {suc i} {suc j} i≢j = remove-punchOut (tail xs) (i≢j ∘ cong suc)
-
-remove-insert : ∀ {n} (xs : Vector A n) (i : Fin (suc n)) (v : A) →
-                remove i (insert xs i v) ≗ xs
-remove-insert xs zero    v j       = refl
-remove-insert xs (suc i) v zero    = refl
-remove-insert xs (suc i) v (suc j) = remove-insert (tail xs) i v j
-
-insert-remove : ∀ {n} (xs : Vector A (suc n)) (i : Fin (suc n)) →
-                insert (remove i xs) i (xs i) ≗ xs
-insert-remove {n = n}     xs zero    zero    = refl
-insert-remove {n = n}     xs zero    (suc j) = refl
-insert-remove {n = suc n} xs (suc i) zero    = refl
-insert-remove {n = suc n} xs (suc i) (suc j) = insert-remove (tail xs) i j
+                  removeAt xs i (punchOut i≢j) ≡ xs j
+removeAt-punchOut {n = n}     xs {zero}  {zero}  i≢j = ⊥-elim (i≢j refl)
+removeAt-punchOut {n = suc n} xs {zero}  {suc j} i≢j = refl
+removeAt-punchOut {n = suc n} xs {suc i} {zero}  i≢j = refl
+removeAt-punchOut {n = suc n} xs {suc i} {suc j} i≢j = removeAt-punchOut (tail xs) (i≢j ∘ cong suc)
+
+removeAt-insertAt : ∀ (xs : Vector A n) (i : Fin (suc n)) (v : A) →
+                    removeAt (insertAt xs i v) i ≗ xs
+removeAt-insertAt xs zero    v j       = refl
+removeAt-insertAt xs (suc i) v zero    = refl
+removeAt-insertAt xs (suc i) v (suc j) = removeAt-insertAt (tail xs) i v j
+
+insertAt-removeAt : ∀ (xs : Vector A (suc n)) (i : Fin (suc n)) →
+                    insertAt (removeAt xs i) i (xs i) ≗ xs
+insertAt-removeAt {n = n}     xs zero    zero    = refl
+insertAt-removeAt {n = n}     xs zero    (suc j) = refl
+insertAt-removeAt {n = suc n} xs (suc i) zero    = refl
+insertAt-removeAt {n = suc n} xs (suc i) (suc j) = insertAt-removeAt (tail xs) i j
 
 ------------------------------------------------------------------------
 -- Conversion functions
 
-toVec∘fromVec : ∀ {n} → (xs : Vec A n) → toVec (fromVec xs) ≡ xs
+toVec∘fromVec : (xs : Vec A n) → toVec (fromVec xs) ≡ xs
 toVec∘fromVec = Vₚ.tabulate∘lookup
 
-fromVec∘toVec : ∀ {n} → (xs : Vector A n) → fromVec (toVec xs) ≗ xs
+fromVec∘toVec : (xs : Vector A n) → fromVec (toVec xs) ≗ xs
 fromVec∘toVec = Vₚ.lookup∘tabulate
 
 toList∘fromList : (xs : List A) → toList (fromList xs) ≡ xs
@@ -279,21 +276,44 @@ updateAt-id-relative = updateAt-id-local
 "Warning: updateAt-id-relative was deprecated in v2.0.
 Please use updateAt-id-local instead."
 #-}
-
-updateAt-compose-relative = updateAt-∘-local
+updateAt-compose-relative = updateAt-updateAt-local
 {-# WARNING_ON_USAGE updateAt-compose-relative
 "Warning: updateAt-compose-relative was deprecated in v2.0.
-Please use updateAt-∘-local instead."
+Please use updateAt-updateAt-local instead."
 #-}
-
-updateAt-compose = updateAt-∘
+updateAt-compose = updateAt-updateAt
 {-# WARNING_ON_USAGE updateAt-compose
 "Warning: updateAt-compose was deprecated in v2.0.
-Please use updateAt-∘ instead."
+Please use updateAt-updateAt instead."
 #-}
-
 updateAt-cong-relative = updateAt-cong-local
 {-# WARNING_ON_USAGE updateAt-cong-relative
 "Warning: updateAt-cong-relative was deprecated in v2.0.
 Please use updateAt-cong-local instead."
 #-}
+
+insert-lookup = insertAt-lookup
+{-# WARNING_ON_USAGE insert-lookup
+"Warning: insert-lookup was deprecated in v2.0.
+Please use insertAt-lookup instead."
+#-}
+insert-punchIn = insertAt-punchIn
+{-# WARNING_ON_USAGE insert-punchIn
+"Warning: insert-punchIn was deprecated in v2.0.
+Please use insertAt-punchIn instead."
+#-}
+remove-punchOut = removeAt-punchOut
+{-# WARNING_ON_USAGE remove-punchOut
+"Warning: remove-punchOut was deprecated in v2.0.
+Please use removeAt-punchOut instead."
+#-}
+remove-insert = removeAt-insertAt
+{-# WARNING_ON_USAGE remove-insert
+"Warning: remove-insert was deprecated in v2.0.
+Please use removeAt-insertAt instead."
+#-}
+insert-remove = insertAt-removeAt
+{-# WARNING_ON_USAGE insert-remove
+"Warning: insert-remove was deprecated in v2.0.
+Please use insertAt-removeAt instead."
+#-}
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index c5b92ec2a1..3d0e3bf590 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -51,7 +51,7 @@ private
 ∷-injectiveʳ : x ∷ xs ≡ y ∷ ys → xs ≡ ys
 ∷-injectiveʳ refl = refl
 
-∷-injective : (x ∷ xs) ≡ (y ∷ ys) → x ≡ y × xs ≡ ys
+∷-injective : x ∷ xs ≡ y ∷ ys → x ≡ y × xs ≡ ys
 ∷-injective refl = refl , refl
 
 ≡-dec : DecidableEquality A → DecidableEquality (Vec A n)
@@ -205,14 +205,14 @@ lookup-take-inject≤ {m = m} {n = n} xs i = begin
 -- (+) updateAt i actually updates the element at index i.
 
 updateAt-updates : ∀ (i : Fin n) {f : A → A} (xs : Vec A n) →
-                   xs [ i ]= x → (updateAt i f xs) [ i ]= f x
+                   xs [ i ]= x → (updateAt xs i f) [ i ]= f x
 updateAt-updates zero    (x ∷ xs) here        = here
 updateAt-updates (suc i) (x ∷ xs) (there loc) = there (updateAt-updates i xs loc)
 
 -- (-) updateAt i does not touch the elements at other indices.
 
 updateAt-minimal : ∀ (i j : Fin n) {f : A → A} (xs : Vec A n) →
-                   i ≢ j → xs [ i ]= x → (updateAt j f xs) [ i ]= x
+                   i ≢ j → xs [ i ]= x → (updateAt xs j f) [ i ]= x
 updateAt-minimal zero    zero    (x ∷ xs) 0≢0 here        = contradiction refl 0≢0
 updateAt-minimal zero    (suc j) (x ∷ xs) _   here        = here
 updateAt-minimal (suc i) zero    (x ∷ xs) _   (there loc) = there loc
@@ -235,29 +235,29 @@ updateAt-minimal (suc i) (suc j) (x ∷ xs) i≢j (there loc) =
 
 updateAt-id-local : ∀ (i : Fin n) {f : A → A} (xs : Vec A n) →
                     f (lookup xs i) ≡ lookup xs i →
-                    updateAt i f xs ≡ xs
+                    updateAt xs i f ≡ xs
 updateAt-id-local zero    (x ∷ xs) eq = cong (_∷ xs) eq
 updateAt-id-local (suc i) (x ∷ xs) eq = cong (x ∷_) (updateAt-id-local i xs eq)
 
 -- 1b. identity:  updateAt i id ≗ id
 
-updateAt-id : ∀ (i : Fin n) (xs : Vec A n) → updateAt i id xs ≡ xs
+updateAt-id : ∀ (i : Fin n) (xs : Vec A n) → updateAt xs i id ≡ xs
 updateAt-id i xs = updateAt-id-local i xs refl
 
 -- 2a. local composition:  f ∘ g = h ↾ (lookup xs i)
 --                implies  updateAt i f ∘ updateAt i g = updateAt i h ↾ xs
 
-updateAt-∘-local : ∀ (i : Fin n) {f g h : A → A} (xs : Vec A n) →
-                         f (g (lookup xs i)) ≡ h (lookup xs i) →
-                         updateAt i f (updateAt i g xs) ≡ updateAt i h xs
-updateAt-∘-local zero    (x ∷ xs) fg=h = cong (_∷ xs) fg=h
-updateAt-∘-local (suc i) (x ∷ xs) fg=h = cong (x ∷_) (updateAt-∘-local i xs fg=h)
+updateAt-updateAt-local : ∀ (i : Fin n) {f g h : A → A} (xs : Vec A n) →
+                          f (g (lookup xs i)) ≡ h (lookup xs i) →
+                          updateAt (updateAt xs i g) i f ≡ updateAt xs i h
+updateAt-updateAt-local zero    (x ∷ xs) fg=h = cong (_∷ xs) fg=h
+updateAt-updateAt-local (suc i) (x ∷ xs) fg=h = cong (x ∷_) (updateAt-updateAt-local i xs fg=h)
 
 -- 2b. composition:  updateAt i f ∘ updateAt i g ≗ updateAt i (f ∘ g)
 
-updateAt-∘ : ∀ (i : Fin n) {f g : A → A} →
-                   updateAt i f ∘ updateAt i g ≗ updateAt i (f ∘ g)
-updateAt-∘ i xs = updateAt-∘-local i xs refl
+updateAt-updateAt : ∀ (i : Fin n) {f g : A → A} (xs : Vec A n) →
+                    updateAt (updateAt xs i g) i f ≡ updateAt xs i (f ∘ g)
+updateAt-updateAt i xs = updateAt-updateAt-local i xs refl
 
 -- 3. congruence:  updateAt i  is a congruence wrt. extensional equality.
 
@@ -266,14 +266,14 @@ updateAt-∘ i xs = updateAt-∘-local i xs refl
 
 updateAt-cong-local : ∀ (i : Fin n) {f g : A → A} (xs : Vec A n) →
                       f (lookup xs i) ≡ g (lookup xs i) →
-                      updateAt i f xs ≡ updateAt i g xs
+                      updateAt xs i f ≡ updateAt xs i g
 updateAt-cong-local zero    (x ∷ xs) f=g = cong (_∷ xs) f=g
 updateAt-cong-local (suc i) (x ∷ xs) f=g = cong (x ∷_) (updateAt-cong-local i xs f=g)
 
 -- 3b. congruence:  f ≗ g → updateAt i f ≗ updateAt i g
 
-updateAt-cong : ∀ (i : Fin n) {f g : A → A} →
-                f ≗ g → updateAt i f ≗ updateAt i g
+updateAt-cong : ∀ (i : Fin n) {f g : A → A} → f ≗ g → (xs : Vec A n) →
+                updateAt xs i f ≡ updateAt xs i g
 updateAt-cong i f≗g xs = updateAt-cong-local i xs (f≗g (lookup xs i))
 
 -- The order of updates at different indices i ≢ j does not matter.
@@ -281,13 +281,13 @@ updateAt-cong i f≗g xs = updateAt-cong-local i xs (f≗g (lookup xs i))
 -- This a consequence of updateAt-updates and updateAt-minimal
 -- but easier to prove inductively.
 
-updateAt-commutes : ∀ (i j : Fin n) {f g : A → A} → i ≢ j →
-                    updateAt i f ∘ updateAt j g ≗ updateAt j g ∘ updateAt i f
+updateAt-commutes : ∀ (i j : Fin n) {f g : A → A} → i ≢ j → (xs : Vec A n) →
+                    updateAt (updateAt xs j g) i f ≡ updateAt (updateAt xs i f) j g
 updateAt-commutes zero    zero    0≢0 (x ∷ xs) = contradiction refl 0≢0
 updateAt-commutes zero    (suc j) i≢j (x ∷ xs) = refl
 updateAt-commutes (suc i) zero    i≢j (x ∷ xs) = refl
 updateAt-commutes (suc i) (suc j) i≢j (x ∷ xs) =
-  cong (x ∷_) (updateAt-commutes i j (i≢j ∘ cong suc) xs)
+   cong (x ∷_) (updateAt-commutes i j (i≢j ∘ cong suc) xs)
 
 -- lookup after updateAt reduces.
 
@@ -295,14 +295,14 @@ updateAt-commutes (suc i) (suc j) i≢j (x ∷ xs) =
 -- using []=↔lookup.
 
 lookup∘updateAt : ∀ (i : Fin n) {f : A → A} xs →
-                  lookup (updateAt i f xs) i ≡ f (lookup xs i)
+                  lookup (updateAt xs i f) i ≡ f (lookup xs i)
 lookup∘updateAt i xs =
   []=⇒lookup (updateAt-updates i xs (lookup⇒[]= i _ refl))
 
 -- For different indices it easily follows from updateAt-minimal.
 
 lookup∘updateAt′ : ∀ (i j : Fin n) {f : A → A} → i ≢ j → ∀ xs →
-                   lookup (updateAt j f xs) i ≡ lookup xs i
+                   lookup (updateAt xs j f) i ≡ lookup xs i
 lookup∘updateAt′ i j xs i≢j =
   []=⇒lookup (updateAt-minimal i j i≢j xs (lookup⇒[]= i _ refl))
 
@@ -312,10 +312,10 @@ lookup∘updateAt′ i j xs i≢j =
 []%=-id xs i = updateAt-id i xs
 
 []%=-∘ : ∀ (xs : Vec A n) (i : Fin n) {f g : A → A} →
-     xs [ i ]%= f
-        [ i ]%= g
-   ≡ xs [ i ]%= g ∘ f
-[]%=-∘ xs i = updateAt-∘ i xs
+      xs [ i ]%= f
+         [ i ]%= g
+    ≡ xs [ i ]%= g ∘ f
+[]%=-∘ xs i = updateAt-updateAt i xs
 
 
 ------------------------------------------------------------------------
@@ -325,11 +325,11 @@ lookup∘updateAt′ i j xs i≢j =
 -- are special cases of the ones for updateAt.
 
 []≔-idempotent : ∀ (xs : Vec A n) (i : Fin n) →
-                 (xs [ i ]≔ x) [ i ]≔ y ≡ xs [ i ]≔ y
-[]≔-idempotent xs i = updateAt-∘ i xs
+                  (xs [ i ]≔ x) [ i ]≔ y ≡ xs [ i ]≔ y
+[]≔-idempotent xs i = updateAt-updateAt i xs
 
 []≔-commutes : ∀ (xs : Vec A n) (i j : Fin n) → i ≢ j →
-               (xs [ i ]≔ x) [ j ]≔ y ≡ (xs [ j ]≔ y) [ i ]≔ x
+                (xs [ i ]≔ x) [ j ]≔ y ≡ (xs [ j ]≔ y) [ i ]≔ x
 []≔-commutes xs i j i≢j = updateAt-commutes j i (i≢j ∘ sym) xs
 
 []≔-updates : ∀ (xs : Vec A n) (i : Fin n) → (xs [ i ]≔ x) [ i ]= x
@@ -423,15 +423,15 @@ lookup-map (suc i) f (x ∷ xs) = lookup-map i f xs
 map-updateAt : ∀ {f : A → B} {g : A → A} {h : B → B}
                (xs : Vec A n) (i : Fin n) →
                f (g (lookup xs i)) ≡ h (f (lookup xs i)) →
-               map f (updateAt i g xs) ≡ updateAt i h (map f xs)
+               map f (updateAt xs i g) ≡ updateAt (map f xs) i h
 map-updateAt (x ∷ xs) zero    eq = cong (_∷ _) eq
 map-updateAt (x ∷ xs) (suc i) eq = cong (_ ∷_) (map-updateAt xs i eq)
 
-map-insert : ∀ (f : A → B) (x : A) (xs : Vec A n) (i : Fin (suc n)) →
-             map f (insert xs i x) ≡ insert (map f xs) i (f x)
-map-insert f _ []        zero    = refl
-map-insert f _ (x' ∷ xs) zero    = refl
-map-insert f x (x' ∷ xs) (suc i) = cong (_ ∷_) (map-insert f x xs i)
+map-insertAt : ∀ (f : A → B) (x : A) (xs : Vec A n) (i : Fin (suc n)) →
+             map f (insertAt xs i x) ≡ insertAt (map f xs) i (f x)
+map-insertAt f _ []        Fin.zero = refl
+map-insertAt f _ (x' ∷ xs) Fin.zero = refl
+map-insertAt f x (x' ∷ xs) (Fin.suc i) = cong (_ ∷_) (map-insertAt f x xs i)
 
 map-[]≔ : ∀ (f : A → B) (xs : Vec A n) (i : Fin n) →
           map f (xs [ i ]≔ x) ≡ map f xs [ i ]≔ f x
@@ -1152,42 +1152,57 @@ module _ {P : Pred A p} (P? : Decidable P) where
   ... | false = m≤n⇒m≤1+n (count≤n xs)
 
 ------------------------------------------------------------------------
--- insert
-
-insert-lookup : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →
-                lookup (insert xs i v) i ≡ v
-insert-lookup xs       zero     v = refl
-insert-lookup (x ∷ xs) (suc i)  v = insert-lookup xs i v
-
-insert-punchIn : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) (j : Fin n) →
-                 lookup (insert xs i v) (Fin.punchIn i j) ≡ lookup xs j
-insert-punchIn xs       zero     v j       = refl
-insert-punchIn (x ∷ xs) (suc i)  v zero    = refl
-insert-punchIn (x ∷ xs) (suc i)  v (suc j) = insert-punchIn xs i v j
-
-remove-punchOut : ∀ (xs : Vec A (suc n)) {i} {j} (i≢j : i ≢ j) →
-                  lookup (remove xs i) (Fin.punchOut i≢j) ≡ lookup xs j
-remove-punchOut (x ∷ xs)     {zero}  {zero}  i≢j = contradiction refl i≢j
-remove-punchOut (x ∷ xs)     {zero}  {suc j} i≢j = refl
-remove-punchOut (x ∷ y ∷ xs) {suc i} {zero}  i≢j = refl
-remove-punchOut (x ∷ y ∷ xs) {suc i} {suc j} i≢j =
-  remove-punchOut (y ∷ xs) (i≢j ∘ cong suc)
+-- length
+
+length-toList : (xs : Vec A n) → List.length (toList xs) ≡ length xs
+length-toList []       = refl
+length-toList (x ∷ xs) = cong suc (length-toList xs)
+
+------------------------------------------------------------------------
+-- insertAt
+
+insertAt-lookup : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →
+                  lookup (insertAt xs i v) i ≡ v
+insertAt-lookup xs       zero     v = refl
+insertAt-lookup (x ∷ xs) (suc i)  v = insertAt-lookup xs i v
+
+insertAt-punchIn : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) (j : Fin n) →
+                   lookup (insertAt xs i v) (Fin.punchIn i j) ≡ lookup xs j
+insertAt-punchIn xs       zero     v j       = refl
+insertAt-punchIn (x ∷ xs) (suc i)  v zero    = refl
+insertAt-punchIn (x ∷ xs) (suc i)  v (suc j) = insertAt-punchIn xs i v j
+
+toList-insertAt : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →
+                  toList (insertAt xs i v) ≡ List.insertAt (toList xs) (Fin.cast (cong suc (sym (length-toList xs))) i) v
+toList-insertAt xs       zero    v = refl
+toList-insertAt (x ∷ xs) (suc i) v = cong (_ List.∷_) (toList-insertAt xs i v)
+
+------------------------------------------------------------------------
+-- removeAt
+
+removeAt-punchOut : ∀ (xs : Vec A (suc n)) {i} {j} (i≢j : i ≢ j) →
+                  lookup (removeAt xs i) (Fin.punchOut i≢j) ≡ lookup xs j
+removeAt-punchOut (x ∷ xs)     {zero}  {zero}  i≢j = contradiction refl i≢j
+removeAt-punchOut (x ∷ xs)     {zero}  {suc j} i≢j = refl
+removeAt-punchOut (x ∷ y ∷ xs) {suc i} {zero}  i≢j = refl
+removeAt-punchOut (x ∷ y ∷ xs) {suc i} {suc j} i≢j =
+  removeAt-punchOut (y ∷ xs) (i≢j ∘ cong suc)
 
 ------------------------------------------------------------------------
--- remove
+-- insertAt and removeAt
 
-remove-insert : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →
-                remove (insert xs i v) i ≡ xs
-remove-insert xs           zero           v = refl
-remove-insert (x ∷ xs)     (suc zero)     v = refl
-remove-insert (x ∷ y ∷ xs) (suc (suc i))  v =
-  cong (x ∷_) (remove-insert (y ∷ xs) (suc i) v)
+removeAt-insertAt : ∀ (xs : Vec A n) (i : Fin (suc n)) (v : A) →
+                    removeAt (insertAt xs i v) i ≡ xs
+removeAt-insertAt xs               zero           v = refl
+removeAt-insertAt (x ∷ xs)         (suc zero)     v = refl
+removeAt-insertAt (x ∷ xs@(_ ∷ _)) (suc (suc i))  v =
+  cong (x ∷_) (removeAt-insertAt xs (suc i) v)
 
-insert-remove : ∀ (xs : Vec A (suc n)) (i : Fin (suc n)) →
-                insert (remove xs i) i (lookup xs i) ≡ xs
-insert-remove (x ∷ xs)     zero     = refl
-insert-remove (x ∷ y ∷ xs) (suc i)  =
-  cong (x ∷_) (insert-remove (y ∷ xs) i)
+insertAt-removeAt : ∀ (xs : Vec A (suc n)) (i : Fin (suc n)) →
+                    insertAt (removeAt xs i) i (lookup xs i) ≡ xs
+insertAt-removeAt (x ∷ xs)         zero     = refl
+insertAt-removeAt (x ∷ xs@(_ ∷ _)) (suc i)  =
+  cong (x ∷_) (insertAt-removeAt xs i)
 
 ------------------------------------------------------------------------
 -- Conversion function
@@ -1234,16 +1249,16 @@ updateAt-id-relative = updateAt-id-local
 Please use updateAt-id-local instead."
 #-}
 
-updateAt-compose-relative = updateAt-∘-local
+updateAt-compose-relative = updateAt-updateAt-local
 {-# WARNING_ON_USAGE updateAt-compose-relative
 "Warning: updateAt-compose-relative was deprecated in v2.0.
-Please use updateAt-∘-local instead."
+Please use updateAt-updateAt-local instead."
 #-}
 
-updateAt-compose = updateAt-∘
+updateAt-compose = updateAt-updateAt
 {-# WARNING_ON_USAGE updateAt-compose
 "Warning: updateAt-compose was deprecated in v2.0.
-Please use updateAt-∘ instead."
+Please use updateAt-updateAt instead."
 #-}
 
 updateAt-cong-relative = updateAt-cong-local
@@ -1300,6 +1315,38 @@ drop-distr-map = drop-map
 "Warning: drop-distr-map was deprecated in v2.0.
 Please use drop-map instead."
 #-}
+
+map-insert = map-insertAt
+{-# WARNING_ON_USAGE map-insert
+"Warning: map-insert was deprecated in v2.0.
+Please use map-insertAt instead."
+#-}
+insert-lookup = insertAt-lookup
+{-# WARNING_ON_USAGE insert-lookup
+"Warning: insert-lookup was deprecated in v2.0.
+Please use insertAt-lookup instead."
+#-}
+insert-punchIn = insertAt-punchIn
+{-# WARNING_ON_USAGE insert-punchIn
+"Warning: insert-punchIn was deprecated in v2.0.
+Please use insertAt-punchIn instead."
+#-}
+remove-PunchOut = removeAt-punchOut
+{-# WARNING_ON_USAGE remove-PunchOut
+"Warning: remove-PunchOut was deprecated in v2.0.
+Please use removeAt-punchOut instead."
+#-}
+remove-insert = removeAt-insertAt
+{-# WARNING_ON_USAGE remove-insert
+"Warning: remove-insert was deprecated in v2.0.
+Please use removeAt-insertAt instead."
+#-}
+insert-remove = insertAt-removeAt
+{-# WARNING_ON_USAGE insert-remove
+"Warning: insert-remove was deprecated in v2.0.
+Please use insertAt-removeAt instead."
+#-}
+
 lookup-inject≤-take : ∀ m (m≤m+n : m ≤ m + n) (i : Fin m) (xs : Vec A (m + n)) →
                       lookup xs (Fin.inject≤ i m≤m+n) ≡ lookup (take m xs) i
 lookup-inject≤-take m m≤m+n i xs = sym (begin
@@ -1313,4 +1360,3 @@ lookup-inject≤-take m m≤m+n i xs = sym (begin
 "Warning: lookup-inject≤-take was deprecated in v2.0.
 Please use lookup-take-inject≤ or lookup-truncate, take≡truncate instead."
 #-}
-

From a2341cce2ff7f23f13a6ecf5e8509ab60edc5adc Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 4 Oct 2023 08:30:52 +0100
Subject: [PATCH 138/316] =?UTF-8?q?De-symmetrising=20`Relation.Binary.Bund?=
 =?UTF-8?q?les.Preorder.=5F=E2=88=BC=5F`=20(#2099)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                                  | 33 +++++++++++++++++-
 src/Data/Container/Related.agda               | 22 ++++++++++--
 .../Relation/Unary/Any/Properties.agda        |  8 ++---
 .../Relation/Binary/BagAndSetEquality.agda    |  4 +--
 src/Effect/Monad/Partiality.agda              |  2 +-
 src/Function/Related.agda                     |  4 +--
 src/Function/Related/Propositional.agda       |  4 +--
 src/Relation/Binary/Bundles.agda              | 30 +++++++++++++---
 .../ReflexiveTransitive/Properties.agda       |  2 +-
 .../Binary/HeterogeneousEquality.agda         |  2 +-
 .../Binary/Indexed/Heterogeneous/Bundles.agda | 23 +++++++++++--
 .../Indexed/Heterogeneous/Construct/At.agda   |  8 ++---
 .../Indexed/Heterogeneous/Structures.agda     |  8 ++---
 src/Relation/Binary/Morphism/Bundles.agda     |  2 +-
 .../Binary/Morphism/OrderMonomorphism.agda    | 34 +++++++++----------
 src/Relation/Binary/Morphism/Structures.agda  | 18 +++++-----
 src/Relation/Binary/Properties/Preorder.agda  |  7 ++--
 src/Relation/Binary/Structures.agda           | 14 ++++----
 18 files changed, 156 insertions(+), 69 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index f787065436..7975dcc77b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -797,6 +797,23 @@ Non-backwards compatible changes
   IO.Instances
   ```
 
+### (Issue #2096) Introduction of flipped relation symbol for `Relation.Binary.Bundles.Preorder`
+
+* Previously, the relation symbol `_∼_`  was (notationally) symmetric, so that its
+  converse relation could only be discussed *semantically* in terms of `flip _∼_`
+  in `Relation.Binary.Properties.Preorder`, `Relation.Binary.Construct.Flip.{Ord|EqAndOrd}`
+
+* Now, the symbol `_∼_` has been renamed to a new symbol `_≲_`, with `_≳_`
+  introduced as a definition in `Relation.Binary.Bundles.Preorder` whose properties
+  in `Relation.Binary.Properties.Preorder` now refer to it. Partial backwards compatible
+  has been achieved by redeclaring a deprecated version of the old name in the record.
+  Therefore, only _declarations_ of `PartialOrder` records will need their field names
+  updating.
+
+* NB (issues #1214 #2098) the corresponding situation regarding the `flip`ped
+  relation symbols `_≥_`, `_>_` (and their negated versions!) has not (yet)
+  been addressed.
+
 ### Standardisation of `insertAt`/`updateAt`/`removeAt`
 
 * Previously, the names and argument order of index-based insertion, update and removal functions for
@@ -840,7 +857,6 @@ Non-backwards compatible changes
   
 * The old names (and the names of all proofs about these functions) have been deprecated appropriately.
 
-
 ### Changes to triple reasoning interface
 
 * The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof that the strict
@@ -1251,6 +1267,11 @@ Deprecated names
   push-function-into-if ↦ if-float
   ```
 
+* In `Data.Container.Related`:
+  ```
+  _∼[_]_  ↦  _≲[_]_
+  ```
+
 * In `Data.Fin.Base`: two new, hopefully more memorable, names `↑ˡ` `↑ʳ`
   for the 'left', resp. 'right' injection of a Fin m into a 'larger' type,
   `Fin (m + n)`, resp. `Fin (n + m)`, with argument order to reflect the
@@ -1661,6 +1682,16 @@ Deprecated names
   fromForeign ↦ Foreign.Haskell.Coerce.coerce
   ```
 
+* In `Relation.Binary.Bundles.Preorder`:
+  ```
+  _∼_  ↦  _≲_
+  ```
+
+* In `Relation.Binary.Indexed.Heterogeneous.Bundles.Preorder`:
+  ```
+  _∼_  ↦  _≲_
+  ```
+
 * In `Relation.Binary.Properties.Preorder`:
   ```
   invIsPreorder ↦ converse-isPreorder
diff --git a/src/Data/Container/Related.agda b/src/Data/Container/Related.agda
index a5af9e334f..1fc9dc5665 100644
--- a/src/Data/Container/Related.agda
+++ b/src/Data/Container/Related.agda
@@ -33,7 +33,23 @@ open Related public
                Setoid (s ⊔ p ⊔ ℓ) (p ⊔ ℓ)
 [ k ]-Equality C X = Related.InducedEquivalence₂ k (_∈_ {C = C} {X = X})
 
-infix 4 _∼[_]_
-_∼[_]_ : ∀ {s p x} {C : Container s p} {X : Set x} →
+infix 4 _≲[_]_
+_≲[_]_ : ∀ {s p x} {C : Container s p} {X : Set x} →
          ⟦ C ⟧ X → Kind → ⟦ C ⟧ X → Set (p ⊔ x)
-_∼[_]_ {C = C} {X} xs k ys = Preorder._∼_ ([ k ]-Order C X) xs ys
+_≲[_]_ {C = C} {X} xs k ys = Preorder._≲_ ([ k ]-Order C X) xs ys
+
+
+
+------------------------------------------------------------------------
+-- DEPRECATED
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+infix 4 _∼[_]_
+_∼[_]_ = _≲[_]_
+{-# WARNING_ON_USAGE _∼[_]_
+"Warning: _∼[_]_ was deprecated in v2.0. Please use _≲[_]_ instead. "
+#-}
diff --git a/src/Data/Container/Relation/Unary/Any/Properties.agda b/src/Data/Container/Relation/Unary/Any/Properties.agda
index 7bf97642a4..d3265acefb 100644
--- a/src/Data/Container/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Container/Relation/Unary/Any/Properties.agda
@@ -64,7 +64,7 @@ module _ {s p} {C : Container s p} {x} {X : Set x}
 -- ◇ is a congruence for bag and set equality and related preorders.
 
   cong : ∀ {k} {xs₁ xs₂ : ⟦ C ⟧ X} →
-         (∀ x → Related k (P₁ x) (P₂ x)) → xs₁ ∼[ k ] xs₂ →
+         (∀ x → Related k (P₁ x) (P₂ x)) → xs₁ ≲[ k ] xs₂ →
          Related k (◇ C P₁ xs₁) (◇ C P₂ xs₂)
   cong {k} {xs₁} {xs₂} P₁↔P₂ xs₁≈xs₂ =
     ◇ C P₁ xs₁               ↔⟨ ↔∈ C ⟩
@@ -185,8 +185,8 @@ module _ {s p} (C : Container s p) {x y} {X : Set x} {Y : Set y}
          {ℓ} (P : Pred Y ℓ) where
 
   map-cong : ∀ {k} {f₁ f₂ : X → Y} {xs₁ xs₂ : ⟦ C ⟧ X} →
-             f₁ ≗ f₂ → xs₁ ∼[ k ] xs₂ →
-             map f₁ xs₁ ∼[ k ] map f₂ xs₂
+             f₁ ≗ f₂ → xs₁ ≲[ k ] xs₂ →
+             map f₁ xs₁ ≲[ k ] map f₂ xs₂
   map-cong {f₁ = f₁} {f₂} {xs₁} {xs₂} f₁≗f₂ xs₁≈xs₂ {x} =
     x ∈ map f₁ xs₁           ↔⟨ map↔∘ C (_≡_ x) f₁ ⟩
     ◇ C (λ y → x ≡ f₁ y) xs₁ ∼⟨ cong (Related.↔⇒ ∘ helper) xs₁≈xs₂ ⟩
@@ -286,7 +286,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
 
 module _ {s p} {C : Container s p} {x} {X : Set x} where
 
-  linear-identity : ∀ {xs : ⟦ C ⟧ X} (m : C ⊸ C) → ⟪ m ⟫⊸ xs ∼[ bag ] xs
+  linear-identity : ∀ {xs : ⟦ C ⟧ X} (m : C ⊸ C) → ⟪ m ⟫⊸ xs ≲[ bag ] xs
   linear-identity {xs} m {x} =
     x ∈ ⟪ m ⟫⊸ xs  ↔⟨ remove-linear (_≡_ x) m ⟩
     x ∈        xs  ∎
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 982f018367..889efe96d9 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -72,8 +72,8 @@ open Related public using (Kind; SymmetricKind) renaming
 
 infix 4 _∼[_]_
 
-_∼[_]_ : List A → Kind → List A → Set _
-_∼[_]_ {A = A} xs k ys = Preorder._∼_ ([ k ]-Order A) xs ys
+_∼[_]_ : ∀ {a} {A : Set a} → List A → Kind → List A → Set _
+_∼[_]_ {A = A} xs k ys = Preorder._≲_ ([ k ]-Order A) xs ys
 
 private
   module Eq  {k a} {A : Set a} = Setoid ([ k ]-Equality A)
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index ed0e3efbc1..44cf13464d 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -301,7 +301,7 @@ module _ {A : Set a} {_∼_ : A → A → Set ℓ} where
   preorder pre k = record
     { Carrier    = A ⊥
     ; _≈_        = _≡_
-    ; _∼_        = Rel k
+    ; _≲_        = Rel k
     ; isPreorder = record
       { isEquivalence = P.isEquivalence
       ; reflexive     = refl′
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index ce67d9580b..9846b4d034 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -404,7 +404,7 @@ InducedPreorder₁ : Kind → ∀ {a s} {A : Set a} →
                    (A → Set s) → Preorder _ _ _
 InducedPreorder₁ k S = record
   { _≈_        = _≡_
-  ; _∼_        = InducedRelation₁ k S
+  ; _≲_        = InducedRelation₁ k S
   ; isPreorder = record
     { isEquivalence = P.isEquivalence
     ; reflexive     = reflexive ∘
@@ -437,7 +437,7 @@ InducedPreorder₂ : Kind → ∀ {a b s} {A : Set a} {B : Set b} →
                    (A → B → Set s) → Preorder _ _ _
 InducedPreorder₂ k _S_ = record
   { _≈_        = _≡_
-  ; _∼_        = InducedRelation₂ k _S_
+  ; _≲_        = InducedRelation₂ k _S_
   ; isPreorder = record
     { isEquivalence = P.isEquivalence
     ; reflexive     = λ x≡y {z} →
diff --git a/src/Function/Related/Propositional.agda b/src/Function/Related/Propositional.agda
index 2eb6aa91c9..ad6054af46 100644
--- a/src/Function/Related/Propositional.agda
+++ b/src/Function/Related/Propositional.agda
@@ -339,7 +339,7 @@ InducedRelation₁ k P = λ x y → P x ∼[ k ] P y
 InducedPreorder₁ : Kind → (P : A → Set p) → Preorder _ _ _
 InducedPreorder₁ k P = record
   { _≈_        = _≡_
-  ; _∼_        = InducedRelation₁ k P
+  ; _≲_        = InducedRelation₁ k P
   ; isPreorder = record
     { isEquivalence = P.isEquivalence
     ; reflexive     = reflexive ∘
@@ -369,7 +369,7 @@ InducedRelation₂ k _S_ = λ x y → ∀ {z} → (z S x) ∼[ k ] (z S y)
 InducedPreorder₂ : Kind → ∀ {s} → (A → B → Set s) → Preorder _ _ _
 InducedPreorder₂ k _S_ = record
   { _≈_        = _≡_
-  ; _∼_        = InducedRelation₂ k _S_
+  ; _≲_        = InducedRelation₂ k _S_
   ; isPreorder = record
     { isEquivalence = P.isEquivalence
     ; reflexive     = λ x≡y {z} →
diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index a774d45d69..968cb78b2e 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -10,6 +10,7 @@
 
 module Relation.Binary.Bundles where
 
+open import Function.Base using (flip)
 open import Level
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Binary.Core
@@ -73,12 +74,12 @@ record DecSetoid c ℓ : Set (suc (c ⊔ ℓ)) where
 ------------------------------------------------------------------------
 
 record Preorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix 4 _≈_ _∼_
+  infix 4 _≈_ _≲_
   field
     Carrier    : Set c
     _≈_        : Rel Carrier ℓ₁  -- The underlying equality.
-    _∼_        : Rel Carrier ℓ₂  -- The relation.
-    isPreorder : IsPreorder _≈_ _∼_
+    _≲_        : Rel Carrier ℓ₂  -- The relation.
+    isPreorder : IsPreorder _≈_ _≲_
 
   open IsPreorder isPreorder public
     hiding (module Eq)
@@ -91,6 +92,12 @@ record Preorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
 
     open Setoid setoid public
 
+  infix 4 _≳_
+  _≳_ = flip _≲_
+
+  infix 4 _∼_ -- for deprecation
+  _∼_ = _≲_
+
 
 record TotalPreorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   infix 4 _≈_ _≲_
@@ -107,7 +114,7 @@ record TotalPreorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   preorder = record { isPreorder = isPreorder }
 
   open Preorder preorder public
-    using (module Eq)
+    using (module Eq; _≳_)
 
 
 ------------------------------------------------------------------------
@@ -331,3 +338,18 @@ record ApartnessRelation c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) w
     isApartnessRelation : IsApartnessRelation _≈_ _#_
 
   open IsApartnessRelation isApartnessRelation public
+
+
+
+
+------------------------------------------------------------------------
+-- DEPRECATED
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+{-# WARNING_ON_USAGE Preorder._∼_
+"Warning: Preorder._∼_ was deprecated in v2.0. Please use Preorder._≲_ instead. "
+#-}
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index b2c173c190..f3912de253 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -104,7 +104,7 @@ module _ {i t} {I : Set i} (T : Rel I t) where
   preorder : Preorder _ _ _
   preorder = record
     { _≈_        = _≡_
-    ; _∼_        = Star T
+    ; _≲_        = Star T
     ; isPreorder = isPreorder
     }
 
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 8c7d26cb2a..fc073add44 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -219,7 +219,7 @@ preorder : Set ℓ → Preorder ℓ ℓ ℓ
 preorder A = record
   { Carrier    = A
   ; _≈_        = _≡_
-  ; _∼_        = λ x y → x ≅ y
+  ; _≲_        = λ x y → x ≅ y
   ; isPreorder = isPreorder-≡
   }
 
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
index 86dc3ccdaa..29d4db5253 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
@@ -33,11 +33,28 @@ record IndexedSetoid {i} (I : Set i) c ℓ : Set (suc (i ⊔ c ⊔ ℓ)) where
 
 record IndexedPreorder {i} (I : Set i) c ℓ₁ ℓ₂ :
                        Set (suc (i ⊔ c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix 4 _≈_ _∼_
+  infix 4 _≈_ _≲_
   field
     Carrier    : I → Set c
     _≈_        : IRel Carrier ℓ₁  -- The underlying equality.
-    _∼_        : IRel Carrier ℓ₂  -- The relation.
-    isPreorder : IsIndexedPreorder Carrier _≈_ _∼_
+    _≲_        : IRel Carrier ℓ₂  -- The relation.
+    isPreorder : IsIndexedPreorder Carrier _≈_ _≲_
 
   open IsIndexedPreorder isPreorder public
+
+  infix 4 _∼_
+  _∼_ = _≲_
+
+
+
+------------------------------------------------------------------------
+-- DEPRECATED
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+{-# WARNING_ON_USAGE IndexedPreorder._∼_
+"Warning: IndexedPreorder._∼_ was deprecated in v2.0. Please use IndexedPreorder._≲_ instead. "
+#-}
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda b/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
index 2bd79000e8..441ef849bc 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Construct/At.agda
@@ -27,9 +27,9 @@ module _ {a i} {I : Set i} {A : I → Set a} where
     }
     where open IsIndexedEquivalence isEq
 
-  isPreorder : ∀ {ℓ₁ ℓ₂} {_≈_ : IRel A ℓ₁} {_∼_ : IRel A ℓ₂} →
-               IsIndexedPreorder A _≈_ _∼_ →
-               (index : I) → IsPreorder (_≈_ {index}) _∼_
+  isPreorder : ∀ {ℓ₁ ℓ₂} {_≈_ : IRel A ℓ₁} {_≲_ : IRel A ℓ₂} →
+               IsIndexedPreorder A _≈_ _≲_ →
+               (index : I) → IsPreorder (_≈_ {index}) _≲_
   isPreorder isPreorder index = record
     { isEquivalence = isEquivalence O.isEquivalence index
     ; reflexive     = O.reflexive
@@ -54,7 +54,7 @@ module _ {a i} {I : Set i} where
   preorder O index = record
     { Carrier    = O.Carrier index
     ; _≈_        = O._≈_
-    ; _∼_        = O._∼_
+    ; _≲_        = O._≲_
     ; isPreorder = isPreorder O.isPreorder index
     }
     where module O = IndexedPreorder O
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda b/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
index 7e27aab07f..4569ef2fb0 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
@@ -34,13 +34,13 @@ record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where
   reflexive P.refl = refl
 
 
-record IsIndexedPreorder {ℓ₂} (_∼_ : IRel A ℓ₂) : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where
+record IsIndexedPreorder {ℓ₂} (_≲_ : IRel A ℓ₂) : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where
   field
     isEquivalence : IsIndexedEquivalence
-    reflexive     : ∀ {i j} → (_≈_ {i} {j}) ⟨ _⇒_ ⟩ _∼_
-    trans         : Transitive A _∼_
+    reflexive     : ∀ {i j} → (_≈_ {i} {j}) ⟨ _⇒_ ⟩ _≲_
+    trans         : Transitive A _≲_
 
   module Eq = IsIndexedEquivalence isEquivalence
 
-  refl : Reflexive A _∼_
+  refl : Reflexive A _≲_
   refl = reflexive Eq.refl
diff --git a/src/Relation/Binary/Morphism/Bundles.agda b/src/Relation/Binary/Morphism/Bundles.agda
index e8c1c84041..8cd52d3341 100644
--- a/src/Relation/Binary/Morphism/Bundles.agda
+++ b/src/Relation/Binary/Morphism/Bundles.agda
@@ -70,7 +70,7 @@ record PreorderHomomorphism (S₁ : Preorder ℓ₁ ℓ₂ ℓ₃)
   open Preorder
   field
     ⟦_⟧                 : Carrier S₁ → Carrier S₂
-    isOrderHomomorphism : IsOrderHomomorphism (_≈_ S₁) (_≈_ S₂) (_∼_ S₁) (_∼_ S₂) ⟦_⟧
+    isOrderHomomorphism : IsOrderHomomorphism (_≈_ S₁) (_≈_ S₂) (_≲_ S₁) (_≲_ S₂) ⟦_⟧
 
   open IsOrderHomomorphism isOrderHomomorphism public
 
diff --git a/src/Relation/Binary/Morphism/OrderMonomorphism.agda b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
index 49d7c4e2e3..ca3ce3677a 100644
--- a/src/Relation/Binary/Morphism/OrderMonomorphism.agda
+++ b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
@@ -23,9 +23,9 @@ import Relation.Binary.Morphism.RelMonomorphism as RawRelation
 module Relation.Binary.Morphism.OrderMonomorphism
   {a b ℓ₁ ℓ₂ ℓ₃ ℓ₄} {A : Set a} {B : Set b}
   {_≈₁_ : Rel A ℓ₁} {_≈₂_ : Rel B ℓ₃}
-  {_∼₁_ : Rel A ℓ₂} {_∼₂_ : Rel B ℓ₄}
+  {_≲₁_ : Rel A ℓ₂} {_≲₂_ : Rel B ℓ₄}
   {⟦_⟧ : A → B}
-  (isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _∼₁_ _∼₂_ ⟦_⟧)
+  (isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧)
   where
 
 open IsOrderMonomorphism isOrderMonomorphism
@@ -40,61 +40,61 @@ open RawRelation isRelMonomorphism public
 ------------------------------------------------------------------------
 -- Properties
 
-reflexive : _≈₂_ ⇒ _∼₂_ → _≈₁_ ⇒ _∼₁_
+reflexive : _≈₂_ ⇒ _≲₂_ → _≈₁_ ⇒ _≲₁_
 reflexive refl x≈y = cancel (refl (cong x≈y))
 
-irrefl : Irreflexive _≈₂_ _∼₂_ → Irreflexive _≈₁_ _∼₁_
+irrefl : Irreflexive _≈₂_ _≲₂_ → Irreflexive _≈₁_ _≲₁_
 irrefl irrefl x≈y x∼y = irrefl (cong x≈y) (mono x∼y)
 
-antisym : Antisymmetric _≈₂_ _∼₂_ → Antisymmetric _≈₁_ _∼₁_
+antisym : Antisymmetric _≈₂_ _≲₂_ → Antisymmetric _≈₁_ _≲₁_
 antisym antisym x∼y y∼x = injective (antisym (mono x∼y) (mono y∼x))
 
-compare : Trichotomous _≈₂_ _∼₂_ → Trichotomous _≈₁_ _∼₁_
+compare : Trichotomous _≈₂_ _≲₂_ → Trichotomous _≈₁_ _≲₁_
 compare compare x y with compare ⟦ x ⟧ ⟦ y ⟧
 ... | tri< a ¬b ¬c = tri< (cancel a) (¬b ∘ cong) (¬c ∘ mono)
 ... | tri≈ ¬a b ¬c = tri≈ (¬a ∘ mono) (injective b) (¬c ∘ mono)
 ... | tri> ¬a ¬b c = tri> (¬a ∘ mono) (¬b ∘ cong) (cancel c)
 
-respˡ : _∼₂_ Respectsˡ _≈₂_ → _∼₁_ Respectsˡ _≈₁_
+respˡ : _≲₂_ Respectsˡ _≈₂_ → _≲₁_ Respectsˡ _≈₁_
 respˡ resp x≈y x∼z = cancel (resp (cong x≈y) (mono x∼z))
 
-respʳ : _∼₂_ Respectsʳ _≈₂_ → _∼₁_ Respectsʳ _≈₁_
+respʳ : _≲₂_ Respectsʳ _≈₂_ → _≲₁_ Respectsʳ _≈₁_
 respʳ resp x≈y y∼z = cancel (resp (cong x≈y) (mono y∼z))
 
-resp : _∼₂_ Respects₂ _≈₂_ → _∼₁_ Respects₂ _≈₁_
+resp : _≲₂_ Respects₂ _≈₂_ → _≲₁_ Respects₂ _≈₁_
 resp = map respʳ respˡ
 
 ------------------------------------------------------------------------
 -- Structures
 
-isPreorder : IsPreorder _≈₂_ _∼₂_ → IsPreorder _≈₁_ _∼₁_
+isPreorder : IsPreorder _≈₂_ _≲₂_ → IsPreorder _≈₁_ _≲₁_
 isPreorder O = record
   { isEquivalence = EqM.isEquivalence O.isEquivalence
   ; reflexive     = reflexive O.reflexive
   ; trans         = trans O.trans
   } where module O = IsPreorder O
 
-isPartialOrder : IsPartialOrder _≈₂_ _∼₂_ → IsPartialOrder _≈₁_ _∼₁_
+isPartialOrder : IsPartialOrder _≈₂_ _≲₂_ → IsPartialOrder _≈₁_ _≲₁_
 isPartialOrder O = record
   { isPreorder = isPreorder O.isPreorder
   ; antisym    = antisym O.antisym
   } where module O = IsPartialOrder O
 
-isTotalOrder : IsTotalOrder _≈₂_ _∼₂_ → IsTotalOrder _≈₁_ _∼₁_
+isTotalOrder : IsTotalOrder _≈₂_ _≲₂_ → IsTotalOrder _≈₁_ _≲₁_
 isTotalOrder O = record
   { isPartialOrder = isPartialOrder O.isPartialOrder
   ; total          = total O.total
   } where module O = IsTotalOrder O
 
-isDecTotalOrder : IsDecTotalOrder _≈₂_ _∼₂_ → IsDecTotalOrder _≈₁_ _∼₁_
+isDecTotalOrder : IsDecTotalOrder _≈₂_ _≲₂_ → IsDecTotalOrder _≈₁_ _≲₁_
 isDecTotalOrder O = record
   { isTotalOrder = isTotalOrder O.isTotalOrder
   ; _≟_          = EqM.dec O._≟_
   ; _≤?_         = dec O._≤?_
   } where module O = IsDecTotalOrder O
 
-isStrictPartialOrder : IsStrictPartialOrder _≈₂_ _∼₂_ →
-                       IsStrictPartialOrder _≈₁_ _∼₁_
+isStrictPartialOrder : IsStrictPartialOrder _≈₂_ _≲₂_ →
+                       IsStrictPartialOrder _≈₁_ _≲₁_
 isStrictPartialOrder O = record
   { isEquivalence = EqM.isEquivalence O.isEquivalence
   ; irrefl        = irrefl O.irrefl
@@ -102,8 +102,8 @@ isStrictPartialOrder O = record
   ; <-resp-≈      = resp O.<-resp-≈
   } where module O = IsStrictPartialOrder O
 
-isStrictTotalOrder : IsStrictTotalOrder _≈₂_ _∼₂_ →
-                     IsStrictTotalOrder _≈₁_ _∼₁_
+isStrictTotalOrder : IsStrictTotalOrder _≈₂_ _≲₂_ →
+                     IsStrictTotalOrder _≈₁_ _≲₁_
 isStrictTotalOrder O = record
   { isEquivalence = EqM.isEquivalence O.isEquivalence
   ; trans         = trans O.trans
diff --git a/src/Relation/Binary/Morphism/Structures.agda b/src/Relation/Binary/Morphism/Structures.agda
index 86eceac6e4..081008fbd1 100644
--- a/src/Relation/Binary/Morphism/Structures.agda
+++ b/src/Relation/Binary/Morphism/Structures.agda
@@ -57,29 +57,29 @@ record IsRelIsomorphism (_∼₁_ : Rel A ℓ₁) (_∼₂_ : Rel B ℓ₂)
 ------------------------------------------------------------------------
 
 record IsOrderHomomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)
-                           (_∼₁_ : Rel A ℓ₃) (_∼₂_ : Rel B ℓ₄)
+                           (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)
                            (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)
                            where
   field
     cong  : Homomorphic₂ _≈₁_ _≈₂_ ⟦_⟧
-    mono  : Homomorphic₂ _∼₁_ _∼₂_ ⟦_⟧
+    mono  : Homomorphic₂ _≲₁_ _≲₂_ ⟦_⟧
 
   module Eq where
     isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧
     isRelHomomorphism = record { cong = cong }
 
-  isRelHomomorphism : IsRelHomomorphism _∼₁_ _∼₂_ ⟦_⟧
+  isRelHomomorphism : IsRelHomomorphism _≲₁_ _≲₂_ ⟦_⟧
   isRelHomomorphism = record { cong = mono }
 
 
 record IsOrderMonomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)
-                           (_∼₁_ : Rel A ℓ₃) (_∼₂_ : Rel B ℓ₄)
+                           (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)
                            (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)
                            where
   field
-    isOrderHomomorphism : IsOrderHomomorphism _≈₁_ _≈₂_ _∼₁_ _∼₂_ ⟦_⟧
+    isOrderHomomorphism : IsOrderHomomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧
     injective           : Injective _≈₁_ _≈₂_ ⟦_⟧
-    cancel              : Injective _∼₁_ _∼₂_ ⟦_⟧
+    cancel              : Injective _≲₁_ _≲₂_ ⟦_⟧
 
   open IsOrderHomomorphism isOrderHomomorphism public
     hiding (module Eq)
@@ -91,7 +91,7 @@ record IsOrderMonomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)
       ; injective      = injective
       }
 
-  isRelMonomorphism : IsRelMonomorphism _∼₁_ _∼₂_ ⟦_⟧
+  isRelMonomorphism : IsRelMonomorphism _≲₁_ _≲₂_ ⟦_⟧
   isRelMonomorphism = record
     { isHomomorphism = isRelHomomorphism
     ; injective      = cancel
@@ -99,11 +99,11 @@ record IsOrderMonomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)
 
 
 record IsOrderIsomorphism (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂)
-                          (_∼₁_ : Rel A ℓ₃) (_∼₂_ : Rel B ℓ₄)
+                          (_≲₁_ : Rel A ℓ₃) (_≲₂_ : Rel B ℓ₄)
                           (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃ ⊔ ℓ₄)
                           where
   field
-    isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _∼₁_ _∼₂_ ⟦_⟧
+    isOrderMonomorphism : IsOrderMonomorphism _≈₁_ _≈₂_ _≲₁_ _≲₂_ ⟦_⟧
     surjective          : Surjective _≈₁_ _≈₂_ ⟦_⟧
 
   open IsOrderMonomorphism isOrderMonomorphism public
diff --git a/src/Relation/Binary/Properties/Preorder.agda b/src/Relation/Binary/Properties/Preorder.agda
index dc9bd2913b..b2be296828 100644
--- a/src/Relation/Binary/Properties/Preorder.agda
+++ b/src/Relation/Binary/Properties/Preorder.agda
@@ -18,10 +18,11 @@ import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 
 open Preorder P
 
+
 ------------------------------------------------------------------------
--- The inverse relation is also a preorder.
+-- The converse relation is also a preorder.
 
-converse-isPreorder : IsPreorder _≈_ (flip _∼_)
+converse-isPreorder : IsPreorder _≈_ _≳_
 converse-isPreorder = EqAndOrd.isPreorder isPreorder
 
 converse-preorder : Preorder p₁ p₂ p₃
@@ -32,7 +33,7 @@ converse-preorder = EqAndOrd.preorder P
 
 InducedEquivalence : Setoid _ _
 InducedEquivalence = record
-  { _≈_           = λ x y → x ∼ y × y ∼ x
+  { _≈_           = λ x y → x ≲ y × x ≳ y
   ; isEquivalence = record
     { refl  = (refl , refl)
     ; sym   = swap
diff --git a/src/Relation/Binary/Structures.agda b/src/Relation/Binary/Structures.agda
index 5b59e900be..751c774da3 100644
--- a/src/Relation/Binary/Structures.agda
+++ b/src/Relation/Binary/Structures.agda
@@ -73,25 +73,25 @@ record IsDecEquivalence : Set (a ⊔ ℓ) where
 -- Preorders
 ------------------------------------------------------------------------
 
-record IsPreorder (_∼_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
+record IsPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   field
     isEquivalence : IsEquivalence
     -- Reflexivity is expressed in terms of the underlying equality:
-    reflexive     : _≈_ ⇒ _∼_
-    trans         : Transitive _∼_
+    reflexive     : _≈_ ⇒ _≲_
+    trans         : Transitive _≲_
 
   module Eq = IsEquivalence isEquivalence
 
-  refl : Reflexive _∼_
+  refl : Reflexive _≲_
   refl = reflexive Eq.refl
 
-  ∼-respˡ-≈ : _∼_ Respectsˡ _≈_
+  ∼-respˡ-≈ : _≲_ Respectsˡ _≈_
   ∼-respˡ-≈ x≈y x∼z = trans (reflexive (Eq.sym x≈y)) x∼z
 
-  ∼-respʳ-≈ : _∼_ Respectsʳ _≈_
+  ∼-respʳ-≈ : _≲_ Respectsʳ _≈_
   ∼-respʳ-≈ x≈y z∼x = trans z∼x (reflexive x≈y)
 
-  ∼-resp-≈ : _∼_ Respects₂ _≈_
+  ∼-resp-≈ : _≲_ Respects₂ _≈_
   ∼-resp-≈ = ∼-respʳ-≈ , ∼-respˡ-≈
 
 

From 225745067ef431d212dba636668b82ed5f35d3b2 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 4 Oct 2023 09:58:37 +0100
Subject: [PATCH 139/316] =?UTF-8?q?Redefines=20`Data.Nat.Base.=5F=E2=89=A4?=
 =?UTF-8?q?=E2=80=B3=5F`=20(#1948)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                 |  28 +++++++++
 src/Data/Nat/Base.agda       | 113 +++++++++++++++++++----------------
 src/Data/Nat/Properties.agda |  21 ++++---
 3 files changed, 101 insertions(+), 61 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7975dcc77b..612e45673e 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -585,6 +585,32 @@ Non-backwards compatible changes
   Prime n = ∀ {d} → 2 ≤ d → d < n → d ∤ n
   ```
 
+### Change in the definition of `_≤″_` (issue #1919)
+
+* The definition of `_≤″_` in `Data.Nat.Base` was previously:
+  ```agda
+  record _≤″_ (m n : ℕ) : Set where
+    constructor less-than-or-equal
+    field
+      {k}   : ℕ
+      proof : m + k ≡ n
+  ```
+  which introduced a spurious additional definition, when this is in fact, modulo
+  field names and implicit/explicit qualifiers, equivalent to the definition of left-
+  divisibility, `_∣ˡ_` for the `RawMagma` structure of `_+_`. Since the addition of
+  raw bundles to `Data.X.Base`, this definition can now be made directly. Knock-on
+  consequences include the need to retain the old constructor name, now introduced
+  as a pattern synonym, and introduction of (a function equivalent to) the former
+  field name/projection function `proof` as `≤″-proof` in `Data.Nat.Properties`. 
+
+* Accordingly, the definition has been changed to:
+  ```agda
+  _≤″_ : (m n : ℕ)  → Set
+  _≤″_ = _∣ˡ_ +-rawMagma
+
+  pattern less-than-or-equal {k} prf = k , prf
+  ```
+
 ### Renaming of `Reflection` modules
 
 * Under the `Reflection` module, there were various impending name clashes
@@ -2584,6 +2610,8 @@ Additions to existing modules
   <⇒<′ : m < n → m <′ n
   <′⇒< : m <′ n → m < n
 
+  ≤″-proof : (le : m ≤″ n) → let less-than-or-equal {k} _ = le in m + k ≡ n
+
   m+n≤p⇒m≤p∸n         : m + n ≤ p → m ≤ p ∸ n
   m≤p∸n⇒m+n≤p         : n ≤ p → m ≤ p ∸ n → m + n ≤ p
 
diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index 4ae0acc2dd..b0a0eafcd6 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -12,8 +12,10 @@
 module Data.Nat.Base where
 
 open import Algebra.Bundles.Raw using (RawMagma; RawMonoid; RawNearSemiring; RawSemiring)
+open import Algebra.Definitions.RawMagma using (_∣ˡ_)
 open import Data.Bool.Base using (Bool; true; false; T; not)
 open import Data.Parity.Base using (Parity; 0ℙ; 1ℙ)
+open import Data.Product.Base using (_,_)
 open import Level using (0ℓ)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core
@@ -120,11 +122,59 @@ instance
 >-nonZero⁻¹ (suc n) = z<s
 
 ------------------------------------------------------------------------
--- Arithmetic
+-- Raw bundles
 
 open import Agda.Builtin.Nat public
   using (_+_; _*_) renaming (_-_ to _∸_)
 
++-rawMagma : RawMagma 0ℓ 0ℓ
++-rawMagma = record
+  { _≈_ = _≡_
+  ; _∙_ = _+_
+  }
+
++-0-rawMonoid : RawMonoid 0ℓ 0ℓ
++-0-rawMonoid = record
+  { _≈_ = _≡_
+  ; _∙_ = _+_
+  ; ε   = 0
+  }
+
+*-rawMagma : RawMagma 0ℓ 0ℓ
+*-rawMagma = record
+  { _≈_ = _≡_
+  ; _∙_ = _*_
+  }
+
+*-1-rawMonoid : RawMonoid 0ℓ 0ℓ
+*-1-rawMonoid = record
+  { _≈_ = _≡_
+  ; _∙_ = _*_
+  ; ε = 1
+  }
+
++-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
++-*-rawNearSemiring = record
+  { Carrier = _
+  ; _≈_ = _≡_
+  ; _+_ = _+_
+  ; _*_ = _*_
+  ; 0# = 0
+  }
+
++-*-rawSemiring : RawSemiring 0ℓ 0ℓ
++-*-rawSemiring = record
+  { Carrier = _
+  ; _≈_ = _≡_
+  ; _+_ = _+_
+  ; _*_ = _*_
+  ; 0# = 0
+  ; 1# = 1
+  }
+
+------------------------------------------------------------------------
+-- Arithmetic
+
 open import Agda.Builtin.Nat
   using (div-helper; mod-helper)
 
@@ -260,14 +310,12 @@ m >′ n = n <′ m
 
 ------------------------------------------------------------------------
 -- Another alternative definition of _≤_
+infix 4 _≤″_ _<″_ _≥″_ _>″_
 
-record _≤″_ (m n : ℕ) : Set where
-  constructor less-than-or-equal
-  field
-    {k}   : ℕ
-    proof : m + k ≡ n
+_≤″_ : (m n : ℕ)  → Set
+_≤″_ = _∣ˡ_ +-rawMagma
 
-infix 4 _≤″_ _<″_ _≥″_ _>″_
+pattern less-than-or-equal {k} proof = k , proof
 
 _<″_ : Rel ℕ 0ℓ
 m <″ n = suc m ≤″ n
@@ -316,51 +364,12 @@ compare (suc m) (suc n) with compare m n
 ... | equal   m   = equal (suc m)
 ... | greater n k = greater (suc n) k
 
-------------------------------------------------------------------------
--- Raw bundles
-
-+-rawMagma : RawMagma 0ℓ 0ℓ
-+-rawMagma = record
-  { _≈_ = _≡_
-  ; _∙_ = _+_
-  }
-
-+-0-rawMonoid : RawMonoid 0ℓ 0ℓ
-+-0-rawMonoid = record
-  { _≈_ = _≡_
-  ; _∙_ = _+_
-  ; ε   = 0
-  }
-
-*-rawMagma : RawMagma 0ℓ 0ℓ
-*-rawMagma = record
-  { _≈_ = _≡_
-  ; _∙_ = _*_
-  }
-
-*-1-rawMonoid : RawMonoid 0ℓ 0ℓ
-*-1-rawMonoid = record
-  { _≈_ = _≡_
-  ; _∙_ = _*_
-  ; ε = 1
-  }
 
-+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
-+-*-rawNearSemiring = record
-  { Carrier = _
-  ; _≈_ = _≡_
-  ; _+_ = _+_
-  ; _*_ = _*_
-  ; 0# = 0
-  }
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
 
-+-*-rawSemiring : RawSemiring 0ℓ 0ℓ
-+-*-rawSemiring = record
-  { Carrier = _
-  ; _≈_ = _≡_
-  ; _+_ = _+_
-  ; _*_ = _*_
-  ; 0# = 0
-  ; 1# = 1
-  }
+-- Version 2.0
 
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 5ed9b64e43..507dfe0a08 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -2065,6 +2065,11 @@ m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)
 <″⇒<ᵇ : ∀ {m n} → m <″ n → T (m <ᵇ n)
 <″⇒<ᵇ {m} (less-than-or-equal refl) = <⇒<ᵇ (m≤m+n (suc m) _)
 
+-- equivalence to the old definition of _≤″_
+
+≤″-proof : ∀ {m n} (le : m ≤″ n) → let less-than-or-equal {k} _ = le in m + k ≡ n
+≤″-proof (less-than-or-equal prf) = prf
+
 -- equivalence to _≤_
 
 ≤″⇒≤ : _≤″_ ⇒ _≤_
@@ -2099,8 +2104,8 @@ _>″?_ = flip _<″?_
 ≤″-irrelevant : Irrelevant _≤″_
 ≤″-irrelevant {m} (less-than-or-equal eq₁)
                   (less-than-or-equal eq₂)
-  with +-cancelˡ-≡ m _ _ (trans eq₁ (sym eq₂))
-... | refl = cong less-than-or-equal (≡-irrelevant eq₁ eq₂)
+  with refl ← +-cancelˡ-≡ m _ _ (trans eq₁ (sym eq₂))
+  = cong less-than-or-equal (≡-irrelevant eq₁ eq₂)
 
 <″-irrelevant : Irrelevant _<″_
 <″-irrelevant = ≤″-irrelevant
@@ -2116,17 +2121,15 @@ _>″?_ = flip _<″?_
 ------------------------------------------------------------------------
 
 ≤‴⇒≤″ : ∀{m n} → m ≤‴ n → m ≤″ n
-≤‴⇒≤″ {m = m} ≤‴-refl     = less-than-or-equal {k = 0} (+-identityʳ m)
-≤‴⇒≤″ {m = m} (≤‴-step x) = less-than-or-equal (trans (+-suc m _) (_≤″_.proof ind)) where
-  ind = ≤‴⇒≤″ x
+≤‴⇒≤″ {m = m} ≤‴-refl       = less-than-or-equal {k = 0} (+-identityʳ m)
+≤‴⇒≤″ {m = m} (≤‴-step m≤n) = less-than-or-equal (trans (+-suc m _) (≤″-proof (≤‴⇒≤″ m≤n)))
 
 m≤‴m+k : ∀{m n k} → m + k ≡ n → m ≤‴ n
-m≤‴m+k {m} {k = zero} refl = subst (λ z → m ≤‴ z) (sym (+-identityʳ m)) (≤‴-refl {m})
-m≤‴m+k {m} {k = suc k} proof
-  = ≤‴-step (m≤‴m+k {k = k} (trans (sym (+-suc m _)) proof))
+m≤‴m+k {m} {k = zero}  refl = subst (λ z → m ≤‴ z) (sym (+-identityʳ m)) (≤‴-refl {m})
+m≤‴m+k {m} {k = suc k} prf  = ≤‴-step (m≤‴m+k {k = k} (trans (sym (+-suc m _)) prf))
 
 ≤″⇒≤‴ : ∀{m n} → m ≤″ n → m ≤‴ n
-≤″⇒≤‴ (less-than-or-equal {k} proof) = m≤‴m+k proof
+≤″⇒≤‴ m≤n = m≤‴m+k (≤″-proof m≤n)
 
 0≤‴n : ∀{n} → 0 ≤‴ n
 0≤‴n {n} = m≤‴m+k refl

From 9b10e00a5ad0b7861903704d6cc9e5edc1d346a1 Mon Sep 17 00:00:00 2001
From: Saransh Chopra <saransh0701@gmail.com>
Date: Thu, 5 Oct 2023 05:27:45 +0530
Subject: [PATCH 140/316] Sync `iterate` and `replicate` for `List` and `Vec`
 (#2051)

---
 CHANGELOG.md                              | 30 +++++++++++--
 src/Codata/Guarded/Stream/Properties.agda |  2 +-
 src/Data/List/Base.agda                   |  4 ++
 src/Data/List/Properties.agda             | 54 ++++++++++++++++++-----
 src/Data/Vec/Base.agda                    |  6 +--
 src/Data/Vec/Properties.agda              | 24 ++++++++++
 6 files changed, 102 insertions(+), 18 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 612e45673e..7bfefd20e9 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1058,17 +1058,20 @@ Non-backwards compatible changes
     Tactic.RingSolver
     Tactic.RingSolver.Core.NatSet
     ```
+
   * Moved & renamed from `Data.Vec.Relation.Unary.All`
     to `Data.Vec.Relation.Unary.All.Properties`:
     ```
     lookup ↦ lookup⁺
     tabulate ↦ lookup⁻
     ```
+
   * Renamed in `Data.Vec.Relation.Unary.Linked.Properties`
     and `Codata.Guarded.Stream.Relation.Binary.Pointwise`:
     ```
     lookup ↦ lookup⁺
     ```
+
   * Added the following new definitions to `Data.Vec.Relation.Unary.All`:
     ```
     lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
@@ -1076,9 +1079,16 @@ Non-backwards compatible changes
     lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
     lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
     ```
+
   * `excluded-middle` in `Relation.Nullary.Decidable.Core` has been renamed to
     `¬¬-excluded-middle`.
 
+  * `iterate` in `Data.Vec.Base` now takes `n` (the length of `Vec`) as an
+    explicit argument.
+    ```agda
+    iterate : (A → A) → A → ∀ n → Vec A n
+    ```
+
 Major improvements
 ------------------
 
@@ -2441,7 +2451,7 @@ Additions to existing modules
   gcd-zero  : Zero 1ℤ gcd
   ```
 
-* Added new functions in `Data.List.Base`:
+* Added new functions and definitions to `Data.List.Base`:
   ```agda
   takeWhileᵇ   : (A → Bool) → List A → List A
   dropWhileᵇ   : (A → Bool) → List A → List A
@@ -2466,6 +2476,7 @@ Additions to existing modules
   ++-rawMagma     : Set a → RawMagma a _
   ++-[]-rawMonoid : Set a → RawMonoid a _
 
+  iterate : (A → A) → A → ℕ → List A
   insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
   updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
   ```
@@ -2527,14 +2538,21 @@ Additions to existing modules
   drop-take-suc-tabulate : drop m (take (suc m) (tabulate f)) ≡ [ f i ]
 
   take-all : n ≥ length xs → take n xs ≡ xs
+  drop-all : n ≥ length xs → drop n xs ≡ []
 
   take-[] : take m [] ≡ []
   drop-[] : drop m [] ≡ []
 
-  map-replicate : map f (replicate n x) ≡ replicate n (f x)
-
   drop-drop : drop n (drop m xs) ≡ drop (m + n) xs
 
+  lookup-replicate  : lookup (replicate n x) i ≡ x
+  map-replicate     : map f (replicate n x) ≡ replicate n (f x)
+  zipWith-replicate : zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)
+
+  length-iterate : length (iterate f x n) ≡ n
+  iterate-id     : iterate id x n ≡ replicate n x
+  lookup-iterate : lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)
+
   length-insertAt   : length (insertAt xs i v) ≡ suc (length xs)
   length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
   removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
@@ -2964,6 +2982,12 @@ Additions to existing modules
   lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
   cast-reverse  : cast eq ∘ reverse ≗ reverse ∘ cast eq
 
+  iterate-id     : iterate id x n ≡ replicate x
+  take-iterate   : take n (iterate f x (n + m)) ≡ iterate f x n
+  drop-iterate   : drop n (iterate f x n) ≡ []
+  lookup-iterate : lookup (iterate f x n) i ≡ ℕ.iterate f x (toℕ i)
+  toList-iterate : toList (iterate f x n) ≡ List.iterate f x n
+
   zipwith-++ : zipWith f (xs ++ ys) (xs' ++ ys') ≡ zipWith f xs xs' ++ zipWith f ys ys'
 
   ++-assoc     : cast eq ((xs ++ ys) ++ zs) ≡ xs ++ (ys ++ zs)
diff --git a/src/Codata/Guarded/Stream/Properties.agda b/src/Codata/Guarded/Stream/Properties.agda
index a07a614b59..41b19ec947 100644
--- a/src/Codata/Guarded/Stream/Properties.agda
+++ b/src/Codata/Guarded/Stream/Properties.agda
@@ -255,7 +255,7 @@ lookup-interleave-odd (suc n) as bs = lookup-interleave-odd n (as .tail) (bs .ta
 ------------------------------------------------------------------------
 -- Properties of take
 
-take-iterate : ∀ n f (x : A) → take n (iterate f x) ≡ Vec.iterate f x
+take-iterate : ∀ n f (x : A) → take n (iterate f x) ≡ Vec.iterate f x n
 take-iterate zero    f x = P.refl
 take-iterate (suc n) f x = cong (x ∷_) (take-iterate n f (f x))
 
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index b96a0451cc..58b51e4893 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -188,6 +188,10 @@ replicate : ℕ → A → List A
 replicate zero    x = []
 replicate (suc n) x = x ∷ replicate n x
 
+iterate : (A → A) → A → ℕ → List A
+iterate f e zero    = []
+iterate f e (suc n) = e ∷ iterate f (f e) n
+
 inits : List A → List (List A)
 inits []       = [] ∷ []
 inits (x ∷ xs) = [] ∷ map (x ∷_) (inits xs)
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 48f533f08d..5afce0cd54 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -43,6 +43,7 @@ open import Relation.Nullary using (¬_; Dec; does; _because_; yes; no; contradi
 open import Relation.Nullary.Decidable as Decidable using (isYes; map′; ⌊_⌋; ¬?; _×-dec_)
 open import Relation.Unary using (Pred; Decidable; ∁)
 open import Relation.Unary.Properties using (∁?)
+import Data.Nat.GeneralisedArithmetic as ℕ
 
 
 open ≡-Reasoning
@@ -118,10 +119,6 @@ map-injective finj {x ∷ xs} {y ∷ ys} eq =
   let fx≡fy , fxs≡fys = ∷-injective eq in
   cong₂ _∷_ (finj fx≡fy) (map-injective finj fxs≡fys)
 
-map-replicate : ∀ (f : A → B) n x → map f (replicate n x) ≡ replicate n (f x)
-map-replicate f zero    x = refl
-map-replicate f (suc n) x = cong (_ ∷_) (map-replicate f n x)
-
 ------------------------------------------------------------------------
 -- mapMaybe
 
@@ -621,13 +618,6 @@ sum-++ (x ∷ xs) ys = begin
 ∈⇒∣product {n} {n ∷ ns} (here  refl) = divides (product ns) (*-comm n (product ns))
 ∈⇒∣product {n} {m ∷ ns} (there n∈ns) = ∣n⇒∣m*n m (∈⇒∣product n∈ns)
 
-------------------------------------------------------------------------
--- replicate
-
-length-replicate : ∀ n {x : A} → length (replicate n x) ≡ n
-length-replicate zero    = refl
-length-replicate (suc n) = cong suc (length-replicate n)
-
 ------------------------------------------------------------------------
 -- scanr
 
@@ -858,6 +848,48 @@ drop-drop zero n xs = refl
 drop-drop (suc m) n [] = drop-[] n
 drop-drop (suc m) n (x ∷ xs) = drop-drop m n xs
 
+drop-all : (n : ℕ) (xs : List A) → n ≥ length xs → drop n xs ≡ []
+drop-all n       []       _ = drop-[] n
+drop-all (suc n) (x ∷ xs) p = drop-all n xs (≤-pred p)
+
+------------------------------------------------------------------------
+-- replicate
+
+length-replicate : ∀ n {x : A} → length (replicate n x) ≡ n
+length-replicate zero    = refl
+length-replicate (suc n) = cong suc (length-replicate n)
+
+lookup-replicate : ∀ n (x : A) (i : Fin n) →
+                   lookup (replicate n x) (cast (sym (length-replicate n)) i) ≡ x
+lookup-replicate (suc n) x zero    = refl
+lookup-replicate (suc n) x (suc i) = lookup-replicate n x i
+
+map-replicate :  ∀ (f : A → B) n (x : A) →
+                 map f (replicate n x) ≡ replicate n (f x)
+map-replicate f zero    x = refl
+map-replicate f (suc n) x = cong (_ ∷_) (map-replicate f n x)
+
+zipWith-replicate : ∀ n (_⊕_ : A → B → C) (x : A) (y : B) →
+                    zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)
+zipWith-replicate zero    _⊕_ x y = refl
+zipWith-replicate (suc n) _⊕_ x y = cong (x ⊕ y ∷_) (zipWith-replicate n _⊕_ x y)
+
+------------------------------------------------------------------------
+-- iterate
+
+length-iterate : ∀ f (x : A) n → length (iterate f x n) ≡ n
+length-iterate f x zero    = refl
+length-iterate f x (suc n) = cong suc (length-iterate f (f x) n)
+
+iterate-id : ∀ (x : A) n → iterate id x n ≡ replicate n x
+iterate-id x zero    = refl
+iterate-id x (suc n) = cong (_ ∷_) (iterate-id x n)
+
+lookup-iterate : ∀ f (x : A) n (i : Fin n) →
+  lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)
+lookup-iterate f x (suc n) zero    = refl
+lookup-iterate f x (suc n) (suc i) = lookup-iterate f (f x) n i
+
 ------------------------------------------------------------------------
 -- splitAt
 
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 469549e1ed..e142db9d00 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -60,9 +60,9 @@ lookup : Vec A n → Fin n → A
 lookup (x ∷ xs) zero    = x
 lookup (x ∷ xs) (suc i) = lookup xs i
 
-iterate : (A → A) → A → ∀ {n} → Vec A n
-iterate s z {zero}  = []
-iterate s z {suc n} = z ∷ iterate s (s z)
+iterate : (A → A) → A → ∀ n → Vec A n
+iterate s z zero    = []
+iterate s z (suc n) = z ∷ iterate s (s z) n
 
 insertAt : Vec A n → Fin (suc n) → A → Vec A (suc n)
 insertAt xs       zero     v = v ∷ xs
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 3d0e3bf590..1357e4e441 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -31,6 +31,7 @@ open import Relation.Binary.PropositionalEquality
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable.Core using (Dec; does; yes; no; _×-dec_; map′)
 open import Relation.Nullary.Negation.Core using (contradiction)
+import Data.Nat.GeneralisedArithmetic as ℕ
 
 open ≡-Reasoning
 
@@ -1098,6 +1099,29 @@ toList-replicate : ∀ (n : ℕ) (x : A) →
 toList-replicate zero    x = refl
 toList-replicate (suc n) x = cong (_ List.∷_) (toList-replicate n x)
 
+------------------------------------------------------------------------
+-- iterate
+
+iterate-id : ∀ (x : A) n → iterate id x n ≡ replicate x
+iterate-id x zero    = refl
+iterate-id x (suc n) = cong (_ ∷_) (iterate-id (id x) n)
+
+take-iterate : ∀ n f (x : A) → take n (iterate f x (n + m)) ≡ iterate f x n
+take-iterate zero    f x = refl
+take-iterate (suc n) f x = cong (_ ∷_) (take-iterate n f (f x))
+
+drop-iterate : ∀ n f (x : A) → drop n (iterate f x (n + zero)) ≡ []
+drop-iterate zero    f x = refl
+drop-iterate (suc n) f x = drop-iterate n f (f x)
+
+lookup-iterate :  ∀ f (x : A) (i : Fin n) → lookup (iterate f x n) i ≡ ℕ.iterate f x (toℕ i)
+lookup-iterate f x zero    = refl
+lookup-iterate f x (suc i) = lookup-iterate f (f x) i
+
+toList-iterate : ∀ f (x : A) n → toList (iterate f x n) ≡ List.iterate f x n
+toList-iterate f x zero    = refl
+toList-iterate f x (suc n) = cong (_ List.∷_) (toList-iterate f (f x) n)
+
 ------------------------------------------------------------------------
 -- tabulate
 

From e5f603016d0e7455a67cceb950eeb9b036dc4d57 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 5 Oct 2023 00:59:44 +0100
Subject: [PATCH 141/316] Changes explicit argument `y` to implicit in
 `Induction.WellFounded.WfRec` (#2084)

---
 CHANGELOG.md                                  | 15 +++++
 README/Data/Nat/Induction.agda                | 24 +++++---
 src/Codata/Musical/Colist/Infinite-merge.agda |  2 +-
 src/Data/Bool/Properties.agda                 |  4 +-
 src/Data/Digit.agda                           |  7 +--
 src/Data/Fin/Induction.agda                   | 23 ++++----
 src/Data/Graph/Acyclic.agda                   |  2 +-
 .../Binary/Permutation/Setoid/Properties.agda |  8 +--
 src/Data/List/Sort/MergeSort.agda             | 13 +++--
 src/Data/Nat/DivMod.agda                      |  2 +-
 src/Data/Nat/GCD.agda                         | 26 ++++-----
 src/Data/Nat/Induction.agda                   |  6 +-
 src/Data/Nat/Logarithm/Core.agda              |  4 +-
 src/Data/Nat/Properties.agda                  |  2 +-
 .../Product/Relation/Binary/Lex/Strict.agda   |  8 +--
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  3 +-
 src/Induction/WellFounded.agda                | 55 ++++++++++---------
 .../Binary/Construct/Closure/Transitive.agda  |  4 +-
 src/Relation/Binary/Construct/On.agda         |  2 +-
 src/Relation/Binary/Rewriting.agda            |  6 +-
 .../RingSolver/Core/Polynomial/Base.agda      | 14 ++---
 .../Homomorphism/Multiplication.agda          | 52 +++++++++---------
 .../Polynomial/Homomorphism/Negation.agda     |  8 +--
 23 files changed, 158 insertions(+), 132 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7bfefd20e9..68c596e074 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -585,6 +585,21 @@ Non-backwards compatible changes
   Prime n = ∀ {d} → 2 ≤ d → d < n → d ∤ n
   ```
 
+### Change to the definition of `Induction.WellFounded.WfRec` (issue #2083)
+
+* Previously, the following definition was adopted
+  ```agda
+  WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
+  WfRec _<_ P x = ∀ y → y < x → P y
+  ```
+  with the consequence that all arguments involving about accesibility and
+  wellfoundedness proofs were polluted by almost-always-inferrable explicit
+  arguments for the `y` position. The definition has now been changed to
+  make that argument *implicit*, as 
+  ```agda
+  WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
+  WfRec _<_ P x = ∀ {y} → y < x → P y
+
 ### Change in the definition of `_≤″_` (issue #1919)
 
 * The definition of `_≤″_` in `Data.Nat.Base` was previously:
diff --git a/README/Data/Nat/Induction.agda b/README/Data/Nat/Induction.agda
index 52b802b2b8..6f47ddc07f 100644
--- a/README/Data/Nat/Induction.agda
+++ b/README/Data/Nat/Induction.agda
@@ -14,6 +14,14 @@ open import Function.Base using (_∘_)
 open import Induction.WellFounded
 open import Relation.Binary.PropositionalEquality
 
+private
+
+  n<′1+n : ∀ {n} → n <′ suc n
+  n<′1+n = ≤′-refl
+
+  n<′2+n : ∀ {n} → n <′ suc (suc n)
+  n<′2+n = ≤′-step ≤′-refl
+
 -- Doubles its input.
 
 twice : ℕ → ℕ
@@ -46,7 +54,7 @@ mutual
   half₂-step = λ
     { zero          _   → zero
     ; (suc zero)    _   → zero
-    ; (suc (suc n)) rec → suc (rec n (≤′-step ≤′-refl))
+    ; (suc (suc n)) rec → suc (rec n<′2+n)
     }
 
   half₂ : ℕ → ℕ
@@ -92,21 +100,21 @@ half₂-2+ n = begin
 
   half₂-step (2 + n) (<′-recBuilder _ half₂-step (2 + n))     ≡⟨⟩
 
-  1 + <′-recBuilder _ half₂-step (2 + n) n (≤′-step ≤′-refl)  ≡⟨⟩
+  1 + <′-recBuilder _ half₂-step (2 + n) n<′2+n  ≡⟨⟩
 
   1 + Some.wfRecBuilder _ half₂-step (2 + n)
-        (<′-wellFounded (2 + n)) n (≤′-step ≤′-refl)          ≡⟨⟩
+        (<′-wellFounded (2 + n)) n<′2+n          ≡⟨⟩
 
   1 + Some.wfRecBuilder _ half₂-step (2 + n)
-        (acc (<′-wellFounded′ (2 + n))) n (≤′-step ≤′-refl)   ≡⟨⟩
+        (acc (<′-wellFounded′ (2 + n))) n<′2+n   ≡⟨⟩
 
   1 + half₂-step n
         (Some.wfRecBuilder _ half₂-step n
-           (<′-wellFounded′ (2 + n) n (≤′-step ≤′-refl)))     ≡⟨⟩
+           (<′-wellFounded′ (2 + n) n<′2+n))     ≡⟨⟩
 
   1 + half₂-step n
         (Some.wfRecBuilder _ half₂-step n
-           (<′-wellFounded′ (1 + n) n ≤′-refl))               ≡⟨⟩
+           (<′-wellFounded′ (1 + n) n<′1+n))               ≡⟨⟩
 
   1 + half₂-step n
         (Some.wfRecBuilder _ half₂-step n (<′-wellFounded n)) ≡⟨⟩
@@ -146,7 +154,7 @@ half₁-+₂ = <′-rec _ λ
   { zero          _   → refl
   ; (suc zero)    _   → refl
   ; (suc (suc n)) rec →
-      cong (suc ∘ suc) (rec n (≤′-step ≤′-refl))
+      cong (suc ∘ suc) (rec n<′2+n)
   }
 
 half₂-+₂ : ∀ n → half₂ (twice n) ≡ n
@@ -154,6 +162,6 @@ half₂-+₂ = <′-rec _ λ
   { zero          _   → refl
   ; (suc zero)    _   → refl
   ; (suc (suc n)) rec →
-      cong (suc ∘ suc) (rec n (≤′-step ≤′-refl))
+      cong (suc ∘ suc) (rec n<′2+n)
   }
 
diff --git a/src/Codata/Musical/Colist/Infinite-merge.agda b/src/Codata/Musical/Colist/Infinite-merge.agda
index 769c0ccc94..cbc142ff5a 100644
--- a/src/Codata/Musical/Colist/Infinite-merge.agda
+++ b/src/Codata/Musical/Colist/Infinite-merge.agda
@@ -195,7 +195,7 @@ Any-merge {P = P} xss = mk↔ₛ′ (proj₁ ∘ to xss) from to∘from (proj₂
     ... | inj₂ q | P.refl | q≤p =
       Prod.map there
                (P.cong (there ∘ (Inverse.from (Any-⋎P xs)) ∘ inj₂))
-               (rec (♭ xss , q) (s≤′s q≤p))
+               (rec (s≤′s q≤p))
 
   to∘from = λ p → from-injective _ _ (proj₂ (to xss (from p)))
 
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index 0449279a7d..3c1c25d245 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -195,8 +195,8 @@ true  <? _     = no  (λ())
 <-wellFounded : WellFounded _<_
 <-wellFounded _ = acc <-acc
   where
-    <-acc : ∀ {x} y → y < x → Acc _<_ y
-    <-acc false f<t = acc (λ _ → λ())
+    <-acc : ∀ {x y} → y < x → Acc _<_ y
+    <-acc f<t = acc λ ()
 
 -- Structures
 
diff --git a/src/Data/Digit.agda b/src/Data/Digit.agda
index 60557b142a..3f11c1c3af 100644
--- a/src/Data/Digit.agda
+++ b/src/Data/Digit.agda
@@ -56,7 +56,7 @@ toNatDigits base@(suc (suc _)) n = aux (<-wellFounded-fast n) []
   aux {zero}        _      xs =  (0 ∷ xs)
   aux {n@(suc _)} (acc wf) xs with does (0 <? n / base)
   ... | false = (n % base) ∷ xs
-  ... | true  = aux (wf (n / base) q<n) ((n % base) ∷ xs)
+  ... | true  = aux (wf q<n) ((n % base) ∷ xs)
     where
     q<n : n / base < n
     q<n = m/n<m n base (s<s z<s)
@@ -107,9 +107,8 @@ toDigits base@(suc (suc k)) n = <′-rec Pred helper n
 
   helper : ∀ n → <′-Rec Pred n → Pred n
   helper n                       rec with n divMod base
-  helper .(toℕ r + 0     * base) rec | result zero    r refl = ([ r ] , refl)
-  helper .(toℕ r + suc x * base) rec | result (suc x) r refl =
-    cons r (rec (suc x) (lem x k (toℕ r)))
+  ... | result zero    r eq = ([ r ] , P.sym eq)
+  ... | result (suc x) r refl = cons r (rec (lem x k (toℕ r)))
 
 ------------------------------------------------------------------------
 -- Showing digits
diff --git a/src/Data/Fin/Induction.agda b/src/Data/Fin/Induction.agda
index a15287a48f..47e37446bf 100644
--- a/src/Data/Fin/Induction.agda
+++ b/src/Data/Fin/Induction.agda
@@ -65,7 +65,7 @@ open WF public using (Acc; acc)
   induct {suc j} (acc rs) (tri< (s≤s i≤j) _ _) _ = Pᵢ⇒Pᵢ₊₁ j P[1+j]
     where
     toℕj≡toℕinjJ = sym $ toℕ-inject₁ j
-    P[1+j] = induct (rs _ (s≤s (subst (ℕ._≤ toℕ j) toℕj≡toℕinjJ ≤-refl)))
+    P[1+j] = induct (rs (s≤s (subst (ℕ._≤ toℕ j) toℕj≡toℕinjJ ≤-refl)))
       (<-cmp i $ inject₁ j) (subst (toℕ i ℕ.≤_) toℕj≡toℕinjJ i≤j)
 
 <-weakInduction : (P : Pred (Fin (suc n)) ℓ) →
@@ -80,8 +80,7 @@ open WF public using (Acc; acc)
 
 private
   acc-map : ∀ {x : Fin n} → Acc ℕ._<_ (n ∸ toℕ x) → Acc _>_ x
-  acc-map {n} (acc rs) = acc λ y y>x →
-    acc-map (rs (n ∸ toℕ y) (ℕ.∸-monoʳ-< y>x (toℕ≤n y)))
+  acc-map (acc rs) = acc λ y>x → acc-map (rs (ℕ.∸-monoʳ-< y>x (toℕ≤n _)))
 
 >-wellFounded : WellFounded {A = Fin n} _>_
 >-wellFounded {n} x = acc-map (ℕ.<-wellFounded (n ∸ toℕ x))
@@ -96,7 +95,7 @@ private
   induct {i} (acc rec) with n ℕ.≟ toℕ i
   ... | yes n≡i = subst P (toℕ-injective (trans (toℕ-fromℕ n) n≡i)) Pₙ
   ... | no  n≢i = subst P (inject₁-lower₁ i n≢i) (Pᵢ₊₁⇒Pᵢ _ Pᵢ₊₁)
-    where Pᵢ₊₁ = induct (rec _ (ℕ.≤-reflexive (cong suc (sym (toℕ-lower₁ i n≢i)))))
+    where Pᵢ₊₁ = induct (rec (ℕ.≤-reflexive (cong suc (sym (toℕ-lower₁ i n≢i)))))
 
 ------------------------------------------------------------------------
 -- Well-foundedness of other (strict) partial orders on Fin
@@ -112,20 +111,20 @@ module _ {_≈_ : Rel (Fin n) ℓ} where
 
   spo-wellFounded : ∀ {r} {_⊏_ : Rel (Fin n) r} →
                     IsStrictPartialOrder _≈_ _⊏_ → WellFounded _⊏_
-  spo-wellFounded {_} {_⊏_} isSPO i = go n i pigeon where
+  spo-wellFounded {_} {_⊏_} isSPO i = go n pigeon where
 
     module ⊏ = IsStrictPartialOrder isSPO
 
-    go : ∀ m i →
-         ((xs : Vec (Fin n) m) → Linked (flip _⊏_) (i ∷ xs) → WellFounded _⊏_) →
+    go : ∀ m {i} →
+         ({xs : Vec (Fin n) m} → Linked (flip _⊏_) (i ∷ xs) → WellFounded _⊏_) →
          Acc _⊏_ i
-    go zero    i k = k [] [-] i
-    go (suc m) i k = acc $ λ j j⊏i → go m j (λ xs i∷xs↑ → k (j ∷ xs) (j⊏i ∷ i∷xs↑))
+    go zero    k = k [-] _
+    go (suc m) k = acc λ j⊏i → go m λ i∷xs↑ → k (j⊏i ∷ i∷xs↑)
 
-    pigeon : (xs : Vec (Fin n) n) → Linked (flip _⊏_) (i ∷ xs) → WellFounded _⊏_
-    pigeon xs i∷xs↑ =
+    pigeon : {xs : Vec (Fin n) n} → Linked (flip _⊏_) (i ∷ xs) → WellFounded _⊏_
+    pigeon {xs} i∷xs↑ =
       let (i₁ , i₂ , i₁<i₂ , xs[i₁]≡xs[i₂]) = pigeonhole (n<1+n n) (Vec.lookup (i ∷ xs)) in
-      let xs[i₁]⊏xs[i₂] = Linkedₚ.lookup⁺ (Ord.transitive _⊏_ ⊏.trans) {xs = i ∷ xs} i∷xs↑ i₁<i₂ in
+      let xs[i₁]⊏xs[i₂] = Linkedₚ.lookup⁺ (Ord.transitive _⊏_ ⊏.trans) i∷xs↑ i₁<i₂ in
       let xs[i₁]⊏xs[i₁] = ⊏.<-respʳ-≈ (⊏.Eq.reflexive xs[i₁]≡xs[i₂]) xs[i₁]⊏xs[i₂] in
       contradiction xs[i₁]⊏xs[i₁] (⊏.irrefl ⊏.Eq.refl)
 
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index d40d078bd3..db9c7286fa 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -313,7 +313,7 @@ module _ {ℓ e} {N : Set ℓ} {E : Set e} where
     expand n rec (c & g) =
       node (label c)
            (List.map
-              (Prod.map id (λ i → rec (n - suc i) (lemma n i) (g [ i ])))
+              (Prod.map id (λ i → rec (lemma n i) (g [ i ])))
               (successors c))
 
 -- Performs the toTree expansion once for each node.
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index ce16a7528a..3ea0cbb3c0 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -384,15 +384,15 @@ split v as bs p = helper as bs p (<-wellFounded (steps p))
   helper (a ∷ [])     bs (refl eq)    _ = [ a ]      , bs , eq
   helper (a ∷ b ∷ as) bs (refl eq)    _ = a ∷ b ∷ as , bs , eq
   helper []           bs (prep v≈x _) _ = [] , _ , v≈x ∷ ≋-refl
-  helper (a ∷ as)     bs (prep eq as↭xs) (acc rec) with helper as bs as↭xs (rec _ ≤-refl)
+  helper (a ∷ as)     bs (prep eq as↭xs) (acc rec) with helper as bs as↭xs (rec ≤-refl)
   ... | (ps , qs , eq₂) = a ∷ ps , qs , eq ∷ eq₂
   helper [] (b ∷ bs)     (swap x≈b y≈v _) _ = [ b ] , _     , x≈b ∷ y≈v ∷ ≋-refl
   helper (a ∷ [])     bs (swap x≈v y≈a ↭) _ = []    , a ∷ _ , x≈v ∷ y≈a ∷ ≋-refl
-  helper (a ∷ b ∷ as) bs (swap x≈b y≈a as↭xs) (acc rec) with helper as bs as↭xs (rec _ ≤-refl)
+  helper (a ∷ b ∷ as) bs (swap x≈b y≈a as↭xs) (acc rec) with helper as bs as↭xs (rec ≤-refl)
   ... | (ps , qs , eq) = b ∷ a ∷ ps , qs , x≈b ∷ y≈a ∷ eq
-  helper as           bs (trans ↭₁ ↭₂) (acc rec) with helper as bs ↭₂ (rec _ (m<n+m (steps ↭₂) (0<steps ↭₁)))
+  helper as           bs (trans ↭₁ ↭₂) (acc rec) with helper as bs ↭₂ (rec (m<n+m (steps ↭₂) (0<steps ↭₁)))
   ... | (ps , qs , eq) = helper ps qs (↭-respʳ-≋ eq ↭₁)
-    (rec _ (subst (_< _) (sym (steps-respʳ eq ↭₁)) (m<m+n (steps ↭₁) (0<steps ↭₂))))
+      (rec (subst (_< _) (sym (steps-respʳ eq ↭₁)) (m<m+n (steps ↭₁) (0<steps ↭₂))))
 
 ------------------------------------------------------------------------
 -- filter
diff --git a/src/Data/List/Sort/MergeSort.agda b/src/Data/List/Sort/MergeSort.agda
index 9648718d0d..9abab0823e 100644
--- a/src/Data/List/Sort/MergeSort.agda
+++ b/src/Data/List/Sort/MergeSort.agda
@@ -44,20 +44,21 @@ open PermutationReasoning
 -- Definition
 
 mergePairs : List (List A) → List (List A)
-mergePairs (xs ∷ ys ∷ xss) = merge _≤?_ xs ys ∷ mergePairs xss
+mergePairs (xs ∷ ys ∷ yss) = merge _≤?_ xs ys ∷ mergePairs yss
 mergePairs xss             = xss
 
 private
-  length-mergePairs : ∀ xs ys xss → length (mergePairs (xs ∷ ys ∷ xss)) < length (xs ∷ ys ∷ xss)
+  length-mergePairs : ∀ xs ys yss → let zss = xs ∷ ys ∷ yss in
+                      length (mergePairs zss) < length zss
   length-mergePairs _ _ []              = s<s z<s
-  length-mergePairs _ _ (xss ∷ [])      = s<s (s<s z<s)
-  length-mergePairs _ _ (xs ∷ ys ∷ xss) = s<s (m<n⇒m<1+n (length-mergePairs xs ys xss))
+  length-mergePairs _ _ (xs ∷ [])       = s<s (s<s z<s)
+  length-mergePairs _ _ (xs ∷ ys ∷ yss) = s<s (m<n⇒m<1+n (length-mergePairs xs ys yss))
 
 mergeAll : (xss : List (List A)) → Acc _<_ (length xss) → List A
 mergeAll []        _               = []
 mergeAll (xs ∷ []) _               = xs
-mergeAll (xs ∷ ys ∷ xss) (acc rec) = mergeAll
-  (mergePairs (xs ∷ ys ∷ xss)) (rec _ (length-mergePairs xs ys xss))
+mergeAll xss@(xs ∷ ys ∷ yss) (acc rec) = mergeAll
+  (mergePairs xss) (rec (length-mergePairs xs ys yss))
 
 sort : List A → List A
 sort xs = mergeAll (map [_] xs) (<-wellFounded-fast _)
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index c879cec098..abc21e7058 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -294,7 +294,7 @@ m*n/m*o≡n/o m@(suc _) n o = helper (<-wellFounded n)
   ... | no  n≮o = begin-equality
     (m * n) / (m * o)             ≡⟨  m/n≡1+[m∸n]/n (*-monoʳ-≤ m (≮⇒≥ n≮o)) ⟩
     1 + (m * n ∸ m * o) / (m * o) ≡˘⟨ cong (λ v → 1 + v / (m * o)) (*-distribˡ-∸ m n o) ⟩
-    1 + (m * (n ∸ o)) / (m * o)   ≡⟨  cong suc (helper (rec (n ∸ o) n∸o<n)) ⟩
+    1 + (m * (n ∸ o)) / (m * o)   ≡⟨  cong suc (helper (rec n∸o<n)) ⟩
     1 + (n ∸ o) / o               ≡˘⟨ cong₂ _+_ (n/n≡1 o) refl ⟩
     o / o + (n ∸ o) / o           ≡˘⟨ +-distrib-/-∣ˡ (n ∸ o) (divides 1 ((sym (*-identityˡ o)))) ⟩
     (o + (n ∸ o)) / o             ≡⟨  cong (_/ o) (m+[n∸m]≡n (≮⇒≥ n≮o)) ⟩
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index 2aa87002ca..555a902df6 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -43,7 +43,7 @@ open import Algebra.Definitions {A = ℕ} _≡_ as Algebra
 
 gcd′ : ∀ m n → Acc _<_ m → n < m → ℕ
 gcd′ m zero      _         _   = m
-gcd′ m n@(suc _) (acc rec) n<m = gcd′ n (m % n) (rec _ n<m) (m%n<n m n)
+gcd′ m n@(suc _) (acc rec) n<m = gcd′ n (m % n) (rec n<m) (m%n<n m n)
 
 gcd : ℕ → ℕ → ℕ
 gcd m n with <-cmp m n
@@ -55,15 +55,15 @@ gcd m n with <-cmp m n
 -- Core properties of gcd′
 
 gcd′[m,n]∣m,n : ∀ {m n} rec n<m → gcd′ m n rec n<m ∣ m × gcd′ m n rec n<m ∣ n
-gcd′[m,n]∣m,n {m} {zero}  rec       n<m = ∣-refl , m ∣0
-gcd′[m,n]∣m,n {m} {suc n} (acc rec) n<m
-  with gcd∣n , gcd∣m%n ← gcd′[m,n]∣m,n (rec _ n<m) (m%n<n m (suc n))
+gcd′[m,n]∣m,n {m} {zero}      rec       n<m = ∣-refl , m ∣0
+gcd′[m,n]∣m,n {m} {n@(suc _)} (acc rec) n<m
+  with gcd∣n , gcd∣m%n ← gcd′[m,n]∣m,n (rec n<m) (m%n<n m n)
   = ∣n∣m%n⇒∣m gcd∣n gcd∣m%n , gcd∣n
 
 gcd′-greatest : ∀ {m n c} rec n<m → c ∣ m → c ∣ n → c ∣ gcd′ m n rec n<m
-gcd′-greatest {m} {zero}  rec       n<m c∣m c∣n = c∣m
-gcd′-greatest {m} {suc n} (acc rec) n<m c∣m c∣n =
-  gcd′-greatest (rec _ n<m) (m%n<n m (suc n)) c∣n (%-presˡ-∣ c∣m c∣n)
+gcd′-greatest {m} {zero}      rec       n<m c∣m c∣n = c∣m
+gcd′-greatest {m} {n@(suc _)} (acc rec) n<m c∣m c∣n =
+  gcd′-greatest (rec n<m) (m%n<n m n) c∣n (%-presˡ-∣ c∣m c∣n)
 
 ------------------------------------------------------------------------
 -- Core properties of gcd
@@ -387,13 +387,13 @@ module Bézout where
     P (m , n) = Lemma m n
 
     gcd″ : ∀ p → (<′-Rec ⊗ <′-Rec) P p → P p
-    gcd″ (zero  , n)     rec = Lemma.base n
-    gcd″ (suc m , zero)  rec = Lemma.sym (Lemma.base (suc m))
-    gcd″ (suc m , suc n) rec with compare m n
-    ... | equal m     = Lemma.refl (suc m)
-    ... | less m k    = Lemma.stepˡ $ proj₁ rec (suc k) (lem₁ k m)
+    gcd″ (zero      , n)     rec = Lemma.base n
+    gcd″ (m@(suc _) , zero)  rec = Lemma.sym (Lemma.base m)
+    gcd″ (m′@(suc m) , n′@(suc n)) rec with compare m n
+    ... | equal m     = Lemma.refl m′
+    ... | less m k    = Lemma.stepˡ $ proj₁ rec (lem₁ k m)
                       -- "gcd (suc m) (suc k)"
-    ... | greater n k = Lemma.stepʳ $ proj₂ rec (suc k) (lem₁ k n) (suc n)
+    ... | greater n k = Lemma.stepʳ $ proj₂ rec (lem₁ k n) n′
                       -- "gcd (suc k) (suc n)"
 
   -- Bézout's identity can be recovered from the GCD.
diff --git a/src/Data/Nat/Induction.agda b/src/Data/Nat/Induction.agda
index efdd57fe3d..e1554074c4 100644
--- a/src/Data/Nat/Induction.agda
+++ b/src/Data/Nat/Induction.agda
@@ -67,8 +67,8 @@ cRec = build cRecBuilder
 
 <′-wellFounded n = acc (<′-wellFounded′ n)
 
-<′-wellFounded′ (suc n) n <′-base       = <′-wellFounded n
-<′-wellFounded′ (suc n) m (<′-step m<n) = <′-wellFounded′ n m m<n
+<′-wellFounded′ (suc n) <′-base       = <′-wellFounded n
+<′-wellFounded′ (suc n) (<′-step m<n) = <′-wellFounded′ n m<n
 
 module _ {ℓ : Level} where
   open WF.All <′-wellFounded ℓ public
@@ -100,7 +100,7 @@ module _ {ℓ : Level} where
   <-wellFounded-skip : ∀ (k : ℕ) → WellFounded _<_
   <-wellFounded-skip zero    n       = <-wellFounded n
   <-wellFounded-skip (suc k) zero    = <-wellFounded 0
-  <-wellFounded-skip (suc k) (suc n) = acc (λ m _ → <-wellFounded-skip k m)
+  <-wellFounded-skip (suc k) (suc n) = acc λ {m} _ → <-wellFounded-skip k m
 
 module _ {ℓ : Level} where
   open WF.All <-wellFounded ℓ public
diff --git a/src/Data/Nat/Logarithm/Core.agda b/src/Data/Nat/Logarithm/Core.agda
index 6a7cdb21e5..397dd3b3b9 100644
--- a/src/Data/Nat/Logarithm/Core.agda
+++ b/src/Data/Nat/Logarithm/Core.agda
@@ -23,7 +23,7 @@ open import Data.Unit
 ⌊log2⌋ : ∀ n → Acc _<_ n → ℕ
 ⌊log2⌋ 0          _        = 0
 ⌊log2⌋ 1          _        = 0
-⌊log2⌋ (suc n′@(suc n)) (acc rs) = 1 + ⌊log2⌋ (suc ⌊ n /2⌋) (rs _ (⌊n/2⌋<n n′))
+⌊log2⌋ (suc n′@(suc n)) (acc rs) = 1 + ⌊log2⌋ (suc ⌊ n /2⌋) (rs (⌊n/2⌋<n n′))
 
 
 -- Ceil version
@@ -31,7 +31,7 @@ open import Data.Unit
 ⌈log2⌉ : ∀ n → Acc _<_ n → ℕ
 ⌈log2⌉ 0                _        = 0
 ⌈log2⌉ 1                _        = 0
-⌈log2⌉ (suc (suc n)) (acc rs) = 1 + ⌈log2⌉ (suc ⌈ n /2⌉) (rs _ (⌈n/2⌉<n n))
+⌈log2⌉ (suc (suc n)) (acc rs) = 1 + ⌈log2⌉ (suc ⌈ n /2⌉) (rs (⌈n/2⌉<n n))
 
 ------------------------------------------------------------------------
 -- Properties of ⌊log2⌋
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 507dfe0a08..ac598c11a4 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -486,7 +486,7 @@ m<1+n⇒m≤n (s≤s m≤n) = m≤n
 ∀[m<n⇒m≢o]⇒n≤o (suc n) (suc o) m<n⇒m≢o = s≤s (∀[m<n⇒m≢o]⇒n≤o n o rec)
   where
   rec : ∀ {m} → m < n → m ≢ o
-  rec x<m refl = m<n⇒m≢o (s≤s x<m) refl
+  rec o<n refl = m<n⇒m≢o (s<s o<n) refl
 
 ------------------------------------------------------------------------
 -- A module for reasoning about the _≤_ and _<_ relations
diff --git a/src/Data/Product/Relation/Binary/Lex/Strict.agda b/src/Data/Product/Relation/Binary/Lex/Strict.agda
index 742c642d60..96adc9b9e1 100644
--- a/src/Data/Product/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/Strict.agda
@@ -202,13 +202,13 @@ module _ {_≈₁_ : Rel A ℓ₁} {_<₁_ : Rel A ℓ₂} {_<₂_ : Rel B ℓ
     ×-acc : ∀ {x y} →
             Acc _<₁_ x → Acc _<₂_ y →
             WfRec _<ₗₑₓ_ (Acc _<ₗₑₓ_) (x , y)
-    ×-acc (acc rec₁) acc₂ (u , v) (inj₁ u<x)
-      = acc (×-acc (rec₁ u u<x) (wf₂ v))
-    ×-acc acc₁ (acc rec₂) (u , v) (inj₂ (u≈x , v<y))
+    ×-acc (acc rec₁) acc₂ (inj₁ u<x)
+      = acc (×-acc (rec₁ u<x) (wf₂ _))
+    ×-acc acc₁ (acc rec₂) (inj₂ (u≈x , v<y))
       = Acc-resp-flip-≈
         (×-respectsʳ {_<₁_ = _<₁_} {_<₂_ = _<₂_} trans resp (≡.respʳ _<₂_))
         (u≈x , ≡.refl)
-        (acc (×-acc acc₁ (rec₂ v v<y)))
+        (acc (×-acc acc₁ (rec₂ v<y)))
 
 
 module _ {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel B ℓ₂} where
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 8c41fdfbd4..944edb4927 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -134,7 +134,8 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
     where
 
     <-wellFounded : ∀ {n} → WellFounded (_<_ {n})
-    <-wellFounded {0}     [] = acc λ ys ys<[] → ⊥-elim (xs≮[] ys<[])
+    <-wellFounded {0}     [] = acc λ ys<[] → ⊥-elim (xs≮[] ys<[])
+
     <-wellFounded {suc n} xs = Subrelation.wellFounded <⇒uncons-Lex uncons-Lex-wellFounded xs
       where
         <⇒uncons-Lex : {xs ys : Vec A (suc n)} → xs < ys → (×-Lex _≈_ _≺_ _<_ on uncons) xs ys
diff --git a/src/Induction/WellFounded.agda b/src/Induction/WellFounded.agda
index 7615c99337..61ba4ded19 100644
--- a/src/Induction/WellFounded.agda
+++ b/src/Induction/WellFounded.agda
@@ -32,7 +32,7 @@ private
 -- well-founded.
 
 WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
-WfRec _<_ P x = ∀ y → y < x → P y
+WfRec _<_ P x = ∀ {y} → y < x → P y
 
 -- The accessibility predicate: x is accessible if everything which is
 -- smaller than x is also accessible (inductively).
@@ -51,13 +51,13 @@ WellFounded _<_ = ∀ x → Acc _<_ x
 -- Basic properties
 
 acc-inverse : ∀ {_<_ : Rel A ℓ} {x : A} (q : Acc _<_ x) →
-              (y : A) → y < x → Acc _<_ y
-acc-inverse (acc rs) y y<x = rs y y<x
+              WfRec _<_ (Acc _<_) x
+acc-inverse (acc rs) y<x = rs y<x
 
 module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where
 
   Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
-  Acc-resp-flip-≈ respʳ x≈y (acc rec) = acc λ z z<y → rec z (respʳ x≈y z<y)
+  Acc-resp-flip-≈ respʳ x≈y (acc rec) = acc λ z<y → rec (respʳ x≈y z<y)
 
   Acc-resp-≈ : Symmetric _≈_ → _<_ Respectsʳ _≈_ → (Acc _<_) Respects _≈_
   Acc-resp-≈ sym respʳ x≈y wf = Acc-resp-flip-≈ (respʳ ∘ sym) x≈y wf
@@ -68,14 +68,13 @@ module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where
 module Some {_<_ : Rel A r} {ℓ} where
 
   wfRecBuilder : SubsetRecursorBuilder (Acc _<_) (WfRec _<_ {ℓ = ℓ})
-  wfRecBuilder P f x (acc rs) = λ y y<x →
-    f y (wfRecBuilder P f y (rs y y<x))
+  wfRecBuilder P f x (acc rs) = λ y<x → f _ (wfRecBuilder P f _ (rs y<x))
 
   wfRec : SubsetRecursor (Acc _<_) (WfRec _<_)
   wfRec = subsetBuild wfRecBuilder
 
   unfold-wfRec : (P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P) {x : A} (q : Acc _<_ x) →
-                 wfRec P f x q ≡ f x (λ y y<x → wfRec P f y (acc-inverse q y y<x))
+                 wfRec P f x q ≡ f x λ y<x → wfRec P f _ (acc-inverse q y<x)
   unfold-wfRec P f (acc rs) = refl
 
 
@@ -96,22 +95,26 @@ module All {_<_ : Rel A r} (wf : WellFounded _<_) ℓ where
 module FixPoint
   {_<_ : Rel A r} (wf : WellFounded _<_)
   (P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P)
-  (f-ext : (x : A) {IH IH′ : WfRec _<_ P x} → (∀ {y} y<x → IH y y<x ≡ IH′ y y<x) → f x IH ≡ f x IH′)
+  (f-ext : (x : A) {IH IH′ : WfRec _<_ P x} →
+           (∀ {y} y<x → IH {y} y<x ≡ IH′ y<x) →
+           f x IH ≡ f x IH′)
   where
 
-  some-wfRec-irrelevant : ∀ x → (q q′ : Acc _<_ x) → Some.wfRec P f x q ≡ Some.wfRec P f x q′
-  some-wfRec-irrelevant = All.wfRec wf _
-                                   (λ x → (q q′ : Acc _<_ x) → Some.wfRec P f x q ≡ Some.wfRec P f x q′)
-                                   (λ { x IH (acc rs) (acc rs′) → f-ext x (λ y<x → IH _ y<x (rs _ y<x) (rs′ _ y<x)) })
+  some-wfrec-Irrelevant : Pred A _
+  some-wfrec-Irrelevant x = ∀ q q′ → Some.wfRec P f x q ≡ Some.wfRec P f x q′
+
+  some-wfRec-irrelevant : ∀ x → some-wfrec-Irrelevant x
+  some-wfRec-irrelevant = All.wfRec wf _ some-wfrec-Irrelevant
+    λ { x IH (acc rs) (acc rs′) → f-ext x λ y<x → IH y<x (rs y<x) (rs′ y<x) }
 
   open All wf ℓ
-  wfRecBuilder-wfRec : ∀ {x y} y<x → wfRecBuilder P f x y y<x ≡ wfRec P f y
-  wfRecBuilder-wfRec {x} {y} y<x with wf x
-  ... | acc rs = some-wfRec-irrelevant y (rs y y<x) (wf y)
 
-  unfold-wfRec : ∀ {x} → wfRec P f x ≡ f x (λ y _ → wfRec P f y)
-  unfold-wfRec {x} = f-ext x wfRecBuilder-wfRec
+  wfRecBuilder-wfRec : ∀ {x y} y<x → wfRecBuilder P f x y<x ≡ wfRec P f y
+  wfRecBuilder-wfRec {x} {y} y<x with acc rs ← wf x
+   = some-wfRec-irrelevant y (rs y<x) (wf y)
 
+  unfold-wfRec : ∀ {x} → wfRec P f x ≡ f x λ _ → wfRec P f _
+  unfold-wfRec {x} = f-ext x wfRecBuilder-wfRec
 
 ------------------------------------------------------------------------
 -- It might be useful to establish proofs of Acc or Well-founded using
@@ -123,7 +126,7 @@ module Subrelation {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel A ℓ₂}
                    (<₁⇒<₂ : ∀ {x y} → x <₁ y → x <₂ y) where
 
   accessible : Acc _<₂_ ⊆ Acc _<₁_
-  accessible (acc rs) = acc λ y y<x → accessible (rs y (<₁⇒<₂ y<x))
+  accessible (acc rs) = acc λ y<x → accessible (rs (<₁⇒<₂ y<x))
 
   wellFounded : WellFounded _<₂_ → WellFounded _<₁_
   wellFounded wf = λ x → accessible (wf x)
@@ -134,7 +137,7 @@ module Subrelation {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel A ℓ₂}
 module InverseImage {_<_ : Rel B ℓ} (f : A → B) where
 
   accessible : ∀ {x} → Acc _<_ (f x) → Acc (_<_ on f) x
-  accessible (acc rs) = acc λ y fy<fx → accessible (rs (f y) fy<fx)
+  accessible (acc rs) = acc λ fy<fx → accessible (rs fy<fx)
 
   wellFounded : WellFounded _<_ → WellFounded (_<_ on f)
   wellFounded wf = λ x → accessible (wf (f x))
@@ -161,7 +164,7 @@ module TransitiveClosure {A : Set a} (_<_ : Rel A ℓ) where
     trans : ∀ {x y z} (x<y : x <⁺ y) (y<z : y <⁺ z) → x <⁺ z
 
   downwardsClosed : ∀ {x y} → Acc _<⁺_ y → x <⁺ y → Acc _<⁺_ x
-  downwardsClosed (acc rs) x<y = acc λ z z<x → rs z (trans z<x x<y)
+  downwardsClosed (acc rs) x<y = acc λ z<x → rs (trans z<x x<y)
 
   mutual
 
@@ -169,9 +172,9 @@ module TransitiveClosure {A : Set a} (_<_ : Rel A ℓ) where
     accessible acc-x = acc (accessible′ acc-x)
 
     accessible′ : ∀ {x} → Acc _<_ x → WfRec _<⁺_ (Acc _<⁺_) x
-    accessible′ (acc rs) y [ y<x ]         = accessible (rs y y<x)
-    accessible′ acc-x    y (trans y<z z<x) =
-      downwardsClosed (accessible′ acc-x _ z<x) y<z
+    accessible′ (acc rs) [ y<x ]         = accessible (rs y<x)
+    accessible′ acc-x    (trans y<z z<x) =
+      downwardsClosed (accessible′ acc-x z<x) y<z
 
   wellFounded : WellFounded _<_ → WellFounded _<⁺_
   wellFounded wf = λ x → accessible (wf x)
@@ -205,9 +208,9 @@ module Lexicographic {A : Set a} {B : A → Set b}
       ∀ {x y} →
       Acc RelA x → Acc (RelB x) y → (∀ {x} → WellFounded (RelB x)) →
       WfRec _<_ (Acc _<_) (x , y)
-    accessible′ (acc rsA) _    wfB ._ (left  x′<x) = accessible (rsA _ x′<x) wfB
-    accessible′ accA (acc rsB) wfB ._ (right y′<y) =
-      acc (accessible′ accA (rsB _ y′<y) wfB)
+    accessible′ (acc rsA) _    wfB (left  x′<x) = accessible (rsA x′<x) wfB
+    accessible′ accA (acc rsB) wfB (right y′<y) =
+      acc (accessible′ accA (rsB y′<y) wfB)
 
   wellFounded : WellFounded RelA → (∀ {x} → WellFounded (RelB x)) →
                 WellFounded _<_
diff --git a/src/Relation/Binary/Construct/Closure/Transitive.agda b/src/Relation/Binary/Construct/Closure/Transitive.agda
index 7c12fa7bcd..e4154d49cd 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive.agda
@@ -80,8 +80,8 @@ module _ (_∼_ : Rel A ℓ) where
   accessible acc[x] = acc (wf-acc acc[x])
     where
     wf-acc : ∀ {x} → Acc _∼_ x → WfRec _∼⁺_ (Acc _∼⁺_) x
-    wf-acc (acc rec) _ [ y∼x ]   = acc (wf-acc (rec _ y∼x))
-    wf-acc acc[x] _ (y∼z ∷ z∼⁺x) = acc-inverse (wf-acc acc[x] _ z∼⁺x) _ [ y∼z ]
+    wf-acc (acc rec) [ y∼x ]   = acc (wf-acc (rec y∼x))
+    wf-acc acc[x] (y∼z ∷ z∼⁺x) = acc-inverse (wf-acc acc[x] z∼⁺x) [ y∼z ]
 
   wellFounded : WellFounded _∼_ → WellFounded _∼⁺_
   wellFounded wf x = accessible (wf x)
diff --git a/src/Relation/Binary/Construct/On.agda b/src/Relation/Binary/Construct/On.agda
index 72959ac922..c124f98dd8 100644
--- a/src/Relation/Binary/Construct/On.agda
+++ b/src/Relation/Binary/Construct/On.agda
@@ -74,7 +74,7 @@ module _ (f : B → A) where
   trichotomous _ _ compare x y = compare (f x) (f y)
 
   accessible : ∀ {∼ : Rel A ℓ} {x} → Acc ∼ (f x) → Acc (∼ on f) x
-  accessible (acc rs) = acc (λ y fy<fx → accessible (rs (f y) fy<fx))
+  accessible (acc rs) = acc (λ fy<fx → accessible (rs fy<fx))
 
   wellFounded : {∼ : Rel A ℓ} → WellFounded ∼ → WellFounded (∼ on f)
   wellFounded wf x = accessible (wf (f x))
diff --git a/src/Relation/Binary/Rewriting.agda b/src/Relation/Binary/Rewriting.agda
index 10d760e23b..903ab9018e 100644
--- a/src/Relation/Binary/Rewriting.agda
+++ b/src/Relation/Binary/Rewriting.agda
@@ -117,8 +117,8 @@ module _ {a ℓ} {A : Set a} {_⟶_ : Rel A ℓ} where
         jToInner = proj₁ (proj₂ wcrProof)
         kToInner = proj₂ (proj₂ wcrProof)
 
-        lhs = helper (g _ [ toJ ]) fst jToInner
-        rhs = helper (g _ [ toK ]) snd kToInner
+        lhs = helper (g [ toJ ]) fst jToInner
+        rhs = helper (g [ toK ]) snd kToInner
 
         fromAB = proj₁ (proj₂ lhs)
         fromInnerB = proj₂ (proj₂ lhs)
@@ -129,7 +129,7 @@ module _ {a ℓ} {A : Set a} {_⟶_ : Rel A ℓ} where
         aToInner : _ ⟶₊ innerPoint
         aToInner = starToPlus toJ jToInner
 
-        finalRecursion = helper (g innerPoint aToInner) fromInnerB fromInnerC
+        finalRecursion = helper (g aToInner) fromInnerB fromInnerC
 
         bMidToDest = proj₁ (proj₂ finalRecursion)
         cMidToDest = proj₂ (proj₂ finalRecursion)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Base.agda b/src/Tactic/RingSolver/Core/Polynomial/Base.agda
index 0a027d6d3f..6de64bf839 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Base.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Base.agda
@@ -316,7 +316,7 @@ mutual
 
 ⊟-step : ∀ {n} → Acc _<′_ n → Poly n → Poly n
 ⊟-step (acc wf) (Κ x  ⊐ i≤n) = Κ (- x) ⊐ i≤n
-⊟-step (acc wf) (⅀ xs ⊐ i≤n) = poly-map (⊟-step (wf _ i≤n)) xs ⊐↓ i≤n
+⊟-step (acc wf) (⅀ xs ⊐ i≤n) = poly-map (⊟-step (wf i≤n)) xs ⊐↓ i≤n
 
 ⊟_ : ∀ {n} → Poly n → Poly n
 ⊟_ = ⊟-step (<′-wellFounded _)
@@ -335,12 +335,12 @@ mutual
 
   ⊠-Κ : ∀ {n} → Acc _<′_ n → Carrier → Poly n → Poly n
   ⊠-Κ (acc _ ) x (Κ y  ⊐ i≤n) = Κ (x * y) ⊐ i≤n
-  ⊠-Κ (acc wf) x (⅀ xs ⊐ i≤n) = ⊠-Κ-inj (wf _ i≤n) x xs ⊐↓ i≤n
+  ⊠-Κ (acc wf) x (⅀ xs ⊐ i≤n) = ⊠-Κ-inj (wf i≤n) x xs ⊐↓ i≤n
   {-# INLINE ⊠-Κ #-}
 
   ⊠-⅀ : ∀ {i n} → Acc _<′_ n → Coeff i + → i <′ n → Poly n → Poly n
   ⊠-⅀ (acc wf) xs i≤n (⅀ ys ⊐ j≤n) = ⊠-match  (acc wf) (inj-compare i≤n j≤n) xs ys
-  ⊠-⅀ (acc wf) xs i≤n (Κ y ⊐ _)    = ⊠-Κ-inj (wf _ i≤n) y xs ⊐↓ i≤n
+  ⊠-⅀ (acc wf) xs i≤n (Κ y ⊐ _)    = ⊠-Κ-inj (wf i≤n) y xs ⊐↓ i≤n
 
   ⊠-Κ-inj : ∀ {i}  → Acc _<′_ i → Carrier → Coeff i + → Coeff i *
   ⊠-Κ-inj a x xs = poly-map (⊠-Κ a x) (xs)
@@ -352,7 +352,7 @@ mutual
           → Poly k
           → Poly k
   ⊠-⅀-inj (acc wf) i≤k x (⅀ y ⊐ j≤k) = ⊠-match (acc wf) (inj-compare i≤k j≤k) x y
-  ⊠-⅀-inj (acc wf) i≤k x (Κ y ⊐ j≤k) = ⊠-Κ-inj (wf _ i≤k) y x ⊐↓ i≤k
+  ⊠-⅀-inj (acc wf) i≤k x (Κ y ⊐ j≤k) = ⊠-Κ-inj (wf i≤k) y x ⊐↓ i≤k
 
   ⊠-match : ∀ {i j n}
           → Acc _<′_ n
@@ -362,9 +362,9 @@ mutual
           → Coeff i +
           → Coeff j +
           → Poly n
-  ⊠-match (acc wf) (inj-eq i&j≤n)     xs ys = ⊠-coeffs (wf _ i&j≤n) xs ys               ⊐↓ i&j≤n
-  ⊠-match (acc wf) (inj-lt i≤j-1 j≤n) xs ys = poly-map (⊠-⅀-inj (wf _ j≤n) i≤j-1 xs) (ys) ⊐↓ j≤n
-  ⊠-match (acc wf) (inj-gt i≤n j≤i-1) xs ys = poly-map (⊠-⅀-inj (wf _ i≤n) j≤i-1 ys) (xs) ⊐↓ i≤n
+  ⊠-match (acc wf) (inj-eq i&j≤n)     xs ys = ⊠-coeffs (wf i&j≤n) xs ys               ⊐↓ i&j≤n
+  ⊠-match (acc wf) (inj-lt i≤j-1 j≤n) xs ys = poly-map (⊠-⅀-inj (wf j≤n) i≤j-1 xs) (ys) ⊐↓ j≤n
+  ⊠-match (acc wf) (inj-gt i≤n j≤i-1) xs ys = poly-map (⊠-⅀-inj (wf i≤n) j≤i-1 ys) (xs) ⊐↓ i≤n
 
   ⊠-coeffs : ∀ {n} → Acc _<′_ n → Coeff n + → Coeff n + → Coeff n *
   ⊠-coeffs a (xs) (y ≠0 Δ j & [])   = poly-map (⊠-step′ a y) (xs) ⍓* j
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
index c4e1ef756b..07e157389c 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Multiplication.agda
@@ -58,10 +58,10 @@ mutual
   ⊠-Κ-hom (acc _)  x (Κ y  ⊐ i≤n) ρ = *-homo x y
   ⊠-Κ-hom (acc wf) x (⅀ xs ⊐ i≤n) ρ =
     begin
-      ⟦ ⊠-Κ-inj (wf _ i≤n) x xs ⊐↓ i≤n ⟧ ρ
-    ≈⟨ ⊐↓-hom (⊠-Κ-inj (wf _ i≤n) x xs) i≤n ρ ⟩
-      ⅀?⟦ ⊠-Κ-inj (wf _ i≤n) x xs ⟧ (drop-1 i≤n ρ)
-    ≈⟨ ⊠-Κ-inj-hom (wf _ i≤n) x xs (drop-1 i≤n ρ) ⟩
+      ⟦ ⊠-Κ-inj (wf i≤n) x xs ⊐↓ i≤n ⟧ ρ
+    ≈⟨ ⊐↓-hom (⊠-Κ-inj (wf i≤n) x xs) i≤n ρ ⟩
+      ⅀?⟦ ⊠-Κ-inj (wf i≤n) x xs ⟧ (drop-1 i≤n ρ)
+    ≈⟨ ⊠-Κ-inj-hom (wf i≤n) x xs (drop-1 i≤n ρ) ⟩
       ⟦ x ⟧ᵣ * ⅀⟦ xs ⟧ (drop-1 i≤n ρ)
     ∎
 
@@ -94,10 +94,10 @@ mutual
   ⊠-⅀-hom (acc wf) xs i<n (⅀ ys ⊐ j≤n) = ⊠-match-hom (acc wf) (inj-compare i<n j≤n) xs ys
   ⊠-⅀-hom (acc wf) xs i<n (Κ y  ⊐ _) ρ =
     begin
-      ⟦ ⊠-Κ-inj (wf _ i<n) y xs ⊐↓ i<n ⟧ ρ
-    ≈⟨ ⊐↓-hom (⊠-Κ-inj (wf _ i<n) y xs) i<n ρ ⟩
-      ⅀?⟦ ⊠-Κ-inj (wf _ i<n) y xs ⟧ (drop-1 i<n ρ)
-    ≈⟨ ⊠-Κ-inj-hom (wf _ i<n) y xs (drop-1 i<n ρ) ⟩
+      ⟦ ⊠-Κ-inj (wf i<n) y xs ⊐↓ i<n ⟧ ρ
+    ≈⟨ ⊐↓-hom (⊠-Κ-inj (wf i<n) y xs) i<n ρ ⟩
+      ⅀?⟦ ⊠-Κ-inj (wf i<n) y xs ⟧ (drop-1 i<n ρ)
+    ≈⟨ ⊠-Κ-inj-hom (wf i<n) y xs (drop-1 i<n ρ) ⟩
       ⟦ y ⟧ᵣ * ⅀⟦ xs ⟧ (drop-1 i<n ρ)
     ≈⟨ *-comm _ _ ⟩
       ⅀⟦ xs ⟧ (drop-1 i<n ρ) * ⟦ y ⟧ᵣ
@@ -113,10 +113,10 @@ mutual
   ⊠-⅀-inj-hom (acc wf) i<k x (⅀ ys ⊐ j≤k) = ⊠-match-hom (acc wf) (inj-compare i<k j≤k) x ys
   ⊠-⅀-inj-hom (acc wf) i<k x (Κ y ⊐ j≤k) ρ =
     begin
-      ⟦ ⊠-Κ-inj (wf _ i<k) y x ⊐↓ i<k ⟧ ρ
-    ≈⟨ ⊐↓-hom (⊠-Κ-inj (wf _ i<k) y x) i<k ρ ⟩
-      ⅀?⟦ ⊠-Κ-inj (wf _ i<k) y x ⟧ (drop-1 i<k ρ)
-    ≈⟨ ⊠-Κ-inj-hom (wf _ i<k) y x (drop-1 i<k ρ) ⟩
+      ⟦ ⊠-Κ-inj (wf i<k) y x ⊐↓ i<k ⟧ ρ
+    ≈⟨ ⊐↓-hom (⊠-Κ-inj (wf i<k) y x) i<k ρ ⟩
+      ⅀?⟦ ⊠-Κ-inj (wf i<k) y x ⟧ (drop-1 i<k ρ)
+    ≈⟨ ⊠-Κ-inj-hom (wf i<k) y x (drop-1 i<k ρ) ⟩
       ⟦ y ⟧ᵣ * ⅀⟦ x ⟧ (drop-1 i<k ρ)
     ≈⟨ *-comm _ _ ⟩
       ⅀⟦ x ⟧ (drop-1 i<k ρ) * ⟦ y ⟧ᵣ
@@ -137,15 +137,15 @@ mutual
         xs′ = ⅀⟦ xs ⟧ (drop-1 (≤′-trans (≤′-step i≤j-1) j≤n) Ρ′)
     in
     begin
-      ⟦ poly-map ( (⊠-⅀-inj (wf _ j≤n) i≤j-1 xs)) ys ⊐↓ j≤n ⟧ Ρ′
-    ≈⟨ ⊐↓-hom (poly-map ( (⊠-⅀-inj (wf _ j≤n) i≤j-1 xs)) ys) j≤n Ρ′ ⟩
-      ⅀?⟦ poly-map ( (⊠-⅀-inj (wf _ j≤n) i≤j-1 xs)) ys ⟧ (ρ , Ρ)
-    ≈⟨ poly-mapR ρ Ρ (⊠-⅀-inj (wf _ j≤n) i≤j-1 xs)
+      ⟦ poly-map ( (⊠-⅀-inj (wf j≤n) i≤j-1 xs)) ys ⊐↓ j≤n ⟧ Ρ′
+    ≈⟨ ⊐↓-hom (poly-map ( (⊠-⅀-inj (wf j≤n) i≤j-1 xs)) ys) j≤n Ρ′ ⟩
+      ⅀?⟦ poly-map ( (⊠-⅀-inj (wf j≤n) i≤j-1 xs)) ys ⟧ (ρ , Ρ)
+    ≈⟨ poly-mapR ρ Ρ (⊠-⅀-inj (wf j≤n) i≤j-1 xs)
                      (_ *_)
                      (*-cong refl)
                      reassoc
                      (distribˡ _)
-                     (λ y → ⊠-⅀-inj-hom (wf _ j≤n) i≤j-1 xs y _)
+                     (λ y → ⊠-⅀-inj-hom (wf j≤n) i≤j-1 xs y _)
                      (zeroʳ _) ys ⟩
        ⅀⟦ xs ⟧ (drop-1 i≤j-1 Ρ) * ⅀⟦ ys ⟧ (ρ , Ρ)
     ≈⟨ ≪* trans-join-coeffs-hom i≤j-1 j≤n xs Ρ′ ⟩
@@ -156,15 +156,15 @@ mutual
         ys′ = ⅀⟦ ys ⟧ (drop-1 (≤′-step j≤i-1 ⟨ ≤′-trans ⟩ i≤n) Ρ′)
     in
     begin
-      ⟦ poly-map ( (⊠-⅀-inj (wf _ i≤n) j≤i-1 ys)) xs ⊐↓ i≤n ⟧ Ρ′
-    ≈⟨ ⊐↓-hom (poly-map ( (⊠-⅀-inj (wf _ i≤n) j≤i-1 ys)) xs) i≤n Ρ′ ⟩
-      ⅀?⟦ poly-map ( (⊠-⅀-inj (wf _ i≤n) j≤i-1 ys)) xs ⟧ (ρ , Ρ)
-    ≈⟨ poly-mapR ρ Ρ (⊠-⅀-inj (wf _ i≤n) j≤i-1 ys)
+      ⟦ poly-map ( (⊠-⅀-inj (wf i≤n) j≤i-1 ys)) xs ⊐↓ i≤n ⟧ Ρ′
+    ≈⟨ ⊐↓-hom (poly-map ( (⊠-⅀-inj (wf i≤n) j≤i-1 ys)) xs) i≤n Ρ′ ⟩
+      ⅀?⟦ poly-map ( (⊠-⅀-inj (wf i≤n) j≤i-1 ys)) xs ⟧ (ρ , Ρ)
+    ≈⟨ poly-mapR ρ Ρ (⊠-⅀-inj (wf i≤n) j≤i-1 ys)
                      (_ *_)
                      (*-cong refl)
                      reassoc
                      (distribˡ _)
-                     (λ x → ⊠-⅀-inj-hom (wf _ i≤n) j≤i-1 ys x _)
+                     (λ x → ⊠-⅀-inj-hom (wf i≤n) j≤i-1 ys x _)
                      (zeroʳ _) xs ⟩
       ⅀⟦ ys ⟧ (drop-1 j≤i-1 Ρ) * ⅀⟦ xs ⟧ (ρ , Ρ)
     ≈⟨ ≪* trans-join-coeffs-hom j≤i-1 i≤n ys Ρ′ ⟩
@@ -174,10 +174,10 @@ mutual
     ∎
   ⊠-match-hom (acc wf) (inj-eq ij≤n) xs ys Ρ =
     begin
-      ⟦ ⊠-coeffs (wf _ ij≤n) xs ys ⊐↓ ij≤n ⟧ Ρ
-    ≈⟨ ⊐↓-hom (⊠-coeffs (wf _ ij≤n) xs ys) ij≤n Ρ ⟩
-      ⅀?⟦ ⊠-coeffs (wf _ ij≤n) xs ys ⟧ (drop-1 ij≤n Ρ)
-    ≈⟨ ⊠-coeffs-hom (wf _ ij≤n) xs ys (drop-1 ij≤n Ρ) ⟩
+      ⟦ ⊠-coeffs (wf ij≤n) xs ys ⊐↓ ij≤n ⟧ Ρ
+    ≈⟨ ⊐↓-hom (⊠-coeffs (wf ij≤n) xs ys) ij≤n Ρ ⟩
+      ⅀?⟦ ⊠-coeffs (wf ij≤n) xs ys ⟧ (drop-1 ij≤n Ρ)
+    ≈⟨ ⊠-coeffs-hom (wf ij≤n) xs ys (drop-1 ij≤n Ρ) ⟩
       ⅀⟦ xs ⟧ (drop-1 ij≤n Ρ) * ⅀⟦ ys ⟧ (drop-1 ij≤n Ρ)
     ∎
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
index 604f02c533..b2a4486d6a 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
@@ -42,10 +42,10 @@ open import Tactic.RingSolver.Core.Polynomial.Semantics homo
         ∎
   in
   begin
-    ⟦ poly-map (⊟-step (wf _ i≤n)) xs ⊐↓ i≤n ⟧ ρ′
-  ≈⟨ ⊐↓-hom (poly-map (⊟-step (wf _ i≤n)) xs) i≤n ρ′ ⟩
-    ⅀?⟦ poly-map (⊟-step  (wf _ i≤n)) xs ⟧ (ρ , ρs)
-  ≈⟨ poly-mapR ρ ρs (⊟-step (wf _ i≤n)) -_ (-‿cong) (λ x y → *-comm x (- y) ⟨ trans ⟩ -‿*-distribˡ y x ⟨ trans ⟩ -‿cong (*-comm _ _)) (λ x y → sym (-‿+-comm x y)) (flip (⊟-step-hom (wf _ i≤n)) ρs) (sym neg-zero ) xs ⟩
+    ⟦ poly-map (⊟-step (wf i≤n)) xs ⊐↓ i≤n ⟧ ρ′
+  ≈⟨ ⊐↓-hom (poly-map (⊟-step (wf i≤n)) xs) i≤n ρ′ ⟩
+    ⅀?⟦ poly-map (⊟-step  (wf i≤n)) xs ⟧ (ρ , ρs)
+  ≈⟨ poly-mapR ρ ρs (⊟-step (wf i≤n)) -_ (-‿cong) (λ x y → *-comm x (- y) ⟨ trans ⟩ -‿*-distribˡ y x ⟨ trans ⟩ -‿cong (*-comm _ _)) (λ x y → sym (-‿+-comm x y)) (flip (⊟-step-hom (wf i≤n)) ρs) (sym neg-zero ) xs ⟩
     - ⅀⟦ xs ⟧ (ρ , ρs)
   ∎
 

From 9a1b9c9b4daa43f52727b31d03360a19dfc207c0 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 5 Oct 2023 12:17:32 +0900
Subject: [PATCH 142/316] Re-export numeric subclass instances

---
 src/Data/Integer/Base.agda               |  5 ++++-
 src/Data/Rational/Base.agda              | 15 ++++++++++++++-
 src/Data/Rational/Unnormalised/Base.agda |  6 ++++++
 3 files changed, 24 insertions(+), 2 deletions(-)

diff --git a/src/Data/Integer/Base.agda b/src/Data/Integer/Base.agda
index 338656956b..9d7e689db0 100644
--- a/src/Data/Integer/Base.agda
+++ b/src/Data/Integer/Base.agda
@@ -14,7 +14,7 @@ module Data.Integer.Base where
 open import Algebra.Bundles.Raw
   using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring; RawRing)
 open import Data.Bool.Base using (Bool; T; true; false)
-open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s)
+open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s) hiding (module ℕ)
 open import Data.Sign.Base as Sign using (Sign)
 open import Level using (0ℓ)
 open import Relation.Binary.Core using (Rel)
@@ -140,6 +140,9 @@ record Negative (i : ℤ) : Set where
 
 -- Instances
 
+open ℕ public
+  using (nonZero)
+
 instance
   pos : ∀ {n} → Positive +[1+ n ]
   pos = _
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 5f5a35e6f9..64bf05327e 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -10,7 +10,9 @@ module Data.Rational.Base where
 
 open import Algebra.Bundles.Raw
 open import Data.Bool.Base using (Bool; true; false; if_then_else_)
-open import Data.Integer.Base as ℤ using (ℤ; +_; +0; +[1+_]; -[1+_])
+open import Data.Integer.Base as ℤ
+  using (ℤ; +_; +0; +[1+_]; -[1+_])
+  hiding (module ℤ)
 open import Data.Nat.GCD
 open import Data.Nat.Coprimality as C
   using (Coprime; Bézout-coprime; coprime-/gcd; coprime?; ¬0-coprimeTo-2+)
@@ -176,6 +178,11 @@ NonPositive p = ℚᵘ.NonPositive (toℚᵘ p)
 NonNegative : Pred ℚ 0ℓ
 NonNegative p = ℚᵘ.NonNegative (toℚᵘ p)
 
+-- Instances
+
+open ℤ public
+  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)
+
 -- Constructors
 
 ≢-nonZero : ∀ {p} → p ≢ 0ℚ → NonZero p
@@ -202,6 +209,12 @@ nonPositive {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonPositive {toℚᵘ p} (
 nonNegative : ∀ {p} → p ≥ 0ℚ → NonNegative p
 nonNegative {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonNegative {toℚᵘ p} (ℚᵘ.*≤* p≤q)
 
+-- Re-export base instances so that users don't have to open
+-- Data.Nat.Base.
+
+open ℕ public
+  using (nonZero)
+
 ------------------------------------------------------------------------
 -- Operations on rationals
 
diff --git a/src/Data/Rational/Unnormalised/Base.agda b/src/Data/Rational/Unnormalised/Base.agda
index dc5bd144dc..c9fcb8d54d 100644
--- a/src/Data/Rational/Unnormalised/Base.agda
+++ b/src/Data/Rational/Unnormalised/Base.agda
@@ -12,6 +12,7 @@ open import Algebra.Bundles.Raw
 open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Integer.Base as ℤ
   using (ℤ; +_; +0; +[1+_]; -[1+_]; +<+; +≤+)
+  hiding (module ℤ)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Level using (0ℓ)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
@@ -149,6 +150,11 @@ NonPositive p = ℤ.NonPositive (↥ p)
 NonNegative : Pred ℚᵘ 0ℓ
 NonNegative p = ℤ.NonNegative (↥ p)
 
+-- Instances
+
+open ℤ public
+  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)
+
 -- Constructors and destructors
 
 -- Note: these could be proved more elegantly using the constructors

From e17f477ed92a05ec5c93f8705c33a2cc3ab2524e Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 5 Oct 2023 12:18:02 +0900
Subject: [PATCH 143/316] Revert "Re-export numeric subclass instances"

This reverts commit 91fd951c117311b2beb2db4a582c4f152eac787d.
---
 src/Data/Integer/Base.agda               |  5 +----
 src/Data/Rational/Base.agda              | 15 +--------------
 src/Data/Rational/Unnormalised/Base.agda |  6 ------
 3 files changed, 2 insertions(+), 24 deletions(-)

diff --git a/src/Data/Integer/Base.agda b/src/Data/Integer/Base.agda
index 9d7e689db0..338656956b 100644
--- a/src/Data/Integer/Base.agda
+++ b/src/Data/Integer/Base.agda
@@ -14,7 +14,7 @@ module Data.Integer.Base where
 open import Algebra.Bundles.Raw
   using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring; RawRing)
 open import Data.Bool.Base using (Bool; T; true; false)
-open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s) hiding (module ℕ)
+open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s)
 open import Data.Sign.Base as Sign using (Sign)
 open import Level using (0ℓ)
 open import Relation.Binary.Core using (Rel)
@@ -140,9 +140,6 @@ record Negative (i : ℤ) : Set where
 
 -- Instances
 
-open ℕ public
-  using (nonZero)
-
 instance
   pos : ∀ {n} → Positive +[1+ n ]
   pos = _
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 64bf05327e..5f5a35e6f9 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -10,9 +10,7 @@ module Data.Rational.Base where
 
 open import Algebra.Bundles.Raw
 open import Data.Bool.Base using (Bool; true; false; if_then_else_)
-open import Data.Integer.Base as ℤ
-  using (ℤ; +_; +0; +[1+_]; -[1+_])
-  hiding (module ℤ)
+open import Data.Integer.Base as ℤ using (ℤ; +_; +0; +[1+_]; -[1+_])
 open import Data.Nat.GCD
 open import Data.Nat.Coprimality as C
   using (Coprime; Bézout-coprime; coprime-/gcd; coprime?; ¬0-coprimeTo-2+)
@@ -178,11 +176,6 @@ NonPositive p = ℚᵘ.NonPositive (toℚᵘ p)
 NonNegative : Pred ℚ 0ℓ
 NonNegative p = ℚᵘ.NonNegative (toℚᵘ p)
 
--- Instances
-
-open ℤ public
-  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)
-
 -- Constructors
 
 ≢-nonZero : ∀ {p} → p ≢ 0ℚ → NonZero p
@@ -209,12 +202,6 @@ nonPositive {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonPositive {toℚᵘ p} (
 nonNegative : ∀ {p} → p ≥ 0ℚ → NonNegative p
 nonNegative {p@(mkℚ _ _ _)} (*≤* p≤q) = ℚᵘ.nonNegative {toℚᵘ p} (ℚᵘ.*≤* p≤q)
 
--- Re-export base instances so that users don't have to open
--- Data.Nat.Base.
-
-open ℕ public
-  using (nonZero)
-
 ------------------------------------------------------------------------
 -- Operations on rationals
 
diff --git a/src/Data/Rational/Unnormalised/Base.agda b/src/Data/Rational/Unnormalised/Base.agda
index c9fcb8d54d..dc5bd144dc 100644
--- a/src/Data/Rational/Unnormalised/Base.agda
+++ b/src/Data/Rational/Unnormalised/Base.agda
@@ -12,7 +12,6 @@ open import Algebra.Bundles.Raw
 open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Integer.Base as ℤ
   using (ℤ; +_; +0; +[1+_]; -[1+_]; +<+; +≤+)
-  hiding (module ℤ)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Level using (0ℓ)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
@@ -150,11 +149,6 @@ NonPositive p = ℤ.NonPositive (↥ p)
 NonNegative : Pred ℚᵘ 0ℓ
 NonNegative p = ℤ.NonNegative (↥ p)
 
--- Instances
-
-open ℤ public
-  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)
-
 -- Constructors and destructors
 
 -- Note: these could be proved more elegantly using the constructors

From 595f4e865f3a8108bca5a3fdb0c784d9aa1e2654 Mon Sep 17 00:00:00 2001
From: shuhung <shhyou@users.noreply.github.com>
Date: Thu, 5 Oct 2023 01:52:29 -0500
Subject: [PATCH 144/316] Add (propositional) equational reasoning combinators
 for vectors (#2067)

---
 CHANGELOG.md                                  |  13 +
 README.agda                                   |   2 +-
 README/Data.agda                              |   5 +
 .../Vec/Relation/Binary/Equality/Cast.agda    | 254 ++++++++++++++++++
 src/Data/Vec/Base.agda                        |   2 +
 src/Data/Vec/Properties.agda                  | 132 ++++++---
 .../Vec/Relation/Binary/Equality/Cast.agda    | 129 +++++++++
 7 files changed, 502 insertions(+), 35 deletions(-)
 create mode 100644 README/Data/Vec/Relation/Binary/Equality/Cast.agda
 create mode 100644 src/Data/Vec/Relation/Binary/Equality/Cast.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 68c596e074..d145333c44 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -2024,6 +2024,11 @@ New modules
   Function.Indexed.Bundles
   ```
 
+* Combinators for propositional equational reasoning on vectors with different indices
+  ```
+  Data.Vec.Relation.Binary.Equality.Cast
+  ```
+
 Additions to existing modules
 -----------------------------
 
@@ -2976,6 +2981,7 @@ Additions to existing modules
   last-∷ʳ   : last (xs ∷ʳ x) ≡ x
   cast-∷ʳ   : cast eq (xs ∷ʳ x) ≡ (cast (cong pred eq) xs) ∷ʳ x
   ++-∷ʳ     : cast eq ((xs ++ ys) ∷ʳ z) ≡ xs ++ (ys ∷ʳ z)
+  ∷ʳ-++     : cast eq ((xs ∷ʳ a) ++ ys) ≡ xs ++ (a ∷ ys)
 
   reverse-∷          : reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
   reverse-involutive : Involutive _≡_ reverse
@@ -2996,6 +3002,8 @@ Additions to existing modules
   lookup-cast₁  : lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)
   lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
   cast-reverse  : cast eq ∘ reverse ≗ reverse ∘ cast eq
+  cast-++ˡ      : cast (cong (_+ n) eq) (xs ++ ys) ≡ cast eq xs ++ ys
+  cast-++ʳ      : cast (cong (m +_) eq) (xs ++ ys) ≡ xs ++ cast eq ys
 
   iterate-id     : iterate id x n ≡ replicate x
   take-iterate   : take n (iterate f x (n + m)) ≡ iterate f x n
@@ -3015,6 +3023,11 @@ Additions to existing modules
   cast-fromList : cast _ (fromList xs) ≡ fromList ys
   fromList-map  : cast _ (fromList (List.map f xs)) ≡ map f (fromList xs)
   fromList-++   : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
+  fromList-reverse : cast (Listₚ.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)
+
+  ∷-ʳ++   : cast eq ((a ∷ xs) ʳ++ ys) ≡ xs ʳ++ (a ∷ ys)
+  ++-ʳ++  : cast eq ((xs ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ʳ++ zs)
+  ʳ++-ʳ++ : cast eq ((xs ʳ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ++ zs)
 
   length-toList   : List.length (toList xs) ≡ length xs
   toList-insertAt : toList (insertAt xs i v) ≡ List.insertAt (toList xs) (Fin.cast (cong suc (sym (length-toList xs))) i) v
diff --git a/README.agda b/README.agda
index 6fee0aeffa..97b8ab3903 100644
--- a/README.agda
+++ b/README.agda
@@ -15,7 +15,7 @@ module README where
 -- James McKinna, Sergei Meshveliani, Eric Mertens, Darin Morrison,
 -- Guilhem Moulin, Shin-Cheng Mu, Ulf Norell, Noriyuki Ohkawa,
 -- Nicolas Pouillard, Andrés Sicard-Ramírez, Lex van der Stoep,
--- Sandro Stucki, Milo Turner, Noam Zeilberger
+-- Sandro Stucki, Milo Turner, Noam Zeilberger, Shu-Hung You
 -- and other anonymous contributors.
 ------------------------------------------------------------------------
 
diff --git a/README/Data.agda b/README/Data.agda
index be35a5bfc4..e7d8a13886 100644
--- a/README/Data.agda
+++ b/README/Data.agda
@@ -209,6 +209,11 @@ import README.Data.Record
 
 import README.Data.Trie.NonDependent
 
+-- Examples of equational reasoning about vectors of non-definitionally
+-- equal lengths.
+
+import README.Data.Vec.Relation.Binary.Equality.Cast
+
 -- Examples how (indexed) containers and constructions over them (free
 -- monad, least fixed point, etc.) can be used
 
diff --git a/README/Data/Vec/Relation/Binary/Equality/Cast.agda b/README/Data/Vec/Relation/Binary/Equality/Cast.agda
new file mode 100644
index 0000000000..fb76544ed7
--- /dev/null
+++ b/README/Data/Vec/Relation/Binary/Equality/Cast.agda
@@ -0,0 +1,254 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- An equational reasoning library for propositional equality over
+-- vectors of different indices using cast.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module README.Data.Vec.Relation.Binary.Equality.Cast where
+
+open import Agda.Primitive
+open import Data.List.Base as L using (List)
+import Data.List.Properties as Lₚ
+open import Data.Nat.Base
+open import Data.Nat.Properties
+open import Data.Vec.Base
+open import Data.Vec.Properties
+open import Data.Vec.Relation.Binary.Equality.Cast
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; trans; sym; cong; subst; module ≡-Reasoning)
+
+private variable
+  a : Level
+  A : Set a
+  l m n o : ℕ
+  xs ys zs ws : Vec A n
+
+
+-- To see example usages of this library, scroll to the combinators
+-- section.
+
+
+------------------------------------------------------------------------
+-- Motivation
+--
+-- The `cast` function is the computational variant of `subst` for
+-- vectors. Since `cast` computes under vector constructors, it
+-- enables reasoning about vectors with non-definitionally equal indices
+-- by induction. See, e.g., Jacques Carette's comment in issue #1668.
+-- <https://github.com/agda/agda-stdlib/pull/1668#issuecomment-1003449509>
+--
+-- Suppose we want to prove that ‘xs ++ [] ≡ xs’. Because `xs ++ []`
+-- has type `Vec A (n + 0)` while `xs` has type `Vec A n`, they cannot
+-- be directly related by homogeneous equality.
+-- To resolve the issue, `++-right-identity` uses `cast` to recast
+-- `xs ++ []` as a vector in `Vec A n`.
+--
+++-right-identity : ∀ .(eq : n + 0 ≡ n) (xs : Vec A n) → cast eq (xs ++ []) ≡ xs
+++-right-identity eq []       = refl
+++-right-identity eq (x ∷ xs) = cong (x ∷_) (++-right-identity (cong pred eq) xs)
+--
+-- When the input is `x ∷ xs`, because `cast eq (x ∷ _)` equals
+-- `x ∷ cast (cong pred eq) _`, the proof obligation
+--     cast eq (x ∷ xs ++ []) ≡ x ∷ xs
+-- simplifies to
+--     x :: cast (cong pred eq) (xs ++ []) ≡ x ∷ xs
+
+
+-- Although `cast` makes it possible to prove vector identities by ind-
+-- uction, the explicit type-casting nature poses a significant barrier
+-- to code reuse in larger proofs. For example, consider the identity
+-- ‘fromList (xs L.∷ʳ x) ≡ (fromList xs) ∷ʳ x’ where `L._∷ʳ_` is the
+-- snoc function of lists. We have
+--
+--     fromList (xs L.∷ʳ x)            : Vec A (L.length (xs L.∷ʳ x))
+--   =   {- by definition -}
+--     fromList (xs L.++ L.[ x ])      : Vec A (L.length (xs L.++ L.[ x ]))
+--   =   {- by fromList-++ -}
+--     fromList xs ++ fromList L.[ x ] : Vec A (L.length xs + L.length [ x ])
+--   =   {- by definition -}
+--     fromList xs ++ [ x ]            : Vec A (L.length xs + 1)
+--   =   {- by unfold-∷ʳ -}
+--     fromList xs ∷ʳ x                : Vec A (suc (L.length xs))
+-- where
+--     fromList-++ : cast _ (fromList (xs L.++ ys)) ≡ fromList xs ++ fromList ys
+--     unfold-∷ʳ   : cast _ (xs ∷ʳ x) ≡ xs ++ [ x ]
+--
+-- Although the identity itself is simple, the reasoning process changes
+-- the index in the type twice. Consequently, its Agda translation must
+-- insert two `cast`s in the proof. Moreover, the proof first has to
+-- rearrange (the Agda version of) the identity into one with two
+-- `cast`s, resulting in lots of boilerplate code as demonstrated by
+-- `example1a-fromList-∷ʳ`.
+example1a-fromList-∷ʳ : ∀ (x : A) xs .(eq : L.length (xs L.∷ʳ x) ≡ suc (L.length xs)) →
+                        cast eq (fromList (xs L.∷ʳ x)) ≡ fromList xs ∷ʳ x
+example1a-fromList-∷ʳ x xs eq = begin
+  cast eq (fromList (xs L.∷ʳ x))                   ≡⟨⟩
+  cast eq (fromList (xs L.++ L.[ x ]))             ≡˘⟨ cast-trans eq₁ eq₂ (fromList (xs L.++ L.[ x ])) ⟩
+  cast eq₂ (cast eq₁ (fromList (xs L.++ L.[ x ]))) ≡⟨ cong (cast eq₂) (fromList-++ xs) ⟩
+  cast eq₂ (fromList xs ++ [ x ])                  ≡⟨ ≈-sym (unfold-∷ʳ (sym eq₂) x (fromList xs)) ⟩
+  fromList xs ∷ʳ x                                 ∎
+  where
+  open ≡-Reasoning
+  eq₁ = Lₚ.length-++ xs {L.[ x ]}
+  eq₂ = +-comm (L.length xs) 1
+
+-- The `cast`s are irrelevant to core of the proof. At the same time,
+-- they can be inferred from the lemmas used during the reasoning steps
+-- (e.g. `fromList-++` and `unfold-∷ʳ`). To eliminate the boilerplate,
+-- this library provides a set of equational reasoning combinators for
+-- equality of the form `cast eq xs ≡ ys`.
+example1b-fromList-∷ʳ : ∀ (x : A) xs .(eq : L.length (xs L.∷ʳ x) ≡ suc (L.length xs)) →
+                        cast eq (fromList (xs L.∷ʳ x)) ≡ fromList xs ∷ʳ x
+example1b-fromList-∷ʳ x xs eq = begin
+  fromList (xs L.∷ʳ x)       ≈⟨⟩
+  fromList (xs L.++ L.[ x ]) ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ [ x ]       ≈˘⟨ unfold-∷ʳ (+-comm 1 (L.length xs)) x (fromList xs) ⟩
+  fromList xs ∷ʳ x           ∎
+  where open CastReasoning
+
+
+------------------------------------------------------------------------
+-- Combinators
+--
+-- Let `xs ≈[ m≡n ] ys` denote `cast m≡n xs ≡ ys`. We have reflexivity,
+-- symmetry and transitivity:
+--     ≈-reflexive : xs ≈[ refl ] xs
+--     ≈-sym       : xs ≈[ m≡n ] ys → ys ≈[ sym m≡n ] xs
+--     ≈-trans     : xs ≈[ m≡n ] ys → ys ≈[ n≡o ] zs → xs ≈[ trans m≡n n≡o ] zs
+-- Accordingly, `_≈[_]_` admits the standard set of equational reasoning
+-- combinators. Suppose `≈-eqn : xs ≈[ m≡n ] ys`,
+--     xs ≈⟨ ≈-eqn  ⟩   -- `_≈⟨_⟩_` takes a `_≈[_]_` step, adjusting
+--     ys               -- the index at the same time
+--
+--     ys ≈˘⟨ ≈-eqn ⟩   -- `_≈˘⟨_⟩_` takes a symmetric `_≈[_]_` step
+--     xs
+example2a : ∀ .(eq : suc m + n ≡ m + suc n) (xs : Vec A m) a ys →
+            cast eq ((reverse xs ∷ʳ a) ++ ys) ≡ reverse xs ++ (a ∷ ys)
+example2a eq xs a ys = begin
+  (reverse xs ∷ʳ a) ++ ys ≈⟨ ∷ʳ-++ eq a (reverse xs) ⟩ -- index: suc m + n
+  reverse xs ++ (a ∷ ys)  ∎                            -- index: m + suc n
+  where open CastReasoning
+
+-- To interoperate with `_≡_`, this library provides `_≂⟨_⟩_` (\-~) for
+-- taking a `_≡_` step during equational reasoning.
+-- Let `≡-eqn : xs ≡ ys`, then
+--     xs ≂⟨ ≡-eqn  ⟩    -- Takes a `_≡_` step; no change to the index
+--     ys
+--
+--     ys ≂˘⟨ ≡-eqn ⟩    -- Takes a symmetric `_≡_` step
+--     xs
+-- Equivalently, `≈-reflexive` injects `_≡_` into `_≈[_]_`. That is,
+-- `xs ≂⟨ ≡-eqn ⟩ ys` is the same as `xs ≈⟨ ≈-reflexive ≡-eqn ⟩ ys`.
+-- Extending `example2a`, we have:
+example2b : ∀ .(eq : suc m + n ≡ m + suc n) (xs : Vec A m) a ys →
+            cast eq ((a ∷ xs) ʳ++ ys) ≡ xs ʳ++ (a ∷ ys)
+example2b eq xs a ys = begin
+  (a ∷ xs) ʳ++ ys         ≂⟨ unfold-ʳ++ (a ∷ xs) ys ⟩          -- index: suc m + n
+  reverse (a ∷ xs) ++ ys  ≂⟨ cong (_++ ys) (reverse-∷ a xs) ⟩  -- index: suc m + n
+  (reverse xs ∷ʳ a) ++ ys ≈⟨ ∷ʳ-++ eq a (reverse xs) ⟩         -- index: suc m + n
+  reverse xs ++ (a ∷ ys)  ≂˘⟨ unfold-ʳ++ xs (a ∷ ys) ⟩         -- index: m + suc n
+  xs ʳ++ (a ∷ ys)         ∎                                    -- index: m + suc n
+  where open CastReasoning
+
+-- Oftentimes index-changing identities apply to only part of the proof
+-- term. When reasoning about `_≡_`, `cong` shifts the focus to the
+-- subterm of interest. In this library, `≈-cong` does a similar job.
+-- Suppose `f : A → B`, `xs : B`, `ys zs : A`, `ys≈zs : ys ≈[ _ ] zs`
+-- and `xs≈f⟨c·ys⟩ : xs ≈[ _ ] f (cast _ ys)`, we have
+--     xs ≈⟨ ≈-cong f xs≈f⟨c·ys⟩ ys≈zs ⟩
+--     f zs
+-- The reason for having the extra argument `xs≈f⟨c·ys⟩` is to expose
+-- `cast` in the subterm in order to apply the step `ys≈zs`. When using
+-- ordinary `cong` the proof has to explicitly push `cast` inside:
+--     xs            ≈⟨ xs≈f⟨c·ys⟩ ⟩
+--     f (cast _ ys) ≂⟨ cong f ys≈zs ⟩
+--     f zs
+-- Note. Technically, `A` and `B` should be vectors of different length
+-- and that `ys`, `zs` are vectors of non-definitionally equal index.
+example3a-fromList-++-++ : {xs ys zs : List A} →
+                           .(eq : L.length (xs L.++ ys L.++ zs) ≡
+                                  L.length xs + (L.length ys + L.length zs)) →
+                           cast eq (fromList (xs L.++ ys L.++ zs)) ≡
+                                   fromList xs ++ fromList ys ++ fromList zs
+example3a-fromList-++-++ {xs = xs} {ys} {zs} eq = begin
+  fromList (xs L.++ ys L.++ zs)             ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ fromList (ys L.++ zs)      ≈⟨ ≈-cong (fromList xs ++_) (cast-++ʳ (Lₚ.length-++ ys) (fromList xs))
+                                                      (fromList-++ ys) ⟩
+  fromList xs ++ fromList ys ++ fromList zs ∎
+  where open CastReasoning
+
+-- As an alternative, one can manually apply `cast-++ʳ` to expose `cast`
+-- in the subterm. However, this unavoidably duplicates the proof term.
+example3b-fromList-++-++′ : {xs ys zs : List A} →
+                            .(eq : L.length (xs L.++ ys L.++ zs) ≡
+                                   L.length xs + (L.length ys + L.length zs)) →
+                            cast eq (fromList (xs L.++ ys L.++ zs)) ≡
+                                    fromList xs ++ fromList ys ++ fromList zs
+example3b-fromList-++-++′ {xs = xs} {ys} {zs} eq = begin
+  fromList (xs L.++ ys L.++ zs)                 ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ fromList (ys L.++ zs)          ≈⟨ cast-++ʳ (Lₚ.length-++ ys) (fromList xs) ⟩
+  fromList xs ++ cast _ (fromList (ys L.++ zs)) ≂⟨ cong (fromList xs ++_) (fromList-++ ys) ⟩
+  fromList xs ++ fromList ys ++ fromList zs     ∎
+  where open CastReasoning
+
+-- `≈-cong` can be chained together much like how `cong` can be nested.
+-- In this example, `unfold-∷ʳ` is applied to the term `xs ++ [ a ]`
+-- in `(_++ ys)` inside of `reverse`. Thus the proof employs two
+-- `≈-cong`.
+example4-cong² : ∀ .(eq : (m + 1) + n ≡ n + suc m) a (xs : Vec A m) ys →
+          cast eq (reverse ((xs ++ [ a ]) ++ ys)) ≡ ys ʳ++ reverse (xs ∷ʳ a)
+example4-cong² {m = m} {n} eq a xs ys = begin
+  reverse ((xs ++ [ a ]) ++ ys)   ≈˘⟨ ≈-cong reverse (cast-reverse (cong (_+ n) (+-comm 1 m)) ((xs ∷ʳ a) ++ ys))
+                                             (≈-cong (_++ ys) (cast-++ˡ (+-comm 1 m) (xs ∷ʳ a))
+                                                     (unfold-∷ʳ _ a xs)) ⟩
+  reverse ((xs ∷ʳ a) ++ ys)       ≈⟨ reverse-++ (+-comm (suc m) n) (xs ∷ʳ a) ys ⟩
+  reverse ys ++ reverse (xs ∷ʳ a) ≂˘⟨ unfold-ʳ++ ys (reverse (xs ∷ʳ a)) ⟩
+  ys ʳ++ reverse (xs ∷ʳ a)        ∎
+  where open CastReasoning
+
+------------------------------------------------------------------------
+-- Interoperation between `_≈[_]_` and `_≡_`
+--
+-- This library is designed to interoperate with `_≡_`. Examples in the
+-- combinators section showed how to apply `_≂⟨_⟩_` to take an `_≡_`
+-- step during equational reasoning about `_≈[_]_`. Recall that
+-- `xs ≈[ m≡n ] ys` is a shorthand for `cast m≡n xs ≡ ys`, the
+-- combinator is essentially the composition of `_≡_` on the left-hand
+-- side of `_≈[_]_`. Dually, the combinator `_≃⟨_⟩_` composes `_≡_` on
+-- the right-hand side of `_≈[_]_`. Thus `_≃⟨_⟩_` intuitively ends the
+-- reasoning system of `_≈[_]_` and switches back to the reasoning
+-- system of `_≡_`.
+example5-fromList-++-++′ : {xs ys zs : List A} →
+                           .(eq : L.length (xs L.++ ys L.++ zs) ≡
+                                  L.length xs + (L.length ys + L.length zs)) →
+                           cast eq (fromList (xs L.++ ys L.++ zs)) ≡
+                                   fromList xs ++ fromList ys ++ fromList zs
+example5-fromList-++-++′ {xs = xs} {ys} {zs} eq = begin
+  fromList (xs L.++ ys L.++ zs)                 ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ fromList (ys L.++ zs)          ≃⟨ cast-++ʳ (Lₚ.length-++ ys) (fromList xs) ⟩
+  fromList xs ++ cast _ (fromList (ys L.++ zs)) ≡⟨ cong (fromList xs ++_) (fromList-++ ys) ⟩
+  fromList xs ++ fromList ys ++ fromList zs     ≡-∎
+  where open CastReasoning
+
+-- Of course, it is possible to start with the reasoning system of `_≡_`
+-- and then switch to the reasoning system of `_≈[_]_`.
+example6a-reverse-∷ʳ : ∀ x (xs : Vec A n) → reverse (xs ∷ʳ x) ≡ x ∷ reverse xs
+example6a-reverse-∷ʳ {n = n} x xs = begin-≡
+  reverse (xs ∷ʳ x)     ≡˘⟨ ≈-reflexive refl ⟩
+  reverse (xs ∷ʳ x)     ≈⟨ ≈-cong reverse (cast-reverse _ _) (unfold-∷ʳ (+-comm 1 n) x xs) ⟩
+  reverse (xs ++ [ x ]) ≈⟨ reverse-++ (+-comm n 1) xs [ x ] ⟩
+  x ∷ reverse xs        ∎
+  where open CastReasoning
+
+example6b-reverse-∷ʳ-by-induction : ∀ x (xs : Vec A n) → reverse (xs ∷ʳ x) ≡ x ∷ reverse xs
+example6b-reverse-∷ʳ-by-induction x []       = refl
+example6b-reverse-∷ʳ-by-induction x (y ∷ xs) = begin
+  reverse (y ∷ (xs ∷ʳ x)) ≡⟨ reverse-∷ y (xs ∷ʳ x) ⟩
+  reverse (xs ∷ʳ x) ∷ʳ y  ≡⟨ cong (_∷ʳ y) (example6b-reverse-∷ʳ-by-induction x xs) ⟩
+  (x ∷ reverse xs) ∷ʳ y   ≡⟨⟩
+  x ∷ (reverse xs ∷ʳ y)   ≡˘⟨ cong (x ∷_) (reverse-∷ y xs) ⟩
+  x ∷ reverse (y ∷ xs)    ∎
+  where open ≡-Reasoning
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index e142db9d00..be668897cb 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -91,6 +91,8 @@ xs [ i ]≔ y = xs [ i ]%= const y
 ------------------------------------------------------------------------
 -- Operations for transforming vectors
 
+-- See README.Data.Vec.Relation.Binary.Equality.Cast for the reasoning
+-- system of `cast`-ed equality.
 cast : .(eq : m ≡ n) → Vec A m → Vec A n
 cast {n = zero}  eq []       = []
 cast {n = suc _} eq (x ∷ xs) = x ∷ cast (cong pred eq) xs
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 1357e4e441..c48abc6f57 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -22,6 +22,8 @@ open import Data.Product.Base as Prod
 open import Data.Sum.Base using ([_,_]′)
 open import Data.Sum.Properties using ([,]-map)
 open import Data.Vec.Base
+open import Data.Vec.Relation.Binary.Equality.Cast as VecCast
+  using (_≈[_]_; ≈-sym; module CastReasoning)
 open import Function.Base
 open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
@@ -33,8 +35,6 @@ open import Relation.Nullary.Decidable.Core using (Dec; does; yes; no; _×-dec_;
 open import Relation.Nullary.Negation.Core using (contradiction)
 import Data.Nat.GeneralisedArithmetic as ℕ
 
-open ≡-Reasoning
-
 private
   variable
     a b c d p : Level
@@ -365,9 +365,8 @@ lookup∘update′ {i = i} {j} i≢j xs y = lookup∘updateAt′ i j i≢j xs
 ------------------------------------------------------------------------
 -- cast
 
-cast-is-id : .(eq : m ≡ m) (xs : Vec A m) → cast eq xs ≡ xs
-cast-is-id eq []       = refl
-cast-is-id eq (x ∷ xs) = cong (x ∷_) (cast-is-id (suc-injective eq) xs)
+open VecCast public
+  using (cast-is-id; cast-trans)
 
 subst-is-cast : (eq : m ≡ n) (xs : Vec A m) → subst (Vec A) eq xs ≡ cast eq xs
 subst-is-cast refl xs = sym (cast-is-id refl xs)
@@ -379,12 +378,6 @@ cast-sym eq {xs = x ∷ xs} {ys = y ∷ ys} xxs[eq]≡yys =
   let x≡y , xs[eq]≡ys = ∷-injective xxs[eq]≡yys
   in cong₂ _∷_ (sym x≡y) (cast-sym (suc-injective eq) xs[eq]≡ys)
 
-cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (xs : Vec A m) →
-             cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
-cast-trans {m = zero}  {n = zero}  {o = zero}  eq₁ eq₂ [] = refl
-cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (x ∷ xs) =
-  cong (x ∷_) (cast-trans (suc-injective eq₁) (suc-injective eq₂) xs)
-
 ------------------------------------------------------------------------
 -- map
 
@@ -477,6 +470,16 @@ toList-map f (x ∷ xs) = cong (f x List.∷_) (toList-map f xs)
 ++-identityʳ eq []       = refl
 ++-identityʳ eq (x ∷ xs) = cong (x ∷_) (++-identityʳ (cong pred eq) xs)
 
+cast-++ˡ : ∀ .(eq : m ≡ o) (xs : Vec A m) {ys : Vec A n} →
+           cast (cong (_+ n) eq) (xs ++ ys) ≡ cast eq xs ++ ys
+cast-++ˡ {o = zero}  eq []       {ys} = cast-is-id refl (cast eq [] ++ ys)
+cast-++ˡ {o = suc o} eq (x ∷ xs) {ys} = cong (x ∷_) (cast-++ˡ (cong pred eq) xs)
+
+cast-++ʳ : ∀ .(eq : n ≡ o) (xs : Vec A m) {ys : Vec A n} →
+           cast (cong (m +_) eq) (xs ++ ys) ≡ xs ++ cast eq ys
+cast-++ʳ {m = zero}  eq []       {ys} = refl
+cast-++ʳ {m = suc m} eq (x ∷ xs) {ys} = cong (x ∷_) (cast-++ʳ eq xs)
+
 lookup-++-< : ∀ (xs : Vec A m) (ys : Vec A n) →
               ∀ i (i<m : toℕ i < m) →
               lookup (xs ++ ys) i  ≡ lookup xs (Fin.fromℕ< i<m)
@@ -753,6 +756,7 @@ zipWith-is-⊛ f (x ∷ xs) (y ∷ ys) = cong (_ ∷_) (zipWith-is-⊛ f xs ys)
   diagonal (map (flip map xs) fs)                  ≡⟨⟩
   diagonal (map (tail ∘ flip map (x ∷ xs)) fs)     ≡⟨ cong diagonal (map-∘ _ _ _) ⟩
   diagonal (map tail (map (flip map (x ∷ xs)) fs)) ∎
+  where open ≡-Reasoning
 
 ------------------------------------------------------------------------
 -- _⊛*_
@@ -780,6 +784,7 @@ foldl-universal B f e h base step (x ∷ xs) = begin
   h (B ∘ suc) f (f e x) xs     ≡⟨ foldl-universal _ f (f e x) h base step xs ⟩
   foldl (B ∘ suc) f (f e x) xs ≡⟨⟩
   foldl B f e (x ∷ xs)         ∎
+  where open ≡-Reasoning
 
 foldl-fusion : ∀ {B : ℕ → Set b} {C : ℕ → Set c}
                (h : ∀ {n} → B n → C n) →
@@ -792,6 +797,7 @@ foldl-fusion h {f} {d} {g} {e} base fuse []       = base
 foldl-fusion h {f} {d} {g} {e} base fuse (x ∷ xs) =
   foldl-fusion h eq fuse xs
   where
+    open ≡-Reasoning
     eq : h (f d x) ≡ g e x
     eq = begin
       h (f d x) ≡⟨ fuse d x ⟩
@@ -819,6 +825,7 @@ module _ (B : ℕ → Set b) (f : FoldrOp A B) {e : B zero} where
     h (x ∷ xs)           ≡⟨ step x xs ⟩
     f x (h xs)           ≡⟨ cong (f x) (foldr-universal h base step xs) ⟩
     f x (foldr B f e xs) ∎
+    where open ≡-Reasoning
 
   foldr-[] : foldr B f e [] ≡ e
   foldr-[] = refl
@@ -911,6 +918,11 @@ cast-∷ʳ {m = suc m} eq x (y ∷ xs) = cong (y ∷_) (cast-∷ʳ (cong pred eq
 ++-∷ʳ {m = zero}  eq z []       (y ∷ ys) = cong (y ∷_) (++-∷ʳ refl z [] ys)
 ++-∷ʳ {m = suc m} eq z (x ∷ xs) ys       = cong (x ∷_) (++-∷ʳ (cong pred eq) z xs ys)
 
+∷ʳ-++ : ∀ .(eq : (suc n) + m ≡ n + suc m) a (xs : Vec A n) {ys} →
+        cast eq ((xs ∷ʳ a) ++ ys) ≡ xs ++ (a ∷ ys)
+∷ʳ-++ eq a []       {ys} = cong (a ∷_) (cast-is-id (cong pred eq) ys)
+∷ʳ-++ eq a (x ∷ xs) {ys} = cong (x ∷_) (∷ʳ-++ (cong pred eq) a xs)
+
 ------------------------------------------------------------------------
 -- reverse
 
@@ -930,6 +942,7 @@ foldl-reverse B f {e} (x ∷ xs) = begin
   f (foldl B f e (reverse xs)) x ≡⟨ cong (flip f x) (foldl-reverse B f xs) ⟩
   f (foldr B (flip f) e xs) x    ≡⟨⟩
   foldr B (flip f) e (x ∷ xs)    ∎
+  where open ≡-Reasoning
 
 -- foldr after a reverse is a foldl
 
@@ -944,12 +957,14 @@ reverse-involutive xs = begin
   reverse (reverse xs)    ≡⟨  foldl-reverse (Vec _) (flip _∷_) xs ⟩
   foldr (Vec _) _∷_ [] xs ≡˘⟨ id-is-foldr xs ⟩
   xs                      ∎
+  where open ≡-Reasoning
 
 reverse-reverse : reverse xs ≡ ys → reverse ys ≡ xs
 reverse-reverse {xs = xs} {ys} eq =  begin
   reverse ys           ≡˘⟨ cong reverse eq ⟩
   reverse (reverse xs) ≡⟨  reverse-involutive xs ⟩
   xs                   ∎
+  where open ≡-Reasoning
 
 -- reverse is injective.
 
@@ -958,6 +973,7 @@ reverse-injective {xs = xs} {ys} eq = begin
   xs                   ≡˘⟨ reverse-reverse eq ⟩
   reverse (reverse ys) ≡⟨  reverse-involutive ys ⟩
   ys                   ∎
+  where open ≡-Reasoning
 
 -- init and last of reverse
 
@@ -966,12 +982,14 @@ init-reverse (x ∷ xs) = begin
   init (reverse (x ∷ xs))   ≡⟨ cong init (reverse-∷ x xs) ⟩
   init (reverse xs ∷ʳ x)    ≡⟨ init-∷ʳ x (reverse xs) ⟩
   reverse xs                ∎
+  where open ≡-Reasoning
 
 last-reverse : last ∘ reverse ≗ head {A = A} {n = n}
 last-reverse (x ∷ xs) = begin
   last (reverse (x ∷ xs))   ≡⟨ cong last (reverse-∷ x xs) ⟩
   last (reverse xs ∷ʳ x)    ≡⟨ last-∷ʳ x (reverse xs) ⟩
   x                         ∎
+  where open ≡-Reasoning
 
 -- map and reverse
 
@@ -985,39 +1003,32 @@ map-reverse f (x ∷ xs) = begin
   reverse (map f xs) ∷ʳ f x ≡˘⟨ reverse-∷ (f x) (map f xs) ⟩
   reverse (f x ∷ map f xs)  ≡⟨⟩
   reverse (map f (x ∷ xs))  ∎
+  where open ≡-Reasoning
 
 -- append and reverse
 
 reverse-++ : ∀ .(eq : m + n ≡ n + m) (xs : Vec A m) (ys : Vec A n) →
              cast eq (reverse (xs ++ ys)) ≡ reverse ys ++ reverse xs
-reverse-++ {m = zero}  {n = n} eq []       ys = begin
-  cast _ (reverse ys)                ≡˘⟨ cong (cast eq) (++-identityʳ (sym eq) (reverse ys)) ⟩
-  cast _ (cast _ (reverse ys ++ [])) ≡⟨ cast-trans (sym eq) eq (reverse ys ++ []) ⟩
-  cast _ (reverse ys ++ [])          ≡⟨ cast-is-id (trans (sym eq) eq) (reverse ys ++ []) ⟩
-  reverse ys ++ []                   ≡⟨⟩
-  reverse ys ++ reverse []           ∎
+reverse-++ {m = zero}  {n = n} eq []       ys = ≈-sym (++-identityʳ (sym eq) (reverse ys))
 reverse-++ {m = suc m} {n = n} eq (x ∷ xs) ys = begin
-  cast eq (reverse (x ∷ xs ++ ys))                ≡⟨ cong (cast eq) (reverse-∷ x (xs ++ ys)) ⟩
-  cast eq (reverse (xs ++ ys) ∷ʳ x)               ≡˘⟨ cast-trans eq₂ eq₁ (reverse (xs ++ ys) ∷ʳ x) ⟩
-  (cast eq₁ ∘ cast eq₂) (reverse (xs ++ ys) ∷ʳ x) ≡⟨ cong (cast eq₁) (cast-∷ʳ _ x (reverse (xs ++ ys))) ⟩
-  cast eq₁ ((cast eq₃ (reverse (xs ++ ys))) ∷ʳ x) ≡⟨ cong (cast eq₁) (cong (_∷ʳ x) (reverse-++ _ xs ys)) ⟩
-  cast eq₁ ((reverse ys ++ reverse xs)      ∷ʳ x) ≡⟨ ++-∷ʳ _ x (reverse ys) (reverse xs) ⟩
-  reverse ys ++ (reverse xs ∷ʳ x)                 ≡˘⟨ cong (reverse ys ++_) (reverse-∷ x xs) ⟩
-  reverse ys ++ (reverse (x ∷ xs))                ∎
-  where
-  eq₁ = sym (+-suc n m)
-  eq₂ = cong suc (+-comm m n)
-  eq₃ = cong pred eq₂
+  reverse (x ∷ xs ++ ys)              ≂⟨ reverse-∷ x (xs ++ ys) ⟩
+  reverse (xs ++ ys) ∷ʳ x             ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (+-comm m n)) x (reverse (xs ++ ys)))
+                                                (reverse-++ _ xs ys) ⟩
+  (reverse ys ++ reverse xs) ∷ʳ x     ≈⟨ ++-∷ʳ (sym (+-suc n m)) x (reverse ys) (reverse xs) ⟩
+  reverse ys ++ (reverse xs ∷ʳ x)     ≂˘⟨ cong (reverse ys ++_) (reverse-∷ x xs) ⟩
+  reverse ys ++ (reverse (x ∷ xs))    ∎
+  where open CastReasoning
 
 cast-reverse : ∀ .(eq : m ≡ n) → cast eq ∘ reverse {A = A} {n = m} ≗ reverse ∘ cast eq
 cast-reverse {n = zero}  eq []       = refl
 cast-reverse {n = suc n} eq (x ∷ xs) = begin
-  cast eq (reverse (x ∷ xs))              ≡⟨ cong (cast eq) (reverse-∷ x xs) ⟩
-  cast eq (reverse xs ∷ʳ x)               ≡⟨ cast-∷ʳ eq x (reverse xs) ⟩
-  (cast (cong pred eq) (reverse xs)) ∷ʳ x ≡⟨ cong (_∷ʳ x) (cast-reverse (cong pred eq) xs) ⟩
-  (reverse (cast (cong pred eq) xs)) ∷ʳ x ≡˘⟨ reverse-∷ x (cast (cong pred eq) xs) ⟩
-  reverse (x ∷ cast (cong pred eq) xs)    ≡⟨⟩
-  reverse (cast eq (x ∷ xs))              ∎
+  reverse (x ∷ xs)           ≂⟨ reverse-∷ x xs ⟩
+  reverse xs ∷ʳ x            ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ eq x (reverse xs))
+                                       (cast-reverse (cong pred eq) xs) ⟩
+  reverse (cast _ xs) ∷ʳ x   ≂˘⟨ reverse-∷ x (cast (cong pred eq) xs) ⟩
+  reverse (x ∷ cast _ xs)    ≈⟨⟩
+  reverse (cast eq (x ∷ xs)) ∎
+  where open CastReasoning
 
 ------------------------------------------------------------------------
 -- _ʳ++_
@@ -1045,6 +1056,42 @@ map-ʳ++ {ys = ys} f xs = begin
   map f (reverse xs) ++ map f ys ≡⟨  cong (_++ map f ys) (map-reverse f xs) ⟩
   reverse (map f xs) ++ map f ys ≡˘⟨ unfold-ʳ++ (map f xs) (map f ys) ⟩
   map f xs ʳ++ map f ys          ∎
+  where open ≡-Reasoning
+
+∷-ʳ++ : ∀ .(eq : (suc m) + n ≡ m + suc n) a (xs : Vec A m) {ys} →
+        cast eq ((a ∷ xs) ʳ++ ys) ≡ xs ʳ++ (a ∷ ys)
+∷-ʳ++ eq a xs {ys} = begin
+  (a ∷ xs) ʳ++ ys         ≂⟨ unfold-ʳ++ (a ∷ xs) ys ⟩
+  reverse (a ∷ xs) ++ ys  ≂⟨ cong (_++ ys) (reverse-∷ a xs) ⟩
+  (reverse xs ∷ʳ a) ++ ys ≈⟨ ∷ʳ-++ eq a (reverse xs) ⟩
+  reverse xs ++ (a ∷ ys)  ≂˘⟨ unfold-ʳ++ xs (a ∷ ys) ⟩
+  xs ʳ++ (a ∷ ys)         ∎
+  where open CastReasoning
+
+++-ʳ++ : ∀ .(eq : m + n + o ≡ n + (m + o)) (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →
+         cast eq ((xs ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ʳ++ zs)
+++-ʳ++ {m = m} {n} {o} eq xs {ys} {zs} = begin
+  ((xs ++ ys) ʳ++ zs)              ≂⟨ unfold-ʳ++ (xs ++ ys) zs ⟩
+  reverse (xs ++ ys) ++ zs         ≈⟨ ≈-cong (_++ zs) (cast-++ˡ (+-comm m n) (reverse (xs ++ ys)))
+                                             (reverse-++ (+-comm m n) xs ys) ⟩
+  (reverse ys ++ reverse xs) ++ zs ≈⟨ ++-assoc (trans (cong (_+ o) (+-comm n m)) eq) (reverse ys) (reverse xs) zs ⟩
+  reverse ys ++ (reverse xs ++ zs) ≂˘⟨ cong (reverse ys ++_) (unfold-ʳ++ xs zs) ⟩
+  reverse ys ++ (xs ʳ++ zs)        ≂˘⟨ unfold-ʳ++ ys (xs ʳ++ zs) ⟩
+  ys ʳ++ (xs ʳ++ zs)               ∎
+  where open CastReasoning
+
+ʳ++-ʳ++ : ∀ .(eq : (m + n) + o ≡ n + (m + o)) (xs : Vec A m) {ys : Vec A n} {zs} →
+          cast eq ((xs ʳ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ++ zs)
+ʳ++-ʳ++ {m = m} {n} {o} eq xs {ys} {zs} = begin
+  (xs ʳ++ ys) ʳ++ zs                         ≂⟨ cong (_ʳ++ zs) (unfold-ʳ++ xs ys) ⟩
+  (reverse xs ++ ys) ʳ++ zs                  ≂⟨ unfold-ʳ++ (reverse xs ++ ys) zs ⟩
+  reverse (reverse xs ++ ys) ++ zs           ≈⟨ ≈-cong (_++ zs) (cast-++ˡ (+-comm m n) (reverse (reverse xs ++ ys)))
+                                                       (reverse-++ (+-comm m n) (reverse xs) ys) ⟩
+  (reverse ys ++ reverse (reverse xs)) ++ zs ≂⟨ cong ((_++ zs) ∘ (reverse ys ++_)) (reverse-involutive xs) ⟩
+  (reverse ys ++ xs) ++ zs                   ≈⟨ ++-assoc (+-assoc n m o) (reverse ys) xs zs ⟩
+  reverse ys ++ (xs ++ zs)                   ≂˘⟨ unfold-ʳ++ ys (xs ++ zs) ⟩
+  ys ʳ++ (xs ++ zs)                          ∎
+  where open CastReasoning
 
 ------------------------------------------------------------------------
 -- sum
@@ -1055,6 +1102,7 @@ sum-++ {ys = ys} (x ∷ xs) = begin
   x + sum (xs ++ ys)     ≡⟨  cong (x +_) (sum-++ xs) ⟩
   x + (sum xs + sum ys)  ≡˘⟨ +-assoc x (sum xs) (sum ys) ⟩
   sum (x ∷ xs) + sum ys  ∎
+  where open ≡-Reasoning
 
 ------------------------------------------------------------------------
 -- replicate
@@ -1076,6 +1124,7 @@ transpose-replicate {n = suc n} xs = begin
   (replicate _∷_ ⊛ xs ⊛ transpose (replicate xs)) ≡⟨ cong₂ _⊛_ (sym (map-is-⊛ _∷_ xs)) (transpose-replicate xs) ⟩
   (map _∷_ xs ⊛ map replicate xs)                 ≡⟨ map-⊛ _∷_ replicate xs ⟩
   map replicate xs                                ∎
+  where open ≡-Reasoning
 
 zipWith-replicate : ∀ (_⊕_ : A → B → C) (x : A) (y : B) →
                     zipWith {n = n} _⊕_ (replicate x) (replicate y) ≡ replicate (x ⊕ y)
@@ -1163,6 +1212,7 @@ map-lookup-allFin {n = n} xs = begin
   map (lookup xs) (allFin n) ≡˘⟨ tabulate-∘ (lookup xs) id ⟩
   tabulate (lookup xs)       ≡⟨ tabulate∘lookup xs ⟩
   xs                         ∎
+  where open ≡-Reasoning
 
 ------------------------------------------------------------------------
 -- count
@@ -1248,6 +1298,7 @@ cast-fromList {xs = x List.∷ xs} {ys = y List.∷ ys} eq =
   x ∷ cast (cong (pred ∘ List.length) eq) (fromList xs) ≡⟨ cong (_ ∷_) (cast-fromList xs≡ys) ⟩
   x ∷ fromList ys                                       ≡⟨ cong (_∷ _) x≡y ⟩
   y ∷ fromList ys                                       ∎
+  where open ≡-Reasoning
 
 fromList-map : ∀ (f : A → B) (xs : List A) →
                cast (Listₚ.length-map f xs) (fromList (List.map f xs)) ≡ map f (fromList xs)
@@ -1259,6 +1310,19 @@ fromList-++ : ∀ (xs : List A) {ys : List A} →
 fromList-++ List.[]       {ys} = cast-is-id refl (fromList ys)
 fromList-++ (x List.∷ xs) {ys} = cong (x ∷_) (fromList-++ xs)
 
+fromList-reverse : (xs : List A) → cast (Listₚ.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)
+fromList-reverse List.[] = refl
+fromList-reverse (x List.∷ xs) = begin
+  fromList (List.reverse (x List.∷ xs))         ≈⟨ cast-fromList (Listₚ.ʳ++-defn xs) ⟩
+  fromList (List.reverse xs List.++ List.[ x ]) ≈⟨ fromList-++ (List.reverse xs) ⟩
+  fromList (List.reverse xs) ++ [ x ]           ≈˘⟨ unfold-∷ʳ (+-comm 1 _) x (fromList (List.reverse xs)) ⟩
+  fromList (List.reverse xs) ∷ʳ x               ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (Listₚ.length-reverse xs)) _ _)
+                                                          (fromList-reverse xs) ⟩
+  reverse (fromList xs) ∷ʳ x                    ≂˘⟨ reverse-∷ x (fromList xs) ⟩
+  reverse (x ∷ fromList xs)                     ≈⟨⟩
+  reverse (fromList (x List.∷ xs))              ∎
+  where open CastReasoning
+
 ------------------------------------------------------------------------
 -- DEPRECATED NAMES
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Relation/Binary/Equality/Cast.agda b/src/Data/Vec/Relation/Binary/Equality/Cast.agda
new file mode 100644
index 0000000000..adca0a42f2
--- /dev/null
+++ b/src/Data/Vec/Relation/Binary/Equality/Cast.agda
@@ -0,0 +1,129 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- An equational reasoning library for propositional equality over
+-- vectors of different indices using cast.
+--
+-- See README.Data.Vec.Relation.Binary.Equality.Cast for
+-- documentation and examples.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Relation.Binary.Equality.Cast {a} {A : Set a} where
+
+open import Data.Nat.Base using (ℕ; zero; suc)
+open import Data.Nat.Properties using (suc-injective)
+open import Data.Vec.Base
+open import Relation.Binary.Core using (REL; _⇒_)
+open import Relation.Binary.Definitions using (Sym; Trans)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; trans; sym; cong; module ≡-Reasoning)
+
+private
+  variable
+    l m n o : ℕ
+    xs ys zs : Vec A n
+
+
+cast-is-id : .(eq : m ≡ m) (xs : Vec A m) → cast eq xs ≡ xs
+cast-is-id eq []       = refl
+cast-is-id eq (x ∷ xs) = cong (x ∷_) (cast-is-id (suc-injective eq) xs)
+
+cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (xs : Vec A m) →
+             cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
+cast-trans {m = zero}  {n = zero}  {o = zero}  eq₁ eq₂ [] = refl
+cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (x ∷ xs) =
+  cong (x ∷_) (cast-trans (suc-injective eq₁) (suc-injective eq₂) xs)
+
+
+infix 3 _≈[_]_
+
+_≈[_]_ : ∀ {n m} → Vec A n → .(eq : n ≡ m) → Vec A m → Set a
+xs ≈[ eq ] ys = cast eq xs ≡ ys
+
+------------------------------------------------------------------------
+-- _≈[_]_ is ‘reflexive’, ‘symmetric’ and ‘transitive’
+
+≈-reflexive : ∀ {n} → _≡_ ⇒ (λ xs ys → _≈[_]_ {n} xs refl ys)
+≈-reflexive {x = x} eq = trans (cast-is-id refl x) eq
+
+≈-sym : .{m≡n : m ≡ n} → Sym _≈[ m≡n ]_ _≈[ sym m≡n ]_
+≈-sym {m≡n = m≡n} {xs} {ys} xs≈ys = begin
+  cast (sym m≡n) ys             ≡˘⟨ cong (cast (sym m≡n)) xs≈ys ⟩
+  cast (sym m≡n) (cast m≡n xs)  ≡⟨ cast-trans m≡n (sym m≡n) xs ⟩
+  cast (trans m≡n (sym m≡n)) xs ≡⟨ cast-is-id (trans m≡n (sym m≡n)) xs ⟩
+  xs                            ∎
+  where open ≡-Reasoning
+
+≈-trans : ∀ .{m≡n : m ≡ n} .{n≡o : n ≡ o} → Trans _≈[ m≡n ]_ _≈[ n≡o ]_ _≈[ trans m≡n n≡o ]_
+≈-trans {m≡n = m≡n} {n≡o} {xs} {ys} {zs} xs≈ys ys≈zs = begin
+  cast (trans m≡n n≡o) xs ≡˘⟨ cast-trans m≡n n≡o xs ⟩
+  cast n≡o (cast m≡n xs)  ≡⟨ cong (cast n≡o) xs≈ys ⟩
+  cast n≡o ys             ≡⟨ ys≈zs ⟩
+  zs                      ∎
+  where open ≡-Reasoning
+
+------------------------------------------------------------------------
+-- Reasoning combinators
+
+module CastReasoning where
+
+  open ≡-Reasoning public
+    renaming (begin_ to begin-≡_; _∎ to _≡-∎)
+
+  begin_ : ∀ .{m≡n : m ≡ n} {xs : Vec A m} {ys} → xs ≈[ m≡n ] ys → cast m≡n xs ≡ ys
+  begin xs≈ys = xs≈ys
+
+  _∎ : (xs : Vec A n) → cast refl xs ≡ xs
+  _∎ xs = ≈-reflexive refl
+
+  _≈⟨⟩_ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys} → xs ≈[ m≡n ] ys → xs ≈[ m≡n ] ys
+  xs ≈⟨⟩ xs≈ys = xs≈ys
+
+  -- composition of _≈[_]_
+  step-≈ : ∀ .{m≡n : m ≡ n}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →
+           ys ≈[ trans (sym m≡n) m≡o ] zs → xs ≈[ m≡n ] ys → xs ≈[ m≡o ] zs
+  step-≈ xs ys≈zs xs≈ys = ≈-trans xs≈ys ys≈zs
+
+  -- composition of the equality type on the right-hand side of _≈[_]_,
+  -- or escaping to ordinary _≡_
+  step-≃ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → xs ≈[ m≡n ] ys → xs ≈[ m≡n ] zs
+  step-≃ xs ys≡zs xs≈ys = ≈-trans xs≈ys (≈-reflexive ys≡zs)
+
+  -- composition of the equality type on the left-hand side of _≈[_]_
+  step-≂ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → xs ≡ ys → xs ≈[ m≡n ] zs
+  step-≂ xs ys≈zs xs≡ys = ≈-trans (≈-reflexive xs≡ys) ys≈zs
+
+  -- `cong` after a `_≈[_]_` step that exposes the `cast` to the `cong`
+  -- operation
+  ≈-cong : ∀ .{l≡o : l ≡ o} .{m≡n : m ≡ n} {xs : Vec A m} {ys zs} (f : Vec A o → Vec A n) →
+           xs ≈[ m≡n ] f (cast l≡o ys) → ys ≈[ l≡o ] zs → xs ≈[ m≡n ] f zs
+  ≈-cong f xs≈fys ys≈zs = trans xs≈fys (cong f ys≈zs)
+
+
+  -- symmetric version of each of the operator
+  step-≈˘ : ∀ .{n≡m : n ≡ m}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →
+           ys ≈[ trans n≡m m≡o ] zs → ys ≈[ n≡m ] xs → xs ≈[ m≡o ] zs
+  step-≈˘ xs ys≈zs ys≈xs = step-≈ xs ys≈zs (≈-sym ys≈xs)
+
+  step-≃˘ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → ys ≈[ sym m≡n ] xs → xs ≈[ m≡n ] zs
+  step-≃˘ xs ys≡zs ys≈xs = step-≃ xs ys≡zs (≈-sym ys≈xs)
+
+  step-≂˘ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → ys ≡ xs → xs ≈[ m≡n ] zs
+  step-≂˘ xs ys≈zs ys≡xs = step-≂ xs ys≈zs (sym ys≡xs)
+
+
+  ------------------------------------------------------------------------
+  -- convenient syntax for ‘equational’ reasoning
+
+  infix 1 begin_
+  infixr 2 step-≃ step-≂ step-≃˘ step-≂˘ step-≈ step-≈˘ _≈⟨⟩_ ≈-cong
+  infix 3 _∎
+
+  syntax step-≃  xs ys≡zs xs≈ys  = xs ≃⟨  xs≈ys ⟩ ys≡zs
+  syntax step-≃˘ xs ys≡zs xs≈ys  = xs ≃˘⟨ xs≈ys ⟩ ys≡zs
+  syntax step-≂  xs ys≈zs xs≡ys  = xs ≂⟨  xs≡ys ⟩ ys≈zs
+  syntax step-≂˘ xs ys≈zs ys≡xs  = xs ≂˘⟨ ys≡xs ⟩ ys≈zs
+  syntax step-≈  xs ys≈zs xs≈ys  = xs ≈⟨  xs≈ys ⟩ ys≈zs
+  syntax step-≈˘ xs ys≈zs ys≈xs  = xs ≈˘⟨ ys≈xs ⟩ ys≈zs

From fe803f3948472263c16729653b90d4a76b82194a Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 5 Oct 2023 10:49:46 +0100
Subject: [PATCH 145/316] Strict and non-strict transitive property names
 (#2089)

---
 CHANGELOG.md                                  |   8 +-
 src/Data/Integer/Properties.agda              |  16 +-
 .../Binary/Permutation/Setoid/Properties.agda |   2 +-
 src/Data/Nat/Coprimality.agda                 |   2 +-
 src/Data/Nat/DivMod.agda                      |  40 ++--
 src/Data/Nat/DivMod/Core.agda                 |   6 +-
 src/Data/Nat/Properties.agda                  | 226 ++++++++++--------
 src/Relation/Binary/Definitions.agda          |   6 +
 8 files changed, 168 insertions(+), 138 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d145333c44..2c9dd0e092 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1413,7 +1413,7 @@ Deprecated names
   m<n⇒m≤pred[n]  ↦  i<j⇒i≤pred[j]
   -1*n≡-n        ↦  -1*i≡-i
   m*n≡0⇒m≡0∨n≡0  ↦  i*j≡0⇒i≡0∨j≡0
-  ∣m*n∣≡∣m∣*∣n∣     ↦  ∣i*j∣≡∣i∣*∣j∣
+  ∣m*n∣≡∣m∣*∣n∣  ↦  ∣i*j∣≡∣i∣*∣j∣
   m≤m+n          ↦  i≤i+j
   n≤m+n          ↦  i≤j+i
   m-n≤m          ↦  i≤i-j
@@ -1521,6 +1521,9 @@ Deprecated names
   ≤-stepsˡ        ↦  m≤n⇒m≤o+n
   ≤-stepsʳ        ↦  m≤n⇒m≤n+o
   <-step          ↦  m<n⇒m<1+n
+
+  <-transʳ        ↦  ≤-<-trans
+  <-transˡ        ↦  <-≤-trans
   ```
 
 * In `Data.Rational.Unnormalised.Base`:
@@ -3261,6 +3264,9 @@ Additions to existing modules
 
 * Added new definitions in `Relation.Binary.Definitions`:
   ```
+  RightTrans R S = Trans R S R
+  LeftTrans  S R = Trans S R R
+
   Dense        _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y
   Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)
   Tight    _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 5760b050ec..8dfa60de79 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -33,7 +33,7 @@ open import Relation.Binary.Bundles using
 open import Relation.Binary.Structures
   using (IsPreorder; IsTotalPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
 open import Relation.Binary.Definitions
-  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri<; tri>)
+  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; LeftTrans; RightTrans; Trichotomous; tri≈; tri<; tri>)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary using (yes; no; ¬_)
 import Relation.Nullary.Reflects as Reflects
@@ -276,17 +276,17 @@ drop‿-<- (-<- n<m) = n<m
 <-asym (-<- n<m) = ℕ.<-asym n<m ∘ drop‿-<-
 <-asym (+<+ m<n) = ℕ.<-asym m<n ∘ drop‿+<+
 
-≤-<-trans : Trans _≤_ _<_ _<_
-≤-<-trans (-≤- n≤m) (-<- o<n) = -<- (ℕ.<-transˡ o<n n≤m)
+≤-<-trans : LeftTrans _≤_ _<_
+≤-<-trans (-≤- n≤m) (-<- o<n) = -<- (ℕ.<-≤-trans o<n n≤m)
 ≤-<-trans (-≤- n≤m) -<+       = -<+
 ≤-<-trans -≤+       (+<+ m<o) = -<+
-≤-<-trans (+≤+ m≤n) (+<+ n<o) = +<+ (ℕ.<-transʳ m≤n n<o)
+≤-<-trans (+≤+ m≤n) (+<+ n<o) = +<+ (ℕ.≤-<-trans m≤n n<o)
 
-<-≤-trans : Trans _<_ _≤_ _<_
-<-≤-trans (-<- n<m) (-≤- o≤n) = -<- (ℕ.<-transʳ o≤n n<m)
+<-≤-trans : RightTrans _<_ _≤_
+<-≤-trans (-<- n<m) (-≤- o≤n) = -<- (ℕ.≤-<-trans o≤n n<m)
 <-≤-trans (-<- n<m) -≤+       = -<+
 <-≤-trans -<+       (+≤+ m≤n) = -<+
-<-≤-trans (+<+ m<n) (+≤+ n≤o) = +<+ (ℕ.<-transˡ m<n n≤o)
+<-≤-trans (+<+ m<n) (+≤+ n≤o) = +<+ (ℕ.<-≤-trans m<n n≤o)
 
 <-trans : Transitive _<_
 <-trans m<n n<p = ≤-<-trans (<⇒≤ m<n) n<p
@@ -614,7 +614,7 @@ n⊖n≡0 n with n ℕ.<ᵇ n in leq
 
 ⊖-≥ : m ℕ.≥ n → m ⊖ n ≡ + (m ∸ n)
 ⊖-≥ {m} {n} p with m ℕ.<ᵇ n | Reflects.invert (ℕ.<ᵇ-reflects-< m n)
-... | true  | q = contradiction (ℕ.<-transʳ p q) (ℕ.<-irrefl refl)
+... | true  | q = contradiction (ℕ.≤-<-trans p q) (ℕ.<-irrefl refl)
 ... | false | q = refl
 
 ≤-⊖ : m ℕ.≤ n → n ⊖ m ≡ + (n ∸ m)
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index 3ea0cbb3c0..e09251caf5 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -121,7 +121,7 @@ Unique-resp-↭ = AllPairs-resp-↭ (_∘ ≈-sym) ≉-resp₂
 0<steps (prep eq xs↭ys)      = m<n⇒m<1+n (0<steps xs↭ys)
 0<steps (swap eq₁ eq₂ xs↭ys) = m<n⇒m<1+n (0<steps xs↭ys)
 0<steps (trans xs↭ys xs↭ys₁) =
-  <-transˡ (0<steps xs↭ys) (m≤m+n (steps xs↭ys) (steps xs↭ys₁))
+  <-≤-trans (0<steps xs↭ys) (m≤m+n (steps xs↭ys) (steps xs↭ys₁))
 
 steps-respˡ : ∀ {xs ys zs} (ys≋xs : ys ≋ xs) (ys↭zs : ys ↭ zs) →
               steps (↭-respˡ-≋ ys≋xs ys↭zs) ≡ steps ys↭zs
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 68237de411..f6f91b44e5 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -146,4 +146,4 @@ prime⇒coprime (suc (suc _)) p _ _ _ {0} (0∣m , _) =
 prime⇒coprime (suc (suc _)) _ _ _ _ {1} _         = refl
 prime⇒coprime (suc (suc _)) p (suc _) _ n<m {(suc (suc _))} (d∣m , d∣n) =
   contradiction d∣m (p 2≤d d<m)
-  where 2≤d = s≤s (s≤s z≤n); d<m = <-transˡ (s≤s (∣⇒≤ d∣n)) n<m
+  where 2≤d = s≤s (s≤s z≤n); d<m = <-≤-trans (s≤s (∣⇒≤ d∣n)) n<m
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index abc21e7058..ad46644442 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -24,6 +24,10 @@ open import Relation.Nullary.Decidable using (False; toWitnessFalse)
 
 open ≤-Reasoning
 
+private
+  variable
+    m n o p : ℕ
+
 ------------------------------------------------------------------------
 -- Definitions
 
@@ -46,10 +50,10 @@ m%n≡m∸m/n*n m n = begin-equality
 ------------------------------------------------------------------------
 -- Properties of _%_
 
-%-congˡ : ∀ {m n o} .⦃ _ : NonZero o ⦄ → m ≡ n → m % o ≡ n % o
+%-congˡ : .⦃ _ : NonZero o ⦄ → m ≡ n → m % o ≡ n % o
 %-congˡ refl = refl
 
-%-congʳ : ∀ {m n o} .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ≡ n →
+%-congʳ : .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ≡ n →
           o % m ≡ o % n
 %-congʳ refl = refl
 
@@ -73,7 +77,7 @@ m%n%n≡m%n m (suc n-1) = modₕ-idem 0 m n-1
   (m + n)           % n ≡⟨ [m+n]%n≡m%n m n ⟩
   m                 % n ∎
 
-m≤n⇒[n∸m]%m≡n%m : ∀ {m n} .⦃ _ : NonZero m ⦄ → m ≤ n →
+m≤n⇒[n∸m]%m≡n%m : .⦃ _ : NonZero m ⦄ → m ≤ n →
                   (n ∸ m) % m ≡ n % m
 m≤n⇒[n∸m]%m≡n%m {m} {n} m≤n = begin-equality
   (n ∸ m) % m     ≡˘⟨ [m+n]%n≡m%n (n ∸ m) m ⟩
@@ -108,7 +112,7 @@ m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
     o                    ∎
 
 m*n%n≡0 : ∀ m n .{{_ : NonZero n}} → (m * n) % n ≡ 0
-m*n%n≡0 m (suc n-1) = [m+kn]%n≡m%n 0 m (suc n-1)
+m*n%n≡0 m n@(suc _) = [m+kn]%n≡m%n 0 m n
 
 m%n<n : ∀ m n .{{_ : NonZero n}} → m % n < n
 m%n<n m (suc n-1) = s≤s (a[modₕ]n<n 0 m n-1)
@@ -119,12 +123,12 @@ m%n≤n m n = <⇒≤ (m%n<n m n)
 m%n≤m : ∀ m n .{{_ : NonZero n}} → m % n ≤ m
 m%n≤m m (suc n-1) = a[modₕ]n≤a 0 m n-1
 
-m≤n⇒m%n≡m : ∀ {m n} → m ≤ n → m % suc n ≡ m
-m≤n⇒m%n≡m {m} {n} m≤n with ≤⇒≤″ m≤n
-... | less-than-or-equal {k} refl = a≤n⇒a[modₕ]n≡a 0 (m + k) m k
+m≤n⇒m%n≡m : m ≤ n → m % suc n ≡ m
+m≤n⇒m%n≡m {m = m} m≤n with less-than-or-equal {k} refl ← ≤⇒≤″ m≤n
+  = a≤n⇒a[modₕ]n≡a 0 (m + k) m k
 
-m<n⇒m%n≡m : ∀ {m n} .⦃ _ : NonZero n ⦄ → m < n → m % n ≡ m
-m<n⇒m%n≡m {m} {suc n} m<n = m≤n⇒m%n≡m (<⇒≤pred m<n)
+m<n⇒m%n≡m : .⦃ _ : NonZero n ⦄ → m < n → m % n ≡ m
+m<n⇒m%n≡m {n = suc _} m<n = m≤n⇒m%n≡m (<⇒≤pred m<n)
 
 %-pred-≡0 : ∀ {m n} .{{_ : NonZero n}} → (suc m % n) ≡ 0 → (m % n) ≡ n ∸ 1
 %-pred-≡0 {m} {suc n-1} eq = a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 n-1 m eq
@@ -182,12 +186,10 @@ m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-
 ------------------------------------------------------------------------
 -- Properties of _/_
 
-/-congˡ : ∀ {m n o : ℕ} .{{_ : NonZero o}} →
-          m ≡ n → m / o ≡ n / o
+/-congˡ : .{{_ : NonZero o}} → m ≡ n → m / o ≡ n / o
 /-congˡ refl = refl
 
-/-congʳ : ∀ {m n o : ℕ} .{{_ : NonZero n}} .{{_ : NonZero o}} →
-          n ≡ o → m / n ≡ m / o
+/-congʳ : .{{_ : NonZero n}} .{{_ : NonZero o}} → n ≡ o → m / n ≡ m / o
 /-congʳ refl = refl
 
 0/n≡0 : ∀ n .{{_ : NonZero n}} → 0 / n ≡ 0
@@ -205,7 +207,7 @@ m*n/n≡m m (suc n-1) = a*n[divₕ]n≡a 0 m n-1
 m/n*n≡m : ∀ {m n} .{{_ : NonZero n}} → n ∣ m → m / n * n ≡ m
 m/n*n≡m {_} {n@(suc _)} (divides-refl q) = cong (_* n) (m*n/n≡m q n)
 
-m*[n/m]≡n : ∀ {m n} .{{_ : NonZero m}} → m ∣ n → m * (n / m) ≡ n
+m*[n/m]≡n : .{{_ : NonZero m}} → m ∣ n → m * (n / m) ≡ n
 m*[n/m]≡n {m} m∣n = trans (*-comm m (_ / m)) (m/n*n≡m m∣n)
 
 m/n*n≤m : ∀ m n .{{_ : NonZero n}} → (m / n) * n ≤ m
@@ -227,11 +229,11 @@ m/n<m m n n≥2 = *-cancelʳ-< _ (m / n) m (begin-strict
   m           <⟨ m<m*n m n n≥2 ⟩
   m * n       ∎)
 
-/-mono-≤ : ∀ {m n o p} .{{_ : NonZero o}} .{{_ : NonZero p}} →
+/-mono-≤ : .{{_ : NonZero o}} .{{_ : NonZero p}} →
            m ≤ n → o ≥ p → m / o ≤ n / p
 /-mono-≤ m≤n (s≤s o≥p) = divₕ-mono-≤ 0 m≤n o≥p
 
-/-monoˡ-≤ : ∀ {m n} o .{{_ : NonZero o}} → m ≤ n → m / o ≤ n / o
+/-monoˡ-≤ : ∀ o .{{_ : NonZero o}} → m ≤ n → m / o ≤ n / o
 /-monoˡ-≤ o m≤n = /-mono-≤ m≤n (≤-refl {o})
 
 /-monoʳ-≤ : ∀ m {n o} .{{_ : NonZero n}} .{{_ : NonZero o}} →
@@ -330,7 +332,7 @@ m<n*o⇒m/o<n {m} {suc n} {o} m<n*o with m <? o
 [m∸n]/n≡m/n∸1 : ∀ m n .⦃ _ : NonZero n ⦄ → (m ∸ n) / n ≡ pred (m / n)
 [m∸n]/n≡m/n∸1 m n with m <? n
 ... | yes m<n = begin-equality
-  (m ∸ n) / n  ≡⟨ m<n⇒m/n≡0 (<-transʳ (m∸n≤m m n) m<n) ⟩
+  (m ∸ n) / n  ≡⟨ m<n⇒m/n≡0 (≤-<-trans (m∸n≤m m n) m<n) ⟩
   0            ≡⟨⟩
   0 ∸ 1        ≡˘⟨ cong (_∸ 1) (m<n⇒m/n≡0 m<n) ⟩
   m / n ∸ 1    ≡⟨⟩
@@ -463,11 +465,11 @@ _div_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ
 _div_ = _/_
 
 _mod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → Fin divisor
-m mod (suc n) = fromℕ< (m%n<n m (suc n))
+m mod n@(suc _) = fromℕ< (m%n<n m n)
 
 _divMod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} →
            DivMod dividend divisor
-m divMod n@(suc n-1) = result (m / n) (m mod n) (begin-equality
+m divMod n@(suc _) = result (m / n) (m mod n) (begin-equality
   m                                   ≡⟨  m≡m%n+[m/n]*n m n ⟩
   m % n                    + [m/n]*n  ≡˘⟨ cong (_+ [m/n]*n) (toℕ-fromℕ< (m%n<n m n)) ⟩
   toℕ (fromℕ< (m%n<n m n)) + [m/n]*n  ∎)
diff --git a/src/Data/Nat/DivMod/Core.agda b/src/Data/Nat/DivMod/Core.agda
index 8ada6d77b2..a49318fbcd 100644
--- a/src/Data/Nat/DivMod/Core.agda
+++ b/src/Data/Nat/DivMod/Core.agda
@@ -154,14 +154,14 @@ private
   divₕ-offsetEq d (suc n) zero    zero    j≤d k≤d (inj₂′ (refl , _)) =
     divₕ-offsetEq d n d d ≤-refl ≤-refl (inj₂′ (refl , a[modₕ]n<n 0 n d , ≤-refl))
   divₕ-offsetEq d (suc n) zero    zero    j≤d k≤d (inj₃ (_ , 0<mod , mod≤0)) =
-    contradiction (<-transˡ 0<mod mod≤0) λ()
+    contradiction (<-≤-trans 0<mod mod≤0) λ()
   -- (0 , suc) cases
   divₕ-offsetEq d (suc n) zero (suc k)    j≤d k≤d (inj₁  (refl , _ , 1+k<mod)) =
     divₕ-offsetEq d n d k ≤-refl (<⇒≤ k≤d) (inj₃ (refl , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d 1+k<mod , a[modₕ]n<n 0 n d))
   divₕ-offsetEq d (suc n) zero (suc k)    j≤d k≤d (inj₂′ (refl , mod≤0 , _)) =
     divₕ-offsetEq d n d k ≤-refl (<⇒≤ k≤d) (inj₃ (refl , subst (k <_) (sym (a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 d n (n≤0⇒n≡0 mod≤0))) k≤d , a[modₕ]n<n 0 n d))
   divₕ-offsetEq d (suc n) zero (suc k)    j≤d k≤d (inj₃  (_ , 1+k<mod , mod≤0)) =
-    contradiction (<-transˡ 1+k<mod mod≤0) λ()
+    contradiction (<-≤-trans 1+k<mod mod≤0) λ()
   -- (suc , 0) cases
   divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₁  (_ , () , _))
   divₕ-offsetEq d (suc n) (suc j) zero j≤d k≤d (inj₂′ (_ , _ , ()))
@@ -174,7 +174,7 @@ private
   ... | yes mod≡0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₁ (eq , j≤k , subst (k <_) (sym (a+1[modₕ]n≡0⇒a[modₕ]n≡n-1 0 d n mod≡0)) k≤d))
   ... | no  mod≢0 = divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₂′ (eq , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (n≢0⇒n>0 mod≢0) mod≤1+j , j≤k))
   divₕ-offsetEq d (suc n) (suc j) (suc k) j≤d k≤d (inj₃  (eq , k<mod , mod≤1+j)) =
-    divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₃ (eq , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d k<mod , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (<-transʳ z≤n k<mod) mod≤1+j))
+    divₕ-offsetEq d n j k (<⇒≤ j≤d) (<⇒≤ k≤d) (inj₃ (eq , k<1+a[modₕ]n⇒k≤a[modₕ]n 0 (suc k) n d k<mod , 1+a[modₕ]n≤1+k⇒a[modₕ]n≤k 0 j n d (≤-<-trans z≤n k<mod) mod≤1+j))
 
 ------------------------------------------------------------------------
 -- Lemmas for divₕ that also have an interpretation for _/_
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index ac598c11a4..cf6e2ab0cd 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -21,7 +21,6 @@ import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
 import Algebra.Properties.CommutativeSemigroup as CommSemigroupProperties
 open import Data.Bool.Base using (Bool; false; true; T)
 open import Data.Bool.Properties using (T?)
-open import Data.Empty using (⊥)
 open import Data.Nat.Base
 open import Data.Product.Base using (∃; _×_; _,_)
 open import Data.Sum.Base as Sum
@@ -38,7 +37,7 @@ open import Relation.Binary.Bundles
 open import Relation.Binary.Structures
   using (IsDecEquivalence; IsPreorder; IsTotalPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
 open import Relation.Binary.Definitions
-  using (DecidableEquality; Irrelevant; Reflexive; Antisymmetric; Transitive; Total; Decidable; Connex; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri>; tri<; _Respects₂_)
+  using (DecidableEquality; Irrelevant; Reflexive; Antisymmetric; Transitive; Total; Decidable; Connex; Irreflexive; Asymmetric; LeftTrans; RightTrans; Trichotomous; tri≈; tri>; tri<; _Respects₂_)
 open import Relation.Binary.Consequences using (flip-Connex)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary hiding (Irrelevant)
@@ -52,6 +51,12 @@ open import Algebra.Definitions
   using (LeftCancellative; RightCancellative; Cancellative)
 open import Algebra.Structures {A = ℕ} _≡_
 
+
+private
+  variable
+    m n o k : ℕ
+
+
 ------------------------------------------------------------------------
 -- Properties of NonZero
 ------------------------------------------------------------------------
@@ -64,8 +69,8 @@ nonZero? (suc n) = yes _
 -- Properties of _≡_
 ------------------------------------------------------------------------
 
-suc-injective : ∀ {m n} → suc m ≡ suc n → m ≡ n
-suc-injective refl = refl
+suc-injective : suc m ≡ suc n → m ≡ n
+suc-injective = cong pred
 
 ≡ᵇ⇒≡ : ∀ m n → T (m ≡ᵇ n) → m ≡ n
 ≡ᵇ⇒≡ zero    zero    _  = refl
@@ -88,7 +93,7 @@ m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 ≡-irrelevant : Irrelevant {A = ℕ} _≡_
 ≡-irrelevant = Decidable⇒UIP.≡-irrelevant _≟_
 
-≟-diag : ∀ {m n} (eq : m ≡ n) → (m ≟ n) ≡ yes eq
+≟-diag : (eq : m ≡ n) → (m ≟ n) ≡ yes eq
 ≟-diag = ≡-≟-identity _≟_
 
 ≡-isDecEquivalence : IsDecEquivalence (_≡_ {A = ℕ})
@@ -104,13 +109,13 @@ m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
   ; isDecEquivalence = ≡-isDecEquivalence
   }
 
-0≢1+n : ∀ {n} → 0 ≢ suc n
+0≢1+n : 0 ≢ suc n
 0≢1+n ()
 
-1+n≢0 : ∀ {n} → suc n ≢ 0
+1+n≢0 : suc n ≢ 0
 1+n≢0 ()
 
-1+n≢n : ∀ {n} → suc n ≢ n
+1+n≢n : suc n ≢ n
 1+n≢n {suc n} = 1+n≢n ∘ suc-injective
 
 ------------------------------------------------------------------------
@@ -121,7 +126,7 @@ m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 <ᵇ⇒< zero    (suc n) m<n = z<s
 <ᵇ⇒< (suc m) (suc n) m<n = s<s (<ᵇ⇒< m n m<n)
 
-<⇒<ᵇ : ∀ {m n} → m < n → T (m <ᵇ n)
+<⇒<ᵇ : m < n → T (m <ᵇ n)
 <⇒<ᵇ z<s               = tt
 <⇒<ᵇ (s<s m<n@(s≤s _)) = <⇒<ᵇ m<n
 
@@ -136,7 +141,7 @@ m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 ≤ᵇ⇒≤ zero    n m≤n = z≤n
 ≤ᵇ⇒≤ (suc m) n m≤n = <ᵇ⇒< m n m≤n
 
-≤⇒≤ᵇ : ∀ {m n} → m ≤ n → T (m ≤ᵇ n)
+≤⇒≤ᵇ : m ≤ n → T (m ≤ᵇ n)
 ≤⇒≤ᵇ z≤n         = tt
 ≤⇒≤ᵇ m≤n@(s≤s _) = <⇒<ᵇ m≤n
 
@@ -256,26 +261,26 @@ _≥?_ = flip _≤?_
 ------------------------------------------------------------------------
 -- Other properties of _≤_
 
-s≤s-injective : ∀ {m n} {p q : m ≤ n} → s≤s p ≡ s≤s q → p ≡ q
+s≤s-injective : {p q : m ≤ n} → s≤s p ≡ s≤s q → p ≡ q
 s≤s-injective refl = refl
 
-≤-pred : ∀ {m n} → suc m ≤ suc n → m ≤ n
+≤-pred : suc m ≤ suc n → m ≤ n
 ≤-pred (s≤s m≤n) = m≤n
 
-m≤n⇒m≤1+n : ∀ {m n} → m ≤ n → m ≤ 1 + n
+m≤n⇒m≤1+n : m ≤ n → m ≤ 1 + n
 m≤n⇒m≤1+n z≤n       = z≤n
 m≤n⇒m≤1+n (s≤s m≤n) = s≤s (m≤n⇒m≤1+n m≤n)
 
 n≤1+n : ∀ n → n ≤ 1 + n
 n≤1+n _ = m≤n⇒m≤1+n ≤-refl
 
-1+n≰n : ∀ {n} → 1 + n ≰ n
+1+n≰n : 1 + n ≰ n
 1+n≰n (s≤s 1+n≤n) = 1+n≰n 1+n≤n
 
-n≤0⇒n≡0 : ∀ {n} → n ≤ 0 → n ≡ 0
+n≤0⇒n≡0 : n ≤ 0 → n ≡ 0
 n≤0⇒n≡0 z≤n = refl
 
-n≤1⇒n≡0∨n≡1 : ∀ {n : ℕ} → n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
+n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
 n≤1⇒n≡0∨n≡1 z≤n       = inj₁ refl
 n≤1⇒n≡0∨n≡1 (s≤s z≤n) = inj₂ refl
 
@@ -355,11 +360,11 @@ n≤1⇒n≡0∨n≡1 (s≤s z≤n) = inj₂ refl
 <-trans : Transitive _<_
 <-trans (s≤s i≤j) (s≤s j<k) = s≤s (≤-trans i≤j (≤-trans (n≤1+n _) j<k))
 
-<-transʳ : Trans _≤_ _<_ _<_
-<-transʳ m≤n (s<s n≤o) = s≤s (≤-trans m≤n n≤o)
+≤-<-trans : LeftTrans _≤_ _<_
+≤-<-trans m≤n (s<s n≤o) = s≤s (≤-trans m≤n n≤o)
 
-<-transˡ : Trans _<_ _≤_ _<_
-<-transˡ (s<s m≤n) (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)
+<-≤-trans : RightTrans _<_ _≤_
+<-≤-trans (s<s m≤n) (s≤s n≤o) = s≤s (≤-trans m≤n n≤o)
 
 -- NB: we use the builtin function `_<ᵇ_` here so that the function
 -- quickly decides which constructor to return. It still takes a
@@ -418,58 +423,58 @@ _>?_ = flip _<?_
 ------------------------------------------------------------------------
 -- Other properties of _<_
 
-s<s-injective : ∀ {m n} {p q : m < n} → s<s p ≡ s<s q → p ≡ q
+s<s-injective : {p q : m < n} → s<s p ≡ s<s q → p ≡ q
 s<s-injective refl = refl
 
-<-pred : ∀ {m n} → suc m < suc n → m < n
+<-pred : suc m < suc n → m < n
 <-pred (s<s m<n) = m<n
 
-m<n⇒m<1+n : ∀ {m n} → m < n → m < 1 + n
+m<n⇒m<1+n : m < n → m < 1 + n
 m<n⇒m<1+n z<s               = z<s
 m<n⇒m<1+n (s<s m<n@(s≤s _)) = s<s (m<n⇒m<1+n m<n)
 
-n≮0 : ∀ {n} → n ≮ 0
+n≮0 : n ≮ 0
 n≮0 ()
 
-n≮n : ∀ n → n ≮ n
+n≮n : ∀ n → n ≮ n -- implicit?
 n≮n n = <-irrefl (refl {x = n})
 
-0<1+n : ∀ {n} → 0 < suc n
+0<1+n : 0 < suc n
 0<1+n = z<s
 
 n<1+n : ∀ n → n < suc n
 n<1+n n = ≤-refl
 
-n<1⇒n≡0 : ∀ {n} → n < 1 → n ≡ 0
+n<1⇒n≡0 : n < 1 → n ≡ 0
 n<1⇒n≡0 (s≤s n≤0) = n≤0⇒n≡0 n≤0
 
-n>0⇒n≢0 : ∀ {n} → n > 0 → n ≢ 0
+n>0⇒n≢0 : n > 0 → n ≢ 0
 n>0⇒n≢0 {suc n} _ ()
 
-n≢0⇒n>0 : ∀ {n} → n ≢ 0 → n > 0
+n≢0⇒n>0 : n ≢ 0 → n > 0
 n≢0⇒n>0 {zero}  0≢0 =  contradiction refl 0≢0
 n≢0⇒n>0 {suc n} _   =  0<1+n
 
-m<n⇒0<n : ∀ {m n} → m < n → 0 < n
+m<n⇒0<n : m < n → 0 < n
 m<n⇒0<n = ≤-trans 0<1+n
 
-m<n⇒n≢0 : ∀ {m n} → m < n → n ≢ 0
+m<n⇒n≢0 : m < n → n ≢ 0
 m<n⇒n≢0 (s≤s m≤n) ()
 
-m<n⇒m≤1+n : ∀ {m n} → m < n → m ≤ suc n
+m<n⇒m≤1+n : m < n → m ≤ suc n
 m<n⇒m≤1+n = m≤n⇒m≤1+n ∘ <⇒≤
 
-m<1+n⇒m<n∨m≡n :  ∀ {m n} → m < suc n → m < n ⊎ m ≡ n
+m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
 m<1+n⇒m<n∨m≡n {0}     {0}     _          =  inj₂ refl
 m<1+n⇒m<n∨m≡n {0}     {suc n} _          =  inj₁ 0<1+n
 m<1+n⇒m<n∨m≡n {suc m} {suc n} (s<s m<1+n)  with m<1+n⇒m<n∨m≡n m<1+n
 ... | inj₂ m≡n = inj₂ (cong suc m≡n)
 ... | inj₁ m<n = inj₁ (s<s m<n)
 
-m≤n⇒m<n∨m≡n :  ∀ {m n} → m ≤ n → m < n ⊎ m ≡ n
+m≤n⇒m<n∨m≡n : m ≤ n → m < n ⊎ m ≡ n
 m≤n⇒m<n∨m≡n m≤n = m<1+n⇒m<n∨m≡n (s≤s m≤n)
 
-m<1+n⇒m≤n : ∀ {m n} → m < suc n → m ≤ n
+m<1+n⇒m≤n : m < suc n → m ≤ n
 m<1+n⇒m≤n (s≤s m≤n) = m≤n
 
 ∀[m≤n⇒m≢o]⇒n<o : ∀ n o → (∀ {m} → m ≤ n → m ≢ o) → n < o
@@ -499,8 +504,8 @@ module ≤-Reasoning where
     <-trans
     (resp₂ _<_)
     <⇒≤
-    <-transˡ
-    <-transʳ
+    <-≤-trans
+    ≤-<-trans
     public
     hiding (step-≈; step-≈˘)
 
@@ -661,11 +666,11 @@ m+n≡0⇒n≡0 m {n} m+n≡0 = m+n≡0⇒m≡0 n (trans (+-comm n m) (m+n≡0))
 +-cancel-< : Cancellative _<_ _+_
 +-cancel-< = +-cancelˡ-< , +-cancelʳ-<
 
-m≤n⇒m≤o+n : ∀ {m n} o → m ≤ n → m ≤ o + n
+m≤n⇒m≤o+n : ∀ o → m ≤ n → m ≤ o + n
 m≤n⇒m≤o+n zero    m≤n = m≤n
 m≤n⇒m≤o+n (suc o) m≤n = m≤n⇒m≤1+n (m≤n⇒m≤o+n o m≤n)
 
-m≤n⇒m≤n+o : ∀ {m n} o → m ≤ n → m ≤ n + o
+m≤n⇒m≤n+o : ∀ o → m ≤ n → m ≤ n + o
 m≤n⇒m≤n+o {m} o m≤n = subst (m ≤_) (+-comm o _) (m≤n⇒m≤o+n o m≤n)
 
 m≤m+n : ∀ m n → m ≤ m + n
@@ -934,7 +939,7 @@ m*n≡1⇒n≡1 m n eq = m*n≡1⇒m≡1 n m (trans (*-comm n m) eq)
 *-cancelʳ-≤ (suc m) (suc n) (suc o) le =
   s≤s (*-cancelʳ-≤ m n (suc o) (+-cancelˡ-≤ _ _ _ le))
 
-*-cancelˡ-≤ : ∀ {m n} o .{{_ : NonZero o}} → o * m ≤ o * n → m ≤ n
+*-cancelˡ-≤ : ∀ o .{{_ : NonZero o}} → o * m ≤ o * n → m ≤ n
 *-cancelˡ-≤ {m} {n} o rewrite *-comm o m | *-comm o n = *-cancelʳ-≤ m n o
 
 *-mono-≤ : _*_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
@@ -980,8 +985,8 @@ m<m*n m@(suc m-1) n@(suc (suc n-2)) (s≤s (s≤s _)) = begin-strict
   n + m-1 * n ≡⟨⟩
   m * n       ∎
 
-m<n⇒m<n*o : ∀ {m n} o .{{_ : NonZero o}} → m < n → m < n * o
-m<n⇒m<n*o {m} {n} o m<n = <-transˡ m<n (m≤m*n n o)
+m<n⇒m<n*o : ∀ o .{{_ : NonZero o}} → m < n → m < n * o
+m<n⇒m<n*o {n = n} o m<n = <-≤-trans m<n (m≤m*n n o)
 
 m<n⇒m<o*n : ∀ {m n} o .{{_ : NonZero o}} → m < n → m < o * n
 m<n⇒m<o*n {m} {n} o m<n = begin-strict
@@ -1078,20 +1083,20 @@ m^n>0 m n = >-nonZero⁻¹ (m ^ n) {{m^n≢0 m n}}
 ------------------------------------------------------------------------
 -- Basic specification in terms of _≤_
 
-m≤n⇒m⊔n≡n : ∀ {m n} → m ≤ n → m ⊔ n ≡ n
+m≤n⇒m⊔n≡n : m ≤ n → m ⊔ n ≡ n
 m≤n⇒m⊔n≡n {zero}  _         = refl
 m≤n⇒m⊔n≡n {suc m} (s≤s m≤n) = cong suc (m≤n⇒m⊔n≡n m≤n)
 
-m≥n⇒m⊔n≡m : ∀ {m n} → m ≥ n → m ⊔ n ≡ m
+m≥n⇒m⊔n≡m : m ≥ n → m ⊔ n ≡ m
 m≥n⇒m⊔n≡m {zero}  {zero}  z≤n       = refl
 m≥n⇒m⊔n≡m {suc m} {zero}  z≤n       = refl
 m≥n⇒m⊔n≡m {suc m} {suc n} (s≤s m≥n) = cong suc (m≥n⇒m⊔n≡m m≥n)
 
-m≤n⇒m⊓n≡m : ∀ {m n} → m ≤ n → m ⊓ n ≡ m
+m≤n⇒m⊓n≡m : m ≤ n → m ⊓ n ≡ m
 m≤n⇒m⊓n≡m {zero}  z≤n       = refl
 m≤n⇒m⊓n≡m {suc m} (s≤s m≤n) = cong suc (m≤n⇒m⊓n≡m m≤n)
 
-m≥n⇒m⊓n≡n : ∀ {m n} → m ≥ n → m ⊓ n ≡ n
+m≥n⇒m⊓n≡n : m ≥ n → m ⊓ n ≡ n
 m≥n⇒m⊓n≡n {zero}  {zero}  z≤n       = refl
 m≥n⇒m⊓n≡n {suc m} {zero}  z≤n       = refl
 m≥n⇒m⊓n≡n {suc m} {suc n} (s≤s m≤n) = cong suc (m≥n⇒m⊓n≡n m≤n)
@@ -1286,23 +1291,23 @@ antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≤_ ⟶ _≥_ →
                        ∀ m n → f (m ⊔ n) ≡ f m ⊓ f n
 antimono-≤-distrib-⊔ {f} = ⊓-⊔-properties.antimono-≤-distrib-⊔ (cong f)
 
-m<n⇒m<n⊔o : ∀ {m n} o → m < n → m < n ⊔ o
+m<n⇒m<n⊔o : ∀ o → m < n → m < n ⊔ o
 m<n⇒m<n⊔o = m≤n⇒m≤n⊔o
 
-m<n⇒m<o⊔n : ∀ {m n} o → m < n → m < o ⊔ n
+m<n⇒m<o⊔n : ∀ o → m < n → m < o ⊔ n
 m<n⇒m<o⊔n = m≤n⇒m≤o⊔n
 
 m⊔n<o⇒m<o : ∀ m n {o} → m ⊔ n < o → m < o
-m⊔n<o⇒m<o m n m⊔n<o = <-transʳ (m≤m⊔n m n) m⊔n<o
+m⊔n<o⇒m<o m n m⊔n<o = ≤-<-trans (m≤m⊔n m n) m⊔n<o
 
 m⊔n<o⇒n<o : ∀ m n {o} → m ⊔ n < o → n < o
-m⊔n<o⇒n<o m n m⊔n<o = <-transʳ (m≤n⊔m m n) m⊔n<o
+m⊔n<o⇒n<o m n m⊔n<o = ≤-<-trans (m≤n⊔m m n) m⊔n<o
 
 ⊔-mono-< : _⊔_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
 ⊔-mono-< = ⊔-mono-≤
 
-⊔-pres-<m : ∀ {m n o} → n < m → o < m → n ⊔ o < m
-⊔-pres-<m {m} n<m o<m = subst (_ <_) (⊔-idem m) (⊔-mono-< n<m o<m)
+⊔-pres-<m : n < m → o < m → n ⊔ o < m
+⊔-pres-<m {m = m} n<m o<m = subst (_ <_) (⊔-idem m) (⊔-mono-< n<m o<m)
 
 ------------------------------------------------------------------------
 -- Other properties of _⊔_ and _+_
@@ -1394,22 +1399,22 @@ m⊔n≤m+n m n with ⊔-sel m n
 ------------------------------------------------------------------------
 -- Other properties of _⊓_ and _≤_/_<_
 
-m<n⇒m⊓o<n : ∀ {m n} o → m < n → m ⊓ o < n
-m<n⇒m⊓o<n o m<n = <-transʳ (m⊓n≤m _ o) m<n
+m<n⇒m⊓o<n : ∀ o → m < n → m ⊓ o < n
+m<n⇒m⊓o<n o m<n = ≤-<-trans (m⊓n≤m _ o) m<n
 
-m<n⇒o⊓m<n : ∀ {m n} o → m < n → o ⊓ m < n
-m<n⇒o⊓m<n o m<n = <-transʳ (m⊓n≤n o _) m<n
+m<n⇒o⊓m<n : ∀ o → m < n → o ⊓ m < n
+m<n⇒o⊓m<n o m<n = ≤-<-trans (m⊓n≤n o _) m<n
 
-m<n⊓o⇒m<n : ∀ {m} n o → m < n ⊓ o → m < n
+m<n⊓o⇒m<n : ∀ n o → m < n ⊓ o → m < n
 m<n⊓o⇒m<n = m≤n⊓o⇒m≤n
 
-m<n⊓o⇒m<o : ∀ {m} n o → m < n ⊓ o → m < o
+m<n⊓o⇒m<o : ∀ n o → m < n ⊓ o → m < o
 m<n⊓o⇒m<o = m≤n⊓o⇒m≤o
 
 ⊓-mono-< : _⊓_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
 ⊓-mono-< = ⊓-mono-≤
 
-⊓-pres-m< : ∀ {m n o} → m < n → m < o → m < n ⊓ o
+⊓-pres-m< : m < n → m < o → m < n ⊓ o
 ⊓-pres-m< {m} m<n m<o = subst (_< _) (⊓-idem m) (⊓-mono-< m<n m<o)
 
 ------------------------------------------------------------------------
@@ -1501,10 +1506,10 @@ m≮m∸n (suc m) (suc n) = m≮m∸n m n ∘ ≤-trans (n≤1+n (suc m))
 ∸-mono (s≤s m₁≤m₂) (s≤s n₁≥n₂)    = ∸-mono m₁≤m₂ n₁≥n₂
 ∸-mono m₁≤m₂       (z≤n {n = n₁}) = ≤-trans (m∸n≤m _ n₁) m₁≤m₂
 
-∸-monoˡ-≤ : ∀ {m n} o → m ≤ n → m ∸ o ≤ n ∸ o
+∸-monoˡ-≤ : ∀ o → m ≤ n → m ∸ o ≤ n ∸ o
 ∸-monoˡ-≤ o m≤n = ∸-mono {u = o} m≤n ≤-refl
 
-∸-monoʳ-≤ : ∀ {m n} o → m ≤ n → o ∸ m ≥ o ∸ n
+∸-monoʳ-≤ : ∀ o → m ≤ n → o ∸ m ≥ o ∸ n
 ∸-monoʳ-≤ _ m≤n = ∸-mono ≤-refl m≤n
 
 ∸-monoˡ-< : ∀ {m n o} → m < o → n ≤ m → m ∸ n < o ∸ n
@@ -1525,37 +1530,37 @@ m≮m∸n (suc m) (suc n) = m≮m∸n m n ∘ ≤-trans (n≤1+n (suc m))
 ∸-cancelʳ-< {suc m} {zero}  {_}     o∸n<o∸m = 0<1+n
 ∸-cancelʳ-< {suc m} {suc n} {suc o} o∸n<o∸m = s≤s (∸-cancelʳ-< o∸n<o∸m)
 
-∸-cancelˡ-≡ :  ∀ {m n o} → n ≤ m → o ≤ m → m ∸ n ≡ m ∸ o → n ≡ o
+∸-cancelˡ-≡ :  n ≤ m → o ≤ m → m ∸ n ≡ m ∸ o → n ≡ o
 ∸-cancelˡ-≡ {_}         z≤n       z≤n       _  = refl
 ∸-cancelˡ-≡ {o = suc o} z≤n       (s≤s _)   eq = contradiction eq (1+m≢m∸n o)
 ∸-cancelˡ-≡ {n = suc n} (s≤s _)   z≤n       eq = contradiction (sym eq) (1+m≢m∸n n)
 ∸-cancelˡ-≡ {_}         (s≤s n≤m) (s≤s o≤m) eq = cong suc (∸-cancelˡ-≡ n≤m o≤m eq)
 
-∸-cancelʳ-≡ :  ∀ {m n o} → o ≤ m → o ≤ n → m ∸ o ≡ n ∸ o → m ≡ n
+∸-cancelʳ-≡ :  o ≤ m → o ≤ n → m ∸ o ≡ n ∸ o → m ≡ n
 ∸-cancelʳ-≡  z≤n       z≤n      eq = eq
 ∸-cancelʳ-≡ (s≤s o≤m) (s≤s o≤n) eq = cong suc (∸-cancelʳ-≡ o≤m o≤n eq)
 
-m∸n≡0⇒m≤n : ∀ {m n} → m ∸ n ≡ 0 → m ≤ n
+m∸n≡0⇒m≤n : m ∸ n ≡ 0 → m ≤ n
 m∸n≡0⇒m≤n {zero}  {_}    _   = z≤n
 m∸n≡0⇒m≤n {suc m} {suc n} eq = s≤s (m∸n≡0⇒m≤n eq)
 
-m≤n⇒m∸n≡0 : ∀ {m n} → m ≤ n → m ∸ n ≡ 0
+m≤n⇒m∸n≡0 : m ≤ n → m ∸ n ≡ 0
 m≤n⇒m∸n≡0 {n = n} z≤n      = 0∸n≡0 n
 m≤n⇒m∸n≡0 {_}    (s≤s m≤n) = m≤n⇒m∸n≡0 m≤n
 
-m<n⇒0<n∸m : ∀ {m n} → m < n → 0 < n ∸ m
+m<n⇒0<n∸m : m < n → 0 < n ∸ m
 m<n⇒0<n∸m {zero}  {suc n} _         = 0<1+n
 m<n⇒0<n∸m {suc m} {suc n} (s≤s m<n) = m<n⇒0<n∸m m<n
 
-m∸n≢0⇒n<m : ∀ {m n} → m ∸ n ≢ 0 → n < m
+m∸n≢0⇒n<m : m ∸ n ≢ 0 → n < m
 m∸n≢0⇒n<m {m} {n} m∸n≢0 with n <? m
 ... | yes n<m = n<m
 ... | no  n≮m = contradiction (m≤n⇒m∸n≡0 (≮⇒≥ n≮m)) m∸n≢0
 
-m>n⇒m∸n≢0 : ∀ {m n} → m > n → m ∸ n ≢ 0
+m>n⇒m∸n≢0 : m > n → m ∸ n ≢ 0
 m>n⇒m∸n≢0 {n = suc n} (s≤s m>n) = m>n⇒m∸n≢0 m>n
 
-m≤n⇒n∸m≤n : ∀ {m n} → m ≤ n → n ∸ m ≤ n
+m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n
 m≤n⇒n∸m≤n z≤n       = ≤-refl
 m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
 
@@ -1605,7 +1610,7 @@ m+n∸n≡m m n = begin-equality
 m+n∸m≡n : ∀ m n → m + n ∸ m ≡ n
 m+n∸m≡n m n = trans (cong (_∸ m) (+-comm m n)) (m+n∸n≡m n m)
 
-m+[n∸m]≡n : ∀ {m n} → m ≤ n → m + (n ∸ m) ≡ n
+m+[n∸m]≡n : m ≤ n → m + (n ∸ m) ≡ n
 m+[n∸m]≡n {m} {n} m≤n = begin-equality
   m + (n ∸ m)  ≡⟨ sym $ +-∸-assoc m m≤n ⟩
   (m + n) ∸ m  ≡⟨ cong (_∸ m) (+-comm m n) ⟩
@@ -1689,19 +1694,19 @@ m⊓n+n∸m≡n (suc m) (suc n) = cong suc $ m⊓n+n∸m≡n m n
 pred-mono : pred Preserves _≤_ ⟶ _≤_
 pred-mono m≤n = ∸-mono m≤n (≤-refl {1})
 
-pred[n]≤n : ∀ {n} → pred n ≤ n
+pred[n]≤n : pred n ≤ n
 pred[n]≤n {zero}  = z≤n
 pred[n]≤n {suc n} = n≤1+n n
 
-≤pred⇒≤ : ∀ {m n} → m ≤ pred n → m ≤ n
-≤pred⇒≤ {m} {zero}  le = le
-≤pred⇒≤ {m} {suc n} le = m≤n⇒m≤1+n le
+≤pred⇒≤ : m ≤ pred n → m ≤ n
+≤pred⇒≤ {n = zero}  le = le
+≤pred⇒≤ {n = suc n} le = m≤n⇒m≤1+n le
 
-≤⇒pred≤ : ∀ {m n} → m ≤ n → pred m ≤ n
+≤⇒pred≤ : m ≤ n → pred m ≤ n
 ≤⇒pred≤ {zero}  le = le
 ≤⇒pred≤ {suc m} le = ≤-trans (n≤1+n m) le
 
-<⇒≤pred : ∀ {m n} → m < n → m ≤ pred n
+<⇒≤pred : m < n → m ≤ pred n
 <⇒≤pred (s≤s le) = le
 
 suc-pred : ∀ n .{{_ : NonZero n}} → suc (pred n) ≡ n
@@ -1714,25 +1719,25 @@ suc-pred (suc n) = refl
 ------------------------------------------------------------------------
 -- Basic
 
-m≡n⇒∣m-n∣≡0 : ∀ {m n} → m ≡ n → ∣ m - n ∣ ≡ 0
+m≡n⇒∣m-n∣≡0 : m ≡ n → ∣ m - n ∣ ≡ 0
 m≡n⇒∣m-n∣≡0 {zero}  refl = refl
 m≡n⇒∣m-n∣≡0 {suc m} refl = m≡n⇒∣m-n∣≡0 {m} refl
 
-∣m-n∣≡0⇒m≡n : ∀ {m n} → ∣ m - n ∣ ≡ 0 → m ≡ n
+∣m-n∣≡0⇒m≡n :  ∣ m - n ∣ ≡ 0 → m ≡ n
 ∣m-n∣≡0⇒m≡n {zero}  {zero}  eq = refl
 ∣m-n∣≡0⇒m≡n {suc m} {suc n} eq = cong suc (∣m-n∣≡0⇒m≡n eq)
 
-m≤n⇒∣n-m∣≡n∸m : ∀ {m n} → m ≤ n → ∣ n - m ∣ ≡ n ∸ m
-m≤n⇒∣n-m∣≡n∸m {_} {zero}  z≤n       = refl
-m≤n⇒∣n-m∣≡n∸m {_} {suc m} z≤n       = refl
-m≤n⇒∣n-m∣≡n∸m {_} {_}     (s≤s m≤n) = m≤n⇒∣n-m∣≡n∸m m≤n
+m≤n⇒∣n-m∣≡n∸m : m ≤ n → ∣ n - m ∣ ≡ n ∸ m
+m≤n⇒∣n-m∣≡n∸m {n = zero}  z≤n       = refl
+m≤n⇒∣n-m∣≡n∸m {n = suc n} z≤n       = refl
+m≤n⇒∣n-m∣≡n∸m {n = _}     (s≤s m≤n) = m≤n⇒∣n-m∣≡n∸m m≤n
 
-m≤n⇒∣m-n∣≡n∸m : ∀ {m n} → m ≤ n → ∣ m - n ∣ ≡ n ∸ m
-m≤n⇒∣m-n∣≡n∸m {_} {zero}  z≤n       = refl
-m≤n⇒∣m-n∣≡n∸m {_} {suc n} z≤n       = refl
-m≤n⇒∣m-n∣≡n∸m {_} {_}     (s≤s m≤n) = m≤n⇒∣m-n∣≡n∸m m≤n
+m≤n⇒∣m-n∣≡n∸m : m ≤ n → ∣ m - n ∣ ≡ n ∸ m
+m≤n⇒∣m-n∣≡n∸m {n = zero}  z≤n       = refl
+m≤n⇒∣m-n∣≡n∸m {n = suc n} z≤n       = refl
+m≤n⇒∣m-n∣≡n∸m {n = _}     (s≤s m≤n) = m≤n⇒∣m-n∣≡n∸m m≤n
 
-∣m-n∣≡m∸n⇒n≤m : ∀ {m n} → ∣ m - n ∣ ≡ m ∸ n → n ≤ m
+∣m-n∣≡m∸n⇒n≤m : ∣ m - n ∣ ≡ m ∸ n → n ≤ m
 ∣m-n∣≡m∸n⇒n≤m {zero}  {zero}  eq = z≤n
 ∣m-n∣≡m∸n⇒n≤m {suc m} {zero}  eq = z≤n
 ∣m-n∣≡m∸n⇒n≤m {suc m} {suc n} eq = s≤s (∣m-n∣≡m∸n⇒n≤m eq)
@@ -1971,11 +1976,11 @@ m !* n !≢0 = m*n≢0 _ _ {{m !≢0}} {{n !≢0}}
 ≤′-trans m≤n ≤′-refl       = m≤n
 ≤′-trans m≤n (≤′-step n≤o) = ≤′-step (≤′-trans m≤n n≤o)
 
-z≤′n : ∀ {n} → zero ≤′ n
+z≤′n : zero ≤′ n
 z≤′n {zero}  = ≤′-refl
 z≤′n {suc n} = ≤′-step z≤′n
 
-s≤′s : ∀ {m n} → m ≤′ n → suc m ≤′ suc n
+s≤′s : m ≤′ n → suc m ≤′ suc n
 s≤′s ≤′-refl        = ≤′-refl
 s≤′s (≤′-step m≤′n) = ≤′-step (s≤′s m≤′n)
 
@@ -1987,30 +1992,30 @@ s≤′s (≤′-step m≤′n) = ≤′-step (s≤′s m≤′n)
 ≤⇒≤′ z≤n       = z≤′n
 ≤⇒≤′ (s≤s m≤n) = s≤′s (≤⇒≤′ m≤n)
 
-≤′-step-injective : ∀ {m n} {p q : m ≤′ n} → ≤′-step p ≡ ≤′-step q → p ≡ q
+≤′-step-injective : {p q : m ≤′ n} → ≤′-step p ≡ ≤′-step q → p ≡ q
 ≤′-step-injective refl = refl
 
 ------------------------------------------------------------------------
 -- Properties of _<′_ and _<_
 ------------------------------------------------------------------------
 
-z<′s : ∀ {n} → zero <′ suc n
+z<′s : zero <′ suc n
 z<′s {zero}  = <′-base
 z<′s {suc n} = <′-step (z<′s {n})
 
-s<′s : ∀ {m n} → m <′ n → suc m <′ suc n
+s<′s : m <′ n → suc m <′ suc n
 s<′s <′-base        = <′-base
 s<′s (<′-step m<′n) = <′-step (s<′s m<′n)
 
-<⇒<′ : ∀ {m n} → m < n → m <′ n
+<⇒<′ : m < n → m <′ n
 <⇒<′ z<s               = z<′s
 <⇒<′ (s<s m<n@(s≤s _)) = s<′s (<⇒<′ m<n)
 
-<′⇒< : ∀ {m n} → m <′ n → m < n
+<′⇒< : m <′ n → m < n
 <′⇒< <′-base        = n<1+n _
 <′⇒< (<′-step m<′n) = m<n⇒m<1+n (<′⇒< m<′n)
 
-m<1+n⇒m<n∨m≡n′ : ∀ {m n} → m < suc n → m < n ⊎ m ≡ n
+m<1+n⇒m<n∨m≡n′ : m < suc n → m < n ⊎ m ≡ n
 m<1+n⇒m<n∨m≡n′ m<n with <⇒<′ m<n
 ... | <′-base      = inj₂ refl
 ... | <′-step m<′n = inj₁ (<′⇒< m<′n)
@@ -2059,10 +2064,10 @@ m<ᵇn⇒1+m+[n-1+m]≡n m n lt = m+[n∸m]≡n (<ᵇ⇒< m n lt)
 m<ᵇ1+m+n : ∀ m {n} → T (m <ᵇ suc (m + n))
 m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)
 
-<ᵇ⇒<″ : ∀ {m n} → T (m <ᵇ n) → m <″ n
+<ᵇ⇒<″ : T (m <ᵇ n) → m <″ n
 <ᵇ⇒<″ {m} {n} leq = less-than-or-equal (m+[n∸m]≡n (<ᵇ⇒< m n leq))
 
-<″⇒<ᵇ : ∀ {m n} → m <″ n → T (m <ᵇ n)
+<″⇒<ᵇ : m <″ n → T (m <ᵇ n)
 <″⇒<ᵇ {m} (less-than-or-equal refl) = <⇒<ᵇ (m≤m+n (suc m) _)
 
 -- equivalence to the old definition of _≤″_
@@ -2131,11 +2136,11 @@ m≤‴m+k {m} {k = suc k} prf  = ≤‴-step (m≤‴m+k {k = k} (trans (sym (+
 ≤″⇒≤‴ : ∀{m n} → m ≤″ n → m ≤‴ n
 ≤″⇒≤‴ m≤n = m≤‴m+k (≤″-proof m≤n)
 
-0≤‴n : ∀{n} → 0 ≤‴ n
-0≤‴n {n} = m≤‴m+k refl
+0≤‴n : 0 ≤‴ n
+0≤‴n = m≤‴m+k refl
 
-<ᵇ⇒<‴ : ∀ {m n} → T (m <ᵇ n) → m <‴ n
-<ᵇ⇒<‴ {m} {n} leq = ≤″⇒≤‴ (<ᵇ⇒<″ leq)
+<ᵇ⇒<‴ : T (m <ᵇ n) → m <‴ n
+<ᵇ⇒<‴ leq = ≤″⇒≤‴ (<ᵇ⇒<″ leq)
 
 <‴⇒<ᵇ : ∀ {m n} → m <‴ n → T (m <ᵇ n)
 <‴⇒<ᵇ leq = <″⇒<ᵇ (≤‴⇒≤″ leq)
@@ -2183,7 +2188,7 @@ module _ {p} {P : Pred ℕ p} (P? : U.Decidable P) where
   ... | _      | yes (n , n<v , Pn) = yes (n , m≤n⇒m≤1+n n<v , Pn)
   ... | no ¬Pv | no ¬Pn<v           = no ¬Pn<1+v
     where
-    ¬Pn<1+v : (∃ λ n → n < suc v × P n) → ⊥
+    ¬Pn<1+v : ¬ (∃ λ n → n < suc v × P n)
     ¬Pn<1+v (n , s≤s n≤v , Pn) with n ≟ v
     ... | yes refl = ¬Pv Pn
     ... | no  n≢v  = ¬Pn<v (n , ≤∧≢⇒< n≤v n≢v , Pn)
@@ -2191,8 +2196,8 @@ module _ {p} {P : Pred ℕ p} (P? : U.Decidable P) where
   allUpTo? : ∀ v → Dec (∀ {n} → n < v → P n)
   allUpTo? zero    = yes λ()
   allUpTo? (suc v) with P? v | allUpTo? v
-  ... | no ¬Pv | _        = no (λ prf → ¬Pv   (prf ≤-refl))
-  ... | _      | no ¬Pn<v = no (λ prf → ¬Pn<v (prf ∘ m≤n⇒m≤1+n))
+  ... | no ¬Pv | _        = no λ prf → ¬Pv   (prf ≤-refl)
+  ... | _      | no ¬Pn<v = no λ prf → ¬Pn<v (prf ∘ m≤n⇒m≤1+n)
   ... | yes Pn | yes Pn<v = yes Pn<1+v
     where
       Pn<1+v : ∀ {n} → n < suc v → P n
@@ -2300,7 +2305,7 @@ n≤m⊔n = m≤n⊔m
 
 -- Version 2.0
 
-suc[pred[n]]≡n : ∀ {n} → n ≢ 0 → suc (pred n) ≡ n
+suc[pred[n]]≡n : n ≢ 0 → suc (pred n) ≡ n
 suc[pred[n]]≡n {zero}  0≢0 = contradiction refl 0≢0
 suc[pred[n]]≡n {suc n} _   = refl
 {-# WARNING_ON_USAGE suc[pred[n]]≡n
@@ -2330,3 +2335,14 @@ suc[pred[n]]≡n {suc n} _   = refl
 {- issue1844/issue1755: raw bundles have moved to `Data.X.Base` -}
 open Data.Nat.Base public
   using (*-rawMagma; *-1-rawMonoid)
+
+<-transʳ = ≤-<-trans
+{-# WARNING_ON_USAGE <-transʳ
+"Warning: <-transʳ was deprecated in v2.0. Please use ≤-<-trans instead. "
+#-}
+
+<-transˡ = <-≤-trans
+{-# WARNING_ON_USAGE <-transˡ
+"Warning: <-transˡ was deprecated in v2.0. Please use ≤-<-trans instead. "
+#-}
+
diff --git a/src/Relation/Binary/Definitions.agda b/src/Relation/Binary/Definitions.agda
index 6854074986..547d4e7b48 100644
--- a/src/Relation/Binary/Definitions.agda
+++ b/src/Relation/Binary/Definitions.agda
@@ -58,6 +58,12 @@ Symmetric _∼_ = Sym _∼_ _∼_
 Trans : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _
 Trans P Q R = ∀ {i j k} → P i j → Q j k → R i k
 
+RightTrans : REL A B ℓ₁ → REL B B ℓ₂ → Set _
+RightTrans R S = Trans R S R
+
+LeftTrans : REL A A ℓ₁ → REL A B ℓ₂ → Set _
+LeftTrans S R = Trans S R R
+
 -- A flipped variant of generalised transitivity.
 
 TransFlip : REL A B ℓ₁ → REL B C ℓ₂ → REL A C ℓ₃ → Set _

From 413aa824746092be5eb346b408ee855c477a871d Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 6 Oct 2023 09:50:25 +0900
Subject: [PATCH 146/316] Re-export numeric subclass instances (#2122)

---
 src/Data/Integer/Base.agda               | 5 ++++-
 src/Data/Rational/Base.agda              | 9 ++++++++-
 src/Data/Rational/Unnormalised/Base.agda | 6 ++++++
 3 files changed, 18 insertions(+), 2 deletions(-)

diff --git a/src/Data/Integer/Base.agda b/src/Data/Integer/Base.agda
index 338656956b..9d7e689db0 100644
--- a/src/Data/Integer/Base.agda
+++ b/src/Data/Integer/Base.agda
@@ -14,7 +14,7 @@ module Data.Integer.Base where
 open import Algebra.Bundles.Raw
   using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring; RawRing)
 open import Data.Bool.Base using (Bool; T; true; false)
-open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s)
+open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s) hiding (module ℕ)
 open import Data.Sign.Base as Sign using (Sign)
 open import Level using (0ℓ)
 open import Relation.Binary.Core using (Rel)
@@ -140,6 +140,9 @@ record Negative (i : ℤ) : Set where
 
 -- Instances
 
+open ℕ public
+  using (nonZero)
+
 instance
   pos : ∀ {n} → Positive +[1+ n ]
   pos = _
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 5f5a35e6f9..ebb5ad0ac9 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -10,7 +10,9 @@ module Data.Rational.Base where
 
 open import Algebra.Bundles.Raw
 open import Data.Bool.Base using (Bool; true; false; if_then_else_)
-open import Data.Integer.Base as ℤ using (ℤ; +_; +0; +[1+_]; -[1+_])
+open import Data.Integer.Base as ℤ
+  using (ℤ; +_; +0; +[1+_]; -[1+_])
+  hiding (module ℤ)
 open import Data.Nat.GCD
 open import Data.Nat.Coprimality as C
   using (Coprime; Bézout-coprime; coprime-/gcd; coprime?; ¬0-coprimeTo-2+)
@@ -176,6 +178,11 @@ NonPositive p = ℚᵘ.NonPositive (toℚᵘ p)
 NonNegative : Pred ℚ 0ℓ
 NonNegative p = ℚᵘ.NonNegative (toℚᵘ p)
 
+-- Instances
+
+open ℤ public
+  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)
+
 -- Constructors
 
 ≢-nonZero : ∀ {p} → p ≢ 0ℚ → NonZero p
diff --git a/src/Data/Rational/Unnormalised/Base.agda b/src/Data/Rational/Unnormalised/Base.agda
index dc5bd144dc..c9fcb8d54d 100644
--- a/src/Data/Rational/Unnormalised/Base.agda
+++ b/src/Data/Rational/Unnormalised/Base.agda
@@ -12,6 +12,7 @@ open import Algebra.Bundles.Raw
 open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Integer.Base as ℤ
   using (ℤ; +_; +0; +[1+_]; -[1+_]; +<+; +≤+)
+  hiding (module ℤ)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Level using (0ℓ)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
@@ -149,6 +150,11 @@ NonPositive p = ℤ.NonPositive (↥ p)
 NonNegative : Pred ℚᵘ 0ℓ
 NonNegative p = ℤ.NonNegative (↥ p)
 
+-- Instances
+
+open ℤ public
+  using (nonZero; pos; nonNeg; nonPos0; nonPos; neg)
+
 -- Constructors and destructors
 
 -- Note: these could be proved more elegantly using the constructors

From ba789ee9ba8d5ef27287cb22378c84d9e7ddf5de Mon Sep 17 00:00:00 2001
From: Ioannis Markakis <70938217+jmarkakis@users.noreply.github.com>
Date: Fri, 6 Oct 2023 01:51:32 +0100
Subject: [PATCH 147/316] Added Irreflexivity and Asymmetry of WellFounded
 Relations (#2119)

---
 CHANGELOG.md                   |  5 +++++
 src/Induction/WellFounded.agda | 22 ++++++++++++++++++++--
 2 files changed, 25 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2c9dd0e092..60e5ce7066 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3764,6 +3764,11 @@ This is a full list of proofs that have changed form to use irrelevant instance
 * Added new proof to `Induction.WellFounded`
   ```agda
   Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
+
+  acc⇒asym : Acc _<_ x → x < y → ¬ (y < x)
+  wf⇒asym : WellFounded _<_ → Asymmetric _<_
+  wf⇒irrefl : _<_ Respects₂ _≈_ → Symmetric _≈_ →
+              WellFounded _<_ → Irreflexive _≈_ _<_
   ```
 
 * Added new file `Relation.Binary.Reasoning.Base.Apartness`
diff --git a/src/Induction/WellFounded.agda b/src/Induction/WellFounded.agda
index 61ba4ded19..fb3ed14255 100644
--- a/src/Induction/WellFounded.agda
+++ b/src/Induction/WellFounded.agda
@@ -8,15 +8,18 @@
 
 module Induction.WellFounded where
 
-open import Data.Product.Base using (Σ; _,_; proj₁)
+open import Data.Product.Base using (Σ; _,_; proj₁; proj₂)
 open import Function.Base using (_∘_; flip; _on_)
 open import Induction
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions
-  using (Symmetric; _Respectsʳ_; _Respects_)
+  using (Symmetric; Asymmetric; Irreflexive; _Respects₂_; 
+    _Respectsʳ_; _Respects_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
+open import Relation.Binary.Consequences using (asym⇒irr)
 open import Relation.Unary
+open import Relation.Nullary.Negation.Core using (¬_)
 
 private
   variable
@@ -116,6 +119,21 @@ module FixPoint
   unfold-wfRec : ∀ {x} → wfRec P f x ≡ f x λ _ → wfRec P f _
   unfold-wfRec {x} = f-ext x wfRecBuilder-wfRec
 
+------------------------------------------------------------------------
+-- Well-founded relations are asymmetric and irreflexive.
+
+module _ {_<_ : Rel A r} where
+  acc⇒asym : ∀ {x y} → Acc _<_ x → x < y → ¬ (y < x)
+  acc⇒asym {x} hx =
+    Some.wfRec (λ x → ∀ {y} → x < y → ¬ (y < x)) (λ _ hx x<y y<x → hx y<x y<x x<y) _ hx
+
+  wf⇒asym : WellFounded _<_ → Asymmetric _<_
+  wf⇒asym wf = acc⇒asym (wf _)
+
+  wf⇒irrefl : {_≈_ : Rel A ℓ} → _<_ Respects₂ _≈_ → 
+              Symmetric _≈_ → WellFounded _<_ → Irreflexive _≈_ _<_
+  wf⇒irrefl r s wf = asym⇒irr r s (wf⇒asym wf)
+
 ------------------------------------------------------------------------
 -- It might be useful to establish proofs of Acc or Well-founded using
 -- combinators such as the ones below (see, for instance,

From 5ec53e6f4367f5c32813e97d89d75e3bd325e002 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 6 Oct 2023 11:49:50 +0900
Subject: [PATCH 148/316] Fix argument order of composition operators (#2121)

---
 CHANGELOG.md                            |  4 +++
 src/Data/Fin/Permutation.agda           |  2 +-
 src/Function/Construct/Composition.agda | 36 +++++++++++++------------
 src/Function/Properties/Inverse.agda    |  2 +-
 src/Function/Related/Propositional.agda | 16 +++++------
 5 files changed, 33 insertions(+), 27 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 60e5ce7066..d4f4d9051b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -29,6 +29,10 @@ Highlights
 Bug-fixes
 ---------
 
+* The combinators `_⟶-∘_`, `_↣-∘_`, `_↠-∘_`, `_⤖-∘_`, `_⇔-∘_`, `_↩-∘_`, `_↪-∘_`, `_↔-∘_`
+  in `Function.Construct.Composition` had their arguments in the wrong order. They have
+  been flipped so they can actually be used as a composition operator.
+
 * The following operators were missing a fixity declaration, which has now
   been fixed -
   ```
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 00b65247fe..59a5c6abdf 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -104,7 +104,7 @@ reverse = permutation opposite opposite PC.reverse-involutive PC.reverse-involut
 
 infixr 9 _∘ₚ_
 _∘ₚ_ : Permutation m n → Permutation n o → Permutation m o
-π₁ ∘ₚ π₂ = π₁ ↔-∘ π₂
+π₁ ∘ₚ π₂ = π₂ ↔-∘ π₁
 
 -- Flip
 
diff --git a/src/Function/Construct/Composition.agda b/src/Function/Construct/Composition.agda
index 1d25753a9c..10289cc8c9 100644
--- a/src/Function/Construct/Composition.agda
+++ b/src/Function/Construct/Composition.agda
@@ -9,7 +9,7 @@
 module Function.Construct.Composition where
 
 open import Data.Product.Base as Product using (_,_)
-open import Function.Base using (_∘_)
+open import Function.Base using (_∘_; flip)
 open import Function.Bundles
 open import Function.Definitions
 open import Function.Structures
@@ -199,31 +199,33 @@ module _ {R : Setoid a ℓ₁} {S : Setoid b ℓ₂} {T : Setoid c ℓ₃} where
 ------------------------------------------------------------------------
 -- Propositional bundles
 
+-- Notice the flipped order of the arguments to mirror composition.
+
 infix 8 _⟶-∘_ _↣-∘_ _↠-∘_ _⤖-∘_ _⇔-∘_ _↩-∘_ _↪-∘_ _↔-∘_
 
-_⟶-∘_ : (A ⟶ B) → (B ⟶ C) → (A ⟶ C)
-_⟶-∘_ = function
+_⟶-∘_ : (B ⟶ C) → (A ⟶ B) → (A ⟶ C)
+_⟶-∘_ = flip function
 
-_↣-∘_ : A ↣ B → B ↣ C → A ↣ C
-_↣-∘_ = injection
+_↣-∘_ : B ↣ C → A ↣ B → A ↣ C
+_↣-∘_ = flip injection
 
-_↠-∘_ : A ↠ B → B ↠ C → A ↠ C
-_↠-∘_ = surjection
+_↠-∘_ : B ↠ C → A ↠ B → A ↠ C
+_↠-∘_ = flip surjection
 
-_⤖-∘_ : A ⤖ B → B ⤖ C → A ⤖ C
-_⤖-∘_ = bijection
+_⤖-∘_ : B ⤖ C → A ⤖ B → A ⤖ C
+_⤖-∘_ = flip bijection
 
-_⇔-∘_ : A ⇔ B → B ⇔ C → A ⇔ C
-_⇔-∘_ = equivalence
+_⇔-∘_ : B ⇔ C → A ⇔ B → A ⇔ C
+_⇔-∘_ = flip equivalence
 
-_↩-∘_ : A ↩ B → B ↩ C → A ↩ C
-_↩-∘_ = leftInverse
+_↩-∘_ : B ↩ C → A ↩ B → A ↩ C
+_↩-∘_ = flip leftInverse
 
-_↪-∘_  : A ↪ B → B ↪ C → A ↪ C
-_↪-∘_ = rightInverse
+_↪-∘_  : B ↪ C → A ↪ B → A ↪ C
+_↪-∘_ = flip rightInverse
 
-_↔-∘_ : A ↔ B → B ↔ C → A ↔ C
-_↔-∘_ = inverse
+_↔-∘_ : B ↔ C → A ↔ B → A ↔ C
+_↔-∘_ = flip inverse
 
 
 ------------------------------------------------------------------------
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index e25f010a3c..87ce778fec 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -55,7 +55,7 @@ isEquivalence = record
 ↔-sym = Symmetry.↔-sym
 
 ↔-trans : A ↔ B → B ↔ C → A ↔ C
-↔-trans = Composition._↔-∘_
+↔-trans = Composition.inverse
 
 -- need to η-expand for everything to line up properly
 ↔-isEquivalence : IsEquivalence {ℓ = ℓ} _↔_
diff --git a/src/Function/Related/Propositional.agda b/src/Function/Related/Propositional.agda
index ad6054af46..96b50667df 100644
--- a/src/Function/Related/Propositional.agda
+++ b/src/Function/Related/Propositional.agda
@@ -256,14 +256,14 @@ K-reflexive P.refl = K-refl
 K-trans : Trans (Related {a} {b} k)
                 (Related {b} {c} k)
                 (Related {a} {c} k)
-K-trans {k = implication}        = Composition._⟶-∘_
-K-trans {k = reverseImplication} = flip Composition._⟶-∘_
-K-trans {k = equivalence}        = Composition._⇔-∘_
-K-trans {k = injection}          = Composition._↣-∘_
-K-trans {k = reverseInjection}   = flip Composition._↣-∘_
-K-trans {k = leftInverse}        = Composition._↪-∘_
-K-trans {k = surjection}         = Composition._↠-∘_
-K-trans {k = bijection}          = Composition._↔-∘_
+K-trans {k = implication}        = flip Composition._⟶-∘_
+K-trans {k = reverseImplication} = Composition._⟶-∘_
+K-trans {k = equivalence}        = flip Composition._⇔-∘_
+K-trans {k = injection}          = flip Composition._↣-∘_
+K-trans {k = reverseInjection}   = Composition._↣-∘_
+K-trans {k = leftInverse}        = flip Composition._↪-∘_
+K-trans {k = surjection}         = flip Composition._↠-∘_
+K-trans {k = bijection}          = flip Composition._↔-∘_
 
 SK-sym : ∀ {k} → Sym (Related {a} {b} ⌊ k ⌋)
                      (Related {b} {a} ⌊ k ⌋)

From 2d45ecb9f9416660ffc3fd8944b867cc747d841f Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 6 Oct 2023 19:56:09 +0900
Subject: [PATCH 149/316] Make size parameter on 'replicate' explicit (#2120)

---
 CHANGELOG.md                                  | 102 +++++++++---------
 src/Algebra/Properties/Monoid/Sum.agda        |   6 +-
 src/Algebra/Solver/CommutativeMonoid.agda     |  32 +++---
 .../Solver/IdempotentCommutativeMonoid.agda   |  28 ++---
 src/Codata/Guarded/Stream/Properties.agda     |  10 +-
 src/Codata/Sized/Stream/Properties.agda       |   4 +-
 src/Data/Fin/Subset.agda                      |   4 +-
 src/Data/String.agda                          |   7 +-
 src/Data/Vec/Base.agda                        |  12 +--
 src/Data/Vec/Bounded/Base.agda                |  12 +--
 src/Data/Vec/Effectful.agda                   |   4 +-
 src/Data/Vec/Effectful/Transformer.agda       |   4 +-
 src/Data/Vec/Functional.agda                  |   4 +-
 .../Relation/Binary/Pointwise/Properties.agda |   2 +-
 .../Relation/Unary/All/Properties.agda        |  62 +++++------
 src/Data/Vec/Properties.agda                  |  44 ++++----
 src/Data/Vec/Recursive.agda                   |   2 +-
 .../Vec/Relation/Binary/Equality/Setoid.agda  |   4 +-
 18 files changed, 173 insertions(+), 170 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d4f4d9051b..3d99a6eb2b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -603,6 +603,7 @@ Non-backwards compatible changes
   ```agda
   WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
   WfRec _<_ P x = ∀ {y} → y < x → P y
+  ```
 
 ### Change in the definition of `_≤″_` (issue #1919)
 
@@ -924,7 +925,7 @@ Non-backwards compatible changes
   Data.Vec.Relation.Binary.Lex.NonStrict
   Relation.Binary.Reasoning.StrictPartialOrder
   Relation.Binary.Reasoning.PartialOrder
-  ```
+  ```	
 
 ### Other
 
@@ -1053,60 +1054,61 @@ Non-backwards compatible changes
   `List` module i.e. `lookup` takes its container first and the index (whose type may
   depend on the container value) second.
   This affects modules:
-    ```
-    Codata.Guarded.Stream
-    Codata.Guarded.Stream.Relation.Binary.Pointwise
-    Codata.Musical.Colist.Base
-    Codata.Musical.Colist.Relation.Unary.Any.Properties
-    Codata.Musical.Covec
-    Codata.Musical.Stream
-    Codata.Sized.Colist
-    Codata.Sized.Covec
-    Codata.Sized.Stream
-    Data.Vec.Relation.Unary.All
-    Data.Star.Environment
-    Data.Star.Pointer
-    Data.Star.Vec
-    Data.Trie
-    Data.Trie.NonEmpty
-    Data.Tree.AVL
-    Data.Tree.AVL.Indexed
-    Data.Tree.AVL.Map
-    Data.Tree.AVL.NonEmpty
-    Data.Vec.Recursive
-    Tactic.RingSolver
-    Tactic.RingSolver.Core.NatSet
-    ```
+  ```
+  Codata.Guarded.Stream
+  Codata.Guarded.Stream.Relation.Binary.Pointwise
+  Codata.Musical.Colist.Base
+  Codata.Musical.Colist.Relation.Unary.Any.Properties
+  Codata.Musical.Covec
+  Codata.Musical.Stream
+  Codata.Sized.Colist
+  Codata.Sized.Covec
+  Codata.Sized.Stream
+  Data.Vec.Relation.Unary.All
+  Data.Star.Environment
+  Data.Star.Pointer
+  Data.Star.Vec
+  Data.Trie
+  Data.Trie.NonEmpty
+  Data.Tree.AVL
+  Data.Tree.AVL.Indexed
+  Data.Tree.AVL.Map
+  Data.Tree.AVL.NonEmpty
+  Data.Vec.Recursive
+  Tactic.RingSolver
+  Tactic.RingSolver.Core.NatSet
+  ```
 
-  * Moved & renamed from `Data.Vec.Relation.Unary.All`
-    to `Data.Vec.Relation.Unary.All.Properties`:
-    ```
-    lookup ↦ lookup⁺
-    tabulate ↦ lookup⁻
-    ```
+* Moved & renamed from `Data.Vec.Relation.Unary.All`
+  to `Data.Vec.Relation.Unary.All.Properties`:
+  ```
+  lookup ↦ lookup⁺
+  tabulate ↦ lookup⁻
+  ```
 
-  * Renamed in `Data.Vec.Relation.Unary.Linked.Properties`
-    and `Codata.Guarded.Stream.Relation.Binary.Pointwise`:
-    ```
-    lookup ↦ lookup⁺
-    ```
+* Renamed in `Data.Vec.Relation.Unary.Linked.Properties`
+  and `Codata.Guarded.Stream.Relation.Binary.Pointwise`:
+  ```
+  lookup ↦ lookup⁺
+  ```
 
-  * Added the following new definitions to `Data.Vec.Relation.Unary.All`:
-    ```
-    lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
-    lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
-    lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
-    lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
-    ```
+* Added the following new definitions to `Data.Vec.Relation.Unary.All`:
+  ```
+  lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
+  lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
+  lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
+  lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
+  ```
 
-  * `excluded-middle` in `Relation.Nullary.Decidable.Core` has been renamed to
-    `¬¬-excluded-middle`.
+* `excluded-middle` in `Relation.Nullary.Decidable.Core` has been renamed to
+  `¬¬-excluded-middle`.
 
-  * `iterate` in `Data.Vec.Base` now takes `n` (the length of `Vec`) as an
-    explicit argument.
-    ```agda
-    iterate : (A → A) → A → ∀ n → Vec A n
-    ```
+* `iterate` and `replicate` in `Data.Vec.Base` and `Data.Vec.Functional` 
+  now take the length of vector, `n`, as an explicit rather than an implicit argument.
+  ```agda
+  iterate : (A → A) → A → ∀ n → Vec A n
+  replicate : ∀ n → A → Vec A n
+  ```
 
 Major improvements
 ------------------
diff --git a/src/Algebra/Properties/Monoid/Sum.agda b/src/Algebra/Properties/Monoid/Sum.agda
index 2d8605a2b0..166cba4462 100644
--- a/src/Algebra/Properties/Monoid/Sum.agda
+++ b/src/Algebra/Properties/Monoid/Sum.agda
@@ -64,15 +64,15 @@ sum-cong-≗ : ∀ {n} → sum {n} Preserves _≗_ ⟶ _≡_
 sum-cong-≗ {zero}  xs≗ys = P.refl
 sum-cong-≗ {suc n} xs≗ys = P.cong₂ _+_ (xs≗ys zero) (sum-cong-≗ (xs≗ys ∘ suc))
 
-sum-replicate : ∀ n {x} → sum {n} (replicate x) ≈ n × x
+sum-replicate : ∀ n {x} → sum (replicate n x) ≈ n × x
 sum-replicate zero    = refl
 sum-replicate (suc n) = +-congˡ (sum-replicate n)
 
 sum-replicate-idem : ∀ {x} → _+_ IdempotentOn x →
-                     ∀ n → .{{_ : NonZero n}} → sum {n} (replicate x) ≈ x
+                     ∀ n → .{{_ : NonZero n}} → sum (replicate n x) ≈ x
 sum-replicate-idem idem n = trans (sum-replicate n) (×-idem idem n)
 
-sum-replicate-zero : ∀ n → sum {n} (replicate 0#) ≈ 0#
+sum-replicate-zero : ∀ n → sum (replicate n 0#) ≈ 0#
 sum-replicate-zero zero    = refl
 sum-replicate-zero (suc n) = sum-replicate-idem (+-identityˡ 0#) (suc n)
 
diff --git a/src/Algebra/Solver/CommutativeMonoid.agda b/src/Algebra/Solver/CommutativeMonoid.agda
index 7e4b14b18c..8062ff34df 100644
--- a/src/Algebra/Solver/CommutativeMonoid.agda
+++ b/src/Algebra/Solver/CommutativeMonoid.agda
@@ -33,6 +33,10 @@ open import Relation.Nullary.Decidable using (Dec)
 open CommutativeMonoid M
 open EqReasoning setoid
 
+private
+  variable
+    n : ℕ
+
 ------------------------------------------------------------------------
 -- Monoid expressions
 
@@ -55,7 +59,7 @@ Env n = Vec Carrier n
 -- The semantics of an expression is a function from an environment to
 -- a value.
 
-⟦_⟧ : ∀ {n} → Expr n → Env n → Carrier
+⟦_⟧ : Expr n → Env n → Carrier
 ⟦ var x   ⟧ ρ = lookup ρ x
 ⟦ id      ⟧ ρ = ε
 ⟦ e₁ ⊕ e₂ ⟧ ρ = ⟦ e₁ ⟧ ρ ∙ ⟦ e₂ ⟧ ρ
@@ -70,27 +74,27 @@ Normal n = Vec ℕ n
 
 -- The semantics of a normal form.
 
-⟦_⟧⇓ : ∀ {n} → Normal n → Env n → Carrier
+⟦_⟧⇓ : Normal n → Env n → Carrier
 ⟦ []    ⟧⇓ _ = ε
-⟦ n ∷ v ⟧⇓ (a ∷ ρ) = fold (⟦ v ⟧⇓ ρ) (λ b → a ∙ b) n
+⟦ n ∷ v ⟧⇓ (a ∷ ρ) = fold (⟦ v ⟧⇓ ρ) (a ∙_) n
 
 ------------------------------------------------------------------------
 -- Constructions on normal forms
 
 -- The empty bag.
 
-empty : ∀{n} → Normal n
-empty = replicate 0
+empty : Normal n
+empty = replicate _ 0
 
 -- A singleton bag.
 
-sg : ∀{n} (i : Fin n) → Normal n
+sg : (i : Fin n) → Normal n
 sg zero    = 1 ∷ empty
 sg (suc i) = 0 ∷ sg i
 
 -- The composition of normal forms.
 
-_•_  : ∀{n} (v w : Normal n) → Normal n
+_•_  : (v w : Normal n) → Normal n
 []      • []      = []
 (l ∷ v) • (m ∷ w) = l + m ∷ v • w
 
@@ -99,13 +103,13 @@ _•_  : ∀{n} (v w : Normal n) → Normal n
 
 -- The empty bag stands for the unit ε.
 
-empty-correct : ∀{n} (ρ : Env n) → ⟦ empty ⟧⇓ ρ ≈ ε
+empty-correct : (ρ : Env n) → ⟦ empty ⟧⇓ ρ ≈ ε
 empty-correct [] = refl
 empty-correct (a ∷ ρ) = empty-correct ρ
 
 -- The singleton bag stands for a single variable.
 
-sg-correct : ∀{n} (x : Fin n) (ρ : Env n) →  ⟦ sg x ⟧⇓ ρ ≈ lookup ρ x
+sg-correct : (x : Fin n) (ρ : Env n) →  ⟦ sg x ⟧⇓ ρ ≈ lookup ρ x
 sg-correct zero (x ∷ ρ) = begin
     x ∙ ⟦ empty ⟧⇓ ρ   ≈⟨ ∙-congˡ (empty-correct ρ) ⟩
     x ∙ ε              ≈⟨ identityʳ _ ⟩
@@ -114,7 +118,7 @@ sg-correct (suc x) (m ∷ ρ) = sg-correct x ρ
 
 -- Normal form composition corresponds to the composition of the monoid.
 
-comp-correct : ∀ {n} (v w : Normal n) (ρ : Env n) →
+comp-correct : (v w : Normal n) (ρ : Env n) →
               ⟦ v • w ⟧⇓ ρ ≈ (⟦ v ⟧⇓ ρ ∙ ⟦ w ⟧⇓ ρ)
 comp-correct [] [] ρ =  sym (identityˡ _)
 comp-correct (l ∷ v) (m ∷ w) (a ∷ ρ) = lemma l m (comp-correct v w ρ)
@@ -136,14 +140,14 @@ comp-correct (l ∷ v) (m ∷ w) (a ∷ ρ) = lemma l m (comp-correct v w ρ)
 
 -- A normaliser.
 
-normalise : ∀ {n} → Expr n → Normal n
+normalise : Expr n → Normal n
 normalise (var x)   = sg x
 normalise id        = empty
 normalise (e₁ ⊕ e₂) = normalise e₁ • normalise e₂
 
 -- The normaliser preserves the semantics of the expression.
 
-normalise-correct : ∀ {n} (e : Expr n) (ρ : Env n) →
+normalise-correct : (e : Expr n) (ρ : Env n) →
     ⟦ normalise e ⟧⇓ ρ ≈ ⟦ e ⟧ ρ
 normalise-correct (var x)   ρ = sg-correct x ρ
 normalise-correct id        ρ = empty-correct ρ
@@ -171,14 +175,14 @@ open module R = Reflection
 
 infix 5 _≟_
 
-_≟_ : ∀ {n} (nf₁ nf₂ : Normal n) → Dec (nf₁ ≡ nf₂)
+_≟_ : (nf₁ nf₂ : Normal n) → Dec (nf₁ ≡ nf₂)
 nf₁ ≟ nf₂ = Dec.map Pointwise-≡↔≡ (decidable ℕ._≟_ nf₁ nf₂)
   where open Pointwise
 
 -- We can also give a sound, but not necessarily complete, procedure
 -- for determining if two expressions have the same semantics.
 
-prove′ : ∀ {n} (e₁ e₂ : Expr n) → Maybe (∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
+prove′ : (e₁ e₂ : Expr n) → Maybe (∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
 prove′ e₁ e₂ =
   Maybe.map lemma (decToMaybe (normalise e₁ ≟ normalise e₂))
   where
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
index fdba81a838..fc65652787 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
@@ -34,6 +34,10 @@ module Algebra.Solver.IdempotentCommutativeMonoid
 open IdempotentCommutativeMonoid M
 open EqReasoning setoid
 
+private
+  variable
+    n : ℕ
+
 ------------------------------------------------------------------------
 -- Monoid expressions
 
@@ -71,7 +75,7 @@ Normal n = Vec Bool n
 
 -- The semantics of a normal form.
 
-⟦_⟧⇓ : ∀ {n} → Normal n → Env n → Carrier
+⟦_⟧⇓ : Normal n → Env n → Carrier
 ⟦ []    ⟧⇓ _ = ε
 ⟦ b ∷ v ⟧⇓ (a ∷ ρ) = if b then a ∙ (⟦ v ⟧⇓ ρ) else (⟦ v ⟧⇓ ρ)
 
@@ -80,18 +84,18 @@ Normal n = Vec Bool n
 
 -- The empty set.
 
-empty : ∀{n} → Normal n
-empty = replicate false
+empty : Normal n
+empty = replicate _ false
 
 -- A singleton set.
 
-sg : ∀{n} (i : Fin n) → Normal n
+sg : (i : Fin n) → Normal n
 sg zero    = true ∷ empty
 sg (suc i) = false ∷ sg i
 
 -- The composition of normal forms.
 
-_•_  : ∀{n} (v w : Normal n) → Normal n
+_•_ : (v w : Normal n) → Normal n
 []      • []      = []
 (l ∷ v) • (m ∷ w) = (l ∨ m) ∷ v • w
 
@@ -100,13 +104,13 @@ _•_  : ∀{n} (v w : Normal n) → Normal n
 
 -- The empty set stands for the unit ε.
 
-empty-correct : ∀{n} (ρ : Env n) → ⟦ empty ⟧⇓ ρ ≈ ε
+empty-correct : (ρ : Env n) → ⟦ empty ⟧⇓ ρ ≈ ε
 empty-correct []      = refl
 empty-correct (a ∷ ρ) = empty-correct ρ
 
 -- The singleton set stands for a single variable.
 
-sg-correct : ∀{n} (x : Fin n) (ρ : Env n) →  ⟦ sg x ⟧⇓ ρ ≈ lookup ρ x
+sg-correct : (x : Fin n) (ρ : Env n) → ⟦ sg x ⟧⇓ ρ ≈ lookup ρ x
 sg-correct zero (x ∷ ρ) = begin
     x ∙ ⟦ empty ⟧⇓ ρ   ≈⟨ ∙-congˡ (empty-correct ρ) ⟩
     x ∙ ε              ≈⟨ identityʳ _ ⟩
@@ -132,7 +136,7 @@ distr a b c = begin
     a ∙ (b ∙ (a ∙ c))  ≈⟨ sym (assoc _ _ _) ⟩
     (a ∙ b) ∙ (a ∙ c)  ∎
 
-comp-correct : ∀ {n} (v w : Normal n) (ρ : Env n) →
+comp-correct : ∀ (v w : Normal n) (ρ : Env n) →
               ⟦ v • w ⟧⇓ ρ ≈ (⟦ v ⟧⇓ ρ ∙ ⟦ w ⟧⇓ ρ)
 comp-correct [] [] ρ = sym (identityˡ _)
 comp-correct (true ∷ v) (true ∷ w) (a ∷ ρ) =
@@ -149,14 +153,14 @@ comp-correct (false ∷ v) (false ∷ w) (a ∷ ρ) =
 
 -- A normaliser.
 
-normalise : ∀ {n} → Expr n → Normal n
+normalise : Expr n → Normal n
 normalise (var x)   = sg x
 normalise id        = empty
 normalise (e₁ ⊕ e₂) = normalise e₁ • normalise e₂
 
 -- The normaliser preserves the semantics of the expression.
 
-normalise-correct : ∀ {n} (e : Expr n) (ρ : Env n) →
+normalise-correct : (e : Expr n) (ρ : Env n) →
     ⟦ normalise e ⟧⇓ ρ ≈ ⟦ e ⟧ ρ
 normalise-correct (var x)   ρ = sg-correct x ρ
 normalise-correct id        ρ = empty-correct ρ
@@ -184,14 +188,14 @@ open module R = Reflection
 
 infix 5 _≟_
 
-_≟_ : ∀ {n} (nf₁ nf₂ : Normal n) → Dec (nf₁ ≡ nf₂)
+_≟_ : (nf₁ nf₂ : Normal n) → Dec (nf₁ ≡ nf₂)
 nf₁ ≟ nf₂ = Dec.map Pointwise-≡↔≡ (decidable Bool._≟_ nf₁ nf₂)
   where open Pointwise
 
 -- We can also give a sound, but not necessarily complete, procedure
 -- for determining if two expressions have the same semantics.
 
-prove′ : ∀ {n} (e₁ e₂ : Expr n) → Maybe (∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
+prove′ : (e₁ e₂ : Expr n) → Maybe (∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
 prove′ e₁ e₂ =
   Maybe.map lemma (decToMaybe (normalise e₁ ≟ normalise e₂))
   where
diff --git a/src/Codata/Guarded/Stream/Properties.agda b/src/Codata/Guarded/Stream/Properties.agda
index 41b19ec947..bd41d1967e 100644
--- a/src/Codata/Guarded/Stream/Properties.agda
+++ b/src/Codata/Guarded/Stream/Properties.agda
@@ -80,11 +80,11 @@ lookup-repeat : ∀ n (a : A) → lookup (repeat a) n ≡ a
 lookup-repeat zero    a = P.refl
 lookup-repeat (suc n) a = lookup-repeat n a
 
-splitAt-repeat : ∀ n (a : A) → splitAt n (repeat a) ≡ (Vec.replicate a , repeat a)
+splitAt-repeat : ∀ n (a : A) → splitAt n (repeat a) ≡ (Vec.replicate n a , repeat a)
 splitAt-repeat zero    a = P.refl
 splitAt-repeat (suc n) a = cong (Prod.map₁ (a ∷_)) (splitAt-repeat n a)
 
-take-repeat : ∀ n (a : A) → take n (repeat a) ≡ Vec.replicate a
+take-repeat : ∀ n (a : A) → take n (repeat a) ≡ Vec.replicate n a
 take-repeat n a = cong proj₁ (splitAt-repeat n a)
 
 drop-repeat : ∀ n (a : A) → drop n (repeat a) ≡ repeat a
@@ -115,17 +115,17 @@ zipWith-repeat : ∀ (f : A → B → C) a b →
 zipWith-repeat f a b .head = P.refl
 zipWith-repeat f a b .tail = zipWith-repeat f a b
 
-chunksOf-repeat : ∀ n (a : A) → chunksOf n (repeat a) ≈ repeat (Vec.replicate a)
+chunksOf-repeat : ∀ n (a : A) → chunksOf n (repeat a) ≈ repeat (Vec.replicate n a)
 chunksOf-repeat n a = begin go where
 
   open ≈-Reasoning
 
-  go : chunksOf n (repeat a) ≈∞ repeat (Vec.replicate a)
+  go : chunksOf n (repeat a) ≈∞ repeat (Vec.replicate n a)
   go .head = take-repeat n a
   go .tail =
     chunksOf n (drop n (repeat a)) ≡⟨ P.cong (chunksOf n) (drop-repeat n a) ⟩
     chunksOf n (repeat a)          ↺⟨ go ⟩
-    repeat (Vec.replicate a)       ∎
+    repeat (Vec.replicate n a)     ∎
 
 ------------------------------------------------------------------------
 -- Properties of map
diff --git a/src/Codata/Sized/Stream/Properties.agda b/src/Codata/Sized/Stream/Properties.agda
index 69685e12b2..d2cf4e3205 100644
--- a/src/Codata/Sized/Stream/Properties.agda
+++ b/src/Codata/Sized/Stream/Properties.agda
@@ -41,11 +41,11 @@ lookup-repeat-identity : (n : ℕ) (a : A) → lookup (repeat a) n ≡ a
 lookup-repeat-identity zero    a = P.refl
 lookup-repeat-identity (suc n) a = lookup-repeat-identity n a
 
-take-repeat-identity : (n : ℕ) (a : A) → take n (repeat a) ≡ Vec.replicate a
+take-repeat-identity : (n : ℕ) (a : A) → take n (repeat a) ≡ Vec.replicate n a
 take-repeat-identity zero    a = P.refl
 take-repeat-identity (suc n) a = P.cong (a Vec.∷_) (take-repeat-identity n a)
 
-splitAt-repeat-identity : (n : ℕ) (a : A) → splitAt n (repeat a) ≡ (Vec.replicate a , repeat a)
+splitAt-repeat-identity : (n : ℕ) (a : A) → splitAt n (repeat a) ≡ (Vec.replicate n a , repeat a)
 splitAt-repeat-identity zero    a = P.refl
 splitAt-repeat-identity (suc n) a = P.cong (Prod.map₁ (a ∷_)) (splitAt-repeat-identity n a)
 
diff --git a/src/Data/Fin/Subset.agda b/src/Data/Fin/Subset.agda
index b3d53106ff..2e44f69762 100644
--- a/src/Data/Fin/Subset.agda
+++ b/src/Data/Fin/Subset.agda
@@ -43,11 +43,11 @@ Subset = Vec Side
 
 -- The empty subset
 ⊥ : Subset n
-⊥ = replicate outside
+⊥ = replicate _ outside
 
 -- The full subset
 ⊤ : Subset n
-⊤ = replicate inside
+⊤ = replicate _ inside
 
 -- A singleton subset, containing just the given element.
 ⁅_⁆ : Fin n → Subset n
diff --git a/src/Data/String.agda b/src/Data/String.agda
index 0ad40779fa..f8ca8385e4 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -23,7 +23,6 @@ open import Relation.Nullary.Decidable.Core using (does)
 open import Data.List.Membership.DecPropositional Char._≟_
 
 
-
 ------------------------------------------------------------------------
 -- Re-export contents of base, and decidability of equality
 
@@ -62,10 +61,10 @@ rectangle pads cells = Vec.zipWith (λ p c → p width c) pads cells where
 -- Special cases for left, center, and right alignment
 
 rectangleˡ : ∀ {n} → Char → Vec String n → Vec String n
-rectangleˡ c = rectangle (Vec.replicate $ padLeft c)
+rectangleˡ c = rectangle (Vec.replicate _ $ padLeft c)
 
 rectangleʳ : ∀ {n} → Char → Vec String n → Vec String n
-rectangleʳ c = rectangle (Vec.replicate $ padRight c)
+rectangleʳ c = rectangle (Vec.replicate _ $ padRight c)
 
 rectangleᶜ : ∀ {n} → Char → Char → Vec String n → Vec String n
-rectangleᶜ cₗ cᵣ = rectangle (Vec.replicate $ padBoth cₗ cᵣ)
+rectangleᶜ cₗ cᵣ = rectangle (Vec.replicate _ $ padBoth cₗ cᵣ)
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index be668897cb..0f56c9fbd7 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -243,9 +243,9 @@ count P? = countᵇ (does ∘ P?)
 [_] : A → Vec A 1
 [ x ] = x ∷ []
 
-replicate : A → Vec A n
-replicate {n = zero}  x = []
-replicate {n = suc n} x = x ∷ replicate x
+replicate : (n : ℕ) → A → Vec A n
+replicate zero    x = []
+replicate (suc n) x = x ∷ replicate n x
 
 tabulate : (Fin n → A) → Vec A n
 tabulate {n = zero}  f = []
@@ -295,7 +295,7 @@ truncate (s≤s le) (x ∷ xs) = x ∷ (truncate le xs)
 
 -- Pad out a vector with extra elements.
 padRight : ∀ {m n} → m ≤ n → A → Vec A m → Vec A n
-padRight z≤n      a xs       = replicate a
+padRight z≤n      a xs       = replicate _ a
 padRight (s≤s le) a (x ∷ xs) = x ∷ padRight le a xs
 
 ------------------------------------------------------------------------
@@ -350,8 +350,8 @@ last xs = proj₁ (proj₂ (initLast xs))
 -- Other operations
 
 transpose : Vec (Vec A n) m → Vec (Vec A m) n
-transpose []         = replicate []
-transpose (as ∷ ass) = replicate _∷_ ⊛ as ⊛ transpose ass
+transpose {n = n} []          = replicate n []
+transpose {n = n} (as ∷ ass) = ((replicate n _∷_) ⊛ as) ⊛ transpose ass
 
 ------------------------------------------------------------------------
 -- DEPRECATED
diff --git a/src/Data/Vec/Bounded/Base.agda b/src/Data/Vec/Bounded/Base.agda
index 17b33ef7be..c857ee6642 100644
--- a/src/Data/Vec/Bounded/Base.agda
+++ b/src/Data/Vec/Bounded/Base.agda
@@ -49,14 +49,14 @@ fromVec v = v , ℕₚ.≤-refl
 padRight : ∀ {n} → A → Vec≤ A n → Vec A n
 padRight a (vs , m≤n)
   with recompute (_ ℕₚ.≤″? _) (ℕₚ.≤⇒≤″ m≤n)
-... | less-than-or-equal refl = vs Vec.++ Vec.replicate a
+... | less-than-or-equal refl = vs Vec.++ Vec.replicate  _ a
 
 padLeft : ∀ {n} → A → Vec≤ A n → Vec A n
 padLeft a as@(vs , m≤n)
   with recompute (_ ℕₚ.≤″? _) (ℕₚ.≤⇒≤″ m≤n)
 ... | less-than-or-equal {k} ∣as∣+k≡n
   with P.trans (ℕₚ.+-comm k (Vec≤.length as)) ∣as∣+k≡n
-... | refl = Vec.replicate a Vec.++ vs
+...   | refl = Vec.replicate k a Vec.++ vs
 
 private
   split : ∀ {m n} k → m + k ≡ n → ⌊ k /2⌋ + (m + ⌈ k /2⌉) ≡ n
@@ -72,10 +72,10 @@ padBoth : ∀ {n} → A → A → Vec≤ A n → Vec A n
 padBoth aₗ aᵣ as@(vs , m≤n)
   with recompute (_ ℕₚ.≤″? _) (ℕₚ.≤⇒≤″ m≤n)
 ... | less-than-or-equal {k} ∣as∣+k≡n
-  with split k ∣as∣+k≡n
-... | refl = Vec.replicate {n = ⌊ k /2⌋} aₗ
+    with split k ∣as∣+k≡n
+... | refl = Vec.replicate ⌊ k /2⌋ aₗ
       Vec.++ vs
-      Vec.++ Vec.replicate {n = ⌈ k /2⌉} aᵣ
+      Vec.++ Vec.replicate ⌈ k /2⌉ aᵣ
 
 
 fromList : (as : List A) → Vec≤ A (List.length as)
@@ -88,7 +88,7 @@ toList = Vec.toList ∘ Vec≤.vec
 -- Creating new Vec≤ vectors
 
 replicate : ∀ {m n} .(m≤n : m ≤ n) → A → Vec≤ A n
-replicate m≤n a = Vec.replicate a , m≤n
+replicate m≤n a = Vec.replicate _ a , m≤n
 
 [] : ∀ {n} → Vec≤ A n
 [] = Vec.[] , z≤n
diff --git a/src/Data/Vec/Effectful.agda b/src/Data/Vec/Effectful.agda
index 3d4c1a75e7..1ff96bd03c 100644
--- a/src/Data/Vec/Effectful.agda
+++ b/src/Data/Vec/Effectful.agda
@@ -34,9 +34,9 @@ functor = record
   }
 
 applicative : RawApplicative (λ (A : Set a) → Vec A n)
-applicative = record
+applicative {n = n} = record
   { rawFunctor = functor
-  ; pure = replicate
+  ; pure = replicate n
   ; _<*>_  = Vec._⊛_
   }
 
diff --git a/src/Data/Vec/Effectful/Transformer.agda b/src/Data/Vec/Effectful/Transformer.agda
index cf491fdb84..637ec0b2ec 100644
--- a/src/Data/Vec/Effectful/Transformer.agda
+++ b/src/Data/Vec/Effectful/Transformer.agda
@@ -40,7 +40,7 @@ functor M = record
 applicative : RawApplicative M → RawApplicative {f} (VecT n M)
 applicative M = record
   { rawFunctor = functor rawFunctor
-  ; pure = mkVecT ∘′ pure ∘′ Vec.replicate
+  ; pure = mkVecT ∘′ pure ∘′ Vec.replicate _
   ; _<*>_  = λ mf ma → mkVecT (Vec.zipWith _$_ <$> runVecT mf <*> runVecT ma)
   } where open RawApplicative M
 
@@ -55,6 +55,6 @@ monad {f} {g} M = record
 
 monadT : RawMonadT {f} {g} (VecT n)
 monadT M = record
-  { lift = mkVecT ∘′ (Vec.replicate <$>_)
+  { lift = mkVecT ∘′ (Vec.replicate _ <$>_)
   ; rawMonad = monad M
   } where open RawMonad M
diff --git a/src/Data/Vec/Functional.agda b/src/Data/Vec/Functional.agda
index 7b217a415b..776fedee3b 100644
--- a/src/Data/Vec/Functional.agda
+++ b/src/Data/Vec/Functional.agda
@@ -77,8 +77,8 @@ tail xs = xs ∘ suc
 uncons : Vector A (suc n) → A × Vector A n
 uncons xs = head xs , tail xs
 
-replicate : A → Vector A n
-replicate = const
+replicate : (n : ℕ) → A → Vector A n
+replicate n = const
 
 insertAt : Vector A n → Fin (suc n) → A → Vector A (suc n)
 insertAt {n = n}     xs zero    v zero    = v
diff --git a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
index 148fcf346f..ced1999e3a 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
@@ -142,7 +142,7 @@ module _ (R : REL A B r) where
 
 module _ {R : REL A B r} {x y n} where
 
-  replicate⁺ : R x y → Pointwise R {n = n} (replicate x) (replicate y)
+  replicate⁺ : R x y → Pointwise R {n = n} (replicate n x) (replicate n y)
   replicate⁺ = const
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda b/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
index 85b5075142..be031a1070 100644
--- a/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Unary/All/Properties.agda
@@ -8,6 +8,7 @@
 
 module Data.Vec.Functional.Relation.Unary.All.Properties where
 
+open import Data.Nat.Base using (ℕ)
 open import Data.Fin.Base using (zero; suc; _↑ˡ_; _↑ʳ_; splitAt)
 open import Data.Fin.Properties using (splitAt-↑ˡ; splitAt-↑ʳ)
 open import Data.Product.Base as Σ using (_×_; _,_; proj₁; proj₂; uncurry)
@@ -20,80 +21,73 @@ open import Relation.Unary
 
 private
   variable
-    a b c p q r ℓ : Level
-    A : Set a
-    B : Set b
-    C : Set c
+    a p ℓ : Level
+    A B C : Set a
+    P Q R : Pred A p
+    m n : ℕ
+    x y : A
+    xs ys : Vector A n
 
 ------------------------------------------------------------------------
 -- map
 
-module _ {P : Pred A p} {Q : Pred B q} {f : A → B} where
-
-  map⁺ :  (∀ {x} → P x → Q (f x)) →
-          ∀ {n xs} → All P {n = n} xs → All Q (VF.map f xs)
-  map⁺ pq ps i = pq (ps i)
+map⁺ :  ∀ {f : A → B} → (∀ {x} → P x → Q (f x)) →
+        All P xs → All Q (VF.map f xs)
+map⁺ pq ps i = pq (ps i)
 
 ------------------------------------------------------------------------
 -- replicate
 
-module _ {P : Pred A p} {x : A} {n} where
-
-  replicate⁺ : P x → All P (replicate {n = n} x)
-  replicate⁺ = const
+replicate⁺ : P x → All P (replicate n x)
+replicate⁺ = const
 
 ------------------------------------------------------------------------
 -- _⊛_
 
-module _ {P : Pred A p} {Q : Pred B q} where
-
-  ⊛⁺ : ∀ {n} {fs : Vector (A → B) n} {xs} →
-       All (λ f → ∀ {x} → P x → Q (f x)) fs → All P xs → All Q (fs ⊛ xs)
-  ⊛⁺ pqs ps i = (pqs i) (ps i)
+⊛⁺ : ∀ {fs : Vector (A → B) n} →
+     All (λ f → ∀ {x} → P x → Q (f x)) fs →
+     All P xs → All Q (fs ⊛ xs)
+⊛⁺ pqs ps i = (pqs i) (ps i)
 
 ------------------------------------------------------------------------
 -- zipWith
 
-module _ {P : Pred A p} {Q : Pred B q} {R : Pred C r} where
-
-  zipWith⁺ : ∀ {f} → (∀ {x y} → P x → Q y → R (f x y)) → ∀ {n xs ys} →
-             All P xs → All Q ys → All R (zipWith f {n = n} xs ys)
-  zipWith⁺ pqr ps qs i = pqr (ps i) (qs i)
+zipWith⁺ : ∀ {f} → (∀ {x y} → P x → Q y → R (f x y)) →
+           All P xs → All Q ys → All R (zipWith f xs ys)
+zipWith⁺ pqr ps qs i = pqr (ps i) (qs i)
 
 ------------------------------------------------------------------------
 -- zip
 
-module _ {P : Pred A p} {Q : Pred B q} {n} {xs : Vector A n} {ys} where
-
-  zip⁺ : All P xs → All Q ys → All (P ⟨×⟩ Q) (zip xs ys)
-  zip⁺ ps qs i = ps i , qs i
+zip⁺ : All P xs → All Q ys → All (P ⟨×⟩ Q) (zip xs ys)
+zip⁺ ps qs i = ps i , qs i
 
-  zip⁻ : All (P ⟨×⟩ Q) (zip xs ys) → All P xs × All Q ys
-  zip⁻ pqs = proj₁ ∘ pqs , proj₂ ∘ pqs
+zip⁻ : All (P ⟨×⟩ Q) (zip xs ys) → All P xs × All Q ys
+zip⁻ pqs = proj₁ ∘ pqs , proj₂ ∘ pqs
 
 ------------------------------------------------------------------------
 -- head
 
-head⁺ : ∀ (P : Pred A p) {n v} → All P v → P (head {n = n} v)
+head⁺ : ∀ (P : Pred A p) → All P xs → P (head xs)
 head⁺ P ps = ps zero
 
 ------------------------------------------------------------------------
 -- tail
 
-tail⁺ : ∀ (P : Pred A p) {n v} → All P v → All P (tail {n = n} v)
-tail⁺ P ps = ps ∘ suc
+tail⁺ : ∀ (P : Pred A p) → All P xs → All P (tail xs)
+tail⁺ P xs = xs ∘ suc
 
 ------------------------------------------------------------------------
 -- ++
 
-module _ (P : Pred A p) {m n} {xs : Vector A m} {ys : Vector A n} where
+module _ (P : Pred A p) {xs : Vector A m} {ys : Vector A n} where
 
   ++⁺ : All P xs → All P ys → All P (xs ++ ys)
   ++⁺ pxs pys i with splitAt m i
   ... | inj₁ i′ = pxs i′
   ... | inj₂ j′ = pys j′
 
-module _ (P : Pred A p) {m n} (xs : Vector A m) {ys : Vector A n} where
+module _ (P : Pred A p) (xs : Vector A m) {ys : Vector A n} where
 
   ++⁻ˡ : All P (xs ++ ys) → All P xs
   ++⁻ˡ ps i with ps (i ↑ˡ n)
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index c48abc6f57..27df432105 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -143,7 +143,7 @@ padRight-refl : (a : A) (xs : Vec A n) → padRight ≤-refl a xs ≡ xs
 padRight-refl a []       = refl
 padRight-refl a (x ∷ xs) = cong (x ∷_) (padRight-refl a xs)
 
-padRight-replicate : (m≤n : m ≤ n) (a : A) → replicate a ≡ padRight m≤n a (replicate a)
+padRight-replicate : (m≤n : m ≤ n) (a : A) → replicate n a ≡ padRight m≤n a (replicate m a)
 padRight-replicate z≤n       a = refl
 padRight-replicate (s≤s m≤n) a = cong (a ∷_) (padRight-replicate m≤n a)
 
@@ -385,7 +385,7 @@ map-id : map id ≗ id {A = Vec A n}
 map-id []       = refl
 map-id (x ∷ xs) = cong (x ∷_) (map-id xs)
 
-map-const : ∀ (xs : Vec A n) (y : B) → map (const y) xs ≡ replicate y
+map-const : ∀ (xs : Vec A n) (y : B) → map (const y) xs ≡ replicate n y
 map-const []       _ = refl
 map-const (_ ∷ xs) y = cong (y ∷_) (map-const xs y)
 
@@ -569,25 +569,25 @@ module _ {f : A → A → A} where
 module _ {f : A → A → A} {e : A} where
 
   zipWith-identityˡ : LeftIdentity _≡_ e f →
-                      LeftIdentity _≡_ (replicate e) (zipWith {n = n} f)
+                      LeftIdentity _≡_ (replicate n e) (zipWith f)
   zipWith-identityˡ idˡ []       = refl
   zipWith-identityˡ idˡ (x ∷ xs) =
     cong₂ _∷_ (idˡ x) (zipWith-identityˡ idˡ xs)
 
   zipWith-identityʳ : RightIdentity _≡_ e f →
-                      RightIdentity _≡_ (replicate e) (zipWith {n = n} f)
+                      RightIdentity _≡_ (replicate n e) (zipWith f)
   zipWith-identityʳ idʳ []       = refl
   zipWith-identityʳ idʳ (x ∷ xs) =
     cong₂ _∷_ (idʳ x) (zipWith-identityʳ idʳ xs)
 
   zipWith-zeroˡ : LeftZero _≡_ e f →
-                  LeftZero _≡_ (replicate e) (zipWith {n = n} f)
+                  LeftZero _≡_ (replicate n e) (zipWith f)
   zipWith-zeroˡ zeˡ []       = refl
   zipWith-zeroˡ zeˡ (x ∷ xs) =
     cong₂ _∷_ (zeˡ x) (zipWith-zeroˡ zeˡ xs)
 
   zipWith-zeroʳ : RightZero _≡_ e f →
-                  RightZero _≡_ (replicate e) (zipWith {n = n} f)
+                  RightZero _≡_ (replicate n e) (zipWith f)
   zipWith-zeroʳ zeʳ []       = refl
   zipWith-zeroʳ zeʳ (x ∷ xs) =
     cong₂ _∷_ (zeʳ x) (zipWith-zeroʳ zeʳ xs)
@@ -595,13 +595,13 @@ module _ {f : A → A → A} {e : A} where
 module _ {f : A → A → A} {e : A} {⁻¹ : A → A} where
 
   zipWith-inverseˡ : LeftInverse _≡_ e ⁻¹ f →
-                     LeftInverse _≡_ (replicate {n = n} e) (map ⁻¹) (zipWith f)
+                     LeftInverse _≡_ (replicate n e) (map ⁻¹) (zipWith f)
   zipWith-inverseˡ invˡ []       = refl
   zipWith-inverseˡ invˡ (x ∷ xs) =
     cong₂ _∷_ (invˡ x) (zipWith-inverseˡ invˡ xs)
 
   zipWith-inverseʳ : RightInverse _≡_ e ⁻¹ f →
-                     RightInverse _≡_ (replicate {n = n} e) (map ⁻¹) (zipWith f)
+                     RightInverse _≡_ (replicate n e) (map ⁻¹) (zipWith f)
   zipWith-inverseʳ invʳ []       = refl
   zipWith-inverseʳ invʳ (x ∷ xs) =
     cong₂ _∷_ (invʳ x) (zipWith-inverseʳ invʳ xs)
@@ -734,7 +734,7 @@ lookup-⊛ zero    (f ∷ fs) (x ∷ xs) = refl
 lookup-⊛ (suc i) (f ∷ fs) (x ∷ xs) = lookup-⊛ i fs xs
 
 map-is-⊛ : ∀ (f : A → B) (xs : Vec A n) →
-           map f xs ≡ (replicate f ⊛ xs)
+           map f xs ≡ (replicate n f ⊛ xs)
 map-is-⊛ f []       = refl
 map-is-⊛ f (x ∷ xs) = cong (_ ∷_) (map-is-⊛ f xs)
 
@@ -744,7 +744,7 @@ map-is-⊛ f (x ∷ xs) = cong (_ ∷_) (map-is-⊛ f xs)
 ⊛-is-zipWith (f ∷ fs) (x ∷ xs) = cong (f x ∷_) (⊛-is-zipWith fs xs)
 
 zipWith-is-⊛ : ∀ (f : A → B → C) (xs : Vec A n) (ys : Vec B n) →
-               zipWith f xs ys ≡ (replicate f ⊛ xs ⊛ ys)
+               zipWith f xs ys ≡ (replicate n f ⊛ xs ⊛ ys)
 zipWith-is-⊛ f []       []       = refl
 zipWith-is-⊛ f (x ∷ xs) (y ∷ ys) = cong (_ ∷_) (zipWith-is-⊛ f xs ys)
 
@@ -1107,51 +1107,51 @@ sum-++ {ys = ys} (x ∷ xs) = begin
 ------------------------------------------------------------------------
 -- replicate
 
-lookup-replicate : ∀ (i : Fin n) (x : A) → lookup (replicate x) i ≡ x
+lookup-replicate : ∀ (i : Fin n) (x : A) → lookup (replicate n x) i ≡ x
 lookup-replicate zero    x = refl
 lookup-replicate (suc i) x = lookup-replicate i x
 
 map-replicate :  ∀ (f : A → B) (x : A) n →
-                 map f (replicate x) ≡ replicate {n = n} (f x)
+                 map f (replicate n x) ≡ replicate n (f x)
 map-replicate f x zero = refl
 map-replicate f x (suc n) = cong (f x ∷_) (map-replicate f x n)
 
 transpose-replicate : ∀ (xs : Vec A m) →
-                      transpose (replicate {n = n} xs) ≡ map replicate xs
+                      transpose (replicate n xs) ≡ map (replicate n) xs
 transpose-replicate {n = zero}  _  = sym (map-const _ [])
 transpose-replicate {n = suc n} xs = begin
-  transpose (replicate xs)                        ≡⟨⟩
-  (replicate _∷_ ⊛ xs ⊛ transpose (replicate xs)) ≡⟨ cong₂ _⊛_ (sym (map-is-⊛ _∷_ xs)) (transpose-replicate xs) ⟩
-  (map _∷_ xs ⊛ map replicate xs)                 ≡⟨ map-⊛ _∷_ replicate xs ⟩
-  map replicate xs                                ∎
+  transpose (replicate (suc n) xs)                    ≡⟨⟩
+  (replicate _ _∷_ ⊛ xs ⊛ transpose (replicate _ xs)) ≡⟨ cong₂ _⊛_ (sym (map-is-⊛ _∷_ xs)) (transpose-replicate xs) ⟩
+  (map _∷_ xs ⊛ map (replicate n) xs)                 ≡⟨ map-⊛ _∷_ (replicate n) xs ⟩
+  map (replicate (suc n)) xs                          ∎
   where open ≡-Reasoning
 
 zipWith-replicate : ∀ (_⊕_ : A → B → C) (x : A) (y : B) →
-                    zipWith {n = n} _⊕_ (replicate x) (replicate y) ≡ replicate (x ⊕ y)
+                    zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)
 zipWith-replicate {n = zero}  _⊕_ x y = refl
 zipWith-replicate {n = suc n} _⊕_ x y = cong (x ⊕ y ∷_) (zipWith-replicate _⊕_ x y)
 
 zipWith-replicate₁ : ∀ (_⊕_ : A → B → C) (x : A) (ys : Vec B n) →
-                     zipWith _⊕_ (replicate x) ys ≡ map (x ⊕_) ys
+                     zipWith _⊕_ (replicate n x) ys ≡ map (x ⊕_) ys
 zipWith-replicate₁ _⊕_ x []       = refl
 zipWith-replicate₁ _⊕_ x (y ∷ ys) =
   cong (x ⊕ y ∷_) (zipWith-replicate₁ _⊕_ x ys)
 
 zipWith-replicate₂ : ∀ (_⊕_ : A → B → C) (xs : Vec A n) (y : B) →
-                     zipWith _⊕_ xs (replicate y) ≡ map (_⊕ y) xs
+                     zipWith _⊕_ xs (replicate n y) ≡ map (_⊕ y) xs
 zipWith-replicate₂ _⊕_ []       y = refl
 zipWith-replicate₂ _⊕_ (x ∷ xs) y =
   cong (x ⊕ y ∷_) (zipWith-replicate₂ _⊕_ xs y)
 
 toList-replicate : ∀ (n : ℕ) (x : A) →
-                   toList (replicate {n = n} a) ≡ List.replicate n a
+                   toList (replicate n a) ≡ List.replicate n a
 toList-replicate zero    x = refl
 toList-replicate (suc n) x = cong (_ List.∷_) (toList-replicate n x)
 
 ------------------------------------------------------------------------
 -- iterate
 
-iterate-id : ∀ (x : A) n → iterate id x n ≡ replicate x
+iterate-id : ∀ (x : A) n → iterate id x n ≡ replicate n x
 iterate-id x zero    = refl
 iterate-id x (suc n) = cong (_ ∷_) (iterate-id (id x) n)
 
diff --git a/src/Data/Vec/Recursive.agda b/src/Data/Vec/Recursive.agda
index baaf4eca37..ea7d04285f 100644
--- a/src/Data/Vec/Recursive.agda
+++ b/src/Data/Vec/Recursive.agda
@@ -90,7 +90,7 @@ lookup as (zero {n})  = head n as
 lookup as (suc {n} k) = lookup (tail n as) k
 
 replicate : ∀ n → A → A ^ n
-replicate n a = fromVec (Vec.replicate a)
+replicate n a = fromVec (Vec.replicate n a)
 
 tabulate : ∀ n → (Fin n → A) → A ^ n
 tabulate n f = fromVec (Vec.tabulate f)
diff --git a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
index 7e2e743088..28f022454b 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
@@ -96,8 +96,8 @@ open PW public using (concat⁺; concat⁻)
 -- replicate
 
 replicate-shiftʳ : ∀ {m} n x (xs : Vec A m) →
-                  replicate {n = n}     x ++ (x ∷ xs) ≋
-                  replicate {n = 1 + n} x ++      xs
+                  replicate n       x ++ (x ∷ xs) ≋
+                  replicate (1 + n) x ++       xs
 replicate-shiftʳ zero    x xs = ≋-refl
 replicate-shiftʳ (suc n) x xs = refl ∷ (replicate-shiftʳ n x xs)
 

From 9bd3b431b742db8d31efe7abc603929ce218c8e6 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 11 Oct 2023 03:00:44 +0100
Subject: [PATCH 150/316] [fixes #2130] Moving `Properties.HeytingAlgebra` from
 `Relation.Binary` to `Relation.Binary.Lattice` (#2131)

---
 CHANGELOG.md                                  |   1 +
 .../Lattice/Properties/HeytingAlgebra.agda    | 199 ++++++++++++++++++
 .../Binary/Properties/HeytingAlgebra.agda     | 195 +----------------
 3 files changed, 208 insertions(+), 187 deletions(-)
 create mode 100644 src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 3d99a6eb2b..81781828c2 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1253,6 +1253,7 @@ Deprecated modules
   Relation.Binary.Properties.BoundedLattice.agda          ↦ Relation.Binary.Lattice.Properties.BoundedLattice.agda
   Relation.Binary.Properties.BoundedMeetSemilattice.agda  ↦ Relation.Binary.Lattice.Properties.BoundedMeetSemilattice.agda
   Relation.Binary.Properties.DistributiveLattice.agda     ↦ Relation.Binary.Lattice.Properties.DistributiveLattice.agda
+  Relation.Binary.Properties.HeytingAlgebra.agda         ↦ Relation.Binary.Lattice.Properties.HeytingAlgebra.agda
   Relation.Binary.Properties.JoinSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.JoinSemilattice.agda
   Relation.Binary.Properties.Lattice.agda                 ↦ Relation.Binary.Lattice.Properties.Lattice.agda
   Relation.Binary.Properties.MeetSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.MeetSemilattice.agda
diff --git a/src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda b/src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda
new file mode 100644
index 0000000000..e7090dc6c5
--- /dev/null
+++ b/src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda
@@ -0,0 +1,199 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties satisfied by Heyting Algebra
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Lattice
+
+module Relation.Binary.Lattice.Properties.HeytingAlgebra
+  {c ℓ₁ ℓ₂} (L : HeytingAlgebra c ℓ₁ ℓ₂) where
+
+open HeytingAlgebra L
+
+open import Algebra.Core
+open import Algebra.Definitions _≈_
+open import Data.Product.Base using (_,_)
+open import Function.Base using (_$_; flip; _∘_)
+open import Level using (_⊔_)
+open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
+import Relation.Binary.Reasoning.PartialOrder as POR
+open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice
+open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
+import Relation.Binary.Lattice.Properties.BoundedMeetSemilattice boundedMeetSemilattice as BM
+open import Relation.Binary.Lattice.Properties.Lattice lattice
+open import Relation.Binary.Lattice.Properties.BoundedLattice boundedLattice
+import Relation.Binary.Reasoning.Setoid as EqReasoning
+
+------------------------------------------------------------------------
+-- Useful lemmas
+
+⇨-eval : ∀ {x y} → (x ⇨ y) ∧ x ≤ y
+⇨-eval {x} {y} = transpose-∧ refl
+
+swap-transpose-⇨ : ∀ {x y w} → x ∧ w ≤ y → w ≤ x ⇨ y
+swap-transpose-⇨ x∧w≤y = transpose-⇨ $ trans (reflexive $ ∧-comm _ _) x∧w≤y
+
+------------------------------------------------------------------------
+-- Properties of exponential
+
+⇨-unit : ∀ {x} → x ⇨ x ≈ ⊤
+⇨-unit = antisym (maximum _) (transpose-⇨ $ reflexive $ BM.identityˡ _)
+
+y≤x⇨y : ∀ {x y} → y ≤ x ⇨ y
+y≤x⇨y = transpose-⇨ (x∧y≤x _ _)
+
+⇨-drop : ∀ {x y} → (x ⇨ y) ∧ y ≈ y
+⇨-drop = antisym (x∧y≤y _ _) (∧-greatest y≤x⇨y refl)
+
+⇨-app : ∀ {x y} → (x ⇨ y) ∧ x ≈ y ∧ x
+⇨-app = antisym (∧-greatest ⇨-eval (x∧y≤y _ _)) (∧-monotonic y≤x⇨y refl)
+
+⇨ʳ-covariant : ∀ {x} → (x ⇨_) Preserves _≤_ ⟶ _≤_
+⇨ʳ-covariant y≤z = transpose-⇨ (trans ⇨-eval y≤z)
+
+⇨ˡ-contravariant : ∀ {x} → (_⇨ x) Preserves (flip _≤_) ⟶ _≤_
+⇨ˡ-contravariant z≤y = transpose-⇨ (trans (∧-monotonic refl z≤y) ⇨-eval)
+
+⇨-relax : _⇨_ Preserves₂ (flip _≤_) ⟶ _≤_ ⟶ _≤_
+⇨-relax {x} {y} {u} {v} y≤x u≤v = begin
+  x ⇨ u ≤⟨ ⇨ʳ-covariant u≤v ⟩
+  x ⇨ v ≤⟨ ⇨ˡ-contravariant y≤x ⟩
+  y ⇨ v ∎
+  where open POR poset
+
+⇨-cong : _⇨_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_
+⇨-cong x≈y u≈v = antisym (⇨-relax (reflexive $ Eq.sym x≈y) (reflexive u≈v))
+                         (⇨-relax (reflexive x≈y) (reflexive $ Eq.sym u≈v))
+
+⇨-applyˡ : ∀ {w x y} → w ≤ x → (x ⇨ y) ∧ w ≤ y
+⇨-applyˡ = transpose-∧ ∘ ⇨ˡ-contravariant
+
+⇨-applyʳ : ∀ {w x y} → w ≤ x → w ∧ (x ⇨ y) ≤ y
+⇨-applyʳ w≤x = trans (reflexive (∧-comm _ _)) (⇨-applyˡ w≤x)
+
+⇨-curry : ∀ {x y z} → x ∧ y ⇨ z ≈ x ⇨ y ⇨ z
+⇨-curry = antisym (transpose-⇨ $ transpose-⇨ $ trans (reflexive $ ∧-assoc _ _ _) ⇨-eval)
+                  (transpose-⇨ $ trans (reflexive $ Eq.sym $ ∧-assoc _ _ _)
+                                       (transpose-∧ $ ⇨-applyˡ refl))
+
+------------------------------------------------------------------------
+-- Various proofs of distributivity
+
+∧-distribˡ-∨-≤ : ∀ x y z → x ∧ (y ∨ z) ≤ x ∧ y ∨ x ∧ z
+∧-distribˡ-∨-≤ x y z = trans (reflexive $ ∧-comm _ _)
+  (transpose-∧ $ ∨-least (swap-transpose-⇨ (x≤x∨y _ _)) $ swap-transpose-⇨ (y≤x∨y _ _))
+
+∧-distribˡ-∨-≥ : ∀ x y z → x ∧ y ∨ x ∧ z ≤ x ∧ (y ∨ z)
+∧-distribˡ-∨-≥ x y z = let
+    x∧y≤x , x∧y≤y , _ = infimum x y
+    x∧z≤x , x∧z≤z , _ = infimum x z
+    y≤y∨z , z≤y∨z , _ = supremum y z
+  in ∧-greatest (∨-least x∧y≤x x∧z≤x)
+                (∨-least (trans x∧y≤y y≤y∨z) (trans x∧z≤z z≤y∨z))
+
+∧-distribˡ-∨ : _∧_ DistributesOverˡ _∨_
+∧-distribˡ-∨ x y z = antisym (∧-distribˡ-∨-≤ x y z) (∧-distribˡ-∨-≥ x y z)
+
+⇨-distribˡ-∧-≤ : ∀ x y z → x ⇨ y ∧ z ≤ (x ⇨ y) ∧ (x ⇨ z)
+⇨-distribˡ-∧-≤ x y z = let
+     y∧z≤y , y∧z≤z , _ = infimum y z
+   in ∧-greatest (transpose-⇨ $ trans ⇨-eval y∧z≤y)
+                 (transpose-⇨ $ trans ⇨-eval y∧z≤z)
+
+⇨-distribˡ-∧-≥ : ∀ x y z → (x ⇨ y) ∧ (x ⇨ z) ≤ x ⇨ y ∧ z
+⇨-distribˡ-∧-≥ x y z = transpose-⇨ (begin
+  (((x ⇨ y) ∧ (x ⇨ z)) ∧ x)      ≈⟨ ∧-cong Eq.refl $ Eq.sym $ ∧-idempotent _ ⟩
+  (((x ⇨ y) ∧ (x ⇨ z)) ∧ x  ∧ x) ≈⟨ Eq.sym $ ∧-assoc _ _ _ ⟩
+  (((x ⇨ y) ∧ (x ⇨ z)) ∧ x) ∧ x  ≈⟨ ∧-cong (∧-assoc _ _ _) Eq.refl ⟩
+  (((x ⇨ y) ∧ (x ⇨ z)  ∧ x) ∧ x) ≈⟨ ∧-cong (∧-cong Eq.refl $ ∧-comm _ _) Eq.refl ⟩
+  (((x ⇨ y) ∧ x  ∧ (x ⇨ z)) ∧ x) ≈⟨ ∧-cong (Eq.sym $ ∧-assoc _ _ _) Eq.refl ⟩
+  (((x ⇨ y) ∧ x) ∧ (x ⇨ z)) ∧ x  ≈⟨ ∧-assoc _ _ _ ⟩
+  (((x ⇨ y) ∧ x) ∧ (x ⇨ z)  ∧ x) ≤⟨ ∧-monotonic ⇨-eval ⇨-eval ⟩
+  y ∧ z                          ∎)
+  where open POR poset
+
+⇨-distribˡ-∧ : _⇨_ DistributesOverˡ _∧_
+⇨-distribˡ-∧ x y z = antisym (⇨-distribˡ-∧-≤ x y z) (⇨-distribˡ-∧-≥ x y z)
+
+⇨-distribˡ-∨-∧-≤ : ∀ x y z → x ∨ y ⇨ z ≤ (x ⇨ z) ∧ (y ⇨ z)
+⇨-distribˡ-∨-∧-≤ x y z = let x≤x∨y , y≤x∨y , _ = supremum x y
+   in ∧-greatest (transpose-⇨ $ trans (∧-monotonic refl x≤x∨y) ⇨-eval)
+                 (transpose-⇨ $ trans (∧-monotonic refl y≤x∨y) ⇨-eval)
+
+⇨-distribˡ-∨-∧-≥ : ∀ x y z → (x ⇨ z) ∧ (y ⇨ z) ≤ x ∨ y ⇨ z
+⇨-distribˡ-∨-∧-≥ x y z = transpose-⇨ (trans (reflexive $ ∧-distribˡ-∨ _ _ _)
+  (∨-least (trans (transpose-∧ (x∧y≤x _ _)) refl)
+           (trans (transpose-∧ (x∧y≤y _ _)) refl)))
+
+⇨-distribˡ-∨-∧ : ∀ x y z → x ∨ y ⇨ z ≈ (x ⇨ z) ∧ (y ⇨ z)
+⇨-distribˡ-∨-∧ x y z = antisym (⇨-distribˡ-∨-∧-≤ x y z) (⇨-distribˡ-∨-∧-≥ x y z)
+
+------------------------------------------------------------------------
+-- Heyting algebras are distributive lattices
+
+isDistributiveLattice : IsDistributiveLattice _≈_ _≤_ _∨_ _∧_
+isDistributiveLattice = record
+  { isLattice    = isLattice
+  ; ∧-distribˡ-∨ = ∧-distribˡ-∨
+  }
+
+distributiveLattice : DistributiveLattice _ _ _
+distributiveLattice = record
+  { isDistributiveLattice = isDistributiveLattice
+  }
+
+------------------------------------------------------------------------
+-- Heyting algebras can define pseudo-complement
+
+infix 8 ¬_
+
+¬_ : Op₁ Carrier
+¬ x = x ⇨ ⊥
+
+x≤¬¬x : ∀ x → x ≤ ¬ ¬ x
+x≤¬¬x x = transpose-⇨ (trans (reflexive (∧-comm _ _)) ⇨-eval)
+
+------------------------------------------------------------------------
+-- De-Morgan laws
+
+de-morgan₁ : ∀ x y → ¬ (x ∨ y) ≈ ¬ x ∧ ¬ y
+de-morgan₁ x y = ⇨-distribˡ-∨-∧ _ _ _
+
+de-morgan₂-≤ : ∀ x y → ¬ (x ∧ y) ≤ ¬ ¬ (¬ x ∨ ¬ y)
+de-morgan₂-≤ x y = transpose-⇨ $ begin
+  ¬ (x ∧ y) ∧ ¬ (¬ x ∨ ¬ y)     ≈⟨ ∧-cong ⇨-curry (de-morgan₁ _ _) ⟩
+  (x ⇨ ¬ y) ∧ ¬ ¬ x ∧ ¬ ¬ y     ≈⟨ ∧-cong Eq.refl (∧-comm _ _) ⟩
+  (x ⇨ ¬ y) ∧ ¬ ¬ y ∧ ¬ ¬ x     ≈⟨ Eq.sym $ ∧-assoc _ _ _ ⟩
+  ((x ⇨ ¬ y) ∧ ¬ ¬ y) ∧ ¬ ¬ x   ≤⟨ ⇨-applyʳ $ transpose-⇨ $
+    begin
+      ((x ⇨ ¬ y) ∧ ¬ ¬ y) ∧ x   ≈⟨ ∧-cong (∧-comm _ _) Eq.refl ⟩
+      ((¬ ¬ y) ∧ (x ⇨ ¬ y)) ∧ x ≈⟨ ∧-assoc _ _ _ ⟩
+      (¬ ¬ y) ∧ (x ⇨ ¬ y) ∧ x   ≤⟨ ∧-monotonic refl ⇨-eval ⟩
+      ¬ ¬ y ∧ ¬ y               ≤⟨ ⇨-eval ⟩
+      ⊥                         ∎ ⟩
+  ⊥                             ∎
+  where open POR poset
+
+de-morgan₂-≥ : ∀ x y → ¬ ¬ (¬ x ∨ ¬ y) ≤ ¬ (x ∧ y)
+de-morgan₂-≥ x y = transpose-⇨ $ ⇨-applyˡ $ transpose-⇨ $ begin
+  (x ∧ y) ∧ (¬ x ∨ ¬ y)         ≈⟨ ∧-distribˡ-∨ _ _ _ ⟩
+  (x ∧ y) ∧ ¬ x ∨ (x ∧ y) ∧ ¬ y ≤⟨ ∨-monotonic (⇨-applyʳ (x∧y≤x _ _))
+                                               (⇨-applyʳ (x∧y≤y _ _)) ⟩
+  ⊥ ∨ ⊥                         ≈⟨ ∨-idempotent _ ⟩
+  ⊥                             ∎
+  where open POR poset
+
+de-morgan₂ : ∀ x y → ¬ (x ∧ y) ≈ ¬ ¬ (¬ x ∨ ¬ y)
+de-morgan₂ x y = antisym (de-morgan₂-≤ x y) (de-morgan₂-≥ x y)
+
+weak-lem : ∀ {x} → ¬ ¬ (¬ x ∨ x) ≈ ⊤
+weak-lem {x} = begin
+  ¬ ¬ (¬ x ∨ x)   ≈⟨ ⇨-cong (de-morgan₁ _ _) Eq.refl ⟩
+  ¬ (¬ ¬ x ∧ ¬ x) ≈⟨ ⇨-cong ⇨-app Eq.refl ⟩
+  ⊥ ∧ (x ⇨ ⊥) ⇨ ⊥ ≈⟨ ⇨-cong (∧-zeroˡ _) Eq.refl ⟩
+  ⊥ ⇨ ⊥           ≈⟨ ⇨-unit ⟩
+  ⊤               ∎
+  where open EqReasoning setoid
diff --git a/src/Relation/Binary/Properties/HeytingAlgebra.agda b/src/Relation/Binary/Properties/HeytingAlgebra.agda
index 87cb01899d..7ec65c0675 100644
--- a/src/Relation/Binary/Properties/HeytingAlgebra.agda
+++ b/src/Relation/Binary/Properties/HeytingAlgebra.agda
@@ -1,199 +1,20 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Properties satisfied by Heyting Algebra
+-- This module is DEPRECATED. Please use
+-- `Relation.Binary.Lattice.Properties.HeytingAlgebra` instead.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Lattice
+open import Relation.Binary.Lattice using (HeytingAlgebra)
 
 module Relation.Binary.Properties.HeytingAlgebra
   {c ℓ₁ ℓ₂} (L : HeytingAlgebra c ℓ₁ ℓ₂) where
 
-open HeytingAlgebra L
+open import Relation.Binary.Lattice.Properties.HeytingAlgebra L public
 
-open import Algebra.Core
-open import Algebra.Definitions _≈_
-open import Data.Product.Base using (_,_)
-open import Function.Base using (_$_; flip; _∘_)
-open import Level using (_⊔_)
-open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-import Relation.Binary.Reasoning.PartialOrder as POR
-open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice
-open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
-import Relation.Binary.Lattice.Properties.BoundedMeetSemilattice boundedMeetSemilattice as BM
-open import Relation.Binary.Lattice.Properties.Lattice lattice
-open import Relation.Binary.Lattice.Properties.BoundedLattice boundedLattice
-import Relation.Binary.Reasoning.Setoid as EqReasoning
-
-------------------------------------------------------------------------
--- Useful lemmas
-
-⇨-eval : ∀ {x y} → (x ⇨ y) ∧ x ≤ y
-⇨-eval {x} {y} = transpose-∧ refl
-
-swap-transpose-⇨ : ∀ {x y w} → x ∧ w ≤ y → w ≤ x ⇨ y
-swap-transpose-⇨ x∧w≤y = transpose-⇨ $ trans (reflexive $ ∧-comm _ _) x∧w≤y
-
-------------------------------------------------------------------------
--- Properties of exponential
-
-⇨-unit : ∀ {x} → x ⇨ x ≈ ⊤
-⇨-unit = antisym (maximum _) (transpose-⇨ $ reflexive $ BM.identityˡ _)
-
-y≤x⇨y : ∀ {x y} → y ≤ x ⇨ y
-y≤x⇨y = transpose-⇨ (x∧y≤x _ _)
-
-⇨-drop : ∀ {x y} → (x ⇨ y) ∧ y ≈ y
-⇨-drop = antisym (x∧y≤y _ _) (∧-greatest y≤x⇨y refl)
-
-⇨-app : ∀ {x y} → (x ⇨ y) ∧ x ≈ y ∧ x
-⇨-app = antisym (∧-greatest ⇨-eval (x∧y≤y _ _)) (∧-monotonic y≤x⇨y refl)
-
-⇨ʳ-covariant : ∀ {x} → (x ⇨_) Preserves _≤_ ⟶ _≤_
-⇨ʳ-covariant y≤z = transpose-⇨ (trans ⇨-eval y≤z)
-
-⇨ˡ-contravariant : ∀ {x} → (_⇨ x) Preserves (flip _≤_) ⟶ _≤_
-⇨ˡ-contravariant z≤y = transpose-⇨ (trans (∧-monotonic refl z≤y) ⇨-eval)
-
-⇨-relax : _⇨_ Preserves₂ (flip _≤_) ⟶ _≤_ ⟶ _≤_
-⇨-relax {x} {y} {u} {v} y≤x u≤v = begin
-  x ⇨ u ≤⟨ ⇨ʳ-covariant u≤v ⟩
-  x ⇨ v ≤⟨ ⇨ˡ-contravariant y≤x ⟩
-  y ⇨ v ∎
-  where open POR poset
-
-⇨-cong : _⇨_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_
-⇨-cong x≈y u≈v = antisym (⇨-relax (reflexive $ Eq.sym x≈y) (reflexive u≈v))
-                         (⇨-relax (reflexive x≈y) (reflexive $ Eq.sym u≈v))
-
-⇨-applyˡ : ∀ {w x y} → w ≤ x → (x ⇨ y) ∧ w ≤ y
-⇨-applyˡ = transpose-∧ ∘ ⇨ˡ-contravariant
-
-⇨-applyʳ : ∀ {w x y} → w ≤ x → w ∧ (x ⇨ y) ≤ y
-⇨-applyʳ w≤x = trans (reflexive (∧-comm _ _)) (⇨-applyˡ w≤x)
-
-⇨-curry : ∀ {x y z} → x ∧ y ⇨ z ≈ x ⇨ y ⇨ z
-⇨-curry = antisym (transpose-⇨ $ transpose-⇨ $ trans (reflexive $ ∧-assoc _ _ _) ⇨-eval)
-                  (transpose-⇨ $ trans (reflexive $ Eq.sym $ ∧-assoc _ _ _)
-                                       (transpose-∧ $ ⇨-applyˡ refl))
-
-------------------------------------------------------------------------
--- Various proofs of distributivity
-
-∧-distribˡ-∨-≤ : ∀ x y z → x ∧ (y ∨ z) ≤ x ∧ y ∨ x ∧ z
-∧-distribˡ-∨-≤ x y z = trans (reflexive $ ∧-comm _ _)
-  (transpose-∧ $ ∨-least (swap-transpose-⇨ (x≤x∨y _ _)) $ swap-transpose-⇨ (y≤x∨y _ _))
-
-∧-distribˡ-∨-≥ : ∀ x y z → x ∧ y ∨ x ∧ z ≤ x ∧ (y ∨ z)
-∧-distribˡ-∨-≥ x y z = let
-    x∧y≤x , x∧y≤y , _ = infimum x y
-    x∧z≤x , x∧z≤z , _ = infimum x z
-    y≤y∨z , z≤y∨z , _ = supremum y z
-  in ∧-greatest (∨-least x∧y≤x x∧z≤x)
-                (∨-least (trans x∧y≤y y≤y∨z) (trans x∧z≤z z≤y∨z))
-
-∧-distribˡ-∨ : _∧_ DistributesOverˡ _∨_
-∧-distribˡ-∨ x y z = antisym (∧-distribˡ-∨-≤ x y z) (∧-distribˡ-∨-≥ x y z)
-
-⇨-distribˡ-∧-≤ : ∀ x y z → x ⇨ y ∧ z ≤ (x ⇨ y) ∧ (x ⇨ z)
-⇨-distribˡ-∧-≤ x y z = let
-     y∧z≤y , y∧z≤z , _ = infimum y z
-   in ∧-greatest (transpose-⇨ $ trans ⇨-eval y∧z≤y)
-                 (transpose-⇨ $ trans ⇨-eval y∧z≤z)
-
-⇨-distribˡ-∧-≥ : ∀ x y z → (x ⇨ y) ∧ (x ⇨ z) ≤ x ⇨ y ∧ z
-⇨-distribˡ-∧-≥ x y z = transpose-⇨ (begin
-  (((x ⇨ y) ∧ (x ⇨ z)) ∧ x)      ≈⟨ ∧-cong Eq.refl $ Eq.sym $ ∧-idempotent _ ⟩
-  (((x ⇨ y) ∧ (x ⇨ z)) ∧ x  ∧ x) ≈⟨ Eq.sym $ ∧-assoc _ _ _ ⟩
-  (((x ⇨ y) ∧ (x ⇨ z)) ∧ x) ∧ x  ≈⟨ ∧-cong (∧-assoc _ _ _) Eq.refl ⟩
-  (((x ⇨ y) ∧ (x ⇨ z)  ∧ x) ∧ x) ≈⟨ ∧-cong (∧-cong Eq.refl $ ∧-comm _ _) Eq.refl ⟩
-  (((x ⇨ y) ∧ x  ∧ (x ⇨ z)) ∧ x) ≈⟨ ∧-cong (Eq.sym $ ∧-assoc _ _ _) Eq.refl ⟩
-  (((x ⇨ y) ∧ x) ∧ (x ⇨ z)) ∧ x  ≈⟨ ∧-assoc _ _ _ ⟩
-  (((x ⇨ y) ∧ x) ∧ (x ⇨ z)  ∧ x) ≤⟨ ∧-monotonic ⇨-eval ⇨-eval ⟩
-  y ∧ z                          ∎)
-  where open POR poset
-
-⇨-distribˡ-∧ : _⇨_ DistributesOverˡ _∧_
-⇨-distribˡ-∧ x y z = antisym (⇨-distribˡ-∧-≤ x y z) (⇨-distribˡ-∧-≥ x y z)
-
-⇨-distribˡ-∨-∧-≤ : ∀ x y z → x ∨ y ⇨ z ≤ (x ⇨ z) ∧ (y ⇨ z)
-⇨-distribˡ-∨-∧-≤ x y z = let x≤x∨y , y≤x∨y , _ = supremum x y
-   in ∧-greatest (transpose-⇨ $ trans (∧-monotonic refl x≤x∨y) ⇨-eval)
-                 (transpose-⇨ $ trans (∧-monotonic refl y≤x∨y) ⇨-eval)
-
-⇨-distribˡ-∨-∧-≥ : ∀ x y z → (x ⇨ z) ∧ (y ⇨ z) ≤ x ∨ y ⇨ z
-⇨-distribˡ-∨-∧-≥ x y z = transpose-⇨ (trans (reflexive $ ∧-distribˡ-∨ _ _ _)
-  (∨-least (trans (transpose-∧ (x∧y≤x _ _)) refl)
-           (trans (transpose-∧ (x∧y≤y _ _)) refl)))
-
-⇨-distribˡ-∨-∧ : ∀ x y z → x ∨ y ⇨ z ≈ (x ⇨ z) ∧ (y ⇨ z)
-⇨-distribˡ-∨-∧ x y z = antisym (⇨-distribˡ-∨-∧-≤ x y z) (⇨-distribˡ-∨-∧-≥ x y z)
-
-------------------------------------------------------------------------
--- Heyting algebras are distributive lattices
-
-isDistributiveLattice : IsDistributiveLattice _≈_ _≤_ _∨_ _∧_
-isDistributiveLattice = record
-  { isLattice    = isLattice
-  ; ∧-distribˡ-∨ = ∧-distribˡ-∨
-  }
-
-distributiveLattice : DistributiveLattice _ _ _
-distributiveLattice = record
-  { isDistributiveLattice = isDistributiveLattice
-  }
-
-------------------------------------------------------------------------
--- Heyting algebras can define pseudo-complement
-
-infix 8 ¬_
-
-¬_ : Op₁ Carrier
-¬ x = x ⇨ ⊥
-
-x≤¬¬x : ∀ x → x ≤ ¬ ¬ x
-x≤¬¬x x = transpose-⇨ (trans (reflexive (∧-comm _ _)) ⇨-eval)
-
-------------------------------------------------------------------------
--- De-Morgan laws
-
-de-morgan₁ : ∀ x y → ¬ (x ∨ y) ≈ ¬ x ∧ ¬ y
-de-morgan₁ x y = ⇨-distribˡ-∨-∧ _ _ _
-
-de-morgan₂-≤ : ∀ x y → ¬ (x ∧ y) ≤ ¬ ¬ (¬ x ∨ ¬ y)
-de-morgan₂-≤ x y = transpose-⇨ $ begin
-  ¬ (x ∧ y) ∧ ¬ (¬ x ∨ ¬ y)     ≈⟨ ∧-cong ⇨-curry (de-morgan₁ _ _) ⟩
-  (x ⇨ ¬ y) ∧ ¬ ¬ x ∧ ¬ ¬ y     ≈⟨ ∧-cong Eq.refl (∧-comm _ _) ⟩
-  (x ⇨ ¬ y) ∧ ¬ ¬ y ∧ ¬ ¬ x     ≈⟨ Eq.sym $ ∧-assoc _ _ _ ⟩
-  ((x ⇨ ¬ y) ∧ ¬ ¬ y) ∧ ¬ ¬ x   ≤⟨ ⇨-applyʳ $ transpose-⇨ $
-    begin
-      ((x ⇨ ¬ y) ∧ ¬ ¬ y) ∧ x   ≈⟨ ∧-cong (∧-comm _ _) Eq.refl ⟩
-      ((¬ ¬ y) ∧ (x ⇨ ¬ y)) ∧ x ≈⟨ ∧-assoc _ _ _ ⟩
-      (¬ ¬ y) ∧ (x ⇨ ¬ y) ∧ x   ≤⟨ ∧-monotonic refl ⇨-eval ⟩
-      ¬ ¬ y ∧ ¬ y               ≤⟨ ⇨-eval ⟩
-      ⊥                         ∎ ⟩
-  ⊥                             ∎
-  where open POR poset
-
-de-morgan₂-≥ : ∀ x y → ¬ ¬ (¬ x ∨ ¬ y) ≤ ¬ (x ∧ y)
-de-morgan₂-≥ x y = transpose-⇨ $ ⇨-applyˡ $ transpose-⇨ $ begin
-  (x ∧ y) ∧ (¬ x ∨ ¬ y)         ≈⟨ ∧-distribˡ-∨ _ _ _ ⟩
-  (x ∧ y) ∧ ¬ x ∨ (x ∧ y) ∧ ¬ y ≤⟨ ∨-monotonic (⇨-applyʳ (x∧y≤x _ _))
-                                               (⇨-applyʳ (x∧y≤y _ _)) ⟩
-  ⊥ ∨ ⊥                         ≈⟨ ∨-idempotent _ ⟩
-  ⊥                             ∎
-  where open POR poset
-
-de-morgan₂ : ∀ x y → ¬ (x ∧ y) ≈ ¬ ¬ (¬ x ∨ ¬ y)
-de-morgan₂ x y = antisym (de-morgan₂-≤ x y) (de-morgan₂-≥ x y)
-
-weak-lem : ∀ {x} → ¬ ¬ (¬ x ∨ x) ≈ ⊤
-weak-lem {x} = begin
-  ¬ ¬ (¬ x ∨ x)   ≈⟨ ⇨-cong (de-morgan₁ _ _) Eq.refl ⟩
-  ¬ (¬ ¬ x ∧ ¬ x) ≈⟨ ⇨-cong ⇨-app Eq.refl ⟩
-  ⊥ ∧ (x ⇨ ⊥) ⇨ ⊥ ≈⟨ ⇨-cong (∧-zeroˡ _) Eq.refl ⟩
-  ⊥ ⇨ ⊥           ≈⟨ ⇨-unit ⟩
-  ⊤               ∎
-  where open EqReasoning setoid
+{-# WARNING_ON_IMPORT
+"Relation.Binary.Properties.HeytingAlgebra was deprecated in v2.0.
+Use Relation.Binary.Lattice.Properties.HeytingAlgebra instead."
+#-}

From ae5b67ed1b91dfc6610d66326b9e19ea047ac8f2 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 11 Oct 2023 03:01:16 +0100
Subject: [PATCH 151/316] [fixes #2127] Fixes #1930 `import` bug (#2128)

---
 README.agda         | 5 +++++
 README/Inspect.agda | 9 +++------
 2 files changed, 8 insertions(+), 6 deletions(-)

diff --git a/README.agda b/README.agda
index 97b8ab3903..ced0f11596 100644
--- a/README.agda
+++ b/README.agda
@@ -237,6 +237,11 @@ import README.Debug.Trace
 
 import README.Nary
 
+-- Explaining the inspect idiom: use case, equivalent handwritten
+-- auxiliary definitions, and implementation details.
+
+import README.Inspect
+
 -- Explaining how to use the automatic solvers
 
 import README.Tactic.MonoidSolver
diff --git a/README/Inspect.agda b/README/Inspect.agda
index 9f40a8d65a..2199a499c8 100644
--- a/README/Inspect.agda
+++ b/README/Inspect.agda
@@ -1,22 +1,19 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- This module is DEPRECATED.
+-- Explaining how to use the inspect idiom and elaborating on the way
+-- it is implemented in the standard library.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module README.Inspect where
 
-{-# WARNING_ON_IMPORT
-"README.Inspect was deprecated in v2.0."
-#-}
-
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Product.Base using (_×_; _,_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
-open import Relation.Binary.PropositionalEquality using (inspect)
+open import Relation.Binary.PropositionalEquality using (inspect; [_])
 
 ------------------------------------------------------------------------
 -- Using inspect

From 5d532cf2c9407dfa1cc6fa315b0916b7819dee97 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 12 Oct 2023 02:52:31 +0100
Subject: [PATCH 152/316] [fixes #1214] Add negated ordering relation symbols
 systematically to `Relation.Binary.*` (#2095)

---
 CHANGELOG.md                                  | 57 ++++++++++++-------
 src/Data/List/Sort/MergeSort.agda             |  4 +-
 src/Relation/Binary/Bundles.agda              | 48 +++++++++-------
 .../Binary/Properties/DecTotalOrder.agda      | 11 ++--
 src/Relation/Binary/Properties/Poset.agda     | 12 ++--
 .../Binary/Properties/StrictPartialOrder.agda |  2 +-
 .../Binary/Properties/StrictTotalOrder.agda   |  2 +-
 .../Binary/Properties/TotalOrder.agda         |  5 +-
 8 files changed, 79 insertions(+), 62 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 81781828c2..5942e648a0 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -621,7 +621,7 @@ Non-backwards compatible changes
   raw bundles to `Data.X.Base`, this definition can now be made directly. Knock-on
   consequences include the need to retain the old constructor name, now introduced
   as a pattern synonym, and introduction of (a function equivalent to) the former
-  field name/projection function `proof` as `≤″-proof` in `Data.Nat.Properties`. 
+  field name/projection function `proof` as `≤″-proof` in `Data.Nat.Properties`.
 
 * Accordingly, the definition has been changed to:
   ```agda
@@ -842,7 +842,6 @@ Non-backwards compatible changes
   IO.Effectful
   IO.Instances
   ```
-
 ### (Issue #2096) Introduction of flipped relation symbol for `Relation.Binary.Bundles.Preorder`
 
 * Previously, the relation symbol `_∼_`  was (notationally) symmetric, so that its
@@ -856,9 +855,25 @@ Non-backwards compatible changes
   Therefore, only _declarations_ of `PartialOrder` records will need their field names
   updating.
 
-* NB (issues #1214 #2098) the corresponding situation regarding the `flip`ped
-  relation symbols `_≥_`, `_>_` (and their negated versions!) has not (yet)
-  been addressed.
+### (Issue #1214) Reorganisation of the introduction of negated relation symbols under `Relation.Binary`
+
+* Previously, negated relation symbols `_≰_` (for `Poset`) and `_≮_` (`TotalOrder`)
+  were introduced in the corresponding `Relation.Binary.Properties` modules, for re-export.
+
+* Now they are introduced as definitions in the corresponding `Relation.Binary.Bundles`,
+  together with, for uniformity's sake, an additional negated symbol `_⋦_` for `Preorder`,
+  with their (obvious) intended semantics:
+  ```agda
+  infix 4 _⋦_ _≰_ _≮_
+  Preorder._⋦_            : Rel Carrier _
+  StrictPartialOrder._≮_  : Rel Carrier _
+  ```
+  Additionally, `Poset._≰_` is defined by renaming public export of `Preorder._⋦_`
+
+* Backwards compatibility has been maintained, with deprecated definitions in the
+  corresponding `Relation.Binary.Properties` modules, and the corresponding client
+  client module `import`s being adjusted accordingly.
+
 
 ### Standardisation of `insertAt`/`updateAt`/`removeAt`
 
@@ -879,12 +894,12 @@ Non-backwards compatible changes
   ```
   _─_ ↦ removeAt
   ```
-  
+
 * In `Data.Vec.Base`:
   ```agda
   insert ↦ insertAt
   remove ↦ removeAt
-  
+
   updateAt : Fin n → (A → A) → Vec A n → Vec A n
     ↦
   updateAt : Vec A n → Fin n → (A → A) → Vec A n
@@ -895,12 +910,12 @@ Non-backwards compatible changes
   remove : Fin (suc n) → Vector A (suc n) → Vector A n
     ↦
   removeAt : Vector A (suc n) → Fin (suc n) → Vector A n
- 
+
   updateAt : Fin n → (A → A) → Vector A n → Vector A n
     ↦
   updateAt : Vector A n → Fin n → (A → A) → Vector A n
   ```
-  
+
 * The old names (and the names of all proofs about these functions) have been deprecated appropriately.
 
 ### Changes to triple reasoning interface
@@ -1659,7 +1674,7 @@ Deprecated names
   take-drop-id  ↦  take++drop≡id
 
   map-insert       ↦  map-insertAt
-  
+
   insert-lookup    ↦  insertAt-lookup
   insert-punchIn   ↦  insertAt-punchIn
   remove-PunchOut  ↦  removeAt-punchOut
@@ -1685,7 +1700,7 @@ Deprecated names
   updateAt-cong-relative    ↦  updateAt-cong-local
 
   map-updateAt              ↦  map-updateAt-local
-  
+
   insert-lookup             ↦  insertAt-lookup
   insert-punchIn            ↦  insertAt-punchIn
   remove-punchOut           ↦  removeAt-punchOut
@@ -1743,11 +1758,6 @@ Deprecated names
   fromForeign ↦ Foreign.Haskell.Coerce.coerce
   ```
 
-* In `Relation.Binary.Bundles.Preorder`:
-  ```
-  _∼_  ↦  _≲_
-  ```
-
 * In `Relation.Binary.Indexed.Heterogeneous.Bundles.Preorder`:
   ```
   _∼_  ↦  _≲_
@@ -1759,6 +1769,15 @@ Deprecated names
   invPreorder   ↦ converse-preorder
   ```
 
+* Moved negated relation symbol from `Relation.Binary.Properties.Poset`:
+  ```
+  _≰_   ↦ Relation.Binary.Bundles.Poset._≰_
+  ```
+
+* Moved negated relation symbol from `Relation.Binary.Properties.TotalOrder`:
+  ```
+  _≮_   ↦ Relation.Binary.Bundles.StrictPartialOrder._≮_
+
 * In `Relation.Nullary.Decidable.Core`:
   ```
   excluded-middle  ↦  ¬¬-excluded-middle
@@ -3301,9 +3320,9 @@ Additions to existing modules
 
 * Added new proofs to `Relation.Binary.Consequences`:
   ```
-  sym⇒¬-sym       : Symmetric _∼_ → Symmetric _≁_
-  cotrans⇒¬-trans : Cotransitive _∼_ → Transitive _≁_
-  irrefl⇒¬-refl   : Reflexive _≈_ → Irreflexive _≈_ _∼_ →  Reflexive _≁_
+  sym⇒¬-sym       : Symmetric _∼_ → Symmetric (¬_ ∘₂ _∼_)
+  cotrans⇒¬-trans : Cotransitive _∼_ → Transitive (¬_ ∘₂ _∼_)
+  irrefl⇒¬-refl   : Reflexive _≈_ → Irreflexive _≈_ _∼_ →  Reflexive (¬_ ∘₂ _∼_)
   mono₂⇒cong₂     : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ → ∀ {f} →
                     f Preserves₂ ≤₁ ⟶ ≤₁ ⟶ ≤₂ →
                     f Preserves₂ ≈₁ ⟶ ≈₁ ⟶ ≈₂
diff --git a/src/Data/List/Sort/MergeSort.agda b/src/Data/List/Sort/MergeSort.agda
index 9abab0823e..bded259428 100644
--- a/src/Data/List/Sort/MergeSort.agda
+++ b/src/Data/List/Sort/MergeSort.agda
@@ -21,7 +21,7 @@ open import Data.List.Relation.Binary.Permutation.Propositional
 import Data.List.Relation.Binary.Permutation.Propositional.Properties as Perm
 open import Data.Maybe.Base using (just)
 open import Relation.Nullary.Decidable using (does)
-open import Data.Nat using (_<_; _>_; z<s; s<s)
+open import Data.Nat.Base using (_<_; _>_; z<s; s<s)
 open import Data.Nat.Induction
 open import Data.Nat.Properties using (m<n⇒m<1+n)
 open import Data.Product.Base as Prod using (_,_)
@@ -36,7 +36,7 @@ open DecTotalOrder O renaming (Carrier to A)
 
 open import Data.List.Sort.Base totalOrder
 open import Data.List.Relation.Unary.Sorted.TotalOrder totalOrder hiding (head)
-open import Relation.Binary.Properties.DecTotalOrder O using (_≰_; ≰⇒≥; ≰-respˡ-≈)
+open import Relation.Binary.Properties.DecTotalOrder O using (≰⇒≥; ≰-respˡ-≈)
 
 open PermutationReasoning
 
diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index 968cb78b2e..90e3c7e2ac 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -92,11 +92,20 @@ record Preorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
 
     open Setoid setoid public
 
+  infix 4 _⋦_
+  _⋦_ : Rel Carrier _
+  x ⋦ y = ¬ (x ≲ y)
+
   infix 4 _≳_
   _≳_ = flip _≲_
 
-  infix 4 _∼_ -- for deprecation
+  -- Deprecated.
+  infix 4 _∼_ 
   _∼_ = _≲_
+  {-# WARNING_ON_USAGE _∼_
+  "Warning: _∼_ was deprecated in v2.0.
+  Please use _≲_ instead. "
+  #-}
 
 
 record TotalPreorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
@@ -114,7 +123,7 @@ record TotalPreorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   preorder = record { isPreorder = isPreorder }
 
   open Preorder preorder public
-    using (module Eq; _≳_)
+    using (module Eq; _≳_; _⋦_)
 
 
 ------------------------------------------------------------------------
@@ -139,6 +148,7 @@ record Poset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
 
   open Preorder preorder public
     using (module Eq)
+    renaming (_⋦_ to _≰_)
 
 
 record DecPoset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
@@ -159,7 +169,7 @@ record DecPoset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     }
 
   open Poset poset public
-    using (preorder)
+    using (preorder; _≰_)
 
   module Eq where
     decSetoid : DecSetoid c ℓ₁
@@ -189,6 +199,10 @@ record StrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂))
 
     open Setoid setoid public
 
+  infix 4 _≮_
+  _≮_ : Rel Carrier _
+  x ≮ y = ¬ (x < y)
+
 
 record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   infix 4 _≈_ _<_
@@ -207,6 +221,9 @@ record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂
     { isStrictPartialOrder = isStrictPartialOrder
     }
 
+  open StrictPartialOrder strictPartialOrder public
+    using (_≮_)
+
   module Eq where
 
     decSetoid : DecSetoid c ℓ₁
@@ -238,7 +255,7 @@ record TotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     }
 
   open Poset poset public
-    using (module Eq; preorder)
+    using (module Eq; preorder; _≰_)
 
   totalPreorder : TotalPreorder c ℓ₁ ℓ₂
   totalPreorder = record
@@ -263,14 +280,16 @@ record DecTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     { isTotalOrder = isTotalOrder
     }
 
-  open TotalOrder totalOrder public using (poset; preorder)
+  open TotalOrder totalOrder public
+    using (poset; preorder; _≰_)
 
   decPoset : DecPoset c ℓ₁ ℓ₂
   decPoset = record
     { isDecPartialOrder = isDecPartialOrder
     }
 
-  open DecPoset decPoset public using (module Eq)
+  open DecPoset decPoset public
+    using (module Eq)
 
 
 -- Note that these orders are decidable. The current implementation
@@ -295,7 +314,7 @@ record StrictTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) wh
     }
 
   open StrictPartialOrder strictPartialOrder public
-    using (module Eq)
+    using (module Eq; _≮_)
 
   decSetoid : DecSetoid c ℓ₁
   decSetoid = record
@@ -338,18 +357,3 @@ record ApartnessRelation c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) w
     isApartnessRelation : IsApartnessRelation _≈_ _#_
 
   open IsApartnessRelation isApartnessRelation public
-
-
-
-
-------------------------------------------------------------------------
--- DEPRECATED
-------------------------------------------------------------------------
--- Please use the new names as continuing support for the old names is
--- not guaranteed.
-
--- Version 2.0
-
-{-# WARNING_ON_USAGE Preorder._∼_
-"Warning: Preorder._∼_ was deprecated in v2.0. Please use Preorder._≲_ instead. "
-#-}
diff --git a/src/Relation/Binary/Properties/DecTotalOrder.agda b/src/Relation/Binary/Properties/DecTotalOrder.agda
index 2923a0d047..d7b3c48cd9 100644
--- a/src/Relation/Binary/Properties/DecTotalOrder.agda
+++ b/src/Relation/Binary/Properties/DecTotalOrder.agda
@@ -12,9 +12,9 @@ open import Relation.Binary.Bundles
   using (DecTotalOrder; StrictTotalOrder)
 
 module Relation.Binary.Properties.DecTotalOrder
-  {d₁ d₂ d₃} (DT : DecTotalOrder d₁ d₂ d₃) where
+  {d₁ d₂ d₃} (DTO : DecTotalOrder d₁ d₂ d₃) where
 
-open DecTotalOrder DT hiding (trans)
+open DecTotalOrder DTO hiding (trans)
 
 import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_ as ToStrict
@@ -80,19 +80,18 @@ open TotalOrderProperties public
   }
 
 open StrictTotalOrder <-strictTotalOrder public
-  using () renaming (compare to <-compare)
+  using (_≮_) renaming (compare to <-compare)
 
 ------------------------------------------------------------------------
 -- _≰_ - the negated order
 
 open TotalOrderProperties public
   using
-  ( _≰_
-  ; ≰-respʳ-≈
+  ( ≰-respʳ-≈
   ; ≰-respˡ-≈
   ; ≰⇒>
   ; ≰⇒≥
   )
 
-≮⇒≥ : ∀ {x y} → ¬ (x < y) → y ≤ x
+≮⇒≥ : ∀ {x y} → x ≮ y → y ≤ x
 ≮⇒≥ = ToStrict.≮⇒≥ Eq.sym _≟_ reflexive total
diff --git a/src/Relation/Binary/Properties/Poset.agda b/src/Relation/Binary/Properties/Poset.agda
index add0e20b28..beb4ddc8aa 100644
--- a/src/Relation/Binary/Properties/Poset.agda
+++ b/src/Relation/Binary/Properties/Poset.agda
@@ -6,6 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
+open import Data.Product.Base using (_,_)
 open import Function.Base using (flip; _∘_)
 open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
 open import Relation.Binary.Bundles using (Poset; StrictPartialOrder)
@@ -62,11 +63,6 @@ open Poset ≥-poset public
 ------------------------------------------------------------------------
 -- Negated order
 
-infix 4 _≰_
-
-_≰_ : Rel A p₃
-x ≰ y = ¬ (x ≤ y)
-
 ≰-respˡ-≈ : _≰_ Respectsˡ _≈_
 ≰-respˡ-≈ x≈y = _∘ ≤-respˡ-≈ (Eq.sym x≈y)
 
@@ -90,7 +86,7 @@ _<_ = ToStrict._<_
   }
 
 open StrictPartialOrder <-strictPartialOrder public
-  using ( <-resp-≈; <-respʳ-≈; <-respˡ-≈)
+  using (_≮_; <-resp-≈; <-respʳ-≈; <-respˡ-≈)
   renaming
   ( irrefl to <-irrefl
   ; asym   to <-asym
@@ -103,10 +99,10 @@ open StrictPartialOrder <-strictPartialOrder public
 ≤∧≉⇒< : ∀ {x y} → x ≤ y → x ≉ y → x < y
 ≤∧≉⇒< = ToStrict.≤∧≉⇒<
 
-<⇒≱ : ∀ {x y} → x < y → ¬ (y ≤ x)
+<⇒≱ : ∀ {x y} → x < y → y ≰ x
 <⇒≱ = ToStrict.<⇒≱ antisym
 
-≤⇒≯ : ∀ {x y} → x ≤ y → ¬ (y < x)
+≤⇒≯ : ∀ {x y} → x ≤ y → y ≮ x
 ≤⇒≯ = ToStrict.≤⇒≯ antisym
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Properties/StrictPartialOrder.agda b/src/Relation/Binary/Properties/StrictPartialOrder.agda
index 7420a2e24f..680323c770 100644
--- a/src/Relation/Binary/Properties/StrictPartialOrder.agda
+++ b/src/Relation/Binary/Properties/StrictPartialOrder.agda
@@ -17,7 +17,7 @@ import Relation.Binary.Construct.StrictToNonStrict
 open StrictPartialOrder SPO
 
 ------------------------------------------------------------------------
--- The inverse relation is also a strict partial order.
+-- The converse relation is also a strict partial order.
 
 >-strictPartialOrder : StrictPartialOrder s₁ s₂ s₃
 >-strictPartialOrder = EqAndOrd.strictPartialOrder SPO
diff --git a/src/Relation/Binary/Properties/StrictTotalOrder.agda b/src/Relation/Binary/Properties/StrictTotalOrder.agda
index f8b0062474..e6d34ab8d1 100644
--- a/src/Relation/Binary/Properties/StrictTotalOrder.agda
+++ b/src/Relation/Binary/Properties/StrictTotalOrder.agda
@@ -18,7 +18,7 @@ import Relation.Binary.Properties.StrictPartialOrder as SPO
 open import Relation.Binary.Consequences
 
 ------------------------------------------------------------------------
--- _<_ - the strict version is a strict total order
+-- _<_ - the strict version is a decidable total order
 
 decTotalOrder : DecTotalOrder _ _ _
 decTotalOrder = record
diff --git a/src/Relation/Binary/Properties/TotalOrder.agda b/src/Relation/Binary/Properties/TotalOrder.agda
index 540de19905..48f788ab7f 100644
--- a/src/Relation/Binary/Properties/TotalOrder.agda
+++ b/src/Relation/Binary/Properties/TotalOrder.agda
@@ -59,7 +59,7 @@ open PosetProperties public
   }
 
 open TotalOrder ≥-totalOrder public
-  using () renaming (total to ≥-total)
+  using () renaming (total to ≥-total; _≰_ to _≱_)
 
 ------------------------------------------------------------------------
 -- _<_ - the strict version is a strict partial order
@@ -90,8 +90,7 @@ open PosetProperties public
 
 open PosetProperties public
   using
-  ( _≰_
-  ; ≰-respʳ-≈
+  ( ≰-respʳ-≈
   ; ≰-respˡ-≈
   )
 

From c111b02c080c3dc0944a1be88c3bd776f8f37b69 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 12 Oct 2023 03:05:52 +0100
Subject: [PATCH 153/316] Refactoring (inversion) properties of `_<_` on `Nat`,
 plus consequences (#2000)

---
 CHANGELOG.md                                  |  17 +-
 src/Data/Digit/Properties.agda                |   2 +-
 src/Data/Fin/Base.agda                        |  44 +++--
 src/Data/Fin/Properties.agda                  | 170 +++++++++---------
 src/Data/Integer/Properties.agda              |  26 +--
 src/Data/List/Properties.agda                 |   2 +-
 .../Relation/Unary/Linked/Properties.agda     |   2 +-
 src/Data/Nat.agda                             |   3 +
 src/Data/Nat/Base.agda                        |  36 +++-
 src/Data/Nat/Binary/Properties.agda           |  10 +-
 src/Data/Nat/DivMod.agda                      |  14 +-
 src/Data/Nat/DivMod/WithK.agda                |  18 +-
 src/Data/Nat/Properties.agda                  |  52 +++---
 src/Data/Nat/Properties/Core.agda             |   6 +-
 src/Data/Nat/Show/Properties.agda             |  16 +-
 src/Data/Vec/Base.agda                        |   2 +-
 src/Data/Vec/Bounded/Base.agda                |   4 +-
 src/Data/Vec/Properties.agda                  |  14 +-
 .../Vec/Relation/Unary/Linked/Properties.agda |  11 +-
 19 files changed, 244 insertions(+), 205 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5942e648a0..988355bd5a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1276,7 +1276,7 @@ Deprecated modules
 
 ### Deprecation of `Data.Nat.Properties.Core`
 
-* The module `Data.Nat.Properties.Core` has been deprecated, and its one entry moved to `Data.Nat.Properties`
+* The module `Data.Nat.Properties.Core` has been deprecated, and its one lemma moved to `Data.Nat.Base`, renamed as `s≤s⁻¹`
 
 ### Deprecation of `Data.Fin.Substitution.Example`
 
@@ -1543,6 +1543,7 @@ Deprecated names
   ≤-stepsˡ        ↦  m≤n⇒m≤o+n
   ≤-stepsʳ        ↦  m≤n⇒m≤n+o
   <-step          ↦  m<n⇒m<1+n
+  pred-mono       ↦  pred-mono-≤
 
   <-transʳ        ↦  ≤-<-trans
   <-transˡ        ↦  <-≤-trans
@@ -2613,9 +2614,18 @@ Additions to existing modules
   pattern z<s {n}         = s≤s (z≤n {n})
   pattern s<s {m} {n} m<n = s≤s {m} {n} m<n
 
+  s≤s⁻¹ : suc m ≤ suc n → m ≤ n
+  s<s⁻¹ : suc m < suc n → m < n
+
   pattern <′-base          = ≤′-refl
   pattern <′-step {n} m<′n = ≤′-step {n} m<′n
 
+  pattern ≤″-offset k = less-than-or-equal {k} refl
+  pattern <″-offset k = ≤″-offset k
+
+  s≤″s⁻¹ : ∀ {m n} → suc m ≤″ suc n → m ≤″ n
+  s<″s⁻¹ : ∀ {m n} → suc m <″ suc n → m <″ n
+
   _⊔′_ : ℕ → ℕ → ℕ
   _⊓′_ : ℕ → ℕ → ℕ
   ∣_-_∣′ : ℕ → ℕ → ℕ
@@ -2666,12 +2676,13 @@ Additions to existing modules
   m*n≢0     : .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
   m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n
 
-  ≤-pred        : suc m ≤ suc n → m ≤ n
   s<s-injective : ∀ {p q : m < n} → s<s p ≡ s<s q → p ≡ q
-  <-pred        : suc m < suc n → m < n
   <-step        : m < n → m < 1 + n
   m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
 
+  pred-mono-≤   : m ≤ n → pred m ≤ pred n
+  pred-mono-<   : .⦃ _ : NonZero m ⦄ → m < n → pred m < pred n
+
   z<′s : zero <′ suc n
   s<′s : m <′ n → suc m <′ suc n
   <⇒<′ : m < n → m <′ n
diff --git a/src/Data/Digit/Properties.agda b/src/Data/Digit/Properties.agda
index 113bdde6ca..fb52d2af72 100644
--- a/src/Data/Digit/Properties.agda
+++ b/src/Data/Digit/Properties.agda
@@ -15,7 +15,7 @@ open import Data.Product.Base using (_,_; proj₁)
 open import Data.Vec.Relation.Unary.Unique.Propositional using (Unique)
 import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Uniqueₚ
 open import Data.Vec.Relation.Unary.AllPairs using (allPairs?)
-open import Relation.Nullary.Decidable using (True; from-yes; ¬?)
+open import Relation.Nullary.Decidable.Core using (True; from-yes; ¬?)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Function.Base using (_∘_)
 
diff --git a/src/Data/Fin/Base.agda b/src/Data/Fin/Base.agda
index 2d5b0f638e..9b4c303f2f 100644
--- a/src/Data/Fin/Base.agda
+++ b/src/Data/Fin/Base.agda
@@ -11,17 +11,16 @@
 
 module Data.Fin.Base where
 
-open import Data.Bool.Base using (Bool; true; false; T; not)
-open import Data.Nat.Base as ℕ using (ℕ; zero; suc; z≤n; s≤s; z<s; s<s; _^_)
+open import Data.Bool.Base using (Bool; T)
+open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (id; _∘_; _on_; flip)
 open import Level using (0ℓ)
-open import Relation.Nullary.Negation.Core using (contradiction)
-open import Relation.Nullary.Decidable.Core using (yes; no; True; toWitness)
 open import Relation.Binary.Core
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; refl; cong)
 open import Relation.Binary.Indexed.Heterogeneous.Core using (IRel)
+open import Relation.Nullary.Negation.Core using (contradiction)
 
 private
   variable
@@ -68,16 +67,15 @@ fromℕ (suc n) = suc (fromℕ n)
 
 -- fromℕ< {m} _ = "m".
 
-fromℕ< : m ℕ.< n → Fin n
-fromℕ< {zero}  {suc n} z<s = zero
-fromℕ< {suc m} {suc n} (s<s m<n) = suc (fromℕ< m<n)
+fromℕ< : .(m ℕ.< n) → Fin n
+fromℕ< {zero}  {suc _} _   = zero
+fromℕ< {suc m} {suc _} m<n = suc (fromℕ< (ℕ.s<s⁻¹ m<n))
 
 -- fromℕ<″ m _ = "m".
 
-fromℕ<″ : ∀ m {n} → m ℕ.<″ n → Fin n
-fromℕ<″ zero    (ℕ.less-than-or-equal refl) = zero
-fromℕ<″ (suc m) (ℕ.less-than-or-equal refl) =
-  suc (fromℕ<″ m (ℕ.less-than-or-equal refl))
+fromℕ<″ : ∀ m {n} → .(m ℕ.<″ n) → Fin n
+fromℕ<″ zero    {suc _} _    = zero
+fromℕ<″ (suc m) {suc _} m<″n = suc (fromℕ<″ m (ℕ.s<″s⁻¹ m<″n))
 
 -- canonical liftings of i:Fin m to larger index
 
@@ -95,9 +93,9 @@ zero    ↑ʳ i = i
 
 -- reduce≥ "m + i" _ = "i".
 
-reduce≥ : ∀ (i : Fin (m ℕ.+ n)) (i≥m : toℕ i ℕ.≥ m) → Fin n
-reduce≥ {zero}  i       i≥m       = i
-reduce≥ {suc m} (suc i) (s≤s i≥m) = reduce≥ i i≥m
+reduce≥ : ∀ (i : Fin (m ℕ.+ n)) → .(m ℕ.≤ toℕ i) → Fin n
+reduce≥ {zero}  i       _   = i
+reduce≥ {suc _} (suc i) m≤i = reduce≥ i (ℕ.s≤s⁻¹ m≤i)
 
 -- inject⋆ m "i" = "i".
 
@@ -106,16 +104,16 @@ inject {i = suc i} zero    = zero
 inject {i = suc i} (suc j) = suc (inject j)
 
 inject! : ∀ {i : Fin (suc n)} → Fin′ i → Fin n
-inject! {n = suc _} {i = suc _}  zero    = zero
-inject! {n = suc _} {i = suc _}  (suc j) = suc (inject! j)
+inject! {n = suc _} {i = suc _} zero    = zero
+inject! {n = suc _} {i = suc _} (suc j) = suc (inject! j)
 
 inject₁ : Fin n → Fin (suc n)
 inject₁ zero    = zero
 inject₁ (suc i) = suc (inject₁ i)
 
-inject≤ : Fin m → m ℕ.≤ n → Fin n
-inject≤ {_} {suc n} zero    _         =  zero
-inject≤ {_} {suc n} (suc i) (s≤s m≤n) = suc (inject≤ i m≤n)
+inject≤ : Fin m → .(m ℕ.≤ n) → Fin n
+inject≤ {n = suc _} zero    _   = zero
+inject≤ {n = suc _} (suc i) m≤n = suc (inject≤ i (ℕ.s≤s⁻¹ m≤n))
 
 -- lower₁ "i" _ = "i".
 
@@ -166,12 +164,12 @@ combine {suc m} {n} zero    j = j ↑ˡ (m ℕ.* n)
 combine {suc m} {n} (suc i) j = n ↑ʳ (combine i j)
 
 -- Next in progression after splitAt and remQuot
-finToFun : Fin (m ^ n) → (Fin n → Fin m)
-finToFun {m} {suc n} i zero    = quotient (m ^ n) i
-finToFun {m} {suc n} i (suc j) = finToFun (remainder {m} (m ^ n) i) j
+finToFun : Fin (m ℕ.^ n) → (Fin n → Fin m)
+finToFun {m} {suc n} i zero    = quotient (m ℕ.^ n) i
+finToFun {m} {suc n} i (suc j) = finToFun (remainder {m} (m ℕ.^ n) i) j
 
 -- inverse of above function
-funToFin : (Fin m → Fin n) → Fin (n ^ m)
+funToFin : (Fin m → Fin n) → Fin (n ℕ.^ m)
 funToFin {zero}  f = zero
 funToFin {suc m} f = combine (f zero) (funToFin (f ∘ suc))
 
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index ef2073269f..ea7c43c90f 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -19,7 +19,7 @@ open import Data.Empty using (⊥; ⊥-elim)
 open import Data.Fin.Base
 open import Data.Fin.Patterns
 open import Data.Nat.Base as ℕ
-  using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; _∸_; _^_)
+  using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; s<s⁻¹; _∸_; _^_)
 import Data.Nat.Properties as ℕₚ
 open import Data.Nat.Solver
 open import Data.Unit using (⊤; tt)
@@ -43,10 +43,10 @@ open import Relation.Binary.Structures
   using (IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_; module ≡-Reasoning)
-open import Relation.Nullary
-  using (Reflects; ofʸ; ofⁿ; Dec; _because_; does; proof; yes; no; ¬_; _×-dec_; _⊎-dec_; contradiction)
-open import Relation.Nullary.Reflects
-open import Relation.Nullary.Decidable as Dec using (map′)
+open import Relation.Nullary.Decidable as Dec
+  using (Dec; _because_; yes; no; _×-dec_; _⊎-dec_; map′)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
+open import Relation.Nullary.Reflects using (Reflects; invert)
 open import Relation.Unary as U
   using (U; Pred; Decidable; _⊆_; Satisfiable; Universal)
 open import Relation.Unary.Properties using (U?)
@@ -162,6 +162,10 @@ toℕ-↑ʳ (suc n) i = cong suc (toℕ-↑ʳ n i)
 -- toℕ and the ordering relations
 ------------------------------------------------------------------------
 
+toℕ<n : ∀ (i : Fin n) → toℕ i ℕ.< n
+toℕ<n {n = suc _} zero    = z<s
+toℕ<n {n = suc _} (suc i) = s<s (toℕ<n i)
+
 toℕ≤pred[n] : ∀ (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n
 toℕ≤pred[n] zero                 = z≤n
 toℕ≤pred[n] (suc {n = suc n} i)  = s≤s (toℕ≤pred[n] i)
@@ -169,9 +173,6 @@ toℕ≤pred[n] (suc {n = suc n} i)  = s≤s (toℕ≤pred[n] i)
 toℕ≤n : ∀ (i : Fin n) → toℕ i ℕ.≤ n
 toℕ≤n {suc n} i = ℕₚ.m≤n⇒m≤1+n (toℕ≤pred[n] i)
 
-toℕ<n : ∀ (i : Fin n) → toℕ i ℕ.< n
-toℕ<n {suc n} i = s<s (toℕ≤pred[n] i)
-
 -- A simpler implementation of toℕ≤pred[n],
 -- however, with a different reduction behavior.
 -- If no one needs the reduction behavior of toℕ≤pred[n],
@@ -203,37 +204,38 @@ fromℕ-toℕ : ∀ (i : Fin n) → fromℕ (toℕ i) ≡ strengthen i
 fromℕ-toℕ zero    = refl
 fromℕ-toℕ (suc i) = cong suc (fromℕ-toℕ i)
 
-≤fromℕ : ∀ (i : Fin (ℕ.suc n)) → i ≤ fromℕ n
-≤fromℕ i = subst (toℕ i ℕ.≤_) (sym (toℕ-fromℕ _)) (toℕ≤pred[n] i)
+≤fromℕ : ∀ (i : Fin (suc n)) → i ≤ fromℕ n
+≤fromℕ {n = n} i rewrite toℕ-fromℕ n = ℕ.s≤s⁻¹ (toℕ<n i)
 
 ------------------------------------------------------------------------
 -- fromℕ<
 ------------------------------------------------------------------------
 
-fromℕ<-toℕ : ∀ (i : Fin n) (i<n : toℕ i ℕ.< n) → fromℕ< i<n ≡ i
-fromℕ<-toℕ zero    z<s       = refl
-fromℕ<-toℕ (suc i) (s<s i<n) = cong suc (fromℕ<-toℕ i i<n)
+fromℕ<-toℕ : ∀ (i : Fin n) .(i<n : toℕ i ℕ.< n) → fromℕ< i<n ≡ i
+fromℕ<-toℕ zero    _   = refl
+fromℕ<-toℕ (suc i) i<n = cong suc (fromℕ<-toℕ i (ℕ.s<s⁻¹ i<n))
 
-toℕ-fromℕ< : ∀ (m<n : m ℕ.< n) → toℕ (fromℕ< m<n) ≡ m
-toℕ-fromℕ< z<s               = refl
-toℕ-fromℕ< (s<s m<n@(s≤s _)) = cong suc (toℕ-fromℕ< m<n)
+toℕ-fromℕ< : ∀ .(m<n : m ℕ.< n) → toℕ (fromℕ< m<n) ≡ m
+toℕ-fromℕ< {m = zero}  {n = suc _} _   = refl
+toℕ-fromℕ< {m = suc m} {n = suc _} m<n = cong suc (toℕ-fromℕ< (ℕ.s<s⁻¹ m<n))
 
 -- fromℕ is a special case of fromℕ<.
 fromℕ-def : ∀ n → fromℕ n ≡ fromℕ< ℕₚ.≤-refl
 fromℕ-def zero    = refl
 fromℕ-def (suc n) = cong suc (fromℕ-def n)
 
-fromℕ<-cong : ∀ m n {o} → m ≡ n → (m<o : m ℕ.< o) (n<o : n ℕ.< o) →
+fromℕ<-cong : ∀ m n {o} → m ≡ n → .(m<o : m ℕ.< o) .(n<o : n ℕ.< o) →
               fromℕ< m<o ≡ fromℕ< n<o
-fromℕ<-cong 0       0       r z<s       z<s  = refl
-fromℕ<-cong (suc _) (suc _) r (s<s m<n) (s<s n<o)
-  = cong suc (fromℕ<-cong _ _ (ℕₚ.suc-injective r) m<n n<o)
+fromℕ<-cong 0       0                   _ _   _   = refl
+fromℕ<-cong (suc _) (suc _) {o = suc _} r m<n n<o
+  = cong suc (fromℕ<-cong _ _ (ℕₚ.suc-injective r) (ℕ.s<s⁻¹ m<n) (ℕ.s<s⁻¹ n<o))
 
-fromℕ<-injective : ∀ m n {o} → (m<o : m ℕ.< o) (n<o : n ℕ.< o) →
+fromℕ<-injective : ∀ m n {o} → .(m<o : m ℕ.< o) .(n<o : n ℕ.< o) →
                    fromℕ< m<o ≡ fromℕ< n<o → m ≡ n
-fromℕ<-injective 0       0       z<s               z<s r = refl
-fromℕ<-injective (suc _) (suc _) (s<s m<n@(s≤s _)) (s<s n<o@(s≤s _)) r
-  = cong suc (fromℕ<-injective _ _ m<n n<o (suc-injective r))
+fromℕ<-injective 0       0                   _   _   _  = refl
+fromℕ<-injective 0       (suc _) {o = suc _} _ _ ()
+fromℕ<-injective (suc _) (suc _) {o = suc _} m<n n<o r
+  = cong suc (fromℕ<-injective _ _ (ℕ.s<s⁻¹ m<n) (ℕ.s<s⁻¹ n<o) (suc-injective r))
 
 ------------------------------------------------------------------------
 -- fromℕ<″
@@ -241,9 +243,9 @@ fromℕ<-injective (suc _) (suc _) (s<s m<n@(s≤s _)) (s<s n<o@(s≤s _)) r
 
 fromℕ<≡fromℕ<″ : ∀ (m<n : m ℕ.< n) (m<″n : m ℕ.<″ n) →
                  fromℕ< m<n ≡ fromℕ<″ m m<″n
-fromℕ<≡fromℕ<″ z<s               (ℕ.less-than-or-equal refl) = refl
-fromℕ<≡fromℕ<″ (s<s m<n@(s≤s _)) (ℕ.less-than-or-equal refl) =
-  cong suc (fromℕ<≡fromℕ<″ m<n (ℕ.less-than-or-equal refl))
+fromℕ<≡fromℕ<″ {m = zero}  m<n (ℕ.<″-offset _) = refl
+fromℕ<≡fromℕ<″ {m = suc m} m<n (ℕ.<″-offset _)
+  = cong suc (fromℕ<≡fromℕ<″ (ℕ.s<s⁻¹ m<n) (ℕ.<″-offset _))
 
 toℕ-fromℕ<″ : ∀ (m<n : m ℕ.<″ n) → toℕ (fromℕ<″ m m<n) ≡ m
 toℕ-fromℕ<″ {m} {n} m<n = begin
@@ -375,9 +377,9 @@ m <? n = suc (toℕ m) ℕₚ.≤? toℕ n
 <-cmp zero    (suc j) = tri< z<s   (λ()) (λ())
 <-cmp (suc i) zero    = tri> (λ()) (λ()) z<s
 <-cmp (suc i) (suc j) with <-cmp i j
-... | tri< i<j i≢j j≮i = tri< (s<s i<j)         (i≢j ∘ suc-injective) (j≮i ∘ ℕₚ.≤-pred)
-... | tri> i≮j i≢j j<i = tri> (i≮j ∘ ℕₚ.≤-pred) (i≢j ∘ suc-injective) (s<s j<i)
-... | tri≈ i≮j i≡j j≮i = tri≈ (i≮j ∘ ℕₚ.≤-pred) (cong suc i≡j)        (j≮i ∘ ℕₚ.≤-pred)
+... | tri< i<j i≢j j≮i = tri< (s<s i<j)     (i≢j ∘ suc-injective) (j≮i ∘ s<s⁻¹)
+... | tri> i≮j i≢j j<i = tri> (i≮j ∘ s<s⁻¹) (i≢j ∘ suc-injective) (s<s j<i)
+... | tri≈ i≮j i≡j j≮i = tri≈ (i≮j ∘ s<s⁻¹) (cong suc i≡j)        (j≮i ∘ s<s⁻¹)
 
 <-respˡ-≡ : (_<_ {m} {n}) Respectsˡ _≡_
 <-respˡ-≡ refl x≤y = x≤y
@@ -432,10 +434,10 @@ i<1+i = ℕₚ.n<1+n ∘ toℕ
 <⇒≢ i<i refl = ℕₚ.n≮n _ i<i
 
 ≤∧≢⇒< : i ≤ j → i ≢ j → i < j
-≤∧≢⇒< {i = zero}  {zero}  _         0≢0     = contradiction refl 0≢0
-≤∧≢⇒< {i = zero}  {suc j} _         _       = z<s
-≤∧≢⇒< {i = suc i} {suc j} (s≤s i≤j) 1+i≢1+j =
-  s<s (≤∧≢⇒< i≤j (1+i≢1+j ∘ (cong suc)))
+≤∧≢⇒< {i = zero}  {zero}  _         0≢0   = contradiction refl 0≢0
+≤∧≢⇒< {i = zero}  {suc j} _         _     = z<s
+≤∧≢⇒< {i = suc i} {suc j} 1+i≤1+j 1+i≢1+j =
+  s<s (≤∧≢⇒< (ℕ.s≤s⁻¹ 1+i≤1+j) (1+i≢1+j ∘ (cong suc)))
 
 ------------------------------------------------------------------------
 -- inject
@@ -474,11 +476,11 @@ inject₁ℕ≤ = ℕₚ.<⇒≤ ∘ inject₁ℕ<
 ≤̄⇒inject₁< {i = i} i≤j rewrite sym (toℕ-inject₁ i) = s<s i≤j
 
 ℕ<⇒inject₁< : ∀ {i : Fin (ℕ.suc n)} {j : Fin n} → j < i → inject₁ j < i
-ℕ<⇒inject₁< {i = suc i} (s≤s j≤i) = ≤̄⇒inject₁< j≤i
+ℕ<⇒inject₁< {i = suc i} j≤i = ≤̄⇒inject₁< (ℕ.s≤s⁻¹ j≤i)
 
 i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j
-i≤inject₁[j]⇒i≤1+j {i = zero} i≤j = z≤n
-i≤inject₁[j]⇒i≤1+j {i = suc i} {j = suc j} (s≤s i≤j) = s≤s (ℕₚ.m≤n⇒m≤1+n (subst (toℕ i ℕ.≤_) (toℕ-inject₁ j) i≤j))
+i≤inject₁[j]⇒i≤1+j {i = zero}              _   = z≤n
+i≤inject₁[j]⇒i≤1+j {i = suc i} {j = suc j} i≤j = s≤s (ℕₚ.m≤n⇒m≤1+n (subst (toℕ i ℕ.≤_) (toℕ-inject₁ j) (ℕ.s≤s⁻¹ i≤j)))
 
 ------------------------------------------------------------------------
 -- lower₁
@@ -491,8 +493,8 @@ toℕ-lower₁ {ℕ.suc m} (suc i) p = cong ℕ.suc (toℕ-lower₁ i (p ∘ con
 
 lower₁-injective : ∀ {n≢i : n ≢ toℕ i} {n≢j : n ≢ toℕ j} →
                    lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
-lower₁-injective {zero}  {zero}  {_}     {n≢i} {_}   _    = ⊥-elim (n≢i refl)
-lower₁-injective {zero}  {_}     {zero}  {_}   {n≢j} _    = ⊥-elim (n≢j refl)
+lower₁-injective {zero}  {zero}  {_}     {n≢i} {_}   _    = contradiction refl n≢i
+lower₁-injective {zero}  {_}     {zero}  {_}   {n≢j} _    = contradiction refl n≢j
 lower₁-injective {suc n} {zero}  {zero}  {_}   {_}   refl = refl
 lower₁-injective {suc n} {suc i} {suc j} {n≢i} {n≢j} eq   =
   cong suc (lower₁-injective (suc-injective eq))
@@ -532,34 +534,34 @@ inject₁≡⇒lower₁≡ n≢j i≡j = inject₁-injective (trans (inject₁-l
 -- inject≤
 ------------------------------------------------------------------------
 
-toℕ-inject≤ : ∀ i (m≤n : m ℕ.≤ n) → toℕ (inject≤ i m≤n) ≡ toℕ i
-toℕ-inject≤ {_} {suc n} zero    _         = refl
-toℕ-inject≤ {_} {suc n} (suc i) (s≤s m≤n) = cong suc (toℕ-inject≤ i m≤n)
+toℕ-inject≤ : ∀ i .(m≤n : m ℕ.≤ n) → toℕ (inject≤ i m≤n) ≡ toℕ i
+toℕ-inject≤ {_} {suc n} zero    _ = refl
+toℕ-inject≤ {_} {suc n} (suc i) _ = cong suc (toℕ-inject≤ i _)
 
-inject≤-refl : ∀ i (n≤n : n ℕ.≤ n) → inject≤ i n≤n ≡ i
-inject≤-refl {suc n} zero    _         = refl
-inject≤-refl {suc n} (suc i) (s≤s n≤n) = cong suc (inject≤-refl i n≤n)
+inject≤-refl : ∀ i .(n≤n : n ℕ.≤ n) → inject≤ i n≤n ≡ i
+inject≤-refl {suc n} zero    _   = refl
+inject≤-refl {suc n} (suc i) _ = cong suc (inject≤-refl i _)
 
 inject≤-idempotent : ∀ (i : Fin m)
-                     (m≤n : m ℕ.≤ n) (n≤o : n ℕ.≤ o) (m≤o : m ℕ.≤ o) →
+                     .(m≤n : m ℕ.≤ n) .(n≤o : n ℕ.≤ o) .(m≤o : m ℕ.≤ o) →
                      inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i m≤o
-inject≤-idempotent {_} {suc n} {suc o} zero    _   _   _ = refl
-inject≤-idempotent {_} {suc n} {suc o} (suc i) (s≤s m≤n) (s≤s n≤o) (s≤s m≤o) =
-  cong suc (inject≤-idempotent i m≤n n≤o m≤o)
+inject≤-idempotent {_} {suc n} {suc o} zero    _ _ _ = refl
+inject≤-idempotent {_} {suc n} {suc o} (suc i) _ _ _ =
+  cong suc (inject≤-idempotent i _ _ _)
 
-inject≤-trans : ∀ (i : Fin m) (m≤n : m ℕ.≤ n) (n≤o : n ℕ.≤ o) →
+inject≤-trans : ∀ (i : Fin m) .(m≤n : m ℕ.≤ n) .(n≤o : n ℕ.≤ o) →
                 inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (ℕₚ.≤-trans m≤n n≤o)
-inject≤-trans i m≤n n≤o = inject≤-idempotent i m≤n n≤o _
+inject≤-trans i _ _ = inject≤-idempotent i _ _ _
 
-inject≤-injective : ∀ (m≤n m≤n′ : m ℕ.≤ n) i j →
+inject≤-injective : ∀ .(m≤n m≤n′ : m ℕ.≤ n) i j →
                     inject≤ i m≤n ≡ inject≤ j m≤n′ → i ≡ j
-inject≤-injective (s≤s p) (s≤s q) zero    zero    eq = refl
-inject≤-injective (s≤s p) (s≤s q) (suc i) (suc j) eq =
-  cong suc (inject≤-injective p q i j (suc-injective eq))
+inject≤-injective {n = suc _} _ _ zero    zero    eq = refl
+inject≤-injective {n = suc _} _ _ (suc i) (suc j) eq =
+  cong suc (inject≤-injective _ _ i j (suc-injective eq))
 
-inject≤-irrelevant : ∀ (m≤n m≤n′ : m ℕ.≤ n) i →
+inject≤-irrelevant : ∀ .(m≤n m≤n′ : m ℕ.≤ n) i →
                     inject≤ i m≤n ≡ inject≤ i m≤n′
-inject≤-irrelevant m≤n m≤n′ i =  cong (inject≤ i) (ℕₚ.≤-irrelevant m≤n m≤n′)
+inject≤-irrelevant _ _ i = refl
 
 ------------------------------------------------------------------------
 -- pred
@@ -581,8 +583,10 @@ splitAt-↑ˡ (suc m) (suc i) n rewrite splitAt-↑ˡ m i n = refl
 
 splitAt⁻¹-↑ˡ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i
 splitAt⁻¹-↑ˡ {suc m} {n} {0F} {.0F} refl = refl
-splitAt⁻¹-↑ˡ {suc m} {n} {suc i} {j} eq with splitAt m i in splitAt[m][i]≡inj₁[j]
-... | inj₁ k with refl ← eq = cong suc (splitAt⁻¹-↑ˡ {i = i} {j = k} splitAt[m][i]≡inj₁[j])
+splitAt⁻¹-↑ˡ {suc m} {n} {suc i} {j} eq
+  with inj₁ k ← splitAt m i in splitAt[m][i]≡inj₁[j]
+  with refl ← eq
+  = cong suc (splitAt⁻¹-↑ˡ {i = i} {j = k} splitAt[m][i]≡inj₁[j])
 
 splitAt-↑ʳ : ∀ m n i → splitAt m (m ↑ʳ i) ≡ inj₂ {B = Fin n} i
 splitAt-↑ʳ zero    n i = refl
@@ -590,8 +594,10 @@ splitAt-↑ʳ (suc m) n i rewrite splitAt-↑ʳ m n i = refl
 
 splitAt⁻¹-↑ʳ : ∀ {m} {n} {i} {j} → splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i
 splitAt⁻¹-↑ʳ {zero}  {n} {i} {j} refl = refl
-splitAt⁻¹-↑ʳ {suc m} {n} {suc i} {j} eq with splitAt m i in splitAt[m][i]≡inj₂[k]
-... | inj₂ k with refl ← eq = cong suc (splitAt⁻¹-↑ʳ {i = i} {j = k} splitAt[m][i]≡inj₂[k])
+splitAt⁻¹-↑ʳ {suc m} {n} {suc i} {j} eq
+  with inj₂ k ← splitAt m i in splitAt[m][i]≡inj₂[k]
+  with refl ← eq
+  = cong suc (splitAt⁻¹-↑ʳ {i = i} {j = k} splitAt[m][i]≡inj₂[k])
 
 splitAt-join : ∀ m n i → splitAt m (join m n i) ≡ i
 splitAt-join m n (inj₁ x) = splitAt-↑ˡ m x n
@@ -609,17 +615,17 @@ join-splitAt (suc m) n (suc i) = begin
 
 -- splitAt "m" "i" ≡ inj₁ "i" if i < m
 
-splitAt-< : ∀ m {n} (i : Fin (m ℕ.+ n)) (i<m : toℕ i ℕ.< m) →
+splitAt-< : ∀ m {n} (i : Fin (m ℕ.+ n)) .(i<m : toℕ i ℕ.< m) →
             splitAt m i ≡ inj₁ (fromℕ< i<m)
-splitAt-< (suc m) zero    z<s               = refl
-splitAt-< (suc m) (suc i) (s<s i<m) = cong (Sum.map suc id) (splitAt-< m i i<m)
+splitAt-< (suc m) zero    _   = refl
+splitAt-< (suc m) (suc i) i<m = cong (Sum.map suc id) (splitAt-< m i (ℕ.s<s⁻¹ i<m))
 
 -- splitAt "m" "i" ≡ inj₂ "i - m" if i ≥ m
 
-splitAt-≥ : ∀ m {n} (i : Fin (m ℕ.+ n)) (i≥m : toℕ i ℕ.≥ m) →
+splitAt-≥ : ∀ m {n} (i : Fin (m ℕ.+ n)) .(i≥m : toℕ i ℕ.≥ m) →
             splitAt m i ≡ inj₂ (reduce≥ i i≥m)
-splitAt-≥ zero    i       _         = refl
-splitAt-≥ (suc m) (suc i) (s≤s i≥m) = cong (Sum.map suc id) (splitAt-≥ m i i≥m)
+splitAt-≥ zero    i       _   = refl
+splitAt-≥ (suc m) (suc i) i≥m = cong (Sum.map suc id) (splitAt-≥ m i (ℕ.s≤s⁻¹ i≥m))
 
 ------------------------------------------------------------------------
 -- Bundles
@@ -691,8 +697,9 @@ combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ with <-cmp i k
 
 combine-injectiveʳ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
                      combine i j ≡ combine k l → j ≡ l
-combine-injectiveʳ {m} {n} i j k l cᵢⱼ≡cₖₗ with combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ
-... | refl = toℕ-injective (ℕₚ.+-cancelˡ-≡ (n ℕ.* toℕ i) _ _ (begin
+combine-injectiveʳ {m} {n} i j k l cᵢⱼ≡cₖₗ
+  with refl ← combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ
+  = toℕ-injective (ℕₚ.+-cancelˡ-≡ (n ℕ.* toℕ i) _ _ (begin
   n ℕ.* toℕ i ℕ.+ toℕ j ≡˘⟨ toℕ-combine i j ⟩
   toℕ (combine i j)     ≡⟨ cong toℕ cᵢⱼ≡cₖₗ ⟩
   toℕ (combine i l)     ≡⟨ toℕ-combine i l ⟩
@@ -706,8 +713,8 @@ combine-injective i j k l cᵢⱼ≡cₖₗ =
   combine-injectiveʳ i j k l cᵢⱼ≡cₖₗ
 
 combine-surjective : ∀ (i : Fin (m ℕ.* n)) → ∃₂ λ j k → combine j k ≡ i
-combine-surjective {m} {n} i with remQuot {m} n i in eq
-... | j , k = j , k , (begin
+combine-surjective {m} {n} i with j , k ← remQuot {m} n i in eq
+  = j , k , (begin
   combine j k                       ≡˘⟨ uncurry (cong₂ combine) (,-injective eq) ⟩
   uncurry combine (remQuot {m} n i) ≡⟨ combine-remQuot {m} n i ⟩
   i                                 ∎)
@@ -891,10 +898,10 @@ pinch-surjective zero    (suc j) = suc (suc j) , λ { refl → refl }
 pinch-surjective (suc i) (suc j) = map suc (λ {f refl → cong suc (f refl)}) (pinch-surjective i j)
 
 pinch-mono-≤ : ∀ (i : Fin n) → (pinch i) Preserves _≤_ ⟶ _≤_
-pinch-mono-≤ 0F      {0F}    {k}     0≤n       = z≤n
-pinch-mono-≤ 0F      {suc j} {suc k} (s≤s j≤k) = j≤k
-pinch-mono-≤ (suc i) {0F}    {k}     0≤n       = z≤n
-pinch-mono-≤ (suc i) {suc j} {suc k} (s≤s j≤k) = s≤s (pinch-mono-≤ i j≤k)
+pinch-mono-≤ 0F      {0F}    {k}     0≤n = z≤n
+pinch-mono-≤ 0F      {suc j} {suc k} j≤k = ℕ.s≤s⁻¹ j≤k
+pinch-mono-≤ (suc i) {0F}    {k}     0≤n = z≤n
+pinch-mono-≤ (suc i) {suc j} {suc k} j≤k = s≤s (pinch-mono-≤ i (ℕ.s≤s⁻¹ j≤k))
 
 pinch-injective : ∀ {i : Fin n} {j k : Fin (ℕ.suc n)} →
                   suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k
@@ -963,8 +970,8 @@ private
   -- The boolean component of the result is exactly the
   -- obvious fold of boolean tests (`foldr _∧_ true`).
   note : ∀ {p} {P : Pred (Fin 3) p} (P? : Decidable P) →
-         ∃ λ z → does (all? P?) ≡ z
-  note P? = does (P? 0F) ∧ does (P? 1F) ∧ does (P? 2F) ∧ true
+         ∃ λ z → Dec.does (all? P?) ≡ z
+  note P? = Dec.does (P? 0F) ∧ Dec.does (P? 1F) ∧ Dec.does (P? 2F) ∧ true
           , refl
 
 -- If a decidable predicate P over a finite set is sometimes false,
@@ -995,10 +1002,9 @@ pigeonhole : m ℕ.< n → (f : Fin n → Fin m) → ∃₂ λ i j → i < j ×
 pigeonhole z<s               f = contradiction (f zero) λ()
 pigeonhole (s<s m<n@(s≤s _)) f with any? (λ k → f zero ≟ f (suc k))
 ... | yes (j , f₀≡fⱼ) = zero , suc j , z<s , f₀≡fⱼ
-... | no  f₀≢fₖ with pigeonhole m<n (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))
-...   | (i , j , i<j , fᵢ≡fⱼ) =
-  suc i , suc j , s<s i<j ,
-  punchOut-injective (f₀≢fₖ ∘ (i ,_)) _ fᵢ≡fⱼ
+... | no  f₀≢fₖ
+  with i , j , i<j , fᵢ≡fⱼ ← pigeonhole m<n (λ j → punchOut (f₀≢fₖ ∘ (j ,_ )))
+  = suc i , suc j , s<s i<j , punchOut-injective (f₀≢fₖ ∘ (i ,_)) _ fᵢ≡fⱼ
 
 injective⇒≤ : ∀ {f : Fin m → Fin n} → Injective _≡_ _≡_ f → m ℕ.≤ n
 injective⇒≤ {zero}  {_}     {f} _   = z≤n
@@ -1087,7 +1093,7 @@ opposite-suc {n} i = begin
   toℕ (opposite (suc i))     ≡⟨ opposite-prop (suc i) ⟩
   suc n ∸ suc (toℕ (suc i))  ≡⟨⟩
   n ∸ toℕ (suc i)            ≡⟨⟩
-  n ∸ suc (toℕ i)            ≡⟨ sym (opposite-prop i) ⟩
+  n ∸ suc (toℕ i)            ≡˘⟨ opposite-prop i ⟩
   toℕ (opposite i)           ∎
   where open ≡-Reasoning
 
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 8dfa60de79..21b3f3d71c 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -16,8 +16,8 @@ import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
 import Algebra.Properties.AbelianGroup
 open import Data.Bool.Base using (T; true; false)
 open import Data.Integer.Base renaming (suc to sucℤ)
-open import Data.Nat as ℕ
-  using (ℕ; suc; zero; _∸_; s≤s; z≤n; s<s; z<s)
+open import Data.Nat.Base as ℕ
+  using (ℕ; suc; zero; _∸_; s≤s; z≤n; s<s; z<s; s≤s⁻¹; s<s⁻¹)
   hiding (module ℕ)
 import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
@@ -302,9 +302,9 @@ drop‿-<- (-<- n<m) = n<m
 ... | tri≈ m≮n m≡n n≯m = tri≈ (n≯m ∘ drop‿-<-) (cong -[1+_] m≡n) (m≮n ∘ drop‿-<-)
 ... | tri> m≮n m≢n n>m = tri< (-<- n>m) (m≢n ∘ -[1+-injective) (m≮n ∘ drop‿-<-)
 <-cmp +[1+ m ] +[1+ n ] with ℕ.<-cmp m n
-... | tri< m<n m≢n n≯m = tri< (+<+ (s<s m<n))              (m≢n ∘ +[1+-injective) (n≯m ∘ ℕ.≤-pred ∘ drop‿+<+)
-... | tri≈ m≮n m≡n n≯m = tri≈ (m≮n ∘ ℕ.≤-pred ∘ drop‿+<+) (cong (+_ ∘ suc) m≡n)  (n≯m ∘ ℕ.≤-pred ∘ drop‿+<+)
-... | tri> m≮n m≢n n>m = tri> (m≮n ∘ ℕ.≤-pred ∘ drop‿+<+) (m≢n ∘ +[1+-injective) (+<+ (s<s n>m))
+... | tri< m<n m≢n n≯m = tri< (+<+ (s<s m<n))              (m≢n ∘ +[1+-injective) (n≯m ∘ s<s⁻¹ ∘ drop‿+<+)
+... | tri≈ m≮n m≡n n≯m = tri≈ (m≮n ∘ s<s⁻¹ ∘ drop‿+<+) (cong (+_ ∘ suc) m≡n)  (n≯m ∘ s<s⁻¹ ∘ drop‿+<+)
+... | tri> m≮n m≢n n>m = tri> (m≮n ∘ s<s⁻¹ ∘ drop‿+<+) (m≢n ∘ +[1+-injective) (+<+ (s<s n>m))
 
 infix 4 _<?_
 _<?_ : Decidable _<_
@@ -433,7 +433,7 @@ neg-mono-< { -[1+ _ ]} { -[1+ _ ]} (-<- n<m) = +<+ (s<s n<m)
 neg-mono-< { -[1+ _ ]} { +0}       -<+       = +<+ z<s
 neg-mono-< { -[1+ _ ]} { +[1+ n ]} -<+       = -<+
 neg-mono-< { +0}       { +[1+ n ]} (+<+ _)   = -<+
-neg-mono-< { +[1+ m ]} { +[1+ n ]} (+<+ m<n) = -<- (ℕ.≤-pred m<n)
+neg-mono-< { +[1+ m ]} { +[1+ n ]} (+<+ m<n) = -<- (s<s⁻¹ m<n)
 
 neg-cancel-< : - i < - j → i > j
 neg-cancel-< { +[1+ m ]} { +[1+ n ]} (-<- n<m)       = +<+ (s<s n<m)
@@ -442,7 +442,7 @@ neg-cancel-< { +[1+ m ]} { -[1+ n ]}  -<+            = -<+
 neg-cancel-< { +0}       { +0}       (+<+ ())
 neg-cancel-< { +0}       { -[1+ n ]} _               = -<+
 neg-cancel-< { -[1+ m ]} { +0}       (+<+ ())
-neg-cancel-< { -[1+ m ]} { -[1+ n ]} (+<+ (s<s m<n)) = -<- m<n
+neg-cancel-< { -[1+ m ]} { -[1+ n ]} (+<+ m<n) = -<- (s<s⁻¹ m<n)
 
 ------------------------------------------------------------------------
 -- Properties of ∣_∣
@@ -1233,12 +1233,12 @@ suc[i]≤j⇒i<j : sucℤ i ≤ j → i < j
 suc[i]≤j⇒i<j {+ i}           {+ _}       (+≤+ i≤j) = +<+ i≤j
 suc[i]≤j⇒i<j { -[1+ 0 ]}     {+ j}       p         = -<+
 suc[i]≤j⇒i<j { -[1+ suc i ]} {+ j}       -≤+       = -<+
-suc[i]≤j⇒i<j { -[1+ suc i ]} { -[1+ j ]} (-≤- j≤i) = -<- (ℕ.s≤s j≤i)
+suc[i]≤j⇒i<j { -[1+ suc i ]} { -[1+ j ]} (-≤- j≤i) = -<- (s≤s j≤i)
 
 i<j⇒suc[i]≤j : i < j → sucℤ i ≤ j
 i<j⇒suc[i]≤j {+ _}           {+ _}       (+<+ i<j) = +≤+ i<j
 i<j⇒suc[i]≤j { -[1+ 0 ]}     {+ _}       -<+       = +≤+ z≤n
-i<j⇒suc[i]≤j { -[1+ suc i ]} { -[1+ _ ]} (-<- j<i) = -≤- (ℕ.≤-pred j<i)
+i<j⇒suc[i]≤j { -[1+ suc i ]} { -[1+ _ ]} (-<- j<i) = -≤- (s≤s⁻¹ j<i)
 i<j⇒suc[i]≤j { -[1+ suc i ]} {+ _}       -<+       = -≤+
 
 ------------------------------------------------------------------------
@@ -1288,7 +1288,7 @@ i≤pred[j]⇒i<j {_} { -[1+ n ]} leq = ≤-<-trans leq (-<- ℕ.≤-refl)
 i<j⇒i≤pred[j] : i < j → i ≤ pred j
 i<j⇒i≤pred[j] {_} { +0}       -<+       = -≤- z≤n
 i<j⇒i≤pred[j] {_} { +[1+ n ]} -<+       = -≤+
-i<j⇒i≤pred[j] {_} { +[1+ n ]} (+<+ m<n) = +≤+ (ℕ.≤-pred m<n)
+i<j⇒i≤pred[j] {_} { +[1+ n ]} (+<+ m<n) = +≤+ (s≤s⁻¹ m<n)
 i<j⇒i≤pred[j] {_} { -[1+ n ]} (-<- n<m) = -≤- n<m
 
 i≤j⇒pred[i]≤j : i ≤ j → pred i ≤ j
@@ -1723,7 +1723,7 @@ neg-distribʳ-* i j = begin
 
 *-cancelʳ-≤-pos : ∀ i j k .{{_ : Positive k}} → i * k ≤ j * k → i ≤ j
 *-cancelʳ-≤-pos -[1+ m ] -[1+ n ] +[1+ o ] (-≤- n≤m) =
-  -≤- (ℕ.≤-pred (ℕ.*-cancelʳ-≤ (suc n) (suc m) (suc o) (s≤s n≤m)))
+  -≤- (s≤s⁻¹ (ℕ.*-cancelʳ-≤ (suc n) (suc m) (suc o) (s≤s n≤m)))
 *-cancelʳ-≤-pos -[1+ _ ] (+ _)    +[1+ o ] _         = -≤+
 *-cancelʳ-≤-pos +0       +0       +[1+ o ] _         = +≤+ z≤n
 *-cancelʳ-≤-pos +0       +[1+ _ ] +[1+ o ] _         = +≤+ z≤n
@@ -1738,7 +1738,7 @@ neg-distribʳ-* i j = begin
 *-monoʳ-≤-nonNeg +0 {i} {j} i≤j rewrite *-zeroʳ i | *-zeroʳ j = +≤+ z≤n
 *-monoʳ-≤-nonNeg +[1+ n ] (-≤+ {n = 0})         = -≤+
 *-monoʳ-≤-nonNeg +[1+ n ] (-≤+ {n = suc _})     = -≤+
-*-monoʳ-≤-nonNeg +[1+ n ] (-≤- n≤m) = -≤- (ℕ.≤-pred (ℕ.*-mono-≤ (s≤s n≤m) (ℕ.≤-refl {x = suc n})))
+*-monoʳ-≤-nonNeg +[1+ n ] (-≤- n≤m) = -≤- (s≤s⁻¹ (ℕ.*-mono-≤ (s≤s n≤m) (ℕ.≤-refl {x = suc n})))
 *-monoʳ-≤-nonNeg +[1+ n ] {+0}       {+0}       (+≤+ m≤n) = +≤+ m≤n
 *-monoʳ-≤-nonNeg +[1+ n ] {+0}       {+[1+ _ ]} (+≤+ m≤n) = +≤+ z≤n
 *-monoʳ-≤-nonNeg +[1+ n ] {+[1+ _ ]} {+[1+ _ ]} (+≤+ m≤n) = +≤+ (ℕ.*-monoˡ-≤ (suc n) m≤n)
@@ -1788,7 +1788,7 @@ neg-distribʳ-* i j = begin
 *-cancelˡ-<-nonNeg {+ i}       {+ j}       (+ n) leq = +<+ (ℕ.*-cancelˡ-< n _ _ (+◃-cancel-< leq))
 *-cancelˡ-<-nonNeg {+ i}       { -[1+ j ]} (+ n) leq = contradiction leq +◃≮-◃
 *-cancelˡ-<-nonNeg { -[1+ i ]} {+ j}       (+ n)leq = -<+
-*-cancelˡ-<-nonNeg { -[1+ i ]} { -[1+ j ]} (+ n) leq = -<- (ℕ.≤-pred (ℕ.*-cancelˡ-< n _ _ (neg◃-cancel-< leq)))
+*-cancelˡ-<-nonNeg { -[1+ i ]} { -[1+ j ]} (+ n) leq = -<- (s<s⁻¹ (ℕ.*-cancelˡ-< n _ _ (neg◃-cancel-< leq)))
 
 *-cancelʳ-<-nonNeg : ∀ k .{{_ : NonNegative k}} → i * k < j * k → i < j
 *-cancelʳ-<-nonNeg {i} {j} k rewrite *-comm i k | *-comm j k = *-cancelˡ-<-nonNeg k
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 5afce0cd54..78fc03f865 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -850,7 +850,7 @@ drop-drop (suc m) n (x ∷ xs) = drop-drop m n xs
 
 drop-all : (n : ℕ) (xs : List A) → n ≥ length xs → drop n xs ≡ []
 drop-all n       []       _ = drop-[] n
-drop-all (suc n) (x ∷ xs) p = drop-all n xs (≤-pred p)
+drop-all (suc n) (x ∷ xs) p = drop-all n xs (s≤s⁻¹ p)
 
 ------------------------------------------------------------------------
 -- replicate
diff --git a/src/Data/List/Relation/Unary/Linked/Properties.agda b/src/Data/List/Relation/Unary/Linked/Properties.agda
index 10ece2fdb3..563448730c 100644
--- a/src/Data/List/Relation/Unary/Linked/Properties.agda
+++ b/src/Data/List/Relation/Unary/Linked/Properties.agda
@@ -17,7 +17,7 @@ open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Linked as Linked
   using (Linked; []; [-]; _∷_)
 open import Data.Nat.Base using (zero; suc; _<_; z<s; s<s)
-open import Data.Nat.Properties using (≤-refl; ≤-pred; m≤n⇒m≤1+n)
+open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)
 open import Data.Maybe.Relation.Binary.Connected
   using (Connected; just; nothing; just-nothing; nothing-just)
 open import Level using (Level)
diff --git a/src/Data/Nat.agda b/src/Data/Nat.agda
index 1abd7d9d7c..8811565c11 100644
--- a/src/Data/Nat.agda
+++ b/src/Data/Nat.agda
@@ -42,5 +42,8 @@ open import Data.Nat.Properties public
 
 -- Version 0.17
 
+-- Version 2.0
+
+-- solely for the re-export of this name, formerly in `Data.Nat.Properties.Core`
 open import Data.Nat.Properties public
   using (≤-pred)
diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index b0a0eafcd6..f2755c55fc 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -62,6 +62,15 @@ m < n = suc m ≤ n
 pattern z<s {n}         = s≤s (z≤n {n})
 pattern s<s {m} {n} m<n = s≤s {m} {n} m<n
 
+-- Smart destructors of _≤_, _<_
+
+s≤s⁻¹ : ∀ {m n} → suc m ≤ suc n → m ≤ n
+s≤s⁻¹ (s≤s m≤n) = m≤n
+
+s<s⁻¹ : ∀ {m n} → suc m < suc n → m < n
+s<s⁻¹ (s<s m<n) = m<n
+
+
 ------------------------------------------------------------------------
 -- other ordering relations
 
@@ -283,10 +292,10 @@ zero  ! = 1
 suc n ! = suc n * n !
 
 ------------------------------------------------------------------------
--- Alternative definition of _≤_
+-- Extensionally equivalent alternative definitions of _≤_/_<_ etc.
 
--- The following definition of _≤_ is more suitable for well-founded
--- induction (see Data.Nat.Induction)
+-- _≤′_: this definition is more suitable for well-founded induction
+-- (see Data.Nat.Induction)
 
 infix 4 _≤′_ _<′_ _≥′_ _>′_
 
@@ -308,8 +317,9 @@ m ≥′ n = n ≤′ m
 _>′_ : Rel ℕ 0ℓ
 m >′ n = n <′ m
 
-------------------------------------------------------------------------
--- Another alternative definition of _≤_
+-- _≤″_: this definition of _≤_ is used for proof-irrelevant ‵DivMod`
+-- and is a specialised instance of a general algebraic construction
+
 infix 4 _≤″_ _<″_ _≥″_ _>″_
 
 _≤″_ : (m n : ℕ)  → Set
@@ -326,10 +336,20 @@ m ≥″ n = n ≤″ m
 _>″_ : Rel ℕ 0ℓ
 m >″ n = n <″ m
 
-------------------------------------------------------------------------
--- Another alternative definition of _≤_
+-- Smart constructors of _≤″_ and _<″_
+
+pattern ≤″-offset k = less-than-or-equal {k = k} refl
+pattern <″-offset k = ≤″-offset k
+
+-- Smart destructors of _<″_
+
+s≤″s⁻¹ : ∀ {m n} → suc m ≤″ suc n → m ≤″ n
+s≤″s⁻¹ (≤″-offset k) = ≤″-offset k
+
+s<″s⁻¹ : ∀ {m n} → suc m <″ suc n → m <″ n
+s<″s⁻¹ (<″-offset k) = <″-offset k
 
--- Useful for induction when you have an upper bound.
+-- _≤‴_: this definition is useful for induction with an upper bound.
 
 data _≤‴_ : ℕ → ℕ → Set where
   ≤‴-refl : ∀{m} → m ≤‴ m
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 638482131f..6109106b26 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -15,7 +15,7 @@ open import Algebra.Consequences.Propositional
 open import Data.Bool.Base using (if_then_else_; Bool; true; false)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Nat.Binary.Base
-open import Data.Nat as ℕ using (ℕ; z≤n; s≤s)
+open import Data.Nat as ℕ using (ℕ; z≤n; s≤s; s<s⁻¹)
 open import Data.Nat.DivMod using (_%_; _/_; m/n≤m; +-distrib-/-∣ˡ)
 open import Data.Nat.Divisibility using (∣-refl)
 import Data.Nat.Base as ℕᵇ
@@ -317,17 +317,17 @@ toℕ-cancel-< : ∀ {x y} → toℕ x ℕ.< toℕ y → x < y
 toℕ-cancel-< {zero}     {2[1+ y ]} x<y = 0<even
 toℕ-cancel-< {zero}     {1+[2 y ]} x<y = 0<odd
 toℕ-cancel-< {2[1+ x ]} {2[1+ y ]} x<y =
-  even<even (toℕ-cancel-< (ℕ.≤-pred (ℕₚ.*-cancelˡ-< 2 _ _ x<y)))
+  even<even (toℕ-cancel-< (s<s⁻¹ (ℕₚ.*-cancelˡ-< 2 _ _ x<y)))
 toℕ-cancel-< {2[1+ x ]} {1+[2 y ]} x<y
   rewrite ℕₚ.*-distribˡ-+ 2 1 (toℕ x) =
-  even<odd (toℕ-cancel-< (ℕₚ.*-cancelˡ-< 2 _ _ (ℕₚ.≤-trans (s≤s (ℕₚ.n≤1+n _)) (ℕₚ.≤-pred x<y))))
+  even<odd (toℕ-cancel-< (ℕₚ.*-cancelˡ-< 2 _ _ (ℕₚ.≤-trans (s≤s (ℕₚ.n≤1+n _)) (s<s⁻¹ x<y))))
 toℕ-cancel-< {1+[2 x ]} {2[1+ y ]} x<y with toℕ x ℕₚ.≟ toℕ y
 ... | yes x≡y = odd<even (inj₂ (toℕ-injective x≡y))
 ... | no  x≢y
   rewrite ℕₚ.+-suc (toℕ y) (toℕ y ℕ.+ 0) =
   odd<even (inj₁ (toℕ-cancel-< (ℕₚ.≤∧≢⇒< (ℕₚ.*-cancelˡ-≤ 2 (ℕₚ.+-cancelˡ-≤ 2 _ _ x<y)) x≢y)))
 toℕ-cancel-< {1+[2 x ]} {1+[2 y ]} x<y =
-  odd<odd (toℕ-cancel-< (ℕₚ.*-cancelˡ-< 2 _ _ (ℕ.≤-pred x<y)))
+  odd<odd (toℕ-cancel-< (ℕₚ.*-cancelˡ-< 2 _ _ (s<s⁻¹ x<y)))
 
 fromℕ-cancel-< : ∀ {x y} → fromℕ x < fromℕ y → x ℕ.< y
 fromℕ-cancel-< = subst₂ ℕ._<_ (toℕ-fromℕ _) (toℕ-fromℕ _) ∘ toℕ-mono-<
@@ -1459,7 +1459,7 @@ pred-mono-≤ : pred Preserves _≤_ ⟶ _≤_
 pred-mono-≤ {x} {y} x≤y = begin
   pred x             ≡⟨ cong pred (sym (fromℕ-toℕ x)) ⟩
   pred (fromℕ m)     ≡⟨ sym (fromℕ-pred m) ⟩
-  fromℕ (ℕ.pred m)   ≤⟨ fromℕ-mono-≤ (ℕₚ.pred-mono (toℕ-mono-≤ x≤y)) ⟩
+  fromℕ (ℕ.pred m)   ≤⟨ fromℕ-mono-≤ (ℕₚ.pred-mono-≤ (toℕ-mono-≤ x≤y)) ⟩
   fromℕ (ℕ.pred n)   ≡⟨ fromℕ-pred n ⟩
   pred (fromℕ n)     ≡⟨ cong pred (fromℕ-toℕ y) ⟩
   pred y             ∎
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index ad46644442..dbe99a30e6 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -20,7 +20,6 @@ open import Data.Nat.Properties
 open import Function.Base using (_$_)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
-open import Relation.Nullary.Decidable using (False; toWitnessFalse)
 
 open ≤-Reasoning
 
@@ -465,12 +464,13 @@ _div_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ
 _div_ = _/_
 
 _mod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → Fin divisor
-m mod n@(suc _) = fromℕ< (m%n<n m n)
+m mod n = fromℕ< (m%n<n m n)
 
 _divMod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} →
            DivMod dividend divisor
-m divMod n@(suc _) = result (m / n) (m mod n) (begin-equality
-  m                                   ≡⟨  m≡m%n+[m/n]*n m n ⟩
-  m % n                    + [m/n]*n  ≡˘⟨ cong (_+ [m/n]*n) (toℕ-fromℕ< (m%n<n m n)) ⟩
-  toℕ (fromℕ< (m%n<n m n)) + [m/n]*n  ∎)
-  where [m/n]*n = m / n * n
+m divMod n = result (m / n) (m mod n) $ begin-equality
+  m                               ≡⟨  m≡m%n+[m/n]*n m n ⟩
+  m % n                + [m/n]*n  ≡˘⟨ cong (_+ [m/n]*n) (toℕ-fromℕ< [m%n]<n) ⟩
+  toℕ (fromℕ< [m%n]<n) + [m/n]*n  ∎
+  where [m/n]*n = m / n * n ; [m%n]<n = m%n<n m n
+
diff --git a/src/Data/Nat/DivMod/WithK.agda b/src/Data/Nat/DivMod/WithK.agda
index 458a7eef33..3f6047df60 100644
--- a/src/Data/Nat/DivMod/WithK.agda
+++ b/src/Data/Nat/DivMod/WithK.agda
@@ -8,7 +8,7 @@
 
 module Data.Nat.DivMod.WithK where
 
-open import Data.Nat using (ℕ; NonZero; _+_; _*_; _≟_)
+open import Data.Nat using (ℕ; NonZero; _+_; _*_)
 open import Data.Nat.DivMod hiding (_mod_; _divMod_)
 open import Data.Nat.Properties using (≤⇒≤″)
 open import Data.Nat.WithK
@@ -16,7 +16,7 @@ open import Data.Fin.Base using (Fin; toℕ; fromℕ<″)
 open import Data.Fin.Properties using (toℕ-fromℕ<″)
 open import Function.Base using (_$_)
 open import Relation.Binary.PropositionalEquality
-  using (refl; sym; cong; module ≡-Reasoning)
+  using (cong; module ≡-Reasoning)
 open import Relation.Binary.PropositionalEquality.WithK
 
 open ≡-Reasoning
@@ -27,17 +27,15 @@ infixl 7 _mod_ _divMod_
 -- Certified modulus
 
 _mod_ : (dividend divisor : ℕ) → .{{ _ : NonZero divisor }} → Fin divisor
-a mod n = fromℕ<″ (a % n) (≤″-erase (≤⇒≤″ (m%n<n a n)))
+m mod n = fromℕ<″ (m % n) (≤″-erase (≤⇒≤″ (m%n<n m n)))
 
 ------------------------------------------------------------------------
 -- Returns modulus and division result with correctness proof
 
 _divMod_ : (dividend divisor : ℕ) → .{{ NonZero divisor }} →
            DivMod dividend divisor
-a divMod n = result (a / n) (a mod n) $ ≡-erase $ begin
-  a                                 ≡⟨ m≡m%n+[m/n]*n a n ⟩
-  a % n              + [a/n]*n      ≡⟨ cong (_+ [a/n]*n) (sym (toℕ-fromℕ<″ lemma′)) ⟩
-  toℕ (fromℕ<″ _ lemma′) + [a/n]*n  ∎
-  where
-  lemma′ = ≤″-erase (≤⇒≤″ (m%n<n a n))
-  [a/n]*n = a / n * n
+m divMod n = result (m / n) (m mod n) $ ≡-erase $ begin
+  m                                 ≡⟨ m≡m%n+[m/n]*n m n ⟩
+  m % n                  + [m/n]*n  ≡˘⟨ cong (_+ [m/n]*n) (toℕ-fromℕ<″ lemma″) ⟩
+  toℕ (fromℕ<″ _ lemma″) + [m/n]*n  ∎
+  where [m/n]*n = m / n * n ; lemma″ = ≤″-erase (≤⇒≤″ (m%n<n m n))
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index cf6e2ab0cd..dcc6746f93 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -42,7 +42,7 @@ open import Relation.Binary.Consequences using (flip-Connex)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary hiding (Irrelevant)
 open import Relation.Nullary.Decidable using (True; via-injection; map′)
-open import Relation.Nullary.Negation using (contradiction; contradiction₂)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Nullary.Reflects using (fromEquivalence)
 
 open import Algebra.Definitions {A = ℕ} _≡_
@@ -173,9 +173,7 @@ m ≟ n = map′ (≡ᵇ⇒≡ m n) (≡⇒≡ᵇ m n) (T? (m ≡ᵇ n))
 ≤-total : Total _≤_
 ≤-total zero    _       = inj₁ z≤n
 ≤-total _       zero    = inj₂ z≤n
-≤-total (suc m) (suc n) with ≤-total m n
-... | inj₁ m≤n = inj₁ (s≤s m≤n)
-... | inj₂ n≤m = inj₂ (s≤s n≤m)
+≤-total (suc m) (suc n) = Sum.map s≤s s≤s (≤-total m n)
 
 ≤-irrelevant : Irrelevant _≤_
 ≤-irrelevant z≤n        z≤n        = refl
@@ -265,7 +263,7 @@ s≤s-injective : {p q : m ≤ n} → s≤s p ≡ s≤s q → p ≡ q
 s≤s-injective refl = refl
 
 ≤-pred : suc m ≤ suc n → m ≤ n
-≤-pred (s≤s m≤n) = m≤n
+≤-pred = s≤s⁻¹
 
 m≤n⇒m≤1+n : m ≤ n → m ≤ 1 + n
 m≤n⇒m≤1+n z≤n       = z≤n
@@ -427,7 +425,7 @@ s<s-injective : {p q : m < n} → s<s p ≡ s<s q → p ≡ q
 s<s-injective refl = refl
 
 <-pred : suc m < suc n → m < n
-<-pred (s<s m<n) = m<n
+<-pred = s<s⁻¹
 
 m<n⇒m<1+n : m < n → m < 1 + n
 m<n⇒m<1+n z<s               = z<s
@@ -464,12 +462,10 @@ m<n⇒n≢0 (s≤s m≤n) ()
 m<n⇒m≤1+n : m < n → m ≤ suc n
 m<n⇒m≤1+n = m≤n⇒m≤1+n ∘ <⇒≤
 
-m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
-m<1+n⇒m<n∨m≡n {0}     {0}     _          =  inj₂ refl
-m<1+n⇒m<n∨m≡n {0}     {suc n} _          =  inj₁ 0<1+n
-m<1+n⇒m<n∨m≡n {suc m} {suc n} (s<s m<1+n)  with m<1+n⇒m<n∨m≡n m<1+n
-... | inj₂ m≡n = inj₂ (cong suc m≡n)
-... | inj₁ m<n = inj₁ (s<s m<n)
+m<1+n⇒m<n∨m≡n :  ∀ {m n} → m < suc n → m < n ⊎ m ≡ n
+m<1+n⇒m<n∨m≡n {0}     {0}     _           = inj₂ refl
+m<1+n⇒m<n∨m≡n {0}     {suc n} _           = inj₁ 0<1+n
+m<1+n⇒m<n∨m≡n {suc m} {suc n} (s<s m<1+n) = Sum.map s<s (cong suc) (m<1+n⇒m<n∨m≡n m<1+n)
 
 m≤n⇒m<n∨m≡n : m ≤ n → m < n ⊎ m ≡ n
 m≤n⇒m<n∨m≡n m≤n = m<1+n⇒m<n∨m≡n (s≤s m≤n)
@@ -717,7 +713,7 @@ m+n≤o⇒n≤o (suc m) m+n<o = m+n≤o⇒n≤o m (<⇒≤ m+n<o)
 +-monoʳ-< (suc n) m≤o = s≤s (+-monoʳ-< n m≤o)
 
 m+1+n≰m : ∀ m {n} → m + suc n ≰ m
-m+1+n≰m (suc m) (s≤s m+1+n≤m) = m+1+n≰m m m+1+n≤m
+m+1+n≰m (suc m) m+1+n≤m = m+1+n≰m m (s≤s⁻¹ m+1+n≤m)
 
 m<m+n : ∀ m {n} → n > 0 → m < m + n
 m<m+n zero    n>0 = n>0
@@ -727,8 +723,8 @@ m<n+m : ∀ m {n} → n > 0 → m < n + m
 m<n+m m {n} n>0 rewrite +-comm n m = m<m+n m n>0
 
 m+n≮n : ∀ m n → m + n ≮ n
-m+n≮n zero    n                   = n≮n n
-m+n≮n (suc m) (suc n) (s<s m+n<n) = m+n≮n m (suc n) (m<n⇒m<1+n m+n<n)
+m+n≮n zero    n                = n≮n n
+m+n≮n (suc m) n@(suc _) sm+n<n = m+n≮n m n (m<n⇒m<1+n (s<s⁻¹ sm+n<n))
 
 m+n≮m : ∀ m n → m + n ≮ m
 m+n≮m m n = subst (_≮ m) (+-comm n m) (m+n≮n n m)
@@ -1691,9 +1687,6 @@ m⊓n+n∸m≡n (suc m) (suc n) = cong suc $ m⊓n+n∸m≡n m n
 -- Properties of pred
 ------------------------------------------------------------------------
 
-pred-mono : pred Preserves _≤_ ⟶ _≤_
-pred-mono m≤n = ∸-mono m≤n (≤-refl {1})
-
 pred[n]≤n : pred n ≤ n
 pred[n]≤n {zero}  = z≤n
 pred[n]≤n {suc n} = n≤1+n n
@@ -1712,6 +1705,13 @@ pred[n]≤n {suc n} = n≤1+n n
 suc-pred : ∀ n .{{_ : NonZero n}} → suc (pred n) ≡ n
 suc-pred (suc n) = refl
 
+pred-mono-≤ : pred Preserves _≤_ ⟶ _≤_
+pred-mono-≤ {zero}          _   = z≤n
+pred-mono-≤ {suc _} {suc _} m≤n = s≤s⁻¹ m≤n
+
+pred-mono-< : .⦃ _ : NonZero m ⦄ → m < n → pred m < pred n
+pred-mono-< {m = suc _} {n = suc _} = s<s⁻¹
+
 ------------------------------------------------------------------------
 -- Properties of ∣_-_∣
 ------------------------------------------------------------------------
@@ -2067,8 +2067,8 @@ m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)
 <ᵇ⇒<″ : T (m <ᵇ n) → m <″ n
 <ᵇ⇒<″ {m} {n} leq = less-than-or-equal (m+[n∸m]≡n (<ᵇ⇒< m n leq))
 
-<″⇒<ᵇ : m <″ n → T (m <ᵇ n)
-<″⇒<ᵇ {m} (less-than-or-equal refl) = <⇒<ᵇ (m≤m+n (suc m) _)
+<″⇒<ᵇ : ∀ {m n} → m <″ n → T (m <ᵇ n)
+<″⇒<ᵇ {m} (<″-offset k) = <⇒<ᵇ (m≤m+n (suc m) k)
 
 -- equivalence to the old definition of _≤″_
 
@@ -2078,9 +2078,8 @@ m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)
 -- equivalence to _≤_
 
 ≤″⇒≤ : _≤″_ ⇒ _≤_
-≤″⇒≤ {zero}  (less-than-or-equal refl) = z≤n
-≤″⇒≤ {suc m} (less-than-or-equal refl) =
-  s≤s (≤″⇒≤ (less-than-or-equal refl))
+≤″⇒≤ {zero}  (≤″-offset k) = z≤n {k}
+≤″⇒≤ {suc m} (≤″-offset k) = s≤s (≤″⇒≤ (≤″-offset k))
 
 ≤⇒≤″ : _≤_ ⇒ _≤″_
 ≤⇒≤″ = less-than-or-equal ∘ m+[n∸m]≡n
@@ -2097,7 +2096,7 @@ _<″?_ : Decidable _<″_
 m <″? n = map′ <ᵇ⇒<″ <″⇒<ᵇ (T? (m <ᵇ n))
 
 _≤″?_ : Decidable _≤″_
-zero  ≤″? n = yes (less-than-or-equal refl)
+zero  ≤″? n = yes (≤″-offset n)
 suc m ≤″? n = m <″? n
 
 _≥″?_ : Decidable _≥″_
@@ -2332,6 +2331,11 @@ suc[pred[n]]≡n {suc n} _   = refl
 "Warning: <-step was deprecated in v2.0. Please use m<n⇒m<1+n instead. "
 #-}
 
+pred-mono = pred-mono-≤
+{-# WARNING_ON_USAGE pred-mono
+"Warning: pred-mono was deprecated in v2.0. Please use pred-mono-≤ instead. "
+#-}
+
 {- issue1844/issue1755: raw bundles have moved to `Data.X.Base` -}
 open Data.Nat.Base public
   using (*-rawMagma; *-1-rawMonoid)
diff --git a/src/Data/Nat/Properties/Core.agda b/src/Data/Nat/Properties/Core.agda
index 94031fdc37..66bfdaa32b 100644
--- a/src/Data/Nat/Properties/Core.agda
+++ b/src/Data/Nat/Properties/Core.agda
@@ -2,7 +2,7 @@
 -- The Agda standard library
 --
 -- This module is DEPRECATED. Please use
--- `Data.Nat.Properties` directly.
+-- `Data.Nat.Base directly.
 --
 ------------------------------------------------------------------------
 
@@ -12,7 +12,7 @@ module Data.Nat.Properties.Core where
 
 {-# WARNING_ON_IMPORT
 "Data.Nat.Properties.Core was deprecated in v2.0.
-Use Data.Nat.Properties instead."
+Use Data.Nat.Base instead."
 #-}
 
 open import Data.Nat.Base
@@ -22,4 +22,4 @@ open import Data.Nat.Base
 ------------------------------------------------------------------------
 
 ≤-pred : ∀ {m n} → suc m ≤ suc n → m ≤ n
-≤-pred (s≤s m≤n) = m≤n
+≤-pred = s≤s⁻¹
diff --git a/src/Data/Nat/Show/Properties.agda b/src/Data/Nat/Show/Properties.agda
index 564264946c..13dd34ace2 100644
--- a/src/Data/Nat/Show/Properties.agda
+++ b/src/Data/Nat/Show/Properties.agda
@@ -6,21 +6,25 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Digit using (showDigit; toDigits)
 open import Data.Digit.Properties using (toDigits-injective; showDigit-injective)
-open import Function.Base using (_∘_)
 import Data.List.Properties as Listₚ
 open import Data.Nat.Base using (ℕ)
 open import Data.Nat.Properties using (_≤?_)
+open import Data.Nat.Show using (charsInBase)
+open import Function.Base using (_∘_)
 open import Relation.Nullary.Decidable using (True)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
-open import Data.Nat.Show using (charsInBase)
 
 module Data.Nat.Show.Properties where
 
 module _ (base : ℕ) {base≥2 : True (2 ≤? base)} {base≤16 : True (base ≤? 16)} where
 
-  charsInBase-injective : ∀ n m → charsInBase base {base≥2} {base≤16} n ≡ charsInBase base {base≥2} {base≤16} m → n ≡ m
-  charsInBase-injective n m = toDigits-injective base {base≥2} _ _
+  private
+    charsInBase-base         = charsInBase base {base≥2} {base≤16}
+    toDigits-injective-base  = toDigits-injective base {base≥2} {base≥2}
+    showDigit-injective-base = showDigit-injective base {base≤16} {base≤16}
+
+  charsInBase-injective : ∀ n m →  charsInBase-base n ≡ charsInBase-base m → n ≡ m
+  charsInBase-injective n m = toDigits-injective-base _ _
                             ∘ Listₚ.reverse-injective
-                            ∘ Listₚ.map-injective (showDigit-injective _ _ _)
+                            ∘ Listₚ.map-injective (showDigit-injective-base _ _)
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 0f56c9fbd7..1ecf91b9e7 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -8,7 +8,7 @@
 
 module Data.Vec.Base where
 
-open import Data.Bool.Base using (Bool; true; false; if_then_else_)
+open import Data.Bool.Base using (Bool; if_then_else_)
 open import Data.Nat.Base
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List)
diff --git a/src/Data/Vec/Bounded/Base.agda b/src/Data/Vec/Bounded/Base.agda
index c857ee6642..63649e940e 100644
--- a/src/Data/Vec/Bounded/Base.agda
+++ b/src/Data/Vec/Bounded/Base.agda
@@ -131,12 +131,12 @@ align = alignWith id
 take : ∀ {m} n → Vec≤ A m → Vec≤ A (n ⊓ m)
 take             zero    _                = []
 take             (suc n) (Vec.[] , p)     = []
-take {m = suc m} (suc n) (a Vec.∷ as , p) = a ∷ take n (as , ℕₚ.≤-pred p)
+take {m = suc m} (suc n) (a Vec.∷ as , p) = a ∷ take n (as , s≤s⁻¹ p)
 
 drop : ∀ {m} n → Vec≤ A m → Vec≤ A (m ∸ n)
 drop             zero    v                = v
 drop             (suc n) (Vec.[] , p)     = []
-drop {m = suc m} (suc n) (a Vec.∷ as , p) = drop n (as , ℕₚ.≤-pred p)
+drop {m = suc m} (suc n) (a Vec.∷ as , p) = drop n (as , s≤s⁻¹ p)
 
 ------------------------------------------------------------------------
 -- Lifting a collection of bounded vectors to the same size
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 27df432105..38793a9038 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -31,7 +31,7 @@ open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; _≢_; _≗_; refl; sym; trans; cong; cong₂; subst; module ≡-Reasoning)
 open import Relation.Unary using (Pred; Decidable)
-open import Relation.Nullary.Decidable.Core using (Dec; does; yes; no; _×-dec_; map′)
+open import Relation.Nullary.Decidable.Core using (Dec; does; yes; _×-dec_; map′)
 open import Relation.Nullary.Negation.Core using (contradiction)
 import Data.Nat.GeneralisedArithmetic as ℕ
 
@@ -483,14 +483,14 @@ cast-++ʳ {m = suc m} eq (x ∷ xs) {ys} = cong (x ∷_) (cast-++ʳ eq xs)
 lookup-++-< : ∀ (xs : Vec A m) (ys : Vec A n) →
               ∀ i (i<m : toℕ i < m) →
               lookup (xs ++ ys) i  ≡ lookup xs (Fin.fromℕ< i<m)
-lookup-++-< (x ∷ xs) ys zero    z<s       = refl
-lookup-++-< (x ∷ xs) ys (suc i) (s<s i<m) = lookup-++-< xs ys i i<m
+lookup-++-< (x ∷ xs) ys zero    _     = refl
+lookup-++-< (x ∷ xs) ys (suc i) si<sm = lookup-++-< xs ys i (s<s⁻¹ si<sm)
 
 lookup-++-≥ : ∀ (xs : Vec A m) (ys : Vec A n) →
               ∀ i (i≥m : toℕ i ≥ m) →
               lookup (xs ++ ys) i ≡ lookup ys (Fin.reduce≥ i i≥m)
-lookup-++-≥ []       ys i       i≥m       = refl
-lookup-++-≥ (x ∷ xs) ys (suc i) (s≤s i≥m) = lookup-++-≥ xs ys i i≥m
+lookup-++-≥ []       ys i       _     = refl
+lookup-++-≥ (x ∷ xs) ys (suc i) si≥sm = lookup-++-≥ xs ys i (s≤s⁻¹ si≥sm)
 
 lookup-++ˡ : ∀ (xs : Vec A m) (ys : Vec A n) i →
              lookup (xs ++ ys) (i ↑ˡ n) ≡ lookup xs i
@@ -1396,7 +1396,7 @@ Please use take-map instead."
 drop-distr-zipWith = drop-zipWith
 {-# WARNING_ON_USAGE drop-distr-zipWith
 "Warning: drop-distr-zipWith was deprecated in v2.0.
-Please use tdrop-zipWith instead."
+Please use drop-zipWith instead."
 #-}
 drop-distr-map = drop-map
 {-# WARNING_ON_USAGE drop-distr-map
@@ -1441,7 +1441,7 @@ lookup-inject≤-take m m≤m+n i xs = sym (begin
   lookup (take m xs) i
     ≡⟨ lookup-take-inject≤ xs i ⟩
   lookup xs (Fin.inject≤ i _)
-    ≡⟨ cong ((lookup xs) ∘ (Fin.inject≤ i)) (≤-irrelevant _ _) ⟩
+    ≡⟨⟩
   lookup xs (Fin.inject≤ i m≤m+n)
     ∎) where open ≡-Reasoning
 {-# WARNING_ON_USAGE lookup-inject≤-take
diff --git a/src/Data/Vec/Relation/Unary/Linked/Properties.agda b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
index c32e4c8df6..8ff5a4d2e3 100644
--- a/src/Data/Vec/Relation/Unary/Linked/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
@@ -8,23 +8,18 @@
 
 module Data.Vec.Relation.Unary.Linked.Properties where
 
-open import Data.Bool.Base using (true; false)
 open import Data.Vec.Base as Vec
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 import Data.Vec.Relation.Unary.All.Properties as Allₚ
 open import Data.Vec.Relation.Unary.Linked as Linked
   using (Linked; []; [-]; _∷_)
 open import Data.Fin.Base using (zero; suc; _<_)
-open import Data.Nat.Base using (ℕ; zero; suc; NonZero)
-open import Data.Nat.Properties using (<-pred)
+open import Data.Nat.Base using (ℕ; zero; suc; s<s⁻¹)
 open import Level using (Level)
-open import Function.Base using (_∘_; flip; _on_)
+open import Function.Base using (_on_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Transitive)
-open import Relation.Binary.Bundles using (DecSetoid)
-open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable)
-open import Relation.Nullary.Decidable using (yes; no; does)
 
 private
   variable
@@ -49,7 +44,7 @@ module _ (trans : Transitive R) where
            Linked R xs → i < j →
            R (lookup xs i) (lookup xs j)
   lookup⁺ {i = zero}  {j = suc j} (rx ∷ rxs) i<j = Allₚ.lookup⁺ (Linked⇒All rx rxs) j
-  lookup⁺ {i = suc i} {j = suc j} (_  ∷ rxs) i<j = lookup⁺ rxs (<-pred i<j)
+  lookup⁺ {i = suc i} {j = suc j} (_  ∷ rxs) i<j = lookup⁺ rxs (s<s⁻¹ i<j)
 
 ------------------------------------------------------------------------
 -- Introduction (⁺) and elimination (⁻) rules for vec operations

From 2efd5f9b5d1e8f080231ccf8a0c863dca4be663f Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 12 Oct 2023 13:01:14 +0900
Subject: [PATCH 154/316] Bump CI tests to Agda-2.6.4 (#2134)

---
 .github/workflows/ci-ubuntu.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index 85e9fe1365..ebafe70fa8 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -74,7 +74,7 @@ jobs:
           if [[ '${{ github.ref }}' == 'refs/heads/master' \
              || '${{ github.base_ref }}' == 'master' ]]; then
             # Pick Agda version for master
-            echo "AGDA_COMMIT=tags/v2.6.3" >> $GITHUB_ENV;
+            echo "AGDA_COMMIT=tags/v2.6.4" >> $GITHUB_ENV;
             echo "AGDA_HTML_DIR=html" >> $GITHUB_ENV
           elif [[ '${{ github.ref }}' == 'refs/heads/experimental' \
                || '${{ github.base_ref }}' == 'experimental' ]]; then

From dea8bf7a0cfb49309231b89ae0731376cb1734f1 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 12 Oct 2023 22:42:47 +0900
Subject: [PATCH 155/316] Remove `Algebra.Ordered` (#2133)

---
 CHANGELOG.md                                  |   7 -
 src/Algebra/Ordered.agda                      |  14 -
 src/Algebra/Ordered/Bundles.agda              | 438 ------------------
 src/Algebra/Ordered/Structures.agda           | 391 ----------------
 .../Properties/BoundedJoinSemilattice.agda    |  25 +-
 .../Lattice/Properties/JoinSemilattice.agda   |  22 -
 .../Lattice/Properties/MeetSemilattice.agda   |   2 +-
 7 files changed, 2 insertions(+), 897 deletions(-)
 delete mode 100644 src/Algebra/Ordered.agda
 delete mode 100644 src/Algebra/Ordered/Bundles.agda
 delete mode 100644 src/Algebra/Ordered/Structures.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 988355bd5a..6bfd659aa9 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1832,13 +1832,6 @@ New modules
   Algebra.Morphism.Construct.Identity
   ```
 
-* Ordered algebraic structures (pomonoids, posemigroups, etc.)
-  ```
-  Algebra.Ordered
-  Algebra.Ordered.Bundles
-  Algebra.Ordered.Structures
-  ```
-
 * 'Optimised' tail-recursive exponentiation properties:
   ```
   Algebra.Properties.Semiring.Exp.TailRecursiveOptimised
diff --git a/src/Algebra/Ordered.agda b/src/Algebra/Ordered.agda
deleted file mode 100644
index 5eb99aec36..0000000000
--- a/src/Algebra/Ordered.agda
+++ /dev/null
@@ -1,14 +0,0 @@
-------------------------------------------------------------------------
--- The Agda standard library
---
--- Definitions of ordered algebraic structures like promonoids and
--- posemigroups (packed in records together with sets, orders,
--- operations, etc.)
-------------------------------------------------------------------------
-
-{-# OPTIONS --cubical-compatible --safe #-}
-
-module Algebra.Ordered where
-
-open import Algebra.Ordered.Structures public
-open import Algebra.Ordered.Bundles public
diff --git a/src/Algebra/Ordered/Bundles.agda b/src/Algebra/Ordered/Bundles.agda
deleted file mode 100644
index 896f9d4d1f..0000000000
--- a/src/Algebra/Ordered/Bundles.agda
+++ /dev/null
@@ -1,438 +0,0 @@
-------------------------------------------------------------------------
--- The Agda standard library
---
--- Definitions of ordered algebraic structures like promonoids and
--- posemigroups (packed in records together with sets, orders,
--- operations, etc.)
-------------------------------------------------------------------------
-
--- The contents of this module should be accessed via `Algebra.Order`.
-
-{-# OPTIONS --cubical-compatible --safe #-}
-
-module Algebra.Ordered.Bundles where
-
-open import Algebra.Core
-open import Algebra.Bundles
-open import Algebra.Ordered.Structures
-open import Level using (suc; _⊔_)
-open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.Bundles using (Preorder; Poset)
-
-------------------------------------------------------------------------
--- Bundles of preordered structures
-
--- Preordered magmas (promagmas)
-
-record Promagma c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier    : Set c
-    _≈_        : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_        : Rel Carrier ℓ₂  -- The preorder.
-    _∙_        : Op₂ Carrier     -- Multiplication.
-    isPromagma : IsPromagma _≈_ _≤_ _∙_
-
-  open IsPromagma isPromagma public
-
-  preorder : Preorder c ℓ₁ ℓ₂
-  preorder = record { isPreorder = isPreorder }
-
-  magma : Magma c ℓ₁
-  magma = record { isMagma = isMagma }
-
--- Preordered semigroups (prosemigroups)
-
-record Prosemigroup c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier        : Set c
-    _≈_            : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_            : Rel Carrier ℓ₂  -- The preorder.
-    _∙_            : Op₂ Carrier     -- Multiplication.
-    isProsemigroup : IsProsemigroup _≈_ _≤_ _∙_
-
-  open IsProsemigroup isProsemigroup public
-
-  promagma : Promagma c ℓ₁ ℓ₂
-  promagma = record { isPromagma = isPromagma }
-
-  open Promagma promagma public using (preorder; magma)
-
-  semigroup : Semigroup c ℓ₁
-  semigroup = record { isSemigroup = isSemigroup }
-
--- Preordered monoids (promonoids)
-
-record Promonoid c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier     : Set c
-    _≈_         : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_         : Rel Carrier ℓ₂  -- The preorder.
-    _∙_         : Op₂ Carrier     -- The monoid multiplication.
-    ε           : Carrier         -- The monoid unit.
-    isPromonoid : IsPromonoid _≈_ _≤_ _∙_ ε
-
-  open IsPromonoid isPromonoid public
-
-  prosemigroup : Prosemigroup c ℓ₁ ℓ₂
-  prosemigroup = record { isProsemigroup = isProsemigroup }
-
-  open Prosemigroup prosemigroup public
-    using (preorder; magma; promagma; semigroup)
-
-  monoid : Monoid c ℓ₁
-  monoid = record { isMonoid = isMonoid }
-
--- Preordered commutative monoids (commutative promonoids)
-
-record CommutativePromonoid c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier : Set c
-    _≈_     : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_     : Rel Carrier ℓ₂  -- The preorder.
-    _∙_     : Op₂ Carrier     -- The monoid multiplication.
-    ε       : Carrier         -- The monoid unit.
-    isCommutativePromonoid : IsCommutativePromonoid _≈_ _≤_ _∙_ ε
-
-  open IsCommutativePromonoid isCommutativePromonoid public
-
-  promonoid : Promonoid c ℓ₁ ℓ₂
-  promonoid = record { isPromonoid = isPromonoid }
-
-  open Promonoid promonoid public
-    using (preorder; magma; promagma; semigroup; prosemigroup; monoid)
-
-  commutativeMonoid : CommutativeMonoid c ℓ₁
-  commutativeMonoid = record { isCommutativeMonoid = isCommutativeMonoid }
-
-  open CommutativeMonoid commutativeMonoid public using (commutativeSemigroup)
-
--- Preordered semirings (prosemirings)
-
-record Prosemiring c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _*_
-  infixl 6 _+_
-  field
-    Carrier       : Set c
-    _≈_           : Rel Carrier ℓ₁
-    _≤_           : Rel Carrier ℓ₂
-    _+_           : Op₂ Carrier
-    _*_           : Op₂ Carrier
-    0#            : Carrier
-    1#            : Carrier
-    isProsemiring : IsProsemiring _≈_ _≤_ _+_ _*_ 0# 1#
-
-  open IsProsemiring isProsemiring public
-
-  +-commutativePromonoid : CommutativePromonoid c ℓ₁ ℓ₂
-  +-commutativePromonoid = record
-    { isCommutativePromonoid = +-isCommutativePromonoid }
-
-  open CommutativePromonoid +-commutativePromonoid public
-    using (preorder)
-    renaming
-    ( magma                to +-magma
-    ; promagma             to +-promagma
-    ; semigroup            to +-semigroup
-    ; prosemigroup         to +-prosemigroup
-    ; monoid               to +-monoid
-    ; promonoid            to +-promonoid
-    ; commutativeSemigroup to +-commutativeSemigroup
-    ; commutativeMonoid    to +-commutativeMonoid
-    )
-
-  *-promonoid : Promonoid c ℓ₁ ℓ₂
-  *-promonoid = record { isPromonoid = *-isPromonoid }
-
-  open Promonoid *-promonoid public
-    using ()
-    renaming
-    ( magma        to *-magma
-    ; promagma     to *-promagma
-    ; semigroup    to *-semigroup
-    ; prosemigroup to *-prosemigroup
-    ; monoid       to *-monoid
-    )
-
-  semiring : Semiring c ℓ₁
-  semiring = record { isSemiring = isSemiring }
-
--- Preordered IdempotentSemiring (IdempotentProsemiring)
-
-record IdempotentProsemiring c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _*_
-  infixl 6 _+_
-  field
-    Carrier       : Set c
-    _≈_           : Rel Carrier ℓ₁
-    _≤_           : Rel Carrier ℓ₂
-    _+_           : Op₂ Carrier
-    _*_           : Op₂ Carrier
-    0#            : Carrier
-    1#            : Carrier
-    isIdempotentProsemiring : IsIdempotentProsemiring _≈_ _≤_ _+_ _*_ 0# 1#
-
-  open IsIdempotentProsemiring isIdempotentProsemiring public
-
-  idempotentSemiring : IdempotentSemiring c ℓ₁
-  idempotentSemiring = record { isIdempotentSemiring = isIdempotentSemiring }
-
--- Preordered KleeneAlgebra (proKleeneAlgebra)
-
-record ProKleeneAlgebra c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infix  8 _⋆
-  infixl 7 _*_
-  infixl 6 _+_
-  field
-    Carrier       : Set c
-    _≈_           : Rel Carrier ℓ₁
-    _≤_           : Rel Carrier ℓ₂
-    _+_           : Op₂ Carrier
-    _*_           : Op₂ Carrier
-    _⋆            : Op₁ Carrier
-    0#            : Carrier
-    1#            : Carrier
-    isProKleeneAlgebra : IsProKleeneAlgebra _≈_ _≤_ _+_ _*_ _⋆ 0# 1#
-
-  open IsProKleeneAlgebra isProKleeneAlgebra public
-
-  kleeneAlgebra : KleeneAlgebra c ℓ₁
-  kleeneAlgebra = record { isKleeneAlgebra = isKleeneAlgebra }
-
-------------------------------------------------------------------------
--- Bundles of partially ordered structures
-
--- Partially ordered magmas (pomagmas)
-
-record Pomagma c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier   : Set c
-    _≈_       : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_       : Rel Carrier ℓ₂  -- The partial order.
-    _∙_       : Op₂ Carrier     -- Multiplication.
-    isPomagma : IsPomagma _≈_ _≤_ _∙_
-
-  open IsPomagma isPomagma public
-
-  poset : Poset c ℓ₁ ℓ₂
-  poset = record { isPartialOrder = isPartialOrder }
-
-  promagma : Promagma c ℓ₁ ℓ₂
-  promagma = record { isPromagma = isPromagma }
-
-  open Promagma promagma public using (preorder; magma)
-
--- Partially ordered semigroups (posemigroups)
-
-record Posemigroup c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier       : Set c
-    _≈_           : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_           : Rel Carrier ℓ₂  -- The partial order.
-    _∙_           : Op₂ Carrier     -- Multiplication.
-    isPosemigroup : IsPosemigroup _≈_ _≤_ _∙_
-
-  open IsPosemigroup isPosemigroup public
-
-  pomagma : Pomagma c ℓ₁ ℓ₂
-  pomagma = record { isPomagma = isPomagma }
-
-  open Pomagma pomagma public using (preorder; poset; magma; promagma)
-
-  prosemigroup : Prosemigroup c ℓ₁ ℓ₂
-  prosemigroup = record { isProsemigroup = isProsemigroup }
-
-  open Prosemigroup prosemigroup public using (semigroup)
-
--- Partially ordered monoids (pomonoids)
-
-record Pomonoid c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier    : Set c
-    _≈_        : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_        : Rel Carrier ℓ₂  -- The partial order.
-    _∙_        : Op₂ Carrier     -- The monoid multiplication.
-    ε          : Carrier         -- The monoid unit.
-    isPomonoid : IsPomonoid _≈_ _≤_ _∙_ ε
-
-  open IsPomonoid isPomonoid public
-
-  posemigroup : Posemigroup c ℓ₁ ℓ₂
-  posemigroup = record { isPosemigroup = isPosemigroup }
-
-  open Posemigroup posemigroup public using
-    ( preorder
-    ; poset
-    ; magma
-    ; promagma
-    ; pomagma
-    ; semigroup
-    ; prosemigroup
-    )
-
-  promonoid : Promonoid c ℓ₁ ℓ₂
-  promonoid = record { isPromonoid = isPromonoid }
-
-  open Promonoid promonoid public using (monoid)
-
--- Partially ordered commutative monoids (commutative pomonoids)
-
-record CommutativePomonoid c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _∙_
-  field
-    Carrier               : Set c
-    _≈_                   : Rel Carrier ℓ₁  -- The underlying equality.
-    _≤_                   : Rel Carrier ℓ₂  -- The partial order.
-    _∙_                   : Op₂ Carrier     -- The monoid multiplication.
-    ε                     : Carrier         -- The monoid unit.
-    isCommutativePomonoid : IsCommutativePomonoid _≈_ _≤_ _∙_ ε
-
-  open IsCommutativePomonoid isCommutativePomonoid public
-
-  pomonoid : Pomonoid c ℓ₁ ℓ₂
-  pomonoid = record { isPomonoid = isPomonoid }
-
-  open Pomonoid pomonoid public using
-    ( preorder
-    ; poset
-    ; magma
-    ; promagma
-    ; pomagma
-    ; semigroup
-    ; prosemigroup
-    ; posemigroup
-    ; monoid
-    ; promonoid
-    )
-
-  commutativePromonoid : CommutativePromonoid c ℓ₁ ℓ₂
-  commutativePromonoid =
-    record { isCommutativePromonoid = isCommutativePromonoid }
-
-  open CommutativePromonoid commutativePromonoid public
-    using (commutativeSemigroup; commutativeMonoid)
-
--- Partially ordered semirings (posemirings)
-
-record Posemiring c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _*_
-  infixl 6 _+_
-  field
-    Carrier      : Set c
-    _≈_          : Rel Carrier ℓ₁
-    _≤_          : Rel Carrier ℓ₂
-    _+_          : Op₂ Carrier
-    _*_          : Op₂ Carrier
-    0#           : Carrier
-    1#           : Carrier
-    isPosemiring : IsPosemiring _≈_ _≤_ _+_ _*_ 0# 1#
-
-  open IsPosemiring isPosemiring public
-
-  +-commutativePomonoid : CommutativePomonoid c ℓ₁ ℓ₂
-  +-commutativePomonoid = record
-    { isCommutativePomonoid = +-isCommutativePomonoid }
-
-  open CommutativePomonoid +-commutativePomonoid public
-    using (preorder)
-    renaming
-    ( magma                to +-magma
-    ; promagma             to +-promagma
-    ; pomagma              to +-pomagma
-    ; semigroup            to +-semigroup
-    ; prosemigroup         to +-prosemigroup
-    ; posemigroup          to +-posemigroup
-    ; monoid               to +-monoid
-    ; promonoid            to +-promonoid
-    ; pomonoid             to +-pomonoid
-    ; commutativeSemigroup to +-commutativeSemigroup
-    ; commutativeMonoid    to +-commutativeMonoid
-    ; commutativePromonoid to +-commutativePromonoid
-    )
-
-  *-pomonoid : Pomonoid c ℓ₁ ℓ₂
-  *-pomonoid = record { isPomonoid = *-isPomonoid }
-
-  open Pomonoid *-pomonoid public
-    using ()
-    renaming
-    ( magma        to *-magma
-    ; promagma     to *-promagma
-    ; pomagma      to *-pomagma
-    ; semigroup    to *-semigroup
-    ; prosemigroup to *-prosemigroup
-    ; posemigroup  to *-posemigroup
-    ; monoid       to *-monoid
-    ; promonoid    to *-promonoid
-    )
-
-  prosemiring : Prosemiring c ℓ₁ ℓ₂
-  prosemiring = record { isProsemiring = isProsemiring }
-
-  open Prosemiring prosemiring public using (semiring)
-
--- Partially ordered idempotentSemiring (IdempotentPosemiring)
-
-record IdempotentPosemiring c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infixl 7 _*_
-  infixl 6 _+_
-  field
-    Carrier       : Set c
-    _≈_           : Rel Carrier ℓ₁
-    _≤_           : Rel Carrier ℓ₂
-    _+_           : Op₂ Carrier
-    _*_           : Op₂ Carrier
-    0#            : Carrier
-    1#            : Carrier
-    isIdempotentPosemiring : IsIdempotentPosemiring _≈_ _≤_ _+_ _*_ 0# 1#
-
-  open IsIdempotentPosemiring isIdempotentPosemiring public
-
-  idempotentProsemiring : IdempotentProsemiring c ℓ₁ ℓ₂
-  idempotentProsemiring = record { isIdempotentProsemiring = isIdempotentProsemiring }
-
-  open IdempotentProsemiring idempotentProsemiring public using (idempotentSemiring; +-idem)
-
--- Partially ordered KleeneAlgebra (PoKleeneAlgebra)
-
-record PoKleeneAlgebra c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  infix  4 _≈_ _≤_
-  infix  8 _⋆
-  infixl 7 _*_
-  infixl 6 _+_
-  field
-    Carrier       : Set c
-    _≈_           : Rel Carrier ℓ₁
-    _≤_           : Rel Carrier ℓ₂
-    _+_           : Op₂ Carrier
-    _*_           : Op₂ Carrier
-    _⋆            : Op₁ Carrier
-    0#            : Carrier
-    1#            : Carrier
-    isPoKleeneAlgebra : IsPoKleeneAlgebra _≈_ _≤_ _+_ _*_ _⋆ 0# 1#
-
-  open IsPoKleeneAlgebra isPoKleeneAlgebra public
-
-  proKleeneAlgebra : ProKleeneAlgebra c ℓ₁ ℓ₂
-  proKleeneAlgebra = record { isProKleeneAlgebra = isProKleeneAlgebra }
-
-  open ProKleeneAlgebra proKleeneAlgebra public using (starExpansive; starDestructive)
diff --git a/src/Algebra/Ordered/Structures.agda b/src/Algebra/Ordered/Structures.agda
deleted file mode 100644
index 638d8783f0..0000000000
--- a/src/Algebra/Ordered/Structures.agda
+++ /dev/null
@@ -1,391 +0,0 @@
-------------------------------------------------------------------------
--- The Agda standard library
---
--- Ordered algebraic structures (not packed up with sets, orders,
--- operations, etc.)
-------------------------------------------------------------------------
-
--- The contents of this module should be accessed via
--- `Algebra.Ordered`.
-
-{-# OPTIONS --cubical-compatible --safe #-}
-
-open import Relation.Binary.Core using (Rel; _⇒_)
-
-module Algebra.Ordered.Structures
-  {a ℓ₁ ℓ₂} {A : Set a}  -- The underlying set
-  (_≈_ : Rel A ℓ₁)       -- The underlying equality relation
-  (_≤_ : Rel A ℓ₂)       -- The order
-  where
-
-open import Algebra.Core
-open import Algebra.Definitions _≈_
-open import Algebra.Structures _≈_
-open import Data.Product.Base using (proj₁; proj₂)
-open import Function.Base using (flip)
-open import Level using (_⊔_)
-open import Relation.Binary.Definitions using (Transitive; Monotonic₁; Monotonic₂)
-open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
-open import Relation.Binary.Consequences using (mono₂⇒cong₂)
-
-------------------------------------------------------------------------
--- Preordered structures
-
--- Preordered magmas (promagmas)
-
-record IsPromagma (∙ : Op₂ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPreorder  : IsPreorder _≈_ _≤_
-    ∙-cong      : Congruent₂ ∙
-    mono        : Monotonic₂ _≤_ _≤_ _≤_ ∙
-
-  open IsPreorder isPreorder public
-
-  mono₁ : ∀ {x} → Monotonic₁ _≤_ _≤_ (flip ∙ x)
-  mono₁ y≈z = mono y≈z refl
-
-  mono₂ : ∀ {x} → Monotonic₁ _≤_ _≤_ (∙ x)
-  mono₂ y≈z = mono refl y≈z
-
-  isMagma : IsMagma ∙
-  isMagma = record { isEquivalence = isEquivalence ; ∙-cong = ∙-cong }
-
-  open IsMagma isMagma public using (setoid; ∙-congˡ; ∙-congʳ)
-
--- Preordered semigroups (prosemigroups)
-
-record IsProsemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPromagma  : IsPromagma ∙
-    assoc       : Associative ∙
-
-  open IsPromagma isPromagma public
-
-  isSemigroup : IsSemigroup ∙
-  isSemigroup = record { isMagma = isMagma ; assoc = assoc }
-
--- Preordered monoids (promonoids)
-
-record IsPromonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isProsemigroup : IsProsemigroup ∙
-    identity       : Identity ε ∙
-
-  open IsProsemigroup isProsemigroup public
-
-  isMonoid : IsMonoid ∙ ε
-  isMonoid = record { isSemigroup = isSemigroup ; identity = identity }
-
-  open IsMonoid isMonoid public using (identityˡ; identityʳ)
-
--- Preordered commutative monoids (commutative promonoids)
-
-record IsCommutativePromonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPromonoid : IsPromonoid ∙ ε
-    comm        : Commutative ∙
-
-  open IsPromonoid isPromonoid public
-
-  isCommutativeMonoid : IsCommutativeMonoid ∙ ε
-  isCommutativeMonoid = record { isMonoid = isMonoid ; comm = comm }
-
-  open IsCommutativeMonoid isCommutativeMonoid public
-    using (isCommutativeSemigroup)
-
--- Preordered semirings (prosemirings)
-
-record IsProsemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    +-isCommutativePromonoid : IsCommutativePromonoid + 0#
-    *-cong                   : Congruent₂ *
-    *-mono                   : Monotonic₂ _≤_ _≤_ _≤_ *
-    *-assoc                  : Associative *
-    *-identity               : Identity 1# *
-    distrib                  : * DistributesOver +
-    zero                     : Zero 0# *
-
-  open IsCommutativePromonoid +-isCommutativePromonoid public
-    renaming
-    ( assoc                  to +-assoc
-    ; ∙-cong                 to +-cong
-    ; ∙-congˡ                to +-congˡ
-    ; ∙-congʳ                to +-congʳ
-    ; mono                   to +-mono
-    ; mono₁                  to +-mono₁
-    ; mono₂                  to +-mono₂
-    ; identity               to +-identity
-    ; identityˡ              to +-identityˡ
-    ; identityʳ              to +-identityʳ
-    ; comm                   to +-comm
-    ; isMagma                to +-isMagma
-    ; isSemigroup            to +-isSemigroup
-    ; isMonoid               to +-isMonoid
-    ; isCommutativeSemigroup to +-isCommutativeSemigroup
-    ; isCommutativeMonoid    to +-isCommutativeMonoid
-    )
-
-  *-isPromonoid : IsPromonoid * 1#
-  *-isPromonoid = record
-    { isProsemigroup = record
-      { isPromagma   = record
-        { isPreorder = isPreorder
-        ; ∙-cong     = *-cong
-        ; mono       = *-mono
-        }
-      ; assoc        = *-assoc
-      }
-    ; identity       = *-identity
-    }
-
-  open IsPromonoid *-isPromonoid public
-    using ()
-    renaming
-    ( ∙-congˡ     to *-congˡ
-    ; ∙-congʳ     to *-congʳ
-    ; mono₁       to *-mono₁
-    ; mono₂       to *-mono₂
-    ; identityˡ   to *-identityˡ
-    ; identityʳ   to *-identityʳ
-    ; isMagma     to *-isMagma
-    ; isSemigroup to *-isSemigroup
-    ; isMonoid    to *-isMonoid
-    )
-
-  isSemiring : IsSemiring + * 0# 1#
-  isSemiring = record
-    { isSemiringWithoutAnnihilatingZero = record
-      { +-isCommutativeMonoid           = +-isCommutativeMonoid
-      ; *-cong                          = *-cong
-      ; *-assoc                         = *-assoc
-      ; *-identity                      = *-identity
-      ; distrib                         = distrib
-      }
-    ; zero                              = zero
-    }
-
-  open IsSemiring isSemiring public using (distribˡ; distribʳ; zeroˡ; zeroʳ)
-
--- Preordered IdempotentSemiring (IdempotentProsemiring)
-
-record IsIdempotentProsemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isProsemiring  : IsProsemiring + * 0# 1#
-    +-idem         : Idempotent +
-
-  open IsProsemiring isProsemiring public
-
-  isIdempotentSemiring : IsIdempotentSemiring + * 0# 1#
-  isIdempotentSemiring = record { isSemiring = isSemiring ; +-idem = +-idem }
-
-  open IsIdempotentSemiring isIdempotentSemiring public using (+-idem)
-
--- Preordered KleeneAlgebra (proKleeneAlgebra)
-
-record IsProKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isIdempotentProsemiring : IsIdempotentProsemiring + * 0# 1#
-    starExpansive           : StarExpansive 1# + * ⋆
-    starDestructive         : StarDestructive + * ⋆
-
-  open IsIdempotentProsemiring isIdempotentProsemiring public
-
-  isKleeneAlgebra : IsKleeneAlgebra + * ⋆ 0# 1#
-  isKleeneAlgebra = record
-    { isIdempotentSemiring = isIdempotentSemiring
-    ; starExpansive        = starExpansive
-    ; starDestructive      = starDestructive
-    }
-
-  open IsKleeneAlgebra isKleeneAlgebra public using (starExpansive; starDestructive)
-
-------------------------------------------------------------------------
--- Partially ordered structures
-
--- Partially ordered magmas (pomagmas)
-
-record IsPomagma (∙ : Op₂ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPartialOrder : IsPartialOrder _≈_ _≤_
-    mono           : Monotonic₂ _≤_ _≤_ _≤_ ∙
-
-  open IsPartialOrder isPartialOrder public
-
-  ∙-cong : Congruent₂ ∙
-  ∙-cong = mono₂⇒cong₂ _≈_ _≈_ Eq.sym reflexive antisym mono
-
-  isPromagma : IsPromagma ∙
-  isPromagma = record
-    { isPreorder = isPreorder
-    ; ∙-cong     = ∙-cong
-    ; mono       = mono
-    }
-
-  open IsPromagma isPromagma public
-    using (setoid; ∙-congˡ; ∙-congʳ; mono₁; mono₂; isMagma)
-
--- Partially ordered semigroups (posemigroups)
-
-record IsPosemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPomagma : IsPomagma ∙
-    assoc     : Associative ∙
-
-  open IsPomagma isPomagma public
-
-  isProsemigroup : IsProsemigroup ∙
-  isProsemigroup = record { isPromagma = isPromagma ; assoc = assoc }
-
-  open IsProsemigroup isProsemigroup public using (isSemigroup)
-
--- Partially ordered monoids (pomonoids)
-
-record IsPomonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPosemigroup : IsPosemigroup ∙
-    identity      : Identity ε ∙
-
-  open IsPosemigroup isPosemigroup public
-
-  isPromonoid : IsPromonoid ∙ ε
-  isPromonoid = record
-    { isProsemigroup = isProsemigroup
-    ; identity       = identity
-    }
-
-  open IsPromonoid isPromonoid public
-    using (isMonoid; identityˡ; identityʳ)
-
--- Partially ordered commutative monoids (commutative pomonoids)
-
-record IsCommutativePomonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPomonoid : IsPomonoid ∙ ε
-    comm       : Commutative ∙
-
-  open IsPomonoid isPomonoid public
-
-  isCommutativePromonoid : IsCommutativePromonoid ∙ ε
-  isCommutativePromonoid = record { isPromonoid = isPromonoid ; comm = comm }
-
-  open IsCommutativePromonoid isCommutativePromonoid public
-    using (isCommutativeMonoid; isCommutativeSemigroup)
-
--- Partially ordered semirings (posemirings)
-
-record IsPosemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    +-isCommutativePomonoid : IsCommutativePomonoid + 0#
-    *-mono                  : Monotonic₂ _≤_ _≤_ _≤_ *
-    *-assoc                 : Associative *
-    *-identity              : Identity 1# *
-    distrib                 : * DistributesOver +
-    zero                    : Zero 0# *
-
-  open IsCommutativePomonoid +-isCommutativePomonoid public
-    renaming
-    ( assoc                  to +-assoc
-    ; ∙-cong                 to +-cong
-    ; ∙-congˡ                to +-congˡ
-    ; ∙-congʳ                to +-congʳ
-    ; mono                   to +-mono
-    ; mono₁                  to +-mono₁
-    ; mono₂                  to +-mono₂
-    ; identity               to +-identity
-    ; identityˡ              to +-identityˡ
-    ; identityʳ              to +-identityʳ
-    ; comm                   to +-comm
-    ; isMagma                to +-isMagma
-    ; isPromagma             to +-isPromagma
-    ; isPomagma              to +-isPomagma
-    ; isSemigroup            to +-isSemigroup
-    ; isProsemigroup         to +-isProsemigroup
-    ; isPosemigroup          to +-isPosemigroup
-    ; isMonoid               to +-isMonoid
-    ; isPromonoid            to +-isPromonoid
-    ; isPomonoid             to +-isPomonoid
-    ; isCommutativeSemigroup to +-isCommutativeSemigroup
-    ; isCommutativeMonoid    to +-isCommutativeMonoid
-    ; isCommutativePromonoid to +-isCommutativePromonoid
-    )
-
-  *-isPomonoid : IsPomonoid * 1#
-  *-isPomonoid = record
-    { isPosemigroup      = record
-      { isPomagma        = record
-        { isPartialOrder = isPartialOrder
-        ; mono           = *-mono
-        }
-      ; assoc            = *-assoc
-      }
-    ; identity           = *-identity
-    }
-
-  open IsPomonoid *-isPomonoid public
-    using ()
-    renaming
-    ( ∙-cong         to *-cong
-    ; ∙-congˡ        to *-congˡ
-    ; ∙-congʳ        to *-congʳ
-    ; mono₁          to *-mono₁
-    ; mono₂          to *-mono₂
-    ; identityˡ      to *-identityˡ
-    ; identityʳ      to *-identityʳ
-    ; isMagma        to *-isMagma
-    ; isPromagma     to *-isPromagma
-    ; isPomagma      to *-isPomagma
-    ; isSemigroup    to *-isSemigroup
-    ; isProsemigroup to *-isProsemigroup
-    ; isPosemigroup  to *-isPosemigroup
-    ; isMonoid       to *-isMonoid
-    ; isPromonoid    to *-isPromonoid
-    )
-
-  isProsemiring : IsProsemiring + * 0# 1#
-  isProsemiring = record
-    { +-isCommutativePromonoid = +-isCommutativePromonoid
-    ; *-cong                   = *-cong
-    ; *-mono                   = *-mono
-    ; *-assoc                  = *-assoc
-    ; *-identity               = *-identity
-    ; distrib                  = distrib
-    ; zero                     = zero
-    }
-
-  open IsProsemiring isProsemiring public
-    using (isSemiring; distribˡ; distribʳ; zeroˡ; zeroʳ)
-
--- Partially ordered idempotentSemiring (IdempotentPosemiring)
-
-record IsIdempotentPosemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isPosemiring  : IsPosemiring + * 0# 1#
-    +-idem      : Idempotent +
-
-  open IsPosemiring isPosemiring public
-
-  isIdempotentProsemiring : IsIdempotentProsemiring + * 0# 1#
-  isIdempotentProsemiring = record { isProsemiring = isProsemiring ; +-idem = +-idem }
-
-  open IsIdempotentProsemiring isIdempotentProsemiring public
-    using (isIdempotentSemiring; +-idem)
-
--- Partially ordered KleeneAlgebra (PoKleeneAlgebra)
-
-record IsPoKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  field
-    isIdempotentPosemiring  : IsIdempotentPosemiring + * 0# 1#
-    starExpansive           : StarExpansive 1# + * ⋆
-    starDestructive         : StarDestructive + * ⋆
-
-  open IsIdempotentPosemiring isIdempotentPosemiring public
-
-  isProKleeneAlgebra : IsProKleeneAlgebra + * ⋆ 0# 1#
-  isProKleeneAlgebra = record
-    { isIdempotentProsemiring = isIdempotentProsemiring
-    ; starExpansive           = starExpansive
-    ; starDestructive         = starDestructive
-    }
-
-  open IsProKleeneAlgebra isProKleeneAlgebra public
-    using (isKleeneAlgebra; starExpansive; starDestructive)
diff --git a/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
index 276e3a9705..4d089b6b0f 100644
--- a/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/BoundedJoinSemilattice.agda
@@ -14,13 +14,11 @@ module Relation.Binary.Lattice.Properties.BoundedJoinSemilattice
 open BoundedJoinSemilattice J
 
 open import Algebra.Definitions _≈_
-open import Algebra.Ordered.Structures using (IsCommutativePomonoid)
-open import Algebra.Ordered.Bundles using (CommutativePomonoid)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; flip)
 open import Relation.Binary.Properties.Poset poset
 open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
-  using (isPosemigroup; ∨-comm)
+  using (∨-comm)
 
 -- Bottom is an identity of the meet operation.
 
@@ -54,24 +52,3 @@ dualBoundedMeetSemilattice = record
   { ⊤                        = ⊥
   ; isBoundedMeetSemilattice = dualIsBoundedMeetSemilattice
   }
-
--- Every bounded semilattice gives rise to a commutative pomonoid
-
-isCommutativePomonoid : IsCommutativePomonoid _≈_ _≤_ _∨_ ⊥
-isCommutativePomonoid = record
-  { isPomonoid          = record
-    { isPosemigroup     = isPosemigroup
-    ; identity          = identity
-    }
-  ; comm                = ∨-comm
-  }
-
-commutativePomonoid : CommutativePomonoid c ℓ₁ ℓ₂
-commutativePomonoid = record
-  { Carrier               = Carrier
-  ; _≈_                   = _≈_
-  ; _≤_                   = _≤_
-  ; _∙_                   = _∨_
-  ; ε                     = ⊥
-  ; isCommutativePomonoid = isCommutativePomonoid
-  }
diff --git a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
index c68467f577..c94c3bad16 100644
--- a/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/JoinSemilattice.agda
@@ -16,8 +16,6 @@ open JoinSemilattice J
 import Algebra.Lattice as Alg
 import Algebra.Structures as Alg
 open import Algebra.Definitions _≈_
-open import Algebra.Ordered.Structures using (IsPosemigroup)
-open import Algebra.Ordered.Bundles using (Posemigroup)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; flip)
 open import Relation.Binary.Core using (_Preserves₂_⟶_⟶_)
@@ -97,26 +95,6 @@ isAlgSemilattice = record
 algSemilattice : Alg.Semilattice c ℓ₁
 algSemilattice = record { isSemilattice = isAlgSemilattice }
 
--- Every semilattice gives rise to a posemigroup
-
-isPosemigroup : IsPosemigroup _≈_ _≤_ _∨_
-isPosemigroup = record
-  { isPomagma        = record
-    { isPartialOrder = isPartialOrder
-    ; mono           = ∨-monotonic
-    }
-  ; assoc            = ∨-assoc
-  }
-
-posemigroup : Posemigroup c ℓ₁ ℓ₂
-posemigroup = record
-  { Carrier           = Carrier
-  ; _≈_               = _≈_
-  ; _≤_               = _≤_
-  ; _∙_               = _∨_
-  ; isPosemigroup     = isPosemigroup
-  }
-
 ------------------------------------------------------------------------
 -- The dual construction is a meet semilattice.
 
diff --git a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
index a78d4fea6a..f8fca492b7 100644
--- a/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/MeetSemilattice.agda
@@ -35,7 +35,7 @@ dualJoinSemilattice = record
   }
 
 open J dualJoinSemilattice public
-  using (isAlgSemilattice; algSemilattice; isPosemigroup; posemigroup)
+  using (isAlgSemilattice; algSemilattice)
   renaming
     ( ∨-monotonic  to ∧-monotonic
     ; ∨-cong       to ∧-cong

From dac1b2b5951228f19be5990ea2403dd20920a7a6 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Fri, 13 Oct 2023 00:39:15 +0100
Subject: [PATCH 156/316] [ fix ] missing name in LICENCE file (#2139)

---
 LICENCE | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/LICENCE b/LICENCE
index 75026ebb47..df31f24985 100644
--- a/LICENCE
+++ b/LICENCE
@@ -7,7 +7,8 @@ Eric Mertens, Joachim Breitner, Liyang Hu, Noam Zeilberger, Érdi Gergő,
 Stevan Andjelkovic, Helmut Grohne, Guilhem Moulin, Noriyuki Ohkawa,
 Evgeny Kotelnikov, James Chapman, Wen Kokke, Matthew Daggitt, Jason Hu,
 Sandro Stucki, Milo Turner, Zack Grannan, Lex van der Stoep,
-Jacques Carette, James McKinna and some anonymous contributors.
+Jacques Carette, James McKinna, Guillaume Allais
+and some anonymous contributors.
 
 Permission is hereby granted, free of charge, to any person obtaining a
 copy of this software and associated documentation files (the

From 9a68858b66a4e97b0e12a1e08d11aa085120da84 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Am=C3=A9lia?= <me@amelia.how>
Date: Fri, 13 Oct 2023 00:41:38 -0300
Subject: [PATCH 157/316] Add new blocking primitives to `Reflection.TCM`
 (#1972)

---
 CHANGELOG.md            | 9 +++++++++
 src/Reflection/TCM.agda | 7 +++++--
 2 files changed, 14 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6bfd659aa9..d012d754a2 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3801,6 +3801,15 @@ This is a full list of proofs that have changed form to use irrelevant instance
               WellFounded _<_ → Irreflexive _≈_ _<_
   ```
 
+* Added new types and operations to `Reflection.TCM`:
+  ```
+  Blocker : Set
+  blockerMeta : Meta → Blocker
+  blockerAny : List Blocker → Blocker
+  blockerAll : List Blocker → Blocker
+  blockTC : Blocker → TC A
+  ```
+  
 * Added new file `Relation.Binary.Reasoning.Base.Apartness`
 
   This is how to use it:
diff --git a/src/Reflection/TCM.agda b/src/Reflection/TCM.agda
index a3df1f1bc3..d762dab2c5 100644
--- a/src/Reflection/TCM.agda
+++ b/src/Reflection/TCM.agda
@@ -8,8 +8,9 @@
 
 module Reflection.TCM where
 
-open import Reflection.AST.Term
 import Agda.Builtin.Reflection as Builtin
+
+open import Reflection.AST.Term
 import Reflection.TCM.Format as Format
 
 ------------------------------------------------------------------------
@@ -29,7 +30,9 @@ open Builtin public
   ; getContext; extendContext; inContext; freshName
   ; declareDef; declarePostulate; defineFun; getType; getDefinition
   ; blockOnMeta; commitTC; isMacro; withNormalisation
-  ; debugPrint; noConstraints; runSpeculative)
+  ; debugPrint; noConstraints; runSpeculative
+  ; Blocker; blockerMeta; blockerAny; blockerAll; blockTC
+  )
   renaming (returnTC to pure)
 
 open Format public

From 3f6f36a6bdbb0597cb3222f3e5c25172eb35800f Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 13 Oct 2023 18:22:01 +0900
Subject: [PATCH 158/316] Change definition of `IsStrictTotalOrder` (#2137)

---
 CHANGELOG.md                                  | 40 ++++++++++++++
 src/Data/Bool/Properties.agda                 |  5 +-
 src/Data/Char/Properties.agda                 |  5 +-
 src/Data/Fin/Properties.agda                  |  5 +-
 src/Data/Integer/Properties.agda              |  5 +-
 src/Data/List/Relation/Binary/Lex/Strict.agda |  5 +-
 src/Data/Nat/Binary/Properties.agda           |  5 +-
 src/Data/Nat/Properties.agda                  |  9 +---
 .../Product/Relation/Binary/Lex/Strict.agda   |  9 ++--
 src/Data/Rational/Properties.agda             |  5 +-
 .../Rational/Unnormalised/Properties.agda     |  5 +-
 src/Data/Sum/Relation/Binary/LeftOrder.agda   |  5 +-
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  3 +-
 src/Induction/WellFounded.agda                |  4 +-
 src/Relation/Binary.agda                      |  1 +
 src/Relation/Binary/Bundles.agda              |  2 +-
 .../Binary/Construct/Add/Infimum/Strict.agda  | 10 ++--
 .../Binary/Construct/Add/Supremum/Strict.agda | 10 ++--
 .../Binary/Construct/Flip/EqAndOrd.agda       |  5 +-
 src/Relation/Binary/Construct/Flip/Ord.agda   |  5 +-
 .../Binary/Construct/NonStrictToStrict.agda   |  5 +-
 src/Relation/Binary/Construct/On.agda         |  5 +-
 .../Binary/Morphism/OrderMonomorphism.agda    |  5 +-
 src/Relation/Binary/Structures.agda           | 53 ++++++++++---------
 src/Relation/Binary/Structures/Biased.agda    | 49 +++++++++++++++++
 25 files changed, 164 insertions(+), 96 deletions(-)
 create mode 100644 src/Relation/Binary/Structures/Biased.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d012d754a2..9934771c1c 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -918,6 +918,46 @@ Non-backwards compatible changes
 
 * The old names (and the names of all proofs about these functions) have been deprecated appropriately.
 
+### Changes to definition of `IsStrictTotalOrder`
+
+* The previous definition of the record `IsStrictTotalOrder` did not build upon `IsStrictPartialOrder`
+  as would be expected. Instead it omitted several fields like irreflexivity as they were derivable from the
+  proof of trichotomy. However, this led to problems further up the hierarchy where bundles such as `StrictTotalOrder`
+  which contained multiple distinct proofs of `IsStrictPartialOrder`.
+  
+* To remedy this the definition of `IsStrictTotalOrder` has been changed to so that it builds upon 
+  `IsStrictPartialOrder` as would be expected. 
+
+* To aid migration, the old record definition has been moved to `Relation.Binary.Structures.Biased`
+  which contains the `isStrictTotalOrderᶜ` smart constructor (which is re-exported by `Relation.Binary`) . 
+  Therefore the old code:
+  ```agda
+  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
+  <-isStrictTotalOrder = record
+	{ isEquivalence = isEquivalence
+	; trans         = <-trans
+	; compare       = <-cmp
+	}
+  ```
+  can be migrated either by updating to the new record fields if you have a proof of `IsStrictPartialOrder`
+  available:
+  ```agda
+  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
+  <-isStrictTotalOrder = record
+	{ isStrictPartialOrder = <-isStrictPartialOrder
+	; compare              = <-cmp
+	}
+  ```
+  or simply applying the smart constructor to the record definition as follows:
+  ```agda
+  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
+  <-isStrictTotalOrder = isStrictTotalOrderᶜ record
+	{ isEquivalence = isEquivalence
+	; trans         = <-trans
+	; compare       = <-cmp
+	}
+  ```
+  
 ### Changes to triple reasoning interface
 
 * The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof that the strict
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index 3c1c25d245..e911390366 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -210,9 +210,8 @@ true  <? _     = no  (λ())
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
 <-isStrictTotalOrder = record
-  { isEquivalence = isEquivalence
-  ; trans         = <-trans
-  ; compare       = <-cmp
+  { isStrictPartialOrder = <-isStrictPartialOrder
+  ; compare              = <-cmp
   }
 
 -- Bundles
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index 970eb6249a..c399bf30e3 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -123,9 +123,8 @@ _<?_ = On.decidable toℕ ℕ._<_ ℕₚ._<?_
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
 <-isStrictTotalOrder = record
-  { isEquivalence = PropEq.isEquivalence
-  ; trans         = λ {a} {b} {c} → <-trans {a} {b} {c}
-  ; compare       = <-cmp
+  { isStrictPartialOrder = <-isStrictPartialOrder
+  ; compare              = <-cmp
   }
 
 <-strictPartialOrder : StrictPartialOrder _ _ _
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index ea7c43c90f..eb5a7f5011 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -406,9 +406,8 @@ m <? n = suc (toℕ m) ℕₚ.≤? toℕ n
 
 <-isStrictTotalOrder : IsStrictTotalOrder {A = Fin n} _≡_ _<_
 <-isStrictTotalOrder = record
-  { isEquivalence = P.isEquivalence
-  ; trans         = <-trans
-  ; compare       = <-cmp
+  { isStrictPartialOrder = <-isStrictPartialOrder
+  ; compare              = <-cmp
   }
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 21b3f3d71c..bfe29c4079 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -331,9 +331,8 @@ _<?_ : Decidable _<_
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
 <-isStrictTotalOrder = record
-  { isEquivalence = isEquivalence
-  ; trans         = <-trans
-  ; compare       = <-cmp
+  { isStrictPartialOrder = <-isStrictPartialOrder
+  ; compare              = <-cmp
   }
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Lex/Strict.agda b/src/Data/List/Relation/Binary/Lex/Strict.agda
index 7b7f5d2baf..277996594f 100644
--- a/src/Data/List/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/List/Relation/Binary/Lex/Strict.agda
@@ -121,9 +121,8 @@ module _ {a ℓ₁ ℓ₂} {A : Set a} where
     <-isStrictTotalOrder : IsStrictTotalOrder _≈_ _≺_ →
                            IsStrictTotalOrder _≋_ _<_
     <-isStrictTotalOrder sto = record
-      { isEquivalence = Pointwise.isEquivalence isEquivalence
-      ; trans         = <-transitive isEquivalence <-resp-≈ trans
-      ; compare       = <-compare Eq.sym compare
+      { isStrictPartialOrder = <-isStrictPartialOrder isStrictPartialOrder
+      ; compare              = <-compare Eq.sym compare
       } where open IsStrictTotalOrder sto
 
 <-strictPartialOrder : ∀ {a ℓ₁ ℓ₂} → StrictPartialOrder a ℓ₁ ℓ₂ →
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 6109106b26..8c08974636 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -414,9 +414,8 @@ _<?_ = tri⇒dec< <-cmp
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
 <-isStrictTotalOrder = record
-  { isEquivalence = isEquivalence
-  ; trans         = <-trans
-  ; compare       = <-cmp
+  { isStrictPartialOrder = <-isStrictPartialOrder
+  ; compare              = <-cmp
   }
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index dcc6746f93..915d1d7d1b 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -32,12 +32,7 @@ open import Level using (0ℓ)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary.Core
   using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.Bundles
-  using (DecSetoid; Preorder; TotalPreorder; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
-open import Relation.Binary.Structures
-  using (IsDecEquivalence; IsPreorder; IsTotalPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
-open import Relation.Binary.Definitions
-  using (DecidableEquality; Irrelevant; Reflexive; Antisymmetric; Transitive; Total; Decidable; Connex; Irreflexive; Asymmetric; LeftTrans; RightTrans; Trichotomous; tri≈; tri>; tri<; _Respects₂_)
+open import Relation.Binary
 open import Relation.Binary.Consequences using (flip-Connex)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary hiding (Irrelevant)
@@ -402,7 +397,7 @@ _>?_ = flip _<?_
   }
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
-<-isStrictTotalOrder = record
+<-isStrictTotalOrder = isStrictTotalOrderᶜ record
   { isEquivalence = isEquivalence
   ; trans         = <-trans
   ; compare       = <-cmp
diff --git a/src/Data/Product/Relation/Binary/Lex/Strict.agda b/src/Data/Product/Relation/Binary/Lex/Strict.agda
index 96adc9b9e1..42d139893d 100644
--- a/src/Data/Product/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/Strict.agda
@@ -269,12 +269,9 @@ module _ {_≈₁_ : Rel A ℓ₁} {_<₁_ : Rel A ℓ₂}
                          IsStrictTotalOrder _≋_ _<ₗₑₓ_
   ×-isStrictTotalOrder spo₁ spo₂ =
     record
-      { isEquivalence = Pointwise.×-isEquivalence
-                          (isEquivalence spo₁) (isEquivalence spo₂)
-      ; trans         = ×-transitive {_<₁_ = _<₁_} {_<₂_ = _<₂_}
-                          (isEquivalence spo₁)
-                          (<-resp-≈ spo₁) (trans spo₁)
-                          (trans spo₂)
+      { isStrictPartialOrder = ×-isStrictPartialOrder
+                                 (isStrictPartialOrder spo₁)
+                                 (isStrictPartialOrder spo₂)
       ; compare       = ×-compare (Eq.sym spo₁) (compare spo₁)
                                                 (compare spo₂)
       }
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index eab0c0cf2c..79d34870ef 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -702,9 +702,8 @@ _>?_ = flip _<?_
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
 <-isStrictTotalOrder = record
-  { isEquivalence = isEquivalence
-  ; trans         = <-trans
-  ; compare       = <-cmp
+  { isStrictPartialOrder = <-isStrictPartialOrder
+  ; compare              = <-cmp
   }
 
 <-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 4e86729725..11b0b4a5d7 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -542,9 +542,8 @@ _>?_ = flip _<?_
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≃_ _<_
 <-isStrictTotalOrder = record
-  { isEquivalence = ≃-isEquivalence
-  ; trans         = <-trans
-  ; compare       = <-cmp
+  { isStrictPartialOrder = <-isStrictPartialOrder
+  ; compare              = <-cmp
   }
 
 <-isDenseLinearOrder : IsDenseLinearOrder _≃_ _<_
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index 397bc598dc..416d71f15a 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -192,9 +192,8 @@ module _ {a₁ a₂} {A₁ : Set a₁} {A₂ : Set a₂}
                            IsStrictTotalOrder ≈₂ ∼₂ →
                            IsStrictTotalOrder (Pointwise ≈₁ ≈₂) (∼₁ ⊎-< ∼₂)
   ⊎-<-isStrictTotalOrder sto₁ sto₂ = record
-    { isEquivalence = PW.⊎-isEquivalence (isEquivalence sto₁) (isEquivalence sto₂)
-    ; trans         = ⊎-<-transitive (trans sto₁) (trans sto₂)
-    ; compare       = ⊎-<-trichotomous (compare sto₁) (compare sto₂)
+    { isStrictPartialOrder = ⊎-<-isStrictPartialOrder (isStrictPartialOrder sto₁) (isStrictPartialOrder sto₂)
+    ; compare              = ⊎-<-trichotomous (compare sto₁) (compare sto₂)
     }
     where open IsStrictTotalOrder
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 944edb4927..0cd7fde477 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -168,8 +168,7 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
   <-isStrictTotalOrder : IsStrictTotalOrder _≈_ _≺_ →
                          ∀ {n} → IsStrictTotalOrder (_≋_ {n} {n}) _<_
   <-isStrictTotalOrder ≺-isStrictTotalOrder {n} = record
-    { isEquivalence = Pointwise.isEquivalence O.isEquivalence n
-    ; trans         = <-trans O.Eq.isPartialEquivalence O.<-resp-≈ O.trans
+    { isStrictPartialOrder = <-isStrictPartialOrder O.isStrictPartialOrder
     ; compare       = <-cmp O.Eq.sym O.compare
     } where module O = IsStrictTotalOrder ≺-isStrictTotalOrder
 
diff --git a/src/Induction/WellFounded.agda b/src/Induction/WellFounded.agda
index fb3ed14255..1dac16c059 100644
--- a/src/Induction/WellFounded.agda
+++ b/src/Induction/WellFounded.agda
@@ -14,7 +14,7 @@ open import Induction
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions
-  using (Symmetric; Asymmetric; Irreflexive; _Respects₂_; 
+  using (Symmetric; Asymmetric; Irreflexive; _Respects₂_;
     _Respectsʳ_; _Respects_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Binary.Consequences using (asym⇒irr)
@@ -130,7 +130,7 @@ module _ {_<_ : Rel A r} where
   wf⇒asym : WellFounded _<_ → Asymmetric _<_
   wf⇒asym wf = acc⇒asym (wf _)
 
-  wf⇒irrefl : {_≈_ : Rel A ℓ} → _<_ Respects₂ _≈_ → 
+  wf⇒irrefl : {_≈_ : Rel A ℓ} → _<_ Respects₂ _≈_ →
               Symmetric _≈_ → WellFounded _<_ → Irreflexive _≈_ _<_
   wf⇒irrefl r s wf = asym⇒irr r s (wf⇒asym wf)
 
diff --git a/src/Relation/Binary.agda b/src/Relation/Binary.agda
index 95b2ca002a..efdd9921f9 100644
--- a/src/Relation/Binary.agda
+++ b/src/Relation/Binary.agda
@@ -14,4 +14,5 @@ module Relation.Binary where
 open import Relation.Binary.Core public
 open import Relation.Binary.Definitions public
 open import Relation.Binary.Structures public
+open import Relation.Binary.Structures.Biased public
 open import Relation.Binary.Bundles public
diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index 90e3c7e2ac..eb7975f376 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -100,7 +100,7 @@ record Preorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   _≳_ = flip _≲_
 
   -- Deprecated.
-  infix 4 _∼_ 
+  infix 4 _∼_
   _∼_ = _≲_
   {-# WARNING_ON_USAGE _∼_
   "Warning: _∼_ was deprecated in v2.0.
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
index ae96849340..5f3bdd7622 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
@@ -153,9 +153,8 @@ module _ {e} {_≈_ : Rel A e} where
 <₋-isStrictTotalOrder-≡ : IsStrictTotalOrder _≡_ _<_ →
                           IsStrictTotalOrder _≡_ _<₋_
 <₋-isStrictTotalOrder-≡ strictot = record
-  { isEquivalence = P.isEquivalence
-  ; trans         = <₋-trans trans
-  ; compare       = <₋-cmp-≡ compare
+  { isStrictPartialOrder = <₋-isStrictPartialOrder-≡ isStrictPartialOrder
+  ; compare              = <₋-cmp-≡ compare
   } where open IsStrictTotalOrder strictot
 
 ------------------------------------------------------------------------
@@ -185,7 +184,6 @@ module _ {e} {_≈_ : Rel A e} where
   <₋-isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_ →
                           IsStrictTotalOrder _≈₋_ _<₋_
   <₋-isStrictTotalOrder strictot = record
-    { isEquivalence = ≈₋-isEquivalence isEquivalence
-    ; trans         = <₋-trans trans
-    ; compare       = <₋-cmp compare
+    { isStrictPartialOrder = <₋-isStrictPartialOrder isStrictPartialOrder
+    ; compare              = <₋-cmp compare
     } where open IsStrictTotalOrder strictot
diff --git a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
index d927cf45c4..733981deb0 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
@@ -154,9 +154,8 @@ module _ {e} {_≈_ : Rel A e} where
 <⁺-isStrictTotalOrder-≡ : IsStrictTotalOrder _≡_ _<_ →
                           IsStrictTotalOrder _≡_ _<⁺_
 <⁺-isStrictTotalOrder-≡ strictot = record
-  { isEquivalence = P.isEquivalence
-  ; trans         = <⁺-trans trans
-  ; compare       = <⁺-cmp-≡ compare
+  { isStrictPartialOrder = <⁺-isStrictPartialOrder-≡ isStrictPartialOrder
+  ; compare              = <⁺-cmp-≡ compare
   } where open IsStrictTotalOrder strictot
 
 ------------------------------------------------------------------------
@@ -186,7 +185,6 @@ module _ {e} {_≈_ : Rel A e} where
   <⁺-isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_ →
                           IsStrictTotalOrder _≈⁺_ _<⁺_
   <⁺-isStrictTotalOrder strictot = record
-    { isEquivalence = ≈⁺-isEquivalence isEquivalence
-    ; trans         = <⁺-trans trans
-    ; compare       = <⁺-cmp compare
+    { isStrictPartialOrder = <⁺-isStrictPartialOrder isStrictPartialOrder
+    ; compare              = <⁺-cmp compare
     } where open IsStrictTotalOrder strictot
diff --git a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
index 78671dfbdc..122842f967 100644
--- a/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
+++ b/src/Relation/Binary/Construct/Flip/EqAndOrd.agda
@@ -145,9 +145,8 @@ isStrictPartialOrder {< = <} O = record
 isStrictTotalOrder : IsStrictTotalOrder ≈ < →
                      IsStrictTotalOrder ≈ (flip <)
 isStrictTotalOrder {< = <} O = record
-  { isEquivalence = O.isEquivalence
-  ; trans         = trans < O.trans
-  ; compare       = compare _ O.compare
+  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder
+  ; compare              = compare _ O.compare
   } where module O = IsStrictTotalOrder O
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Construct/Flip/Ord.agda b/src/Relation/Binary/Construct/Flip/Ord.agda
index 742a315cc6..64d41fdb77 100644
--- a/src/Relation/Binary/Construct/Flip/Ord.agda
+++ b/src/Relation/Binary/Construct/Flip/Ord.agda
@@ -139,9 +139,8 @@ isStrictPartialOrder {≈ = ≈} {< = <} O = record
 isStrictTotalOrder : IsStrictTotalOrder ≈ < →
                      IsStrictTotalOrder (flip ≈) (flip <)
 isStrictTotalOrder {≈ = ≈} {< = <} O = record
-  { isEquivalence = isEquivalence O.isEquivalence
-  ; trans         = transitive < O.trans
-  ; compare       = trichotomous ≈ < O.compare
+  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder
+  ; compare              = trichotomous ≈ < O.compare
   } where module O = IsStrictTotalOrder O
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Construct/NonStrictToStrict.agda b/src/Relation/Binary/Construct/NonStrictToStrict.agda
index c2eb114004..f0402d7fb9 100644
--- a/src/Relation/Binary/Construct/NonStrictToStrict.agda
+++ b/src/Relation/Binary/Construct/NonStrictToStrict.agda
@@ -138,9 +138,8 @@ x < y = x ≤ y × x ≉ y
 <-isStrictTotalOrder₁ : Decidable _≈_ → IsTotalOrder _≈_ _≤_ →
                         IsStrictTotalOrder _≈_ _<_
 <-isStrictTotalOrder₁ ≟ tot = record
-  { isEquivalence = isEquivalence
-  ; trans         = <-trans isPartialOrder
-  ; compare       = <-trichotomous Eq.sym ≟ antisym total
+  { isStrictPartialOrder = <-isStrictPartialOrder isPartialOrder
+  ; compare              = <-trichotomous Eq.sym ≟ antisym total
   } where open IsTotalOrder tot
 
 <-isStrictTotalOrder₂ : IsDecTotalOrder _≈_ _≤_ →
diff --git a/src/Relation/Binary/Construct/On.agda b/src/Relation/Binary/Construct/On.agda
index c124f98dd8..5544edbf4a 100644
--- a/src/Relation/Binary/Construct/On.agda
+++ b/src/Relation/Binary/Construct/On.agda
@@ -150,9 +150,8 @@ module _ (f : B → A) {≈ : Rel A ℓ₁} {∼ : Rel A ℓ₂} where
   isStrictTotalOrder : IsStrictTotalOrder ≈ ∼ →
                        IsStrictTotalOrder (≈ on f) (∼ on f)
   isStrictTotalOrder sto = record
-    { isEquivalence = isEquivalence f Sto.isEquivalence
-    ; trans         = transitive f ∼ Sto.trans
-    ; compare       = trichotomous f ≈ ∼ Sto.compare
+    { isStrictPartialOrder = isStrictPartialOrder Sto.isStrictPartialOrder
+    ; compare              = trichotomous _ _ _ Sto.compare
     } where module Sto = IsStrictTotalOrder sto
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Morphism/OrderMonomorphism.agda b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
index ca3ce3677a..12a80d1f31 100644
--- a/src/Relation/Binary/Morphism/OrderMonomorphism.agda
+++ b/src/Relation/Binary/Morphism/OrderMonomorphism.agda
@@ -105,7 +105,6 @@ isStrictPartialOrder O = record
 isStrictTotalOrder : IsStrictTotalOrder _≈₂_ _≲₂_ →
                      IsStrictTotalOrder _≈₁_ _≲₁_
 isStrictTotalOrder O = record
-  { isEquivalence = EqM.isEquivalence O.isEquivalence
-  ; trans         = trans O.trans
-  ; compare       = compare O.compare
+  { isStrictPartialOrder = isStrictPartialOrder O.isStrictPartialOrder
+  ; compare              = compare O.compare
   } where module O = IsStrictTotalOrder O
diff --git a/src/Relation/Binary/Structures.agda b/src/Relation/Binary/Structures.agda
index 751c774da3..bf11d67250 100644
--- a/src/Relation/Binary/Structures.agda
+++ b/src/Relation/Binary/Structures.agda
@@ -235,16 +235,20 @@ record IsDecTotalOrder (_≤_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
 
 
 -- Note that these orders are decidable. The current implementation
--- of `Trichotomous` subsumes irreflexivity and asymmetry. Any reasonable
--- definition capturing these three properties implies decidability
--- as `Trichotomous` necessarily separates out the equality case.
+-- of `Trichotomous` subsumes irreflexivity and asymmetry. See
+-- `Relation.Binary.Structures.Biased` for ways of constructing this
+-- record without having to prove `isStrictPartialOrder`.
 
 record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   field
-    isEquivalence : IsEquivalence
-    trans         : Transitive _<_
-    compare       : Trichotomous _≈_ _<_
+    isStrictPartialOrder : IsStrictPartialOrder _<_
+    compare              : Trichotomous _≈_ _<_
+
+  open IsStrictPartialOrder isStrictPartialOrder public
+    hiding (module Eq)
 
+  -- `Trichotomous` necessarily separates out the equality case so
+  --  it implies decidability.
   infix 4 _≟_ _<?_
 
   _≟_ : Decidable _≈_
@@ -253,22 +257,6 @@ record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) wher
   _<?_ : Decidable _<_
   _<?_ = tri⇒dec< compare
 
-  isDecEquivalence : IsDecEquivalence
-  isDecEquivalence = record
-    { isEquivalence = isEquivalence
-    ; _≟_           = _≟_
-    }
-
-  module Eq = IsDecEquivalence isDecEquivalence
-
-  isStrictPartialOrder : IsStrictPartialOrder _<_
-  isStrictPartialOrder = record
-    { isEquivalence = isEquivalence
-    ; irrefl        = tri⇒irr compare
-    ; trans         = trans
-    ; <-resp-≈      = trans∧tri⇒resp Eq.sym Eq.trans trans compare
-    }
-
   isDecStrictPartialOrder : IsDecStrictPartialOrder _<_
   isDecStrictPartialOrder = record
     { isStrictPartialOrder = isStrictPartialOrder
@@ -276,9 +264,26 @@ record IsStrictTotalOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) wher
     ; _<?_                 = _<?_
     }
 
-  open IsStrictPartialOrder isStrictPartialOrder public
-    using (irrefl; asym; <-respʳ-≈; <-respˡ-≈; <-resp-≈)
+  -- Redefine the `Eq` module to include decidability proofs
+  module Eq where
+
+    isDecEquivalence : IsDecEquivalence
+    isDecEquivalence = record
+      { isEquivalence = isEquivalence
+      ; _≟_           = _≟_
+      }
 
+    open IsDecEquivalence isDecEquivalence public
+
+  isDecEquivalence : IsDecEquivalence
+  isDecEquivalence = record
+    { isEquivalence = isEquivalence
+    ; _≟_           = _≟_
+    }
+  {-# WARNING_ON_USAGE isDecEquivalence
+  "Warning: isDecEquivalence was deprecated in v2.0.
+  Please use Eq.isDecEquivalence instead. "
+  #-}
 
 record IsDenseLinearOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   field
diff --git a/src/Relation/Binary/Structures/Biased.agda b/src/Relation/Binary/Structures/Biased.agda
new file mode 100644
index 0000000000..83636aa5d6
--- /dev/null
+++ b/src/Relation/Binary/Structures/Biased.agda
@@ -0,0 +1,49 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Ways to give instances of certain structures where some fields can
+-- be given in terms of others
+------------------------------------------------------------------------
+
+-- The contents of this module should be accessed via `Relation.Binary`.
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Core
+
+module Relation.Binary.Structures.Biased
+  {a ℓ} {A : Set a} -- The underlying set
+  (_≈_ : Rel A ℓ)   -- The underlying equality relation
+  where
+
+open import Level using (Level; _⊔_)
+open import Relation.Binary.Consequences
+open import Relation.Binary.Definitions
+open import Relation.Binary.Structures _≈_
+
+private
+  variable
+    ℓ₂ : Level
+
+-- To construct a StrictTotalOrder you only need to prove transitivity and
+-- trichotomy as the current implementation of `Trichotomous` subsumes
+-- irreflexivity and asymmetry.
+record IsStrictTotalOrderᶜ (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
+  field
+    isEquivalence : IsEquivalence
+    trans         : Transitive _<_
+    compare       : Trichotomous _≈_ _<_
+
+  isStrictTotalOrderᶜ : IsStrictTotalOrder _<_
+  isStrictTotalOrderᶜ = record
+    { isStrictPartialOrder = record
+      { isEquivalence = isEquivalence
+      ; irrefl = tri⇒irr compare
+      ; trans = trans
+      ; <-resp-≈ = trans∧tri⇒resp Eq.sym Eq.trans trans compare
+      }
+    ; compare = compare
+    } where module Eq = IsEquivalence isEquivalence
+
+open IsStrictTotalOrderᶜ public
+  using (isStrictTotalOrderᶜ)

From 60a93de9f68d1846a5d5c66ebc69effd4907a8e2 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Sat, 14 Oct 2023 09:30:28 +0900
Subject: [PATCH 159/316] Add _<$>_ operator for Function bundle (#2144)

---
 CHANGELOG.md              |  6 ++++++
 src/Function/Bundles.agda | 13 +++++++++++++
 2 files changed, 19 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9934771c1c..1fb43bc2a1 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3154,6 +3154,12 @@ Additions to existing modules
   execState : State s a → s → s
   ```
 
+* Added new application operator synonym to `Function.Bundles`:
+  ```agda
+  _⟨$⟩_ : Func From To → Carrier From → Carrier To
+  _⟨$⟩_ = Func.to 
+  ```
+  
 * Added new proofs in `Function.Construct.Symmetry`:
   ```
   bijective     : Bijective ≈₁ ≈₂ f → Symmetric ≈₂ → Transitive ≈₂ → Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index 1e479cf46a..a51dc4ea84 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -66,6 +66,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
     open IsCongruent isCongruent public
       using (module Eq₁; module Eq₂)
 
+
   record Injection : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
       to          : A → B
@@ -472,3 +473,15 @@ module _ {A : Set a} {B : Set b} where
     , strictlyInverseʳ⇒inverseʳ to invʳ
     )
 
+------------------------------------------------------------------------
+-- Other
+------------------------------------------------------------------------
+
+-- Alternative syntax for the application of functions
+
+module _ {From : Setoid a ℓ₁} {To : Setoid b ℓ₂} where
+  open Setoid
+  
+  infixl 5 _⟨$⟩_
+  _⟨$⟩_ : Func From To → Carrier From → Carrier To
+  _⟨$⟩_ = Func.to

From 77a394a51ecc6f2b56f5b9569d37135f94692e83 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Sat, 14 Oct 2023 08:48:13 +0100
Subject: [PATCH 160/316] [ fix #2086 ] new web deployment strategy (#2147)

---
 .github/workflows/ci-ubuntu.yml |  6 +++++-
 travis/agda-logo.svg            | 16 ++++++++++++++++
 travis/landing-bottom.html      |  4 ++++
 travis/landing-top.html         | 24 ++++++++++++++++++++++++
 travis/landing.sh               | 17 +++++++++++++++++
 5 files changed, 66 insertions(+), 1 deletion(-)
 create mode 100644 travis/agda-logo.svg
 create mode 100644 travis/landing-bottom.html
 create mode 100644 travis/landing-top.html
 create mode 100755 travis/landing.sh

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index ebafe70fa8..2ecc969589 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -75,7 +75,7 @@ jobs:
              || '${{ github.base_ref }}' == 'master' ]]; then
             # Pick Agda version for master
             echo "AGDA_COMMIT=tags/v2.6.4" >> $GITHUB_ENV;
-            echo "AGDA_HTML_DIR=html" >> $GITHUB_ENV
+            echo "AGDA_HTML_DIR=html/master" >> $GITHUB_ENV
           elif [[ '${{ github.ref }}' == 'refs/heads/experimental' \
                || '${{ github.base_ref }}' == 'experimental' ]]; then
             # Pick Agda version for experimental
@@ -125,6 +125,7 @@ jobs:
         run: cabal update
 
       - name: Install alex & happy
+        if: steps.cache-cabal.outputs.cache-hit != 'true'
         run: |
           ${{ env.CABAL_INSTALL }} alex
           ${{ env.CABAL_INSTALL }} happy
@@ -177,6 +178,9 @@ jobs:
           rm -f '${{ env.AGDA_HTML_DIR }}'/*.css
           ${{ env.AGDA }} --html --html-dir ${{ env.AGDA_HTML_DIR }} index.agda
 
+          cp travis/* .
+          ./landing.sh
+
       - name: Deploy HTML
         uses: JamesIves/github-pages-deploy-action@4.1.3
         if: ${{ success() && env.AGDA_DEPLOY }}
diff --git a/travis/agda-logo.svg b/travis/agda-logo.svg
new file mode 100644
index 0000000000..0fe434e8ed
--- /dev/null
+++ b/travis/agda-logo.svg
@@ -0,0 +1,16 @@
+<svg version="1.1" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" viewBox="-43.48 -43.48 1286.97 1086.97">
+    <title>logotype</title>
+    <desc>Created with Sketch.</desc>
+    <g id="Page-1" stroke="none" stroke-width="1" fill="none" fill-rule="evenodd">
+      <g id="logo">
+        <circle id="left-eye" fill="#000000" cx="240" cy="270" r="27"/>
+        <circle id="right-eye" fill="#000000" cx="340" cy="270" r="27"/>
+        <path d="M1000,0 L600,400" id="wing" stroke="#000000" stroke-width="36" stroke-linecap="round" stroke-linejoin="round"/>
+        <path d="M1200,100 L1000,300" id="tail-middle" stroke="#000000" stroke-width="36" stroke-linecap="round" stroke-linejoin="round"/>
+        <path d="M1200,0 L1000,200" id="tail-top" stroke="#000000" stroke-width="36" stroke-linecap="round" stroke-linejoin="round"/>
+        <path d="M500,0 L300,200" id="head-middle" stroke="#000000" stroke-width="36" stroke-linecap="round" stroke-linejoin="round"/>
+        <path d="M600,0 L400,200" id="head-right" stroke="#000000" stroke-width="36" stroke-linecap="round" stroke-linejoin="round"/>
+        <path d="M400,-2.84217094e-13 L0,400 L200,400 L200,600 C200,820 380,1000 600,1000 C820,1000 1000,820 1000,600 L1000,400 L1200,200" id="body" stroke="#000000" stroke-width="36" stroke-linecap="round" stroke-linejoin="round"/>
+      </g>
+    </g>
+</svg>
\ No newline at end of file
diff --git a/travis/landing-bottom.html b/travis/landing-bottom.html
new file mode 100644
index 0000000000..9cd52d7c69
--- /dev/null
+++ b/travis/landing-bottom.html
@@ -0,0 +1,4 @@
+      </ul>
+    </div>
+  </body>
+</html>
diff --git a/travis/landing-top.html b/travis/landing-top.html
new file mode 100644
index 0000000000..81b03304b9
--- /dev/null
+++ b/travis/landing-top.html
@@ -0,0 +1,24 @@
+<html>
+
+  <head>
+    <title>Documention for the Agda standard library</title>
+  </head>
+
+  <body>
+
+    <div id="container" style="width:50%;min-width:500px;margin:auto">
+      <img src="agda-logo.svg" style="width:80px;float:right" />
+      <h1>Documention for the Agda standard library</h1>
+
+      <hr />
+
+      <h2>Development versions</h2>
+
+      <ul>
+        <li><a href="master">master</a></li>
+        <li><a href="experimental">experimental</a></li>
+      </ul>
+
+      <h2>Released versions</h2>
+
+      <ul>
diff --git a/travis/landing.sh b/travis/landing.sh
new file mode 100755
index 0000000000..55ebe0c8fb
--- /dev/null
+++ b/travis/landing.sh
@@ -0,0 +1,17 @@
+set -eu
+set -o pipefail
+
+rm html/index.html
+
+cat landing-top.html >> landing.html
+
+find html/ -name "index.html" \
+  | grep -v "master\|experimental" \
+  | sort -r \
+  | sed 's|html/\([^\/]*\)/index.html|        <li><a href="\1">\1</a></li>|g' \
+  >> landing.html
+
+cat landing-bottom.html >> landing.html
+
+mv landing.html html/index.html
+mv agda-logo.svg html/

From be281080f7f063f976bd924db770301836c3aae8 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Sun, 15 Oct 2023 04:59:09 +0100
Subject: [PATCH 161/316] [ admin ] fix sorting logic (#2151)

With the previous script we were sorting entries of the form
html/vX.Y.Z/index.html but the order is such that vX.Y/ < vX.Y.Z/
and so we were ending up with v1.7 coming after v1.7.3.

This fixes that by using sed to get rid of the html/ prefix
and the /index.html suffix before the sorting phase.
---
 travis/landing.sh | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/travis/landing.sh b/travis/landing.sh
index 55ebe0c8fb..fcc042da46 100755
--- a/travis/landing.sh
+++ b/travis/landing.sh
@@ -7,8 +7,9 @@ cat landing-top.html >> landing.html
 
 find html/ -name "index.html" \
   | grep -v "master\|experimental" \
+  | sed 's|html/\([^\/]*\)/index.html|\1|g' \
   | sort -r \
-  | sed 's|html/\([^\/]*\)/index.html|        <li><a href="\1">\1</a></li>|g' \
+  | sed 's|^\(.*\)$|        <li><a href="\1">\1</a></li>|g' \
   >> landing.html
 
 cat landing-bottom.html >> landing.html

From 7cdd4a556539ae3cfb887e3b159d8d0d4ca8c7fb Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Mon, 16 Oct 2023 05:13:47 +0200
Subject: [PATCH 162/316] Add coincidence law to modules (#1898)

---
 CHANGELOG.md                                   |  5 +++++
 .../Module/Construct/DirectProduct.agda        |  2 ++
 src/Algebra/Module/Construct/TensorUnit.agda   |  2 ++
 src/Algebra/Module/Definitions.agda            |  2 ++
 .../Module/Definitions/Bi/Simultaneous.agda    | 18 ++++++++++++++++++
 src/Algebra/Module/Structures.agda             | 18 +++++++++++-------
 src/Algebra/Module/Structures/Biased.agda      | 12 ++++++++++--
 7 files changed, 50 insertions(+), 9 deletions(-)
 create mode 100644 src/Algebra/Module/Definitions/Bi/Simultaneous.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1fb43bc2a1..a3adeca602 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1843,6 +1843,11 @@ New modules
   Algebra.Module.Core
   ```
 
+* Definitions for module-like algebraic structures with left- and right- multiplication over the same scalars:
+  ```
+  Algebra.Module.Definitions.Bi.Simultaneous
+  ```
+
 * Constructive algebraic structures with apartness relations:
   ```
   Algebra.Apartness
diff --git a/src/Algebra/Module/Construct/DirectProduct.agda b/src/Algebra/Module/Construct/DirectProduct.agda
index 840cc42cab..429a2d11ca 100644
--- a/src/Algebra/Module/Construct/DirectProduct.agda
+++ b/src/Algebra/Module/Construct/DirectProduct.agda
@@ -97,6 +97,7 @@ semimodule M N = record
   { isSemimodule = record
     { isBisemimodule = Bisemimodule.isBisemimodule
       (bisemimodule M.bisemimodule N.bisemimodule)
+    ; *ₗ-*ᵣ-coincident = λ x m → M.*ₗ-*ᵣ-coincident x (proj₁ m) , N.*ₗ-*ᵣ-coincident x (proj₂ m)
     }
   } where module M = Semimodule M; module N = Semimodule N
 
@@ -155,5 +156,6 @@ bimodule M N = record
 ⟨module⟩ M N = record
   { isModule = record
     { isBimodule = Bimodule.isBimodule (bimodule M.bimodule N.bimodule)
+    ; *ₗ-*ᵣ-coincident = λ x m → M.*ₗ-*ᵣ-coincident x (proj₁ m) , N.*ₗ-*ᵣ-coincident x (proj₂ m)
     }
   } where module M = Module M; module N = Module N
diff --git a/src/Algebra/Module/Construct/TensorUnit.agda b/src/Algebra/Module/Construct/TensorUnit.agda
index 6277210520..c76f2a39ba 100644
--- a/src/Algebra/Module/Construct/TensorUnit.agda
+++ b/src/Algebra/Module/Construct/TensorUnit.agda
@@ -77,6 +77,7 @@ semimodule : {R : CommutativeSemiring c ℓ} → Semimodule R c ℓ
 semimodule {R = commutativeSemiring} = record
   { isSemimodule = record
     { isBisemimodule = Bisemimodule.isBisemimodule bisemimodule
+    ; *ₗ-*ᵣ-coincident = *-comm
     }
   } where open CommutativeSemiring commutativeSemiring
 
@@ -113,5 +114,6 @@ bimodule {R = ring} = record
 ⟨module⟩ {R = commutativeRing} = record
   { isModule = record
     { isBimodule = Bimodule.isBimodule bimodule
+    ; *ₗ-*ᵣ-coincident = *-comm
     }
   } where open CommutativeRing commutativeRing
diff --git a/src/Algebra/Module/Definitions.agda b/src/Algebra/Module/Definitions.agda
index 19c1abdb46..949aa45119 100644
--- a/src/Algebra/Module/Definitions.agda
+++ b/src/Algebra/Module/Definitions.agda
@@ -12,7 +12,9 @@ module Algebra.Module.Definitions where
   import Algebra.Module.Definitions.Left as L
   import Algebra.Module.Definitions.Right as R
   import Algebra.Module.Definitions.Bi as B
+  import Algebra.Module.Definitions.Bi.Simultaneous as BS
 
   module LeftDefs = L
   module RightDefs = R
   module BiDefs = B
+  module SimultaneousBiDefs = BS
diff --git a/src/Algebra/Module/Definitions/Bi/Simultaneous.agda b/src/Algebra/Module/Definitions/Bi/Simultaneous.agda
new file mode 100644
index 0000000000..686f4ff06c
--- /dev/null
+++ b/src/Algebra/Module/Definitions/Bi/Simultaneous.agda
@@ -0,0 +1,18 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties connecting left-scaling and right-scaling over the same scalars
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary
+
+module Algebra.Module.Definitions.Bi.Simultaneous
+  {a b ℓb} (A : Set a) {B : Set b} (_≈_ : Rel B ℓb)
+  where
+
+open import Algebra.Module.Core
+
+Coincident : Opₗ A B → Opᵣ A B → Set _
+Coincident _∙ₗ_ _∙ᵣ_ = ∀ x m → (x ∙ₗ m) ≈ (m ∙ᵣ x)
diff --git a/src/Algebra/Module/Structures.agda b/src/Algebra/Module/Structures.agda
index d8a291bd13..64873252bd 100644
--- a/src/Algebra/Module/Structures.agda
+++ b/src/Algebra/Module/Structures.agda
@@ -166,14 +166,16 @@ module _ (commutativeSemiring : CommutativeSemiring r ℓr)
   open CommutativeSemiring commutativeSemiring renaming (Carrier to R)
 
   -- An R-semimodule is an R-R-bisemimodule where R is commutative.
-  -- This means that *ₗ and *ᵣ coincide up to mathematical equality,
-  -- though it may be that they do not coincide up to definitional
-  -- equality.
+  -- We enforce that *ₗ and *ᵣ coincide up to mathematical equality, though it
+  -- may be that they do not coincide up to definitional equality.
+
+  open SimultaneousBiDefs R ≈ᴹ
 
   record IsSemimodule (*ₗ : Opₗ R M) (*ᵣ : Opᵣ R M)
                       : Set (r ⊔ m ⊔ ℓr ⊔ ℓm) where
     field
       isBisemimodule : IsBisemimodule semiring semiring ≈ᴹ +ᴹ 0ᴹ *ₗ *ᵣ
+      *ₗ-*ᵣ-coincident : Coincident *ₗ *ᵣ
 
     open IsBisemimodule isBisemimodule public
 
@@ -282,15 +284,17 @@ module _ (commutativeRing : CommutativeRing r ℓr)
   open CommutativeRing commutativeRing renaming (Carrier to R)
 
   -- An R-module is an R-R-bimodule where R is commutative.
-  -- This means that *ₗ and *ᵣ coincide up to mathematical equality,
-  -- though it may be that they do not coincide up to definitional
-  -- equality.
+  -- We enforce that *ₗ and *ᵣ coincide up to mathematical equality, though it
+  -- may be that they do not coincide up to definitional equality.
+
+  open SimultaneousBiDefs R ≈ᴹ
 
   record IsModule (*ₗ : Opₗ R M) (*ᵣ : Opᵣ R M) : Set (r ⊔ m ⊔ ℓr ⊔ ℓm) where
     field
       isBimodule : IsBimodule ring ring ≈ᴹ +ᴹ 0ᴹ -ᴹ *ₗ *ᵣ
+      *ₗ-*ᵣ-coincident : Coincident *ₗ *ᵣ
 
     open IsBimodule isBimodule public
 
     isSemimodule : IsSemimodule commutativeSemiring ≈ᴹ +ᴹ 0ᴹ *ₗ *ᵣ
-    isSemimodule = record { isBisemimodule = isBisemimodule }
+    isSemimodule = record { isBisemimodule = isBisemimodule; *ₗ-*ᵣ-coincident = *ₗ-*ᵣ-coincident }
diff --git a/src/Algebra/Module/Structures/Biased.agda b/src/Algebra/Module/Structures/Biased.agda
index 99e48695ae..764745c54c 100644
--- a/src/Algebra/Module/Structures/Biased.agda
+++ b/src/Algebra/Module/Structures/Biased.agda
@@ -57,7 +57,10 @@ module _ (commutativeSemiring : CommutativeSemiring r ℓr) where
       }
 
     isSemimodule : IsSemimodule commutativeSemiring ≈ᴹ +ᴹ 0ᴹ *ₗ (flip *ₗ)
-    isSemimodule = record { isBisemimodule = isBisemimodule }
+    isSemimodule = record
+      { isBisemimodule = isBisemimodule
+      ; *ₗ-*ᵣ-coincident = λ _ _ → ≈ᴹ-refl
+      }
 
   -- Similarly, a right semimodule over a commutative semiring
   -- is already a semimodule.
@@ -88,7 +91,10 @@ module _ (commutativeSemiring : CommutativeSemiring r ℓr) where
       }
 
     isSemimodule : IsSemimodule commutativeSemiring ≈ᴹ +ᴹ 0ᴹ (flip *ᵣ) *ᵣ
-    isSemimodule = record { isBisemimodule = isBisemimodule }
+    isSemimodule = record
+      { isBisemimodule = isBisemimodule
+      ; *ₗ-*ᵣ-coincident = λ _ _ → ≈ᴹ-refl
+      }
 
 
 module _ (commutativeRing : CommutativeRing r ℓr) where
@@ -113,6 +119,7 @@ module _ (commutativeRing : CommutativeRing r ℓr) where
         ; -ᴹ‿cong = -ᴹ‿cong
         ; -ᴹ‿inverse = -ᴹ‿inverse
         }
+      ; *ₗ-*ᵣ-coincident = λ _ _ → ≈ᴹ-refl
       }
 
   -- Similarly, a right module over a commutative ring is already a module.
@@ -134,4 +141,5 @@ module _ (commutativeRing : CommutativeRing r ℓr) where
         ; -ᴹ‿cong = -ᴹ‿cong
         ; -ᴹ‿inverse = -ᴹ‿inverse
         }
+      ; *ₗ-*ᵣ-coincident = λ _ _ → ≈ᴹ-refl
       }

From 24c05d2128493d91da29d5ec1545b36aa7c3cf33 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Mon, 16 Oct 2023 20:28:06 +0900
Subject: [PATCH 163/316] Make reasoning modular by adding new
 `Reasoning.Syntax` module (#2152)

---
 CHANGELOG.md                                  |  28 ++
 .../Stream/Relation/Binary/Pointwise.agda     |   4 +
 src/Codata/Musical/Colist.agda                |  32 +-
 .../Relation/Unary/Any/Properties.agda        |  19 +-
 src/Data/Integer/Divisibility/Signed.agda     |  11 +-
 src/Data/Integer/Properties.agda              |   2 +-
 .../Relation/Binary/BagAndSetEquality.agda    |  29 +-
 .../Binary/Permutation/Propositional.agda     |  16 +-
 .../Relation/Binary/Permutation/Setoid.agda   |  26 +-
 .../Binary/Subset/Setoid/Properties.agda      |  29 +-
 src/Data/Nat/Binary/Properties.agda           |  13 +-
 src/Data/Nat/Divisibility.agda                |  11 +-
 src/Data/Nat/Properties.agda                  |   2 +-
 src/Data/Rational/Base.agda                   |   7 +-
 src/Data/Rational/Properties.agda             |  58 +--
 .../Rational/Unnormalised/Properties.agda     |  24 +-
 .../Vec/Relation/Binary/Lex/NonStrict.agda    |   2 +-
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  28 +-
 src/Function/Related/Propositional.agda       |  53 ++-
 .../ReflexiveTransitive/Properties.agda       |  19 +-
 .../Binary/HeterogeneousEquality.agda         |  48 +--
 .../Binary/Reasoning/Base/Apartness.agda      | 113 +++---
 .../Binary/Reasoning/Base/Double.agda         | 110 ++----
 .../Binary/Reasoning/Base/Partial.agda        |  84 ++--
 .../Binary/Reasoning/Base/Single.agda         |  83 ++--
 .../Binary/Reasoning/Base/Triple.agda         | 164 +++-----
 .../Binary/Reasoning/MultiSetoid.agda         |  65 ++--
 .../Binary/Reasoning/PartialOrder.agda        |   2 +-
 .../Binary/Reasoning/PartialSetoid.agda       |  27 +-
 src/Relation/Binary/Reasoning/Setoid.agda     |  23 +-
 .../Binary/Reasoning/StrictPartialOrder.agda  |   2 +-
 src/Relation/Binary/Reasoning/Syntax.agda     | 366 ++++++++++++++++++
 32 files changed, 856 insertions(+), 644 deletions(-)
 create mode 100644 src/Relation/Binary/Reasoning/Syntax.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index a3adeca602..a1fb08efcf 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -33,6 +33,9 @@ Bug-fixes
   in `Function.Construct.Composition` had their arguments in the wrong order. They have
   been flipped so they can actually be used as a composition operator.
 
+* The combinators `_≃⟨_⟩_` and `_≃˘⟨_⟩_` in `Data.Rational.Properties.≤-Reasoning`
+  now correctly accepts proofs of type `_≃_` instead of the previous proofs of type `_≡_`.
+
 * The following operators were missing a fixity declaration, which has now
   been fixed -
   ```
@@ -536,6 +539,10 @@ Non-backwards compatible changes
   3. Finally, if the above approaches are not viable then you may be forced to explicitly
      use `cong` combined with a lemma that proves the old reduction behaviour.
 
+* Similarly, in order to prevent reduction, the equality `_≃_` in `Data.Rational.Base` 
+  has been made into a data type with the single constructor `*≡*`. The destructor `drop-*≡*`
+  has been added to `Data.Rational.Properties`.
+
 ### Change to the definition of `LeftCancellative` and `RightCancellative`
 
 * The definitions of the types for cancellativity in `Algebra.Definitions` previously
@@ -1222,6 +1229,15 @@ Major improvements
 
 * In `Relation.Binary.Reasoning.Base.Triple`, added a new parameter `<-irrefl : Irreflexive _≈_ _<_`
 
+### More modular design of equational reasoning.
+
+* Have introduced a new module `Relation.Binary.Reasoning.Syntax` which exports 
+  a range of modules containing pre-existing reasoning combinator syntax.
+
+* This makes it possible to add new or rename existing reasoning combinators to a 
+  pre-existing `Reasoning` module in just a couple of lines 
+  (e.g. see `∣-Reasoning` in `Data.Nat.Divisibility`)
+
 Deprecated modules
 ------------------
 
@@ -1793,6 +1809,11 @@ Deprecated names
   sym-↔   ↦   ↔-sym
   ```
 
+* In `Function.Related.Propositional.Reasoning`:
+  ```agda
+  _↔⟨⟩_  ↦  _≡⟨⟩_
+  ```
+  
 * In `Foreign.Haskell.Either` and `Foreign.Haskell.Pair`:
   ```
   toForeign   ↦ Foreign.Haskell.Coerce.coerce
@@ -2685,6 +2706,8 @@ Additions to existing modules
   toℕ-inverseˡ  : Inverseˡ _≡_ _≡_ toℕ fromℕ
   toℕ-inverseʳ  : Inverseʳ _≡_ _≡_ toℕ fromℕ
   toℕ-inverseᵇ  : Inverseᵇ _≡_ _≡_ toℕ fromℕ
+  
+  <-asym : Asymmetric _<_
   ```
 
 * Added a new pattern synonym to `Data.Nat.Divisibility.Core`:
@@ -2854,6 +2877,7 @@ Additions to existing modules
   ```agda
   ↥ᵘ-toℚᵘ : ↥ᵘ (toℚᵘ p) ≡ ↥ p
   ↧ᵘ-toℚᵘ : ↧ᵘ (toℚᵘ p) ≡ ↧ p
+  ↥p≡↥q≡0⇒p≡q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≡ q
 
   _≥?_ : Decidable _≥_
   _>?_ : Decidable _>_
@@ -2878,6 +2902,10 @@ Additions to existing modules
 
 * Added new definitions in `Data.Rational.Unnormalised.Properties`:
   ```agda
+  ↥p≡0⇒p≃0 : ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ
+  p≃0⇒↥p≡0 : p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ
+  ↥p≡↥q≡0⇒p≃q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≃ q
+
   +-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
   +-*-rawSemiring     : RawSemiring 0ℓ 0ℓ
 
diff --git a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
index acc7153f52..9f573cc711 100644
--- a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
+++ b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
@@ -127,10 +127,14 @@ module _ {A : Set a} where
 
 ------------------------------------------------------------------------
 -- Pointwise DSL
+--
 -- A guardedness check does not play well with compositional proofs.
 -- Luckily we can learn from Nils Anders Danielsson's
 -- Beating the Productivity Checker Using Embedded Languages
 -- and design a little compositional DSL to define such proofs
+--
+-- NOTE: also because of the guardedness check we can't use the standard
+-- `Relation.Binary.Reasoning.Syntax` approach.
 
 module pw-Reasoning (S : Setoid a ℓ) where
   private module S = Setoid S
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index 2f59082e22..aeff3076dc 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -34,6 +34,7 @@ import Relation.Binary.Construct.FromRel as Ind
 import Relation.Binary.Reasoning.Preorder as PreR
 import Relation.Binary.Reasoning.PartialOrder as POR
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.Reasoning.Syntax
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary
 open import Relation.Nullary.Negation
@@ -165,12 +166,11 @@ Any-∈ {P = P} = mk↔ₛ′
 module ⊑-Reasoning {a} {A : Set a} where
   private module Base = POR (⊑-Poset A)
 
-  open Base public
-    hiding (step-<; begin-strict_; step-≤)
+  open Base public hiding (step-<; step-≤)
+
+  open ⊑-syntax _IsRelatedTo_ _IsRelatedTo_ Base.≤-go public
+  open ⊏-syntax _IsRelatedTo_ _IsRelatedTo_ Base.<-go public
 
-  infixr 2 step-⊑
-  step-⊑ = Base.step-≤
-  syntax step-⊑ x ys⊑zs xs⊑ys = x ⊑⟨ xs⊑ys ⟩ ys⊑zs
 
 -- The subset relation forms a preorder.
 
@@ -179,22 +179,22 @@ module ⊑-Reasoning {a} {A : Set a} where
                  (λ xs≈ys → ⊑⇒⊆ (⊑P.reflexive xs≈ys))
   where module ⊑P = Poset (⊑-Poset A)
 
+-- Example uses:
+--
+--    x∈zs : x ∈ zs
+--    x∈zs =
+--      x  ∈⟨ x∈xs ⟩
+--      xs ⊆⟨ xs⊆ys ⟩
+--      ys ≡⟨ ys≡zs ⟩
+--      zs  ∎
 module ⊆-Reasoning {A : Set a} where
   private module Base = PreR (⊆-Preorder A)
 
-  open Base public
-    hiding (step-∼)
-
-  infixr 2 step-⊆
-  infix  1 step-∈
-
-  step-⊆ = Base.step-∼
+  open Base public hiding (step-∼)
 
-  step-∈ : ∀ (x : A) {xs ys} → xs IsRelatedTo ys → x ∈ xs → x ∈ ys
-  step-∈ x xs⊆ys x∈xs = (begin xs⊆ys) x∈xs
+  open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → begin x)
+  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
 
-  syntax step-⊆ xs ys⊆zs xs⊆ys = xs ⊆⟨ xs⊆ys ⟩ ys⊆zs
-  syntax step-∈ x  xs⊆ys x∈xs  = x  ∈⟨ x∈xs  ⟩ xs⊆ys
 
 -- take returns a prefix.
 take-⊑ : ∀ n (xs : Colist A) →
diff --git a/src/Data/Container/Relation/Unary/Any/Properties.agda b/src/Data/Container/Relation/Unary/Any/Properties.agda
index d3265acefb..0a4340fa7b 100644
--- a/src/Data/Container/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Container/Relation/Unary/Any/Properties.agda
@@ -26,7 +26,6 @@ open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; _≗_; refl)
 
-open Related.EquationalReasoning hiding (_≡⟨_⟩_)
 private
   module ×⊎ {k ℓ} = CommutativeSemiring (×-⊎-commutativeSemiring k ℓ)
 
@@ -71,6 +70,7 @@ module _ {s p} {C : Container s p} {x} {X : Set x}
     (∃ λ x → x ∈ xs₁ × P₁ x) ∼⟨ Σ.cong ↔-refl (xs₁≈xs₂ ×-cong P₁↔P₂ _) ⟩
     (∃ λ x → x ∈ xs₂ × P₂ x) ↔⟨ SK-sym (↔∈ C) ⟩
     ◇ C P₂ xs₂               ∎
+    where open Related.EquationalReasoning
 
 -- Nested occurrences of ◇ can sometimes be swapped.
 
@@ -95,6 +95,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
     (∃ λ y → y ∈ ys × ∃ λ x → x ∈ xs × P x y)  ↔⟨ Σ.cong ↔-refl (Σ.cong ↔-refl (SK-sym (↔∈ C₁))) ⟩
     (∃ λ y → y ∈ ys × ◇ _ (flip P y) xs)       ↔⟨ SK-sym (↔∈ C₂) ⟩
     ◇ _ (λ y → ◇ _ (flip P y) xs) ys           ∎
+    where open Related.EquationalReasoning
 
 -- Nested occurrences of ◇ can sometimes be flattened.
 
@@ -162,9 +163,10 @@ module _ {s p} (C : Container s p) {x y} {X : Set x} {Y : Set y}
   map↔∘ : ∀ {xs : ⟦ C ⟧ X} (f : X → Y) → ◇ C P (map f xs) ↔ ◇ C (P ∘′ f) xs
   map↔∘ {xs} f =
    ◇ C P (map f xs)          ↔⟨ ↔Σ C ⟩
-   ∃ (P ∘′ proj₂ (map f xs)) ↔⟨⟩
+   ∃ (P ∘′ proj₂ (map f xs)) ≡⟨⟩
    ∃ (P ∘′ f ∘′ proj₂ xs)    ↔⟨ SK-sym (↔Σ C) ⟩
    ◇ C (P ∘′ f) xs           ∎
+   where open Related.EquationalReasoning
 
 -- Membership in a mapped container can be expressed without reference
 -- to map.
@@ -178,6 +180,7 @@ module _ {s p} (C : Container s p) {x y} {X : Set x} {Y : Set y}
     y ∈ map f xs               ↔⟨ map↔∘ C (y ≡_) f ⟩
     ◇ C (λ x → y ≡ f x) xs     ↔⟨ ↔∈ C ⟩
     ∃ (λ x → x ∈ xs × y ≡ f x) ∎
+    where open Related.EquationalReasoning
 
 -- map is a congruence for bag and set equality and related preorders.
 
@@ -193,6 +196,7 @@ module _ {s p} (C : Container s p) {x y} {X : Set x} {Y : Set y}
     ◇ C (λ y → x ≡ f₂ y) xs₂ ↔⟨ SK-sym (map↔∘ C (_≡_ x) f₂) ⟩
     x ∈ map f₂ xs₂           ∎
     where
+    open Related.EquationalReasoning
     helper : ∀ y → (x ≡ f₁ y) ↔ (x ≡ f₂ y)
     helper y rewrite f₁≗f₂ y = ↔-refl
 
@@ -205,7 +209,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
   remove-linear {xs} m = mk↔ₛ′ t f t∘f f∘t
     where
     open _≃_
-    open P.≡-Reasoning renaming (_∎ to _∎′)
+    open P.≡-Reasoning
 
     position⊸m : ∀ {s} → Position C₂ (shape⊸ m s) ≃ Position C₁ s
     position⊸m = ↔⇒≃ (position⊸ m)
@@ -259,7 +263,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
 
          P.subst (P ∘ proj₂ xs) P.refl p                        ≡⟨⟩
 
-        p                                                       ∎′)
+        p                                                       ∎)
       )
 
     t∘f : t ∘ f ≗ id
@@ -279,7 +283,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
                                                                      (P.trans-symˡ (right-inverse-of position⊸m _)) ⟩
          P.subst (P ∘ proj₂ xs) P.refl p                        ≡⟨⟩
 
-        p                                                       ∎′)
+        p                                                       ∎)
       )
 
 -- Linear endomorphisms are identity functions if bag equality is used.
@@ -290,6 +294,7 @@ module _ {s p} {C : Container s p} {x} {X : Set x} where
   linear-identity {xs} m {x} =
     x ∈ ⟪ m ⟫⊸ xs  ↔⟨ remove-linear (_≡_ x) m ⟩
     x ∈        xs  ∎
+    where open Related.EquationalReasoning
 
 -- If join can be expressed using a linear morphism (in a certain
 -- way), then it can be absorbed by the predicate.
@@ -307,4 +312,6 @@ module _ {s₁ s₂ s₃ p₁ p₂ p₃}
     ◇ C₃ P (⟪ join ⟫⊸ xss′) ↔⟨ remove-linear P join ⟩
     ◇ (C₁ C.∘ C₂) P xss′    ↔⟨ SK-sym $ flatten P xss ⟩
     ◇ C₁ (◇ C₂ P) xss       ∎
-    where xss′ = Inverse.from (Composition.correct C₁ C₂) xss
+    where
+    open Related.EquationalReasoning
+    xss′ = Inverse.from (Composition.correct C₁ C₂) xss
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index af06d57486..9a4b938950 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -30,6 +30,7 @@ open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Nullary.Decidable using (yes; no)
 import Relation.Nullary.Decidable as DEC
+open import Relation.Binary.Reasoning.Syntax
 
 ------------------------------------------------------------------------
 -- Type
@@ -118,15 +119,11 @@ open _∣_ using (quotient) public
 -- Divisibility reasoning
 
 module ∣-Reasoning where
-  private
-    module Base = PreorderReasoning ∣-preorder
+  private module Base = PreorderReasoning ∣-preorder
 
-  open Base public
-    hiding (step-∼; step-≈; step-≈˘)
+  open Base public hiding (step-∼; step-≈; step-≈˘)
 
-  infixr 2 step-∣
-  step-∣ = Base.step-∼
-  syntax step-∣ x y∣z x∣y = x ∣⟨ x∣y ⟩ y∣z
+  open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public
 
 ------------------------------------------------------------------------
 -- Other properties of _∣_
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index bfe29c4079..13ed3deab8 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -364,7 +364,7 @@ i≮i = <-irrefl refl
 module ≤-Reasoning where
   open import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
-    <-irrefl
+    <-asym
     <-trans
     (resp₂ _<_)
     <⇒≤
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 889efe96d9..0acef52dcb 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -37,11 +37,13 @@ open import Function.Related.TypeIsomorphisms
 open import Function.Properties.Inverse using (↔-sym; ↔-trans; to-from)
 open import Level using (Level)
 open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Definitions using (Trans)
 open import Relation.Binary.Bundles using (Preorder; Setoid)
 import Relation.Binary.Reasoning.Setoid as EqR
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; _≢_; _≗_; refl)
+open import Relation.Binary.Reasoning.Syntax
 open import Relation.Nullary
 open import Data.List.Membership.Propositional.Properties
 
@@ -90,29 +92,22 @@ bag-=⇒ xs≈ys = ↔⇒ xs≈ys
 ------------------------------------------------------------------------
 -- "Equational" reasoning for _⊆_ along with an additional relatedness
 
-module ⊆-Reasoning where
-  private
-    module PreOrder {a} {A : Set a} = PreorderReasoning (⊆-preorder A)
+module ⊆-Reasoning {A : Set a} where
+  private module Base = PreorderReasoning (⊆-preorder A)
 
-  open PreOrder public
+  open Base public
     hiding (step-≈; step-≈˘; step-∼)
+    renaming (∼-go to ⊆-go)
 
-  infixr 2 step-∼ step-⊆
-  infix  1 step-∈
+  open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → begin x) public
+  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ ⊆-go public
 
-  step-⊆ = PreOrder.step-∼
+  module _ {k : Related.ForwardKind} where
+    ∼-go : Trans _∼[ ⌊ k ⌋→ ]_ _IsRelatedTo_ _IsRelatedTo_
+    ∼-go eq = ⊆-go (⇒→ eq)
 
-  step-∈ : ∀ x {xs ys : List A} →
-           xs IsRelatedTo ys → x ∈ xs → x ∈ ys
-  step-∈ x xs⊆ys x∈xs = (begin xs⊆ys) x∈xs
+    open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public
 
-  step-∼ : ∀ {k} xs {ys zs : List A} →
-           ys IsRelatedTo zs → xs ∼[ ⌊ k ⌋→ ] ys → xs IsRelatedTo zs
-  step-∼ xs ys⊆zs xs≈ys = step-⊆ xs ys⊆zs (⇒→ xs≈ys)
-
-  syntax step-∈ x  xs⊆ys x∈xs  = x ∈⟨ x∈xs ⟩ xs⊆ys
-  syntax step-∼ xs ys⊆zs xs≈ys = xs ∼⟨ xs≈ys ⟩ ys⊆zs
-  syntax step-⊆ xs ys⊆zs xs⊆ys = xs ⊆⟨ xs⊆ys ⟩ ys⊆zs
 
 ------------------------------------------------------------------------
 -- Congruence lemmas
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional.agda b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
index aaea103279..28cf25d9ed 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
@@ -16,6 +16,7 @@ open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions using (Reflexive; Transitive)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.Reasoning.Setoid as EqReasoning
+open import Relation.Binary.Reasoning.Syntax
 
 ------------------------------------------------------------------------
 -- An inductive definition of permutation
@@ -73,16 +74,15 @@ data _↭_ : Rel (List A) a where
 
 module PermutationReasoning where
 
-  private
-    module Base = EqReasoning ↭-setoid
+  private module Base = EqReasoning ↭-setoid
 
-  open EqReasoning ↭-setoid public
-    hiding (step-≈; step-≈˘)
+  open Base public hiding (step-≈; step-≈˘)
 
-  infixr 2 step-↭  step-↭˘ step-swap step-prep
+  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go ↭-sym public
 
-  step-↭  = Base.step-≈
-  step-↭˘ = Base.step-≈˘
+  -- Some extra combinators that allow us to skip certain elements
+
+  infixr 2 step-swap step-prep
 
   -- Skip reasoning on the first element
   step-prep : ∀ x xs {ys zs : List A} → (x ∷ ys) IsRelatedTo zs →
@@ -94,7 +94,5 @@ module PermutationReasoning where
               xs ↭ ys → (x ∷ y ∷ xs) IsRelatedTo zs
   step-swap x y xs rel xs↭ys = relTo (trans (swap x y xs↭ys) (begin rel))
 
-  syntax step-↭  x y↭z x↭y = x ↭⟨  x↭y ⟩ y↭z
-  syntax step-↭˘ x y↭z y↭x = x ↭˘⟨  y↭x ⟩ y↭z
   syntax step-prep x xs y↭z x↭y = x ∷ xs <⟨ x↭y ⟩ y↭z
   syntax step-swap x y xs y↭z x↭y = x ∷ y ∷ xs <<⟨ x↭y ⟩ y↭z
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid.agda b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
index fe3f095ee5..0d375226db 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
@@ -6,11 +6,13 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
+open import Function.Base using (_∘′_)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
+open import Relation.Binary.Reasoning.Syntax
 
 module Data.List.Relation.Binary.Permutation.Setoid
   {a ℓ} (S : Setoid a ℓ) where
@@ -86,24 +88,16 @@ steps (trans xs↭ys ys↭zs) = steps xs↭ys + steps ys↭zs
 
 module PermutationReasoning where
 
-  private
-    module Base = SetoidReasoning ↭-setoid
+  private module Base = SetoidReasoning ↭-setoid
 
-  open SetoidReasoning ↭-setoid public
-    hiding (step-≈; step-≈˘)
+  open Base public hiding (step-≈; step-≈˘)
 
-  infixr 2 step-↭  step-↭˘ step-≋ step-≋˘ step-swap step-prep
+  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go ↭-sym public
+  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ (Base.∼-go ∘′ refl) ≋-sym public
 
-  step-↭  = Base.step-≈
-  step-↭˘ = Base.step-≈˘
+  -- Some extra combinators that allow us to skip certain elements
 
-  -- Step with pointwise list equality
-  step-≋ : ∀ x {y z} → y IsRelatedTo z → x ≋ y → x IsRelatedTo z
-  step-≋ x (relTo y↔z) x≋y = relTo (trans (refl x≋y) y↔z)
-
-  -- Step with flipped pointwise list equality
-  step-≋˘ : ∀ x {y z} → y IsRelatedTo z → y ≋ x → x IsRelatedTo z
-  step-≋˘ x y↭z y≋x = x ≋⟨ ≋-sym y≋x ⟩ y↭z
+  infixr 2 step-swap step-prep
 
   -- Skip reasoning on the first element
   step-prep : ∀ x xs {ys zs : List A} → (x ∷ ys) IsRelatedTo zs →
@@ -115,9 +109,5 @@ module PermutationReasoning where
               xs ↭ ys → (x ∷ y ∷ xs) IsRelatedTo zs
   step-swap x y xs rel xs↭ys = relTo (trans (swap Eq.refl Eq.refl xs↭ys) (begin rel))
 
-  syntax step-↭  x y↭z x↭y = x ↭⟨  x↭y ⟩ y↭z
-  syntax step-↭˘ x y↭z y↭x = x ↭˘⟨  y↭x ⟩ y↭z
-  syntax step-≋  x y↭z x≋y = x ≋⟨  x≋y ⟩ y↭z
-  syntax step-≋˘ x y↭z y≋x = x ≋˘⟨  y≋x ⟩ y↭z
   syntax step-prep x xs y↭z x↭y = x ∷ xs <⟨ x↭y ⟩ y↭z
   syntax step-swap x y xs y↭z x↭y = x ∷ y ∷ xs <<⟨ x↭y ⟩ y↭z
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index abf5bdc86f..0d11079ad0 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -32,6 +32,7 @@ open import Relation.Binary.Definitions
 open import Relation.Binary.Bundles using (Setoid; Preorder)
 open import Relation.Binary.Structures using (IsPreorder)
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
+open import Relation.Binary.Reasoning.Syntax
 
 open Setoid using (Carrier)
 
@@ -112,31 +113,15 @@ module _ (S : Setoid a ℓ) where
 ------------------------------------------------------------------------
 
 module ⊆-Reasoning (S : Setoid a ℓ) where
+  open Membership S using (_∈_)
 
-  open Setoid S renaming (Carrier to A)
-  open Subset S
-  open Membership S
-
-  private
-    module Base = PreorderReasoning (⊆-preorder S)
-
-  open Base public
-    hiding (step-∼; step-≈; step-≈˘)
-
-  infixr 2 step-⊆ step-≋ step-≋˘
-  infix 1 step-∈
-
-  step-∈ : ∀ x {xs ys} → xs IsRelatedTo ys → x ∈ xs → x ∈ ys
-  step-∈ x xs⊆ys x∈xs = (begin xs⊆ys) x∈xs
+  private module Base = PreorderReasoning (⊆-preorder S)
 
-  step-⊆  = Base.step-∼
-  step-≋  = Base.step-≈
-  step-≋˘ = Base.step-≈˘
+  open Base public hiding (step-∼; step-≈; step-≈˘)
 
-  syntax step-∈  x  xs⊆ys x∈xs  = x  ∈⟨  x∈xs  ⟩ xs⊆ys
-  syntax step-⊆  xs ys⊆zs xs⊆ys = xs ⊆⟨  xs⊆ys ⟩ ys⊆zs
-  syntax step-≋  xs ys⊆zs xs≋ys = xs ≋⟨  xs≋ys ⟩ ys⊆zs
-  syntax step-≋˘ xs ys⊆zs xs≋ys = xs ≋˘⟨ xs≋ys ⟩ ys⊆zs
+  open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → Base.begin x) public
+  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
+  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ Base.≈-go public
 
 ------------------------------------------------------------------------
 -- Relationship with other binary relations
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 8c08974636..9e8976842e 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -27,13 +27,7 @@ open import Function.Base using (_∘_; _$_; id)
 open import Function.Definitions
 open import Function.Consequences.Propositional
 open import Level using (0ℓ)
-open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.Bundles
-  using (Setoid; DecSetoid; StrictPartialOrder; StrictTotalOrder; Preorder; Poset; TotalOrder; DecTotalOrder)
-open import Relation.Binary.Definitions
-  using (Decidable; Irreflexive; Transitive; Reflexive; Antisymmetric; Total; Trichotomous; tri≈; tri<; tri>)
-open import Relation.Binary.Structures
-  using (IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
+open import Relation.Binary
 open import Relation.Binary.Consequences
 open import Relation.Binary.Morphism
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphism
@@ -371,6 +365,9 @@ toℕ-isMonomorphism-< = record
 <-trans (odd<odd x<y) (odd<even (inj₂ refl))   =  odd<even (inj₁ x<y)
 <-trans (odd<odd x<y) (odd<odd y<z)            =  odd<odd (<-trans x<y y<z)
 
+<-asym : Asymmetric _<_
+<-asym {x} {y} = trans∧irr⇒asym refl <-trans <-irrefl {x} {y}
+
 -- Should not be implemented via the morphism `toℕ` in order to
 -- preserve O(log n) time requirement.
 <-cmp : Trichotomous _≡_ _<_
@@ -609,7 +606,7 @@ module ≤-Reasoning where
 
   open import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
-    <-irrefl
+    <-asym
     <-trans
     (resp₂ _<_)
     <⇒≤
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index ea4814b964..5a0491e498 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -27,6 +27,7 @@ open import Relation.Binary.Definitions
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst)
+open import Relation.Binary.Reasoning.Syntax
 import Relation.Binary.PropositionalEquality.Properties as PropEq
 
 ------------------------------------------------------------------------
@@ -117,15 +118,11 @@ suc n ∣? m      = Dec.map (m%n≡0⇔n∣m m (suc n)) (m % suc n ≟ 0)
 -- A reasoning module for the _∣_ relation
 
 module ∣-Reasoning where
-  private
-    module Base = PreorderReasoning ∣-preorder
+  private module Base = PreorderReasoning ∣-preorder
 
-  open Base public
-    hiding (step-≈; step-≈˘; step-∼)
+  open Base public hiding (step-≈; step-≈˘; step-∼)
 
-  infixr 2 step-∣
-  step-∣ = Base.step-∼
-  syntax step-∣ x y∣z x∣y = x ∣⟨ x∣y ⟩ y∣z
+  open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
 
 ------------------------------------------------------------------------
 -- Simple properties of _∣_
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 915d1d7d1b..74350e1f6b 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -491,7 +491,7 @@ m<1+n⇒m≤n (s≤s m≤n) = m≤n
 module ≤-Reasoning where
   open import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
-    <-irrefl
+    <-asym
     <-trans
     (resp₂ _<_)
     <⇒≤
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index ebb5ad0ac9..1e49894eb6 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -66,8 +66,11 @@ mkℚ+ n (suc d) coprime = mkℚ (+ n) d coprime
 
 infix 4 _≃_
 
-_≃_ : Rel ℚ 0ℓ
-p ≃ q = (↥ p ℤ.* ↧ q) ≡ (↥ q ℤ.* ↧ p)
+data _≃_ : Rel ℚ 0ℓ where
+  *≡* : ∀ {p q} → (↥ p ℤ.* ↧ q) ≡ (↥ q ℤ.* ↧ p) → p ≃ q
+
+_≄_ : Rel ℚ 0ℓ
+p ≄ q = ¬ (p ≃ q)
 
 ------------------------------------------------------------------------
 -- Ordering of rationals
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 79d34870ef..b43b109943 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -49,19 +49,14 @@ import Data.Sign as S
 open import Function.Base using (_∘_; _∘′_; _∘₂_; _$_; flip)
 open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
-open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.Bundles
-  using (Setoid; DecSetoid; TotalPreorder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder; DenseLinearOrder)
-open import Relation.Binary.Structures
-  using (IsPreorder; IsTotalOrder; IsTotalPreorder; IsPartialOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder; IsDenseLinearOrder)
-open import Relation.Binary.Definitions
-  using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; Dense; Trans; Trichotomous; tri<; tri>; tri≈; _Respectsʳ_; _Respectsˡ_; _Respects₂_)
+open import Relation.Binary
 open import Relation.Binary.PropositionalEquality
 open import Relation.Binary.Morphism.Structures
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms
 open import Relation.Nullary.Decidable.Core as Dec
   using (yes; no; recompute; map′; _×-dec_)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
+open import Relation.Binary.Reasoning.Syntax
 
 open import Algebra.Definitions {A = ℚ} _≡_
 open import Algebra.Structures  {A = ℚ} _≡_
@@ -136,11 +131,14 @@ mkℚ+-pos (suc n) (suc d) = _
 -- Numerator and denominator equality
 ------------------------------------------------------------------------
 
+drop-*≡* : p ≃ q → ↥ p ℤ.* ↧ q ≡ ↥ q ℤ.* ↧ p
+drop-*≡* (*≡* eq) = eq
+
 ≡⇒≃ : _≡_ ⇒ _≃_
-≡⇒≃ refl = refl
+≡⇒≃ refl = *≡* refl
 
 ≃⇒≡ : _≃_ ⇒ _≡_
-≃⇒≡ {x = mkℚ n₁ d₁ c₁} {y = mkℚ n₂ d₂ c₂} eq = helper
+≃⇒≡ {x = mkℚ n₁ d₁ c₁} {y = mkℚ n₂ d₂ c₂} (*≡* eq) = helper
   where
   open ≡-Reasoning
 
@@ -165,6 +163,9 @@ mkℚ+-pos (suc n) (suc d) = _
   ... | refl with ℤ.*-cancelʳ-≡ n₁ n₂ (+ suc d₁) eq
   ...   | refl = refl
 
+≃-sym : Symmetric _≃_
+≃-sym = ≡⇒≃ ∘′ sym ∘′ ≃⇒≡ 
+
 ------------------------------------------------------------------------
 -- Properties of ↥
 ------------------------------------------------------------------------
@@ -176,6 +177,9 @@ mkℚ+-pos (suc n) (suc d) = _
 p≡0⇒↥p≡0 : ∀ p → p ≡ 0ℚ → ↥ p ≡ 0ℤ
 p≡0⇒↥p≡0 p refl = refl
 
+↥p≡↥q≡0⇒p≡q : ∀ p q → ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≡ q
+↥p≡↥q≡0⇒p≡q p q ↥p≡0 ↥q≡0 = trans (↥p≡0⇒p≡0 p ↥p≡0) (sym (↥p≡0⇒p≡0 q ↥q≡0))
+
 ------------------------------------------------------------------------
 -- Basic properties of sign predicates
 ------------------------------------------------------------------------
@@ -418,7 +422,7 @@ private
 ↧ᵘ-toℚᵘ p@record{} = refl
 
 toℚᵘ-injective : Injective _≡_ _≃ᵘ_ toℚᵘ
-toℚᵘ-injective {x@record{}} {y@record{}} (*≡* eq) = ≃⇒≡ eq
+toℚᵘ-injective {x@record{}} {y@record{}} (*≡* eq) = ≃⇒≡ (*≡* eq)
 
 fromℚᵘ-injective : Injective _≃ᵘ_ _≡_ fromℚᵘ
 fromℚᵘ-injective {p@record{}} {q@record{}} = /-injective-≃ p q
@@ -497,7 +501,7 @@ private
 ≤-trans = ≤-Monomorphism.trans ℚᵘ.≤-trans
 
 ≤-antisym : Antisymmetric _≡_ _≤_
-≤-antisym (*≤* le₁) (*≤* le₂) = ≃⇒≡ (ℤ.≤-antisym le₁ le₂)
+≤-antisym (*≤* le₁) (*≤* le₂) = ≃⇒≡ (*≡* (ℤ.≤-antisym le₁ le₂))
 
 ≤-total : Total _≤_
 ≤-total p q = [ inj₁ ∘ *≤* , inj₂ ∘ *≤* ]′ (ℤ.≤-total (↥ p ℤ.* ↧ q) (↥ q ℤ.* ↧ p))
@@ -606,7 +610,7 @@ toℚᵘ-isOrderMonomorphism-< = record
 ≰⇒> {p} {q} p≰q = *<* (ℤ.≰⇒> (p≰q ∘ *≤*))
 
 <⇒≢ : _<_ ⇒ _≢_
-<⇒≢ {p} {q} (*<* p<q) = ℤ.<⇒≢ p<q ∘ ≡⇒≃
+<⇒≢ {p} {q} (*<* p<q) = ℤ.<⇒≢ p<q ∘ drop-*≡* ∘ ≡⇒≃
 
 <-irrefl : Irreflexive _≡_ _<_
 <-irrefl refl (*<* p<p) = ℤ.<-irrefl refl p<p
@@ -673,9 +677,9 @@ _>?_ = flip _<?_
 
 <-cmp : Trichotomous _≡_ _<_
 <-cmp p q with ℤ.<-cmp (↥ p ℤ.* ↧ q) (↥ q ℤ.* ↧ p)
-... | tri< < ≢ ≯ = tri< (*<* <)        (≢ ∘ ≡⇒≃) (≯ ∘ drop-*<*)
-... | tri≈ ≮ ≡ ≯ = tri≈ (≮ ∘ drop-*<*) (≃⇒≡ ≡)   (≯ ∘ drop-*<*)
-... | tri> ≮ ≢ > = tri> (≮ ∘ drop-*<*) (≢ ∘ ≡⇒≃) (*<* >)
+... | tri< < ≢ ≯ = tri< (*<* <)        (≢ ∘ drop-*≡* ∘ ≡⇒≃) (≯ ∘ drop-*<*)
+... | tri≈ ≮ ≡ ≯ = tri≈ (≮ ∘ drop-*<*) (≃⇒≡ (*≡* ≡))   (≯ ∘ drop-*<*)
+... | tri> ≮ ≢ > = tri> (≮ ∘ drop-*<*) (≢ ∘ drop-*≡* ∘ ≡⇒≃) (*<* >)
 
 <-irrelevant : Irrelevant _<_
 <-irrelevant (*<* p<q₁) (*<* p<q₂) = cong *<* (ℤ.<-irrelevant p<q₁ p<q₂)
@@ -737,7 +741,7 @@ _>?_ = flip _<?_
 module ≤-Reasoning where
   import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
-    <-irrefl
+    <-asym
     <-trans
     (resp₂ _<_)
     <⇒≤
@@ -745,16 +749,12 @@ module ≤-Reasoning where
     ≤-<-trans
     as Triple
 
-  open Triple public
-    hiding (step-≈; step-≈˘)
-
-  infixr 2 step-≃ step-≃˘
-
-  step-≃  = Triple.step-≈
-  step-≃˘ = Triple.step-≈˘
+  open Triple public hiding (step-≈; step-≈˘)
 
-  syntax step-≃  x y∼z x≃y = x ≃⟨  x≃y ⟩ y∼z
-  syntax step-≃˘ x y∼z y≃x = x ≃˘⟨ y≃x ⟩ y∼z
+  ≃-go : Trans _≃_ _IsRelatedTo_ _IsRelatedTo_
+  ≃-go = Triple.≈-go ∘′ ≃⇒≡
+  
+  open ≃-syntax _IsRelatedTo_ _IsRelatedTo_ ≃-go (λ {x y} → ≃-sym {x} {y}) public
 
 ------------------------------------------------------------------------
 -- Properties of Positive/NonPositive/Negative/NonNegative and _≤_/_<_
@@ -848,14 +848,14 @@ toℚᵘ-homo-+ p@record{} q@record{} with +-nf p q ℤ.≟ 0ℤ
   eq : ↥ (p + q) ≡ 0ℤ
   eq rewrite eq2 = cong ↥_ (0/n≡0 (↧ₙ p ℕ.* ↧ₙ q))
 
-... | no  nf[p,q]≢0 = *≡* $ ℤ.*-cancelʳ-≡ _ _ (+-nf p q) {{ℤ.≢-nonZero nf[p,q]≢0}} $ begin
+... | no  nf[p,q]≢0 = *≡* (ℤ.*-cancelʳ-≡ _ _ (+-nf p q) {{ℤ.≢-nonZero nf[p,q]≢0}} $ begin
     (↥ᵘ (toℚᵘ (p + q))) ℤ.* ↧+ᵘ p q  ℤ.* +-nf p q ≡⟨ cong (λ v → v ℤ.* ↧+ᵘ p q ℤ.* +-nf p q) (↥ᵘ-toℚᵘ (p + q)) ⟩
     ↥ (p + q) ℤ.* ↧+ᵘ p q ℤ.* +-nf p q            ≡⟨ xy∙z≈xz∙y (↥ (p + q)) _ _ ⟩
     ↥ (p + q) ℤ.* +-nf p q ℤ.* ↧+ᵘ p q            ≡⟨ cong (ℤ._* ↧+ᵘ p q) (↥-+ p q) ⟩
     ↥+ᵘ p q ℤ.* ↧+ᵘ p q                           ≡⟨ cong (↥+ᵘ p q ℤ.*_) (sym (↧-+ p q)) ⟩
     ↥+ᵘ p q ℤ.* (↧ (p + q) ℤ.* +-nf p q)          ≡⟨ x∙yz≈xy∙z (↥+ᵘ p q) _ _ ⟩
     ↥+ᵘ p q ℤ.* ↧ (p + q)  ℤ.* +-nf p q           ≡˘⟨ cong (λ v → ↥+ᵘ p q ℤ.* v ℤ.* +-nf p q) (↧ᵘ-toℚᵘ (p + q)) ⟩
-    ↥+ᵘ p q ℤ.* ↧ᵘ (toℚᵘ (p + q)) ℤ.* +-nf p q    ∎
+    ↥+ᵘ p q ℤ.* ↧ᵘ (toℚᵘ (p + q)) ℤ.* +-nf p q    ∎)
   where open ≡-Reasoning; open CommSemigroupProperties ℤ.*-commutativeSemigroup
 
 toℚᵘ-isMagmaHomomorphism-+ : IsMagmaHomomorphism +-rawMagma ℚᵘ.+-rawMagma toℚᵘ
@@ -1059,14 +1059,14 @@ toℚᵘ-homo-* p@record{} q@record{} with *-nf p q ℤ.≟ 0ℤ
 
   eq : ↥ (p * q) ≡ 0ℤ
   eq rewrite eq2 = cong ↥_ (0/n≡0 (↧ₙ p ℕ.* ↧ₙ q))
-... | no nf[p,q]≢0 = *≡* $ ℤ.*-cancelʳ-≡ _ _ (*-nf p q) {{ℤ.≢-nonZero nf[p,q]≢0}} $ begin
+... | no nf[p,q]≢0 = *≡* (ℤ.*-cancelʳ-≡ _ _ (*-nf p q) {{ℤ.≢-nonZero nf[p,q]≢0}} $ begin
   ↥ᵘ (toℚᵘ (p * q)) ℤ.* (↧ p ℤ.* ↧ q) ℤ.* *-nf p q     ≡⟨ cong (λ v → v ℤ.* (↧ p ℤ.* ↧ q) ℤ.* *-nf p q) (↥ᵘ-toℚᵘ (p * q)) ⟩
   ↥ (p * q)         ℤ.* (↧ p ℤ.* ↧ q) ℤ.* *-nf p q     ≡⟨ xy∙z≈xz∙y (↥ (p * q)) _ _ ⟩
   ↥ (p * q)         ℤ.* *-nf p q ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨ cong (ℤ._* (↧ p ℤ.* ↧ q)) (↥-* p q) ⟩
   (↥ p ℤ.* ↥ q)     ℤ.* (↧ p ℤ.* ↧ q)                  ≡⟨ cong ((↥ p ℤ.* ↥ q) ℤ.*_) (sym (↧-* p q)) ⟩
   (↥ p ℤ.* ↥ q)     ℤ.* (↧ (p * q) ℤ.* *-nf p q)       ≡⟨ x∙yz≈xy∙z (↥ p ℤ.* ↥ q) _ _ ⟩
   (↥ p ℤ.* ↥ q)     ℤ.* ↧ (p * q)  ℤ.* *-nf p q        ≡˘⟨ cong (λ v → (↥ p ℤ.* ↥ q) ℤ.* v ℤ.* *-nf p q) (↧ᵘ-toℚᵘ (p * q)) ⟩
-  (↥ p ℤ.* ↥ q)     ℤ.* ↧ᵘ (toℚᵘ (p * q)) ℤ.* *-nf p q ∎
+  (↥ p ℤ.* ↥ q)     ℤ.* ↧ᵘ (toℚᵘ (p * q)) ℤ.* *-nf p q ∎)
   where open ≡-Reasoning; open CommSemigroupProperties ℤ.*-commutativeSemigroup
 
 toℚᵘ-homo-1/ : ∀ p .{{_ : NonZero p}} → toℚᵘ (1/ p) ℚᵘ.≃ (ℚᵘ.1/ toℚᵘ p)
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 11b0b4a5d7..dffd2529b0 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -43,6 +43,7 @@ import Relation.Binary.Consequences as BC
 open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Properties.Poset as PosetProperties
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+open import Relation.Binary.Reasoning.Syntax
 
 open import Algebra.Properties.CommutativeSemigroup ℤ.*-commutativeSemigroup
 
@@ -154,6 +155,19 @@ proj₂ (≄-tight p q) p≃q p≄q = p≄q p≃q
 
 module ≃-Reasoning = SetoidReasoning ≃-setoid
 
+↥p≡0⇒p≃0 : ∀ p → ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ
+↥p≡0⇒p≃0 p ↥p≡0 = *≡* (cong (ℤ._* (↧ 0ℚᵘ)) ↥p≡0)
+
+p≃0⇒↥p≡0 : ∀ p → p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ
+p≃0⇒↥p≡0 p (*≡* eq) = begin
+  ↥ p          ≡˘⟨ ℤ.*-identityʳ (↥ p) ⟩
+  ↥ p ℤ.* 1ℤ  ≡⟨ eq ⟩
+  0ℤ           ∎
+  where open ≡-Reasoning
+  
+↥p≡↥q≡0⇒p≃q : ∀ p q → ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≃ q
+↥p≡↥q≡0⇒p≃q p q ↥p≡0 ↥q≡0 = ≃-trans (↥p≡0⇒p≃0 p ↥p≡0) (≃-sym (↥p≡0⇒p≃0 _ ↥q≡0))
+
 ------------------------------------------------------------------------
 -- Properties of -_
 ------------------------------------------------------------------------
@@ -582,7 +596,7 @@ _>?_ = flip _<?_
 module ≤-Reasoning where
   import Relation.Binary.Reasoning.Base.Triple
     ≤-isPreorder
-    <-irrefl
+    <-asym
     <-trans
     <-resp-≃
     <⇒≤
@@ -593,13 +607,7 @@ module ≤-Reasoning where
   open Triple public
     hiding (step-≈; step-≈˘)
 
-  infixr 2 step-≃ step-≃˘
-
-  step-≃  = Triple.step-≈
-  step-≃˘ = Triple.step-≈˘
-
-  syntax step-≃  x y∼z x≃y = x ≃⟨  x≃y ⟩ y∼z
-  syntax step-≃˘ x y∼z y≃x = x ≃˘⟨ y≃x ⟩ y∼z
+  open ≃-syntax _IsRelatedTo_ _IsRelatedTo_  Triple.≈-go ≃-sym public
 
 ------------------------------------------------------------------------
 -- Properties of ↥_/↧_
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index ff1837b85d..b273029052 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -261,7 +261,7 @@ module ≤-Reasoning  {_≈_ : Rel A ℓ₁} {_≼_ : Rel A ℓ₂}
 
   open import Relation.Binary.Reasoning.Base.Triple
     (≤-isPreorder ≼-po {n})
-    <-irrefl
+    (<-asym isEquivalence ≤-resp-≈ antisym)
     (<-trans ≼-po)
     (<-resp₂ isEquivalence ≤-resp-≈)
     <⇒≤
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 0cd7fde477..83608e4456 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -324,23 +324,21 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
 -- Equational Reasoning
 ------------------------------------------------------------------------
 
-module ≤-Reasoning {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂}
-                   (≈-isEquivalence : IsEquivalence _≈_)
-                   (≺-irrefl : Irreflexive _≈_ _≺_)
-                   (≺-trans : Transitive _≺_)
-                   (≺-resp-≈ : _≺_ Respects₂ _≈_)
-                   (n : ℕ)
-                   where
+module ≤-Reasoning
+  {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂}
+  (≺-isStrictPartialOrder : IsStrictPartialOrder _≈_ _≺_)
+  (n : ℕ)
+  where
 
-  private
-    ≈-isPartialEquivalence = IsEquivalence.isPartialEquivalence ≈-isEquivalence
+  open IsStrictPartialOrder ≺-isStrictPartialOrder
 
   open import Relation.Binary.Reasoning.Base.Triple
-    (≤-isPreorder ≈-isEquivalence ≺-trans ≺-resp-≈)
-    (<-irrefl ≺-irrefl)
-    (<-trans ≈-isPartialEquivalence ≺-resp-≈ ≺-trans)
-    (<-respects₂ ≈-isPartialEquivalence ≺-resp-≈)
+    (≤-isPreorder isEquivalence trans <-resp-≈)
+    (<-asym Eq.sym <-resp-≈ asym)
+    (<-trans Eq.isPartialEquivalence <-resp-≈ trans)
+    (<-respects₂ Eq.isPartialEquivalence <-resp-≈)
     (<⇒≤ {m = n})
-    (<-transˡ ≈-isPartialEquivalence ≺-resp-≈ ≺-trans)
-    (<-transʳ ≈-isPartialEquivalence ≺-resp-≈ ≺-trans)
+    (<-transˡ Eq.isPartialEquivalence <-resp-≈ trans)
+    (<-transʳ Eq.isPartialEquivalence <-resp-≈ trans)
     public
+
diff --git a/src/Function/Related/Propositional.agda b/src/Function/Related/Propositional.agda
index 96b50667df..d7f0610e78 100644
--- a/src/Function/Related/Propositional.agda
+++ b/src/Function/Related/Propositional.agda
@@ -10,11 +10,12 @@ module Function.Related.Propositional where
 
 open import Level
 open import Relation.Binary
-  using (Sym; Reflexive; Trans; IsEquivalence; Setoid; IsPreorder; Preorder)
+  using (Rel; REL; Sym; Reflexive; Trans; IsEquivalence; Setoid; IsPreorder; Preorder)
 open import Function.Bundles
 open import Function.Base
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.Reasoning.Syntax
 
 open import Function.Properties.Surjection   using (↠⇒↪; ↠⇒⇔)
 open import Function.Properties.RightInverse using (↪⇒↠)
@@ -293,41 +294,39 @@ K-preorder k ℓ = record { isPreorder = K-isPreorder k }
 ------------------------------------------------------------------------
 -- Equational reasoning
 
--- Equational reasoning for related things.
+-- Equational reasoning for related things. Note that we don't use
+-- the `Relation.Binary.Reasoning.Syntax` for this as this relation
+-- is heterogeneous.
 
-module EquationalReasoning where
+module EquationalReasoning {k : Kind} where
 
-  infix  3 _∎
-  infixr 2 _∼⟨_⟩_ _⤖⟨_⟩_ _↔⟨_⟩_ _↔⟨⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_
-  infix  1 begin_
+  -- Combinators with one heterogeneous relation
+  module _ {a b : Level} where
+    open begin-syntax (Related {a} {b} k) id public
+    open ≡-noncomputing-syntax (Related {a} {b} k) public
 
-  begin_ : ∀ {k} → A ∼[ k ] B → A ∼[ k ] B
-  begin_ x∼y = x∼y
+  -- Combinators with two heterogeneous relations
+  module _ {a b c : Level} where
+    private
+      rel1 = Related {b} {c} k
+      rel2 = Related {a} {c} k
 
-  _∼⟨_⟩_ : (A : Set a) → A ∼[ k ] B → B ∼[ k ] C → A ∼[ k ] C
-  _ ∼⟨ A↝B ⟩ B↝C = K-trans A↝B B↝C
+    open ∼-syntax rel1 rel2 K-trans public
+    open ⤖-syntax rel1 rel2 (K-trans ∘′ ↔⇒ ∘′ ⤖⇒↔) Symmetry.⤖-sym public
+    open ↔-syntax rel1 rel2 (K-trans ∘′ ↔⇒) Symmetry.↔-sym public
 
-  -- Isomorphisms and bijections can be combined with any other kind of
-  -- relatedness.
+  -- Combinators with homogeneous relations
+  module _ {a : Level} where
+    open end-syntax (Related {a} k) K-refl public
 
-  _⤖⟨_⟩_ : (A : Set a) → A ⤖ B → B ∼[ k ] C → A ∼[ k ] C
-  A ⤖⟨ A⤖B ⟩ B⇔C = A ∼⟨ ↔⇒ (⤖⇒↔ A⤖B) ⟩ B⇔C
-
-  _↔⟨_⟩_ : (A : Set a) → A ↔ B → B ∼[ k ] C → A ∼[ k ] C
-  A ↔⟨ A↔B ⟩ B⇔C = A ∼⟨ ↔⇒ A↔B ⟩ B⇔C
 
+  infixr 2 _↔⟨⟩_
   _↔⟨⟩_ : (A : Set a) → A ∼[ k ] B → A ∼[ k ] B
   A ↔⟨⟩ A⇔B = A⇔B
-
-  _≡⟨_⟩_ : (A : Set a) → A ≡ B → B ∼[ k ] C → A ∼[ k ] C
-  A ≡⟨ A≡B ⟩ B⇔C = A ∼⟨ ≡⇒ A≡B ⟩ B⇔C
-
-  _≡˘⟨_⟩_ : (A : Set a) → B ≡ A → B ∼[ k ] C → A ∼[ k ] C
-  A ≡˘⟨ B≡A ⟩ B⇔C = A ∼⟨ ≡⇒ (P.sym B≡A) ⟩ B⇔C
-
-  _∎ : (A : Set a) → A ∼[ k ] A
-  A ∎ = K-refl
-
+  {-# WARNING_ON_USAGE _↔⟨⟩_
+  "Warning: _↔⟨⟩_ was deprecated in v2.0.
+  Please use _≡⟨⟩_ instead. "
+  #-}
 
 ------------------------------------------------------------------------
 -- Every unary relation induces a preorder and, for symmetric kinds,
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index f3912de253..c8ebd64b7c 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -17,7 +17,8 @@ open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; refl; sym; cong; cong₂)
 import Relation.Binary.PropositionalEquality.Properties as PropEq
-import Relation.Binary.Reasoning.Preorder as PreR
+import Relation.Binary.Reasoning.Preorder as PreorderReasoning
+open import Relation.Binary.Reasoning.Syntax
 
 ------------------------------------------------------------------------
 -- _◅◅_
@@ -112,17 +113,9 @@ module _ {i t} {I : Set i} (T : Rel I t) where
 -- Preorder reasoning for Star
 
 module StarReasoning {i t} {I : Set i} (T : Rel I t) where
-  private module Base = PreR (preorder T)
+  private module Base = PreorderReasoning (preorder T)
 
-  open Base public
-    hiding (step-≈; step-∼)
+  open Base public hiding (step-≈; step-∼)
 
-  infixr 2 step-⟶ step-⟶⋆
-
-  step-⟶⋆ = Base.step-∼
-
-  step-⟶ : ∀ x {y z} → y IsRelatedTo z → T x y → x IsRelatedTo z
-  step-⟶ x y⟶⋆z x⟶y = step-⟶⋆ x y⟶⋆z (x⟶y ◅ ε)
-
-  syntax step-⟶⋆ x y⟶⋆z x⟶⋆y = x ⟶⋆⟨ x⟶⋆y ⟩ y⟶⋆z
-  syntax step-⟶  x y⟶⋆z x⟶y  = x ⟶⟨ x⟶y ⟩ y⟶⋆z
+  open ⟶-syntax _IsRelatedTo_ _IsRelatedTo_ (Base.∼-go ∘ (_◅ ε)) public
+  open ⟶*-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index fc073add44..4581514feb 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -19,15 +19,16 @@ open import Relation.Unary using (Pred)
 open import Relation.Binary.Core using (Rel; REL; _⇒_)
 open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder)
 open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)
-open import Relation.Binary.Definitions using (Substitutive; Irrelevant; Decidable; _Respects₂_)
+open import Relation.Binary.Definitions using (Substitutive; Irrelevant; Decidable; _Respects₂_; Trans; Reflexive)
 open import Relation.Binary.Consequences
 open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
   using (_atₛ_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.Reasoning.Syntax
 
+import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.HeterogeneousEquality.Core as Core
 
 private
@@ -231,17 +232,30 @@ module ≅-Reasoning where
   -- The code in `Relation.Binary.Reasoning.Setoid` cannot handle
   -- heterogeneous equalities, hence the code duplication here.
 
-  infix  4 _IsRelatedTo_
-  infix  3 _∎
-  infixr 2 _≅⟨_⟩_ _≅˘⟨_⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_ _≡⟨⟩_
-  infix  1 begin_
+  infix 4 _IsRelatedTo_
 
-  data _IsRelatedTo_ {A : Set ℓ} (x : A) {B : Set ℓ} (y : B) :
-                     Set ℓ where
+  data _IsRelatedTo_ {A : Set ℓ} {B : Set ℓ} (x : A) (y : B) : Set ℓ where
     relTo : (x≅y : x ≅ y) → x IsRelatedTo y
 
-  begin_ : ∀ {x : A} {y : B} → x IsRelatedTo y → x ≅ y
-  begin relTo x≅y = x≅y
+  start : ∀ {x : A} {y : B} → x IsRelatedTo y → x ≅ y
+  start (relTo x≅y) = x≅y
+
+  ≡-go : ∀ {A : Set a} → Trans {A = A} {C = A} _≡_ _IsRelatedTo_ _IsRelatedTo_
+  ≡-go x≡y (relTo y≅z) = relTo (trans (reflexive x≡y) y≅z)
+
+  -- Combinators with one heterogeneous relation
+  module _ {A : Set ℓ} {B : Set ℓ} where
+    open begin-syntax (_IsRelatedTo_ {A = A} {B}) start public
+
+  -- Combinators with homogeneous relations
+  module _ {A : Set ℓ} where
+    open ≡-syntax (_IsRelatedTo_ {A = A}) ≡-go public
+    open end-syntax (_IsRelatedTo_ {A = A}) (relTo refl) public
+
+  -- Can't create syntax in the standard `Syntax` module for
+  -- heterogeneous steps because it would force that module to use
+  -- the `--with-k` option.
+  infixr 2 _≅⟨_⟩_ _≅˘⟨_⟩_
 
   _≅⟨_⟩_ : ∀ (x : A) {y : B} {z : C} →
            x ≅ y → y IsRelatedTo z → x IsRelatedTo z
@@ -251,20 +265,6 @@ module ≅-Reasoning where
             y ≅ x → y IsRelatedTo z → x IsRelatedTo z
   _ ≅˘⟨ y≅x ⟩ relTo y≅z = relTo (trans (sym y≅x) y≅z)
 
-  _≡⟨_⟩_ : ∀ (x : A) {y : A} {z : C} →
-           x ≡ y → y IsRelatedTo z → x IsRelatedTo z
-  _ ≡⟨ x≡y ⟩ relTo y≅z = relTo (trans (reflexive x≡y) y≅z)
-
-  _≡˘⟨_⟩_ : ∀ (x : A) {y : A} {z : C} →
-            y ≡ x → y IsRelatedTo z → x IsRelatedTo z
-  _ ≡˘⟨ y≡x ⟩ relTo y≅z = relTo (trans (sym (reflexive y≡x)) y≅z)
-
-  _≡⟨⟩_ : ∀ (x : A) {y : B} → x IsRelatedTo y → x IsRelatedTo y
-  _ ≡⟨⟩ x≅y = x≅y
-
-  _∎ : ∀ (x : A) → x IsRelatedTo x
-  _∎ _ = relTo refl
-
 ------------------------------------------------------------------------
 -- Inspect
 
diff --git a/src/Relation/Binary/Reasoning/Base/Apartness.agda b/src/Relation/Binary/Reasoning/Base/Apartness.agda
index 6e0e239a02..3d4d72f2c8 100644
--- a/src/Relation/Binary/Reasoning/Base/Apartness.agda
+++ b/src/Relation/Binary/Reasoning/Base/Apartness.agda
@@ -7,9 +7,15 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
+open import Level using (Level; _⊔_)
+open import Function using (case_of_)
+open import Relation.Nullary.Decidable using (Dec; yes; no)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Binary.Definitions using (Transitive; Symmetric; Trans)
+open import Relation.Binary.Definitions using (Reflexive; Transitive; Symmetric; Trans)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.Reasoning.Syntax
+
 
 module Relation.Binary.Reasoning.Base.Apartness {a ℓ₁ ℓ₂} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_#_ : Rel A ℓ₂}
@@ -18,18 +24,10 @@ module Relation.Binary.Reasoning.Base.Apartness {a ℓ₁ ℓ₂} {A : Set a}
   (#-≈-trans : Trans _#_ _≈_ _#_) (≈-#-trans : Trans _≈_ _#_ _#_)
   where
 
-open import Level using (Level; _⊔_)
-open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; sym)
-open import Relation.Nullary.Decidable using (Dec; yes; no)
-open import Relation.Nullary.Decidable using (True; toWitness)
+module Eq = IsEquivalence ≈-equiv
 
-open IsEquivalence ≈-equiv
-  renaming
-  ( refl  to ≈-refl
-  ; sym   to ≈-sym
-  ; trans to ≈-trans
-  )
+------------------------------------------------------------------------
+-- A datatype to hide the current relation type
 
 infix 4 _IsRelatedTo_
 
@@ -38,6 +36,27 @@ data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
   apartness : (x#y : x # y) → x IsRelatedTo y
   equals    : (x≈y : x ≈ y) → x IsRelatedTo y
 
+≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
+≡-go x≡y nothing = nothing
+≡-go x≡y (apartness y#z) = apartness (case x≡y of λ where P.refl → y#z)
+≡-go x≡y (equals y≈z) = equals (case x≡y of λ where P.refl → y≈z)
+
+≈-go  : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_
+≈-go x≈y nothing         = nothing
+≈-go x≈y (apartness y#z) = apartness (≈-#-trans x≈y y#z)
+≈-go x≈y (equals    y≈z) = equals    (Eq.trans   x≈y y≈z)
+
+#-go : Trans _#_ _IsRelatedTo_ _IsRelatedTo_
+#-go x#y nothing         = nothing
+#-go x#y (apartness y#z) = nothing
+#-go x#y (equals    y≈z) = apartness (#-≈-trans x#y y≈z)
+
+stop : Reflexive _IsRelatedTo_
+stop = equals Eq.refl
+
+------------------------------------------------------------------------
+-- Apartness subrelation
+
 data IsApartness {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
   isApartness : ∀ x#y → IsApartness (apartness x#y)
 
@@ -49,6 +68,16 @@ IsApartness? (equals x≈y)    = no (λ ())
 extractApartness : ∀ {x y} {x#y : x IsRelatedTo y} → IsApartness x#y → x # y
 extractApartness (isApartness x#y) = x#y
 
+apartnessRelation : SubRelation _IsRelatedTo_ _ _
+apartnessRelation = record
+  { IsS = IsApartness
+  ; IsS? = IsApartness?
+  ; extract = extractApartness
+  }
+
+------------------------------------------------------------------------
+-- Equality subrelation
+
 data IsEquality {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
   isEquality : ∀ x≈y → IsEquality (equals x≈y)
 
@@ -60,49 +89,19 @@ IsEquality? (equals  x≈y) = yes (isEquality x≈y)
 extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y
 extractEquality (isEquality x≈y) = x≈y
 
-infix  1 begin-apartness_ begin-equality_
-infixr 2 step-≈ step-≈˘ step-≡ step-≡˘ step-# step-#˘ _≡⟨⟩_
-infix  3 _∎
-
-begin-apartness_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsApartness? r)} → x # y
-begin-apartness_ _ {s} = extractApartness (toWitness s)
-
-begin-equality_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsEquality? r)} → x ≈ y
-begin-equality_ _ {s} = extractEquality (toWitness s)
-
-step-# : ∀ (x : A) {y z} → y IsRelatedTo z → x # y → x IsRelatedTo z
-step-# x nothing  _          = nothing
-step-# x (apartness y#z) x#y = nothing
-step-# x (equals    y≈z) x#y = apartness (#-≈-trans x#y y≈z)
-
-step-#˘ : ∀ (x : A) {y z} → y IsRelatedTo z → y # x → x IsRelatedTo z
-step-#˘ x y-z y#x = step-# x y-z (#-sym y#x)
-
-step-≈  : ∀ (x : A) {y z} → y IsRelatedTo z → x ≈ y → x IsRelatedTo z
-step-≈ x nothing         x≈y = nothing
-step-≈ x (apartness y#z) x≈y = apartness (≈-#-trans x≈y y#z)
-step-≈ x (equals    y≈z) x≈y = equals    (≈-trans   x≈y y≈z)
+eqRelation : SubRelation _IsRelatedTo_ _ _
+eqRelation = record
+  { IsS = IsEquality
+  ; IsS? = IsEquality?
+  ; extract = extractEquality
+  }
 
-step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z
-step-≈˘ x y∼z x≈y = step-≈ x y∼z (≈-sym x≈y)
-
-step-≡ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z
-step-≡ x nothing _            = nothing
-step-≡ x (apartness x#y) refl = apartness x#y
-step-≡ x (equals    x≈y) refl = equals    x≈y
-
-step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z
-step-≡˘ x y∼z x≡y = step-≡ x y∼z (sym x≡y)
-
-_≡⟨⟩_ : ∀ (x : A) {y} → x IsRelatedTo y → x IsRelatedTo y
-x ≡⟨⟩ x-y = x-y
-
-_∎ : ∀ x → x IsRelatedTo x
-x ∎ = equals ≈-refl
-
-syntax step-#  x y∼z x#y = x #⟨  x#y ⟩ y∼z
-syntax step-#˘ x y∼z x#y = x #˘⟨ x#y ⟩ y∼z
-syntax step-≈  x y∼z x≈y = x ≈⟨  x≈y ⟩ y∼z
-syntax step-≈˘ x y∼z y≈x = x ≈˘⟨ y≈x ⟩ y∼z
-syntax step-≡  x y∼z x≡y = x ≡⟨  x≡y ⟩ y∼z
-syntax step-≡˘ x y∼z y≡x = x ≡˘⟨ y≡x ⟩ y∼z
+------------------------------------------------------------------------
+-- Reasoning combinators
+
+open begin-apartness-syntax _IsRelatedTo_ apartnessRelation public
+open begin-equality-syntax _IsRelatedTo_ eqRelation public
+open ≡-syntax _IsRelatedTo_ ≡-go public
+open #-syntax _IsRelatedTo_ _IsRelatedTo_ #-go #-sym public
+open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go Eq.sym public
+open end-syntax _IsRelatedTo_ stop public
diff --git a/src/Relation/Binary/Reasoning/Base/Double.agda b/src/Relation/Binary/Reasoning/Base/Double.agda
index aa61052b0c..91203437f6 100644
--- a/src/Relation/Binary/Reasoning/Base/Double.agda
+++ b/src/Relation/Binary/Reasoning/Base/Double.agda
@@ -10,21 +10,20 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Core using (Rel)
+open import Level using (_⊔_)
+open import Function using (case_of_)
+open import Relation.Nullary.Decidable.Core using (Dec; yes; no)
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Definitions using (Reflexive; Trans)
 open import Relation.Binary.Structures using (IsPreorder)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.Reasoning.Syntax
+
 
 module Relation.Binary.Reasoning.Base.Double {a ℓ₁ ℓ₂} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} (isPreorder : IsPreorder _≈_ _∼_)
   where
 
-open import Data.Product.Base using (proj₁; proj₂)
-open import Level using (Level; _⊔_; Lift; lift)
-open import Function.Base using (case_of_; id)
-open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; sym)
-open import Relation.Nullary.Decidable.Core
-  using (Dec; yes; no; True; toWitness)
-
 open IsPreorder isPreorder
 
 ------------------------------------------------------------------------
@@ -36,6 +35,25 @@ data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
   nonstrict : (x∼y : x ∼ y) → x IsRelatedTo y
   equals    : (x≈y : x ≈ y) → x IsRelatedTo y
 
+start : _IsRelatedTo_ ⇒ _∼_
+start (equals x≈y) = reflexive x≈y
+start (nonstrict x∼y) = x∼y
+
+≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
+≡-go x≡y (equals y≈z) = equals (case x≡y of λ where P.refl → y≈z)
+≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where P.refl → y≤z)
+
+∼-go : Trans _∼_ _IsRelatedTo_ _IsRelatedTo_
+∼-go x∼y (equals y≈z) = nonstrict (∼-respʳ-≈ y≈z x∼y)
+∼-go x∼y (nonstrict y∼z) = nonstrict (trans x∼y y∼z)
+
+≈-go : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_
+≈-go x≈y (equals y≈z) = equals (Eq.trans x≈y y≈z)
+≈-go x≈y (nonstrict y∼z) = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y∼z)
+
+stop : Reflexive _IsRelatedTo_
+stop = equals Eq.refl
+
 ------------------------------------------------------------------------
 -- A record that is used to ensure that the final relation proved by the
 -- chain of reasoning can be converted into the required relation.
@@ -50,67 +68,19 @@ IsEquality? (equals x≈y)  = yes (isEquality x≈y)
 extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y
 extractEquality (isEquality x≈y) = x≈y
 
+equalitySubRelation : SubRelation  _IsRelatedTo_ _ _
+equalitySubRelation = record
+  { IsS = IsEquality
+  ; IsS? = IsEquality?
+  ; extract = extractEquality
+  }
+
 ------------------------------------------------------------------------
 -- Reasoning combinators
 
--- See `Relation.Binary.Reasoning.Base.Partial` for the design decisions
--- behind these combinators.
-
-infix  1 begin_ begin-equality_
-infixr 2 step-∼ step-≈ step-≈˘ step-≡ step-≡˘ _≡⟨⟩_
-infix  3 _∎
-
--- Beginnings of various types of proofs
-
-begin_ : ∀ {x y} (r : x IsRelatedTo y) → x ∼ y
-begin (nonstrict x∼y) = x∼y
-begin (equals    x≈y) = reflexive x≈y
-
-begin-equality_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsEquality? r)} → x ≈ y
-begin-equality_ r {s} = extractEquality (toWitness s)
-
--- Step with the main relation
-
-step-∼ : ∀ (x : A) {y z} → y IsRelatedTo z → x ∼ y → x IsRelatedTo z
-step-∼ x (nonstrict y∼z) x∼y = nonstrict (trans x∼y y∼z)
-step-∼ x (equals    y≈z) x∼y = nonstrict (∼-respʳ-≈ y≈z x∼y)
-
--- Step with the setoid equality
-
-step-≈ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≈ y → x IsRelatedTo z
-step-≈ x (nonstrict y∼z) x≈y = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y∼z)
-step-≈ x (equals    y≈z) x≈y = equals    (Eq.trans x≈y y≈z)
-
--- Flipped step with the setoid equality
-
-step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z
-step-≈˘ x y∼z x≈y = step-≈ x y∼z (Eq.sym x≈y)
-
--- Step with non-trivial propositional equality
-
-step-≡ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z
-step-≡ x (nonstrict y∼z) x≡y = nonstrict (case x≡y of λ where refl → y∼z)
-step-≡ x (equals    y≈z) x≡y = equals    (case x≡y of λ where refl → y≈z)
-
--- Flipped step with non-trivial propositional equality
-
-step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z
-step-≡˘ x y∼z x≡y = step-≡ x y∼z (sym x≡y)
-
--- Step with trivial propositional equality
-
-_≡⟨⟩_ : ∀ (x : A) {y} → x IsRelatedTo y → x IsRelatedTo y
-x ≡⟨⟩ x≲y = x≲y
-
--- Termination step
-
-_∎ : ∀ x → x IsRelatedTo x
-x ∎ = equals Eq.refl
-
--- Syntax declarations
-
-syntax step-∼  x y∼z x∼y = x ∼⟨  x∼y ⟩ y∼z
-syntax step-≈  x y∼z x≈y = x ≈⟨  x≈y ⟩ y∼z
-syntax step-≈˘ x y∼z y≈x = x ≈˘⟨ y≈x ⟩ y∼z
-syntax step-≡  x y∼z x≡y = x ≡⟨  x≡y ⟩ y∼z
-syntax step-≡˘ x y∼z y≡x = x ≡˘⟨ y≡x ⟩ y∼z
+open begin-syntax  _IsRelatedTo_ start public
+open begin-equality-syntax  _IsRelatedTo_ equalitySubRelation public
+open ≡-syntax _IsRelatedTo_ ≡-go public
+open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public
+open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go Eq.sym public
+open end-syntax _IsRelatedTo_ stop public
diff --git a/src/Relation/Binary/Reasoning/Base/Partial.agda b/src/Relation/Binary/Reasoning/Base/Partial.agda
index e53f7e4efe..bd1fcd8693 100644
--- a/src/Relation/Binary/Reasoning/Base/Partial.agda
+++ b/src/Relation/Binary/Reasoning/Base/Partial.agda
@@ -6,13 +6,14 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
+open import Function using (case_of_)
+open import Level using (_⊔_)
 open import Relation.Binary.Core
 open import Relation.Binary.Definitions
-open import Level using (_⊔_)
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_)
 open import Relation.Nullary.Decidable using (Dec; yes; no)
-open import Relation.Nullary.Decidable using (True)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.Reasoning.Syntax
+
 
 module Relation.Binary.Reasoning.Base.Partial
   {a ℓ} {A : Set a} (_∼_ : Rel A ℓ) (trans : Transitive _∼_)
@@ -30,6 +31,13 @@ data _IsRelatedTo_ : A → A → Set (a ⊔ ℓ) where
   singleStep : ∀ x → x IsRelatedTo x
   multiStep  : ∀ {x y} (x∼y : x ∼ y) → x IsRelatedTo y
 
+∼-go : Trans _∼_ _IsRelatedTo_ _IsRelatedTo_
+∼-go x∼y (singleStep y) = multiStep x∼y
+∼-go x∼y (multiStep y∼z) = multiStep (trans x∼y y∼z)
+
+stop : Reflexive _IsRelatedTo_
+stop = singleStep _
+
 ------------------------------------------------------------------------
 -- Types that are used to ensure that the final relation proved by the
 -- chain of reasoning can be converted into the required relation.
@@ -41,61 +49,23 @@ IsMultiStep? : ∀ {x y} (x∼y : x IsRelatedTo y) → Dec (IsMultiStep x∼y)
 IsMultiStep? (multiStep x<y) = yes (isMultiStep x<y)
 IsMultiStep? (singleStep _)  = no λ()
 
-------------------------------------------------------------------------
--- Reasoning combinators
-
--- Note that the arguments to the `step`s are not provided in their
--- "natural" order and syntax declarations are later used to re-order
--- them. This is because the `step` ordering allows the type-checker to
--- better infer the middle argument `y` from the `_IsRelatedTo_`
--- argument (see issue 622).
---
--- This has two practical benefits. First it speeds up type-checking by
--- approximately a factor of 5. Secondly it allows the combinators to be
--- used with macros that use reflection, e.g. `Tactic.RingSolver`, where
--- they need to be able to extract `y` using reflection.
-
-infix  1 begin_
-infixr 2 step-∼ step-≡ step-≡˘
-infixr 2 _≡⟨⟩_
-infix  3 _∎⟨_⟩ _∎
-
--- Beginning of a proof
-
-begin_ : ∀ {x y} (x∼y : x IsRelatedTo y) → {True (IsMultiStep? x∼y)} → x ∼ y
-begin (multiStep x∼y) = x∼y
+extractMultiStep : ∀ {x y} {x∼y : x IsRelatedTo y} → IsMultiStep x∼y → x ∼ y
+extractMultiStep (isMultiStep x≈y) = x≈y
 
--- Standard step with the relation
+multiStepSubRelation : SubRelation _IsRelatedTo_ _ _
+multiStepSubRelation = record
+  { IsS = IsMultiStep
+  ; IsS? = IsMultiStep?
+  ; extract = extractMultiStep
+  }
 
-step-∼ : ∀ x {y z} → y IsRelatedTo z → x ∼ y → x IsRelatedTo z
-step-∼ _ (singleStep _) x∼y  = multiStep x∼y
-step-∼ _ (multiStep y∼z) x∼y = multiStep (trans x∼y y∼z)
-
--- Step with a non-trivial propositional equality
-
-step-≡ : ∀ x {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z
-step-≡ _ x∼z P.refl = x∼z
-
--- Step with a flipped non-trivial propositional equality
-
-step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z
-step-≡˘ _ x∼z P.refl = x∼z
-
--- -- Step with a trivial propositional equality
-
-_≡⟨⟩_ : ∀ x {y} → x IsRelatedTo y → x IsRelatedTo y
-_ ≡⟨⟩ x∼y = x∼y
-
--- Termination
-
-_∎ : ∀ x → x IsRelatedTo x
-_∎ = singleStep
-
--- Syntax declarations
+------------------------------------------------------------------------
+-- Reasoning combinators
 
-syntax step-∼  x y∼z x∼y = x ∼⟨  x∼y ⟩ y∼z
-syntax step-≡  x y≡z x≡y = x ≡⟨  x≡y ⟩ y≡z
-syntax step-≡˘ x y≡z y≡x = x ≡˘⟨ y≡x ⟩ y≡z
+open begin-subrelation-syntax _IsRelatedTo_ multiStepSubRelation public
+open ≡-noncomputing-syntax _IsRelatedTo_ public
+open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public
+open end-syntax _IsRelatedTo_ stop public
 
 
 ------------------------------------------------------------------------
@@ -106,6 +76,8 @@ syntax step-≡˘ x y≡z y≡x = x ≡˘⟨ y≡x ⟩ y≡z
 
 -- Version 1.6
 
+infix  3 _∎⟨_⟩
+
 _∎⟨_⟩ : ∀ x → x ∼ x → x IsRelatedTo x
 _ ∎⟨ x∼x ⟩ = multiStep x∼x
 {-# WARNING_ON_USAGE _∎⟨_⟩
diff --git a/src/Relation/Binary/Reasoning/Base/Single.agda b/src/Relation/Binary/Reasoning/Base/Single.agda
index 43b553c7d1..ccbf63dda3 100644
--- a/src/Relation/Binary/Reasoning/Base/Single.agda
+++ b/src/Relation/Binary/Reasoning/Base/Single.agda
@@ -6,84 +6,45 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.Definitions using (Reflexive; Transitive)
+open import Level using (_⊔_)
+open import Function.Base using (case_of_)
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Definitions using (Reflexive; Transitive; Trans)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.Reasoning.Syntax
 
 module Relation.Binary.Reasoning.Base.Single
   {a ℓ} {A : Set a} (_∼_ : Rel A ℓ)
   (refl : Reflexive _∼_) (trans : Transitive _∼_)
   where
 
--- TODO: the following part is copied from Relation.Binary.Reasoning.Base.Partial
--- in order to avoid larger refactors. We will refactor this part later
--- so taht we use the same framework as Relation.Binary.Reasoning.Base.Partial.
-
-open import Level using (_⊔_)
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_)
-
-infix  4 _IsRelatedTo_
-
 ------------------------------------------------------------------------
 -- Definition of "related to"
 
 -- This seemingly unnecessary type is used to make it possible to
 -- infer arguments even if the underlying equality evaluates.
 
+infix 4 _IsRelatedTo_
+
 data _IsRelatedTo_ (x y : A) : Set ℓ where
   relTo : (x∼y : x ∼ y) → x IsRelatedTo y
 
-------------------------------------------------------------------------
--- Reasoning combinators
-
--- Note that the arguments to the `step`s are not provided in their
--- "natural" order and syntax declarations are later used to re-order
--- them. This is because the `step` ordering allows the type-checker to
--- better infer the middle argument `y` from the `_IsRelatedTo_`
--- argument (see issue 622).
---
--- This has two practical benefits. First it speeds up type-checking by
--- approximately a factor of 5. Secondly it allows the combinators to be
--- used with macros that use reflection, e.g. `Tactic.RingSolver`, where
--- they need to be able to extract `y` using reflection.
-
-infix  1 begin_
-infixr 2 step-∼ step-≡ step-≡˘
-infixr 2 _≡⟨⟩_
-infix  3 _∎
+start : _IsRelatedTo_ ⇒ _∼_
+start (relTo x∼y) = x∼y
 
--- Beginning of a proof
+∼-go : Trans _∼_ _IsRelatedTo_ _IsRelatedTo_
+∼-go x∼y (relTo y∼z) = relTo (trans x∼y y∼z)
 
-begin_ : ∀ {x y} → x IsRelatedTo y → x ∼ y
-begin relTo x∼y = x∼y
+≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
+≡-go x≡y (relTo y∼z) = relTo (case x≡y of λ where P.refl → y∼z)
 
--- Standard step with the relation
+stop : Reflexive _IsRelatedTo_
+stop = relTo refl
 
-step-∼ : ∀ x {y z} → y IsRelatedTo z → x ∼ y → x IsRelatedTo z
-step-∼ _ (relTo y∼z) x∼y = relTo (trans x∼y y∼z)
-
--- Step with a non-trivial propositional equality
-
-step-≡ : ∀ x {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z
-step-≡ _ x∼z P.refl = x∼z
-
--- Step with a flipped non-trivial propositional equality
-
-step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z
-step-≡˘ _ x∼z P.refl = x∼z
-
--- Step with a trivial propositional equality
-
-_≡⟨⟩_ : ∀ x {y} → x IsRelatedTo y → x IsRelatedTo y
-_ ≡⟨⟩ x∼y = x∼y
-
--- Termination
-
-_∎ : ∀ x → x IsRelatedTo x
-x ∎ = relTo refl
-
--- Syntax declarations
+------------------------------------------------------------------------
+-- Reasoning combinators
 
-syntax step-∼  x y∼z x∼y = x ∼⟨  x∼y ⟩ y∼z
-syntax step-≡  x y≡z x≡y = x ≡⟨  x≡y ⟩ y≡z
-syntax step-≡˘ x y≡z y≡x = x ≡˘⟨ y≡x ⟩ y≡z
+open begin-syntax _IsRelatedTo_ start public
+open ≡-syntax _IsRelatedTo_ ≡-go public
+open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public
+open end-syntax _IsRelatedTo_ stop public
diff --git a/src/Relation/Binary/Reasoning/Base/Triple.agda b/src/Relation/Binary/Reasoning/Base/Triple.agda
index 75aeaaef17..460115eff9 100644
--- a/src/Relation/Binary/Reasoning/Base/Triple.agda
+++ b/src/Relation/Binary/Reasoning/Base/Triple.agda
@@ -10,35 +10,27 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
+open import Data.Product.Base using (proj₁; proj₂)
+open import Level using (_⊔_)
+open import Function using (case_of_)
+open import Relation.Nullary.Decidable.Core
+  using (Dec; yes; no)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Structures using (IsPreorder)
 open import Relation.Binary.Definitions
-  using (Transitive; _Respects₂_; Trans; Irreflexive)
+  using (Transitive; _Respects₂_; Reflexive; Trans; Irreflexive; Asymmetric)
+open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.Reasoning.Syntax
 
 module Relation.Binary.Reasoning.Base.Triple {a ℓ₁ ℓ₂ ℓ₃} {A : Set a}
   {_≈_ : Rel A ℓ₁} {_≤_ : Rel A ℓ₂} {_<_ : Rel A ℓ₃}
   (isPreorder : IsPreorder _≈_ _≤_)
-  (<-irrefl : Irreflexive _≈_ _<_) (<-trans : Transitive _<_) (<-resp-≈ : _<_ Respects₂ _≈_)
+  (<-asym : Asymmetric _<_) (<-trans : Transitive _<_) (<-resp-≈ : _<_ Respects₂ _≈_)
   (<⇒≤ : _<_ ⇒ _≤_)
   (<-≤-trans : Trans _<_ _≤_ _<_) (≤-<-trans : Trans _≤_ _<_ _<_)
   where
 
-open import Data.Product.Base using (proj₁; proj₂)
-open import Function.Base using (case_of_; id)
-open import Level using (Level; _⊔_; Lift; lift)
-open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; sym)
-open import Relation.Nullary using (¬_)
-open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Nullary.Decidable.Core
-  using (Dec; yes; no; True; toWitness)
-
 open IsPreorder isPreorder
-  renaming
-  ( reflexive to ≤-reflexive
-  ; trans     to ≤-trans
-  ; ∼-resp-≈  to ≤-resp-≈
-  )
 
 ------------------------------------------------------------------------
 -- A datatype to abstract over the current relation
@@ -50,6 +42,35 @@ data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where
   nonstrict : (x≤y : x ≤ y) → x IsRelatedTo y
   equals    : (x≈y : x ≈ y) → x IsRelatedTo y
 
+start : _IsRelatedTo_ ⇒ _≤_
+start (equals x≈y) = reflexive x≈y
+start (nonstrict x≤y) = x≤y
+start (strict x<y) = <⇒≤ x<y
+
+≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
+≡-go x≡y (equals y≈z) = equals (case x≡y of λ where P.refl → y≈z)
+≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where P.refl → y≤z)
+≡-go x≡y (strict y<z) = strict (case x≡y of λ where P.refl → y<z)
+
+≈-go : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_
+≈-go x≈y (equals y≈z) = equals (Eq.trans x≈y y≈z)
+≈-go x≈y (nonstrict y≤z) = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y≤z)
+≈-go x≈y (strict y<z) = strict (proj₂ <-resp-≈ (Eq.sym x≈y) y<z)
+
+≤-go : Trans _≤_ _IsRelatedTo_ _IsRelatedTo_
+≤-go x≤y (equals y≈z) = nonstrict (∼-respʳ-≈ y≈z x≤y)
+≤-go x≤y (nonstrict y≤z) = nonstrict (trans x≤y y≤z)
+≤-go x≤y (strict y<z) = strict (≤-<-trans x≤y y<z)
+
+<-go : Trans _<_ _IsRelatedTo_ _IsRelatedTo_
+<-go x<y (equals y≈z) = strict (proj₁ <-resp-≈ y≈z x<y)
+<-go x<y (nonstrict y≤z) = strict (<-≤-trans x<y y≤z)
+<-go x<y (strict y<z) = strict (<-trans x<y y<z)
+
+stop : Reflexive _IsRelatedTo_
+stop = equals Eq.refl
+
+
 ------------------------------------------------------------------------
 -- Types that are used to ensure that the final relation proved by the
 -- chain of reasoning can be converted into the required relation.
@@ -65,6 +86,16 @@ IsStrict? (equals    _)   = no λ()
 extractStrict : ∀ {x y} {x≲y : x IsRelatedTo y} → IsStrict x≲y → x < y
 extractStrict (isStrict x<y) = x<y
 
+strictRelation : SubRelation _IsRelatedTo_ _ _
+strictRelation = record
+  { IsS = IsStrict
+  ; IsS? = IsStrict?
+  ; extract = extractStrict
+  }
+
+------------------------------------------------------------------------
+-- Equality sub-relation
+
 data IsEquality {x y} : x IsRelatedTo y → Set (a ⊔ ℓ₁ ⊔ ℓ₂ ⊔ ℓ₃) where
   isEquality : ∀ x≈y → IsEquality (equals x≈y)
 
@@ -76,89 +107,22 @@ IsEquality? (equals x≈y)  = yes (isEquality x≈y)
 extractEquality : ∀ {x y} {x≲y : x IsRelatedTo y} → IsEquality x≲y → x ≈ y
 extractEquality (isEquality x≈y) = x≈y
 
+eqRelation : SubRelation _IsRelatedTo_ _ _
+eqRelation = record
+  { IsS = IsEquality
+  ; IsS? = IsEquality?
+  ; extract = extractEquality
+  }
+
 ------------------------------------------------------------------------
 -- Reasoning combinators
 
--- See `Relation.Binary.Reasoning.Base.Partial` for the design decisions
--- behind these combinators.
-
-infix  1 begin_ begin-strict_ begin-equality_ begin-contradiction_
-infixr 2 step-< step-≤ step-≈ step-≈˘ step-≡ step-≡˘ _≡⟨⟩_
-infix  3 _∎
-
--- Beginnings of various types of proofs
-
-begin_ : ∀ {x y} → x IsRelatedTo y → x ≤ y
-begin (strict    x<y) = <⇒≤ x<y
-begin (nonstrict x≤y) = x≤y
-begin (equals    x≈y) = ≤-reflexive x≈y
-
-begin-strict_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsStrict? r)} → x < y
-begin-strict_ r {s} = extractStrict (toWitness s)
-
-begin-equality_ : ∀ {x y} (r : x IsRelatedTo y) → {s : True (IsEquality? r)} → x ≈ y
-begin-equality_ r {s} = extractEquality (toWitness s)
-
-begin-contradiction_ : ∀ {x} (r : x IsRelatedTo x) {s : True (IsStrict? r)} →
-                      ∀ {a} {A : Set a} → A
-begin-contradiction_ {x} r {s} = contradiction x<x (<-irrefl Eq.refl)
-  where
-  x<x : x < x
-  x<x = extractStrict (toWitness s)
-
--- Step with the strict relation
-
-step-< : ∀ (x : A) {y z} → y IsRelatedTo z → x < y → x IsRelatedTo z
-step-< x (strict    y<z) x<y = strict (<-trans x<y y<z)
-step-< x (nonstrict y≤z) x<y = strict (<-≤-trans x<y y≤z)
-step-< x (equals    y≈z) x<y = strict (proj₁ <-resp-≈ y≈z x<y)
-
--- Step with the non-strict relation
-
-step-≤ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≤ y → x IsRelatedTo z
-step-≤ x (strict    y<z) x≤y = strict    (≤-<-trans x≤y y<z)
-step-≤ x (nonstrict y≤z) x≤y = nonstrict (≤-trans x≤y y≤z)
-step-≤ x (equals    y≈z) x≤y = nonstrict (proj₁ ≤-resp-≈ y≈z x≤y)
-
--- Step with the setoid equality
-
-step-≈  : ∀ (x : A) {y z} → y IsRelatedTo z → x ≈ y → x IsRelatedTo z
-step-≈ x (strict    y<z) x≈y = strict    (proj₂ <-resp-≈ (Eq.sym x≈y) y<z)
-step-≈ x (nonstrict y≤z) x≈y = nonstrict (proj₂ ≤-resp-≈ (Eq.sym x≈y) y≤z)
-step-≈ x (equals    y≈z) x≈y = equals    (Eq.trans x≈y y≈z)
-
--- Flipped step with the setoid equality
-
-step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z
-step-≈˘ x y∼z x≈y = step-≈ x y∼z (Eq.sym x≈y)
-
--- Step with non-trivial propositional equality
-
-step-≡ : ∀ (x : A) {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z
-step-≡ x (strict    y<z) x≡y  = strict    (case x≡y of λ where refl → y<z)
-step-≡ x (nonstrict y≤z) x≡y  = nonstrict (case x≡y of λ where refl → y≤z)
-step-≡ x (equals    y≈z) x≡y  = equals    (case x≡y of λ where refl → y≈z)
-
--- Flipped step with non-trivial propositional equality
-
-step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z
-step-≡˘ x y∼z x≡y = step-≡ x y∼z (sym x≡y)
-
--- Step with trivial propositional equality
-
-_≡⟨⟩_ : ∀ (x : A) {y} → x IsRelatedTo y → x IsRelatedTo y
-x ≡⟨⟩ x≲y = x≲y
-
--- Termination step
-
-_∎ : ∀ x → x IsRelatedTo x
-x ∎ = equals Eq.refl
-
--- Syntax declarations
-
-syntax step-<  x y∼z x<y = x <⟨  x<y ⟩ y∼z
-syntax step-≤  x y∼z x≤y = x ≤⟨  x≤y ⟩ y∼z
-syntax step-≈  x y∼z x≈y = x ≈⟨  x≈y ⟩ y∼z
-syntax step-≈˘ x y∼z y≈x = x ≈˘⟨ y≈x ⟩ y∼z
-syntax step-≡  x y∼z x≡y = x ≡⟨  x≡y ⟩ y∼z
-syntax step-≡˘ x y∼z y≡x = x ≡˘⟨ y≡x ⟩ y∼z
+open begin-syntax _IsRelatedTo_ start public
+open begin-equality-syntax _IsRelatedTo_ eqRelation public
+open begin-strict-syntax _IsRelatedTo_ strictRelation public
+open begin-contradiction-syntax _IsRelatedTo_ strictRelation <-asym public
+open ≡-syntax _IsRelatedTo_ ≡-go public
+open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go Eq.sym public
+open ≤-syntax _IsRelatedTo_ _IsRelatedTo_ ≤-go public
+open <-syntax _IsRelatedTo_ _IsRelatedTo_ <-go public
+open end-syntax _IsRelatedTo_ stop public
diff --git a/src/Relation/Binary/Reasoning/MultiSetoid.agda b/src/Relation/Binary/Reasoning/MultiSetoid.agda
index 194e7ae410..aff53dd984 100644
--- a/src/Relation/Binary/Reasoning/MultiSetoid.agda
+++ b/src/Relation/Binary/Reasoning/MultiSetoid.agda
@@ -26,56 +26,51 @@
 
 module Relation.Binary.Reasoning.MultiSetoid where
 
-open import Function.Base using (flip)
-open import Level using (_⊔_)
+open import Level using (Level; _⊔_)
+open import Function.Base using (case_of_)
+open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Definitions using (Trans; Reflexive)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.Reasoning.Syntax
 
-import Relation.Binary.Reasoning.Setoid as EqR
+private
+  variable
+    a ℓ : Level
 
 ------------------------------------------------------------------------
 -- Combinators that take the current setoid as an explicit argument.
 
-module _ {c ℓ} (S : Setoid c ℓ) where
+module _ (S : Setoid a ℓ) where
   open Setoid S
 
-  data IsRelatedTo (x y : _) : Set (c ⊔ ℓ) where
-    relTo : (x∼y : x ≈ y) → IsRelatedTo x y
+  data IsRelatedTo (x y : _) : Set (a ⊔ ℓ) where
+    relTo : (x≈y : x ≈ y) → IsRelatedTo x y
 
-  infix 1 begin⟨_⟩_
+  start : IsRelatedTo ⇒ _≈_
+  start (relTo x≈y) = x≈y
 
-  begin⟨_⟩_ : ∀ {x y} → IsRelatedTo x y → x ≈ y
-  begin⟨_⟩_ (relTo eq) = eq
+  ≡-go : Trans _≡_ IsRelatedTo IsRelatedTo
+  ≡-go x≡y (relTo y∼z) = relTo (case x≡y of λ where P.refl → y∼z)
 
-------------------------------------------------------------------------
--- Combinators that take the current setoid as an implicit argument.
-
-module _ {c ℓ} {S : Setoid c ℓ} where
-
-  open Setoid S renaming (_≈_ to _≈_)
-
-  infixr 2 step-≈ step-≈˘ step-≡ step-≡˘ _≡⟨⟩_
-  infix 3 _∎
+  ≈-go : Trans _≈_ IsRelatedTo IsRelatedTo
+  ≈-go x≈y (relTo y≈z) = relTo (trans x≈y y≈z)
 
-  step-≈ : ∀ x {y z} → IsRelatedTo S y z → x ≈ y → IsRelatedTo S x z
-  step-≈ x (relTo y∼z) x∼y = relTo (trans x∼y y∼z)
+  end : Reflexive IsRelatedTo
+  end = relTo refl
 
-  step-≈˘ : ∀ x {y z} → IsRelatedTo S y z → y ≈ x → IsRelatedTo S x z
-  step-≈˘ x y∼z x≈y = step-≈ x y∼z (sym x≈y)
-
-  step-≡ : ∀ x {y z} → IsRelatedTo S y z → x ≡ y → IsRelatedTo S x z
-  step-≡ _ x∼z P.refl = x∼z
+------------------------------------------------------------------------
+-- Reasoning combinators
 
-  step-≡˘ : ∀ x {y z} → IsRelatedTo S y z → y ≡ x → IsRelatedTo S x z
-  step-≡˘ _ x∼z P.refl = x∼z
+-- Those that take the current setoid as an explicit argument.
+  open begin-syntax IsRelatedTo start public
+    renaming (begin_ to begin⟨_⟩_)
 
-  _≡⟨⟩_ : ∀ x {y} → IsRelatedTo S x y → IsRelatedTo S x y
-  _ ≡⟨⟩ x∼y = x∼y
 
-  _∎ : ∀ x → IsRelatedTo S x x
-  _∎ _ = relTo refl
+-- Those that take the current setoid as an implicit argument.
+module _ {S : Setoid a ℓ} where
+  open Setoid S
 
-  syntax step-≈  x y∼z x≈y = x ≈⟨  x≈y ⟩ y∼z
-  syntax step-≈˘ x y∼z y≈x = x ≈˘⟨ y≈x ⟩ y∼z
-  syntax step-≡  x y≡z x≡y = x ≡⟨  x≡y ⟩ y≡z
-  syntax step-≡˘ x y≡z y≡x = x ≡˘⟨ y≡x ⟩ y≡z
+  open ≡-syntax (IsRelatedTo S) (≡-go S)
+  open ≈-syntax (IsRelatedTo S) (IsRelatedTo S) (≈-go S) sym public
+  open end-syntax (IsRelatedTo S) (end S) public
diff --git a/src/Relation/Binary/Reasoning/PartialOrder.agda b/src/Relation/Binary/Reasoning/PartialOrder.agda
index a6cb213e1a..9dbf2fc712 100644
--- a/src/Relation/Binary/Reasoning/PartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/PartialOrder.agda
@@ -54,7 +54,7 @@ open import Relation.Binary.Construct.NonStrictToStrict _≈_ _≤_
 
 open import Relation.Binary.Reasoning.Base.Triple
   isPreorder
-  Strict.<-irrefl
+  (Strict.<-asym antisym)
   (Strict.<-trans isPartialOrder)
   (Strict.<-resp-≈ isEquivalence ≤-resp-≈)
   Strict.<⇒≤
diff --git a/src/Relation/Binary/Reasoning/PartialSetoid.agda b/src/Relation/Binary/Reasoning/PartialSetoid.agda
index e679474016..1149f1a298 100644
--- a/src/Relation/Binary/Reasoning/PartialSetoid.agda
+++ b/src/Relation/Binary/Reasoning/PartialSetoid.agda
@@ -7,31 +7,22 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary.Bundles using (PartialSetoid)
+open import Relation.Binary.Reasoning.Syntax
 
 module Relation.Binary.Reasoning.PartialSetoid
   {s₁ s₂} (S : PartialSetoid s₁ s₂) where
 
 open PartialSetoid S
-import Relation.Binary.Reasoning.Base.Partial _≈_ trans as Base
 
-------------------------------------------------------------------------
--- Re-export the contents of the base module
-
-open Base public
-  hiding (step-∼)
+import Relation.Binary.Reasoning.Base.Partial _≈_ trans
+  as PartialReasoning
 
 ------------------------------------------------------------------------
--- Additional reasoning combinators
-
-infixr 2 step-≈ step-≈˘
-
--- A step using an equality
-
-step-≈ = Base.step-∼
-syntax step-≈ x y≈z x≈y = x ≈⟨ x≈y ⟩ y≈z
+-- Reasoning combinators
 
--- A step using a symmetric equality
+-- Export the combinators for partial relation reasoning, hiding the
+-- single misnamed combinator.
+open PartialReasoning public hiding (step-∼)
 
-step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z
-step-≈˘ x y∼z y≈x = x ≈⟨ sym y≈x ⟩ y∼z
-syntax step-≈˘ x y≈z y≈x = x ≈˘⟨ y≈x ⟩ y≈z
+-- Re-export the equality-based combinators instead
+open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ PartialReasoning.∼-go sym public
diff --git a/src/Relation/Binary/Reasoning/Setoid.agda b/src/Relation/Binary/Reasoning/Setoid.agda
index 5b731767eb..bce0a3075c 100644
--- a/src/Relation/Binary/Reasoning/Setoid.agda
+++ b/src/Relation/Binary/Reasoning/Setoid.agda
@@ -21,26 +21,21 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Reasoning.Syntax
 
 module Relation.Binary.Reasoning.Setoid {s₁ s₂} (S : Setoid s₁ s₂) where
 
 open Setoid S
 
+import Relation.Binary.Reasoning.Base.Single _≈_ refl trans
+  as SingleRelReasoning
+
 ------------------------------------------------------------------------
 -- Reasoning combinators
 
-open import Relation.Binary.Reasoning.Base.Single _≈_ refl trans as Base public
-  hiding (step-∼)
-
-infixr 2 step-≈ step-≈˘
-
--- A step using an equality
-
-step-≈ = Base.step-∼
-syntax step-≈ x y≈z x≈y = x ≈⟨ x≈y ⟩ y≈z
-
--- A step using a symmetric equality
+-- Export the combinators for single relation reasoning, hiding the
+-- single misnamed combinator.
+open SingleRelReasoning public hiding (step-∼)
 
-step-≈˘ : ∀ x {y z} → y IsRelatedTo z → y ≈ x → x IsRelatedTo z
-step-≈˘ x y∼z y≈x = x ≈⟨ sym y≈x ⟩ y∼z
-syntax step-≈˘ x y≈z y≈x = x ≈˘⟨ y≈x ⟩ y≈z
+-- Re-export the equality-based combinators instead
+open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ SingleRelReasoning.∼-go sym public
diff --git a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
index 71511d5192..2317ba4216 100644
--- a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
@@ -53,7 +53,7 @@ import Relation.Binary.Construct.StrictToNonStrict _≈_ _<_ as NonStrict
 
 open import Relation.Binary.Reasoning.Base.Triple
   (NonStrict.isPreorder₂ isStrictPartialOrder)
-  irrefl
+  asym
   trans
   <-resp-≈
   NonStrict.<⇒≤
diff --git a/src/Relation/Binary/Reasoning/Syntax.agda b/src/Relation/Binary/Reasoning/Syntax.agda
new file mode 100644
index 0000000000..8bd7899770
--- /dev/null
+++ b/src/Relation/Binary/Reasoning/Syntax.agda
@@ -0,0 +1,366 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Syntax for the building blocks of equational reasoning modules 
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Level using (Level; _⊔_; suc)
+open import Relation.Nullary.Decidable.Core
+  using (Dec; True; toWitness)
+open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Binary.Core using (Rel; REL; _⇒_)
+open import Relation.Binary.Definitions
+open import Relation.Binary.PropositionalEquality.Core as P
+  using (_≡_)
+
+-- List of `Reasoning` modules that do not use this framework and so
+-- need to be updated manually if the syntax changes.
+--
+--   Data/Vec/Relation/Binary/Equality/Cast
+--   Relation/Binary/HeterogeneousEquality
+--   Effect/Monad/Partiality
+--   Effect/Monad/Partiality/All
+--   Relation/Binary/PropositionalEquality/Core
+--   Codata/Guarded/Stream/Relation/Binary/Pointwise
+--   Codata/Sized/Stream/Bisimilarity
+--   Function/Reasoning
+ 
+module Relation.Binary.Reasoning.Syntax where
+
+private
+  variable
+    a ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level
+    A B C : Set a
+    x y z : A
+
+------------------------------------------------------------------------
+-- Syntax for beginning a reasoning chain
+------------------------------------------------------------------------
+
+------------------------------------------------------------------------
+-- Basic begin syntax
+
+module begin-syntax
+  (R : REL A B ℓ₁)
+  {S : REL A B ℓ₂}
+  (reflexive : R ⇒ S)
+  where
+
+  infix 1 begin_
+
+  begin_ : R x y → S x y
+  begin_ = reflexive
+
+------------------------------------------------------------------------
+-- Begin subrelation syntax
+
+-- Sometimes we want to support sub-relations with the
+-- same reasoning operators as the main relations (e.g. perform equality
+-- proofs with non-strict reasoning operators). This record bundles all
+-- the parts needed to extract the sub-relation proofs.
+record SubRelation {A : Set a} (R : Rel A ℓ₁) ℓ₂ ℓ₃ : Set (a ⊔ ℓ₁ ⊔ suc ℓ₂ ⊔ suc ℓ₃) where
+  field
+    S : Rel A ℓ₂
+    IsS : R x y → Set ℓ₃
+    IsS? : ∀ (xRy : R x y) → Dec (IsS xRy)
+    extract : ∀ {xRy : R x y} → IsS xRy → S x y
+
+module begin-subrelation-syntax
+  (R : Rel A ℓ₁)
+  (sub : SubRelation R ℓ₂ ℓ₃)
+  where
+  open SubRelation sub
+
+  infix 1 begin_
+
+  begin_ : ∀ {x y} (xRy : R x y) → {s : True (IsS? xRy)} → S x y
+  begin_ r {s} = extract (toWitness s)
+
+-- Begin equality syntax
+module begin-equality-syntax
+  (R : Rel A ℓ₁)
+  (sub : SubRelation R ℓ₂ ℓ₃) where
+
+  open begin-subrelation-syntax R sub public
+    renaming (begin_ to begin-equality_)
+
+-- Begin apartness syntax
+module begin-apartness-syntax
+  (R : Rel A ℓ₁)
+  (sub : SubRelation R ℓ₂ ℓ₃) where
+
+  open begin-subrelation-syntax R sub public
+    renaming (begin_ to begin-apartness_)
+
+-- Begin strict syntax
+module begin-strict-syntax
+  (R : Rel A ℓ₁)
+  (sub : SubRelation R ℓ₂ ℓ₃) where
+
+  open begin-subrelation-syntax R sub public
+    renaming (begin_ to begin-strict_)
+
+------------------------------------------------------------------------
+-- Begin membership syntax
+
+module begin-membership-syntax
+  (R : Rel A ℓ₁)
+  (_∈_ : REL B A ℓ₂)
+  (resp : _∈_ Respectsʳ R) where
+
+  infix  1 step-∈
+
+  step-∈ : ∀ (x : B) {xs ys} → R xs ys → x ∈ xs → x ∈ ys
+  step-∈ x = resp
+
+  syntax step-∈ x  xs⊆ys x∈xs  = x ∈⟨ x∈xs ⟩ xs⊆ys
+
+------------------------------------------------------------------------
+-- Begin contradiction syntax
+
+-- Used with asymmetric subrelations to derive a contradiction from a
+-- proof that an element is related to itself.
+module begin-contradiction-syntax
+  (R : Rel A ℓ₁)
+  (sub : SubRelation R ℓ₂ ℓ₃)
+  (asym : Asymmetric (SubRelation.S sub))
+  where
+
+  open SubRelation sub
+
+  infix 1 begin-contradiction_
+
+  begin-contradiction_ : ∀ (xRx : R x x) {s : True (IsS? xRx)} →
+                         ∀ {b} {B : Set b} → B
+  begin-contradiction_ {x} r {s} = contradiction x<x (asym x<x)
+    where
+    x<x : S x x
+    x<x = extract (toWitness s)
+
+------------------------------------------------------------------------
+-- Syntax for continuing a chain of reasoning steps
+------------------------------------------------------------------------
+
+-- Note that the arguments to the `step`s are not provided in their
+-- "natural" order and syntax declarations are later used to re-order
+-- them. This is because the `step` ordering allows the type-checker to
+-- better infer the middle argument `y` from the `_IsRelatedTo_`
+-- argument (see issue 622).
+--
+-- This has two practical benefits. First it speeds up type-checking by
+-- approximately a factor of 5. Secondly it allows the combinators to be
+-- used with macros that use reflection, e.g. `Tactic.RingSolver`, where
+-- they need to be able to extract `y` using reflection.
+
+------------------------------------------------------------------------
+-- Syntax for unidirectional relations
+
+-- See https://github.com/agda/agda-stdlib/issues/2150 for a possible
+-- simplification.
+
+module _
+  {R : REL A B ℓ₂}
+  (S : REL B C ℓ₁)
+  (T : REL A C ℓ₃)
+  (step : Trans R S T)
+  where
+
+  forward : ∀ (x : A) {y z} → S y z → R x y → T x z
+  forward x yRz x∼y = step {x} x∼y yRz
+
+
+  -- Preorder syntax
+  module ∼-syntax where
+    infixr 2 step-∼
+    step-∼ = forward
+    syntax step-∼ x yRz x∼y = x ∼⟨ x∼y ⟩ yRz
+
+
+  -- Partial order syntax
+  module ≤-syntax where
+    infixr 2 step-≤
+    step-≤ = forward
+    syntax step-≤ x yRz x≤y = x ≤⟨ x≤y ⟩ yRz
+
+
+  -- Strict partial order syntax
+  module <-syntax where
+    infixr 2 step-<
+    step-< = forward
+    syntax step-< x yRz x<y = x <⟨ x<y ⟩ yRz
+
+
+  -- Subset order syntax
+  module ⊆-syntax where
+    infixr 2 step-⊆
+    step-⊆ = forward
+    syntax step-⊆ x yRz x⊆y = x ⊆⟨ x⊆y ⟩ yRz
+
+
+  -- Strict subset order syntax
+  module ⊂-syntax where
+    infixr 2 step-⊂
+    step-⊂ = forward
+    syntax step-⊂ x yRz x⊂y = x ⊂⟨ x⊂y ⟩ yRz
+
+
+  -- Square subset order syntax
+  module ⊑-syntax where
+    infixr 2 step-⊑
+    step-⊑ = forward
+    syntax step-⊑ x yRz x⊑y = x ⊑⟨ x⊑y ⟩ yRz
+
+
+  -- Strict square subset order syntax
+  module ⊏-syntax where
+    infixr 2 step-⊏
+    step-⊏ = forward
+    syntax step-⊏ x yRz x⊏y = x ⊏⟨ x⊏y ⟩ yRz
+
+
+  -- Divisibility syntax
+  module ∣-syntax where
+    infixr 2 step-∣
+    step-∣ = forward
+    syntax step-∣ x yRz x∣y = x ∣⟨ x∣y ⟩ yRz
+
+
+  -- Single-step syntax
+  module ⟶-syntax where
+    infixr 2 step-⟶
+    step-⟶ = forward
+    syntax step-⟶ x yRz x∣y = x ⟶⟨ x∣y ⟩ yRz
+
+
+  -- Multi-step syntax
+  module ⟶*-syntax where
+    infixr 2 step-⟶*
+    step-⟶* = forward
+    syntax step-⟶* x yRz x∣y = x ⟶*⟨ x∣y ⟩ yRz
+
+
+------------------------------------------------------------------------
+-- Syntax for bidirectional relations
+
+  module _
+    {U : REL B A ℓ₄}
+    (sym : Sym U R)
+    where
+
+    backward : ∀ x {y z} → S y z → U y x → T x z
+    backward x yRz x≈y = forward x yRz (sym x≈y)
+
+    -- Setoid equality syntax
+    module ≈-syntax where
+      infixr 2 step-≈ step-≈˘
+      step-≈ = forward
+      step-≈˘ = backward
+      syntax step-≈  x yRz x≈y = x ≈⟨ x≈y ⟩ yRz
+      syntax step-≈˘ x yRz y≈x = x ≈˘⟨ y≈x ⟩ yRz
+
+
+    -- Container equality syntax
+    module ≋-syntax where
+      infixr 2 step-≋ step-≋˘
+      step-≋ = forward
+      step-≋˘ = backward
+      syntax step-≋  x yRz x≋y = x ≋⟨ x≋y ⟩ yRz
+      syntax step-≋˘ x yRz y≋x = x ≋˘⟨ y≋x ⟩ yRz
+
+
+    -- Other equality syntax
+    module ≃-syntax where
+      infixr 2 step-≃ step-≃˘
+      step-≃ = forward
+      step-≃˘ = backward
+      syntax step-≃  x yRz x≃y = x ≃⟨ x≃y ⟩ yRz
+      syntax step-≃˘ x yRz y≃x = x ≃˘⟨ y≃x ⟩ yRz
+
+
+    -- Apartness relation syntax
+    module #-syntax where
+      infixr 2 step-# step-#˘
+      step-# = forward
+      step-#˘ = backward
+      syntax step-#  x yRz x#y = x #⟨ x#y ⟩ yRz
+      syntax step-#˘ x yRz y#x = x #˘⟨ y#x ⟩ yRz
+
+
+    -- Bijection syntax
+    module ⤖-syntax where
+      infixr 2 step-⤖ step-⬻
+      step-⤖ = forward
+      step-⬻ = backward
+      syntax step-⤖ x yRz x⤖y = x ⤖⟨ x⤖y ⟩ yRz
+      syntax step-⬻ x yRz y⤖x = x ⬻⟨ y⤖x ⟩ yRz
+
+
+    -- Inverse syntax
+    module ↔-syntax where
+      infixr 2 step-↔ step-↔-sym
+      step-↔ = forward
+      step-↔-sym = backward
+      syntax step-↔ x yRz x↔y = x ↔⟨ x↔y ⟩ yRz
+      syntax step-↔-sym x yRz y↔x = x ↔˘⟨ y↔x ⟩ yRz
+
+
+    -- Inverse syntax
+    module ↭-syntax where
+      infixr 2 step-↭ step-↭-sym
+      step-↭ = forward
+      step-↭-sym = backward
+      syntax step-↭ x yRz x↭y = x ↭⟨ x↭y ⟩ yRz
+      syntax step-↭-sym x yRz y↭x = x ↭˘⟨ y↭x ⟩ yRz
+
+------------------------------------------------------------------------
+-- Propositional equality
+
+-- Crucially often the step function cannot just be `subst` or pattern
+-- match on `refl` as we often want to compute which constructor the
+-- relation begins with, in order for the implicit subrelation
+-- arguments to resolve. See `≡-noncomputable-syntax` below if this
+-- is not required.
+module ≡-syntax
+  (R : REL A B ℓ₁)
+  (step : Trans _≡_ R R)
+  where
+
+  infixr 2 step-≡ step-≡-refl step-≡˘
+  step-≡ = forward R R step
+  step-≡˘ = backward R R step P.sym
+
+  step-≡-refl : ∀ x {y} → R x y → R x y
+  step-≡-refl x xRy = xRy
+
+  syntax step-≡-refl x xRy     = x ≡⟨⟩ xRy
+  syntax step-≡˘     x yRz y≡x = x ≡˘⟨ y≡x ⟩ yRz
+  syntax step-≡      x yRz x≡y = x ≡⟨ x≡y ⟩ yRz
+
+
+-- Unlike ≡-syntax above, chains of reasoning using this syntax will not
+-- reduce when proofs of propositional equality which are not definitionally
+-- equal to `refl` are passed.
+module ≡-noncomputing-syntax (R : REL A B ℓ₁) where
+
+  private
+    step : Trans _≡_ R R
+    step P.refl xRy = xRy
+
+  open ≡-syntax R step public
+
+------------------------------------------------------------------------
+-- Syntax for ending a chain of reasoning
+------------------------------------------------------------------------
+
+module end-syntax
+  (R : Rel A ℓ₁)
+  (reflexive : Reflexive R)
+  where
+
+  infix 3 _∎
+
+  _∎ : ∀ x → R x x
+  x ∎ = reflexive
+

From 019e8ae83b6f8272f79ccf740b19d3c429bec8a0 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 17 Oct 2023 00:18:02 +0100
Subject: [PATCH 164/316] Fixes typos identified in #2154 (#2158)

---
 CHANGELOG.md              | 58 +++++++++++++++++++--------------------
 src/Function/Bundles.agda |  6 ++--
 2 files changed, 32 insertions(+), 32 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index a1fb08efcf..b7369380f5 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -539,7 +539,7 @@ Non-backwards compatible changes
   3. Finally, if the above approaches are not viable then you may be forced to explicitly
      use `cong` combined with a lemma that proves the old reduction behaviour.
 
-* Similarly, in order to prevent reduction, the equality `_≃_` in `Data.Rational.Base` 
+* Similarly, in order to prevent reduction, the equality `_≃_` in `Data.Rational.Base`
   has been made into a data type with the single constructor `*≡*`. The destructor `drop-*≡*`
   has been added to `Data.Rational.Properties`.
 
@@ -606,7 +606,7 @@ Non-backwards compatible changes
   with the consequence that all arguments involving about accesibility and
   wellfoundedness proofs were polluted by almost-always-inferrable explicit
   arguments for the `y` position. The definition has now been changed to
-  make that argument *implicit*, as 
+  make that argument *implicit*, as
   ```agda
   WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
   WfRec _<_ P x = ∀ {y} → y < x → P y
@@ -931,40 +931,40 @@ Non-backwards compatible changes
   as would be expected. Instead it omitted several fields like irreflexivity as they were derivable from the
   proof of trichotomy. However, this led to problems further up the hierarchy where bundles such as `StrictTotalOrder`
   which contained multiple distinct proofs of `IsStrictPartialOrder`.
-  
-* To remedy this the definition of `IsStrictTotalOrder` has been changed to so that it builds upon 
-  `IsStrictPartialOrder` as would be expected. 
+
+* To remedy this the definition of `IsStrictTotalOrder` has been changed to so that it builds upon
+  `IsStrictPartialOrder` as would be expected.
 
 * To aid migration, the old record definition has been moved to `Relation.Binary.Structures.Biased`
-  which contains the `isStrictTotalOrderᶜ` smart constructor (which is re-exported by `Relation.Binary`) . 
+  which contains the `isStrictTotalOrderᶜ` smart constructor (which is re-exported by `Relation.Binary`) .
   Therefore the old code:
   ```agda
   <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
   <-isStrictTotalOrder = record
-	{ isEquivalence = isEquivalence
-	; trans         = <-trans
-	; compare       = <-cmp
-	}
+        { isEquivalence = isEquivalence
+        ; trans         = <-trans
+        ; compare       = <-cmp
+        }
   ```
   can be migrated either by updating to the new record fields if you have a proof of `IsStrictPartialOrder`
   available:
   ```agda
   <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
   <-isStrictTotalOrder = record
-	{ isStrictPartialOrder = <-isStrictPartialOrder
-	; compare              = <-cmp
-	}
+        { isStrictPartialOrder = <-isStrictPartialOrder
+        ; compare              = <-cmp
+        }
   ```
   or simply applying the smart constructor to the record definition as follows:
   ```agda
   <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
   <-isStrictTotalOrder = isStrictTotalOrderᶜ record
-	{ isEquivalence = isEquivalence
-	; trans         = <-trans
-	; compare       = <-cmp
-	}
+        { isEquivalence = isEquivalence
+        ; trans         = <-trans
+        ; compare       = <-cmp
+        }
   ```
-  
+
 ### Changes to triple reasoning interface
 
 * The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof that the strict
@@ -987,7 +987,7 @@ Non-backwards compatible changes
   Data.Vec.Relation.Binary.Lex.NonStrict
   Relation.Binary.Reasoning.StrictPartialOrder
   Relation.Binary.Reasoning.PartialOrder
-  ```	
+  ```
 
 ### Other
 
@@ -1165,7 +1165,7 @@ Non-backwards compatible changes
 * `excluded-middle` in `Relation.Nullary.Decidable.Core` has been renamed to
   `¬¬-excluded-middle`.
 
-* `iterate` and `replicate` in `Data.Vec.Base` and `Data.Vec.Functional` 
+* `iterate` and `replicate` in `Data.Vec.Base` and `Data.Vec.Functional`
   now take the length of vector, `n`, as an explicit rather than an implicit argument.
   ```agda
   iterate : (A → A) → A → ∀ n → Vec A n
@@ -1231,11 +1231,11 @@ Major improvements
 
 ### More modular design of equational reasoning.
 
-* Have introduced a new module `Relation.Binary.Reasoning.Syntax` which exports 
+* Have introduced a new module `Relation.Binary.Reasoning.Syntax` which exports
   a range of modules containing pre-existing reasoning combinator syntax.
 
-* This makes it possible to add new or rename existing reasoning combinators to a 
-  pre-existing `Reasoning` module in just a couple of lines 
+* This makes it possible to add new or rename existing reasoning combinators to a
+  pre-existing `Reasoning` module in just a couple of lines
   (e.g. see `∣-Reasoning` in `Data.Nat.Divisibility`)
 
 Deprecated modules
@@ -1813,7 +1813,7 @@ Deprecated names
   ```agda
   _↔⟨⟩_  ↦  _≡⟨⟩_
   ```
-  
+
 * In `Foreign.Haskell.Either` and `Foreign.Haskell.Pair`:
   ```
   toForeign   ↦ Foreign.Haskell.Coerce.coerce
@@ -2706,7 +2706,7 @@ Additions to existing modules
   toℕ-inverseˡ  : Inverseˡ _≡_ _≡_ toℕ fromℕ
   toℕ-inverseʳ  : Inverseʳ _≡_ _≡_ toℕ fromℕ
   toℕ-inverseᵇ  : Inverseᵇ _≡_ _≡_ toℕ fromℕ
-  
+
   <-asym : Asymmetric _<_
   ```
 
@@ -3163,7 +3163,7 @@ Additions to existing modules
                 ∀ {m n} → _Respectsʳ_ (_<_ {m} {n}) _≋_
   <-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ →
                   ∀ {n} → WellFounded (_<_ {n})
-```
+  ```
 
 * Added new functions in `Data.Vec.Relation.Unary.Any`:
   ```
@@ -3190,9 +3190,9 @@ Additions to existing modules
 * Added new application operator synonym to `Function.Bundles`:
   ```agda
   _⟨$⟩_ : Func From To → Carrier From → Carrier To
-  _⟨$⟩_ = Func.to 
+  _⟨$⟩_ = Func.to
   ```
-  
+
 * Added new proofs in `Function.Construct.Symmetry`:
   ```
   bijective     : Bijective ≈₁ ≈₂ f → Symmetric ≈₂ → Transitive ≈₂ → Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹
@@ -3888,7 +3888,7 @@ This is a full list of proofs that have changed form to use irrelevant instance
   blockerAll : List Blocker → Blocker
   blockTC : Blocker → TC A
   ```
-  
+
 * Added new file `Relation.Binary.Reasoning.Base.Apartness`
 
   This is how to use it:
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index a51dc4ea84..0aef27fc57 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -182,7 +182,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
 
     open Func toFunction public
       using (module Eq₁; module Eq₂)
-      renaming (isCongruent to to-isCongrunet)
+      renaming (isCongruent to to-isCongruent)
 
     fromFunction : Func To From
     fromFunction = record
@@ -192,7 +192,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
 
     open Func fromFunction public
       using ()
-      renaming (isCongruent to from-isCongrunet)
+      renaming (isCongruent to from-isCongruent)
 
 
   record LeftInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
@@ -481,7 +481,7 @@ module _ {A : Set a} {B : Set b} where
 
 module _ {From : Setoid a ℓ₁} {To : Setoid b ℓ₂} where
   open Setoid
-  
+
   infixl 5 _⟨$⟩_
   _⟨$⟩_ : Func From To → Carrier From → Carrier To
   _⟨$⟩_ = Func.to

From 3fe4163e01d3f5e56b2c6b39f92ff7a2b3874dd9 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 17 Oct 2023 00:18:55 +0100
Subject: [PATCH 165/316] tackles #2124 as regards `case_return_of_` (#2157)

---
 CHANGELOG.md           |  5 +++++
 README/Case.agda       |  4 ++--
 src/Function/Base.agda | 24 +++++++++++++++++++-----
 3 files changed, 26 insertions(+), 7 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b7369380f5..89e22ce60e 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1776,6 +1776,11 @@ Deprecated names
   map-++-commute ↦  map-++
   ```
 
+* In `Function.Base`:
+  ```
+  case_return_of_   ↦   case_returning_of_
+  ```
+
 * In `Function.Construct.Composition`:
   ```
   _∘-⟶_   ↦   _⟶-∘_
diff --git a/README/Case.agda b/README/Case.agda
index 30ce0221aa..62fec18c0f 100644
--- a/README/Case.agda
+++ b/README/Case.agda
@@ -12,7 +12,7 @@ module README.Case where
 open import Data.Fin   hiding (pred)
 open import Data.Maybe hiding (from-just)
 open import Data.Nat   hiding (pred)
-open import Function.Base using (case_of_; case_return_of_)
+open import Function.Base using (case_of_; case_returning_of_)
 open import Relation.Nullary
 
 ------------------------------------------------------------------------
@@ -35,7 +35,7 @@ pred n = case n of λ
 -- where-introduced and indentation-identified block of list of clauses
 
 from-just : ∀ {a} {A : Set a} (x : Maybe A) → From-just x
-from-just x = case x return From-just of λ where
+from-just x = case x returning From-just of λ where
   (just x) → x
   nothing  → _
 
diff --git a/src/Function/Base.agda b/src/Function/Base.agda
index 9bdb996dc8..122e276469 100644
--- a/src/Function/Base.agda
+++ b/src/Function/Base.agda
@@ -110,10 +110,10 @@ f $- = f _
 -- Case expressions (to be used with pattern-matching lambdas, see
 -- README.Case).
 
-case_return_of_ : ∀ {A : Set a} (x : A) (B : A → Set b) →
+case_returning_of_ : ∀ {A : Set a} (x : A) (B : A → Set b) →
                   ((x : A) → B x) → B x
-case x return B of f = f x
-{-# INLINE case_return_of_ #-}
+case x returning B of f = f x
+{-# INLINE case_returning_of_ #-}
 
 ------------------------------------------------------------------------
 -- Non-dependent versions of dependent operations
@@ -156,7 +156,7 @@ _|>′_ = _|>_
 -- README.Case).
 
 case_of_ : A → (A → B) → B
-case x of f = case x return _ of f
+case x of f = case x returning _ of f
 {-# INLINE case_of_ #-}
 
 ------------------------------------------------------------------------
@@ -247,10 +247,24 @@ _*_ on f = f -⟨ _*_ ⟩- f
 
 
 ------------------------------------------------------------------------
--- Deprecated
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 1.4
 
 _-[_]-_ = _-⟪_⟫-_
 {-# WARNING_ON_USAGE _-[_]-_
 "Warning: Function._-[_]-_ was deprecated in v1.4.
 Please use _-⟪_⟫-_ instead."
 #-}
+
+-- Version 2.0
+
+case_return_of_ = case_returning_of_
+{-# WARNING_ON_USAGE case_return_of_
+"case_return_of_ was deprecated in v2.0.
+Please use case_returning_of_ instead."
+#-}
+

From a3235b962d1d88efd34c2353270896c80fc30acb Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Tue, 17 Oct 2023 08:19:32 +0900
Subject: [PATCH 166/316] Rename preorder ~ relation reasoning combinators
 (#2156)

---
 src/Codata/Musical/Colist.agda                |  6 ++--
 src/Data/Integer/Divisibility/Signed.agda     |  6 ++--
 .../Relation/Binary/BagAndSetEquality.agda    |  4 +--
 .../Binary/Subset/Setoid/Properties.agda      |  8 +++--
 src/Data/Nat/Divisibility.agda                |  6 ++--
 src/Data/Rational/Properties.agda             |  4 +--
 .../Rational/Unnormalised/Properties.agda     |  2 +-
 .../ReflexiveTransitive/Properties.agda       |  8 +++--
 .../Binary/Reasoning/Base/Double.agda         | 33 ++++++++++++++-----
 src/Relation/Binary/Reasoning/Syntax.agda     | 14 +++++---
 10 files changed, 61 insertions(+), 30 deletions(-)

diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index aeff3076dc..3699ef64be 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -190,10 +190,12 @@ module ⊑-Reasoning {a} {A : Set a} where
 module ⊆-Reasoning {A : Set a} where
   private module Base = PreR (⊆-Preorder A)
 
-  open Base public hiding (step-∼)
+  open Base public
+    hiding (step-≲; step-∼)
+    renaming (≲-go to ⊆-go)
 
   open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → begin x)
-  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
+  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ ⊆-go public
 
 
 -- take returns a prefix.
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index 9a4b938950..5149871534 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -121,9 +121,11 @@ open _∣_ using (quotient) public
 module ∣-Reasoning where
   private module Base = PreorderReasoning ∣-preorder
 
-  open Base public hiding (step-∼; step-≈; step-≈˘)
+  open Base public
+    hiding (step-≲; step-∼; step-≈; step-≈˘)
+    renaming (≲-go to ∣-go)
 
-  open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public
+  open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∣-go public
 
 ------------------------------------------------------------------------
 -- Other properties of _∣_
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 0acef52dcb..f842ae8087 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -96,8 +96,8 @@ module ⊆-Reasoning {A : Set a} where
   private module Base = PreorderReasoning (⊆-preorder A)
 
   open Base public
-    hiding (step-≈; step-≈˘; step-∼)
-    renaming (∼-go to ⊆-go)
+    hiding (step-≈; step-≈˘; step-∼; step-≲)
+    renaming (≲-go to ⊆-go)
 
   open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → begin x) public
   open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ ⊆-go public
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index 0d11079ad0..bb20fd417f 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -117,11 +117,13 @@ module ⊆-Reasoning (S : Setoid a ℓ) where
 
   private module Base = PreorderReasoning (⊆-preorder S)
 
-  open Base public hiding (step-∼; step-≈; step-≈˘)
+  open Base public
+    hiding (step-≲; step-≈; step-≈˘; step-∼)
+    renaming (≲-go to ⊆-go; ≈-go to ≋-go)
 
   open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → Base.begin x) public
-  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
-  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ Base.≈-go public
+  open ⊆-syntax _IsRelatedTo_ _IsRelatedTo_ ⊆-go public
+  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ ≋-go public
 
 ------------------------------------------------------------------------
 -- Relationship with other binary relations
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index 5a0491e498..63037a49b2 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -120,9 +120,11 @@ suc n ∣? m      = Dec.map (m%n≡0⇔n∣m m (suc n)) (m % suc n ≟ 0)
 module ∣-Reasoning where
   private module Base = PreorderReasoning ∣-preorder
 
-  open Base public hiding (step-≈; step-≈˘; step-∼)
+  open Base public
+    hiding (step-≈; step-≈˘; step-∼; step-≲)
+    renaming (≲-go to ∣-go)
 
-  open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
+  open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∣-go public
 
 ------------------------------------------------------------------------
 -- Simple properties of _∣_
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index b43b109943..915eb6e058 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -164,7 +164,7 @@ drop-*≡* (*≡* eq) = eq
   ...   | refl = refl
 
 ≃-sym : Symmetric _≃_
-≃-sym = ≡⇒≃ ∘′ sym ∘′ ≃⇒≡ 
+≃-sym = ≡⇒≃ ∘′ sym ∘′ ≃⇒≡
 
 ------------------------------------------------------------------------
 -- Properties of ↥
@@ -753,7 +753,7 @@ module ≤-Reasoning where
 
   ≃-go : Trans _≃_ _IsRelatedTo_ _IsRelatedTo_
   ≃-go = Triple.≈-go ∘′ ≃⇒≡
-  
+
   open ≃-syntax _IsRelatedTo_ _IsRelatedTo_ ≃-go (λ {x y} → ≃-sym {x} {y}) public
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index dffd2529b0..6348a6ef46 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -164,7 +164,7 @@ p≃0⇒↥p≡0 p (*≡* eq) = begin
   ↥ p ℤ.* 1ℤ  ≡⟨ eq ⟩
   0ℤ           ∎
   where open ≡-Reasoning
-  
+
 ↥p≡↥q≡0⇒p≃q : ∀ p q → ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≃ q
 ↥p≡↥q≡0⇒p≃q p q ↥p≡0 ↥q≡0 = ≃-trans (↥p≡0⇒p≃0 p ↥p≡0) (≃-sym (↥p≡0⇒p≃0 _ ↥q≡0))
 
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index c8ebd64b7c..58b2ee198a 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -115,7 +115,9 @@ module _ {i t} {I : Set i} (T : Rel I t) where
 module StarReasoning {i t} {I : Set i} (T : Rel I t) where
   private module Base = PreorderReasoning (preorder T)
 
-  open Base public hiding (step-≈; step-∼)
+  open Base public
+    hiding (step-≈; step-∼; step-≲)
+    renaming (≲-go to ⟶-go)
 
-  open ⟶-syntax _IsRelatedTo_ _IsRelatedTo_ (Base.∼-go ∘ (_◅ ε)) public
-  open ⟶*-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go public
+  open ⟶-syntax _IsRelatedTo_ _IsRelatedTo_ (⟶-go ∘ (_◅ ε)) public
+  open ⟶*-syntax _IsRelatedTo_ _IsRelatedTo_ ⟶-go public
diff --git a/src/Relation/Binary/Reasoning/Base/Double.agda b/src/Relation/Binary/Reasoning/Base/Double.agda
index 91203437f6..6fa901d573 100644
--- a/src/Relation/Binary/Reasoning/Base/Double.agda
+++ b/src/Relation/Binary/Reasoning/Base/Double.agda
@@ -21,7 +21,7 @@ open import Relation.Binary.Reasoning.Syntax
 
 
 module Relation.Binary.Reasoning.Base.Double {a ℓ₁ ℓ₂} {A : Set a}
-  {_≈_ : Rel A ℓ₁} {_∼_ : Rel A ℓ₂} (isPreorder : IsPreorder _≈_ _∼_)
+  {_≈_ : Rel A ℓ₁} {_≲_ : Rel A ℓ₂} (isPreorder : IsPreorder _≈_ _≲_)
   where
 
 open IsPreorder isPreorder
@@ -32,24 +32,24 @@ open IsPreorder isPreorder
 infix 4 _IsRelatedTo_
 
 data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
-  nonstrict : (x∼y : x ∼ y) → x IsRelatedTo y
+  nonstrict : (x≲y : x ≲ y) → x IsRelatedTo y
   equals    : (x≈y : x ≈ y) → x IsRelatedTo y
 
-start : _IsRelatedTo_ ⇒ _∼_
+start : _IsRelatedTo_ ⇒ _≲_
 start (equals x≈y) = reflexive x≈y
-start (nonstrict x∼y) = x∼y
+start (nonstrict x≲y) = x≲y
 
 ≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
 ≡-go x≡y (equals y≈z) = equals (case x≡y of λ where P.refl → y≈z)
 ≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where P.refl → y≤z)
 
-∼-go : Trans _∼_ _IsRelatedTo_ _IsRelatedTo_
-∼-go x∼y (equals y≈z) = nonstrict (∼-respʳ-≈ y≈z x∼y)
-∼-go x∼y (nonstrict y∼z) = nonstrict (trans x∼y y∼z)
+≲-go : Trans _≲_ _IsRelatedTo_ _IsRelatedTo_
+≲-go x≲y (equals y≈z) = nonstrict (∼-respʳ-≈ y≈z x≲y)
+≲-go x≲y (nonstrict y≲z) = nonstrict (trans x≲y y≲z)
 
 ≈-go : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_
 ≈-go x≈y (equals y≈z) = equals (Eq.trans x≈y y≈z)
-≈-go x≈y (nonstrict y∼z) = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y∼z)
+≈-go x≈y (nonstrict y≲z) = nonstrict (∼-respˡ-≈ (Eq.sym x≈y) y≲z)
 
 stop : Reflexive _IsRelatedTo_
 stop = equals Eq.refl
@@ -81,6 +81,21 @@ equalitySubRelation = record
 open begin-syntax  _IsRelatedTo_ start public
 open begin-equality-syntax  _IsRelatedTo_ equalitySubRelation public
 open ≡-syntax _IsRelatedTo_ ≡-go public
-open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ∼-go public
 open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go Eq.sym public
+open ≲-syntax _IsRelatedTo_ _IsRelatedTo_ ≲-go public
 open end-syntax _IsRelatedTo_ stop public
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+open ∼-syntax _IsRelatedTo_ _IsRelatedTo_ ≲-go public
+{-# WARNING_ON_USAGE step-∼
+"Warning: step-∼ and _∼⟨_⟩_ syntax was deprecated in v2.0.
+Please use step-≲ and _≲⟨_⟩_ instead. "
+#-}
diff --git a/src/Relation/Binary/Reasoning/Syntax.agda b/src/Relation/Binary/Reasoning/Syntax.agda
index 8bd7899770..4837fe33f4 100644
--- a/src/Relation/Binary/Reasoning/Syntax.agda
+++ b/src/Relation/Binary/Reasoning/Syntax.agda
@@ -1,7 +1,7 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Syntax for the building blocks of equational reasoning modules 
+-- Syntax for the building blocks of equational reasoning modules
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -26,7 +26,7 @@ open import Relation.Binary.PropositionalEquality.Core as P
 --   Codata/Guarded/Stream/Relation/Binary/Pointwise
 --   Codata/Sized/Stream/Bisimilarity
 --   Function/Reasoning
- 
+
 module Relation.Binary.Reasoning.Syntax where
 
 private
@@ -170,14 +170,20 @@ module _
   forward : ∀ (x : A) {y z} → S y z → R x y → T x z
   forward x yRz x∼y = step {x} x∼y yRz
 
-
-  -- Preorder syntax
+  -- Arbitrary relation syntax
   module ∼-syntax where
     infixr 2 step-∼
     step-∼ = forward
     syntax step-∼ x yRz x∼y = x ∼⟨ x∼y ⟩ yRz
 
 
+  -- Preorder syntax
+  module ≲-syntax where
+    infixr 2 step-≲
+    step-≲ = forward
+    syntax step-≲ x yRz x≲y = x ≲⟨ x≲y ⟩ yRz
+
+
   -- Partial order syntax
   module ≤-syntax where
     infixr 2 step-≤

From b391a3d3958f80087e7653ba2290584174354a8f Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 18 Oct 2023 07:20:25 +0900
Subject: [PATCH 167/316] =?UTF-8?q?Move=20=E2=89=A1-Reasoning=20from=20Cor?=
 =?UTF-8?q?e=20to=20Properties=20and=20implement=20using=20syntax=20(#2159?=
 =?UTF-8?q?)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 CHANGELOG.md                                  |  6 ++++
 src/Codata/Guarded/Stream/Properties.agda     |  6 ++--
 src/Codata/Musical/Colist/Infinite-merge.agda |  4 ++-
 src/Codata/Sized/Stream/Properties.agda       |  4 ++-
 src/Data/Fin/Relation/Unary/Top.agda          |  2 ++
 src/Data/Fin/Substitution/Lemmas.agda         |  4 ++-
 src/Data/List/Countdown.agda                  |  2 +-
 .../Example/UniqueBoundVariables.agda         |  3 +-
 .../Ternary/Interleaving/Properties.agda      |  4 ++-
 .../List/Relation/Unary/Any/Properties.agda   |  4 ++-
 src/Data/Nat/Coprimality.agda                 |  2 +-
 src/Data/Nat/GCD.agda                         |  4 ++-
 src/Data/Nat/LCM.agda                         |  4 ++-
 src/Data/String/Unsafe.agda                   |  5 ++-
 src/Data/Vec/Bounded/Base.agda                |  4 ++-
 src/Data/Vec/Recursive/Properties.agda        |  2 ++
 .../Vec/Relation/Binary/Equality/Cast.agda    |  4 ++-
 src/Effect/Applicative/Indexed.agda           |  4 ++-
 src/Function/Related/TypeIsomorphisms.agda    |  4 ++-
 .../Binary/PropositionalEquality/Core.agda    | 32 -------------------
 .../PropositionalEquality/Properties.agda     | 15 +++++++++
 src/Relation/Binary/Reasoning/Syntax.agda     |  1 -
 22 files changed, 70 insertions(+), 50 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 89e22ce60e..2dcdb3d924 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1238,6 +1238,12 @@ Major improvements
   pre-existing `Reasoning` module in just a couple of lines
   (e.g. see `∣-Reasoning` in `Data.Nat.Divisibility`)
 
+* One pre-requisite for that is that `≡-Reasoning` has been moved from
+  `Relation.Binary.PropositionalEquality.Core` (which shouldn't be
+  imported anyway as it's a `Core` module) to 
+  `Relation.Binary.PropositionalEquality.Properties`.
+  It is still exported by `Relation.Binary.PropositionalEquality`.
+  
 Deprecated modules
 ------------------
 
diff --git a/src/Codata/Guarded/Stream/Properties.agda b/src/Codata/Guarded/Stream/Properties.agda
index bd41d1967e..3742aab856 100644
--- a/src/Codata/Guarded/Stream/Properties.agda
+++ b/src/Codata/Guarded/Stream/Properties.agda
@@ -21,6 +21,8 @@ open import Data.Vec.Base as Vec using (Vec; _∷_)
 open import Function.Base using (const; flip; id; _∘′_; _$′_; _⟨_⟩_; _∘₂′_)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
@@ -218,7 +220,7 @@ lookup-transpose n (as ∷ ass) = begin
   lookup as n ∷ lookup (transpose ass) n     ≡⟨ cong (lookup as n ∷_) (lookup-transpose n ass) ⟩
   lookup as n ∷ List.map (flip lookup n) ass ≡⟨⟩
   List.map (flip lookup n) (as ∷ ass)        ∎
-  where open P.≡-Reasoning
+  where open ≡-Reasoning
 
 lookup-transpose⁺ : ∀ n (ass : List⁺ (Stream A)) →
                     lookup (transpose⁺ ass) n ≡ List⁺.map (flip lookup n) ass
@@ -228,7 +230,7 @@ lookup-transpose⁺ n (as ∷ ass) = begin
   lookup as n ∷ lookup (transpose ass) n     ≡⟨ cong (lookup as n ∷_) (lookup-transpose n ass) ⟩
   lookup as n ∷ List.map (flip lookup n) ass ≡⟨⟩
   List⁺.map (flip lookup n) (as ∷ ass)       ∎
-  where open P.≡-Reasoning
+  where open ≡-Reasoning
 
 lookup-tails : ∀ n (as : Stream A) → lookup (tails as) n ≈ ℕ.iterate tail as n
 lookup-tails zero    as = B.refl
diff --git a/src/Codata/Musical/Colist/Infinite-merge.agda b/src/Codata/Musical/Colist/Infinite-merge.agda
index cbc142ff5a..584880a768 100644
--- a/src/Codata/Musical/Colist/Infinite-merge.agda
+++ b/src/Codata/Musical/Colist/Infinite-merge.agda
@@ -26,6 +26,8 @@ open import Level
 open import Relation.Unary using (Pred)
 import Induction.WellFounded as WF
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 import Relation.Binary.Construct.On as On
 
 private
@@ -132,7 +134,7 @@ merge xss = ⟦ merge′ xss ⟧P
 Any-merge : ∀ xss → Any P (merge xss) ↔ Any (λ { (x , xs) → P x ⊎ Any P xs }) xss
 Any-merge {P = P} xss = mk↔ₛ′ (proj₁ ∘ to xss) from to∘from (proj₂ ∘ to xss)
   where
-  open P.≡-Reasoning
+  open ≡-Reasoning
 
   -- The from function.
 
diff --git a/src/Codata/Sized/Stream/Properties.agda b/src/Codata/Sized/Stream/Properties.agda
index d2cf4e3205..40cb3fc961 100644
--- a/src/Codata/Sized/Stream/Properties.agda
+++ b/src/Codata/Sized/Stream/Properties.agda
@@ -25,6 +25,8 @@ open import Data.Vec.Base as Vec using (_∷_)
 
 open import Function.Base using (id; _$_; _∘′_; const)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; _≢_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
@@ -116,7 +118,7 @@ lookup-iterate-identity (suc n)  f a = begin
   lookup (iterate f (f a)) n   ≡⟨ lookup-iterate-identity n f (f a) ⟩
   fold (f a) f n               ≡⟨ fold-pull a f (const ∘′ f) (f a) P.refl (λ _ → P.refl) n ⟩
   f (fold a f n)               ≡⟨⟩
-  fold a f (suc n)             ∎ where open P.≡-Reasoning
+  fold a f (suc n)             ∎ where open ≡-Reasoning
 
 ------------------------------------------------------------------------
 -- DEPRECATED
diff --git a/src/Data/Fin/Relation/Unary/Top.agda b/src/Data/Fin/Relation/Unary/Top.agda
index b6a567a9a8..a884a09243 100644
--- a/src/Data/Fin/Relation/Unary/Top.agda
+++ b/src/Data/Fin/Relation/Unary/Top.agda
@@ -16,6 +16,8 @@ module Data.Fin.Relation.Unary.Top where
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.Fin.Base using (Fin; zero; suc; fromℕ; inject₁)
 open import Relation.Binary.PropositionalEquality.Core
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
diff --git a/src/Data/Fin/Substitution/Lemmas.agda b/src/Data/Fin/Substitution/Lemmas.agda
index ffa26e64fb..8b4d0b176a 100644
--- a/src/Data/Fin/Substitution/Lemmas.agda
+++ b/src/Data/Fin/Substitution/Lemmas.agda
@@ -16,9 +16,11 @@ import Data.Vec.Properties as VecProp
 open import Function.Base as Fun using (_∘_; _$_; flip)
 open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; refl; sym; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
   using (Star; ε; _◅_; _▻_)
-open PropEq.≡-Reasoning
+open ≡-Reasoning
 open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred)
 
diff --git a/src/Data/List/Countdown.agda b/src/Data/List/Countdown.agda
index 8d6761de27..dbfd75b332 100644
--- a/src/Data/List/Countdown.agda
+++ b/src/Data/List/Countdown.agda
@@ -31,8 +31,8 @@ open import Relation.Nullary
 open import Relation.Nullary.Decidable using (dec-true; dec-false)
 open import Relation.Binary.PropositionalEquality.Core as PropEq
   using (_≡_; _≢_; refl; cong)
-open PropEq.≡-Reasoning
 import Relation.Binary.PropositionalEquality.Properties as PropEq
+open PropEq.≡-Reasoning
 
 private
   open module D = DecSetoid D
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
index 39a6fe65a1..163818b6e8 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
@@ -10,7 +10,8 @@
 
 module Data.List.Relation.Binary.Sublist.Propositional.Example.UniqueBoundVariables (Base : Set) where
 
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; sym; cong; subst; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; sym; cong; subst; module ≡-Reasoning)
 open ≡-Reasoning
 
 open import Data.List.Base using (List; []; _∷_; [_])
diff --git a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
index dff26bb6b2..5317d5a7c5 100644
--- a/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving/Properties.agda
@@ -16,7 +16,9 @@ open import Data.List.Relation.Ternary.Interleaving hiding (map)
 open import Function.Base using (_$_)
 open import Relation.Binary.Core using (REL)
 open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; sym; cong; module ≡-Reasoning)
+  using (_≡_; refl; sym; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index aba1a777ee..b88bdfbe95 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -43,6 +43,8 @@ open import Relation.Binary.Core using (Rel; REL)
 open import Relation.Binary.Definitions as B
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Unary as U
   using (Pred; _⟨×⟩_; _⟨→⟩_) renaming (_⊆_ to _⋐_)
 open import Relation.Nullary using (¬_; _because_; does; ofʸ; ofⁿ; yes; no)
@@ -219,7 +221,7 @@ Any-×⁻ pq with Prod.map₂ (Prod.map₂ find) (find pq)
      (Any P xs × Any Q ys) ↔ Any (λ x → Any (λ y → P x × Q y) ys) xs
 ×↔ {P = P} {Q = Q} {xs} {ys} = mk↔ₛ′ Any-×⁺ Any-×⁻ to∘from from∘to
   where
-  open P.≡-Reasoning
+  open ≡-Reasoning
 
   from∘to : ∀ pq → Any-×⁻ (Any-×⁺ pq) ≡ pq
   from∘to (p , q) =
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index f6f91b44e5..1df320ea96 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -20,7 +20,7 @@ open import Data.Product.Base as Prod
 open import Function.Base using (_∘_)
 open import Level using (0ℓ)
 open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; _≢_; refl; trans; cong; subst; module ≡-Reasoning)
+  using (_≡_; _≢_; refl; trans; cong; subst)
 open import Relation.Nullary as Nullary hiding (recompute)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Binary.Core using (Rel)
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index 555a902df6..b88947c707 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -24,6 +24,8 @@ open import Induction.Lexicographic using (_⊗_; [_⊗_])
 open import Relation.Binary.Definitions using (tri<; tri>; tri≈; Symmetric)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; subst; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Nullary.Decidable using (Dec)
 open import Relation.Nullary.Negation using (contradiction)
 import Relation.Nullary.Decidable as Dec
@@ -190,7 +192,7 @@ c*gcd[m,n]≡gcd[cm,cn] c@(suc _) m n = begin
   c * gcd m n                   ≡⟨ cong (c *_) (P.sym (gcd[cm,cn]/c≡gcd[m,n] c m n)) ⟩
   c * (gcd (c * m) (c * n) / c) ≡⟨ m*[n/m]≡n (gcd-greatest (m∣m*n m) (m∣m*n n)) ⟩
   gcd (c * m) (c * n)           ∎
-  where open P.≡-Reasoning
+  where open ≡-Reasoning
 
 gcd[m,n]≤n : ∀ m n .{{_ : NonZero n}} → gcd m n ≤ n
 gcd[m,n]≤n m n = ∣⇒≤ (gcd[m,n]∣n m n)
diff --git a/src/Data/Nat/LCM.agda b/src/Data/Nat/LCM.agda
index 545870574f..5dff2eb530 100644
--- a/src/Data/Nat/LCM.agda
+++ b/src/Data/Nat/LCM.agda
@@ -18,7 +18,9 @@ open import Data.Nat.GCD
 open import Data.Product.Base using (_×_; _,_; uncurry′; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl; sym; trans; cong; cong₂; module ≡-Reasoning)
+  using (_≡_; refl; sym; trans; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Nullary.Decidable using (False; fromWitnessFalse)
 
 private
diff --git a/src/Data/String/Unsafe.agda b/src/Data/String/Unsafe.agda
index ff1ff28408..35b408f131 100644
--- a/src/Data/String/Unsafe.agda
+++ b/src/Data/String/Unsafe.agda
@@ -16,8 +16,11 @@ open import Data.Product.Base using (proj₂)
 open import Data.String.Base
 open import Function.Base using (_∘′_)
 
-open import Relation.Binary.PropositionalEquality.Core; open ≡-Reasoning
+open import Relation.Binary.PropositionalEquality.Core
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Binary.PropositionalEquality.TrustMe using (trustMe)
+open ≡-Reasoning
 
 ------------------------------------------------------------------------
 -- Properties of tail
diff --git a/src/Data/Vec/Bounded/Base.agda b/src/Data/Vec/Bounded/Base.agda
index 63649e940e..c926c9dfb1 100644
--- a/src/Data/Vec/Bounded/Base.agda
+++ b/src/Data/Vec/Bounded/Base.agda
@@ -22,6 +22,8 @@ open import Function.Base using (_∘_; id; _$_)
 open import Level using (Level)
 open import Relation.Nullary.Decidable.Core using (recompute)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
@@ -66,7 +68,7 @@ private
     m + (⌈ k /2⌉ + ⌊ k /2⌋) ≡⟨ P.cong (m +_) (ℕₚ.+-comm ⌈ k /2⌉ ⌊ k /2⌋) ⟩
     m + (⌊ k /2⌋ + ⌈ k /2⌉) ≡⟨ P.cong (m +_) (ℕₚ.⌊n/2⌋+⌈n/2⌉≡n k) ⟩
     m + k                   ≡⟨ eq ⟩
-    n                       ∎ where open P.≡-Reasoning
+    n                       ∎ where open ≡-Reasoning
 
 padBoth : ∀ {n} → A → A → Vec≤ A n → Vec A n
 padBoth aₗ aᵣ as@(vs , m≤n)
diff --git a/src/Data/Vec/Recursive/Properties.agda b/src/Data/Vec/Recursive/Properties.agda
index b012e59a80..d036134a45 100644
--- a/src/Data/Vec/Recursive/Properties.agda
+++ b/src/Data/Vec/Recursive/Properties.agda
@@ -15,6 +15,8 @@ open import Data.Vec.Recursive
 open import Data.Vec.Base using (Vec; _∷_)
 open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 
 private
diff --git a/src/Data/Vec/Relation/Binary/Equality/Cast.agda b/src/Data/Vec/Relation/Binary/Equality/Cast.agda
index adca0a42f2..f65981f7a3 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Cast.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Cast.agda
@@ -18,7 +18,9 @@ open import Data.Vec.Base
 open import Relation.Binary.Core using (REL; _⇒_)
 open import Relation.Binary.Definitions using (Sym; Trans)
 open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; trans; sym; cong; module ≡-Reasoning)
+  using (_≡_; refl; trans; sym; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
diff --git a/src/Effect/Applicative/Indexed.agda b/src/Effect/Applicative/Indexed.agda
index 30e9b632bb..f547e68b27 100644
--- a/src/Effect/Applicative/Indexed.agda
+++ b/src/Effect/Applicative/Indexed.agda
@@ -16,6 +16,8 @@ open import Data.Product.Base using (_×_; _,_)
 open import Function.Base
 open import Level
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
@@ -111,4 +113,4 @@ record Morphism {I : Set i} {F₁ F₂ : IFun I f}
     op (A₁._⊛_ (A₁.pure f) x)       ≡⟨ op-⊛ _ _ ⟩
     A₂._⊛_ (op (A₁.pure f)) (op x)  ≡⟨ P.cong₂ A₂._⊛_ (op-pure _) P.refl ⟩
     A₂._⊛_ (A₂.pure f) (op x)       ∎
-    where open P.≡-Reasoning
+    where open ≡-Reasoning
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index 830bb6f8bb..9fb1fded28 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -29,6 +29,8 @@ open import Function.Related.Propositional
 import Function.Construct.Identity as Identity
 open import Relation.Binary hiding (_⇔_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ)
 import Relation.Nullary.Indexed as I
@@ -284,7 +286,7 @@ private
     from C↔D (to C↔D (f (from A↔B (to A↔B x))))  ≡⟨ strictlyInverseʳ C↔D _ ⟩
     f (from A↔B (to A↔B x))                        ≡⟨ P.cong f $ strictlyInverseʳ A↔B x ⟩
     f x                                              ∎)
-  where open Inverse; open P.≡-Reasoning
+  where open Inverse; open ≡-Reasoning
 
 →-cong :  Extensionality a c → Extensionality b d →
           ∀ {k} {A : Set a} {B : Set b} {C : Set c} {D : Set d} →
diff --git a/src/Relation/Binary/PropositionalEquality/Core.agda b/src/Relation/Binary/PropositionalEquality/Core.agda
index c0c4030b83..848258348b 100644
--- a/src/Relation/Binary/PropositionalEquality/Core.agda
+++ b/src/Relation/Binary/PropositionalEquality/Core.agda
@@ -91,35 +91,3 @@ resp₂ _∼_ = respʳ _∼_ , respˡ _∼_
 
 ≢-sym : Symmetric {A = A} _≢_
 ≢-sym x≢y =  x≢y ∘ sym
-
-------------------------------------------------------------------------
--- Convenient syntax for equational reasoning
-
--- This is a special instance of `Relation.Binary.Reasoning.Setoid`.
--- Rather than instantiating the latter with (setoid A), we reimplement
--- equation chains from scratch since then goals are printed much more
--- readably.
-
-module ≡-Reasoning {A : Set a} where
-
-  infix  3 _∎
-  infixr 2 _≡⟨⟩_ step-≡ step-≡˘
-  infix  1 begin_
-
-  begin_ : ∀{x y : A} → x ≡ y → x ≡ y
-  begin_ x≡y = x≡y
-
-  _≡⟨⟩_ : ∀ (x {y} : A) → x ≡ y → x ≡ y
-  _ ≡⟨⟩ x≡y = x≡y
-
-  step-≡ : ∀ (x {y z} : A) → y ≡ z → x ≡ y → x ≡ z
-  step-≡ _ y≡z x≡y = trans x≡y y≡z
-
-  step-≡˘ : ∀ (x {y z} : A) → y ≡ z → y ≡ x → x ≡ z
-  step-≡˘ _ y≡z y≡x = trans (sym y≡x) y≡z
-
-  _∎ : ∀ (x : A) → x ≡ x
-  _∎ _ = refl
-
-  syntax step-≡  x y≡z x≡y = x ≡⟨  x≡y ⟩ y≡z
-  syntax step-≡˘ x y≡z y≡x = x ≡˘⟨ y≡x ⟩ y≡z
diff --git a/src/Relation/Binary/PropositionalEquality/Properties.agda b/src/Relation/Binary/PropositionalEquality/Properties.agda
index 9515b77700..a420e45b64 100644
--- a/src/Relation/Binary/PropositionalEquality/Properties.agda
+++ b/src/Relation/Binary/PropositionalEquality/Properties.agda
@@ -23,6 +23,8 @@ open import Relation.Binary.Definitions
 import Relation.Binary.Properties.Setoid as Setoid
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Unary using (Pred)
+open import Relation.Binary.Reasoning.Syntax
+
 
 private
   variable
@@ -188,3 +190,16 @@ preorder A = Setoid.≈-preorder (setoid A)
 
 poset : Set a → Poset _ _ _
 poset A = Setoid.≈-poset (setoid A)
+
+------------------------------------------------------------------------
+-- Reasoning
+
+-- This is a special instance of `Relation.Binary.Reasoning.Setoid`.
+-- Rather than instantiating the latter with (setoid A), we reimplement
+-- equation chains from scratch since then goals are printed much more
+-- readably.
+module ≡-Reasoning {a} {A : Set a} where
+
+  open begin-syntax {A = A} _≡_ id public
+  open ≡-syntax {A = A} _≡_ trans public
+  open end-syntax {A = A} _≡_ refl public
diff --git a/src/Relation/Binary/Reasoning/Syntax.agda b/src/Relation/Binary/Reasoning/Syntax.agda
index 4837fe33f4..e8286bee30 100644
--- a/src/Relation/Binary/Reasoning/Syntax.agda
+++ b/src/Relation/Binary/Reasoning/Syntax.agda
@@ -22,7 +22,6 @@ open import Relation.Binary.PropositionalEquality.Core as P
 --   Relation/Binary/HeterogeneousEquality
 --   Effect/Monad/Partiality
 --   Effect/Monad/Partiality/All
---   Relation/Binary/PropositionalEquality/Core
 --   Codata/Guarded/Stream/Relation/Binary/Pointwise
 --   Codata/Sized/Stream/Bisimilarity
 --   Function/Reasoning

From e8dcbfd18b427a8a5555091a425104e30adcb09c Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 18 Oct 2023 22:11:37 +0900
Subject: [PATCH 168/316] Add consistent deprecation warnings to old function
 hierarchy (#2163)

---
 CHANGELOG.md                                  |  1 +
 .../Product/Function/Dependent/Setoid.agda    |  9 ++-
 src/Function/Bijection.agda                   | 27 ++++++++
 src/Function/Construct/Constant.agda          | 64 +++++++++++++++++++
 src/Function/Equality.agda                    | 21 ++++++
 src/Function/Equivalence.agda                 | 35 ++++++++++
 src/Function/Injection.agda                   | 31 +++++++--
 src/Function/Inverse.agda                     | 35 ++++++++++
 src/Function/LeftInverse.agda                 | 34 ++++++++++
 src/Function/Properties/Equivalence.agda      | 40 ++++++++----
 src/Function/Properties/Injection.agda        | 22 +++++--
 src/Function/Properties/Surjection.agda       | 22 +++++--
 src/Function/Surjection.agda                  | 30 +++++++++
 13 files changed, 340 insertions(+), 31 deletions(-)
 create mode 100644 src/Function/Construct/Constant.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2dcdb3d924..e7d43ec18f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1993,6 +1993,7 @@ New modules
   Function.Properties.Bijection
   Function.Properties.RightInverse
   Function.Properties.Surjection
+  Function.Construct.Constant
   ```
 
 * In order to improve modularity, the contents of `Relation.Binary.Lattice` has been
diff --git a/src/Data/Product/Function/Dependent/Setoid.agda b/src/Data/Product/Function/Dependent/Setoid.agda
index 9866971920..f290c7341b 100644
--- a/src/Data/Product/Function/Dependent/Setoid.agda
+++ b/src/Data/Product/Function/Dependent/Setoid.agda
@@ -16,11 +16,10 @@ open import Data.Product.Relation.Binary.Pointwise.Dependent as Σ
 open import Level using (Level)
 open import Function
 open import Function.Consequences
-open import Function.Properties.Injection
-open import Function.Properties.Surjection
-open import Function.Properties.Equivalence
-open import Function.Properties.RightInverse
-import Function.Properties.Inverse as InverseProperties
+open import Function.Properties.Injection using (mkInjection)
+open import Function.Properties.Surjection using (mkSurjection; ↠⇒⇔)
+open import Function.Properties.Equivalence using (mkEquivalence; ⇔⇒⟶; ⇔⇒⟵)
+open import Function.Properties.RightInverse using (mkRightInverse)
 open import Relation.Binary.Core using (_=[_]⇒_)
 open import Relation.Binary.Bundles as B
 open import Relation.Binary.Indexed.Heterogeneous
diff --git a/src/Function/Bijection.agda b/src/Function/Bijection.agda
index bb7f2d2cce..ac32159e7a 100644
--- a/src/Function/Bijection.agda
+++ b/src/Function/Bijection.agda
@@ -38,6 +38,10 @@ record Bijective {f₁ f₂ t₁ t₂}
 
   left-inverse-of : from LeftInverseOf to
   left-inverse-of x = injective (right-inverse-of (to ⟨$⟩ x))
+{-# WARNING_ON_USAGE Bijective
+"Warning: Bijective was deprecated in v2.0.
+Please use Function.(Structures.)IsBijection instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all bijections between two setoids.
@@ -74,6 +78,10 @@ record Bijection {f₁ f₂ t₁ t₂}
     }
 
   open LeftInverse left-inverse public using (to-from)
+{-# WARNING_ON_USAGE Bijection
+"Warning: Bijection was deprecated in v2.0.
+Please use Function.(Bundles.)Bijection instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all bijections between two sets (i.e. bijections with
@@ -83,6 +91,10 @@ infix 3 _⤖_
 
 _⤖_ : ∀ {f t} → Set f → Set t → Set _
 From ⤖ To = Bijection (P.setoid From) (P.setoid To)
+{-# WARNING_ON_USAGE _⤖_
+"Warning: _⤖_ was deprecated in v2.0.
+Please use Function.(Bundles.)mk⤖ instead."
+#-}
 
 bijection : ∀ {f t} {From : Set f} {To : Set t} →
             (to : From → To) (from : To → From) →
@@ -99,6 +111,11 @@ bijection to from inj invʳ = record
       }
     }
   }
+{-# WARNING_ON_USAGE bijection
+"Warning: bijection was deprecated in v2.0.
+Please use either Function.Properties.Bijection.trans or
+Function.Construct.Composition.bijection instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Identity and composition. (Note that these proofs are superfluous,
@@ -112,6 +129,11 @@ id {S = S} = record
     ; surjective = Surjection.surjective (Surj.id {S = S})
     }
   }
+{-# WARNING_ON_USAGE id
+"Warning: id was deprecated in v2.0.
+Please use either Function.Properties.Bijection.refl or
+Function.Construct.Identity.bijection instead."
+#-}
 
 infixr 9 _∘_
 
@@ -125,3 +147,8 @@ f ∘ g = record
     ; surjective = Surjection.surjective (Surj._∘_ (surjection f) (surjection g))
     }
   } where open Bijection
+{-# WARNING_ON_USAGE _∘_
+"Warning: _∘_ was deprecated in v2.0.
+Please use either Function.Properties.Bijection.trans or
+Function.Construct.Composition.bijection instead."
+#-}
diff --git a/src/Function/Construct/Constant.agda b/src/Function/Construct/Constant.agda
new file mode 100644
index 0000000000..2f43929a87
--- /dev/null
+++ b/src/Function/Construct/Constant.agda
@@ -0,0 +1,64 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The constant function
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Construct.Constant where
+
+open import Function.Base using (const)
+open import Function.Bundles
+import Function.Definitions as Definitions
+import Function.Structures as Structures
+open import Level using (Level)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures as B hiding (IsEquivalence)
+
+private
+  variable
+    a b ℓ₁ ℓ₂ : Level
+    A B : Set a
+
+------------------------------------------------------------------------
+-- Properties
+
+module _ (_≈₁_ : Rel A ℓ₁) (_≈₂_ : Rel B ℓ₂) where
+
+  open Definitions
+
+  congruent : ∀ {b} → b ≈₂ b → Congruent _≈₁_ _≈₂_ (const b)
+  congruent refl _ = refl
+
+------------------------------------------------------------------------
+-- Structures
+
+module _
+  {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂}
+  (isEq₁ : B.IsEquivalence ≈₁)
+  (isEq₂ : B.IsEquivalence ≈₂) where
+
+  open Structures ≈₁ ≈₂
+  open B.IsEquivalence
+
+  isCongruent : ∀ b → IsCongruent (const b)
+  isCongruent b = record
+    { cong           = congruent ≈₁ ≈₂ (refl isEq₂)
+    ; isEquivalence₁ = isEq₁
+    ; isEquivalence₂ = isEq₂
+    }
+
+------------------------------------------------------------------------
+-- Setoid bundles
+
+module _ (S : Setoid a ℓ₂) (T : Setoid b ℓ₂) where
+
+  open Setoid
+
+  function : Carrier T → Func S T
+  function b = record
+    { to   = const b
+    ; cong = congruent (_≈_ S) (_≈_ T) (refl T)
+    }
diff --git a/src/Function/Equality.agda b/src/Function/Equality.agda
index 4ce068b45c..7451aaf3f7 100644
--- a/src/Function/Equality.agda
+++ b/src/Function/Equality.agda
@@ -5,6 +5,7 @@
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
+{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.Equality where
 
@@ -34,6 +35,10 @@ record Π {f₁ f₂ t₁ t₂}
   field
     _⟨$⟩_ : (x : Setoid.Carrier From) → IndexedSetoid.Carrier To x
     cong  : Setoid._≈_ From =[ _⟨$⟩_ ]⇒ IndexedSetoid._≈_ To
+{-# WARNING_ON_USAGE Π
+"Warning: Π was deprecated in v2.0.
+Please use Function.Dependent.Bundles.Func instead."
+#-}
 
 open Π public
 
@@ -41,12 +46,20 @@ infixr 0 _⟶_
 
 _⟶_ : ∀ {f₁ f₂ t₁ t₂} → Setoid f₁ f₂ → Setoid t₁ t₂ → Set _
 From ⟶ To = Π From (Trivial.indexedSetoid To)
+{-# WARNING_ON_USAGE _⟶_
+"Warning: _⟶_ was deprecated in v2.0.
+Please use Function.(Bundles.)Func instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Identity and composition.
 
 id : ∀ {a₁ a₂} {A : Setoid a₁ a₂} → A ⟶ A
 id = record { _⟨$⟩_ = Fun.id; cong = Fun.id }
+{-# WARNING_ON_USAGE id
+"Warning: id was deprecated in v2.0.
+Please use Function.Construct.Identity.function instead."
+#-}
 
 infixr 9 _∘_
 
@@ -58,6 +71,10 @@ f ∘ g = record
   { _⟨$⟩_ = Fun._∘_ (_⟨$⟩_ f) (_⟨$⟩_ g)
   ; cong  = Fun._∘_ (cong  f) (cong  g)
   }
+{-# WARNING_ON_USAGE _∘_
+"Warning: _∘_ was deprecated in v2.0.
+Please use Function.Construct.Composition.function instead."
+#-}
 
 -- Constant equality-preserving function.
 
@@ -68,6 +85,10 @@ const {B = B} b = record
   { _⟨$⟩_ = Fun.const b
   ; cong  = Fun.const (Setoid.refl B)
   }
+{-# WARNING_ON_USAGE const
+"Warning: const was deprecated in v2.0.
+Please use Function.Construct.Constant.function instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Function setoids
diff --git a/src/Function/Equivalence.agda b/src/Function/Equivalence.agda
index b25f2a062f..1e3652da7f 100644
--- a/src/Function/Equivalence.agda
+++ b/src/Function/Equivalence.agda
@@ -31,6 +31,10 @@ record Equivalence {f₁ f₂ t₁ t₂}
   field
     to   : From ⟶ To
     from : To ⟶ From
+{-# WARNING_ON_USAGE Equivalence
+"Warning: Equivalence was deprecated in v2.0.
+Please use Function.(Bundles.)Equivalence instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all equivalences between two sets (i.e. equivalences
@@ -40,6 +44,10 @@ infix 3 _⇔_
 
 _⇔_ : ∀ {f t} → Set f → Set t → Set _
 From ⇔ To = Equivalence (P.setoid From) (P.setoid To)
+{-# WARNING_ON_USAGE _⇔_
+"Warning: _⇔_ was deprecated in v2.0.
+Please use Function.(Bundles.)_⇔_ instead."
+#-}
 
 equivalence : ∀ {f t} {From : Set f} {To : Set t} →
               (From → To) → (To → From) → From ⇔ To
@@ -47,6 +55,10 @@ equivalence to from = record
   { to   = →-to-⟶ to
   ; from = →-to-⟶ from
   }
+{-# WARNING_ON_USAGE equivalence
+"Warning: equivalence was deprecated in v2.0.
+Please use Function.Properties.Equivalence.mkEquivalence instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Equivalence is an equivalence relation
@@ -58,6 +70,11 @@ id {x = S} = record
   { to   = F.id
   ; from = F.id
   }
+{-# WARNING_ON_USAGE id
+"Warning: id was deprecated in v2.0.
+Please use Function.Properties.Equivalence.refl or
+Function.Construct.Identity.equivalence instead."
+#-}
 
 infixr 9 _∘_
 
@@ -69,6 +86,11 @@ f ∘ g = record
   { to   = to   f ⟪∘⟫ to   g
   ; from = from g ⟪∘⟫ from f
   } where open Equivalence
+{-# WARNING_ON_USAGE _∘_
+"Warning: _∘_ was deprecated in v2.0.
+Please use Function.Properties.Equivalence.trans or
+Function.Construct.Composition.equivalence instead."
+#-}
 
 -- Symmetry.
 
@@ -79,6 +101,11 @@ sym eq = record
   { from       = to
   ; to         = from
   } where open Equivalence eq
+{-# WARNING_ON_USAGE sym
+"Warning: sym was deprecated in v2.0.
+Please use Function.Properties.Equivalence.sym or
+Function.Construct.Symmetry.equivalence instead."
+#-}
 
 -- For fixed universe levels we can construct setoids.
 
@@ -92,6 +119,10 @@ setoid s₁ s₂ = record
     ; trans = flip _∘_
     }
   }
+{-# WARNING_ON_USAGE setoid
+"Warning: setoid was deprecated in v2.0.
+Please use Function.Properties.Equivalence.setoid instead."
+#-}
 
 ⇔-setoid : (ℓ : Level) → Setoid (suc ℓ) ℓ
 ⇔-setoid ℓ = record
@@ -103,6 +134,10 @@ setoid s₁ s₂ = record
     ; trans = flip _∘_
     }
   }
+{-# WARNING_ON_USAGE ⇔-setoid
+"Warning: ⇔-setoid was deprecated in v2.0.
+Please use Function.Properties.Equivalence.⇔-setoid instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Transformations
diff --git a/src/Function/Injection.agda b/src/Function/Injection.agda
index 910c91aa35..462b8b3428 100644
--- a/src/Function/Injection.agda
+++ b/src/Function/Injection.agda
@@ -7,11 +7,6 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 {-# OPTIONS --warn=noUserWarning #-}
 
--- Note: use of the standard function hierarchy is encouraged. The
--- module `Function` re-exports `Injective`, `IsInjection` and
--- `Injection`. The alternative definitions found in this file will
--- eventually be deprecated.
-
 module Function.Injection where
 
 {-# WARNING_ON_IMPORT
@@ -35,6 +30,10 @@ Injective {A = A} {B} f = ∀ {x y} → f ⟨$⟩ x ≈₂ f ⟨$⟩ y → x ≈
   where
   open Setoid A renaming (_≈_ to _≈₁_)
   open Setoid B renaming (_≈_ to _≈₂_)
+{-# WARNING_ON_USAGE Injective
+"Warning: Injective was deprecated in v2.0.
+Please use Function.(Definitions.)Injective instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all injections between two setoids
@@ -47,6 +46,10 @@ record Injection {f₁ f₂ t₁ t₂}
     injective : Injective to
 
   open Π to public
+{-# WARNING_ON_USAGE Injection
+"Warning: Injection was deprecated in v2.0.
+Please use Function.(Bundles.)Injection instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all injections from one set to another (i.e. injections
@@ -56,6 +59,10 @@ infix 3 _↣_
 
 _↣_ : ∀ {f t} → Set f → Set t → Set _
 From ↣ To = Injection (P.setoid From) (P.setoid To)
+{-# WARNING_ON_USAGE _↣_
+"Warning: _↣_ was deprecated in v2.0.
+Please use Function.(Bundles.)_↣_ instead."
+#-}
 
 injection : ∀ {f t} {From : Set f} {To : Set t} → (to : From → To) →
             (∀ {x y} → to x ≡ to y → x ≡ y) → From ↣ To
@@ -66,6 +73,10 @@ injection to injective = record
     }
   ; injective = injective
   }
+{-# WARNING_ON_USAGE injection
+"Warning: injection was deprecated in v2.0.
+Please use Function.(Bundles.)mk↣ instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Identity and composition.
@@ -77,6 +88,11 @@ id = record
   { to        = F.id
   ; injective = Fun.id
   }
+{-# WARNING_ON_USAGE id
+"Warning: id was deprecated in v2.0.
+Please use Function.Properties.Injection.refl or
+Function.Construct.Identity.injection instead."
+#-}
 
 _∘_ : ∀ {f₁ f₂ m₁ m₂ t₁ t₂}
         {F : Setoid f₁ f₂} {M : Setoid m₁ m₂} {T : Setoid t₁ t₂} →
@@ -85,3 +101,8 @@ f ∘ g = record
   { to        =          to        f  ⟪∘⟫ to        g
   ; injective = (λ {_} → injective g) ⟨∘⟩ injective f
   } where open Injection
+{-# WARNING_ON_USAGE _∘_
+"Warning: _∘_ was deprecated in v2.0.
+Please use Function.Properties.Injection.trans or
+Function.Construct.Composition.injection instead."
+#-}
diff --git a/src/Function/Inverse.agda b/src/Function/Inverse.agda
index e2cb395ea5..e9d796eddf 100644
--- a/src/Function/Inverse.agda
+++ b/src/Function/Inverse.agda
@@ -35,6 +35,10 @@ record _InverseOf_ {f₁ f₂ t₁ t₂}
   field
     left-inverse-of  : from LeftInverseOf  to
     right-inverse-of : from RightInverseOf to
+{-# WARNING_ON_USAGE _InverseOf_
+"Warning: _InverseOf_ was deprecated in v2.0.
+Please use Function.(Structures.)IsInverse instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all inverses between two setoids
@@ -74,6 +78,10 @@ record Inverse {f₁ f₂ t₁ t₂}
   open Bijection bijection public
     using (equivalence; surjective; surjection; right-inverse;
            to-from; from-to)
+{-# WARNING_ON_USAGE Inverse
+"Warning: Inverse was deprecated in v2.0.
+Please use Function.(Bundles.)Inverse instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all inverses between two sets (i.e. inverses with
@@ -83,9 +91,17 @@ infix 3 _↔_ _↔̇_
 
 _↔_ : ∀ {f t} → Set f → Set t → Set _
 From ↔ To = Inverse (P.setoid From) (P.setoid To)
+{-# WARNING_ON_USAGE _↔_
+"Warning: _↔_ was deprecated in v2.0.
+Please use Function.(Bundles.)_↔_ instead."
+#-}
 
 _↔̇_ : ∀ {i f t} {I : Set i} → Pred I f → Pred I t → Set _
 From ↔̇ To = ∀ {i} → From i ↔ To i
+{-# WARNING_ON_USAGE _↔̇_
+"Warning: _↔̇_ was deprecated in v2.0.
+Please use Function.Indexed.(Bundles.)_↔ᵢ_ instead."
+#-}
 
 inverse : ∀ {f t} {From : Set f} {To : Set t} →
           (to : From → To) (from : To → From) →
@@ -100,6 +116,10 @@ inverse to from from∘to to∘from = record
     ; right-inverse-of = to∘from
     }
   }
+{-# WARNING_ON_USAGE inverse
+"Warning: inverse was deprecated in v2.0.
+Please use Function.(Bundles.)mk↔ instead."
+#-}
 
 ------------------------------------------------------------------------
 -- If two setoids are in bijective correspondence, then there is an
@@ -131,6 +151,11 @@ id {x = S} = record
     ; right-inverse-of = LeftInverse.left-inverse-of id′
     }
   } where id′ = Left.id {S = S}
+{-# WARNING_ON_USAGE id
+"Warning: id was deprecated in v2.0.
+Please use either Function.Properties.Inverse.refl or
+Function.Construct.Identity.inverse instead."
+#-}
 
 -- Transitivity
 
@@ -148,6 +173,11 @@ f ∘ g = record
     ; right-inverse-of = LeftInverse.left-inverse-of (Left._∘_ (right-inverse g) (right-inverse f))
     }
   } where open Inverse
+{-# WARNING_ON_USAGE _∘_
+"Warning: _∘_ was deprecated in v2.0.
+Please use either Function.Properties.Inverse.trans or
+Function.Construct.Composition.inverse instead."
+#-}
 
 -- Symmetry.
 
@@ -161,6 +191,11 @@ sym inv = record
     ; right-inverse-of = left-inverse-of
     }
   } where open Inverse inv
+{-# WARNING_ON_USAGE sym
+"Warning: sym was deprecated in v2.0.
+Please use either Function.Properties.Inverse.sym or
+Function.Construct.Symmetry.inverse instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Transformations
diff --git a/src/Function/LeftInverse.agda b/src/Function/LeftInverse.agda
index 7ff0fda218..3824ca95d3 100644
--- a/src/Function/LeftInverse.agda
+++ b/src/Function/LeftInverse.agda
@@ -31,11 +31,19 @@ _LeftInverseOf_ :
   To ⟶ From → From ⟶ To → Set _
 _LeftInverseOf_ {From = From} f g = ∀ x → f ⟨$⟩ (g ⟨$⟩ x) ≈ x
   where open Setoid From
+{-# WARNING_ON_USAGE _LeftInverseOf_
+"Warning: _LeftInverseOf_ was deprecated in v2.0.
+Please use Function.(Structures.)IsRightInverse instead."
+#-}
 
 _RightInverseOf_ :
   ∀ {f₁ f₂ t₁ t₂} {From : Setoid f₁ f₂} {To : Setoid t₁ t₂} →
   To ⟶ From → From ⟶ To → Set _
 f RightInverseOf g = g LeftInverseOf f
+{-# WARNING_ON_USAGE _RightInverseOf_
+"Warning: _RightInverseOf_ was deprecated in v2.0.
+Please use Function.(Structures.)IsLeftInverse instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all left inverses between two setoids.
@@ -74,12 +82,20 @@ record LeftInverse {f₁ f₂ t₁ t₂}
     from ⟨$⟩ y           ≈⟨ Eq.cong from (T.sym to-x≈y) ⟩
     from ⟨$⟩ (to ⟨$⟩ x)  ≈⟨ left-inverse-of x ⟩
     x                    ∎
+{-# WARNING_ON_USAGE LeftInverse
+"Warning: LeftInverse was deprecated in v2.0.
+Please use Function.(Bundles.)RightInverse instead."
+#-}
 
 -- The set of all right inverses between two setoids.
 
 RightInverse : ∀ {f₁ f₂ t₁ t₂}
                (From : Setoid f₁ f₂) (To : Setoid t₁ t₂) → Set _
 RightInverse From To = LeftInverse To From
+{-# WARNING_ON_USAGE RightInverse
+"Warning: RightInverse was deprecated in v2.0.
+Please use Function.(Bundles.)LeftInverse instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all left inverses from one set to another (i.e. left
@@ -91,6 +107,10 @@ infix 3 _↞_
 
 _↞_ : ∀ {f t} → Set f → Set t → Set _
 From ↞ To = LeftInverse (P.setoid From) (P.setoid To)
+{-# WARNING_ON_USAGE _↞_
+"Warning: _↞_ was deprecated in v2.0.
+Please use Function.(Bundles.)_↪_ instead."
+#-}
 
 leftInverse : ∀ {f t} {From : Set f} {To : Set t} →
               (to : From → To) (from : To → From) →
@@ -101,6 +121,10 @@ leftInverse to from invˡ = record
   ; from            = Eq.→-to-⟶ from
   ; left-inverse-of = invˡ
   }
+{-# WARNING_ON_USAGE leftInverse
+"Warning: leftInverse was deprecated in v2.0.
+Please use Function.(Bundles.)mk↪ instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Identity and composition.
@@ -111,6 +135,11 @@ id {S = S} = record
   ; from            = Eq.id
   ; left-inverse-of = λ _ → Setoid.refl S
   }
+{-# WARNING_ON_USAGE id
+"Warning: id was deprecated in v2.0.
+Please use either Function.Properties.RightInverse.refl or
+Function.Construct.Identity.rightInverse instead."
+#-}
 
 infixr 9 _∘_
 
@@ -128,3 +157,8 @@ _∘_ {F = F} f g = record
   where
   open LeftInverse
   open EqReasoning F
+{-# WARNING_ON_USAGE _∘_
+"Warning: _∘_ was deprecated in v2.0.
+Please use either Function.Properties.RightInverse.trans or
+Function.Construct.Composition.rightInverse instead."
+#-}
diff --git a/src/Function/Properties/Equivalence.agda b/src/Function/Properties/Equivalence.agda
index 308a29525d..ad1264ec7c 100644
--- a/src/Function/Properties/Equivalence.agda
+++ b/src/Function/Properties/Equivalence.agda
@@ -11,6 +11,7 @@ module Function.Properties.Equivalence where
 
 open import Function.Bundles
 open import Level
+open import Relation.Binary.Definitions
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 import Relation.Binary.PropositionalEquality.Properties as Eq
@@ -21,7 +22,7 @@ import Function.Construct.Composition as Composition
 
 private
   variable
-    a ℓ : Level
+    a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level
     A B : Set a
     S T : Setoid a ℓ
 
@@ -36,13 +37,37 @@ mkEquivalence f g = record
   ; from-cong = cong g
   } where open Func
 
+⟶×⟵⇒⇔ : A ⟶ B → B ⟶ A → A ⇔ B
+⟶×⟵⇒⇔ = mkEquivalence
+
+------------------------------------------------------------------------
+-- Destructors
+
+⇔⇒⟶ : A ⇔ B → A ⟶ B
+⇔⇒⟶ = Equivalence.toFunction
+
+⇔⇒⟵ : A ⇔ B → B ⟶ A
+⇔⇒⟵ = Equivalence.fromFunction
+
 ------------------------------------------------------------------------
 -- Setoid bundles
 
+refl : Reflexive (Equivalence {a} {ℓ})
+refl {x = x} = Identity.equivalence x
+
+sym : Sym (Equivalence {a} {ℓ₁} {b} {ℓ₂})
+          (Equivalence {b} {ℓ₂} {a} {ℓ₁})
+sym = Symmetry.equivalence
+
+trans : Trans (Equivalence {a} {ℓ₁} {b} {ℓ₂})
+              (Equivalence {b} {ℓ₂} {c} {ℓ₃})
+              (Equivalence {a} {ℓ₁} {c} {ℓ₃})
+trans = Composition.equivalence
+
 isEquivalence : IsEquivalence (Equivalence {a} {ℓ})
 isEquivalence = record
-  { refl = λ {x} → Identity.equivalence x
-  ; sym = Symmetry.equivalence
+  { refl = refl
+  ; sym = sym
   ; trans = Composition.equivalence
   }
 
@@ -69,12 +94,3 @@ setoid s₁ s₂ = record
   ; _≈_           = _⇔_
   ; isEquivalence = ⇔-isEquivalence
   }
-
-⟶×⟵⇒⇔ : A ⟶ B → B ⟶ A → A ⇔ B
-⟶×⟵⇒⇔ = mkEquivalence
-
-⇔⇒⟶ : A ⇔ B → A ⟶ B
-⇔⇒⟶ = Equivalence.toFunction
-
-⇔⇒⟵ : A ⇔ B → B ⟶ A
-⇔⇒⟵ = Equivalence.fromFunction
diff --git a/src/Function/Properties/Injection.agda b/src/Function/Properties/Injection.agda
index bca8ac4dc5..0e773cf9b8 100644
--- a/src/Function/Properties/Injection.agda
+++ b/src/Function/Properties/Injection.agda
@@ -15,13 +15,14 @@ import Function.Construct.Identity as Identity
 import Function.Construct.Composition as Compose
 open import Level using (Level)
 open import Data.Product.Base using (proj₁; proj₂)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.Definitions
+open import Relation.Binary.PropositionalEquality using ()
 open import Relation.Binary using (Setoid)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 private
   variable
-    a b ℓ : Level
+    a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level
     A B : Set a
     T S U : Setoid a ℓ
 
@@ -43,10 +44,21 @@ mkInjection f injective = record
 ↣⇒⟶ = Injection.function
 
 ------------------------------------------------------------------------
--- Properties
+-- Setoid properties
+
+refl : Reflexive (Injection {a} {ℓ})
+refl {x = x} = Identity.injection x
+
+trans : Trans (Injection {a} {ℓ₁} {b} {ℓ₂})
+              (Injection {b} {ℓ₂} {c} {ℓ₃})
+              (Injection {a} {ℓ₁} {c} {ℓ₃})
+trans = Compose.injection
+
+------------------------------------------------------------------------
+-- Propositonal properties
 
 ↣-refl : Injection S S
-↣-refl = Identity.injection _
+↣-refl = refl
 
 ↣-trans : Injection S T → Injection T U → Injection S U
-↣-trans = Compose.injection
+↣-trans = trans
diff --git a/src/Function/Properties/Surjection.agda b/src/Function/Properties/Surjection.agda
index d16670639c..5c2b235506 100644
--- a/src/Function/Properties/Surjection.agda
+++ b/src/Function/Properties/Surjection.agda
@@ -11,15 +11,18 @@ module Function.Properties.Surjection where
 open import Function.Base
 open import Function.Definitions
 open import Function.Bundles
+import Function.Construct.Identity as Identity
+import Function.Construct.Composition as Compose
 open import Level using (Level)
 open import Data.Product.Base using (proj₁; proj₂)
-open import Relation.Binary.PropositionalEquality
-open import Relation.Binary using (Setoid)
+import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.Definitions
+open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 private
   variable
-    a b ℓ : Level
+    a b c ℓ ℓ₁ ℓ₂ ℓ₃ : Level
     A B : Set a
     T S : Setoid a ℓ
 
@@ -41,13 +44,24 @@ mkSurjection f surjective = record
 ↠⇒⟶ = Surjection.function
 
 ↠⇒↪ : A ↠ B → B ↪ A
-↠⇒↪ s = mk↪ {from = to} λ { refl → proj₂ (strictlySurjective _)}
+↠⇒↪ s = mk↪ {from = to} λ { P.refl → proj₂ (strictlySurjective _)}
   where open Surjection s
 
 ↠⇒⇔ : A ↠ B → A ⇔ B
 ↠⇒⇔ s = mk⇔ to (proj₁ ∘ surjective)
   where open Surjection s
 
+------------------------------------------------------------------------
+-- Setoid properties
+
+refl : Reflexive (Surjection {a} {ℓ})
+refl {x = x} = Identity.surjection x
+
+trans : Trans (Surjection {a} {ℓ₁} {b} {ℓ₂})
+              (Surjection {b} {ℓ₂} {c} {ℓ₃})
+              (Surjection {a} {ℓ₁} {c} {ℓ₃})
+trans = Compose.surjection
+
 ------------------------------------------------------------------------
 -- Other
 
diff --git a/src/Function/Surjection.agda b/src/Function/Surjection.agda
index bd96c50efb..2fdfefa9af 100644
--- a/src/Function/Surjection.agda
+++ b/src/Function/Surjection.agda
@@ -33,6 +33,10 @@ record Surjective {f₁ f₂ t₁ t₂}
   field
     from             : To ⟶ From
     right-inverse-of : from RightInverseOf to
+{-# WARNING_ON_USAGE Surjective
+"Warning: Surjective was deprecated in v2.0.
+Please use Function.(Definitions.)Surjective instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all surjections from one setoid to another.
@@ -67,6 +71,10 @@ record Surjection {f₁ f₂ t₁ t₂}
     { to   = to
     ; from = from
     }
+{-# WARNING_ON_USAGE Surjection
+"Warning: Surjection was deprecated in v2.0.
+Please use Function.(Bundles.)Surjection instead."
+#-}
 
 -- Right inverses can be turned into surjections.
 
@@ -80,6 +88,10 @@ fromRightInverse r = record
     ; right-inverse-of = left-inverse-of
     }
   } where open LeftInverse r
+{-# WARNING_ON_USAGE fromRightInverse
+"Warning: fromRightInverse was deprecated in v2.0.
+Please use Function.(Properties.)RightInverse.RightInverse⇒Surjection instead."
+#-}
 
 ------------------------------------------------------------------------
 -- The set of all surjections from one set to another (i.e. sujections
@@ -89,6 +101,10 @@ infix 3 _↠_
 
 _↠_ : ∀ {f t} → Set f → Set t → Set _
 From ↠ To = Surjection (P.setoid From) (P.setoid To)
+{-# WARNING_ON_USAGE _↠_
+"Warning: _↠_ was deprecated in v2.0.
+Please use Function.(Bundles.)_↠_ instead."
+#-}
 
 surjection : ∀ {f t} {From : Set f} {To : Set t} →
              (to : From → To) (from : To → From) →
@@ -101,6 +117,10 @@ surjection to from surjective = record
     ; right-inverse-of = surjective
     }
   }
+{-# WARNING_ON_USAGE surjection
+"Warning: surjection was deprecated in v2.0.
+Please use Function.(Bundles.)mk↠ instead."
+#-}
 
 ------------------------------------------------------------------------
 -- Identity and composition.
@@ -113,6 +133,11 @@ id {S = S} = record
     ; right-inverse-of = LeftInverse.left-inverse-of id′
     }
   } where id′ = Left.id {S = S}
+{-# WARNING_ON_USAGE id
+"Warning: id was deprecated in v2.0.
+Please use Function.Properties.Surjection.refl or
+Function.Construct.Identity.surjection instead."
+#-}
 
 infixr 9 _∘_
 
@@ -129,3 +154,8 @@ f ∘ g = record
   where
   open Surjection
   g∘f = Left._∘_ (right-inverse g) (right-inverse f)
+{-# WARNING_ON_USAGE _∘_
+"Warning: _∘_ was deprecated in v2.0.
+Please use Function.Properties.Surjection.trans or
+Function.Construct.Composition.surjection instead."
+#-}

From daa4437050987baed6de2f59aa1b00ad75e9a5b3 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 19 Oct 2023 09:40:49 +0900
Subject: [PATCH 169/316] Rename symmetric reasoning combinators. Minimal set
 of changes (#2160)

---
 CHANGELOG.md                                  |  63 ++++++--
 README/Data/Fin/Relation/Unary/Top.agda       |   4 +-
 .../Vec/Relation/Binary/Equality/Cast.agda    |  24 +--
 README/Tactic/Cong.agda                       |   6 +-
 .../Properties/HeytingCommutativeRing.agda    |  16 +-
 src/Algebra/Consequences/Setoid.agda          |  10 +-
 src/Algebra/Construct/LexProduct/Inner.agda   |  36 ++---
 src/Algebra/Construct/LiftedChoice.agda       |   6 +-
 .../Construct/NaturalChoice/MinMaxOp.agda     |  16 +-
 .../Construct/NaturalChoice/MinOp.agda        |  22 +--
 .../Lattice/Morphism/LatticeMonomorphism.agda |   4 +-
 .../Lattice/Properties/BooleanAlgebra.agda    |  96 ++++++------
 src/Algebra/Lattice/Properties/Lattice.agda   |   4 +-
 src/Algebra/Module/Consequences.agda          |   8 +-
 src/Algebra/Morphism/Consequences.agda        |   8 +-
 src/Algebra/Morphism/GroupMonomorphism.agda   |   6 +-
 src/Algebra/Morphism/MagmaMonomorphism.agda   |  12 +-
 src/Algebra/Morphism/MonoidMonomorphism.agda  |   4 +-
 src/Algebra/Morphism/RingMonomorphism.agda    |  12 +-
 src/Algebra/Properties/AbelianGroup.agda      |   2 +-
 .../CommutativeMonoid/Mult/TCOptimised.agda   |   2 +-
 .../Properties/CommutativeMonoid/Sum.agda     |   4 +-
 .../Properties/CommutativeSemigroup.agda      |   4 +-
 src/Algebra/Properties/Group.agda             |  16 +-
 src/Algebra/Properties/Monoid/Mult.agda       |   2 +-
 .../Properties/Monoid/Mult/TCOptimised.agda   |   8 +-
 src/Algebra/Properties/Monoid/Sum.agda        |   4 +-
 src/Algebra/Properties/Semiring/Binomial.agda |  34 ++---
 src/Algebra/Properties/Semiring/Exp.agda      |   8 +-
 .../Semiring/Exp/TailRecursiveOptimised.agda  |   2 +-
 src/Algebra/Properties/Semiring/Mult.agda     |   4 +-
 .../Properties/Semiring/Mult/TCOptimised.agda |   2 +-
 .../Solver/Ring/NaturalCoefficients.agda      |   2 +-
 .../Stream/Relation/Binary/Pointwise.agda     |  35 ++++-
 src/Codata/Sized/Colist/Properties.agda       |   4 +-
 src/Data/Fin/Properties.agda                  |  16 +-
 src/Data/Fin/Subset/Properties.agda           |   2 +-
 src/Data/Integer/DivMod.agda                  |   2 +-
 src/Data/Integer/Properties.agda              | 102 ++++++-------
 src/Data/List/Effectful.agda                  |  14 +-
 src/Data/List/Properties.agda                 |   4 +-
 .../Relation/Binary/BagAndSetEquality.agda    |   8 +-
 .../Binary/Permutation/Propositional.agda     |   6 +-
 .../Permutation/Propositional/Properties.agda |  10 +-
 .../Relation/Binary/Permutation/Setoid.agda   |   8 +-
 .../Binary/Permutation/Setoid/Properties.agda |  20 +--
 .../Binary/Subset/Setoid/Properties.agda      |   2 +-
 .../List/Relation/Unary/Any/Properties.agda   |   4 +-
 src/Data/Nat/Binary/Properties.agda           |   8 +-
 src/Data/Nat/Combinatorics.agda               |  30 ++--
 src/Data/Nat/Combinatorics/Specification.agda |   8 +-
 src/Data/Nat/DivMod.agda                      |  50 +++----
 src/Data/Nat/DivMod/WithK.agda                |   2 +-
 src/Data/Nat/Divisibility.agda                |   6 +-
 src/Data/Nat/LCM.agda                         |   6 +-
 src/Data/Nat/Primality.agda                   |   6 +-
 src/Data/Nat/Properties.agda                  |  18 +--
 src/Data/Rational/Properties.agda             |  53 +++----
 .../Rational/Unnormalised/Properties.agda     | 117 +++++++--------
 src/Data/Tree/Binary/Zipper/Properties.agda   |  12 +-
 src/Data/Tree/Rose/Properties.agda            |   4 +-
 src/Data/Vec/Functional/Properties.agda       |   8 +-
 src/Data/Vec/Properties.agda                  |  30 ++--
 .../Vec/Relation/Binary/Equality/Cast.agda    |  53 ++++---
 src/Function/Properties/Surjection.agda       |   2 +-
 .../ReflexiveTransitive/Properties.agda       |   2 +-
 .../Binary/Construct/NaturalOrder/Right.agda  |   2 +-
 .../Binary/HeterogeneousEquality.agda         |  14 +-
 .../Binary/Reasoning/PartialOrder.agda        |   4 +-
 src/Relation/Binary/Reasoning/Preorder.agda   |   2 +-
 src/Relation/Binary/Reasoning/Setoid.agda     |   6 +-
 .../Binary/Reasoning/StrictPartialOrder.agda  |   4 +-
 src/Relation/Binary/Reasoning/Syntax.agda     | 137 ++++++++++++++----
 src/Tactic/Cong.agda                          |   2 +-
 src/Tactic/MonoidSolver.agda                  |   4 +-
 .../Core/Polynomial/Homomorphism/Lemmas.agda  |   8 +-
 src/Text/Pretty/Core.agda                     |   2 +-
 77 files changed, 716 insertions(+), 572 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e7d43ec18f..5649103aee 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -849,6 +849,7 @@ Non-backwards compatible changes
   IO.Effectful
   IO.Instances
   ```
+
 ### (Issue #2096) Introduction of flipped relation symbol for `Relation.Binary.Bundles.Preorder`
 
 * Previously, the relation symbol `_∼_`  was (notationally) symmetric, so that its
@@ -989,6 +990,53 @@ Non-backwards compatible changes
   Relation.Binary.Reasoning.PartialOrder
   ```
 
+### More modular design of equational reasoning.
+
+* Have introduced a new module `Relation.Binary.Reasoning.Syntax` which exports
+  a range of modules containing pre-existing reasoning combinator syntax.
+
+* This makes it possible to add new or rename existing reasoning combinators to a
+  pre-existing `Reasoning` module in just a couple of lines
+  (e.g. see `∣-Reasoning` in `Data.Nat.Divisibility`)
+
+* One pre-requisite for that is that `≡-Reasoning` has been moved from
+  `Relation.Binary.PropositionalEquality.Core` (which shouldn't be
+  imported anyway as it's a `Core` module) to 
+  `Relation.Binary.PropositionalEquality.Properties`.
+  It is still exported by `Relation.Binary.PropositionalEquality`.
+
+### Renaming of symmetric combinators for equational reasoning
+
+* We've had various complaints about the symmetric version of reasoning combinators 
+  that use the syntax `_R˘⟨_⟩_` for some relation `R`, (e.g. `_≡˘⟨_⟩_` and `_≃˘⟨_⟩_`)
+  introduced in `v1.0`. In particular:
+  1. The symbol `˘` is hard to type.
+  2. The symbol `˘` doesn't convey the direction of the equality
+  3. The left brackets aren't vertically aligned with the left brackets of the non-symmetric version.
+  
+* To address these problems we have renamed all the symmetric versions of the 
+  combinators from `_R˘⟨_⟩_` to `_R⟨_⟨_` (the direction of the last bracket is flipped
+  to indicate the quality goes from right to left).
+  
+* The old combinators still exist but have been deprecated. However due to 
+  [Agda issue #5617](https://github.com/agda/agda/issues/5617), the deprecation warnings
+  don't fire correctly. We will not remove the old combinators before the above issue is
+  addressed. However, we still encourage migrating to the new combinators!
+
+* On a Linux-based system, the following command was used to globally migrate all uses of the
+  old combinators to the new ones in the standard library itself. 
+  It *may* be useful when trying to migrate your own code:
+  ```bash
+  find . -type f -name "*.agda" -print0 | xargs -0 sed -i -e 's/˘⟨\(.*\)⟩/⟨\1⟨/g'
+  ```
+  USUAL CAVEATS: It has not been widely tested and the standard library developers are not 
+  responsible for the results of this command. It is strongly recommended you back up your 
+  work before you attempt to run it.
+  
+* NOTE: this refactoring may require some extra bracketing around the operator `_⟨_⟩_` from 
+  `Function.Base` if the `_⟨_⟩_` operator is used within the reasoning proof. The symptom
+  for this is a `Don't know how to parse` error.
+
 ### Other
 
 * In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
@@ -1228,21 +1276,6 @@ Major improvements
   * `RawLattice` has been moved from `Algebra.Lattice.Bundles` to this new module.
 
 * In `Relation.Binary.Reasoning.Base.Triple`, added a new parameter `<-irrefl : Irreflexive _≈_ _<_`
-
-### More modular design of equational reasoning.
-
-* Have introduced a new module `Relation.Binary.Reasoning.Syntax` which exports
-  a range of modules containing pre-existing reasoning combinator syntax.
-
-* This makes it possible to add new or rename existing reasoning combinators to a
-  pre-existing `Reasoning` module in just a couple of lines
-  (e.g. see `∣-Reasoning` in `Data.Nat.Divisibility`)
-
-* One pre-requisite for that is that `≡-Reasoning` has been moved from
-  `Relation.Binary.PropositionalEquality.Core` (which shouldn't be
-  imported anyway as it's a `Core` module) to 
-  `Relation.Binary.PropositionalEquality.Properties`.
-  It is still exported by `Relation.Binary.PropositionalEquality`.
   
 Deprecated modules
 ------------------
diff --git a/README/Data/Fin/Relation/Unary/Top.agda b/README/Data/Fin/Relation/Unary/Top.agda
index e602932827..390a48d58f 100644
--- a/README/Data/Fin/Relation/Unary/Top.agda
+++ b/README/Data/Fin/Relation/Unary/Top.agda
@@ -76,8 +76,8 @@ opposite-prop {suc n} i with view i
 ... | ‵fromℕ  rewrite toℕ-fromℕ n | n∸n≡0 n = refl
 ... | ‵inject₁ j = begin
   suc (toℕ (opposite j)) ≡⟨ cong suc (opposite-prop j) ⟩
-  suc (n ∸ suc (toℕ j))  ≡˘⟨ +-∸-assoc 1 (toℕ<n j) ⟩
-  n ∸ toℕ j              ≡˘⟨ cong (n ∸_) (toℕ-inject₁ j) ⟩
+  suc (n ∸ suc (toℕ j))  ≡⟨ +-∸-assoc 1 (toℕ<n j) ⟨
+  n ∸ toℕ j              ≡⟨ cong (n ∸_) (toℕ-inject₁ j) ⟨
   n ∸ toℕ (inject₁ j)    ∎ where open ≡-Reasoning
 
 ------------------------------------------------------------------------
diff --git a/README/Data/Vec/Relation/Binary/Equality/Cast.agda b/README/Data/Vec/Relation/Binary/Equality/Cast.agda
index fb76544ed7..42cd8bfd3d 100644
--- a/README/Data/Vec/Relation/Binary/Equality/Cast.agda
+++ b/README/Data/Vec/Relation/Binary/Equality/Cast.agda
@@ -86,7 +86,7 @@ example1a-fromList-∷ʳ : ∀ (x : A) xs .(eq : L.length (xs L.∷ʳ x) ≡ suc
                         cast eq (fromList (xs L.∷ʳ x)) ≡ fromList xs ∷ʳ x
 example1a-fromList-∷ʳ x xs eq = begin
   cast eq (fromList (xs L.∷ʳ x))                   ≡⟨⟩
-  cast eq (fromList (xs L.++ L.[ x ]))             ≡˘⟨ cast-trans eq₁ eq₂ (fromList (xs L.++ L.[ x ])) ⟩
+  cast eq (fromList (xs L.++ L.[ x ]))             ≡⟨ cast-trans eq₁ eq₂ (fromList (xs L.++ L.[ x ])) ⟨
   cast eq₂ (cast eq₁ (fromList (xs L.++ L.[ x ]))) ≡⟨ cong (cast eq₂) (fromList-++ xs) ⟩
   cast eq₂ (fromList xs ++ [ x ])                  ≡⟨ ≈-sym (unfold-∷ʳ (sym eq₂) x (fromList xs)) ⟩
   fromList xs ∷ʳ x                                 ∎
@@ -105,7 +105,7 @@ example1b-fromList-∷ʳ : ∀ (x : A) xs .(eq : L.length (xs L.∷ʳ x) ≡ suc
 example1b-fromList-∷ʳ x xs eq = begin
   fromList (xs L.∷ʳ x)       ≈⟨⟩
   fromList (xs L.++ L.[ x ]) ≈⟨ fromList-++ xs ⟩
-  fromList xs ++ [ x ]       ≈˘⟨ unfold-∷ʳ (+-comm 1 (L.length xs)) x (fromList xs) ⟩
+  fromList xs ++ [ x ]       ≈⟨ unfold-∷ʳ (+-comm 1 (L.length xs)) x (fromList xs) ⟨
   fromList xs ∷ʳ x           ∎
   where open CastReasoning
 
@@ -120,10 +120,10 @@ example1b-fromList-∷ʳ x xs eq = begin
 --     ≈-trans     : xs ≈[ m≡n ] ys → ys ≈[ n≡o ] zs → xs ≈[ trans m≡n n≡o ] zs
 -- Accordingly, `_≈[_]_` admits the standard set of equational reasoning
 -- combinators. Suppose `≈-eqn : xs ≈[ m≡n ] ys`,
---     xs ≈⟨ ≈-eqn  ⟩   -- `_≈⟨_⟩_` takes a `_≈[_]_` step, adjusting
---     ys               -- the index at the same time
+--     xs ≈⟨ ≈-eqn ⟩   -- `_≈⟨_⟩_` takes a `_≈[_]_` step, adjusting
+--     ys              -- the index at the same time
 --
---     ys ≈˘⟨ ≈-eqn ⟩   -- `_≈˘⟨_⟩_` takes a symmetric `_≈[_]_` step
+--     ys ≈⟨ ≈-eqn ⟨   -- `_≈⟨_⟨_` takes a symmetric `_≈[_]_` step
 --     xs
 example2a : ∀ .(eq : suc m + n ≡ m + suc n) (xs : Vec A m) a ys →
             cast eq ((reverse xs ∷ʳ a) ++ ys) ≡ reverse xs ++ (a ∷ ys)
@@ -138,7 +138,7 @@ example2a eq xs a ys = begin
 --     xs ≂⟨ ≡-eqn  ⟩    -- Takes a `_≡_` step; no change to the index
 --     ys
 --
---     ys ≂˘⟨ ≡-eqn ⟩    -- Takes a symmetric `_≡_` step
+--     ys ≂⟨ ≡-eqn ⟨    -- Takes a symmetric `_≡_` step
 --     xs
 -- Equivalently, `≈-reflexive` injects `_≡_` into `_≈[_]_`. That is,
 -- `xs ≂⟨ ≡-eqn ⟩ ys` is the same as `xs ≈⟨ ≈-reflexive ≡-eqn ⟩ ys`.
@@ -149,7 +149,7 @@ example2b eq xs a ys = begin
   (a ∷ xs) ʳ++ ys         ≂⟨ unfold-ʳ++ (a ∷ xs) ys ⟩          -- index: suc m + n
   reverse (a ∷ xs) ++ ys  ≂⟨ cong (_++ ys) (reverse-∷ a xs) ⟩  -- index: suc m + n
   (reverse xs ∷ʳ a) ++ ys ≈⟨ ∷ʳ-++ eq a (reverse xs) ⟩         -- index: suc m + n
-  reverse xs ++ (a ∷ ys)  ≂˘⟨ unfold-ʳ++ xs (a ∷ ys) ⟩         -- index: m + suc n
+  reverse xs ++ (a ∷ ys)  ≂⟨ unfold-ʳ++ xs (a ∷ ys) ⟨         -- index: m + suc n
   xs ʳ++ (a ∷ ys)         ∎                                    -- index: m + suc n
   where open CastReasoning
 
@@ -201,11 +201,11 @@ example3b-fromList-++-++′ {xs = xs} {ys} {zs} eq = begin
 example4-cong² : ∀ .(eq : (m + 1) + n ≡ n + suc m) a (xs : Vec A m) ys →
           cast eq (reverse ((xs ++ [ a ]) ++ ys)) ≡ ys ʳ++ reverse (xs ∷ʳ a)
 example4-cong² {m = m} {n} eq a xs ys = begin
-  reverse ((xs ++ [ a ]) ++ ys)   ≈˘⟨ ≈-cong reverse (cast-reverse (cong (_+ n) (+-comm 1 m)) ((xs ∷ʳ a) ++ ys))
+  reverse ((xs ++ [ a ]) ++ ys)   ≈⟨ ≈-cong reverse (cast-reverse (cong (_+ n) (+-comm 1 m)) ((xs ∷ʳ a) ++ ys))
                                              (≈-cong (_++ ys) (cast-++ˡ (+-comm 1 m) (xs ∷ʳ a))
-                                                     (unfold-∷ʳ _ a xs)) ⟩
+                                                     (unfold-∷ʳ _ a xs)) ⟨
   reverse ((xs ∷ʳ a) ++ ys)       ≈⟨ reverse-++ (+-comm (suc m) n) (xs ∷ʳ a) ys ⟩
-  reverse ys ++ reverse (xs ∷ʳ a) ≂˘⟨ unfold-ʳ++ ys (reverse (xs ∷ʳ a)) ⟩
+  reverse ys ++ reverse (xs ∷ʳ a) ≂⟨ unfold-ʳ++ ys (reverse (xs ∷ʳ a)) ⟨
   ys ʳ++ reverse (xs ∷ʳ a)        ∎
   where open CastReasoning
 
@@ -237,7 +237,7 @@ example5-fromList-++-++′ {xs = xs} {ys} {zs} eq = begin
 -- and then switch to the reasoning system of `_≈[_]_`.
 example6a-reverse-∷ʳ : ∀ x (xs : Vec A n) → reverse (xs ∷ʳ x) ≡ x ∷ reverse xs
 example6a-reverse-∷ʳ {n = n} x xs = begin-≡
-  reverse (xs ∷ʳ x)     ≡˘⟨ ≈-reflexive refl ⟩
+  reverse (xs ∷ʳ x)     ≡⟨ ≈-reflexive refl ⟨
   reverse (xs ∷ʳ x)     ≈⟨ ≈-cong reverse (cast-reverse _ _) (unfold-∷ʳ (+-comm 1 n) x xs) ⟩
   reverse (xs ++ [ x ]) ≈⟨ reverse-++ (+-comm n 1) xs [ x ] ⟩
   x ∷ reverse xs        ∎
@@ -249,6 +249,6 @@ example6b-reverse-∷ʳ-by-induction x (y ∷ xs) = begin
   reverse (y ∷ (xs ∷ʳ x)) ≡⟨ reverse-∷ y (xs ∷ʳ x) ⟩
   reverse (xs ∷ʳ x) ∷ʳ y  ≡⟨ cong (_∷ʳ y) (example6b-reverse-∷ʳ-by-induction x xs) ⟩
   (x ∷ reverse xs) ∷ʳ y   ≡⟨⟩
-  x ∷ (reverse xs ∷ʳ y)   ≡˘⟨ cong (x ∷_) (reverse-∷ y xs) ⟩
+  x ∷ (reverse xs ∷ʳ y)   ≡⟨ cong (x ∷_) (reverse-∷ y xs) ⟨
   x ∷ reverse (y ∷ xs)    ∎
   where open ≡-Reasoning
diff --git a/README/Tactic/Cong.agda b/README/Tactic/Cong.agda
index 6503604d71..20cb5ad7d9 100644
--- a/README/Tactic/Cong.agda
+++ b/README/Tactic/Cong.agda
@@ -33,7 +33,7 @@ verbose-example m n eq =
     suc (suc m) + m
   ≡⟨ cong (λ ϕ → suc (suc (ϕ + ϕ))) eq ⟩
     suc (suc n) + n
-  ≡˘⟨ cong (λ ϕ → suc (suc (n + ϕ))) (+-identityʳ n) ⟩
+  ≡⟨ cong (λ ϕ → suc (suc (n + ϕ))) (+-identityʳ n) ⟨
     suc (suc n) + (n + 0)
   ∎
 
@@ -51,7 +51,7 @@ succinct-example m n eq =
     suc (suc m) + m
   ≡⟨ cong! eq ⟩
     suc (suc n) + n
-  ≡˘⟨ cong! (+-identityʳ n) ⟩
+  ≡⟨ cong! (+-identityʳ n) ⟨
     suc (suc n) + (n + 0)
   ∎
 
@@ -124,7 +124,7 @@ module EquationalReasoningTests where
       suc (suc m) + m
     ≡⟨ cong! eq ⟩
       suc (suc n) + n
-    ≡˘⟨ cong! (+-identityʳ n) ⟩
+    ≡⟨ cong! (+-identityʳ n) ⟨
       suc (suc n) + (n + 0)
     ∎
 
diff --git a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
index 7f4f164f42..868224bf0c 100644
--- a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
+++ b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
@@ -56,11 +56,11 @@ x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
     y⁻¹*x⁻¹*x*y≈1 = begin
       y⁻¹ * x⁻¹ * (x * y - 0#)     ≈⟨ *-congˡ (x-0≈x (x * y)) ⟩
       y⁻¹ * x⁻¹ * (x * y)          ≈⟨ *-assoc y⁻¹ x⁻¹ (x * y) ⟩
-      y⁻¹ * (x⁻¹ * (x * y))       ≈˘⟨ *-congˡ (*-assoc x⁻¹ x y) ⟩
-      y⁻¹ * ((x⁻¹ * x) * y)       ≈˘⟨ *-congˡ (*-congʳ (*-congˡ (x-0≈x x))) ⟩
+      y⁻¹ * (x⁻¹ * (x * y))       ≈⟨ *-congˡ (*-assoc x⁻¹ x y) ⟨
+      y⁻¹ * ((x⁻¹ * x) * y)       ≈⟨ *-congˡ (*-congʳ (*-congˡ (x-0≈x x))) ⟨
       y⁻¹ * ((x⁻¹ * (x - 0#)) * y) ≈⟨ *-congˡ (*-congʳ x⁻¹*x≈1) ⟩
       y⁻¹ * (1# * y)               ≈⟨ *-congˡ (*-identityˡ y) ⟩
-      y⁻¹ * y                     ≈˘⟨ *-congˡ (x-0≈x y) ⟩
+      y⁻¹ * y                     ≈⟨ *-congˡ (x-0≈x y) ⟨
       y⁻¹ * (y - 0#)               ≈⟨ y⁻¹*y≈1 ⟩
       1# ∎
 
@@ -75,15 +75,15 @@ x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
 
   y-x≈-[x-y] : y - x ≈ - (x - y)
   y-x≈-[x-y] = begin
-    y - x     ≈˘⟨ +-congʳ (-‿involutive y) ⟩
-    - - y - x ≈˘⟨ -‿anti-homo-+ x (- y) ⟩
+    y - x     ≈⟨ +-congʳ (-‿involutive y) ⟨
+    - - y - x ≈⟨ -‿anti-homo-+ x (- y) ⟨
     - (x - y) ∎
 
   x-y⁻¹*y-x≈1 : (- x-y⁻¹) * (y - x) ≈ 1#
   x-y⁻¹*y-x≈1 = begin
-    (- x-y⁻¹) * (y - x)   ≈˘⟨ -‿distribˡ-* x-y⁻¹ (y - x) ⟩
+    (- x-y⁻¹) * (y - x)   ≈⟨ -‿distribˡ-* x-y⁻¹ (y - x) ⟨
     - (x-y⁻¹ * (y - x))    ≈⟨ -‿cong (*-congˡ y-x≈-[x-y]) ⟩
-    - (x-y⁻¹ * - (x - y)) ≈˘⟨ -‿cong (-‿distribʳ-* x-y⁻¹ (x - y)) ⟩
+    - (x-y⁻¹ * - (x - y)) ≈⟨ -‿cong (-‿distribʳ-* x-y⁻¹ (x - y)) ⟨
     - - (x-y⁻¹ * (x - y))  ≈⟨ -‿involutive (x-y⁻¹ * ((x - y))) ⟩
     x-y⁻¹ * (x - y)        ≈⟨ InvX-Y .proj₂ .proj₁ ⟩
     1# ∎
@@ -100,7 +100,7 @@ x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
 
     x-z⁻¹*y-z≈1 : x-z⁻¹ * (y - z) ≈ 1#
     x-z⁻¹*y-z≈1 = begin
-      x-z⁻¹ * (y - z) ≈˘⟨ *-congˡ (+-congʳ x≈y) ⟩
+      x-z⁻¹ * (y - z) ≈⟨ *-congˡ (+-congʳ x≈y) ⟨
       x-z⁻¹ * (x - z)  ≈⟨ x-z⁻¹*x-z≈1# ⟩
       1# ∎
 
diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index 1aa2b8f4c8..068ecb2d12 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -88,7 +88,7 @@ module _ {f : Op₁ A} (self : SelfInverse f) where
 
   selfInverse⇒injective : Injective _≈_ _≈_ f
   selfInverse⇒injective {x} {y} x≈y = begin
-    x       ≈˘⟨ self x≈y ⟩
+    x       ≈⟨ self x≈y ⟨
     f (f y) ≈⟨ selfInverse⇒involutive y ⟩
     y       ∎
 
@@ -267,12 +267,12 @@ module _ {_•_ _◦_ : Op₂ A}
                              _◦_ DistributesOver _•_ →
                              _•_ DistributesOverˡ _◦_
   distrib+absorbs⇒distribˡ •-absorbs-◦ ◦-absorbs-• (◦-distribˡ-• , ◦-distribʳ-•) x y z = begin
-    x • (y ◦ z)                    ≈˘⟨ •-cong (•-absorbs-◦ _ _) refl ⟩
+    x • (y ◦ z)                    ≈⟨ •-cong (•-absorbs-◦ _ _) refl ⟨
     (x • (x ◦ y)) • (y ◦ z)        ≈⟨  •-cong (•-cong refl (◦-comm _ _)) refl ⟩
     (x • (y ◦ x)) • (y ◦ z)        ≈⟨  •-assoc _ _ _ ⟩
-    x • ((y ◦ x) • (y ◦ z))        ≈˘⟨ •-cong refl (◦-distribˡ-• _ _ _) ⟩
-    x • (y ◦ (x • z))              ≈˘⟨ •-cong (◦-absorbs-• _ _) refl ⟩
-    (x ◦ (x • z)) • (y ◦ (x • z))  ≈˘⟨ ◦-distribʳ-• _ _ _ ⟩
+    x • ((y ◦ x) • (y ◦ z))        ≈⟨ •-cong refl (◦-distribˡ-• _ _ _) ⟨
+    x • (y ◦ (x • z))              ≈⟨ •-cong (◦-absorbs-• _ _) refl ⟨
+    (x ◦ (x • z)) • (y ◦ (x • z))  ≈⟨ ◦-distribʳ-• _ _ _ ⟨
     (x • y) ◦ (x • z)              ∎
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Construct/LexProduct/Inner.agda b/src/Algebra/Construct/LexProduct/Inner.agda
index 9b6e06f0e6..beff9e9e3a 100644
--- a/src/Algebra/Construct/LexProduct/Inner.agda
+++ b/src/Algebra/Construct/LexProduct/Inner.agda
@@ -82,8 +82,8 @@ module NaturalOrder where
 
   ≤∙ˡ-trans : Associative _≈₁_ _∙_ → (a ∙ b) ≈₁ b → (b ∙ c) ≈₁ c → (a ∙ c) ≈₁ c
   ≤∙ˡ-trans {a} {b} {c} ∙-assoc ab≈b bc≈c = begin
-    a ∙ c        ≈˘⟨ ∙-congˡ bc≈c ⟩
-    a ∙ (b ∙ c)  ≈˘⟨ ∙-assoc a b c ⟩
+    a ∙ c        ≈⟨ ∙-congˡ bc≈c ⟨
+    a ∙ (b ∙ c)  ≈⟨ ∙-assoc a b c ⟨
     (a ∙ b) ∙ c  ≈⟨  ∙-congʳ ab≈b ⟩
     b ∙ c        ≈⟨  bc≈c ⟩
     c            ∎
@@ -179,51 +179,51 @@ assoc ∙-assoc ∙-comm ∙-sel ◦-assoc a b c x y z
 ... | no ab≉a  | yes ab≈b | yes bc≈b | yes bc≈c = begin
   lex (a ∙ b) c y z        ≈⟨  cong₁ ab≈b ⟩
   lex b c y z              ≈⟨  case₃ bc≈b bc≈c ⟩
-  y ◦ z                    ≈˘⟨ case₂ ab≉a ab≈b ⟩
-  lex a b x (y ◦ z)        ≈˘⟨ cong₂ bc≈b ⟩
+  y ◦ z                    ≈⟨ case₂ ab≉a ab≈b ⟨
+  lex a b x (y ◦ z)        ≈⟨ cong₂ bc≈b ⟨
   lex a (b ∙ c) x (y ◦ z)  ∎
 ... | no ab≉a  | yes ab≈b | yes bc≈b | no bc≉c = begin
   lex (a ∙ b) c y z        ≈⟨  cong₁ ab≈b ⟩
   lex b c y z              ≈⟨  case₁ bc≈b bc≉c ⟩
-  y                        ≈˘⟨ case₂ ab≉a ab≈b ⟩
-  lex a b x y              ≈˘⟨ cong₂ bc≈b ⟩
+  y                        ≈⟨ case₂ ab≉a ab≈b ⟨
+  lex a b x y              ≈⟨ cong₂ bc≈b ⟨
   lex a (b ∙ c) x y        ∎
 ... | yes ab≈a | yes ab≈b | yes bc≈b | no bc≉c = begin
   lex (a ∙ b) c (x ◦ y) z  ≈⟨  cong₁ ab≈b ⟩
   lex b c (x ◦ y) z        ≈⟨  case₁ bc≈b bc≉c ⟩
-  x ◦ y                    ≈˘⟨ case₃ ab≈a ab≈b ⟩
-  lex a b x y              ≈˘⟨ cong₂ bc≈b ⟩
+  x ◦ y                    ≈⟨ case₃ ab≈a ab≈b ⟨
+  lex a b x y              ≈⟨ cong₂ bc≈b ⟨
   lex a (b ∙ c) x y        ∎
 ... | yes ab≈a | yes ab≈b | yes bc≈b | yes bc≈c = begin
   lex (a ∙ b) c (x ◦ y) z  ≈⟨  cong₁ ab≈b ⟩
   lex b c (x ◦ y) z        ≈⟨  case₃ bc≈b bc≈c  ⟩
   (x ◦ y) ◦ z              ≈⟨  ◦-assoc x y z ⟩
-  x ◦ (y ◦ z)              ≈˘⟨ case₃ ab≈a ab≈b ⟩
-  lex a b x (y ◦ z)        ≈˘⟨ cong₂ bc≈b ⟩
+  x ◦ (y ◦ z)              ≈⟨ case₃ ab≈a ab≈b ⟨
+  lex a b x (y ◦ z)        ≈⟨ cong₂ bc≈b ⟨
   lex a (b ∙ c) x (y ◦ z)  ∎
 ... | yes ab≈a | yes ab≈b | no bc≉b | yes bc≈c = begin
   lex (a ∙ b) c (x ◦ y) z  ≈⟨  cong₁ ab≈b ⟩
   lex b c (x ◦ y) z        ≈⟨  case₂ bc≉b bc≈c ⟩
-  z                        ≈˘⟨ case₂ bc≉b bc≈c ⟩
-  lex b c x z              ≈˘⟨ cong₁₂ (M.trans (M.sym ab≈a) ab≈b) bc≈c ⟩
+  z                        ≈⟨ case₂ bc≉b bc≈c ⟨
+  lex b c x z              ≈⟨ cong₁₂ (M.trans (M.sym ab≈a) ab≈b) bc≈c ⟨
   lex a (b ∙ c) x z        ∎
 ... | yes ab≈a | no ab≉b | yes bc≈b | yes bc≈c = begin
   lex (a ∙ b) c x z        ≈⟨  cong₁₂ ab≈a (M.trans (M.sym bc≈c) bc≈b) ⟩
   lex a b x z              ≈⟨  case₁ ab≈a ab≉b ⟩
-  x                        ≈˘⟨ case₁ ab≈a ab≉b ⟩
-  lex a b x (y ◦ z)        ≈˘⟨ cong₂ bc≈b ⟩
+  x                        ≈⟨ case₁ ab≈a ab≉b ⟨
+  lex a b x (y ◦ z)        ≈⟨ cong₂ bc≈b ⟨
   lex a (b ∙ c) x (y ◦ z)  ∎
 ... | no ab≉a | yes ab≈b | no bc≉b | yes bc≈c = begin
   lex (a ∙ b) c y z        ≈⟨  cong₁ ab≈b ⟩
   lex b c y z              ≈⟨  case₂ bc≉b bc≈c ⟩
-  z                        ≈˘⟨ uncurry′ case₂ (<∙ˡ-trans ∙-assoc ∙-comm ab≈b ab≉a bc≈c) ⟩
-  lex a c x z              ≈˘⟨ cong₂ bc≈c ⟩
+  z                        ≈⟨ uncurry′ case₂ (<∙ˡ-trans ∙-assoc ∙-comm ab≈b ab≉a bc≈c) ⟨
+  lex a c x z              ≈⟨ cong₂ bc≈c ⟨
   lex a (b ∙ c) x z        ∎
 ... | yes ab≈a | no ab≉b | yes bc≈b | no bc≉c = begin
   lex (a ∙ b) c x z        ≈⟨  cong₁ ab≈a ⟩
   lex a c x z              ≈⟨  uncurry′ case₁ (<∙ʳ-trans ∙-assoc ∙-comm ab≈a bc≈b bc≉c) ⟩
-  x                        ≈˘⟨ case₁ ab≈a ab≉b ⟩
-  lex a b x y              ≈˘⟨ cong₂ bc≈b ⟩
+  x                        ≈⟨ case₁ ab≈a ab≉b ⟨
+  lex a b x y              ≈⟨ cong₂ bc≈b ⟨
   lex a (b ∙ c) x y        ∎
 
 comm : Commutative _≈₁_ _∙_ → Commutative _≈₂_ _◦_ →
diff --git a/src/Algebra/Construct/LiftedChoice.agda b/src/Algebra/Construct/LiftedChoice.agda
index 2433c87b9d..f53eecc7f1 100644
--- a/src/Algebra/Construct/LiftedChoice.agda
+++ b/src/Algebra/Construct/LiftedChoice.agda
@@ -100,8 +100,8 @@ module _ {_≈_ : Rel B ℓ} {_∙_ : Op₂ B}
 
     assoc : Associative _≈_ _∙_ → Associative _≈′_ _◦_
     assoc ∙-assoc x y z = f-injective (begin
-      f ((x ◦ y) ◦ z)   ≈˘⟨ distrib f (x ◦ y) z ⟩
-      f (x ◦ y) ∙ f z   ≈˘⟨ ∙-congʳ (distrib f x y) ⟩
+      f ((x ◦ y) ◦ z)   ≈⟨ distrib f (x ◦ y) z ⟨
+      f (x ◦ y) ∙ f z   ≈⟨ ∙-congʳ (distrib f x y) ⟨
       (f x ∙ f y) ∙ f z ≈⟨  ∙-assoc (f x) (f y) (f z) ⟩
       f x ∙ (f y ∙ f z) ≈⟨  ∙-congˡ (distrib f y z) ⟩
       f x ∙ f (y ◦ z)   ≈⟨  distrib f x (y ◦ z) ⟩
@@ -109,7 +109,7 @@ module _ {_≈_ : Rel B ℓ} {_∙_ : Op₂ B}
 
     comm : Commutative _≈_ _∙_ → Commutative _≈′_ _◦_
     comm ∙-comm x y = f-injective (begin
-      f (x ◦ y) ≈˘⟨ distrib f x y ⟩
+      f (x ◦ y) ≈⟨ distrib f x y ⟨
       f x ∙ f y ≈⟨  ∙-comm (f x) (f y) ⟩
       f y ∙ f x ≈⟨  distrib f y x ⟩
       f (y ◦ x) ∎)
diff --git a/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda b/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
index 04866f0ad0..7b64727832 100644
--- a/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MinMaxOp.agda
@@ -51,11 +51,11 @@ open import Algebra.Construct.NaturalChoice.MaxOp maxOp public
 ⊓-distribˡ-⊔ x y z with total y z
 ... | inj₁ y≤z = begin-equality
   x ⊓ (y ⊔ z)       ≈⟨  ⊓-congˡ x (x≤y⇒x⊔y≈y y≤z) ⟩
-  x ⊓ z             ≈˘⟨ x≤y⇒x⊔y≈y (⊓-monoʳ-≤ x y≤z) ⟩
+  x ⊓ z             ≈⟨ x≤y⇒x⊔y≈y (⊓-monoʳ-≤ x y≤z) ⟨
   (x ⊓ y) ⊔ (x ⊓ z) ∎
 ... | inj₂ y≥z = begin-equality
   x ⊓ (y ⊔ z)       ≈⟨  ⊓-congˡ x (x≥y⇒x⊔y≈x y≥z) ⟩
-  x ⊓ y             ≈˘⟨ x≥y⇒x⊔y≈x (⊓-monoʳ-≤ x y≥z) ⟩
+  x ⊓ y             ≈⟨ x≥y⇒x⊔y≈x (⊓-monoʳ-≤ x y≥z) ⟨
   (x ⊓ y) ⊔ (x ⊓ z) ∎
 
 ⊓-distribʳ-⊔ : _⊓_ DistributesOverʳ _⊔_
@@ -68,11 +68,11 @@ open import Algebra.Construct.NaturalChoice.MaxOp maxOp public
 ⊔-distribˡ-⊓ x y z with total y z
 ... | inj₁ y≤z = begin-equality
   x ⊔ (y ⊓ z)       ≈⟨  ⊔-congˡ x (x≤y⇒x⊓y≈x y≤z) ⟩
-  x ⊔ y             ≈˘⟨ x≤y⇒x⊓y≈x (⊔-monoʳ-≤ x y≤z) ⟩
+  x ⊔ y             ≈⟨ x≤y⇒x⊓y≈x (⊔-monoʳ-≤ x y≤z) ⟨
   (x ⊔ y) ⊓ (x ⊔ z) ∎
 ... | inj₂ y≥z = begin-equality
   x ⊔ (y ⊓ z)       ≈⟨  ⊔-congˡ x (x≥y⇒x⊓y≈y y≥z) ⟩
-  x ⊔ z             ≈˘⟨ x≥y⇒x⊓y≈y (⊔-monoʳ-≤ x y≥z) ⟩
+  x ⊔ z             ≈⟨ x≥y⇒x⊓y≈y (⊔-monoʳ-≤ x y≥z) ⟨
   (x ⊔ y) ⊓ (x ⊔ z) ∎
 
 ⊔-distribʳ-⊓ : _⊔_ DistributesOverʳ _⊓_
@@ -119,11 +119,11 @@ antimono-≤-distrib-⊓ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserv
 antimono-≤-distrib-⊓ {f} cong antimono x y with total x y
 ... | inj₁ x≤y = begin-equality
   f (x ⊓ y)  ≈⟨ cong (x≤y⇒x⊓y≈x x≤y) ⟩
-  f x        ≈˘⟨ x≥y⇒x⊔y≈x (antimono x≤y) ⟩
+  f x        ≈⟨ x≥y⇒x⊔y≈x (antimono x≤y) ⟨
   f x ⊔ f y  ∎
 ... | inj₂ y≤x = begin-equality
   f (x ⊓ y)  ≈⟨ cong (x≥y⇒x⊓y≈y y≤x) ⟩
-  f y        ≈˘⟨ x≤y⇒x⊔y≈y (antimono y≤x) ⟩
+  f y        ≈⟨ x≤y⇒x⊔y≈y (antimono y≤x) ⟨
   f x ⊔ f y  ∎
 
 antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserves _≤_ ⟶ _≥_ →
@@ -131,11 +131,11 @@ antimono-≤-distrib-⊔ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserv
 antimono-≤-distrib-⊔ {f} cong antimono x y with total x y
 ... | inj₁ x≤y = begin-equality
   f (x ⊔ y)  ≈⟨ cong (x≤y⇒x⊔y≈y x≤y) ⟩
-  f y        ≈˘⟨ x≥y⇒x⊓y≈y (antimono x≤y) ⟩
+  f y        ≈⟨ x≥y⇒x⊓y≈y (antimono x≤y) ⟨
   f x ⊓ f y  ∎
 ... | inj₂ y≤x = begin-equality
   f (x ⊔ y)  ≈⟨ cong (x≥y⇒x⊔y≈x y≤x) ⟩
-  f x        ≈˘⟨ x≤y⇒x⊓y≈x (antimono y≤x) ⟩
+  f x        ≈⟨ x≤y⇒x⊓y≈x (antimono y≤x) ⟨
   f x ⊓ f y  ∎
 
 x⊓y≤x⊔y : ∀ x y → x ⊓ y ≤ x ⊔ y
diff --git a/src/Algebra/Construct/NaturalChoice/MinOp.agda b/src/Algebra/Construct/NaturalChoice/MinOp.agda
index 744d32e1fe..22d0686f8f 100644
--- a/src/Algebra/Construct/NaturalChoice/MinOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MinOp.agda
@@ -59,17 +59,17 @@ x⊓y≤y x y with total x y
 ⊓-congˡ x {y} {r} y≈r with total x y
 ... | inj₁ x≤y = begin-equality
   x ⊓ y  ≈⟨  x≤y⇒x⊓y≈x x≤y ⟩
-  x      ≈˘⟨ x≤y⇒x⊓y≈x (≤-respʳ-≈ y≈r x≤y) ⟩
+  x      ≈⟨ x≤y⇒x⊓y≈x (≤-respʳ-≈ y≈r x≤y) ⟨
   x ⊓ r  ∎
 ... | inj₂ y≤x = begin-equality
   x ⊓ y  ≈⟨  x≥y⇒x⊓y≈y y≤x ⟩
   y      ≈⟨  y≈r ⟩
-  r      ≈˘⟨ x≥y⇒x⊓y≈y (≤-respˡ-≈ y≈r y≤x) ⟩
+  r      ≈⟨ x≥y⇒x⊓y≈y (≤-respˡ-≈ y≈r y≤x) ⟨
   x ⊓ r  ∎
 
 ⊓-congʳ : ∀ x → Congruent₁ (_⊓ x)
 ⊓-congʳ x {y₁} {y₂} y₁≈y₂ = begin-equality
-  y₁ ⊓ x  ≈˘⟨ ⊓-comm x y₁ ⟩
+  y₁ ⊓ x  ≈⟨ ⊓-comm x y₁ ⟨
   x  ⊓ y₁ ≈⟨  ⊓-congˡ x y₁≈y₂ ⟩
   x  ⊓ y₂ ≈⟨  ⊓-comm x y₂ ⟩
   y₂ ⊓ x  ∎
@@ -82,16 +82,16 @@ x⊓y≤y x y with total x y
 ⊓-assoc x y r | inj₁ x≤y | inj₁ y≤r = begin-equality
   (x ⊓ y) ⊓ r  ≈⟨ ⊓-congʳ r (x≤y⇒x⊓y≈x x≤y) ⟩
   x ⊓ r        ≈⟨ x≤y⇒x⊓y≈x (trans x≤y y≤r) ⟩
-  x            ≈˘⟨ x≤y⇒x⊓y≈x x≤y ⟩
-  x ⊓ y        ≈˘⟨ ⊓-congˡ x (x≤y⇒x⊓y≈x y≤r) ⟩
+  x            ≈⟨ x≤y⇒x⊓y≈x x≤y ⟨
+  x ⊓ y        ≈⟨ ⊓-congˡ x (x≤y⇒x⊓y≈x y≤r) ⟨
   x ⊓ (y ⊓ r)  ∎
 ⊓-assoc x y r | inj₁ x≤y | inj₂ r≤y = begin-equality
   (x ⊓ y) ⊓ r  ≈⟨ ⊓-congʳ r (x≤y⇒x⊓y≈x x≤y) ⟩
-  x ⊓ r        ≈˘⟨ ⊓-congˡ x (x≥y⇒x⊓y≈y r≤y) ⟩
+  x ⊓ r        ≈⟨ ⊓-congˡ x (x≥y⇒x⊓y≈y r≤y) ⟨
   x ⊓ (y ⊓ r)  ∎
 ⊓-assoc x y r | inj₂ y≤x | _ = begin-equality
   (x ⊓ y) ⊓ r  ≈⟨ ⊓-congʳ r (x≥y⇒x⊓y≈y y≤x) ⟩
-  y ⊓ r        ≈˘⟨ x≥y⇒x⊓y≈y (trans (x⊓y≤x y r) y≤x) ⟩
+  y ⊓ r        ≈⟨ x≥y⇒x⊓y≈y (trans (x⊓y≤x y r) y≤x) ⟨
   x ⊓ (y ⊓ r)  ∎
 
 ⊓-idem : Idempotent _⊓_
@@ -215,11 +215,11 @@ mono-≤-distrib-⊓ : ∀ {f} → f Preserves _≈_ ⟶ _≈_ → f Preserves _
 mono-≤-distrib-⊓ {f} cong mono x y with total x y
 ... | inj₁ x≤y = begin-equality
   f (x ⊓ y)  ≈⟨ cong (x≤y⇒x⊓y≈x x≤y) ⟩
-  f x        ≈˘⟨ x≤y⇒x⊓y≈x (mono x≤y) ⟩
+  f x        ≈⟨ x≤y⇒x⊓y≈x (mono x≤y) ⟨
   f x ⊓ f y  ∎
 ... | inj₂ y≤x = begin-equality
   f (x ⊓ y)  ≈⟨ cong (x≥y⇒x⊓y≈y y≤x) ⟩
-  f y        ≈˘⟨ x≥y⇒x⊓y≈y (mono y≤x) ⟩
+  f y        ≈⟨ x≥y⇒x⊓y≈y (mono y≤x) ⟨
   f x ⊓ f y  ∎
 
 x≤y⇒x⊓z≤y : ∀ {x y} z → x ≤ y → x ⊓ z ≤ y
@@ -250,8 +250,8 @@ x≤y⊓z⇒x≤z y z x≤y⊓z = trans x≤y⊓z (x⊓y≤y y z)
 
 ⊓-triangulate : ∀ x y z → x ⊓ y ⊓ z ≈ (x ⊓ y) ⊓ (y ⊓ z)
 ⊓-triangulate x y z = begin-equality
-  x ⊓ y ⊓ z           ≈˘⟨ ⊓-congʳ z (⊓-congˡ x (⊓-idem y)) ⟩
+  x ⊓ y ⊓ z           ≈⟨ ⊓-congʳ z (⊓-congˡ x (⊓-idem y)) ⟨
   x ⊓ (y ⊓ y) ⊓ z     ≈⟨  ⊓-assoc x _ _ ⟩
   x ⊓ ((y ⊓ y) ⊓ z)   ≈⟨  ⊓-congˡ x (⊓-assoc y y z) ⟩
-  x ⊓ (y ⊓ (y ⊓ z))   ≈˘⟨ ⊓-assoc x y (y ⊓ z) ⟩
+  x ⊓ (y ⊓ (y ⊓ z))   ≈⟨ ⊓-assoc x y (y ⊓ z) ⟨
   (x ⊓ y) ⊓ (y ⊓ z)   ∎
diff --git a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
index fe3ceec5be..9eb1d4da24 100644
--- a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
+++ b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
@@ -97,8 +97,8 @@ module _ (⊔-⊓-isLattice : IsLattice _≈₂_ _⊔_ _⊓_) where
     ⟦ y ∧ z ∨ x ⟧                        ≈⟨ ∨-homo (y ∧ z) x ⟩
     ⟦ y ∧ z ⟧ ⊔ ⟦ x ⟧                    ≈⟨ ⊔-congʳ (∧-homo y z) ⟩
     ⟦ y ⟧ ⊓ ⟦ z ⟧ ⊔ ⟦ x ⟧                ≈⟨ distribʳ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩
-    (⟦ y ⟧ ⊔ ⟦ x ⟧) ⊓ (⟦ z ⟧ ⊔ ⟦ x ⟧)    ≈˘⟨ ⊓-cong (∨-homo y x) (∨-homo z x) ⟩
-    ⟦ y ∨ x ⟧ ⊓ ⟦ z ∨ x ⟧                ≈˘⟨ ∧-homo (y ∨ x) (z ∨ x) ⟩
+    (⟦ y ⟧ ⊔ ⟦ x ⟧) ⊓ (⟦ z ⟧ ⊔ ⟦ x ⟧)    ≈⟨ ⊓-cong (∨-homo y x) (∨-homo z x) ⟨
+    ⟦ y ∨ x ⟧ ⊓ ⟦ z ∨ x ⟧                ≈⟨ ∧-homo (y ∨ x) (z ∨ x) ⟨
     ⟦ (y ∨ x) ∧ (z ∨ x) ⟧                ∎)
 
 isLattice : IsLattice _≈₂_ _⊔_ _⊓_ → IsLattice _≈₁_ _∨_ _∧_
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
index 85f22c39be..450527ddc0 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
@@ -76,8 +76,8 @@ open DistribLatticeProperties distributiveLattice public
 
 ∧-zeroʳ : RightZero ⊥ _∧_
 ∧-zeroʳ x = begin
-  x ∧ ⊥          ≈˘⟨ ∧-congˡ (∧-complementʳ x) ⟩
-  x ∧  x  ∧ ¬ x  ≈˘⟨ ∧-assoc x x (¬ x) ⟩
+  x ∧ ⊥          ≈⟨ ∧-congˡ (∧-complementʳ x) ⟨
+  x ∧  x  ∧ ¬ x  ≈⟨ ∧-assoc x x (¬ x) ⟨
   (x ∧ x) ∧ ¬ x  ≈⟨  ∧-congʳ (∧-idem x) ⟩
   x       ∧ ¬ x  ≈⟨  ∧-complementʳ x ⟩
   ⊥              ∎
@@ -90,8 +90,8 @@ open DistribLatticeProperties distributiveLattice public
 
 ∨-zeroʳ : ∀ x → x ∨ ⊤ ≈ ⊤
 ∨-zeroʳ x = begin
-  x ∨ ⊤          ≈˘⟨ ∨-congˡ (∨-complementʳ x) ⟩
-  x ∨  x  ∨ ¬ x  ≈˘⟨ ∨-assoc x x (¬ x) ⟩
+  x ∨ ⊤          ≈⟨ ∨-congˡ (∨-complementʳ x) ⟨
+  x ∨  x  ∨ ¬ x  ≈⟨ ∨-assoc x x (¬ x) ⟨
   (x ∨ x) ∨ ¬ x  ≈⟨ ∨-congʳ (∨-idem x) ⟩
   x       ∨ ¬ x  ≈⟨ ∨-complementʳ x ⟩
   ⊤              ∎
@@ -182,12 +182,12 @@ open DistribLatticeProperties distributiveLattice public
 private
   lemma : ∀ x y → x ∧ y ≈ ⊥ → x ∨ y ≈ ⊤ → ¬ x ≈ y
   lemma x y x∧y=⊥ x∨y=⊤ = begin
-    ¬ x                ≈˘⟨ ∧-identityʳ _ ⟩
-    ¬ x ∧ ⊤            ≈˘⟨ ∧-congˡ x∨y=⊤ ⟩
+    ¬ x                ≈⟨ ∧-identityʳ _ ⟨
+    ¬ x ∧ ⊤            ≈⟨ ∧-congˡ x∨y=⊤ ⟨
     ¬ x ∧ (x ∨ y)      ≈⟨  ∧-distribˡ-∨ _ _ _ ⟩
     ¬ x ∧ x ∨ ¬ x ∧ y  ≈⟨  ∨-congʳ $ ∧-complementˡ _ ⟩
-    ⊥ ∨ ¬ x ∧ y        ≈˘⟨ ∨-congʳ x∧y=⊥ ⟩
-    x ∧ y ∨ ¬ x ∧ y    ≈˘⟨ ∧-distribʳ-∨ _ _ _ ⟩
+    ⊥ ∨ ¬ x ∧ y        ≈⟨ ∨-congʳ x∧y=⊥ ⟨
+    x ∧ y ∨ ¬ x ∧ y    ≈⟨ ∧-distribʳ-∨ _ _ _ ⟨
     (x ∨ ¬ x) ∧ y      ≈⟨  ∧-congʳ $ ∨-complementʳ _ ⟩
     ⊤ ∧ y              ≈⟨  ∧-identityˡ _ ⟩
     y                  ∎
@@ -222,7 +222,7 @@ deMorgan₁ x y = lemma (x ∧ y) (¬ x ∨ ¬ y) lem₁ lem₂
     ¬ x ∨ y                ∎
 
   lem₂ = begin
-    (x ∧ y) ∨ (¬ x ∨ ¬ y)  ≈˘⟨ ∨-assoc _ _ _ ⟩
+    (x ∧ y) ∨ (¬ x ∨ ¬ y)  ≈⟨ ∨-assoc _ _ _ ⟨
     ((x ∧ y) ∨ ¬ x) ∨ ¬ y  ≈⟨ ∨-congʳ lem₃ ⟩
     (¬ x ∨ y) ∨ ¬ y        ≈⟨ ∨-assoc _ _ _ ⟩
     ¬ x ∨ (y ∨ ¬ y)        ≈⟨ ∨-congˡ $ ∨-complementʳ _ ⟩
@@ -231,8 +231,8 @@ deMorgan₁ x y = lemma (x ∧ y) (¬ x ∨ ¬ y) lem₁ lem₂
 
 deMorgan₂ : ∀ x y → ¬ (x ∨ y) ≈ ¬ x ∧ ¬ y
 deMorgan₂ x y = begin
-  ¬ (x ∨ y)          ≈˘⟨ ¬-cong $ ((¬-involutive _) ⟨ ∨-cong ⟩ (¬-involutive _)) ⟩
-  ¬ (¬ ¬ x ∨ ¬ ¬ y)  ≈˘⟨ ¬-cong $ deMorgan₁ _ _ ⟩
+  ¬ (x ∨ y)          ≈⟨ ¬-cong $ ((¬-involutive _) ⟨ ∨-cong ⟩ (¬-involutive _)) ⟨
+  ¬ (¬ ¬ x ∨ ¬ ¬ y)  ≈⟨ ¬-cong $ deMorgan₁ _ _ ⟨
   ¬ ¬ (¬ x ∧ ¬ y)    ≈⟨ ¬-involutive _ ⟩
   ¬ x ∧ ¬ y          ∎
 
@@ -260,14 +260,14 @@ module XorRing
     x ⊕ u                ≈⟨  ⊕-def _ _ ⟩
     (x ∨ u) ∧ ¬ (x ∧ u)  ≈⟨  helper (x≈y ⟨ ∨-cong ⟩ u≈v)
                                     (x≈y ⟨ ∧-cong ⟩ u≈v) ⟩
-    (y ∨ v) ∧ ¬ (y ∧ v)  ≈˘⟨ ⊕-def _ _ ⟩
+    (y ∨ v) ∧ ¬ (y ∧ v)  ≈⟨ ⊕-def _ _ ⟨
     y ⊕ v                ∎
 
   ⊕-comm : Commutative _⊕_
   ⊕-comm x y = begin
     x ⊕ y                ≈⟨  ⊕-def _ _ ⟩
     (x ∨ y) ∧ ¬ (x ∧ y)  ≈⟨  helper (∨-comm _ _) (∧-comm _ _) ⟩
-    (y ∨ x) ∧ ¬ (y ∧ x)  ≈˘⟨ ⊕-def _ _ ⟩
+    (y ∨ x) ∧ ¬ (y ∧ x)  ≈⟨ ⊕-def _ _ ⟨
     y ⊕ x                ∎
 
   ¬-distribˡ-⊕ : ∀ x y → ¬ (x ⊕ y) ≈ ¬ x ⊕ y
@@ -279,7 +279,7 @@ module XorRing
     ¬ ((x ∧ ¬ y) ∨ (y ∧ ¬ x))              ≈⟨ deMorgan₂ _ _ ⟩
     ¬ (x ∧ ¬ y) ∧ ¬ (y ∧ ¬ x)              ≈⟨ ∧-congʳ $ deMorgan₁ _ _ ⟩
     (¬ x ∨ (¬ ¬ y)) ∧ ¬ (y ∧ ¬ x)          ≈⟨ helper (∨-congˡ $ ¬-involutive _) (∧-comm _ _) ⟩
-    (¬ x ∨ y) ∧ ¬ (¬ x ∧ y)                ≈˘⟨ ⊕-def _ _ ⟩
+    (¬ x ∨ y) ∧ ¬ (¬ x ∧ y)                ≈⟨ ⊕-def _ _ ⟨
     ¬ x ⊕ y                                ∎
     where
     lem : ∀ x y → x ∧ ¬ (x ∧ y) ≈ x ∧ ¬ y
@@ -299,7 +299,7 @@ module XorRing
 
   ⊕-annihilates-¬ : ∀ x y → x ⊕ y ≈ ¬ x ⊕ ¬ y
   ⊕-annihilates-¬ x y = begin
-    x ⊕ y        ≈˘⟨ ¬-involutive _ ⟩
+    x ⊕ y        ≈⟨ ¬-involutive _ ⟨
     ¬ ¬ (x ⊕ y)  ≈⟨  ¬-cong $ ¬-distribˡ-⊕ _ _ ⟩
     ¬ (¬ x ⊕ y)  ≈⟨  ¬-distribʳ-⊕ _ _ ⟩
     ¬ x ⊕ ¬ y    ∎
@@ -334,46 +334,46 @@ module XorRing
   ∧-distribˡ-⊕ : _∧_ DistributesOverˡ _⊕_
   ∧-distribˡ-⊕ x y z = begin
     x ∧ (y ⊕ z)                ≈⟨ ∧-congˡ $ ⊕-def _ _ ⟩
-    x ∧ ((y ∨ z) ∧ ¬ (y ∧ z))  ≈˘⟨ ∧-assoc _ _ _ ⟩
+    x ∧ ((y ∨ z) ∧ ¬ (y ∧ z))  ≈⟨ ∧-assoc _ _ _ ⟨
     (x ∧ (y ∨ z)) ∧ ¬ (y ∧ z)  ≈⟨ ∧-congˡ $ deMorgan₁ _ _ ⟩
     (x ∧ (y ∨ z)) ∧
-    (¬ y ∨ ¬ z)                ≈˘⟨ ∨-identityˡ _ ⟩
+    (¬ y ∨ ¬ z)                ≈⟨ ∨-identityˡ _ ⟨
     ⊥ ∨
     ((x ∧ (y ∨ z)) ∧
     (¬ y ∨ ¬ z))               ≈⟨ ∨-congʳ lem₃ ⟩
     ((x ∧ (y ∨ z)) ∧ ¬ x) ∨
     ((x ∧ (y ∨ z)) ∧
-    (¬ y ∨ ¬ z))               ≈˘⟨ ∧-distribˡ-∨ _ _ _ ⟩
+    (¬ y ∨ ¬ z))               ≈⟨ ∧-distribˡ-∨ _ _ _ ⟨
     (x ∧ (y ∨ z)) ∧
-    (¬ x ∨ (¬ y ∨ ¬ z))        ≈˘⟨ ∧-congˡ $ ∨-congˡ (deMorgan₁ _ _) ⟩
+    (¬ x ∨ (¬ y ∨ ¬ z))        ≈⟨ ∧-congˡ $ ∨-congˡ (deMorgan₁ _ _) ⟨
     (x ∧ (y ∨ z)) ∧
-    (¬ x ∨ ¬ (y ∧ z))          ≈˘⟨ ∧-congˡ (deMorgan₁ _ _) ⟩
+    (¬ x ∨ ¬ (y ∧ z))          ≈⟨ ∧-congˡ (deMorgan₁ _ _) ⟨
     (x ∧ (y ∨ z)) ∧
     ¬ (x ∧ (y ∧ z))            ≈⟨ helper refl lem₁ ⟩
     (x ∧ (y ∨ z)) ∧
     ¬ ((x ∧ y) ∧ (x ∧ z))      ≈⟨ ∧-congʳ $ ∧-distribˡ-∨ _ _ _ ⟩
     ((x ∧ y) ∨ (x ∧ z)) ∧
-    ¬ ((x ∧ y) ∧ (x ∧ z))      ≈˘⟨ ⊕-def _ _ ⟩
+    ¬ ((x ∧ y) ∧ (x ∧ z))      ≈⟨ ⊕-def _ _ ⟨
     (x ∧ y) ⊕ (x ∧ z)          ∎
       where
       lem₂ = begin
-        x ∧ (y ∧ z)  ≈˘⟨ ∧-assoc _ _ _ ⟩
+        x ∧ (y ∧ z)  ≈⟨ ∧-assoc _ _ _ ⟨
         (x ∧ y) ∧ z  ≈⟨ ∧-congʳ $ ∧-comm _ _ ⟩
         (y ∧ x) ∧ z  ≈⟨ ∧-assoc _ _ _ ⟩
         y ∧ (x ∧ z)  ∎
 
       lem₁ = begin
-        x ∧ (y ∧ z)        ≈˘⟨ ∧-congʳ (∧-idem _) ⟩
+        x ∧ (y ∧ z)        ≈⟨ ∧-congʳ (∧-idem _) ⟨
         (x ∧ x) ∧ (y ∧ z)  ≈⟨ ∧-assoc _ _ _ ⟩
         x ∧ (x ∧ (y ∧ z))  ≈⟨ ∧-congˡ lem₂ ⟩
-        x ∧ (y ∧ (x ∧ z))  ≈˘⟨ ∧-assoc _ _ _ ⟩
+        x ∧ (y ∧ (x ∧ z))  ≈⟨ ∧-assoc _ _ _ ⟨
         (x ∧ y) ∧ (x ∧ z)  ∎
 
       lem₃ = begin
-        ⊥                      ≈˘⟨ ∧-zeroʳ _ ⟩
-        (y ∨ z) ∧ ⊥            ≈˘⟨ ∧-congˡ (∧-complementʳ _) ⟩
-        (y ∨ z) ∧ (x ∧ ¬ x)    ≈˘⟨ ∧-assoc _ _ _ ⟩
-        ((y ∨ z) ∧ x) ∧ ¬ x    ≈⟨ ∧-comm _ _ ⟨ ∧-cong ⟩ refl  ⟩
+        ⊥                      ≈⟨ ∧-zeroʳ _ ⟨
+        (y ∨ z) ∧ ⊥            ≈⟨ ∧-congˡ (∧-complementʳ _) ⟨
+        (y ∨ z) ∧ (x ∧ ¬ x)    ≈⟨ ∧-assoc _ _ _ ⟨
+        ((y ∨ z) ∧ x) ∧ ¬ x    ≈⟨ ∧-congʳ (∧-comm _ _) ⟩
         (x ∧ (y ∨ z)) ∧ ¬ x    ∎
 
   ∧-distribʳ-⊕ : _∧_ DistributesOverʳ _⊕_
@@ -398,10 +398,10 @@ module XorRing
 
   ⊕-assoc : Associative _⊕_
   ⊕-assoc x y z = sym $ begin
-    x ⊕ (y ⊕ z)                                ≈⟨ refl ⟨ ⊕-cong ⟩ ⊕-def _ _ ⟩
+    x ⊕ (y ⊕ z)                                ≈⟨ ⊕-cong refl (⊕-def _ _) ⟩
     x ⊕ ((y ∨ z) ∧ ¬ (y ∧ z))                  ≈⟨ ⊕-def _ _ ⟩
       (x ∨ ((y ∨ z) ∧ ¬ (y ∧ z))) ∧
-    ¬ (x ∧ ((y ∨ z) ∧ ¬ (y ∧ z)))              ≈⟨ lem₃ ⟨ ∧-cong ⟩ lem₄ ⟩
+    ¬ (x ∧ ((y ∨ z) ∧ ¬ (y ∧ z)))              ≈⟨ ∧-cong lem₃ lem₄ ⟩
     (((x ∨ y) ∨ z) ∧ ((x ∨ ¬ y) ∨ ¬ z)) ∧
     (((¬ x ∨ ¬ y) ∨ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈⟨ ∧-assoc _ _ _ ⟩
     ((x ∨ y) ∨ z) ∧
@@ -409,40 +409,40 @@ module XorRing
      (((¬ x ∨ ¬ y) ∨ z) ∧ ((¬ x ∨ y) ∨ ¬ z)))  ≈⟨ ∧-congˡ lem₅ ⟩
     ((x ∨ y) ∨ z) ∧
     (((¬ x ∨ ¬ y) ∨ z) ∧
-     (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z)))  ≈˘⟨ ∧-assoc _ _ _ ⟩
+     (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z)))  ≈⟨ ∧-assoc _ _ _ ⟨
     (((x ∨ y) ∨ z) ∧ ((¬ x ∨ ¬ y) ∨ z)) ∧
-    (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈⟨ lem₁ ⟨ ∧-cong ⟩ lem₂ ⟩
+    (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈⟨ ∧-cong lem₁ lem₂ ⟩
       (((x ∨ y) ∧ ¬ (x ∧ y)) ∨ z) ∧
-    ¬ (((x ∨ y) ∧ ¬ (x ∧ y)) ∧ z)              ≈˘⟨ ⊕-def _ _ ⟩
-    ((x ∨ y) ∧ ¬ (x ∧ y)) ⊕ z                  ≈˘⟨ ⊕-def _ _ ⟨ ⊕-cong ⟩ refl ⟩
+    ¬ (((x ∨ y) ∧ ¬ (x ∧ y)) ∧ z)              ≈⟨ ⊕-def _ _ ⟨
+    ((x ∨ y) ∧ ¬ (x ∧ y)) ⊕ z                  ≈⟨ ⊕-cong (⊕-def _ _) refl ⟨
     (x ⊕ y) ⊕ z                                ∎
     where
     lem₁ = begin
-      ((x ∨ y) ∨ z) ∧ ((¬ x ∨ ¬ y) ∨ z)  ≈˘⟨ ∨-distribʳ-∧ _ _ _ ⟩
-      ((x ∨ y) ∧ (¬ x ∨ ¬ y)) ∨ z        ≈˘⟨ ∨-congʳ $ ∧-congˡ (deMorgan₁ _ _) ⟩
+      ((x ∨ y) ∨ z) ∧ ((¬ x ∨ ¬ y) ∨ z)  ≈⟨ ∨-distribʳ-∧ _ _ _ ⟨
+      ((x ∨ y) ∧ (¬ x ∨ ¬ y)) ∨ z        ≈⟨ ∨-congʳ $ ∧-congˡ (deMorgan₁ _ _) ⟨
       ((x ∨ y) ∧ ¬ (x ∧ y)) ∨ z          ∎
 
     lem₂′ = begin
-      (x ∨ ¬ y) ∧ (¬ x ∨ y)              ≈˘⟨ ∧-identityˡ _ ⟨ ∧-cong ⟩ ∧-identityʳ _ ⟩
-      (⊤ ∧ (x ∨ ¬ y)) ∧ ((¬ x ∨ y) ∧ ⊤)  ≈˘⟨  (∨-complementˡ _ ⟨ ∧-cong ⟩ ∨-comm _ _)
-                                                ⟨ ∧-cong ⟩
-                                              (∧-congˡ $ ∨-complementˡ _) ⟩
+      (x ∨ ¬ y) ∧ (¬ x ∨ y)              ≈⟨ ∧-cong (∧-identityˡ _) (∧-identityʳ _) ⟨
+      (⊤ ∧ (x ∨ ¬ y)) ∧ ((¬ x ∨ y) ∧ ⊤)  ≈⟨ ∧-cong
+                                              (∧-cong (∨-complementˡ _) (∨-comm _ _))
+                                              (∧-congˡ $ ∨-complementˡ _) ⟨
       ((¬ x ∨ x) ∧ (¬ y ∨ x)) ∧
-      ((¬ x ∨ y) ∧ (¬ y ∨ y))            ≈˘⟨ lemma₂ _ _ _ _ ⟩
-      (¬ x ∧ ¬ y) ∨ (x ∧ y)              ≈˘⟨ deMorgan₂ _ _ ⟨ ∨-cong ⟩ ¬-involutive _ ⟩
-      ¬ (x ∨ y) ∨ ¬ ¬ (x ∧ y)            ≈˘⟨ deMorgan₁ _ _ ⟩
+      ((¬ x ∨ y) ∧ (¬ y ∨ y))            ≈⟨ lemma₂ _ _ _ _ ⟨
+      (¬ x ∧ ¬ y) ∨ (x ∧ y)              ≈⟨ ∨-cong (deMorgan₂ _ _) (¬-involutive _) ⟨
+      ¬ (x ∨ y) ∨ ¬ ¬ (x ∧ y)            ≈⟨ deMorgan₁ _ _ ⟨
       ¬ ((x ∨ y) ∧ ¬ (x ∧ y))            ∎
 
     lem₂ = begin
-      ((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z)  ≈˘⟨ ∨-distribʳ-∧ _ _ _ ⟩
+      ((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ y) ∨ ¬ z)  ≈⟨ ∨-distribʳ-∧ _ _ _ ⟨
       ((x ∨ ¬ y) ∧ (¬ x ∨ y)) ∨ ¬ z          ≈⟨ ∨-congʳ lem₂′ ⟩
-      ¬ ((x ∨ y) ∧ ¬ (x ∧ y)) ∨ ¬ z          ≈˘⟨ deMorgan₁ _ _ ⟩
+      ¬ ((x ∨ y) ∧ ¬ (x ∧ y)) ∨ ¬ z          ≈⟨ deMorgan₁ _ _ ⟨
       ¬ (((x ∨ y) ∧ ¬ (x ∧ y)) ∧ z)          ∎
 
     lem₃ = begin
       x ∨ ((y ∨ z) ∧ ¬ (y ∧ z))          ≈⟨ ∨-congˡ $ ∧-congˡ $ deMorgan₁ _ _ ⟩
       x ∨ ((y ∨ z) ∧ (¬ y ∨ ¬ z))        ≈⟨ ∨-distribˡ-∧ _ _ _ ⟩
-      (x ∨ (y ∨ z)) ∧ (x ∨ (¬ y ∨ ¬ z))  ≈˘⟨ ∨-assoc _ _ _ ⟨ ∧-cong ⟩ ∨-assoc _ _ _ ⟩
+      (x ∨ (y ∨ z)) ∧ (x ∨ (¬ y ∨ ¬ z))  ≈⟨ ∨-assoc _ _ _ ⟨ ∧-cong ⟩ ∨-assoc _ _ _ ⟨
       ((x ∨ y) ∨ z) ∧ ((x ∨ ¬ y) ∨ ¬ z)  ∎
 
     lem₄′ = begin
@@ -462,7 +462,7 @@ module XorRing
       ¬ x ∨ ¬ ((y ∨ z) ∧ ¬ (y ∧ z))  ≈⟨ ∨-congˡ lem₄′ ⟩
       ¬ x ∨ ((y ∨ ¬ z) ∧ (¬ y ∨ z))  ≈⟨ ∨-distribˡ-∧ _ _ _ ⟩
       (¬ x ∨ (y     ∨ ¬ z)) ∧
-      (¬ x ∨ (¬ y ∨ z))              ≈˘⟨ ∨-assoc _ _ _ ⟨ ∧-cong ⟩ ∨-assoc _ _ _ ⟩
+      (¬ x ∨ (¬ y ∨ z))              ≈⟨ ∨-assoc _ _ _ ⟨ ∧-cong ⟩ ∨-assoc _ _ _ ⟨
       ((¬ x ∨ y)     ∨ ¬ z) ∧
       ((¬ x ∨ ¬ y) ∨ z)              ≈⟨ ∧-comm _ _ ⟩
       ((¬ x ∨ ¬ y) ∨ z) ∧
@@ -470,7 +470,7 @@ module XorRing
 
     lem₅ = begin
       ((x ∨ ¬ y) ∨ ¬ z) ∧
-      (((¬ x ∨ ¬ y) ∨ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈˘⟨ ∧-assoc _ _ _ ⟩
+      (((¬ x ∨ ¬ y) ∨ z) ∧ ((¬ x ∨ y) ∨ ¬ z))    ≈⟨ ∧-assoc _ _ _ ⟨
       (((x ∨ ¬ y) ∨ ¬ z) ∧ ((¬ x ∨ ¬ y) ∨ z)) ∧
       ((¬ x ∨ y) ∨ ¬ z)                          ≈⟨ ∧-congʳ $ ∧-comm _ _ ⟩
       (((¬ x ∨ ¬ y) ∨ z) ∧ ((x ∨ ¬ y) ∨ ¬ z)) ∧
diff --git a/src/Algebra/Lattice/Properties/Lattice.agda b/src/Algebra/Lattice/Properties/Lattice.agda
index b83dcdea10..281330ef15 100644
--- a/src/Algebra/Lattice/Properties/Lattice.agda
+++ b/src/Algebra/Lattice/Properties/Lattice.agda
@@ -27,7 +27,7 @@ open import Relation.Binary.Reasoning.Setoid setoid
 
 ∧-idem : Idempotent _∧_
 ∧-idem x = begin
-  x ∧ x            ≈˘⟨ ∧-congˡ (∨-absorbs-∧ _ _) ⟩
+  x ∧ x            ≈⟨ ∧-congˡ (∨-absorbs-∧ _ _) ⟨
   x ∧ (x ∨ x ∧ x)  ≈⟨  ∧-absorbs-∨ _ _ ⟩
   x                ∎
 
@@ -73,7 +73,7 @@ open SemilatticeProperties ∧-semilattice public
 
 ∨-idem : Idempotent _∨_
 ∨-idem x = begin
-  x ∨ x      ≈˘⟨ ∨-congˡ (∧-idem _) ⟩
+  x ∨ x      ≈⟨ ∨-congˡ (∧-idem _) ⟨
   x ∨ x ∧ x  ≈⟨  ∨-absorbs-∧ _ _ ⟩
   x          ∎
 
diff --git a/src/Algebra/Module/Consequences.agda b/src/Algebra/Module/Consequences.agda
index 2e53f96d76..9acd6d6413 100644
--- a/src/Algebra/Module/Consequences.agda
+++ b/src/Algebra/Module/Consequences.agda
@@ -45,7 +45,7 @@ module _ (_≈ᴬ_ : Rel {a} A ℓa) (S : Setoid c ℓ) where
       L.Associative _*_ _*ₗ_ → Commutative _*_ → R.Associative _*_ _*ᵣ_
     *ₗ-assoc+comm⇒*ᵣ-assoc *ₗ-congʳ *ₗ-assoc *-comm m x y = begin
       (m *ᵣ x) *ᵣ y  ≈⟨ refl ⟩
-      y *ₗ (x *ₗ m)  ≈˘⟨ *ₗ-assoc _ _ _ ⟩
+      y *ₗ (x *ₗ m)  ≈⟨ *ₗ-assoc _ _ _ ⟨
       (y * x) *ₗ m   ≈⟨ *ₗ-congʳ (*-comm y x) ⟩
       (x * y) *ₗ m   ≈⟨ refl ⟩
       m *ᵣ (x * y)   ∎
@@ -55,7 +55,7 @@ module _ (_≈ᴬ_ : Rel {a} A ℓa) (S : Setoid c ℓ) where
       L.Associative _*_ _*ₗ_ → Commutative _*_ → B.Associative _*ₗ_ _*ᵣ_
     *ₗ-assoc+comm⇒*ₗ-*ᵣ-assoc *ₗ-congʳ *ₗ-assoc *-comm x m y = begin
       ((x *ₗ m) *ᵣ y)  ≈⟨ refl ⟩
-      (y *ₗ (x *ₗ m))  ≈˘⟨ *ₗ-assoc _ _ _ ⟩
+      (y *ₗ (x *ₗ m))  ≈⟨ *ₗ-assoc _ _ _ ⟨
       ((y * x) *ₗ m)   ≈⟨ *ₗ-congʳ (*-comm y x) ⟩
       ((x * y) *ₗ m)   ≈⟨ *ₗ-assoc _ _ _ ⟩
       (x *ₗ (y *ₗ m))  ≈⟨ refl ⟩
@@ -72,7 +72,7 @@ module _ (_≈ᴬ_ : Rel {a} A ℓa) (S : Setoid c ℓ) where
     *ᵣ-assoc+comm⇒*ₗ-assoc *ᵣ-congˡ *ᵣ-assoc *-comm x y m = begin
       ((x * y) *ₗ m)   ≈⟨ refl ⟩
       (m *ᵣ (x * y))   ≈⟨ *ᵣ-congˡ (*-comm x y) ⟩
-      (m *ᵣ (y * x))   ≈˘⟨ *ᵣ-assoc _ _ _ ⟩
+      (m *ᵣ (y * x))   ≈⟨ *ᵣ-assoc _ _ _ ⟨
       ((m *ᵣ y) *ᵣ x)  ≈⟨ refl ⟩
       (x *ₗ (y *ₗ m))  ∎
 
@@ -83,6 +83,6 @@ module _ (_≈ᴬ_ : Rel {a} A ℓa) (S : Setoid c ℓ) where
       ((x *ₗ m) *ᵣ y)  ≈⟨ refl ⟩
       ((m *ᵣ x) *ᵣ y)  ≈⟨ *ᵣ-assoc _ _ _ ⟩
       (m *ᵣ (x * y))   ≈⟨ *ᵣ-congˡ (*-comm x y) ⟩
-      (m *ᵣ (y * x))   ≈˘⟨ *ᵣ-assoc _ _ _ ⟩
+      (m *ᵣ (y * x))   ≈⟨ *ᵣ-assoc _ _ _ ⟨
       ((m *ᵣ y) *ᵣ x)  ≈⟨ refl ⟩
       (x *ₗ (m *ᵣ y))  ∎
diff --git a/src/Algebra/Morphism/Consequences.agda b/src/Algebra/Morphism/Consequences.agda
index 96175740e6..e297a8dbc9 100644
--- a/src/Algebra/Morphism/Consequences.agda
+++ b/src/Algebra/Morphism/Consequences.agda
@@ -30,8 +30,8 @@ module _ {α α= β β=} (M₁ : Magma α α=) (M₂ : Magma β β=) where
                       Homomorphic₂ _ _ _≈₂_ f _∙₁_ _∙₂_  →
                       Homomorphic₂ _ _ _≈₁_ g _∙₂_ _∙₁_
   homomorphic₂-inv {f} {g} g-cong (invˡ , invʳ) homo x y = begin
-    g (x ∙₂ y)             ≈˘⟨ g-cong (M₂.∙-cong (invˡ M₁.refl) (invˡ M₁.refl)) ⟩
-    g (f (g x) ∙₂ f (g y)) ≈˘⟨ g-cong (homo (g x) (g y)) ⟩
+    g (x ∙₂ y)             ≈⟨ g-cong (M₂.∙-cong (invˡ M₁.refl) (invˡ M₁.refl)) ⟨
+    g (f (g x) ∙₂ f (g y)) ≈⟨ g-cong (homo (g x) (g y)) ⟨
     g (f (g x ∙₁ g y))     ≈⟨ invʳ M₂.refl ⟩
     g x ∙₁ g y             ∎
     where open EqR M₁.setoid
@@ -42,7 +42,7 @@ module _ {α α= β β=} (M₁ : Magma α α=) (M₂ : Magma β β=) where
                       Homomorphic₂ _ _ _≈₁_ g _∙₂_ _∙₁_
   homomorphic₂-inj {f} {g} inj invˡ homo x y = inj (begin
     f (g (x ∙₂ y))      ≈⟨ invˡ M₁.refl ⟩
-    x ∙₂ y              ≈˘⟨ M₂.∙-cong (invˡ M₁.refl) (invˡ M₁.refl) ⟩
-    f (g x) ∙₂ f (g y)  ≈˘⟨ homo (g x) (g y) ⟩
+    x ∙₂ y              ≈⟨ M₂.∙-cong (invˡ M₁.refl) (invˡ M₁.refl) ⟨
+    f (g x) ∙₂ f (g y)  ≈⟨ homo (g x) (g y) ⟨
     f (g x ∙₁ g y)      ∎)
     where open EqR M₂.setoid
diff --git a/src/Algebra/Morphism/GroupMonomorphism.agda b/src/Algebra/Morphism/GroupMonomorphism.agda
index b862559938..51cdcbafe7 100644
--- a/src/Algebra/Morphism/GroupMonomorphism.agda
+++ b/src/Algebra/Morphism/GroupMonomorphism.agda
@@ -48,7 +48,7 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
     ⟦ x ⁻¹₁ ∙ x ⟧     ≈⟨ ∙-homo (x ⁻¹₁ ) x ⟩
     ⟦ x ⁻¹₁ ⟧ ◦ ⟦ x ⟧ ≈⟨ ◦-cong (⁻¹-homo x) refl ⟩
     ⟦ x ⟧ ⁻¹₂ ◦ ⟦ x ⟧ ≈⟨ invˡ ⟦ x ⟧ ⟩
-    ε₂                ≈˘⟨ ε-homo ⟩
+    ε₂                ≈⟨ ε-homo ⟨
     ⟦ ε₁ ⟧ ∎)
 
   inverseʳ : RightInverse _≈₂_ ε₂ _⁻¹₂ _◦_ → RightInverse _≈₁_ ε₁ _⁻¹₁ _∙_
@@ -56,7 +56,7 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
     ⟦ x ∙ x ⁻¹₁ ⟧     ≈⟨ ∙-homo x (x ⁻¹₁) ⟩
     ⟦ x ⟧ ◦ ⟦ x ⁻¹₁ ⟧ ≈⟨ ◦-cong refl (⁻¹-homo x) ⟩
     ⟦ x ⟧ ◦ ⟦ x ⟧ ⁻¹₂ ≈⟨ invʳ ⟦ x ⟧ ⟩
-    ε₂                ≈˘⟨ ε-homo ⟩
+    ε₂                ≈⟨ ε-homo ⟨
     ⟦ ε₁ ⟧ ∎)
 
   inverse : Inverse _≈₂_ ε₂ _⁻¹₂ _◦_ → Inverse _≈₁_ ε₁ _⁻¹₁ _∙_
@@ -66,7 +66,7 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
   ⁻¹-cong ⁻¹-cong {x} {y} x≈y = injective (begin
     ⟦ x ⁻¹₁ ⟧ ≈⟨ ⁻¹-homo x ⟩
     ⟦ x ⟧ ⁻¹₂ ≈⟨ ⁻¹-cong (⟦⟧-cong x≈y) ⟩
-    ⟦ y ⟧ ⁻¹₂ ≈˘⟨ ⁻¹-homo y ⟩
+    ⟦ y ⟧ ⁻¹₂ ≈⟨ ⁻¹-homo y ⟨
     ⟦ y ⁻¹₁ ⟧ ∎)
 
 module _ (◦-isAbelianGroup : IsAbelianGroup _≈₂_ _◦_ ε₂ _⁻¹₂) where
diff --git a/src/Algebra/Morphism/MagmaMonomorphism.agda b/src/Algebra/Morphism/MagmaMonomorphism.agda
index 30f19edb02..87027ba5e1 100644
--- a/src/Algebra/Morphism/MagmaMonomorphism.agda
+++ b/src/Algebra/Morphism/MagmaMonomorphism.agda
@@ -42,7 +42,7 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
   cong {x} {y} {u} {v} x≈y u≈v = injective (begin
     ⟦ x ∙ u ⟧      ≈⟨  homo x u ⟩
     ⟦ x ⟧ ◦ ⟦ u ⟧  ≈⟨  ◦-cong (⟦⟧-cong x≈y) (⟦⟧-cong u≈v) ⟩
-    ⟦ y ⟧ ◦ ⟦ v ⟧  ≈˘⟨ homo y v ⟩
+    ⟦ y ⟧ ◦ ⟦ v ⟧  ≈⟨ homo y v ⟨
     ⟦ y ∙ v ⟧      ∎)
 
   assoc : Associative _≈₂_ _◦_ → Associative _≈₁_ _∙_
@@ -50,15 +50,15 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
     ⟦ (x ∙ y) ∙ z ⟧          ≈⟨  homo (x ∙ y) z ⟩
     ⟦ x ∙ y ⟧ ◦ ⟦ z ⟧        ≈⟨  ◦-cong (homo x y) refl ⟩
     (⟦ x ⟧ ◦ ⟦ y ⟧) ◦ ⟦ z ⟧  ≈⟨  assoc ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩
-    ⟦ x ⟧ ◦ (⟦ y ⟧ ◦ ⟦ z ⟧)  ≈˘⟨ ◦-cong refl (homo y z) ⟩
-    ⟦ x ⟧ ◦ ⟦ y ∙ z ⟧        ≈˘⟨ homo x (y ∙ z) ⟩
+    ⟦ x ⟧ ◦ (⟦ y ⟧ ◦ ⟦ z ⟧)  ≈⟨ ◦-cong refl (homo y z) ⟨
+    ⟦ x ⟧ ◦ ⟦ y ∙ z ⟧        ≈⟨ homo x (y ∙ z) ⟨
     ⟦ x ∙ (y ∙ z) ⟧          ∎)
 
   comm : Commutative _≈₂_ _◦_ → Commutative _≈₁_ _∙_
   comm comm x y = injective (begin
     ⟦ x ∙ y ⟧      ≈⟨  homo x y ⟩
     ⟦ x ⟧ ◦ ⟦ y ⟧  ≈⟨  comm ⟦ x ⟧ ⟦ y ⟧ ⟩
-    ⟦ y ⟧ ◦ ⟦ x ⟧  ≈˘⟨ homo y x ⟩
+    ⟦ y ⟧ ◦ ⟦ x ⟧  ≈⟨ homo y x ⟨
     ⟦ y ∙ x ⟧      ∎)
 
   idem : Idempotent _≈₂_ _◦_ → Idempotent _≈₁_ _∙_
@@ -80,14 +80,14 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
 
   cancelˡ : LeftCancellative _≈₂_ _◦_ → LeftCancellative _≈₁_ _∙_
   cancelˡ cancelˡ x y z x∙y≈x∙z = injective (cancelˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ (begin
-    ⟦ x ⟧ ◦ ⟦ y ⟧  ≈˘⟨ homo x y ⟩
+    ⟦ x ⟧ ◦ ⟦ y ⟧  ≈⟨ homo x y ⟨
     ⟦ x ∙ y ⟧      ≈⟨  ⟦⟧-cong x∙y≈x∙z ⟩
     ⟦ x ∙ z ⟧      ≈⟨  homo x z ⟩
     ⟦ x ⟧ ◦ ⟦ z ⟧  ∎))
 
   cancelʳ : RightCancellative _≈₂_ _◦_ → RightCancellative _≈₁_ _∙_
   cancelʳ cancelʳ x y z y∙x≈z∙x = injective (cancelʳ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ (begin
-    ⟦ y ⟧ ◦ ⟦ x ⟧  ≈˘⟨ homo y x ⟩
+    ⟦ y ⟧ ◦ ⟦ x ⟧  ≈⟨ homo y x ⟨
     ⟦ y ∙ x ⟧      ≈⟨  ⟦⟧-cong y∙x≈z∙x ⟩
     ⟦ z ∙ x ⟧      ≈⟨  homo z x ⟩
     ⟦ z ⟧ ◦ ⟦ x ⟧  ∎))
diff --git a/src/Algebra/Morphism/MonoidMonomorphism.agda b/src/Algebra/Morphism/MonoidMonomorphism.agda
index 4a512b2687..bfb412302e 100644
--- a/src/Algebra/Morphism/MonoidMonomorphism.agda
+++ b/src/Algebra/Morphism/MonoidMonomorphism.agda
@@ -63,7 +63,7 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
     ⟦ ε₁ ∙ x ⟧     ≈⟨  homo ε₁ x ⟩
     ⟦ ε₁ ⟧ ◦ ⟦ x ⟧ ≈⟨  ◦-cong ε-homo refl ⟩
     ε₂   ◦ ⟦ x ⟧   ≈⟨  zeˡ ⟦ x ⟧ ⟩
-    ε₂             ≈˘⟨ ε-homo ⟩
+    ε₂             ≈⟨ ε-homo ⟨
     ⟦ ε₁ ⟧         ∎)
 
   zeroʳ : RightZero _≈₂_ ε₂ _◦_ → RightZero _≈₁_ ε₁ _∙_
@@ -71,7 +71,7 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
     ⟦ x ∙ ε₁ ⟧     ≈⟨  homo x ε₁ ⟩
     ⟦ x ⟧ ◦ ⟦ ε₁ ⟧ ≈⟨  ◦-cong refl ε-homo ⟩
     ⟦ x ⟧ ◦ ε₂     ≈⟨  zeʳ ⟦ x ⟧ ⟩
-    ε₂             ≈˘⟨ ε-homo ⟩
+    ε₂             ≈⟨ ε-homo ⟨
     ⟦ ε₁ ⟧         ∎)
 
   zero : Zero _≈₂_ ε₂ _◦_ → Zero _≈₁_ ε₁ _∙_
diff --git a/src/Algebra/Morphism/RingMonomorphism.agda b/src/Algebra/Morphism/RingMonomorphism.agda
index 4b2c900946..9230c802a9 100644
--- a/src/Algebra/Morphism/RingMonomorphism.agda
+++ b/src/Algebra/Morphism/RingMonomorphism.agda
@@ -90,8 +90,8 @@ module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)
     ⟦ x * (y + z) ⟧               ≈⟨ *-homo x (y + z) ⟩
     ⟦ x ⟧ ⊛ ⟦ y + z ⟧             ≈⟨ ◦-cong refl (+-homo y z) ⟩
     ⟦ x ⟧ ⊛ (⟦ y ⟧ ⊕ ⟦ z ⟧)       ≈⟨ distribˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩
-    ⟦ x ⟧ ⊛ ⟦ y ⟧ ⊕ ⟦ x ⟧ ⊛ ⟦ z ⟧ ≈˘⟨ ∙-cong (*-homo x y) (*-homo x z) ⟩
-    ⟦ x * y ⟧ ⊕ ⟦ x * z ⟧         ≈˘⟨ +-homo (x * y) (x * z) ⟩
+    ⟦ x ⟧ ⊛ ⟦ y ⟧ ⊕ ⟦ x ⟧ ⊛ ⟦ z ⟧ ≈⟨ ∙-cong (*-homo x y) (*-homo x z) ⟨
+    ⟦ x * y ⟧ ⊕ ⟦ x * z ⟧         ≈⟨ +-homo (x * y) (x * z) ⟨
     ⟦ x * y + x * z ⟧ ∎)
 
   distribʳ : _DistributesOverʳ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverʳ_ _≈₁_ _*_ _+_
@@ -99,8 +99,8 @@ module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)
     ⟦ (y + z) * x ⟧               ≈⟨ *-homo (y + z) x ⟩
     ⟦ y + z ⟧ ⊛ ⟦ x ⟧             ≈⟨ ◦-cong (+-homo y z) refl ⟩
     (⟦ y ⟧ ⊕ ⟦ z ⟧) ⊛ ⟦ x ⟧       ≈⟨ distribˡ ⟦ x ⟧ ⟦ y ⟧ ⟦ z ⟧ ⟩
-    ⟦ y ⟧ ⊛ ⟦ x ⟧ ⊕ ⟦ z ⟧ ⊛ ⟦ x ⟧ ≈˘⟨ ∙-cong (*-homo y x) (*-homo z x) ⟩
-    ⟦ y * x ⟧ ⊕ ⟦ z * x ⟧         ≈˘⟨ +-homo (y * x) (z * x) ⟩
+    ⟦ y ⟧ ⊛ ⟦ x ⟧ ⊕ ⟦ z ⟧ ⊛ ⟦ x ⟧ ≈⟨ ∙-cong (*-homo y x) (*-homo z x) ⟨
+    ⟦ y * x ⟧ ⊕ ⟦ z * x ⟧         ≈⟨ +-homo (y * x) (z * x) ⟨
     ⟦ y * x + z * x ⟧ ∎)
 
   distrib : _DistributesOver_ _≈₂_ _⊛_ _⊕_ → _DistributesOver_ _≈₁_ _*_ _+_
@@ -111,7 +111,7 @@ module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)
     ⟦ 0# * x ⟧     ≈⟨ *-homo 0# x ⟩
     ⟦ 0# ⟧ ⊛ ⟦ x ⟧ ≈⟨ ◦-cong 0#-homo refl ⟩
     0#₂ ⊛ ⟦ x ⟧    ≈⟨ zeroˡ ⟦ x ⟧ ⟩
-    0#₂            ≈˘⟨ 0#-homo ⟩
+    0#₂            ≈⟨ 0#-homo ⟨
     ⟦ 0# ⟧         ∎)
 
   zeroʳ : RightZero _≈₂_ 0#₂ _⊛_ → RightZero _≈₁_ 0# _*_
@@ -119,7 +119,7 @@ module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)
     ⟦ x * 0# ⟧     ≈⟨ *-homo x 0# ⟩
     ⟦ x ⟧ ⊛ ⟦ 0# ⟧ ≈⟨ ◦-cong refl 0#-homo ⟩
     ⟦ x ⟧ ⊛ 0#₂    ≈⟨ zeroʳ ⟦ x ⟧ ⟩
-    0#₂            ≈˘⟨ 0#-homo ⟩
+    0#₂            ≈⟨ 0#-homo ⟨
     ⟦ 0# ⟧         ∎)
 
   zero : Zero _≈₂_ 0#₂ _⊛_ → Zero _≈₁_ 0# _*_
diff --git a/src/Algebra/Properties/AbelianGroup.agda b/src/Algebra/Properties/AbelianGroup.agda
index 6846f29d83..87c6b7c6b9 100644
--- a/src/Algebra/Properties/AbelianGroup.agda
+++ b/src/Algebra/Properties/AbelianGroup.agda
@@ -33,6 +33,6 @@ xyx⁻¹≈y x y = begin
 
 ⁻¹-∙-comm : ∀ x y → x ⁻¹ ∙ y ⁻¹ ≈ (x ∙ y) ⁻¹
 ⁻¹-∙-comm x y = begin
-  x ⁻¹ ∙ y ⁻¹ ≈˘⟨ ⁻¹-anti-homo-∙ y x ⟩
+  x ⁻¹ ∙ y ⁻¹ ≈⟨ ⁻¹-anti-homo-∙ y x ⟨
   (y ∙ x) ⁻¹  ≈⟨ ⁻¹-cong $ comm y x ⟩
   (x ∙ y) ⁻¹  ∎
diff --git a/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda b/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda
index 8985fb21cb..0774e5d7a1 100644
--- a/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda
+++ b/src/Algebra/Properties/CommutativeMonoid/Mult/TCOptimised.agda
@@ -40,7 +40,7 @@ open import Algebra.Properties.Monoid.Mult.TCOptimised monoid public
 
 ×-distrib-+ : ∀ x y n → n × (x + y) ≈ n × x + n × y
 ×-distrib-+ x y n = begin
-  n ×  (x + y)    ≈˘⟨ ×ᵤ≈× n (x + y) ⟩
+  n ×  (x + y)    ≈⟨ ×ᵤ≈× n (x + y) ⟨
   n ×ᵤ (x + y)    ≈⟨  U.×-distrib-+ x y n ⟩
   n ×ᵤ x + n ×ᵤ y ≈⟨  +-cong (×ᵤ≈× n x) (×ᵤ≈× n y) ⟩
   n ×  x + n ×  y ∎
diff --git a/src/Algebra/Properties/CommutativeMonoid/Sum.agda b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
index 5873d4eb61..eed54de4ed 100644
--- a/src/Algebra/Properties/CommutativeMonoid/Sum.agda
+++ b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
@@ -90,9 +90,9 @@ sum-permute {zero}  {suc n} f π = contradiction π (Perm.refute λ())
 sum-permute {suc m} {zero}  f π = contradiction π (Perm.refute λ())
 sum-permute {suc m} {suc n} f π = begin
   sum f                                    ≡⟨⟩
-  f 0F  + sum f/0                          ≡˘⟨ P.cong (_+ sum f/0) (P.cong f (Perm.inverseʳ π)) ⟩
+  f 0F  + sum f/0                          ≡⟨ P.cong (_+ sum f/0) (P.cong f (Perm.inverseʳ π)) ⟨
   πf π₀ + sum f/0                          ≈⟨ +-congˡ (sum-permute f/0 (Perm.remove π₀ π)) ⟩
-  πf π₀ + sum (rearrange (π/0 ⟨$⟩ʳ_) f/0)  ≡˘⟨ P.cong (πf π₀ +_) (sum-cong-≗ (P.cong f ∘ Perm.punchIn-permute′ π 0F)) ⟩
+  πf π₀ + sum (rearrange (π/0 ⟨$⟩ʳ_) f/0)  ≡⟨ P.cong (πf π₀ +_) (sum-cong-≗ (P.cong f ∘ Perm.punchIn-permute′ π 0F)) ⟨
   πf π₀ + sum (removeAt πf π₀)             ≈⟨ sym (sum-remove πf) ⟩
   sum πf                                   ∎
   where
diff --git a/src/Algebra/Properties/CommutativeSemigroup.agda b/src/Algebra/Properties/CommutativeSemigroup.agda
index 0c27a5e96b..bcfee1fec2 100644
--- a/src/Algebra/Properties/CommutativeSemigroup.agda
+++ b/src/Algebra/Properties/CommutativeSemigroup.agda
@@ -29,10 +29,10 @@ open import Algebra.Properties.Semigroup semigroup public
 interchange : Interchangable _∙_ _∙_
 interchange a b c d = begin
   (a ∙ b) ∙ (c ∙ d)  ≈⟨  assoc a b (c ∙ d) ⟩
-  a ∙ (b ∙ (c ∙ d))  ≈˘⟨ ∙-congˡ (assoc b c d) ⟩
+  a ∙ (b ∙ (c ∙ d))  ≈⟨ ∙-congˡ (assoc b c d) ⟨
   a ∙ ((b ∙ c) ∙ d)  ≈⟨  ∙-congˡ (∙-congʳ (comm b c)) ⟩
   a ∙ ((c ∙ b) ∙ d)  ≈⟨  ∙-congˡ (assoc c b d) ⟩
-  a ∙ (c ∙ (b ∙ d))  ≈˘⟨ assoc a c (b ∙ d) ⟩
+  a ∙ (c ∙ (b ∙ d))  ≈⟨ assoc a c (b ∙ d) ⟨
   (a ∙ c) ∙ (b ∙ d)  ∎
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Properties/Group.agda b/src/Algebra/Properties/Group.agda
index 7cdf88c9fe..31f9aedf39 100644
--- a/src/Algebra/Properties/Group.agda
+++ b/src/Algebra/Properties/Group.agda
@@ -42,14 +42,14 @@ private
 ∙-cancelˡ x y z eq = begin
               y  ≈⟨ right-helper x y ⟩
   x ⁻¹ ∙ (x ∙ y) ≈⟨ ∙-congˡ eq ⟩
-  x ⁻¹ ∙ (x ∙ z) ≈˘⟨ right-helper x z ⟩
+  x ⁻¹ ∙ (x ∙ z) ≈⟨ right-helper x z ⟨
               z  ∎
 
 ∙-cancelʳ : RightCancellative _∙_
 ∙-cancelʳ x y z eq = begin
   y            ≈⟨ left-helper y x ⟩
   y ∙ x ∙ x ⁻¹ ≈⟨ ∙-congʳ eq ⟩
-  z ∙ x ∙ x ⁻¹ ≈˘⟨ left-helper z x ⟩
+  z ∙ x ∙ x ⁻¹ ≈⟨ left-helper z x ⟨
   z            ∎
 
 ∙-cancel : Cancellative _∙_
@@ -57,22 +57,22 @@ private
 
 ⁻¹-involutive : ∀ x → x ⁻¹ ⁻¹ ≈ x
 ⁻¹-involutive x = begin
-  x ⁻¹ ⁻¹              ≈˘⟨ identityʳ _ ⟩
-  x ⁻¹ ⁻¹ ∙ ε          ≈˘⟨ ∙-congˡ $ inverseˡ _ ⟩
-  x ⁻¹ ⁻¹ ∙ (x ⁻¹ ∙ x) ≈˘⟨ right-helper (x ⁻¹) x ⟩
+  x ⁻¹ ⁻¹              ≈⟨ identityʳ _ ⟨
+  x ⁻¹ ⁻¹ ∙ ε          ≈⟨ ∙-congˡ $ inverseˡ _ ⟨
+  x ⁻¹ ⁻¹ ∙ (x ⁻¹ ∙ x) ≈⟨ right-helper (x ⁻¹) x ⟨
   x                    ∎
 
 ⁻¹-injective : ∀ {x y} → x ⁻¹ ≈ y ⁻¹ → x ≈ y
 ⁻¹-injective {x} {y} eq = ∙-cancelʳ _ _ _ ( begin
   x ∙ x ⁻¹ ≈⟨ inverseʳ x ⟩
-  ε        ≈˘⟨ inverseʳ y ⟩
-  y ∙ y ⁻¹ ≈˘⟨ ∙-congˡ eq ⟩
+  ε        ≈⟨ inverseʳ y ⟨
+  y ∙ y ⁻¹ ≈⟨ ∙-congˡ eq ⟨
   y ∙ x ⁻¹ ∎ )
 
 ⁻¹-anti-homo-∙ : ∀ x y → (x ∙ y) ⁻¹ ≈ y ⁻¹ ∙ x ⁻¹
 ⁻¹-anti-homo-∙ x y = ∙-cancelˡ _ _ _ ( begin
   x ∙ y ∙ (x ∙ y) ⁻¹    ≈⟨ inverseʳ _ ⟩
-  ε                     ≈˘⟨ inverseʳ _ ⟩
+  ε                     ≈⟨ inverseʳ _ ⟨
   x ∙ x ⁻¹              ≈⟨ ∙-congʳ (left-helper x y) ⟩
   (x ∙ y) ∙ y ⁻¹ ∙ x ⁻¹ ≈⟨ assoc (x ∙ y) (y ⁻¹) (x ⁻¹) ⟩
   x ∙ y ∙ (y ⁻¹ ∙ x ⁻¹) ∎ )
diff --git a/src/Algebra/Properties/Monoid/Mult.agda b/src/Algebra/Properties/Monoid/Mult.agda
index 7abdb583ca..88c4258fcb 100644
--- a/src/Algebra/Properties/Monoid/Mult.agda
+++ b/src/Algebra/Properties/Monoid/Mult.agda
@@ -70,5 +70,5 @@ open import Algebra.Definitions.RawMonoid rawMonoid public
 ×-assocˡ x zero    n = refl
 ×-assocˡ x (suc m) n = begin
   n × x + m × n × x     ≈⟨ +-congˡ (×-assocˡ x m n) ⟩
-  n × x + (m ℕ.* n) × x ≈˘⟨ ×-homo-+ x n (m ℕ.* n) ⟩
+  n × x + (m ℕ.* n) × x ≈⟨ ×-homo-+ x n (m ℕ.* n) ⟨
   (suc m ℕ.* n) × x     ∎
diff --git a/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda b/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
index ded5f1339e..c862c5e3f2 100644
--- a/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
+++ b/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
@@ -55,14 +55,14 @@ open import Algebra.Definitions.RawMonoid rawMonoid public
 ×ᵤ≈× 0       x = refl
 ×ᵤ≈× (suc n) x = begin
   x + n ×ᵤ x  ≈⟨ +-congˡ (×ᵤ≈× n x) ⟩
-  x + n ×  x  ≈˘⟨ 1+× n x ⟩
+  x + n ×  x  ≈⟨ 1+× n x ⟨
   suc n ×  x  ∎
 
 -- _×_ is homomorphic with respect to _ℕ.+_/_+_.
 
 ×-homo-+ : ∀ c m n → (m ℕ.+ n) × c ≈ m × c + n × c
 ×-homo-+ c m n = begin
-  (m ℕ.+ n) ×  c   ≈˘⟨ ×ᵤ≈× (m ℕ.+ n) c ⟩
+  (m ℕ.+ n) ×  c   ≈⟨ ×ᵤ≈× (m ℕ.+ n) c ⟨
   (m ℕ.+ n) ×ᵤ c   ≈⟨ U.×-homo-+ c m n ⟩
   m ×ᵤ c + n ×ᵤ c  ≈⟨ +-cong (×ᵤ≈× m c) (×ᵤ≈× n c) ⟩
   m ×  c + n ×  c  ∎
@@ -79,8 +79,8 @@ open import Algebra.Definitions.RawMonoid rawMonoid public
 
 ×-assocˡ : ∀ x m n → m × (n × x) ≈ (m ℕ.* n) × x
 ×-assocˡ x m n = begin
-  m ×  (n ×  x)  ≈˘⟨ ×-congʳ m (×ᵤ≈× n x) ⟩
-  m ×  (n ×ᵤ x)  ≈˘⟨ ×ᵤ≈× m (n ×ᵤ x) ⟩
+  m ×  (n ×  x)  ≈⟨ ×-congʳ m (×ᵤ≈× n x) ⟨
+  m ×  (n ×ᵤ x)  ≈⟨ ×ᵤ≈× m (n ×ᵤ x) ⟨
   m ×ᵤ (n ×ᵤ x)  ≈⟨  U.×-assocˡ x m n ⟩
   (m ℕ.* n) ×ᵤ x ≈⟨  ×ᵤ≈× (m ℕ.* n) x ⟩
   (m ℕ.* n) ×  x ∎
diff --git a/src/Algebra/Properties/Monoid/Sum.agda b/src/Algebra/Properties/Monoid/Sum.agda
index 166cba4462..6ba7d12fc8 100644
--- a/src/Algebra/Properties/Monoid/Sum.agda
+++ b/src/Algebra/Properties/Monoid/Sum.agda
@@ -81,11 +81,11 @@ sum-replicate-zero (suc n) = sum-replicate-idem (+-identityˡ 0#) (suc n)
 sum-init-last : ∀ {n} (t : Vector Carrier (suc n)) → sum t ≈ sum (init t) + last t
 sum-init-last {zero} t  = begin
   t₀ + 0# ≈⟨ +-identityʳ t₀ ⟩
-  t₀      ≈˘⟨ +-identityˡ t₀ ⟩
+  t₀      ≈⟨ +-identityˡ t₀ ⟨
   0# + t₀ ∎ where t₀ = t zero
 sum-init-last {suc n} t = begin
   t₀ + ∑t             ≈⟨ +-congˡ (sum-init-last (tail t)) ⟩
-  t₀ + (∑t′ + tₗ)     ≈˘⟨ +-assoc _ _ _ ⟩
+  t₀ + (∑t′ + tₗ)     ≈⟨ +-assoc _ _ _ ⟨
   (t₀ + ∑t′) + tₗ     ∎
   where
     t₀ = head t
diff --git a/src/Algebra/Properties/Semiring/Binomial.agda b/src/Algebra/Properties/Semiring/Binomial.agda
index 122da9fcb9..6731018af6 100644
--- a/src/Algebra/Properties/Semiring/Binomial.agda
+++ b/src/Algebra/Properties/Semiring/Binomial.agda
@@ -77,7 +77,7 @@ term₂[n,n+1]≈0# n rewrite view-fromℕ (suc n) = refl
 lemma₁ : ∀ n → x * binomialExpansion n ≈ sum₁ n
 lemma₁ n = begin
   x * binomialExpansion n                ≈⟨ *-distribˡ-sum x (binomialTerm n) ⟩
-  ∑[ i ≤ n ] (x * binomialTerm n i)      ≈˘⟨ +-identityˡ _ ⟩
+  ∑[ i ≤ n ] (x * binomialTerm n i)      ≈⟨ +-identityˡ _ ⟨
   0# + ∑[ i ≤ n ] (x * binomialTerm n i) ≡⟨⟩
   0# + ∑[ i ≤ n ] term₁ n (suc i)        ≡⟨⟩
   sum₁ n                                 ∎
@@ -85,9 +85,9 @@ lemma₁ n = begin
 lemma₂ : ∀ n → y * binomialExpansion n ≈ sum₂ n
 lemma₂ n = begin
   y * binomialExpansion n                       ≈⟨ *-distribˡ-sum y (binomialTerm n) ⟩
-  ∑[ i ≤ n ] (y * binomialTerm n i)             ≈˘⟨ +-identityʳ _ ⟩
+  ∑[ i ≤ n ] (y * binomialTerm n i)             ≈⟨ +-identityʳ _ ⟨
   ∑[ i ≤ n ] (y * binomialTerm n i) + 0#        ≈⟨ +-cong (sum-cong-≋ lemma₂-inject₁) (sym (term₂[n,n+1]≈0# n)) ⟩
-  (∑[ i ≤ n ] term₂-inject₁ i) + term₂ n [n+1]  ≈˘⟨ sum-init-last (term₂ n) ⟩
+  (∑[ i ≤ n ] term₂-inject₁ i) + term₂ n [n+1]  ≈⟨ sum-init-last (term₂ n) ⟨
   sum (term₂ n)                                 ≡⟨⟩
   sum₂ n                                        ∎
   where
@@ -106,8 +106,8 @@ x*lemma : ∀ {n} (i : Fin (suc n)) →
           x * binomialTerm n i ≈ (n C toℕ i) × binomial (suc n) (suc i)
 x*lemma {n} i = begin
   x * binomialTerm n i                        ≡⟨⟩
-  x * ((n C k) × (x ^ k * y ^ (n ∸ k)))       ≈˘⟨ *-congˡ (×-assoc-* (n C k) _ _) ⟩
-  x * ((n C k) × x ^ k * y ^ (n ∸ k))         ≈˘⟨ *-assoc x ((n C k) × x ^ k) _ ⟩
+  x * ((n C k) × (x ^ k * y ^ (n ∸ k)))       ≈⟨ *-congˡ (×-assoc-* (n C k) _ _) ⟨
+  x * ((n C k) × x ^ k * y ^ (n ∸ k))         ≈⟨ *-assoc x ((n C k) × x ^ k) _ ⟨
   (x * (n C k) × x ^ k) * y ^ (n ∸ k)         ≈⟨ *-congʳ (×-comm-* (n C k) _ _) ⟩
   (n C k) × x ^ (suc k) * y ^ (n ∸ k)         ≡⟨⟩
   (n C k) × x ^ (suc k) * y ^ (suc n ∸ suc k) ≈⟨ ×-assoc-* (n C k) _ _ ⟩
@@ -125,9 +125,9 @@ y*lemma x*y≈y*x {n} j = begin
   nC[j+1] × (y * binomial n (suc j))
     ≈⟨ ×-congʳ nC[j+1] (y*x^m*y^n≈x^m*y^[n+1] x*y≈y*x [j+1] [n-j-1]) ⟩
   nC[j+1] × (x ^ [j+1] * y ^ [n-j])
-    ≈˘⟨ ×-congʳ nC[j+1] (*-congʳ (^-congʳ x (cong suc k≡j))) ⟩
+    ≈⟨ ×-congʳ nC[j+1] (*-congʳ (^-congʳ x (cong suc k≡j))) ⟨
   nC[j+1] × (x ^ [k+1] * y ^ [n-j])
-    ≈˘⟨ ×-congʳ nC[j+1] (*-congˡ (^-congʳ y [n-k]≡[n-j])) ⟩
+    ≈⟨ ×-congʳ nC[j+1] (*-congˡ (^-congʳ y [n-k]≡[n-j])) ⟨
   nC[j+1] × (x ^ [k+1] * y ^ [n-k])
     ≡⟨⟩
   nC[j+1] × (x ^ [k+1] * y ^ [n+1]-[k+1])
@@ -165,8 +165,8 @@ term₁+term₂≈term x*y≈y*x n 0F = begin
   y * (1 × (1# * y ^ n))       ≈⟨ *-congˡ (+-identityʳ (1# * y ^ n)) ⟩
   y * (1# * y ^ n)             ≈⟨ *-congˡ (*-identityˡ (y ^ n)) ⟩
   y * y ^ n                    ≡⟨⟩
-  y ^ suc n                    ≈˘⟨ *-identityˡ (y ^ suc n) ⟩
-  1# * y ^ suc n               ≈˘⟨ +-identityʳ (1# * y ^ suc n) ⟩
+  y ^ suc n                    ≈⟨ *-identityˡ (y ^ suc n) ⟨
+  1# * y ^ suc n               ≈⟨ +-identityʳ (1# * y ^ suc n) ⟨
   1 × (1# * y ^ suc n)         ≡⟨⟩
   binomialTerm (suc n) 0F      ∎
 
@@ -180,8 +180,8 @@ term₁+term₂≈term x*y≈y*x n (suc i) with view i
     = begin
   x * ((x ^ n * 1#) + 0#) + 0# ≈⟨ +-identityʳ _ ⟩
   x * ((x ^ n * 1#) + 0#)      ≈⟨ *-congˡ (+-identityʳ _) ⟩
-  x * (x ^ n * 1#)             ≈˘⟨ *-assoc _ _ _ ⟩
-  x * x ^ n * 1#               ≈˘⟨ +-identityʳ _ ⟩
+  x * (x ^ n * 1#)             ≈⟨ *-assoc _ _ _ ⟨
+  x * x ^ n * 1#               ≈⟨ +-identityʳ _ ⟨
   1 × (x * x ^ n * 1#)         ∎
 
 ... | ‵inject₁ j
@@ -190,9 +190,9 @@ term₁+term₂≈term x*y≈y*x n (suc i) with view i
   (x * binomialTerm n i) + (y * binomialTerm n (suc j))
     ≈⟨ +-cong (x*lemma i) (y*lemma x*y≈y*x j) ⟩
   (nCk × [x^k+1]*[y^n-k]) + (nC[j+1] × [x^k+1]*[y^n-k])
-    ≈˘⟨ +-congˡ (×-congˡ nC[k+1]≡nC[j+1]) ⟩
+    ≈⟨ +-congˡ (×-congˡ nC[k+1]≡nC[j+1]) ⟨
   (nCk × [x^k+1]*[y^n-k]) + (nC[k+1] × [x^k+1]*[y^n-k])
-    ≈˘⟨ ×-homo-+ [x^k+1]*[y^n-k] nCk nC[k+1] ⟩
+    ≈⟨ ×-homo-+ [x^k+1]*[y^n-k] nCk nC[k+1] ⟨
   (nCk Nat.+ nC[k+1]) × [x^k+1]*[y^n-k]
     ≡⟨ cong (_× [x^k+1]*[y^n-k]) (nCk+nC[k+1]≡[n+1]C[k+1] n k) ⟩
   ((suc n) C (suc k)) × [x^k+1]*[y^n-k]
@@ -216,11 +216,11 @@ term₁+term₂≈term x*y≈y*x n (suc i) with view i
 theorem : x * y ≈ y * x → ∀ n → (x + y) ^ n ≈ binomialExpansion n
 theorem x*y≈y*x zero    = begin
   (x + y) ^ 0                     ≡⟨⟩
-  1#                              ≈˘⟨ 1×-identityʳ 1# ⟩
-  1 × 1#                          ≈˘⟨ *-identityʳ (1 × 1#) ⟩
+  1#                              ≈⟨ 1×-identityʳ 1# ⟨
+  1 × 1#                          ≈⟨ *-identityʳ (1 × 1#) ⟨
   (1 × 1#) * 1#                   ≈⟨ ×-assoc-* 1 1# 1# ⟩
   1 × (1# * 1#)                   ≡⟨⟩
-  1 × (x ^ 0 * y ^ 0)             ≈˘⟨ +-identityʳ _ ⟩
+  1 × (x ^ 0 * y ^ 0)             ≈⟨ +-identityʳ _ ⟨
   1 × (x ^ 0 * y ^ 0) + 0#        ≡⟨⟩
   (0 C 0) × (x ^ 0 * y ^ 0) + 0#  ≡⟨⟩
   binomialExpansion 0             ∎
@@ -229,7 +229,7 @@ theorem x*y≈y*x n+1@(suc n) = begin
   (x + y) * (x + y) ^ n                             ≈⟨ *-congˡ (theorem x*y≈y*x n) ⟩
   (x + y) * binomialExpansion n                     ≈⟨ distribʳ _ _ _ ⟩
   x * binomialExpansion n + y * binomialExpansion n ≈⟨ +-cong (lemma₁ n) (lemma₂ n) ⟩
-  sum₁ n + sum₂ n                                   ≈˘⟨ ∑-distrib-+ (term₁ n) (term₂ n) ⟩
+  sum₁ n + sum₂ n                                   ≈⟨ ∑-distrib-+ (term₁ n) (term₂ n) ⟨
   ∑[ i ≤ n+1 ] (term₁ n i + term₂ n i)              ≈⟨ sum-cong-≋ (term₁+term₂≈term x*y≈y*x n) ⟩
   ∑[ i ≤ n+1 ] binomialTerm n+1 i                   ≡⟨⟩
   binomialExpansion n+1                             ∎
diff --git a/src/Algebra/Properties/Semiring/Exp.agda b/src/Algebra/Properties/Semiring/Exp.agda
index 5da80a99f1..ed8b094520 100644
--- a/src/Algebra/Properties/Semiring/Exp.agda
+++ b/src/Algebra/Properties/Semiring/Exp.agda
@@ -57,16 +57,16 @@ y*x^m*y^n≈x^m*y^[n+1] {x} {y} x*y≈y*x = helper
       y * (x ^ ℕ.zero * y ^ n)  ≡⟨⟩
       y * (1# * y ^ n)          ≈⟨ *-congˡ (*-identityˡ (y ^ n)) ⟩
       y * (y ^ n)               ≡⟨⟩
-      y ^ (suc n)               ≈˘⟨ *-identityˡ (y ^ suc n) ⟩
+      y ^ (suc n)               ≈⟨ *-identityˡ (y ^ suc n) ⟨
       1# * y ^ (suc n)          ≡⟨⟩
       x ^ ℕ.zero * y ^ (suc n)  ∎
     helper (suc m) n = begin
       y * (x ^ suc m * y ^ n)    ≡⟨⟩
       y * ((x * x ^ m) * y ^ n)  ≈⟨ *-congˡ (*-assoc x (x ^ m) (y ^ n)) ⟩
-      y * (x * (x ^ m * y ^ n))  ≈˘⟨ *-assoc y x (x ^ m * y ^ n) ⟩
-      y * x * (x ^ m * y ^ n)    ≈˘⟨ *-congʳ x*y≈y*x ⟩
+      y * (x * (x ^ m * y ^ n))  ≈⟨ *-assoc y x (x ^ m * y ^ n) ⟨
+      y * x * (x ^ m * y ^ n)    ≈⟨ *-congʳ x*y≈y*x ⟨
       x * y * (x ^ m * y ^ n)    ≈⟨ *-assoc x y _ ⟩
       x * (y * (x ^ m * y ^ n))  ≈⟨ *-congˡ (helper m n) ⟩
-      x * (x ^ m * y ^ suc n)    ≈˘⟨ *-assoc x (x ^ m) (y ^ suc n) ⟩
+      x * (x ^ m * y ^ suc n)    ≈⟨ *-assoc x (x ^ m) (y ^ suc n) ⟨
       (x * x ^ m) * y ^ suc n    ≡⟨⟩
       x ^ suc m * y ^ suc n      ∎
diff --git a/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda b/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda
index d661dde6b1..1ad573866d 100644
--- a/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda
+++ b/src/Algebra/Properties/Semiring/Exp/TailRecursiveOptimised.agda
@@ -72,7 +72,7 @@ x^[m+1]≈x*[x^m] x m = x^[m]*[x*y]≈x*x^[m]*y x m 1#
 x^[m+n]≈[x^m]*[x^n] : ∀ x m n → x ^ (m ℕ.+ n) ≈ x ^ m * x ^ n
 x^[m+n]≈[x^m]*[x^n] x m n = begin
   x ^ (m  ℕ.+ n)   ≈⟨ x^[m+n]*y≈x^[m]*x^[n]*y x m n 1# ⟩
-  x ^[ m ]* (x ^ n) ≈˘⟨ x^m*y≈x^[m]*y x m (x ^ n) ⟩
+  x ^[ m ]* (x ^ n) ≈⟨ x^m*y≈x^[m]*y x m (x ^ n) ⟨
   x ^ m * x ^ n    ∎
 
 ^≈^ᵘ : ∀ x m → x ^ m ≈ x ^ᵘ m
diff --git a/src/Algebra/Properties/Semiring/Mult.agda b/src/Algebra/Properties/Semiring/Mult.agda
index 3121e8a8e8..801e4f1966 100644
--- a/src/Algebra/Properties/Semiring/Mult.agda
+++ b/src/Algebra/Properties/Semiring/Mult.agda
@@ -30,8 +30,8 @@ open import Algebra.Properties.Monoid.Mult +-monoid public
 ×1-homo-* (suc m) n = begin
   (n ℕ.+ m ℕ.* n) × 1#                ≈⟨  ×-homo-+ 1# n (m ℕ.* n) ⟩
   n × 1# + (m ℕ.* n) × 1#             ≈⟨  +-congˡ (×1-homo-* m n) ⟩
-  n × 1# + (m × 1#) * (n × 1#)        ≈˘⟨ +-congʳ (*-identityˡ _) ⟩
-  1# * (n × 1#) + (m × 1#) * (n × 1#) ≈˘⟨ distribʳ (n × 1#) 1# (m × 1#) ⟩
+  n × 1# + (m × 1#) * (n × 1#)        ≈⟨ +-congʳ (*-identityˡ _) ⟨
+  1# * (n × 1#) + (m × 1#) * (n × 1#) ≈⟨ distribʳ (n × 1#) 1# (m × 1#) ⟨
   (1# + m × 1#) * (n × 1#)            ∎
 
 -- (1 ×_) is the identity
diff --git a/src/Algebra/Properties/Semiring/Mult/TCOptimised.agda b/src/Algebra/Properties/Semiring/Mult/TCOptimised.agda
index 4ea0e2939b..6f1e538e7d 100644
--- a/src/Algebra/Properties/Semiring/Mult/TCOptimised.agda
+++ b/src/Algebra/Properties/Semiring/Mult/TCOptimised.agda
@@ -30,7 +30,7 @@ open import Algebra.Properties.Monoid.Mult.TCOptimised +-monoid public
 
 ×1-homo-* : ∀ m n → (m ℕ.* n) × 1# ≈ (m × 1#) * (n × 1#)
 ×1-homo-* m n = begin
-  (m ℕ.* n) ×  1#        ≈˘⟨ ×ᵤ≈× (m ℕ.* n) 1# ⟩
+  (m ℕ.* n) ×  1#        ≈⟨ ×ᵤ≈× (m ℕ.* n) 1# ⟨
   (m ℕ.* n) ×ᵤ 1#        ≈⟨  U.×1-homo-* m n ⟩
   (m ×ᵤ 1#) * (n ×ᵤ 1#)  ≈⟨  *-cong (×ᵤ≈× m 1#) (×ᵤ≈× n 1#) ⟩
   (m ×  1#) * (n ×  1#)  ∎
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients.agda b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
index 6d3ab81b37..0c5f807042 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients.agda
@@ -64,7 +64,7 @@ private
     where
     to : m ×ᵤ 1# ≈ n ×ᵤ 1# → m × 1# ≈ n × 1#
     to m≈n = begin
-      m ×  1#  ≈˘⟨ ×ᵤ≈× m 1# ⟩
+      m ×  1#  ≈⟨ ×ᵤ≈× m 1# ⟨
       m ×ᵤ 1#  ≈⟨  m≈n ⟩
       n ×ᵤ 1#  ≈⟨  ×ᵤ≈× n 1# ⟩
       n ×  1#  ∎
diff --git a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
index 9f573cc711..a4908e851f 100644
--- a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
+++ b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
@@ -180,7 +180,7 @@ module pw-Reasoning (S : Setoid a ℓ) where
   run `rs .tail = run (`tail `rs)
 
   infix  1 begin_
-  infixr 2 _↺⟨_⟩_ _↺˘⟨_⟩_ _∼⟨_⟩_ _∼˘⟨_⟩_ _≈⟨_⟩_ _≈˘⟨_⟩_ _≡⟨_⟩_ _≡˘⟨_⟩_ _≡⟨⟩_
+  infixr 2 _↺⟨_⟩_ _↺⟨_⟨_ _∼⟨_⟩_ _∼⟨_⟨_ _≈⟨_⟩_ _≈⟨_⟨_ _≡⟨_⟩_ _≡⟨_⟨_ _≡⟨⟩_
   infix  3 _∎
 
   -- Beginning of a proof
@@ -189,16 +189,41 @@ module pw-Reasoning (S : Setoid a ℓ) where
   (begin `rs) .tail = run (`rs .tail)
 
   pattern _↺⟨_⟩_  as as∼bs bs∼cs = `trans {as = as} (`step as∼bs) bs∼cs
-  pattern _↺˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`step bs∼as)) bs∼cs
+  pattern _↺⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`step bs∼as)) bs∼cs
   pattern _∼⟨_⟩_  as as∼bs bs∼cs = `trans {as = as} (`lift as∼bs) bs∼cs
-  pattern _∼˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`lift bs∼as)) bs∼cs
+  pattern _∼⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`lift bs∼as)) bs∼cs
   pattern _≈⟨_⟩_  as as∼bs bs∼cs = `trans {as = as} (`bisim as∼bs) bs∼cs
-  pattern _≈˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`bisim bs∼as)) bs∼cs
+  pattern _≈⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`bisim bs∼as)) bs∼cs
   pattern _≡⟨_⟩_  as as∼bs bs∼cs = `trans {as = as} (`refl as∼bs) bs∼cs
-  pattern _≡˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`refl bs∼as)) bs∼cs
+  pattern _≡⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`refl bs∼as)) bs∼cs
   pattern _≡⟨⟩_   as as∼bs       = `trans {as = as} (`refl P.refl) as∼bs
   pattern _∎      as             = `refl  {as = as} P.refl
 
+
+  -- Deprecated v2.0
+  infixr 2 _↺˘⟨_⟩_ _∼˘⟨_⟩_ _≈˘⟨_⟩_ _≡˘⟨_⟩_
+  pattern _↺˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`step bs∼as)) bs∼cs
+  pattern _∼˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`lift bs∼as)) bs∼cs
+  pattern _≈˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`bisim bs∼as)) bs∼cs
+  pattern _≡˘⟨_⟩_ as bs∼as bs∼cs = `trans {as = as} (`sym (`refl bs∼as)) bs∼cs
+  {-# WARNING_ON_USAGE _↺˘⟨_⟩_
+  "Warning: _↺˘⟨_⟩_ was deprecated in v2.0.
+  Please use _↺⟨_⟨_ instead."
+  #-}
+  {-# WARNING_ON_USAGE _∼˘⟨_⟩_
+  "Warning: _∼˘⟨_⟩_ was deprecated in v2.0.
+  Please use _∼⟨_⟨_ instead."
+  #-}
+  {-# WARNING_ON_USAGE _≈˘⟨_⟩_
+  "Warning: _≈˘⟨_⟩_ was deprecated in v2.0.
+  Please use _≈⟨_⟨_ instead."
+  #-}
+  {-# WARNING_ON_USAGE _≡˘⟨_⟩_
+  "Warning: _≡˘⟨_⟩_ was deprecated in v2.0.
+  Please use _≡⟨_⟨_ instead."
+  #-}
+
+
 module ≈-Reasoning {a} {A : Set a} where
 
   open pw-Reasoning (P.setoid A) public
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index 2ec1564627..a4196bb952 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -132,7 +132,7 @@ length-scanl c n (a ∷ as) = suc λ { .force → begin
   length (scanl c (c n a) (as .force))
     ≈⟨ length-scanl c (c n a) (as .force) ⟩
   1 ℕ+ length (as .force)
-    ≈˘⟨ length-∷ a as ⟩
+    ≈⟨ length-∷ a as ⟨
   length (a ∷ as) ∎ } where open coℕᵇ.≈-Reasoning
 
 module _ (cons : C → B → C) (alg : A → Maybe (A × B)) where
@@ -153,7 +153,7 @@ module _ (cons : C → B → C) (alg : A → Maybe (A × B)) where
     (cons nil b ∷ _)
      ≈⟨ Eq.refl ∷ (λ where .force → refl) ⟩
     (cons nil b ∷ _)
-     ≈˘⟨ unfold-just (Maybeₚ.map-just eq) ⟩
+     ≈⟨ unfold-just (Maybeₚ.map-just eq) ⟨
     unfold alg′ (a , nil) ∎ } where open ≈-Reasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index eb5a7f5011..cdcbeb4e85 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -607,7 +607,7 @@ join-splitAt zero    n i       = refl
 join-splitAt (suc m) n zero    = refl
 join-splitAt (suc m) n (suc i) = begin
   [ _↑ˡ n , (suc m) ↑ʳ_ ]′ (splitAt (suc m) (suc i)) ≡⟨ [,]-map (splitAt m i) ⟩
-  [ suc ∘ (_↑ˡ n) , suc ∘ (m ↑ʳ_) ]′ (splitAt m i)   ≡˘⟨ [,]-∘ suc (splitAt m i) ⟩
+  [ suc ∘ (_↑ˡ n) , suc ∘ (m ↑ʳ_) ]′ (splitAt m i)   ≡⟨ [,]-∘ suc (splitAt m i) ⟨
   suc ([ _↑ˡ n , m ↑ʳ_ ]′ (splitAt m i))             ≡⟨ cong suc (join-splitAt m n i) ⟩
   suc i                                                         ∎
   where open ≡-Reasoning
@@ -646,13 +646,13 @@ remQuot-combine {suc n} {k} (suc i) j rewrite splitAt-↑ʳ k   (n ℕ.* k) (com
 combine-remQuot : ∀ {n} k (i : Fin (n ℕ.* k)) → uncurry combine (remQuot {n} k i) ≡ i
 combine-remQuot {suc n} k i with splitAt k i in eq
 ... | inj₁ j = begin
-  join k (n ℕ.* k) (inj₁ j)      ≡˘⟨ cong (join k (n ℕ.* k)) eq ⟩
+  join k (n ℕ.* k) (inj₁ j)      ≡⟨ cong (join k (n ℕ.* k)) eq ⟨
   join k (n ℕ.* k) (splitAt k i) ≡⟨ join-splitAt k (n ℕ.* k) i ⟩
   i                              ∎
   where open ≡-Reasoning
 ... | inj₂ j = begin
   k ↑ʳ (uncurry combine (remQuot {n} k j)) ≡⟨ cong (k ↑ʳ_) (combine-remQuot {n} k j) ⟩
-  join k (n ℕ.* k) (inj₂ j)                ≡˘⟨ cong (join k (n ℕ.* k)) eq ⟩
+  join k (n ℕ.* k) (inj₂ j)                ≡⟨ cong (join k (n ℕ.* k)) eq ⟨
   join k (n ℕ.* k) (splitAt k i)           ≡⟨ join-splitAt k (n ℕ.* k) i ⟩
   i                                        ∎
   where open ≡-Reasoning
@@ -662,7 +662,7 @@ toℕ-combine {suc m} {n} i@0F j = begin
   toℕ (combine i j)          ≡⟨⟩
   toℕ (j ↑ˡ (m ℕ.* n))       ≡⟨ toℕ-↑ˡ j (m ℕ.* n) ⟩
   toℕ j                      ≡⟨⟩
-  0 ℕ.+ toℕ j                ≡˘⟨ cong (ℕ._+ toℕ j) (ℕₚ.*-zeroʳ n) ⟩
+  0 ℕ.+ toℕ j                ≡⟨ cong (ℕ._+ toℕ j) (ℕₚ.*-zeroʳ n) ⟨
   n ℕ.* toℕ i ℕ.+ toℕ j      ∎
   where open ≡-Reasoning
 toℕ-combine {suc m} {n} (suc i) j = begin
@@ -683,7 +683,7 @@ combine-monoˡ-< {m} {n} {i} {j} k l i<j = begin-strict
   n ℕ.+ toℕ i ℕ.* n      ≡⟨ ℕₚ.*-comm (suc (toℕ i)) n ⟩
   n ℕ.* suc (toℕ i)      ≤⟨ ℕₚ.*-monoʳ-≤ n (toℕ-mono-< i<j) ⟩
   n ℕ.* toℕ j            ≤⟨ ℕₚ.m≤m+n (n ℕ.* toℕ j) (toℕ l) ⟩
-  n ℕ.* toℕ j ℕ.+ toℕ l  ≡˘⟨ toℕ-combine j l ⟩
+  n ℕ.* toℕ j ℕ.+ toℕ l  ≡⟨ toℕ-combine j l ⟨
   toℕ (combine j l)      ∎
   where open ℕₚ.≤-Reasoning; open +-*-Solver
 
@@ -699,7 +699,7 @@ combine-injectiveʳ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
 combine-injectiveʳ {m} {n} i j k l cᵢⱼ≡cₖₗ
   with refl ← combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ
   = toℕ-injective (ℕₚ.+-cancelˡ-≡ (n ℕ.* toℕ i) _ _ (begin
-  n ℕ.* toℕ i ℕ.+ toℕ j ≡˘⟨ toℕ-combine i j ⟩
+  n ℕ.* toℕ i ℕ.+ toℕ j ≡⟨ toℕ-combine i j ⟨
   toℕ (combine i j)     ≡⟨ cong toℕ cᵢⱼ≡cₖₗ ⟩
   toℕ (combine i l)     ≡⟨ toℕ-combine i l ⟩
   n ℕ.* toℕ i ℕ.+ toℕ l ∎))
@@ -714,7 +714,7 @@ combine-injective i j k l cᵢⱼ≡cₖₗ =
 combine-surjective : ∀ (i : Fin (m ℕ.* n)) → ∃₂ λ j k → combine j k ≡ i
 combine-surjective {m} {n} i with j , k ← remQuot {m} n i in eq
   = j , k , (begin
-  combine j k                       ≡˘⟨ uncurry (cong₂ combine) (,-injective eq) ⟩
+  combine j k                       ≡⟨ uncurry (cong₂ combine) (,-injective eq) ⟨
   uncurry combine (remQuot {m} n i) ≡⟨ combine-remQuot {m} n i ⟩
   i                                 ∎)
   where open ≡-Reasoning
@@ -1092,7 +1092,7 @@ opposite-suc {n} i = begin
   toℕ (opposite (suc i))     ≡⟨ opposite-prop (suc i) ⟩
   suc n ∸ suc (toℕ (suc i))  ≡⟨⟩
   n ∸ toℕ (suc i)            ≡⟨⟩
-  n ∸ suc (toℕ i)            ≡˘⟨ opposite-prop i ⟩
+  n ∸ suc (toℕ i)            ≡⟨ opposite-prop i ⟨
   toℕ (opposite i)           ∎
   where open ≡-Reasoning
 
diff --git a/src/Data/Fin/Subset/Properties.agda b/src/Data/Fin/Subset/Properties.agda
index 2f7246da6a..6029b9a78c 100644
--- a/src/Data/Fin/Subset/Properties.agda
+++ b/src/Data/Fin/Subset/Properties.agda
@@ -784,7 +784,7 @@ p─q─r≡p─r─q : ∀ (p q r : Subset n) → p ─ q ─ r ≡ p ─ r ─
 p─q─r≡p─r─q p q r = begin
   (p ─ q) ─ r  ≡⟨  p─q─r≡p─q∪r p q r ⟩
   p ─ (q ∪ r)  ≡⟨  cong (p ─_) (∪-comm q r) ⟩
-  p ─ (r ∪ q)  ≡˘⟨ p─q─r≡p─q∪r p r q ⟩
+  p ─ (r ∪ q)  ≡⟨ p─q─r≡p─q∪r p r q ⟨
   (p ─ r) ─ q  ∎
   where open ≡-Reasoning
 
diff --git a/src/Data/Integer/DivMod.agda b/src/Data/Integer/DivMod.agda
index 87bbaabbe7..687570ba1b 100644
--- a/src/Data/Integer/DivMod.agda
+++ b/src/Data/Integer/DivMod.agda
@@ -75,7 +75,7 @@ a≡a%ℕn+[a/ℕn]*n n@(-[1+ _ ]) d with ∣ n ∣ ℕ.% d in eq
 [n/ℕd]*d≤n : ∀ n d .{{_ : ℕ.NonZero d}} → (n /ℕ d) * + d ≤ n
 [n/ℕd]*d≤n n d = let q = n /ℕ d; r = n %ℕ d in begin
   q * + d        ≤⟨  i≤j+i _ (+ r) ⟩
-  + r + q * + d  ≡˘⟨ a≡a%ℕn+[a/ℕn]*n n d ⟩
+  + r + q * + d  ≡⟨ a≡a%ℕn+[a/ℕn]*n n d ⟨
   n              ∎
 
 div-pos-is-/ℕ : ∀ n d .{{_ : ℕ.NonZero d}} →
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 13ed3deab8..43152bceee 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -371,7 +371,7 @@ module ≤-Reasoning where
     <-≤-trans
     ≤-<-trans
     public
-    hiding (step-≈; step-≈˘)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
 
 ------------------------------------------------------------------------
 -- Properties of Positive/NonPositive/Negative/NonNegative and _≤_/_<_
@@ -403,7 +403,7 @@ neg-involutive +[1+ n ] = refl
 
 neg-injective : - i ≡ - j → i ≡ j
 neg-injective {i} {j} -i≡-j = begin
-  i     ≡˘⟨ neg-involutive i ⟩
+  i     ≡⟨ neg-involutive i ⟨
   - - i ≡⟨  cong -_ -i≡-j ⟩
   - - j ≡⟨  neg-involutive j ⟩
   j     ∎ where open ≡-Reasoning
@@ -541,7 +541,7 @@ signᵢ◃∣i∣≡i -[1+ n ] = refl
 sign-cong : .{{_ : ℕ.NonZero m}} .{{_ : ℕ.NonZero n}} →
             s ◃ m ≡ t ◃ n → s ≡ t
 sign-cong {n@(suc _)} {m@(suc _)} {s} {t} eq = begin
-  s             ≡˘⟨ sign-◃ s n ⟩
+  s             ≡⟨ sign-◃ s n ⟨
   sign (s ◃ n)  ≡⟨  cong sign eq ⟩
   sign (t ◃ m)  ≡⟨  sign-◃ t m ⟩
   t             ∎ where open ≡-Reasoning
@@ -556,7 +556,7 @@ sign-cong′ {s}       {suc m} {t}       {suc n} eq = inj₁ (sign-cong eq)
 
 abs-cong : s ◃ m ≡ t ◃ n → m ≡ n
 abs-cong {s} {m} {t} {n} eq = begin
-  m          ≡˘⟨ abs-◃ s m ⟩
+  m          ≡⟨ abs-◃ s m ⟨
   ∣ s ◃ m ∣  ≡⟨  cong ∣_∣ eq ⟩
   ∣ t ◃ n ∣  ≡⟨  abs-◃ t n ⟩
   n          ∎ where open ≡-Reasoning
@@ -608,7 +608,7 @@ n⊖n≡0 n with n ℕ.<ᵇ n in leq
 ⊖-swap (suc m) (suc n) = begin
   suc m ⊖ suc n     ≡⟨ [1+m]⊖[1+n]≡m⊖n m n ⟩
   m ⊖ n             ≡⟨ ⊖-swap m n ⟩
-  - (n ⊖ m)         ≡˘⟨ cong -_ ([1+m]⊖[1+n]≡m⊖n n m) ⟩
+  - (n ⊖ m)         ≡⟨ cong -_ ([1+m]⊖[1+n]≡m⊖n n m) ⟨
   - (suc n ⊖ suc m) ∎ where open ≡-Reasoning
 
 ⊖-≥ : m ℕ.≥ n → m ⊖ n ≡ + (m ∸ n)
@@ -666,12 +666,12 @@ m-n≡m⊖n (suc m) (suc n) = refl
 -[n⊖m]≡-m+n m n with m ℕ.<ᵇ n | Reflects.invert (ℕ.<ᵇ-reflects-< m n)
 ... | true  | p = begin
   - (- (+ (n ∸ m))) ≡⟨ neg-involutive (+ (n ∸ m)) ⟩
-  + (n ∸ m)         ≡˘⟨ ⊖-≥ (ℕ.≤-trans (ℕ.m≤n+m m 1) p) ⟩
-  n ⊖ m             ≡˘⟨ -m+n≡n⊖m m n ⟩
+  + (n ∸ m)         ≡⟨ ⊖-≥ (ℕ.≤-trans (ℕ.m≤n+m m 1) p) ⟨
+  n ⊖ m             ≡⟨ -m+n≡n⊖m m n ⟨
   - (+ m) + + n     ∎ where open ≡-Reasoning
 ... | false | p = begin
-  - (+ (m ∸ n))     ≡˘⟨ ⊖-≤ (ℕ.≮⇒≥ p) ⟩
-  n ⊖ m             ≡˘⟨ -m+n≡n⊖m m n ⟩
+  - (+ (m ∸ n))     ≡⟨ ⊖-≤ (ℕ.≮⇒≥ p) ⟨
+  n ⊖ m             ≡⟨ -m+n≡n⊖m m n ⟨
   - (+ m) + + n     ∎ where open ≡-Reasoning
 
 ∣m⊖n∣≡∣n⊖m∣ : ∀ m n → ∣ m ⊖ n ∣ ≡ ∣ n ⊖ m ∣
@@ -712,14 +712,14 @@ m⊖1+n<m (suc m) (suc n) = begin-strict
 -1+m<n⊖m (suc m) (suc n) = begin-strict
   -[1+ suc m ]  <⟨ -<- ℕ.≤-refl ⟩
   -[1+ m ]      <⟨ -1+m<n⊖m m n ⟩
-  n ⊖ m         ≡˘⟨ [1+m]⊖[1+n]≡m⊖n n m ⟩
+  n ⊖ m         ≡⟨ [1+m]⊖[1+n]≡m⊖n n m ⟨
   suc n ⊖ suc m ∎ where open ≤-Reasoning
 
 -[1+m]≤n⊖m+1 : ∀ m n → -[1+ m ] ≤ n ⊖ suc m
 -[1+m]≤n⊖m+1 m zero    = ≤-refl
 -[1+m]≤n⊖m+1 m (suc n) = begin
   -[1+ m ]      ≤⟨ <⇒≤ (-1+m<n⊖m m n) ⟩
-  n ⊖ m         ≡˘⟨ [1+m]⊖[1+n]≡m⊖n n m ⟩
+  n ⊖ m         ≡⟨ [1+m]⊖[1+n]≡m⊖n n m ⟨
   suc n ⊖ suc m ∎ where open ≤-Reasoning
 
 -1+m≤n⊖m : ∀ m n → -[1+ m ] ≤ n ⊖ m
@@ -750,7 +750,7 @@ sign-⊖-≰ = sign-⊖-< ∘ ℕ.≰⇒>
 ⊖-monoʳ-≥-≤ (suc n) {suc m} {suc o} (s≤s m≤o) = begin
   suc n ⊖ suc m ≡⟨  [1+m]⊖[1+n]≡m⊖n n m ⟩
   n ⊖ m         ≤⟨  ⊖-monoʳ-≥-≤ n m≤o ⟩
-  n ⊖ o         ≡˘⟨ [1+m]⊖[1+n]≡m⊖n n o ⟩
+  n ⊖ o         ≡⟨ [1+m]⊖[1+n]≡m⊖n n o ⟨
   suc n ⊖ suc o ∎ where open ≤-Reasoning
 
 ⊖-monoˡ-≤ : ∀ n → (_⊖ n) Preserves ℕ._≤_ ⟶ _≤_
@@ -759,12 +759,12 @@ sign-⊖-≰ = sign-⊖-< ∘ ℕ.≰⇒>
 ⊖-monoˡ-≤ (suc n) {_} {suc o} z≤n = begin
   zero ⊖ suc n  ≤⟨  ⊖-monoʳ-≥-≤ 0 (ℕ.n≤1+n n) ⟩
   zero ⊖ n      ≤⟨  ⊖-monoˡ-≤ n z≤n ⟩
-  o ⊖ n         ≡˘⟨ [1+m]⊖[1+n]≡m⊖n o n ⟩
+  o ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n o n ⟨
   suc o ⊖ suc n ∎ where open ≤-Reasoning
 ⊖-monoˡ-≤ (suc n) {suc m} {suc o} (s≤s m≤o) = begin
   suc m ⊖ suc n ≡⟨  [1+m]⊖[1+n]≡m⊖n m n ⟩
   m ⊖ n         ≤⟨  ⊖-monoˡ-≤ n m≤o ⟩
-  o ⊖ n         ≡˘⟨ [1+m]⊖[1+n]≡m⊖n o n ⟩
+  o ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n o n ⟨
   suc o ⊖ suc n ∎ where open ≤-Reasoning
 
 ⊖-monoʳ->-< : ∀ p → (p ⊖_) Preserves ℕ._>_ ⟶ _<_
@@ -777,19 +777,19 @@ sign-⊖-≰ = sign-⊖-< ∘ ℕ.≰⇒>
 ⊖-monoʳ->-< (suc p) {suc m} {suc n} (s<s m<n@(s≤s _)) = begin-strict
   suc p ⊖ suc m ≡⟨  [1+m]⊖[1+n]≡m⊖n p m ⟩
   p ⊖ m         <⟨  ⊖-monoʳ->-< p m<n ⟩
-  p ⊖ n         ≡˘⟨ [1+m]⊖[1+n]≡m⊖n p n ⟩
+  p ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n p n ⟨
   suc p ⊖ suc n ∎ where open ≤-Reasoning
 
 ⊖-monoˡ-< : ∀ n → (_⊖ n) Preserves ℕ._<_ ⟶ _<_
 ⊖-monoˡ-< zero    m<o             = +<+ m<o
 ⊖-monoˡ-< (suc n) {_} {suc o} z<s = begin-strict
   -[1+ n ]      <⟨  -1+m<n⊖m n _ ⟩
-  o ⊖ n         ≡˘⟨ [1+m]⊖[1+n]≡m⊖n o n ⟩
+  o ⊖ n         ≡⟨ [1+m]⊖[1+n]≡m⊖n o n ⟨
   suc o ⊖ suc n ∎ where open ≤-Reasoning
 ⊖-monoˡ-< (suc n) {suc m} {suc (suc o)} (s<s m<o@(s≤s _)) = begin-strict
   suc m ⊖ suc n       ≡⟨  [1+m]⊖[1+n]≡m⊖n m n ⟩
   m ⊖ n               <⟨  ⊖-monoˡ-< n m<o ⟩
-  suc o ⊖ n           ≡˘⟨ [1+m]⊖[1+n]≡m⊖n (suc o) n ⟩
+  suc o ⊖ n           ≡⟨ [1+m]⊖[1+n]≡m⊖n (suc o) n ⟨
   suc (suc o) ⊖ suc n ∎ where open ≤-Reasoning
 
 ------------------------------------------------------------------------
@@ -822,7 +822,7 @@ distribˡ-⊖-+-pos _ (suc _) zero    = refl
 distribˡ-⊖-+-pos m (suc n) (suc o) = begin
   suc n ⊖ suc o + + m   ≡⟨ cong (_+ + m) ([1+m]⊖[1+n]≡m⊖n n o) ⟩
   n ⊖ o + + m           ≡⟨ distribˡ-⊖-+-pos m n o ⟩
-  n ℕ.+ m ⊖ o           ≡˘⟨ [1+m]⊖[1+n]≡m⊖n (n ℕ.+ m) o ⟩
+  n ℕ.+ m ⊖ o           ≡⟨ [1+m]⊖[1+n]≡m⊖n (n ℕ.+ m) o ⟨
   suc (n ℕ.+ m) ⊖ suc o ∎ where open ≡-Reasoning
 
 distribˡ-⊖-+-neg : ∀ m n o → n ⊖ o + -[1+ m ] ≡ n ⊖ (suc o ℕ.+ m)
@@ -832,7 +832,7 @@ distribˡ-⊖-+-neg _ (suc _) zero    = refl
 distribˡ-⊖-+-neg m (suc n) (suc o) = begin
   suc n ⊖ suc o + -[1+ m ]    ≡⟨ cong (_+ -[1+ m ]) ([1+m]⊖[1+n]≡m⊖n n o) ⟩
   n ⊖ o + -[1+ m ]            ≡⟨ distribˡ-⊖-+-neg m n o ⟩
-  n ⊖ (suc o ℕ.+ m)           ≡˘⟨ [1+m]⊖[1+n]≡m⊖n n (suc o ℕ.+ m) ⟩
+  n ⊖ (suc o ℕ.+ m)           ≡⟨ [1+m]⊖[1+n]≡m⊖n n (suc o ℕ.+ m) ⟨
   suc n ⊖ (suc (suc o) ℕ.+ m) ∎ where open ≡-Reasoning
 
 distribʳ-⊖-+-pos : ∀ m n o → + m + (n ⊖ o) ≡ m ℕ.+ n ⊖ o
@@ -855,18 +855,18 @@ distribʳ-⊖-+-neg m n o = begin
 +-assoc i j +0 rewrite +-identityʳ (i + j) | +-identityʳ  j      = refl
 +-assoc -[1+ m ] -[1+ n ] +[1+ o ] = begin
   suc o ⊖ suc (suc (m ℕ.+ n)) ≡⟨ [1+m]⊖[1+n]≡m⊖n o (suc m ℕ.+ n) ⟩
-  o ⊖ (suc m ℕ.+ n)           ≡˘⟨ distribʳ-⊖-+-neg m o n ⟩
-  -[1+ m ] + (o ⊖ n)          ≡˘⟨ cong (λ z → -[1+ m ] + z) ([1+m]⊖[1+n]≡m⊖n o n) ⟩
+  o ⊖ (suc m ℕ.+ n)           ≡⟨ distribʳ-⊖-+-neg m o n ⟨
+  -[1+ m ] + (o ⊖ n)          ≡⟨ cong (λ z → -[1+ m ] + z) ([1+m]⊖[1+n]≡m⊖n o n) ⟨
   -[1+ m ] + (suc o ⊖ suc n)  ∎ where open ≡-Reasoning
 +-assoc -[1+ m ] +[1+ n ] +[1+ o ] = begin
   suc n ⊖ suc m + +[1+ o ]  ≡⟨ cong (_+ +[1+ o ]) ([1+m]⊖[1+n]≡m⊖n n m) ⟩
   (n ⊖ m) + +[1+ o ]        ≡⟨ distribˡ-⊖-+-pos (suc o) n m ⟩
-  n ℕ.+ suc o ⊖ m           ≡˘⟨ [1+m]⊖[1+n]≡m⊖n (n ℕ.+ suc o) m ⟩
+  n ℕ.+ suc o ⊖ m           ≡⟨ [1+m]⊖[1+n]≡m⊖n (n ℕ.+ suc o) m ⟨
   suc (n ℕ.+ suc o) ⊖ suc m ∎ where open ≡-Reasoning
 +-assoc +[1+ m ] -[1+ n ] -[1+ o ] = begin
   (suc m ⊖ suc n) + -[1+ o ]  ≡⟨ cong (_+ -[1+ o ]) ([1+m]⊖[1+n]≡m⊖n m n) ⟩
   (m ⊖ n) + -[1+ o ]          ≡⟨ distribˡ-⊖-+-neg o m n ⟩
-  m ⊖ suc (n ℕ.+ o)           ≡˘⟨ [1+m]⊖[1+n]≡m⊖n m (suc n ℕ.+ o) ⟩
+  m ⊖ suc (n ℕ.+ o)           ≡⟨ [1+m]⊖[1+n]≡m⊖n m (suc n ℕ.+ o) ⟨
   suc m ⊖ suc (suc (n ℕ.+ o)) ∎ where open ≡-Reasoning
 +-assoc +[1+ m ] -[1+ n ] +[1+ o ]
   rewrite [1+m]⊖[1+n]≡m⊖n m n
@@ -1087,8 +1087,8 @@ neg-minus-pos (suc m) (suc n) = cong (-[1+_] ∘ suc) (ℕ.+-comm (suc m) n)
 +-minus-telescope : ∀ i j k → (i - j) + (j - k) ≡ i - k
 +-minus-telescope i j k = begin
   (i - j) + (j - k)   ≡⟨  +-assoc i (- j) (j - k) ⟩
-  i + (- j + (j - k)) ≡˘⟨ cong (λ v → i + v) (+-assoc (- j) j _) ⟩
-  i + ((- j + j) - k) ≡˘⟨ +-assoc i (- j + j) (- k) ⟩
+  i + (- j + (j - k)) ≡⟨ cong (λ v → i + v) (+-assoc (- j) j _) ⟨
+  i + ((- j + j) - k) ≡⟨ +-assoc i (- j + j) (- k) ⟨
   i + (- j + j) - k   ≡⟨  cong (λ a → i + a - k) (+-inverseˡ j) ⟩
   i + 0ℤ - k          ≡⟨  cong (_- k) (+-identityʳ i) ⟩
   i - k               ∎ where open ≡-Reasoning
@@ -1103,16 +1103,16 @@ neg-minus-pos (suc m) (suc n) = cong (-[1+_] ∘ suc) (ℕ.+-comm (suc m) n)
 ∣i-j∣≡∣j-i∣ -[1+ m ] -[1+ n ] = ∣m⊖n∣≡∣n⊖m∣ (suc n) (suc m)
 ∣i-j∣≡∣j-i∣ -[1+ m ] (+ n)    = begin
   ∣ -[1+ m ] - (+ n) ∣  ≡⟨  cong ∣_∣ (neg-minus-pos m n) ⟩
-  suc (n ℕ.+ m)         ≡˘⟨ ℕ.+-suc n m ⟩
+  suc (n ℕ.+ m)         ≡⟨ ℕ.+-suc n m ⟨
   n ℕ.+ suc m           ∎ where open ≡-Reasoning
 ∣i-j∣≡∣j-i∣ (+ m) -[1+ n ] = begin
   m ℕ.+ suc n          ≡⟨  ℕ.+-suc m n ⟩
-  suc (m ℕ.+ n)        ≡˘⟨ cong ∣_∣ (neg-minus-pos n m) ⟩
+  suc (m ℕ.+ n)        ≡⟨ cong ∣_∣ (neg-minus-pos n m) ⟨
   ∣ -[1+ n ] + - + m ∣ ∎ where open ≡-Reasoning
 ∣i-j∣≡∣j-i∣ (+ m) (+ n) = begin
   ∣ + m - + n ∣  ≡⟨  cong ∣_∣ ([+m]-[+n]≡m⊖n m n) ⟩
   ∣ m ⊖ n ∣      ≡⟨  ∣m⊖n∣≡∣n⊖m∣ m n ⟩
-  ∣ n ⊖ m ∣      ≡˘⟨ cong ∣_∣ ([+m]-[+n]≡m⊖n n m) ⟩
+  ∣ n ⊖ m ∣      ≡⟨ cong ∣_∣ ([+m]-[+n]≡m⊖n n m) ⟨
   ∣ + n - + m ∣  ∎ where open ≡-Reasoning
 
 ∣-∣-≤ : i ≤ j → + ∣ i - j ∣ ≡ j - i
@@ -1140,9 +1140,9 @@ i≡j⇒i-j≡0 {i} refl = +-inverseʳ i
 
 i-j≡0⇒i≡j : ∀ i j → i - j ≡ 0ℤ → i ≡ j
 i-j≡0⇒i≡j i j i-j≡0 = begin
-  i             ≡˘⟨ +-identityʳ i ⟩
-  i + 0ℤ        ≡˘⟨ cong (_+_ i) (+-inverseˡ j) ⟩
-  i + (- j + j) ≡˘⟨ +-assoc i (- j) j ⟩
+  i             ≡⟨ +-identityʳ i ⟨
+  i + 0ℤ        ≡⟨ cong (_+_ i) (+-inverseˡ j) ⟨
+  i + (- j + j) ≡⟨ +-assoc i (- j) j ⟨
   (i - j) + j   ≡⟨  cong (_+ j) i-j≡0 ⟩
   0ℤ + j        ≡⟨  +-identityˡ j ⟩
   j             ∎ where open ≡-Reasoning
@@ -1172,9 +1172,9 @@ i≤j⇒i-j≤0 {+[1+ m ]} {+[1+ n ]} (+≤+ (s≤s m≤n)) = begin
 
 i-j≤0⇒i≤j : i - j ≤ 0ℤ → i ≤ j
 i-j≤0⇒i≤j {i} {j} i-j≤0 = begin
-  i             ≡˘⟨ +-identityʳ i ⟩
-  i + 0ℤ        ≡˘⟨ cong (_+_ i) (+-inverseˡ j) ⟩
-  i + (- j + j) ≡˘⟨ +-assoc i (- j) j ⟩
+  i             ≡⟨ +-identityʳ i ⟨
+  i + 0ℤ        ≡⟨ cong (_+_ i) (+-inverseˡ j) ⟨
+  i + (- j + j) ≡⟨ +-assoc i (- j) j ⟨
   (i - j) + j   ≤⟨  +-monoˡ-≤ j i-j≤0 ⟩
   0ℤ + j        ≡⟨  +-identityˡ j ⟩
   j             ∎
@@ -1182,14 +1182,14 @@ i-j≤0⇒i≤j {i} {j} i-j≤0 = begin
 
 i≤j⇒0≤j-i : i ≤ j → 0ℤ ≤ j - i
 i≤j⇒0≤j-i {i} {j} i≤j = begin
-  0ℤ    ≡˘⟨ +-inverseʳ i ⟩
+  0ℤ    ≡⟨ +-inverseʳ i ⟨
   i - i ≤⟨  +-monoˡ-≤ (- i) i≤j ⟩
   j - i ∎
   where open ≤-Reasoning
 
 0≤i-j⇒j≤i : 0ℤ ≤ i - j → j ≤ i
 0≤i-j⇒j≤i {i} {j} 0≤i-j = begin
-  j             ≡˘⟨ +-identityˡ j ⟩
+  j             ≡⟨ +-identityˡ j ⟨
   0ℤ + j        ≤⟨  +-monoˡ-≤ j 0≤i-j ⟩
   i - j + j     ≡⟨  +-assoc i (- j) j ⟩
   i + (- j + j) ≡⟨  cong (_+_ i) (+-inverseˡ j) ⟩
@@ -1246,19 +1246,19 @@ i<j⇒suc[i]≤j { -[1+ suc i ]} {+ _}       -<+       = -≤+
 
 suc-pred : ∀ i → sucℤ (pred i) ≡ i
 suc-pred i = begin
-  sucℤ (pred i) ≡˘⟨ +-assoc 1ℤ -1ℤ i ⟩
+  sucℤ (pred i) ≡⟨ +-assoc 1ℤ -1ℤ i ⟨
   0ℤ + i        ≡⟨  +-identityˡ i ⟩
   i             ∎ where open ≡-Reasoning
 
 pred-suc : ∀ i → pred (sucℤ i) ≡ i
 pred-suc i = begin
-  pred (sucℤ i) ≡˘⟨ +-assoc -1ℤ 1ℤ i ⟩
+  pred (sucℤ i) ≡⟨ +-assoc -1ℤ 1ℤ i ⟨
   0ℤ + i        ≡⟨  +-identityˡ i ⟩
   i             ∎ where open ≡-Reasoning
 
 +-pred : ∀ i j → i + pred j ≡ pred (i + j)
 +-pred i j = begin
-  i + (-1ℤ + j) ≡˘⟨ +-assoc i -1ℤ j ⟩
+  i + (-1ℤ + j) ≡⟨ +-assoc i -1ℤ j ⟨
   i + -1ℤ + j   ≡⟨  cong (_+ j) (+-comm i -1ℤ) ⟩
   -1ℤ + i + j   ≡⟨  +-assoc -1ℤ i j ⟩
   -1ℤ + (i + j) ∎ where open ≡-Reasoning
@@ -1419,12 +1419,12 @@ private
 *-distribʳ-+ -[1+ m ] -[1+ n ] +[1+ o ] = begin
   (suc o ⊖ suc n) * -[1+ m ]                ≡⟨ cong (_* -[1+ m ]) ([1+m]⊖[1+n]≡m⊖n o n) ⟩
   (o ⊖ n) * -[1+ m ]                        ≡⟨ distrib-lemma m n o ⟩
-  m ℕ.+ n ℕ.* suc m ⊖ (m ℕ.+ o ℕ.* suc m)   ≡˘⟨ [1+m]⊖[1+n]≡m⊖n (m ℕ.+ n ℕ.* suc m) (m ℕ.+ o ℕ.* suc m) ⟩
+  m ℕ.+ n ℕ.* suc m ⊖ (m ℕ.+ o ℕ.* suc m)   ≡⟨ [1+m]⊖[1+n]≡m⊖n (m ℕ.+ n ℕ.* suc m) (m ℕ.+ o ℕ.* suc m) ⟨
   -[1+ n ] * -[1+ m ] + +[1+ o ] * -[1+ m ] ∎ where open ≡-Reasoning
 *-distribʳ-+ -[1+ m ] +[1+ n ] -[1+ o ] = begin
   (+[1+ n ] + -[1+ o ]) * -[1+ m ]          ≡⟨ cong (_* -[1+ m ]) ([1+m]⊖[1+n]≡m⊖n n o) ⟩
   (n ⊖ o) * -[1+ m ]                        ≡⟨ distrib-lemma m o n ⟩
-  m ℕ.+ o ℕ.* suc m ⊖ (m ℕ.+ n ℕ.* suc m)   ≡˘⟨ [1+m]⊖[1+n]≡m⊖n (m ℕ.+ o ℕ.* suc m) (m ℕ.+ n ℕ.* suc m) ⟩
+  m ℕ.+ o ℕ.* suc m ⊖ (m ℕ.+ n ℕ.* suc m)   ≡⟨ [1+m]⊖[1+n]≡m⊖n (m ℕ.+ o ℕ.* suc m) (m ℕ.+ n ℕ.* suc m) ⟨
   +[1+ n ] * -[1+ m ] + -[1+ o ] * -[1+ m ] ∎ where open ≡-Reasoning
 *-distribʳ-+ +[1+ m ] -[1+ n ] +[1+ o ] with n ℕ.≤? o
 ... | yes n≤o
@@ -1690,8 +1690,8 @@ pos-* (suc m) (suc n) = refl
 
 neg-distribˡ-* : ∀ i j → - (i * j) ≡ (- i) * j
 neg-distribˡ-* i j = begin
-  - (i * j)      ≡˘⟨ -1*i≡-i (i * j) ⟩
-  -1ℤ * (i * j)  ≡˘⟨ *-assoc -1ℤ i j ⟩
+  - (i * j)      ≡⟨ -1*i≡-i (i * j) ⟨
+  -1ℤ * (i * j)  ≡⟨ *-assoc -1ℤ i j ⟨
   -1ℤ * i * j    ≡⟨ cong (_* j) (-1*i≡-i i) ⟩
   - i * j        ∎ where open ≡-Reasoning
 
@@ -1747,10 +1747,10 @@ neg-distribʳ-* i j = begin
 
 *-cancelˡ-≤-neg : ∀ i j k .{{_ : Negative i}} → i * j ≤ i * k → j ≥ k
 *-cancelˡ-≤-neg i@(-[1+ _ ]) j k ij≤ik = neg-cancel-≤ (*-cancelˡ-≤-pos (- j) (- k) (- i) (begin
-  - i * - j   ≡˘⟨ neg-distribʳ-* (- i) j ⟩
+  - i * - j   ≡⟨ neg-distribʳ-* (- i) j ⟨
   -(- i * j)  ≡⟨  neg-distribˡ-* (- i) j ⟩
   i * j       ≤⟨  ij≤ik ⟩
-  i * k       ≡˘⟨ neg-distribˡ-* (- i) k ⟩
+  i * k       ≡⟨ neg-distribˡ-* (- i) k ⟨
   -(- i * k)  ≡⟨  neg-distribʳ-* (- i) k ⟩
   - i * - k   ∎))
   where open ≤-Reasoning
@@ -1761,10 +1761,10 @@ neg-distribʳ-* i j = begin
 *-monoˡ-≤-nonPos : ∀ i .{{_ : NonPositive i}} → (i *_) Preserves _≤_ ⟶ _≥_
 *-monoˡ-≤-nonPos +0           {j} {k} j≤k = +≤+ z≤n
 *-monoˡ-≤-nonPos i@(-[1+ m ]) {j} {k} j≤k = begin
-  i * k        ≡˘⟨ neg-distribˡ-* (- i) k ⟩
+  i * k        ≡⟨ neg-distribˡ-* (- i) k ⟨
   -(- i * k)   ≡⟨  neg-distribʳ-* (- i) k ⟩
   - i * - k    ≤⟨  *-monoˡ-≤-nonNeg (- i) (neg-mono-≤ j≤k) ⟩
-  - i * - j    ≡˘⟨ neg-distribʳ-* (- i) j ⟩
+  - i * - j    ≡⟨ neg-distribʳ-* (- i) j ⟨
   -(- i * j)   ≡⟨  neg-distribˡ-* (- i) j ⟩
   i * j        ∎
   where open ≤-Reasoning
@@ -1794,10 +1794,10 @@ neg-distribʳ-* i j = begin
 
 *-monoˡ-<-neg : ∀ i .{{_ : Negative i}} → (i *_) Preserves _<_ ⟶ _>_
 *-monoˡ-<-neg i@(-[1+ _ ]) {j} {k} j<k = begin-strict
-  i * k        ≡˘⟨ neg-distribˡ-* (- i) k ⟩
+  i * k        ≡⟨ neg-distribˡ-* (- i) k ⟨
   -(- i * k)   ≡⟨  neg-distribʳ-* (- i) k ⟩
   - i * - k    <⟨  *-monoˡ-<-pos (- i) (neg-mono-< j<k) ⟩
-  - i * - j    ≡˘⟨ neg-distribʳ-* (- i) j ⟩
+  - i * - j    ≡⟨ neg-distribʳ-* (- i) j ⟨
   - (- i * j)  ≡⟨  neg-distribˡ-* (- i) j ⟩
   i * j        ∎
   where open ≤-Reasoning
@@ -1808,10 +1808,10 @@ neg-distribʳ-* i j = begin
 *-cancelˡ-<-nonPos : ∀ k .{{_ : NonPositive k}} → k * i < k * j → i > j
 *-cancelˡ-<-nonPos {i} {j} +0           (+<+ ())
 *-cancelˡ-<-nonPos {i} {j} k@(-[1+ _ ]) ki<kj = neg-cancel-< (*-cancelˡ-<-nonNeg (- k) (begin-strict
-  - k * - i   ≡˘⟨ neg-distribʳ-* (- k) i ⟩
+  - k * - i   ≡⟨ neg-distribʳ-* (- k) i ⟨
   -(- k * i)  ≡⟨  neg-distribˡ-* (- k) i ⟩
   k * i       <⟨  ki<kj ⟩
-  k * j       ≡˘⟨ neg-distribˡ-* (- k) j ⟩
+  k * j       ≡⟨ neg-distribˡ-* (- k) j ⟨
   -(- k * j)  ≡⟨  neg-distribʳ-* (- k) j ⟩
   - k * - j   ∎))
   where open ≤-Reasoning
diff --git a/src/Data/List/Effectful.agda b/src/Data/List/Effectful.agda
index 3d4e557468..b21ddf7616 100644
--- a/src/Data/List/Effectful.agda
+++ b/src/Data/List/Effectful.agda
@@ -209,7 +209,7 @@ module Applicative where
              (fs ⊛ as) ≡ (fs >>= pam as)
   unfold-⊛ fs as = begin
     fs ⊛ as
-      ≡˘⟨ concatMap-cong (λ f → P.cong (map f) (concatMap-pure as)) fs ⟩
+      ≡⟨ concatMap-cong (λ f → P.cong (map f) (concatMap-pure as)) fs ⟨
     concatMap (λ f → map f (concatMap pure as)) fs
       ≡⟨ concatMap-cong (λ f → map-concatMap f pure as) fs ⟩
     concatMap (λ f → concatMap (λ x → pure (f x)) as) fs
@@ -224,7 +224,7 @@ module Applicative where
   right-distributive fs₁ fs₂ xs = begin
     (fs₁ ∣ fs₂) ⊛ xs                     ≡⟨ unfold-⊛ (fs₁ ∣ fs₂) xs ⟩
     (fs₁ ∣ fs₂ >>= pam xs)               ≡⟨ MonadProperties.right-distributive fs₁ fs₂ (pam xs) ⟩
-    (fs₁ >>= pam xs) ∣ (fs₂ >>= pam xs)  ≡˘⟨ P.cong₂ _∣_ (unfold-⊛ fs₁ xs) (unfold-⊛ fs₂ xs) ⟩
+    (fs₁ >>= pam xs) ∣ (fs₂ >>= pam xs)  ≡⟨ P.cong₂ _∣_ (unfold-⊛ fs₁ xs) (unfold-⊛ fs₂ xs) ⟨
     (fs₁ ⊛ xs ∣ fs₂ ⊛ xs)                ∎
 
   -- _⊛_ does not distribute over _∣_ from the left.
@@ -251,7 +251,7 @@ module Applicative where
                 (xs : List A) (f : A → B) (fs : B → List C) →
                 (pam xs f >>= fs) ≡ (xs >>= λ x → fs (f x))
     pam-lemma xs f fs = begin
-      (pam xs f >>= fs)                 ≡˘⟨ MP.associative xs (pure ∘ f) fs ⟩
+      (pam xs f >>= fs)                 ≡⟨ MP.associative xs (pure ∘ f) fs ⟨
       (xs >>= λ x → pure (f x) >>= fs)  ≡⟨ MP.cong (refl {x = xs}) (λ x → MP.left-identity (f x) fs) ⟩
       (xs >>= λ x → fs (f x))           ∎
 
@@ -272,7 +272,7 @@ module Applicative where
     (pam fs _∘′_ >>= pam gs >>= pam xs)
       ≡⟨ MP.cong (pam-lemma fs _∘′_ (pam gs)) (λ _ → refl) ⟩
     ((fs >>= λ f → pam gs (f ∘′_)) >>= pam xs)
-      ≡˘⟨ MP.associative fs (λ f → pam gs (_∘′_ f)) (pam xs) ⟩
+      ≡⟨ MP.associative fs (λ f → pam gs (_∘′_ f)) (pam xs) ⟨
     (fs >>= λ f → pam gs (f ∘′_) >>= pam xs)
       ≡⟨ MP.cong (refl {x = fs}) (λ f → pam-lemma gs (f ∘′_) (pam xs)) ⟩
     (fs >>= λ f → gs >>= λ g → pam xs (f ∘′ g))
@@ -282,9 +282,9 @@ module Applicative where
     (fs >>= λ f → gs >>= λ g → pam (pam xs g) f)
       ≡⟨ MP.cong (refl {x = fs}) (λ f → MP.associative gs (pam xs) (pure ∘ f)) ⟩
     (fs >>= pam (gs >>= pam xs))
-      ≡˘⟨ unfold-⊛ fs (gs >>= pam xs) ⟩
+      ≡⟨ unfold-⊛ fs (gs >>= pam xs) ⟨
     fs ⊛ (gs >>= pam xs)
-      ≡˘⟨ P.cong (fs ⊛_) (unfold-⊛ gs xs) ⟩
+      ≡⟨ P.cong (fs ⊛_) (unfold-⊛ gs xs) ⟨
     fs ⊛ (gs ⊛ xs)
       ∎
 
@@ -304,5 +304,5 @@ module Applicative where
                                       MP.left-identity x (pure ∘ f)) ⟩
     (fs >>= λ f → pure (f x))  ≡⟨⟩
     (pam fs (_$′ x))           ≡⟨ P.sym $ MP.left-identity (_$′ x) (pam fs) ⟩
-    (pure (_$′ x) >>= pam fs)  ≡˘⟨ unfold-⊛ (pure (_$′ x)) fs ⟩
+    (pure (_$′ x) >>= pam fs)  ≡⟨ unfold-⊛ (pure (_$′ x)) fs ⟨
     pure (_$′ x) ⊛ fs          ∎
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 78fc03f865..5bd47dc763 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -280,7 +280,7 @@ module _ (f : A → B → C) where
   cartesianProductWith-distribʳ-++ (x ∷ xs) ys zs = begin
     prod (x ∷ xs ++ ys) zs ≡⟨⟩
     map (f x) zs ++ prod (xs ++ ys) zs ≡⟨ cong (map (f x) zs ++_) (cartesianProductWith-distribʳ-++ xs ys zs) ⟩
-    map (f x) zs ++ prod xs zs ++ prod ys zs ≡˘⟨ ++-assoc (map (f x) zs) (prod xs zs) (prod ys zs) ⟩
+    map (f x) zs ++ prod xs zs ++ prod ys zs ≡⟨ ++-assoc (map (f x) zs) (prod xs zs) (prod ys zs) ⟨
     (map (f x) zs ++ prod xs zs) ++ prod ys zs ≡⟨⟩
     prod (x ∷ xs) zs ++ prod ys zs ∎
 
@@ -590,7 +590,7 @@ map-concatMap f g xs = begin
   map f (concatMap g xs)
     ≡⟨⟩
   map f (concat (map g xs))
-    ≡˘⟨ concat-map (map g xs) ⟩
+    ≡⟨ concat-map (map g xs) ⟨
   concat (map (map f) (map g xs))
     ≡⟨ cong concat
          {x = map (map f) (map g xs)}
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index f842ae8087..31ab6941ac 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -96,7 +96,7 @@ module ⊆-Reasoning {A : Set a} where
   private module Base = PreorderReasoning (⊆-preorder A)
 
   open Base public
-    hiding (step-≈; step-≈˘; step-∼; step-≲)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)
     renaming (≲-go to ⊆-go)
 
   open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → begin x) public
@@ -199,7 +199,7 @@ module _ {k} {A B : Set a} {fs gs : List (A → B)} {xs ys} where
     x ∈ (fs >>= λ f → xs >>= λ x → pure (f x))
       ∼⟨ >>=-cong fs≈gs (λ f → >>=-cong xs≈ys λ x → K-refl) ⟩
     x ∈ (gs >>= λ g → ys >>= λ y → pure (g y))
-      ≡˘⟨ P.cong (x ∈_) (Applicative.unfold-⊛ gs ys) ⟩
+      ≡⟨ P.cong (x ∈_) (Applicative.unfold-⊛ gs ys) ⟨
     x ∈ (gs ⊛ ys) ∎
     where open Related.EquationalReasoning
 
@@ -284,7 +284,7 @@ empty-unique {xs = _ ∷ _} ∷∼[] with ⇒→ ∷∼[] (here refl)
                 (xs₂ >>= pure ∘ f))       ≈⟨ >>=-left-distributive fs ⟩
 
   (fs >>= λ f → xs₁ >>= pure ∘ f) ++
-  (fs >>= λ f → xs₂ >>= pure ∘ f)         ≡˘⟨ P.cong₂ _++_ (Applicative.unfold-⊛ fs xs₁) (Applicative.unfold-⊛ fs xs₂) ⟩
+  (fs >>= λ f → xs₂ >>= pure ∘ f)         ≡⟨ P.cong₂ _++_ (Applicative.unfold-⊛ fs xs₁) (Applicative.unfold-⊛ fs xs₂) ⟨
 
   (fs ⊛ xs₁) ++ (fs ⊛ xs₂)                ∎
   where open EqR ([ bag ]-Equality B)
@@ -592,7 +592,7 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
 ... | zs₁ , zs₂ , p rewrite p = begin
   x ∷ xs           <⟨ ∼bag⇒↭ (drop-cons (↔-trans eq (comm zs₁ (x ∷ zs₂)))) ⟩
   x ∷ (zs₂ ++ zs₁) <⟨ ++-comm zs₂ zs₁ ⟩
-  x ∷ (zs₁ ++ zs₂) ↭˘⟨ shift x zs₁ zs₂ ⟩
+  x ∷ (zs₁ ++ zs₂) ↭⟨ shift x zs₁ zs₂ ⟨
   zs₁ ++ x ∷ zs₂   ∎
   where
   open CommutativeMonoid (commutativeMonoid bag A)
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional.agda b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
index 28cf25d9ed..c41793d28a 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional.agda
@@ -76,9 +76,11 @@ module PermutationReasoning where
 
   private module Base = EqReasoning ↭-setoid
 
-  open Base public hiding (step-≈; step-≈˘)
+  open Base public
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
+    renaming (≈-go to ↭-go)
 
-  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go ↭-sym public
+  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ ↭-go ↭-sym public
 
   -- Some extra combinators that allow us to skip certain elements
 
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index 1b33a771a0..781de1cc36 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -243,7 +243,7 @@ drop-mid {A = A} {x} ws xs p = drop-mid′ p ws xs refl refl
 ++-comm []       ys = ↭-sym (++-identityʳ ys)
 ++-comm (x ∷ xs) ys = begin
   x ∷ xs ++ ys   <⟨ ++-comm xs ys ⟩
-  x ∷ ys ++ xs   ↭˘⟨ shift x ys xs ⟩
+  x ∷ ys ++ xs   ↭⟨ shift x ys xs ⟨
   ys ++ (x ∷ xs) ∎
 
 ++-isMagma : IsMagma {A = List A} _↭_ _++_
@@ -296,7 +296,7 @@ module _ {a} {A : Set a} where
 
 shifts : ∀ xs ys {zs : List A} → xs ++ ys ++ zs ↭ ys ++ xs ++ zs
 shifts xs ys {zs} = begin
-   xs ++ ys  ++ zs ↭˘⟨ ++-assoc xs ys zs ⟩
+   xs ++ ys  ++ zs ↭⟨ ++-assoc xs ys zs ⟨
   (xs ++ ys) ++ zs ↭⟨ ++⁺ʳ zs (++-comm xs ys) ⟩
   (ys ++ xs) ++ zs ↭⟨ ++-assoc ys xs zs ⟩
    ys ++ xs  ++ zs ∎
@@ -330,7 +330,7 @@ drop-∷ = drop-mid [] []
 ↭-reverse [] = ↭-refl
 ↭-reverse (x ∷ xs) = begin
   reverse (x ∷ xs) ≡⟨ Lₚ.unfold-reverse x xs ⟩
-  reverse xs ∷ʳ x ↭˘⟨ ∷↭∷ʳ x (reverse xs) ⟩
+  reverse xs ∷ʳ x ↭⟨ ∷↭∷ʳ x (reverse xs) ⟨
   x ∷ reverse xs   ↭⟨ prep x (↭-reverse xs) ⟩
   x ∷ xs ∎
   where open PermutationReasoning
@@ -349,7 +349,7 @@ module _ {ℓ} {R : Rel A ℓ} (R? : Decidable R) where
   ... | true  | rec | _   = prep x rec
   ... | false | _   | rec = begin
     y ∷ merge R? (x ∷ xs) ys <⟨ rec ⟩
-    y ∷ x ∷ xs ++ ys         ↭˘⟨ shift y (x ∷ xs) ys ⟩
-    (x ∷ xs) ++ y ∷ ys       ≡˘⟨ Lₚ.++-assoc [ x ] xs (y ∷ ys) ⟩
+    y ∷ x ∷ xs ++ ys         ↭⟨ shift y (x ∷ xs) ys ⟨
+    (x ∷ xs) ++ y ∷ ys       ≡⟨ Lₚ.++-assoc [ x ] xs (y ∷ ys) ⟨
     x ∷ xs ++ y ∷ ys         ∎
     where open PermutationReasoning
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid.agda b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
index 0d375226db..edcf62b6da 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
@@ -90,10 +90,12 @@ module PermutationReasoning where
 
   private module Base = SetoidReasoning ↭-setoid
 
-  open Base public hiding (step-≈; step-≈˘)
+  open Base public
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
+    renaming (≈-go to ↭-go)
 
-  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ Base.∼-go ↭-sym public
-  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ (Base.∼-go ∘′ refl) ≋-sym public
+  open ↭-syntax _IsRelatedTo_ _IsRelatedTo_ ↭-go ↭-sym public
+  open ≋-syntax _IsRelatedTo_ _IsRelatedTo_ (↭-go ∘′ refl) ≋-sym public
 
   -- Some extra combinators that allow us to skip certain elements
 
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index e09251caf5..3af753e826 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -200,7 +200,7 @@ shift {v} {w} v≈w (x ∷ xs) ys = begin
 ++-comm []       ys = ↭-sym (++-identityʳ ys)
 ++-comm (x ∷ xs) ys = begin
   x ∷ xs ++ ys   <⟨ ++-comm xs ys ⟩
-  x ∷ ys ++ xs   ↭˘⟨ ↭-shift ys xs ⟩
+  x ∷ ys ++ xs   ↭⟨ ↭-shift ys xs ⟨
   ys ++ (x ∷ xs) ∎
 
 -- Structures
@@ -262,7 +262,7 @@ inject v ws↭ys xs↭zs = trans (++⁺ˡ _ (↭-prep _ xs↭zs)) (++⁺ʳ _ ws
 
 shifts : ∀ xs ys {zs : List A} → xs ++ ys ++ zs ↭ ys ++ xs ++ zs
 shifts xs ys {zs} = begin
-   xs ++ ys  ++ zs ↭˘⟨ ++-assoc xs ys zs ⟩
+   xs ++ ys  ++ zs ↭⟨ ++-assoc xs ys zs ⟨
   (xs ++ ys) ++ zs ↭⟨ ++⁺ʳ zs (++-comm xs ys) ⟩
   (ys ++ xs) ++ zs ↭⟨ ++-assoc ys xs zs ⟩
    ys ++ xs  ++ zs ∎
@@ -297,19 +297,19 @@ dropMiddleElement {v} ws xs {ys} {zs} p = helper p ws xs ≋-refl ≋-refl
   helper {_ ∷ as} {_ ∷ bs} (prep _ as↭bs) [] [] {ys} {zs} (_ ∷ ys≋as) (_ ∷ zs≋bs) = begin
     ys               ≋⟨  ys≋as ⟩
     as               ↭⟨  as↭bs ⟩
-    bs               ≋˘⟨ zs≋bs ⟩
+    bs               ≋⟨ zs≋bs ⟨
     zs               ∎
   helper {_ ∷ as} {_ ∷ bs} (prep a≈b as↭bs) [] (x ∷ xs) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin
     ys               ≋⟨  ≋₁ ⟩
     as               ↭⟨  as↭bs ⟩
-    bs               ≋˘⟨ ≋₂ ⟩
+    bs               ≋⟨ ≋₂ ⟨
     xs ++ v ∷ zs     ↭⟨  shift (lemma ≈₁ a≈b ≈₂) xs zs ⟩
     x ∷ xs ++ zs     ∎
   helper {_ ∷ as} {_ ∷ bs} (prep v≈w p) (w ∷ ws) [] {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin
     w ∷ ws ++ ys     ↭⟨  ↭-sym (shift (lemma ≈₂ (≈-sym v≈w) ≈₁) ws ys) ⟩
     ws ++ v ∷ ys     ≋⟨  ≋₁ ⟩
     as               ↭⟨  p ⟩
-    bs               ≋˘⟨ ≋₂ ⟩
+    bs               ≋⟨ ≋₂ ⟨
     zs               ∎
   helper {_ ∷ as} {_ ∷ bs} (prep w≈x p) (w ∷ ws) (x ∷ xs) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin
     w ∷ ws ++ ys     ↭⟨ prep (lemma ≈₁ w≈x ≈₂) (helper p ws xs ≋₁ ≋₂) ⟩
@@ -317,12 +317,12 @@ dropMiddleElement {v} ws xs {ys} {zs} p = helper p ws xs ≋-refl ≋-refl
   helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap v≈x y≈v p) [] [] {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin
     ys               ≋⟨  ≋₁ ⟩
     a ∷ as           ↭⟨  prep (≈-trans (≈-trans (≈-trans y≈v (≈-sym ≈₂)) ≈₁) v≈x) p ⟩
-    b ∷ bs           ≋˘⟨ ≋₂ ⟩
+    b ∷ bs           ≋⟨ ≋₂ ⟨
     zs               ∎
   helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap v≈w y≈w p) [] (x ∷ []) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin
     ys               ≋⟨  ≋₁ ⟩
     a ∷ as           ↭⟨  prep y≈w p ⟩
-    _ ∷ bs           ≋˘⟨ ≈₂ ∷ tail ≋₂ ⟩
+    _ ∷ bs           ≋⟨ ≈₂ ∷ tail ≋₂ ⟨
     x ∷ zs           ∎
   helper {_ ∷ a ∷ as} {_ ∷ b ∷ bs} (swap v≈w y≈x p) [] (x ∷ w ∷ xs) {ys} {zs} (≈₁ ∷ ≋₁) (≈₂ ∷ ≋₂) = begin
     ys               ≋⟨ ≋₁ ⟩
@@ -455,8 +455,8 @@ module _ {ℓ} {R : Rel A ℓ} (R? : B.Decidable R) where
   ... | true  | rec | _   = ↭-prep x rec
   ... | false | _   | rec = begin
     y ∷ merge R? (x ∷ xs) ys <⟨ rec ⟩
-    y ∷ x ∷ xs ++ ys         ↭˘⟨ ↭-shift (x ∷ xs) ys ⟩
-    (x ∷ xs) ++ y ∷ ys       ≡˘⟨ Lₚ.++-assoc [ x ] xs (y ∷ ys) ⟩
+    y ∷ x ∷ xs ++ ys         ↭⟨ ↭-shift (x ∷ xs) ys ⟨
+    (x ∷ xs) ++ y ∷ ys       ≡⟨ Lₚ.++-assoc [ x ] xs (y ∷ ys) ⟨
     x ∷ xs ++ y ∷ ys         ∎
     where open PermutationReasoning
 
@@ -497,7 +497,7 @@ module _ {_∙_ : Op₂ A} {ε : A} (isCmonoid : IsCommutativeMonoid _≈_ _∙_
   foldr-commMonoid (prep x≈y xs↭ys) = ∙-cong x≈y (foldr-commMonoid xs↭ys)
   foldr-commMonoid (swap {xs} {ys} {x} {y} {x′} {y′} x≈x′ y≈y′ xs↭ys) = S.begin
     x ∙ (y ∙ foldr _∙_ ε xs)   S.≈⟨ ∙-congˡ (∙-congˡ (foldr-commMonoid xs↭ys)) ⟩
-    x ∙ (y ∙ foldr _∙_ ε ys)   S.≈˘⟨ assoc x y (foldr _∙_ ε ys) ⟩
+    x ∙ (y ∙ foldr _∙_ ε ys)   S.≈⟨ assoc x y (foldr _∙_ ε ys) ⟨
     (x ∙ y) ∙ foldr _∙_ ε ys   S.≈⟨ ∙-congʳ (comm x y) ⟩
     (y ∙ x) ∙ foldr _∙_ ε ys   S.≈⟨ ∙-congʳ (∙-cong y≈y′ x≈x′) ⟩
     (y′ ∙ x′) ∙ foldr _∙_ ε ys S.≈⟨ assoc y′ x′ (foldr _∙_ ε ys) ⟩
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index bb20fd417f..0c344758b7 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -118,7 +118,7 @@ module ⊆-Reasoning (S : Setoid a ℓ) where
   private module Base = PreorderReasoning (⊆-preorder S)
 
   open Base public
-    hiding (step-≲; step-≈; step-≈˘; step-∼)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-≲; step-∼)
     renaming (≲-go to ⊆-go; ≈-go to ≋-go)
 
   open begin-membership-syntax _IsRelatedTo_ _∈_ (λ x → Base.begin x) public
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index b88bdfbe95..e356e9eca1 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -686,7 +686,7 @@ module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where
   Any (λ f → Any (P ∘ f) xs) fs                ↔⟨ Any-cong (λ _ → Any-cong (λ _ → pure↔) (_ ∎)) (_ ∎) ⟩
   Any (λ f → Any (Any P ∘ pure ∘ f) xs) fs     ↔⟨ Any-cong (λ _ → >>=↔ ) (_ ∎) ⟩
   Any (λ f → Any P (xs >>= pure ∘ f)) fs       ↔⟨ >>=↔ ⟩
-  Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡˘⟨ P.cong (Any P) (Listₑ.Applicative.unfold-⊛ fs xs) ⟩
+  Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡⟨ P.cong (Any P) (Listₑ.Applicative.unfold-⊛ fs xs) ⟨
   Any P (fs ⊛ xs)                               ∎
   where open Related.EquationalReasoning
 
@@ -706,7 +706,7 @@ module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where
   Any (λ x → Any (λ y → P (x , y)) ys) xs                           ↔⟨ pure↔ ⟩
   Any (λ _,_ → Any (λ x → Any (λ y → P (x , y)) ys) xs) (pure _,_)  ↔⟨ ⊛↔ ⟩
   Any (λ x, → Any (P ∘ x,) ys) (pure _,_ ⊛ xs)                      ↔⟨ ⊛↔ ⟩
-  Any P (pure _,_ ⊛ xs ⊛ ys)                                        ≡˘⟨ P.cong (Any P ∘′ (_⊛ ys)) (Listₑ.Applicative.unfold-<$> _,_ xs) ⟩
+  Any P (pure _,_ ⊛ xs ⊛ ys)                                        ≡⟨ P.cong (Any P ∘′ (_⊛ ys)) (Listₑ.Applicative.unfold-<$> _,_ xs) ⟨
   Any P (xs ⊗ ys)                                                   ∎
   where open Related.EquationalReasoning
 
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 9e8976842e..f83bbfc12c 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -156,9 +156,9 @@ fromℕ≡fromℕ' n = fromℕ-helper≡fromℕ' n n ℕₚ.≤-refl
   tail-homo ℕ.zero = refl
   tail-homo (ℕ.suc ℕ.zero) = refl
   tail-homo (ℕ.suc (ℕ.suc n)) = begin
-    tail (suc (fromℕ' (ℕ.suc (ℕ.suc n))))  ≡˘⟨ tail-suc (fromℕ' n) ⟩
+    tail (suc (fromℕ' (ℕ.suc (ℕ.suc n))))  ≡⟨ tail-suc (fromℕ' n) ⟨
     suc (tail (suc (fromℕ' n)))            ≡⟨ cong suc (tail-homo n) ⟩
-    fromℕ' (ℕ.suc (n / 2))                 ≡˘⟨ cong fromℕ' (+-distrib-/-∣ˡ {2} n ∣-refl) ⟩
+    fromℕ' (ℕ.suc (n / 2))                 ≡⟨ cong fromℕ' (+-distrib-/-∣ˡ {2} n ∣-refl) ⟨
     fromℕ' (ℕ.suc (ℕ.suc n) / 2)           ∎
 
   fromℕ-helper≡fromℕ' : ∀ n w → n ℕ.≤ w → fromℕ.helper n n w ≡ fromℕ' n
@@ -167,7 +167,7 @@ fromℕ≡fromℕ' n = fromℕ-helper≡fromℕ' n n ℕₚ.≤-refl
     split-injective (begin
       split (fromℕ.helper n (ℕ.suc n) (ℕ.suc w))         ≡⟨ split-if _ _ ⟩
       just (n % 2 ℕ.≡ᵇ 0) , fromℕ.helper n (n / 2) w     ≡⟨ cong (_ ,_) rec-n/2 ⟩
-      just (n % 2 ℕ.≡ᵇ 0) , fromℕ' (n / 2)               ≡˘⟨ cong₂ _,_ (head-homo n) (tail-homo n) ⟩
+      just (n % 2 ℕ.≡ᵇ 0) , fromℕ' (n / 2)               ≡⟨ cong₂ _,_ (head-homo n) (tail-homo n) ⟨
       head (fromℕ' (ℕ.suc n)) , tail (fromℕ' (ℕ.suc n))  ≡⟨⟩
       split (fromℕ' (ℕ.suc n))                           ∎)
     where rec-n/2 = fromℕ-helper≡fromℕ' (n / 2) w (ℕₚ.≤-trans (m/n≤m n 2) n≤w)
@@ -613,7 +613,7 @@ module ≤-Reasoning where
     <-≤-trans
     ≤-<-trans
     public
-    hiding (step-≈; step-≈˘)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
 
 ------------------------------------------------------------------------
 -- Properties of _<ℕ_
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index fc6ef422b4..d07d22db47 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -66,9 +66,9 @@ open Specification public
 nCk≡nC[n∸k] : k ≤ n → n C k ≡ n C (n ∸ k)
 nCk≡nC[n∸k] {k} {n} k≤n = begin-equality
   n C k                               ≡⟨ nCk≡n!/k![n-k]! k≤n ⟩
-  n ! / (k ! * (n ∸ k) !)             ≡˘⟨ /-congʳ (*-comm ((n ∸ k) !) (k !)) ⟩
-  n ! / ((n ∸ k) ! * k !)             ≡˘⟨ /-congʳ (cong ((n ∸ k) ! *_) (cong _! (m∸[m∸n]≡n k≤n))) ⟩
-  n ! / ((n ∸ k) ! * (n ∸ (n ∸ k)) !) ≡˘⟨ nCk≡n!/k![n-k]! (m≤n⇒n∸m≤n k≤n) ⟩
+  n ! / (k ! * (n ∸ k) !)             ≡⟨ /-congʳ (*-comm ((n ∸ k) !) (k !)) ⟨
+  n ! / ((n ∸ k) ! * k !)             ≡⟨ /-congʳ (cong ((n ∸ k) ! *_) (cong _! (m∸[m∸n]≡n k≤n))) ⟨
+  n ! / ((n ∸ k) ! * (n ∸ (n ∸ k)) !) ≡⟨ nCk≡n!/k![n-k]! (m≤n⇒n∸m≤n k≤n) ⟨
   n C (n ∸ k)                         ∎
   where instance
     _ = (n ∸ k) !* k !≢0
@@ -78,9 +78,9 @@ nCk≡nC[n∸k] {k} {n} k≤n = begin-equality
 nCk≡nPk/k! : k ≤ n → n C k ≡ ((n P k) / k !) {{k !≢0}}
 nCk≡nPk/k! {k} {n} k≤n = begin-equality
   n C k                   ≡⟨ nCk≡n!/k![n-k]! k≤n ⟩
-  n ! / (k ! * (n ∸ k) !) ≡˘⟨ /-congʳ (*-comm ((n ∸ k)!) (k !)) ⟩
-  n ! / ((n ∸ k) ! * k !) ≡˘⟨ m/n/o≡m/[n*o] (n !) ((n ∸ k) !) (k !) ⟩
-  (n ! / (n ∸ k) !) / k ! ≡˘⟨ /-congˡ (nPk≡n!/[n∸k]! k≤n) ⟩
+  n ! / (k ! * (n ∸ k) !) ≡⟨ /-congʳ (*-comm ((n ∸ k)!) (k !)) ⟨
+  n ! / ((n ∸ k) ! * k !) ≡⟨ m/n/o≡m/[n*o] (n !) ((n ∸ k) !) (k !) ⟨
+  (n ! / (n ∸ k) !) / k ! ≡⟨ /-congˡ (nPk≡n!/[n∸k]! k≤n) ⟨
   (n P k) / k !           ∎
   where instance
     _ = k !≢0
@@ -128,7 +128,7 @@ module _ {n k} (k<n : k < n) where
       [n-k] * [n-k-1]!
         ≡⟨ cong (_* [n-k-1]!) [n-k]≡1+[n-k-1] ⟩
       (suc [n-k-1]) * [n-k-1]!
-        ≡˘⟨ cong _! [n-k]≡1+[n-k-1] ⟩
+        ≡⟨ cong _! [n-k]≡1+[n-k-1] ⟨
       [n-k]!                ∎
 
   private
@@ -163,39 +163,39 @@ nCk+nC[k+1]≡[n+1]C[k+1] n k with <-cmp k n
   n C k + n C (suc k) ≡⟨ cong (_+ (n C (suc k))) (k>n⇒nCk≡0 k>n) ⟩
   0 + n C (suc k)     ≡⟨⟩
   n C (suc k)         ≡⟨ k>n⇒nCk≡0 (m<n⇒m<1+n k>n) ⟩
-  0                   ≡˘⟨ k>n⇒nCk≡0 (s<s k>n) ⟩
+  0                   ≡⟨ k>n⇒nCk≡0 (s<s k>n) ⟨
   suc n C suc k       ∎
 {- case k≡n, in which case 1+k≡1+n>n -}
 ... | tri≈ _ k≡n _ rewrite k≡n = begin-equality
   n C n + n C (suc n) ≡⟨ cong (n C n +_) (k>n⇒nCk≡0 (n<1+n n)) ⟩
   n C n + 0           ≡⟨ +-identityʳ (n C n) ⟩
   n C n               ≡⟨ nCn≡1 n ⟩
-  1                   ≡˘⟨ nCn≡1 (suc n) ⟩
+  1                   ≡⟨ nCn≡1 (suc n) ⟨
   suc n C suc n       ∎
 {- case k<n, in which case 1+k<1+n and there's arithmetic to perform -}
 ... | tri< k<n _ _ = begin-equality
   n C k + n C (suc k)
     ≡⟨ cong (n C k +_) (nCk≡n!/k![n-k]! k<n)  ⟩
   n C k + (n! / d[k+1])
-    ≡˘⟨ cong (n C k +_) (m*n/m*o≡n/o (n ∸ k) n! d[k+1]) ⟩
+    ≡⟨ cong (n C k +_) (m*n/m*o≡n/o (n ∸ k) n! d[k+1]) ⟨
   n C k + [n-k]*n!/[n-k]*d[k+1]
     ≡⟨ cong (_+ [n-k]*n!/[n-k]*d[k+1]) (nCk≡n!/k![n-k]! k≤n) ⟩
   n! / d[k] + _
-    ≡˘⟨ cong (_+ [n-k]*n!/[n-k]*d[k+1]) (m*n/m*o≡n/o (suc k) n! d[k]) ⟩
+    ≡⟨ cong (_+ [n-k]*n!/[n-k]*d[k+1]) (m*n/m*o≡n/o (suc k) n! d[k]) ⟨
   (suc k * n!) / [k+1]*d[k] + _
     ≡⟨ cong (((suc k * n!) / [k+1]*d[k]) +_) (/-congʳ ([n-k]*d[k+1]≡[k+1]*d[k] k<n)) ⟩
   (suc k * n!) / [k+1]*d[k] + ((n ∸ k) * n! / [k+1]*d[k])
-    ≡˘⟨ +-distrib-/-∣ˡ _ (*-monoʳ-∣ (suc k) (k![n∸k]!∣n! k≤n)) ⟩
+    ≡⟨ +-distrib-/-∣ˡ _ (*-monoʳ-∣ (suc k) (k![n∸k]!∣n! k≤n)) ⟨
   ((suc k) * n! + (n ∸ k) * n!) / [k+1]*d[k]
-    ≡˘⟨ cong (_/ [k+1]*d[k]) (*-distribʳ-+ (n !) (suc k) (n ∸ k)) ⟩
+    ≡⟨ cong (_/ [k+1]*d[k]) (*-distribʳ-+ (n !) (suc k) (n ∸ k)) ⟨
   ((suc k + (n ∸ k)) * n !) / [k+1]*d[k]
     ≡⟨ cong (λ z → z * n ! / [k+1]*d[k]) [k+1]+[n-k]≡[n+1] ⟩
   ((suc n) * n !) / [k+1]*d[k]
-    ≡˘⟨ /-congʳ (*-assoc (suc k) (k !) ((n ∸ k) !)) ⟩
+    ≡⟨ /-congʳ (*-assoc (suc k) (k !) ((n ∸ k) !)) ⟨
   ((suc n) * n !) / (((suc k) * k !) * (n ∸ k) !)
     ≡⟨⟩
   suc n ! / (suc k ! * (suc n ∸ suc k) !)
-    ≡˘⟨ nCk≡n!/k![n-k]! [k+1]≤[n+1] ⟩
+    ≡⟨ nCk≡n!/k![n-k]! [k+1]≤[n+1] ⟨
   suc n C suc k ∎
   where
     k≤n                     : k ≤ n
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index 522e3104ed..b217ea4143 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -40,7 +40,7 @@ nP′k≡n!/[n∸k]! : ∀ {n k} → k ≤ n → n P′ k ≡ (n ! / (n ∸ k) !
 nP′k≡n!/[n∸k]! {n} {zero}  z≤n = sym (n/n≡1 (n !) {{n !≢0}})
 nP′k≡n!/[n∸k]! {n} {suc k} k<n = begin-equality
   (n ∸ k) * (n P′ k)             ≡⟨ cong ((n ∸ k) *_) (nP′k≡n!/[n∸k]! (<⇒≤ k<n)) ⟩
-  (n ∸ k) * (n ! / (n ∸ k) !)    ≡˘⟨ *-/-assoc (n ∸ k) (m≤n⇒m!∣n! (m∸n≤m n k)) ⟩
+  (n ∸ k) * (n ! / (n ∸ k) !)    ≡⟨ *-/-assoc (n ∸ k) (m≤n⇒m!∣n! (m∸n≤m n k)) ⟨
   (((n ∸ k) * n !) / (n ∸ k) !)  ≡⟨ m*n/m!≡n/[m∸1]! (n ∸ k) (n !) ⟩
   (n ! / (pred (n ∸ k) !))       ≡⟨ /-congʳ (cong _! (pred[m∸n]≡m∸[1+n] n k)) ⟩
   (n ! / (n ∸ suc k) !)          ∎
@@ -67,7 +67,7 @@ P′-rec : ∀ {n k} → k ≤ n → .{{NonZero k}} →
         n P′ k ≡ k * (pred n P′ pred k) + pred n P′ k
 P′-rec n@{suc n-1} k@{suc k-1} k≤n = begin-equality
   n P′ k                                        ≡⟨ nP′k≡n[n∸1P′k∸1] n k ⟩
-  n * (n-1 P′ k-1)                              ≡˘⟨ cong (_* (n-1 P′ k-1)) (m+[n∸m]≡n {k} {n} k≤n) ⟩
+  n * (n-1 P′ k-1)                              ≡⟨ cong (_* (n-1 P′ k-1)) (m+[n∸m]≡n {k} {n} k≤n) ⟨
   (k + (n ∸ k)) * (n-1 P′ k-1)                  ≡⟨ *-distribʳ-+ (n-1 P′ k-1) k (n ∸ k) ⟩
   k * (n-1 P′ k-1) + (n-1 ∸ k-1) * (n-1 P′ k-1) ≡⟨⟩
   k * (n-1 P′ k-1) + (n-1 P′ k)                 ∎
@@ -89,7 +89,7 @@ k!∣nP′k n@{suc n-1} k@{suc k-1} k≤n@(s≤s k-1≤n-1) with k-1 ≟ n-1
 ... | no  k≢n  = begin
   k !                           ≡⟨⟩
   k * k-1 !                     ∣⟨ ∣m∣n⇒∣m+n (*-monoʳ-∣ k (k!∣nP′k k-1≤n-1)) ( k!∣nP′k (≤∧≢⇒< k-1≤n-1 k≢n)) ⟩
-  k * (n-1 P′ k-1) + (n-1 P′ k) ≡˘⟨ P′-rec k≤n ⟩
+  k * (n-1 P′ k-1) + (n-1 P′ k) ≡⟨ P′-rec k≤n ⟨
   n P′ k                        ∎
   where open ∣-Reasoning
 
@@ -134,7 +134,7 @@ C′-sym {n} {k} k≤n = begin-equality
   n C′ (n ∸ k)                        ≡⟨ nC′k≡n!/k![n-k]! {n} {n ∸ k} (m≤n⇒n∸m≤n k≤n) ⟩
   n ! / ((n ∸ k) ! * (n ∸ (n ∸ k)) !) ≡⟨ /-congʳ (cong ((n ∸ k) ! *_) (cong _! (m∸[m∸n]≡n k≤n))) ⟩
   n ! / ((n ∸ k) ! * k !)             ≡⟨ /-congʳ (*-comm ((n ∸ k) !) (k !)) ⟩
-  n ! / (k ! * (n ∸ k) !)             ≡˘⟨ nC′k≡n!/k![n-k]! k≤n ⟩
+  n ! / (k ! * (n ∸ k) !)             ≡⟨ nC′k≡n!/k![n-k]! k≤n ⟨
   n C′ k                              ∎
   where
   open ≤-Reasoning
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index dbe99a30e6..bbb8288838 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -41,8 +41,8 @@ m≡m%n+[m/n]*n m (suc n) = div-mod-lemma 0 0 m n
 
 m%n≡m∸m/n*n : ∀ m n .{{_ : NonZero n}} → m % n ≡ m ∸ (m / n) * n
 m%n≡m∸m/n*n m n = begin-equality
-  m % n                  ≡˘⟨ m+n∸n≡m (m % n) m/n*n ⟩
-  m % n + m/n*n ∸ m/n*n  ≡˘⟨ cong (_∸ m/n*n) (m≡m%n+[m/n]*n m n) ⟩
+  m % n                  ≡⟨ m+n∸n≡m (m % n) m/n*n ⟨
+  m % n + m/n*n ∸ m/n*n  ≡⟨ cong (_∸ m/n*n) (m≡m%n+[m/n]*n m n) ⟨
   m ∸ m/n*n              ∎
   where m/n*n = (m / n) * n
 
@@ -79,14 +79,14 @@ m%n%n≡m%n m (suc n-1) = modₕ-idem 0 m n-1
 m≤n⇒[n∸m]%m≡n%m : .⦃ _ : NonZero m ⦄ → m ≤ n →
                   (n ∸ m) % m ≡ n % m
 m≤n⇒[n∸m]%m≡n%m {m} {n} m≤n = begin-equality
-  (n ∸ m) % m     ≡˘⟨ [m+n]%n≡m%n (n ∸ m) m ⟩
+  (n ∸ m) % m     ≡⟨ [m+n]%n≡m%n (n ∸ m) m ⟨
   (n ∸ m + m) % m ≡⟨ cong (_% m) (m∸n+n≡m m≤n) ⟩
   n % m           ∎
 
 m*n≤o⇒[o∸m*n]%n≡o%n : ∀ m {n o} .⦃ _ : NonZero n ⦄ → m * n ≤ o →
                       (o ∸ m * n) % n ≡ o % n
 m*n≤o⇒[o∸m*n]%n≡o%n m {n} {o} m*n≤o = begin-equality
-  (o ∸ m * n) % n         ≡˘⟨ [m+kn]%n≡m%n (o ∸ m * n) m n ⟩
+  (o ∸ m * n) % n         ≡⟨ [m+kn]%n≡m%n (o ∸ m * n) m n ⟨
   (o ∸ m * n + m * n) % n ≡⟨ cong (_% n) (m∸n+n≡m m*n≤o) ⟩
   o % n                   ∎
 
@@ -95,7 +95,7 @@ m∣n⇒o%n%m≡o%m : ∀ m n o .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ 
 m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
   o % n % m                ≡⟨⟩
   o % pm % m               ≡⟨ %-congˡ (m%n≡m∸m/n*n o pm) ⟩
-  (o ∸ o / pm * pm) % m    ≡˘⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟩
+  (o ∸ o / pm * pm) % m    ≡⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟨
   (o ∸ o / pm * p * m) % m ≡⟨ m*n≤o⇒[o∸m*n]%n≡o%n (o / pm * p) lem ⟩
   o % m                    ∎
   where
@@ -107,7 +107,7 @@ m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
     -- Sort out dependencies in this file, then use m/n*n≤m instead.
     o / pm * pm          ≤⟨ m≤m+n (o / pm * pm) (o % pm) ⟩
     o / pm * pm + o % pm ≡⟨ +-comm _ (o % pm) ⟩
-    o % pm + o / pm * pm ≡˘⟨ m≡m%n+[m/n]*n o pm ⟩
+    o % pm + o / pm * pm ≡⟨ m≡m%n+[m/n]*n o pm ⟨
     o                    ∎
 
 m*n%n≡0 : ∀ m n .{{_ : NonZero n}} → (m * n) % n ≡ 0
@@ -168,8 +168,8 @@ m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-
     (m′ + k * d) * (n′ + j * d)                       ≡⟨ *-distribʳ-+ (n′ + j * d) m′ (k * d) ⟩
     m′ * (n′ + j * d) + (k * d) * (n′ + j * d)        ≡⟨ cong₂ _+_ (*-distribˡ-+ m′ n′ (j * d)) (*-comm (k * d) (n′ + j * d)) ⟩
     (m′ * n′ + m′ * (j * d)) + (n′ + j * d) * (k * d) ≡⟨ +-assoc (m′ * n′) (m′ * (j * d)) ((n′ + j * d) * (k * d)) ⟩
-    m′ * n′ + (m′ * (j * d) + (n′ + j * d) * (k * d)) ≡˘⟨ cong (m′ * n′ +_) (cong₂ _+_ (*-assoc m′ j d) (*-assoc (n′ + j * d) k d)) ⟩
-    m′ * n′ + ((m′ * j) * d + ((n′ + j * d) * k) * d) ≡˘⟨ cong (m′ * n′ +_) (*-distribʳ-+ d (m′ * j) ((n′ + j * d) * k)) ⟩
+    m′ * n′ + (m′ * (j * d) + (n′ + j * d) * (k * d)) ≡⟨ cong (m′ * n′ +_) (cong₂ _+_ (*-assoc m′ j d) (*-assoc (n′ + j * d) k d)) ⟨
+    m′ * n′ + ((m′ * j) * d + ((n′ + j * d) * k) * d) ≡⟨ cong (m′ * n′ +_) (*-distribʳ-+ d (m′ * j) ((n′ + j * d) * k)) ⟨
     m′ * n′ + (m′ * j + (n′ + j * d) * k) * d         ∎
 
 %-remove-+ˡ : ∀ {m} n {d} .{{_ : NonZero d}} → d ∣ m → (m + n) % d ≡ n % d
@@ -242,7 +242,7 @@ m/n<m m n n≥2 = *-cancelʳ-< _ (m / n) m (begin-strict
 /-cancelʳ-≡ : ∀ {m n o} .{{_ : NonZero o}} →
               o ∣ m → o ∣ n → m / o ≡ n / o → m ≡ n
 /-cancelʳ-≡ {m} {n} {o} o∣m o∣n m/o≡n/o = begin-equality
-  m           ≡˘⟨ m*[n/m]≡n {o} {m} o∣m ⟩
+  m           ≡⟨ m*[n/m]≡n {o} {m} o∣m ⟨
   o * (m / o) ≡⟨  cong (o *_) m/o≡n/o ⟩
   o * (n / o) ≡⟨  m*[n/m]≡n {o} {n} o∣n ⟩
   n           ∎
@@ -294,10 +294,10 @@ m*n/m*o≡n/o m@(suc _) n o = helper (<-wellFounded n)
   ... | yes n<o = trans (m<n⇒m/n≡0 (*-monoʳ-< m n<o)) (sym (m<n⇒m/n≡0 n<o))
   ... | no  n≮o = begin-equality
     (m * n) / (m * o)             ≡⟨  m/n≡1+[m∸n]/n (*-monoʳ-≤ m (≮⇒≥ n≮o)) ⟩
-    1 + (m * n ∸ m * o) / (m * o) ≡˘⟨ cong (λ v → 1 + v / (m * o)) (*-distribˡ-∸ m n o) ⟩
+    1 + (m * n ∸ m * o) / (m * o) ≡⟨ cong (λ v → 1 + v / (m * o)) (*-distribˡ-∸ m n o) ⟨
     1 + (m * (n ∸ o)) / (m * o)   ≡⟨  cong suc (helper (rec n∸o<n)) ⟩
-    1 + (n ∸ o) / o               ≡˘⟨ cong₂ _+_ (n/n≡1 o) refl ⟩
-    o / o + (n ∸ o) / o           ≡˘⟨ +-distrib-/-∣ˡ (n ∸ o) (divides 1 ((sym (*-identityˡ o)))) ⟩
+    1 + (n ∸ o) / o               ≡⟨ cong₂ _+_ (n/n≡1 o) refl ⟨
+    o / o + (n ∸ o) / o           ≡⟨ +-distrib-/-∣ˡ (n ∸ o) (divides 1 ((sym (*-identityˡ o)))) ⟨
     (o + (n ∸ o)) / o             ≡⟨  cong (_/ o) (m+[n∸m]≡n (≮⇒≥ n≮o)) ⟩
     n / o                         ∎
     where n∸o<n = ∸-monoʳ-< (n≢0⇒n>0 (≢-nonZero⁻¹ o)) (≮⇒≥ n≮o)
@@ -333,19 +333,19 @@ m<n*o⇒m/o<n {m} {suc n} {o} m<n*o with m <? o
 ... | yes m<n = begin-equality
   (m ∸ n) / n  ≡⟨ m<n⇒m/n≡0 (≤-<-trans (m∸n≤m m n) m<n) ⟩
   0            ≡⟨⟩
-  0 ∸ 1        ≡˘⟨ cong (_∸ 1) (m<n⇒m/n≡0 m<n) ⟩
+  0 ∸ 1        ≡⟨ cong (_∸ 1) (m<n⇒m/n≡0 m<n) ⟨
   m / n ∸ 1    ≡⟨⟩
   pred (m / n) ∎
 ... | no m≮n = begin-equality
   (m ∸ n) / n           ≡⟨⟩
-  suc ((m ∸ n) / n) ∸ 1 ≡˘⟨ cong (_∸ 1) (m/n≡1+[m∸n]/n (≮⇒≥ m≮n)) ⟩
+  suc ((m ∸ n) / n) ∸ 1 ≡⟨ cong (_∸ 1) (m/n≡1+[m∸n]/n (≮⇒≥ m≮n)) ⟨
   m / n ∸ 1             ≡⟨⟩
   pred (m / n)          ∎
 
 [m∸n*o]/o≡m/o∸n : ∀ m n o .⦃ _ : NonZero o ⦄ → (m ∸ n * o) / o ≡ m / o ∸ n
 [m∸n*o]/o≡m/o∸n m zero    o = refl
 [m∸n*o]/o≡m/o∸n m (suc n) o = begin-equality
-  (m ∸ (o + n * o)) / o ≡˘⟨ /-congˡ (∸-+-assoc m o (n * o)) ⟩
+  (m ∸ (o + n * o)) / o ≡⟨ /-congˡ (∸-+-assoc m o (n * o)) ⟨
   (m ∸ o ∸ n * o) / o   ≡⟨ [m∸n*o]/o≡m/o∸n (m ∸ o) n o ⟩
   (m ∸ o) / o ∸ n       ≡⟨ cong (_∸ n) ([m∸n]/n≡m/n∸1 m o) ⟩
   m / o ∸ 1 ∸ n         ≡⟨ ∸-+-assoc (m / o) 1 n ⟩
@@ -368,13 +368,13 @@ m/n/o≡m/[n*o] m n o = begin-equality
   lem₁ : n ∣ m / n*o * n*o
   lem₁ = divides (m / n*o * o) $ begin-equality
     m / n*o * n*o   ≡⟨ cong (m / n*o *_) (*-comm n o) ⟩
-    m / n*o * o*n   ≡˘⟨ *-assoc (m / n*o) o n ⟩
+    m / n*o * o*n   ≡⟨ *-assoc (m / n*o) o n ⟨
     m / n*o * o * n ∎
 
   lem₂ : m / n*o * n*o / n ≡ m / n*o * o
   lem₂ = begin-equality
     m / n*o * n*o / n   ≡⟨ cong (λ # → m / n*o * # / n) (*-comm n o) ⟩
-    m / n*o * o*n / n   ≡˘⟨ /-congˡ (*-assoc (m / n*o) o n) ⟩
+    m / n*o * o*n / n   ≡⟨ /-congˡ (*-assoc (m / n*o) o n) ⟨
     m / n*o * o * n / n ≡⟨ m*n/n≡m (m / n*o * o) n ⟩
     m / n*o * o         ∎
 
@@ -395,7 +395,7 @@ m/n/o≡m/[n*o] m n o = begin-equality
                   o ∣ m → p ∣ n → (m * n) / (o * p) ≡ (m / o) * (n / p)
 /-*-interchange {m} {n} {o@(suc _)} {p@(suc _)} o∣m p∣n = *-cancelˡ-≡ _ _ (o * p) (begin-equality
   (o * p) * ((m * n) / (o * p)) ≡⟨  m*[n/m]≡n (*-pres-∣ o∣m p∣n) ⟩
-  m * n                         ≡˘⟨ cong₂ _*_ (m*[n/m]≡n o∣m) (m*[n/m]≡n p∣n) ⟩
+  m * n                         ≡⟨ cong₂ _*_ (m*[n/m]≡n o∣m) (m*[n/m]≡n p∣n) ⟨
   (o * (m / o)) * (p * (n / p)) ≡⟨ [m*n]*[o*p]≡[m*o]*[n*p] o (m / o) p (n / p) ⟩
   (o * p) * ((m / o) * (n / p)) ∎)
 
@@ -407,11 +407,11 @@ m%[n*o]/o≡m/o%n : ∀ m n o .⦃ _ : NonZero n ⦄ .⦃ _ : NonZero o ⦄
                   ⦃ _ : NonZero (n * o) ⦄ → m % (n * o) / o ≡ m / o % n
 m%[n*o]/o≡m/o%n m n o ⦃ _ ⦄ ⦃ _ ⦄ ⦃ n*o≢0 ⦄ = begin-equality
   m % (n * o) / o                   ≡⟨ /-congˡ (m%n≡m∸m/n*n m (n * o)) ⟩
-  (m ∸ (m / (n * o) * (n * o))) / o ≡˘⟨ cong (λ # → (m ∸ #) / o) (*-assoc (m / (n * o)) n o) ⟩
+  (m ∸ (m / (n * o) * (n * o))) / o ≡⟨ cong (λ # → (m ∸ #) / o) (*-assoc (m / (n * o)) n o) ⟨
   (m ∸ (m / (n * o) * n * o)) / o   ≡⟨ [m∸n*o]/o≡m/o∸n m (m / (n * o) * n) o ⟩
   m / o ∸ m / (n * o) * n           ≡⟨ cong (λ # → m / o ∸ # * n) (/-congʳ (*-comm n o)) ⟩
-  m / o ∸ m / (o * n) * n           ≡˘⟨ cong (λ # → m / o ∸ # * n) (m/n/o≡m/[n*o] m o n ) ⟩
-  m / o ∸ m / o / n * n             ≡˘⟨ m%n≡m∸m/n*n (m / o) n ⟩
+  m / o ∸ m / (o * n) * n           ≡⟨ cong (λ # → m / o ∸ # * n) (m/n/o≡m/[n*o] m o n ) ⟨
+  m / o ∸ m / o / n * n             ≡⟨ m%n≡m∸m/n*n (m / o) n ⟨
   m / o % n                         ∎
   where instance o*n≢0 = subst NonZero (*-comm n o) n*o≢0
 
@@ -420,9 +420,9 @@ m%n*o≡m*o%[n*o] : ∀ m n o .⦃ _ : NonZero n ⦄ ⦃ _ : NonZero (n * o) ⦄
 m%n*o≡m*o%[n*o] m n o ⦃ _ ⦄ ⦃ n*o≢0 ⦄ = begin-equality
   m % n * o                         ≡⟨ cong (_* o) (m%n≡m∸m/n*n m n) ⟩
   (m ∸ m / n * n) * o               ≡⟨ *-distribʳ-∸ o m (m / n * n) ⟩
-  m * o ∸ m / n * n * o             ≡˘⟨ cong (λ # → m * o ∸ # * n * o) (m*n/o*n≡m/o m o n) ⟩
+  m * o ∸ m / n * n * o             ≡⟨ cong (λ # → m * o ∸ # * n * o) (m*n/o*n≡m/o m o n) ⟨
   m * o ∸ m * o / (n * o) * n * o   ≡⟨ cong (m * o ∸_) (*-assoc (m * o / (n * o)) n o) ⟩
-  m * o ∸ m * o / (n * o) * (n * o) ≡˘⟨ m%n≡m∸m/n*n (m * o) (n * o) ⟩
+  m * o ∸ m * o / (n * o) * (n * o) ≡⟨ m%n≡m∸m/n*n (m * o) (n * o) ⟨
   m * o % (n * o)                   ∎
 
 [m*n+o]%[p*n]≡[m*n]%[p*n]+o : ∀ m {n o} p ⦃ _ : NonZero (p * n) ⦄ → o < n →
@@ -438,7 +438,7 @@ m%n*o≡m*o%[n*o] m n o ⦃ _ ⦄ ⦃ n*o≢0 ⦄ = begin-equality
 
   lem₁ : mn % pn ≤ p-1 * n
   lem₁ = begin
-    mn % pn     ≡˘⟨ m%n*o≡m*o%[n*o] m p n ⟩
+    mn % pn     ≡⟨ m%n*o≡m*o%[n*o] m p n ⟨
     (m % p) * n ≤⟨ *-monoˡ-≤ n (m<1+n⇒m≤n (m%n<n m p)) ⟩
     p-1 * n     ∎
 
@@ -470,7 +470,7 @@ _divMod_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} →
            DivMod dividend divisor
 m divMod n = result (m / n) (m mod n) $ begin-equality
   m                               ≡⟨  m≡m%n+[m/n]*n m n ⟩
-  m % n                + [m/n]*n  ≡˘⟨ cong (_+ [m/n]*n) (toℕ-fromℕ< [m%n]<n) ⟩
+  m % n                + [m/n]*n  ≡⟨ cong (_+ [m/n]*n) (toℕ-fromℕ< [m%n]<n) ⟨
   toℕ (fromℕ< [m%n]<n) + [m/n]*n  ∎
   where [m/n]*n = m / n * n ; [m%n]<n = m%n<n m n
 
diff --git a/src/Data/Nat/DivMod/WithK.agda b/src/Data/Nat/DivMod/WithK.agda
index 3f6047df60..10ef4c4f21 100644
--- a/src/Data/Nat/DivMod/WithK.agda
+++ b/src/Data/Nat/DivMod/WithK.agda
@@ -36,6 +36,6 @@ _divMod_ : (dividend divisor : ℕ) → .{{ NonZero divisor }} →
            DivMod dividend divisor
 m divMod n = result (m / n) (m mod n) $ ≡-erase $ begin
   m                                 ≡⟨ m≡m%n+[m/n]*n m n ⟩
-  m % n                  + [m/n]*n  ≡˘⟨ cong (_+ [m/n]*n) (toℕ-fromℕ<″ lemma″) ⟩
+  m % n                  + [m/n]*n  ≡⟨ cong (_+ [m/n]*n) (toℕ-fromℕ<″ lemma″) ⟨
   toℕ (fromℕ<″ _ lemma″) + [m/n]*n  ∎
   where [m/n]*n = m / n * n ; lemma″ = ≤″-erase (≤⇒≤″ (m%n<n m n))
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index 63037a49b2..bb30716e01 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -121,7 +121,7 @@ module ∣-Reasoning where
   private module Base = PreorderReasoning ∣-preorder
 
   open Base public
-    hiding (step-≈; step-≈˘; step-∼; step-≲)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)
     renaming (≲-go to ∣-go)
 
   open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∣-go public
@@ -293,9 +293,9 @@ m∣n*o⇒m/n∣o {m@.(p * n)} {n@(suc _)} {o} (divides-refl p) pn∣on = begin
   divides (a ∸ m / n * b) $ begin-equality
     m % n                   ≡⟨  m%n≡m∸m/n*n m n ⟩
     m ∸ m / n * n           ≡⟨  cong (λ v → m ∸ m / n * v) 1+n≡bd ⟩
-    m ∸ m / n * (b * d)     ≡˘⟨ cong (m ∸_) (*-assoc (m / n) b d) ⟩
+    m ∸ m / n * (b * d)     ≡⟨ cong (m ∸_) (*-assoc (m / n) b d) ⟨
     m  ∸ (m / n * b) * d    ≡⟨⟩
-    a * d ∸ (m / n * b) * d ≡˘⟨ *-distribʳ-∸ d a (m / n * b) ⟩
+    a * d ∸ (m / n * b) * d ≡⟨ *-distribʳ-∸ d a (m / n * b) ⟨
     (a ∸ m / n * b) * d     ∎
   where open ≤-Reasoning
 
diff --git a/src/Data/Nat/LCM.agda b/src/Data/Nat/LCM.agda
index 5dff2eb530..aead0fbf01 100644
--- a/src/Data/Nat/LCM.agda
+++ b/src/Data/Nat/LCM.agda
@@ -53,9 +53,9 @@ n∣lcm[m,n] : ∀ m n → n ∣ lcm m n
 n∣lcm[m,n] zero        n = n ∣0
 n∣lcm[m,n] m@(suc m-1) n = begin
   n                 ∣⟨  m∣m*n (m / gcd m n) ⟩
-  n * (m / gcd m n) ≡˘⟨ *-/-assoc n (gcd[m,n]∣m m n) ⟩
+  n * (m / gcd m n) ≡⟨ *-/-assoc n (gcd[m,n]∣m m n) ⟨
   n * m / gcd m n   ≡⟨  cong (_/ gcd m n) (*-comm n m) ⟩
-  m * n / gcd m n   ≡˘⟨ rearrange m n ⟩
+  m * n / gcd m n   ≡⟨ rearrange m n ⟨
   m * (n / gcd m n) ∎
   where open ∣-Reasoning; instance _ = gcd≢0ˡ {m} {n}
 
@@ -74,7 +74,7 @@ lcm-least {m@(suc _)} {n} {c} m∣c n∣c = P.subst (_∣ c) (sym (rearrange m n
   mn∣c*gcd : m * n ∣ c * gcd m n
   mn∣c*gcd = begin
     m * n               ∣⟨  gcd-greatest (P.subst (_∣ c * m) (*-comm n m) (*-monoˡ-∣ m n∣c)) (*-monoˡ-∣ n m∣c) ⟩
-    gcd (c * m) (c * n) ≡˘⟨ c*gcd[m,n]≡gcd[cm,cn] c m n ⟩
+    gcd (c * m) (c * n) ≡⟨ c*gcd[m,n]≡gcd[cm,cn] c m n ⟨
     c * gcd m n         ∎
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index 8e72b32be0..361df42b73 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -99,7 +99,7 @@ euclidsLemma m n {p@(suc (suc _))} p-prime p∣m*n = result
   p∣rmn r = begin
     p           ∣⟨ p∣m*n ⟩
     m * n       ∣⟨ n∣m*n r ⟩
-    r * (m * n) ≡˘⟨ *-assoc r m n ⟩
+    r * (m * n) ≡⟨ *-assoc r m n ⟨
     r * m * n   ∎
 
   result : p ∣ m ⊎ p ∣ n
@@ -114,14 +114,14 @@ euclidsLemma m n {p@(suc (suc _))} p-prime p∣m*n = result
   ... | Bézout.result 1 _ (Bézout.+- r s 1+sp≡rm) =
     inj₂ (flip ∣m+n∣m⇒∣n (n∣m*n*o s n) (begin
       p             ∣⟨ p∣rmn r ⟩
-      r * m * n     ≡˘⟨ cong (_* n) 1+sp≡rm ⟩
+      r * m * n     ≡⟨ cong (_* n) 1+sp≡rm ⟨
       n + s * p * n ≡⟨ +-comm n (s * p * n) ⟩
       s * p * n + n ∎))
 
   ... | Bézout.result 1 _ (Bézout.-+ r s 1+rm≡sp) =
     inj₂ (flip ∣m+n∣m⇒∣n (p∣rmn r) (begin
       p             ∣⟨ n∣m*n*o s n ⟩
-      s * p * n     ≡˘⟨ cong (_* n) 1+rm≡sp ⟩
+      s * p * n     ≡⟨ cong (_* n) 1+rm≡sp ⟨
       n + r * m * n ≡⟨ +-comm n (r * m * n) ⟩
       r * m * n + n ∎))
 
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 74350e1f6b..e304ed659f 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -498,7 +498,7 @@ module ≤-Reasoning where
     <-≤-trans
     ≤-<-trans
     public
-    hiding (step-≈; step-≈˘)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
 
 open ≤-Reasoning
 
@@ -918,7 +918,7 @@ m*n≡1⇒n≡1 m n eq = m*n≡1⇒m≡1 n m (trans (*-comm n m) eq)
 [m*n]*[o*p]≡[m*o]*[n*p] m n o p = begin-equality
   (m * n) * (o * p) ≡⟨  *-assoc m n (o * p) ⟩
   m * (n * (o * p)) ≡⟨  cong (m *_) (x∙yz≈y∙xz n o p) ⟩
-  m * (o * (n * p)) ≡˘⟨ *-assoc m o (n * p) ⟩
+  m * (o * (n * p)) ≡⟨ *-assoc m o (n * p) ⟨
   (m * o) * (n * p) ∎
   where open CommSemigroupProperties *-commutativeSemigroup
 
@@ -1325,15 +1325,15 @@ m⊔n≤m+n m n with ⊔-sel m n
 *-distribˡ-⊔ m zero o = sym (cong (_⊔ m * o) (*-zeroʳ m))
 *-distribˡ-⊔ m (suc n) zero = begin-equality
   m * (suc n ⊔ zero)         ≡⟨⟩
-  m * suc n                  ≡˘⟨ ⊔-identityʳ (m * suc n) ⟩
-  m * suc n ⊔ zero           ≡˘⟨ cong (m * suc n ⊔_) (*-zeroʳ m) ⟩
+  m * suc n                  ≡⟨ ⊔-identityʳ (m * suc n) ⟨
+  m * suc n ⊔ zero           ≡⟨ cong (m * suc n ⊔_) (*-zeroʳ m) ⟨
   m * suc n ⊔ m * zero       ∎
 *-distribˡ-⊔ m (suc n) (suc o) = begin-equality
   m * (suc n ⊔ suc o)        ≡⟨⟩
   m * suc (n ⊔ o)            ≡⟨ *-suc m (n ⊔ o) ⟩
   m + m * (n ⊔ o)            ≡⟨ cong (m +_) (*-distribˡ-⊔ m n o) ⟩
   m + (m * n ⊔ m * o)        ≡⟨ +-distribˡ-⊔ m (m * n) (m * o) ⟩
-  (m + m * n) ⊔ (m + m * o)  ≡˘⟨ cong₂ _⊔_ (*-suc m n) (*-suc m o) ⟩
+  (m + m * n) ⊔ (m + m * o)  ≡⟨ cong₂ _⊔_ (*-suc m n) (*-suc m o) ⟨
   (m * suc n) ⊔ (m * suc o)  ∎
 
 *-distribʳ-⊔ : _*_ DistributesOverʳ _⊔_
@@ -1434,20 +1434,20 @@ m⊓n≤m+n m n with ⊓-sel m n
   m * (0 ⊓ o)               ≡⟨⟩
   m * 0                     ≡⟨ *-zeroʳ m ⟩
   0                         ≡⟨⟩
-  0 ⊓ (m * o)               ≡˘⟨ cong (_⊓ (m * o)) (*-zeroʳ m) ⟩
+  0 ⊓ (m * o)               ≡⟨ cong (_⊓ (m * o)) (*-zeroʳ m) ⟨
   (m * 0) ⊓ (m * o)         ∎
 *-distribˡ-⊓ m (suc n) 0 = begin-equality
   m * (suc n ⊓ 0)           ≡⟨⟩
   m * 0                     ≡⟨ *-zeroʳ m ⟩
-  0                         ≡˘⟨ ⊓-zeroʳ (m * suc n) ⟩
-  (m * suc n) ⊓ 0           ≡˘⟨ cong (m * suc n ⊓_) (*-zeroʳ m) ⟩
+  0                         ≡⟨ ⊓-zeroʳ (m * suc n) ⟨
+  (m * suc n) ⊓ 0           ≡⟨ cong (m * suc n ⊓_) (*-zeroʳ m) ⟨
   (m * suc n) ⊓ (m * 0)     ∎
 *-distribˡ-⊓ m (suc n) (suc o) = begin-equality
   m * (suc n ⊓ suc o)       ≡⟨⟩
   m * suc (n ⊓ o)           ≡⟨ *-suc m (n ⊓ o) ⟩
   m + m * (n ⊓ o)           ≡⟨ cong (m +_) (*-distribˡ-⊓ m n o) ⟩
   m + (m * n) ⊓ (m * o)     ≡⟨ +-distribˡ-⊓ m (m * n) (m * o) ⟩
-  (m + m * n) ⊓ (m + m * o) ≡˘⟨ cong₂ _⊓_ (*-suc m n) (*-suc m o) ⟩
+  (m + m * n) ⊓ (m + m * o) ≡⟨ cong₂ _⊓_ (*-suc m n) (*-suc m o) ⟨
   (m * suc n) ⊓ (m * suc o) ∎
 
 *-distribʳ-⊓ : _*_ DistributesOverʳ _⊓_
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 915eb6e058..f0a7d11c96 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -314,7 +314,7 @@ normalize-injective-≃ m n c d eq = ℕ./-cancelʳ-≡
   (begin
     (m ℕ.* d) ℕ./ (gcd[m,c] ℕ.* gcd[n,d]) ≡⟨  ℕ./-*-interchange gcd[m,c]∣m gcd[n,d]∣d ⟩
     (m ℕ./ gcd[m,c]) ℕ.* (d ℕ./ gcd[n,d]) ≡⟨  cong₂ ℕ._*_ m/gcd[m,c]≡n/gcd[n,d] (sym c/gcd[m,c]≡d/gcd[n,d]) ⟩
-    (n ℕ./ gcd[n,d]) ℕ.* (c ℕ./ gcd[m,c]) ≡˘⟨ ℕ./-*-interchange gcd[n,d]∣n gcd[m,c]∣c ⟩
+    (n ℕ./ gcd[n,d]) ℕ.* (c ℕ./ gcd[m,c]) ≡⟨ ℕ./-*-interchange gcd[n,d]∣n gcd[m,c]∣c ⟨
     (n ℕ.* c) ℕ./ (gcd[n,d] ℕ.* gcd[m,c]) ≡⟨  ℕ./-congʳ (ℕ.*-comm gcd[n,d] gcd[m,c]) ⟩
     (n ℕ.* c) ℕ./ (gcd[m,c] ℕ.* gcd[n,d]) ∎)
   where
@@ -441,7 +441,7 @@ fromℚᵘ-cong : fromℚᵘ Preserves _≃ᵘ_ ⟶ _≡_
 fromℚᵘ-cong {p} {q} p≃q = toℚᵘ-injective (begin-equality
   toℚᵘ (fromℚᵘ p)  ≃⟨  toℚᵘ-fromℚᵘ p ⟩
   p                ≃⟨  p≃q ⟩
-  q                ≃˘⟨ toℚᵘ-fromℚᵘ q ⟩
+  q                ≃⟨ toℚᵘ-fromℚᵘ q ⟨
   toℚᵘ (fromℚᵘ q)  ∎)
   where open ℚᵘ.≤-Reasoning
 
@@ -749,7 +749,8 @@ module ≤-Reasoning where
     ≤-<-trans
     as Triple
 
-  open Triple public hiding (step-≈; step-≈˘)
+  open Triple public
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
 
   ≃-go : Trans _≃_ _IsRelatedTo_ _IsRelatedTo_
   ≃-go = Triple.≈-go ∘′ ≃⇒≡
@@ -854,7 +855,7 @@ toℚᵘ-homo-+ p@record{} q@record{} with +-nf p q ℤ.≟ 0ℤ
     ↥ (p + q) ℤ.* +-nf p q ℤ.* ↧+ᵘ p q            ≡⟨ cong (ℤ._* ↧+ᵘ p q) (↥-+ p q) ⟩
     ↥+ᵘ p q ℤ.* ↧+ᵘ p q                           ≡⟨ cong (↥+ᵘ p q ℤ.*_) (sym (↧-+ p q)) ⟩
     ↥+ᵘ p q ℤ.* (↧ (p + q) ℤ.* +-nf p q)          ≡⟨ x∙yz≈xy∙z (↥+ᵘ p q) _ _ ⟩
-    ↥+ᵘ p q ℤ.* ↧ (p + q)  ℤ.* +-nf p q           ≡˘⟨ cong (λ v → ↥+ᵘ p q ℤ.* v ℤ.* +-nf p q) (↧ᵘ-toℚᵘ (p + q)) ⟩
+    ↥+ᵘ p q ℤ.* ↧ (p + q)  ℤ.* +-nf p q           ≡⟨ cong (λ v → ↥+ᵘ p q ℤ.* v ℤ.* +-nf p q) (↧ᵘ-toℚᵘ (p + q)) ⟨
     ↥+ᵘ p q ℤ.* ↧ᵘ (toℚᵘ (p + q)) ℤ.* +-nf p q    ∎)
   where open ≡-Reasoning; open CommSemigroupProperties ℤ.*-commutativeSemigroup
 
@@ -1065,7 +1066,7 @@ toℚᵘ-homo-* p@record{} q@record{} with *-nf p q ℤ.≟ 0ℤ
   ↥ (p * q)         ℤ.* *-nf p q ℤ.* (↧ p ℤ.* ↧ q)     ≡⟨ cong (ℤ._* (↧ p ℤ.* ↧ q)) (↥-* p q) ⟩
   (↥ p ℤ.* ↥ q)     ℤ.* (↧ p ℤ.* ↧ q)                  ≡⟨ cong ((↥ p ℤ.* ↥ q) ℤ.*_) (sym (↧-* p q)) ⟩
   (↥ p ℤ.* ↥ q)     ℤ.* (↧ (p * q) ℤ.* *-nf p q)       ≡⟨ x∙yz≈xy∙z (↥ p ℤ.* ↥ q) _ _ ⟩
-  (↥ p ℤ.* ↥ q)     ℤ.* ↧ (p * q)  ℤ.* *-nf p q        ≡˘⟨ cong (λ v → (↥ p ℤ.* ↥ q) ℤ.* v ℤ.* *-nf p q) (↧ᵘ-toℚᵘ (p * q)) ⟩
+  (↥ p ℤ.* ↥ q)     ℤ.* ↧ (p * q)  ℤ.* *-nf p q        ≡⟨ cong (λ v → (↥ p ℤ.* ↥ q) ℤ.* v ℤ.* *-nf p q) (↧ᵘ-toℚᵘ (p * q)) ⟨
   (↥ p ℤ.* ↥ q)     ℤ.* ↧ᵘ (toℚᵘ (p * q)) ℤ.* *-nf p q ∎)
   where open ≡-Reasoning; open CommSemigroupProperties ℤ.*-commutativeSemigroup
 
@@ -1242,7 +1243,7 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 
 *-cancelʳ-≤-pos : ∀ r .{{_ : Positive r}} → p * r ≤ q * r → p ≤ q
 *-cancelʳ-≤-pos {p} {q} r pr≤qr = toℚᵘ-cancel-≤ (ℚᵘ.*-cancelʳ-≤-pos (toℚᵘ r) (begin
-  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃˘⟨ toℚᵘ-homo-* p r ⟩
+  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨
   toℚᵘ (p * r)        ≤⟨  toℚᵘ-mono-≤ pr≤qr ⟩
   toℚᵘ (q * r)        ≃⟨  toℚᵘ-homo-* q r ⟩
   toℚᵘ q ℚᵘ.* toℚᵘ r  ∎))
@@ -1255,7 +1256,7 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 *-monoʳ-≤-nonNeg r {p} {q} p≤q = toℚᵘ-cancel-≤ (begin
   toℚᵘ (p * r)        ≃⟨  toℚᵘ-homo-* p r ⟩
   toℚᵘ p ℚᵘ.* toℚᵘ r  ≤⟨  ℚᵘ.*-monoˡ-≤-nonNeg (toℚᵘ r) (toℚᵘ-mono-≤ p≤q) ⟩
-  toℚᵘ q ℚᵘ.* toℚᵘ r  ≃˘⟨ toℚᵘ-homo-* q r ⟩
+  toℚᵘ q ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* q r ⟨
   toℚᵘ (q * r)        ∎)
   where open ℚᵘ.≤-Reasoning
 
@@ -1266,7 +1267,7 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 *-monoʳ-≤-nonPos r {p} {q} p≤q = toℚᵘ-cancel-≤ (begin
   toℚᵘ (q * r)        ≃⟨ toℚᵘ-homo-* q r ⟩
   toℚᵘ q ℚᵘ.* toℚᵘ r  ≤⟨ ℚᵘ.*-monoˡ-≤-nonPos (toℚᵘ r) (toℚᵘ-mono-≤ p≤q) ⟩
-  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃˘⟨ toℚᵘ-homo-* p r ⟩
+  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨
   toℚᵘ (p * r)        ∎)
   where open ℚᵘ.≤-Reasoning
 
@@ -1275,9 +1276,9 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 
 *-cancelʳ-≤-neg : ∀ r .{{_ : Negative r}} → p * r ≤ q * r → p ≥ q
 *-cancelʳ-≤-neg {p} {q} r pr≤qr = toℚᵘ-cancel-≤ (ℚᵘ.*-cancelʳ-≤-neg _ (begin
-  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃˘⟨ toℚᵘ-homo-* p r ⟩
-  toℚᵘ (p * r)        ≤⟨  toℚᵘ-mono-≤ pr≤qr ⟩
-  toℚᵘ (q * r)        ≃⟨  toℚᵘ-homo-* q r ⟩
+  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨
+  toℚᵘ (p * r)        ≤⟨ toℚᵘ-mono-≤ pr≤qr ⟩
+  toℚᵘ (q * r)        ≃⟨ toℚᵘ-homo-* q r ⟩
   toℚᵘ q ℚᵘ.* toℚᵘ r  ∎))
   where open ℚᵘ.≤-Reasoning
 
@@ -1291,7 +1292,7 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 *-monoˡ-<-pos r {p} {q} p<q = toℚᵘ-cancel-< (begin-strict
   toℚᵘ (p * r)        ≃⟨ toℚᵘ-homo-* p r ⟩
   toℚᵘ p ℚᵘ.* toℚᵘ r  <⟨ ℚᵘ.*-monoˡ-<-pos (toℚᵘ r) (toℚᵘ-mono-< p<q) ⟩
-  toℚᵘ q ℚᵘ.* toℚᵘ r  ≃˘⟨ toℚᵘ-homo-* q r ⟩
+  toℚᵘ q ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* q r ⟨
   toℚᵘ (q * r)        ∎)
   where open ℚᵘ.≤-Reasoning
 
@@ -1300,7 +1301,7 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 
 *-cancelˡ-<-nonNeg : ∀ r .{{_ : NonNegative r}} → ∀ {p q} → r * p < r * q → p < q
 *-cancelˡ-<-nonNeg r {p} {q} rp<rq = toℚᵘ-cancel-< (ℚᵘ.*-cancelˡ-<-nonNeg (toℚᵘ r) (begin-strict
-  toℚᵘ r ℚᵘ.* toℚᵘ p  ≃˘⟨ toℚᵘ-homo-* r p ⟩
+  toℚᵘ r ℚᵘ.* toℚᵘ p  ≃⟨ toℚᵘ-homo-* r p ⟨
   toℚᵘ (r * p)        <⟨ toℚᵘ-mono-< rp<rq ⟩
   toℚᵘ (r * q)        ≃⟨ toℚᵘ-homo-* r q ⟩
   toℚᵘ r ℚᵘ.* toℚᵘ q  ∎))
@@ -1313,7 +1314,7 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 *-monoˡ-<-neg r {p} {q} p<q = toℚᵘ-cancel-< (begin-strict
   toℚᵘ (q * r)        ≃⟨ toℚᵘ-homo-* q r ⟩
   toℚᵘ q ℚᵘ.* toℚᵘ r  <⟨ ℚᵘ.*-monoˡ-<-neg (toℚᵘ r) (toℚᵘ-mono-< p<q) ⟩
-  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃˘⟨ toℚᵘ-homo-* p r ⟩
+  toℚᵘ p ℚᵘ.* toℚᵘ r  ≃⟨ toℚᵘ-homo-* p r ⟨
   toℚᵘ (p * r)        ∎)
   where open ℚᵘ.≤-Reasoning
 
@@ -1322,7 +1323,7 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
 
 *-cancelˡ-<-nonPos : ∀ r .{{_ : NonPositive r}} → r * p < r * q → p > q
 *-cancelˡ-<-nonPos {p} {q} r rp<rq = toℚᵘ-cancel-< (ℚᵘ.*-cancelˡ-<-nonPos (toℚᵘ r) (begin-strict
-  toℚᵘ r ℚᵘ.* toℚᵘ p  ≃˘⟨ toℚᵘ-homo-* r p ⟩
+  toℚᵘ r ℚᵘ.* toℚᵘ p  ≃⟨ toℚᵘ-homo-* r p ⟨
   toℚᵘ (r * p)        <⟨  toℚᵘ-mono-< rp<rq ⟩
   toℚᵘ (r * q)        ≃⟨  toℚᵘ-homo-* r q ⟩
   toℚᵘ r ℚᵘ.* toℚᵘ q  ∎))
@@ -1488,17 +1489,17 @@ mono-<-distrib-⊓ : ∀ {f} → f Preserves _<_ ⟶ _<_ →
 mono-<-distrib-⊓ {f} f-mono-< p q with <-cmp p q
 ... | tri< p<q p≢r  p≯q = begin
   f (p ⊓ q)  ≡⟨ cong f (p≤q⇒p⊓q≡p (<⇒≤ p<q)) ⟩
-  f p        ≡˘⟨ p≤q⇒p⊓q≡p (<⇒≤ (f-mono-< p<q)) ⟩
+  f p        ≡⟨ p≤q⇒p⊓q≡p (<⇒≤ (f-mono-< p<q)) ⟨
   f p ⊓ f q  ∎
   where open ≡-Reasoning
 ... | tri≈ p≮q refl p≯q = begin
   f (p ⊓ q)  ≡⟨ cong f (⊓-idem p) ⟩
-  f p        ≡˘⟨ ⊓-idem (f p) ⟩
+  f p        ≡⟨ ⊓-idem (f p) ⟨
   f p ⊓ f q  ∎
   where open ≡-Reasoning
 ... | tri> p≮q p≡r  p>q = begin
   f (p ⊓ q)  ≡⟨ cong f (p≥q⇒p⊓q≡q (<⇒≤ p>q)) ⟩
-  f q        ≡˘⟨ p≥q⇒p⊓q≡q (<⇒≤ (f-mono-< p>q)) ⟩
+  f q        ≡⟨ p≥q⇒p⊓q≡q (<⇒≤ (f-mono-< p>q)) ⟨
   f p ⊓ f q  ∎
   where open ≡-Reasoning
 
@@ -1507,17 +1508,17 @@ mono-<-distrib-⊔ : ∀ {f} → f Preserves _<_ ⟶ _<_ →
 mono-<-distrib-⊔ {f} f-mono-< p q with <-cmp p q
 ... | tri< p<q p≢r  p≯q = begin
   f (p ⊔ q)  ≡⟨ cong f (p≤q⇒p⊔q≡q (<⇒≤ p<q)) ⟩
-  f q        ≡˘⟨ p≤q⇒p⊔q≡q (<⇒≤ (f-mono-< p<q)) ⟩
+  f q        ≡⟨ p≤q⇒p⊔q≡q (<⇒≤ (f-mono-< p<q)) ⟨
   f p ⊔ f q  ∎
   where open ≡-Reasoning
 ... | tri≈ p≮q refl p≯q = begin
   f (p ⊔ q)  ≡⟨ cong f (⊔-idem p) ⟩
-  f q        ≡˘⟨ ⊔-idem (f p) ⟩
+  f q        ≡⟨ ⊔-idem (f p) ⟨
   f p ⊔ f q  ∎
   where open ≡-Reasoning
 ... | tri> p≮q p≡r  p>q = begin
   f (p ⊔ q)  ≡⟨ cong f (p≥q⇒p⊔q≡p (<⇒≤ p>q)) ⟩
-  f p        ≡˘⟨ p≥q⇒p⊔q≡p (<⇒≤ (f-mono-< p>q)) ⟩
+  f p        ≡⟨ p≥q⇒p⊔q≡p (<⇒≤ (f-mono-< p>q)) ⟨
   f p ⊔ f q  ∎
   where open ≡-Reasoning
 
@@ -1605,7 +1606,7 @@ toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl
 0≤∣p∣ p@record{} = *≤* (begin
   (↥ 0ℚ) ℤ.* (↧ ∣ p ∣)  ≡⟨ ℤ.*-zeroˡ (↧ ∣ p ∣) ⟩
   0ℤ                    ≤⟨ ℤ.+≤+ ℕ.z≤n ⟩
-  ↥ ∣ p ∣               ≡˘⟨ ℤ.*-identityʳ (↥ ∣ p ∣) ⟩
+  ↥ ∣ p ∣               ≡⟨ ℤ.*-identityʳ (↥ ∣ p ∣) ⟨
   ↥ ∣ p ∣ ℤ.* 1ℤ        ∎)
   where open ℤ.≤-Reasoning
 
@@ -1633,8 +1634,8 @@ toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl
   toℚᵘ ∣ p + q ∣                    ≃⟨ toℚᵘ-homo-∣-∣ (p + q) ⟩
   ℚᵘ.∣ toℚᵘ (p + q) ∣               ≃⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-+ p q) ⟩
   ℚᵘ.∣ toℚᵘ p ℚᵘ.+ toℚᵘ q ∣         ≤⟨ ℚᵘ.∣p+q∣≤∣p∣+∣q∣ (toℚᵘ p) (toℚᵘ q) ⟩
-  ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.+ ℚᵘ.∣ toℚᵘ q ∣  ≃˘⟨ ℚᵘ.+-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟩
-  toℚᵘ ∣ p ∣ ℚᵘ.+ toℚᵘ ∣ q ∣        ≃˘⟨ toℚᵘ-homo-+ ∣ p ∣ ∣ q ∣ ⟩
+  ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.+ ℚᵘ.∣ toℚᵘ q ∣  ≃⟨ ℚᵘ.+-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟨
+  toℚᵘ ∣ p ∣ ℚᵘ.+ toℚᵘ ∣ q ∣        ≃⟨ toℚᵘ-homo-+ ∣ p ∣ ∣ q ∣ ⟨
   toℚᵘ (∣ p ∣ + ∣ q ∣)              ∎)
   where open ℚᵘ.≤-Reasoning
 
@@ -1650,8 +1651,8 @@ toℚᵘ-homo-∣-∣ (mkℚ -[1+ _ ] _ _) = *≡* refl
   toℚᵘ ∣ p * q ∣                    ≃⟨ toℚᵘ-homo-∣-∣ (p * q) ⟩
   ℚᵘ.∣ toℚᵘ (p * q) ∣               ≃⟨ ℚᵘ.∣-∣-cong (toℚᵘ-homo-* p q) ⟩
   ℚᵘ.∣ toℚᵘ p ℚᵘ.* toℚᵘ q ∣         ≃⟨ ℚᵘ.∣p*q∣≃∣p∣*∣q∣ (toℚᵘ p) (toℚᵘ q) ⟩
-  ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.* ℚᵘ.∣ toℚᵘ q ∣  ≃˘⟨ ℚᵘ.*-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟩
-  toℚᵘ ∣ p ∣ ℚᵘ.* toℚᵘ ∣ q ∣        ≃˘⟨ toℚᵘ-homo-* ∣ p ∣ ∣ q ∣ ⟩
+  ℚᵘ.∣ toℚᵘ p ∣ ℚᵘ.* ℚᵘ.∣ toℚᵘ q ∣  ≃⟨ ℚᵘ.*-cong (toℚᵘ-homo-∣-∣ p) (toℚᵘ-homo-∣-∣ q) ⟨
+  toℚᵘ ∣ p ∣ ℚᵘ.* toℚᵘ ∣ q ∣        ≃⟨ toℚᵘ-homo-* ∣ p ∣ ∣ q ∣ ⟨
   toℚᵘ (∣ p ∣ * ∣ q ∣)              ∎)
   where open ℚᵘ.≤-Reasoning
 
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 6348a6ef46..dce11c3f21 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -160,7 +160,7 @@ module ≃-Reasoning = SetoidReasoning ≃-setoid
 
 p≃0⇒↥p≡0 : ∀ p → p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ
 p≃0⇒↥p≡0 p (*≡* eq) = begin
-  ↥ p          ≡˘⟨ ℤ.*-identityʳ (↥ p) ⟩
+  ↥ p          ≡⟨ ℤ.*-identityʳ (↥ p) ⟨
   ↥ p ℤ.* 1ℤ  ≡⟨ eq ⟩
   0ℤ           ∎
   where open ≡-Reasoning
@@ -180,14 +180,14 @@ neg-involutive p rewrite neg-involutive-≡ p = ≃-refl
 
 -‿cong : Congruent₁ _≃_ (-_)
 -‿cong {p@record{}} {q@record{}} (*≡* p≡q) = *≡* (begin
-  ↥(- p) ℤ.* ↧ q            ≡˘⟨ ℤ.*-identityˡ (ℤ.- (↥ p) ℤ.* ↧ q) ⟩
-  1ℤ ℤ.* (↥(- p) ℤ.* ↧ q)   ≡˘⟨ ℤ.*-assoc 1ℤ (↥ (- p)) (↧ q) ⟩
-  (1ℤ ℤ.* ℤ.-(↥ p)) ℤ.* ↧ q ≡˘⟨ cong (ℤ._* ↧ q) (ℤ.neg-distribʳ-* 1ℤ (↥ p)) ⟩
+  ↥(- p) ℤ.* ↧ q            ≡⟨ ℤ.*-identityˡ (ℤ.- (↥ p) ℤ.* ↧ q) ⟨
+  1ℤ ℤ.* (↥(- p) ℤ.* ↧ q)   ≡⟨ ℤ.*-assoc 1ℤ (↥ (- p)) (↧ q) ⟨
+  (1ℤ ℤ.* ℤ.-(↥ p)) ℤ.* ↧ q ≡⟨ cong (ℤ._* ↧ q) (ℤ.neg-distribʳ-* 1ℤ (↥ p)) ⟨
   ℤ.-(1ℤ ℤ.* ↥ p) ℤ.* ↧ q   ≡⟨  cong (ℤ._* ↧ q) (ℤ.neg-distribˡ-* 1ℤ (↥ p)) ⟩
   (-1ℤ ℤ.* ↥ p) ℤ.* ↧ q     ≡⟨  ℤ.*-assoc ℤ.-1ℤ (↥ p) (↧ q) ⟩
   -1ℤ ℤ.* (↥ p ℤ.* ↧ q)     ≡⟨  cong (ℤ.-1ℤ ℤ.*_) p≡q ⟩
-  -1ℤ ℤ.* (↥ q ℤ.* ↧ p)     ≡˘⟨ ℤ.*-assoc ℤ.-1ℤ (↥ q) (↧ p) ⟩
-  (-1ℤ ℤ.* ↥ q) ℤ.* ↧ p     ≡˘⟨ cong (ℤ._* ↧ p) (ℤ.neg-distribˡ-* 1ℤ (↥ q)) ⟩
+  -1ℤ ℤ.* (↥ q ℤ.* ↧ p)     ≡⟨ ℤ.*-assoc ℤ.-1ℤ (↥ q) (↧ p) ⟨
+  (-1ℤ ℤ.* ↥ q) ℤ.* ↧ p     ≡⟨ cong (ℤ._* ↧ p) (ℤ.neg-distribˡ-* 1ℤ (↥ q)) ⟨
   ℤ.-(1ℤ ℤ.* ↥ q) ℤ.* ↧ p   ≡⟨  cong (ℤ._* ↧ p) (ℤ.neg-distribʳ-* 1ℤ (↥ q)) ⟩
   (1ℤ ℤ.* ↥(- q)) ℤ.* ↧ p   ≡⟨  ℤ.*-assoc 1ℤ (ℤ.- (↥ q)) (↧ p) ⟩
   1ℤ ℤ.* (↥(- q) ℤ.* ↧ p)   ≡⟨  ℤ.*-identityˡ (↥ (- q) ℤ.* ↧ p) ⟩
@@ -196,7 +196,7 @@ neg-involutive p rewrite neg-involutive-≡ p = ≃-refl
 
 neg-mono-< : -_ Preserves  _<_ ⟶ _>_
 neg-mono-< {p@record{}} {q@record{}} (*<* p<q) = *<* $ begin-strict
-  ℤ.-  ↥ q ℤ.* ↧ p     ≡˘⟨ ℤ.neg-distribˡ-* (↥ q) (↧ p) ⟩
+  ℤ.-  ↥ q ℤ.* ↧ p     ≡⟨ ℤ.neg-distribˡ-* (↥ q) (↧ p) ⟨
   ℤ.- (↥ q ℤ.* ↧ p)    <⟨ ℤ.neg-mono-< p<q ⟩
   ℤ.- (↥ p ℤ.* ↧ q)    ≡⟨ ℤ.neg-distribˡ-* (↥ p) (↧ q) ⟩
   ↥ (- p) ℤ.* ↧ (- q)  ∎
@@ -204,10 +204,10 @@ neg-mono-< {p@record{}} {q@record{}} (*<* p<q) = *<* $ begin-strict
 
 neg-cancel-< : ∀ {p q} → - p < - q → q < p
 neg-cancel-< {p@record{}} {q@record{}} (*<* -↥p↧q<-↥q↧p) = *<* $ begin-strict
-  ↥ q ℤ.* ↧ p              ≡˘⟨ ℤ.neg-involutive (↥ q ℤ.* ↧ p) ⟩
+  ↥ q ℤ.* ↧ p              ≡⟨ ℤ.neg-involutive (↥ q ℤ.* ↧ p) ⟨
   ℤ.- ℤ.- (↥ q ℤ.* ↧ p)    ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ q) (↧ p)) ⟩
   ℤ.- ((ℤ.- ↥ q) ℤ.* ↧ p)  <⟨ ℤ.neg-mono-< -↥p↧q<-↥q↧p ⟩
-  ℤ.- ((ℤ.- ↥ p) ℤ.* ↧ q)  ≡˘⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ p) (↧ q)) ⟩
+  ℤ.- ((ℤ.- ↥ p) ℤ.* ↧ q)  ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ p) (↧ q)) ⟨
   ℤ.- ℤ.- (↥ p ℤ.* ↧ q)    ≡⟨ ℤ.neg-involutive (↥ p ℤ.* ↧ q) ⟩
   ↥ p ℤ.* ↧ q              ∎
   where open ℤ.≤-Reasoning
@@ -235,12 +235,12 @@ drop-*≤* (*≤* pq≤qp) = pq≤qp
   ℤ.*-cancelʳ-≤-pos (n₁ ℤ.* d₃) (n₃ ℤ.* d₁) d₂ $ begin
   (n₁  ℤ.* d₃) ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩
   n₁   ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩
-  n₁   ℤ.* (d₂ ℤ.* d₃) ≡˘⟨ ℤ.*-assoc n₁ d₂ d₃ ⟩
+  n₁   ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨
   (n₁  ℤ.* d₂) ℤ.* d₃  ≤⟨ ℤ.*-monoʳ-≤-nonNeg d₃ eq₁ ⟩
   (n₂  ℤ.* d₁) ℤ.* d₃  ≡⟨ cong (ℤ._* d₃) (ℤ.*-comm n₂ d₁) ⟩
   (d₁ ℤ.* n₂)  ℤ.* d₃  ≡⟨ ℤ.*-assoc d₁ n₂ d₃ ⟩
   d₁  ℤ.* (n₂  ℤ.* d₃) ≤⟨ ℤ.*-monoˡ-≤-nonNeg d₁ eq₂ ⟩
-  d₁  ℤ.* (n₃  ℤ.* d₂) ≡˘⟨ ℤ.*-assoc d₁ n₃ d₂ ⟩
+  d₁  ℤ.* (n₃  ℤ.* d₂) ≡⟨ ℤ.*-assoc d₁ n₃ d₂ ⟨
   (d₁ ℤ.* n₃)  ℤ.* d₂  ≡⟨ cong (ℤ._* d₂) (ℤ.*-comm d₁ n₃) ⟩
   (n₃  ℤ.* d₁) ℤ.* d₂  ∎ where open ℤ.≤-Reasoning
 
@@ -439,7 +439,7 @@ drop-*<* (*<* pq<qp) = pq<qp
     ↥ p ℤ.* ↧ p ℤ.+ ↥ p ℤ.* ↧ q
       <⟨ ℤ.+-monoʳ-< (↥ p ℤ.* ↧ p) p<q ⟩
     ↥ p ℤ.* ↧ p ℤ.+ ↥ q ℤ.* ↧ p
-      ≡˘⟨ ℤ.*-distribʳ-+ (↧ p) (↥ p) (↥ q) ⟩
+      ≡⟨ ℤ.*-distribʳ-+ (↧ p) (↥ p) (↥ q) ⟨
     (↥ p ℤ.+ ↥ q) ℤ.* ↧ p
       ≡⟨⟩
     ↥ m ℤ.* ↧ p ∎)
@@ -451,7 +451,7 @@ drop-*<* (*<* pq<qp) = pq<qp
     ↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ q
       <⟨ ℤ.+-monoˡ-< (↥ q ℤ.* ↧ q) p<q ⟩
     ↥ q ℤ.* ↧ p ℤ.+ ↥ q ℤ.* ↧ q
-      ≡˘⟨ ℤ.*-distribˡ-+ (↥ q) (↧ p) (↧ q) ⟩
+      ≡⟨ ℤ.*-distribˡ-+ (↥ q) (↧ p) (↧ q) ⟨
     ↥ q ℤ.* (↧ p ℤ.+ ↧ q)
       ≡⟨⟩
     ↥ q ℤ.* ↧ m ∎)
@@ -461,12 +461,12 @@ drop-*<* (*<* pq<qp) = pq<qp
   ℤ.*-cancelʳ-<-nonNeg _ $ begin-strict
   n₁ ℤ.*  d₃ ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩
   n₁ ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩
-  n₁ ℤ.* (d₂ ℤ.* d₃) ≡˘⟨ ℤ.*-assoc n₁ d₂ d₃ ⟩
+  n₁ ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨
   n₁ ℤ.*  d₂ ℤ.* d₃  ≤⟨ ℤ.*-monoʳ-≤-nonNeg (↧ r) p≤q ⟩
   n₂ ℤ.*  d₁ ℤ.* d₃  ≡⟨ cong (ℤ._* d₃) (ℤ.*-comm n₂ d₁) ⟩
   d₁ ℤ.*  n₂ ℤ.* d₃  ≡⟨ ℤ.*-assoc d₁ n₂ d₃ ⟩
   d₁ ℤ.* (n₂ ℤ.* d₃) <⟨ ℤ.*-monoˡ-<-pos (↧ p) q<r ⟩
-  d₁ ℤ.* (n₃ ℤ.* d₂) ≡˘⟨ ℤ.*-assoc d₁ n₃ d₂ ⟩
+  d₁ ℤ.* (n₃ ℤ.* d₂) ≡⟨ ℤ.*-assoc d₁ n₃ d₂ ⟨
   d₁ ℤ.*  n₃ ℤ.* d₂  ≡⟨ cong (ℤ._* d₂) (ℤ.*-comm d₁ n₃) ⟩
   n₃ ℤ.*  d₁ ℤ.* d₂  ∎
   where open ℤ.≤-Reasoning
@@ -477,12 +477,12 @@ drop-*<* (*<* pq<qp) = pq<qp
   ℤ.*-cancelʳ-<-nonNeg _ $ begin-strict
   n₁ ℤ.*  d₃ ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩
   n₁ ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩
-  n₁ ℤ.* (d₂ ℤ.* d₃) ≡˘⟨ ℤ.*-assoc n₁ d₂ d₃ ⟩
+  n₁ ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨
   n₁ ℤ.*  d₂ ℤ.* d₃  <⟨ ℤ.*-monoʳ-<-pos (↧ r) p<q ⟩
   n₂ ℤ.*  d₁ ℤ.* d₃  ≡⟨ cong (ℤ._* d₃) (ℤ.*-comm n₂ d₁) ⟩
   d₁ ℤ.*  n₂ ℤ.* d₃  ≡⟨ ℤ.*-assoc d₁ n₂ d₃ ⟩
   d₁ ℤ.* (n₂ ℤ.* d₃) ≤⟨ ℤ.*-monoˡ-≤-nonNeg (↧ p) q≤r ⟩
-  d₁ ℤ.* (n₃ ℤ.* d₂) ≡˘⟨ ℤ.*-assoc d₁ n₃ d₂ ⟩
+  d₁ ℤ.* (n₃ ℤ.* d₂) ≡⟨ ℤ.*-assoc d₁ n₃ d₂ ⟨
   d₁ ℤ.*  n₃ ℤ.* d₂  ≡⟨ cong (ℤ._* d₂) (ℤ.*-comm d₁ n₃) ⟩
   n₃ ℤ.*  d₁ ℤ.* d₂  ∎
   where open ℤ.≤-Reasoning
@@ -513,15 +513,15 @@ _>?_ = flip _<?_
   ℤ.*-cancelʳ-<-nonNeg _ $ begin-strict
   n₁ ℤ.*  d₃ ℤ.* d₂  ≡⟨ ℤ.*-assoc n₁ d₃ d₂ ⟩
   n₁ ℤ.* (d₃ ℤ.* d₂) ≡⟨ cong (n₁ ℤ.*_) (ℤ.*-comm d₃ d₂) ⟩
-  n₁ ℤ.* (d₂ ℤ.* d₃) ≡˘⟨ ℤ.*-assoc n₁ d₂ d₃ ⟩
+  n₁ ℤ.* (d₂ ℤ.* d₃) ≡⟨ ℤ.*-assoc n₁ d₂ d₃ ⟨
   n₁ ℤ.*  d₂ ℤ.* d₃  <⟨ ℤ.*-monoʳ-<-pos (↧ r) p<q ⟩
   n₂ ℤ.*  d₁ ℤ.* d₃  ≡⟨ ℤ.*-assoc n₂ d₁ d₃ ⟩
   n₂ ℤ.* (d₁ ℤ.* d₃) ≡⟨ cong (n₂ ℤ.*_) (ℤ.*-comm d₁ d₃) ⟩
-  n₂ ℤ.* (d₃ ℤ.* d₁) ≡˘⟨ ℤ.*-assoc n₂ d₃ d₁ ⟩
+  n₂ ℤ.* (d₃ ℤ.* d₁) ≡⟨ ℤ.*-assoc n₂ d₃ d₁ ⟨
   n₂ ℤ.*  d₃ ℤ.* d₁  ≡⟨ cong (ℤ._* d₁) q≡r ⟩
   n₃ ℤ.*  d₂ ℤ.* d₁  ≡⟨ ℤ.*-assoc n₃ d₂ d₁ ⟩
   n₃ ℤ.* (d₂ ℤ.* d₁) ≡⟨ cong (n₃ ℤ.*_) (ℤ.*-comm d₂ d₁) ⟩
-  n₃ ℤ.* (d₁ ℤ.* d₂) ≡˘⟨ ℤ.*-assoc n₃ d₁ d₂ ⟩
+  n₃ ℤ.* (d₁ ℤ.* d₂) ≡⟨ ℤ.*-assoc n₃ d₁ d₂ ⟨
   n₃ ℤ.*  d₁ ℤ.* d₂  ∎
   where open ℤ.≤-Reasoning
         n₁ = ↥ p; n₂ = ↥ q; n₃ = ↥ r; d₁ = ↧ p; d₂ = ↧ q; d₃ = ↧ r
@@ -605,9 +605,10 @@ module ≤-Reasoning where
     as Triple
 
   open Triple public
-    hiding (step-≈; step-≈˘)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
+    renaming (≈-go to ≃-go)
 
-  open ≃-syntax _IsRelatedTo_ _IsRelatedTo_  Triple.≈-go ≃-sym public
+  open ≃-syntax _IsRelatedTo_ _IsRelatedTo_ ≃-go ≃-sym public
 
 ------------------------------------------------------------------------
 -- Properties of ↥_/↧_
@@ -760,14 +761,14 @@ neg⇒nonZero (mkℚᵘ (-[1+ _ ]) _) = _
 +-inverseˡ : LeftInverse _≃_ 0ℚᵘ -_ _+_
 +-inverseˡ p@record{} = *≡* let n = ↥ p; d = ↧ p in
   ((ℤ.- n) ℤ.* d ℤ.+ n ℤ.* d) ℤ.* 1ℤ ≡⟨ ℤ.*-identityʳ ((ℤ.- n) ℤ.* d ℤ.+ n ℤ.* d) ⟩
-  (ℤ.- n) ℤ.* d ℤ.+ n ℤ.* d          ≡˘⟨ cong (ℤ._+ (n ℤ.* d)) (ℤ.neg-distribˡ-* n d) ⟩
+  (ℤ.- n) ℤ.* d ℤ.+ n ℤ.* d          ≡⟨ cong (ℤ._+ (n ℤ.* d)) (ℤ.neg-distribˡ-* n d) ⟨
   ℤ.- (n ℤ.* d) ℤ.+ n ℤ.* d          ≡⟨ ℤ.+-inverseˡ (n ℤ.* d) ⟩
   0ℤ                                 ∎ where open ≡-Reasoning
 
 +-inverseʳ : RightInverse _≃_ 0ℚᵘ -_ _+_
 +-inverseʳ p@record{} = *≡* let n = ↥ p; d = ↧ p in
   (n ℤ.* d ℤ.+ (ℤ.- n) ℤ.* d) ℤ.* 1ℤ ≡⟨ ℤ.*-identityʳ (n ℤ.* d ℤ.+ (ℤ.- n) ℤ.* d) ⟩
-  n ℤ.* d ℤ.+ (ℤ.- n) ℤ.* d          ≡˘⟨ cong (λ n+d → n ℤ.* d ℤ.+ n+d) (ℤ.neg-distribˡ-* n d) ⟩
+  n ℤ.* d ℤ.+ (ℤ.- n) ℤ.* d          ≡⟨ cong (λ n+d → n ℤ.* d ℤ.+ n+d) (ℤ.neg-distribˡ-* n d) ⟨
   n ℤ.* d ℤ.+ ℤ.- (n ℤ.* d)          ≡⟨ ℤ.+-inverseʳ (n ℤ.* d) ⟩
   0ℤ                                 ∎ where open ≡-Reasoning
 
@@ -776,9 +777,9 @@ neg⇒nonZero (mkℚᵘ (-[1+ _ ]) _) = _
 
 +-cancelˡ : ∀ {r p q} → r + p ≃ r + q → p ≃ q
 +-cancelˡ {r} {p} {q} r+p≃r+q = begin-equality
-  p            ≃˘⟨ +-identityʳ p ⟩
-  p + 0ℚᵘ      ≃˘⟨ +-congʳ p (+-inverseʳ r) ⟩
-  p + (r - r)  ≃˘⟨ +-assoc p r (- r) ⟩
+  p            ≃⟨ +-identityʳ p ⟨
+  p + 0ℚᵘ      ≃⟨ +-congʳ p (+-inverseʳ r) ⟨
+  p + (r - r)  ≃⟨ +-assoc p r (- r) ⟨
   (p + r) - r  ≃⟨ +-congˡ (- r) (+-comm p r) ⟩
   (r + p) - r  ≃⟨ +-congˡ (- r) r+p≃r+q ⟩
   (r + q) - r  ≃⟨ +-congˡ (- r) (+-comm r q) ⟩
@@ -915,26 +916,26 @@ p≃q⇒p-q≃0 p q p≃q = begin-equality
 
 p-q≃0⇒p≃q : ∀ p q → p - q ≃ 0ℚᵘ → p ≃ q
 p-q≃0⇒p≃q p q p-q≃0 = begin-equality
-  p             ≡˘⟨ +-identityʳ-≡ p ⟩
+  p             ≡⟨ +-identityʳ-≡ p ⟨
   p + 0ℚᵘ       ≃⟨ +-congʳ p (≃-sym (+-inverseˡ q)) ⟩
-  p + (- q + q) ≡˘⟨ +-assoc-≡ p (- q) q ⟩
+  p + (- q + q) ≡⟨ +-assoc-≡ p (- q) q ⟨
   (p - q) + q   ≃⟨ +-congˡ q p-q≃0 ⟩
   0ℚᵘ + q       ≡⟨ +-identityˡ-≡ q ⟩
   q             ∎ where open ≤-Reasoning
 
 neg-mono-≤ : -_ Preserves _≤_ ⟶ _≥_
 neg-mono-≤ {p@record{}} {q@record{}} (*≤* p≤q) = *≤* $ begin
-  ℤ.- ↥ q ℤ.* ↧ p   ≡˘⟨ ℤ.neg-distribˡ-* (↥ q) (↧ p) ⟩
+  ℤ.- ↥ q ℤ.* ↧ p   ≡⟨ ℤ.neg-distribˡ-* (↥ q) (↧ p) ⟨
   ℤ.- (↥ q ℤ.* ↧ p) ≤⟨ ℤ.neg-mono-≤ p≤q ⟩
   ℤ.- (↥ p ℤ.* ↧ q) ≡⟨ ℤ.neg-distribˡ-* (↥ p) (↧ q) ⟩
   ℤ.- ↥ p ℤ.* ↧ q   ∎ where open ℤ.≤-Reasoning
 
 neg-cancel-≤ : ∀ {p q} → - p ≤ - q → q ≤ p
 neg-cancel-≤ {p@record{}} {q@record{}} (*≤* -↥p↧q≤-↥q↧p) = *≤* $ begin
-  ↥ q ℤ.* ↧ p             ≡˘⟨ ℤ.neg-involutive (↥ q ℤ.* ↧ p) ⟩
+  ↥ q ℤ.* ↧ p             ≡⟨ ℤ.neg-involutive (↥ q ℤ.* ↧ p) ⟨
   ℤ.- ℤ.- (↥ q ℤ.* ↧ p)   ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ q) (↧ p)) ⟩
   ℤ.- ((ℤ.- ↥ q) ℤ.* ↧ p) ≤⟨ ℤ.neg-mono-≤ -↥p↧q≤-↥q↧p ⟩
-  ℤ.- ((ℤ.- ↥ p) ℤ.* ↧ q) ≡˘⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ p) (↧ q)) ⟩
+  ℤ.- ((ℤ.- ↥ p) ℤ.* ↧ q) ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ p) (↧ q)) ⟨
   ℤ.- ℤ.- (↥ p ℤ.* ↧ q)   ≡⟨ ℤ.neg-involutive (↥ p ℤ.* ↧ q) ⟩
   ↥ p ℤ.* ↧ q             ∎
   where
@@ -948,9 +949,9 @@ p≤q⇒p-q≤0 {p} {q} p≤q = begin
 
 p-q≤0⇒p≤q : ∀ {p q} → p - q ≤ 0ℚᵘ → p ≤ q
 p-q≤0⇒p≤q {p} {q} p-q≤0 = begin
-  p             ≡˘⟨ +-identityʳ-≡ p ⟩
+  p             ≡⟨ +-identityʳ-≡ p ⟨
   p + 0ℚᵘ       ≃⟨ +-congʳ p (≃-sym (+-inverseˡ q)) ⟩
-  p + (- q + q) ≡˘⟨ +-assoc-≡ p (- q) q ⟩
+  p + (- q + q) ≡⟨ +-assoc-≡ p (- q) q ⟨
   (p - q) + q   ≤⟨ +-monoˡ-≤ q p-q≤0 ⟩
   0ℚᵘ + q       ≡⟨ +-identityˡ-≡ q ⟩
   q             ∎ where open ≤-Reasoning
@@ -963,7 +964,7 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
 
 0≤q-p⇒p≤q : ∀ {p q} → 0ℚᵘ ≤ q - p → p ≤ q
 0≤q-p⇒p≤q {p} {q} 0≤p-q = begin
-  p             ≡˘⟨ +-identityˡ-≡ p ⟩
+  p             ≡⟨ +-identityˡ-≡ p ⟨
   0ℚᵘ + p       ≤⟨ +-monoˡ-≤ p 0≤p-q ⟩
   q - p + p     ≡⟨ +-assoc-≡ q (- p) p ⟩
   q + (- p + p) ≃⟨ +-congʳ q (+-inverseˡ p) ⟩
@@ -1125,7 +1126,7 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
   (n ℕ.+ d ℕ.* suc n)       ≡⟨ cong (n ℕ.+_) (ℕ.*-suc d n) ⟩
   (n ℕ.+ (d ℕ.+ d ℕ.* n))   ≡⟨ solve 2 (λ n d → n :+ (d :+ d :* n) := d :+ (n :+ n :* d)) refl n d ⟩
   (d ℕ.+ (n ℕ.+ n ℕ.* d))   ≡⟨ cong (d ℕ.+_) (sym (ℕ.*-suc n d)) ⟩
-  d ℕ.+ n ℕ.* suc d         ≡˘⟨ ℕ.+-identityʳ _ ⟩
+  d ℕ.+ n ℕ.* suc d         ≡⟨ ℕ.+-identityʳ _ ⟨
   d ℕ.+ n ℕ.* suc d ℕ.+ 0   ∎
   where open ≡-Reasoning; open ℕ-solver
 
@@ -1149,8 +1150,8 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
 
 invertible⇒≄ : Invertible _≃_ 1ℚᵘ _*_ (p - q) → p ≄ q
 invertible⇒≄ {p} {q} (1/p-q , 1/x*x≃1 , x*1/x≃1) p≃q = 0≄1 (begin
-  0ℚᵘ             ≈˘⟨ *-zeroˡ 1/p-q ⟩
-  0ℚᵘ * 1/p-q     ≈˘⟨ *-congʳ (p≃q⇒p-q≃0 p q p≃q) ⟩
+  0ℚᵘ             ≈⟨ *-zeroˡ 1/p-q ⟨
+  0ℚᵘ * 1/p-q     ≈⟨ *-congʳ (p≃q⇒p-q≃0 p q p≃q) ⟨
   (p - q) * 1/p-q ≈⟨ x*1/x≃1 ⟩
   1ℚᵘ             ∎)
   where open ≃-Reasoning
@@ -1194,9 +1195,9 @@ neg-distribʳ-* p@record{} q@record{} =
   (↥ ((ℤ.+ p ℤ.* q) / (p ℕ.* r))) ℤ.* (↧ (q / r)) ≡⟨  cong (ℤ._* ↧ (q / r)) (↥[n/d]≡n (ℤ.+ p ℤ.* q) (p ℕ.* r)) ⟩
   (ℤ.+ p ℤ.* q) ℤ.* (↧ (q / r))                   ≡⟨  cong ((ℤ.+ p ℤ.* q) ℤ.*_) (↧[n/d]≡d q r) ⟩
   (ℤ.+ p ℤ.* q) ℤ.* ℤ.+ r                         ≡⟨  xy∙z≈y∙xz (ℤ.+ p) q (ℤ.+ r) ⟩
-  (q ℤ.* (ℤ.+ p ℤ.* ℤ.+ r))                       ≡˘⟨ cong (ℤ._* (ℤ.+ p ℤ.* ℤ.+ r)) (↥[n/d]≡n q r) ⟩
-  (↥ (q / r)) ℤ.* (ℤ.+ p ℤ.* ℤ.+ r)               ≡˘⟨  cong (↥ (q / r) ℤ.*_) (ℤ.pos-* p r) ⟩
-  (↥ (q / r)) ℤ.* (ℤ.+ (p ℕ.* r))                 ≡˘⟨ cong (↥ (q / r) ℤ.*_) (↧[n/d]≡d (ℤ.+ p ℤ.* q) (p ℕ.* r)) ⟩
+  (q ℤ.* (ℤ.+ p ℤ.* ℤ.+ r))                       ≡⟨ cong (ℤ._* (ℤ.+ p ℤ.* ℤ.+ r)) (↥[n/d]≡n q r) ⟨
+  (↥ (q / r)) ℤ.* (ℤ.+ p ℤ.* ℤ.+ r)               ≡⟨  cong (↥ (q / r) ℤ.*_) (ℤ.pos-* p r) ⟨
+  (↥ (q / r)) ℤ.* (ℤ.+ (p ℕ.* r))                 ≡⟨ cong (↥ (q / r) ℤ.*_) (↧[n/d]≡d (ℤ.+ p ℤ.* q) (p ℕ.* r)) ⟨
   (↥ (q / r)) ℤ.* (↧ ((ℤ.+ p ℤ.* q) / (p ℕ.* r))) ∎)
   where open ℤ.≤-Reasoning
 
@@ -1233,11 +1234,11 @@ private
 
 *-cancelʳ-≤-neg : ∀ r .{{_ : Negative r}} → p * r ≤ q * r → q ≤ p
 *-cancelʳ-≤-neg {p} {q} r@(mkℚᵘ -[1+ _ ] _) pr≤qr = neg-cancel-≤ (*-cancelʳ-≤-pos (- r) (begin
-  - p * - r    ≃˘⟨ neg-distribˡ-* p (- r) ⟩
-  - (p * - r)  ≃˘⟨ -‿cong (neg-distribʳ-* p r) ⟩
+  - p * - r    ≃⟨ neg-distribˡ-* p (- r) ⟨
+  - (p * - r)  ≃⟨ -‿cong (neg-distribʳ-* p r) ⟨
   - - (p * r)  ≃⟨ neg-involutive (p * r) ⟩
   p * r        ≤⟨ pr≤qr ⟩
-  q * r        ≃˘⟨ neg-involutive (q * r) ⟩
+  q * r        ≃⟨ neg-involutive (q * r) ⟨
   - - (q * r)  ≃⟨ -‿cong (neg-distribʳ-* q r) ⟩
   - (q * - r)  ≃⟨ neg-distribˡ-* q (- r) ⟩
   - q * - r    ∎))
@@ -1249,7 +1250,7 @@ private
 *-monoˡ-≤-nonNeg : ∀ r .{{_ : NonNegative r}} → (_* r) Preserves _≤_ ⟶ _≤_
 *-monoˡ-≤-nonNeg r@(mkℚᵘ (ℤ.+ n) _) {p@record{}} {q@record{}} (*≤* x<y) = *≤* $ begin
   ↥ p ℤ.* ↥ r ℤ.* (↧ q   ℤ.* ↧ r)  ≡⟨  reorder₂ (↥ p) _ _ _ ⟩
-  l₁          ℤ.* (ℤ.+ n ℤ.* ↧ r)  ≡˘⟨  cong (l₁ ℤ.*_) (ℤ.pos-* n _) ⟩
+  l₁          ℤ.* (ℤ.+ n ℤ.* ↧ r)  ≡⟨  cong (l₁ ℤ.*_) (ℤ.pos-* n _) ⟨
   l₁          ℤ.* ℤ.+ (n ℕ.* ↧ₙ r) ≤⟨  ℤ.*-monoʳ-≤-nonNeg (ℤ.+ (n ℕ.* ↧ₙ r)) x<y ⟩
   l₂          ℤ.* ℤ.+ (n ℕ.* ↧ₙ r) ≡⟨ cong (l₂ ℤ.*_) (ℤ.pos-* n _) ⟩
   l₂          ℤ.* (ℤ.+ n ℤ.* ↧ r)  ≡⟨  reorder₂ (↥ q) _ _ _ ⟩
@@ -1269,10 +1270,10 @@ private
 
 *-monoˡ-≤-nonPos : ∀ r .{{_ : NonPositive r}} → (_* r) Preserves _≤_ ⟶ _≥_
 *-monoˡ-≤-nonPos r {p} {q} p≤q = begin
-  q * r        ≃˘⟨ neg-involutive (q * r) ⟩
+  q * r        ≃⟨ neg-involutive (q * r) ⟨
   - - (q * r)  ≃⟨  -‿cong (neg-distribʳ-* q r) ⟩
   - (q * - r)  ≤⟨  neg-mono-≤ (*-monoˡ-≤-nonNeg (- r) {{ -r≥0}} p≤q) ⟩
-  - (p * - r)  ≃˘⟨ -‿cong (neg-distribʳ-* p r) ⟩
+  - (p * - r)  ≃⟨ -‿cong (neg-distribʳ-* p r) ⟨
   - - (p * r)  ≃⟨  neg-involutive (p * r) ⟩
   p * r        ∎
   where open ≤-Reasoning; -r≥0 = nonNegative (neg-mono-≤ (nonPositive⁻¹ r))
@@ -1287,7 +1288,7 @@ private
 *-monoˡ-<-pos r@record{} {p@record{}} {q@record{}} (*<* x<y) = *<* $ begin-strict
   ↥ p ℤ.*  ↥ r ℤ.* (↧ q  ℤ.* ↧ r) ≡⟨ reorder₁ (↥ p) _ _ _ ⟩
   ↥ p ℤ.*  ↧ q ℤ.*  ↥ r  ℤ.* ↧ r  <⟨ ℤ.*-monoʳ-<-pos (↧ r) (ℤ.*-monoʳ-<-pos (↥ r) x<y) ⟩
-  ↥ q ℤ.*  ↧ p ℤ.*  ↥ r  ℤ.* ↧ r  ≡˘⟨ reorder₁ (↥ q) _ _ _ ⟩
+  ↥ q ℤ.*  ↧ p ℤ.*  ↥ r  ℤ.* ↧ r  ≡⟨ reorder₁ (↥ q) _ _ _ ⟨
   ↥ q ℤ.*  ↥ r ℤ.* (↧ p  ℤ.* ↧ r) ∎ where open ℤ.≤-Reasoning
 
 *-monoʳ-<-pos : ∀ r .{{_ : Positive r}} → (r *_) Preserves _<_ ⟶ _<_
@@ -1314,10 +1315,10 @@ private
 
 *-monoˡ-<-neg : ∀ r .{{_ :  Negative r}} → (_* r) Preserves _<_ ⟶ _>_
 *-monoˡ-<-neg r {p} {q} p<q = begin-strict
-  q * r        ≃˘⟨ neg-involutive (q * r) ⟩
+  q * r        ≃⟨ neg-involutive (q * r) ⟨
   - - (q * r)  ≃⟨ -‿cong (neg-distribʳ-* q r) ⟩
   - (q * - r)  <⟨ neg-mono-< (*-monoˡ-<-pos (- r) {{ -r>0}} p<q) ⟩
-  - (p * - r)  ≃˘⟨ -‿cong (neg-distribʳ-* p r) ⟩
+  - (p * - r)  ≃⟨ -‿cong (neg-distribʳ-* p r) ⟨
   - - (p * r)  ≃⟨ neg-involutive (p * r) ⟩
   p * r        ∎
   where open ≤-Reasoning; -r>0 = positive (neg-mono-< (negative⁻¹ r))
@@ -1328,7 +1329,7 @@ private
 *-cancelˡ-<-nonPos : ∀ r .{{_ : NonPositive r}} → r * p < r * q → q < p
 *-cancelˡ-<-nonPos {p} {q} r rp<rq =
   *-cancelˡ-<-nonNeg (- r) {{ -r≥0}} $ begin-strict
-    - r * q    ≃˘⟨ neg-distribˡ-* r q ⟩
+    - r * q    ≃⟨ neg-distribˡ-* r q ⟨
     - (r * q)  <⟨ neg-mono-< rp<rq ⟩
     - (r * p)  ≃⟨ neg-distribˡ-* r p ⟩
     - r * p    ∎
@@ -1495,8 +1496,8 @@ p>1⇒1/p<1 {p} p>1 = lemma′ p (p>1⇒p≢0 p>1) p>1
 1/-antimono-≤-pos : ∀ {p q} .{{_ : Positive p}} .{{_ : Positive q}} →
                     p ≤ q → (1/ q) {{pos⇒nonZero q}} ≤ (1/ p) {{pos⇒nonZero p}}
 1/-antimono-≤-pos {p} {q} p≤q = begin
-  1/q              ≃˘⟨ *-identityˡ 1/q ⟩
-  1ℚᵘ * 1/q        ≃˘⟨ *-congʳ (*-inverseˡ p) ⟩
+  1/q              ≃⟨ *-identityˡ 1/q ⟨
+  1ℚᵘ * 1/q        ≃⟨ *-congʳ (*-inverseˡ p) ⟨
   (1/p * p) * 1/q  ≤⟨  *-monoˡ-≤-nonNeg 1/q (*-monoʳ-≤-nonNeg 1/p p≤q) ⟩
   (1/p * q) * 1/q  ≃⟨  *-assoc 1/p q 1/q ⟩
   1/p * (q * 1/q)  ≃⟨  *-congˡ {1/p} (*-inverseʳ q) ⟩
@@ -1775,7 +1776,7 @@ pos⊔pos⇒pos p q = positive (⊔-mono-< (positive⁻¹ p) (positive⁻¹ q))
   ℤ.- ℤ.- (↥ ∣ p ∣ ℤ.* ↧ q)  ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ ∣ p ∣) (↧ q)) ⟩
   ℤ.- (↥ p ℤ.* ↧ q)          ≡⟨ cong ℤ.-_ ↥p↧q≡↥q↧p ⟩
   ℤ.- (↥ q ℤ.* ↧ p)          ≡⟨ cong ℤ.-_ (ℤ.neg-distribˡ-* (↥ ∣ q ∣) (↧ p)) ⟩
-  ℤ.- ℤ.- (↥ ∣ q ∣ ℤ.* ↧ p)  ≡˘⟨ ℤ.neg-involutive _ ⟩
+  ℤ.- ℤ.- (↥ ∣ q ∣ ℤ.* ↧ p)  ≡⟨ ℤ.neg-involutive _ ⟨
   ↥ ∣ q ∣ ℤ.* ↧ p            ∎)
   where open ≡-Reasoning
 
@@ -1807,7 +1808,7 @@ pos⊔pos⇒pos p q = positive (⊔-mono-< (positive⁻¹ p) (positive⁻¹ q))
 ∣p∣≡p⇒0≤p {mkℚᵘ (ℤ.+ n) d-1} ∣p∣≡p = *≤* (begin
   0ℤ ℤ.* +[1+ d-1 ]  ≡⟨ ℤ.*-zeroˡ (ℤ.+ d-1) ⟩
   0ℤ                 ≤⟨ ℤ.+≤+ ℕ.z≤n ⟩
-  ℤ.+ n              ≡˘⟨ ℤ.*-identityʳ (ℤ.+ n) ⟩
+  ℤ.+ n              ≡⟨ ℤ.*-identityʳ (ℤ.+ n) ⟨
   ℤ.+ n ℤ.* 1ℤ       ∎)
   where open ℤ.≤-Reasoning
 
@@ -1826,7 +1827,7 @@ pos⊔pos⇒pos p q = positive (⊔-mono-< (positive⁻¹ p) (positive⁻¹ q))
   ↥ ∣ (↥p↧q ℤ.+ ↥q↧p) / ↧p↧q ∣ ℤ.* ℤ.+ ↧p↧q        ≡⟨⟩
   ↥ (ℤ.+ ℤ.∣ ↥p↧q ℤ.+ ↥q↧p ∣ / ↧p↧q) ℤ.* ℤ.+ ↧p↧q  ≡⟨ cong (ℤ._* ℤ.+ ↧p↧q) (↥[n/d]≡n (ℤ.+ ℤ.∣ ↥p↧q ℤ.+ ↥q↧p ∣) ↧p↧q) ⟩
   ℤ.+ ℤ.∣ ↥p↧q ℤ.+ ↥q↧p ∣ ℤ.* ℤ.+ ↧p↧q             ≤⟨ ℤ.*-monoʳ-≤-nonNeg (ℤ.+ ↧p↧q) (ℤ.+≤+ (ℤ.∣i+j∣≤∣i∣+∣j∣ ↥p↧q ↥q↧p)) ⟩
-  (ℤ.+ ℤ.∣ ↥p↧q ∣ ℤ.+ ℤ.+ ℤ.∣ ↥q↧p ∣) ℤ.* ℤ.+ ↧p↧q ≡˘⟨ cong₂ (λ h₁ h₂ → (h₁ ℤ.+ h₂) ℤ.* ℤ.+ ↧p↧q) ∣↥p∣↧q≡∣↥p↧q∣ ∣↥q∣↧p≡∣↥q↧p∣ ⟩
+  (ℤ.+ ℤ.∣ ↥p↧q ∣ ℤ.+ ℤ.+ ℤ.∣ ↥q↧p ∣) ℤ.* ℤ.+ ↧p↧q ≡⟨ cong₂ (λ h₁ h₂ → (h₁ ℤ.+ h₂) ℤ.* ℤ.+ ↧p↧q) ∣↥p∣↧q≡∣↥p↧q∣ ∣↥q∣↧p≡∣↥q↧p∣ ⟨
   (∣↥p∣↧q ℤ.+ ∣↥q∣↧p) ℤ.* ℤ.+ ↧p↧q                 ≡⟨⟩
   (↥∣p∣↧q ℤ.+ ↥∣q∣↧p) ℤ.* ℤ.+ ↧p↧q                 ≡⟨ cong (ℤ._* ℤ.+ ↧p↧q) (↥[n/d]≡n (↥∣p∣↧q ℤ.+ ↥∣q∣↧p) ↧p↧q) ⟩
   ↥ ((↥∣p∣↧q ℤ.+ ↥∣q∣↧p) / ↧p↧q) ℤ.* ℤ.+ ↧p↧q      ≡⟨⟩
@@ -1843,9 +1844,9 @@ pos⊔pos⇒pos p q = positive (⊔-mono-< (positive⁻¹ p) (positive⁻¹ q))
     ∣m∣n≡∣mn∣ : ∀ m n → ℤ.+ ℤ.∣ m ∣ ℤ.* ℤ.+ n ≡ ℤ.+ ℤ.∣ m ℤ.* ℤ.+ n ∣
     ∣m∣n≡∣mn∣ m n = begin-equality
       ℤ.+ ℤ.∣ m ∣ ℤ.* ℤ.+ n                        ≡⟨⟩
-      ℤ.+ ℤ.∣ m ∣ ℤ.* ℤ.+ ℤ.∣ ℤ.+ n ∣              ≡˘⟨ ℤ.pos-* ℤ.∣ m ∣ ℤ.∣ ℤ.+ n ∣ ⟩
+      ℤ.+ ℤ.∣ m ∣ ℤ.* ℤ.+ ℤ.∣ ℤ.+ n ∣              ≡⟨ ℤ.pos-* ℤ.∣ m ∣ ℤ.∣ ℤ.+ n ∣ ⟨
       ℤ.+ (ℤ.∣ m ∣ ℕ.* n)                          ≡⟨⟩
-      ℤ.+ (ℤ.∣ m ∣ ℕ.* ℤ.∣ ℤ.+ n ∣)                ≡˘⟨ cong ℤ.+_ (ℤ.∣i*j∣≡∣i∣*∣j∣ m (ℤ.+ n)) ⟩
+      ℤ.+ (ℤ.∣ m ∣ ℕ.* ℤ.∣ ℤ.+ n ∣)                ≡⟨ cong ℤ.+_ (ℤ.∣i*j∣≡∣i∣*∣j∣ m (ℤ.+ n)) ⟨
       ℤ.+ (ℤ.∣ m ℤ.* ℤ.+ n ∣)                      ∎
     ∣↥p∣↧q≡∣↥p↧q∣ : ∣↥p∣↧q ≡ ℤ.+ ℤ.∣ ↥p↧q ∣
     ∣↥p∣↧q≡∣↥p↧q∣ = ∣m∣n≡∣mn∣ (↥ p) (↧ₙ q)
diff --git a/src/Data/Tree/Binary/Zipper/Properties.agda b/src/Data/Tree/Binary/Zipper/Properties.agda
index 107981bcbf..fa2f09c190 100644
--- a/src/Data/Tree/Binary/Zipper/Properties.agda
+++ b/src/Data/Tree/Binary/Zipper/Properties.agda
@@ -60,9 +60,9 @@ toTree-#nodes (mkZipper c v) = helper c v
       #ctx = sum (List.map (suc ∘ BT.#nodes ∘ getTree) ctx)
       #foc = BT.#nodes foc
       #l = BT.#nodes l in begin
-      #foc + (1 + (#l + #ctx))             ≡˘⟨ +-assoc #foc 1 (#l + #ctx) ⟩
+      #foc + (1 + (#l + #ctx))             ≡⟨ +-assoc #foc 1 (#l + #ctx) ⟨
       #foc + 1 + (#l + #ctx)               ≡⟨ cong (_+ (#l + #ctx)) (+-comm #foc 1) ⟩
-      1 + #foc + (#l + #ctx)               ≡˘⟨ +-assoc (1 + #foc) #l #ctx ⟩
+      1 + #foc + (#l + #ctx)               ≡⟨ +-assoc (1 + #foc) #l #ctx ⟨
       1 + #foc + #l + #ctx                 ≡⟨ cong (_+ #ctx) (+-comm (1 + #foc) #l) ⟩
       #nodes (mkZipper ctx (node l m foc)) ≡⟨ helper ctx (node l m foc) ⟩
       BT.#nodes (toTree (mkZipper cs foc)) ∎
@@ -70,8 +70,8 @@ toTree-#nodes (mkZipper c v) = helper c v
       #ctx = sum (List.map (suc ∘ BT.#nodes ∘ getTree) ctx)
       #foc = BT.#nodes foc
       #r = BT.#nodes r in begin
-      #foc + (1 + (#r + #ctx))             ≡˘⟨ cong (#foc +_) (+-assoc 1 #r #ctx) ⟩
-      #foc + (1 + #r + #ctx)               ≡˘⟨ +-assoc #foc (suc #r) #ctx ⟩
+      #foc + (1 + (#r + #ctx))             ≡⟨ cong (#foc +_) (+-assoc 1 #r #ctx) ⟨
+      #foc + (1 + #r + #ctx)               ≡⟨ +-assoc #foc (suc #r) #ctx ⟨
       #nodes (mkZipper ctx (node foc m r)) ≡⟨ helper ctx (node foc m r) ⟩
       BT.#nodes (toTree (mkZipper cs foc)) ∎
 
@@ -86,7 +86,7 @@ toTree-#leaves (mkZipper c v) = helper c v
       #ctx = sum (List.map (BT.#leaves ∘ getTree) ctx)
       #foc = BT.#leaves foc
       #l = BT.#leaves l in begin
-      #foc + (#l + #ctx)                    ≡˘⟨ +-assoc #foc #l #ctx ⟩
+      #foc + (#l + #ctx)                    ≡⟨ +-assoc #foc #l #ctx ⟨
       #foc + #l + #ctx                      ≡⟨ cong (_+ #ctx) (+-comm #foc #l) ⟩
       #leaves (mkZipper ctx (node l m foc)) ≡⟨ helper ctx (node l m foc) ⟩
       BT.#leaves (toTree (mkZipper cs foc)) ∎
@@ -94,7 +94,7 @@ toTree-#leaves (mkZipper c v) = helper c v
       #ctx = sum (List.map (BT.#leaves ∘ getTree) ctx)
       #foc = BT.#leaves foc
       #r = BT.#leaves r in begin
-      #foc + (#r + #ctx)                    ≡˘⟨ +-assoc #foc #r #ctx ⟩
+      #foc + (#r + #ctx)                    ≡⟨ +-assoc #foc #r #ctx ⟨
       #leaves (mkZipper ctx (node foc m r)) ≡⟨ helper ctx (node foc m r) ⟩
       BT.#leaves (toTree (mkZipper cs foc)) ∎
 
diff --git a/src/Data/Tree/Rose/Properties.agda b/src/Data/Tree/Rose/Properties.agda
index 749bdb5748..054103fdd1 100644
--- a/src/Data/Tree/Rose/Properties.agda
+++ b/src/Data/Tree/Rose/Properties.agda
@@ -39,7 +39,7 @@ map-∘ f g (node a ts) = cong (node (g (f a))) $ begin
 
 depth-map : (f : A → B) (t : Rose A i) → depth {i = i} (map {i = i} f t) ≡ depth {i = i} t
 depth-map f (node a ts) = cong (suc ∘′ max 0) $ begin
-  List.map depth (List.map (map f) ts) ≡˘⟨ Listₚ.map-∘ ts ⟩
+  List.map depth (List.map (map f) ts) ≡⟨ Listₚ.map-∘ ts ⟨
   List.map (depth ∘′ map f) ts         ≡⟨ Listₚ.map-cong (depth-map f) ts ⟩
   List.map depth ts ∎
 
@@ -49,7 +49,7 @@ depth-map f (node a ts) = cong (suc ∘′ max 0) $ begin
 foldr-map : (f : A → B) (n : B → List C → C) (ts : Rose A i) →
             foldr {i = i} n (map {i = i} f ts) ≡ foldr {i = i} (n ∘′ f) ts
 foldr-map f n (node a ts) = cong (n (f a)) $ begin
-  List.map (foldr n) (List.map (map f) ts) ≡˘⟨ Listₚ.map-∘ ts ⟩
+  List.map (foldr n) (List.map (map f) ts) ≡⟨ Listₚ.map-∘ ts ⟨
   List.map (foldr n ∘′ map f) ts           ≡⟨ Listₚ.map-cong (foldr-map f n) ts ⟩
   List.map (foldr (n ∘′ f)) ts             ∎
 
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index 93f95b423b..d1d5f8c628 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -181,20 +181,20 @@ module _ {ys ys′ : Vector A m} where
   ... | yes i<n = begin
     (xs ++ ys) i      ≡⟨ lookup-++-< xs ys i i<n ⟩
     xs (fromℕ< i<n)   ≡⟨ eq₁ (fromℕ< i<n) ⟩
-    xs′ (fromℕ< i<n)  ≡˘⟨ lookup-++-< xs′ ys′ i i<n ⟩
+    xs′ (fromℕ< i<n)  ≡⟨ lookup-++-< xs′ ys′ i i<n ⟨
     (xs′ ++ ys′) i    ∎
     where open ≡-Reasoning
   ... | no i≮n = begin
     (xs ++ ys) i               ≡⟨ lookup-++-≥ xs ys i (ℕₚ.≮⇒≥ i≮n) ⟩
     ys (reduce≥ i (ℕₚ.≮⇒≥ i≮n))   ≡⟨ eq₂ (reduce≥ i (ℕₚ.≮⇒≥ i≮n)) ⟩
-    ys′ (reduce≥ i (ℕₚ.≮⇒≥ i≮n))  ≡˘⟨ lookup-++-≥ xs′ ys′ i (ℕₚ.≮⇒≥ i≮n) ⟩
+    ys′ (reduce≥ i (ℕₚ.≮⇒≥ i≮n))  ≡⟨ lookup-++-≥ xs′ ys′ i (ℕₚ.≮⇒≥ i≮n) ⟨
     (xs′ ++ ys′) i             ∎
     where open ≡-Reasoning
 
   ++-injectiveˡ : ∀ (xs xs′ : Vector A n) →
                   xs ++ ys ≗ xs′ ++ ys′ → xs ≗ xs′
   ++-injectiveˡ xs xs′ eq i = begin
-    xs i                   ≡˘⟨ lookup-++ˡ xs ys i ⟩
+    xs i                   ≡⟨ lookup-++ˡ xs ys i ⟨
     (xs ++ ys) (i ↑ˡ m)    ≡⟨ eq (i ↑ˡ m) ⟩
     (xs′ ++ ys′) (i ↑ˡ m)  ≡⟨ lookup-++ˡ xs′ ys′ i ⟩
     xs′ i                  ∎
@@ -202,7 +202,7 @@ module _ {ys ys′ : Vector A m} where
 
   ++-injectiveʳ : ∀ (xs xs′ : Vector A n) → xs ++ ys ≗ xs′ ++ ys′ → ys ≗ ys′
   ++-injectiveʳ {n} xs xs′ eq i = begin
-    ys i                   ≡˘⟨ lookup-++ʳ xs ys i ⟩
+    ys i                   ≡⟨ lookup-++ʳ xs ys i ⟨
     (xs ++ ys) (n ↑ʳ i)    ≡⟨ eq (n ↑ʳ i)   ⟩
     (xs′ ++ ys′) (n ↑ʳ i)  ≡⟨ lookup-++ʳ xs′ ys′ i ⟩
     ys′ i                  ∎
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 38793a9038..4c5801fd7b 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -955,13 +955,13 @@ foldr-reverse B f {e} xs = foldl-fusion (foldr B f e) refl (λ _ _ → refl) xs
 reverse-involutive : Involutive {A = Vec A n} _≡_ reverse
 reverse-involutive xs = begin
   reverse (reverse xs)    ≡⟨  foldl-reverse (Vec _) (flip _∷_) xs ⟩
-  foldr (Vec _) _∷_ [] xs ≡˘⟨ id-is-foldr xs ⟩
+  foldr (Vec _) _∷_ [] xs ≡⟨ id-is-foldr xs ⟨
   xs                      ∎
   where open ≡-Reasoning
 
 reverse-reverse : reverse xs ≡ ys → reverse ys ≡ xs
 reverse-reverse {xs = xs} {ys} eq =  begin
-  reverse ys           ≡˘⟨ cong reverse eq ⟩
+  reverse ys           ≡⟨ cong reverse eq ⟨
   reverse (reverse xs) ≡⟨  reverse-involutive xs ⟩
   xs                   ∎
   where open ≡-Reasoning
@@ -970,7 +970,7 @@ reverse-reverse {xs = xs} {ys} eq =  begin
 
 reverse-injective : reverse xs ≡ reverse ys → xs ≡ ys
 reverse-injective {xs = xs} {ys} eq = begin
-  xs                   ≡˘⟨ reverse-reverse eq ⟩
+  xs                   ≡⟨ reverse-reverse eq ⟨
   reverse (reverse ys) ≡⟨  reverse-involutive ys ⟩
   ys                   ∎
   where open ≡-Reasoning
@@ -1000,7 +1000,7 @@ map-reverse f (x ∷ xs) = begin
   map f (reverse (x ∷ xs))  ≡⟨  cong (map f) (reverse-∷ x xs) ⟩
   map f (reverse xs ∷ʳ x)   ≡⟨  map-∷ʳ f x (reverse xs) ⟩
   map f (reverse xs) ∷ʳ f x ≡⟨  cong (_∷ʳ f x) (map-reverse f xs ) ⟩
-  reverse (map f xs) ∷ʳ f x ≡˘⟨ reverse-∷ (f x) (map f xs) ⟩
+  reverse (map f xs) ∷ʳ f x ≡⟨ reverse-∷ (f x) (map f xs) ⟨
   reverse (f x ∷ map f xs)  ≡⟨⟩
   reverse (map f (x ∷ xs))  ∎
   where open ≡-Reasoning
@@ -1015,7 +1015,7 @@ reverse-++ {m = suc m} {n = n} eq (x ∷ xs) ys = begin
   reverse (xs ++ ys) ∷ʳ x             ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (+-comm m n)) x (reverse (xs ++ ys)))
                                                 (reverse-++ _ xs ys) ⟩
   (reverse ys ++ reverse xs) ∷ʳ x     ≈⟨ ++-∷ʳ (sym (+-suc n m)) x (reverse ys) (reverse xs) ⟩
-  reverse ys ++ (reverse xs ∷ʳ x)     ≂˘⟨ cong (reverse ys ++_) (reverse-∷ x xs) ⟩
+  reverse ys ++ (reverse xs ∷ʳ x)     ≂⟨ cong (reverse ys ++_) (reverse-∷ x xs) ⟨
   reverse ys ++ (reverse (x ∷ xs))    ∎
   where open CastReasoning
 
@@ -1025,7 +1025,7 @@ cast-reverse {n = suc n} eq (x ∷ xs) = begin
   reverse (x ∷ xs)           ≂⟨ reverse-∷ x xs ⟩
   reverse xs ∷ʳ x            ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ eq x (reverse xs))
                                        (cast-reverse (cong pred eq) xs) ⟩
-  reverse (cast _ xs) ∷ʳ x   ≂˘⟨ reverse-∷ x (cast (cong pred eq) xs) ⟩
+  reverse (cast _ xs) ∷ʳ x   ≂⟨ reverse-∷ x (cast (cong pred eq) xs) ⟨
   reverse (x ∷ cast _ xs)    ≈⟨⟩
   reverse (cast eq (x ∷ xs)) ∎
   where open CastReasoning
@@ -1054,7 +1054,7 @@ map-ʳ++ {ys = ys} f xs = begin
   map f (xs ʳ++ ys)              ≡⟨  cong (map f) (unfold-ʳ++ xs ys) ⟩
   map f (reverse xs ++ ys)       ≡⟨  map-++ f (reverse xs) ys ⟩
   map f (reverse xs) ++ map f ys ≡⟨  cong (_++ map f ys) (map-reverse f xs) ⟩
-  reverse (map f xs) ++ map f ys ≡˘⟨ unfold-ʳ++ (map f xs) (map f ys) ⟩
+  reverse (map f xs) ++ map f ys ≡⟨ unfold-ʳ++ (map f xs) (map f ys) ⟨
   map f xs ʳ++ map f ys          ∎
   where open ≡-Reasoning
 
@@ -1064,7 +1064,7 @@ map-ʳ++ {ys = ys} f xs = begin
   (a ∷ xs) ʳ++ ys         ≂⟨ unfold-ʳ++ (a ∷ xs) ys ⟩
   reverse (a ∷ xs) ++ ys  ≂⟨ cong (_++ ys) (reverse-∷ a xs) ⟩
   (reverse xs ∷ʳ a) ++ ys ≈⟨ ∷ʳ-++ eq a (reverse xs) ⟩
-  reverse xs ++ (a ∷ ys)  ≂˘⟨ unfold-ʳ++ xs (a ∷ ys) ⟩
+  reverse xs ++ (a ∷ ys)  ≂⟨ unfold-ʳ++ xs (a ∷ ys) ⟨
   xs ʳ++ (a ∷ ys)         ∎
   where open CastReasoning
 
@@ -1075,8 +1075,8 @@ map-ʳ++ {ys = ys} f xs = begin
   reverse (xs ++ ys) ++ zs         ≈⟨ ≈-cong (_++ zs) (cast-++ˡ (+-comm m n) (reverse (xs ++ ys)))
                                              (reverse-++ (+-comm m n) xs ys) ⟩
   (reverse ys ++ reverse xs) ++ zs ≈⟨ ++-assoc (trans (cong (_+ o) (+-comm n m)) eq) (reverse ys) (reverse xs) zs ⟩
-  reverse ys ++ (reverse xs ++ zs) ≂˘⟨ cong (reverse ys ++_) (unfold-ʳ++ xs zs) ⟩
-  reverse ys ++ (xs ʳ++ zs)        ≂˘⟨ unfold-ʳ++ ys (xs ʳ++ zs) ⟩
+  reverse ys ++ (reverse xs ++ zs) ≂⟨ cong (reverse ys ++_) (unfold-ʳ++ xs zs) ⟨
+  reverse ys ++ (xs ʳ++ zs)        ≂⟨ unfold-ʳ++ ys (xs ʳ++ zs) ⟨
   ys ʳ++ (xs ʳ++ zs)               ∎
   where open CastReasoning
 
@@ -1089,7 +1089,7 @@ map-ʳ++ {ys = ys} f xs = begin
                                                        (reverse-++ (+-comm m n) (reverse xs) ys) ⟩
   (reverse ys ++ reverse (reverse xs)) ++ zs ≂⟨ cong ((_++ zs) ∘ (reverse ys ++_)) (reverse-involutive xs) ⟩
   (reverse ys ++ xs) ++ zs                   ≈⟨ ++-assoc (+-assoc n m o) (reverse ys) xs zs ⟩
-  reverse ys ++ (xs ++ zs)                   ≂˘⟨ unfold-ʳ++ ys (xs ++ zs) ⟩
+  reverse ys ++ (xs ++ zs)                   ≂⟨ unfold-ʳ++ ys (xs ++ zs) ⟨
   ys ʳ++ (xs ++ zs)                          ∎
   where open CastReasoning
 
@@ -1100,7 +1100,7 @@ sum-++ : ∀ (xs : Vec ℕ m) → sum (xs ++ ys) ≡ sum xs + sum ys
 sum-++ {_}       []       = refl
 sum-++ {ys = ys} (x ∷ xs) = begin
   x + sum (xs ++ ys)     ≡⟨  cong (x +_) (sum-++ xs) ⟩
-  x + (sum xs + sum ys)  ≡˘⟨ +-assoc x (sum xs) (sum ys) ⟩
+  x + (sum xs + sum ys)  ≡⟨ +-assoc x (sum xs) (sum ys) ⟨
   sum (x ∷ xs) + sum ys  ∎
   where open ≡-Reasoning
 
@@ -1209,7 +1209,7 @@ tabulate-allFin f = tabulate-∘ f id
 
 map-lookup-allFin : ∀ (xs : Vec A n) → map (lookup xs) (allFin n) ≡ xs
 map-lookup-allFin {n = n} xs = begin
-  map (lookup xs) (allFin n) ≡˘⟨ tabulate-∘ (lookup xs) id ⟩
+  map (lookup xs) (allFin n) ≡⟨ tabulate-∘ (lookup xs) id ⟨
   tabulate (lookup xs)       ≡⟨ tabulate∘lookup xs ⟩
   xs                         ∎
   where open ≡-Reasoning
@@ -1315,10 +1315,10 @@ fromList-reverse List.[] = refl
 fromList-reverse (x List.∷ xs) = begin
   fromList (List.reverse (x List.∷ xs))         ≈⟨ cast-fromList (Listₚ.ʳ++-defn xs) ⟩
   fromList (List.reverse xs List.++ List.[ x ]) ≈⟨ fromList-++ (List.reverse xs) ⟩
-  fromList (List.reverse xs) ++ [ x ]           ≈˘⟨ unfold-∷ʳ (+-comm 1 _) x (fromList (List.reverse xs)) ⟩
+  fromList (List.reverse xs) ++ [ x ]           ≈⟨ unfold-∷ʳ (+-comm 1 _) x (fromList (List.reverse xs)) ⟨
   fromList (List.reverse xs) ∷ʳ x               ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (Listₚ.length-reverse xs)) _ _)
                                                           (fromList-reverse xs) ⟩
-  reverse (fromList xs) ∷ʳ x                    ≂˘⟨ reverse-∷ x (fromList xs) ⟩
+  reverse (fromList xs) ∷ʳ x                    ≂⟨ reverse-∷ x (fromList xs) ⟨
   reverse (x ∷ fromList xs)                     ≈⟨⟩
   reverse (fromList (x List.∷ xs))              ∎
   where open CastReasoning
diff --git a/src/Data/Vec/Relation/Binary/Equality/Cast.agda b/src/Data/Vec/Relation/Binary/Equality/Cast.agda
index f65981f7a3..cc18556ea7 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Cast.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Cast.agda
@@ -52,7 +52,7 @@ xs ≈[ eq ] ys = cast eq xs ≡ ys
 
 ≈-sym : .{m≡n : m ≡ n} → Sym _≈[ m≡n ]_ _≈[ sym m≡n ]_
 ≈-sym {m≡n = m≡n} {xs} {ys} xs≈ys = begin
-  cast (sym m≡n) ys             ≡˘⟨ cong (cast (sym m≡n)) xs≈ys ⟩
+  cast (sym m≡n) ys             ≡⟨ cong (cast (sym m≡n)) xs≈ys ⟨
   cast (sym m≡n) (cast m≡n xs)  ≡⟨ cast-trans m≡n (sym m≡n) xs ⟩
   cast (trans m≡n (sym m≡n)) xs ≡⟨ cast-is-id (trans m≡n (sym m≡n)) xs ⟩
   xs                            ∎
@@ -60,7 +60,7 @@ xs ≈[ eq ] ys = cast eq xs ≡ ys
 
 ≈-trans : ∀ .{m≡n : m ≡ n} .{n≡o : n ≡ o} → Trans _≈[ m≡n ]_ _≈[ n≡o ]_ _≈[ trans m≡n n≡o ]_
 ≈-trans {m≡n = m≡n} {n≡o} {xs} {ys} {zs} xs≈ys ys≈zs = begin
-  cast (trans m≡n n≡o) xs ≡˘⟨ cast-trans m≡n n≡o xs ⟩
+  cast (trans m≡n n≡o) xs ≡⟨ cast-trans m≡n n≡o xs ⟨
   cast n≡o (cast m≡n xs)  ≡⟨ cong (cast n≡o) xs≈ys ⟩
   cast n≡o ys             ≡⟨ ys≈zs ⟩
   zs                      ∎
@@ -84,18 +84,28 @@ module CastReasoning where
   xs ≈⟨⟩ xs≈ys = xs≈ys
 
   -- composition of _≈[_]_
-  step-≈ : ∀ .{m≡n : m ≡ n}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →
+  step-≈-⟩ : ∀ .{m≡n : m ≡ n}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →
            ys ≈[ trans (sym m≡n) m≡o ] zs → xs ≈[ m≡n ] ys → xs ≈[ m≡o ] zs
-  step-≈ xs ys≈zs xs≈ys = ≈-trans xs≈ys ys≈zs
+  step-≈-⟩ xs ys≈zs xs≈ys = ≈-trans xs≈ys ys≈zs
+
+  step-≈-⟨ : ∀ .{n≡m : n ≡ m}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →
+           ys ≈[ trans n≡m m≡o ] zs → ys ≈[ n≡m ] xs → xs ≈[ m≡o ] zs
+  step-≈-⟨ xs ys≈zs ys≈xs = step-≈-⟩ xs ys≈zs (≈-sym ys≈xs)
 
   -- composition of the equality type on the right-hand side of _≈[_]_,
   -- or escaping to ordinary _≡_
-  step-≃ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → xs ≈[ m≡n ] ys → xs ≈[ m≡n ] zs
-  step-≃ xs ys≡zs xs≈ys = ≈-trans xs≈ys (≈-reflexive ys≡zs)
+  step-≃-⟩ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → xs ≈[ m≡n ] ys → xs ≈[ m≡n ] zs
+  step-≃-⟩ xs ys≡zs xs≈ys = ≈-trans xs≈ys (≈-reflexive ys≡zs)
+
+  step-≃-⟨ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → ys ≈[ sym m≡n ] xs → xs ≈[ m≡n ] zs
+  step-≃-⟨ xs ys≡zs ys≈xs = step-≃-⟩ xs ys≡zs (≈-sym ys≈xs)
 
   -- composition of the equality type on the left-hand side of _≈[_]_
-  step-≂ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → xs ≡ ys → xs ≈[ m≡n ] zs
-  step-≂ xs ys≈zs xs≡ys = ≈-trans (≈-reflexive xs≡ys) ys≈zs
+  step-≂-⟩ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → xs ≡ ys → xs ≈[ m≡n ] zs
+  step-≂-⟩ xs ys≈zs xs≡ys = ≈-trans (≈-reflexive xs≡ys) ys≈zs
+
+  step-≂-⟨ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → ys ≡ xs → xs ≈[ m≡n ] zs
+  step-≂-⟨ xs ys≈zs ys≡xs = step-≂-⟩ xs ys≈zs (sym ys≡xs)
 
   -- `cong` after a `_≈[_]_` step that exposes the `cast` to the `cong`
   -- operation
@@ -103,29 +113,16 @@ module CastReasoning where
            xs ≈[ m≡n ] f (cast l≡o ys) → ys ≈[ l≡o ] zs → xs ≈[ m≡n ] f zs
   ≈-cong f xs≈fys ys≈zs = trans xs≈fys (cong f ys≈zs)
 
-
-  -- symmetric version of each of the operator
-  step-≈˘ : ∀ .{n≡m : n ≡ m}.{m≡o : m ≡ o} (xs : Vec A m) {ys : Vec A n} {zs : Vec A o} →
-           ys ≈[ trans n≡m m≡o ] zs → ys ≈[ n≡m ] xs → xs ≈[ m≡o ] zs
-  step-≈˘ xs ys≈zs ys≈xs = step-≈ xs ys≈zs (≈-sym ys≈xs)
-
-  step-≃˘ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≡ zs → ys ≈[ sym m≡n ] xs → xs ≈[ m≡n ] zs
-  step-≃˘ xs ys≡zs ys≈xs = step-≃ xs ys≡zs (≈-sym ys≈xs)
-
-  step-≂˘ : ∀ .{m≡n : m ≡ n} (xs : Vec A m) {ys zs} → ys ≈[ m≡n ] zs → ys ≡ xs → xs ≈[ m≡n ] zs
-  step-≂˘ xs ys≈zs ys≡xs = step-≂ xs ys≈zs (sym ys≡xs)
-
-
   ------------------------------------------------------------------------
   -- convenient syntax for ‘equational’ reasoning
 
   infix 1 begin_
-  infixr 2 step-≃ step-≂ step-≃˘ step-≂˘ step-≈ step-≈˘ _≈⟨⟩_ ≈-cong
+  infixr 2 step-≃-⟩ step-≃-⟨ step-≂-⟩ step-≂-⟨ step-≈-⟩ step-≈-⟨ _≈⟨⟩_ ≈-cong
   infix 3 _∎
 
-  syntax step-≃  xs ys≡zs xs≈ys  = xs ≃⟨  xs≈ys ⟩ ys≡zs
-  syntax step-≃˘ xs ys≡zs xs≈ys  = xs ≃˘⟨ xs≈ys ⟩ ys≡zs
-  syntax step-≂  xs ys≈zs xs≡ys  = xs ≂⟨  xs≡ys ⟩ ys≈zs
-  syntax step-≂˘ xs ys≈zs ys≡xs  = xs ≂˘⟨ ys≡xs ⟩ ys≈zs
-  syntax step-≈  xs ys≈zs xs≈ys  = xs ≈⟨  xs≈ys ⟩ ys≈zs
-  syntax step-≈˘ xs ys≈zs ys≈xs  = xs ≈˘⟨ ys≈xs ⟩ ys≈zs
+  syntax step-≃-⟩ xs ys≡zs xs≈ys  = xs ≃⟨ xs≈ys ⟩ ys≡zs
+  syntax step-≃-⟨ xs ys≡zs xs≈ys  = xs ≃⟨ xs≈ys ⟨ ys≡zs
+  syntax step-≂-⟩ xs ys≈zs xs≡ys  = xs ≂⟨ xs≡ys ⟩ ys≈zs
+  syntax step-≂-⟨ xs ys≈zs ys≡xs  = xs ≂⟨ ys≡xs ⟨ ys≈zs
+  syntax step-≈-⟩ xs ys≈zs xs≈ys  = xs ≈⟨ xs≈ys ⟩ ys≈zs
+  syntax step-≈-⟨ xs ys≈zs ys≈xs  = xs ≈⟨ ys≈xs ⟨ ys≈zs
diff --git a/src/Function/Properties/Surjection.agda b/src/Function/Properties/Surjection.agda
index 5c2b235506..6f0bc11e6b 100644
--- a/src/Function/Properties/Surjection.agda
+++ b/src/Function/Properties/Surjection.agda
@@ -72,7 +72,7 @@ injective⇒to⁻-cong : (surj : Surjection S T) →
 injective⇒to⁻-cong {T = T} surj injective {x} {y} x≈y = injective $ begin
   to (to⁻ x) ≈⟨ to∘to⁻ x ⟩
   x          ≈⟨ x≈y ⟩
-  y          ≈˘⟨ to∘to⁻ y ⟩
+  y          ≈⟨ to∘to⁻ y ⟨
   to (to⁻ y) ∎
   where
   open SetoidReasoning T
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index 58b2ee198a..a78d14938b 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -116,7 +116,7 @@ module StarReasoning {i t} {I : Set i} (T : Rel I t) where
   private module Base = PreorderReasoning (preorder T)
 
   open Base public
-    hiding (step-≈; step-∼; step-≲)
+    hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)
     renaming (≲-go to ⟶-go)
 
   open ⟶-syntax _IsRelatedTo_ _IsRelatedTo_ (⟶-go ∘ (_◅ ε)) public
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Right.agda b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
index 80901f5939..3c2aa348d6 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Right.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
@@ -53,7 +53,7 @@ antisym : IsEquivalence _≈_ → Commutative _∙_ → Antisymmetric _≈_ _≤
 antisym isEq comm {x} {y} x≤y y≤x = begin
   x     ≈⟨  x≤y ⟩
   y ∙ x ≈⟨  comm y x ⟩
-  x ∙ y ≈˘⟨ y≤x ⟩
+  x ∙ y ≈⟨ y≤x ⟨
   y     ∎
   where open EqReasoning (record { isEquivalence = isEq })
 
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 4581514feb..9181229eec 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -255,15 +255,23 @@ module ≅-Reasoning where
   -- Can't create syntax in the standard `Syntax` module for
   -- heterogeneous steps because it would force that module to use
   -- the `--with-k` option.
-  infixr 2 _≅⟨_⟩_ _≅˘⟨_⟩_
+  infixr 2 _≅⟨_⟩_ _≅⟨_⟨_
 
   _≅⟨_⟩_ : ∀ (x : A) {y : B} {z : C} →
            x ≅ y → y IsRelatedTo z → x IsRelatedTo z
   _ ≅⟨ x≅y ⟩ relTo y≅z = relTo (trans x≅y y≅z)
 
-  _≅˘⟨_⟩_ : ∀ (x : A) {y : B} {z : C} →
+  _≅⟨_⟨_ : ∀ (x : A) {y : B} {z : C} →
             y ≅ x → y IsRelatedTo z → x IsRelatedTo z
-  _ ≅˘⟨ y≅x ⟩ relTo y≅z = relTo (trans (sym y≅x) y≅z)
+  _ ≅⟨ y≅x ⟨ relTo y≅z = relTo (trans (sym y≅x) y≅z)
+
+  -- Deprecated
+  infixr 2 _≅˘⟨_⟩_
+  _≅˘⟨_⟩_ = _≅⟨_⟨_
+  {-# WARNING_ON_USAGE _≅˘⟨_⟩_
+  "Warning: _≅˘⟨_⟩_ was deprecated in v2.0.
+  Please use _≅⟨_⟨_ instead."
+  #-}
 
 ------------------------------------------------------------------------
 -- Inspect
diff --git a/src/Relation/Binary/Reasoning/PartialOrder.agda b/src/Relation/Binary/Reasoning/PartialOrder.agda
index 9dbf2fc712..b633f1f548 100644
--- a/src/Relation/Binary/Reasoning/PartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/PartialOrder.agda
@@ -27,8 +27,8 @@
 --      w  <⟨ w<x ⟩
 --      x  ≤⟨ x≤y ⟩
 --      y  <⟨ y<z ⟩
---      z  ≡˘⟨ d≡z ⟩
---      d  ≈˘⟨ e≈d ⟩
+--      z  ≡⟨ d≡z ⟨
+--      d  ≈⟨ e≈d ⟨
 --      e  ∎
 --
 --    u≈w : u ≈ w
diff --git a/src/Relation/Binary/Reasoning/Preorder.agda b/src/Relation/Binary/Reasoning/Preorder.agda
index 933e8f0a45..aca729acc0 100644
--- a/src/Relation/Binary/Reasoning/Preorder.agda
+++ b/src/Relation/Binary/Reasoning/Preorder.agda
@@ -18,7 +18,7 @@
 --    u≈w = begin-equality
 --      u  ≈⟨ u≈v ⟩
 --      v  ≡⟨ v≡w ⟩
---      w  ≡˘⟨ x≡w ⟩
+--      w  ≡⟨ x≡w ⟨
 --      x  ∎
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Relation/Binary/Reasoning/Setoid.agda b/src/Relation/Binary/Reasoning/Setoid.agda
index bce0a3075c..ba6be9f79e 100644
--- a/src/Relation/Binary/Reasoning/Setoid.agda
+++ b/src/Relation/Binary/Reasoning/Setoid.agda
@@ -35,7 +35,9 @@ import Relation.Binary.Reasoning.Base.Single _≈_ refl trans
 
 -- Export the combinators for single relation reasoning, hiding the
 -- single misnamed combinator.
-open SingleRelReasoning public hiding (step-∼)
+open SingleRelReasoning public
+  hiding (step-∼)
+  renaming (∼-go to ≈-go)
 
 -- Re-export the equality-based combinators instead
-open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ SingleRelReasoning.∼-go sym public
+open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-go sym public
diff --git a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
index 2317ba4216..ad23a68481 100644
--- a/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
+++ b/src/Relation/Binary/Reasoning/StrictPartialOrder.agda
@@ -28,8 +28,8 @@
 --      w  <⟨ w<x ⟩
 --      x  ≤⟨ x≤y ⟩
 --      y  <⟨ y<z ⟩
---      z  ≡˘⟨ d≡z ⟩
---      d  ≈˘⟨ e≈d ⟩
+--      z  ≡⟨ d≡z ⟨
+--      d  ≈⟨ e≈d ⟨
 --      e  ∎
 --
 --    u≈w : u ≈ w
diff --git a/src/Relation/Binary/Reasoning/Syntax.agda b/src/Relation/Binary/Reasoning/Syntax.agda
index e8286bee30..4806512296 100644
--- a/src/Relation/Binary/Reasoning/Syntax.agda
+++ b/src/Relation/Binary/Reasoning/Syntax.agda
@@ -23,7 +23,6 @@ open import Relation.Binary.PropositionalEquality.Core as P
 --   Effect/Monad/Partiality
 --   Effect/Monad/Partiality/All
 --   Codata/Guarded/Stream/Relation/Binary/Pointwise
---   Codata/Sized/Stream/Bisimilarity
 --   Function/Reasoning
 
 module Relation.Binary.Reasoning.Syntax where
@@ -259,37 +258,80 @@ module _
 
     -- Setoid equality syntax
     module ≈-syntax where
+      infixr 2 step-≈-⟩ step-≈-⟨
+      step-≈-⟩ = forward
+      step-≈-⟨ = backward
+      syntax step-≈-⟩ x yRz x≈y = x ≈⟨ x≈y ⟩ yRz
+      syntax step-≈-⟨ x yRz y≈x = x ≈⟨ y≈x ⟨ yRz
+
+      -- Deprecated
       infixr 2 step-≈ step-≈˘
-      step-≈ = forward
-      step-≈˘ = backward
-      syntax step-≈  x yRz x≈y = x ≈⟨ x≈y ⟩ yRz
+      step-≈ = step-≈-⟩
+      {-# WARNING_ON_USAGE step-≈
+      "Warning: step-≈ was deprecated in v2.0.
+      Please use step-≈-⟩ instead."
+      #-}
+      step-≈˘ = step-≈-⟨
+      {-# WARNING_ON_USAGE step-≈˘
+      "Warning: step-≈˘ and _≈˘⟨_⟩_ was deprecated in v2.0.
+      Please use step-≈-⟨ and _≈⟨_⟨_ instead."
+      #-}
       syntax step-≈˘ x yRz y≈x = x ≈˘⟨ y≈x ⟩ yRz
 
 
     -- Container equality syntax
     module ≋-syntax where
+      infixr 2 step-≋-⟩ step-≋-⟨
+      step-≋-⟩ = forward
+      step-≋-⟨ = backward
+      syntax step-≋-⟩ x yRz x≋y = x ≋⟨ x≋y ⟩ yRz
+      syntax step-≋-⟨ x yRz y≋x = x ≋⟨ y≋x ⟨ yRz
+
+
+      -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.
       infixr 2 step-≋ step-≋˘
-      step-≋ = forward
-      step-≋˘ = backward
-      syntax step-≋  x yRz x≋y = x ≋⟨ x≋y ⟩ yRz
+      step-≋ = step-≋-⟩
+      {-# WARNING_ON_USAGE step-≋
+      "Warning: step-≋ was deprecated in v2.0.
+      Please use step-≋-⟩ instead."
+      #-}
+      step-≋˘ = step-≋-⟨
+      {-# WARNING_ON_USAGE step-≋˘
+      "Warning: step-≋˘ and _≋˘⟨_⟩_ was deprecated in v2.0.
+      Please use step-≋-⟨ and _≋⟨_⟨_ instead."
+      #-}
       syntax step-≋˘ x yRz y≋x = x ≋˘⟨ y≋x ⟩ yRz
 
 
     -- Other equality syntax
     module ≃-syntax where
-      infixr 2 step-≃ step-≃˘
-      step-≃ = forward
-      step-≃˘ = backward
-      syntax step-≃  x yRz x≃y = x ≃⟨ x≃y ⟩ yRz
-      syntax step-≃˘ x yRz y≃x = x ≃˘⟨ y≃x ⟩ yRz
+      infixr 2 step-≃-⟩ step-≃-⟨
+      step-≃-⟩ = forward
+      step-≃-⟨ = backward
+      syntax step-≃-⟩ x yRz x≃y = x ≃⟨ x≃y ⟩ yRz
+      syntax step-≃-⟨ x yRz y≃x = x ≃⟨ y≃x ⟨ yRz
 
 
     -- Apartness relation syntax
     module #-syntax where
+      infixr 2 step-#-⟩ step-#-⟨
+      step-#-⟩ = forward
+      step-#-⟨ = backward
+      syntax step-#-⟩ x yRz x#y = x #⟨ x#y ⟩ yRz
+      syntax step-#-⟨ x yRz y#x = x #⟨ y#x ⟨ yRz
+
+      -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.
       infixr 2 step-# step-#˘
-      step-# = forward
-      step-#˘ = backward
-      syntax step-#  x yRz x#y = x #⟨ x#y ⟩ yRz
+      step-# = step-#-⟩
+      {-# WARNING_ON_USAGE step-#
+      "Warning: step-# was deprecated in v2.0.
+      Please use step-#-⟩ instead."
+      #-}
+      step-#˘ = step-#-⟨
+      {-# WARNING_ON_USAGE step-#˘
+      "Warning: step-#˘ and _#˘⟨_⟩_ was deprecated in v2.0.
+      Please use step-#-⟨ and _#⟨_⟨_ instead."
+      #-}
       syntax step-#˘ x yRz y#x = x #˘⟨ y#x ⟩ yRz
 
 
@@ -304,20 +346,35 @@ module _
 
     -- Inverse syntax
     module ↔-syntax where
-      infixr 2 step-↔ step-↔-sym
-      step-↔ = forward
-      step-↔-sym = backward
-      syntax step-↔ x yRz x↔y = x ↔⟨ x↔y ⟩ yRz
-      syntax step-↔-sym x yRz y↔x = x ↔˘⟨ y↔x ⟩ yRz
+      infixr 2 step-↔-⟩ step-↔-⟨
+      step-↔-⟩ = forward
+      step-↔-⟨ = backward
+      syntax step-↔-⟩ x yRz x↔y = x ↔⟨ x↔y ⟩ yRz
+      syntax step-↔-⟨ x yRz y↔x = x ↔⟨ y↔x ⟨ yRz
 
 
     -- Inverse syntax
     module ↭-syntax where
-      infixr 2 step-↭ step-↭-sym
+      infixr 2 step-↭-⟩ step-↭-⟨
+      step-↭-⟩ = forward
+      step-↭-⟨ = backward
+      syntax step-↭-⟩ x yRz x↭y = x ↭⟨ x↭y ⟩ yRz
+      syntax step-↭-⟨ x yRz y↭x = x ↭⟨ y↭x ⟨ yRz
+
+
+      -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.
+      infixr 2 step-↭ step-↭˘
       step-↭ = forward
-      step-↭-sym = backward
-      syntax step-↭ x yRz x↭y = x ↭⟨ x↭y ⟩ yRz
-      syntax step-↭-sym x yRz y↭x = x ↭˘⟨ y↭x ⟩ yRz
+      {-# WARNING_ON_USAGE step-↭
+      "Warning: step-↭ was deprecated in v2.0.
+      Please use step-↭-⟩ instead."
+      #-}
+      step-↭˘ = backward
+      {-# WARNING_ON_USAGE step-↭˘
+      "Warning: step-↭˘ and _↭˘⟨_⟩_ was deprecated in v2.0.
+      Please use step-↭-⟨ and _↭⟨_⟨_ instead."
+      #-}
+      syntax step-↭˘ x yRz y↭x = x ↭˘⟨ y↭x ⟩ yRz
 
 ------------------------------------------------------------------------
 -- Propositional equality
@@ -332,16 +389,32 @@ module ≡-syntax
   (step : Trans _≡_ R R)
   where
 
-  infixr 2 step-≡ step-≡-refl step-≡˘
-  step-≡ = forward R R step
-  step-≡˘ = backward R R step P.sym
+  infixr 2 step-≡-⟩  step-≡-∣ step-≡-⟨
+  step-≡-⟩ = forward R R step
+
+  step-≡-∣ : ∀ x {y} → R x y → R x y
+  step-≡-∣ x xRy = xRy
+
+  step-≡-⟨ = backward R R step P.sym
+
+  syntax step-≡-⟩ x yRz x≡y = x ≡⟨ x≡y ⟩ yRz
+  syntax step-≡-∣ x xRy     = x ≡⟨⟩ xRy
+  syntax step-≡-⟨ x yRz y≡x = x ≡⟨ y≡x ⟨ yRz
 
-  step-≡-refl : ∀ x {y} → R x y → R x y
-  step-≡-refl x xRy = xRy
 
-  syntax step-≡-refl x xRy     = x ≡⟨⟩ xRy
-  syntax step-≡˘     x yRz y≡x = x ≡˘⟨ y≡x ⟩ yRz
-  syntax step-≡      x yRz x≡y = x ≡⟨ x≡y ⟩ yRz
+  -- Don't remove until https://github.com/agda/agda/issues/5617 fixed.
+  infixr 2 step-≡ step-≡˘
+  step-≡ = step-≡-⟩
+  {-# WARNING_ON_USAGE step-≡
+  "Warning: step-≡ was deprecated in v2.0.
+  Please use step-≡-⟩ instead."
+  #-}
+  step-≡˘ = step-≡-⟨
+  {-# WARNING_ON_USAGE step-≡˘
+  "Warning: step-≡˘ and _≡˘⟨_⟩_ was deprecated in v2.0.
+  Please use step-≡-⟨ and _≡⟨_⟨_ instead."
+  #-}
+  syntax step-≡˘ x yRz y≡x = x ≡˘⟨ y≡x ⟩ yRz
 
 
 -- Unlike ≡-syntax above, chains of reasoning using this syntax will not
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index b7accfcc08..2c1b4196b1 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -15,7 +15,7 @@
 --     suc (suc m) + m
 --   ≡⟨ cong! eq ⟩
 --     suc (suc n) + n
---   ≡˘⟨ cong! (+-identityʳ n) ⟩
+--   ≡⟨ cong! (+-identityʳ n) ⟨
 --     suc (suc n) + (n + 0)
 --   ∎
 ------------------------------------------------------------------------
diff --git a/src/Tactic/MonoidSolver.agda b/src/Tactic/MonoidSolver.agda
index c137e44990..c62ea9dccf 100644
--- a/src/Tactic/MonoidSolver.agda
+++ b/src/Tactic/MonoidSolver.agda
@@ -132,7 +132,7 @@ module _ {m₁ m₂} (monoid : Monoid m₁ m₂) where
   homo′ [ x ↑] y   = ∙-congʳ (identityʳ x)
   homo′ (x ∙′ y) z = begin
     [ x ∙′ y ⇓] ∙ z       ≡⟨⟩
-    [ x ⇓]′ [ y ⇓] ∙ z    ≈˘⟨ ∙-congʳ (homo′ x [ y ⇓]) ⟩
+    [ x ⇓]′ [ y ⇓] ∙ z    ≈⟨ ∙-congʳ (homo′ x [ y ⇓]) ⟨
     ([ x ⇓] ∙ [ y ⇓]) ∙ z ≈⟨ assoc [ x ⇓] [ y ⇓] z ⟩
     [ x ⇓] ∙ ([ y ⇓] ∙ z) ≈⟨ ∙-congˡ (homo′ y z) ⟩
     [ x ⇓] ∙ ([ y ⇓]′ z)  ≈⟨ homo′ x ([ y ⇓]′ z) ⟩
@@ -143,7 +143,7 @@ module _ {m₁ m₂} (monoid : Monoid m₁ m₂) where
   homo [ x ↑]   = identityʳ x
   homo (x ∙′ y) = begin
     [ x ∙′ y ⇓]     ≡⟨⟩
-    [ x ⇓]′ [ y ⇓]  ≈˘⟨ homo′ x [ y ⇓] ⟩
+    [ x ⇓]′ [ y ⇓]  ≈⟨ homo′ x [ y ⇓] ⟨
     [ x ⇓] ∙ [ y ⇓] ≈⟨ ∙-cong (homo x) (homo y) ⟩
     [ x ↓] ∙ [ y ↓] ∎
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index 4bdfd74381..f014d54c9f 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -246,7 +246,7 @@ poly-foldR ρ ρs f e cng dist step base (x ≠0 Δ suc i & []) =
     ⅀?⟦ y Δ suc i ∷↓ ys ⟧ (ρ , ρs)
   ≈⟨ ∷↓-hom-s y i ys ρ ρs ⟩
     (ρ ^ suc i) * ((y , ys) ⟦∷⟧ (ρ , ρs))
-  ≈˘⟨ *≫ ⟦∷⟧?-hom y ys ρ ρs ⟩
+  ≈⟨ *≫ ⟦∷⟧?-hom y ys ρ ρs ⟨
     (ρ ^ suc i) * ((y , ys) ⟦∷⟧? (ρ , ρs))
   ≈⟨ *≫ step x [] (sym base) ⟩
     (ρ ^ suc i) * e ((x , []) ⟦∷⟧? (ρ , ρs))
@@ -265,7 +265,7 @@ poly-foldR ρ ρs f e cng dist step base (x ≠0 Δ suc i & ∹ xs) =
     ⅀?⟦ y Δ suc i ∷↓ zs ⟧ (ρ , ρs)
   ≈⟨ ∷↓-hom-s y i zs ρ ρs ⟩
     (ρ ^ suc i) * ((y , zs) ⟦∷⟧ (ρ , ρs))
-  ≈˘⟨ *≫ ⟦∷⟧?-hom y zs ρ ρs ⟩
+  ≈⟨ *≫ ⟦∷⟧?-hom y zs ρ ρs ⟨
     (ρ ^ suc i) * ((y , zs) ⟦∷⟧? (ρ , ρs))
   ≈⟨ *≫ step x (∹ xs) (poly-foldR ρ ρs f e cng dist step base xs) ⟩
     (ρ ^ suc i) * e ((x , (∹ xs )) ⟦∷⟧? (ρ , ρs))
@@ -283,7 +283,7 @@ poly-foldR ρ ρs f e cng dist step base (x ≠0 Δ ℕ.zero & []) =
     ⅀?⟦ y Δ ℕ.zero ∷↓ zs ⟧ (ρ , ρs)
   ≈⟨ ∷↓-hom-0 y zs ρ ρs ⟩
     ((y , zs) ⟦∷⟧ (ρ , ρs))
-  ≈˘⟨ ⟦∷⟧?-hom y zs ρ ρs ⟩
+  ≈⟨ ⟦∷⟧?-hom y zs ρ ρs ⟨
     ((y , zs) ⟦∷⟧? (ρ , ρs))
   ≈⟨ step x [] (sym base) ⟩
     e ((x , []) ⟦∷⟧? (ρ , ρs))
@@ -300,7 +300,7 @@ poly-foldR ρ ρs f e cng dist step base (x ≠0 Δ ℕ.zero & (∹ xs )) =
     ⅀?⟦ y Δ ℕ.zero ∷↓ zs ⟧ (ρ , ρs)
   ≈⟨ ∷↓-hom-0 y zs ρ ρs ⟩
     ((y , zs) ⟦∷⟧ (ρ , ρs))
-  ≈˘⟨ ⟦∷⟧?-hom y zs ρ ρs ⟩
+  ≈⟨ ⟦∷⟧?-hom y zs ρ ρs ⟨
     ((y , zs) ⟦∷⟧? (ρ , ρs))
   ≈⟨ step x (∹ xs ) (poly-foldR ρ ρs f e cng dist step base xs) ⟩
     e ((x , (∹ xs )) ⟦∷⟧ (ρ , ρs))
diff --git a/src/Text/Pretty/Core.agda b/src/Text/Pretty/Core.agda
index 974b1af823..96415feb7f 100644
--- a/src/Text/Pretty/Core.agda
+++ b/src/Text/Pretty/Core.agda
@@ -190,7 +190,7 @@ private
     isBlock ∣x∣ ∣y∣ with blocky
     ... | nothing        = begin
       size blockx         ≡⟨ ∣x∣ ⟩
-      x.height            ≡˘⟨ +-identityʳ x.height ⟩
+      x.height            ≡⟨ +-identityʳ x.height ⟨
       x.height + 0        ≡⟨ cong (_ +_) ∣y∣ ⟩
       x.height + y.height ∎ where open ≡-Reasoning
     ... | just (hd , tl) = begin

From 9352cf595012f8c230a18c6c39e923897a44204a Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 19 Oct 2023 10:35:34 +0900
Subject: [PATCH 170/316] Restore 'return' as an alias for 'pure' (#2164)

---
 CHANGELOG.md                |  3 +--
 src/Effect/Applicative.agda |  4 ++++
 src/IO/Primitive.agda       | 17 +++++++++++++----
 3 files changed, 18 insertions(+), 6 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5649103aee..f91f4dbace 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -774,8 +774,7 @@ Non-backwards compatible changes
   This is needed for level-increasing functors like `IO : Set l → Set (suc l)`.
 
 * `RawApplicative` is now `RawFunctor + pure + _<*>_` and `RawMonad` is now
-  `RawApplicative` + `_>>=_` and so `return` is not used anywhere anymore.
-  This fixes the conflict with `case_return_of` (#356)
+  `RawApplicative` + `_>>=_`.
   This reorganisation means in particular that the functor/applicative of a monad
   are not computed using `_>>=_`. This may break proofs.
 
diff --git a/src/Effect/Applicative.agda b/src/Effect/Applicative.agda
index 729966752c..a2263c6e0a 100644
--- a/src/Effect/Applicative.agda
+++ b/src/Effect/Applicative.agda
@@ -53,6 +53,10 @@ record RawApplicative (F : Set f → Set g) : Set (suc f ⊔ g) where
   zip : F A → F B → F (A × B)
   zip = zipWith _,_
 
+  -- Haskell-style alternative name for pure
+  return : A → F A
+  return = pure
+  
   -- backwards compatibility: unicode variants
   _⊛_ : F (A → B) → F A → F B
   _⊛_ = _<*>_
diff --git a/src/IO/Primitive.agda b/src/IO/Primitive.agda
index 220e862fb4..8014a2bd7c 100644
--- a/src/IO/Primitive.agda
+++ b/src/IO/Primitive.agda
@@ -10,20 +10,29 @@
 
 module IO.Primitive where
 
-open import Agda.Builtin.IO
+open import Level using (Level)
+private
+  variable
+    a : Level
+    A B : Set a
 
 ------------------------------------------------------------------------
 -- The IO monad
 
-open import Agda.Builtin.IO public using (IO)
+open import Agda.Builtin.IO public
+  using (IO)
 
 infixl 1 _>>=_
 
 postulate
-  pure : ∀ {a} {A : Set a} → A → IO A
-  _>>=_  : ∀ {a b} {A : Set a} {B : Set b} → IO A → (A → IO B) → IO B
+  pure : A → IO A
+  _>>=_  : IO A → (A → IO B) → IO B
 
 {-# COMPILE GHC pure = \_ _ -> return    #-}
 {-# COMPILE GHC _>>=_  = \_ _ _ _ -> (>>=) #-}
 {-# COMPILE UHC pure = \_ _ x -> UHC.Agda.Builtins.primReturn x #-}
 {-# COMPILE UHC _>>=_  = \_ _ _ _ x y -> UHC.Agda.Builtins.primBind x y #-}
+
+-- Haskell-style alternative syntax
+return : A → IO A
+return = pure

From 31880ec5be034805178512867e1200e1a9c76b53 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Thu, 19 Oct 2023 02:35:54 +0100
Subject: [PATCH 171/316] [ fix #2153 ] Properly re-export specialised
 combinators (#2161)

---
 CHANGELOG.md                         | 3 +++
 src/Data/List/Membership/Setoid.agda | 5 +++--
 2 files changed, 6 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index f91f4dbace..9485368502 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -33,6 +33,9 @@ Bug-fixes
   in `Function.Construct.Composition` had their arguments in the wrong order. They have
   been flipped so they can actually be used as a composition operator.
 
+* The operations `_∷=_` and `_─_` exported from `Data.List.Membership.Setoid` 
+  had an extraneous `{A : Set a}` parameter. This has been removed.
+
 * The combinators `_≃⟨_⟩_` and `_≃˘⟨_⟩_` in `Data.Rational.Properties.≤-Reasoning`
   now correctly accepts proofs of type `_≃_` instead of the previous proofs of type `_≡_`.
 
diff --git a/src/Data/List/Membership/Setoid.agda b/src/Data/List/Membership/Setoid.agda
index 1b8a67fb7d..1a19169992 100644
--- a/src/Data/List/Membership/Setoid.agda
+++ b/src/Data/List/Membership/Setoid.agda
@@ -13,7 +13,7 @@ module Data.List.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where
 
 open import Function.Base using (_∘_; id; flip)
 open import Data.List.Base as List using (List; []; _∷_; length; lookup)
-open import Data.List.Relation.Unary.Any
+open import Data.List.Relation.Unary.Any as Any
   using (Any; index; map; here; there)
 open import Data.Product.Base as Prod using (∃; _×_; _,_)
 open import Relation.Unary using (Pred)
@@ -35,7 +35,8 @@ x ∉ xs = ¬ x ∈ xs
 ------------------------------------------------------------------------
 -- Operations
 
-open Data.List.Relation.Unary.Any using (_∷=_; _─_) public
+_∷=_ = Any._∷=_ {A = A}
+_─_ = Any._─_ {A = A}
 
 mapWith∈ : ∀ {b} {B : Set b}
            (xs : List A) → (∀ {x} → x ∈ xs → B) → List B

From c00fba6545afa613ca3f468549e2abc6f4661f2b Mon Sep 17 00:00:00 2001
From: James Wood <laMudri@gmail.com>
Date: Thu, 19 Oct 2023 03:39:33 +0100
Subject: [PATCH 172/316] Connect `LeftInverse` with (`Split`)`Surjection`
 (#2054)

---
 CHANGELOG.md                 | 13 +++++++++
 src/Function/Bundles.agda    | 55 +++++++++++++++++++++++++++++++++++-
 src/Function/Structures.agda | 20 +++++++++++++
 3 files changed, 87 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9485368502..24480da16f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3233,6 +3233,13 @@ Additions to existing modules
   evalState : State s a → s → a
   execState : State s a → s → s
   ```
+  
+* Added new proofs and definitions in `Function.Bundles`:
+  ```agda
+  LeftInverse.isSplitSurjection : LeftInverse → IsSplitSurjection to
+  LeftInverse.surjection        : LeftInverse → Surjection
+  SplitSurjection               = LeftInverse
+  ```
 
 * Added new application operator synonym to `Function.Bundles`:
   ```agda
@@ -3255,6 +3262,12 @@ Additions to existing modules
   _!|>_  : (a : A) → (∀ a → B a) → B a
   _!|>′_ : A → (A → B) → B
   ```
+  
+* Added new proof and record in `Function.Structures`:
+  ```agda
+  IsLeftInverse.isSurjection : IsLeftInverse to from → IsSurjection to
+  record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  ```
 
 * Added new definition to the `Surjection` module in `Function.Related.Surjection`:
   ```
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index 0aef27fc57..9e18c30fe5 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -218,7 +218,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       }
 
     open IsLeftInverse isLeftInverse public
-      using (module Eq₁; module Eq₂; strictlyInverseˡ)
+      using (module Eq₁; module Eq₂; strictlyInverseˡ; isSurjection)
 
     equivalence : Equivalence
     equivalence = record
@@ -226,6 +226,20 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       ; from-cong = from-cong
       }
 
+    isSplitSurjection : IsSplitSurjection to
+    isSplitSurjection = record
+      { from = from
+      ; isLeftInverse = isLeftInverse
+      }
+
+    surjection : Surjection From To
+    surjection = record
+      { to = to
+      ; cong = to-cong
+      ; surjective = λ y → from y , inverseˡ
+      }
+
+
 
   record RightInverse : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
     field
@@ -346,6 +360,45 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
       ; from₂-cong = from₂-cong
       }
 
+------------------------------------------------------------------------
+-- Other
+
+  -- A left inverse is also known as a “split surjection”.
+  --
+  -- As the name implies, a split surjection is a special kind of
+  -- surjection where the witness generated in the domain in the
+  -- function for elements `x₁` and `x₂` are equal if `x₁ ≈ x₂` .
+  --
+  -- The difference is the `from-cong` law --- generally, the section
+  -- (called `Surjection.to⁻` or `SplitSurjection.from`) of a surjection
+  -- need not respect equality, whereas it must in a split surjection.
+  --
+  -- The two notions coincide when the equivalence relation on `B` is
+  -- propositional equality (because all functions respect propositional
+  -- equality).
+  --
+  -- For further background on (split) surjections, one may consult any
+  -- general mathematical references which work without the principle
+  -- of choice. For example:
+  -- 
+  --   https://ncatlab.org/nlab/show/split+epimorphism.
+  --
+  -- The connection to set-theoretic notions with the same names is
+  -- justified by the setoid type theory/homotopy type theory
+  -- observation/definition that (∃x : A. P) = ∥ Σx : A. P ∥ --- i.e.,
+  -- we can read set-theoretic ∃ as squashed/propositionally truncated Σ.
+  --
+  -- We see working with setoids as working in the MLTT model of a setoid
+  -- type theory, in which ∥ X ∥ is interpreted as the setoid with carrier
+  -- set X and the equivalence relation that relates all elements.
+  -- All maps into ∥ X ∥ respect equality, so in the idiomatic definitions
+  -- here, we drop the corresponding trivial `cong` field completely.
+
+  SplitSurjection : Set _
+  SplitSurjection = LeftInverse
+
+  module SplitSurjection (splitSurjection : SplitSurjection) =
+    LeftInverse splitSurjection
 
 ------------------------------------------------------------------------
 -- Bundles specialised for propositional equality
diff --git a/src/Function/Structures.agda b/src/Function/Structures.agda
index a1cc0d6ceb..baf33dddf0 100644
--- a/src/Function/Structures.agda
+++ b/src/Function/Structures.agda
@@ -107,6 +107,12 @@ record IsLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ 
   strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from
   strictlyInverseˡ x = inverseˡ Eq₁.refl
 
+  isSurjection : IsSurjection to
+  isSurjection = record
+    { isCongruent = isCongruent
+    ; surjective = λ y → from y , inverseˡ
+    }
+
 
 record IsRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
   field
@@ -168,3 +174,17 @@ record IsBiInverse
 
   open IsCongruent to-isCongruent public
     renaming (cong to to-cong)
+
+
+------------------------------------------------------------------------
+-- Other
+------------------------------------------------------------------------
+
+-- See the comment on `SplitSurjection` in `Function.Bundles` for an
+-- explanation of (split) surjections.
+record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+  field
+    from : B → A
+    isLeftInverse : IsLeftInverse f from
+
+  open IsLeftInverse isLeftInverse public

From b7358c857f58444065721c675e994ee4945103ba Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 19 Oct 2023 11:39:49 +0900
Subject: [PATCH 173/316] Added remaining flipped and negated relations to
 binary relation bundles (#2162)

---
 CHANGELOG.md                                  | 30 ++++---
 README/Design/Hierarchies.agda                | 37 ++++++++
 .../Lattice/Properties/Semilattice.agda       |  4 +-
 .../Product/Relation/Binary/Lex/Strict.agda   |  2 +-
 src/Relation/Binary/Bundles.agda              | 88 +++++++++++++------
 .../Binary/Properties/DecTotalOrder.agda      |  3 +-
 src/Relation/Binary/Properties/Poset.agda     |  5 --
 .../Binary/Properties/TotalOrder.agda         |  5 +-
 src/Relation/Binary/Structures.agda           | 33 ++++---
 9 files changed, 144 insertions(+), 63 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 24480da16f..7b6d9fa069 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -852,18 +852,24 @@ Non-backwards compatible changes
   IO.Instances
   ```
 
-### (Issue #2096) Introduction of flipped relation symbol for `Relation.Binary.Bundles.Preorder`
-
-* Previously, the relation symbol `_∼_`  was (notationally) symmetric, so that its
-  converse relation could only be discussed *semantically* in terms of `flip _∼_`
-  in `Relation.Binary.Properties.Preorder`, `Relation.Binary.Construct.Flip.{Ord|EqAndOrd}`
-
-* Now, the symbol `_∼_` has been renamed to a new symbol `_≲_`, with `_≳_`
-  introduced as a definition in `Relation.Binary.Bundles.Preorder` whose properties
-  in `Relation.Binary.Properties.Preorder` now refer to it. Partial backwards compatible
-  has been achieved by redeclaring a deprecated version of the old name in the record.
-  Therefore, only _declarations_ of `PartialOrder` records will need their field names
-  updating.
+### (Issue #2096) Introduction of flipped and negated relation symbols to bundles in `Relation.Binary.Bundles`
+
+* Previously, bundles such as `Preorder`, `Poset`, `TotalOrder` etc. did not have the flipped 
+  and negated versions of the operators exposed. In some cases they could obtained by opening the
+  relevant `Relation.Binary.Properties.X` file but usually they had to be redefined every time.
+  
+* To fix this, these bundles now all export all 4 versions of the operator: normal, converse, negated,
+  converse-negated. Accordingly they are no longer exported from the corresponding `Properties` file.
+
+* To make this work for `Preorder`, it was necessary to change the name of the relation symbol.
+  Previously, the symbol was `_∼_`  which is (notationally) symmetric, so that its
+  converse relation could only be discussed *semantically* in terms of `flip _∼_`.
+
+* Now, the `Preorder` record field `_∼_` has been renamed to `_≲_`, with `_≳_`
+  introduced as a definition in `Relation.Binary.Bundles.Preorder`. 
+  Partial backwards compatible has been achieved by redeclaring a deprecated version 
+  of the old symbol in the record. Therefore, only _declarations_ of `PartialOrder` records will 
+  need their field names updating.
 
 ### (Issue #1214) Reorganisation of the introduction of negated relation symbols under `Relation.Binary`
 
diff --git a/README/Design/Hierarchies.agda b/README/Design/Hierarchies.agda
index 09d5504a91..bc9220ba23 100644
--- a/README/Design/Hierarchies.agda
+++ b/README/Design/Hierarchies.agda
@@ -249,6 +249,43 @@ record Semigroup : Set (suc (a ⊔ ℓ)) where
 -- differently, as a function with an unknown domain an codomain is
 -- of little use.
 
+-------------------------
+-- Bundle re-exporting --
+-------------------------
+
+-- In general ensuring that bundles re-export everything in their
+-- sub-bundles can get a little tricky.
+
+-- Imagine we have the following general scenario where bundle A is a
+-- direct refinement of bundle C (i.e. the record `IsA` has a `IsC` field)
+-- but is also morally a refinement of bundles B and D.
+
+--   Structures               Bundles
+--   ==========               =======
+--       IsA                     A
+--    /  ||   \               /  ||  \
+-- IsB   IsC   IsD          B    C    D
+ 
+-- The procedure for re-exports in the bundles is as follows:
+
+-- 1. `open IsA isA public using (IsC, M)` where `M` is everything
+--    exported by `IsA` that is not exported by `IsC`.
+
+-- 2. Construct `c : C` via the `isC` obtained in step 1.
+
+-- 3. `open C c public hiding (N)` where `N` is the list of fields
+--    shared by both `A` and `C`.
+
+-- 4. Construct `b : B` via the `isB` obtained in step 1.
+
+-- 5. `open B b public using (O)` where `O` is everything exported
+--    by `B` but not exported by `IsA`.
+ 
+-- 6. Construct `d : D` via the `isC` obtained in step 1.
+
+-- 7. `open D d public using (P)` where `P` is everything exported
+--    by `D` but not exported by `IsA`
+
 ------------------------------------------------------------------------
 -- Other hierarchy modules
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Lattice/Properties/Semilattice.agda b/src/Algebra/Lattice/Properties/Semilattice.agda
index 38554aedfa..7b8135ced7 100644
--- a/src/Algebra/Lattice/Properties/Semilattice.agda
+++ b/src/Algebra/Lattice/Properties/Semilattice.agda
@@ -27,8 +27,8 @@ import Relation.Binary.Construct.NaturalOrder.Left _≈_ _∧_
 poset : Poset c ℓ ℓ
 poset = LeftNaturalOrder.poset isSemilattice
 
-open Poset poset using (_≤_; isPartialOrder)
-open PosetProperties poset using (_≥_; ≥-isPartialOrder)
+open Poset poset using (_≤_; _≥_; isPartialOrder)
+open PosetProperties poset using (≥-isPartialOrder)
 
 ------------------------------------------------------------------------
 -- Every algebraic semilattice can be turned into an order-theoretic one.
diff --git a/src/Data/Product/Relation/Binary/Lex/Strict.agda b/src/Data/Product/Relation/Binary/Lex/Strict.agda
index 42d139893d..39d86602f2 100644
--- a/src/Data/Product/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Product/Relation/Binary/Lex/Strict.agda
@@ -240,7 +240,7 @@ module _ {_≈₁_ : Rel A ℓ₁} {_<₁_ : Rel A ℓ₂}
                           (isEquivalence pre₁) (isEquivalence pre₂)
       ; reflexive     = ×-reflexive _≈₁_ _<₁_ _<₂_ (reflexive pre₂)
       ; trans         = ×-transitive {_<₂_ = _<₂_}
-                          (isEquivalence pre₁) (∼-resp-≈ pre₁)
+                          (isEquivalence pre₁) (≲-resp-≈ pre₁)
                           (trans pre₁) (trans pre₂)
       }
     where open IsPreorder
diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index eb7975f376..641803f8ef 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -43,13 +43,15 @@ record Setoid c ℓ : Set (suc (c ⊔ ℓ)) where
     isEquivalence : IsEquivalence _≈_
 
   open IsEquivalence isEquivalence public
+    using (refl; reflexive; isPartialEquivalence)
 
   partialSetoid : PartialSetoid c ℓ
   partialSetoid = record
     { isPartialEquivalence = isPartialEquivalence
     }
 
-  open PartialSetoid partialSetoid public using (_≉_)
+  open PartialSetoid partialSetoid public
+    hiding (Carrier; _≈_; isPartialEquivalence)
 
 
 record DecSetoid c ℓ : Set (suc (c ⊔ ℓ)) where
@@ -60,14 +62,15 @@ record DecSetoid c ℓ : Set (suc (c ⊔ ℓ)) where
     isDecEquivalence : IsDecEquivalence _≈_
 
   open IsDecEquivalence isDecEquivalence public
+    using (_≟_; isEquivalence)
 
   setoid : Setoid c ℓ
   setoid = record
     { isEquivalence = isEquivalence
     }
 
-  open Setoid setoid public using (partialSetoid; _≉_)
-
+  open Setoid setoid public
+    hiding (Carrier; _≈_; isEquivalence)
 
 ------------------------------------------------------------------------
 -- Preorders
@@ -99,6 +102,9 @@ record Preorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   infix 4 _≳_
   _≳_ = flip _≲_
 
+  infix 4 _⋧_
+  _⋧_ = flip _⋦_
+
   -- Deprecated.
   infix 4 _∼_
   _∼_ = _≲_
@@ -117,14 +123,15 @@ record TotalPreorder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     isTotalPreorder : IsTotalPreorder _≈_ _≲_
 
   open IsTotalPreorder isTotalPreorder public
-    hiding (module Eq)
+    using (total; isPreorder)
 
   preorder : Preorder c ℓ₁ ℓ₂
-  preorder = record { isPreorder = isPreorder }
+  preorder = record
+    { isPreorder = isPreorder
+    }
 
   open Preorder preorder public
-    using (module Eq; _≳_; _⋦_)
-
+    hiding (Carrier; _≈_; _≲_; isPreorder)
 
 ------------------------------------------------------------------------
 -- Partial orders
@@ -139,7 +146,7 @@ record Poset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     isPartialOrder : IsPartialOrder _≈_ _≤_
 
   open IsPartialOrder isPartialOrder public
-    hiding (module Eq)
+    using (antisym; isPreorder)
 
   preorder : Preorder c ℓ₁ ℓ₂
   preorder = record
@@ -147,8 +154,13 @@ record Poset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     }
 
   open Preorder preorder public
-    using (module Eq)
-    renaming (_⋦_ to _≰_)
+    hiding (Carrier; _≈_; _≲_; isPreorder)
+    renaming
+    ( _⋦_ to _≰_; _≳_ to _≥_; _⋧_ to _≱_
+    ; ≲-respˡ-≈ to ≤-respˡ-≈
+    ; ≲-respʳ-≈ to ≤-respʳ-≈
+    ; ≲-resp-≈  to ≤-resp-≈
+    )
 
 
 record DecPoset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
@@ -159,9 +171,10 @@ record DecPoset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     _≤_               : Rel Carrier ℓ₂
     isDecPartialOrder : IsDecPartialOrder _≈_ _≤_
 
-  private
-    module DPO = IsDecPartialOrder isDecPartialOrder
-  open DPO public hiding (module Eq)
+  private module DPO = IsDecPartialOrder isDecPartialOrder
+
+  open DPO public
+    using (_≟_; _≤?_; isPartialOrder)
 
   poset : Poset c ℓ₁ ℓ₂
   poset = record
@@ -169,7 +182,7 @@ record DecPoset c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     }
 
   open Poset poset public
-    using (preorder; _≰_)
+    hiding (Carrier; _≈_; _≤_; isPartialOrder; module Eq)
 
   module Eq where
     decSetoid : DecSetoid c ℓ₁
@@ -203,6 +216,12 @@ record StrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂))
   _≮_ : Rel Carrier _
   x ≮ y = ¬ (x < y)
 
+  infix 4 _>_
+  _>_ = flip _<_
+
+  infix 4 _≯_
+  _≯_ = flip _≮_
+
 
 record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   infix 4 _≈_ _<_
@@ -212,9 +231,10 @@ record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂
     _<_                     : Rel Carrier ℓ₂
     isDecStrictPartialOrder : IsDecStrictPartialOrder _≈_ _<_
 
-  private
-    module DSPO = IsDecStrictPartialOrder isDecStrictPartialOrder
-  open DSPO public hiding (module Eq)
+  private module DSPO = IsDecStrictPartialOrder isDecStrictPartialOrder
+
+  open DSPO public
+    using (_<?_; _≟_; isStrictPartialOrder)
 
   strictPartialOrder : StrictPartialOrder c ℓ₁ ℓ₂
   strictPartialOrder = record
@@ -222,7 +242,7 @@ record DecStrictPartialOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂
     }
 
   open StrictPartialOrder strictPartialOrder public
-    using (_≮_)
+    hiding (Carrier; _≈_; _<_; isStrictPartialOrder; module Eq)
 
   module Eq where
 
@@ -247,7 +267,7 @@ record TotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     isTotalOrder : IsTotalOrder _≈_ _≤_
 
   open IsTotalOrder isTotalOrder public
-    hiding (module Eq)
+    using (total; isPartialOrder; isTotalPreorder)
 
   poset : Poset c ℓ₁ ℓ₂
   poset = record
@@ -255,7 +275,7 @@ record TotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     }
 
   open Poset poset public
-    using (module Eq; preorder; _≰_)
+    hiding (Carrier; _≈_; _≤_; isPartialOrder)
 
   totalPreorder : TotalPreorder c ℓ₁ ℓ₂
   totalPreorder = record
@@ -271,9 +291,10 @@ record DecTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     _≤_             : Rel Carrier ℓ₂
     isDecTotalOrder : IsDecTotalOrder _≈_ _≤_
 
-  private
-    module DTO = IsDecTotalOrder isDecTotalOrder
-  open DTO public hiding (module Eq)
+  private module DTO = IsDecTotalOrder isDecTotalOrder
+
+  open DTO public
+    using (_≟_; _≤?_; isTotalOrder; isDecPartialOrder)
 
   totalOrder : TotalOrder c ℓ₁ ℓ₂
   totalOrder = record
@@ -281,7 +302,7 @@ record DecTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
     }
 
   open TotalOrder totalOrder public
-    using (poset; preorder; _≰_)
+    hiding (Carrier; _≈_; _≤_; isTotalOrder; module Eq)
 
   decPoset : DecPoset c ℓ₁ ℓ₂
   decPoset = record
@@ -306,7 +327,10 @@ record StrictTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) wh
     isStrictTotalOrder : IsStrictTotalOrder _≈_ _<_
 
   open IsStrictTotalOrder isStrictTotalOrder public
-    hiding (module Eq)
+    using
+    ( _≟_; _<?_; compare; isStrictPartialOrder
+    ; isDecStrictPartialOrder; isDecEquivalence
+    )
 
   strictPartialOrder : StrictPartialOrder c ℓ₁ ℓ₂
   strictPartialOrder = record
@@ -314,17 +338,26 @@ record StrictTotalOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) wh
     }
 
   open StrictPartialOrder strictPartialOrder public
-    using (module Eq; _≮_)
+    hiding (Carrier; _≈_; _<_; isStrictPartialOrder; module Eq)
+
+  decStrictPartialOrder : DecStrictPartialOrder c ℓ₁ ℓ₂
+  decStrictPartialOrder = record
+    { isDecStrictPartialOrder = isDecStrictPartialOrder
+    }
+
+  open DecStrictPartialOrder decStrictPartialOrder public
+    using (module Eq)
 
   decSetoid : DecSetoid c ℓ₁
   decSetoid = record
-    { isDecEquivalence = isDecEquivalence
+    { isDecEquivalence = Eq.isDecEquivalence
     }
   {-# WARNING_ON_USAGE decSetoid
   "Warning: decSetoid was deprecated in v1.3.
   Please use Eq.decSetoid instead."
   #-}
 
+
 record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
   infix 4 _≈_ _<_
   field
@@ -342,6 +375,7 @@ record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) wh
     }
 
   open StrictTotalOrder strictTotalOrder public
+    hiding (Carrier; _≈_; _<_; isStrictTotalOrder)
 
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Properties/DecTotalOrder.agda b/src/Relation/Binary/Properties/DecTotalOrder.agda
index d7b3c48cd9..38df65dc94 100644
--- a/src/Relation/Binary/Properties/DecTotalOrder.agda
+++ b/src/Relation/Binary/Properties/DecTotalOrder.agda
@@ -26,8 +26,7 @@ open import Relation.Nullary.Negation using (¬_)
 
 open TotalOrderProperties public
   using
-  ( _≥_
-  ; ≥-refl
+  ( ≥-refl
   ; ≥-reflexive
   ; ≥-trans
   ; ≥-antisym
diff --git a/src/Relation/Binary/Properties/Poset.agda b/src/Relation/Binary/Properties/Poset.agda
index beb4ddc8aa..d25f3af79a 100644
--- a/src/Relation/Binary/Properties/Poset.agda
+++ b/src/Relation/Binary/Properties/Poset.agda
@@ -30,11 +30,6 @@ open Eq using (_≉_)
 ------------------------------------------------------------------------
 -- The _≥_ relation is also a poset.
 
-infix 4 _≥_
-
-_≥_ : Rel A p₃
-x ≥ y = y ≤ x
-
 open PreorderProperties public
   using () renaming
   ( converse-isPreorder to ≥-isPreorder
diff --git a/src/Relation/Binary/Properties/TotalOrder.agda b/src/Relation/Binary/Properties/TotalOrder.agda
index 48f788ab7f..b949f9b361 100644
--- a/src/Relation/Binary/Properties/TotalOrder.agda
+++ b/src/Relation/Binary/Properties/TotalOrder.agda
@@ -39,8 +39,7 @@ decTotalOrder ≟ = record
 
 open PosetProperties public
   using
-  ( _≥_
-  ; ≥-refl
+  ( ≥-refl
   ; ≥-reflexive
   ; ≥-trans
   ; ≥-antisym
@@ -59,7 +58,7 @@ open PosetProperties public
   }
 
 open TotalOrder ≥-totalOrder public
-  using () renaming (total to ≥-total; _≰_ to _≱_)
+  using () renaming (total to ≥-total)
 
 ------------------------------------------------------------------------
 -- _<_ - the strict version is a strict partial order
diff --git a/src/Relation/Binary/Structures.agda b/src/Relation/Binary/Structures.agda
index bf11d67250..0dc5187228 100644
--- a/src/Relation/Binary/Structures.agda
+++ b/src/Relation/Binary/Structures.agda
@@ -85,14 +85,30 @@ record IsPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
   refl : Reflexive _≲_
   refl = reflexive Eq.refl
 
-  ∼-respˡ-≈ : _≲_ Respectsˡ _≈_
-  ∼-respˡ-≈ x≈y x∼z = trans (reflexive (Eq.sym x≈y)) x∼z
+  ≲-respˡ-≈ : _≲_ Respectsˡ _≈_
+  ≲-respˡ-≈ x≈y x∼z = trans (reflexive (Eq.sym x≈y)) x∼z
 
-  ∼-respʳ-≈ : _≲_ Respectsʳ _≈_
-  ∼-respʳ-≈ x≈y z∼x = trans z∼x (reflexive x≈y)
+  ≲-respʳ-≈ : _≲_ Respectsʳ _≈_
+  ≲-respʳ-≈ x≈y z∼x = trans z∼x (reflexive x≈y)
 
-  ∼-resp-≈ : _≲_ Respects₂ _≈_
-  ∼-resp-≈ = ∼-respʳ-≈ , ∼-respˡ-≈
+  ≲-resp-≈ : _≲_ Respects₂ _≈_
+  ≲-resp-≈ = ≲-respʳ-≈ , ≲-respˡ-≈
+
+  ∼-respˡ-≈ = ≲-respˡ-≈
+  {-# WARNING_ON_USAGE ∼-respˡ-≈
+  "Warning: ∼-respˡ-≈ was deprecated in v2.0.
+  Please use ≲-respˡ-≈ instead. "
+  #-}
+  ∼-respʳ-≈ = ≲-respʳ-≈
+  {-# WARNING_ON_USAGE ∼-respʳ-≈
+  "Warning: ∼-respʳ-≈ was deprecated in v2.0.
+  Please use ≲-respʳ-≈ instead. "
+  #-}
+  ∼-resp-≈ = ≲-resp-≈
+  {-# WARNING_ON_USAGE ∼-resp-≈
+  "Warning: ∼-resp-≈ was deprecated in v2.0.
+  Please use ≲-resp-≈ instead. "
+  #-}
 
 
 record IsTotalPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
@@ -101,11 +117,6 @@ record IsTotalPreorder (_≲_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
     total      : Total _≲_
 
   open IsPreorder isPreorder public
-    renaming
-    ( ∼-respˡ-≈ to ≲-respˡ-≈
-    ; ∼-respʳ-≈ to ≲-respʳ-≈
-    ; ∼-resp-≈  to ≲-resp-≈
-    )
 
 
 ------------------------------------------------------------------------

From c42635a4fd6c3649b9146e997fda56addea5a8f4 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 19 Oct 2023 19:19:07 +0900
Subject: [PATCH 174/316] Tidy up CHANGELOG in preparation for release
 candidate (#2165)

---
 CHANGELOG.md | 3151 ++++++++++++++++++++++++--------------------------
 1 file changed, 1483 insertions(+), 1668 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7b6d9fa069..6e03140b75 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,259 +1,156 @@
-Version 2.0-dev
+Version 2.0-rc1
 ===============
 
-The library has been tested using Agda 2.6.3.
+The library has been tested using Agda 2.6.4.
+
+NOTE: Version `2.0` contains various breaking changes and is not backwards 
+compatible with code written with version `1.X` of the library.
 
 Highlights
 ----------
 
-* A golden testing library in `Test.Golden`. This allows you to run a set
-  of tests and make sure their output matches an expected `golden` value.
-  The test runner has many options: filtering tests by name, dumping the
-  list of failures to a file, timing the runs, coloured output, etc.
-  Cf. the comments in `Test.Golden` and the standard library's own tests
-  in `tests/` for documentation on how to use the library.
-
 * A new tactic `cong!` available from `Tactic.Cong` which automatically
   infers the argument to `cong` for you via anti-unification.
 
 * Improved the `solve` tactic in `Tactic.RingSolver` to work in a much
   wider range of situations.
 
-* Added `⌊log₂_⌋` and `⌈log₂_⌉` on Natural Numbers.
-
-* A massive refactoring of the unindexed Functor/Applicative/Monad hierarchy
-  and the MonadReader / MonadState type classes. These are now usable with
+* A massive refactoring of the unindexed `Functor`/`Applicative`/`Monad` hierarchy
+  and the `MonadReader` / `MonadState` type classes. These are now usable with
   instance arguments as demonstrated in the tests/monad examples.
 
+* Significant tightening of `import` statements internally in the library,
+  drastically decreasing the dependencies and hence load time of many key
+  modules.
+
+* A golden testing library in `Test.Golden`. This allows you to run a set
+  of tests and make sure their output matches an expected `golden` value.
+  The test runner has many options: filtering tests by name, dumping the
+  list of failures to a file, timing the runs, coloured output, etc.
+  Cf. the comments in `Test.Golden` and the standard library's own tests
+  in `tests/` for documentation on how to use the library.
 
 Bug-fixes
 ---------
 
-* The combinators `_⟶-∘_`, `_↣-∘_`, `_↠-∘_`, `_⤖-∘_`, `_⇔-∘_`, `_↩-∘_`, `_↪-∘_`, `_↔-∘_`
-  in `Function.Construct.Composition` had their arguments in the wrong order. They have
-  been flipped so they can actually be used as a composition operator.
+* In `Algebra.Definitions.RawSemiring` the record `Prime` did not
+  enforce that the number was not divisible by `1#`. To fix this 
+  `p∤1 : p ∤ 1#` hsa been added as a field.
 
-* The operations `_∷=_` and `_─_` exported from `Data.List.Membership.Setoid` 
+* In `Data.Container.FreeMonad`, we give a direct definition of `_⋆_` as an inductive
+  type rather than encoding it as an instance of `μ`. This ensures Agda notices that
+  `C ⋆ X` is strictly positive in `X` which in turn allows us to use the free monad
+  when defining auxiliary (co)inductive types (cf. the `Tap` example in
+  `README.Data.Container.FreeMonad`).
+
+* In `Data.Fin.Properties` the `i` argument to `opposite-suc` was implicit
+  but could not be inferred in general. It has been made explicit.
+  
+* In `Data.List.Membership.Setoid` the operations `_∷=_` and `_─_` 
   had an extraneous `{A : Set a}` parameter. This has been removed.
 
-* The combinators `_≃⟨_⟩_` and `_≃˘⟨_⟩_` in `Data.Rational.Properties.≤-Reasoning`
-  now correctly accepts proofs of type `_≃_` instead of the previous proofs of type `_≡_`.
+* In `Data.List.Relation.Ternary.Appending.Setoid` the constructors 
+  were re-exported in their full generality which lead to unsolved meta 
+  variables at their use sites. Now versions of the constructors specialised
+  to use the setoid's carrier set are re-exported.
+  
+* In `Data.Nat.DivMod` the parameter `o` in the proof `/-monoˡ-≤` was 
+  implicit but not inferrable. It has been changed to be explicit.
 
-* The following operators were missing a fixity declaration, which has now
-  been fixed -
-  ```
-  infixr  5 _∷_                                              (Codata.Guarded.Stream)
-  infix   4 _[_]                                             (Codata.Guarded.Stream)
-  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Binary.Pointwise)
-  infix   4 _≈∞_                                             (Codata.Guarded.Stream.Relation.Binary.Pointwise)
-  infixr  5 _∷_                                              (Codata.Musical.Colist)
-  infix   4 _≈_                                              (Codata.Musical.Conat)
-  infixr  5 _∷_                                              (Codata.Musical.Colist.Bisimilarity)
-  infixr  5 _∷_                                              (Codata.Musical.Colist.Relation.Unary.All)
-  infixr  5 _∷_                                              (Codata.Sized.Colist)
-  infixr  5 _∷_                                              (Codata.Sized.Covec)
-  infixr  5 _∷_                                              (Codata.Sized.Cowriter)
-  infixl  1 _>>=_                                            (Codata.Sized.Cowriter)
-  infixr  5 _∷_                                              (Codata.Sized.Stream)
-  infixr  5 _∷_                                              (Codata.Sized.Colist.Bisimilarity)
-  infix   4 _ℕ≤?_                                            (Codata.Sized.Conat.Properties)
-  infixr  5 _∷_                                              (Codata.Sized.Covec.Bisimilarity)
-  infixr  5 _∷_                                              (Codata.Sized.Cowriter.Bisimilarity)
-  infixr  5 _∷_                                              (Codata.Sized.Stream.Bisimilarity)
-  infixr  8 _⇒_ _⊸_                                          (Data.Container.Core)
-  infixr -1 _<$>_ _<*>_                                      (Data.Container.FreeMonad)
-  infixl  1 _>>=_                                            (Data.Container.FreeMonad)
-  infix   5 _▷_                                              (Data.Container.Indexed)
-  infix   4 _≈_                                              (Data.Float.Base)
-  infixl  7 _⊓′_                                             (Data.Nat.Base)
-  infixl  6 _⊔′_                                             (Data.Nat.Base)
-  infixr  8 _^_                                              (Data.Nat.Base)
-  infix   4 _!≢0 _!*_!≢0                                     (Data.Nat.Properties)
-  infix   4 _≃?_                                             (Data.Rational.Unnormalised.Properties)
-  infix   4 _≈ₖᵥ_                                            (Data.Tree.AVL.Map.Membership.Propositional)
-  infix   4 _<_                                              (Induction.WellFounded)
-  infix  -1 _$ⁿ_                                             (Data.Vec.N-ary)
-  infix   4 _≋_                                              (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
-  infix   4 _≟_                                              (Reflection.AST.Definition)
-  infix   4 _≡ᵇ_                                             (Reflection.AST.Literal)
-  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Meta)
-  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Name)
-  infix   4 _≟-Telescope_                                    (Reflection.AST.Term)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Information)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Modality)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Quantity)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Relevance)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Visibility)
-  infixr  8 _^_                                              (Function.Endomorphism.Propositional)
-  infixr  8 _^_                                              (Function.Endomorphism.Setoid)
-  infix   4 _≃_                                              (Function.HalfAdjointEquivalence)
-  infix   4 _≈_ _≈ᵢ_ _≤_                                     (Function.Metric.Bundles)
-  infixl  6 _∙_                                              (Function.Metric.Bundles)
-  infix   4 _≈_                                              (Function.Metric.Nat.Bundles)
-  infix   3 _←_ _↢_                                          (Function.Related)
-  infix   4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
-  infix   6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
-  infixl  4 _+ _*                                            (Data.List.Kleene.Base)
-  infixr  4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
-  infix   4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
-  infix   4 _≢∈_                                             (Data.List.Membership.Propositional)
-  infixr  5 _`∷_                                             (Data.List.Reflection)
-  infix   4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
-  infixr  5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
-  infixr  5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
-  infixr  5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
-  infixr  5 _∷=_                                             (Data.List.Relation.Unary.Any)
-  infixr  5 _++_                                             (Data.List.Ternary.Appending)
-  infixr  2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
-  infixr  2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
-  infixr  2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
-  infixr  2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
-  infixr  1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
-  infixr  1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
-  infixr  1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
-  infixr  1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
-  infix   8 _⁻¹                                              (Data.Parity.Base)
-  infixr  5 _`∷_                                             (Data.Vec.Reflection)
-  infixr  5 _∷=_                                             (Data.Vec.Membership.Setoid)
-  infix   4 _≟H_ _≟N_                                        (Algebra.Solver.Ring)
-  infix   4 _≈_                                              (Relation.Binary.Bundles)
-  infixl  6 _∩_                                              (Relation.Binary.Construct.Intersection)
-  infix   4 _<₋_                                             (Relation.Binary.Construct.Add.Infimum.Strict)
-  infix   4 _≈∙_                                             (Relation.Binary.Construct.Add.Point.Equality)
-  infix   4 _≤⁺_ _≤⊤⁺                                        (Relation.Binary.Construct.Add.Supremum.NonStrict)
-  infixr  5 _∷ʳ_                                             (Relation.Binary.Construct.Closure.Transitive)
-  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Unary.All)
-  infixr  5 _∷_                                              (Foreign.Haskell.List.NonEmpty)
-  infix   4 _≈_                                              (Function.Metric.Rational.Bundles)
-  infixl  6 _ℕ+_                                             (Level.Literals)
-  infixr  6 _∪_                                              (Relation.Binary.Construct.Union)
-  infixl  6 _+ℤ_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
-  infix   4 _≉_ _≈ᵢ_ _≤ᵢ_                                    (Relation.Binary.Indexed.Homogeneous.Bundles)
-  infixr  5 _∷ᴹ_ _∷⁻¹ᴹ_                                      (Text.Regex.Search)
-  infixr  4 _,_                                              (Data.Refinement)
-  infixr  4 _,_                                              (Data.Container.Relation.Binary.Pointwise)
-  infixr  4 _,_                                              (Data.Tree.AVL.Value)
-  infixr  4 _,_                                              (Foreign.Haskell.Pair)
-  infixr  4 _,_                                              (Reflection.AnnotatedAST)
-  infixr  4 _,_                                              (Reflection.AST.Traversal)
-  infixl  6.5 _P′_ _P_ _C′_ _C_                              (Data.Nat.Combinatorics.Base)
-  infixl  1 _>>=-cong_ _≡->>=-cong_                          (Effect.Monad.Partiality)
-  infixl  1 _?>=′_                                           (Effect.Monad.Predicate)
-  infixl  1 _>>=-cong_ _>>=-congP_                           (Effect.Monad.Partiality.All)
-  infix   4 _∈FV_                                            (Reflection.AST.DeBruijn)
-  infixr  9 _;_                                              (Relation.Binary.Construct.Composition)
-  infixl  6 _+²_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
-  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Heterogeneous.Construct.At)
-  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Homogeneous.Construct.At)
-  infix   4 _∈_ _∉_                                          (Relation.Unary.Indexed)
-  infixr  9 _⍮_                                              (Relation.Unary.PredicateTransformer)
-  infix   8 ∼_                                               (Relation.Unary.PredicateTransformer)
-  infix   2 _×?_ _⊙?_                                        (Relation.Unary.Properties)
-  infix   10 _~?                                             (Relation.Unary.Properties)
-  infixr  1 _⊎?_                                             (Relation.Unary.Properties)
-  infixr  7 _∩?_                                             (Relation.Unary.Properties)
-  infixr  6 _∪?_                                             (Relation.Unary.Properties)
-  infixl  6 _`⊜_                                             (Tactic.RingSolver)
-  infix   8 ⊝_                                               (Tactic.RingSolver.Core.Expression)
-  infix   4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]                  (Text.Regex.Properties)
-  infix   4 _∈?_ _∉?_                                        (Text.Regex.Derivative.Brzozowski)
-  infix   4 _∈_ _∉_ _∈?_ _∉?_                                (Text.Regex.String.Unsafe)
-  ```
+* In `Data.Nat.DivMod` the parameter `m` in the proof `+-distrib-/-∣ʳ` was 
+  implicit but not inferrable, while `n` is explicit but inferrable.
+  They have been to explicit and implicit respectively.
 
-* In `System.Exit`, the `ExitFailure` constructor is now carrying an integer
-  rather than a natural. The previous binding was incorrectly assuming that
-  all exit codes where non-negative.
+* In `Data.Nat.GeneralisedArithmetic` the `s` and `z` arguments to the 
+  following functions were implicit but no inferrable:
+  `fold-+`, `fold-k`, `fold-*`, `fold-pull`. They have been made explicit.
 
-* In `/-monoˡ-≤` in `Data.Nat.DivMod` the parameter `o` was implicit but not inferrable.
-  It has been changed to be explicit.
+* In `Data.Rational(.Unnormalised).Properties` the module `≤-Reasoning`
+  exported equality combinators using the generic setoid symbol `_≈_`. They
+  have been renamed to use the same `_≃_` symbol used for non-propositional
+  equality over `Rational`s, i.e.
+  ```agda
+  step-≈  ↦  step-≃
+  step-≈˘ ↦  step-≃˘
+  ```
+  with corresponding associated syntax:
+  ```agda
+  _≈⟨_⟩_ ↦ _≃⟨_⟩_
+  _≈⟨_⟨_ ↦ _≃⟨_⟨_
+  ```
 
-* In `+-distrib-/-∣ʳ` in `Data.Nat.DivMod` the parameter `m` was implicit but not inferrable,
-  while `n` is explicit but inferrable.  They have been changed.
+* In `Function.Construct.Composition` the combinators 
+  `_⟶-∘_`, `_↣-∘_`, `_↠-∘_`, `_⤖-∘_`, `_⇔-∘_`, `_↩-∘_`, `_↪-∘_`, `_↔-∘_`
+  had their arguments in the wrong order. They have been flipped so they can 
+  actually be  used as a composition operator.
 
 * In `Function.Definitions` the definitions of `Surjection`, `Inverseˡ`,
   `Inverseʳ` were not being re-exported correctly and therefore had an unsolved
   meta-variable whenever this module was explicitly parameterised. This has
   been fixed.
 
-* Add module `Algebra.Module` that re-exports the contents of
-  `Algebra.Module.(Definitions/Structures/Bundles)`
-
-* Various module-like bundles in `Algebra.Module.Bundles` were missing a fixity
-  declaration for `_*ᵣ_`. This has been fixed.
-
-* In `Algebra.Definitions.RawSemiring` the record `prime` add `p∤1 : p ∤ 1#` to the field.
-
-* In `Data.List.Relation.Ternary.Appending.Setoid` we re-export specialised versions of
-  the constructors using the setoid's carrier set. Prior to that, the constructor were
-  re-exported in their full generality which would lead to unsolved meta variables at
-  their use sites.
-
-* In `Data.Container.FreeMonad`, we give a direct definition of `_⋆_` as an inductive
-  type rather than encoding it as an instance of `μ`. This ensures Agda notices that
-  `C ⋆ X` is strictly positive in `X` which in turn allows us to use the free monad
-  when defining auxiliary (co)inductive types (cf. the `Tap` example in
-  `README.Data.Container.FreeMonad`).
-
-* In `Data.Maybe.Base` the fixity declaration of `_<∣>_` was missing. This has been fixed.
+* In `System.Exit` the `ExitFailure` constructor is now carrying an integer
+  rather than a natural. The previous binding was incorrectly assuming that
+  all exit codes where non-negative.
 
 Non-backwards compatible changes
 --------------------------------
 
-#### Removed deprecated names
+### Removed deprecated names
 
-* All modules and names that were deprecated in v1.2 and before have been removed.
+* All modules and names that were deprecated in `v1.2` and before have 
+  been removed.
 
-#### Moved `Codata` modules to `Codata.Sized`
+### Changes to `LeftCancellative` and `RightCancellative` in `Algebra.Definitions`
 
-* Due to the change in Agda 2.6.2 where sized types are no longer compatible
-  with the `--safe` flag, it has become clear that a third variant of codata
-  will be needed using coinductive records. Therefore all existing modules in
-  `Codata` which used sized types have been moved inside a new folder named
-  `Sized`, e.g. `Codata.Stream` has become `Codata.Sized.Stream`.
+* The definitions of the types for cancellativity in `Algebra.Definitions` previously
+  made some of their arguments implicit. This was under the assumption that the operators were
+  defined by pattern matching on the left argument so that Agda could always infer the
+  argument on the RHS.
 
-#### Moved `Category` modules to `Effect`
+* Although many of the operators defined in the library follow this convention, this is not
+  always true and cannot be assumed in user's code.
 
-* As observed by Wen Kokke in Issue #1636, it no longer really makes sense
-  to group the modules which correspond to the variety of concepts of
-  (effectful) type constructor arising in functional programming (esp. in haskell)
-  such as `Monad`, `Applicative`, `Functor`, etc, under `Category.*`, as this
-  obstructs the importing of the `agda-categories` development into the Standard Library,
-  and moreover needlessly restricts the applicability of categorical concepts to this
-  (highly specific) mode of use. Correspondingly, client modules grouped under `*.Categorical.*` which exploit such structure for effectful programming have been renamed `*.Effectful`, with the originals being deprecated.
+* Therefore the definitions have been changed as follows to make all their arguments explicit:
+  - `LeftCancellative _•_`
+    - From: `∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z`
+    - To: `∀ x y z → (x • y) ≈ (x • z) → y ≈ z`
 
-### Improvements to pretty printing and regexes
+  - `RightCancellative _•_`
+    - From: `∀ {x} y z → (y • x) ≈ (z • x) → y ≈ z`
+    - To: `∀ x y z → (y • x) ≈ (z • x) → y ≈ z`
 
-* In `Text.Pretty`, `Doc` is now a record rather than a type alias. This
-  helps Agda reconstruct the `width` parameter when the module is opened
-  without it being applied. In particular this allows users to write
-  width-generic pretty printers and only pick a width when calling the
-  renderer by using this import style:
-  ```
-  open import Text.Pretty using (Doc; render)
-  --                      ^-- no width parameter for Doc & render
-  open module Pretty {w} = Text.Pretty w hiding (Doc; render)
-  --                 ^-- implicit width parameter for the combinators
+  - `AlmostLeftCancellative e _•_`
+    - From: `∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
 
-  pretty : Doc w
-  pretty = ? -- you can use the combinators here and there won't be any
-             -- issues related to the fact that `w` cannot be reconstructed
-             -- anymore
+  - `AlmostRightCancellative e _•_`
+    - From: `∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
 
-  main = do
-    -- you can now use the same pretty with different widths:
-    putStrLn $ render 40 pretty
-    putStrLn $ render 80 pretty
-  ```
+* Correspondingly some proofs of the above types will need additional arguments passed explicitly.
+  Instances can easily be fixed by adding additional underscores, e.g.
+  - `∙-cancelˡ x` to `∙-cancelˡ x _ _`
+  - `∙-cancelʳ y z` to `∙-cancelʳ _ y z`
 
-* In `Text.Regex.Search` the `searchND` function finding infix matches has
-  been tweaked to make sure the returned solution is a local maximum in terms
-  of length. It used to be that we would make the match start as late as
-  possible and finish as early as possible. It's now the reverse.
+### Changes to ring structures in `Algebra`
 
-  So `[a-zA-Z]+.agdai?` run on "the path _build/Main.agdai corresponds to"
-  will return "Main.agdai" when it used to be happy to just return "n.agda".
+* Several ring-like structures now have the multiplicative structure defined by
+  its laws rather than as a substructure, to avoid repeated proofs that the
+  underlying relation is an equivalence. These are:
+  * `IsNearSemiring`
+  * `IsSemiringWithoutOne`
+  * `IsSemiringWithoutAnnihilatingZero`
+  * `IsRing`
+
+* To aid with migration, structures matching the old style ones have been added
+  to `Algebra.Structures.Biased`, with conversion functions:
+  * `IsNearSemiring*` and `isNearSemiring*`
+  * `IsSemiringWithoutOne*` and `isSemiringWithoutOne*`
+  * `IsSemiringWithoutAnnihilatingZero*` and `isSemiringWithoutAnnihilatingZero*`
+  * `IsRing*` and `isRing*`
 
-### Refactoring of algebraic lattice hierarchy
+### Refactoring of lattices in `Algebra.Structures/Bundles` hierarchy
 
 * In order to improve modularity and consistency with `Relation.Binary.Lattice`,
   the structures & bundles for `Semilattice`, `Lattice`, `DistributiveLattice`
@@ -265,9 +162,9 @@ Non-backwards compatible changes
   `Algebra.Lattice.Properties.Semilattice` or `Algebra.Lattice.Morphism.Lattice`). See
   the `Deprecated modules` section below for full details.
 
-* Changed definition of `IsDistributiveLattice` and `IsBooleanAlgebra` so that they are
-  no longer right-biased which hinders compositionality. More concretely, `IsDistributiveLattice`
-  now has fields:
+* The definition of `IsDistributiveLattice` and `IsBooleanAlgebra` have changed so 
+  that they are no longer right-biased which hindered compositionality. 
+  More concretely, `IsDistributiveLattice` now has fields:
   ```agda
   ∨-distrib-∧ : ∨ DistributesOver ∧
   ∧-distrib-∨ : ∧ DistributesOver ∨
@@ -277,7 +174,7 @@ Non-backwards compatible changes
   ∨-distribʳ-∧ : ∨ DistributesOverʳ ∧
   ```
   and `IsBooleanAlgebra` now has fields:
-  ```
+  ```agda
   ∨-complement : Inverse ⊤ ¬ ∨
   ∧-complement : Inverse ⊥ ¬ ∧
   ```
@@ -310,140 +207,326 @@ Non-backwards compatible changes
 * Added new aliases `Is(Meet/Join)(Bounded)Semilattice` for `Is(Bounded)Semilattice`
   which can be used to indicate meet/join-ness of the original structures.
 
-#### Removal of the old function hierarchy
-
-* The switch to the new function hierarchy is complete and the following modules
-  have been completely switched over to use the new definitions:
-  ```
-  Data.Sum.Function.Setoid
-  Data.Sum.Function.Propositional
-  ```
-
-* Additionally the following proofs now use the new definitions instead of the old ones:
-  * `Algebra.Lattice.Properties.BooleanAlgebra`
-  * `Algebra.Properties.CommutativeMonoid.Sum`
-  * `Algebra.Properties.Lattice`
-    * `replace-equality`
-  * `Data.Fin.Properties`
-    * `∀-cons-⇔`
-    * `⊎⇔∃`
-  * `Data.Fin.Subset.Properties`
-    * `out⊆-⇔`
-    * `in⊆in-⇔`
-    * `out⊂in-⇔`
-    * `out⊂out-⇔`
-    * `in⊂in-⇔`
-    * `x∈⁅y⁆⇔x≡y`
-    * `∩⇔×`
-    * `∪⇔⊎`
-    * `∃-Subset-[]-⇔`
-    * `∃-Subset-∷-⇔`
-  * `Data.List.Countdown`
-    * `empty`
-  * `Data.List.Fresh.Relation.Unary.Any`
-    * `⊎⇔Any`
-  * `Data.List.NonEmpty`
-  * `Data.List.Relation.Binary.Lex`
-    * `[]<[]-⇔`
-    * `∷<∷-⇔`
-  * `Data.List.Relation.Binary.Sublist.Heterogeneous.Properties`
-    * `∷⁻¹`
-    * `∷ʳ⁻¹`
-    * `Sublist-[x]-bijection`
-  * `Data.List.Relation.Binary.Sublist.Setoid.Properties`
-    * `∷⁻¹`
-    * `∷ʳ⁻¹`
-    * `[x]⊆xs⤖x∈xs`
-  * `Data.List.Relation.Unary.Grouped.Properties`
-  * `Data.Maybe.Relation.Binary.Connected`
-    * `just-equivalence`
-  * `Data.Maybe.Relation.Binary.Pointwise`
-    * `just-equivalence`
-  * `Data.Maybe.Relation.Unary.All`
-    * `just-equivalence`
-  * `Data.Maybe.Relation.Unary.Any`
-    * `just-equivalence`
-  * `Data.Nat.Divisibility`
-    * `m%n≡0⇔n∣m`
-  * `Data.Nat.Properties`
-    * `eq?`
-  * `Data.Vec.N-ary`
-    * `uncurry-∀ⁿ`
-    * `uncurry-∃ⁿ`
-  * `Data.Vec.Relation.Binary.Lex.Core`
-    * `P⇔[]<[]`
-    * `∷<∷-⇔`
-  * `Data.Vec.Relation.Binary.Pointwise.Extensional`
-    * `equivalent`
-    * `Pointwise-≡↔≡`
-  * `Data.Vec.Relation.Binary.Pointwise.Inductive`
-    * `Pointwise-≡↔≡`
-  * `Effect.Monad.Partiality`
-    * `correct`
-  * `Relation.Binary.Construct.Closure.Reflexive.Properties`
-    * `⊎⇔Refl`
-  * `Relation.Binary.Construct.Closure.Transitive`
-    * `equivalent`
-  * `Relation.Binary.Reflection`
-    * `solve₁`
-  * `Relation.Nullary.Decidable`
-    * `map`
-
-
-#### Changes to the new function hierarchy
-
-* The names of the fields in `Function.Bundles` have been
-  changed from `f`, `g`, `cong₁` and `cong₂` to `to`, `from`, `to-cong`, `from-cong`.
+* Finally, the following auxiliary files have been moved:
+  ```agda
+  Algebra.Properties.Semilattice               ↦ Algebra.Lattice.Properties.Semilattice
+  Algebra.Properties.Lattice                   ↦ Algebra.Lattice.Properties.Lattice
+  Algebra.Properties.DistributiveLattice       ↦ Algebra.Lattice.Properties.DistributiveLattice
+  Algebra.Properties.BooleanAlgebra            ↦ Algebra.Lattice.Properties.BooleanAlgebra
+  Algebra.Properties.BooleanAlgebra.Expression ↦ Algebra.Lattice.Properties.BooleanAlgebra.Expression
+  Algebra.Morphism.LatticeMonomorphism         ↦ Algebra.Lattice.Morphism.LatticeMonomorphism
+  ```
 
-* The module `Function.Definitions` no longer has two equalities as module arguments, as
-  they did not interact as intended with the re-exports from `Function.Definitions.(Core1/Core2)`.
-  The latter have been removed and their definitions folded into `Function.Definitions`.
+#### Changes to `Algebra.Morphism.Structures`
 
-* In `Function.Definitions` the types of `Surjective`, `Injective` and `Surjective`
-  have been changed from:
+* Previously the record definitions:
   ```
-  Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
-  Inverseˡ f g = ∀ y → f (g y) ≈₂ y
-  Inverseʳ f g = ∀ x → g (f x) ≈₁ x
+  IsNearSemiringHomomorphism
+  IsSemiringHomomorphism
+  IsRingHomomorphism
   ```
-  to:
+  all had two separate proofs of `IsRelHomomorphism` within them. 
+  
+* To fix this they have all been redefined to build up from `IsMonoidHomomorphism`, 
+  `IsNearSemiringHomomorphism`, and `IsSemiringHomomorphism` respectively, 
+  adding a single property at each step. 
+  
+* Similarly, `IsLatticeHomomorphism` is now built as
+  `IsRelHomomorphism` along with proofs that `_∧_` and `_∨_` are homomorphic.
+
+* Finally `⁻¹-homo` in `IsRingHomomorphism` has been renamed to `-‿homo`.
+
+#### Renamed `Category` modules to `Effect`
+
+* As observed by Wen Kokke in issue #1636, it no longer really makes sense
+  to group the modules which correspond to the variety of concepts of
+  (effectful) type constructor arising in functional programming (esp. in Haskell)
+  such as `Monad`, `Applicative`, `Functor`, etc, under `Category.*`, 
+  as this obstructs the importing of the `agda-categories` development into 
+  the Standard Library, and moreover needlessly restricts the applicability of 
+  categorical concepts to this (highly specific) mode of use. 
+  
+* Correspondingly, client modules grouped under  `*.Categorical.*` which 
+  exploit such structure  for effectful programming have been  renamed 
+  `*.Effectful`, with the originals being deprecated.
+
+* Full list of moved modules:
+  ```agda
+  Codata.Sized.Colist.Categorical            ↦ Codata.Sized.Colist.Effectful
+  Codata.Sized.Covec.Categorical             ↦ Codata.Sized.Covec.Effectful
+  Codata.Sized.Delay.Categorical             ↦ Codata.Sized.Delay.Effectful
+  Codata.Sized.Stream.Categorical            ↦ Codata.Sized.Stream.Effectful
+  Data.List.Categorical                      ↦ Data.List.Effectful
+  Data.List.Categorical.Transformer          ↦ Data.List.Effectful.Transformer
+  Data.List.NonEmpty.Categorical             ↦ Data.List.NonEmpty.Effectful
+  Data.List.NonEmpty.Categorical.Transformer ↦ Data.List.NonEmpty.Effectful.Transformer
+  Data.Maybe.Categorical                     ↦ Data.Maybe.Effectful
+  Data.Maybe.Categorical.Transformer         ↦ Data.Maybe.Effectful.Transformer
+  Data.Product.Categorical.Examples          ↦ Data.Product.Effectful.Examples
+  Data.Product.Categorical.Left              ↦ Data.Product.Effectful.Left
+  Data.Product.Categorical.Left.Base         ↦ Data.Product.Effectful.Left.Base
+  Data.Product.Categorical.Right             ↦ Data.Product.Effectful.Right
+  Data.Product.Categorical.Right.Base        ↦ Data.Product.Effectful.Right.Base
+  Data.Sum.Categorical.Examples              ↦ Data.Sum.Effectful.Examples
+  Data.Sum.Categorical.Left                  ↦ Data.Sum.Effectful.Left
+  Data.Sum.Categorical.Left.Transformer      ↦ Data.Sum.Effectful.Left.Transformer
+  Data.Sum.Categorical.Right                 ↦ Data.Sum.Effectful.Right
+  Data.Sum.Categorical.Right.Transformer     ↦ Data.Sum.Effectful.Right.Transformer
+  Data.These.Categorical.Examples            ↦ Data.These.Effectful.Examples
+  Data.These.Categorical.Left                ↦ Data.These.Effectful.Left
+  Data.These.Categorical.Left.Base           ↦ Data.These.Effectful.Left.Base
+  Data.These.Categorical.Right               ↦ Data.These.Effectful.Right
+  Data.These.Categorical.Right.Base          ↦ Data.These.Effectful.Right.Base
+  Data.Vec.Categorical                       ↦ Data.Vec.Effectful
+  Data.Vec.Categorical.Transformer           ↦ Data.Vec.Effectful.Transformer
+  Data.Vec.Recursive.Categorical             ↦ Data.Vec.Recursive.Effectful
+  Function.Identity.Categorical              ↦ Function.Identity.Effectful
+  IO.Categorical                             ↦ IO.Effectful
+  Reflection.TCM.Categorical                 ↦ Reflection.TCM.Effectful
   ```
-  Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y
-  Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
-  Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
+  
+* Full list of new modules:
+  ```
+  Algebra.Construct.Initial
+  Algebra.Construct.Terminal
+  Data.List.Effectful.Transformer
+  Data.List.NonEmpty.Effectful.Transformer
+  Data.Maybe.Effectful.Transformer
+  Data.Sum.Effectful.Left.Transformer
+  Data.Sum.Effectful.Right.Transformer
+  Data.Vec.Effectful.Transformer
+  Effect.Empty
+  Effect.Choice
+  Effect.Monad.Error.Transformer
+  Effect.Monad.Identity
+  Effect.Monad.IO
+  Effect.Monad.IO.Instances
+  Effect.Monad.Reader.Indexed
+  Effect.Monad.Reader.Instances
+  Effect.Monad.Reader.Transformer
+  Effect.Monad.Reader.Transformer.Base
+  Effect.Monad.State.Indexed
+  Effect.Monad.State.Instances
+  Effect.Monad.State.Transformer
+  Effect.Monad.State.Transformer.Base
+  Effect.Monad.Writer
+  Effect.Monad.Writer.Indexed
+  Effect.Monad.Writer.Instances
+  Effect.Monad.Writer.Transformer
+  Effect.Monad.Writer.Transformer.Base
+  IO.Effectful
+  IO.Instances
   ```
-  This is for several reasons: i) the new definitions compose much more easily, ii) Agda
-  can better infer the equalities used.
 
-  To ease backwards compatibility:
-   - the old definitions have been moved to the new names  `StrictlySurjective`,
-         `StrictlyInverseˡ` and `StrictlyInverseʳ`.
-   - The records in  `Function.Structures` and `Function.Bundles` export proofs
-         of these under the names `strictlySurjective`, `strictlyInverseˡ` and
-         `strictlyInverseʳ`,
-   - Conversion functions have been added in both directions to
-         `Function.Consequences(.Propositional)`.
+### Refactoring of the unindexed Functor/Applicative/Monad hierarchy in `Effect`
+
+* The unindexed versions are not defined in terms of the named versions anymore.
+
+* The `RawApplicative` and `RawMonad` type classes have been relaxed so that the underlying
+  functors do not need their domain and codomain to live at the same Set level.
+  This is needed for level-increasing functors like `IO : Set l → Set (suc l)`.
+
+* `RawApplicative` is now `RawFunctor + pure + _<*>_` and `RawMonad` is now
+  `RawApplicative` + `_>>=_`.
+  This reorganisation means in particular that the functor/applicative of a monad
+  are not computed using `_>>=_`. This may break proofs.
+
+* When `F : Set f → Set f` we moreover have a definable join/μ operator
+  `join : (M : RawMonad F) → F (F A) → F A`.
+
+* We now have `RawEmpty` and `RawChoice` respectively packing `empty : M A` and
+  `(<|>) : M A → M A → M A`. `RawApplicativeZero`, `RawAlternative`, `RawMonadZero`,
+  `RawMonadPlus` are all defined in terms of these.
+
+* `MonadT T` now returns a `MonadTd` record that packs both a proof that the
+  `Monad M` transformed by `T` is a monad and that we can `lift` a computation
+  `M A` to a transformed computation `T M A`.
+
+* The monad transformer are not mere aliases anymore, they are record-wrapped
+  which allows constraints such as `MonadIO (StateT S (ReaderT R IO))` to be
+  discharged by instance arguments.
+
+* The mtl-style type classes (`MonadState`, `MonadReader`) do not contain a proof
+  that the underlying functor is a `Monad` anymore. This ensures we do not have
+  conflicting `Monad M` instances from a pair of `MonadState S M` & `MonadReader R M`
+  constraints.
+
+* `MonadState S M` is now defined in terms of
+  ```agda
+  gets : (S → A) → M A
+  modify : (S → S) → M ⊤
+  ```
+  with `get` and `put` defined as derived notions.
+  This is needed because `MonadState S M` does not pack a `Monad M` instance anymore
+  and so we cannot define `modify f` as `get >>= λ s → put (f s)`.
+
+* `MonadWriter 𝕎 M` is defined similarly:
+   ```agda
+   writer : W × A → M A
+   listen : M A → M (W × A)
+   pass   : M ((W → W) × A) → M A
+   ```
+   with `tell` defined as a derived notion.
+   Note that `𝕎` is a `RawMonoid`, not a `Set` and `W` is the carrier of the monoid.
 
-#### Proofs of non-zeroness/positivity/negativity as instance arguments
+
+#### Moved `Codata` modules to `Codata.Sized`
+
+* Due to the change in Agda 2.6.2 where sized types are no longer compatible
+  with the `--safe` flag, it has become clear that a third variant of codata
+  will be needed using coinductive records. 
+  
+* Therefore all existing modules in `Codata` which used sized types have been 
+  moved inside a new folder named `Codata.Sized`,  e.g. `Codata.Stream` 
+  has become `Codata.Sized.Stream`.
+
+### New proof-irrelevant for empty type in `Data.Empty`
+
+* The definition of `⊥` has been changed to
+  ```agda
+  private
+    data Empty : Set where
+
+  ⊥ : Set
+  ⊥ = Irrelevant Empty
+  ```
+  in order to make ⊥ proof irrelevant. Any two proofs of `⊥` or of a negated
+  statements are now *judgmentally* equal to each other.
+
+* Consequently the following two definitions have been modified:
+
+  + In `Relation.Nullary.Decidable.Core`, the type of `dec-no` has changed
+    ```agda
+    dec-no : (p? : Dec P) → ¬ P → ∃ λ ¬p′ → p? ≡ no ¬p′
+      ↦ 
+    dec-no : (p? : Dec P) (¬p : ¬ P) → p? ≡ no ¬p
+    ```
+
+  + In `Relation.Binary.PropositionalEquality`, the type of `≢-≟-identity` has changed
+    ```agda
+    ≢-≟-identity : x ≢ y → ∃ λ ¬eq → x ≟ y ≡ no ¬eq
+      ↦ 
+    ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y
+    ```
+
+### Deprecation of `_≺_` in `Data.Fin.Base`
+
+* In `Data.Fin.Base` the relation `_≺_` and its single constructor `_≻toℕ_`
+  have been deprecated in favour of their extensional equivalent `_<_`
+  but omitting the inversion principle which pattern matching on `_≻toℕ_`
+  would achieve; this instead is proxied by the property `Data.Fin.Properties.toℕ<`.
+
+* Consequently in `Data.Fin.Induction`:
+  ```
+  ≺-Rec
+  ≺-wellFounded
+  ≺-recBuilder
+  ≺-rec
+  ```
+  these functions are also deprecated.
+  
+* Likewise in `Data.Fin.Properties` the proofs `≺⇒<′` and `<′⇒≺` have been deprecated
+  in favour of their proxy counterparts `<⇒<′` and `<′⇒<`.
+
+### Standardisation of `insertAt`/`updateAt`/`removeAt` in `Data.List`/`Data.Vec`
+
+* Previously, the names and argument order of index-based insertion, update and removal functions for
+  various types of lists and vectors were inconsistent.
+
+* To fix this the names have all been standardised to `insertAt`/`updateAt`/`removeAt`.
+
+* Correspondingly the following changes have occurred:
+
+* In `Data.List.Base` the following have been added:
+  ```agda
+  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
+  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
+  removeAt : (xs : List A) → Fin (length xs) → List A
+  ```
+  and the following has been deprecated
+  ```
+  _─_ ↦ removeAt
+  ```
+
+* In `Data.Vec.Base`:
+  ```agda
+  insert ↦ insertAt
+  remove ↦ removeAt
+
+  updateAt : Fin n → (A → A) → Vec A n → Vec A n
+    ↦
+  updateAt : Vec A n → Fin n → (A → A) → Vec A n
+  ```
+
+* In `Data.Vec.Functional`:
+  ```agda
+  remove : Fin (suc n) → Vector A (suc n) → Vector A n
+    ↦
+  removeAt : Vector A (suc n) → Fin (suc n) → Vector A n
+
+  updateAt : Fin n → (A → A) → Vector A n → Vector A n
+    ↦
+  updateAt : Vector A n → Fin n → (A → A) → Vector A n
+  ```
+
+* The old names (and the names of all proofs about these functions) have been deprecated appropriately.
+
+#### Standardisation of `lookup` in `Data.(List/Vec/...)`
+
+* All the types of `lookup` functions (and variants) in the following modules 
+  have been changed to match the argument convention adopted in the `List` module (i.e. 
+  `lookup` takes its container first and the index, whose type may depend on the 
+  container value, second):
+  ```
+  Codata.Guarded.Stream
+  Codata.Guarded.Stream.Relation.Binary.Pointwise
+  Codata.Musical.Colist.Base
+  Codata.Musical.Colist.Relation.Unary.Any.Properties
+  Codata.Musical.Covec
+  Codata.Musical.Stream
+  Codata.Sized.Colist
+  Codata.Sized.Covec
+  Codata.Sized.Stream
+  Data.Vec.Relation.Unary.All
+  Data.Star.Environment
+  Data.Star.Pointer
+  Data.Star.Vec
+  Data.Trie
+  Data.Trie.NonEmpty
+  Data.Tree.AVL
+  Data.Tree.AVL.Indexed
+  Data.Tree.AVL.Map
+  Data.Tree.AVL.NonEmpty
+  Data.Vec.Recursive
+  ```
+
+* To accomodate this in in `Data.Vec.Relation.Unary.Linked.Properties`
+  and `Codata.Guarded.Stream.Relation.Binary.Pointwise`, the proofs
+  called `lookup` have been renamed `lookup⁺`.
+
+#### Changes to `Data.(Nat/Integer/Rational)` proofs of `NonZero`/`Positive`/`Negative` to instance arguments
 
 * Many numeric operations in the library require their arguments to be non-zero,
   and various proofs require their arguments to be non-zero/positive/negative etc.
   As discussed on the [mailing list](https://lists.chalmers.se/pipermail/agda/2021/012693.html),
-  the previous way of constructing and passing round these proofs was extremely clunky and lead
-  to messy and difficult to read code. We have therefore changed every occurrence where we need
-  a proof of non-zeroness/positivity/etc. to take it as an irrelevant
+  the previous way of constructing and passing round these proofs was extremely 
+  clunky and lead to messy and difficult to read code.
+  
+* We have therefore changed every occurrence where we need a proof of 
+  non-zeroness/positivity/etc. to take it as an irrelevant 
   [instance argument](https://agda.readthedocs.io/en/latest/language/instance-arguments.html).
-  See the mailing list for a fuller explanation of the motivation and implementation.
+  See the mailing list discussion for a fuller explanation of the motivation and implementation.
 
-* For example, whereas the type signature of division used to be:
-  ```
+* For example, whereas the type of division over `ℕ` used to be:
+  ```agda
   _/_ : (dividend divisor : ℕ) {≢0 : False (divisor ≟ 0)} → ℕ
   ```
   it is now:
-  ```
+  ```agda
   _/_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ
   ```
-  which means that as long as an instance of `NonZero n` is in scope then you can write
+
+* This means that as long as an instance of `NonZero n` is in scope then you can write
   `m / n` without having to explicitly provide a proof, as instance search will fill it in
   for you. The full list of such operations changed is as follows:
     - In `Data.Nat.DivMod`: `_/_`, `_%_`, `_div_`, `_mod_`
@@ -454,62 +537,120 @@ Non-backwards compatible changes
 
 * At the moment, there are 4 different ways such instance arguments can be provided,
   listed in order of convenience and clarity:
-    1. *Automatic basic instances* - the standard library provides instances based on the constructors of each
-       numeric type in `Data.X.Base`. For example, `Data.Nat.Base` constrains an instance of `NonZero (suc n)` for any `n`
-       and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`. Consequently,
-       if the argument is of the required form, these instances will always be filled in by instance search
-       automatically, e.g.
-       ```
-       0/n≡0 : 0 / suc n ≡ 0
-       ```
-    2. *Take the instance as an argument* - You can provide the instance argument as a parameter to your function
-       and Agda's instance search will automatically use it in the correct place without you having to
-       explicitly pass it, e.g.
-       ```
-       0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
-       ```
-    3. *Define the instance locally* - You can define an instance argument in scope (e.g. in a `where` clause)
-       and Agda's instance search will again find it automatically, e.g.
-       ```
-       instance
-         n≢0 : NonZero n
-         n≢0 = ...
-
-       0/n≡0 : 0 / n ≡ 0
-       ```
-    4. *Pass the instance argument explicitly* - Finally, if all else fails you can pass the
-       instance argument explicitly into the function using `{{ }}`, e.g.
-       ```
-       0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
-       ```
-       Suitable constructors for `NonZero`/`Positive` etc. can be found in `Data.X.Base`.
-
-* A full list of proofs that have changed to use instance arguments is available at the end of this file.
-  Notable changes to proofs are now discussed below.
-
-* Previously one of the hacks used in proofs was to explicitly refer to everything in the correct form,
-  e.g. if the argument `n` had to be non-zero then you would refer to the argument as `suc n` everywhere
-  instead of `n`, e.g.
+
+  1. *Automatic basic instances* - the standard library provides instances based on 
+	 the constructors of each numeric type in `Data.X.Base`. For example, 
+	 `Data.Nat.Base` constrains an instance of `NonZero (suc n)` for any `n`
+	 and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`. 
+	 Consequently, if the argument is of the required form, these instances will always 
+	 be filled in by instance search automatically, e.g.
+	```agda
+	0/n≡0 : 0 / suc n ≡ 0
+	```
+
+  2. *Add an instance argument parameter* - You can provide the instance argument as 
+	 a parameter to your function and Agda's instance search will automatically use it 
+	 in the correct place without you having to explicitly pass it, e.g.
+    ```agda
+    0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
+    ```
+
+  3. *Define the instance locally* - You can define an instance argument in scope 
+	 (e.g. in a `where` clause) and Agda's instance search will again find it automatically, 
+	 e.g.
+	```agda
+	instance
+	  n≢0 : NonZero n
+	  n≢0 = ...
+
+	0/n≡0 : 0 / n ≡ 0
+	```
+
+  4. *Pass the instance argument explicitly* - Finally, if all else fails you can pass the
+	 instance argument explicitly into the function using `{{ }}`, e.g.
+	 ```
+	 0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
+	 ```
+
+* Suitable constructors for `NonZero`/`Positive` etc. can be found in `Data.X.Base`.
+
+* A full list of proofs that have changed to use instance arguments is available 
+  at the end of this file. Notable changes to proofs are now discussed below.
+
+* Previously one of the hacks used in proofs was to explicitly refer to everything
+  in the correct form, e.g. if the argument `n` had to be non-zero then you would 
+  refer to the argument as `suc n` everywhere instead of `n`, e.g.
   ```
   n/n≡1 : ∀ n → suc n / suc n ≡ 1
   ```
-  This made the proofs extremely difficult to use if your term wasn't in the right form.
+  This made the proofs extremely difficult to use if your term wasn't in the form `suc n`.
   After being updated to use instance arguments instead, the proof above becomes:
   ```
   n/n≡1 : ∀ n {{_ : NonZero n}} → n / n ≡ 1
   ```
-  However, note that this means that if you passed in the value `x` to these proofs before, then you
-  will now have to pass in `suc x`. The proofs for which the arguments have changed form in this way
-  are highlighted in the list at the bottom of the file.
+  However, note that this means that if you passed in the value `x` to these proofs 
+  before, then you will now have to pass in `suc x`. The proofs for which the 
+  arguments have changed form in this way are highlighted in the list at the bottom 
+  of the file.
 
-* Finally, the definition of `_≢0` has been removed from `Data.Rational.Unnormalised.Base`
-  and the following proofs about it have been removed from `Data.Rational.Unnormalised.Properties`:
+* Finally, in `Data.Rational.Unnormalised.Base` the definition of `_≢0` is now
+  redundent and so has been removed. Additionally the following proofs about it have 
+  also been removed from `Data.Rational.Unnormalised.Properties`:
   ```
   p≄0⇒∣↥p∣≢0 : ∀ p → p ≠ 0ℚᵘ → ℤ.∣ (↥ p) ∣ ≢0
   ∣↥p∣≢0⇒p≄0 : ∀ p → ℤ.∣ (↥ p) ∣ ≢0 → p ≠ 0ℚᵘ
   ```
 
-### Change in reduction behaviour of rationals
+### Changes to the definition of `_≤″_` in `Data.Nat.Base` (issue #1919)
+
+* The definition of `_≤″_` was previously:
+  ```agda
+  record _≤″_ (m n : ℕ) : Set where
+    constructor less-than-or-equal
+    field
+      {k}   : ℕ
+      proof : m + k ≡ n
+  ```
+  which introduced a spurious additional definition, when this is in fact, modulo
+  field names and implicit/explicit qualifiers, equivalent to the definition of left-
+  divisibility, `_∣ˡ_` for the `RawMagma` structure of `_+_`. 
+  
+* Since the addition of raw bundles to `Data.X.Base`, this definition can now be 
+  made directly. Accordingly, the definition has been changed to:
+  ```agda
+  _≤″_ : (m n : ℕ)  → Set
+  _≤″_ = _∣ˡ_ +-rawMagma
+
+  pattern less-than-or-equal {k} prf = k , prf
+  ```
+
+* Knock-on consequences include the need to retain the old constructor
+  name, now introduced as a pattern synonym, and introduction of (a function 
+  equivalent to) the former field name/projection function `proof` as 
+  `≤″-proof` in `Data.Nat.Properties`.
+
+### Changes to definition of `Prime` in `Data.Nat.Primality`
+
+* The definition of `Prime` was:
+  ```agda
+  Prime 0             = ⊥
+  Prime 1             = ⊥
+  Prime (suc (suc n)) = (i : Fin n) → 2 + toℕ i ∤ 2 + n
+  ```
+  which was very hard to reason about as, not only did it involve conversion
+  to and from the `Fin` type, it also required that the divisor was of the form
+  `2 + toℕ i`, which has exactly the same problem as the `suc n` hack described
+  above used for non-zeroness.
+
+* To make it easier to use, reason about and read, the definition has been
+  changed to:
+  ```agda
+  Prime 0 = ⊥
+  Prime 1 = ⊥
+  Prime n = ∀ {d} → 2 ≤ d → d < n → d ∤ n
+  ```
+
+### Changes to operation reduction behaviour in `Data.Rational(.Unnormalised)`
 
 * Currently arithmetic expressions involving rationals (both normalised and
   unnormalised) undergo disastrous exponential normalisation. For example,
@@ -530,118 +671,114 @@ Non-backwards compatible changes
      p@record{} + q@record{} = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)
      ```
 
-* As a consequence of this, some proofs that relied on this reduction behaviour
-  or on eta-equality may no longer go through. There are several ways to fix this:
+* As a consequence of this, some proofs that relied either on this reduction behaviour
+  or on eta-equality may no longer type-check. 
+  
+* There are several ways to fix this:
+  
   1. The principled way is to not rely on this reduction behaviour in the first place.
      The `Properties` files for rational numbers have been greatly expanded in `v1.7`
      and `v2.0`, and we believe most proofs should be able to be built up from existing
      proofs contained within these files.
+  
   2. Alternatively, annotating any rational arguments to a proof with either
      `@record{}` or `@(mkℚ _ _ _)` should restore the old reduction behaviour for any
      terms involving those parameters.
+  
   3. Finally, if the above approaches are not viable then you may be forced to explicitly
      use `cong` combined with a lemma that proves the old reduction behaviour.
 
 * Similarly, in order to prevent reduction, the equality `_≃_` in `Data.Rational.Base`
-  has been made into a data type with the single constructor `*≡*`. The destructor `drop-*≡*`
-  has been added to `Data.Rational.Properties`.
+  has been made into a data type with the single constructor `*≡*`. The destructor 
+  `drop-*≡*` has been added to `Data.Rational.Properties`.
 
-### Change to the definition of `LeftCancellative` and `RightCancellative`
-
-* The definitions of the types for cancellativity in `Algebra.Definitions` previously
-  made some of their arguments implicit. This was under the assumption that the operators were
-  defined by pattern matching on the left argument so that Agda could always infer the
-  argument on the RHS.
+#### Deprecation of old `Function` hierarchy
 
-* Although many of the operators defined in the library follow this convention, this is not
-  always true and cannot be assumed in user's code.
+* The new `Function` hierarchy was introduced in `v1.2` which follows
+  the same `Core`/`Definitions`/`Bundles`/`Structures` as all the other
+  hierarchies in the library.
+  
+* At the time the old function hierarchy in:
+  ```
+  Function.Equality
+  Function.Injection
+  Function.Surjection
+  Function.Bijection
+  Function.LeftInverse
+  Function.Inverse
+  Function.HalfAdjointEquivalence
+  Function.Related
+  ```
+  was unofficially deprecated, but could not be officially deprecated because
+  of it's widespread use in the rest of the library.
+  
+* Now, the old hierarchy modules have all been officially deprecated. All 
+  uses of them in the rest of the library have been switched to use the
+  new hierarachy. 
+  
+* The latter is unfortunately a relatively big breaking change, but was judged
+  to be unavoidable given how widely used the old hierarchy was.
 
-* Therefore the definitions have been changed as follows to make all their arguments explicit:
-  - `LeftCancellative _•_`
-    - From: `∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z`
-    - To: `∀ x y z → (x • y) ≈ (x • z) → y ≈ z`
+#### Changes to the new `Function` hierarchy
 
-  - `RightCancellative _•_`
-    - From: `∀ {x} y z → (y • x) ≈ (z • x) → y ≈ z`
-    - To: `∀ x y z → (y • x) ≈ (z • x) → y ≈ z`
+* In `Function.Bundles` the names of the fields in the records have been
+  changed from `f`, `g`, `cong₁` and `cong₂` to `to`, `from`, `to-cong`, `from-cong`.
 
-  - `AlmostLeftCancellative e _•_`
-    - From: `∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
-    - To: `∀ x y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
+* In `Function.Definitions`, the module no longer has two equalities as 
+  module arguments, as they did not interact as intended with the re-exports 
+  from `Function.Definitions.(Core1/Core2)`. The latter two modules have 
+  been removed and their definitions folded into `Function.Definitions`.
 
-  - `AlmostRightCancellative e _•_`
-    - From: `∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
-    - To: `∀ x y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
+* In `Function.Definitions` the following definitions have been changed from:
+  ```
+  Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
+  Inverseˡ f g = ∀ y → f (g y) ≈₂ y
+  Inverseʳ f g = ∀ x → g (f x) ≈₁ x
+  ```
+  to:
+  ```
+  Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y
+  Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
+  Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
+  ```
+  This is for several reasons: 
+	  i) the new definitions compose much more easily, 
+	  ii) Agda can better infer the equalities used.
 
-* Correspondingly some proofs of the above types will need additional arguments passed explicitly.
-  Instances can easily be fixed by adding additional underscores, e.g.
-  - `∙-cancelˡ x` to `∙-cancelˡ x _ _`
-  - `∙-cancelʳ y z` to `∙-cancelʳ _ y z`
+* To ease backwards compatibility:
+   - the old definitions have been moved to the new names  `StrictlySurjective`,
+         `StrictlyInverseˡ` and `StrictlyInverseʳ`.
+   - The records in  `Function.Structures` and `Function.Bundles` export proofs
+         of these under the names `strictlySurjective`, `strictlyInverseˡ` and
+         `strictlyInverseʳ`,
+   - Conversion functions have been added in both directions to
+         `Function.Consequences(.Propositional)`.
 
-### Change in the definition of `Prime`
+### New `Function.Strict`
 
-* The definition of `Prime` in `Data.Nat.Primality` was:
-  ```agda
-  Prime 0             = ⊥
-  Prime 1             = ⊥
-  Prime (suc (suc n)) = (i : Fin n) → 2 + toℕ i ∤ 2 + n
-  ```
-  which was very hard to reason about as not only did it involve conversion
-  to and from the `Fin` type, it also required that the divisor was of the form
-  `2 + toℕ i`, which has exactly the same problem as the `suc n` hack described
-  above used for non-zeroness.
+* The module `Strict` has been deprecated in favour of `Function.Strict`
+  and the definitions of strict application, `_$!_` and `_$!′_`, have been
+  moved from `Function.Base` to `Function.Strict`.
 
-* To make it easier to use, reason about and read, the definition has been
-  changed to:
-  ```agda
-  Prime 0 = ⊥
-  Prime 1 = ⊥
-  Prime n = ∀ {d} → 2 ≤ d → d < n → d ∤ n
-  ```
+* The contents of `Function.Strict` is now re-exported by `Function`.
 
-### Change to the definition of `Induction.WellFounded.WfRec` (issue #2083)
+### Change to the definition of `WfRec` in `Induction.WellFounded` (issue #2083)
 
-* Previously, the following definition was adopted
+* Previously, the definition of `WfRec` was:
   ```agda
   WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
   WfRec _<_ P x = ∀ y → y < x → P y
   ```
-  with the consequence that all arguments involving about accesibility and
-  wellfoundedness proofs were polluted by almost-always-inferrable explicit
-  arguments for the `y` position. The definition has now been changed to
-  make that argument *implicit*, as
+  which meant that all arguments involving accessibility and wellfoundedness proofs 
+  were polluted by almost-always-inferrable explicit arguments for the `y` position. 
+  
+* The definition has now been changed to make that argument *implicit*, as
   ```agda
   WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
   WfRec _<_ P x = ∀ {y} → y < x → P y
   ```
 
-### Change in the definition of `_≤″_` (issue #1919)
-
-* The definition of `_≤″_` in `Data.Nat.Base` was previously:
-  ```agda
-  record _≤″_ (m n : ℕ) : Set where
-    constructor less-than-or-equal
-    field
-      {k}   : ℕ
-      proof : m + k ≡ n
-  ```
-  which introduced a spurious additional definition, when this is in fact, modulo
-  field names and implicit/explicit qualifiers, equivalent to the definition of left-
-  divisibility, `_∣ˡ_` for the `RawMagma` structure of `_+_`. Since the addition of
-  raw bundles to `Data.X.Base`, this definition can now be made directly. Knock-on
-  consequences include the need to retain the old constructor name, now introduced
-  as a pattern synonym, and introduction of (a function equivalent to) the former
-  field name/projection function `proof` as `≤″-proof` in `Data.Nat.Properties`.
-
-* Accordingly, the definition has been changed to:
-  ```agda
-  _≤″_ : (m n : ℕ)  → Set
-  _≤″_ = _∣ˡ_ +-rawMagma
-
-  pattern less-than-or-equal {k} prf = k , prf
-  ```
-
-### Renaming of `Reflection` modules
+### Reorganisation of `Reflection` modules
 
 * Under the `Reflection` module, there were various impending name clashes
   between the core AST as exposed by Agda and the annotated AST defined in
@@ -681,176 +818,40 @@ Non-backwards compatible changes
 * A new module `Reflection.AST` that re-exports the contents of the
   submodules has been added.
 
-### Implementation of division and modulus for `ℤ`
-
-* The previous implementations of `_divℕ_`, `_div_`, `_modℕ_`, `_mod_`
-  in `Data.Integer.DivMod` internally converted to the unary `Fin` datatype
-  resulting in poor performance. The implementation has been updated to
-  use the corresponding operations from `Data.Nat.DivMod` which are
-  efficiently implemented using the Haskell backend.
-
-### Strict functions
-
-* The module `Strict` has been deprecated in favour of `Function.Strict`
-  and the definitions of strict application, `_$!_` and `_$!′_`, have been
-  moved from `Function.Base` to `Function.Strict`.
-
-* The contents of `Function.Strict` is now re-exported by `Function`.
-
-### Changes to ring structures
-
-* Several ring-like structures now have the multiplicative structure defined by
-  its laws rather than as a substructure, to avoid repeated proofs that the
-  underlying relation is an equivalence. These are:
-  * `IsNearSemiring`
-  * `IsSemiringWithoutOne`
-  * `IsSemiringWithoutAnnihilatingZero`
-  * `IsRing`
-* To aid with migration, structures matching the old style ones have been added
-  to `Algebra.Structures.Biased`, with conversion functions:
-  * `IsNearSemiring*` and `isNearSemiring*`
-  * `IsSemiringWithoutOne*` and `isSemiringWithoutOne*`
-  * `IsSemiringWithoutAnnihilatingZero*` and `isSemiringWithoutAnnihilatingZero*`
-  * `IsRing*` and `isRing*`
-
-### Proof-irrelevant empty type
-
-* The definition of ⊥ has been changed to
-  ```agda
-  private
-    data Empty : Set where
-
-  ⊥ : Set
-  ⊥ = Irrelevant Empty
-  ```
-  in order to make ⊥ proof irrelevant. Any two proofs of `⊥` or of a negated
-  statements are now *judgmentally* equal to each other.
-
-* Consequently we have modified the following definitions:
-  + In `Relation.Nullary.Decidable.Core`, the type of `dec-no` has changed
-    ```agda
-    dec-no : (p? : Dec P) → ¬ P → ∃ λ ¬p′ → p? ≡ no ¬p′
-      ↦ dec-no : (p? : Dec P) (¬p : ¬ P) → p? ≡ no ¬p
-    ```
-  + In `Relation.Binary.PropositionalEquality`, the type of `≢-≟-identity` has changed
-    ```agda
-    ≢-≟-identity : x ≢ y → ∃ λ ¬eq → x ≟ y ≡ no ¬eq
-      ↦ ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y
-    ```
 
 ### Reorganisation of the `Relation.Nullary` hierarchy
 
-* It was very difficult to use the `Relation.Nullary` modules, as `Relation.Nullary`
-  contained the basic definitions of negation, decidability etc., and the operations and
-  proofs were smeared over `Relation.Nullary.(Negation/Product/Sum/Implication etc.)`.
-
-* In order to fix this:
-  - the definition of `Dec` and `recompute` have been moved to `Relation.Nullary.Decidable.Core`
-  - the definition of `Reflects` has been moved to `Relation.Nullary.Reflects`
-  - the definition of `¬_` has been moved to `Relation.Nullary.Negation.Core`
-
-* Backwards compatibility has been maintained, as `Relation.Nullary` still re-exports these publicly.
-
-* The modules:
-  ```
-  Relation.Nullary.Product
-  Relation.Nullary.Sum
-  Relation.Nullary.Implication
-  ```
-  have been deprecated and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`
-  however all their contents is re-exported by `Relation.Nullary` which is the easiest way to access
-  it now.
-
-* In order to facilitate this reorganisation the following breaking moves have occurred:
-  - `¬?` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Decidable.Core`
-  - `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to `Relation.Nullary.Reflects`.
-  - `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved from `Relation.Nullary.Negation`
-    to `Relation.Nullary.Decidable`.
-  - `fromDec` and `toDec` have been moved from `Data.Sum.Base` to `Data.Sum`.
-
-### Refactoring of the unindexed Functor/Applicative/Monad hierarchy
-
-* The unindexed versions are not defined in terms of the named versions anymore
-
-* The `RawApplicative` and `RawMonad` type classes have been relaxed so that the underlying
-  functors do not need their domain and codomain to live at the same Set level.
-  This is needed for level-increasing functors like `IO : Set l → Set (suc l)`.
-
-* `RawApplicative` is now `RawFunctor + pure + _<*>_` and `RawMonad` is now
-  `RawApplicative` + `_>>=_`.
-  This reorganisation means in particular that the functor/applicative of a monad
-  are not computed using `_>>=_`. This may break proofs.
-
-* When `F : Set f → Set f` we moreover have a definable join/μ operator
-  `join : (M : RawMonad F) → F (F A) → F A`.
-
-* We now have `RawEmpty` and `RawChoice` respectively packing `empty : M A` and
-  `(<|>) : M A → M A → M A`. `RawApplicativeZero`, `RawAlternative`, `RawMonadZero`,
-  `RawMonadPlus` are all defined in terms of these.
-
-* `MonadT T` now returns a `MonadTd` record that packs both a proof that the
-  `Monad M` transformed by `T` is a monad and that we can `lift` a computation
-  `M A` to a transformed computation `T M A`.
-
-* The monad transformer are not mere aliases anymore, they are record-wrapped
-  which allows constraints such as `MonadIO (StateT S (ReaderT R IO))` to be
-  discharged by instance arguments.
-
-* The mtl-style type classes (`MonadState`, `MonadReader`) do not contain a proof
-  that the underlying functor is a `Monad` anymore. This ensures we do not have
-  conflicting `Monad M` instances from a pair of `MonadState S M` & `MonadReader R M`
-  constraints.
-
-* `MonadState S M` is now defined in terms of
-  ```agda
-  gets : (S → A) → M A
-  modify : (S → S) → M ⊤
-  ```
-  with `get` and `put` defined as derived notions.
-  This is needed because `MonadState S M` does not pack a `Monad M` instance anymore
-  and so we cannot define `modify f` as `get >>= λ s → put (f s)`.
-
-* `MonadWriter 𝕎 M` is defined similarly:
-   ```agda
-   writer : W × A → M A
-   listen : M A → M (W × A)
-   pass   : M ((W → W) × A) → M A
-   ```
-   with `tell` defined as a derived notion.
-   Note that `𝕎` is a `RawMonoid`, not a `Set` and `W` is the carrier of the monoid.
+* It was very difficult to use the `Relation.Nullary` modules, as 
+  `Relation.Nullary` contained the basic definitions of negation, decidability etc., 
+  and the operations and proofs about these definitions were spread over 
+  `Relation.Nullary.(Negation/Product/Sum/Implication etc.)`.
+
+* To fix this all the contents of the latter is now exported by `Relation.Nullary`.
+
+* In order to achieve this the following backwards compatible changes have been made:
+  
+  1. the definition of `Dec` and `recompute` have been moved to `Relation.Nullary.Decidable.Core`
+  
+  2. the definition of `Reflects` has been moved to `Relation.Nullary.Reflects`
+  
+  3. the definition of `¬_` has been moved to `Relation.Nullary.Negation.Core`
+
+  4. The modules `Relation.Nullary.(Product/Sum/Implication)` have been deprecated 
+	 and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`.
+
+  as well as the following breaking changes:
+
+  1. `¬?` has been moved from `Relation.Nullary.Negation.Core` to 
+	 `Relation.Nullary.Decidable.Core`
+  
+  2. `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to 
+	 `Relation.Nullary.Reflects`.
+  
+  3. `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved 
+	 from `Relation.Nullary.Negation` to `Relation.Nullary.Decidable`.
+  
+  4. `fromDec` and `toDec` have been moved from `Data.Sum.Base` to `Data.Sum`.
 
-* New modules:
-  ```
-  Algebra.Construct.Initial
-  Algebra.Construct.Terminal
-  Data.List.Effectful.Transformer
-  Data.List.NonEmpty.Effectful.Transformer
-  Data.Maybe.Effectful.Transformer
-  Data.Sum.Effectful.Left.Transformer
-  Data.Sum.Effectful.Right.Transformer
-  Data.Vec.Effectful.Transformer
-  Effect.Empty
-  Effect.Choice
-  Effect.Monad.Error.Transformer
-  Effect.Monad.Identity
-  Effect.Monad.IO
-  Effect.Monad.IO.Instances
-  Effect.Monad.Reader.Indexed
-  Effect.Monad.Reader.Instances
-  Effect.Monad.Reader.Transformer
-  Effect.Monad.Reader.Transformer.Base
-  Effect.Monad.State.Indexed
-  Effect.Monad.State.Instances
-  Effect.Monad.State.Transformer
-  Effect.Monad.State.Transformer.Base
-  Effect.Monad.Writer
-  Effect.Monad.Writer.Indexed
-  Effect.Monad.Writer.Instances
-  Effect.Monad.Writer.Transformer
-  Effect.Monad.Writer.Transformer.Base
-  IO.Effectful
-  IO.Instances
-  ```
 
 ### (Issue #2096) Introduction of flipped and negated relation symbols to bundles in `Relation.Binary.Bundles`
 
@@ -871,82 +872,22 @@ Non-backwards compatible changes
   of the old symbol in the record. Therefore, only _declarations_ of `PartialOrder` records will 
   need their field names updating.
 
-### (Issue #1214) Reorganisation of the introduction of negated relation symbols under `Relation.Binary`
-
-* Previously, negated relation symbols `_≰_` (for `Poset`) and `_≮_` (`TotalOrder`)
-  were introduced in the corresponding `Relation.Binary.Properties` modules, for re-export.
-
-* Now they are introduced as definitions in the corresponding `Relation.Binary.Bundles`,
-  together with, for uniformity's sake, an additional negated symbol `_⋦_` for `Preorder`,
-  with their (obvious) intended semantics:
-  ```agda
-  infix 4 _⋦_ _≰_ _≮_
-  Preorder._⋦_            : Rel Carrier _
-  StrictPartialOrder._≮_  : Rel Carrier _
-  ```
-  Additionally, `Poset._≰_` is defined by renaming public export of `Preorder._⋦_`
-
-* Backwards compatibility has been maintained, with deprecated definitions in the
-  corresponding `Relation.Binary.Properties` modules, and the corresponding client
-  client module `import`s being adjusted accordingly.
-
-
-### Standardisation of `insertAt`/`updateAt`/`removeAt`
-
-* Previously, the names and argument order of index-based insertion, update and removal functions for
-  various types of lists and vectors were inconsistent.
-
-* To fix this the names have all been standardised to `insertAt`/`updateAt`/`removeAt`.
-
-* Correspondingly the following changes have occurred:
-
-* In `Data.List.Base` the following have been added:
-  ```agda
-  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
-  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
-  removeAt : (xs : List A) → Fin (length xs) → List A
-  ```
-  and the following has been deprecated
-  ```
-  _─_ ↦ removeAt
-  ```
-
-* In `Data.Vec.Base`:
-  ```agda
-  insert ↦ insertAt
-  remove ↦ removeAt
-
-  updateAt : Fin n → (A → A) → Vec A n → Vec A n
-    ↦
-  updateAt : Vec A n → Fin n → (A → A) → Vec A n
-  ```
-
-* In `Data.Vec.Functional`:
-  ```agda
-  remove : Fin (suc n) → Vector A (suc n) → Vector A n
-    ↦
-  removeAt : Vector A (suc n) → Fin (suc n) → Vector A n
-
-  updateAt : Fin n → (A → A) → Vector A n → Vector A n
-    ↦
-  updateAt : Vector A n → Fin n → (A → A) → Vector A n
-  ```
-
-* The old names (and the names of all proofs about these functions) have been deprecated appropriately.
 
-### Changes to definition of `IsStrictTotalOrder`
+### Changes to definition of `IsStrictTotalOrder` in `Relation.Binary.Structures`
 
-* The previous definition of the record `IsStrictTotalOrder` did not build upon `IsStrictPartialOrder`
-  as would be expected. Instead it omitted several fields like irreflexivity as they were derivable from the
-  proof of trichotomy. However, this led to problems further up the hierarchy where bundles such as `StrictTotalOrder`
-  which contained multiple distinct proofs of `IsStrictPartialOrder`.
+* The previous definition of the record `IsStrictTotalOrder` did not 
+  build upon `IsStrictPartialOrder` as would be expected. 
+  Instead it omitted several fields like irreflexivity as they were derivable from the
+  proof of trichotomy. However, this led to problems further up the hierarchy where 
+  bundles such as `StrictTotalOrder` which contained multiple distinct proofs of 
+  `IsStrictPartialOrder`.
 
-* To remedy this the definition of `IsStrictTotalOrder` has been changed to so that it builds upon
-  `IsStrictPartialOrder` as would be expected.
+* To remedy this the definition of `IsStrictTotalOrder` has been changed to so 
+  that it builds upon `IsStrictPartialOrder` as would be expected.
 
-* To aid migration, the old record definition has been moved to `Relation.Binary.Structures.Biased`
-  which contains the `isStrictTotalOrderᶜ` smart constructor (which is re-exported by `Relation.Binary`) .
-  Therefore the old code:
+* To aid migration, the old record definition has been moved to 
+  `Relation.Binary.Structures.Biased` which contains the `isStrictTotalOrderᶜ` 
+  smart constructor (which is re-exported by `Relation.Binary`) . Therefore the old code:
   ```agda
   <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
   <-isStrictTotalOrder = record
@@ -974,18 +915,20 @@ Non-backwards compatible changes
         }
   ```
 
-### Changes to triple reasoning interface
+### Changes to the interface of `Relation.Binary.Reasoning.Triple`
 
-* The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof that the strict
-  relation is irreflexive.
+* The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof 
+  that the strict relation is irreflexive.
 
-* This allows the following new proof combinator:
+* This allows the addition of the following new proof combinator:
   ```agda
   begin-contradiction : (r : x IsRelatedTo x) → {s : True (IsStrict? r)} → A
   ```
-  that takes a proof that a value is strictly less than itself and then applies the principle of explosion.
+  that takes a proof that a value is strictly less than itself and then applies the 
+  principle of explosion to derive anything.
 
-* Specialised versions of this combinator are available in the following derived modules:
+* Specialised versions of this combinator are available in the `Reasoning` modules
+  exported by the following modules:
   ```
   Data.Nat.Properties
   Data.Nat.Binary.Properties
@@ -998,10 +941,14 @@ Non-backwards compatible changes
   Relation.Binary.Reasoning.PartialOrder
   ```
 
-### More modular design of equational reasoning.
+### A more modular design for `Relation.Binary.Reasoning`
 
-* Have introduced a new module `Relation.Binary.Reasoning.Syntax` which exports
-  a range of modules containing pre-existing reasoning combinator syntax.
+* Previously, every `Reasoning` module in the library tended to roll it's own set
+  of syntax for the combinators. This hindered consistency and maintainability.
+  
+* To improve the situation, a new module `Relation.Binary.Reasoning.Syntax` 
+  has been introduced which exports a wide range of sub-modules containing 
+  pre-existing reasoning combinator syntax.
 
 * This makes it possible to add new or rename existing reasoning combinators to a
   pre-existing `Reasoning` module in just a couple of lines
@@ -1013,7 +960,7 @@ Non-backwards compatible changes
   `Relation.Binary.PropositionalEquality.Properties`.
   It is still exported by `Relation.Binary.PropositionalEquality`.
 
-### Renaming of symmetric combinators for equational reasoning
+### Renaming of symmetric combinators in `Reasoning` modules
 
 * We've had various complaints about the symmetric version of reasoning combinators 
   that use the syntax `_R˘⟨_⟩_` for some relation `R`, (e.g. `_≡˘⟨_⟩_` and `_≃˘⟨_⟩_`)
@@ -1045,59 +992,86 @@ Non-backwards compatible changes
   `Function.Base` if the `_⟨_⟩_` operator is used within the reasoning proof. The symptom
   for this is a `Don't know how to parse` error.
 
+### Improvements to `Text.Pretty` and `Text.Regex`
+
+* In `Text.Pretty`, `Doc` is now a record rather than a type alias. This
+  helps Agda reconstruct the `width` parameter when the module is opened
+  without it being applied. In particular this allows users to write
+  width-generic pretty printers and only pick a width when calling the
+  renderer by using this import style:
+  ```
+  open import Text.Pretty using (Doc; render)
+  --                      ^-- no width parameter for Doc & render
+  open module Pretty {w} = Text.Pretty w hiding (Doc; render)
+  --                 ^-- implicit width parameter for the combinators
+
+  pretty : Doc w
+  pretty = ? -- you can use the combinators here and there won't be any
+             -- issues related to the fact that `w` cannot be reconstructed
+             -- anymore
+
+  main = do
+    -- you can now use the same pretty with different widths:
+    putStrLn $ render 40 pretty
+    putStrLn $ render 80 pretty
+  ```
+
+* In `Text.Regex.Search` the `searchND` function finding infix matches has
+  been tweaked to make sure the returned solution is a local maximum in terms
+  of length. It used to be that we would make the match start as late as
+  possible and finish as early as possible. It's now the reverse.
+
+  So `[a-zA-Z]+.agdai?` run on "the path _build/Main.agdai corresponds to"
+  will return "Main.agdai" when it used to be happy to just return "n.agda".
+
 ### Other
 
 * In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
   the `--without-K` flag now use the `--cubical-compatible` flag instead.
 
-* To avoid _large indices_ that are by default no longer allowed in Agda 2.6.4,
-  universe levels have been increased in the following definitions:
-  - `Data.Star.Decoration.DecoratedWith`
-  - `Data.Star.Pointer.Pointer`
-  - `Reflection.AnnotatedAST.Typeₐ`
-  - `Reflection.AnnotatedAST.AnnotationFun`
+* In `Algebra.Core` the operations `Opₗ` and `Opᵣ` have moved to `Algebra.Module.Core`.
 
-* The first two arguments of `m≡n⇒m-n≡0` (now `i≡j⇒i-j≡0`) in `Data.Integer.Base`
-  have been made implicit.
+* In `Codata.Guarded.Stream` the following functions have been modified to have simpler definitions:
+  * `cycle`
+  * `interleave⁺`
+  * `cantor`
+  Furthermore, the direction of interleaving of `cantor` has changed. Precisely, 
+  suppose `pair` is the cantor pairing function, then `lookup (pair i j) (cantor xss)` 
+  according to the old definition corresponds to `lookup (pair j i) (cantor xss)` 
+  according to the new definition. 
+  For a concrete example see the one included at the end of the module.
 
-* The relations `_≤_` `_≥_` `_<_` `_>_` in `Data.Fin.Base` have been
+* In `Data.Fin.Base` the relations `_≤_` `_≥_` `_<_` `_>_`  have been
   generalised so they can now relate `Fin` terms with different indices.
   Should be mostly backwards compatible, but very occasionally when proving
   properties about the orderings themselves the second index must be provided
   explicitly.
 
-* The argument `xs` in `xs≮[]` in `Data.{List|Vec}.Relation.Binary.Lex.Strict`
-  introduced in PRs #1648 and #1672 has now been made implicit.
+* In `Data.Fin.Properties` the proof `inj⇒≟` that an injection from a type 
+  `A` into `Fin n` induces a decision procedure for `_≡_` on `A` has been 
+  generalized to other equivalences over `A` (i.e. to arbitrary setoids), and 
+  renamed from `eq?` to the more descriptive  and `inj⇒decSetoid`.
 
-* Issue #2075 (Johannes Waldmann): wellfoundedness of the lexicographic ordering
-  on products, defined in `Data.Product.Relation.Binary.Lex.Strict`, no longer
-  requires the assumption of symmetry for the first equality relation `_≈₁_`,
-  leading to a new lemma `Induction.WellFounded.Acc-resp-flip-≈`, and refactoring
-  of the previous proof `Induction.WellFounded.Acc-resp-≈`.
+* In `Data.Fin.Properties` the proof `pigeonhole` has been strengthened
+  so that the a proof that `i < j` rather than a mere `i ≢ j` is returned.
 
-* The operation `SymClosure` on relations in
-  `Relation.Binary.Construct.Closure.Symmetric` has been reimplemented
-  as a data type `SymClosure _⟶_ a b` that is parameterized by the
-  input relation `_⟶_` (as well as the elements `a` and `b` of the
-  domain) so that `_⟶_` can be inferred, which it could not from the
-  previous implementation using the sum type `a ⟶ b ⊎ b ⟶ a`.
+* In `Data.Fin.Substitution.TermSubst` the fixity of `_/Var_` has been changed
+  from `infix 8` to `infixl 8`.
 
-* In `Algebra.Morphism.Structures`, `IsNearSemiringHomomorphism`,
-  `IsSemiringHomomorphism`, and `IsRingHomomorphism` have been redesigned to
-  build up from `IsMonoidHomomorphism`, `IsNearSemiringHomomorphism`, and
-  `IsSemiringHomomorphism` respectively, adding a single property at each step.
-  This means that they no longer need to have two separate proofs of
-  `IsRelHomomorphism`. Similarly, `IsLatticeHomomorphism` is now built as
-  `IsRelHomomorphism` along with proofs that `_∧_` and `_∨_` are homomorphic.
+* In `Data.Integer.DivMod` the previous implementations of 
+  `_divℕ_`, `_div_`, `_modℕ_`, `_mod_` internally converted to the unary 
+  `Fin` datatype resulting in poor performance. The implementation has been 
+  updated to use the corresponding operations from `Data.Nat.DivMod` which are
+  efficiently implemented using the Haskell backend.
 
-  Also, `⁻¹-homo` in `IsRingHomomorphism` has been renamed to `-‿homo`.
+* In `Data.Integer.Properties` the first two arguments of `m≡n⇒m-n≡0` 
+  (now renamed `i≡j⇒i-j≡0`) have been made implicit.
 
-* Moved definition of `_>>=_` under `Data.Vec.Base` to its submodule `CartesianBind`
-  in order to keep another new definition of `_>>=_`, located in `DiagonalBind`
-  which is also a submodule of `Data.Vec.Base`.
+* In `Data.(List|Vec).Relation.Binary.Lex.Strict` the argument `xs` in 
+  `xs≮[]` in introduced in PRs #1648 and #1672 has now been made implicit.
 
-* The functions `split`, `flatten` and `flatten-split` have been removed from
-  `Data.List.NonEmpty`. In their place `groupSeqs` and `ungroupSeqs`
+* In `Data.List.NonEmpty` the functions `split`, `flatten` and `flatten-split` 
+  have been removed from. In their place `groupSeqs` and `ungroupSeqs`
   have been added to `Data.List.NonEmpty.Base` which morally perform the same
   operations but without computing the accompanying proofs. The proofs can be
   found in `Data.List.NonEmpty.Properties` under the names `groupSeqs-groups`
@@ -1107,138 +1081,89 @@ Non-backwards compatible changes
   have had their preconditions weakened so the equivalences no longer require congruence
   proofs.
 
-* The constructors `+0` and `+[1+_]` from `Data.Integer.Base` are no longer
-  exported by `Data.Rational.Base`. You will have to open `Data.Integer(.Base)`
+* In `Data.Nat.Divisibility` the proof `m/n/o≡m/[n*o]` has been removed. In it's
+  place a new more general proof `m/n/o≡m/[n*o]` has been added to `Data.Nat.DivMod` 
+  that doesn't require the `n * o ∣ m` pre-condition.
+
+* In `Data.Product.Relation.Binary.Lex.Strict` the proof of wellfoundedness 
+  of the lexicographic ordering on products, no longer
+  requires the assumption of symmetry for the first equality relation `_≈₁_`.
+
+* In `Data.Rational.Base` the constructors `+0` and `+[1+_]` from `Data.Integer.Base` 
+  are no longer re-exported by `Data.Rational.Base`. You will have to open `Data.Integer(.Base)`
   directly to use them.
 
-* The names of the (in)equational reasoning combinators defined in the internal
-  modules `Data.Rational(.Unnormalised).Properties.≤-Reasoning` have been renamed
-  (issue #1437) to conform with the defined setoid equality `_≃_` on `Rational`s:
-  ```agda
-  step-≈  ↦  step-≃
-  step-≈˘ ↦  step-≃˘
-  ```
-  with corresponding associated syntax:
+* In `Data.Rational(.Unnormalised).Properties` the types of the proofs 
+  `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` have been switched, 
+  as the previous naming scheme didn't correctly generalise to e.g. `pos+pos⇒pos`. 
+  For example the types of `pos⇒1/pos`/`1/pos⇒pos` were:
   ```agda
-  _≈⟨_⟩_  ↦  _≃⟨_⟩_
-  _≈˘⟨_⟩_ ↦  _≃˘⟨_⟩_
-  ```
-  NB. It is not possible to rename or deprecate `syntax` declarations, so Agda will
-  only issue a "Could not parse the application `begin ...` when scope checking"
-  warning if the old combinators are used.
-
-* The types of the proofs `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` in
-  `Data.Rational(.Unnormalised).Properties` have been switched, as the previous
-  naming scheme didn't correctly generalise to e.g. `pos+pos⇒pos`. For example
-  the types of `pos⇒1/pos`/`1/pos⇒pos` were:
-  ```
   pos⇒1/pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
   1/pos⇒pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
   ```
   but are now:
-  ```
+  ```agda
   pos⇒1/pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
   1/pos⇒pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
   ```
-* `Opₗ` and `Opᵣ` have moved from `Algebra.Core` to `Algebra.Module.Core`.
-
-* In `Data.Nat.GeneralisedArithmetic`, the `s` and `z` arguments to the following
-  functions have been made explicit:
-  * `fold-+`
-  * `fold-k`
-  * `fold-*`
-  * `fold-pull`
-
-* In `Data.Fin.Properties`:
-  + the `i` argument to `opposite-suc` has been made explicit;
-  + `pigeonhole` has been strengthened: wlog, we return a proof that
-    `i < j` rather than a mere `i ≢ j`.
 
 * In `Data.Sum.Base` the definitions `fromDec` and `toDec` have been moved to `Data.Sum`.
 
-* In `Data.Vec.Base`: the definitions `init` and `last` have been changed from the `initLast`
-  view-derived implementation to direct recursive definitions.
-
-* In `Codata.Guarded.Stream` the following functions have been modified to have simpler definitions:
-  * `cycle`
-  * `interleave⁺`
-  * `cantor`
-  Furthermore, the direction of interleaving of `cantor` has changed. Precisely, suppose `pair` is the cantor pairing function, then `lookup (pair i j) (cantor xss)` according to the old definition corresponds to `lookup (pair j i) (cantor xss)` according to the new definition. For a concrete example see the one included at the end of the module.
-
-* Removed `m/n/o≡m/[n*o]` from `Data.Nat.Divisibility` and added a more general
-  `m/n/o≡m/[n*o]` to `Data.Nat.DivMod` that doesn't require `n * o ∣ m`.
-
-* Updated `lookup` functions (and variants) to match the conventions adopted in the
-  `List` module i.e. `lookup` takes its container first and the index (whose type may
-  depend on the container value) second.
-  This affects modules:
-  ```
-  Codata.Guarded.Stream
-  Codata.Guarded.Stream.Relation.Binary.Pointwise
-  Codata.Musical.Colist.Base
-  Codata.Musical.Colist.Relation.Unary.Any.Properties
-  Codata.Musical.Covec
-  Codata.Musical.Stream
-  Codata.Sized.Colist
-  Codata.Sized.Covec
-  Codata.Sized.Stream
-  Data.Vec.Relation.Unary.All
-  Data.Star.Environment
-  Data.Star.Pointer
-  Data.Star.Vec
-  Data.Trie
-  Data.Trie.NonEmpty
-  Data.Tree.AVL
-  Data.Tree.AVL.Indexed
-  Data.Tree.AVL.Map
-  Data.Tree.AVL.NonEmpty
-  Data.Vec.Recursive
-  Tactic.RingSolver
-  Tactic.RingSolver.Core.NatSet
-  ```
-
-* Moved & renamed from `Data.Vec.Relation.Unary.All`
-  to `Data.Vec.Relation.Unary.All.Properties`:
-  ```
-  lookup ↦ lookup⁺
-  tabulate ↦ lookup⁻
-  ```
+* In `Data.Vec.Base` the definition of `_>>=_` under `Data.Vec.Base` has been
+  moved to the submodule `CartesianBind` in order to avoid clashing with the
+  new, alternative definition of `_>>=_`, located in the second new submodule 
+  `DiagonalBind`.
 
-* Renamed in `Data.Vec.Relation.Unary.Linked.Properties`
-  and `Codata.Guarded.Stream.Relation.Binary.Pointwise`:
-  ```
-  lookup ↦ lookup⁺
-  ```
+* In `Data.Vec.Base` the definitions `init` and `last` have been changed from the `initLast`
+  view-derived implementation to direct recursive definitions.
 
-* Added the following new definitions to `Data.Vec.Relation.Unary.All`:
+* In `Data.Vec.Properties` the type of the proof `zipWith-comm` has been generalised from:
+  ```agda
+  zipWith-comm : ∀ {f : A → A → B} (comm : ∀ x y → f x y ≡ f y x) (xs ys : Vec A n) → 
+                 zipWith f xs ys ≡ zipWith f ys xs
   ```
-  lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
-  lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
-  lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
-  lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
+  to
+  ```agda
+  zipWith-comm : ∀ {f : A → B → C} {g : B → A → C}  (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys → 
+                 zipWith f xs ys ≡ zipWith g ys xs
   ```
+  
+* In `Data.Vec.Relation.Unary.All` the functions `lookup` and `tabulate` have
+  been moved to `Data.Vec.Relation.Unary.All.Properties` and renamed
+  `lookup⁺` and `lookup⁻` respectively.
 
-* `excluded-middle` in `Relation.Nullary.Decidable.Core` has been renamed to
-  `¬¬-excluded-middle`.
-
-* `iterate` and `replicate` in `Data.Vec.Base` and `Data.Vec.Functional`
-  now take the length of vector, `n`, as an explicit rather than an implicit argument.
+* In `Data.Vec.Base` and `Data.Vec.Functional` the functions `iterate` and `replicate`
+  now take the length of vector, `n`, as an explicit rather than an implicit argument, i.e.
+  the new types are:
   ```agda
   iterate : (A → A) → A → ∀ n → Vec A n
   replicate : ∀ n → A → Vec A n
   ```
+  
+* In `Relation.Binary.Construct.Closure.Symmetric` the operation 
+  `SymClosure` on relations in has been reimplemented
+  as a data type `SymClosure _⟶_ a b` that is parameterized by the
+  input relation `_⟶_` (as well as the elements `a` and `b` of the
+  domain) so that `_⟶_` can be inferred, which it could not from the
+  previous implementation using the sum type `a ⟶ b ⊎ b ⟶ a`.
+
+* In `Relation.Nullary.Decidable.Core` the name `excluded-middle` has been 
+  renamed to `¬¬-excluded-middle`.
 
-Major improvements
-------------------
+
+Other major improvements
+------------------------
 
 ### Improvements to ring solver tactic
 
 * The ring solver tactic has been greatly improved. In particular:
+
   1. When the solver is used for concrete ring types, e.g. ℤ, the equality can now use
      all the ring operations defined natively for that type, rather than having
      to use the operations defined in `AlmostCommutativeRing`. For example
      previously you could not use `Data.Integer.Base._*_` but instead had to
      use `AlmostCommutativeRing._*_`.
+
   2. The solver now supports use of the subtraction operator `_-_` whenever
      it is defined immediately in terms of `_+_` and `-_`. This is the case for
      `Data.Integer` and `Data.Rational`.
@@ -1256,47 +1181,41 @@ Major improvements
 * Beneficiaries of this change include `Data.Rational.Unnormalised.Base` whose
   dependencies are now significantly smaller.
 
-### Moved raw bundles from Data.X.Properties to Data.X.Base
-
-* As mentioned by MatthewDaggitt in Issue #1755, Raw bundles defined in Data.X.Properties
-  should be defined in Data.X.Base as they don't require any properties.
-  * Moved raw bundles From `Data.Nat.Properties` to `Data.Nat.Base`
-  * Moved raw bundles From `Data.Nat.Binary.Properties` to `Data.Nat.Binary.Base`
-  * Moved raw bundles From `Data.Rational.Properties` to `Data.Rational.Base`
-  * Moved raw bundles From `Data.Rational.Unnormalised.Properties` to `Data.Rational.Unnormalised.Base`
-
-### Moved the definition of RawX from `Algebra.X.Bundles` to `Algebra.X.Bundles.Raw`
-
-* A new module `Algebra.Bundles.Raw` containing the definitions of the raw bundles
-  can be imported at much lower cost from `Data.X.Base`.
-  The following definitions have been moved:
-  * `RawMagma`
-  * `RawMonoid`
-  * `RawGroup`
-  * `RawNearSemiring`
-  * `RawSemiring`
-  * `RawRingWithoutOne`
-  * `RawRing`
-  * `RawQuasigroup`
-  * `RawLoop`
-  * `RawKleeneAlgebra`
-* A new module `Algebra.Lattice.Bundles.Raw` is also introduced.
-  * `RawLattice` has been moved from `Algebra.Lattice.Bundles` to this new module.
-
-* In `Relation.Binary.Reasoning.Base.Triple`, added a new parameter `<-irrefl : Irreflexive _≈_ _<_`
+### Moved raw bundles from `Data.X.Properties` to `Data.X.Base`
+
+* Raw bundles by design don't contain any proofs so should in theory be able to live
+  in `Data.X.Base` rather than `Data.X.Bundles`.
   
+* To achieve this while keeping the dependency footprint small, the definitions of
+  raw bundles (`RawMagma`, `RawMonoid` etc.) have been moved from `Algebra(.Lattice)Bundles` to 
+  a new module `Algebra(.Lattice).Bundles.Raw` which can be imported at much lower cost 
+  from `Data.X.Base`.
+
+* We have then moved raw bundles defined in `Data.X.Properties` to `Data.X.Base` for
+  `X` = `Nat`/`Nat.Binary`/`Integer`/`Rational`/`Rational.Unnormalised`.
+
 Deprecated modules
 ------------------
 
-### Moving `Relation.Binary.Construct.(Converse/Flip)`
+### Moving `Data.Erased` to `Data.Irrelevant`
 
-* The following files have been moved:
-  ```agda
-  Relation.Binary.Construct.Converse               ↦ Relation.Binary.Construct.Flip.EqAndOrd
-  Relation.Binary.Construct.Flip                   ↦ Relation.Binary.Construct.Flip.Ord
-  ```
+* This fixes the fact we had picked the wrong name originally. The erased modality
+  corresponds to `@0` whereas the irrelevance one corresponds to `.`.
+  
+### Deprecation of `Data.Fin.Substitution.Example`
+
+* The module `Data.Fin.Substitution.Example` has been deprecated, and moved to `README.Data.Fin.Substitution.UntypedLambda`
+
+### Deprecation of `Data.Nat.Properties.Core`
+
+* The module `Data.Nat.Properties.Core` has been deprecated, and its one lemma moved to `Data.Nat.Base`, renamed as `s≤s⁻¹`
+
+### Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`
+
+* This module has been deprecated, as none of its contents actually depended on axiom K. 
+  The contents has been moved to `Data.Product.Function.Dependent.Setoid`.
 
-### Deprecation of old function hierarchy
+### Moving `Function.Related`
 
 * The module `Function.Related` has been deprecated in favour of `Function.Related.Propositional`
   whose code uses the new function hierarchy. This also opens up the possibility of a more
@@ -1314,56 +1233,15 @@ Deprecated modules
   Equivalence-kind    ↦ EquivalenceKind
   ```
 
-### Moving `Algebra.Lattice` files
-
-* As discussed above the following files have been moved:
-  ```agda
-  Algebra.Properties.Semilattice               ↦ Algebra.Lattice.Properties.Semilattice
-  Algebra.Properties.Lattice                   ↦ Algebra.Lattice.Properties.Lattice
-  Algebra.Properties.DistributiveLattice       ↦ Algebra.Lattice.Properties.DistributiveLattice
-  Algebra.Properties.BooleanAlgebra            ↦ Algebra.Lattice.Properties.BooleanAlgebra
-  Algebra.Properties.BooleanAlgebra.Expression ↦ Algebra.Lattice.Properties.BooleanAlgebra.Expression
-  Algebra.Morphism.LatticeMonomorphism         ↦ Algebra.Lattice.Morphism.LatticeMonomorphism
-  ```
-
-### Moving `*.Catgeorical.*` files
+### Moving `Relation.Binary.Construct.(Converse/Flip)`
 
-* As discussed above the following files have been moved:
+* The following files have been moved:
   ```agda
-  Codata.Sized.Colist.Categorical            ↦ Codata.Sized.Colist.Effectful
-  Codata.Sized.Covec.Categorical             ↦ Codata.Sized.Covec.Effectful
-  Codata.Sized.Delay.Categorical             ↦ Codata.Sized.Delay.Effectful
-  Codata.Sized.Stream.Categorical            ↦ Codata.Sized.Stream.Effectful
-  Data.List.Categorical                      ↦ Data.List.Effectful
-  Data.List.Categorical.Transformer          ↦ Data.List.Effectful.Transformer
-  Data.List.NonEmpty.Categorical             ↦ Data.List.NonEmpty.Effectful
-  Data.List.NonEmpty.Categorical.Transformer ↦ Data.List.NonEmpty.Effectful.Transformer
-  Data.Maybe.Categorical                     ↦ Data.Maybe.Effectful
-  Data.Maybe.Categorical.Transformer         ↦ Data.Maybe.Effectful.Transformer
-  Data.Product.Categorical.Examples          ↦ Data.Product.Effectful.Examples
-  Data.Product.Categorical.Left              ↦ Data.Product.Effectful.Left
-  Data.Product.Categorical.Left.Base         ↦ Data.Product.Effectful.Left.Base
-  Data.Product.Categorical.Right             ↦ Data.Product.Effectful.Right
-  Data.Product.Categorical.Right.Base        ↦ Data.Product.Effectful.Right.Base
-  Data.Sum.Categorical.Examples              ↦ Data.Sum.Effectful.Examples
-  Data.Sum.Categorical.Left                  ↦ Data.Sum.Effectful.Left
-  Data.Sum.Categorical.Left.Transformer      ↦ Data.Sum.Effectful.Left.Transformer
-  Data.Sum.Categorical.Right                 ↦ Data.Sum.Effectful.Right
-  Data.Sum.Categorical.Right.Transformer     ↦ Data.Sum.Effectful.Right.Transformer
-  Data.These.Categorical.Examples            ↦ Data.These.Effectful.Examples
-  Data.These.Categorical.Left                ↦ Data.These.Effectful.Left
-  Data.These.Categorical.Left.Base           ↦ Data.These.Effectful.Left.Base
-  Data.These.Categorical.Right               ↦ Data.These.Effectful.Right
-  Data.These.Categorical.Right.Base          ↦ Data.These.Effectful.Right.Base
-  Data.Vec.Categorical                       ↦ Data.Vec.Effectful
-  Data.Vec.Categorical.Transformer           ↦ Data.Vec.Effectful.Transformer
-  Data.Vec.Recursive.Categorical             ↦ Data.Vec.Recursive.Effectful
-  Function.Identity.Categorical              ↦ Function.Identity.Effectful
-  IO.Categorical                             ↦ IO.Effectful
-  Reflection.TCM.Categorical                 ↦ Reflection.TCM.Effectful
+  Relation.Binary.Construct.Converse ↦ Relation.Binary.Construct.Flip.EqAndOrd
+  Relation.Binary.Construct.Flip     ↦ Relation.Binary.Construct.Flip.Ord
   ```
 
-### Moving `Relation.Binary.Properties.XLattice` files
+### Moving `Relation.Binary.Properties.XLattice`
 
 * The following files have been moved:
   ```agda
@@ -1377,18 +1255,6 @@ Deprecated modules
   Relation.Binary.Properties.MeetSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.MeetSemilattice.agda
   ```
 
-### Deprecation of `Data.Nat.Properties.Core`
-
-* The module `Data.Nat.Properties.Core` has been deprecated, and its one lemma moved to `Data.Nat.Base`, renamed as `s≤s⁻¹`
-
-### Deprecation of `Data.Fin.Substitution.Example`
-
-* The module `Data.Fin.Substitution.Example` has been deprecated, and moved to `README.Data.Fin.Substitution.UntypedLambda`
-
-### Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`
-
-* This module has been deprecated, as none of its contents actually depended on axiom K. The contents has been moved to `Data.Product.Function.Dependent.Setoid`.
-
 Deprecated names
 ----------------
 
@@ -1438,16 +1304,16 @@ Deprecated names
   map-map-fusion  ↦  map-∘
   ```
 
-* In `Data.Bool.Properties` (Issue #2046):
-  ```
-  push-function-into-if ↦ if-float
-  ```
-
 * In `Data.Container.Related`:
-  ```
+  ```agda
   _∼[_]_  ↦  _≲[_]_
   ```
 
+* In `Data.Bool.Properties` (Issue #2046):
+  ```agda
+  push-function-into-if ↦ if-float
+  ```
+
 * In `Data.Fin.Base`: two new, hopefully more memorable, names `↑ˡ` `↑ʳ`
   for the 'left', resp. 'right' injection of a Fin m into a 'larger' type,
   `Fin (m + n)`, resp. `Fin (n + m)`, with argument order to reflect the
@@ -1457,24 +1323,6 @@ Deprecated names
   raise    ↦  _↑ʳ_
   ```
 
-  Issue #1726: the relation `_≺_` and its single constructor `_≻toℕ_`
-  have been deprecated in favour of their extensional equivalent `_<_`
-  but omitting the inversion principle which pattern matching on `_≻toℕ_`
-  would achieve; this instead is proxied by the property `Data.Fin.Properties.toℕ<`.
-
-* In `Data.Fin.Induction`:
-  ```
-  ≺-Rec
-  ≺-wellFounded
-  ≺-recBuilder
-  ≺-rec
-  ```
-
-  As with Issue #1726 above: the deprecation of relation `_≺_` means that these definitions
-  associated with wf-recursion are deprecated in favour of their `_<_` counterparts.
-  But it's not quite sensible to say that these definitions should be *renamed* to *anything*,
-  least of all those counterparts... the type confusion would be intolerable.
-
 * In `Data.Fin.Properties`:
   ```
   toℕ-raise        ↦ toℕ-↑ʳ
@@ -1485,9 +1333,6 @@ Deprecated names
   eq?              ↦ inj⇒≟
   ```
 
-  Likewise under issue #1726: the properties `≺⇒<′` and `<′⇒≺` have been deprecated
-  in favour of their proxy counterparts `<⇒<′` and `<′⇒<`.
-
 * In `Data.Fin.Permutation.Components`:
   ```
   reverse            ↦ Data.Fin.Base.opposite
@@ -1563,7 +1408,7 @@ Deprecated names
   ```
 
 * In `Data.List.Base`:
-  ```
+  ```agda
   _─_  ↦  removeAt
   ```
 
@@ -1598,12 +1443,26 @@ Deprecated names
 
 * In `Data.List.Relation.Unary.All.Properties`:
   ```agda
+  gmap  ↦  gmap⁺
+  
   updateAt-id-relative      ↦  updateAt-id-local
   updateAt-compose-relative ↦  updateAt-updateAt-local
   updateAt-compose          ↦  updateAt-updateAt
   updateAt-cong-relative    ↦  updateAt-cong-local
   ```
 
+* In `Data.List.Relation.Unary.Any.Properties`:
+  ```agda
+  map-with-∈⁺  ↦  mapWith∈⁺
+  map-with-∈⁻  ↦  mapWith∈⁻
+  map-with-∈↔  ↦  mapWith∈↔
+  ```
+
+* In `Data.List.Relation.Binary.Subset.Propositional.Properties`:
+  ```agda
+  map-with-∈⁺  ↦  mapWith∈⁺
+  ```
+
 * In `Data.List.Zipper.Properties`:
   ```agda
   toList-reverse-commute ↦  toList-reverse
@@ -1621,26 +1480,10 @@ Deprecated names
   map-compose    ↦  map-∘
 
   map-<∣>-commute ↦  map-<∣>
-
-* In `Data.List.Relation.Binary.Subset.Propositional.Properties`:
-  ```
-  map-with-∈⁺  ↦  mapWith∈⁺
-  ```
-
-* In `Data.List.Relation.Unary.Any.Properties`:
-  ```
-  map-with-∈⁺  ↦  mapWith∈⁺
-  map-with-∈⁻  ↦  mapWith∈⁻
-  map-with-∈↔  ↦  mapWith∈↔
-  ```
-
-* In `Data.List.Relation.Unary.All.Properties`:
-  ```
-  gmap  ↦  gmap⁺
   ```
 
 * In `Data.Nat.Properties`:
-  ```
+  ```agda
   suc[pred[n]]≡n  ↦  suc-pred
   ≤-step          ↦  m≤n⇒m≤1+n
   ≤-stepsˡ        ↦  m≤n⇒m≤o+n
@@ -1653,14 +1496,14 @@ Deprecated names
   ```
 
 * In `Data.Rational.Unnormalised.Base`:
-  ```
+  ```agda
   _≠_  ↦  _≄_
   +-rawMonoid ↦ +-0-rawMonoid
   *-rawMonoid ↦ *-1-rawMonoid
   ```
 
 * In `Data.Rational.Unnormalised.Properties`:
-  ```
+  ```agda
   ↥[p/q]≡p         ↦  ↥[n/d]≡n
   ↧[p/q]≡q         ↦  ↧[n/d]≡d
   *-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
@@ -1677,13 +1520,13 @@ Deprecated names
   ```
 
 * In `Data.Rational.Base`:
-  ```
+  ```agda
   +-rawMonoid ↦ +-0-rawMonoid
   *-rawMonoid ↦ *-1-rawMonoid
   ```
 
 * In `Data.Rational.Properties`:
-  ```
+  ```agda
   *-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
   *-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
   *-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
@@ -1697,13 +1540,13 @@ Deprecated names
   ```
 
 * In `Data.Rational.Unnormalised.Base`:
-  ```
+  ```agda
   +-rawMonoid ↦ +-0-rawMonoid
   *-rawMonoid ↦ *-1-rawMonoid
   ```
 
 * In `Data.Rational.Unnormalised.Properties`:
-  ```
+  ```agda
   ≤-steps  ↦  p≤q⇒p≤r+q
   ```
 
@@ -1716,27 +1559,27 @@ Deprecated names
   ```
 
 * In `Data.Tree.AVL`:
-  ```
+  ```agda
   _∈?_ ↦ member
   ```
 
 * In `Data.Tree.AVL.IndexedMap`:
-  ```
+  ```agda
   _∈?_ ↦ member
   ```
 
 * In `Data.Tree.AVL.Map`:
-  ```
+  ```agda
   _∈?_ ↦ member
   ```
 
 * In `Data.Tree.AVL.Sets`:
-  ```
+  ```agda
   _∈?_ ↦ member
   ```
 
 * In `Data.Tree.Binary.Zipper.Properties`:
-  ```
+  ```agda
   toTree-#nodes-commute   ↦  toTree-#nodes
   toTree-#leaves-commute  ↦  toTree-#leaves
   toTree-map-commute      ↦  toTree-map
@@ -1751,13 +1594,13 @@ Deprecated names
   ```
 
 * In `Data.Vec.Base`:
-  ```
+  ```agda
   remove  ↦  removeAt
   insert  ↦  insertAt
   ```
 
 * In `Data.Vec.Properties`:
-  ```
+  ```agda
   take-distr-zipWith ↦  take-zipWith
   take-distr-map     ↦  take-map
   drop-distr-zipWith ↦  drop-zipWith
@@ -1787,17 +1630,9 @@ Deprecated names
 
   lookup-inject≤-take ↦ lookup-take-inject≤
   ```
-  and the type of the proof `zipWith-comm` has been generalised from:
-  ```
-  zipWith-comm : ∀ {f : A → A → B} (comm : ∀ x y → f x y ≡ f y x) (xs ys : Vec A n) → zipWith f xs ys ≡ zipWith f ys xs
-  ```
-  to
-  ```
-  zipWith-comm : ∀ {f : A → B → C} {g : B → A → C}  (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys → zipWith f xs ys ≡ zipWith g ys xs
-  ```
 
 * In `Data.Vec.Functional.Properties`:
-  ```
+  ```agda
   updateAt-id-relative      ↦  updateAt-id-local
   updateAt-compose-relative ↦  updateAt-updateAt-local
   updateAt-compose          ↦  updateAt-updateAt
@@ -1814,7 +1649,7 @@ Deprecated names
   NB. This last one is complicated by the *addition* of a 'global' property `map-updateAt`
 
 * In `Data.Vec.Recursive.Properties`:
-  ```
+  ```agda
   append-cons-commute  ↦  append-cons
   ```
 
@@ -1824,12 +1659,12 @@ Deprecated names
   ```
 
 * In `Function.Base`:
-  ```
+  ```agda
   case_return_of_   ↦   case_returning_of_
   ```
 
 * In `Function.Construct.Composition`:
-  ```
+  ```agda
   _∘-⟶_   ↦   _⟶-∘_
   _∘-↣_   ↦   _↣-∘_
   _∘-↠_   ↦   _↠-∘_
@@ -1841,7 +1676,7 @@ Deprecated names
   ```
 
   * In `Function.Construct.Identity`:
-  ```
+  ```agda
   id-⟶   ↦   ⟶-id
   id-↣   ↦   ↣-id
   id-↠   ↦   ↠-id
@@ -1853,7 +1688,7 @@ Deprecated names
   ```
 
 * In `Function.Construct.Symmetry`:
-  ```
+  ```agda
   sym-⤖   ↦   ⤖-sym
   sym-⇔   ↦   ⇔-sym
   sym-↩   ↦   ↩-sym
@@ -1867,60 +1702,142 @@ Deprecated names
   ```
 
 * In `Foreign.Haskell.Either` and `Foreign.Haskell.Pair`:
-  ```
+  ```agda
   toForeign   ↦ Foreign.Haskell.Coerce.coerce
   fromForeign ↦ Foreign.Haskell.Coerce.coerce
   ```
 
-* In `Relation.Binary.Indexed.Heterogeneous.Bundles.Preorder`:
-  ```
-  _∼_  ↦  _≲_
-  ```
-
 * In `Relation.Binary.Properties.Preorder`:
-  ```
+  ```agda
   invIsPreorder ↦ converse-isPreorder
   invPreorder   ↦ converse-preorder
   ```
 
-* Moved negated relation symbol from `Relation.Binary.Properties.Poset`:
-  ```
-  _≰_   ↦ Relation.Binary.Bundles.Poset._≰_
-  ```
-
-* Moved negated relation symbol from `Relation.Binary.Properties.TotalOrder`:
-  ```
-  _≮_   ↦ Relation.Binary.Bundles.StrictPartialOrder._≮_
+## Missing fixity declarations added
 
-* In `Relation.Nullary.Decidable.Core`:
+* An effort has been made to add sensible fixities for many declarations:
   ```
-  excluded-middle  ↦  ¬¬-excluded-middle
+  infix   4 _≟H_ _≟N_                                        (Algebra.Solver.Ring)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream)
+  infix   4 _[_]                                             (Codata.Guarded.Stream)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Binary.Pointwise)
+  infix   4 _≈∞_                                             (Codata.Guarded.Stream.Relation.Binary.Pointwise)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Unary.All)
+  infixr  5 _∷_                                              (Codata.Musical.Colist)
+  infix   4 _≈_                                              (Codata.Musical.Conat)
+  infixr  5 _∷_                                              (Codata.Musical.Colist.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Musical.Colist.Relation.Unary.All)
+  infixr  5 _∷_                                              (Codata.Sized.Colist)
+  infixr  5 _∷_                                              (Codata.Sized.Covec)
+  infixr  5 _∷_                                              (Codata.Sized.Cowriter)
+  infixl  1 _>>=_                                            (Codata.Sized.Cowriter)
+  infixr  5 _∷_                                              (Codata.Sized.Stream)
+  infixr  5 _∷_                                              (Codata.Sized.Colist.Bisimilarity)
+  infix   4 _ℕ≤?_                                            (Codata.Sized.Conat.Properties)
+  infixr  5 _∷_                                              (Codata.Sized.Covec.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Sized.Cowriter.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Sized.Stream.Bisimilarity)
+  infix   4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
+  infix   6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
+  infixr  8 _⇒_ _⊸_                                          (Data.Container.Core)
+  infixr -1 _<$>_ _<*>_                                      (Data.Container.FreeMonad)
+  infixl  1 _>>=_                                            (Data.Container.FreeMonad)
+  infix   5 _▷_                                              (Data.Container.Indexed)
+  infixr  4 _,_                                              (Data.Container.Relation.Binary.Pointwise)
+  infix   4 _≈_                                              (Data.Float.Base)
+  infixl  4 _+ _*                                            (Data.List.Kleene.Base)
+  infixr  4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
+  infix   4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
+  infix   4 _≢∈_                                             (Data.List.Membership.Propositional)
+  infixr  5 _`∷_                                             (Data.List.Reflection)
+  infix   4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
+  infixr  5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
+  infixr  5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
+  infixr  5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
+  infixr  5 _∷=_                                             (Data.List.Relation.Unary.Any)
+  infixr  5 _++_                                             (Data.List.Ternary.Appending)
+  infixl  7 _⊓′_                                             (Data.Nat.Base)
+  infixl  6 _⊔′_                                             (Data.Nat.Base)
+  infixr  8 _^_                                              (Data.Nat.Base)
+  infix   4 _!≢0 _!*_!≢0                                     (Data.Nat.Properties)
+  infixl  6.5 _P′_ _P_ _C′_ _C_                              (Data.Nat.Combinatorics.Base)
+  infix   8 _⁻¹                                              (Data.Parity.Base)
+  infixr  2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
+  infixr  2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
+  infixr  2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
+  infixr  2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
+  infix   4 _≃?_                                             (Data.Rational.Unnormalised.Properties)
+  infixr  4 _,_                                              (Data.Refinement)
+  infixr  1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
+  infixr  1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
+  infixr  1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
+  infixr  1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
+  infixr  4 _,_                                              (Data.Tree.AVL.Value)
+  infix   4 _≈ₖᵥ_                                            (Data.Tree.AVL.Map.Membership.Propositional)
+  infixr  5 _`∷_                                             (Data.Vec.Reflection)
+  infixr  5 _∷=_                                             (Data.Vec.Membership.Setoid)
+  infix  -1 _$ⁿ_                                             (Data.Vec.N-ary)
+  infix   4 _≋_                                              (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
+  infixl  1 _>>=-cong_ _≡->>=-cong_                          (Effect.Monad.Partiality)
+  infixl  1 _?>=′_                                           (Effect.Monad.Predicate)
+  infixl  1 _>>=-cong_ _>>=-congP_                           (Effect.Monad.Partiality.All)
+  infixr  5 _∷_                                              (Foreign.Haskell.List.NonEmpty)
+  infixr  4 _,_                                              (Foreign.Haskell.Pair)
+  infixr  8 _^_                                              (Function.Endomorphism.Propositional)
+  infixr  8 _^_                                              (Function.Endomorphism.Setoid)
+  infix   4 _≃_                                              (Function.HalfAdjointEquivalence)
+  infix   4 _≈_ _≈ᵢ_ _≤_                                     (Function.Metric.Bundles)
+  infixl  6 _∙_                                              (Function.Metric.Bundles)
+  infix   4 _≈_                                              (Function.Metric.Nat.Bundles)
+  infix   4 _≈_                                              (Function.Metric.Rational.Bundles)
+  infix   3 _←_ _↢_                                          (Function.Related)
+  infix   4 _<_                                              (Induction.WellFounded)
+  infixl  6 _ℕ+_                                             (Level.Literals)
+  infixr  4 _,_                                              (Reflection.AnnotatedAST)
+  infix   4 _≟_                                              (Reflection.AST.Definition)
+  infix   4 _≡ᵇ_                                             (Reflection.AST.Literal)
+  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Meta)
+  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Name)
+  infix   4 _≟-Telescope_                                    (Reflection.AST.Term)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Information)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Modality)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Quantity)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Relevance)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Visibility)
+  infixr  4 _,_                                              (Reflection.AST.Traversal)
+  infix   4 _∈FV_                                            (Reflection.AST.DeBruijn)
+  infixr  9 _;_                                              (Relation.Binary.Construct.Composition)
+  infixl  6 _+²_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
+  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Heterogeneous.Construct.At)
+  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Homogeneous.Construct.At)
+  infix   4 _∈_ _∉_                                          (Relation.Unary.Indexed)
+  infix   4 _≈_                                              (Relation.Binary.Bundles)
+  infixl  6 _∩_                                              (Relation.Binary.Construct.Intersection)
+  infix   4 _<₋_                                             (Relation.Binary.Construct.Add.Infimum.Strict)
+  infix   4 _≈∙_                                             (Relation.Binary.Construct.Add.Point.Equality)
+  infix   4 _≤⁺_ _≤⊤⁺                                        (Relation.Binary.Construct.Add.Supremum.NonStrict)
+  infixr  5 _∷ʳ_                                             (Relation.Binary.Construct.Closure.Transitive)
+  infixr  6 _∪_                                              (Relation.Binary.Construct.Union)
+  infixl  6 _+ℤ_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
+  infix   4 _≉_ _≈ᵢ_ _≤ᵢ_                                    (Relation.Binary.Indexed.Homogeneous.Bundles)
+  infixr  9 _⍮_                                              (Relation.Unary.PredicateTransformer)
+  infix   8 ∼_                                               (Relation.Unary.PredicateTransformer)
+  infix   2 _×?_ _⊙?_                                        (Relation.Unary.Properties)
+  infix   10 _~?                                             (Relation.Unary.Properties)
+  infixr  1 _⊎?_                                             (Relation.Unary.Properties)
+  infixr  7 _∩?_                                             (Relation.Unary.Properties)
+  infixr  6 _∪?_                                             (Relation.Unary.Properties)
+  infixr  5 _∷ᴹ_ _∷⁻¹ᴹ_                                      (Text.Regex.Search)
+  infixl  6 _`⊜_                                             (Tactic.RingSolver)
+  infix   8 ⊝_                                               (Tactic.RingSolver.Core.Expression)
+  infix   4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]                  (Text.Regex.Properties)
+  infix   4 _∈?_ _∉?_                                        (Text.Regex.Derivative.Brzozowski)
+  infix   4 _∈_ _∉_ _∈?_ _∉?_                                (Text.Regex.String.Unsafe)
   ```
 
-### Renamed Data.Erased to Data.Irrelevant
-
-* This fixes the fact we had picked the wrong name originally. The erased modality
-  corresponds to @0 whereas the irrelevance one corresponds to `.`.
-
-
 New modules
 -----------
 
-* Algebraic structures when freely adding an identity element:
-  ```
-  Algebra.Construct.Add.Identity
-  ```
-
-* Operations for module-like algebraic structures:
-  ```
-  Algebra.Module.Core
-  ```
-
-* Definitions for module-like algebraic structures with left- and right- multiplication over the same scalars:
-  ```
-  Algebra.Module.Definitions.Bi.Simultaneous
-  ```
-
 * Constructive algebraic structures with apartness relations:
   ```
   Algebra.Apartness
@@ -1935,11 +1852,20 @@ New modules
   Algebra.Construct.Flip.Op
   ```
 
-* Morphisms between module-like algebraic structures:
+* Algebraic structures when freely adding an identity element:
+  ```
+  Algebra.Construct.Add.Identity
+  ```
+
+* Definitions for algebraic modules:
   ```
+  Algebra.Module
+  Algebra.Module.Core
+  Algebra.Module.Definitions.Bi.Simultaneous
   Algebra.Module.Morphism.Construct.Composition
   Algebra.Module.Morphism.Construct.Identity
   Algebra.Module.Morphism.Definitions
+  Algebra.Module.Morphism.ModuleHomomorphism
   Algebra.Module.Morphism.Structures
   Algebra.Module.Properties
   ```
@@ -1950,11 +1876,36 @@ New modules
   Algebra.Morphism.Construct.Identity
   ```
 
+* Properties of various new algebraic structures:
+  ```
+  Algebra.Properties.MoufangLoop
+  Algebra.Properties.Quasigroup
+  Algebra.Properties.MiddleBolLoop
+  Algebra.Properties.Loop
+  Algebra.Properties.KleeneAlgebra
+  ```
+
+* Properties of rings without a unit
+  ```
+  Algebra.Properties.RingWithoutOne`
+  ```
+
+* Proof of the Binomial Theorem for semirings
+  ```
+  Algebra.Properties.Semiring.Binomial
+  Algebra.Properties.CommutativeSemiring.Binomial
+  ```
+
 * 'Optimised' tail-recursive exponentiation properties:
   ```
   Algebra.Properties.Semiring.Exp.TailRecursiveOptimised
   ```
 
+* An implementation of M-types with `--guardedness` flag:
+  ```
+  Codata.Guarded.M
+  ```
+
 * A definition of infinite streams using coinductive records:
   ```
   Codata.Guarded.Stream
@@ -2015,6 +1966,17 @@ New modules
   Data.List.NonEmpty.Relation.Unary.All
   ```
 
+* Some n-ary functions manipulating lists
+  ```
+  Data.List.Nary.NonDependent
+  ```
+
+* Added Logarithm base 2 on natural numbers:
+  ```
+  Data.Nat.Logarithm.Core
+  Data.Nat.Logarithm
+  ```
+
 * Show module for unnormalised rationals:
   ```
   Data.Rational.Unnormalised.Show
@@ -2028,7 +1990,23 @@ New modules
   Data.Tree.AVL.Sets.Membership.Properties
   ```
 
-* Properties of bijections:
+* Port of `Linked` to `Vec`:
+  ```
+  Data.Vec.Relation.Unary.Linked
+  Data.Vec.Relation.Unary.Linked.Properties
+  ```
+
+* Combinators for propositional equational reasoning on vectors with different indices
+  ```
+  Data.Vec.Relation.Binary.Equality.Cast
+  ```
+
+* Relations on indexed sets
+  ```
+  Function.Indexed.Bundles
+  ```
+
+* Properties of various types of functions:
   ```
   Function.Consequences
   Function.Properties.Bijection
@@ -2037,6 +2015,11 @@ New modules
   Function.Construct.Constant
   ```
 
+* New interface for `NonEmpty` Haskell lists:
+  ```
+  Foreign.Haskell.List.NonEmpty
+  ```
+
 * In order to improve modularity, the contents of `Relation.Binary.Lattice` has been
   split out into the standard:
   ```
@@ -2046,6 +2029,12 @@ New modules
   ```
   All contents is re-exported by `Relation.Binary.Lattice` as before.
 
+
+* Added relational reasoning over apartness relations:
+  ```
+  Relation.Binary.Reasoning.Base.Apartness`
+  ```
+
 * Algebraic properties of `_∩_` and `_∪_` for predicates
   ```
   Relation.Unary.Algebra
@@ -2072,11 +2061,6 @@ New modules
   Reflection.AST.AlphaEquality
   ```
 
-* `cong!` tactic for deriving arguments to `cong`
-  ```
-  Tactic.Cong
-  ```
-
 * Various system types and primitives:
   ```
   System.Clock.Primitive
@@ -2090,161 +2074,94 @@ New modules
   System.Process
   ```
 
-* A golden testing library with test pools, an options parser, a runner:
-  ```
-  Test.Golden
-  ```
-
-* New interfaces for using Haskell datatypes:
-  ```
-  Foreign.Haskell.List.NonEmpty
-  ```
-* Added new module `Algebra.Properties.RingWithoutOne`:
-  ```
-  -‿distribˡ-* : ∀ x y → - (x * y) ≈ - x * y
-  -‿distribʳ-* : ∀ x y → - (x * y) ≈ x * - y
-  ```
-* An implementation of M-types with `--guardedness` flag:
-  ```
-  Codata.Guarded.M
-  ```
-
-* Port of `Linked` to `Vec`:
-  ```
-  Data.Vec.Relation.Unary.Linked
-  Data.Vec.Relation.Unary.Linked.Properties
-  ```
-* Added Logarithm base 2 on Natural Numbers:
-  ```
-  Data.Nat.Logarithm.Core
-  Data.Nat.Logarithm
-  ```
-
-* Proofs of some axioms of linearity:
-  ```
-  Algebra.Module.Morphism.ModuleHomomorphism
-  Algebra.Module.Properties
-  ```
-* Properties of MoufangLoop
-  ```
-  Algebra.Properties.MoufangLoop
-  ```
-
-* Properties of Quasigroup
-  ```
-  Algebra.Properties.Quasigroup
-  ```
-
-* Properties of MiddleBolLoop
-  ```
-  Algebra.Properties.MiddleBolLoop
-  ```
-
-* Properties of Loop
-  ```
-  Algebra.Properties.Loop
-  ```
-
-* Properties of (Commutative)Semiring: the Binomial Theorem
-  ```
-  Algebra.Properties.CommutativeSemiring.Binomial
-  Algebra.Properties.Semiring.Binomial
-  ```
-
-* Some n-ary functions manipulating lists
-  ```
-  Data.List.Nary.NonDependent
-  ```
-
-* Properties of KleeneAlgebra
-  ```
-  Algebra.Properties.KleeneAlgebra
-  ```
-
-* Relations on indexed sets
+* A new `cong!` tactic for automatically deriving arguments to `cong`
   ```
-  Function.Indexed.Bundles
+  Tactic.Cong
   ```
 
-* Combinators for propositional equational reasoning on vectors with different indices
+* A golden testing library with test pools, an options parser, a runner:
   ```
-  Data.Vec.Relation.Binary.Equality.Cast
+  Test.Golden
   ```
 
 Additions to existing modules
 -----------------------------
 
-* Added new proof to `Data.Maybe.Properties`
-  ```agda
-    <∣>-idem : Idempotent _<∣>_
-  ```
-
 * The module `Algebra` now publicly re-exports the contents of
   `Algebra.Structures.Biased`.
 
 * Added new definitions to `Algebra.Bundles`:
   ```agda
-  record UnitalMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record InvertibleMagma c ℓ : Set (suc (c ⊔ ℓ))
+  record UnitalMagma           c ℓ : Set (suc (c ⊔ ℓ))
+  record InvertibleMagma       c ℓ : Set (suc (c ⊔ ℓ))
   record InvertibleUnitalMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record RawQuasiGroup c ℓ : Set (suc (c ⊔ ℓ))
-  record Quasigroup c ℓ : Set (suc (c ⊔ ℓ))
-  record RawLoop  c ℓ : Set (suc (c ⊔ ℓ))
-  record Loop  c ℓ : Set (suc (c ⊔ ℓ))
-  record RingWithoutOne c ℓ : Set (suc (c ⊔ ℓ))
-  record IdempotentSemiring c ℓ : Set (suc (c ⊔ ℓ))
-  record KleeneAlgebra c ℓ : Set (suc (c ⊔ ℓ))
-  record RawRingWithoutOne c ℓ : Set (suc (c ⊔ ℓ))
-  record Quasiring c ℓ : Set (suc (c ⊔ ℓ)) where
-  record Nearring c ℓ : Set (suc (c ⊔ ℓ)) where
-  record IdempotentMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record AlternateMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record FlexibleMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record MedialMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record SemimedialMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record LeftBolLoop c ℓ : Set (suc (c ⊔ ℓ))
-  record RightBolLoop c ℓ : Set (suc (c ⊔ ℓ))
-  record MoufangLoop c ℓ : Set (suc (c ⊔ ℓ))
-  record NonAssociativeRing c ℓ : Set (suc (c ⊔ ℓ))
-  record MiddleBolLoop c ℓ : Set (suc (c ⊔ ℓ))
-  ```
-  and the existing record `Lattice` now provides
+  record Quasigroup            c ℓ : Set (suc (c ⊔ ℓ))
+  record Loop                  c ℓ : Set (suc (c ⊔ ℓ))
+  record RingWithoutOne        c ℓ : Set (suc (c ⊔ ℓ))
+  record IdempotentSemiring    c ℓ : Set (suc (c ⊔ ℓ))
+  record KleeneAlgebra         c ℓ : Set (suc (c ⊔ ℓ))
+  record Quasiring             c ℓ : Set (suc (c ⊔ ℓ))
+  record Nearring              c ℓ : Set (suc (c ⊔ ℓ))
+  record IdempotentMagma       c ℓ : Set (suc (c ⊔ ℓ))
+  record AlternateMagma        c ℓ : Set (suc (c ⊔ ℓ))
+  record FlexibleMagma         c ℓ : Set (suc (c ⊔ ℓ))
+  record MedialMagma           c ℓ : Set (suc (c ⊔ ℓ))
+  record SemimedialMagma       c ℓ : Set (suc (c ⊔ ℓ))
+  record LeftBolLoop           c ℓ : Set (suc (c ⊔ ℓ))
+  record RightBolLoop          c ℓ : Set (suc (c ⊔ ℓ))
+  record MoufangLoop           c ℓ : Set (suc (c ⊔ ℓ))
+  record NonAssociativeRing    c ℓ : Set (suc (c ⊔ ℓ))
+  record MiddleBolLoop         c ℓ : Set (suc (c ⊔ ℓ))
+  ```
+
+* Added new definitions to `Algebra.Bundles.Raw`:
+  ```agda
+  record RawLoop               c ℓ : Set (suc (c ⊔ ℓ))
+  record RawQuasiGroup         c ℓ : Set (suc (c ⊔ ℓ))
+  record RawRingWithoutOne     c ℓ : Set (suc (c ⊔ ℓ))
+  ```
+
+* Added new definitions to `Algebra.Structures`:
   ```agda
-  ∨-commutativeSemigroup : CommutativeSemigroup c ℓ
-  ∧-commutativeSemigroup : CommutativeSemigroup c ℓ
+  record IsUnitalMagma (_∙_ : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsInvertibleMagma (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
+  record IsInvertibleUnitalMagma (_∙_ : Op₂ A) (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
+  record IsQuasigroup (∙ \\ // : Op₂ A) : Set (a ⊔ ℓ)
+  record IsLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ)
+  record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ)
+  record IsKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
+  record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
+  record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
+  record IsIdempotentMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsAlternativeMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsFlexibleMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsMedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsSemimedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsLeftBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsRightBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsMoufangLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
+  record IsMiddleBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
   ```
-  and their corresponding algebraic subbundles.
 
-* Added new proofs to `Algebra.Consequences.Base`:
+* Added new proof to `Algebra.Consequences.Base`:
   ```agda
-  reflexive+selfInverse⇒involutive : Reflexive _≈_ →
-                                     SelfInverse _≈_ f →
-                                     Involutive _≈_ f
+  reflexive+selfInverse⇒involutive : Reflexive _≈_ → SelfInverse _≈_ f → Involutive _≈_ f
   ```
 
 * Added new proofs to `Algebra.Consequences.Propositional`:
   ```agda
-  comm+assoc⇒middleFour     : Commutative _•_ →
-                              Associative _•_ →
-                              _•_ MiddleFourExchange _•_
-  identity+middleFour⇒assoc : Identity e _•_ →
-                              _•_ MiddleFourExchange _•_ →
-                              Associative _•_
-  identity+middleFour⇒comm  : Identity e _+_ →
-                              _•_ MiddleFourExchange _+_ →
-                              Commutative _•_
+  comm+assoc⇒middleFour     : Commutative _•_ → Associative _•_ → _•_ MiddleFourExchange _•_
+  identity+middleFour⇒assoc : Identity e _•_ → _•_ MiddleFourExchange _•_ → Associative _•_
+  identity+middleFour⇒comm  : Identity e _+_ → _•_ MiddleFourExchange _+_ → Commutative _•_
   ```
 
 * Added new proofs to `Algebra.Consequences.Setoid`:
   ```agda
-  comm+assoc⇒middleFour     : Congruent₂ _•_ → Commutative _•_ → Associative _•_ →
-                              _•_ MiddleFourExchange _•_
-  identity+middleFour⇒assoc : Congruent₂ _•_ → Identity e _•_ →
-                              _•_ MiddleFourExchange _•_ →
-                              Associative _•_
-  identity+middleFour⇒comm  : Congruent₂ _•_ → Identity e _+_ →
-                              _•_ MiddleFourExchange _+_ →
-                              Commutative _•_
+  comm+assoc⇒middleFour     : Congruent₂ _•_ → Commutative _•_ → Associative _•_ → _•_ MiddleFourExchange _•_
+  identity+middleFour⇒assoc : Congruent₂ _•_ → Identity e _•_ → _•_ MiddleFourExchange _•_ → Associative _•_
+  identity+middleFour⇒comm  : Congruent₂ _•_ → Identity e _+_ → _•_ MiddleFourExchange _+_ → Commutative _•_
 
   involutive⇒surjective  : Involutive f  → Surjective f
   selfInverse⇒involutive : SelfInverse f → Involutive f
@@ -2267,63 +2184,60 @@ Additions to existing modules
 
 * Added new functions to `Algebra.Construct.DirectProduct`:
   ```agda
-  rawSemiring : RawSemiring a ℓ₁ → RawSemiring b ℓ₂ → RawSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawRing : RawRing a ℓ₁ → RawRing b ℓ₂ → RawRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawSemiring                     : RawSemiring a ℓ₁ → RawSemiring b ℓ₂ → RawSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawRing                         : RawRing a ℓ₁ → RawRing b ℓ₂ → RawRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero a ℓ₁ →
                                     SemiringWithoutAnnihilatingZero b ℓ₂ →
                                     SemiringWithoutAnnihilatingZero (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  semiring : Semiring a ℓ₁ → Semiring b ℓ₂ → Semiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  commutativeSemiring : CommutativeSemiring a ℓ₁ → CommutativeSemiring b ℓ₂ →
-                        CommutativeSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  ring : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  commutativeRing : CommutativeRing a ℓ₁ → CommutativeRing b ℓ₂ →
-                    CommutativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawQuasigroup : RawQuasigroup a ℓ₁ → RawQuasigroup b ℓ₂ →
-                  RawQuasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawLoop : RawLoop a ℓ₁ → RawLoop b ℓ₂ → RawLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  unitalMagma : UnitalMagma a ℓ₁ → UnitalMagma b ℓ₂ →
-                UnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  invertibleMagma : InvertibleMagma a ℓ₁ → InvertibleMagma b ℓ₂ →
-                    InvertibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  invertibleUnitalMagma : InvertibleUnitalMagma a ℓ₁ → InvertibleUnitalMagma b ℓ₂ →
-                          InvertibleUnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  quasigroup : Quasigroup a ℓ₁ → Quasigroup b ℓ₂ → Quasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  loop : Loop a ℓ₁ → Loop b ℓ₂ → Loop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  idempotentSemiring : IdempotentSemiring a ℓ₁ → IdempotentSemiring b ℓ₂ →
-                       IdempotentSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  idempotentMagma : IdempotentMagma a ℓ₁ → IdempotentMagma b ℓ₂ →
-                    IdempotentMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  alternativeMagma : AlternativeMagma a ℓ₁ → AlternativeMagma b ℓ₂ →
-                     AlternativeMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  flexibleMagma : FlexibleMagma a ℓ₁ → FlexibleMagma b ℓ₂ →
-                  FlexibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  medialMagma : MedialMagma a ℓ₁ → MedialMagma b ℓ₂ → MedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  semimedialMagma : SemimedialMagma a ℓ₁ → SemimedialMagma b ℓ₂ →
-                    SemimedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  kleeneAlgebra : KleeneAlgebra a ℓ₁ → KleeneAlgebra b ℓ₂ → KleeneAlgebra (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  leftBolLoop : LeftBolLoop a ℓ₁ → LeftBolLoop b ℓ₂ → LeftBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rightBolLoop : RightBolLoop a ℓ₁ → RightBolLoop b ℓ₂ → RightBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  middleBolLoop : MiddleBolLoop a ℓ₁ → MiddleBolLoop b ℓ₂ → MiddleBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  moufangLoop : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawRingWithoutOne : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  ringWithoutOne : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  nonAssociativeRing : NonAssociativeRing a ℓ₁ → NonAssociativeRing b ℓ₂ → NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  quasiring : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  nearring : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
- ```
+  semiring                        : Semiring a ℓ₁ → Semiring b ℓ₂ → Semiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  commutativeSemiring             : CommutativeSemiring a ℓ₁ → 
+	                                CommutativeSemiring b ℓ₂ →
+                                    CommutativeSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  ring                            : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  commutativeRing                 : CommutativeRing a ℓ₁ → CommutativeRing b ℓ₂ → CommutativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawQuasigroup                   : RawQuasigroup a ℓ₁ → RawQuasigroup b ℓ₂ → RawQuasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawLoop                         : RawLoop a ℓ₁ → RawLoop b ℓ₂ → RawLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  unitalMagma                     : UnitalMagma a ℓ₁ → UnitalMagma b ℓ₂ → UnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  invertibleMagma                 : InvertibleMagma a ℓ₁ → InvertibleMagma b ℓ₂ → InvertibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  invertibleUnitalMagma           : InvertibleUnitalMagma a ℓ₁ → 
+	                                InvertibleUnitalMagma b ℓ₂ →
+                                    InvertibleUnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  quasigroup                      : Quasigroup a ℓ₁ → Quasigroup b ℓ₂ → Quasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  loop                            : Loop a ℓ₁ → Loop b ℓ₂ → Loop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  idempotentSemiring              : IdempotentSemiring a ℓ₁ → 
+                                    IdempotentSemiring b ℓ₂ → 
+									IdempotentSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  idempotentMagma                 : IdempotentMagma a ℓ₁ → IdempotentMagma b ℓ₂ → IdempotentMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  alternativeMagma                : AlternativeMagma a ℓ₁ → AlternativeMagma b ℓ₂ → AlternativeMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  flexibleMagma                   : FlexibleMagma a ℓ₁ → FlexibleMagma b ℓ₂ → FlexibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  medialMagma                     : MedialMagma a ℓ₁ → MedialMagma b ℓ₂ → MedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  semimedialMagma                 : SemimedialMagma a ℓ₁ → SemimedialMagma b ℓ₂ → SemimedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  kleeneAlgebra                   : KleeneAlgebra a ℓ₁ → KleeneAlgebra b ℓ₂ → KleeneAlgebra (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  leftBolLoop                     : LeftBolLoop a ℓ₁ → LeftBolLoop b ℓ₂ → LeftBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rightBolLoop                    : RightBolLoop a ℓ₁ → RightBolLoop b ℓ₂ → RightBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  middleBolLoop                   : MiddleBolLoop a ℓ₁ → MiddleBolLoop b ℓ₂ → MiddleBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  moufangLoop                     : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawRingWithoutOne               : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  ringWithoutOne                  : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  nonAssociativeRing              : NonAssociativeRing a ℓ₁ → 
+		                            NonAssociativeRing b ℓ₂ → 
+									NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  quasiring                       : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  nearring                        : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  ```
 
 * Added new definition to `Algebra.Definitions`:
   ```agda
-  _MiddleFourExchange_ : Op₂ A → Op₂ A → Set _
+  _*_ MiddleFourExchange _+_ = ∀ w x y z → ((w + x) * (y + z)) ≈ ((w + y) * (x + z))
 
-  SelfInverse : Op₁ A → Set _
+  SelfInverse f = ∀ {x y} → f x ≈ y → f y ≈ x
 
-  LeftDividesˡ  : Op₂ A → Op₂ A → Set _
-  LeftDividesʳ  : Op₂ A → Op₂ A → Set _
-  RightDividesˡ : Op₂ A → Op₂ A → Set _
-  RightDividesʳ : Op₂ A → Op₂ A → Set _
-  LeftDivides   : Op₂ A → Op₂ A → Set _
-  RightDivides  : Op₂ A → Op₂ A → Set _
+  LeftDividesˡ  _∙_ _\\_ = ∀ x y → (x ∙ (x \\ y)) ≈ y
+  LeftDividesʳ  _∙_ _\\_ = ∀ x y → (x \\ (x ∙ y)) ≈ y
+  RightDividesˡ _∙_ _//_ = ∀ x y → ((y // x) ∙ x) ≈ y
+  RightDividesʳ _∙_ _//_ = ∀ x y → ((y ∙ x) // x) ≈ y
+  LeftDivides    ∙   \\ = (LeftDividesˡ ∙ \\) × (LeftDividesʳ ∙ \\)
+  RightDivides   ∙   // = (RightDividesˡ ∙ //) × (RightDividesʳ ∙ //)
 
   LeftInvertible  e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e
   RightInvertible e _∙_ x = ∃[ x⁻¹ ] (x ∙ x⁻¹) ≈ e
@@ -2350,24 +2264,64 @@ Additions to existing modules
 * Added new functions to `Algebra.Definitions.RawSemiring`:
   ```agda
   _^[_]*_ : A → ℕ → A → A
-  _^ᵗ_     : A → ℕ → A
+  _^ᵗ_    : A → ℕ → A
+  ```
+
+* In `Algebra.Bundles.Lattice` the existing record `Lattice` now provides
+  ```agda
+  ∨-commutativeSemigroup : CommutativeSemigroup c ℓ
+  ∧-commutativeSemigroup : CommutativeSemigroup c ℓ
+  ```
+  and their corresponding algebraic sub-bundles.
+
+* In `Algebra.Lattice.Structures` the record `IsLattice` now provides
+  ```
+  ∨-isCommutativeSemigroup : IsCommutativeSemigroup ∨
+  ∧-isCommutativeSemigroup : IsCommutativeSemigroup ∧
   ```
+  and their corresponding algebraic substructures.
 
-* `Algebra.Properties.Magma.Divisibility` now re-exports operations
+* The module `Algebra.Properties.Magma.Divisibility` now re-exports operations
   `_∣ˡ_`, `_∤ˡ_`, `_∣ʳ_`, `_∤ʳ_` from `Algebra.Definitions.Magma`.
 
+* Added new records to `Algebra.Morphism.Structures`:
+  ```agda
+  record IsQuasigroupHomomorphism     (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsQuasigroupMonomorphism     (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsQuasigroupIsomorphism      (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsLoopHomomorphism           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsLoopMonomorphism           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsLoopIsomorphism            (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsRingWithoutOneIsoMorphism  (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraHomomorphism  (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraMonomorphism  (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraIsomorphism   (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  ```
+
 * Added new proofs to `Algebra.Properties.CommutativeSemigroup`:
   ```agda
+  xy∙xx≈x∙yxx : (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))
+  
   interchange : Interchangable _∙_ _∙_
-  xy∙xx≈x∙yxx : ∀ x y → (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))
   leftSemimedial : LeftSemimedial _∙_
   rightSemimedial : RightSemimedial _∙_
-  middleSemimedial : ∀ x y z → (x ∙ y) ∙ (z ∙ x) ≈ (x ∙ z) ∙ (y ∙ x)
+  middleSemimedial : (x ∙ y) ∙ (z ∙ x) ≈ (x ∙ z) ∙ (y ∙ x)
   semimedial : Semimedial _∙_
   ```
+
+* Added new functions to `Algebra.Properties.CommutativeMonoid`
+  ```agda
+  invertibleˡ⇒invertibleʳ : LeftInvertible  _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x
+  invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible  _≈_ 0# _+_ x
+  invertibleˡ⇒invertible  : LeftInvertible  _≈_ 0# _+_ x → Invertible      _≈_ 0# _+_ x
+  invertibleʳ⇒invertible  : RightInvertible _≈_ 0# _+_ x → Invertible      _≈_ 0# _+_ x
+  ```
+
 * Added new proof to `Algebra.Properties.Monoid.Mult`:
   ```agda
-  ×-congˡ : ∀ {x} → (_× x) Preserves _≡_ ⟶ _≈_
+  ×-congˡ : (_× x) Preserves _≡_ ⟶ _≈_
   ```
 
 * Added new proof to `Algebra.Properties.Monoid.Sum`:
@@ -2377,16 +2331,16 @@ Additions to existing modules
 
 * Added new proofs to `Algebra.Properties.Semigroup`:
   ```agda
-  leftAlternative : LeftAlternative _∙_
+  leftAlternative  : LeftAlternative _∙_
   rightAlternative : RightAlternative _∙_
-  alternative : Alternative _∙_
-  flexible : Flexible _∙_
+  alternative      : Alternative _∙_
+  flexible         : Flexible _∙_
   ```
 
 * Added new proofs to `Algebra.Properties.Semiring.Exp`:
   ```agda
   ^-congʳ               : (x ^_) Preserves _≡_ ⟶ _≈_
-  y*x^m*y^n≈x^m*y^[n+1] : (x * y ≈ y * x) → y * (x ^ m * y ^ n) ≈ x ^ m * y ^ suc n
+  y*x^m*y^n≈x^m*y^[n+1] : x * y ≈ y * x → y * (x ^ m * y ^ n) ≈ x ^ m * y ^ suc n
   ```
 
 * Added new proofs to `Algebra.Properties.Semiring.Mult`:
@@ -2398,56 +2352,25 @@ Additions to existing modules
 
 * Added new proofs to `Algebra.Properties.Ring`:
   ```agda
-  -1*x≈-x : ∀ x → - 1# * x ≈ - x
-  x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#
-  x[y-z]≈xy-xz : ∀ x y z → x * (y - z) ≈ x * y - x * z
-  [y-z]x≈yx-zx : ∀ x y z → (y - z) * x ≈ (y * x) - (z * x)
+  -1*x≈-x      : - 1# * x ≈ - x
+  x+x≈x⇒x≈0    : x + x ≈ x → x ≈ 0#
+  x[y-z]≈xy-xz : x * (y - z) ≈ x * y - x * z
+  [y-z]x≈yx-zx : (y - z) * x ≈ (y * x) - (z * x)
   ```
 
-* Added new definitions to `Algebra.Structures`:
-  ```agda
-  record IsUnitalMagma (_∙_ : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsInvertibleMagma (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
-  record IsInvertibleUnitalMagma (_∙_ : Op₂ A) (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
-  record IsQuasigroup (∙ \\ // : Op₂ A) : Set (a ⊔ ℓ)
-  record IsLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ)
-  record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ)
-  record IsKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
-  record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
-  record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
-  record IsIdempotentMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsAlternativeMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsFlexibleMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsMedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsSemimedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsLeftBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsRightBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsMoufangLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
-  record IsMiddleBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  ```
-  and the existing record `IsLattice` now provides
+* Added new functions in `Codata.Guarded.Stream`:
   ```
-  ∨-isCommutativeSemigroup : IsCommutativeSemigroup ∨
-  ∧-isCommutativeSemigroup : IsCommutativeSemigroup ∧
+  transpose  : List (Stream A) → Stream (List A)
+  transpose⁺ : List⁺ (Stream A) → Stream (List⁺ A)
+  concat     : Stream (List⁺ A) → Stream A
   ```
-  and their corresponding algebraic substructures.
 
-* Added new records to `Algebra.Morphism.Structures`:
-  ```agda
-  record IsQuasigroupHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsQuasigroupMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsQuasigroupIsomorphism  (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsLoopHomomorphism       (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsLoopMonomorphism       (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsLoopIsomorphism        (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsRingWithoutOneIsoMorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsKleeneAlgebraHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsKleeneAlgebraMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsKleeneAlgebraIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+* Added new proofs in `Codata.Guarded.Stream.Properties`:
+  ```
+  cong-concat       : ass ≈ bss → concat ass ≈ concat bss
+  map-concat        : map f (concat ass) ≈ concat (map (List⁺.map f) ass)
+  lookup-transpose  : lookup n (transpose ass) ≡ List.map (lookup n) ass
+  lookup-transpose⁺ : lookup n (transpose⁺ ass) ≡ List⁺.map (lookup n) ass
   ```
 
 * Added new proofs in `Data.Bool.Properties`:
@@ -2478,6 +2401,25 @@ Additions to existing modules
   xor-annihilates-not : (not x) xor (not y) ≡ x xor y
   ```
 
+* Exposed container combinator conversion functions from `Data.Container.Combinator.Properties` 
+  separately from their correctness proofs in `Data.Container.Combinator`:
+  ```
+  to-id      : F.id A → ⟦ id ⟧ A
+  from-id    : ⟦ id ⟧ A → F.id A
+  to-const   : A → ⟦ const A ⟧ B
+  from-const : ⟦ const A ⟧ B → A
+  to-∘       : ⟦ C₁ ⟧ (⟦ C₂ ⟧ A) → ⟦ C₁ ∘ C₂ ⟧ A
+  from-∘     : ⟦ C₁ ∘ C₂ ⟧ A → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A)
+  to-×       : ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A → ⟦ C₁ × C₂ ⟧ A
+  from-×     : ⟦ C₁ × C₂ ⟧ A → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A
+  to-Π       : (∀ i → ⟦ Cᵢ i ⟧ A) → ⟦ Π I Cᵢ ⟧ A
+  from-Π     : ⟦ Π I Cᵢ ⟧ A → ∀ i → ⟦ Cᵢ i ⟧ A
+  to-⊎       : ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A → ⟦ C₁ ⊎ C₂ ⟧ A
+  from-⊎     : ⟦ C₁ ⊎ C₂ ⟧ A → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A
+  to-Σ       : (∃ λ i → ⟦ C i ⟧ A) → ⟦ Σ I C ⟧ A
+  from-Σ     : ⟦ Σ I C ⟧ A → ∃ λ i → ⟦ C i ⟧ A
+  ```
+
 * Added new functions in `Data.Fin.Base`:
   ```
   finToFun  : Fin (m ^ n) → (Fin n → Fin m)
@@ -2487,11 +2429,11 @@ Additions to existing modules
   ```
 
 * Added new proofs in `Data.Fin.Induction`:
-  every (strict) partial order is well-founded and Noetherian.
-
   ```agda
-  spo-wellFounded : ∀ {r} {_⊏_ : Rel (Fin n) r} → IsStrictPartialOrder _≈_ _⊏_ → WellFounded _⊏_
-  spo-noetherian  : ∀ {r} {_⊏_ : Rel (Fin n) r} → IsStrictPartialOrder _≈_ _⊏_ → WellFounded (flip _⊏_)
+  spo-wellFounded : IsStrictPartialOrder _≈_ _⊏_ → WellFounded _⊏_
+  spo-noetherian  : IsStrictPartialOrder _≈_ _⊏_ → WellFounded (flip _⊏_)
+  
+  <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
   ```
 
 * Added new definitions and proofs in `Data.Fin.Permutation`:
@@ -2502,15 +2444,10 @@ Additions to existing modules
   remove-insert  : remove i (insert i j π) ≈ π
   ```
 
-* In `Data.Fin.Properties`:
-  the proof that an injection from a type `A` into `Fin n` induces a
-  decision procedure for `_≡_` on `A` has been generalized to other
-  equivalences over `A` (i.e. to arbitrary setoids), and renamed from
-  `eq?` to the more descriptive `inj⇒≟` and `inj⇒decSetoid`.
-
 * Added new proofs in `Data.Fin.Properties`:
   ```
   1↔⊤                : Fin 1 ↔ ⊤
+  2↔Bool             : Fin 2 ↔ Bool
 
   0≢1+n              : zero ≢ suc i
 
@@ -2555,31 +2492,24 @@ Additions to existing modules
 
   inject≤-trans      : inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (≤-trans m≤n n≤o)
   inject≤-irrelevant : inject≤ i m≤n ≡ inject≤ i m≤n′
-  ```
-
-* Changed the fixity of `Data.Fin.Substitution.TermSubst._/Var_`.
-  ```agda
-  infix 8 ↦ infixl 8
+  i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j
   ```
 
 * Added new lemmas in `Data.Fin.Substitution.Lemmas.TermLemmas`:
   ```
-  map-var≡ : {ρ₁ : Sub Fin m n} {ρ₂ : Sub T m n} {f : Fin m → Fin n} →
-             (∀ x → lookup ρ₁ x ≡ f x) →
-             (∀ x → lookup ρ₂ x ≡ T.var (f x)) →
-             map T.var ρ₁ ≡ ρ₂
-  wk≡wk : map T.var VarSubst.wk ≡ T.wk {n = n}
-  id≡id : map T.var VarSubst.id ≡ T.id {n = n}
-  sub≡sub : {x : Fin n} → map T.var (VarSubst.sub x) ≡ T.sub (T.var x)
-  ↑≡↑ : {ρ : Sub Fin m n} → map T.var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
-  /Var≡/ : {ρ : Sub Fin m n} {t} → t /Var ρ ≡ t T./ map T.var ρ
-  sub-renaming-commutes : {ρ : Sub T m n} →
-    t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
-  sub-commutes-with-renaming : {ρ : Sub Fin m n} →
-    t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)
-
-* Added new functions in `Data.Integer.Base`:
+  map-var≡ : (∀ x → lookup ρ₁ x ≡ f x) → (∀ x → lookup ρ₂ x ≡ T.var (f x)) → map var ρ₁ ≡ ρ₂
+  wk≡wk    : map var VarSubst.wk ≡ T.wk {n = n}
+  id≡id    : map var VarSubst.id ≡ T.id {n = n}
+  sub≡sub  : map var (VarSubst.sub x) ≡ T.sub (T.var x)
+  ↑≡↑      : map var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
+  /Var≡/   : t /Var ρ ≡ t T./ map T.var ρ
+  
+  sub-renaming-commutes : t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
+  sub-commutes-with-renaming : t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)
   ```
+
+* Added new functions and definitions in `Data.Integer.Base`:
+  ```agda
   _^_ : ℤ → ℕ → ℤ
 
   +-0-rawGroup  : Rawgroup 0ℓ 0ℓ
@@ -2654,11 +2584,32 @@ Additions to existing modules
   ∈-resp-[σ⁻¹∘σ]   : (σ : xs ↭ ys) → (ix : x ∈ xs)   → ∈-resp-↭   (trans σ (↭-sym σ)) ix ≡ ix
   ```
 
+* In `Data.List.Relation.Binary.Permutation.Setoid.Properties`:
+  ```agda
+  foldr-commMonoid : xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys
+  ```
+
+* Added new function to `Data.List.Relation.Binary.Permutation.Propositional.Properties`
+  ```agda
+  ↭-reverse : reverse xs ↭ xs
+  ```
+
+* Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
+  ```
+  ⊆-mergeˡ : xs ⊆ merge _≤?_ xs ys
+  ⊆-mergeʳ : ys ⊆ merge _≤?_ xs ys  
+  ```
+  
 * Added new functions in `Data.List.Relation.Unary.All`:
   ```
   decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
   ```
 
+* Added new proof to `Data.List.Relation.Unary.All.Properties`:
+  ```agda
+  gmap⁻ : Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P
+  ```
+
 * Added new functions in `Data.List.Fresh.Relation.Unary.All`:
   ```
   decide :  Π[ P ∪ Q ] → Π[ All {R = R} P ∪ Any Q ]
@@ -2672,15 +2623,22 @@ Additions to existing modules
 
 * Added new proofs to `Data.List.Membership.Setoid.Properties`:
   ```agda
-  mapWith∈-id  : mapWith∈ xs (λ {x} _ → x) ≡ xs
-  map-mapWith∈ : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
+  mapWith∈-id     : mapWith∈ xs (λ {x} _ → x) ≡ xs
+  map-mapWith∈    : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
   index-injective : index x₁∈xs ≡ index x₂∈xs → x₁ ≈ x₂
+  
+  ∈-++⁺∘++⁻         : (p : v ∈ xs ++ ys) → [ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ (∈-++⁻ xs p) ≡ p
+  ∈-++⁻∘++⁺         : (p : v ∈ xs ⊎ v ∈ ys) → ∈-++⁻ xs ([ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ p) ≡ p
+  ∈-++↔             : (v ∈ xs ⊎ v ∈ ys) ↔ v ∈ xs ++ ys
+  ∈-++-comm         : v ∈ xs ++ ys → v ∈ ys ++ xs
+  ∈-++-comm∘++-comm : (p : v ∈ xs ++ ys) → ∈-++-comm ys xs (∈-++-comm xs ys p) ≡ p
+  ∈-++↔++           : v ∈ xs ++ ys ↔ v ∈ ys ++ xs
   ```
 
 * Add new proofs in `Data.List.Properties`:
   ```agda
   ∈⇒∣product : n ∈ ns → n ∣ product ns
-  ∷ʳ-++ : xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
+  ∷ʳ-++      : xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
 
   concatMap-cong : f ≗ g → concatMap f ≗ concatMap g
   concatMap-pure : concatMap [_] ≗ id
@@ -2711,14 +2669,46 @@ Additions to existing modules
   map-replicate     : map f (replicate n x) ≡ replicate n (f x)
   zipWith-replicate : zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)
 
-  length-iterate : length (iterate f x n) ≡ n
-  iterate-id     : iterate id x n ≡ replicate n x
-  lookup-iterate : lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)
+  length-iterate : length (iterate f x n) ≡ n
+  iterate-id     : iterate id x n ≡ replicate n x
+  lookup-iterate : lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)
+
+  length-insertAt   : length (insertAt xs i v) ≡ suc (length xs)
+  length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
+  removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
+  insertAt-removeAt : insertAt (removeAt xs i) (cast (sym (lengthAt-removeAt xs i)) i) (lookup xs i) ≡ xs
+  
+  cartesianProductWith-zeroˡ       : cartesianProductWith f [] ys ≡ []
+  cartesianProductWith-zeroʳ       : cartesianProductWith f xs [] ≡ []
+  cartesianProductWith-distribʳ-++ : cartesianProductWith f (xs ++ ys) zs ≡
+                                     cartesianProductWith f xs zs ++ cartesianProductWith f ys zs
+									 
+  foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
+  foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
+  ```
+
+* In `Data.List.NonEmpty.Base`:
+  ```agda
+  drop+ : ℕ → List⁺ A → List⁺ A
+  ```
+
+* Added new proofs in `Data.List.NonEmpty.Properties`:
+  ```agda
+  length-++⁺      : length (xs ++⁺ ys) ≡ length xs + length ys
+  length-++⁺-tail : length (xs ++⁺ ys) ≡ suc (length xs + length (tail ys))
+  ++-++⁺          : (xs ++ ys) ++⁺ zs ≡ xs ++⁺ ys ++⁺ zs
+  ++⁺-cancelˡ′    : xs ++⁺ zs ≡ ys ++⁺ zs′ → List.length xs ≡ List.length ys → zs ≡ zs′
+  ++⁺-cancelˡ     : xs ++⁺ ys ≡ xs ++⁺ zs → ys ≡ zs
+  drop+-++⁺       : drop+ (length xs) (xs ++⁺ ys) ≡ ys
+  map-++⁺-commute : map f (xs ++⁺ ys) ≡ map f xs ++⁺ map f ys
+  length-map      : length (map f xs) ≡ length xs
+  map-cong        : f ≗ g → map f ≗ map g
+  map-compose     : map (g ∘ f) ≗ map g ∘ map f
+  ```
 
-  length-insertAt   : length (insertAt xs i v) ≡ suc (length xs)
-  length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
-  removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
-  insertAt-removeAt : insertAt (removeAt xs i) (cast (sym (lengthAt-removeAt xs i)) i) (lookup xs i) ≡ xs
+* Added new proof to `Data.Maybe.Properties`
+  ```agda
+  <∣>-idem : Idempotent _<∣>_
   ```
 
 * Added new patterns and definitions to `Data.Nat.Base`:
@@ -2735,8 +2725,8 @@ Additions to existing modules
   pattern ≤″-offset k = less-than-or-equal {k} refl
   pattern <″-offset k = ≤″-offset k
 
-  s≤″s⁻¹ : ∀ {m n} → suc m ≤″ suc n → m ≤″ n
-  s<″s⁻¹ : ∀ {m n} → suc m <″ suc n → m <″ n
+  s≤″s⁻¹ : suc m ≤″ suc n → m ≤″ n
+  s<″s⁻¹ : suc m <″ suc n → m <″ n
 
   _⊔′_ : ℕ → ℕ → ℕ
   _⊓′_ : ℕ → ℕ → ℕ
@@ -2790,7 +2780,7 @@ Additions to existing modules
   m*n≢0     : .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
   m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n
 
-  s<s-injective : ∀ {p q : m < n} → s<s p ≡ s<s q → p ≡ q
+  s<s-injective : s<s p ≡ s<s q → p ≡ q
   <-step        : m < n → m < 1 + n
   m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
 
@@ -2816,12 +2806,12 @@ Additions to existing modules
 
   n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
 
-  m^n>0 : ∀ m .{{_ : NonZero m}} n → m ^ n > 0
+  m^n>0 : .{{_ : NonZero m}} n → m ^ n > 0
 
-  ^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
-  ^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
-  ^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
-  ^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_
+  ^-monoˡ-≤ : (_^ n) Preserves _≤_ ⟶ _≤_
+  ^-monoʳ-≤ : .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
+  ^-monoˡ-< : .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
+  ^-monoʳ-< : 1 < m → (m ^_) Preserves _<_ ⟶ _<_
 
   n≡⌊n+n/2⌋ : n ≡ ⌊ n + n /2⌋
   n≡⌈n+n/2⌉ : n ≡ ⌈ n + n /2⌉
@@ -2835,10 +2825,7 @@ Additions to existing modules
   ⊔≡⊔′ : m ⊔ n ≡ m ⊔′ n
   ⊓≡⊓′ : m ⊓ n ≡ m ⊓′ n
   ∣-∣≡∣-∣′ : ∣ m - n ∣ ≡ ∣ m - n ∣′
-  ```
-
-* Re-exported additional functions from `Data.Nat`:
-  ```agda
+  
   nonZero? : Decidable NonZero
   eq? : A ↣ ℕ → DecidableEquality A
   ≤-<-connex : Connex _≤_ _<_
@@ -2854,7 +2841,7 @@ Additions to existing modules
   ```agda
   [n-k]*[n-k-1]!≡[n-k]!   : k < n → (n ∸ k) * (n ∸ suc k) ! ≡ (n ∸ k) !
   [n-k]*d[k+1]≡[k+1]*d[k] : k < n → (n ∸ k) * ((suc k) ! * (n ∸ suc k) !) ≡ (suc k) * (k ! * (n ∸ k) !)
-  k![n∸k]!∣n!              : k ≤ n → k ! * (n ∸ k) ! ∣ n !
+  k![n∸k]!∣n!             : k ≤ n → k ! * (n ∸ k) ! ∣ n !
   nP1≡n                   : n P 1 ≡ n
   nC1≡n                   : n C 1 ≡ n
   nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
@@ -2909,6 +2896,12 @@ Additions to existing modules
   pattern `suc x = con (quote ℕ.suc) (x ⟨∷⟩ [])
   ```
 
+* Added new functions and proofs in `Data.Nat.GeneralisedArithmetic`:
+  ```agda
+  iterate         : (A → A) → A → ℕ → A
+  iterate-is-fold : fold z s m ≡ iterate s z m
+  ```
+
 * Added new proofs in `Data.Parity.Properties`:
   ```agda
   suc-homo-⁻¹ : (parity (suc n)) ⁻¹ ≡ parity n
@@ -2928,8 +2921,8 @@ Additions to existing modules
 
 * Added new definitions and proofs in `Data.Rational.Properties`:
   ```agda
-  ↥ᵘ-toℚᵘ : ↥ᵘ (toℚᵘ p) ≡ ↥ p
-  ↧ᵘ-toℚᵘ : ↧ᵘ (toℚᵘ p) ≡ ↧ p
+  ↥ᵘ-toℚᵘ     : ↥ᵘ (toℚᵘ p) ≡ ↥ p
+  ↧ᵘ-toℚᵘ     : ↧ᵘ (toℚᵘ p) ≡ ↧ p
   ↥p≡↥q≡0⇒p≡q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≡ q
 
   _≥?_ : Decidable _≥_
@@ -2945,6 +2938,10 @@ Additions to existing modules
   pos⇒nonZero       : .{{Positive p}} → NonZero p
   neg⇒nonZero       : .{{Negative p}} → NonZero p
   nonZero⇒1/nonZero : .{{_ : NonZero p}} → NonZero (1/ p)
+  
+  <-dense              : Dense _<_
+  <-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
+  <-denseLinearOrder   : DenseLinearOrder 0ℓ 0ℓ 0ℓ
   ```
 
 * Added new rounding functions in `Data.Rational.Unnormalised.Base`:
@@ -2955,8 +2952,8 @@ Additions to existing modules
 
 * Added new definitions in `Data.Rational.Unnormalised.Properties`:
   ```agda
-  ↥p≡0⇒p≃0 : ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ
-  p≃0⇒↥p≡0 : p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ
+  ↥p≡0⇒p≃0    : ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ
+  p≃0⇒↥p≡0    : p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ
   ↥p≡↥q≡0⇒p≃q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≃ q
 
   +-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
@@ -2982,12 +2979,29 @@ Additions to existing modules
   pos⊓pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊓ q)
   pos⊔pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊔ q)
   1/nonZero⇒nonZero    : .{{_ : NonZero p}} → NonZero (1/ p)
+  
+  0≄1             : 0ℚᵘ ≄ 1ℚᵘ
+  ≃-≄-irreflexive : Irreflexive _≃_ _≄_
+  ≄-symmetric     : Symmetric _≄_
+  ≄-cotransitive  : Cotransitive _≄_
+  ≄⇒invertible    : p ≄ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
+
+  <-dense         : Dense _<_
+  
+  <-isDenseLinearOrder         : IsDenseLinearOrder _≃_ _<_
+  +-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
+  +-*-isHeytingField           : IsHeytingField _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
+  
+  <-denseLinearOrder           : DenseLinearOrder 0ℓ 0ℓ 0ℓ
+  +-*-heytingCommutativeRing   : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
+  +-*-heytingField             : HeytingField 0ℓ 0ℓ 0ℓ
+  
+  module ≃-Reasoning = SetoidReasoning ≃-setoid
   ```
 
 * Added new functions to `Data.Product.Nary.NonDependent`:
   ```agda
-  zipWith : (∀ k → Projₙ as k → Projₙ bs k → Projₙ cs k) →
-            Product n as → Product n bs → Product n cs
+  zipWith : (∀ k → Projₙ as k → Projₙ bs k → Projₙ cs k) → Product n as → Product n bs → Product n cs
   ```
 
 * Added new proof to `Data.Product.Properties`:
@@ -2995,8 +3009,7 @@ Additions to existing modules
   map-cong : f ≗ g → h ≗ i → map f h ≗ map g i
   ```
 
-* Added new definitions to `Data.Product.Properties` and
-  `Function.Related.TypeIsomorphisms` for convenience:
+* Added new definitions to `Data.Product.Properties`:
   ```
   Σ-≡,≡→≡ : (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂) → p₁ ≡ p₂
   Σ-≡,≡←≡ : p₁ ≡ p₂ → (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂)
@@ -3006,12 +3019,9 @@ Additions to existing modules
 
 * Added new proofs to `Data.Product.Relation.Binary.Lex.Strict`
   ```agda
-  ×-respectsʳ : Transitive _≈₁_ →
-                _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_
-  ×-respectsˡ : Symmetric _≈₁_ → Transitive _≈₁_ →
-                 _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_
-  ×-wellFounded' : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ →
-                   WellFounded _<₁_ → WellFounded _<₂_ → WellFounded _<ₗₑₓ_
+  ×-respectsʳ : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_
+  ×-respectsˡ : Symmetric _≈₁_ → Transitive _≈₁_ → _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_
+  ×-wellFounded' : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ → WellFounded _<₁_ → WellFounded _<₂_ → WellFounded _<ₗₑₓ_
   ```
 
 * Added new definitions to `Data.Sign.Base`:
@@ -3042,21 +3052,19 @@ Additions to existing modules
   ```
 
 * Added new proofs in `Data.String.Properties`:
-  ```
+  ```agda
   ≤-isDecTotalOrder-≈ : IsDecTotalOrder _≈_ _≤_
-  ≤-decTotalOrder-≈   :  DecTotalOrder _ _ _
+  ≤-decTotalOrder-≈   : DecTotalOrder _ _ _
   ```
 * Added new definitions in `Data.Sum.Properties`:
-  ```
+  ```agda
   swap-↔ : (A ⊎ B) ↔ (B ⊎ A)
   ```
 
 * Added new proofs in `Data.Sum.Properties`:
-  ```
-  map-assocˡ : (f : A → C) (g : B → D) (h : C → F) →
-    map (map f g) h ∘ assocˡ ≗ assocˡ ∘ map f (map g h)
-  map-assocʳ : (f : A → C) (g : B → D) (h : C → F) →
-    map f (map g h) ∘ assocʳ ≗ assocʳ ∘ map (map f g) h
+  ```agda
+  map-assocˡ : map (map f g) h ∘ assocˡ ≗ assocˡ ∘ map f (map g h)
+  map-assocʳ : map f (map g h) ∘ assocʳ ≗ assocʳ ∘ map (map f g) h
   ```
 
 * Made `Map` public in `Data.Tree.AVL.IndexedMap`
@@ -3066,8 +3074,8 @@ Additions to existing modules
   truncate : m ≤ n → Vec A n → Vec A m
   pad      : m ≤ n → A → Vec A m → Vec A n
 
-  FoldrOp A B = ∀ {n} → A → B n → B (suc n)
-  FoldlOp A B = ∀ {n} → B n → A → B (suc n)
+  FoldrOp A B = A → B n → B (suc n)
+  FoldlOp A B = B n → A → B (suc n)
 
   foldr′ : (A → B → B) → B → Vec A n → B
   foldl′ : (B → A → B) → B → Vec A n → B
@@ -3116,7 +3124,7 @@ Additions to existing modules
   []≔-++-↑ʳ   : (xs ++ ys) [ m ↑ʳ i ]≔ y ≡ xs ++ (ys [ i ]≔ y)
   unfold-ʳ++  : xs ʳ++ ys ≡ reverse xs ++ ys
 
-  foldl-universal : ∀ (h : ∀ {c} (C : ℕ → Set c) (g : FoldlOp A C) (e : C zero) → ∀ {n} → Vec A n → C n) →
+  foldl-universal : (e : C zero) → ∀ {n} → Vec A n → C n) →
                     (∀ ... → h C g e [] ≡ e) →
                     (∀ ... → h C g e ∘ (x ∷_) ≗ h (C ∘ suc) g (g e x)) →
                     h B f e ≗ foldl B f e
@@ -3202,6 +3210,11 @@ Additions to existing modules
   index-∈-fromList⁺ : Any.index (∈-fromList⁺ v∈xs) ≡ indexₗ v∈xs
   ```
 
+* Added new isomorphisms to `Data.Unit.Polymorphic.Properties`:
+  ```agda
+  ⊤↔⊤* : ⊤ ↔ ⊤*
+  ```
+
 * Added new proofs in `Data.Vec.Functional.Properties`:
   ```
   map-updateAt : f ∘ g ≗ h ∘ f → map f (updateAt i g xs) ≗ updateAt i h (map f xs)
@@ -3209,13 +3222,21 @@ Additions to existing modules
 
 * Added new proofs in `Data.Vec.Relation.Binary.Lex.Strict`:
   ```agda
-  xs≮[] : {xs : Vec A n} → ¬ xs < []
-  <-respectsˡ : IsPartialEquivalence _≈_ → _≺_ Respectsˡ _≈_ →
-                ∀ {m n} → _Respectsˡ_ (_<_ {m} {n}) _≋_
-  <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ →
-                ∀ {m n} → _Respectsʳ_ (_<_ {m} {n}) _≋_
-  <-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ →
-                  ∀ {n} → WellFounded (_<_ {n})
+  xs≮[] : ¬ xs < []
+  
+  <-respectsˡ : IsPartialEquivalence _≈_ → _≺_ Respectsˡ _≈_ → _<_ Respectsˡ _≋_
+  <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ → _<_ _Respectsʳ _≋_
+  <-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ → WellFounded _<_
+  ```
+
+* Added new function to `Data.Vec.Relation.Binary.Pointwise.Inductive`
+  ```agda
+  cong-[_]≔ : Pointwise _∼_ xs ys → Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)
+  ```
+
+* Added new function to `Data.Vec.Relation.Binary.Equality.Setoid`
+  ```agda
+  map-[]≔ : map f (xs [ i ]≔ p) ≋ map f xs [ i ]≔ f p
   ```
 
 * Added new functions in `Data.Vec.Relation.Unary.Any`:
@@ -3226,12 +3247,28 @@ Additions to existing modules
 * Added new functions in `Data.Vec.Relation.Unary.All`:
   ```
   decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
+  
+  lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
+  lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
+  lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
+  lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
   ```
 
 * Added vector associativity proof to  `Data.Vec.Relation.Binary.Equality.Setoid`:
   ```
   ++-assoc : (xs ++ ys) ++ zs ≋ xs ++ (ys ++ zs)
   ```
+  
+* Added new isomorphisms to `Data.Vec.N-ary`:
+  ```agda
+  Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B
+  ```
+
+* Added new isomorphisms to `Data.Vec.Recursive`:
+  ```agda
+  lift↔             : A ↔ B → A ^ n ↔ B ^ n
+  Fin[m^n]↔Fin[m]^n : Fin (m ^ n) ↔ Fin m Vec.^ n
+  ```
 
 * Added new functions in `Effect.Monad.State`:
   ```
@@ -3240,17 +3277,29 @@ Additions to existing modules
   execState : State s a → s → s
   ```
   
+* Added a non-dependent version of `flip` in `Function.Base`:
+  ```agda
+  flip′ : (A → B → C) → (B → A → C)
+  ```
+
 * Added new proofs and definitions in `Function.Bundles`:
   ```agda
   LeftInverse.isSplitSurjection : LeftInverse → IsSplitSurjection to
   LeftInverse.surjection        : LeftInverse → Surjection
   SplitSurjection               = LeftInverse
+  _⟨$⟩_ = Func.to
   ```
 
-* Added new application operator synonym to `Function.Bundles`:
+* Added new proofs to `Function.Properties.Inverse`:
   ```agda
-  _⟨$⟩_ : Func From To → Carrier From → Carrier To
-  _⟨$⟩_ = Func.to
+  ↔-refl  : Reflexive _↔_
+  ↔-sym   : Symmetric _↔_
+  ↔-trans : Transitive _↔_
+  
+  ↔⇒↣   : A ↔ B → A ↣ B
+  ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)
+  
+  Inverse⇒Injection : Inverse S T → Injection S T
   ```
 
 * Added new proofs in `Function.Construct.Symmetry`:
@@ -3271,7 +3320,6 @@ Additions to existing modules
   
 * Added new proof and record in `Function.Structures`:
   ```agda
-  IsLeftInverse.isSurjection : IsLeftInverse to from → IsSurjection to
   record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
   ```
 
@@ -3280,6 +3328,14 @@ Additions to existing modules
   f⁻ = proj₁ ∘ surjective
   ```
 
+* Added new proof to `Induction.WellFounded`
+  ```agda
+  Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
+  acc⇒asym        : Acc _<_ x → x < y → ¬ (y < x)
+  wf⇒asym         : WellFounded _<_ → Asymmetric _<_
+  wf⇒irrefl       : _<_ Respects₂ _≈_ → Symmetric _≈_ → WellFounded _<_ → Irreflexive _≈_ _<_
+  ```
+
 * Added new operations in `IO`:
   ```
   Colist.forM  : Colist A → (A → IO B) → IO (Colist B)
@@ -3313,6 +3369,15 @@ Additions to existing modules
   prependVLams : List String → Term → Term
   ```
 
+* Added new types and operations to `Reflection.TCM`:
+  ```
+  Blocker : Set
+  blockerMeta : Meta → Blocker
+  blockerAny : List Blocker → Blocker
+  blockerAll : List Blocker → Blocker
+  blockTC : Blocker → TC A
+  ```
+
 * Added new operations in `Relation.Binary.Construct.Closure.Equivalence`:
   ```
   fold   : IsEquivalence _∼_ → _⟶_ ⇒ _∼_ → EqClosure _⟶_ ⇒ _∼_
@@ -3333,21 +3398,21 @@ Additions to existing modules
 
 * Added new proofs to `Relation.Binary.Lattice.Properties.{Join,Meet}Semilattice`:
   ```agda
-  isPosemigroup : IsPosemigroup _≈_ _≤_ _∨_
-  posemigroup : Posemigroup c ℓ₁ ℓ₂
-  ≈-dec⇒≤-dec : Decidable _≈_ → Decidable _≤_
+  isPosemigroup           : IsPosemigroup _≈_ _≤_ _∨_
+  posemigroup             : Posemigroup c ℓ₁ ℓ₂
+  ≈-dec⇒≤-dec             : Decidable _≈_ → Decidable _≤_
   ≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_
   ```
 
 * Added new proofs to `Relation.Binary.Lattice.Properties.Bounded{Join,Meet}Semilattice`:
   ```agda
   isCommutativePomonoid : IsCommutativePomonoid _≈_ _≤_ _∨_ ⊥
-  commutativePomonoid : CommutativePomonoid c ℓ₁ ℓ₂
+  commutativePomonoid   : CommutativePomonoid c ℓ₁ ℓ₂
   ```
 
 * Added new proofs to `Relation.Binary.Properties.Poset`:
   ```agda
-  ≤-dec⇒≈-dec : Decidable _≤_ → Decidable _≈_
+  ≤-dec⇒≈-dec             : Decidable _≤_ → Decidable _≈_
   ≤-dec⇒isDecPartialOrder : Decidable _≤_ → IsDecPartialOrder _≈_ _≤_
   ```
 
@@ -3359,7 +3424,7 @@ Additions to existing modules
 * Added new proofs in `Relation.Binary.PropositionalEquality.Properties`:
   ```
   subst-application′ : subst Q eq (f x p) ≡ f y (subst P eq p)
-  sym-cong : sym (cong f p) ≡ cong f (sym p)
+  sym-cong           : sym (cong f p) ≡ cong f (sym p)
   ```
 
 * Added new proofs in `Relation.Binary.HeterogeneousEquality`:
@@ -3377,49 +3442,51 @@ Additions to existing modules
 * Added new proofs in `Relation.Unary.Properties`:
   ```
   ⊆-reflexive : _≐_ ⇒ _⊆_
-  ⊆-antisym : Antisymmetric _≐_ _⊆_
-  ⊆-min : Min _⊆_ ∅
-  ⊆-max : Max _⊆_ U
-  ⊂⇒⊆ : _⊂_ ⇒ _⊆_
-  ⊂-trans : Trans _⊂_ _⊂_ _⊂_
-  ⊂-⊆-trans : Trans _⊂_ _⊆_ _⊂_
-  ⊆-⊂-trans : Trans _⊆_ _⊂_ _⊂_
-  ⊂-respʳ-≐ : _⊂_ Respectsʳ _≐_
-  ⊂-respˡ-≐ : _⊂_ Respectsˡ _≐_
-  ⊂-resp-≐ : _⊂_ Respects₂ _≐_
-  ⊂-irrefl : Irreflexive _≐_ _⊂_
-  ⊂-antisym : Antisymmetric _≐_ _⊂_
-  ∅-⊆′ : (P : Pred A ℓ) → ∅ ⊆′ P
-  ⊆′-U : (P : Pred A ℓ) → P ⊆′ U
-  ⊆′-refl : Reflexive {A = Pred A ℓ} _⊆′_
+  ⊆-antisym   : Antisymmetric _≐_ _⊆_
+  ⊆-min       : Min _⊆_ ∅
+  ⊆-max       : Max _⊆_ U
+  ⊂⇒⊆         : _⊂_ ⇒ _⊆_
+  ⊂-trans     : Trans _⊂_ _⊂_ _⊂_
+  ⊂-⊆-trans   : Trans _⊂_ _⊆_ _⊂_
+  ⊆-⊂-trans   : Trans _⊆_ _⊂_ _⊂_
+  ⊂-respʳ-≐   : _⊂_ Respectsʳ _≐_
+  ⊂-respˡ-≐   : _⊂_ Respectsˡ _≐_
+  ⊂-resp-≐    : _⊂_ Respects₂ _≐_
+  ⊂-irrefl    : Irreflexive _≐_ _⊂_
+  ⊂-antisym   : Antisymmetric _≐_ _⊂_
+  
+  ∅-⊆′         : (P : Pred A ℓ) → ∅ ⊆′ P
+  ⊆′-U         : (P : Pred A ℓ) → P ⊆′ U
+  ⊆′-refl      : Reflexive {A = Pred A ℓ} _⊆′_
   ⊆′-reflexive : _≐′_ ⇒ _⊆′_
-  ⊆′-trans : Trans _⊆′_ _⊆′_ _⊆′_
-  ⊆′-antisym : Antisymmetric _≐′_ _⊆′_
-  ⊆′-min : Min _⊆′_ ∅
-  ⊆′-max : Max _⊆′_ U
-  ⊂′⇒⊆′ : _⊂′_ ⇒ _⊆′_
-  ⊂′-trans : Trans _⊂′_ _⊂′_ _⊂′_
-  ⊂′-⊆′-trans : Trans _⊂′_ _⊆′_ _⊂′_
-  ⊆′-⊂′-trans : Trans _⊆′_ _⊂′_ _⊂′_
-  ⊂′-respʳ-≐′ : _⊂′_ Respectsʳ _≐′_
-  ⊂′-respˡ-≐′ : _⊂′_ Respectsˡ _≐′_
-  ⊂′-resp-≐′ : _⊂′_ Respects₂ _≐′_
-  ⊂′-irrefl : Irreflexive _≐′_ _⊂′_
-  ⊂′-antisym : Antisymmetric _≐′_ _⊂′_
-  ⊆⇒⊆′ : _⊆_ ⇒ _⊆′_
-  ⊆′⇒⊆ : _⊆′_ ⇒ _⊆_
-  ⊂⇒⊂′ : _⊂_ ⇒ _⊂′_
-  ⊂′⇒⊂ : _⊂′_ ⇒ _⊂_
-  ≐-refl : Reflexive _≐_
-  ≐-sym : Sym _≐_ _≐_
-  ≐-trans : Trans _≐_ _≐_ _≐_
-  ≐′-refl : Reflexive _≐′_
-  ≐′-sym : Sym _≐′_ _≐′_
+  ⊆′-trans     : Trans _⊆′_ _⊆′_ _⊆′_
+  ⊆′-antisym   : Antisymmetric _≐′_ _⊆′_
+  ⊆′-min       : Min _⊆′_ ∅
+  ⊆′-max       : Max _⊆′_ U
+  ⊂′-trans     : Trans _⊂′_ _⊂′_ _⊂′_
+  ⊂′-⊆′-trans  : Trans _⊂′_ _⊆′_ _⊂′_
+  ⊆′-⊂′-trans  : Trans _⊆′_ _⊂′_ _⊂′_
+  ⊂′-respʳ-≐′  : _⊂′_ Respectsʳ _≐′_
+  ⊂′-respˡ-≐′  : _⊂′_ Respectsˡ _≐′_
+  ⊂′-resp-≐′   : _⊂′_ Respects₂ _≐′_
+  ⊂′-irrefl    : Irreflexive _≐′_ _⊂′_
+  ⊂′-antisym   : Antisymmetric _≐′_ _⊂′_
+  ⊂′⇒⊆′        : _⊂′_ ⇒ _⊆′_
+  ⊆⇒⊆′         : _⊆_ ⇒ _⊆′_
+  ⊆′⇒⊆         : _⊆′_ ⇒ _⊆_
+  ⊂⇒⊂′         : _⊂_ ⇒ _⊂′_
+  ⊂′⇒⊂         : _⊂′_ ⇒ _⊂_
+  
+  ≐-refl   : Reflexive _≐_
+  ≐-sym    : Sym _≐_ _≐_
+  ≐-trans  : Trans _≐_ _≐_ _≐_
+  ≐′-refl  : Reflexive _≐′_
+  ≐′-sym   : Sym _≐′_ _≐′_
   ≐′-trans : Trans _≐′_ _≐′_ _≐′_
-  ≐⇒≐′ : _≐_ ⇒ _≐′_
-  ≐′⇒≐ : _≐′_ ⇒ _≐_
+  ≐⇒≐′     : _≐_ ⇒ _≐′_
+  ≐′⇒≐     : _≐′_ ⇒ _≐_
 
-  U-irrelevant : Irrelevant {A = A} U
+  U-irrelevant : Irrelevant U
   ∁-irrelevant : (P : Pred A ℓ) → Irrelevant (∁ P)
   ```
 
@@ -3442,9 +3509,9 @@ Additions to existing modules
   RightTrans R S = Trans R S R
   LeftTrans  S R = Trans S R R
 
-  Dense        _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y
-  Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)
-  Tight    _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)
+  Dense        _<_ = x < y → ∃[ z ] x < z × z < y
+  Cotransitive _#_ = x # y → ∀ z → (x # z) ⊎ (z # y)
+  Tight    _≈_ _#_ = (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)
 
   Monotonic₁         _≤_ _⊑_ f     = f Preserves _≤_ ⟶ _⊑_
   Antitonic₁         _≤_ _⊑_ f     = f Preserves (flip _≤_) ⟶ _⊑_
@@ -3452,7 +3519,7 @@ Additions to existing modules
   MonotonicAntitonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ (flip _⊑_) ⟶ _≼_
   AntitonicMonotonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ _⊑_ ⟶ _≼_
   Antitonic₂         _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ (flip _⊑_) ⟶ _≼_
-  Adjoint            _≤_ _⊑_ f g   = ∀ {x y} → (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)
+  Adjoint            _≤_ _⊑_ f g   = (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)
   ```
 
 * Added new definitions in `Relation.Binary.Bundles`:
@@ -3479,9 +3546,9 @@ Additions to existing modules
 
 * Added new proofs to `Relation.Binary.Construct.Closure.Transitive`:
   ```
-  accessible⁻ : ∀ {x} → Acc _∼⁺_ x → Acc _∼_ x
+  accessible⁻  : Acc _∼⁺_ x → Acc _∼_ x
   wellFounded⁻ : WellFounded _∼⁺_ → WellFounded _∼_
-  accessible : ∀ {x} → Acc _∼_ x → Acc _∼⁺_ x
+  accessible   : Acc _∼_ x → Acc _∼⁺_ x
   ```
 
 * Added new operations in `Relation.Binary.PropositionalEquality.Properties`:
@@ -3502,65 +3569,6 @@ Additions to existing modules
   isFailure : ExitCode → Bool
   ```
 
-* Added new functions in `Codata.Guarded.Stream`:
-  ```
-  transpose : List (Stream A) → Stream (List A)
-  transpose⁺ : List⁺ (Stream A) → Stream (List⁺ A)
-  concat : Stream (List⁺ A) → Stream A
-  ```
-
-* Added new proofs in `Codata.Guarded.Stream.Properties`:
-  ```
-  cong-concat : {ass bss : Stream (List⁺.List⁺ A)} → ass ≈ bss → concat ass ≈ concat bss
-  map-concat : ∀ (f : A → B) ass → map f (concat ass) ≈ concat (map (List⁺.map f) ass)
-  lookup-transpose : ∀ n (ass : List (Stream A)) → lookup n (transpose ass) ≡ List.map (lookup n) ass
-  lookup-transpose⁺ : ∀ n (ass : List⁺ (Stream A)) → lookup n (transpose⁺ ass) ≡ List⁺.map (lookup n) ass
-  ```
-
-* Added new corollaries in `Data.List.Membership.Setoid.Properties`:
-  ```
-  ∈-++⁺∘++⁻ : ∀ {v} xs {ys} (p : v ∈ xs ++ ys) → [ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ (∈-++⁻ xs p) ≡ p
-  ∈-++⁻∘++⁺ : ∀ {v} xs {ys} (p : v ∈ xs ⊎ v ∈ ys) → ∈-++⁻ xs ([ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ p) ≡ p
-  ∈-++↔ : ∀ {v xs ys} → (v ∈ xs ⊎ v ∈ ys) ↔ v ∈ xs ++ ys
-  ∈-++-comm : ∀ {v} xs ys → v ∈ xs ++ ys → v ∈ ys ++ xs
-  ∈-++-comm∘++-comm : ∀ {v} xs {ys} (p : v ∈ xs ++ ys) → ∈-++-comm ys xs (∈-++-comm xs ys p) ≡ p
-  ∈-++↔++ : ∀ {v} xs ys → v ∈ xs ++ ys ↔ v ∈ ys ++ xs
-  ```
-
-* Exposed container combinator conversion functions from `Data.Container.Combinator.Properties` separate from their correctness proofs in `Data.Container.Combinator`:
-  ```
-  to-id      : F.id A → ⟦ id ⟧ A
-  from-id    : ⟦ id ⟧ A → F.id A
-  to-const   : (A : Set s) → A → ⟦ const A ⟧ B
-  from-const : (A : Set s) → ⟦ const A ⟧ B → A
-  to-∘       : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A) → ⟦ C₁ ∘ C₂ ⟧ A
-  from-∘     : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ ∘ C₂ ⟧ A → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A)
-  to-×       : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A → ⟦ C₁ × C₂ ⟧ A
-  from-×     : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ × C₂ ⟧ A → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A
-  to-Π       : (I : Set i) (Cᵢ : I → Container s p) → (∀ i → ⟦ Cᵢ i ⟧ A) → ⟦ Π I Cᵢ ⟧ A
-  from-Π     : (I : Set i) (Cᵢ : I → Container s p) → ⟦ Π I Cᵢ ⟧ A → ∀ i → ⟦ Cᵢ i ⟧ A
-  to-⊎       : (C₁ : Container s₁ p) (C₂ : Container s₂ p) → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A → ⟦ C₁ ⊎ C₂ ⟧ A
-  from-⊎     : (C₁ : Container s₁ p) (C₂ : Container s₂ p) → ⟦ C₁ ⊎ C₂ ⟧ A → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A
-  to-Σ       : (I : Set i) (C : I → Container s p) → (∃ λ i → ⟦ C i ⟧ A) → ⟦ Σ I C ⟧ A
-  from-Σ     : (I : Set i) (C : I → Container s p) → ⟦ Σ I C ⟧ A → ∃ λ i → ⟦ C i ⟧ A
-  ```
-
-* Added a non-dependent version of `Function.Base.flip` due to an issue noted in
-  Pull Request #1812:
-  ```agda
-  flip′ : (A → B → C) → (B → A → C)
-  ```
-
-* Added new proofs to `Data.List.Properties`
-  ```agda
-  cartesianProductWith-zeroˡ       : cartesianProductWith f [] ys ≡ []
-  cartesianProductWith-zeroʳ       : cartesianProductWith f xs [] ≡ []
-  cartesianProductWith-distribʳ-++ : cartesianProductWith f (xs ++ ys) zs ≡
-                                     cartesianProductWith f xs zs ++ cartesianProductWith f ys zs
-  foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
-  foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
-  ```
-
 NonZero/Positive/Negative changes
 ---------------------------------
 
@@ -3773,196 +3781,3 @@ This is a full list of proofs that have changed form to use irrelevant instance
   1/pos⇒pos : ∀ p .{{_ : NonZero p}} → (1/p : Positive (1/ p)) → Positive p
   1/neg⇒neg : ∀ p .{{_ : NonZero p}} → (1/p : Negative (1/ p)) → Negative p
   ```
-
-* In `Data.List.NonEmpty.Base`:
-  ```agda
-  drop+ : ℕ → List⁺ A → List⁺ A
-  ```
-  When drop+ping more than the size of the length of the list, the last element remains.
-
-* Added new proofs in `Data.List.NonEmpty.Properties`:
-  ```agda
-  length-++⁺ : length (xs ++⁺ ys) ≡ length xs + length ys
-  length-++⁺-tail : length (xs ++⁺ ys) ≡ suc (length xs + length (tail ys))
-  ++-++⁺ : (xs ++ ys) ++⁺ zs ≡ xs ++⁺ ys ++⁺ zs
-  ++⁺-cancelˡ′ : xs ++⁺ zs ≡ ys ++⁺ zs′ → List.length xs ≡ List.length ys → zs ≡ zs′
-  ++⁺-cancelˡ : xs ++⁺ ys ≡ xs ++⁺ zs → ys ≡ zs
-  drop+-++⁺ : drop+ (length xs) (xs ++⁺ ys) ≡ ys
-  map-++⁺-commute : map f (xs ++⁺ ys) ≡ map f xs ++⁺ map f ys
-  length-map : length (map f xs) ≡ length xs
-  map-cong : f ≗ g → map f ≗ map g
-  map-compose : map (g ∘ f) ≗ map g ∘ map f
-  ```
-
-* Added new functions and proofs in `Data.Nat.GeneralisedArithmetic`:
-  ```agda
-  iterate : (A → A) → A → ℕ → A
-  iterate-is-fold : ∀ (z : A) s m → fold z s m ≡ iterate s z m
-  ```
-
-* Added new proofs to `Function.Properties.Inverse`:
-  ```agda
-  Inverse⇒Injection : Inverse S T → Injection S T
-  ↔⇒↣ : A ↔ B → A ↣ B
-  ```
-
-* Added a new isomorphism to `Data.Fin.Properties`:
-  ```agda
-  2↔Bool : Fin 2 ↔ Bool
-  ```
-
-* Added new isomorphisms to `Data.Unit.Polymorphic.Properties`:
-  ```agda
-  ⊤↔⊤* : ⊤ {ℓ} ↔ ⊤*
-  ```
-
-* Added new isomorphisms to `Data.Vec.N-ary`:
-  ```agda
-  Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B
-  ```
-
-* Added new isomorphisms to `Data.Vec.Recursive`:
-  ```agda
-  lift↔ : ∀ n → A ↔ B → A ^ n ↔ B ^ n
-  Fin[m^n]↔Fin[m]^n : ∀ m n → Fin (m ^ n) ↔ Fin m Vec.^ n
-  ```
-
-* Added new functions to `Function.Properties.Inverse`:
-  ```agda
-  ↔-refl  : Reflexive _↔_
-  ↔-sym   : Symmetric _↔_
-  ↔-trans : Transitive _↔_
-  ```
-
-* Added new isomorphisms to `Function.Properties.Inverse`:
-  ```agda
-  ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)
-  ```
-
-* Added new function to `Data.Fin.Properties`
-  ```agda
-  i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j
-  ```
-
-* Added new function to `Data.Fin.Induction`
-  ```agda
-  <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
-  ```
-
-* In `Data.List.Relation.Binary.Permutation.Setoid.Properties`:
-  ```agda
-  foldr-commMonoid : xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys
-  ```
-
-* Added new proof, structure, and bundle to `Data.Rational.Properties`
-  ```agda
-  <-dense              : Dense _<_
-  <-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
-  <-denseLinearOrder   : DenseLinearOrder 0ℓ 0ℓ 0ℓ
-  ```
-
-* Added new module to `Data.Rational.Unnormalised.Properties`
-  ```agda
-  module ≃-Reasoning = SetoidReasoning ≃-setoid
-  ```
-
-* Added new functions to `Data.Rational.Unnormalised.Properties`
-  ```agda
-  0≠1 : 0ℚᵘ ≠ 1ℚᵘ
-  ≃-≠-irreflexive : Irreflexive _≃_ _≠_
-  ≠-symmetric : Symmetric _≠_
-  ≠-cotransitive : Cotransitive _≠_
-  ≠⇒invertible : p ≠ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
-
-  <-dense : Dense _<_
-  ```
-
-* Added new structures to `Data.Rational.Unnormalised.Properties`
-  ```agda
-  <-isDenseLinearOrder : IsDenseLinearOrder _≃_ _<_
-  +-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
-  +-*-isHeytingField : IsHeytingField _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
-  ```
-
-* Added new bundles to `Data.Rational.Unnormalised.Properties`
-  ```agda
-  <-denseLinearOrder : DenseLinearOrder 0ℓ 0ℓ 0ℓ
-  +-*-heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
-  +-*-heytingField : HeytingField 0ℓ 0ℓ 0ℓ
-  ```
-
-* Added new function to `Data.Vec.Relation.Binary.Pointwise.Inductive`
-  ```agda
-  cong-[_]≔ : Pointwise _∼_ xs ys → Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)
-  ```
-
-* Added new function to `Data.Vec.Relation.Binary.Equality.Setoid`
-  ```agda
-  map-[]≔ : map f (xs [ i ]≔ p) ≋ map f xs [ i ]≔ f p
-  ```
-
-* Added new function to `Data.List.Relation.Binary.Permutation.Propositional.Properties`
-  ```agda
-  ↭-reverse : (xs : List A) → reverse xs ↭ xs
-  ```
-
-* Added new functions to `Algebra.Properties.CommutativeMonoid`
-  ```agda
-  invertibleˡ⇒invertibleʳ : LeftInvertible _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x
-  invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible _≈_ 0# _+_ x
-  invertibleˡ⇒invertible  : LeftInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x
-  invertibleʳ⇒invertible  : RightInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x
-  ```
-
-* Added new functions to `Algebra.Apartness.Bundles`
-  ```agda
-  invertibleˡ⇒# : LeftInvertible _≈_ 1# _*_ (x - y) → x # y
-  invertibleʳ⇒# : RightInvertible _≈_ 1# _*_ (x - y) → x # y
-  x#0y#0→xy#0   : x # 0# → y # 0# → x * y # 0#
-  #-sym         : Symmetric _#_
-  #-congʳ       : x ≈ y → x # z → y # z
-  #-congˡ       : y ≈ z → x # y → x # z
-  ```
-
-* Added new proof to `Data.List.Relation.Unary.All.Properties`:
-  ```agda
-  gmap⁻ : Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P
-  ```
-
-* Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`
-  and `Data.List.Relation.Unary.Sorted.TotalOrder.Properties`.
-  ```agda
-  ⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys
-  ⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys
-  ```
-
-* Added new proof to `Induction.WellFounded`
-  ```agda
-  Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
-
-  acc⇒asym : Acc _<_ x → x < y → ¬ (y < x)
-  wf⇒asym : WellFounded _<_ → Asymmetric _<_
-  wf⇒irrefl : _<_ Respects₂ _≈_ → Symmetric _≈_ →
-              WellFounded _<_ → Irreflexive _≈_ _<_
-  ```
-
-* Added new types and operations to `Reflection.TCM`:
-  ```
-  Blocker : Set
-  blockerMeta : Meta → Blocker
-  blockerAny : List Blocker → Blocker
-  blockerAll : List Blocker → Blocker
-  blockTC : Blocker → TC A
-  ```
-
-* Added new file `Relation.Binary.Reasoning.Base.Apartness`
-
-  This is how to use it:
-    ```agda
-    _ : a # d
-    _ = begin-apartness
-      a ≈⟨ a≈b ⟩
-      b #⟨ b#c ⟩
-      c ≈⟨ c≈d ⟩
-      d ∎
-    ```

From 22355361175cc763e8ecbc9ec0120a391ea29b13 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Fri, 20 Oct 2023 01:12:38 +0100
Subject: [PATCH 175/316] Spellcheck CHANGELOG (#2167)

* spellcheck; `fix-whitespace`; unfixed: a few alignment issues; typo `predications` to `predicates` in `Relation.Unary.Relation.Binary.Equality`
---
 CHANGELOG.md | 430 +++++++++++++++++++++++++--------------------------
 1 file changed, 215 insertions(+), 215 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6e03140b75..9d9f403d64 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -3,7 +3,7 @@ Version 2.0-rc1
 
 The library has been tested using Agda 2.6.4.
 
-NOTE: Version `2.0` contains various breaking changes and is not backwards 
+NOTE: Version `2.0` contains various breaking changes and is not backwards
 compatible with code written with version `1.X` of the library.
 
 Highlights
@@ -34,8 +34,8 @@ Bug-fixes
 ---------
 
 * In `Algebra.Definitions.RawSemiring` the record `Prime` did not
-  enforce that the number was not divisible by `1#`. To fix this 
-  `p∤1 : p ∤ 1#` hsa been added as a field.
+  enforce that the number was not divisible by `1#`. To fix this
+  `p∤1 : p ∤ 1#` has been added as a field.
 
 * In `Data.Container.FreeMonad`, we give a direct definition of `_⋆_` as an inductive
   type rather than encoding it as an instance of `μ`. This ensures Agda notices that
@@ -45,24 +45,24 @@ Bug-fixes
 
 * In `Data.Fin.Properties` the `i` argument to `opposite-suc` was implicit
   but could not be inferred in general. It has been made explicit.
-  
-* In `Data.List.Membership.Setoid` the operations `_∷=_` and `_─_` 
+
+* In `Data.List.Membership.Setoid` the operations `_∷=_` and `_─_`
   had an extraneous `{A : Set a}` parameter. This has been removed.
 
-* In `Data.List.Relation.Ternary.Appending.Setoid` the constructors 
-  were re-exported in their full generality which lead to unsolved meta 
+* In `Data.List.Relation.Ternary.Appending.Setoid` the constructors
+  were re-exported in their full generality which lead to unsolved meta
   variables at their use sites. Now versions of the constructors specialised
   to use the setoid's carrier set are re-exported.
-  
-* In `Data.Nat.DivMod` the parameter `o` in the proof `/-monoˡ-≤` was 
+
+* In `Data.Nat.DivMod` the parameter `o` in the proof `/-monoˡ-≤` was
   implicit but not inferrable. It has been changed to be explicit.
 
-* In `Data.Nat.DivMod` the parameter `m` in the proof `+-distrib-/-∣ʳ` was 
+* In `Data.Nat.DivMod` the parameter `m` in the proof `+-distrib-/-∣ʳ` was
   implicit but not inferrable, while `n` is explicit but inferrable.
   They have been to explicit and implicit respectively.
 
-* In `Data.Nat.GeneralisedArithmetic` the `s` and `z` arguments to the 
-  following functions were implicit but no inferrable:
+* In `Data.Nat.GeneralisedArithmetic` the `s` and `z` arguments to the
+  following functions were implicit but not inferrable:
   `fold-+`, `fold-k`, `fold-*`, `fold-pull`. They have been made explicit.
 
 * In `Data.Rational(.Unnormalised).Properties` the module `≤-Reasoning`
@@ -79,9 +79,9 @@ Bug-fixes
   _≈⟨_⟨_ ↦ _≃⟨_⟨_
   ```
 
-* In `Function.Construct.Composition` the combinators 
+* In `Function.Construct.Composition` the combinators
   `_⟶-∘_`, `_↣-∘_`, `_↠-∘_`, `_⤖-∘_`, `_⇔-∘_`, `_↩-∘_`, `_↪-∘_`, `_↔-∘_`
-  had their arguments in the wrong order. They have been flipped so they can 
+  had their arguments in the wrong order. They have been flipped so they can
   actually be  used as a composition operator.
 
 * In `Function.Definitions` the definitions of `Surjection`, `Inverseˡ`,
@@ -98,7 +98,7 @@ Non-backwards compatible changes
 
 ### Removed deprecated names
 
-* All modules and names that were deprecated in `v1.2` and before have 
+* All modules and names that were deprecated in `v1.2` and before have
   been removed.
 
 ### Changes to `LeftCancellative` and `RightCancellative` in `Algebra.Definitions`
@@ -162,8 +162,8 @@ Non-backwards compatible changes
   `Algebra.Lattice.Properties.Semilattice` or `Algebra.Lattice.Morphism.Lattice`). See
   the `Deprecated modules` section below for full details.
 
-* The definition of `IsDistributiveLattice` and `IsBooleanAlgebra` have changed so 
-  that they are no longer right-biased which hindered compositionality. 
+* The definition of `IsDistributiveLattice` and `IsBooleanAlgebra` have changed so
+  that they are no longer right-biased which hindered compositionality.
   More concretely, `IsDistributiveLattice` now has fields:
   ```agda
   ∨-distrib-∧ : ∨ DistributesOver ∧
@@ -225,12 +225,12 @@ Non-backwards compatible changes
   IsSemiringHomomorphism
   IsRingHomomorphism
   ```
-  all had two separate proofs of `IsRelHomomorphism` within them. 
-  
-* To fix this they have all been redefined to build up from `IsMonoidHomomorphism`, 
-  `IsNearSemiringHomomorphism`, and `IsSemiringHomomorphism` respectively, 
-  adding a single property at each step. 
-  
+  all had two separate proofs of `IsRelHomomorphism` within them.
+
+* To fix this they have all been redefined to build up from `IsMonoidHomomorphism`,
+  `IsNearSemiringHomomorphism`, and `IsSemiringHomomorphism` respectively,
+  adding a single property at each step.
+
 * Similarly, `IsLatticeHomomorphism` is now built as
   `IsRelHomomorphism` along with proofs that `_∧_` and `_∨_` are homomorphic.
 
@@ -241,13 +241,13 @@ Non-backwards compatible changes
 * As observed by Wen Kokke in issue #1636, it no longer really makes sense
   to group the modules which correspond to the variety of concepts of
   (effectful) type constructor arising in functional programming (esp. in Haskell)
-  such as `Monad`, `Applicative`, `Functor`, etc, under `Category.*`, 
-  as this obstructs the importing of the `agda-categories` development into 
-  the Standard Library, and moreover needlessly restricts the applicability of 
-  categorical concepts to this (highly specific) mode of use. 
-  
-* Correspondingly, client modules grouped under  `*.Categorical.*` which 
-  exploit such structure  for effectful programming have been  renamed 
+  such as `Monad`, `Applicative`, `Functor`, etc, under `Category.*`,
+  as this obstructs the importing of the `agda-categories` development into
+  the Standard Library, and moreover needlessly restricts the applicability of
+  categorical concepts to this (highly specific) mode of use.
+
+* Correspondingly, client modules grouped under  `*.Categorical.*` which
+  exploit such structure  for effectful programming have been  renamed
   `*.Effectful`, with the originals being deprecated.
 
 * Full list of moved modules:
@@ -284,7 +284,7 @@ Non-backwards compatible changes
   IO.Categorical                             ↦ IO.Effectful
   Reflection.TCM.Categorical                 ↦ Reflection.TCM.Effectful
   ```
-  
+
 * Full list of new modules:
   ```
   Algebra.Construct.Initial
@@ -374,10 +374,10 @@ Non-backwards compatible changes
 
 * Due to the change in Agda 2.6.2 where sized types are no longer compatible
   with the `--safe` flag, it has become clear that a third variant of codata
-  will be needed using coinductive records. 
-  
-* Therefore all existing modules in `Codata` which used sized types have been 
-  moved inside a new folder named `Codata.Sized`,  e.g. `Codata.Stream` 
+  will be needed using coinductive records.
+
+* Therefore all existing modules in `Codata` which used sized types have been
+  moved inside a new folder named `Codata.Sized`,  e.g. `Codata.Stream`
   has become `Codata.Sized.Stream`.
 
 ### New proof-irrelevant for empty type in `Data.Empty`
@@ -398,14 +398,14 @@ Non-backwards compatible changes
   + In `Relation.Nullary.Decidable.Core`, the type of `dec-no` has changed
     ```agda
     dec-no : (p? : Dec P) → ¬ P → ∃ λ ¬p′ → p? ≡ no ¬p′
-      ↦ 
+      ↦
     dec-no : (p? : Dec P) (¬p : ¬ P) → p? ≡ no ¬p
     ```
 
   + In `Relation.Binary.PropositionalEquality`, the type of `≢-≟-identity` has changed
     ```agda
     ≢-≟-identity : x ≢ y → ∃ λ ¬eq → x ≟ y ≡ no ¬eq
-      ↦ 
+      ↦
     ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y
     ```
 
@@ -424,7 +424,7 @@ Non-backwards compatible changes
   ≺-rec
   ```
   these functions are also deprecated.
-  
+
 * Likewise in `Data.Fin.Properties` the proofs `≺⇒<′` and `<′⇒≺` have been deprecated
   in favour of their proxy counterparts `<⇒<′` and `<′⇒<`.
 
@@ -473,9 +473,9 @@ Non-backwards compatible changes
 
 #### Standardisation of `lookup` in `Data.(List/Vec/...)`
 
-* All the types of `lookup` functions (and variants) in the following modules 
-  have been changed to match the argument convention adopted in the `List` module (i.e. 
-  `lookup` takes its container first and the index, whose type may depend on the 
+* All the types of `lookup` functions (and variants) in the following modules
+  have been changed to match the argument convention adopted in the `List` module (i.e.
+  `lookup` takes its container first and the index, whose type may depend on the
   container value, second):
   ```
   Codata.Guarded.Stream
@@ -500,7 +500,7 @@ Non-backwards compatible changes
   Data.Vec.Recursive
   ```
 
-* To accomodate this in in `Data.Vec.Relation.Unary.Linked.Properties`
+* To accommodate this in in `Data.Vec.Relation.Unary.Linked.Properties`
   and `Codata.Guarded.Stream.Relation.Binary.Pointwise`, the proofs
   called `lookup` have been renamed `lookup⁺`.
 
@@ -509,11 +509,11 @@ Non-backwards compatible changes
 * Many numeric operations in the library require their arguments to be non-zero,
   and various proofs require their arguments to be non-zero/positive/negative etc.
   As discussed on the [mailing list](https://lists.chalmers.se/pipermail/agda/2021/012693.html),
-  the previous way of constructing and passing round these proofs was extremely 
+  the previous way of constructing and passing round these proofs was extremely
   clunky and lead to messy and difficult to read code.
-  
-* We have therefore changed every occurrence where we need a proof of 
-  non-zeroness/positivity/etc. to take it as an irrelevant 
+
+* We have therefore changed every occurrence where we need a proof of
+  non-zeroness/positivity/etc. to take it as an irrelevant
   [instance argument](https://agda.readthedocs.io/en/latest/language/instance-arguments.html).
   See the mailing list discussion for a fuller explanation of the motivation and implementation.
 
@@ -538,47 +538,47 @@ Non-backwards compatible changes
 * At the moment, there are 4 different ways such instance arguments can be provided,
   listed in order of convenience and clarity:
 
-  1. *Automatic basic instances* - the standard library provides instances based on 
-	 the constructors of each numeric type in `Data.X.Base`. For example, 
-	 `Data.Nat.Base` constrains an instance of `NonZero (suc n)` for any `n`
-	 and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`. 
-	 Consequently, if the argument is of the required form, these instances will always 
-	 be filled in by instance search automatically, e.g.
-	```agda
-	0/n≡0 : 0 / suc n ≡ 0
-	```
-
-  2. *Add an instance argument parameter* - You can provide the instance argument as 
-	 a parameter to your function and Agda's instance search will automatically use it 
-	 in the correct place without you having to explicitly pass it, e.g.
+  1. *Automatic basic instances* - the standard library provides instances based on
+         the constructors of each numeric type in `Data.X.Base`. For example,
+         `Data.Nat.Base` constrains an instance of `NonZero (suc n)` for any `n`
+         and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`.
+         Consequently, if the argument is of the required form, these instances will always
+         be filled in by instance search automatically, e.g.
+        ```agda
+        0/n≡0 : 0 / suc n ≡ 0
+        ```
+
+  2. *Add an instance argument parameter* - You can provide the instance argument as
+         a parameter to your function and Agda's instance search will automatically use it
+         in the correct place without you having to explicitly pass it, e.g.
     ```agda
     0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
     ```
 
-  3. *Define the instance locally* - You can define an instance argument in scope 
-	 (e.g. in a `where` clause) and Agda's instance search will again find it automatically, 
-	 e.g.
-	```agda
-	instance
-	  n≢0 : NonZero n
-	  n≢0 = ...
+  3. *Define the instance locally* - You can define an instance argument in scope
+         (e.g. in a `where` clause) and Agda's instance search will again find it automatically,
+         e.g.
+        ```agda
+        instance
+          n≢0 : NonZero n
+          n≢0 = ...
 
-	0/n≡0 : 0 / n ≡ 0
-	```
+        0/n≡0 : 0 / n ≡ 0
+        ```
 
   4. *Pass the instance argument explicitly* - Finally, if all else fails you can pass the
-	 instance argument explicitly into the function using `{{ }}`, e.g.
-	 ```
-	 0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
-	 ```
+         instance argument explicitly into the function using `{{ }}`, e.g.
+         ```
+         0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
+         ```
 
 * Suitable constructors for `NonZero`/`Positive` etc. can be found in `Data.X.Base`.
 
-* A full list of proofs that have changed to use instance arguments is available 
+* A full list of proofs that have changed to use instance arguments is available
   at the end of this file. Notable changes to proofs are now discussed below.
 
 * Previously one of the hacks used in proofs was to explicitly refer to everything
-  in the correct form, e.g. if the argument `n` had to be non-zero then you would 
+  in the correct form, e.g. if the argument `n` had to be non-zero then you would
   refer to the argument as `suc n` everywhere instead of `n`, e.g.
   ```
   n/n≡1 : ∀ n → suc n / suc n ≡ 1
@@ -588,13 +588,13 @@ Non-backwards compatible changes
   ```
   n/n≡1 : ∀ n {{_ : NonZero n}} → n / n ≡ 1
   ```
-  However, note that this means that if you passed in the value `x` to these proofs 
-  before, then you will now have to pass in `suc x`. The proofs for which the 
-  arguments have changed form in this way are highlighted in the list at the bottom 
+  However, note that this means that if you passed in the value `x` to these proofs
+  before, then you will now have to pass in `suc x`. The proofs for which the
+  arguments have changed form in this way are highlighted in the list at the bottom
   of the file.
 
 * Finally, in `Data.Rational.Unnormalised.Base` the definition of `_≢0` is now
-  redundent and so has been removed. Additionally the following proofs about it have 
+  redundant and so has been removed. Additionally the following proofs about it have
   also been removed from `Data.Rational.Unnormalised.Properties`:
   ```
   p≄0⇒∣↥p∣≢0 : ∀ p → p ≠ 0ℚᵘ → ℤ.∣ (↥ p) ∣ ≢0
@@ -613,9 +613,9 @@ Non-backwards compatible changes
   ```
   which introduced a spurious additional definition, when this is in fact, modulo
   field names and implicit/explicit qualifiers, equivalent to the definition of left-
-  divisibility, `_∣ˡ_` for the `RawMagma` structure of `_+_`. 
-  
-* Since the addition of raw bundles to `Data.X.Base`, this definition can now be 
+  divisibility, `_∣ˡ_` for the `RawMagma` structure of `_+_`.
+
+* Since the addition of raw bundles to `Data.X.Base`, this definition can now be
   made directly. Accordingly, the definition has been changed to:
   ```agda
   _≤″_ : (m n : ℕ)  → Set
@@ -625,8 +625,8 @@ Non-backwards compatible changes
   ```
 
 * Knock-on consequences include the need to retain the old constructor
-  name, now introduced as a pattern synonym, and introduction of (a function 
-  equivalent to) the former field name/projection function `proof` as 
+  name, now introduced as a pattern synonym, and introduction of (a function
+  equivalent to) the former field name/projection function `proof` as
   `≤″-proof` in `Data.Nat.Properties`.
 
 ### Changes to definition of `Prime` in `Data.Nat.Primality`
@@ -672,24 +672,24 @@ Non-backwards compatible changes
      ```
 
 * As a consequence of this, some proofs that relied either on this reduction behaviour
-  or on eta-equality may no longer type-check. 
-  
+  or on eta-equality may no longer type-check.
+
 * There are several ways to fix this:
-  
+
   1. The principled way is to not rely on this reduction behaviour in the first place.
      The `Properties` files for rational numbers have been greatly expanded in `v1.7`
      and `v2.0`, and we believe most proofs should be able to be built up from existing
      proofs contained within these files.
-  
+
   2. Alternatively, annotating any rational arguments to a proof with either
      `@record{}` or `@(mkℚ _ _ _)` should restore the old reduction behaviour for any
      terms involving those parameters.
-  
+
   3. Finally, if the above approaches are not viable then you may be forced to explicitly
      use `cong` combined with a lemma that proves the old reduction behaviour.
 
 * Similarly, in order to prevent reduction, the equality `_≃_` in `Data.Rational.Base`
-  has been made into a data type with the single constructor `*≡*`. The destructor 
+  has been made into a data type with the single constructor `*≡*`. The destructor
   `drop-*≡*` has been added to `Data.Rational.Properties`.
 
 #### Deprecation of old `Function` hierarchy
@@ -697,7 +697,7 @@ Non-backwards compatible changes
 * The new `Function` hierarchy was introduced in `v1.2` which follows
   the same `Core`/`Definitions`/`Bundles`/`Structures` as all the other
   hierarchies in the library.
-  
+
 * At the time the old function hierarchy in:
   ```
   Function.Equality
@@ -711,11 +711,11 @@ Non-backwards compatible changes
   ```
   was unofficially deprecated, but could not be officially deprecated because
   of it's widespread use in the rest of the library.
-  
-* Now, the old hierarchy modules have all been officially deprecated. All 
+
+* Now, the old hierarchy modules have all been officially deprecated. All
   uses of them in the rest of the library have been switched to use the
-  new hierarachy. 
-  
+  new hierarchy.
+
 * The latter is unfortunately a relatively big breaking change, but was judged
   to be unavoidable given how widely used the old hierarchy was.
 
@@ -724,9 +724,9 @@ Non-backwards compatible changes
 * In `Function.Bundles` the names of the fields in the records have been
   changed from `f`, `g`, `cong₁` and `cong₂` to `to`, `from`, `to-cong`, `from-cong`.
 
-* In `Function.Definitions`, the module no longer has two equalities as 
-  module arguments, as they did not interact as intended with the re-exports 
-  from `Function.Definitions.(Core1/Core2)`. The latter two modules have 
+* In `Function.Definitions`, the module no longer has two equalities as
+  module arguments, as they did not interact as intended with the re-exports
+  from `Function.Definitions.(Core1/Core2)`. The latter two modules have
   been removed and their definitions folded into `Function.Definitions`.
 
 * In `Function.Definitions` the following definitions have been changed from:
@@ -741,9 +741,9 @@ Non-backwards compatible changes
   Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
   Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
   ```
-  This is for several reasons: 
-	  i) the new definitions compose much more easily, 
-	  ii) Agda can better infer the equalities used.
+  This is for several reasons:
+          i) the new definitions compose much more easily,
+          ii) Agda can better infer the equalities used.
 
 * To ease backwards compatibility:
    - the old definitions have been moved to the new names  `StrictlySurjective`,
@@ -769,9 +769,9 @@ Non-backwards compatible changes
   WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
   WfRec _<_ P x = ∀ y → y < x → P y
   ```
-  which meant that all arguments involving accessibility and wellfoundedness proofs 
-  were polluted by almost-always-inferrable explicit arguments for the `y` position. 
-  
+  which meant that all arguments involving accessibility and wellfoundedness proofs
+  were polluted by almost-always-inferrable explicit arguments for the `y` position.
+
 * The definition has now been changed to make that argument *implicit*, as
   ```agda
   WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
@@ -821,44 +821,44 @@ Non-backwards compatible changes
 
 ### Reorganisation of the `Relation.Nullary` hierarchy
 
-* It was very difficult to use the `Relation.Nullary` modules, as 
-  `Relation.Nullary` contained the basic definitions of negation, decidability etc., 
-  and the operations and proofs about these definitions were spread over 
+* It was very difficult to use the `Relation.Nullary` modules, as
+  `Relation.Nullary` contained the basic definitions of negation, decidability etc.,
+  and the operations and proofs about these definitions were spread over
   `Relation.Nullary.(Negation/Product/Sum/Implication etc.)`.
 
 * To fix this all the contents of the latter is now exported by `Relation.Nullary`.
 
 * In order to achieve this the following backwards compatible changes have been made:
-  
+
   1. the definition of `Dec` and `recompute` have been moved to `Relation.Nullary.Decidable.Core`
-  
+
   2. the definition of `Reflects` has been moved to `Relation.Nullary.Reflects`
-  
+
   3. the definition of `¬_` has been moved to `Relation.Nullary.Negation.Core`
 
-  4. The modules `Relation.Nullary.(Product/Sum/Implication)` have been deprecated 
-	 and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`.
+  4. The modules `Relation.Nullary.(Product/Sum/Implication)` have been deprecated
+         and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`.
 
   as well as the following breaking changes:
 
-  1. `¬?` has been moved from `Relation.Nullary.Negation.Core` to 
-	 `Relation.Nullary.Decidable.Core`
-  
-  2. `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to 
-	 `Relation.Nullary.Reflects`.
-  
-  3. `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved 
-	 from `Relation.Nullary.Negation` to `Relation.Nullary.Decidable`.
-  
+  1. `¬?` has been moved from `Relation.Nullary.Negation.Core` to
+         `Relation.Nullary.Decidable.Core`
+
+  2. `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to
+         `Relation.Nullary.Reflects`.
+
+  3. `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved
+         from `Relation.Nullary.Negation` to `Relation.Nullary.Decidable`.
+
   4. `fromDec` and `toDec` have been moved from `Data.Sum.Base` to `Data.Sum`.
 
 
 ### (Issue #2096) Introduction of flipped and negated relation symbols to bundles in `Relation.Binary.Bundles`
 
-* Previously, bundles such as `Preorder`, `Poset`, `TotalOrder` etc. did not have the flipped 
+* Previously, bundles such as `Preorder`, `Poset`, `TotalOrder` etc. did not have the flipped
   and negated versions of the operators exposed. In some cases they could obtained by opening the
   relevant `Relation.Binary.Properties.X` file but usually they had to be redefined every time.
-  
+
 * To fix this, these bundles now all export all 4 versions of the operator: normal, converse, negated,
   converse-negated. Accordingly they are no longer exported from the corresponding `Properties` file.
 
@@ -867,26 +867,26 @@ Non-backwards compatible changes
   converse relation could only be discussed *semantically* in terms of `flip _∼_`.
 
 * Now, the `Preorder` record field `_∼_` has been renamed to `_≲_`, with `_≳_`
-  introduced as a definition in `Relation.Binary.Bundles.Preorder`. 
-  Partial backwards compatible has been achieved by redeclaring a deprecated version 
-  of the old symbol in the record. Therefore, only _declarations_ of `PartialOrder` records will 
+  introduced as a definition in `Relation.Binary.Bundles.Preorder`.
+  Partial backwards compatible has been achieved by redeclaring a deprecated version
+  of the old symbol in the record. Therefore, only _declarations_ of `PartialOrder` records will
   need their field names updating.
 
 
 ### Changes to definition of `IsStrictTotalOrder` in `Relation.Binary.Structures`
 
-* The previous definition of the record `IsStrictTotalOrder` did not 
-  build upon `IsStrictPartialOrder` as would be expected. 
+* The previous definition of the record `IsStrictTotalOrder` did not
+  build upon `IsStrictPartialOrder` as would be expected.
   Instead it omitted several fields like irreflexivity as they were derivable from the
-  proof of trichotomy. However, this led to problems further up the hierarchy where 
-  bundles such as `StrictTotalOrder` which contained multiple distinct proofs of 
+  proof of trichotomy. However, this led to problems further up the hierarchy where
+  bundles such as `StrictTotalOrder` which contained multiple distinct proofs of
   `IsStrictPartialOrder`.
 
-* To remedy this the definition of `IsStrictTotalOrder` has been changed to so 
+* To remedy this the definition of `IsStrictTotalOrder` has been changed to so
   that it builds upon `IsStrictPartialOrder` as would be expected.
 
-* To aid migration, the old record definition has been moved to 
-  `Relation.Binary.Structures.Biased` which contains the `isStrictTotalOrderᶜ` 
+* To aid migration, the old record definition has been moved to
+  `Relation.Binary.Structures.Biased` which contains the `isStrictTotalOrderᶜ`
   smart constructor (which is re-exported by `Relation.Binary`) . Therefore the old code:
   ```agda
   <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
@@ -917,14 +917,14 @@ Non-backwards compatible changes
 
 ### Changes to the interface of `Relation.Binary.Reasoning.Triple`
 
-* The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof 
+* The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof
   that the strict relation is irreflexive.
 
 * This allows the addition of the following new proof combinator:
   ```agda
   begin-contradiction : (r : x IsRelatedTo x) → {s : True (IsStrict? r)} → A
   ```
-  that takes a proof that a value is strictly less than itself and then applies the 
+  that takes a proof that a value is strictly less than itself and then applies the
   principle of explosion to derive anything.
 
 * Specialised versions of this combinator are available in the `Reasoning` modules
@@ -945,9 +945,9 @@ Non-backwards compatible changes
 
 * Previously, every `Reasoning` module in the library tended to roll it's own set
   of syntax for the combinators. This hindered consistency and maintainability.
-  
-* To improve the situation, a new module `Relation.Binary.Reasoning.Syntax` 
-  has been introduced which exports a wide range of sub-modules containing 
+
+* To improve the situation, a new module `Relation.Binary.Reasoning.Syntax`
+  has been introduced which exports a wide range of sub-modules containing
   pre-existing reasoning combinator syntax.
 
 * This makes it possible to add new or rename existing reasoning combinators to a
@@ -956,39 +956,39 @@ Non-backwards compatible changes
 
 * One pre-requisite for that is that `≡-Reasoning` has been moved from
   `Relation.Binary.PropositionalEquality.Core` (which shouldn't be
-  imported anyway as it's a `Core` module) to 
+  imported anyway as it's a `Core` module) to
   `Relation.Binary.PropositionalEquality.Properties`.
   It is still exported by `Relation.Binary.PropositionalEquality`.
 
 ### Renaming of symmetric combinators in `Reasoning` modules
 
-* We've had various complaints about the symmetric version of reasoning combinators 
+* We've had various complaints about the symmetric version of reasoning combinators
   that use the syntax `_R˘⟨_⟩_` for some relation `R`, (e.g. `_≡˘⟨_⟩_` and `_≃˘⟨_⟩_`)
   introduced in `v1.0`. In particular:
   1. The symbol `˘` is hard to type.
   2. The symbol `˘` doesn't convey the direction of the equality
   3. The left brackets aren't vertically aligned with the left brackets of the non-symmetric version.
-  
-* To address these problems we have renamed all the symmetric versions of the 
+
+* To address these problems we have renamed all the symmetric versions of the
   combinators from `_R˘⟨_⟩_` to `_R⟨_⟨_` (the direction of the last bracket is flipped
   to indicate the quality goes from right to left).
-  
-* The old combinators still exist but have been deprecated. However due to 
+
+* The old combinators still exist but have been deprecated. However due to
   [Agda issue #5617](https://github.com/agda/agda/issues/5617), the deprecation warnings
   don't fire correctly. We will not remove the old combinators before the above issue is
   addressed. However, we still encourage migrating to the new combinators!
 
 * On a Linux-based system, the following command was used to globally migrate all uses of the
-  old combinators to the new ones in the standard library itself. 
+  old combinators to the new ones in the standard library itself.
   It *may* be useful when trying to migrate your own code:
   ```bash
   find . -type f -name "*.agda" -print0 | xargs -0 sed -i -e 's/˘⟨\(.*\)⟩/⟨\1⟨/g'
   ```
-  USUAL CAVEATS: It has not been widely tested and the standard library developers are not 
-  responsible for the results of this command. It is strongly recommended you back up your 
+  USUAL CAVEATS: It has not been widely tested and the standard library developers are not
+  responsible for the results of this command. It is strongly recommended you back up your
   work before you attempt to run it.
-  
-* NOTE: this refactoring may require some extra bracketing around the operator `_⟨_⟩_` from 
+
+* NOTE: this refactoring may require some extra bracketing around the operator `_⟨_⟩_` from
   `Function.Base` if the `_⟨_⟩_` operator is used within the reasoning proof. The symptom
   for this is a `Don't know how to parse` error.
 
@@ -1035,10 +1035,10 @@ Non-backwards compatible changes
   * `cycle`
   * `interleave⁺`
   * `cantor`
-  Furthermore, the direction of interleaving of `cantor` has changed. Precisely, 
-  suppose `pair` is the cantor pairing function, then `lookup (pair i j) (cantor xss)` 
-  according to the old definition corresponds to `lookup (pair j i) (cantor xss)` 
-  according to the new definition. 
+  Furthermore, the direction of interleaving of `cantor` has changed. Precisely,
+  suppose `pair` is the cantor pairing function, then `lookup (pair i j) (cantor xss)`
+  according to the old definition corresponds to `lookup (pair j i) (cantor xss)`
+  according to the new definition.
   For a concrete example see the one included at the end of the module.
 
 * In `Data.Fin.Base` the relations `_≤_` `_≥_` `_<_` `_>_`  have been
@@ -1047,9 +1047,9 @@ Non-backwards compatible changes
   properties about the orderings themselves the second index must be provided
   explicitly.
 
-* In `Data.Fin.Properties` the proof `inj⇒≟` that an injection from a type 
-  `A` into `Fin n` induces a decision procedure for `_≡_` on `A` has been 
-  generalized to other equivalences over `A` (i.e. to arbitrary setoids), and 
+* In `Data.Fin.Properties` the proof `inj⇒≟` that an injection from a type
+  `A` into `Fin n` induces a decision procedure for `_≡_` on `A` has been
+  generalised to other equivalences over `A` (i.e. to arbitrary setoids), and
   renamed from `eq?` to the more descriptive  and `inj⇒decSetoid`.
 
 * In `Data.Fin.Properties` the proof `pigeonhole` has been strengthened
@@ -1058,19 +1058,19 @@ Non-backwards compatible changes
 * In `Data.Fin.Substitution.TermSubst` the fixity of `_/Var_` has been changed
   from `infix 8` to `infixl 8`.
 
-* In `Data.Integer.DivMod` the previous implementations of 
-  `_divℕ_`, `_div_`, `_modℕ_`, `_mod_` internally converted to the unary 
-  `Fin` datatype resulting in poor performance. The implementation has been 
+* In `Data.Integer.DivMod` the previous implementations of
+  `_divℕ_`, `_div_`, `_modℕ_`, `_mod_` internally converted to the unary
+  `Fin` datatype resulting in poor performance. The implementation has been
   updated to use the corresponding operations from `Data.Nat.DivMod` which are
   efficiently implemented using the Haskell backend.
 
-* In `Data.Integer.Properties` the first two arguments of `m≡n⇒m-n≡0` 
+* In `Data.Integer.Properties` the first two arguments of `m≡n⇒m-n≡0`
   (now renamed `i≡j⇒i-j≡0`) have been made implicit.
 
-* In `Data.(List|Vec).Relation.Binary.Lex.Strict` the argument `xs` in 
+* In `Data.(List|Vec).Relation.Binary.Lex.Strict` the argument `xs` in
   `xs≮[]` in introduced in PRs #1648 and #1672 has now been made implicit.
 
-* In `Data.List.NonEmpty` the functions `split`, `flatten` and `flatten-split` 
+* In `Data.List.NonEmpty` the functions `split`, `flatten` and `flatten-split`
   have been removed from. In their place `groupSeqs` and `ungroupSeqs`
   have been added to `Data.List.NonEmpty.Base` which morally perform the same
   operations but without computing the accompanying proofs. The proofs can be
@@ -1082,20 +1082,20 @@ Non-backwards compatible changes
   proofs.
 
 * In `Data.Nat.Divisibility` the proof `m/n/o≡m/[n*o]` has been removed. In it's
-  place a new more general proof `m/n/o≡m/[n*o]` has been added to `Data.Nat.DivMod` 
+  place a new more general proof `m/n/o≡m/[n*o]` has been added to `Data.Nat.DivMod`
   that doesn't require the `n * o ∣ m` pre-condition.
 
-* In `Data.Product.Relation.Binary.Lex.Strict` the proof of wellfoundedness 
+* In `Data.Product.Relation.Binary.Lex.Strict` the proof of wellfoundedness
   of the lexicographic ordering on products, no longer
   requires the assumption of symmetry for the first equality relation `_≈₁_`.
 
-* In `Data.Rational.Base` the constructors `+0` and `+[1+_]` from `Data.Integer.Base` 
+* In `Data.Rational.Base` the constructors `+0` and `+[1+_]` from `Data.Integer.Base`
   are no longer re-exported by `Data.Rational.Base`. You will have to open `Data.Integer(.Base)`
   directly to use them.
 
-* In `Data.Rational(.Unnormalised).Properties` the types of the proofs 
-  `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` have been switched, 
-  as the previous naming scheme didn't correctly generalise to e.g. `pos+pos⇒pos`. 
+* In `Data.Rational(.Unnormalised).Properties` the types of the proofs
+  `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` have been switched,
+  as the previous naming scheme didn't correctly generalise to e.g. `pos+pos⇒pos`.
   For example the types of `pos⇒1/pos`/`1/pos⇒pos` were:
   ```agda
   pos⇒1/pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
@@ -1111,7 +1111,7 @@ Non-backwards compatible changes
 
 * In `Data.Vec.Base` the definition of `_>>=_` under `Data.Vec.Base` has been
   moved to the submodule `CartesianBind` in order to avoid clashing with the
-  new, alternative definition of `_>>=_`, located in the second new submodule 
+  new, alternative definition of `_>>=_`, located in the second new submodule
   `DiagonalBind`.
 
 * In `Data.Vec.Base` the definitions `init` and `last` have been changed from the `initLast`
@@ -1119,15 +1119,15 @@ Non-backwards compatible changes
 
 * In `Data.Vec.Properties` the type of the proof `zipWith-comm` has been generalised from:
   ```agda
-  zipWith-comm : ∀ {f : A → A → B} (comm : ∀ x y → f x y ≡ f y x) (xs ys : Vec A n) → 
+  zipWith-comm : ∀ {f : A → A → B} (comm : ∀ x y → f x y ≡ f y x) (xs ys : Vec A n) →
                  zipWith f xs ys ≡ zipWith f ys xs
   ```
   to
   ```agda
-  zipWith-comm : ∀ {f : A → B → C} {g : B → A → C}  (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys → 
+  zipWith-comm : ∀ {f : A → B → C} {g : B → A → C}  (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys →
                  zipWith f xs ys ≡ zipWith g ys xs
   ```
-  
+
 * In `Data.Vec.Relation.Unary.All` the functions `lookup` and `tabulate` have
   been moved to `Data.Vec.Relation.Unary.All.Properties` and renamed
   `lookup⁺` and `lookup⁻` respectively.
@@ -1139,15 +1139,15 @@ Non-backwards compatible changes
   iterate : (A → A) → A → ∀ n → Vec A n
   replicate : ∀ n → A → Vec A n
   ```
-  
-* In `Relation.Binary.Construct.Closure.Symmetric` the operation 
+
+* In `Relation.Binary.Construct.Closure.Symmetric` the operation
   `SymClosure` on relations in has been reimplemented
   as a data type `SymClosure _⟶_ a b` that is parameterized by the
   input relation `_⟶_` (as well as the elements `a` and `b` of the
   domain) so that `_⟶_` can be inferred, which it could not from the
   previous implementation using the sum type `a ⟶ b ⊎ b ⟶ a`.
 
-* In `Relation.Nullary.Decidable.Core` the name `excluded-middle` has been 
+* In `Relation.Nullary.Decidable.Core` the name `excluded-middle` has been
   renamed to `¬¬-excluded-middle`.
 
 
@@ -1185,10 +1185,10 @@ Other major improvements
 
 * Raw bundles by design don't contain any proofs so should in theory be able to live
   in `Data.X.Base` rather than `Data.X.Bundles`.
-  
+
 * To achieve this while keeping the dependency footprint small, the definitions of
-  raw bundles (`RawMagma`, `RawMonoid` etc.) have been moved from `Algebra(.Lattice)Bundles` to 
-  a new module `Algebra(.Lattice).Bundles.Raw` which can be imported at much lower cost 
+  raw bundles (`RawMagma`, `RawMonoid` etc.) have been moved from `Algebra(.Lattice)Bundles` to
+  a new module `Algebra(.Lattice).Bundles.Raw` which can be imported at much lower cost
   from `Data.X.Base`.
 
 * We have then moved raw bundles defined in `Data.X.Properties` to `Data.X.Base` for
@@ -1201,7 +1201,7 @@ Deprecated modules
 
 * This fixes the fact we had picked the wrong name originally. The erased modality
   corresponds to `@0` whereas the irrelevance one corresponds to `.`.
-  
+
 ### Deprecation of `Data.Fin.Substitution.Example`
 
 * The module `Data.Fin.Substitution.Example` has been deprecated, and moved to `README.Data.Fin.Substitution.UntypedLambda`
@@ -1212,7 +1212,7 @@ Deprecated modules
 
 ### Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`
 
-* This module has been deprecated, as none of its contents actually depended on axiom K. 
+* This module has been deprecated, as none of its contents actually depended on axiom K.
   The contents has been moved to `Data.Product.Function.Dependent.Setoid`.
 
 ### Moving `Function.Related`
@@ -1444,7 +1444,7 @@ Deprecated names
 * In `Data.List.Relation.Unary.All.Properties`:
   ```agda
   gmap  ↦  gmap⁺
-  
+
   updateAt-id-relative      ↦  updateAt-id-local
   updateAt-compose-relative ↦  updateAt-updateAt-local
   updateAt-compose          ↦  updateAt-updateAt
@@ -2040,7 +2040,7 @@ New modules
   Relation.Unary.Algebra
   ```
 
-* Both versions of equality on predications are equivalences
+* Both versions of equality on predicates are equivalences
   ```
   Relation.Unary.Relation.Binary.Equality
   ```
@@ -2190,8 +2190,8 @@ Additions to existing modules
                                     SemiringWithoutAnnihilatingZero b ℓ₂ →
                                     SemiringWithoutAnnihilatingZero (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   semiring                        : Semiring a ℓ₁ → Semiring b ℓ₂ → Semiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  commutativeSemiring             : CommutativeSemiring a ℓ₁ → 
-	                                CommutativeSemiring b ℓ₂ →
+  commutativeSemiring             : CommutativeSemiring a ℓ₁ →
+                                        CommutativeSemiring b ℓ₂ →
                                     CommutativeSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   ring                            : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   commutativeRing                 : CommutativeRing a ℓ₁ → CommutativeRing b ℓ₂ → CommutativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
@@ -2199,14 +2199,14 @@ Additions to existing modules
   rawLoop                         : RawLoop a ℓ₁ → RawLoop b ℓ₂ → RawLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   unitalMagma                     : UnitalMagma a ℓ₁ → UnitalMagma b ℓ₂ → UnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   invertibleMagma                 : InvertibleMagma a ℓ₁ → InvertibleMagma b ℓ₂ → InvertibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  invertibleUnitalMagma           : InvertibleUnitalMagma a ℓ₁ → 
-	                                InvertibleUnitalMagma b ℓ₂ →
+  invertibleUnitalMagma           : InvertibleUnitalMagma a ℓ₁ →
+                                        InvertibleUnitalMagma b ℓ₂ →
                                     InvertibleUnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   quasigroup                      : Quasigroup a ℓ₁ → Quasigroup b ℓ₂ → Quasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   loop                            : Loop a ℓ₁ → Loop b ℓ₂ → Loop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  idempotentSemiring              : IdempotentSemiring a ℓ₁ → 
-                                    IdempotentSemiring b ℓ₂ → 
-									IdempotentSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  idempotentSemiring              : IdempotentSemiring a ℓ₁ →
+                                    IdempotentSemiring b ℓ₂ →
+                                                                        IdempotentSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   idempotentMagma                 : IdempotentMagma a ℓ₁ → IdempotentMagma b ℓ₂ → IdempotentMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   alternativeMagma                : AlternativeMagma a ℓ₁ → AlternativeMagma b ℓ₂ → AlternativeMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   flexibleMagma                   : FlexibleMagma a ℓ₁ → FlexibleMagma b ℓ₂ → FlexibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
@@ -2219,9 +2219,9 @@ Additions to existing modules
   moufangLoop                     : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   rawRingWithoutOne               : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   ringWithoutOne                  : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  nonAssociativeRing              : NonAssociativeRing a ℓ₁ → 
-		                            NonAssociativeRing b ℓ₂ → 
-									NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  nonAssociativeRing              : NonAssociativeRing a ℓ₁ →
+                                            NonAssociativeRing b ℓ₂ →
+                                                                        NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   quasiring                       : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   nearring                        : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
   ```
@@ -2303,7 +2303,7 @@ Additions to existing modules
 * Added new proofs to `Algebra.Properties.CommutativeSemigroup`:
   ```agda
   xy∙xx≈x∙yxx : (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))
-  
+
   interchange : Interchangable _∙_ _∙_
   leftSemimedial : LeftSemimedial _∙_
   rightSemimedial : RightSemimedial _∙_
@@ -2401,7 +2401,7 @@ Additions to existing modules
   xor-annihilates-not : (not x) xor (not y) ≡ x xor y
   ```
 
-* Exposed container combinator conversion functions from `Data.Container.Combinator.Properties` 
+* Exposed container combinator conversion functions from `Data.Container.Combinator.Properties`
   separately from their correctness proofs in `Data.Container.Combinator`:
   ```
   to-id      : F.id A → ⟦ id ⟧ A
@@ -2432,7 +2432,7 @@ Additions to existing modules
   ```agda
   spo-wellFounded : IsStrictPartialOrder _≈_ _⊏_ → WellFounded _⊏_
   spo-noetherian  : IsStrictPartialOrder _≈_ _⊏_ → WellFounded (flip _⊏_)
-  
+
   <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
   ```
 
@@ -2503,7 +2503,7 @@ Additions to existing modules
   sub≡sub  : map var (VarSubst.sub x) ≡ T.sub (T.var x)
   ↑≡↑      : map var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
   /Var≡/   : t /Var ρ ≡ t T./ map T.var ρ
-  
+
   sub-renaming-commutes : t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
   sub-commutes-with-renaming : t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)
   ```
@@ -2597,9 +2597,9 @@ Additions to existing modules
 * Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
   ```
   ⊆-mergeˡ : xs ⊆ merge _≤?_ xs ys
-  ⊆-mergeʳ : ys ⊆ merge _≤?_ xs ys  
+  ⊆-mergeʳ : ys ⊆ merge _≤?_ xs ys
   ```
-  
+
 * Added new functions in `Data.List.Relation.Unary.All`:
   ```
   decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
@@ -2626,7 +2626,7 @@ Additions to existing modules
   mapWith∈-id     : mapWith∈ xs (λ {x} _ → x) ≡ xs
   map-mapWith∈    : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
   index-injective : index x₁∈xs ≡ index x₂∈xs → x₁ ≈ x₂
-  
+
   ∈-++⁺∘++⁻         : (p : v ∈ xs ++ ys) → [ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ (∈-++⁻ xs p) ≡ p
   ∈-++⁻∘++⁺         : (p : v ∈ xs ⊎ v ∈ ys) → ∈-++⁻ xs ([ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ p) ≡ p
   ∈-++↔             : (v ∈ xs ⊎ v ∈ ys) ↔ v ∈ xs ++ ys
@@ -2677,12 +2677,12 @@ Additions to existing modules
   length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
   removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
   insertAt-removeAt : insertAt (removeAt xs i) (cast (sym (lengthAt-removeAt xs i)) i) (lookup xs i) ≡ xs
-  
+
   cartesianProductWith-zeroˡ       : cartesianProductWith f [] ys ≡ []
   cartesianProductWith-zeroʳ       : cartesianProductWith f xs [] ≡ []
   cartesianProductWith-distribʳ-++ : cartesianProductWith f (xs ++ ys) zs ≡
                                      cartesianProductWith f xs zs ++ cartesianProductWith f ys zs
-									 
+
   foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
   foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
   ```
@@ -2825,7 +2825,7 @@ Additions to existing modules
   ⊔≡⊔′ : m ⊔ n ≡ m ⊔′ n
   ⊓≡⊓′ : m ⊓ n ≡ m ⊓′ n
   ∣-∣≡∣-∣′ : ∣ m - n ∣ ≡ ∣ m - n ∣′
-  
+
   nonZero? : Decidable NonZero
   eq? : A ↣ ℕ → DecidableEquality A
   ≤-<-connex : Connex _≤_ _<_
@@ -2938,7 +2938,7 @@ Additions to existing modules
   pos⇒nonZero       : .{{Positive p}} → NonZero p
   neg⇒nonZero       : .{{Negative p}} → NonZero p
   nonZero⇒1/nonZero : .{{_ : NonZero p}} → NonZero (1/ p)
-  
+
   <-dense              : Dense _<_
   <-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
   <-denseLinearOrder   : DenseLinearOrder 0ℓ 0ℓ 0ℓ
@@ -2979,7 +2979,7 @@ Additions to existing modules
   pos⊓pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊓ q)
   pos⊔pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊔ q)
   1/nonZero⇒nonZero    : .{{_ : NonZero p}} → NonZero (1/ p)
-  
+
   0≄1             : 0ℚᵘ ≄ 1ℚᵘ
   ≃-≄-irreflexive : Irreflexive _≃_ _≄_
   ≄-symmetric     : Symmetric _≄_
@@ -2987,15 +2987,15 @@ Additions to existing modules
   ≄⇒invertible    : p ≄ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
 
   <-dense         : Dense _<_
-  
+
   <-isDenseLinearOrder         : IsDenseLinearOrder _≃_ _<_
   +-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
   +-*-isHeytingField           : IsHeytingField _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
-  
+
   <-denseLinearOrder           : DenseLinearOrder 0ℓ 0ℓ 0ℓ
   +-*-heytingCommutativeRing   : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
   +-*-heytingField             : HeytingField 0ℓ 0ℓ 0ℓ
-  
+
   module ≃-Reasoning = SetoidReasoning ≃-setoid
   ```
 
@@ -3223,7 +3223,7 @@ Additions to existing modules
 * Added new proofs in `Data.Vec.Relation.Binary.Lex.Strict`:
   ```agda
   xs≮[] : ¬ xs < []
-  
+
   <-respectsˡ : IsPartialEquivalence _≈_ → _≺_ Respectsˡ _≈_ → _<_ Respectsˡ _≋_
   <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ → _<_ _Respectsʳ _≋_
   <-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ → WellFounded _<_
@@ -3247,7 +3247,7 @@ Additions to existing modules
 * Added new functions in `Data.Vec.Relation.Unary.All`:
   ```
   decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
-  
+
   lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
   lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
   lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
@@ -3258,7 +3258,7 @@ Additions to existing modules
   ```
   ++-assoc : (xs ++ ys) ++ zs ≋ xs ++ (ys ++ zs)
   ```
-  
+
 * Added new isomorphisms to `Data.Vec.N-ary`:
   ```agda
   Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B
@@ -3276,7 +3276,7 @@ Additions to existing modules
   evalState : State s a → s → a
   execState : State s a → s → s
   ```
-  
+
 * Added a non-dependent version of `flip` in `Function.Base`:
   ```agda
   flip′ : (A → B → C) → (B → A → C)
@@ -3295,10 +3295,10 @@ Additions to existing modules
   ↔-refl  : Reflexive _↔_
   ↔-sym   : Symmetric _↔_
   ↔-trans : Transitive _↔_
-  
+
   ↔⇒↣   : A ↔ B → A ↣ B
   ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)
-  
+
   Inverse⇒Injection : Inverse S T → Injection S T
   ```
 
@@ -3317,7 +3317,7 @@ Additions to existing modules
   _!|>_  : (a : A) → (∀ a → B a) → B a
   _!|>′_ : A → (A → B) → B
   ```
-  
+
 * Added new proof and record in `Function.Structures`:
   ```agda
   record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
@@ -3454,7 +3454,7 @@ Additions to existing modules
   ⊂-resp-≐    : _⊂_ Respects₂ _≐_
   ⊂-irrefl    : Irreflexive _≐_ _⊂_
   ⊂-antisym   : Antisymmetric _≐_ _⊂_
-  
+
   ∅-⊆′         : (P : Pred A ℓ) → ∅ ⊆′ P
   ⊆′-U         : (P : Pred A ℓ) → P ⊆′ U
   ⊆′-refl      : Reflexive {A = Pred A ℓ} _⊆′_
@@ -3476,7 +3476,7 @@ Additions to existing modules
   ⊆′⇒⊆         : _⊆′_ ⇒ _⊆_
   ⊂⇒⊂′         : _⊂_ ⇒ _⊂′_
   ⊂′⇒⊂         : _⊂′_ ⇒ _⊂_
-  
+
   ≐-refl   : Reflexive _≐_
   ≐-sym    : Sym _≐_ _≐_
   ≐-trans  : Trans _≐_ _≐_ _≐_

From d7a393ea7a808e5884508349d3eb4d2233723e7e Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 21 Oct 2023 08:37:35 +0100
Subject: [PATCH 176/316] Fixed Agda version typo in README (#2176)

---
 README.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/README.agda b/README.agda
index ced0f11596..cc436d0848 100644
--- a/README.agda
+++ b/README.agda
@@ -19,7 +19,7 @@ module README where
 -- and other anonymous contributors.
 ------------------------------------------------------------------------
 
--- This version of the library has been tested using Agda 2.6.3.
+-- This version of the library has been tested using Agda 2.6.4.
 
 -- The library comes with a .agda-lib file, for use with the library
 -- management system.

From 5bb2a861b405b404cdbc9c3a9a70c36a3545015e Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 21 Oct 2023 08:38:19 +0100
Subject: [PATCH 177/316] =?UTF-8?q?Fixed=20in=20deprecation=20warning=20fo?=
 =?UTF-8?q?r=20`<-trans=CB=A1`=20(#2173)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

---
 src/Data/Nat/Properties.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index e304ed659f..37b1f33ee0 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -2342,6 +2342,6 @@ open Data.Nat.Base public
 
 <-transˡ = <-≤-trans
 {-# WARNING_ON_USAGE <-transˡ
-"Warning: <-transˡ was deprecated in v2.0. Please use ≤-<-trans instead. "
+"Warning: <-transˡ was deprecated in v2.0. Please use <-≤-trans instead. "
 #-}
 

From cedde4c4f0e39b18c99ec6e9f5d82c51b1f66cf8 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Wed, 1 Nov 2023 03:31:40 +0100
Subject: [PATCH 178/316] Bump Haskell CI (original!) to latest minor GHC
 versions (#2187)

---
 .github/workflows/ci-ubuntu.yml  | 14 ++++++------
 .github/workflows/haskell-ci.yml | 38 +++++++++-----------------------
 agda-stdlib-utils.cabal          |  6 ++---
 3 files changed, 21 insertions(+), 37 deletions(-)

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index 2ecc969589..51c5f7b09c 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -8,6 +8,7 @@ on:
     branches:
       - master
       - experimental
+  merge_group:
 
 ########################################################################
 ## CONFIGURATION
@@ -71,16 +72,15 @@ jobs:
 
       - name: Initialise variables
         run: |
-          if [[ '${{ github.ref }}' == 'refs/heads/master' \
-             || '${{ github.base_ref }}' == 'master' ]]; then
-            # Pick Agda version for master
-            echo "AGDA_COMMIT=tags/v2.6.4" >> $GITHUB_ENV;
-            echo "AGDA_HTML_DIR=html/master" >> $GITHUB_ENV
-          elif [[ '${{ github.ref }}' == 'refs/heads/experimental' \
-               || '${{ github.base_ref }}' == 'experimental' ]]; then
+          if [[ '${{ github.ref }}' == 'refs/heads/experimental' \
+             || '${{ github.base_ref }}' == 'experimental' ]]; then
             # Pick Agda version for experimental
             echo "AGDA_COMMIT=4d36cb37f8bfb765339b808b13356d760aa6f0ec" >> $GITHUB_ENV;
             echo "AGDA_HTML_DIR=html/experimental" >> $GITHUB_ENV
+          else
+            # Pick Agda version for master
+            echo "AGDA_COMMIT=tags/v2.6.4" >> $GITHUB_ENV;
+            echo "AGDA_HTML_DIR=html/master" >> $GITHUB_ENV
           fi
 
           if [[ '${{ github.ref }}' == 'refs/heads/master' \
diff --git a/.github/workflows/haskell-ci.yml b/.github/workflows/haskell-ci.yml
index b8c55c19be..5f9fa86b50 100644
--- a/.github/workflows/haskell-ci.yml
+++ b/.github/workflows/haskell-ci.yml
@@ -8,9 +8,9 @@
 #
 # For more information, see https://github.com/haskell-CI/haskell-ci
 #
-# version: 0.17.20230824
+# version: 0.17.20231010
 #
-# REGENDATA ("0.17.20230824",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
+# REGENDATA ("0.17.20231010",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
 #
 name: Haskell-CI
 on:
@@ -32,6 +32,7 @@ on:
       - agda-stdlib-utils.cabal
       - cabal.haskell-ci
       - "*.hs"
+  merge_group:
 jobs:
   linux:
     name: Haskell-CI - Linux - ${{ matrix.compiler }}
@@ -44,19 +45,19 @@ jobs:
     strategy:
       matrix:
         include:
-          - compiler: ghc-9.8.0.20230822
+          - compiler: ghc-9.8.1
             compilerKind: ghc
-            compilerVersion: 9.8.0.20230822
+            compilerVersion: 9.8.1
             setup-method: ghcup
-            allow-failure: true
-          - compiler: ghc-9.6.2
+            allow-failure: false
+          - compiler: ghc-9.6.3
             compilerKind: ghc
-            compilerVersion: 9.6.2
+            compilerVersion: 9.6.3
             setup-method: ghcup
             allow-failure: false
-          - compiler: ghc-9.4.6
+          - compiler: ghc-9.4.7
             compilerKind: ghc
-            compilerVersion: 9.4.6
+            compilerVersion: 9.4.7
             setup-method: ghcup
             allow-failure: false
           - compiler: ghc-9.2.8
@@ -94,7 +95,6 @@ jobs:
             mkdir -p "$HOME/.ghcup/bin"
             curl -sL https://downloads.haskell.org/ghcup/0.1.19.5/x86_64-linux-ghcup-0.1.19.5 > "$HOME/.ghcup/bin/ghcup"
             chmod a+x "$HOME/.ghcup/bin/ghcup"
-            "$HOME/.ghcup/bin/ghcup" config add-release-channel https://raw.githubusercontent.com/haskell/ghcup-metadata/master/ghcup-prereleases-0.0.7.yaml;
             "$HOME/.ghcup/bin/ghcup" install ghc "$HCVER" || (cat "$HOME"/.ghcup/logs/*.* && false)
             "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
           else
@@ -104,7 +104,6 @@ jobs:
             mkdir -p "$HOME/.ghcup/bin"
             curl -sL https://downloads.haskell.org/ghcup/0.1.19.5/x86_64-linux-ghcup-0.1.19.5 > "$HOME/.ghcup/bin/ghcup"
             chmod a+x "$HOME/.ghcup/bin/ghcup"
-            "$HOME/.ghcup/bin/ghcup" config add-release-channel https://raw.githubusercontent.com/haskell/ghcup-metadata/master/ghcup-prereleases-0.0.7.yaml;
             "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
           fi
         env:
@@ -138,7 +137,7 @@ jobs:
           echo "HCNUMVER=$HCNUMVER" >> "$GITHUB_ENV"
           echo "ARG_TESTS=--enable-tests" >> "$GITHUB_ENV"
           echo "ARG_BENCH=--enable-benchmarks" >> "$GITHUB_ENV"
-          if [ $((HCNUMVER >= 90800)) -ne 0 ] ; then echo "HEADHACKAGE=true" >> "$GITHUB_ENV" ; else echo "HEADHACKAGE=false" >> "$GITHUB_ENV" ; fi
+          echo "HEADHACKAGE=false" >> "$GITHUB_ENV"
           echo "ARG_COMPILER=--$HCKIND --with-compiler=$HC" >> "$GITHUB_ENV"
           echo "GHCJSARITH=0" >> "$GITHUB_ENV"
         env:
@@ -167,18 +166,6 @@ jobs:
           repository hackage.haskell.org
             url: http://hackage.haskell.org/
           EOF
-          if $HEADHACKAGE; then
-          cat >> $CABAL_CONFIG <<EOF
-          repository head.hackage.ghc.haskell.org
-             url: https://ghc.gitlab.haskell.org/head.hackage/
-             secure: True
-             root-keys: 7541f32a4ccca4f97aea3b22f5e593ba2c0267546016b992dfadcd2fe944e55d
-                        26021a13b401500c8eb2761ca95c61f2d625bfef951b939a8124ed12ecf07329
-                        f76d08be13e9a61a377a85e2fb63f4c5435d40f8feb3e12eb05905edb8cdea89
-             key-threshold: 3
-          active-repositories: hackage.haskell.org, head.hackage.ghc.haskell.org:override
-          EOF
-          fi
           cat >> $CABAL_CONFIG <<EOF
           program-default-options
             ghc-options: $GHCJOBS +RTS -M3G -RTS
@@ -230,9 +217,6 @@ jobs:
           echo "    ghc-options: -Werror=missing-methods" >> cabal.project
           cat >> cabal.project <<EOF
           EOF
-          if $HEADHACKAGE; then
-          echo "allow-newer: $($HCPKG list --simple-output | sed -E 's/([a-zA-Z-]+)-[0-9.]+/*:\1,/g')" >> cabal.project
-          fi
           $HCPKG list --simple-output --names-only | perl -ne 'for (split /\s+/) { print "constraints: $_ installed\n" unless /^(agda-stdlib-utils)$/; }' >> cabal.project.local
           cat cabal.project
           cat cabal.project.local
diff --git a/agda-stdlib-utils.cabal b/agda-stdlib-utils.cabal
index a482bee069..3accc6972e 100644
--- a/agda-stdlib-utils.cabal
+++ b/agda-stdlib-utils.cabal
@@ -6,9 +6,9 @@ description:     Helper programs for setting up the Agda standard library.
 license:         MIT
 
 tested-with:
-  GHC == 9.8.0
-  GHC == 9.6.2
-  GHC == 9.4.6
+  GHC == 9.8.1
+  GHC == 9.6.3
+  GHC == 9.4.7
   GHC == 9.2.8
   GHC == 9.0.2
   GHC == 8.10.7

From d8cd956df9ca5305fcc50fbdfd9915466373c525 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 1 Nov 2023 02:32:30 +0000
Subject: [PATCH 179/316] [fixes #2175] Documentation misc. typos etc. for RC1 
 (#2183)

* missing comma!

* corrected reference to `README.Design.Decidability`

* typo: capitalisation

* updated `installation-guide`

* added word to `NonZero` section heading

* Run workflows on any PR

* Add merge-group trigger to workflows

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                             |  4 ++--
 README.agda                              |  2 +-
 notes/installation-guide.md              | 12 ++++++------
 src/Relation/Nullary/Decidable/Core.agda |  4 ++--
 4 files changed, 11 insertions(+), 11 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9d9f403d64..ff21ab1c98 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -504,7 +504,7 @@ Non-backwards compatible changes
   and `Codata.Guarded.Stream.Relation.Binary.Pointwise`, the proofs
   called `lookup` have been renamed `lookup⁺`.
 
-#### Changes to `Data.(Nat/Integer/Rational)` proofs of `NonZero`/`Positive`/`Negative` to instance arguments
+#### Changes to `Data.(Nat/Integer/Rational)` proofs of `NonZero`/`Positive`/`Negative` to use instance arguments
 
 * Many numeric operations in the library require their arguments to be non-zero,
   and various proofs require their arguments to be non-zero/positive/negative etc.
@@ -2761,7 +2761,7 @@ Additions to existing modules
 * Added new definitions and proofs to `Data.Nat.Primality`:
   ```agda
   Composite        : ℕ → Set
-  composite?       : Decidable composite
+  composite?       : Decidable Composite
   composite⇒¬prime : Composite n → ¬ Prime n
   ¬composite⇒prime : 2 ≤ n → ¬ Composite n → Prime n
   prime⇒¬composite : Prime n → ¬ Composite n
diff --git a/README.agda b/README.agda
index cc436d0848..41b536a0d0 100644
--- a/README.agda
+++ b/README.agda
@@ -7,7 +7,7 @@ module README where
 --
 -- Authors: Nils Anders Danielsson, Matthew Daggitt, Guillaume Allais
 -- with contributions from Andreas Abel, Stevan Andjelkovic,
--- Jean-Philippe Bernardy, Peter Berry, Bradley Hardy Joachim Breitner,
+-- Jean-Philippe Bernardy, Peter Berry, Bradley Hardy, Joachim Breitner,
 -- Samuel Bronson, Daniel Brown, Jacques Carette, James Chapman,
 -- Liang-Ting Chen, Dominique Devriese, Dan Doel, Érdi Gergő,
 -- Zack Grannan, Helmut Grohne, Simon Foster, Liyang Hu, Jason Hu,
diff --git a/notes/installation-guide.md b/notes/installation-guide.md
index f05f21ed90..67d1685ad9 100644
--- a/notes/installation-guide.md
+++ b/notes/installation-guide.md
@@ -3,19 +3,19 @@ Installation instructions
 
 Note: the full story on installing Agda libraries can be found at [readthedocs](http://agda.readthedocs.io/en/latest/tools/package-system.html).
 
-Use version v1.7.2 of the standard library with Agda 2.6.3.
+Use version v2.0 of the standard library with Agda 2.6.4.
 
 1. Navigate to a suitable directory `$HERE` (replace appropriately) where
    you would like to install the library.
 
-2. Download the tarball of v1.7.2 of the standard library. This can either be
+2. Download the tarball of v2.0 of the standard library. This can either be
    done manually by visiting the Github repository for the library, or via the
    command line as follows:
    ```
-   wget -O agda-stdlib.tar https://github.com/agda/agda-stdlib/archive/v1.7.2.tar.gz
+   wget -O agda-stdlib.tar https://github.com/agda/agda-stdlib/archive/v2.0.tar.gz
    ```
    Note that you can replace `wget` with other popular tools such as `curl` and that
-   you can replace `1.7.2` with any other version of the library you desire.
+   you can replace `2.0` with any other version of the library you desire.
 
 3. Extract the standard library from the tarball. Again this can either be
    done manually or via the command line as follows:
@@ -26,7 +26,7 @@ Use version v1.7.2 of the standard library with Agda 2.6.3.
 4. [ OPTIONAL ] If using [cabal](https://www.haskell.org/cabal/) then run
    the commands to install via cabal:
    ```
-   cd agda-stdlib-1.7.2
+   cd agda-stdlib-2.0
    cabal install
    ```
 
@@ -40,7 +40,7 @@ Use version v1.7.2 of the standard library with Agda 2.6.3.
 6. Register the standard library with Agda's package system by adding
    the following line to `$HOME/.agda/libraries`:
    ```
-   $HERE/agda-stdlib-1.7.2/standard-library.agda-lib
+   $HERE/agda-stdlib-2.0/standard-library.agda-lib
    ```
 
 Now, the standard library is ready to be used either:
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index 166b1f264e..ff73c6af20 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -35,8 +35,8 @@ private
 -- reflects the boolean result. This definition allows the boolean
 -- part of the decision procedure to compute independently from the
 -- proof. This leads to better computational behaviour when we only care
--- about the result and not the proof. See README.Decidability for
--- further details.
+-- about the result and not the proof. See README.Design.Decidability
+-- for further details.
 
 infix 2 _because_
 

From 67ff824c07bbde0b1d5866577c8f22a0757d9e0d Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 1 Nov 2023 02:33:25 +0000
Subject: [PATCH 180/316] [fixes #2178] regularise and specify/systematise, the
 conventions for symbol usage (#2185)

* regularise and specify/systematise, the conventions for symbol usage

* typos/revisions
---
 CHANGELOG.md                                |  54 ++---
 notes/style-guide.md                        |  44 +++-
 src/Algebra/Consequences/Propositional.agda |  54 ++---
 src/Algebra/Consequences/Setoid.agda        | 242 ++++++++++----------
 4 files changed, 216 insertions(+), 178 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ff21ab1c98..ef730a4546 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -112,21 +112,21 @@ Non-backwards compatible changes
   always true and cannot be assumed in user's code.
 
 * Therefore the definitions have been changed as follows to make all their arguments explicit:
-  - `LeftCancellative _•_`
-    - From: `∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z`
-    - To: `∀ x y z → (x • y) ≈ (x • z) → y ≈ z`
+  - `LeftCancellative _∙_`
+    - From: `∀ x {y z} → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
+    - To: `∀ x y z → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
 
-  - `RightCancellative _•_`
-    - From: `∀ {x} y z → (y • x) ≈ (z • x) → y ≈ z`
-    - To: `∀ x y z → (y • x) ≈ (z • x) → y ≈ z`
+  - `RightCancellative _∙_`
+    - From: `∀ {x} y z → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
+    - To: `∀ x y z → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
 
-  - `AlmostLeftCancellative e _•_`
-    - From: `∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
-    - To: `∀ x y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z`
+  - `AlmostLeftCancellative e _∙_`
+    - From: `∀ {x} y z → ¬ x ≈ e → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
 
-  - `AlmostRightCancellative e _•_`
-    - From: `∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
-    - To: `∀ x y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z`
+  - `AlmostRightCancellative e _∙_`
+    - From: `∀ {x} y z → ¬ x ≈ e → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
 
 * Correspondingly some proofs of the above types will need additional arguments passed explicitly.
   Instances can easily be fixed by adding additional underscores, e.g.
@@ -2152,16 +2152,16 @@ Additions to existing modules
 
 * Added new proofs to `Algebra.Consequences.Propositional`:
   ```agda
-  comm+assoc⇒middleFour     : Commutative _•_ → Associative _•_ → _•_ MiddleFourExchange _•_
-  identity+middleFour⇒assoc : Identity e _•_ → _•_ MiddleFourExchange _•_ → Associative _•_
-  identity+middleFour⇒comm  : Identity e _+_ → _•_ MiddleFourExchange _+_ → Commutative _•_
+  comm+assoc⇒middleFour     : Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
+  identity+middleFour⇒assoc : Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
+  identity+middleFour⇒comm  : Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
   ```
 
 * Added new proofs to `Algebra.Consequences.Setoid`:
   ```agda
-  comm+assoc⇒middleFour     : Congruent₂ _•_ → Commutative _•_ → Associative _•_ → _•_ MiddleFourExchange _•_
-  identity+middleFour⇒assoc : Congruent₂ _•_ → Identity e _•_ → _•_ MiddleFourExchange _•_ → Associative _•_
-  identity+middleFour⇒comm  : Congruent₂ _•_ → Identity e _+_ → _•_ MiddleFourExchange _+_ → Commutative _•_
+  comm+assoc⇒middleFour     : Congruent₂ _∙_ → Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
+  identity+middleFour⇒assoc : Congruent₂ _∙_ → Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
+  identity+middleFour⇒comm  : Congruent₂ _∙_ → Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
 
   involutive⇒surjective  : Involutive f  → Surjective f
   selfInverse⇒involutive : SelfInverse f → Involutive f
@@ -2171,15 +2171,15 @@ Additions to existing modules
   selfInverse⇒injective  : SelfInverse f → Injective f
   selfInverse⇒bijective  : SelfInverse f → Bijective f
 
-  comm+idˡ⇒id              : Commutative _•_ → LeftIdentity  e _•_ → Identity e _•_
-  comm+idʳ⇒id              : Commutative _•_ → RightIdentity e _•_ → Identity e _•_
-  comm+zeˡ⇒ze              : Commutative _•_ → LeftZero      e _•_ → Zero     e _•_
-  comm+zeʳ⇒ze              : Commutative _•_ → RightZero     e _•_ → Zero     e _•_
-  comm+invˡ⇒inv            : Commutative _•_ → LeftInverse  e _⁻¹ _•_ → Inverse e _⁻¹ _•_
-  comm+invʳ⇒inv            : Commutative _•_ → RightInverse e _⁻¹ _•_ → Inverse e _⁻¹ _•_
-  comm+distrˡ⇒distr        : Commutative _•_ → _•_ DistributesOverˡ _◦_ → _•_ DistributesOver _◦_
-  comm+distrʳ⇒distr        : Commutative _•_ → _•_ DistributesOverʳ _◦_ → _•_ DistributesOver _◦_
-  distrib+absorbs⇒distribˡ : Associative _•_ → Commutative _◦_ → _•_ Absorbs _◦_ → _◦_ Absorbs _•_ → _◦_ DistributesOver _•_ → _•_ DistributesOverˡ _◦_
+  comm+idˡ⇒id              : Commutative _∙_ → LeftIdentity  e _∙_ → Identity e _∙_
+  comm+idʳ⇒id              : Commutative _∙_ → RightIdentity e _∙_ → Identity e _∙_
+  comm+zeˡ⇒ze              : Commutative _∙_ → LeftZero      e _∙_ → Zero     e _∙_
+  comm+zeʳ⇒ze              : Commutative _∙_ → RightZero     e _∙_ → Zero     e _∙_
+  comm+invˡ⇒inv            : Commutative _∙_ → LeftInverse  e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm+invʳ⇒inv            : Commutative _∙_ → RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm+distrˡ⇒distr        : Commutative _∙_ → _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
+  comm+distrʳ⇒distr        : Commutative _∙_ → _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
+  distrib+absorbs⇒distribˡ : Associative _∙_ → Commutative _◦_ → _∙_ Absorbs _◦_ → _◦_ Absorbs _∙_ → _◦_ DistributesOver _∙_ → _∙_ DistributesOverˡ _◦_
   ```
 
 * Added new functions to `Algebra.Construct.DirectProduct`:
diff --git a/notes/style-guide.md b/notes/style-guide.md
index d6201abe88..95ef7ba2d0 100644
--- a/notes/style-guide.md
+++ b/notes/style-guide.md
@@ -347,12 +347,12 @@ line of code, indented by two spaces.
 
 #### Variables
 
-* `Level` and `Set`s can always be generalized using the keyword `variable`.
+* `Level` and `Set`s can always be generalised using the keyword `variable`.
 
 * A file may only declare variables of other types if those types are used
   in the definition of the main type that the file concerns itself with.
-  At the moment the policy is *not* to generalize over any other types to
-  minimize the amount of information that users have to keep in their head
+  At the moment the policy is *not* to generalise over any other types to
+  minimise the amount of information that users have to keep in their head
   concurrently.
 
 * Example 1: the main type in `Data.List.Properties` is `List A` where `A : Set a`.
@@ -443,6 +443,44 @@ word within a compound word is capitalized except for the first word.
   relations should be used over the `¬` symbol (e.g. `m+n≮n` should be
   used instead of `¬m+n<n`).
 
+#### Symbols for operators and relations
+
+* The stdlib aims to use a consistent set of notations, governed by a
+  consistent set of conventions, but sometimes, different
+  Unicode/emacs-input-method symbols nevertheless can be rendered by
+  identical-*seeming* symbols, so this is an attempt to document these.
+
+* The typical binary operator in the `Algebra` hierarchy, inheriting
+  from the root `Structure`/`Bundle` `isMagma`/`Magma`, is written as
+  infix `∙`, obtained as `\.`, NOT as `\bu2`. Nevertheless, there is
+  also a 'generic' operator, written as infix `·`, obtained as
+  `\cdot`. Do NOT attempt to use related, but typographically
+  indistinguishable, symbols.
+
+* Similarly, 'primed' names and symbols, used to standardise names
+  apart, or to provide (more) simply-typed versions of
+  dependently-typed operations, should be written using `\'`, NOT the
+  unmarked `'` character.
+
+* Likewise, standard infix symbols for eg, divisibility on numeric
+  datatypes/algebraic structure, should be written `\|`, NOT the
+  unmarked `|` character. An exception to this is the *strict*
+  ordering relation, written using `<`, NOT `\<` as might be expected.
+
+* Since v2.0, the `Algebra` hierarchy systematically introduces
+  consistent symbolic notation for the negated versions of the usual
+  binary predicates for equality, ordering etc. These are obtained
+  from the corresponding input sequence by adding `n` to the symbol
+  name, so that `≤`, obtained as `\le`, becomes `≰` obtained as
+  `\len`, etc.
+
+* Correspondingly, the flipped symbols (and their negations) for the
+  converse relations are systematically introduced, eg `≥` as `\ge`
+  and `≱` as `\gen`.
+
+* Any exceptions to these conventions should be flagged on the GitHub
+  `agda-stdlib` issue tracker in the usual way.
+
 #### Functions and relations over specific datatypes
 
 * When defining a new relation `P` over a datatype `X` in a `Data.X.Relation` module,
diff --git a/src/Algebra/Consequences/Propositional.agda b/src/Algebra/Consequences/Propositional.agda
index 6f2626889c..1adcb12afc 100644
--- a/src/Algebra/Consequences/Propositional.agda
+++ b/src/Algebra/Consequences/Propositional.agda
@@ -44,16 +44,16 @@ open Base public
 ------------------------------------------------------------------------
 -- Group-like structures
 
-module _ {_•_ _⁻¹ ε} where
+module _ {_∙_ _⁻¹ ε} where
 
-  assoc+id+invʳ⇒invˡ-unique : Associative _•_ → Identity ε _•_ →
-                              RightInverse ε _⁻¹ _•_ →
-                              ∀ x y → (x • y) ≡ ε → x ≡ (y ⁻¹)
+  assoc+id+invʳ⇒invˡ-unique : Associative _∙_ → Identity ε _∙_ →
+                              RightInverse ε _⁻¹ _∙_ →
+                              ∀ x y → (x ∙ y) ≡ ε → x ≡ (y ⁻¹)
   assoc+id+invʳ⇒invˡ-unique = Base.assoc+id+invʳ⇒invˡ-unique (cong₂ _)
 
-  assoc+id+invˡ⇒invʳ-unique : Associative _•_ → Identity ε _•_ →
-                              LeftInverse ε _⁻¹ _•_ →
-                              ∀ x y → (x • y) ≡ ε → y ≡ (x ⁻¹)
+  assoc+id+invˡ⇒invʳ-unique : Associative _∙_ → Identity ε _∙_ →
+                              LeftInverse ε _⁻¹ _∙_ →
+                              ∀ x y → (x ∙ y) ≡ ε → y ≡ (x ⁻¹)
   assoc+id+invˡ⇒invʳ-unique = Base.assoc+id+invˡ⇒invʳ-unique (cong₂ _)
 
 ------------------------------------------------------------------------
@@ -76,43 +76,43 @@ module _ {_+_ _*_ -_ 0#} where
 ------------------------------------------------------------------------
 -- Bisemigroup-like structures
 
-module _ {_•_ _◦_ : Op₂ A} (•-comm : Commutative _•_) where
+module _ {_∙_ _◦_ : Op₂ A} (∙-comm : Commutative _∙_) where
 
-  comm+distrˡ⇒distrʳ : _•_ DistributesOverˡ _◦_ → _•_ DistributesOverʳ _◦_
-  comm+distrˡ⇒distrʳ = Base.comm+distrˡ⇒distrʳ (cong₂ _) •-comm
+  comm+distrˡ⇒distrʳ : _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_
+  comm+distrˡ⇒distrʳ = Base.comm+distrˡ⇒distrʳ (cong₂ _) ∙-comm
 
-  comm+distrʳ⇒distrˡ : _•_ DistributesOverʳ _◦_ → _•_ DistributesOverˡ _◦_
-  comm+distrʳ⇒distrˡ = Base.comm+distrʳ⇒distrˡ (cong₂ _) •-comm
+  comm+distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_
+  comm+distrʳ⇒distrˡ = Base.comm+distrʳ⇒distrˡ (cong₂ _) ∙-comm
 
-  comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y • z)) ≡ ((x ◦ y) • (x ◦ z)))
-  comm⇒sym[distribˡ] = Base.comm⇒sym[distribˡ] (cong₂ _◦_) •-comm
+  comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y ∙ z)) ≡ ((x ◦ y) ∙ (x ◦ z)))
+  comm⇒sym[distribˡ] = Base.comm⇒sym[distribˡ] (cong₂ _◦_) ∙-comm
 
 ------------------------------------------------------------------------
 -- Selectivity
 
-module _ {_•_ : Op₂ A} where
+module _ {_∙_ : Op₂ A} where
 
-  sel⇒idem : Selective _•_ → Idempotent _•_
+  sel⇒idem : Selective _∙_ → Idempotent _∙_
   sel⇒idem = Base.sel⇒idem _≡_
 
 ------------------------------------------------------------------------
 -- Middle-Four Exchange
 
-module _ {_•_ : Op₂ A} where
+module _ {_∙_ : Op₂ A} where
 
-  comm+assoc⇒middleFour : Commutative _•_ → Associative _•_ →
-                          _•_ MiddleFourExchange _•_
-  comm+assoc⇒middleFour = Base.comm+assoc⇒middleFour (cong₂ _•_)
+  comm+assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →
+                          _∙_ MiddleFourExchange _∙_
+  comm+assoc⇒middleFour = Base.comm+assoc⇒middleFour (cong₂ _∙_)
 
-  identity+middleFour⇒assoc : {e : A} → Identity e _•_ →
-                              _•_ MiddleFourExchange _•_ →
-                              Associative _•_
-  identity+middleFour⇒assoc = Base.identity+middleFour⇒assoc (cong₂ _•_)
+  identity+middleFour⇒assoc : {e : A} → Identity e _∙_ →
+                              _∙_ MiddleFourExchange _∙_ →
+                              Associative _∙_
+  identity+middleFour⇒assoc = Base.identity+middleFour⇒assoc (cong₂ _∙_)
 
   identity+middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
-                             _•_ MiddleFourExchange _+_ →
-                             Commutative _•_
-  identity+middleFour⇒comm = Base.identity+middleFour⇒comm (cong₂ _•_)
+                             _∙_ MiddleFourExchange _+_ →
+                             Commutative _∙_
+  identity+middleFour⇒comm = Base.identity+middleFour⇒comm (cong₂ _∙_)
 
 ------------------------------------------------------------------------
 -- Without Loss of Generality
diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index 068ecb2d12..f90a801fe4 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -35,36 +35,36 @@ open import Algebra.Consequences.Base public
 ------------------------------------------------------------------------
 -- MiddleFourExchange
 
-module _ {_•_ : Op₂ A} (cong : Congruent₂ _•_) where
+module _ {_∙_ : Op₂ A} (cong : Congruent₂ _∙_) where
 
-  comm+assoc⇒middleFour : Commutative _•_ → Associative _•_ →
-                          _•_ MiddleFourExchange _•_
+  comm+assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →
+                          _∙_ MiddleFourExchange _∙_
   comm+assoc⇒middleFour comm assoc w x y z = begin
-    (w • x) • (y • z) ≈⟨ assoc w x (y • z) ⟩
-    w • (x • (y • z)) ≈⟨ cong refl (sym (assoc x y z)) ⟩
-    w • ((x • y) • z) ≈⟨ cong refl (cong (comm x y) refl) ⟩
-    w • ((y • x) • z) ≈⟨ cong refl (assoc y x z) ⟩
-    w • (y • (x • z)) ≈⟨ sym (assoc w y (x • z)) ⟩
-    (w • y) • (x • z) ∎
-
-  identity+middleFour⇒assoc : {e : A} → Identity e _•_ →
-                              _•_ MiddleFourExchange _•_ →
-                              Associative _•_
+    (w ∙ x) ∙ (y ∙ z) ≈⟨ assoc w x (y ∙ z) ⟩
+    w ∙ (x ∙ (y ∙ z)) ≈⟨ cong refl (sym (assoc x y z)) ⟩
+    w ∙ ((x ∙ y) ∙ z) ≈⟨ cong refl (cong (comm x y) refl) ⟩
+    w ∙ ((y ∙ x) ∙ z) ≈⟨ cong refl (assoc y x z) ⟩
+    w ∙ (y ∙ (x ∙ z)) ≈⟨ sym (assoc w y (x ∙ z)) ⟩
+    (w ∙ y) ∙ (x ∙ z) ∎
+
+  identity+middleFour⇒assoc : {e : A} → Identity e _∙_ →
+                              _∙_ MiddleFourExchange _∙_ →
+                              Associative _∙_
   identity+middleFour⇒assoc {e} (identityˡ , identityʳ) middleFour x y z = begin
-    (x • y) • z       ≈⟨ cong refl (sym (identityˡ z)) ⟩
-    (x • y) • (e • z) ≈⟨ middleFour x y e z ⟩
-    (x • e) • (y • z) ≈⟨ cong (identityʳ x) refl ⟩
-    x • (y • z)       ∎
+    (x ∙ y) ∙ z       ≈⟨ cong refl (sym (identityˡ z)) ⟩
+    (x ∙ y) ∙ (e ∙ z) ≈⟨ middleFour x y e z ⟩
+    (x ∙ e) ∙ (y ∙ z) ≈⟨ cong (identityʳ x) refl ⟩
+    x ∙ (y ∙ z)       ∎
 
   identity+middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
-                             _•_ MiddleFourExchange _+_ →
-                             Commutative _•_
+                             _∙_ MiddleFourExchange _+_ →
+                             Commutative _∙_
   identity+middleFour⇒comm {_+_} {e} (identityˡ , identityʳ) middleFour x y
     = begin
-    x • y             ≈⟨ sym (cong (identityˡ x) (identityʳ y)) ⟩
-    (e + x) • (y + e) ≈⟨ middleFour e x y e ⟩
-    (e + y) • (x + e) ≈⟨ cong (identityˡ y) (identityʳ x) ⟩
-    y • x             ∎
+    x ∙ y             ≈⟨ sym (cong (identityˡ x) (identityʳ y)) ⟩
+    (e + x) ∙ (y + e) ≈⟨ middleFour e x y e ⟩
+    (e + y) ∙ (x + e) ≈⟨ cong (identityˡ y) (identityʳ x) ⟩
+    y ∙ x             ∎
 
 ------------------------------------------------------------------------
 -- SelfInverse
@@ -98,182 +98,182 @@ module _ {f : Op₁ A} (self : SelfInverse f) where
 ------------------------------------------------------------------------
 -- Magma-like structures
 
-module _ {_•_ : Op₂ A} (comm : Commutative _•_) where
+module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) where
 
-  comm+cancelˡ⇒cancelʳ : LeftCancellative _•_ → RightCancellative _•_
+  comm+cancelˡ⇒cancelʳ : LeftCancellative _∙_ → RightCancellative _∙_
   comm+cancelˡ⇒cancelʳ cancelˡ x y z eq = cancelˡ x y z $ begin
-    x • y ≈⟨ comm x y ⟩
-    y • x ≈⟨ eq ⟩
-    z • x ≈⟨ comm z x ⟩
-    x • z ∎
+    x ∙ y ≈⟨ comm x y ⟩
+    y ∙ x ≈⟨ eq ⟩
+    z ∙ x ≈⟨ comm z x ⟩
+    x ∙ z ∎
 
-  comm+cancelʳ⇒cancelˡ : RightCancellative _•_ → LeftCancellative _•_
+  comm+cancelʳ⇒cancelˡ : RightCancellative _∙_ → LeftCancellative _∙_
   comm+cancelʳ⇒cancelˡ cancelʳ x y z eq = cancelʳ x y z $ begin
-    y • x ≈⟨ comm y x ⟩
-    x • y ≈⟨ eq ⟩
-    x • z ≈⟨ comm x z ⟩
-    z • x ∎
+    y ∙ x ≈⟨ comm y x ⟩
+    x ∙ y ≈⟨ eq ⟩
+    x ∙ z ≈⟨ comm x z ⟩
+    z ∙ x ∎
 
 ------------------------------------------------------------------------
 -- Monoid-like structures
 
-module _ {_•_ : Op₂ A} (comm : Commutative _•_) {e : A} where
+module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) {e : A} where
 
-  comm+idˡ⇒idʳ : LeftIdentity e _•_ → RightIdentity e _•_
+  comm+idˡ⇒idʳ : LeftIdentity e _∙_ → RightIdentity e _∙_
   comm+idˡ⇒idʳ idˡ x = begin
-    x • e ≈⟨ comm x e ⟩
-    e • x ≈⟨ idˡ x ⟩
+    x ∙ e ≈⟨ comm x e ⟩
+    e ∙ x ≈⟨ idˡ x ⟩
     x     ∎
 
-  comm+idʳ⇒idˡ : RightIdentity e _•_ → LeftIdentity e _•_
+  comm+idʳ⇒idˡ : RightIdentity e _∙_ → LeftIdentity e _∙_
   comm+idʳ⇒idˡ idʳ x = begin
-    e • x ≈⟨ comm e x ⟩
-    x • e ≈⟨ idʳ x ⟩
+    e ∙ x ≈⟨ comm e x ⟩
+    x ∙ e ≈⟨ idʳ x ⟩
     x     ∎
 
-  comm+idˡ⇒id : LeftIdentity e _•_ → Identity e _•_
+  comm+idˡ⇒id : LeftIdentity e _∙_ → Identity e _∙_
   comm+idˡ⇒id idˡ = idˡ , comm+idˡ⇒idʳ idˡ
 
-  comm+idʳ⇒id : RightIdentity e _•_ → Identity e _•_
+  comm+idʳ⇒id : RightIdentity e _∙_ → Identity e _∙_
   comm+idʳ⇒id idʳ = comm+idʳ⇒idˡ idʳ , idʳ
 
-  comm+zeˡ⇒zeʳ : LeftZero e _•_ → RightZero e _•_
+  comm+zeˡ⇒zeʳ : LeftZero e _∙_ → RightZero e _∙_
   comm+zeˡ⇒zeʳ zeˡ x = begin
-    x • e ≈⟨ comm x e ⟩
-    e • x ≈⟨ zeˡ x ⟩
+    x ∙ e ≈⟨ comm x e ⟩
+    e ∙ x ≈⟨ zeˡ x ⟩
     e     ∎
 
-  comm+zeʳ⇒zeˡ : RightZero e _•_ → LeftZero e _•_
+  comm+zeʳ⇒zeˡ : RightZero e _∙_ → LeftZero e _∙_
   comm+zeʳ⇒zeˡ zeʳ x = begin
-    e • x ≈⟨ comm e x ⟩
-    x • e ≈⟨ zeʳ x ⟩
+    e ∙ x ≈⟨ comm e x ⟩
+    x ∙ e ≈⟨ zeʳ x ⟩
     e     ∎
 
-  comm+zeˡ⇒ze : LeftZero e _•_ → Zero e _•_
+  comm+zeˡ⇒ze : LeftZero e _∙_ → Zero e _∙_
   comm+zeˡ⇒ze zeˡ = zeˡ , comm+zeˡ⇒zeʳ zeˡ
 
-  comm+zeʳ⇒ze : RightZero e _•_ → Zero e _•_
+  comm+zeʳ⇒ze : RightZero e _∙_ → Zero e _∙_
   comm+zeʳ⇒ze zeʳ = comm+zeʳ⇒zeˡ zeʳ , zeʳ
 
-  comm+almostCancelˡ⇒almostCancelʳ : AlmostLeftCancellative e _•_ →
-                                     AlmostRightCancellative e _•_
+  comm+almostCancelˡ⇒almostCancelʳ : AlmostLeftCancellative e _∙_ →
+                                     AlmostRightCancellative e _∙_
   comm+almostCancelˡ⇒almostCancelʳ cancelˡ-nonZero x y z x≉e yx≈zx =
     cancelˡ-nonZero x y z x≉e $ begin
-      x • y ≈⟨ comm x y ⟩
-      y • x ≈⟨ yx≈zx ⟩
-      z • x ≈⟨ comm z x ⟩
-      x • z ∎
+      x ∙ y ≈⟨ comm x y ⟩
+      y ∙ x ≈⟨ yx≈zx ⟩
+      z ∙ x ≈⟨ comm z x ⟩
+      x ∙ z ∎
 
-  comm+almostCancelʳ⇒almostCancelˡ : AlmostRightCancellative e _•_ →
-                                     AlmostLeftCancellative e _•_
+  comm+almostCancelʳ⇒almostCancelˡ : AlmostRightCancellative e _∙_ →
+                                     AlmostLeftCancellative e _∙_
   comm+almostCancelʳ⇒almostCancelˡ cancelʳ-nonZero x y z x≉e xy≈xz =
     cancelʳ-nonZero x y z x≉e $ begin
-      y • x ≈⟨ comm y x ⟩
-      x • y ≈⟨ xy≈xz ⟩
-      x • z ≈⟨ comm x z ⟩
-      z • x ∎
+      y ∙ x ≈⟨ comm y x ⟩
+      x ∙ y ≈⟨ xy≈xz ⟩
+      x ∙ z ≈⟨ comm x z ⟩
+      z ∙ x ∎
 
 ------------------------------------------------------------------------
 -- Group-like structures
 
-module _ {_•_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (comm : Commutative _•_) where
+module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (comm : Commutative _∙_) where
 
-  comm+invˡ⇒invʳ : LeftInverse e _⁻¹ _•_ → RightInverse e _⁻¹ _•_
+  comm+invˡ⇒invʳ : LeftInverse e _⁻¹ _∙_ → RightInverse e _⁻¹ _∙_
   comm+invˡ⇒invʳ invˡ x = begin
-    x • (x ⁻¹) ≈⟨ comm x (x ⁻¹) ⟩
-    (x ⁻¹) • x ≈⟨ invˡ x ⟩
+    x ∙ (x ⁻¹) ≈⟨ comm x (x ⁻¹) ⟩
+    (x ⁻¹) ∙ x ≈⟨ invˡ x ⟩
     e          ∎
 
-  comm+invˡ⇒inv : LeftInverse e _⁻¹ _•_ → Inverse e _⁻¹ _•_
+  comm+invˡ⇒inv : LeftInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
   comm+invˡ⇒inv invˡ = invˡ , comm+invˡ⇒invʳ invˡ
 
-  comm+invʳ⇒invˡ : RightInverse e _⁻¹ _•_ → LeftInverse e _⁻¹ _•_
+  comm+invʳ⇒invˡ : RightInverse e _⁻¹ _∙_ → LeftInverse e _⁻¹ _∙_
   comm+invʳ⇒invˡ invʳ x = begin
-    (x ⁻¹) • x ≈⟨ comm (x ⁻¹) x ⟩
-    x • (x ⁻¹) ≈⟨ invʳ x ⟩
+    (x ⁻¹) ∙ x ≈⟨ comm (x ⁻¹) x ⟩
+    x ∙ (x ⁻¹) ≈⟨ invʳ x ⟩
     e          ∎
 
-  comm+invʳ⇒inv : RightInverse e _⁻¹ _•_ → Inverse e _⁻¹ _•_
+  comm+invʳ⇒inv : RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
   comm+invʳ⇒inv invʳ = comm+invʳ⇒invˡ invʳ , invʳ
 
-module _ {_•_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _•_) where
+module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _∙_) where
 
-  assoc+id+invʳ⇒invˡ-unique : Associative _•_ →
-                              Identity e _•_ → RightInverse e _⁻¹ _•_ →
-                              ∀ x y → (x • y) ≈ e → x ≈ (y ⁻¹)
+  assoc+id+invʳ⇒invˡ-unique : Associative _∙_ →
+                              Identity e _∙_ → RightInverse e _⁻¹ _∙_ →
+                              ∀ x y → (x ∙ y) ≈ e → x ≈ (y ⁻¹)
   assoc+id+invʳ⇒invˡ-unique assoc (idˡ , idʳ) invʳ x y eq = begin
     x                ≈⟨ sym (idʳ x) ⟩
-    x • e            ≈⟨ cong refl (sym (invʳ y)) ⟩
-    x • (y • (y ⁻¹)) ≈⟨ sym (assoc x y (y ⁻¹)) ⟩
-    (x • y) • (y ⁻¹) ≈⟨ cong eq refl ⟩
-    e • (y ⁻¹)       ≈⟨ idˡ (y ⁻¹) ⟩
+    x ∙ e            ≈⟨ cong refl (sym (invʳ y)) ⟩
+    x ∙ (y ∙ (y ⁻¹)) ≈⟨ sym (assoc x y (y ⁻¹)) ⟩
+    (x ∙ y) ∙ (y ⁻¹) ≈⟨ cong eq refl ⟩
+    e ∙ (y ⁻¹)       ≈⟨ idˡ (y ⁻¹) ⟩
     y ⁻¹             ∎
 
-  assoc+id+invˡ⇒invʳ-unique : Associative _•_ →
-                              Identity e _•_ → LeftInverse e _⁻¹ _•_ →
-                              ∀ x y → (x • y) ≈ e → y ≈ (x ⁻¹)
+  assoc+id+invˡ⇒invʳ-unique : Associative _∙_ →
+                              Identity e _∙_ → LeftInverse e _⁻¹ _∙_ →
+                              ∀ x y → (x ∙ y) ≈ e → y ≈ (x ⁻¹)
   assoc+id+invˡ⇒invʳ-unique assoc (idˡ , idʳ) invˡ x y eq = begin
     y                ≈⟨ sym (idˡ y) ⟩
-    e • y            ≈⟨ cong (sym (invˡ x)) refl ⟩
-    ((x ⁻¹) • x) • y ≈⟨ assoc (x ⁻¹) x y ⟩
-    (x ⁻¹) • (x • y) ≈⟨ cong refl eq ⟩
-    (x ⁻¹) • e       ≈⟨ idʳ (x ⁻¹) ⟩
+    e ∙ y            ≈⟨ cong (sym (invˡ x)) refl ⟩
+    ((x ⁻¹) ∙ x) ∙ y ≈⟨ assoc (x ⁻¹) x y ⟩
+    (x ⁻¹) ∙ (x ∙ y) ≈⟨ cong refl eq ⟩
+    (x ⁻¹) ∙ e       ≈⟨ idʳ (x ⁻¹) ⟩
     x ⁻¹             ∎
 
 ------------------------------------------------------------------------
 -- Bisemigroup-like structures
 
-module _ {_•_ _◦_ : Op₂ A}
+module _ {_∙_ _◦_ : Op₂ A}
          (◦-cong : Congruent₂ _◦_)
-         (•-comm : Commutative _•_)
+         (∙-comm : Commutative _∙_)
          where
 
-  comm+distrˡ⇒distrʳ :  _•_ DistributesOverˡ _◦_ → _•_ DistributesOverʳ _◦_
+  comm+distrˡ⇒distrʳ :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_
   comm+distrˡ⇒distrʳ distrˡ x y z = begin
-    (y ◦ z) • x       ≈⟨ •-comm (y ◦ z) x ⟩
-    x • (y ◦ z)       ≈⟨ distrˡ x y z ⟩
-    (x • y) ◦ (x • z) ≈⟨ ◦-cong (•-comm x y) (•-comm x z) ⟩
-    (y • x) ◦ (z • x) ∎
+    (y ◦ z) ∙ x       ≈⟨ ∙-comm (y ◦ z) x ⟩
+    x ∙ (y ◦ z)       ≈⟨ distrˡ x y z ⟩
+    (x ∙ y) ◦ (x ∙ z) ≈⟨ ◦-cong (∙-comm x y) (∙-comm x z) ⟩
+    (y ∙ x) ◦ (z ∙ x) ∎
 
-  comm+distrʳ⇒distrˡ : _•_ DistributesOverʳ _◦_ → _•_ DistributesOverˡ _◦_
+  comm+distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_
   comm+distrʳ⇒distrˡ distrˡ x y z = begin
-    x • (y ◦ z)       ≈⟨ •-comm x (y ◦ z) ⟩
-    (y ◦ z) • x       ≈⟨ distrˡ x y z ⟩
-    (y • x) ◦ (z • x) ≈⟨ ◦-cong (•-comm y x) (•-comm z x) ⟩
-    (x • y) ◦ (x • z) ∎
+    x ∙ (y ◦ z)       ≈⟨ ∙-comm x (y ◦ z) ⟩
+    (y ◦ z) ∙ x       ≈⟨ distrˡ x y z ⟩
+    (y ∙ x) ◦ (z ∙ x) ≈⟨ ◦-cong (∙-comm y x) (∙-comm z x) ⟩
+    (x ∙ y) ◦ (x ∙ z) ∎
 
-  comm+distrˡ⇒distr :  _•_ DistributesOverˡ _◦_ → _•_ DistributesOver _◦_
+  comm+distrˡ⇒distr :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
   comm+distrˡ⇒distr distrˡ = distrˡ , comm+distrˡ⇒distrʳ distrˡ
 
-  comm+distrʳ⇒distr :  _•_ DistributesOverʳ _◦_ → _•_ DistributesOver _◦_
+  comm+distrʳ⇒distr :  _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
   comm+distrʳ⇒distr distrʳ = comm+distrʳ⇒distrˡ distrʳ , distrʳ
 
-  comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y • z)) ≈ ((x ◦ y) • (x ◦ z)))
+  comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y ∙ z)) ≈ ((x ◦ y) ∙ (x ◦ z)))
   comm⇒sym[distribˡ] x {y} {z} prf = begin
-    x ◦ (z • y)       ≈⟨ ◦-cong refl (•-comm z y) ⟩
-    x ◦ (y • z)       ≈⟨ prf ⟩
-    (x ◦ y) • (x ◦ z) ≈⟨ •-comm (x ◦ y) (x ◦ z) ⟩
-    (x ◦ z) • (x ◦ y) ∎
+    x ◦ (z ∙ y)       ≈⟨ ◦-cong refl (∙-comm z y) ⟩
+    x ◦ (y ∙ z)       ≈⟨ prf ⟩
+    (x ◦ y) ∙ (x ◦ z) ≈⟨ ∙-comm (x ◦ y) (x ◦ z) ⟩
+    (x ◦ z) ∙ (x ◦ y) ∎
 
 
-module _ {_•_ _◦_ : Op₂ A}
-         (•-cong  : Congruent₂ _•_)
-         (•-assoc : Associative _•_)
+module _ {_∙_ _◦_ : Op₂ A}
+         (∙-cong  : Congruent₂ _∙_)
+         (∙-assoc : Associative _∙_)
          (◦-comm  : Commutative _◦_)
          where
 
-  distrib+absorbs⇒distribˡ : _•_ Absorbs _◦_ →
-                             _◦_ Absorbs _•_ →
-                             _◦_ DistributesOver _•_ →
-                             _•_ DistributesOverˡ _◦_
-  distrib+absorbs⇒distribˡ •-absorbs-◦ ◦-absorbs-• (◦-distribˡ-• , ◦-distribʳ-•) x y z = begin
-    x • (y ◦ z)                    ≈⟨ •-cong (•-absorbs-◦ _ _) refl ⟨
-    (x • (x ◦ y)) • (y ◦ z)        ≈⟨  •-cong (•-cong refl (◦-comm _ _)) refl ⟩
-    (x • (y ◦ x)) • (y ◦ z)        ≈⟨  •-assoc _ _ _ ⟩
-    x • ((y ◦ x) • (y ◦ z))        ≈⟨ •-cong refl (◦-distribˡ-• _ _ _) ⟨
-    x • (y ◦ (x • z))              ≈⟨ •-cong (◦-absorbs-• _ _) refl ⟨
-    (x ◦ (x • z)) • (y ◦ (x • z))  ≈⟨ ◦-distribʳ-• _ _ _ ⟨
-    (x • y) ◦ (x • z)              ∎
+  distrib+absorbs⇒distribˡ : _∙_ Absorbs _◦_ →
+                             _◦_ Absorbs _∙_ →
+                             _◦_ DistributesOver _∙_ →
+                             _∙_ DistributesOverˡ _◦_
+  distrib+absorbs⇒distribˡ ∙-absorbs-◦ ◦-absorbs-∙ (◦-distribˡ-∙ , ◦-distribʳ-∙) x y z = begin
+    x ∙ (y ◦ z)                    ≈⟨ ∙-cong (∙-absorbs-◦ _ _) refl ⟨
+    (x ∙ (x ◦ y)) ∙ (y ◦ z)        ≈⟨  ∙-cong (∙-cong refl (◦-comm _ _)) refl ⟩
+    (x ∙ (y ◦ x)) ∙ (y ◦ z)        ≈⟨  ∙-assoc _ _ _ ⟩
+    x ∙ ((y ◦ x) ∙ (y ◦ z))        ≈⟨ ∙-cong refl (◦-distribˡ-∙ _ _ _) ⟨
+    x ∙ (y ◦ (x ∙ z))              ≈⟨ ∙-cong (◦-absorbs-∙ _ _) refl ⟨
+    (x ◦ (x ∙ z)) ∙ (y ◦ (x ∙ z))  ≈⟨ ◦-distribʳ-∙ _ _ _ ⟨
+    (x ∙ y) ◦ (x ∙ z)              ∎
 
 ------------------------------------------------------------------------
 -- Ring-like structures

From 3941b3965bf591424739d6ce7520f1993473419e Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 1 Nov 2023 20:07:23 +0900
Subject: [PATCH 181/316] Move `T?` to `Relation.Nullary.Decidable.Core`
 (#2189)

* Move `T?` to `Relation.Nullary.Decidable.Core`

* Var name fix

* Some refactoring

* Fix failing tests and address remaining comments

* Fix style-guide
---
 CHANGELOG.md                                  |  9 ++++
 README/Design/Hierarchies.agda                |  4 +-
 notes/style-guide.md                          |  2 +-
 src/Data/Bool.agda                            |  3 --
 src/Data/Bool/Base.agda                       | 18 ++++----
 src/Data/Bool/Properties.agda                 | 43 ++++++++++---------
 .../Relation/Binary/BagAndSetEquality.agda    |  2 +-
 src/Effect/Applicative.agda                   |  2 +-
 src/Function/Bundles.agda                     |  2 +-
 src/Relation/Nullary.agda                     | 17 ++------
 src/Relation/Nullary/Decidable/Core.agda      |  5 ++-
 src/Relation/Nullary/Reflects.agda            | 17 +++++---
 12 files changed, 67 insertions(+), 57 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ef730a4546..479eb395bc 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -838,6 +838,9 @@ Non-backwards compatible changes
 
   4. The modules `Relation.Nullary.(Product/Sum/Implication)` have been deprecated
          and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`.
+		 
+  5. The proof `T?` has been moved from `Data.Bool.Properties` to `Relation.Nullary.Decidable.Core`
+	 (but is still re-exported by the former).
 
   as well as the following breaking changes:
 
@@ -3563,6 +3566,12 @@ Additions to existing modules
   poset          : Set a → Poset _ _ _
   ```
 
+* Added new proof in `Relation.Nullary.Reflects`:
+  ```agda
+  T-reflects : Reflects (T b) b
+  T-reflects-elim : Reflects (T a) b → b ≡ a
+  ```
+
 * Added new operations in `System.Exit`:
   ```
   isSuccess : ExitCode → Bool
diff --git a/README/Design/Hierarchies.agda b/README/Design/Hierarchies.agda
index bc9220ba23..71ca5eed44 100644
--- a/README/Design/Hierarchies.agda
+++ b/README/Design/Hierarchies.agda
@@ -265,7 +265,7 @@ record Semigroup : Set (suc (a ⊔ ℓ)) where
 --       IsA                     A
 --    /  ||   \               /  ||  \
 -- IsB   IsC   IsD          B    C    D
- 
+
 -- The procedure for re-exports in the bundles is as follows:
 
 -- 1. `open IsA isA public using (IsC, M)` where `M` is everything
@@ -280,7 +280,7 @@ record Semigroup : Set (suc (a ⊔ ℓ)) where
 
 -- 5. `open B b public using (O)` where `O` is everything exported
 --    by `B` but not exported by `IsA`.
- 
+
 -- 6. Construct `d : D` via the `isC` obtained in step 1.
 
 -- 7. `open D d public using (P)` where `P` is everything exported
diff --git a/notes/style-guide.md b/notes/style-guide.md
index 95ef7ba2d0..a3c918e6a4 100644
--- a/notes/style-guide.md
+++ b/notes/style-guide.md
@@ -402,7 +402,7 @@ word within a compound word is capitalized except for the first word.
 
 * Rational variables are named `p`, `q`, `r`, ... (default `p`)
 
-* All other variables tend to be named `x`, `y`, `z`.
+* All other variables should be named `x`, `y`, `z`.
 
 * Collections of elements are usually indicated by appending an `s`
   (e.g. if you are naming your variables `x` and `y` then lists
diff --git a/src/Data/Bool.agda b/src/Data/Bool.agda
index 1270125254..55ed146e2c 100644
--- a/src/Data/Bool.agda
+++ b/src/Data/Bool.agda
@@ -8,9 +8,6 @@
 
 module Data.Bool where
 
-open import Relation.Nullary
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
-
 ------------------------------------------------------------------------
 -- The boolean type and some operations
 
diff --git a/src/Data/Bool/Base.agda b/src/Data/Bool/Base.agda
index bc2e67644b..ea99ee880d 100644
--- a/src/Data/Bool/Base.agda
+++ b/src/Data/Bool/Base.agda
@@ -57,17 +57,19 @@ true  xor b = not b
 false xor b = b
 
 ------------------------------------------------------------------------
--- Other operations
-
-infix  0 if_then_else_
-
-if_then_else_ : Bool → A → A → A
-if true  then t else f = t
-if false then t else f = f
+-- Conversion to Set
 
 -- A function mapping true to an inhabited type and false to an empty
 -- type.
-
 T : Bool → Set
 T true  = ⊤
 T false = ⊥
+
+------------------------------------------------------------------------
+-- Other operations
+
+infix 0 if_then_else_
+
+if_then_else_ : Bool → A → A → A
+if true  then t else f = t
+if false then t else f = f
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index e911390366..bb8cda9a0e 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -28,8 +28,7 @@ open import Relation.Binary.Definitions
   using (Decidable; Reflexive; Transitive; Antisymmetric; Minimum; Maximum; Total; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri<; tri>; _Respects₂_)
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Binary.PropositionalEquality.Properties
-open import Relation.Nullary.Reflects using (ofʸ; ofⁿ)
-open import Relation.Nullary.Decidable.Core using (True; does; proof; yes; no)
+open import Relation.Nullary.Decidable.Core using (True; yes; no; fromWitness)
 import Relation.Unary as U
 
 open import Algebra.Definitions {A = Bool} _≡_
@@ -726,15 +725,17 @@ xor-∧-commutativeRing = ⊕-∧-commutativeRing
   open XorRing _xor_ xor-is-ok
 
 ------------------------------------------------------------------------
--- Miscellaneous other properties
+-- Properties of if_then_else_
 
-⇔→≡ : {x y z : Bool} → x ≡ z ⇔ y ≡ z → x ≡ y
-⇔→≡ {true } {true }         hyp = refl
-⇔→≡ {true } {false} {true } hyp = sym (Equivalence.to hyp refl)
-⇔→≡ {true } {false} {false} hyp = Equivalence.from hyp refl
-⇔→≡ {false} {true } {true } hyp = Equivalence.from hyp refl
-⇔→≡ {false} {true } {false} hyp = sym (Equivalence.to hyp refl)
-⇔→≡ {false} {false}         hyp = refl
+if-float : ∀ (f : A → B) b {x y} →
+           f (if b then x else y) ≡ (if b then f x else f y)
+if-float _ true  = refl
+if-float _ false = refl
+
+------------------------------------------------------------------------
+-- Properties of T
+
+open Relation.Nullary.Decidable.Core public using (T?)
 
 T-≡ : ∀ {x} → T x ⇔ x ≡ true
 T-≡ {false} = mk⇔ (λ ())       (λ ())
@@ -757,18 +758,20 @@ T-∨ {false} {false} = mk⇔ inj₁ [ id , id ]
 T-irrelevant : U.Irrelevant T
 T-irrelevant {true}  _  _  = refl
 
-T? : U.Decidable T
-does  (T? b) = b
-proof (T? true ) = ofʸ _
-proof (T? false) = ofⁿ λ()
-
 T?-diag : ∀ b → T b → True (T? b)
-T?-diag true  _ = _
+T?-diag b = fromWitness
+
+------------------------------------------------------------------------
+-- Miscellaneous other properties
+
+⇔→≡ : {x y z : Bool} → x ≡ z ⇔ y ≡ z → x ≡ y
+⇔→≡ {true } {true }         hyp = refl
+⇔→≡ {true } {false} {true } hyp = sym (Equivalence.to hyp refl)
+⇔→≡ {true } {false} {false} hyp = Equivalence.from hyp refl
+⇔→≡ {false} {true } {true } hyp = Equivalence.from hyp refl
+⇔→≡ {false} {true } {false} hyp = sym (Equivalence.to hyp refl)
+⇔→≡ {false} {false}         hyp = refl
 
-if-float : ∀ (f : A → B) b {x y} →
-           f (if b then x else y) ≡ (if b then f x else f y)
-if-float _ true  = refl
-if-float _ false = refl
 
 ------------------------------------------------------------------------
 -- DEPRECATED NAMES
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 31ab6941ac..17af3d8955 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -44,7 +44,7 @@ import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Binary.PropositionalEquality as P
   using (_≡_; _≢_; _≗_; refl)
 open import Relation.Binary.Reasoning.Syntax
-open import Relation.Nullary
+open import Relation.Nullary.Negation using (¬_)
 open import Data.List.Membership.Propositional.Properties
 
 private
diff --git a/src/Effect/Applicative.agda b/src/Effect/Applicative.agda
index a2263c6e0a..95e2debda8 100644
--- a/src/Effect/Applicative.agda
+++ b/src/Effect/Applicative.agda
@@ -56,7 +56,7 @@ record RawApplicative (F : Set f → Set g) : Set (suc f ⊔ g) where
   -- Haskell-style alternative name for pure
   return : A → F A
   return = pure
-  
+
   -- backwards compatibility: unicode variants
   _⊛_ : F (A → B) → F A → F B
   _⊛_ = _<*>_
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index 9e18c30fe5..98aac7a48c 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -380,7 +380,7 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
   -- For further background on (split) surjections, one may consult any
   -- general mathematical references which work without the principle
   -- of choice. For example:
-  -- 
+  --
   --   https://ncatlab.org/nlab/show/split+epimorphism.
   --
   -- The connection to set-theoretic notions with the same names is
diff --git a/src/Relation/Nullary.agda b/src/Relation/Nullary.agda
index b8a4c30470..0b0f83f77b 100644
--- a/src/Relation/Nullary.agda
+++ b/src/Relation/Nullary.agda
@@ -15,20 +15,9 @@ open import Agda.Builtin.Equality
 ------------------------------------------------------------------------
 -- Re-exports
 
-open import Relation.Nullary.Negation.Core public using
-  ( ¬_; _¬-⊎_
-  ; contradiction; contradiction₂; contraposition
-  )
-
-open import Relation.Nullary.Reflects public using
-  ( Reflects; ofʸ; ofⁿ
-  ; _×-reflects_; _⊎-reflects_; _→-reflects_
-  )
-
-open import Relation.Nullary.Decidable.Core public using
-  ( Dec; does; proof; yes; no; _because_; recompute
-  ; ¬?; _×-dec_; _⊎-dec_; _→-dec_
-  )
+open import Relation.Nullary.Negation.Core public
+open import Relation.Nullary.Reflects public
+open import Relation.Nullary.Decidable.Core public
 
 ------------------------------------------------------------------------
 -- Irrelevant types
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index ff73c6af20..f77613ea65 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -12,7 +12,7 @@
 module Relation.Nullary.Decidable.Core where
 
 open import Level using (Level; Lift)
-open import Data.Bool.Base using (Bool; false; true; not; T; _∧_; _∨_)
+open import Data.Bool.Base using (Bool; T; false; true; not; _∧_; _∨_)
 open import Data.Unit.Base using (⊤)
 open import Data.Empty using (⊥)
 open import Data.Empty.Irrelevant using (⊥-elim)
@@ -79,6 +79,9 @@ recompute (no ¬a) a = ⊥-elim (¬a a)
 infixr 1 _⊎-dec_
 infixr 2 _×-dec_ _→-dec_
 
+T? : ∀ x → Dec (T x)
+T? x = x because T-reflects x
+
 ¬? : Dec A → Dec (¬ A)
 does  (¬? a?) = not (does a?)
 proof (¬? a?) = ¬-reflects (proof a?)
diff --git a/src/Relation/Nullary/Reflects.agda b/src/Relation/Nullary/Reflects.agda
index 4f97bd8709..3b301fdbce 100644
--- a/src/Relation/Nullary/Reflects.agda
+++ b/src/Relation/Nullary/Reflects.agda
@@ -11,11 +11,12 @@ module Relation.Nullary.Reflects where
 open import Agda.Builtin.Equality
 
 open import Data.Bool.Base
+open import Data.Unit.Base using (⊤)
 open import Data.Empty
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Level using (Level)
-open import Function.Base using (_$_; _∘_; const)
+open import Function.Base using (_$_; _∘_; const; id)
 
 open import Relation.Nullary.Negation.Core
 
@@ -54,28 +55,31 @@ invert (ofⁿ ¬a) = ¬a
 ------------------------------------------------------------------------
 -- Interaction with negation, product, sums etc.
 
+infixr 1 _⊎-reflects_
+infixr 2 _×-reflects_ _→-reflects_
+
+T-reflects : ∀ b → Reflects (T b) b
+T-reflects true  = of _
+T-reflects false = of id
+
 -- If we can decide A, then we can decide its negation.
 ¬-reflects : ∀ {b} → Reflects A b → Reflects (¬ A) (not b)
 ¬-reflects (ofʸ  a) = ofⁿ (_$ a)
 ¬-reflects (ofⁿ ¬a) = ofʸ ¬a
 
 -- If we can decide A and Q then we can decide their product
-infixr 2 _×-reflects_
 _×-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
                Reflects (A × B) (a ∧ b)
 ofʸ  a ×-reflects ofʸ  b = ofʸ (a , b)
 ofʸ  a ×-reflects ofⁿ ¬b = ofⁿ (¬b ∘ proj₂)
 ofⁿ ¬a ×-reflects _      = ofⁿ (¬a ∘ proj₁)
 
-
-infixr 1 _⊎-reflects_
 _⊎-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
                Reflects (A ⊎ B) (a ∨ b)
 ofʸ  a ⊎-reflects      _ = ofʸ (inj₁ a)
 ofⁿ ¬a ⊎-reflects ofʸ  b = ofʸ (inj₂ b)
 ofⁿ ¬a ⊎-reflects ofⁿ ¬b = ofⁿ (¬a ¬-⊎ ¬b)
 
-infixr 2 _→-reflects_
 _→-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
                 Reflects (A → B) (not a ∨ b)
 ofʸ  a →-reflects ofʸ  b = ofʸ (const b)
@@ -95,3 +99,6 @@ det (ofʸ  a) (ofʸ  _) = refl
 det (ofʸ  a) (ofⁿ ¬a) = contradiction a ¬a
 det (ofⁿ ¬a) (ofʸ  a) = contradiction a ¬a
 det (ofⁿ ¬a) (ofⁿ  _) = refl
+
+T-reflects-elim : ∀ {a b} → Reflects (T a) b → b ≡ a
+T-reflects-elim {a} r = det r (T-reflects a)

From e7d864224aae8d13334aa31efbeadccb3928a554 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 3 Nov 2023 12:43:34 +0900
Subject: [PATCH 182/316] Make decidable versions of sublist functions the
 default (#2186)

* Make decdable versions of sublist functions the default

* Remove imports Bool.Properties

* Address comments
---
 src/Data/List/Base.agda                       | 209 ++++++++++--------
 .../Unary/Unique/DecSetoid/Properties.agda    |   4 +-
 src/Data/String/Base.agda                     |  18 +-
 src/Data/Vec/Base.agda                        |  16 +-
 4 files changed, 133 insertions(+), 114 deletions(-)

diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 58b51e4893..16546110b6 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -23,11 +23,11 @@ open import Data.These.Base as These using (These; this; that; these)
 open import Function.Base
   using (id; _∘_ ; _∘′_; _∘₂_; _$_; const; flip)
 open import Level using (Level)
-open import Relation.Nullary.Decidable.Core using (does; ¬?)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Binary.Core using (Rel)
 import Relation.Binary.Definitions as B
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Nullary.Decidable.Core using (T?; does; ¬?)
 
 private
   variable
@@ -346,141 +346,160 @@ removeAt : (xs : List A) → Fin (length xs) → List A
 removeAt (x ∷ xs) zero     = xs
 removeAt (x ∷ xs) (suc i)  = x ∷ removeAt xs i
 
--- The following are functions which split a list up using boolean
--- predicates. However, in practice they are difficult to use and
--- prove properties about, and are mainly provided for advanced use
--- cases where one wants to minimise dependencies. In most cases
--- you probably want to use the versions defined below based on
--- decidable predicates. e.g. use `takeWhile (_≤? 10) xs`
--- instead of `takeWhileᵇ (_≤ᵇ 10) xs`
+------------------------------------------------------------------------
+-- Operations for filtering lists
+
+-- The following are a variety of functions that can be used to
+-- construct sublists using a predicate.
+--
+-- Each function has two forms. The first main variant uses a
+-- proof-relevant decidable predicate, while the second variant uses
+-- a irrelevant boolean predicate and are suffixed with a `ᵇ` character,
+-- typed as \^b.
+--
+-- The decidable versions have several advantages: 1) easier to prove
+-- properties, 2) better meta-variable inference and 3) most of the rest
+-- of the library is set-up to work with decidable predicates. However,
+-- in rare cases the boolean versions can be useful, mainly when one
+-- wants to minimise dependencies.
+--
+-- In summary, in most cases you probably want to use the decidable
+-- versions over the boolean versions, e.g. use `takeWhile (_≤? 10) xs`
+-- rather than `takeWhileᵇ (_≤ᵇ 10) xs`.
+
+takeWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
+takeWhile P? []       = []
+takeWhile P? (x ∷ xs) with does (P? x)
+... | true  = x ∷ takeWhile P? xs
+... | false = []
 
 takeWhileᵇ : (A → Bool) → List A → List A
-takeWhileᵇ p []       = []
-takeWhileᵇ p (x ∷ xs) = if p x then x ∷ takeWhileᵇ p xs else []
+takeWhileᵇ p = takeWhile (T? ∘ p)
+
+dropWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
+dropWhile P? []       = []
+dropWhile P? (x ∷ xs) with does (P? x)
+... | true  = dropWhile P? xs
+... | false = x ∷ xs
 
 dropWhileᵇ : (A → Bool) → List A → List A
-dropWhileᵇ p []       = []
-dropWhileᵇ p (x ∷ xs) = if p x then dropWhileᵇ p xs else x ∷ xs
+dropWhileᵇ p = dropWhile (T? ∘ p)
+
+filter : ∀ {P : Pred A p} → Decidable P → List A → List A
+filter P? [] = []
+filter P? (x ∷ xs) with does (P? x)
+... | false = filter P? xs
+... | true  = x ∷ filter P? xs
 
 filterᵇ : (A → Bool) → List A → List A
-filterᵇ p []       = []
-filterᵇ p (x ∷ xs) = if p x then x ∷ filterᵇ p xs else filterᵇ p xs
+filterᵇ p = filter (T? ∘ p)
+
+partition : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
+partition P? []       = ([] , [])
+partition P? (x ∷ xs) with does (P? x) | partition P? xs
+... | true  | (ys , zs) = (x ∷ ys , zs)
+... | false | (ys , zs) = (ys , x ∷ zs)
 
 partitionᵇ : (A → Bool) → List A → List A × List A
-partitionᵇ p []       = ([] , [])
-partitionᵇ p (x ∷ xs) = (if p x then Prod.map₁ else Prod.map₂′) (x ∷_) (partitionᵇ p xs)
+partitionᵇ p = partition (T? ∘ p)
+
+span : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
+span P? []       = ([] , [])
+span P? ys@(x ∷ xs) with does (P? x)
+... | true  = Prod.map (x ∷_) id (span P? xs)
+... | false = ([] , ys)
+
 
 spanᵇ : (A → Bool) → List A → List A × List A
-spanᵇ p []       = ([] , [])
-spanᵇ p (x ∷ xs) = if p x
-  then Prod.map₁ (x ∷_) (spanᵇ p xs)
-  else ([] , x ∷ xs)
+spanᵇ p = span (T? ∘ p)
+
+break : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
+break P? = span (¬? ∘ P?)
 
 breakᵇ : (A → Bool) → List A → List A × List A
-breakᵇ p = spanᵇ (not ∘ p)
+breakᵇ p = break (T? ∘ p)
+
+-- The predicate `P` represents the notion of newline character for the
+-- type `A`. It is used to split the input list into a list of lines.
+-- Some lines may be empty if the input contains at least two
+-- consecutive newline characters.
+linesBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A)
+linesBy {A = A} P? = go nothing where
 
-linesByᵇ : (A → Bool) → List A → List (List A)
-linesByᵇ {A = A} p = go nothing
-  where
   go : Maybe (List A) → List A → List (List A)
   go acc []       = maybe′ ([_] ∘′ reverse) [] acc
-  go acc (c ∷ cs) with p c
+  go acc (c ∷ cs) with does (P? c)
   ... | true  = reverse (Maybe.fromMaybe [] acc) ∷ go nothing cs
   ... | false = go (just (c ∷ Maybe.fromMaybe [] acc)) cs
 
-wordsByᵇ : (A → Bool) → List A → List (List A)
-wordsByᵇ {A = A} p = go []
-  where
+linesByᵇ : (A → Bool) → List A → List (List A)
+linesByᵇ p = linesBy (T? ∘ p)
+
+-- The predicate `P` represents the notion of space character for the
+-- type `A`. It is used to split the input list into a list of words.
+-- All the words are non empty and the output does not contain any space
+-- characters.
+wordsBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A)
+wordsBy {A = A} P? = go [] where
+
   cons : List A → List (List A) → List (List A)
   cons [] ass = ass
   cons as ass = reverse as ∷ ass
 
   go : List A → List A → List (List A)
   go acc []       = cons acc []
-  go acc (c ∷ cs) with p c
+  go acc (c ∷ cs) with does (P? c)
   ... | true  = cons acc (go [] cs)
   ... | false = go (c ∷ acc) cs
 
+wordsByᵇ : (A → Bool) → List A → List (List A)
+wordsByᵇ p = wordsBy (T? ∘ p)
+
+derun : ∀ {R : Rel A p} → B.Decidable R → List A → List A
+derun R? [] = []
+derun R? (x ∷ []) = x ∷ []
+derun R? (x ∷ xs@(y ∷ _)) with does (R? x y) | derun R? xs
+... | true  | ys = ys
+... | false | ys = x ∷ ys
+
 derunᵇ : (A → A → Bool) → List A → List A
-derunᵇ r []           = []
-derunᵇ r (x ∷ [])     = x ∷ []
-derunᵇ r (x ∷ y ∷ xs) = if r x y
-  then derunᵇ r (y ∷ xs)
-  else x ∷ derunᵇ r (y ∷ xs)
+derunᵇ r = derun (T? ∘₂ r)
+
+deduplicate : ∀ {R : Rel A p} → B.Decidable R → List A → List A
+deduplicate R? [] = []
+deduplicate R? (x ∷ xs) = x ∷ filter (¬? ∘ R? x) (deduplicate R? xs)
 
 deduplicateᵇ : (A → A → Bool) → List A → List A
-deduplicateᵇ r [] = []
-deduplicateᵇ r (x ∷ xs) = x ∷ filterᵇ (not ∘ r x) (deduplicateᵇ r xs)
+deduplicateᵇ r = deduplicate (T? ∘₂ r)
 
 -- Finds the first element satisfying the boolean predicate
+find : ∀ {P : Pred A p} → Decidable P → List A → Maybe A
+find P? []       = nothing
+find P? (x ∷ xs) = if does (P? x) then just x else find P? xs
+
 findᵇ : (A → Bool) → List A → Maybe A
-findᵇ p []       = nothing
-findᵇ p (x ∷ xs) = if p x then just x else findᵇ p xs
+findᵇ p = find (T? ∘ p)
 
 -- Finds the index of the first element satisfying the boolean predicate
-findIndexᵇ : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)
-findIndexᵇ p []       = nothing
-findIndexᵇ p (x ∷ xs) = if p x
+findIndex : ∀ {P : Pred A p} → Decidable P → (xs : List A) → Maybe $ Fin (length xs)
+findIndex P? [] = nothing
+findIndex P? (x ∷ xs) = if does (P? x)
   then just zero
-  else Maybe.map suc (findIndexᵇ p xs)
+  else Maybe.map suc (findIndex P? xs)
+
+findIndexᵇ : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)
+findIndexᵇ p = findIndex (T? ∘ p)
 
 -- Finds indices of all the elements satisfying the boolean predicate
-findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)
-findIndicesᵇ p []       = []
-findIndicesᵇ p (x ∷ xs) = if p x
+findIndices : ∀ {P : Pred A p} → Decidable P → (xs : List A) → List $ Fin (length xs)
+findIndices P? []       = []
+findIndices P? (x ∷ xs) = if does (P? x)
   then zero ∷ indices
   else indices
-    where indices = map suc (findIndicesᵇ p xs)
-
--- Equivalent functions that use a decidable predicate instead of a
--- boolean function.
-
-takeWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
-takeWhile P? = takeWhileᵇ (does ∘ P?)
-
-dropWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
-dropWhile P? = dropWhileᵇ (does ∘ P?)
+    where indices = map suc (findIndices P? xs)
 
-filter : ∀ {P : Pred A p} → Decidable P → List A → List A
-filter P? = filterᵇ (does ∘ P?)
-
-partition : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
-partition P? = partitionᵇ (does ∘ P?)
-
-span : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
-span P? = spanᵇ (does ∘ P?)
-
-break : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
-break P? = breakᵇ (does ∘ P?)
-
--- The predicate `P` represents the notion of newline character for the
--- type `A`. It is used to split the input list into a list of lines.
--- Some lines may be empty if the input contains at least two
--- consecutive newline characters.
-linesBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A)
-linesBy P? = linesByᵇ (does ∘ P?)
-
--- The predicate `P` represents the notion of space character for the
--- type `A`. It is used to split the input list into a list of words.
--- All the words are non empty and the output does not contain any space
--- characters.
-wordsBy : ∀ {P : Pred A p} → Decidable P → List A → List (List A)
-wordsBy P? = wordsByᵇ (does ∘ P?)
-
-derun : ∀ {R : Rel A p} → B.Decidable R → List A → List A
-derun R? = derunᵇ (does ∘₂ R?)
-
-deduplicate : ∀ {R : Rel A p} → B.Decidable R → List A → List A
-deduplicate R? = deduplicateᵇ (does ∘₂ R?)
-
-find : ∀ {P : Pred A p} → Decidable P → List A → Maybe A
-find P? = findᵇ (does ∘ P?)
-
-findIndex : ∀ {P : Pred A p} → Decidable P → (xs : List A) → Maybe $ Fin (length xs)
-findIndex P? = findIndexᵇ (does ∘ P?)
-
-findIndices : ∀ {P : Pred A p} → Decidable P → (xs : List A) → List $ Fin (length xs)
-findIndices P? = findIndicesᵇ (does ∘ P?)
+findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)
+findIndicesᵇ p = findIndices (T? ∘ p)
 
 ------------------------------------------------------------------------
 -- Actions on single elements
diff --git a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
index 9e97527ff8..73d42b7b8a 100644
--- a/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecSetoid/Properties.agda
@@ -12,7 +12,6 @@ open import Data.List.Relation.Unary.All.Properties using (all-filter)
 open import Data.List.Relation.Unary.Unique.Setoid.Properties
 open import Level
 open import Relation.Binary.Bundles using (DecSetoid)
-open import Relation.Nullary.Decidable using (¬?)
 
 module Data.List.Relation.Unary.Unique.DecSetoid.Properties where
 
@@ -30,5 +29,4 @@ module _ (DS : DecSetoid a ℓ) where
 
   deduplicate-! : ∀ xs → Unique (deduplicate _≟_ xs)
   deduplicate-! []       = []
-  deduplicate-! (x ∷ xs) = all-filter (λ y → ¬? (x ≟ y)) (deduplicate _≟_ xs)
-                         ∷ filter⁺ S  (λ y → ¬? (x ≟ y)) (deduplicate-! xs)
+  deduplicate-! (x ∷ xs) = all-filter _ (deduplicate _≟_ xs) ∷ filter⁺ S _ (deduplicate-! xs)
diff --git a/src/Data/String/Base.agda b/src/Data/String/Base.agda
index 1a56376f59..d583da042c 100644
--- a/src/Data/String/Base.agda
+++ b/src/Data/String/Base.agda
@@ -22,7 +22,7 @@ open import Level using (Level; 0ℓ)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Unary using (Pred; Decidable)
-open import Relation.Nullary.Decidable.Core using (does)
+open import Relation.Nullary.Decidable.Core using (does; T?)
 
 ------------------------------------------------------------------------
 -- From Agda.Builtin: type and renamed primitives
@@ -160,11 +160,11 @@ fromAlignment Right  = padLeft ' '
 ------------------------------------------------------------------------
 -- Splitting strings
 
-wordsByᵇ : (Char → Bool) → String → List String
-wordsByᵇ p = List.map fromList ∘ List.wordsByᵇ p ∘ toList
-
 wordsBy : ∀ {p} {P : Pred Char p} → Decidable P → String → List String
-wordsBy P? = wordsByᵇ (does ∘ P?)
+wordsBy P? = List.map fromList ∘ List.wordsBy P? ∘ toList
+
+wordsByᵇ : (Char → Bool) → String → List String
+wordsByᵇ p = wordsBy (T? ∘ p)
 
 words : String → List String
 words = wordsByᵇ Char.isSpace
@@ -173,11 +173,11 @@ words = wordsByᵇ Char.isSpace
 _ : words " abc  b   " ≡ "abc" ∷ "b" ∷ []
 _ = refl
 
-linesByᵇ : (Char → Bool) → String → List String
-linesByᵇ p = List.map fromList ∘ List.linesByᵇ p ∘ toList
-
 linesBy : ∀ {p} {P : Pred Char p} → Decidable P → String → List String
-linesBy P? = linesByᵇ (does ∘ P?)
+linesBy P? = List.map fromList ∘ List.linesBy P? ∘ toList
+
+linesByᵇ : (Char → Bool) → String → List String
+linesByᵇ p = linesBy (T? ∘ p)
 
 lines : String → List String
 lines = linesByᵇ ('\n' Char.≈ᵇ_)
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 1ecf91b9e7..10684dee30 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -8,7 +8,7 @@
 
 module Data.Vec.Base where
 
-open import Data.Bool.Base using (Bool; if_then_else_)
+open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Nat.Base
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List)
@@ -17,7 +17,7 @@ open import Data.These.Base as These using (These; this; that; these)
 open import Function.Base using (const; _∘′_; id; _∘_)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; trans; cong)
-open import Relation.Nullary.Decidable.Core using (does)
+open import Relation.Nullary.Decidable.Core using (does; T?)
 open import Relation.Unary using (Pred; Decidable)
 
 private
@@ -230,12 +230,14 @@ foldl₁ _⊕_ (x ∷ xs) = foldl _ _⊕_ x xs
 sum : Vec ℕ n → ℕ
 sum = foldr _ _+_ 0
 
-countᵇ : (A → Bool) → Vec A n → ℕ
-countᵇ p []       = zero
-countᵇ p (x ∷ xs) = if p x then suc (countᵇ p xs) else countᵇ p xs
-
 count : ∀ {P : Pred A p} → Decidable P → Vec A n → ℕ
-count P? = countᵇ (does ∘ P?)
+count P? []       = zero
+count P? (x ∷ xs) with does (P? x)
+... | true  = suc (count P? xs)
+... | false = count P? xs
+
+countᵇ : (A → Bool) → Vec A n → ℕ
+countᵇ p = count (T? ∘ p)
 
 ------------------------------------------------------------------------
 -- Operations for building vectors

From 947e300dd91e0de70929643fc1ce93af01a35a8e Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Sun, 5 Nov 2023 01:17:12 +0000
Subject: [PATCH 183/316] [ fix #1743 ] move README to `doc/` directory (#2184)

* [ admin ] dev playground

* [ fix #1743 ] move README to doc/ directory

* [ fix ] whitespace violations

* [ ci ] update to cope with new doc/ directory

* [ cleanup ] remove stale reference to travis.yml

* [ admin ] update README-related instructions

* [ admin ] fix build badges

* [ fix ] `make test` build

* Moved contents of notes/ to doc/

* Added CHANGELOG entry

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
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 rename {README => doc/README}/Data/List/Relation/Binary/Pointwise.agda (100%)
 rename {README => doc/README}/Data/List/Relation/Binary/Subset.agda (100%)
 rename {README => doc/README}/Data/List/Relation/Ternary/Interleaving.agda (100%)
 rename {README => doc/README}/Data/List/Relation/Unary/All.agda (100%)
 rename {README => doc/README}/Data/List/Relation/Unary/Any.agda (100%)
 rename {README => doc/README}/Data/Nat.agda (100%)
 rename {README => doc/README}/Data/Nat/Induction.agda (100%)
 rename {README => doc/README}/Data/Record.agda (100%)
 rename {README => doc/README}/Data/Tree/AVL.agda (100%)
 rename {README => doc/README}/Data/Tree/Binary.agda (100%)
 rename {README => doc/README}/Data/Tree/Rose.agda (100%)
 rename {README => doc/README}/Data/Trie/NonDependent.agda (100%)
 rename {README => doc/README}/Data/Vec/Relation/Binary/Equality/Cast.agda (100%)
 rename {README => doc/README}/Data/Wrap.agda (100%)
 rename {README => doc/README}/Debug/Trace.agda (100%)
 rename {README => doc/README}/Design/Decidability.agda (100%)
 rename {README => doc/README}/Design/Fixity.agda (100%)
 rename {README => doc/README}/Design/Hierarchies.agda (100%)
 rename {README => doc/README}/Foreign/Haskell.agda (100%)
 rename {README => doc/README}/Function/Reasoning.agda (100%)
 rename {README => doc/README}/IO.agda (100%)
 rename {README => doc/README}/Inspect.agda (100%)
 rename {README => doc/README}/Nary.agda (100%)
 rename {README => doc/README}/Reflection/External.agda (100%)
 rename {README => doc/README}/Relation/Binary/TypeClasses.agda (100%)
 rename {README => doc/README}/Tactic/Cong.agda (100%)
 rename {README => doc/README}/Tactic/MonoidSolver.agda (100%)
 rename {README => doc/README}/Tactic/RingSolver.agda (100%)
 rename {README => doc/README}/Text/Pretty.agda (100%)
 rename {README => doc/README}/Text/Printf.agda (100%)
 rename {README => doc/README}/Text/Regex.agda (100%)
 rename {README => doc/README}/Text/Tabular.agda (100%)
 rename {notes => doc}/installation-guide.md (100%)
 rename {notes => doc}/release-guide.txt (85%)
 create mode 100644 doc/standard-library-doc.agda-lib
 rename {notes => doc}/style-guide.md (100%)
 rename {notes => doc}/updating-experimental.txt (100%)

diff --git a/travis/agda-logo.svg b/.github/tooling/agda-logo.svg
similarity index 100%
rename from travis/agda-logo.svg
rename to .github/tooling/agda-logo.svg
diff --git a/travis/index.agda b/.github/tooling/index.agda
similarity index 100%
rename from travis/index.agda
rename to .github/tooling/index.agda
diff --git a/travis/index.sh b/.github/tooling/index.sh
similarity index 100%
rename from travis/index.sh
rename to .github/tooling/index.sh
diff --git a/travis/landing-bottom.html b/.github/tooling/landing-bottom.html
similarity index 100%
rename from travis/landing-bottom.html
rename to .github/tooling/landing-bottom.html
diff --git a/travis/landing-top.html b/.github/tooling/landing-top.html
similarity index 100%
rename from travis/landing-top.html
rename to .github/tooling/landing-top.html
diff --git a/travis/landing.sh b/.github/tooling/landing.sh
similarity index 100%
rename from travis/landing.sh
rename to .github/tooling/landing.sh
diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index 51c5f7b09c..9ac08748df 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -51,7 +51,7 @@ env:
   CABAL_VERSION: 3.6.2.0
   CABAL_INSTALL: cabal v1-install --ghc-options='-O1 +RTS -M6G -RTS'
   # CABAL_INSTALL: cabal install --overwrite-policy=always --ghc-options='-O1 +RTS -M6G -RTS'
-  AGDA: agda -Werror +RTS -M3.5G -H3.5G -A128M -RTS -i. -i src/
+  AGDA: agda -Werror +RTS -M3.5G -H3.5G -A128M -RTS -i. -isrc -idoc
 
 jobs:
   test-stdlib:
@@ -152,7 +152,7 @@ jobs:
       - name: Test stdlib
         run: |
           cabal run GenerateEverything
-          cp travis/* .
+          cp .github/tooling/* .
           ./index.sh
           ${{ env.AGDA }} --safe EverythingSafe.agda
           ${{ env.AGDA }} index.agda
@@ -178,7 +178,7 @@ jobs:
           rm -f '${{ env.AGDA_HTML_DIR }}'/*.css
           ${{ env.AGDA }} --html --html-dir ${{ env.AGDA_HTML_DIR }} index.agda
 
-          cp travis/* .
+          cp .github/tooling/* .
           ./landing.sh
 
       - name: Deploy HTML
diff --git a/CHANGELOG.md b/CHANGELOG.md
index 479eb395bc..562c089ddf 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1196,6 +1196,17 @@ Other major improvements
 
 * We have then moved raw bundles defined in `Data.X.Properties` to `Data.X.Base` for
   `X` = `Nat`/`Nat.Binary`/`Integer`/`Rational`/`Rational.Unnormalised`.
+  
+### Upgrades to `README` sub-library
+
+* The `README` sub-library has been moved to `doc/README` and a new `doc/standard-library-doc.agda-lib` has been added.
+
+* The first consequence is that `README` files now can be type-checked in Emacs
+  using an out-of-the-box standard Agda installation without altering the main
+  `standard-library.agda-lib` file.
+
+* The second is that the `README` files are now their own first-class library 
+  and can be imported like an other library.
 
 Deprecated modules
 ------------------
diff --git a/GNUmakefile b/GNUmakefile
index 685d1c990b..e744f066cc 100644
--- a/GNUmakefile
+++ b/GNUmakefile
@@ -9,11 +9,14 @@ AGDA=$(AGDA_EXEC) $(AGDA_OPTIONS) $(AGDA_RTS_OPTIONS)
 #   cabal install fix-whitespace
 
 test: Everything.agda check-whitespace
-	$(AGDA) -i. -isrc README.agda
+	cd doc && $(AGDA) README.agda
 
 testsuite:
 	$(MAKE) -C tests test AGDA="$(AGDA)" AGDA_EXEC="$(AGDA_EXEC)" only=$(only)
 
+fix-whitespace:
+	cabal exec -- fix-whitespace
+
 check-whitespace:
 	cabal exec -- fix-whitespace --check
 
@@ -21,11 +24,12 @@ setup: Everything.agda
 
 .PHONY: Everything.agda
 Everything.agda:
-	cabal run GenerateEverything
+	cabal run GenerateEverything -- --out-dir doc
 
 .PHONY: listings
 listings: Everything.agda
-	$(AGDA) -i. -isrc --html README.agda -v0
+	cd doc && $(AGDA) --html README.agda -v0
 
 clean :
 	find . -type f -name '*.agdai' -delete
+	rm -f Everything.agda EverythingSafe.agda
diff --git a/GenerateEverything.hs b/GenerateEverything.hs
index ee49679f3a..edea56a0f0 100644
--- a/GenerateEverything.hs
+++ b/GenerateEverything.hs
@@ -1,5 +1,6 @@
 {-# LANGUAGE PatternGuards #-}
 {-# LANGUAGE PatternSynonyms #-}
+{-# LANGUAGE RecordWildCards #-}
 
 import Control.Applicative
 import Control.Monad
@@ -299,18 +300,37 @@ checkFilePaths cat fps = forM_ fps $ \ fp -> do
   unless b $
     die $ fp ++ " is listed as " ++ cat ++ " but does not exist."
 
+data Options = Options
+  { includeDeprecated :: Bool
+  , outputDirectory :: FilePath
+  }
+
+initOptions :: Options
+initOptions = Options
+  { includeDeprecated = False
+  , outputDirectory = "."
+  }
+
+parseOptions :: [String] -> Options -> Maybe Options
+parseOptions [] opts = pure opts
+parseOptions ("--include-deprecated" : rest) opts
+  = parseOptions rest (opts { includeDeprecated = True })
+parseOptions ("--out-dir" : dir : rest) opts
+  = parseOptions rest (opts { outputDirectory = dir })
+parseOptions _ _ = Nothing
+
 ---------------------------------------------------------------------------
 -- Collecting all non-Core library files, analysing them and generating
--- 4 files:
+-- 2 files:
 -- Everything.agda                 all the modules
 -- EverythingSafe.agda             all the safe modules
 
+main :: IO ()
 main = do
   args <- getArgs
-  includeDeprecated <- case args of
-    [] -> return False
-    ["--include-deprecated"] -> return True
-    _  -> hPutStr stderr usage >> exitFailure
+  Options{..} <- case parseOptions args initOptions of
+    Just opts -> pure opts
+    Nothing -> hPutStr stderr usage >> exitFailure
 
   checkFilePaths "unsafe" unsafeModules
   checkFilePaths "using K" withKModules
@@ -325,14 +345,14 @@ main = do
 
   let mkModule str = "module " ++ str ++ " where"
 
-  writeFileUTF8 (allOutputFile ++ ".agda") $
+  writeFileUTF8 (outputDirectory ++ "/" ++ allOutputFile ++ ".agda") $
     unlines [ header
             , "{-# OPTIONS --rewriting --guardedness --sized-types #-}\n"
             , mkModule allOutputFile
             , format libraryfiles
             ]
 
-  writeFileUTF8 (safeOutputFile ++ ".agda") $
+  writeFileUTF8 (outputDirectory ++ "/" ++ safeOutputFile ++ ".agda") $
     unlines [ header
             , "{-# OPTIONS --safe --guardedness #-}\n"
             , mkModule safeOutputFile
@@ -353,6 +373,9 @@ usage = unlines
   , "The program generates documentation for the library by extracting"
   , "headers from library modules. The output is written to " ++ allOutputFile
   , "with the file " ++ headerFile ++ " inserted verbatim at the beginning."
+  , ""
+  , "If the option --out-dir is used then the output is placed in the"
+  , "subdirectory thus selected."
   ]
 
 
diff --git a/HACKING.md b/HACKING.md
index b2d619ec32..b7573188b5 100644
--- a/HACKING.md
+++ b/HACKING.md
@@ -244,23 +244,14 @@ you are never committing anything with a whitespace violation:
 Type-checking the README directory
 ----------------------------------
 
-* By default the README directory is not exported in the
-  `standard-library.agda-lib` file in order to avoid clashing with other people's
-  README files. This means that by default type-checking a file in the README
-  directory fails.
-
-* If you wish to type-check a README file, then you will need to change the line:
-  ```
-  include: src
-  ```
-  to
-  ```
-  include: src .
-  ```
-  in the `standard-library.agda-lib` file.
-
-* Please do not include this change in your pull request.
-
+* By default the README files are not exported in the
+  `standard-library.agda-lib` file in order to avoid
+  clashing with other people's README files.
+
+* If you wish to type-check a README file, then you will
+  need to change the present working directory to `doc/`
+  where an appropriate `standard-library-doc.agda-lib`
+  file is present.
 
 Continuous Integration (CI)
 ===========================
diff --git a/README.md b/README.md
index 16a2f10959..f9489e81f3 100644
--- a/README.md
+++ b/README.md
@@ -1,6 +1,6 @@
-![Travis (.org) branch](https://img.shields.io/travis/agda/agda-stdlib/master?label=master)
-![Travis (.org) branch](https://img.shields.io/travis/agda/agda-stdlib/experimental?label=experimental)
+[![Ubuntu build](https://github.com/agda/agda-stdlib/actions/workflows/ci-ubuntu.yml/badge.svg)](https://github.com/agda/agda-stdlib/actions/workflows/ci-ubuntu.yml)
 
+[![Ubuntu build](https://github.com/agda/agda-stdlib/actions/workflows/ci-ubuntu.yml/badge.svg?branch=experimental)](https://github.com/agda/agda-stdlib/actions/workflows/ci-ubuntu.yml)
 
 The Agda standard library
 =========================
diff --git a/dev/.gitignore b/dev/.gitignore
new file mode 100644
index 0000000000..f59ec20aab
--- /dev/null
+++ b/dev/.gitignore
@@ -0,0 +1 @@
+*
\ No newline at end of file
diff --git a/dev/README.md b/dev/README.md
new file mode 100644
index 0000000000..d97d344209
--- /dev/null
+++ b/dev/README.md
@@ -0,0 +1,4 @@
+# Development playground
+
+This directory allows you to develop modules against the current dev
+version of the stdlib as it currently sits in `src/`.
diff --git a/dev/experimental.agda-lib b/dev/experimental.agda-lib
new file mode 100644
index 0000000000..cb094977d8
--- /dev/null
+++ b/dev/experimental.agda-lib
@@ -0,0 +1,4 @@
+name: standard-library-dev
+include: . ../src
+flags:
+  --warning=noUnsupportedIndexedMatch
diff --git a/README.agda b/doc/README.agda
similarity index 100%
rename from README.agda
rename to doc/README.agda
diff --git a/README/Axiom.agda b/doc/README/Axiom.agda
similarity index 100%
rename from README/Axiom.agda
rename to doc/README/Axiom.agda
diff --git a/README/Case.agda b/doc/README/Case.agda
similarity index 100%
rename from README/Case.agda
rename to doc/README/Case.agda
diff --git a/README/Data.agda b/doc/README/Data.agda
similarity index 100%
rename from README/Data.agda
rename to doc/README/Data.agda
diff --git a/README/Data/Container/FreeMonad.agda b/doc/README/Data/Container/FreeMonad.agda
similarity index 100%
rename from README/Data/Container/FreeMonad.agda
rename to doc/README/Data/Container/FreeMonad.agda
diff --git a/README/Data/Container/Indexed.agda b/doc/README/Data/Container/Indexed.agda
similarity index 100%
rename from README/Data/Container/Indexed.agda
rename to doc/README/Data/Container/Indexed.agda
diff --git a/README/Data/Default.agda b/doc/README/Data/Default.agda
similarity index 100%
rename from README/Data/Default.agda
rename to doc/README/Data/Default.agda
diff --git a/README/Data/Fin/Relation/Unary/Top.agda b/doc/README/Data/Fin/Relation/Unary/Top.agda
similarity index 100%
rename from README/Data/Fin/Relation/Unary/Top.agda
rename to doc/README/Data/Fin/Relation/Unary/Top.agda
diff --git a/README/Data/Fin/Substitution/UntypedLambda.agda b/doc/README/Data/Fin/Substitution/UntypedLambda.agda
similarity index 100%
rename from README/Data/Fin/Substitution/UntypedLambda.agda
rename to doc/README/Data/Fin/Substitution/UntypedLambda.agda
diff --git a/README/Data/Integer.agda b/doc/README/Data/Integer.agda
similarity index 100%
rename from README/Data/Integer.agda
rename to doc/README/Data/Integer.agda
diff --git a/README/Data/List.agda b/doc/README/Data/List.agda
similarity index 100%
rename from README/Data/List.agda
rename to doc/README/Data/List.agda
diff --git a/README/Data/List/Fresh.agda b/doc/README/Data/List/Fresh.agda
similarity index 100%
rename from README/Data/List/Fresh.agda
rename to doc/README/Data/List/Fresh.agda
diff --git a/README/Data/List/Membership.agda b/doc/README/Data/List/Membership.agda
similarity index 100%
rename from README/Data/List/Membership.agda
rename to doc/README/Data/List/Membership.agda
diff --git a/README/Data/List/Relation/Binary/Equality.agda b/doc/README/Data/List/Relation/Binary/Equality.agda
similarity index 100%
rename from README/Data/List/Relation/Binary/Equality.agda
rename to doc/README/Data/List/Relation/Binary/Equality.agda
diff --git a/README/Data/List/Relation/Binary/Permutation.agda b/doc/README/Data/List/Relation/Binary/Permutation.agda
similarity index 100%
rename from README/Data/List/Relation/Binary/Permutation.agda
rename to doc/README/Data/List/Relation/Binary/Permutation.agda
diff --git a/README/Data/List/Relation/Binary/Pointwise.agda b/doc/README/Data/List/Relation/Binary/Pointwise.agda
similarity index 100%
rename from README/Data/List/Relation/Binary/Pointwise.agda
rename to doc/README/Data/List/Relation/Binary/Pointwise.agda
diff --git a/README/Data/List/Relation/Binary/Subset.agda b/doc/README/Data/List/Relation/Binary/Subset.agda
similarity index 100%
rename from README/Data/List/Relation/Binary/Subset.agda
rename to doc/README/Data/List/Relation/Binary/Subset.agda
diff --git a/README/Data/List/Relation/Ternary/Interleaving.agda b/doc/README/Data/List/Relation/Ternary/Interleaving.agda
similarity index 100%
rename from README/Data/List/Relation/Ternary/Interleaving.agda
rename to doc/README/Data/List/Relation/Ternary/Interleaving.agda
diff --git a/README/Data/List/Relation/Unary/All.agda b/doc/README/Data/List/Relation/Unary/All.agda
similarity index 100%
rename from README/Data/List/Relation/Unary/All.agda
rename to doc/README/Data/List/Relation/Unary/All.agda
diff --git a/README/Data/List/Relation/Unary/Any.agda b/doc/README/Data/List/Relation/Unary/Any.agda
similarity index 100%
rename from README/Data/List/Relation/Unary/Any.agda
rename to doc/README/Data/List/Relation/Unary/Any.agda
diff --git a/README/Data/Nat.agda b/doc/README/Data/Nat.agda
similarity index 100%
rename from README/Data/Nat.agda
rename to doc/README/Data/Nat.agda
diff --git a/README/Data/Nat/Induction.agda b/doc/README/Data/Nat/Induction.agda
similarity index 100%
rename from README/Data/Nat/Induction.agda
rename to doc/README/Data/Nat/Induction.agda
diff --git a/README/Data/Record.agda b/doc/README/Data/Record.agda
similarity index 100%
rename from README/Data/Record.agda
rename to doc/README/Data/Record.agda
diff --git a/README/Data/Tree/AVL.agda b/doc/README/Data/Tree/AVL.agda
similarity index 100%
rename from README/Data/Tree/AVL.agda
rename to doc/README/Data/Tree/AVL.agda
diff --git a/README/Data/Tree/Binary.agda b/doc/README/Data/Tree/Binary.agda
similarity index 100%
rename from README/Data/Tree/Binary.agda
rename to doc/README/Data/Tree/Binary.agda
diff --git a/README/Data/Tree/Rose.agda b/doc/README/Data/Tree/Rose.agda
similarity index 100%
rename from README/Data/Tree/Rose.agda
rename to doc/README/Data/Tree/Rose.agda
diff --git a/README/Data/Trie/NonDependent.agda b/doc/README/Data/Trie/NonDependent.agda
similarity index 100%
rename from README/Data/Trie/NonDependent.agda
rename to doc/README/Data/Trie/NonDependent.agda
diff --git a/README/Data/Vec/Relation/Binary/Equality/Cast.agda b/doc/README/Data/Vec/Relation/Binary/Equality/Cast.agda
similarity index 100%
rename from README/Data/Vec/Relation/Binary/Equality/Cast.agda
rename to doc/README/Data/Vec/Relation/Binary/Equality/Cast.agda
diff --git a/README/Data/Wrap.agda b/doc/README/Data/Wrap.agda
similarity index 100%
rename from README/Data/Wrap.agda
rename to doc/README/Data/Wrap.agda
diff --git a/README/Debug/Trace.agda b/doc/README/Debug/Trace.agda
similarity index 100%
rename from README/Debug/Trace.agda
rename to doc/README/Debug/Trace.agda
diff --git a/README/Design/Decidability.agda b/doc/README/Design/Decidability.agda
similarity index 100%
rename from README/Design/Decidability.agda
rename to doc/README/Design/Decidability.agda
diff --git a/README/Design/Fixity.agda b/doc/README/Design/Fixity.agda
similarity index 100%
rename from README/Design/Fixity.agda
rename to doc/README/Design/Fixity.agda
diff --git a/README/Design/Hierarchies.agda b/doc/README/Design/Hierarchies.agda
similarity index 100%
rename from README/Design/Hierarchies.agda
rename to doc/README/Design/Hierarchies.agda
diff --git a/README/Foreign/Haskell.agda b/doc/README/Foreign/Haskell.agda
similarity index 100%
rename from README/Foreign/Haskell.agda
rename to doc/README/Foreign/Haskell.agda
diff --git a/README/Function/Reasoning.agda b/doc/README/Function/Reasoning.agda
similarity index 100%
rename from README/Function/Reasoning.agda
rename to doc/README/Function/Reasoning.agda
diff --git a/README/IO.agda b/doc/README/IO.agda
similarity index 100%
rename from README/IO.agda
rename to doc/README/IO.agda
diff --git a/README/Inspect.agda b/doc/README/Inspect.agda
similarity index 100%
rename from README/Inspect.agda
rename to doc/README/Inspect.agda
diff --git a/README/Nary.agda b/doc/README/Nary.agda
similarity index 100%
rename from README/Nary.agda
rename to doc/README/Nary.agda
diff --git a/README/Reflection/External.agda b/doc/README/Reflection/External.agda
similarity index 100%
rename from README/Reflection/External.agda
rename to doc/README/Reflection/External.agda
diff --git a/README/Relation/Binary/TypeClasses.agda b/doc/README/Relation/Binary/TypeClasses.agda
similarity index 100%
rename from README/Relation/Binary/TypeClasses.agda
rename to doc/README/Relation/Binary/TypeClasses.agda
diff --git a/README/Tactic/Cong.agda b/doc/README/Tactic/Cong.agda
similarity index 100%
rename from README/Tactic/Cong.agda
rename to doc/README/Tactic/Cong.agda
diff --git a/README/Tactic/MonoidSolver.agda b/doc/README/Tactic/MonoidSolver.agda
similarity index 100%
rename from README/Tactic/MonoidSolver.agda
rename to doc/README/Tactic/MonoidSolver.agda
diff --git a/README/Tactic/RingSolver.agda b/doc/README/Tactic/RingSolver.agda
similarity index 100%
rename from README/Tactic/RingSolver.agda
rename to doc/README/Tactic/RingSolver.agda
diff --git a/README/Text/Pretty.agda b/doc/README/Text/Pretty.agda
similarity index 100%
rename from README/Text/Pretty.agda
rename to doc/README/Text/Pretty.agda
diff --git a/README/Text/Printf.agda b/doc/README/Text/Printf.agda
similarity index 100%
rename from README/Text/Printf.agda
rename to doc/README/Text/Printf.agda
diff --git a/README/Text/Regex.agda b/doc/README/Text/Regex.agda
similarity index 100%
rename from README/Text/Regex.agda
rename to doc/README/Text/Regex.agda
diff --git a/README/Text/Tabular.agda b/doc/README/Text/Tabular.agda
similarity index 100%
rename from README/Text/Tabular.agda
rename to doc/README/Text/Tabular.agda
diff --git a/notes/installation-guide.md b/doc/installation-guide.md
similarity index 100%
rename from notes/installation-guide.md
rename to doc/installation-guide.md
diff --git a/notes/release-guide.txt b/doc/release-guide.txt
similarity index 85%
rename from notes/release-guide.txt
rename to doc/release-guide.txt
index 2f3f9d862f..0296bc0fe1 100644
--- a/notes/release-guide.txt
+++ b/doc/release-guide.txt
@@ -61,20 +61,12 @@ procedure should be followed:
 * Announce the release of the new version on the Agda mailing lists
   (users and developers).
 
-* Add v$VERSION to the list of protected directories in the .travis.yml
-  file of BOTH master and experimental. They should look something like:
-
-  >  git checkout HEAD -- v0.16/ v0.17/ v1.0/ v1.1/ (...)
-
-  Commit & push these changes. This will prevent the next step from being
-  overwritten by travis.
-
 * Generate and upload documentation for the released version:
 
-    cp travis/* .
+    cp .github/tooling/* .
     runhaskell GenerateEverything.hs
     ./index.sh
-    agda -i. -isrc --html index.agda
+    agda -i. -idoc -isrc --html index.agda
     mv html v$VERSION
     git checkout gh-pages
     git add v$VERSION/*.html v$VERSION/*.css
diff --git a/doc/standard-library-doc.agda-lib b/doc/standard-library-doc.agda-lib
new file mode 100644
index 0000000000..02aaaba402
--- /dev/null
+++ b/doc/standard-library-doc.agda-lib
@@ -0,0 +1,4 @@
+name: standard-library-doc
+include: . ../src
+flags:
+  --warning=noUnsupportedIndexedMatch
diff --git a/notes/style-guide.md b/doc/style-guide.md
similarity index 100%
rename from notes/style-guide.md
rename to doc/style-guide.md
diff --git a/notes/updating-experimental.txt b/doc/updating-experimental.txt
similarity index 100%
rename from notes/updating-experimental.txt
rename to doc/updating-experimental.txt
diff --git a/fix-whitespace.yaml b/fix-whitespace.yaml
index 6f50570fcf..a12022572e 100644
--- a/fix-whitespace.yaml
+++ b/fix-whitespace.yaml
@@ -5,7 +5,6 @@ included-dirs:
 included-files:
   - "*.agda"
   - "*.md"
-  - ".travis.yml"
 
 excluded-files:
   - "README/Text/Tabular.agda"

From d198697f269ea988f5adda06ddee99319a2ab29a Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 14 Nov 2023 15:00:36 +0000
Subject: [PATCH 184/316] documentation: fix link to `installation-guide`,
 `README.agda`, `README.md`... (#2197)

* fix link to `installation-guide`

* catching another reference to `notes/`

* note on instance brackets

* `HACKING` guide

* added Gurmeet Singh's changes

* [ fix ] links in README.md

---------

Co-authored-by: Guillaume Allais <guillaume.allais@ens-lyon.org>
---
 HACKING.md            |  4 ++--
 README.md             | 12 ++++--------
 doc/release-guide.txt |  4 ++--
 doc/style-guide.md    | 10 ++++++++++
 4 files changed, 18 insertions(+), 12 deletions(-)

diff --git a/HACKING.md b/HACKING.md
index b7573188b5..9b0e7454ac 100644
--- a/HACKING.md
+++ b/HACKING.md
@@ -4,7 +4,7 @@ Contributing to the library
 Thank you for your interest in contributing to the Agda standard library.
 Hopefully this guide should make it easy to do so! Feel free to ask any
 questions on the Agda mailing list. Before you start please read the
-[style-guide](https://github.com/agda/agda-stdlib/blob/master/notes/style-guide.md).
+[style-guide](https://github.com/agda/agda-stdlib/blob/master/doc/style-guide.md).
 
 What is an acceptable contribution?
 ===================================
@@ -124,7 +124,7 @@ git push USER -u new_feature
 
 You can then proceed to make your changes. Please follow existing
 conventions in the library, see
-[style-guide](https://github.com/agda/agda-stdlib/blob/master/notes/style-guide.md).
+[style-guide](https://github.com/agda/agda-stdlib/blob/master/doc/style-guide.md).
 for details. Document your changes in `agda-stdlib-fork/CHANGELOG.md`.
 
 If you are creating new modules, please make sure you are having a
diff --git a/README.md b/README.md
index f9489e81f3..3c66d46ee0 100644
--- a/README.md
+++ b/README.md
@@ -18,23 +18,19 @@ If you're looking to find your way around the library, there are several
 different ways to get started:
 
 - The library's structure and the associated design choices are described
-in the [README.agda](https://github.com/agda/agda-stdlib/tree/master/README.agda).
+in the [README.agda](https://github.com/agda/agda-stdlib/tree/master/doc/README.agda).
 
-- The [README folder](https://github.com/agda/agda-stdlib/tree/master/README),
+- The [README folder](https://github.com/agda/agda-stdlib/tree/master/doc/README),
 which mirrors the structure of the main library, contains examples of how to
 use some of the more common modules. Feel free to [open a new issue](https://github.com/agda/agda-stdlib/issues/new) if there's a particular module you feel could do with
 some more documentation.
 
-- You can [browse the library's source code](https://agda.github.io/agda-stdlib/README.html)
+- You can [browse the library's source code](https://agda.github.io/agda-stdlib/)
 in glorious clickable HTML.
 
-- Finally, you can get an overview of the entire library by looking at the
-[index](https://agda.github.io/agda-stdlib/), which lists all modules
-in the library except the deprecated ones.
-
 ## Installation instructions
 
-See the [installation instructions](https://github.com/agda/agda-stdlib/blob/master/notes/installation-guide.md) for the latest version of the standard library.
+See the [installation instructions](https://github.com/agda/agda-stdlib/blob/master/doc/installation-guide.md) for the latest version of the standard library.
 
 #### Old versions of Agda
 
diff --git a/doc/release-guide.txt b/doc/release-guide.txt
index 0296bc0fe1..0c60b5d4b3 100644
--- a/doc/release-guide.txt
+++ b/doc/release-guide.txt
@@ -3,7 +3,7 @@ procedure should be followed:
 
 #### Pre-release changes
 
-* Update `README.agda` by replacing 'development version' by 'version X.Y' in the title.
+* Update `doc/README.agda` by replacing 'development version' by 'version X.Y' in the title.
 
 * Update the version to `X.Y` in:
   - `agda-stdlib-utils.cabal`
@@ -11,7 +11,7 @@ procedure should be followed:
   - `CITATION.cff`
   - `CHANGELOG.md`
   - `README.md`
-  - `notes/installation-guide.txt`
+  - `doc/installation-guide.md`
 
 * Update the copyright year range in the LICENSE file, if necessary.
 
diff --git a/doc/style-guide.md b/doc/style-guide.md
index a3c918e6a4..5732747cda 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -333,6 +333,16 @@ line of code, indented by two spaces.
   ...                  | false = filter p xs
   ```
 
+* Instance arguments, and their types, should use the vanilla ASCII/UTF-8 `{{_}}`
+  syntax in preference to the Unicode `⦃_⦄` syntax (written using `\{{`/`\}}`),
+  which moreover requires additional whitespace to parse correctly.
+  NB. Even for irrelevant instances, such as typically for `NonZero` arguments,
+  neverthelesss it is necessary to supply an underscore binding `{{_ : NonZero n}}`
+  if subsequent terms occurring in the type rely on that argument to be well-formed:
+  eg in `Data.Nat.DivMod`, in the use of `_/ n` and `_% n`
+  ```agda
+  m≡m%n+[m/n]*n : ∀ m n .{{_ : NonZero n}} → m ≡ m % n + (m / n) * n
+  ```
 
 ## Types
 

From 792a83852c682e2d2dfcbc5562c49b0c8206c4f7 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 15 Nov 2023 11:26:52 +0000
Subject: [PATCH 185/316] easy deprecation; leftover from `v1.6` perhaps?
 (#2203)

---
 CHANGELOG.md                         |  5 +++++
 src/Relation/Unary/Consequences.agda | 18 ++++++++++++++++--
 2 files changed, 21 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 562c089ddf..1d88b1bca5 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1727,6 +1727,11 @@ Deprecated names
   invPreorder   ↦ converse-preorder
   ```
 
+* In `Relation.Unary.Consequences`:
+  ```agda
+  dec⟶recomputable  ↦  dec⇒recomputable
+  ```
+
 ## Missing fixity declarations added
 
 * An effort has been made to add sensible fixities for many declarations:
diff --git a/src/Relation/Unary/Consequences.agda b/src/Relation/Unary/Consequences.agda
index 1b23578161..75c046c375 100644
--- a/src/Relation/Unary/Consequences.agda
+++ b/src/Relation/Unary/Consequences.agda
@@ -11,5 +11,19 @@ module Relation.Unary.Consequences where
 open import Relation.Unary
 open import Relation.Nullary using (recompute)
 
-dec⟶recomputable : {a ℓ : _} {A : Set a} {P : Pred A ℓ} → Decidable P → Recomputable P
-dec⟶recomputable P-dec = recompute (P-dec _)
+dec⇒recomputable : {a ℓ : _} {A : Set a} {P : Pred A ℓ} → Decidable P → Recomputable P
+dec⇒recomputable P-dec = recompute (P-dec _)
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+dec⟶recomputable = dec⇒recomputable
+{-# WARNING_ON_USAGE dec⟶recomputable
+"Warning: dec⟶recomputable was deprecated in v2.0.
+Please use dec⇒recomputable instead."
+#-}

From fd3cac8b0619cd8414e45ccc11c12c98c6a3c09c Mon Sep 17 00:00:00 2001
From: Jesin <jesin00@gmail.com>
Date: Thu, 16 Nov 2023 01:27:58 -0500
Subject: [PATCH 186/316] fix Connex comment (#2204)

Connex allows both relations to hold, so the old comment was wrong.
---
 src/Relation/Binary/Definitions.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Relation/Binary/Definitions.agda b/src/Relation/Binary/Definitions.agda
index 547d4e7b48..887cb7c4a6 100644
--- a/src/Relation/Binary/Definitions.agda
+++ b/src/Relation/Binary/Definitions.agda
@@ -99,7 +99,7 @@ Asymmetric _<_ = ∀ {x y} → x < y → ¬ (y < x)
 Dense : Rel A ℓ → Set _
 Dense _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y
 
--- Generalised connex - exactly one of the two relations holds.
+-- Generalised connex - at least one of the two relations holds.
 
 Connex : REL A B ℓ₁ → REL B A ℓ₂ → Set _
 Connex P Q = ∀ x y → P x y ⊎ Q y x

From 586bca19df02516230d0d6cc381562d67d6f522d Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Fri, 17 Nov 2023 10:54:26 +0900
Subject: [PATCH 187/316] Add `Function.Consequences.Setoid` (#2191)

* Add Function.Consequences.Setoid

* Fix comment

* Fix re-export bug

* Finally fixed bug I hope

* Removed rogue comment

* More tidying up
---
 CHANGELOG.md                                  |  4 +-
 .../Product/Function/Dependent/Setoid.agda    | 10 +-
 src/Function/Consequences.agda                |  4 +-
 src/Function/Consequences/Propositional.agda  | 88 +++++-------------
 src/Function/Consequences/Setoid.agda         | 92 +++++++++++++++++++
 src/Function/Properties/Inverse.agda          | 18 ++--
 src/Function/Structures.agda                  |  1 -
 7 files changed, 136 insertions(+), 81 deletions(-)
 create mode 100644 src/Function/Consequences/Setoid.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1d88b1bca5..ddc3b262fc 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -752,7 +752,7 @@ Non-backwards compatible changes
          of these under the names `strictlySurjective`, `strictlyInverseˡ` and
          `strictlyInverseʳ`,
    - Conversion functions have been added in both directions to
-         `Function.Consequences(.Propositional)`.
+         `Function.Consequences(.Propositional/Setoid)`.
 
 ### New `Function.Strict`
 
@@ -2028,6 +2028,8 @@ New modules
 * Properties of various types of functions:
   ```
   Function.Consequences
+  Function.Consequences.Setoid
+  Function.Consequences.Propositional
   Function.Properties.Bijection
   Function.Properties.RightInverse
   Function.Properties.Surjection
diff --git a/src/Data/Product/Function/Dependent/Setoid.agda b/src/Data/Product/Function/Dependent/Setoid.agda
index f290c7341b..22cb950626 100644
--- a/src/Data/Product/Function/Dependent/Setoid.agda
+++ b/src/Data/Product/Function/Dependent/Setoid.agda
@@ -15,7 +15,7 @@ open import Data.Product.Base using (map; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.Dependent as Σ
 open import Level using (Level)
 open import Function
-open import Function.Consequences
+open import Function.Consequences.Setoid
 open import Function.Properties.Injection using (mkInjection)
 open import Function.Properties.Surjection using (mkSurjection; ↠⇒⇔)
 open import Function.Properties.Equivalence using (mkEquivalence; ⇔⇒⟶; ⇔⇒⟵)
@@ -176,7 +176,7 @@ module _ where
       to∘to⁻ I↠J j , IndexedSetoid.trans B (to∘to⁻ A↠B _) (cast-eq B (to∘to⁻ I↠J j))
 
     surj : Surjective (Func.Eq₁._≈_ func) (Func.Eq₂._≈_ func) (Func.to func)
-    surj = strictlySurjective⇒surjective (trans (J ×ₛ B)) (Func.cong func) strictlySurj
+    surj = strictlySurjective⇒surjective (I ×ₛ A) (J ×ₛ B) (Func.cong func) strictlySurj
 
 ------------------------------------------------------------------------
 -- LeftInverse
@@ -199,7 +199,7 @@ module _ where
     strictlyInvʳ (i , x) = strictlyInverseʳ I↪J i , IndexedSetoid.trans A (strictlyInverseʳ A↪B _) (cast-eq A (strictlyInverseʳ I↪J i))
 
     invʳ : Inverseʳ (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) (Equivalence.to equiv) (Equivalence.from equiv)
-    invʳ = strictlyInverseʳ⇒inverseʳ {f⁻¹ = Equivalence.from equiv} (trans (I ×ₛ A)) (Equivalence.from-cong equiv) strictlyInvʳ
+    invʳ = strictlyInverseʳ⇒inverseʳ (I ×ₛ A) (J ×ₛ B) (Equivalence.from-cong equiv) strictlyInvʳ
 
 
 ------------------------------------------------------------------------
@@ -238,7 +238,7 @@ module _ where
           (cast-eq B (strictlyInverseˡ I↔J i))
 
     invˡ : Inverseˡ (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′ from′
-    invˡ = strictlyInverseˡ⇒inverseˡ {≈₁ = _≈_ (I ×ₛ A)} {f⁻¹ = from′} (trans (J ×ₛ B)) to′-cong strictlyInvˡ
+    invˡ = strictlyInverseˡ⇒inverseˡ (I ×ₛ A) (J ×ₛ B) to′-cong strictlyInvˡ
 
     lem : ∀ {i j} → i ≡ j → ∀ {x : IndexedSetoid.Carrier B (to I↔J i)} {y : IndexedSetoid.Carrier B (to I↔J j)} →
           IndexedSetoid._≈_ B x y →
@@ -250,5 +250,5 @@ module _ where
       IndexedSetoid.trans A (lem (strictlyInverseʳ I↔J _) (cast-eq B (strictlyInverseˡ I↔J _))) (strictlyInverseʳ A↔B _)
 
     invʳ : Inverseʳ (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′ from′
-    invʳ = strictlyInverseʳ⇒inverseʳ {f⁻¹ = from′} (trans (I ×ₛ A)) from′-cong strictlyInvʳ
+    invʳ = strictlyInverseʳ⇒inverseʳ (I ×ₛ A) (J ×ₛ B) from′-cong strictlyInvʳ
 
diff --git a/src/Function/Consequences.agda b/src/Function/Consequences.agda
index 60606cc673..2fda92224e 100644
--- a/src/Function/Consequences.agda
+++ b/src/Function/Consequences.agda
@@ -28,9 +28,9 @@ private
 ------------------------------------------------------------------------
 -- Injective
 
-contraInjective : Injective ≈₁ ≈₂ f →
+contraInjective : ∀ (≈₂ : Rel B ℓ₂) → Injective ≈₁ ≈₂ f →
                   ∀ {x y} → ¬ (≈₁ x y) → ¬ (≈₂ (f x) (f y))
-contraInjective inj p = contraposition inj p
+contraInjective _ inj p = contraposition inj p
 
 ------------------------------------------------------------------------
 -- Inverseˡ
diff --git a/src/Function/Consequences/Propositional.agda b/src/Function/Consequences/Propositional.agda
index d6067a7702..a2558a9c48 100644
--- a/src/Function/Consequences/Propositional.agda
+++ b/src/Function/Consequences/Propositional.agda
@@ -7,85 +7,47 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-module Function.Consequences.Propositional where
+module Function.Consequences.Propositional
+  {a b} {A : Set a} {B : Set b}
+  where
 
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid)
 open import Function.Definitions
-open import Level
 open import Relation.Nullary.Negation.Core using (contraposition)
-open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; _≢_; refl; sym; trans; cong)
 
-import Function.Consequences as C
-
-private
-  variable
-    a b ℓ₁ ℓ₂ : Level
-    A B : Set a
-    f f⁻¹ : A → B
-
-------------------------------------------------------------------------
--- Injective
-
-contraInjective : Injective _≡_ _≡_ f →
-                  ∀ {x y} → x ≢ y → f x ≢ f y
-contraInjective inj p = contraposition inj p
-
-------------------------------------------------------------------------
--- Inverseˡ
-
-inverseˡ⇒surjective : Inverseˡ _≡_ _≡_ f f⁻¹ → Surjective _≡_ _≡_ f
-inverseˡ⇒surjective = C.inverseˡ⇒surjective _≡_
-
-------------------------------------------------------------------------
--- Inverseʳ
-
-inverseʳ⇒injective : ∀ f →
-                     Inverseʳ _≡_ _≡_ f f⁻¹ →
-                     Injective _≡_ _≡_ f
-inverseʳ⇒injective f = C.inverseʳ⇒injective _≡_ f refl sym trans
+import Function.Consequences.Setoid (setoid A) (setoid B) as Setoid
 
 ------------------------------------------------------------------------
--- Inverseᵇ
+-- Re-export setoid properties
 
-inverseᵇ⇒bijective : Inverseᵇ _≡_ _≡_ f f⁻¹ → Bijective _≡_ _≡_ f
-inverseᵇ⇒bijective = C.inverseᵇ⇒bijective _≡_ refl sym trans
+open Setoid public
+  hiding
+  ( strictlySurjective⇒surjective
+  ; strictlyInverseˡ⇒inverseˡ
+  ; strictlyInverseʳ⇒inverseʳ
+  )
 
 ------------------------------------------------------------------------
--- StrictlySurjective
+-- Properties that rely on congruence
 
-surjective⇒strictlySurjective : Surjective _≡_ _≡_ f →
-                                 StrictlySurjective _≡_ f
-surjective⇒strictlySurjective =
-  C.surjective⇒strictlySurjective _≡_ refl
+private
+  variable
+    f : A → B
+    f⁻¹ : B → A
 
 strictlySurjective⇒surjective : StrictlySurjective _≡_ f →
                                  Surjective _≡_ _≡_ f
 strictlySurjective⇒surjective =
-  C.strictlySurjective⇒surjective trans (cong _)
-
-------------------------------------------------------------------------
--- StrictlyInverseˡ
-
-inverseˡ⇒strictlyInverseˡ : Inverseˡ _≡_ _≡_ f f⁻¹ →
-                            StrictlyInverseˡ _≡_ f f⁻¹
-inverseˡ⇒strictlyInverseˡ =
-  C.inverseˡ⇒strictlyInverseˡ _≡_ _≡_ refl
+ Setoid.strictlySurjective⇒surjective (cong _)
 
-strictlyInverseˡ⇒inverseˡ : ∀ f →
-                            StrictlyInverseˡ _≡_ f f⁻¹ →
+strictlyInverseˡ⇒inverseˡ : ∀ f → StrictlyInverseˡ _≡_ f f⁻¹ →
                             Inverseˡ _≡_ _≡_ f f⁻¹
 strictlyInverseˡ⇒inverseˡ f =
-  C.strictlyInverseˡ⇒inverseˡ trans (cong f)
-
-------------------------------------------------------------------------
--- StrictlyInverseʳ
-
-inverseʳ⇒strictlyInverseʳ : Inverseʳ _≡_ _≡_ f f⁻¹ →
-                            StrictlyInverseʳ _≡_ f f⁻¹
-inverseʳ⇒strictlyInverseʳ = C.inverseʳ⇒strictlyInverseʳ _≡_ _≡_ refl
+  Setoid.strictlyInverseˡ⇒inverseˡ (cong _)
 
-strictlyInverseʳ⇒inverseʳ : ∀ f →
-                            StrictlyInverseʳ _≡_ f f⁻¹ →
+strictlyInverseʳ⇒inverseʳ : ∀ f → StrictlyInverseʳ _≡_ f f⁻¹ →
                             Inverseʳ _≡_ _≡_ f f⁻¹
-strictlyInverseʳ⇒inverseʳ {f⁻¹ = f⁻¹} _ =
-  C.strictlyInverseʳ⇒inverseʳ trans (cong f⁻¹)
+strictlyInverseʳ⇒inverseʳ f =
+  Setoid.strictlyInverseʳ⇒inverseʳ (cong _)
diff --git a/src/Function/Consequences/Setoid.agda b/src/Function/Consequences/Setoid.agda
new file mode 100644
index 0000000000..d2c3b948d9
--- /dev/null
+++ b/src/Function/Consequences/Setoid.agda
@@ -0,0 +1,92 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Relationships between properties of functions where the equality
+-- over both the domain and codomain are assumed to be setoids.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Bundles using (Setoid)
+
+module Function.Consequences.Setoid
+  {a b ℓ₁ ℓ₂}
+  (S : Setoid a ℓ₁)
+  (T : Setoid b ℓ₂)
+  where
+
+open import Function.Definitions
+open import Relation.Nullary.Negation.Core
+
+import Function.Consequences as C
+
+private
+  open module S = Setoid S using () renaming (Carrier to A; _≈_ to ≈₁)
+  open module T = Setoid T using () renaming (Carrier to B; _≈_ to ≈₂)
+
+  variable
+    f : A → B
+    f⁻¹ : B → A
+
+------------------------------------------------------------------------
+-- Injective
+
+contraInjective : Injective ≈₁ ≈₂ f →
+                  ∀ {x y} → ¬ (≈₁ x y) → ¬ (≈₂ (f x) (f y))
+contraInjective = C.contraInjective ≈₂
+
+------------------------------------------------------------------------
+-- Inverseˡ
+
+inverseˡ⇒surjective : Inverseˡ ≈₁ ≈₂ f f⁻¹ → Surjective ≈₁ ≈₂ f
+inverseˡ⇒surjective = C.inverseˡ⇒surjective ≈₂
+
+------------------------------------------------------------------------
+-- Inverseʳ
+
+inverseʳ⇒injective : ∀ f → Inverseʳ ≈₁ ≈₂ f f⁻¹ → Injective ≈₁ ≈₂ f
+inverseʳ⇒injective f = C.inverseʳ⇒injective ≈₂ f T.refl S.sym S.trans
+
+------------------------------------------------------------------------
+-- Inverseᵇ
+
+inverseᵇ⇒bijective : Inverseᵇ ≈₁ ≈₂ f f⁻¹ → Bijective ≈₁ ≈₂ f
+inverseᵇ⇒bijective = C.inverseᵇ⇒bijective ≈₂ T.refl S.sym S.trans
+
+------------------------------------------------------------------------
+-- StrictlySurjective
+
+surjective⇒strictlySurjective : Surjective ≈₁ ≈₂ f →
+                                 StrictlySurjective ≈₂ f
+surjective⇒strictlySurjective =
+  C.surjective⇒strictlySurjective ≈₂ S.refl
+
+strictlySurjective⇒surjective : Congruent ≈₁ ≈₂ f →
+                                 StrictlySurjective ≈₂ f →
+                                 Surjective ≈₁ ≈₂ f
+strictlySurjective⇒surjective =
+  C.strictlySurjective⇒surjective T.trans
+
+------------------------------------------------------------------------
+-- StrictlyInverseˡ
+
+inverseˡ⇒strictlyInverseˡ : Inverseˡ ≈₁ ≈₂ f f⁻¹ →
+                            StrictlyInverseˡ ≈₂ f f⁻¹
+inverseˡ⇒strictlyInverseˡ = C.inverseˡ⇒strictlyInverseˡ ≈₁ ≈₂ S.refl
+
+strictlyInverseˡ⇒inverseˡ : Congruent ≈₁ ≈₂ f →
+                            StrictlyInverseˡ ≈₂ f f⁻¹ →
+                            Inverseˡ ≈₁ ≈₂ f f⁻¹
+strictlyInverseˡ⇒inverseˡ = C.strictlyInverseˡ⇒inverseˡ T.trans
+
+------------------------------------------------------------------------
+-- StrictlyInverseʳ
+
+inverseʳ⇒strictlyInverseʳ : Inverseʳ ≈₁ ≈₂ f f⁻¹ →
+                            StrictlyInverseʳ ≈₁ f f⁻¹
+inverseʳ⇒strictlyInverseʳ = C.inverseʳ⇒strictlyInverseʳ ≈₁ ≈₂ T.refl
+
+strictlyInverseʳ⇒inverseʳ : Congruent ≈₂ ≈₁ f⁻¹ →
+                            StrictlyInverseʳ ≈₁ f f⁻¹ →
+                            Inverseʳ ≈₁ ≈₂ f f⁻¹
+strictlyInverseʳ⇒inverseʳ = C.strictlyInverseʳ⇒inverseʳ S.trans
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index 87ce778fec..5ade85d843 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -19,7 +19,7 @@ open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as P
 import Relation.Binary.Reasoning.Setoid as SetoidReasoning
-open import Function.Consequences
+import Function.Consequences.Setoid as Consequences
 
 import Function.Construct.Identity as Identity
 import Function.Construct.Symmetry as Symmetry
@@ -77,25 +77,25 @@ fromFunction I = record { to = from ; cong = from-cong }
   where open Inverse I
 
 Inverse⇒Injection : Inverse S T → Injection S T
-Inverse⇒Injection I = record
+Inverse⇒Injection {S = S} {T = T} I = record
   { to = to
   ; cong = to-cong
-  ; injective = inverseʳ⇒injective Eq₂._≈_ to Eq₂.refl Eq₁.sym Eq₁.trans inverseʳ
-  } where open Inverse I
+  ; injective = inverseʳ⇒injective to inverseʳ
+  } where open Inverse I; open Consequences S T
 
 Inverse⇒Surjection : Inverse S T → Surjection S T
 Inverse⇒Surjection {S = S} {T = T} I = record
   { to = to
   ; cong = to-cong
-  ; surjective = inverseˡ⇒surjective (_≈_ T) inverseˡ
-  } where open Inverse I; open Setoid
+  ; surjective = inverseˡ⇒surjective inverseˡ
+  } where open Inverse I; open Consequences S T
 
 Inverse⇒Bijection : Inverse S T → Bijection S T
-Inverse⇒Bijection I = record
+Inverse⇒Bijection {S = S} {T = T} I = record
   { to        = to
   ; cong      = to-cong
-  ; bijective = inverseᵇ⇒bijective Eq₂._≈_ Eq₂.refl Eq₁.sym Eq₁.trans inverse
-  } where open Inverse I
+  ; bijective = inverseᵇ⇒bijective inverse
+  } where open Inverse I; open Consequences S T
 
 Inverse⇒Equivalence : Inverse S T → Equivalence S T
 Inverse⇒Equivalence I = record
diff --git a/src/Function/Structures.agda b/src/Function/Structures.agda
index baf33dddf0..ae34203fe0 100644
--- a/src/Function/Structures.agda
+++ b/src/Function/Structures.agda
@@ -20,7 +20,6 @@ module Function.Structures {a b ℓ₁ ℓ₂}
 open import Data.Product.Base as Product using (∃; _×_; _,_)
 open import Function.Base
 open import Function.Definitions
-open import Function.Consequences
 open import Level using (_⊔_)
 
 ------------------------------------------------------------------------

From 4220f83c15b28b06c1ae907af7fde2685261dc77 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 21 Nov 2023 01:47:22 +0000
Subject: [PATCH 188/316] Deprecating
 `Relation.Binary.PropositionalEquality.isPropositional` (#2205)

* deprecating `isPropositional`

* tighten `import`s

* bumped Agda version number in comment
---
 CHANGELOG.md                                  |  5 +++
 .../Binary/PropositionalEquality.agda         | 31 +++++++++++++------
 2 files changed, 26 insertions(+), 10 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ddc3b262fc..51966c2176 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1727,6 +1727,11 @@ Deprecated names
   invPreorder   ↦ converse-preorder
   ```
 
+* In `Relation.Binary.PropositionalEquality`:
+  ```agda
+  isPropositional ↦ Relation.Nullary.Irrelevant
+  ```
+
 * In `Relation.Unary.Consequences`:
   ```agda
   dec⟶recomputable  ↦  dec⇒recomputable
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 6ff59d8c28..7ba3825e31 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -8,14 +8,12 @@
 
 module Relation.Binary.PropositionalEquality where
 
-import Axiom.Extensionality.Propositional as Ext
 open import Axiom.UniquenessOfIdentityProofs
 open import Function.Base using (id; _∘_)
 import Function.Dependent.Bundles as Dependent
 open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)
 open import Level using (Level; _⊔_)
-open import Data.Product.Base using (∃)
-
+open import Relation.Nullary using (Irrelevant)
 open import Relation.Nullary.Decidable using (yes; no; dec-yes-irr; dec-no)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (DecidableEquality)
@@ -60,12 +58,6 @@ _≗_ {A = A} {B = B} = Setoid._≈_ (A →-setoid B)
          Dependent.Func (setoid A) (Trivial.indexedSetoid B)
 →-to-⟶ = :→-to-Π
 
-------------------------------------------------------------------------
--- Propositionality
-
-isPropositional : Set a → Set a
-isPropositional A = (a b : A) → a ≡ b
-
 ------------------------------------------------------------------------
 -- More complex rearrangement lemmas
 
@@ -113,7 +105,7 @@ module _ (_≟_ : DecidableEquality A) {x y : A} where
 -- See README.Inspect for an explanation of how/why to use this.
 
 -- Normally (but not always) the new `with ... in` syntax described at
--- https://agda.readthedocs.io/en/v2.6.3/language/with-abstraction.html#with-abstraction-equality
+-- https://agda.readthedocs.io/en/v2.6.4/language/with-abstraction.html#with-abstraction-equality
 -- can be used instead."
 
 record Reveal_·_is_ {A : Set a} {B : A → Set b}
@@ -125,3 +117,22 @@ record Reveal_·_is_ {A : Set a} {B : A → Set b}
 inspect : ∀ {A : Set a} {B : A → Set b}
           (f : (x : A) → B x) (x : A) → Reveal f · x is f x
 inspect f x = [ refl ]
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+isPropositional : Set a → Set a
+isPropositional = Irrelevant
+
+{-# WARNING_ON_USAGE isPropositional
+"Warning: isPropositional was deprecated in v2.0.
+Please use Relation.Nullary.Irrelevant instead. "
+#-}
+
+

From 37b8676dcc14429fd2a8f746057043d1a3a01eed Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Fri, 24 Nov 2023 08:33:41 +0000
Subject: [PATCH 189/316] definition of `Irreducible` and `Rough`; refactoring
 of `Prime` and `Composite` cf. #2180 (#2181)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* definition of `Irreducible`; refactoring of `Prime` and `Composite`

* tidying up old cruft

* knock-on consequences: `Coprimality`

* considerable refactoring of `Primality`

* knock-on consequences: `Coprimality`

* refactoring: no appeal to `Data.Nat.Induction`

* refactoring: renamed `SmoothAt` and its constructor; added pattern synonym; proved `Decidable Irreducible`; misc. tweaks

* knock-on consequences: `Coprimality`

* refactoring: removed `NonZero` analysis; misc. tweaks

* knock-on consequences: `Coprimality`; tightened `import`s

* knock-on consequences: `Coprimality`; tightened `import`s

* refactoring: every number is `1-rough`

* knock-on consequences: `Coprimality`; use of smart constructor

* refactoring: every number is `0-rough`; streamlining; inverse to `prime`; documentation

* attempt to optimise for a nearly-common case pseudo-constructor

* fiddling now

* refactoring: better use of `primality` API

* `Rough` is bounded

* removed unnecessary implicits

* comment

* refactoring: comprehensive shift over to `NonTrivial` instances

* refactoring: oops!

* tidying up: removed infixes

* tidying up: restored `rough⇒≤`

* further refinements; added `NonTrivial` proofs

* tidying up

* moving definitions to `Data.Nat.Base`

* propagated changes; many more explicit s required?

* `NonTrivial` is `Recomputable`

* all instances of `NonTrivial` now irrelevant; weird to need `NonTrivial 2` explicitly

* tidying up `Coprimality`, eg with `variable`s

* `NonTrivial` is `Decidable`

* systematise proofs of `Decidable` properties via the `UpTo` predicates

* explicit quantifier now in `composite≢`

* Nathan's simplification

* interaction of `NonZero` and `NonTrivial` instances

* divisor now a record field; final lemmas: closure under divisibility; plus partial `CHANGELOG` entries

* '(non-)instances' become '(counter-)examples'

* stylistics

* removed `k` as a variable/parameter

* renamed fields and smart constructors

* moved smart constructors; made  a local definition

* tidying up

* refactoring per review comments: equivalence of `UpTo` predicates; making more things `private`

* tidying up: names congruent to ordering relation

* removed variable `k`; removed old proof in favour of new one with `NonZero` instance

* removed `recomputable` in favour of a private lemma

* regularised use of instance brackets

* made instances more explicit

* made instances more explicit

* blank line

* made `nonTrivial⇒nonZero` take an explicit argument in lieu of being able to make it an `instance`

* regularised use of instance brackets

* regularised use of instance brackets

* trimming

* tidied up `Coprimality` entries

* Make HasBoundedNonTrivialDivisor infix

* Make NonTrivial into a record to fix instance resolution bug

* Move HasNonTrivialDivisor to Divisibility.Core and hide UpTo lemmas

* Rearrange file to follow standard ordering of lemmas in the rest of the library

* Move UpTo predicates into decidability proofs

* No-op refactoring to curb excessively long lines

* Make a couple of names consistent with style-guide

* new definition of `Prime`

* renamed fundamental definition

* one last reference in `CHANGELOG`

* more better words; one fewer definition

* tidied up `Definitions` section; rejigged order of proofs of properties to reflect definitional order

* refactored proof of `prime⇒irreducible`

* finishing touches

* missing lemma from irrelevant instance list

* regularised final proof to use `contradiction`

* added fixity `infix 10`

* added fixity `infix 10`; made `composite` a pattern synonym; knock-on improvements

* comprehensive `CHNAGELOG` entry; whitespace fixes

* Rename 1<2+ to sz<ss

* Rename hasNonTrivialDivisor relation

* Updated CHANGELOG

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                        | 104 +++++---
 src/Data/Nat/Base.agda              |  55 +++-
 src/Data/Nat/Coprimality.agda       |  81 +++---
 src/Data/Nat/Divisibility.agda      |  27 +-
 src/Data/Nat/Divisibility/Core.agda |  21 +-
 src/Data/Nat/Primality.agda         | 379 +++++++++++++++++++++++-----
 src/Data/Nat/Properties.agda        |  23 +-
 src/Data/Rational/Base.agda         |  11 +-
 8 files changed, 550 insertions(+), 151 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 51966c2176..0e50316b36 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -645,10 +645,14 @@ Non-backwards compatible changes
 * To make it easier to use, reason about and read, the definition has been
   changed to:
   ```agda
-  Prime 0 = ⊥
-  Prime 1 = ⊥
-  Prime n = ∀ {d} → 2 ≤ d → d < n → d ∤ n
+  record Prime (p : ℕ) : Set where
+    constructor prime
+    field
+      .{{nontrivial}} : NonTrivial p
+      notComposite    : ¬ Composite p
   ```
+  where `Composite` is now defined as the diagonal of the new relation
+  `_HasNonTrivialDivisorLessThan_` in `Data.Nat.Divisibility.Core`.
 
 ### Changes to operation reduction behaviour in `Data.Rational(.Unnormalised)`
 
@@ -2754,6 +2758,18 @@ Additions to existing modules
   s≤″s⁻¹ : suc m ≤″ suc n → m ≤″ n
   s<″s⁻¹ : suc m <″ suc n → m <″ n
 
+  pattern 2+ n = suc (suc n)
+  pattern 1<2+n {n} = s<s (z<s {n})
+
+  NonTrivial            : Pred ℕ 0ℓ
+  instance nonTrivial   : NonTrivial (2+ n)
+  n>1⇒nonTrivial        : 1 < n → NonTrivial n
+  nonZero⇒≢1⇒nonTrivial : .{{NonZero n}} → n ≢ 1 → NonTrivial n
+  recompute-nonTrivial  : .{{NonTrivial n}} → NonTrivial n
+  nonTrivial⇒nonZero    : .{{NonTrivial n}} → NonZero n
+  nonTrivial⇒n>1        : .{{NonTrivial n}} → 1 < n
+  nonTrivial⇒≢1         : .{{NonTrivial n}} → n ≢ 1
+
   _⊔′_ : ℕ → ℕ → ℕ
   _⊓′_ : ℕ → ℕ → ℕ
   ∣_-_∣′ : ℕ → ℕ → ℕ
@@ -2779,20 +2795,42 @@ Additions to existing modules
   <-asym : Asymmetric _<_
   ```
 
-* Added a new pattern synonym to `Data.Nat.Divisibility.Core`:
+* Added a new pattern synonym and a new definition to `Data.Nat.Divisibility.Core`:
   ```agda
   pattern divides-refl q = divides q refl
+  record _HasNonTrivialDivisorLessThan_ (m n : ℕ) : Set where
+  ```
+
+* Added new proofs to `Data.Nat.Divisibility`:
+  ```agda
+  hasNonTrivialDivisor-≢ : .{{NonTrivial d}} → .{{NonZero n}} → d ≢ n → d ∣ n → n HasNonTrivialDivisorLessThan n
+  hasNonTrivialDivisor-∣ : m HasNonTrivialDivisorLessThan n → m ∣ o → o HasNonTrivialDivisorLessThan n
+  hasNonTrivialDivisor-≤ : m HasNonTrivialDivisorLessThan n → n ≤ o → m HasNonTrivialDivisorLessThan o
   ```
 
-* Added new definitions and proofs to `Data.Nat.Primality`:
+* Added new definitions, smart constructors and proofs to `Data.Nat.Primality`:
   ```agda
-  Composite        : ℕ → Set
-  composite?       : Decidable Composite
-  composite⇒¬prime : Composite n → ¬ Prime n
-  ¬composite⇒prime : 2 ≤ n → ¬ Composite n → Prime n
-  prime⇒¬composite : Prime n → ¬ Composite n
-  ¬prime⇒composite : 2 ≤ n → ¬ Prime n → Composite n
-  euclidsLemma     : Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n
+  infix 10 _Rough_
+  _Rough_           : ℕ → Pred ℕ _
+  0-rough           : 0 Rough n
+  1-rough           : 1 Rough n
+  2-rough           : 2 Rough n
+  rough⇒≤           : .{{NonTrivial n}} → m Rough n → m ≤ n
+  ∤⇒rough-suc        : m ∤ n → m Rough n → (suc m) Rough n
+  rough∧∣⇒rough     : m Rough o → n ∣ o → m Rough n
+  Composite         : ℕ → Set
+  composite-≢       : .{{NonTrivial d}} → .{{NonZero n}} → d ≢ n → d ∣ n → Composite n
+  composite-∣       : .{{NonZero n}} → Composite m → m ∣ n → Composite n
+  composite?        : Decidable Composite
+  Irreducible       : ℕ → Set
+  irreducible?      : Decidable Irreducible
+  composite⇒¬prime  : Composite n → ¬ Prime n
+  ¬composite⇒prime  : .{{NonTrivial n} → ¬ Composite n → Prime n
+  prime⇒¬composite  : Prime n → ¬ Composite n
+  ¬prime⇒composite  : .{{NonTrivial n} → ¬ Prime n → Composite n
+  prime⇒irreducible : Prime p → Irreducible p
+  irreducible⇒prime : .{{NonTrivial p}} → Irreducible p → Prime p
+  euclidsLemma      : Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n
   ```
 
 * Added new proofs in `Data.Nat.Properties`:
@@ -2802,8 +2840,12 @@ Additions to existing modules
   n+1+m≢m   : n + suc m ≢ m
   m*n≡0⇒m≡0 : .{{_ : NonZero n}} → m * n ≡ 0 → m ≡ 0
   n>0⇒n≢0   : n > 0 → n ≢ 0
-  m^n≢0     : .{{_ : NonZero m}} → NonZero (m ^ n)
   m*n≢0     : .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
+  m*n≢0⇒m≢0 : .{{NonZero (m * n)}} → NonZero m
+  m*n≢0⇒n≢0 : .{{NonZero (m * n)}} → NonZero n
+  m≢0∧n>1⇒m*n>1 : .{{_ : NonZero m}} .{{_ : NonTrivial n}} → NonTrivial (m * n)
+  n≢0∧m>1⇒m*n>1 : .{{_ : NonZero n}} .{{_ : NonTrivial m}} → NonTrivial (m * n)
+  m^n≢0     : .{{_ : NonZero m}} → NonZero (m ^ n)
   m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n
 
   s<s-injective : s<s p ≡ s<s q → p ≡ q
@@ -2811,7 +2853,7 @@ Additions to existing modules
   m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
 
   pred-mono-≤   : m ≤ n → pred m ≤ pred n
-  pred-mono-<   : .⦃ _ : NonZero m ⦄ → m < n → pred m < pred n
+  pred-mono-<   : .{{_ : NonZero m}} → m < n → pred m < pred n
 
   z<′s : zero <′ suc n
   s<′s : m <′ n → suc m <′ suc n
@@ -2852,7 +2894,7 @@ Additions to existing modules
   ⊓≡⊓′ : m ⊓ n ≡ m ⊓′ n
   ∣-∣≡∣-∣′ : ∣ m - n ∣ ≡ ∣ m - n ∣′
 
-  nonZero? : Decidable NonZero
+  nonTrivial? : Decidable NonTrivial
   eq? : A ↣ ℕ → DecidableEquality A
   ≤-<-connex : Connex _≤_ _<_
   ≥->-connex : Connex _≥_ _>_
@@ -2878,21 +2920,21 @@ Additions to existing modules
   m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n
   m*n/m!≡n/[m∸1]! : .{{_ : NonZero m}} → m * n / m ! ≡ n / (pred m) !
 
-  %-congˡ             : .⦃ _ : NonZero o ⦄ → m ≡ n → m % o ≡ n % o
-  %-congʳ             : .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ≡ n → o % m ≡ o % n
-  m≤n⇒[n∸m]%m≡n%m     : .⦃ _ : NonZero m ⦄ → m ≤ n → (n ∸ m) % m ≡ n % m
-  m*n≤o⇒[o∸m*n]%n≡o%n : .⦃ _ : NonZero n ⦄ → m * n ≤ o → (o ∸ m * n) % n ≡ o % n
-  m∣n⇒o%n%m≡o%m       : .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ∣ n → o % n % m ≡ o % m
-  m<n⇒m%n≡m           : .⦃ _ : NonZero n ⦄ → m < n → m % n ≡ m
-  m*n/o*n≡m/o         : .⦃ _ : NonZero o ⦄ ⦃ _ : NonZero (o * n) ⦄ → m * n / (o * n) ≡ m / o
-  m<n*o⇒m/o<n         : .⦃ _ : NonZero o ⦄ → m < n * o → m / o < n
-  [m∸n]/n≡m/n∸1       : .⦃ _ : NonZero n ⦄ → (m ∸ n) / n ≡ pred (m / n)
-  [m∸n*o]/o≡m/o∸n     : .⦃ _ : NonZero o ⦄ → (m ∸ n * o) / o ≡ m / o ∸ n
-  m/n/o≡m/[n*o]       : .⦃ _ : NonZero n ⦄ .⦃ _ : NonZero o ⦄ .⦃ _ : NonZero (n * o) ⦄ → m / n / o ≡ m / (n * o)
-  m%[n*o]/o≡m/o%n     : .⦃ _ : NonZero n ⦄ .⦃ _ : NonZero o ⦄ ⦃ _ : NonZero (n * o) ⦄ → m % (n * o) / o ≡ m / o % n
-  m%n*o≡m*o%[n*o]     : .⦃ _ : NonZero n ⦄ ⦃ _ : NonZero (n * o) ⦄ → m % n * o ≡ m * o % (n * o)
+  %-congˡ             : .{{_ : NonZero o}} → m ≡ n → m % o ≡ n % o
+  %-congʳ             : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ≡ n → o % m ≡ o % n
+  m≤n⇒[n∸m]%m≡n%m     : .{{_ : NonZero m}} → m ≤ n → (n ∸ m) % m ≡ n % m
+  m*n≤o⇒[o∸m*n]%n≡o%n : .{{_ : NonZero n}} → m * n ≤ o → (o ∸ m * n) % n ≡ o % n
+  m∣n⇒o%n%m≡o%m       : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ∣ n → o % n % m ≡ o % m
+  m<n⇒m%n≡m           : .{{_ : NonZero n}} → m < n → m % n ≡ m
+  m*n/o*n≡m/o         : .{{_ : NonZero o}} {{_ : NonZero (o * n)}} → m * n / (o * n) ≡ m / o
+  m<n*o⇒m/o<n         : .{{_ : NonZero o}} → m < n * o → m / o < n
+  [m∸n]/n≡m/n∸1       : .{{_ : NonZero n}} → (m ∸ n) / n ≡ pred (m / n)
+  [m∸n*o]/o≡m/o∸n     : .{{_ : NonZero o}} → (m ∸ n * o) / o ≡ m / o ∸ n
+  m/n/o≡m/[n*o]       : .{{_ : NonZero n}} .{{_ : NonZero o}} .{{_ : NonZero (n * o)}} → m / n / o ≡ m / (n * o)
+  m%[n*o]/o≡m/o%n     : .{{_ : NonZero n}} .{{_ : NonZero o}} {{_ : NonZero (n * o)}} → m % (n * o) / o ≡ m / o % n
+  m%n*o≡m*o%[n*o]     : .{{_ : NonZero n}} {{_ : NonZero (n * o)}} → m % n * o ≡ m * o % (n * o)
 
-  [m*n+o]%[p*n]≡[m*n]%[p*n]+o : ⦃ _ : NonZero (p * n) ⦄ → o < n → (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o
+  [m*n+o]%[p*n]≡[m*n]%[p*n]+o : {{_ : NonZero (p * n)}} → o < n → (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o
   ```
 
 * Added new proofs in `Data.Nat.Divisibility`:
@@ -3629,7 +3671,9 @@ This is a full list of proofs that have changed form to use irrelevant instance
 
 * In `Data.Nat.Coprimality`:
   ```
-  Bézout-coprime : ∀ {i j d} → Bézout.Identity (suc d) (i * suc d) (j * suc d) → Coprime i j
+  ¬0-coprimeTo-2+ : ∀ {n} → ¬ Coprime 0 (2 + n)
+  Bézout-coprime  : ∀ {i j d} → Bézout.Identity (suc d) (i * suc d) (j * suc d) → Coprime i j
+  prime⇒coprime   : ∀ m → Prime m → ∀ n → 0 < n → n < m → Coprime m n
   ```
 
 * In `Data.Nat.Divisibility`
diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index f2755c55fc..6135cfe01d 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -21,6 +21,7 @@ open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; _≢_; refl)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
+open import Relation.Unary using (Pred)
 
 ------------------------------------------------------------------------
 -- Types
@@ -28,6 +29,9 @@ open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Agda.Builtin.Nat public
   using (zero; suc) renaming (Nat to ℕ)
 
+--smart constructor
+pattern 2+ n = suc (suc n)
+
 ------------------------------------------------------------------------
 -- Boolean equality relation
 
@@ -59,8 +63,9 @@ m < n = suc m ≤ n
 
 -- Smart constructors of _<_
 
-pattern z<s {n}         = s≤s (z≤n {n})
+pattern z<s     {n}     = s≤s (z≤n {n})
 pattern s<s {m} {n} m<n = s≤s {m} {n} m<n
+pattern sz<ss   {n}     = s<s (z<s {n})
 
 -- Smart destructors of _≤_, _<_
 
@@ -72,7 +77,7 @@ s<s⁻¹ (s<s m<n) = m<n
 
 
 ------------------------------------------------------------------------
--- other ordering relations
+-- Other derived ordering relations
 
 _≥_ : Rel ℕ 0ℓ
 m ≥ n = n ≤ m
@@ -95,13 +100,19 @@ a ≯ b = ¬ a > b
 ------------------------------------------------------------------------
 -- Simple predicates
 
--- Defining `NonZero` in terms of `T` and therefore ultimately `⊤` and
--- `⊥` allows Agda to automatically infer nonZero-ness for any natural
--- of the form `suc n`. Consequently in many circumstances this
--- eliminates the need to explicitly pass a proof when the NonZero
--- argument is either an implicit or an instance argument.
+-- Defining these predicates in terms of `T` and therefore ultimately
+-- `⊤` and `⊥` allows Agda to automatically infer them for any natural
+-- of the correct form. Consequently in many circumstances this
+-- eliminates the need to explicitly pass a proof when the predicate
+-- argument is either an implicit or an instance argument. See `_/_`
+-- and `_%_` further down this file for examples.
 --
--- See `Data.Nat.DivMod` for an example.
+-- Furthermore, defining these predicates as single-field records
+-- (rather defining them directly as the type of their field) is
+-- necessary as the current version of Agda is far better at
+-- reconstructing meta-variable values for the record parameters.
+
+-- A predicate saying that a number is not equal to 0.
 
 record NonZero (n : ℕ) : Set where
   field
@@ -130,6 +141,34 @@ instance
 >-nonZero⁻¹ : ∀ n → .{{NonZero n}} → n > 0
 >-nonZero⁻¹ (suc n) = z<s
 
+-- The property of being a non-zero, non-unit
+
+record NonTrivial (n : ℕ) : Set where
+  field
+    nonTrivial : T (1 <ᵇ n)
+
+-- Instances
+
+instance
+  nonTrivial : ∀ {n} → NonTrivial (2+ n)
+  nonTrivial = _
+
+-- Constructors
+
+n>1⇒nonTrivial : ∀ {n} → n > 1 → NonTrivial n
+n>1⇒nonTrivial sz<ss = _
+
+-- Destructors
+
+nonTrivial⇒nonZero : ∀ n → .{{NonTrivial n}} → NonZero n
+nonTrivial⇒nonZero (2+ _) = _
+
+nonTrivial⇒n>1 : ∀ n → .{{NonTrivial n}} → n > 1
+nonTrivial⇒n>1 (2+ _) = sz<ss
+
+nonTrivial⇒≢1 : ∀ {n} → .{{NonTrivial n}} → n ≢ 1
+nonTrivial⇒≢1 {{()}} refl
+
 ------------------------------------------------------------------------
 -- Raw bundles
 
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index 1df320ea96..f385944f50 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -8,7 +8,6 @@
 
 module Data.Nat.Coprimality where
 
-open import Data.Empty
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.GCD
@@ -17,15 +16,18 @@ open import Data.Nat.Primality
 open import Data.Nat.Properties
 open import Data.Nat.DivMod
 open import Data.Product.Base as Prod
+open import Data.Sum.Base as Sum using (inj₁; inj₂)
 open import Function.Base using (_∘_)
 open import Level using (0ℓ)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; trans; cong; subst)
-open import Relation.Nullary as Nullary hiding (recompute)
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary as Nullary using (¬_; contradiction; yes ; no)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Symmetric; Decidable)
 
+private
+  variable d m n o p : ℕ
+
 open ≤-Reasoning
 
 ------------------------------------------------------------------------
@@ -35,22 +37,22 @@ open ≤-Reasoning
 -- prime), i.e. if their only common divisor is 1.
 
 Coprime : Rel ℕ 0ℓ
-Coprime m n = ∀ {i} → i ∣ m × i ∣ n → i ≡ 1
+Coprime m n = ∀ {d} → d ∣ m × d ∣ n → d ≡ 1
 
 ------------------------------------------------------------------------
 -- Relationship between GCD and coprimality
 
-coprime⇒GCD≡1 : ∀ {m n} → Coprime m n → GCD m n 1
-coprime⇒GCD≡1 {m} {n} c = GCD.is (1∣ m , 1∣ n) (∣-reflexive ∘ c)
+coprime⇒GCD≡1 : Coprime m n → GCD m n 1
+coprime⇒GCD≡1 {m} {n} coprime = GCD.is (1∣ m , 1∣ n) (∣-reflexive ∘ coprime)
 
-GCD≡1⇒coprime : ∀ {m n} → GCD m n 1 → Coprime m n
+GCD≡1⇒coprime : GCD m n 1 → Coprime m n
 GCD≡1⇒coprime g cd with divides q eq ← GCD.greatest g cd
   = m*n≡1⇒n≡1 q _ (P.sym eq)
 
-coprime⇒gcd≡1 : ∀ {m n} → Coprime m n → gcd m n ≡ 1
+coprime⇒gcd≡1 : Coprime m n → gcd m n ≡ 1
 coprime⇒gcd≡1 coprime = GCD.unique (gcd-GCD _ _) (coprime⇒GCD≡1 coprime)
 
-gcd≡1⇒coprime : ∀ {m n} → gcd m n ≡ 1 → Coprime m n
+gcd≡1⇒coprime : gcd m n ≡ 1 → Coprime m n
 gcd≡1⇒coprime gcd≡1 = GCD≡1⇒coprime (subst (GCD _ _) gcd≡1 (gcd-GCD _ _))
 
 coprime-/gcd : ∀ m n .{{_ : NonZero (gcd m n)}} →
@@ -67,14 +69,14 @@ private
   0≢1 : 0 ≢ 1
   0≢1 ()
 
-  2+≢1 : ∀ {n} → suc (suc n) ≢ 1
+  2+≢1 : ∀ {n} → 2+ n ≢ 1
   2+≢1 ()
 
 coprime? : Decidable Coprime
-coprime? i j with mkGCD i j
-... | (0           , g) = no  (0≢1  ∘ GCD.unique g ∘ coprime⇒GCD≡1)
-... | (1           , g) = yes (GCD≡1⇒coprime g)
-... | (suc (suc d) , g) = no  (2+≢1 ∘ GCD.unique g ∘ coprime⇒GCD≡1)
+coprime? m n with mkGCD m n
+... | (0    , g) = no  (0≢1  ∘ GCD.unique g ∘ coprime⇒GCD≡1)
+... | (1    , g) = yes (GCD≡1⇒coprime g)
+... | (2+ _ , g) = no  (2+≢1 ∘ GCD.unique g ∘ coprime⇒GCD≡1)
 
 ------------------------------------------------------------------------
 -- Other basic properties
@@ -86,21 +88,21 @@ coprime? i j with mkGCD i j
 
 -- Nothing except for 1 is coprime to 0.
 
-0-coprimeTo-m⇒m≡1 : ∀ {m} → Coprime 0 m → m ≡ 1
-0-coprimeTo-m⇒m≡1 {m} c = c (m ∣0 , ∣-refl)
+0-coprimeTo-m⇒m≡1 : Coprime 0 m → m ≡ 1
+0-coprimeTo-m⇒m≡1 {m} coprime = coprime (m ∣0 , ∣-refl)
 
-¬0-coprimeTo-2+ : ∀ {n} → ¬ Coprime 0 (2 + n)
-¬0-coprimeTo-2+ coprime = contradiction (0-coprimeTo-m⇒m≡1 coprime) λ()
+¬0-coprimeTo-2+ : .{{NonTrivial n}} → ¬ Coprime 0 n
+¬0-coprimeTo-2+ {n} coprime = contradiction (0-coprimeTo-m⇒m≡1 coprime) (nonTrivial⇒≢1 {n})
 
 -- If m and n are coprime, then n + m and n are also coprime.
 
-coprime-+ : ∀ {m n} → Coprime m n → Coprime (n + m) n
-coprime-+ c (d₁ , d₂) = c (∣m+n∣m⇒∣n d₁ d₂ , d₂)
+coprime-+ : Coprime m n → Coprime (n + m) n
+coprime-+ coprime (d₁ , d₂) = coprime (∣m+n∣m⇒∣n d₁ d₂ , d₂)
 
 -- Recomputable
 
-recompute : ∀ {n d} → .(Coprime n d) → Coprime n d
-recompute {n} {d} c = Nullary.recompute (coprime? n d) c
+recompute : .(Coprime n d) → Coprime n d
+recompute {n} {d} coprime = Nullary.recompute (coprime? n d) coprime
 
 ------------------------------------------------------------------------
 -- Relationship with Bezout's lemma
@@ -108,8 +110,8 @@ recompute {n} {d} c = Nullary.recompute (coprime? n d) c
 -- If the "gcd" in Bézout's identity is non-zero, then the "other"
 -- divisors are coprime.
 
-Bézout-coprime : ∀ {i j d} .{{_ : NonZero d}} →
-                 Bézout.Identity d (i * d) (j * d) → Coprime i j
+Bézout-coprime : .{{NonZero d}} →
+                 Bézout.Identity d (m * d) (n * d) → Coprime m n
 Bézout-coprime {d = suc _} (Bézout.+- x y eq) (divides-refl q₁ , divides-refl q₂) =
   lem₁₀ y q₂ x q₁ eq
 Bézout-coprime {d = suc _} (Bézout.-+ x y eq) (divides-refl q₁ , divides-refl q₂) =
@@ -117,21 +119,20 @@ Bézout-coprime {d = suc _} (Bézout.-+ x y eq) (divides-refl q₁ , divides-ref
 
 -- Coprime numbers satisfy Bézout's identity.
 
-coprime-Bézout : ∀ {i j} → Coprime i j → Bézout.Identity 1 i j
+coprime-Bézout : Coprime m n → Bézout.Identity 1 m n
 coprime-Bézout = Bézout.identity ∘ coprime⇒GCD≡1
 
--- If i divides jk and is coprime to j, then it divides k.
+-- If m divides n*o and is coprime to n, then it divides o.
 
-coprime-divisor : ∀ {k i j} → Coprime i j → i ∣ j * k → i ∣ k
-coprime-divisor {k} c (divides q eq′) with coprime-Bézout c
-... | Bézout.+- x y eq = divides (x * k ∸ y * q) (lem₈ x y eq eq′)
-... | Bézout.-+ x y eq = divides (y * q ∸ x * k) (lem₉ x y eq eq′)
+coprime-divisor : Coprime m n → m ∣ n * o → m ∣ o
+coprime-divisor {o = o} c (divides q eq′) with coprime-Bézout c
+... | Bézout.+- x y eq = divides (x * o ∸ y * q) (lem₈ x y eq eq′)
+... | Bézout.-+ x y eq = divides (y * q ∸ x * o) (lem₉ x y eq eq′)
 
--- If d is a common divisor of mk and nk, and m and n are coprime,
--- then d divides k.
+-- If d is a common divisor of m*o and n*o, and m and n are coprime,
+-- then d divides o.
 
-coprime-factors : ∀ {d m n k} →
-                  Coprime m n → d ∣ m * k × d ∣ n * k → d ∣ k
+coprime-factors : Coprime m n → d ∣ m * o × d ∣ n * o → d ∣ o
 coprime-factors c (divides q₁ eq₁ , divides q₂ eq₂) with coprime-Bézout c
 ... | Bézout.+- x y eq = divides (x * q₁ ∸ y * q₂) (lem₁₁ x y eq eq₁ eq₂)
 ... | Bézout.-+ x y eq = divides (y * q₂ ∸ x * q₁) (lem₁₁ y x eq eq₂ eq₁)
@@ -139,11 +140,7 @@ coprime-factors c (divides q₁ eq₁ , divides q₂ eq₂) with coprime-Bézout
 ------------------------------------------------------------------------
 -- Primality implies coprimality.
 
-prime⇒coprime : ∀ m → Prime m →
-                ∀ n → 0 < n → n < m → Coprime m n
-prime⇒coprime (suc (suc _)) p _ _ _ {0} (0∣m , _) =
-  contradiction (0∣⇒≡0 0∣m) λ()
-prime⇒coprime (suc (suc _)) _ _ _ _ {1} _         = refl
-prime⇒coprime (suc (suc _)) p (suc _) _ n<m {(suc (suc _))} (d∣m , d∣n) =
-  contradiction d∣m (p 2≤d d<m)
-  where 2≤d = s≤s (s≤s z≤n); d<m = <-≤-trans (s≤s (∣⇒≤ d∣n)) n<m
+prime⇒coprime : Prime p → .{{NonZero n}} → n < p → Coprime p n
+prime⇒coprime p n<p (d∣p , d∣n) with prime⇒irreducible p d∣p
+... | inj₁ d≡1      = d≡1
+... | inj₂ d≡p@refl = contradiction n<p (≤⇒≯ (∣⇒≤ d∣n))
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index bb30716e01..9083642f80 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -12,11 +12,10 @@ open import Algebra
 open import Data.Nat.Base
 open import Data.Nat.DivMod
 open import Data.Nat.Properties
-open import Data.Unit.Base using (tt)
 open import Function.Base using (_∘′_; _$_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (0ℓ)
-open import Relation.Nullary.Decidable as Dec using (False; yes; no)
+open import Relation.Nullary.Decidable as Dec using (yes; no)
 open import Relation.Nullary.Negation.Core using (contradiction)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Bundles using (Preorder; Poset)
@@ -308,3 +307,27 @@ m≤n⇒m!∣n! m≤n = help (≤⇒≤′ m≤n)
   help : ∀ {m n} → m ≤′ n → m ! ∣ n !
   help {m} {n}     ≤′-refl        = ∣-refl
   help {m} {suc n} (≤′-step m≤′n) = ∣n⇒∣m*n (suc n) (help m≤′n)
+
+------------------------------------------------------------------------
+-- Properties of _BoundedNonTrivialDivisor_
+
+-- Smart constructor
+
+hasNonTrivialDivisor-≢ : ∀ {d n} → .{{NonTrivial d}} → .{{NonZero n}} →
+                         d ≢ n → d ∣ n → n HasNonTrivialDivisorLessThan n
+hasNonTrivialDivisor-≢ d≢n d∣n
+  = hasNonTrivialDivisor (≤∧≢⇒< (∣⇒≤ d∣n) d≢n) d∣n
+
+-- Monotonicity wrt ∣
+
+hasNonTrivialDivisor-∣ : ∀ {m n o} → m HasNonTrivialDivisorLessThan n → m ∣ o →
+                         o HasNonTrivialDivisorLessThan n
+hasNonTrivialDivisor-∣ (hasNonTrivialDivisor d<n d∣m) n∣o
+  = hasNonTrivialDivisor d<n (∣-trans d∣m n∣o)
+
+-- Monotonicity wrt ≤
+
+hasNonTrivialDivisor-≤ : ∀ {m n o} → m HasNonTrivialDivisorLessThan n → n ≤ o →
+                             m HasNonTrivialDivisorLessThan o
+hasNonTrivialDivisor-≤ (hasNonTrivialDivisor d<n d∣m) m≤o =
+  hasNonTrivialDivisor (<-≤-trans d<n m≤o) d∣m
diff --git a/src/Data/Nat/Divisibility/Core.agda b/src/Data/Nat/Divisibility/Core.agda
index 2f8cdbe241..58ef913a5f 100644
--- a/src/Data/Nat/Divisibility/Core.agda
+++ b/src/Data/Nat/Divisibility/Core.agda
@@ -12,7 +12,7 @@
 
 module Data.Nat.Divisibility.Core where
 
-open import Data.Nat.Base using (ℕ; _*_)
+open import Data.Nat.Base using (ℕ; _*_; _<_; NonTrivial)
 open import Data.Nat.Properties
 open import Level using (0ℓ)
 open import Relation.Nullary.Negation using (¬_)
@@ -21,7 +21,7 @@ open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong₂; module ≡-Reasoning)
 
 ------------------------------------------------------------------------
--- Definition
+-- Main definition
 --
 -- m ∣ n is inhabited iff m divides n. Some sources, like Hardy and
 -- Wright's "An Introduction to the Theory of Numbers", require m to
@@ -40,10 +40,25 @@ open _∣_ using (quotient) public
 _∤_ : Rel ℕ 0ℓ
 m ∤ n = ¬ (m ∣ n)
 
--- smart constructor
+-- Smart constructor
 
 pattern divides-refl q = divides q refl
 
+------------------------------------------------------------------------
+-- Restricted divisor relation
+
+-- Relation for having a non-trivial divisor below a given bound.
+-- Useful when reasoning about primality.
+infix 10 _HasNonTrivialDivisorLessThan_
+
+record _HasNonTrivialDivisorLessThan_ (m n : ℕ) : Set where
+  constructor hasNonTrivialDivisor
+  field
+    {divisor}       : ℕ
+    .{{nontrivial}} : NonTrivial divisor
+    divisor-<       : divisor < n
+    divisor-∣       : divisor ∣ m
+
 ------------------------------------------------------------------------
 -- Basic properties
 
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index 361df42b73..620f0c87b0 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -8,92 +8,245 @@
 
 module Data.Nat.Primality where
 
-open import Data.Empty using (⊥)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.GCD using (module GCD; module Bézout)
 open import Data.Nat.Properties
-open import Data.Product.Base using (∃; _×_; map₂; _,_)
-open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
+open import Data.Product.Base using (∃-syntax; _×_; map₂; _,_)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (flip; _∘_; _∘′_)
+open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Nullary.Decidable as Dec
-  using (yes; no; from-yes; ¬?; decidable-stable; _×-dec_; _→-dec_)
-open import Relation.Nullary.Negation using (¬_; contradiction)
-open import Relation.Unary using (Decidable)
+  using (yes; no; from-yes; from-no; ¬?; _×-dec_; _⊎-dec_; _→-dec_; decidable-stable)
+open import Relation.Nullary.Negation using (¬_; contradiction; contradiction₂)
+open import Relation.Unary using (Pred; Decidable)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality
-  using (refl; cong)
+  using (_≡_; _≢_; refl; cong)
 
 private
   variable
-    n : ℕ
+    d m n o p : ℕ
+
+  recompute-nonTrivial : .{{NonTrivial n}} → NonTrivial n
+  recompute-nonTrivial {n} {{nontrivial}} =
+    Dec.recompute (nonTrivial? n) nontrivial
 
 ------------------------------------------------------------------------
 -- Definitions
+------------------------------------------------------------------------
+
+-- The positive/existential relation `BoundedNonTrivialDivisor` is
+-- the basis for the whole development, as it captures the possible
+-- non-trivial divisors of a given number; its complement, `Rough`,
+-- therefore sets *lower* bounds on any possible such divisors.
 
--- Definition of compositeness
+-- The predicate `Composite` is then defined as the 'diagonal' instance
+-- of `BoundedNonTrivialDivisor`, while `Prime` is essentially defined as
+-- the complement of `Composite`. Finally, `Irreducible` is the positive
+-- analogue of `Prime`.
 
-Composite : ℕ → Set
-Composite 0 = ⊥
-Composite 1 = ⊥
-Composite n = ∃ λ d → 2 ≤ d × d < n × d ∣ n
+------------------------------------------------------------------------
+-- Roughness
 
--- Definition of primality.
+-- A number is m-rough if all its non-trivial divisors are bounded below
+-- by m.
+infix 10 _Rough_
 
-Prime : ℕ → Set
-Prime 0 = ⊥
-Prime 1 = ⊥
-Prime n = ∀ {d} → 2 ≤ d → d < n → d ∤ n
+_Rough_ : ℕ → Pred ℕ _
+m Rough n = ¬ (n HasNonTrivialDivisorLessThan m)
 
 ------------------------------------------------------------------------
--- Decidability
+-- Compositeness
 
-composite? : Decidable Composite
-composite? 0               = no λ()
-composite? 1               = no λ()
-composite? n@(suc (suc _)) = Dec.map′
-  (map₂ λ { (a , b , c) → (b , a , c)})
-  (map₂ λ { (a , b , c) → (b , a , c)})
-  (anyUpTo? (λ d → 2 ≤? d ×-dec d ∣? n) n)
+-- A number is composite if it has a proper non-trivial divisor.
+Composite : Pred ℕ _
+Composite n = n HasNonTrivialDivisorLessThan n
 
-prime? : Decidable Prime
-prime? 0               = no λ()
-prime? 1               = no λ()
-prime? n@(suc (suc _)) = Dec.map′
-  (λ f {d} → flip (f {d}))
-  (λ f {d} → flip (f {d}))
-  (allUpTo? (λ d → 2 ≤? d →-dec ¬? (d ∣? n)) n)
+-- A shorter pattern synonym for the record constructor producing a
+-- witness for `Composite`.
+pattern
+  composite {d} d<n d∣n = hasNonTrivialDivisor {divisor = d} d<n d∣n
 
 ------------------------------------------------------------------------
--- Relationships between compositeness and primality
+-- Primality
 
-composite⇒¬prime : Composite n → ¬ Prime n
-composite⇒¬prime {n@(suc (suc _))} (d , 2≤d , d<n , d∣n) n-prime =
-  n-prime 2≤d d<n d∣n
+-- Prime as the complement of Composite (and hence the diagonal of Rough
+-- as defined above). The constructor `prime` takes a proof `notComposite`
+-- that NonTrivial p is not composite and thereby enforces that:
+-- * p is a fortiori NonZero and NonUnit
+-- * p is p-Rough, i.e. any proper divisor must be at least p, i.e. p itself
+record Prime (p : ℕ) : Set where
+  constructor prime
+  field
+    .{{nontrivial}} : NonTrivial p
+    notComposite    : ¬ Composite p
 
-¬composite⇒prime : 2 ≤ n → ¬ Composite n → Prime n
-¬composite⇒prime (s≤s (s≤s _)) ¬n-composite {d} 2≤d d<n d∣n =
-  ¬n-composite (d , 2≤d , d<n , d∣n)
+------------------------------------------------------------------------
+-- Irreducibility
 
-prime⇒¬composite : Prime n → ¬ Composite n
-prime⇒¬composite {n@(suc (suc _))} n-prime (d , 2≤d , d<n , d∣n) =
-  n-prime 2≤d d<n d∣n
+Irreducible : Pred ℕ _
+Irreducible n = ∀ {d} → d ∣ n → d ≡ 1 ⊎ d ≡ n
+
+------------------------------------------------------------------------
+-- Properties
+------------------------------------------------------------------------
+
+------------------------------------------------------------------------
+-- Roughness
+
+-- 1 is always n-rough
+rough-1 : ∀ n → n Rough 1
+rough-1 _ (hasNonTrivialDivisor _ d∣1) =
+  contradiction (∣1⇒≡1 d∣1) nonTrivial⇒≢1
+
+-- Any number is 0-, 1- and 2-rough,
+-- because no proper divisor d can be strictly less than 0, 1, or 2
+0-rough : 0 Rough n
+0-rough (hasNonTrivialDivisor () _)
 
--- note that this has to recompute the factor!
-¬prime⇒composite : 2 ≤ n → ¬ Prime n → Composite n
-¬prime⇒composite {n} 2≤n ¬n-prime =
-  decidable-stable (composite? n) (¬n-prime ∘′ ¬composite⇒prime 2≤n)
+1-rough : 1 Rough n
+1-rough (hasNonTrivialDivisor {{()}} z<s _)
+
+2-rough : 2 Rough n
+2-rough (hasNonTrivialDivisor {{()}} (s<s z<s) _)
+
+-- If a number n > 1 is m-rough, then m ≤ n
+rough⇒≤ : .{{NonTrivial n}} → m Rough n → m ≤ n
+rough⇒≤ rough = ≮⇒≥ n≮m
+  where n≮m = λ m>n → rough (hasNonTrivialDivisor m>n ∣-refl)
+
+-- If a number n is m-rough, and m ∤ n, then n is (suc m)-rough
+∤⇒rough-suc : m ∤ n → m Rough n → (suc m) Rough n
+∤⇒rough-suc m∤n r (hasNonTrivialDivisor d<1+m d∣n)
+  with m<1+n⇒m<n∨m≡n d<1+m
+... | inj₁ d<m      = r (hasNonTrivialDivisor d<m d∣n)
+... | inj₂ d≡m@refl = contradiction d∣n m∤n
+
+-- If a number is m-rough, then so are all of its divisors
+rough∧∣⇒rough : m Rough o → n ∣ o → m Rough n
+rough∧∣⇒rough r n∣o bntd = r (hasNonTrivialDivisor-∣ bntd n∣o)
 
 ------------------------------------------------------------------------
--- Euclid's lemma
+-- Compositeness
+
+-- Smart constructors
+
+composite-≢ : ∀ d → .{{NonTrivial d}} → .{{NonZero n}} →
+              d ≢ n → d ∣ n → Composite n
+composite-≢ d = hasNonTrivialDivisor-≢ {d}
+
+composite-∣ : .{{NonZero n}} → Composite m → m ∣ n → Composite n
+composite-∣ (composite {d} d<m d∣n) m∣n@(divides-refl q)
+  = composite (*-monoʳ-< q d<m) (*-monoʳ-∣ q d∣n)
+  where instance
+    _ = m≢0∧n>1⇒m*n>1 q d
+    _ = m*n≢0⇒m≢0 q
+
+-- Basic (counter-)examples of Composite
+
+¬composite[0] : ¬ Composite 0
+¬composite[0] = 0-rough
+
+¬composite[1] : ¬ Composite 1
+¬composite[1] = 1-rough
+
+composite[4] : Composite 4
+composite[4] = composite-≢ 2 (λ()) (divides-refl 2)
+
+composite[6] : Composite 6
+composite[6] = composite-≢ 3 (λ()) (divides-refl 2)
+
+composite⇒nonZero : Composite n → NonZero n
+composite⇒nonZero {suc _} _ = _
+
+composite⇒nonTrivial : Composite n → NonTrivial n
+composite⇒nonTrivial {1}    composite[1] =
+  contradiction composite[1] ¬composite[1]
+composite⇒nonTrivial {2+ _} _            = _
+
+composite? : Decidable Composite
+composite? n = Dec.map CompositeUpTo⇔Composite (compositeUpTo? n)
+  where
+  -- For technical reasons, in order to be able to prove decidability
+  -- via the `all?` and `any?` combinators for *bounded* predicates on
+  -- `ℕ`, we further define the bounded counterparts to predicates
+  -- `P...` as `P...UpTo` and show the equivalence of the two.
+
+  -- Equivalent bounded predicate definition
+  CompositeUpTo : Pred ℕ _
+  CompositeUpTo n = ∃[ d ] d < n × NonTrivial d × d ∣ n
+
+  -- Proof of equivalence
+  comp-upto⇒comp : CompositeUpTo n → Composite n
+  comp-upto⇒comp (_ , d<n , ntd , d∣n) = composite d<n d∣n
+    where instance _ = ntd
+
+  comp⇒comp-upto : Composite n → CompositeUpTo n
+  comp⇒comp-upto (composite d<n d∣n) = _ , d<n , recompute-nonTrivial , d∣n
+
+  CompositeUpTo⇔Composite : CompositeUpTo n ⇔ Composite n
+  CompositeUpTo⇔Composite = mk⇔ comp-upto⇒comp comp⇒comp-upto
+
+  -- Proof of decidability
+  compositeUpTo? : Decidable CompositeUpTo
+  compositeUpTo? n = anyUpTo? (λ d → nonTrivial? d ×-dec d ∣? n) n
+
+------------------------------------------------------------------------
+-- Primality
+
+-- Basic (counter-)examples
 
--- For p prime, if p ∣ m * n, then either p ∣ m or p ∣ n.
+¬prime[0] : ¬ Prime 0
+¬prime[0] ()
+
+¬prime[1] : ¬ Prime 1
+¬prime[1] ()
+
+prime[2] : Prime 2
+prime[2] = prime 2-rough
+
+prime⇒nonZero : Prime p → NonZero p
+prime⇒nonZero _ = nonTrivial⇒nonZero _
+
+prime⇒nonTrivial : Prime p → NonTrivial p
+prime⇒nonTrivial _ = recompute-nonTrivial
+
+prime? : Decidable Prime
+prime? 0        = no ¬prime[0]
+prime? 1        = no ¬prime[1]
+prime? n@(2+ _) = Dec.map PrimeUpTo⇔Prime (primeUpTo? n)
+  where
+  -- Equivalent bounded predicate definition
+  PrimeUpTo : Pred ℕ _
+  PrimeUpTo n = ∀ {d} → d < n → NonTrivial d → d ∤ n
+
+  -- Proof of equivalence
+  prime⇒prime-upto : Prime n → PrimeUpTo n
+  prime⇒prime-upto (prime p) {d} d<n ntd d∣n
+    = p (composite d<n d∣n) where instance _ = ntd
+
+  prime-upto⇒prime : .{{NonTrivial n}} → PrimeUpTo n → Prime n
+  prime-upto⇒prime upto = prime
+    λ (composite d<n d∣n) → upto d<n recompute-nonTrivial d∣n
+
+  PrimeUpTo⇔Prime : .{{NonTrivial n}} → PrimeUpTo n ⇔ Prime n
+  PrimeUpTo⇔Prime = mk⇔ prime-upto⇒prime prime⇒prime-upto
+
+  -- Proof of decidability
+  primeUpTo? : Decidable PrimeUpTo
+  primeUpTo? n = allUpTo? (λ d → nonTrivial? d →-dec ¬? (d ∣? n)) n
+
+-- Euclid's lemma - for p prime, if p ∣ m * n, then either p ∣ m or p ∣ n.
+--
 -- This demonstrates that the usual definition of prime numbers matches
 -- the ring theoretic definition of a prime element of the semiring ℕ.
 -- This is useful for proving many other theorems involving prime numbers.
 euclidsLemma : ∀ m n {p} → Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n
-euclidsLemma m n {p@(suc (suc _))} p-prime p∣m*n = result
+euclidsLemma m n {p} pp@(prime pr) p∣m*n = result
   where
   open ∣-Reasoning
+  instance _ = prime⇒nonZero pp
 
   p∣rmn : ∀ r → p ∣ r * m * n
   p∣rmn r = begin
@@ -105,10 +258,12 @@ euclidsLemma m n {p@(suc (suc _))} p-prime p∣m*n = result
   result : p ∣ m ⊎ p ∣ n
   result with Bézout.lemma m p
   -- if the GCD of m and p is zero then p must be zero, which is
-  -- impossible as p is a prime
-  ... | Bézout.result 0 g _ = contradiction (0∣⇒≡0 (GCD.gcd∣n g)) λ()
+  -- impossible as p is a prime.
+  -- note: this should be a typechecker-rejectable case!?
+  ... | Bézout.result 0 g _ =
+    contradiction (0∣⇒≡0 (GCD.gcd∣n g)) (≢-nonZero⁻¹ _)
 
-  -- if the GCD of m and p is one then m and p is coprime, and we know
+  -- if the GCD of m and p is one then m and p are coprime, and we know
   -- that for some integers s and r, sm + rp = 1. We can use this fact
   -- to determine that p divides n
   ... | Bézout.result 1 _ (Bézout.+- r s 1+sp≡rm) =
@@ -125,19 +280,123 @@ euclidsLemma m n {p@(suc (suc _))} p-prime p∣m*n = result
       n + r * m * n ≡⟨ +-comm n (r * m * n) ⟩
       r * m * n + n ∎))
 
-  -- if the GCD of m and p is greater than one, then it must be p and hence p ∣ m.
-  ... | Bézout.result d@(suc (suc _)) g _ with d ≟ p
-  ...   | yes refl = inj₁ (GCD.gcd∣m g)
-  ...   | no  d≢p  = contradiction (GCD.gcd∣n g) (p-prime 2≤d d<p)
-    where 2≤d = s≤s (s≤s z≤n); d<p = flip ≤∧≢⇒< d≢p (∣⇒≤ (GCD.gcd∣n g))
+  -- if the GCD of m and p is greater than one, then it must be p and
+  -- hence p ∣ m.
+  ... | Bézout.result d@(2+ _) g _ with d ≟ p
+  ...   | yes d≡p@refl = inj₁ (GCD.gcd∣m g)
+  ...   | no  d≢p = contradiction (composite-≢ d d≢p (GCD.gcd∣n g)) pr
+
+-- Relationship between roughness and primality.
+prime⇒rough : Prime p → p Rough p
+prime⇒rough (prime pr) = pr
+
+-- If a number n is p-rough, and p > 1 divides n, then p must be prime
+rough∧∣⇒prime : .{{NonTrivial p}} → p Rough n → p ∣ n → Prime p
+rough∧∣⇒prime r p∣n = prime (rough∧∣⇒rough r p∣n)
+
+-- Relationship between compositeness and primality.
+composite⇒¬prime : Composite n → ¬ Prime n
+composite⇒¬prime composite[d] (prime p) = p composite[d]
+
+¬composite⇒prime : .{{NonTrivial n}} → ¬ Composite n → Prime n
+¬composite⇒prime = prime
+
+prime⇒¬composite : Prime n → ¬ Composite n
+prime⇒¬composite (prime p) = p
+
+-- Note that this has to recompute the factor!
+¬prime⇒composite : .{{NonTrivial n}} → ¬ Prime n → Composite n
+¬prime⇒composite {n} ¬prime[n] =
+  decidable-stable (composite? n) (¬prime[n] ∘′ ¬composite⇒prime)
+
+------------------------------------------------------------------------
+-- Basic (counter-)examples of Irreducible
+
+¬irreducible[0] : ¬ Irreducible 0
+¬irreducible[0] irr[0] = contradiction₂ 2≡1⊎2≡0 (λ ()) (λ ())
+  where 2≡1⊎2≡0 = irr[0] {2} (divides-refl 0)
+
+irreducible[1] : Irreducible 1
+irreducible[1] m|1 = inj₁ (∣1⇒≡1 m|1)
+
+irreducible[2] : Irreducible 2
+irreducible[2] {zero}  0∣2 with () ← 0∣⇒≡0 0∣2
+irreducible[2] {suc _} d∣2 with ∣⇒≤ d∣2
+... | z<s     = inj₁ refl
+... | s<s z<s = inj₂ refl
+
+irreducible⇒nonZero : Irreducible n → NonZero n
+irreducible⇒nonZero {zero}    = flip contradiction ¬irreducible[0]
+irreducible⇒nonZero {suc _} _ = _
+
+irreducible? : Decidable Irreducible
+irreducible? zero      = no ¬irreducible[0]
+irreducible? n@(suc _) =
+  Dec.map IrreducibleUpTo⇔Irreducible (irreducibleUpTo? n)
+  where
+  -- Equivalent bounded predicate definition
+  IrreducibleUpTo : Pred ℕ _
+  IrreducibleUpTo n = ∀ {d} → d < n → d ∣ n → d ≡ 1 ⊎ d ≡ n
+
+  -- Proof of equivalence
+  irr-upto⇒irr : .{{NonZero n}} → IrreducibleUpTo n → Irreducible n
+  irr-upto⇒irr irr-upto m∣n
+    = [ flip irr-upto m∣n , inj₂ ]′ (m≤n⇒m<n∨m≡n (∣⇒≤ m∣n))
+
+  irr⇒irr-upto : Irreducible n → IrreducibleUpTo n
+  irr⇒irr-upto irr m<n m∣n = irr m∣n
+
+  IrreducibleUpTo⇔Irreducible : .{{NonZero n}} →
+                                 IrreducibleUpTo n ⇔ Irreducible n
+  IrreducibleUpTo⇔Irreducible = mk⇔ irr-upto⇒irr irr⇒irr-upto
+
+  -- Decidability
+  irreducibleUpTo? : Decidable IrreducibleUpTo
+  irreducibleUpTo? n = allUpTo?
+    (λ m → (m ∣? n) →-dec (m ≟ 1 ⊎-dec m ≟ n)) n
+
+-- Relationship between primality and irreducibility.
+prime⇒irreducible : Prime p → Irreducible p
+prime⇒irreducible {p} pp@(prime pr) = irr
+  where
+  instance _ = prime⇒nonZero pp
+  irr : .{{NonZero p}} → Irreducible p
+  irr {0}    0∣p = contradiction (0∣⇒≡0 0∣p) (≢-nonZero⁻¹ p)
+  irr {1}    1∣p = inj₁ refl
+  irr {2+ _} d∣p = inj₂ (≤∧≮⇒≡ (∣⇒≤ d∣p) d≮p)
+    where d≮p = λ d<p → pr (composite d<p d∣p)
+
+
+irreducible⇒prime : .{{NonTrivial p}} → Irreducible p → Prime p
+irreducible⇒prime irr = prime
+  λ (composite d<p d∣p) → [ nonTrivial⇒≢1 , (<⇒≢ d<p) ]′ (irr d∣p)
+
+------------------------------------------------------------------------
+-- Using decidability
+
+-- Once we have the above decision procedures, then instead of
+-- constructing proofs of e.g. Prime-ness by hand, we call the
+-- appropriate function, and use the witness extraction functions
+-- `from-yes`, `from-no` to return the checked proofs.
 
 private
 
-  -- Example: 2 is prime.
+  -- Example: 2 is prime, but not-composite.
   2-is-prime : Prime 2
   2-is-prime = from-yes (prime? 2)
 
+  2-is-not-composite : ¬ Composite 2
+  2-is-not-composite = from-no (composite? 2)
+
+  -- Example: 4 and 6 are composite, hence not-prime
+  4-is-composite : Composite 4
+  4-is-composite = from-yes (composite? 4)
+
+  4-is-not-prime : ¬ Prime 4
+  4-is-not-prime = from-no (prime? 4)
 
-  -- Example: 6 is composite
   6-is-composite : Composite 6
   6-is-composite = from-yes (composite? 6)
+
+  6-is-not-prime : ¬ Prime 6
+  6-is-not-prime = from-no (prime? 6)
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 37b1f33ee0..9df927064a 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -60,6 +60,15 @@ nonZero? : U.Decidable NonZero
 nonZero? zero    = no NonZero.nonZero
 nonZero? (suc n) = yes _
 
+------------------------------------------------------------------------
+-- Properties of NonTrivial
+------------------------------------------------------------------------
+
+nonTrivial? : U.Decidable NonTrivial
+nonTrivial? 0      = no λ()
+nonTrivial? 1      = no λ()
+nonTrivial? (2+ n) = yes _
+
 ------------------------------------------------------------------------
 -- Properties of _≡_
 ------------------------------------------------------------------------
@@ -902,6 +911,12 @@ m*n≡0⇒m≡0∨n≡0 (suc m) {zero}  eq = inj₂ refl
 m*n≢0 : ∀ m n → .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
 m*n≢0 (suc m) (suc n) = _
 
+m*n≢0⇒m≢0 : ∀ m {n} → .{{NonZero (m * n)}} → NonZero m
+m*n≢0⇒m≢0 (suc _) = _
+
+m*n≢0⇒n≢0 : ∀ m {n} → .{{NonZero (m * n)}} → NonZero n
+m*n≢0⇒n≢0 m {n} rewrite *-comm m n = m*n≢0⇒m≢0 n {m}
+
 m*n≡0⇒m≡0 : ∀ m n .{{_ : NonZero n}} → m * n ≡ 0 → m ≡ 0
 m*n≡0⇒m≡0 zero (suc _) eq = refl
 
@@ -922,6 +937,12 @@ m*n≡1⇒n≡1 m n eq = m*n≡1⇒m≡1 n m (trans (*-comm n m) eq)
   (m * o) * (n * p) ∎
   where open CommSemigroupProperties *-commutativeSemigroup
 
+m≢0∧n>1⇒m*n>1 : ∀ m n .{{_ : NonZero m}} .{{_ : NonTrivial n}} → NonTrivial (m * n)
+m≢0∧n>1⇒m*n>1 (suc m) (2+ n) = _
+
+n≢0∧m>1⇒m*n>1 : ∀ m n .{{_ : NonZero n}} .{{_ : NonTrivial m}} → NonTrivial (m * n)
+n≢0∧m>1⇒m*n>1 m n rewrite *-comm m n = m≢0∧n>1⇒m*n>1 n m
+
 ------------------------------------------------------------------------
 -- Other properties of _*_ and _≤_/_<_
 
@@ -1704,7 +1725,7 @@ pred-mono-≤ : pred Preserves _≤_ ⟶ _≤_
 pred-mono-≤ {zero}          _   = z≤n
 pred-mono-≤ {suc _} {suc _} m≤n = s≤s⁻¹ m≤n
 
-pred-mono-< : .⦃ _ : NonZero m ⦄ → m < n → pred m < pred n
+pred-mono-< : .{{NonZero m}} → m < n → pred m < pred n
 pred-mono-< {m = suc _} {n = suc _} = s<s⁻¹
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 1e49894eb6..0948447163 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -16,7 +16,7 @@ open import Data.Integer.Base as ℤ
 open import Data.Nat.GCD
 open import Data.Nat.Coprimality as C
   using (Coprime; Bézout-coprime; coprime-/gcd; coprime?; ¬0-coprimeTo-2+)
-open import Data.Nat.Base as ℕ using (ℕ; zero; suc) hiding (module ℕ)
+open import Data.Nat.Base as ℕ using (ℕ; zero; suc; 2+) hiding (module ℕ)
 open import Data.Rational.Unnormalised.Base as ℚᵘ using (ℚᵘ; mkℚᵘ)
 open import Data.Sum.Base using (inj₂)
 open import Function.Base using (id)
@@ -189,10 +189,11 @@ open ℤ public
 -- Constructors
 
 ≢-nonZero : ∀ {p} → p ≢ 0ℚ → NonZero p
-≢-nonZero {mkℚ -[1+ _ ] _       _} _   = _
-≢-nonZero {mkℚ +[1+ _ ] _       _} _   = _
-≢-nonZero {mkℚ +0       zero    _} p≢0 = contradiction refl p≢0
-≢-nonZero {mkℚ +0       (suc d) c} p≢0 = contradiction (λ {i} → C.recompute c {i}) ¬0-coprimeTo-2+
+≢-nonZero {mkℚ -[1+ _ ] _         _} _   = _
+≢-nonZero {mkℚ +[1+ _ ] _         _} _   = _
+≢-nonZero {mkℚ +0       zero      _} p≢0 = contradiction refl p≢0
+≢-nonZero {mkℚ +0       d@(suc m) c} p≢0 =
+  contradiction (λ {d} → C.recompute c {d}) (¬0-coprimeTo-2+ {{ℕ.nonTrivial {m}}})
 
 >-nonZero : ∀ {p} → p > 0ℚ → NonZero p
 >-nonZero {p@(mkℚ _ _ _)} (*<* p<q) = ℚᵘ.>-nonZero {toℚᵘ p} (ℚᵘ.*<* p<q)

From b58f923d704c7c9ce9e73d0da795b7791bdc661f Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 25 Nov 2023 05:06:09 +0000
Subject: [PATCH 190/316] [fixes #2168] Change names in
 `Algebra.Consequences.*` to reflect `style-guide` conventions (#2206)

* fixes issue #2168

* added more names for deprecation, plus `hiding` them in `Propositional`
---
 CHANGELOG.md                                  |  70 +++--
 src/Algebra/Consequences/Base.agda            |  17 +-
 src/Algebra/Consequences/Propositional.agda   | 123 +++++++--
 src/Algebra/Consequences/Setoid.agda          | 240 +++++++++++++-----
 .../Lattice/Properties/BooleanAlgebra.agda    |  10 +-
 src/Algebra/Lattice/Structures/Biased.agda    |  10 +-
 src/Algebra/Structures.agda                   |   4 +-
 src/Algebra/Structures/Biased.agda            |  16 +-
 src/Data/Integer/Properties.agda              |  10 +-
 src/Data/Nat/Binary/Properties.agda           |   2 +-
 src/Data/Nat/Binary/Subtraction.agda          |   4 +-
 src/Data/Nat/Properties.agda                  |  14 +-
 src/Data/Parity/Properties.agda               |   4 +-
 13 files changed, 383 insertions(+), 141 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0e50316b36..25065935ce 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1276,6 +1276,44 @@ Deprecated modules
 Deprecated names
 ----------------
 
+* In `Algebra.Consequences.Propositional`:
+  ```agda
+  comm+assoc⇒middleFour             ↦  comm∧assoc⇒middleFour
+  identity+middleFour⇒assoc         ↦  identity∧middleFour⇒assoc
+  identity+middleFour⇒comm          ↦  identity∧middleFour⇒comm
+  comm+distrˡ⇒distrʳ                ↦  comm∧distrˡ⇒distrʳ
+  comm+distrʳ⇒distrˡ                ↦  comm∧distrʳ⇒distrˡ
+  assoc+distribʳ+idʳ+invʳ⇒zeˡ       ↦  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
+  assoc+distribˡ+idʳ+invʳ⇒zeʳ       ↦  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
+  assoc+id+invʳ⇒invˡ-unique         ↦  assoc∧id∧invʳ⇒invˡ-unique
+  assoc+id+invˡ⇒invʳ-unique         ↦  assoc∧id∧invˡ⇒invʳ-unique
+  subst+comm⇒sym                    ↦  subst∧comm⇒sym
+  ```
+
+* In `Algebra.Consequences.Setoid`:
+  ```agda
+  comm+assoc⇒middleFour             ↦  comm∧assoc⇒middleFour
+  identity+middleFour⇒assoc         ↦  identity∧middleFour⇒assoc
+  identity+middleFour⇒comm          ↦  identity∧middleFour⇒comm
+  comm+cancelˡ⇒cancelʳ              ↦  comm∧cancelˡ⇒cancelʳ
+  comm+cancelʳ⇒cancelˡ              ↦  comm∧cancelʳ⇒cancelˡ
+  comm+almostCancelˡ⇒almostCancelʳ  ↦  comm∧almostCancelˡ⇒almostCancelʳ
+  comm+almostCancelʳ⇒almostCancelˡ  ↦  comm∧almostCancelʳ⇒almostCancelˡ
+  comm+idˡ⇒idʳ                      ↦  comm∧idˡ⇒idʳ
+  comm+idʳ⇒idˡ                      ↦  comm∧idʳ⇒idˡ
+  comm+zeˡ⇒zeʳ                      ↦  comm∧zeˡ⇒zeʳ
+  comm+zeʳ⇒zeˡ                      ↦  comm∧zeʳ⇒zeˡ
+  comm+invˡ⇒invʳ                    ↦  comm∧invˡ⇒invʳ
+  comm+invʳ⇒invˡ                    ↦  comm∧invʳ⇒invˡ
+  comm+distrˡ⇒distrʳ                ↦  comm∧distrˡ⇒distrʳ
+  comm+distrʳ⇒distrˡ                ↦  comm∧distrʳ⇒distrˡ
+  assoc+distribʳ+idʳ+invʳ⇒zeˡ       ↦  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
+  assoc+distribˡ+idʳ+invʳ⇒zeʳ       ↦  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
+  assoc+id+invʳ⇒invˡ-unique         ↦  assoc∧id∧invʳ⇒invˡ-unique
+  assoc+id+invˡ⇒invʳ-unique         ↦  assoc∧id∧invˡ⇒invʳ-unique
+  subst+comm⇒sym                    ↦  subst∧comm⇒sym
+  ```
+
 * In `Algebra.Construct.Zero`:
   ```agda
   rawMagma   ↦  Algebra.Construct.Terminal.rawMagma
@@ -2177,21 +2215,21 @@ Additions to existing modules
 
 * Added new proof to `Algebra.Consequences.Base`:
   ```agda
-  reflexive+selfInverse⇒involutive : Reflexive _≈_ → SelfInverse _≈_ f → Involutive _≈_ f
+  reflexive∧selfInverse⇒involutive : Reflexive _≈_ → SelfInverse _≈_ f → Involutive _≈_ f
   ```
 
 * Added new proofs to `Algebra.Consequences.Propositional`:
   ```agda
-  comm+assoc⇒middleFour     : Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
-  identity+middleFour⇒assoc : Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
-  identity+middleFour⇒comm  : Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
+  comm∧assoc⇒middleFour     : Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
+  identity∧middleFour⇒assoc : Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
+  identity∧middleFour⇒comm  : Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
   ```
 
 * Added new proofs to `Algebra.Consequences.Setoid`:
   ```agda
-  comm+assoc⇒middleFour     : Congruent₂ _∙_ → Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
-  identity+middleFour⇒assoc : Congruent₂ _∙_ → Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
-  identity+middleFour⇒comm  : Congruent₂ _∙_ → Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
+  comm∧assoc⇒middleFour     : Congruent₂ _∙_ → Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
+  identity∧middleFour⇒assoc : Congruent₂ _∙_ → Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
+  identity∧middleFour⇒comm  : Congruent₂ _∙_ → Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
 
   involutive⇒surjective  : Involutive f  → Surjective f
   selfInverse⇒involutive : SelfInverse f → Involutive f
@@ -2201,15 +2239,15 @@ Additions to existing modules
   selfInverse⇒injective  : SelfInverse f → Injective f
   selfInverse⇒bijective  : SelfInverse f → Bijective f
 
-  comm+idˡ⇒id              : Commutative _∙_ → LeftIdentity  e _∙_ → Identity e _∙_
-  comm+idʳ⇒id              : Commutative _∙_ → RightIdentity e _∙_ → Identity e _∙_
-  comm+zeˡ⇒ze              : Commutative _∙_ → LeftZero      e _∙_ → Zero     e _∙_
-  comm+zeʳ⇒ze              : Commutative _∙_ → RightZero     e _∙_ → Zero     e _∙_
-  comm+invˡ⇒inv            : Commutative _∙_ → LeftInverse  e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
-  comm+invʳ⇒inv            : Commutative _∙_ → RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
-  comm+distrˡ⇒distr        : Commutative _∙_ → _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
-  comm+distrʳ⇒distr        : Commutative _∙_ → _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
-  distrib+absorbs⇒distribˡ : Associative _∙_ → Commutative _◦_ → _∙_ Absorbs _◦_ → _◦_ Absorbs _∙_ → _◦_ DistributesOver _∙_ → _∙_ DistributesOverˡ _◦_
+  comm∧idˡ⇒id              : Commutative _∙_ → LeftIdentity  e _∙_ → Identity e _∙_
+  comm∧idʳ⇒id              : Commutative _∙_ → RightIdentity e _∙_ → Identity e _∙_
+  comm∧zeˡ⇒ze              : Commutative _∙_ → LeftZero      e _∙_ → Zero     e _∙_
+  comm∧zeʳ⇒ze              : Commutative _∙_ → RightZero     e _∙_ → Zero     e _∙_
+  comm∧invˡ⇒inv            : Commutative _∙_ → LeftInverse  e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm∧invʳ⇒inv            : Commutative _∙_ → RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm∧distrˡ⇒distr        : Commutative _∙_ → _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
+  comm∧distrʳ⇒distr        : Commutative _∙_ → _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
+  distrib∧absorbs⇒distribˡ : Associative _∙_ → Commutative _◦_ → _∙_ Absorbs _◦_ → _◦_ Absorbs _∙_ → _◦_ DistributesOver _∙_ → _∙_ DistributesOverˡ _◦_
   ```
 
 * Added new functions to `Algebra.Construct.DirectProduct`:
diff --git a/src/Algebra/Consequences/Base.agda b/src/Algebra/Consequences/Base.agda
index 446a82d868..574ad48a16 100644
--- a/src/Algebra/Consequences/Base.agda
+++ b/src/Algebra/Consequences/Base.agda
@@ -23,8 +23,21 @@ module _ {ℓ} {_•_ : Op₂ A} (_≈_ : Rel A ℓ) where
 
 module _ {ℓ} {f : Op₁ A} (_≈_ : Rel A ℓ) where
 
-  reflexive+selfInverse⇒involutive : Reflexive _≈_ →
+  reflexive∧selfInverse⇒involutive : Reflexive _≈_ →
                                      SelfInverse _≈_ f →
                                      Involutive _≈_ f
-  reflexive+selfInverse⇒involutive refl inv _ = inv refl
+  reflexive∧selfInverse⇒involutive refl inv _ = inv refl
 
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+reflexive+selfInverse⇒involutive = reflexive∧selfInverse⇒involutive
+{-# WARNING_ON_USAGE reflexive+selfInverse⇒involutive
+"Warning: reflexive+selfInverse⇒involutive was deprecated in v2.0.
+Please use reflexive∧selfInverse⇒involutive instead."
+#-}
diff --git a/src/Algebra/Consequences/Propositional.agda b/src/Algebra/Consequences/Propositional.agda
index 1adcb12afc..5c10a9e4cc 100644
--- a/src/Algebra/Consequences/Propositional.agda
+++ b/src/Algebra/Consequences/Propositional.agda
@@ -26,7 +26,21 @@ import Algebra.Consequences.Setoid (setoid A) as Base
 
 open Base public
   hiding
-  ( comm+assoc⇒middleFour
+  ( comm∧assoc⇒middleFour
+  ; identity∧middleFour⇒assoc
+  ; identity∧middleFour⇒comm
+  ; assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
+  ; assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
+  ; assoc∧id∧invʳ⇒invˡ-unique
+  ; assoc∧id∧invˡ⇒invʳ-unique
+  ; comm∧distrˡ⇒distrʳ
+  ; comm∧distrʳ⇒distrˡ
+  ; comm⇒sym[distribˡ]
+  ; subst∧comm⇒sym
+  ; wlog
+  ; sel⇒idem
+-- plus all the deprecated versions of the above
+  ; comm+assoc⇒middleFour
   ; identity+middleFour⇒assoc
   ; identity+middleFour⇒comm
   ; assoc+distribʳ+idʳ+invʳ⇒zeˡ
@@ -35,10 +49,7 @@ open Base public
   ; assoc+id+invˡ⇒invʳ-unique
   ; comm+distrˡ⇒distrʳ
   ; comm+distrʳ⇒distrˡ
-  ; comm⇒sym[distribˡ]
   ; subst+comm⇒sym
-  ; wlog
-  ; sel⇒idem
   )
 
 ------------------------------------------------------------------------
@@ -46,43 +57,43 @@ open Base public
 
 module _ {_∙_ _⁻¹ ε} where
 
-  assoc+id+invʳ⇒invˡ-unique : Associative _∙_ → Identity ε _∙_ →
+  assoc∧id∧invʳ⇒invˡ-unique : Associative _∙_ → Identity ε _∙_ →
                               RightInverse ε _⁻¹ _∙_ →
                               ∀ x y → (x ∙ y) ≡ ε → x ≡ (y ⁻¹)
-  assoc+id+invʳ⇒invˡ-unique = Base.assoc+id+invʳ⇒invˡ-unique (cong₂ _)
+  assoc∧id∧invʳ⇒invˡ-unique = Base.assoc∧id∧invʳ⇒invˡ-unique (cong₂ _)
 
-  assoc+id+invˡ⇒invʳ-unique : Associative _∙_ → Identity ε _∙_ →
+  assoc∧id∧invˡ⇒invʳ-unique : Associative _∙_ → Identity ε _∙_ →
                               LeftInverse ε _⁻¹ _∙_ →
                               ∀ x y → (x ∙ y) ≡ ε → y ≡ (x ⁻¹)
-  assoc+id+invˡ⇒invʳ-unique = Base.assoc+id+invˡ⇒invʳ-unique (cong₂ _)
+  assoc∧id∧invˡ⇒invʳ-unique = Base.assoc∧id∧invˡ⇒invʳ-unique (cong₂ _)
 
 ------------------------------------------------------------------------
 -- Ring-like structures
 
 module _ {_+_ _*_ -_ 0#} where
 
-  assoc+distribʳ+idʳ+invʳ⇒zeˡ : Associative _+_ → _*_ DistributesOverʳ _+_ →
+  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ : Associative _+_ → _*_ DistributesOverʳ _+_ →
                                 RightIdentity 0# _+_ → RightInverse 0# -_ _+_ →
                                 LeftZero 0# _*_
-  assoc+distribʳ+idʳ+invʳ⇒zeˡ =
-    Base.assoc+distribʳ+idʳ+invʳ⇒zeˡ (cong₂ _+_) (cong₂ _*_)
+  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ =
+    Base.assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ (cong₂ _+_) (cong₂ _*_)
 
-  assoc+distribˡ+idʳ+invʳ⇒zeʳ : Associative _+_ → _*_ DistributesOverˡ _+_ →
+  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ : Associative _+_ → _*_ DistributesOverˡ _+_ →
                                 RightIdentity 0# _+_ → RightInverse 0# -_ _+_ →
                                 RightZero 0# _*_
-  assoc+distribˡ+idʳ+invʳ⇒zeʳ =
-    Base.assoc+distribˡ+idʳ+invʳ⇒zeʳ (cong₂ _+_) (cong₂ _*_)
+  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ =
+    Base.assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ (cong₂ _+_) (cong₂ _*_)
 
 ------------------------------------------------------------------------
 -- Bisemigroup-like structures
 
 module _ {_∙_ _◦_ : Op₂ A} (∙-comm : Commutative _∙_) where
 
-  comm+distrˡ⇒distrʳ : _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_
-  comm+distrˡ⇒distrʳ = Base.comm+distrˡ⇒distrʳ (cong₂ _) ∙-comm
+  comm∧distrˡ⇒distrʳ : _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_
+  comm∧distrˡ⇒distrʳ = Base.comm+distrˡ⇒distrʳ (cong₂ _) ∙-comm
 
-  comm+distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_
-  comm+distrʳ⇒distrˡ = Base.comm+distrʳ⇒distrˡ (cong₂ _) ∙-comm
+  comm∧distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_
+  comm∧distrʳ⇒distrˡ = Base.comm∧distrʳ⇒distrˡ (cong₂ _) ∙-comm
 
   comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y ∙ z)) ≡ ((x ◦ y) ∙ (x ◦ z)))
   comm⇒sym[distribˡ] = Base.comm⇒sym[distribˡ] (cong₂ _◦_) ∙-comm
@@ -100,31 +111,91 @@ module _ {_∙_ : Op₂ A} where
 
 module _ {_∙_ : Op₂ A} where
 
-  comm+assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →
+  comm∧assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →
                           _∙_ MiddleFourExchange _∙_
-  comm+assoc⇒middleFour = Base.comm+assoc⇒middleFour (cong₂ _∙_)
+  comm∧assoc⇒middleFour = Base.comm∧assoc⇒middleFour (cong₂ _∙_)
 
-  identity+middleFour⇒assoc : {e : A} → Identity e _∙_ →
+  identity∧middleFour⇒assoc : {e : A} → Identity e _∙_ →
                               _∙_ MiddleFourExchange _∙_ →
                               Associative _∙_
-  identity+middleFour⇒assoc = Base.identity+middleFour⇒assoc (cong₂ _∙_)
+  identity∧middleFour⇒assoc = Base.identity∧middleFour⇒assoc (cong₂ _∙_)
 
-  identity+middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
+  identity∧middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
                              _∙_ MiddleFourExchange _+_ →
                              Commutative _∙_
-  identity+middleFour⇒comm = Base.identity+middleFour⇒comm (cong₂ _∙_)
+  identity∧middleFour⇒comm = Base.identity∧middleFour⇒comm (cong₂ _∙_)
 
 ------------------------------------------------------------------------
 -- Without Loss of Generality
 
 module _ {p} {P : Pred A p} where
 
-  subst+comm⇒sym : ∀ {f} (f-comm : Commutative f) →
+  subst∧comm⇒sym : ∀ {f} (f-comm : Commutative f) →
                    Symmetric (λ a b → P (f a b))
-  subst+comm⇒sym = Base.subst+comm⇒sym {P = P} subst
+  subst∧comm⇒sym = Base.subst∧comm⇒sym {P = P} subst
 
   wlog : ∀ {f} (f-comm : Commutative f) →
          ∀ {r} {_R_ : Rel _ r} → Total _R_ →
          (∀ a b → a R b → P (f a b)) →
          ∀ a b → P (f a b)
   wlog = Base.wlog {P = P} subst
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+comm+assoc⇒middleFour = comm∧assoc⇒middleFour
+{-# WARNING_ON_USAGE comm+assoc⇒middleFour
+"Warning: comm+assoc⇒middleFour was deprecated in v2.0.
+Please use comm∧assoc⇒middleFour instead."
+#-}
+identity+middleFour⇒assoc = identity∧middleFour⇒assoc
+{-# WARNING_ON_USAGE identity+middleFour⇒assoc
+"Warning: identity+middleFour⇒assoc was deprecated in v2.0.
+Please use identity∧middleFour⇒assoc instead."
+#-}
+identity+middleFour⇒comm = identity∧middleFour⇒comm
+{-# WARNING_ON_USAGE identity+middleFour⇒comm
+"Warning: identity+middleFour⇒comm was deprecated in v2.0.
+Please use identity∧middleFour⇒comm instead."
+#-}
+comm+distrˡ⇒distrʳ = comm∧distrˡ⇒distrʳ
+{-# WARNING_ON_USAGE comm+distrˡ⇒distrʳ
+"Warning: comm+distrˡ⇒distrʳ was deprecated in v2.0.
+Please use comm∧distrˡ⇒distrʳ instead."
+#-}
+comm+distrʳ⇒distrˡ = comm∧distrʳ⇒distrˡ
+{-# WARNING_ON_USAGE comm+distrʳ⇒distrˡ
+"Warning: comm+distrʳ⇒distrˡ was deprecated in v2.0.
+Please use comm∧distrʳ⇒distrˡ instead."
+#-}
+assoc+distribʳ+idʳ+invʳ⇒zeˡ = assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
+{-# WARNING_ON_USAGE assoc+distribʳ+idʳ+invʳ⇒zeˡ
+"Warning: assoc+distribʳ+idʳ+invʳ⇒zeˡ was deprecated in v2.0.
+Please use assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ instead."
+#-}
+assoc+distribˡ+idʳ+invʳ⇒zeʳ = assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
+{-# WARNING_ON_USAGE assoc+distribˡ+idʳ+invʳ⇒zeʳ
+"Warning: assoc+distribˡ+idʳ+invʳ⇒zeʳ was deprecated in v2.0.
+Please use assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ instead."
+#-}
+assoc+id+invʳ⇒invˡ-unique = assoc∧id∧invʳ⇒invˡ-unique
+{-# WARNING_ON_USAGE assoc+id+invʳ⇒invˡ-unique
+"Warning: assoc+id+invʳ⇒invˡ-unique was deprecated in v2.0.
+Please use assoc∧id∧invʳ⇒invˡ-unique instead."
+#-}
+assoc+id+invˡ⇒invʳ-unique = assoc∧id∧invˡ⇒invʳ-unique
+{-# WARNING_ON_USAGE assoc+id+invˡ⇒invʳ-unique
+"Warning: assoc+id+invˡ⇒invʳ-unique was deprecated in v2.0.
+Please use assoc∧id∧invˡ⇒invʳ-unique instead."
+#-}
+subst+comm⇒sym = subst∧comm⇒sym
+{-# WARNING_ON_USAGE subst+comm⇒sym
+"Warning: subst+comm⇒sym was deprecated in v2.0.
+Please use subst∧comm⇒sym instead."
+#-}
diff --git a/src/Algebra/Consequences/Setoid.agda b/src/Algebra/Consequences/Setoid.agda
index f90a801fe4..66e441a9e0 100644
--- a/src/Algebra/Consequences/Setoid.agda
+++ b/src/Algebra/Consequences/Setoid.agda
@@ -37,9 +37,9 @@ open import Algebra.Consequences.Base public
 
 module _ {_∙_ : Op₂ A} (cong : Congruent₂ _∙_) where
 
-  comm+assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →
+  comm∧assoc⇒middleFour : Commutative _∙_ → Associative _∙_ →
                           _∙_ MiddleFourExchange _∙_
-  comm+assoc⇒middleFour comm assoc w x y z = begin
+  comm∧assoc⇒middleFour comm assoc w x y z = begin
     (w ∙ x) ∙ (y ∙ z) ≈⟨ assoc w x (y ∙ z) ⟩
     w ∙ (x ∙ (y ∙ z)) ≈⟨ cong refl (sym (assoc x y z)) ⟩
     w ∙ ((x ∙ y) ∙ z) ≈⟨ cong refl (cong (comm x y) refl) ⟩
@@ -47,19 +47,19 @@ module _ {_∙_ : Op₂ A} (cong : Congruent₂ _∙_) where
     w ∙ (y ∙ (x ∙ z)) ≈⟨ sym (assoc w y (x ∙ z)) ⟩
     (w ∙ y) ∙ (x ∙ z) ∎
 
-  identity+middleFour⇒assoc : {e : A} → Identity e _∙_ →
+  identity∧middleFour⇒assoc : {e : A} → Identity e _∙_ →
                               _∙_ MiddleFourExchange _∙_ →
                               Associative _∙_
-  identity+middleFour⇒assoc {e} (identityˡ , identityʳ) middleFour x y z = begin
+  identity∧middleFour⇒assoc {e} (identityˡ , identityʳ) middleFour x y z = begin
     (x ∙ y) ∙ z       ≈⟨ cong refl (sym (identityˡ z)) ⟩
     (x ∙ y) ∙ (e ∙ z) ≈⟨ middleFour x y e z ⟩
     (x ∙ e) ∙ (y ∙ z) ≈⟨ cong (identityʳ x) refl ⟩
     x ∙ (y ∙ z)       ∎
 
-  identity+middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
+  identity∧middleFour⇒comm : {_+_ : Op₂ A} {e : A} → Identity e _+_ →
                              _∙_ MiddleFourExchange _+_ →
                              Commutative _∙_
-  identity+middleFour⇒comm {_+_} {e} (identityˡ , identityʳ) middleFour x y
+  identity∧middleFour⇒comm {_+_} {e} (identityˡ , identityʳ) middleFour x y
     = begin
     x ∙ y             ≈⟨ sym (cong (identityˡ x) (identityʳ y)) ⟩
     (e + x) ∙ (y + e) ≈⟨ middleFour e x y e ⟩
@@ -72,7 +72,7 @@ module _ {_∙_ : Op₂ A} (cong : Congruent₂ _∙_) where
 module _ {f : Op₁ A} (self : SelfInverse f) where
 
   selfInverse⇒involutive : Involutive f
-  selfInverse⇒involutive = reflexive+selfInverse⇒involutive _≈_ refl self
+  selfInverse⇒involutive = reflexive∧selfInverse⇒involutive _≈_ refl self
 
   selfInverse⇒congruent : Congruent _≈_ _≈_ f
   selfInverse⇒congruent {x} {y} x≈y = sym (self (begin
@@ -100,15 +100,15 @@ module _ {f : Op₁ A} (self : SelfInverse f) where
 
 module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) where
 
-  comm+cancelˡ⇒cancelʳ : LeftCancellative _∙_ → RightCancellative _∙_
-  comm+cancelˡ⇒cancelʳ cancelˡ x y z eq = cancelˡ x y z $ begin
+  comm∧cancelˡ⇒cancelʳ : LeftCancellative _∙_ → RightCancellative _∙_
+  comm∧cancelˡ⇒cancelʳ cancelˡ x y z eq = cancelˡ x y z $ begin
     x ∙ y ≈⟨ comm x y ⟩
     y ∙ x ≈⟨ eq ⟩
     z ∙ x ≈⟨ comm z x ⟩
     x ∙ z ∎
 
-  comm+cancelʳ⇒cancelˡ : RightCancellative _∙_ → LeftCancellative _∙_
-  comm+cancelʳ⇒cancelˡ cancelʳ x y z eq = cancelʳ x y z $ begin
+  comm∧cancelʳ⇒cancelˡ : RightCancellative _∙_ → LeftCancellative _∙_
+  comm∧cancelʳ⇒cancelˡ cancelʳ x y z eq = cancelʳ x y z $ begin
     y ∙ x ≈⟨ comm y x ⟩
     x ∙ y ≈⟨ eq ⟩
     x ∙ z ≈⟨ comm x z ⟩
@@ -119,54 +119,54 @@ module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) where
 
 module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) {e : A} where
 
-  comm+idˡ⇒idʳ : LeftIdentity e _∙_ → RightIdentity e _∙_
-  comm+idˡ⇒idʳ idˡ x = begin
+  comm∧idˡ⇒idʳ : LeftIdentity e _∙_ → RightIdentity e _∙_
+  comm∧idˡ⇒idʳ idˡ x = begin
     x ∙ e ≈⟨ comm x e ⟩
     e ∙ x ≈⟨ idˡ x ⟩
     x     ∎
 
-  comm+idʳ⇒idˡ : RightIdentity e _∙_ → LeftIdentity e _∙_
-  comm+idʳ⇒idˡ idʳ x = begin
+  comm∧idʳ⇒idˡ : RightIdentity e _∙_ → LeftIdentity e _∙_
+  comm∧idʳ⇒idˡ idʳ x = begin
     e ∙ x ≈⟨ comm e x ⟩
     x ∙ e ≈⟨ idʳ x ⟩
     x     ∎
 
-  comm+idˡ⇒id : LeftIdentity e _∙_ → Identity e _∙_
-  comm+idˡ⇒id idˡ = idˡ , comm+idˡ⇒idʳ idˡ
+  comm∧idˡ⇒id : LeftIdentity e _∙_ → Identity e _∙_
+  comm∧idˡ⇒id idˡ = idˡ , comm∧idˡ⇒idʳ idˡ
 
-  comm+idʳ⇒id : RightIdentity e _∙_ → Identity e _∙_
-  comm+idʳ⇒id idʳ = comm+idʳ⇒idˡ idʳ , idʳ
+  comm∧idʳ⇒id : RightIdentity e _∙_ → Identity e _∙_
+  comm∧idʳ⇒id idʳ = comm∧idʳ⇒idˡ idʳ , idʳ
 
-  comm+zeˡ⇒zeʳ : LeftZero e _∙_ → RightZero e _∙_
-  comm+zeˡ⇒zeʳ zeˡ x = begin
+  comm∧zeˡ⇒zeʳ : LeftZero e _∙_ → RightZero e _∙_
+  comm∧zeˡ⇒zeʳ zeˡ x = begin
     x ∙ e ≈⟨ comm x e ⟩
     e ∙ x ≈⟨ zeˡ x ⟩
     e     ∎
 
-  comm+zeʳ⇒zeˡ : RightZero e _∙_ → LeftZero e _∙_
-  comm+zeʳ⇒zeˡ zeʳ x = begin
+  comm∧zeʳ⇒zeˡ : RightZero e _∙_ → LeftZero e _∙_
+  comm∧zeʳ⇒zeˡ zeʳ x = begin
     e ∙ x ≈⟨ comm e x ⟩
     x ∙ e ≈⟨ zeʳ x ⟩
     e     ∎
 
-  comm+zeˡ⇒ze : LeftZero e _∙_ → Zero e _∙_
-  comm+zeˡ⇒ze zeˡ = zeˡ , comm+zeˡ⇒zeʳ zeˡ
+  comm∧zeˡ⇒ze : LeftZero e _∙_ → Zero e _∙_
+  comm∧zeˡ⇒ze zeˡ = zeˡ , comm∧zeˡ⇒zeʳ zeˡ
 
-  comm+zeʳ⇒ze : RightZero e _∙_ → Zero e _∙_
-  comm+zeʳ⇒ze zeʳ = comm+zeʳ⇒zeˡ zeʳ , zeʳ
+  comm∧zeʳ⇒ze : RightZero e _∙_ → Zero e _∙_
+  comm∧zeʳ⇒ze zeʳ = comm∧zeʳ⇒zeˡ zeʳ , zeʳ
 
-  comm+almostCancelˡ⇒almostCancelʳ : AlmostLeftCancellative e _∙_ →
+  comm∧almostCancelˡ⇒almostCancelʳ : AlmostLeftCancellative e _∙_ →
                                      AlmostRightCancellative e _∙_
-  comm+almostCancelˡ⇒almostCancelʳ cancelˡ-nonZero x y z x≉e yx≈zx =
+  comm∧almostCancelˡ⇒almostCancelʳ cancelˡ-nonZero x y z x≉e yx≈zx =
     cancelˡ-nonZero x y z x≉e $ begin
       x ∙ y ≈⟨ comm x y ⟩
       y ∙ x ≈⟨ yx≈zx ⟩
       z ∙ x ≈⟨ comm z x ⟩
       x ∙ z ∎
 
-  comm+almostCancelʳ⇒almostCancelˡ : AlmostRightCancellative e _∙_ →
+  comm∧almostCancelʳ⇒almostCancelˡ : AlmostRightCancellative e _∙_ →
                                      AlmostLeftCancellative e _∙_
-  comm+almostCancelʳ⇒almostCancelˡ cancelʳ-nonZero x y z x≉e xy≈xz =
+  comm∧almostCancelʳ⇒almostCancelˡ cancelʳ-nonZero x y z x≉e xy≈xz =
     cancelʳ-nonZero x y z x≉e $ begin
       y ∙ x ≈⟨ comm y x ⟩
       x ∙ y ≈⟨ xy≈xz ⟩
@@ -178,30 +178,30 @@ module _ {_∙_ : Op₂ A} (comm : Commutative _∙_) {e : A} where
 
 module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (comm : Commutative _∙_) where
 
-  comm+invˡ⇒invʳ : LeftInverse e _⁻¹ _∙_ → RightInverse e _⁻¹ _∙_
-  comm+invˡ⇒invʳ invˡ x = begin
+  comm∧invˡ⇒invʳ : LeftInverse e _⁻¹ _∙_ → RightInverse e _⁻¹ _∙_
+  comm∧invˡ⇒invʳ invˡ x = begin
     x ∙ (x ⁻¹) ≈⟨ comm x (x ⁻¹) ⟩
     (x ⁻¹) ∙ x ≈⟨ invˡ x ⟩
     e          ∎
 
-  comm+invˡ⇒inv : LeftInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
-  comm+invˡ⇒inv invˡ = invˡ , comm+invˡ⇒invʳ invˡ
+  comm∧invˡ⇒inv : LeftInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm∧invˡ⇒inv invˡ = invˡ , comm∧invˡ⇒invʳ invˡ
 
-  comm+invʳ⇒invˡ : RightInverse e _⁻¹ _∙_ → LeftInverse e _⁻¹ _∙_
-  comm+invʳ⇒invˡ invʳ x = begin
+  comm∧invʳ⇒invˡ : RightInverse e _⁻¹ _∙_ → LeftInverse e _⁻¹ _∙_
+  comm∧invʳ⇒invˡ invʳ x = begin
     (x ⁻¹) ∙ x ≈⟨ comm (x ⁻¹) x ⟩
     x ∙ (x ⁻¹) ≈⟨ invʳ x ⟩
     e          ∎
 
-  comm+invʳ⇒inv : RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
-  comm+invʳ⇒inv invʳ = comm+invʳ⇒invˡ invʳ , invʳ
+  comm∧invʳ⇒inv : RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm∧invʳ⇒inv invʳ = comm∧invʳ⇒invˡ invʳ , invʳ
 
 module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _∙_) where
 
-  assoc+id+invʳ⇒invˡ-unique : Associative _∙_ →
+  assoc∧id∧invʳ⇒invˡ-unique : Associative _∙_ →
                               Identity e _∙_ → RightInverse e _⁻¹ _∙_ →
                               ∀ x y → (x ∙ y) ≈ e → x ≈ (y ⁻¹)
-  assoc+id+invʳ⇒invˡ-unique assoc (idˡ , idʳ) invʳ x y eq = begin
+  assoc∧id∧invʳ⇒invˡ-unique assoc (idˡ , idʳ) invʳ x y eq = begin
     x                ≈⟨ sym (idʳ x) ⟩
     x ∙ e            ≈⟨ cong refl (sym (invʳ y)) ⟩
     x ∙ (y ∙ (y ⁻¹)) ≈⟨ sym (assoc x y (y ⁻¹)) ⟩
@@ -209,10 +209,10 @@ module _ {_∙_ : Op₂ A} {_⁻¹ : Op₁ A} {e} (cong : Congruent₂ _∙_) wh
     e ∙ (y ⁻¹)       ≈⟨ idˡ (y ⁻¹) ⟩
     y ⁻¹             ∎
 
-  assoc+id+invˡ⇒invʳ-unique : Associative _∙_ →
+  assoc∧id∧invˡ⇒invʳ-unique : Associative _∙_ →
                               Identity e _∙_ → LeftInverse e _⁻¹ _∙_ →
                               ∀ x y → (x ∙ y) ≈ e → y ≈ (x ⁻¹)
-  assoc+id+invˡ⇒invʳ-unique assoc (idˡ , idʳ) invˡ x y eq = begin
+  assoc∧id∧invˡ⇒invʳ-unique assoc (idˡ , idʳ) invˡ x y eq = begin
     y                ≈⟨ sym (idˡ y) ⟩
     e ∙ y            ≈⟨ cong (sym (invˡ x)) refl ⟩
     ((x ⁻¹) ∙ x) ∙ y ≈⟨ assoc (x ⁻¹) x y ⟩
@@ -228,25 +228,25 @@ module _ {_∙_ _◦_ : Op₂ A}
          (∙-comm : Commutative _∙_)
          where
 
-  comm+distrˡ⇒distrʳ :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_
-  comm+distrˡ⇒distrʳ distrˡ x y z = begin
+  comm∧distrˡ⇒distrʳ :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOverʳ _◦_
+  comm∧distrˡ⇒distrʳ distrˡ x y z = begin
     (y ◦ z) ∙ x       ≈⟨ ∙-comm (y ◦ z) x ⟩
     x ∙ (y ◦ z)       ≈⟨ distrˡ x y z ⟩
     (x ∙ y) ◦ (x ∙ z) ≈⟨ ◦-cong (∙-comm x y) (∙-comm x z) ⟩
     (y ∙ x) ◦ (z ∙ x) ∎
 
-  comm+distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_
-  comm+distrʳ⇒distrˡ distrˡ x y z = begin
+  comm∧distrʳ⇒distrˡ : _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOverˡ _◦_
+  comm∧distrʳ⇒distrˡ distrˡ x y z = begin
     x ∙ (y ◦ z)       ≈⟨ ∙-comm x (y ◦ z) ⟩
     (y ◦ z) ∙ x       ≈⟨ distrˡ x y z ⟩
     (y ∙ x) ◦ (z ∙ x) ≈⟨ ◦-cong (∙-comm y x) (∙-comm z x) ⟩
     (x ∙ y) ◦ (x ∙ z) ∎
 
-  comm+distrˡ⇒distr :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
-  comm+distrˡ⇒distr distrˡ = distrˡ , comm+distrˡ⇒distrʳ distrˡ
+  comm∧distrˡ⇒distr :  _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
+  comm∧distrˡ⇒distr distrˡ = distrˡ , comm∧distrˡ⇒distrʳ distrˡ
 
-  comm+distrʳ⇒distr :  _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
-  comm+distrʳ⇒distr distrʳ = comm+distrʳ⇒distrˡ distrʳ , distrʳ
+  comm∧distrʳ⇒distr :  _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
+  comm∧distrʳ⇒distr distrʳ = comm∧distrʳ⇒distrˡ distrʳ , distrʳ
 
   comm⇒sym[distribˡ] : ∀ x → Symmetric (λ y z → (x ◦ (y ∙ z)) ≈ ((x ◦ y) ∙ (x ◦ z)))
   comm⇒sym[distribˡ] x {y} {z} prf = begin
@@ -262,11 +262,11 @@ module _ {_∙_ _◦_ : Op₂ A}
          (◦-comm  : Commutative _◦_)
          where
 
-  distrib+absorbs⇒distribˡ : _∙_ Absorbs _◦_ →
+  distrib∧absorbs⇒distribˡ : _∙_ Absorbs _◦_ →
                              _◦_ Absorbs _∙_ →
                              _◦_ DistributesOver _∙_ →
                              _∙_ DistributesOverˡ _◦_
-  distrib+absorbs⇒distribˡ ∙-absorbs-◦ ◦-absorbs-∙ (◦-distribˡ-∙ , ◦-distribʳ-∙) x y z = begin
+  distrib∧absorbs⇒distribˡ ∙-absorbs-◦ ◦-absorbs-∙ (◦-distribˡ-∙ , ◦-distribʳ-∙) x y z = begin
     x ∙ (y ◦ z)                    ≈⟨ ∙-cong (∙-absorbs-◦ _ _) refl ⟨
     (x ∙ (x ◦ y)) ∙ (y ◦ z)        ≈⟨  ∙-cong (∙-cong refl (◦-comm _ _)) refl ⟩
     (x ∙ (y ◦ x)) ∙ (y ◦ z)        ≈⟨  ∙-assoc _ _ _ ⟩
@@ -284,10 +284,10 @@ module _ {_+_ _*_ : Op₂ A}
          (*-cong : Congruent₂ _*_)
          where
 
-  assoc+distribʳ+idʳ+invʳ⇒zeˡ : Associative _+_ → _*_ DistributesOverʳ _+_ →
+  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ : Associative _+_ → _*_ DistributesOverʳ _+_ →
                                 RightIdentity 0# _+_ → RightInverse 0# _⁻¹ _+_ →
                                 LeftZero 0# _*_
-  assoc+distribʳ+idʳ+invʳ⇒zeˡ +-assoc distribʳ idʳ invʳ  x = begin
+  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ +-assoc distribʳ idʳ invʳ  x = begin
     0# * x                                 ≈⟨ sym (idʳ _) ⟩
     (0# * x) + 0#                          ≈⟨ +-cong refl (sym (invʳ _)) ⟩
     (0# * x) + ((0# * x)  + ((0# * x)⁻¹))  ≈⟨ sym (+-assoc _ _ _) ⟩
@@ -296,10 +296,10 @@ module _ {_+_ _*_ : Op₂ A}
     (0# * x) + ((0# * x)⁻¹)                ≈⟨ invʳ _ ⟩
     0#                                     ∎
 
-  assoc+distribˡ+idʳ+invʳ⇒zeʳ : Associative _+_ → _*_ DistributesOverˡ _+_ →
+  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ : Associative _+_ → _*_ DistributesOverˡ _+_ →
                                 RightIdentity 0# _+_ → RightInverse 0# _⁻¹ _+_ →
                                 RightZero 0# _*_
-  assoc+distribˡ+idʳ+invʳ⇒zeʳ +-assoc distribˡ idʳ invʳ  x = begin
+  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ +-assoc distribˡ idʳ invʳ  x = begin
      x * 0#                                ≈⟨ sym (idʳ _) ⟩
      (x * 0#) + 0#                         ≈⟨ +-cong refl (sym (invʳ _)) ⟩
      (x * 0#) + ((x * 0#) + ((x * 0#)⁻¹))  ≈⟨ sym (+-assoc _ _ _) ⟩
@@ -316,10 +316,130 @@ module _ {p} {f : Op₂ A} {P : Pred A p}
          (comm : Commutative f)
          where
 
-  subst+comm⇒sym : Symmetric (λ a b → P (f a b))
-  subst+comm⇒sym = ≈-subst P (comm _ _)
+  subst∧comm⇒sym : Symmetric (λ a b → P (f a b))
+  subst∧comm⇒sym = ≈-subst P (comm _ _)
 
   wlog : ∀ {r} {_R_ : Rel _ r} → Total _R_ →
          (∀ a b → a R b → P (f a b)) →
          ∀ a b → P (f a b)
-  wlog r-total = Bin.wlog r-total subst+comm⇒sym
+  wlog r-total = Bin.wlog r-total subst∧comm⇒sym
+
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.0
+
+comm+assoc⇒middleFour = comm∧assoc⇒middleFour
+{-# WARNING_ON_USAGE comm+assoc⇒middleFour
+"Warning: comm+assoc⇒middleFour was deprecated in v2.0.
+Please use comm∧assoc⇒middleFour instead."
+#-}
+identity+middleFour⇒assoc = identity∧middleFour⇒assoc
+{-# WARNING_ON_USAGE identity+middleFour⇒assoc
+"Warning: identity+middleFour⇒assoc was deprecated in v2.0.
+Please use identity∧middleFour⇒assoc instead."
+#-}
+identity+middleFour⇒comm = identity∧middleFour⇒comm
+{-# WARNING_ON_USAGE identity+middleFour⇒comm
+"Warning: identity+middleFour⇒comm was deprecated in v2.0.
+Please use identity∧middleFour⇒comm instead."
+#-}
+comm+cancelˡ⇒cancelʳ = comm∧cancelˡ⇒cancelʳ
+{-# WARNING_ON_USAGE comm+cancelˡ⇒cancelʳ
+"Warning: comm+cancelˡ⇒cancelʳ was deprecated in v2.0.
+Please use comm∧cancelˡ⇒cancelʳ instead."
+#-}
+comm+cancelʳ⇒cancelˡ = comm∧cancelʳ⇒cancelˡ
+{-# WARNING_ON_USAGE comm+cancelʳ⇒cancelˡ
+"Warning: comm+cancelʳ⇒cancelˡ was deprecated in v2.0.
+Please use comm∧cancelʳ⇒cancelˡ instead."
+#-}
+comm+idˡ⇒idʳ = comm∧idˡ⇒idʳ
+{-# WARNING_ON_USAGE comm+idˡ⇒idʳ
+"Warning: comm+idˡ⇒idʳ was deprecated in v2.0.
+Please use comm∧idˡ⇒idʳ instead."
+#-}
+comm+idʳ⇒idˡ = comm∧idʳ⇒idˡ
+{-# WARNING_ON_USAGE comm+idʳ⇒idˡ
+"Warning: comm+idʳ⇒idˡ was deprecated in v2.0.
+Please use comm∧idʳ⇒idˡ instead."
+#-}
+comm+zeˡ⇒zeʳ = comm∧zeˡ⇒zeʳ
+{-# WARNING_ON_USAGE comm+zeˡ⇒zeʳ
+"Warning: comm+zeˡ⇒zeʳ was deprecated in v2.0.
+Please use comm∧zeˡ⇒zeʳ instead."
+#-}
+comm+zeʳ⇒zeˡ = comm∧zeʳ⇒zeˡ
+{-# WARNING_ON_USAGE comm+zeʳ⇒zeˡ
+"Warning: comm+zeʳ⇒zeˡ was deprecated in v2.0.
+Please use comm∧zeʳ⇒zeˡ instead."
+#-}
+comm+almostCancelˡ⇒almostCancelʳ = comm∧almostCancelˡ⇒almostCancelʳ
+{-# WARNING_ON_USAGE comm+almostCancelˡ⇒almostCancelʳ
+"Warning: comm+almostCancelˡ⇒almostCancelʳ was deprecated in v2.0.
+Please use comm∧almostCancelˡ⇒almostCancelʳ instead."
+#-}
+comm+almostCancelʳ⇒almostCancelˡ = comm∧almostCancelʳ⇒almostCancelˡ
+{-# WARNING_ON_USAGE comm+almostCancelʳ⇒almostCancelˡ
+"Warning: comm+almostCancelʳ⇒almostCancelˡ was deprecated in v2.0.
+Please use comm∧almostCancelʳ⇒almostCancelˡ instead."
+#-}
+comm+invˡ⇒invʳ = comm∧invˡ⇒invʳ
+{-# WARNING_ON_USAGE comm+invˡ⇒invʳ
+"Warning: comm+invˡ⇒invʳ was deprecated in v2.0.
+Please use comm∧invˡ⇒invʳ instead."
+#-}
+comm+invʳ⇒invˡ = comm∧invʳ⇒invˡ
+{-# WARNING_ON_USAGE comm+invʳ⇒invˡ
+"Warning: comm+invʳ⇒invˡ was deprecated in v2.0.
+Please use comm∧invʳ⇒invˡ instead."
+#-}
+comm+invˡ⇒inv = comm∧invˡ⇒inv
+{-# WARNING_ON_USAGE comm+invˡ⇒inv
+"Warning: comm+invˡ⇒inv was deprecated in v2.0.
+Please use comm∧invˡ⇒inv instead."
+#-}
+comm+invʳ⇒inv = comm∧invʳ⇒inv
+{-# WARNING_ON_USAGE comm+invʳ⇒inv
+"Warning: comm+invʳ⇒inv was deprecated in v2.0.
+Please use comm∧invʳ⇒inv instead."
+#-}
+comm+distrˡ⇒distrʳ = comm∧distrˡ⇒distrʳ
+{-# WARNING_ON_USAGE comm+distrˡ⇒distrʳ
+"Warning: comm+distrˡ⇒distrʳ was deprecated in v2.0.
+Please use comm∧distrˡ⇒distrʳ instead."
+#-}
+comm+distrʳ⇒distrˡ = comm∧distrʳ⇒distrˡ
+{-# WARNING_ON_USAGE comm+distrʳ⇒distrˡ
+"Warning: comm+distrʳ⇒distrˡ was deprecated in v2.0.
+Please use comm∧distrʳ⇒distrˡ instead."
+#-}
+assoc+distribʳ+idʳ+invʳ⇒zeˡ = assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
+{-# WARNING_ON_USAGE assoc+distribʳ+idʳ+invʳ⇒zeˡ
+"Warning: assoc+distribʳ+idʳ+invʳ⇒zeˡ was deprecated in v2.0.
+Please use assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ instead."
+#-}
+assoc+distribˡ+idʳ+invʳ⇒zeʳ = assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
+{-# WARNING_ON_USAGE assoc+distribˡ+idʳ+invʳ⇒zeʳ
+"Warning: assoc+distribˡ+idʳ+invʳ⇒zeʳ was deprecated in v2.0.
+Please use assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ instead."
+#-}
+assoc+id+invʳ⇒invˡ-unique = assoc∧id∧invʳ⇒invˡ-unique
+{-# WARNING_ON_USAGE assoc+id+invʳ⇒invˡ-unique
+"Warning: assoc+id+invʳ⇒invˡ-unique was deprecated in v2.0.
+Please use assoc∧id∧invʳ⇒invˡ-unique instead."
+#-}
+assoc+id+invˡ⇒invʳ-unique = assoc∧id∧invˡ⇒invʳ-unique
+{-# WARNING_ON_USAGE assoc+id+invˡ⇒invʳ-unique
+"Warning: assoc+id+invˡ⇒invʳ-unique was deprecated in v2.0.
+Please use assoc∧id∧invˡ⇒invʳ-unique instead."
+#-}
+subst+comm⇒sym = subst∧comm⇒sym
+{-# WARNING_ON_USAGE subst+comm⇒sym
+"Warning: subst+comm⇒sym was deprecated in v2.0.
+Please use subst∧comm⇒sym instead."
+#-}
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
index 450527ddc0..702e9fa1c8 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
@@ -57,7 +57,7 @@ open DistribLatticeProperties distributiveLattice public
   x              ∎
 
 ∧-identityˡ : LeftIdentity ⊤ _∧_
-∧-identityˡ = comm+idʳ⇒idˡ ∧-comm ∧-identityʳ
+∧-identityˡ = comm∧idʳ⇒idˡ ∧-comm ∧-identityʳ
 
 ∧-identity : Identity ⊤ _∧_
 ∧-identity = ∧-identityˡ , ∧-identityʳ
@@ -69,7 +69,7 @@ open DistribLatticeProperties distributiveLattice public
   x              ∎
 
 ∨-identityˡ : LeftIdentity ⊥ _∨_
-∨-identityˡ = comm+idʳ⇒idˡ ∨-comm ∨-identityʳ
+∨-identityˡ = comm∧idʳ⇒idˡ ∨-comm ∨-identityʳ
 
 ∨-identity : Identity ⊥ _∨_
 ∨-identity = ∨-identityˡ , ∨-identityʳ
@@ -83,7 +83,7 @@ open DistribLatticeProperties distributiveLattice public
   ⊥              ∎
 
 ∧-zeroˡ : LeftZero ⊥ _∧_
-∧-zeroˡ = comm+zeʳ⇒zeˡ ∧-comm ∧-zeroʳ
+∧-zeroˡ = comm∧zeʳ⇒zeˡ ∧-comm ∧-zeroʳ
 
 ∧-zero : Zero ⊥ _∧_
 ∧-zero = ∧-zeroˡ , ∧-zeroʳ
@@ -97,7 +97,7 @@ open DistribLatticeProperties distributiveLattice public
   ⊤              ∎
 
 ∨-zeroˡ : LeftZero ⊤ _∨_
-∨-zeroˡ = comm+zeʳ⇒zeˡ ∨-comm ∨-zeroʳ
+∨-zeroˡ = comm∧zeʳ⇒zeˡ ∨-comm ∨-zeroʳ
 
 ∨-zero : Zero ⊤ _∨_
 ∨-zero = ∨-zeroˡ , ∨-zeroʳ
@@ -377,7 +377,7 @@ module XorRing
         (x ∧ (y ∨ z)) ∧ ¬ x    ∎
 
   ∧-distribʳ-⊕ : _∧_ DistributesOverʳ _⊕_
-  ∧-distribʳ-⊕ = comm+distrˡ⇒distrʳ ⊕-cong ∧-comm ∧-distribˡ-⊕
+  ∧-distribʳ-⊕ = comm∧distrˡ⇒distrʳ ⊕-cong ∧-comm ∧-distribˡ-⊕
 
   ∧-distrib-⊕ : _∧_ DistributesOver _⊕_
   ∧-distrib-⊕ = ∧-distribˡ-⊕ , ∧-distribʳ-⊕
diff --git a/src/Algebra/Lattice/Structures/Biased.agda b/src/Algebra/Lattice/Structures/Biased.agda
index b5e1e6c9fd..5da9714354 100644
--- a/src/Algebra/Lattice/Structures/Biased.agda
+++ b/src/Algebra/Lattice/Structures/Biased.agda
@@ -82,9 +82,9 @@ record IsDistributiveLatticeʳʲᵐ (∨ ∧ : Op₂ A) : Set (a ⊔ ℓ) where
   setoid : Setoid a ℓ
   setoid = record { isEquivalence = isEquivalence }
 
-  ∨-distrib-∧  = comm+distrʳ⇒distr setoid ∧-cong ∨-comm ∨-distribʳ-∧
-  ∧-distribˡ-∨ = distrib+absorbs⇒distribˡ setoid ∧-cong ∧-assoc ∨-comm ∧-absorbs-∨ ∨-absorbs-∧ ∨-distrib-∧
-  ∧-distrib-∨  = comm+distrˡ⇒distr setoid ∨-cong ∧-comm ∧-distribˡ-∨
+  ∨-distrib-∧  = comm∧distrʳ⇒distr setoid ∧-cong ∨-comm ∨-distribʳ-∧
+  ∧-distribˡ-∨ = distrib∧absorbs⇒distribˡ setoid ∧-cong ∧-assoc ∨-comm ∧-absorbs-∨ ∨-absorbs-∧ ∨-distrib-∧
+  ∧-distrib-∨  = comm∧distrˡ⇒distr setoid ∨-cong ∧-comm ∧-distribˡ-∨
 
   isDistributiveLatticeʳʲᵐ : IsDistributiveLattice ∨ ∧
   isDistributiveLatticeʳʲᵐ = record
@@ -115,8 +115,8 @@ record IsBooleanAlgebraʳ
   isBooleanAlgebraʳ : IsBooleanAlgebra  ∨ ∧ ¬ ⊤ ⊥
   isBooleanAlgebraʳ = record
     { isDistributiveLattice = isDistributiveLattice
-    ; ∨-complement          = comm+invʳ⇒inv setoid ∨-comm ∨-complementʳ
-    ; ∧-complement          = comm+invʳ⇒inv setoid ∧-comm ∧-complementʳ
+    ; ∨-complement          = comm∧invʳ⇒inv setoid ∨-comm ∨-complementʳ
+    ; ∧-complement          = comm∧invʳ⇒inv setoid ∧-comm ∧-complementʳ
     ; ¬-cong                = ¬-cong
     }
 
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index ed55226d67..237e50a876 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -264,11 +264,11 @@ record IsGroup (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) w
   inverseʳ = proj₂ inverse
 
   uniqueˡ-⁻¹ : ∀ x y → (x ∙ y) ≈ ε → x ≈ (y ⁻¹)
-  uniqueˡ-⁻¹ = Consequences.assoc+id+invʳ⇒invˡ-unique
+  uniqueˡ-⁻¹ = Consequences.assoc∧id∧invʳ⇒invˡ-unique
                 setoid ∙-cong assoc identity inverseʳ
 
   uniqueʳ-⁻¹ : ∀ x y → (x ∙ y) ≈ ε → y ≈ (x ⁻¹)
-  uniqueʳ-⁻¹ = Consequences.assoc+id+invˡ⇒invʳ-unique
+  uniqueʳ-⁻¹ = Consequences.assoc∧id∧invˡ⇒invʳ-unique
                 setoid ∙-cong assoc identity inverseˡ
 
   isInvertibleMagma : IsInvertibleMagma _∙_ ε _⁻¹
diff --git a/src/Algebra/Structures/Biased.agda b/src/Algebra/Structures/Biased.agda
index c0c6994e3f..cb730e9ba6 100644
--- a/src/Algebra/Structures/Biased.agda
+++ b/src/Algebra/Structures/Biased.agda
@@ -36,7 +36,7 @@ record IsCommutativeMonoidˡ (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
   isCommutativeMonoid = record
     { isMonoid = record
       { isSemigroup = isSemigroup
-      ; identity    = comm+idˡ⇒id setoid comm identityˡ
+      ; identity    = comm∧idˡ⇒id setoid comm identityˡ
       }
     ; comm = comm
     } where open IsSemigroup isSemigroup
@@ -55,7 +55,7 @@ record IsCommutativeMonoidʳ (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
   isCommutativeMonoid = record
     { isMonoid = record
       { isSemigroup = isSemigroup
-      ; identity    = comm+idʳ⇒id setoid comm identityʳ
+      ; identity    = comm∧idʳ⇒id setoid comm identityʳ
       }
     ; comm = comm
     } where open IsSemigroup isSemigroup
@@ -146,9 +146,9 @@ record IsCommutativeSemiringˡ (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) whe
         ; *-cong                = *.∙-cong
         ; *-assoc               = *.assoc
         ; *-identity            = *.identity
-        ; distrib               = comm+distrʳ⇒distr +.setoid +.∙-cong *.comm distribʳ
+        ; distrib               = comm∧distrʳ⇒distr +.setoid +.∙-cong *.comm distribʳ
         }
-      ; zero = comm+zeˡ⇒ze +.setoid *.comm zeroˡ
+      ; zero = comm∧zeˡ⇒ze +.setoid *.comm zeroˡ
       }
     ; *-comm = *.comm
     }
@@ -175,9 +175,9 @@ record IsCommutativeSemiringʳ (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) whe
         ; *-cong                = *.∙-cong
         ; *-assoc               = *.assoc
         ; *-identity            = *.identity
-        ; distrib               = comm+distrˡ⇒distr +.setoid +.∙-cong *.comm distribˡ
+        ; distrib               = comm∧distrˡ⇒distr +.setoid +.∙-cong *.comm distribˡ
         }
-      ; zero = comm+zeʳ⇒ze +.setoid *.comm zeroʳ
+      ; zero = comm∧zeʳ⇒ze +.setoid *.comm zeroʳ
       }
     ; *-comm = *.comm
     }
@@ -207,11 +207,11 @@ record IsRingWithoutAnnihilatingZero (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A)
   open * using ()       renaming (∙-cong to *-cong)
 
   zeroˡ : LeftZero 0# *
-  zeroˡ = assoc+distribʳ+idʳ+invʳ⇒zeˡ setoid
+  zeroˡ = assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ setoid
             +-cong *-cong +.assoc (proj₂ distrib) +.identityʳ +.inverseʳ
 
   zeroʳ : RightZero 0# *
-  zeroʳ = assoc+distribˡ+idʳ+invʳ⇒zeʳ setoid
+  zeroʳ = assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ setoid
             +-cong *-cong +.assoc (proj₁ distrib) +.identityʳ +.inverseʳ
 
   zero : Zero 0# *
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 43152bceee..01bdfcfe0c 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -810,7 +810,7 @@ sign-⊖-≰ = sign-⊖-< ∘ ℕ.≰⇒>
 +-identityˡ (+ _)    = refl
 
 +-identityʳ : RightIdentity +0 _+_
-+-identityʳ = comm+idˡ⇒idʳ +-comm +-identityˡ
++-identityʳ = comm∧idˡ⇒idʳ +-comm +-identityˡ
 
 +-identity : Identity +0 _+_
 +-identity = +-identityˡ , +-identityʳ
@@ -904,7 +904,7 @@ distribʳ-⊖-+-neg m n o = begin
 +-inverseˡ +[1+ n ] = n⊖n≡0 (suc n)
 
 +-inverseʳ : RightInverse +0 -_ _+_
-+-inverseʳ = comm+invˡ⇒invʳ +-comm +-inverseˡ
++-inverseʳ = comm∧invˡ⇒invʳ +-comm +-inverseˡ
 
 +-inverse : Inverse +0 -_ _+_
 +-inverse = +-inverseˡ , +-inverseʳ
@@ -1319,7 +1319,7 @@ pred-mono (+≤+ m≤n)         = ⊖-monoˡ-≤ 1 m≤n
 *-identityˡ +[1+ n ] rewrite ℕ.+-identityʳ n = refl
 
 *-identityʳ : RightIdentity 1ℤ _*_
-*-identityʳ = comm+idˡ⇒idʳ *-comm *-identityˡ
+*-identityʳ = comm∧idˡ⇒idʳ *-comm *-identityˡ
 
 *-identity : Identity 1ℤ _*_
 *-identity = *-identityˡ , *-identityʳ
@@ -1328,7 +1328,7 @@ pred-mono (+≤+ m≤n)         = ⊖-monoˡ-≤ 1 m≤n
 *-zeroˡ _ = refl
 
 *-zeroʳ : RightZero 0ℤ _*_
-*-zeroʳ = comm+zeˡ⇒zeʳ *-comm *-zeroˡ
+*-zeroʳ = comm∧zeˡ⇒zeʳ *-comm *-zeroˡ
 
 *-zero : Zero 0ℤ _*_
 *-zero = *-zeroˡ , *-zeroʳ
@@ -1469,7 +1469,7 @@ private
         = refl
 
 *-distribˡ-+ : _*_ DistributesOverˡ _+_
-*-distribˡ-+ = comm+distrʳ⇒distrˡ *-comm *-distribʳ-+
+*-distribˡ-+ = comm∧distrʳ⇒distrˡ *-comm *-distribʳ-+
 
 *-distrib-+ : _*_ DistributesOver _+_
 *-distrib-+ = *-distribˡ-+ , *-distribʳ-+
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index f83bbfc12c..4f0dc58e73 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -1077,7 +1077,7 @@ private
   where open ≡-Reasoning;  k = toℕ a;  m = toℕ b;  n = toℕ c
 
 *-distribʳ-+ : _*_ DistributesOverʳ _+_
-*-distribʳ-+ = comm+distrˡ⇒distrʳ *-comm *-distribˡ-+
+*-distribʳ-+ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-+
 
 *-distrib-+ : _*_ DistributesOver _+_
 *-distrib-+ = *-distribˡ-+ , *-distribʳ-+
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index 48eb4b4ae7..a4d0ba125d 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -9,7 +9,7 @@
 module Data.Nat.Binary.Subtraction where
 
 open import Algebra using (Op₂; Magma)
-open import Algebra.Consequences.Propositional using (comm+distrˡ⇒distrʳ)
+open import Algebra.Consequences.Propositional using (comm∧distrˡ⇒distrʳ)
 open import Algebra.Morphism.Consequences using (homomorphic₂-inv)
 open import Data.Bool.Base using (true; false)
 open import Data.Nat as ℕ using (ℕ)
@@ -237,7 +237,7 @@ x∸y<x {x} {y} x≢0 y≢0 = begin-strict
   where open ≡-Reasoning;  k = toℕ x;  m = toℕ y;  n = toℕ z
 
 *-distribʳ-∸ : _*_ DistributesOverʳ _∸_
-*-distribʳ-∸ = comm+distrˡ⇒distrʳ *-comm *-distribˡ-∸
+*-distribʳ-∸ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-∸
 
 *-distrib-∸ : _*_ DistributesOver _∸_
 *-distrib-∸ = *-distribˡ-∸ , *-distribʳ-∸
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 9df927064a..ab1678ac9b 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -790,7 +790,7 @@ m+n≮m m n = subst (_≮ m) (+-comm n m) (m+n≮n n m)
   suc n * m + o * m   ∎
 
 *-distribˡ-+ : _*_ DistributesOverˡ _+_
-*-distribˡ-+ = comm+distrʳ⇒distrˡ *-comm *-distribʳ-+
+*-distribˡ-+ = comm∧distrʳ⇒distrˡ *-comm *-distribʳ-+
 
 *-distrib-+ : _*_ DistributesOver _+_
 *-distrib-+ = *-distribˡ-+ , *-distribʳ-+
@@ -1329,7 +1329,7 @@ m⊔n<o⇒n<o m n m⊔n<o = ≤-<-trans (m≤n⊔m m n) m⊔n<o
 +-distribˡ-⊔ (suc m) n o = cong suc (+-distribˡ-⊔ m n o)
 
 +-distribʳ-⊔ : _+_ DistributesOverʳ _⊔_
-+-distribʳ-⊔ = comm+distrˡ⇒distrʳ +-comm +-distribˡ-⊔
++-distribʳ-⊔ = comm∧distrˡ⇒distrʳ +-comm +-distribˡ-⊔
 
 +-distrib-⊔ : _+_ DistributesOver _⊔_
 +-distrib-⊔ = +-distribˡ-⊔ , +-distribʳ-⊔
@@ -1358,7 +1358,7 @@ m⊔n≤m+n m n with ⊔-sel m n
   (m * suc n) ⊔ (m * suc o)  ∎
 
 *-distribʳ-⊔ : _*_ DistributesOverʳ _⊔_
-*-distribʳ-⊔ = comm+distrˡ⇒distrʳ *-comm *-distribˡ-⊔
+*-distribʳ-⊔ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-⊔
 
 *-distrib-⊔ : _*_ DistributesOver _⊔_
 *-distrib-⊔ = *-distribˡ-⊔ , *-distribʳ-⊔
@@ -1437,7 +1437,7 @@ m<n⊓o⇒m<o = m≤n⊓o⇒m≤o
 +-distribˡ-⊓ (suc m) n o = cong suc (+-distribˡ-⊓ m n o)
 
 +-distribʳ-⊓ : _+_ DistributesOverʳ _⊓_
-+-distribʳ-⊓ = comm+distrˡ⇒distrʳ +-comm +-distribˡ-⊓
++-distribʳ-⊓ = comm∧distrˡ⇒distrʳ +-comm +-distribˡ-⊓
 
 +-distrib-⊓ : _+_ DistributesOver _⊓_
 +-distrib-⊓ = +-distribˡ-⊓ , +-distribʳ-⊓
@@ -1472,7 +1472,7 @@ m⊓n≤m+n m n with ⊓-sel m n
   (m * suc n) ⊓ (m * suc o) ∎
 
 *-distribʳ-⊓ : _*_ DistributesOverʳ _⊓_
-*-distribʳ-⊓ = comm+distrˡ⇒distrʳ *-comm *-distribˡ-⊓
+*-distribʳ-⊓ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-⊓
 
 *-distrib-⊓ : _*_ DistributesOver _⊓_
 *-distrib-⊓ = *-distribˡ-⊓ , *-distribʳ-⊓
@@ -1660,7 +1660,7 @@ m∸[m∸n]≡n {suc m} {suc n} (s≤s n≤m) = begin-equality
   m + n * m ∸ (m + o * m) ∎
 
 *-distribˡ-∸ : _*_ DistributesOverˡ _∸_
-*-distribˡ-∸ = comm+distrʳ⇒distrˡ *-comm *-distribʳ-∸
+*-distribˡ-∸ = comm∧distrʳ⇒distrˡ *-comm *-distribʳ-∸
 
 *-distrib-∸ : _*_ DistributesOver _∸_
 *-distrib-∸ = *-distribˡ-∸ , *-distribʳ-∸
@@ -1822,7 +1822,7 @@ private
   ∣ a * m - a * n ∣ ∎
 
 *-distribʳ-∣-∣ : _*_ DistributesOverʳ ∣_-_∣
-*-distribʳ-∣-∣ = comm+distrˡ⇒distrʳ *-comm *-distribˡ-∣-∣
+*-distribʳ-∣-∣ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-∣-∣
 
 *-distrib-∣-∣ : _*_ DistributesOver ∣_-_∣
 *-distrib-∣-∣ = *-distribˡ-∣-∣ , *-distribʳ-∣-∣
diff --git a/src/Data/Parity/Properties.agda b/src/Data/Parity/Properties.agda
index 0d5d29607a..13fdb3ed57 100644
--- a/src/Data/Parity/Properties.agda
+++ b/src/Data/Parity/Properties.agda
@@ -28,7 +28,7 @@ open import Relation.Nullary using (yes; no)
 open import Algebra.Structures {A = Parity} _≡_
 open import Algebra.Definitions {A = Parity} _≡_
 open import Algebra.Consequences.Propositional
-  using (selfInverse⇒involutive; selfInverse⇒injective; comm+distrˡ⇒distrʳ)
+  using (selfInverse⇒involutive; selfInverse⇒injective; comm∧distrˡ⇒distrʳ)
 open import Algebra.Morphism.Structures
 
 ------------------------------------------------------------------------
@@ -254,7 +254,7 @@ p+p≡0ℙ 1ℙ = refl
 *-distribˡ-+ 1ℙ 1ℙ 1ℙ = refl
 
 *-distribʳ-+ : ℙ._*_ DistributesOverʳ ℙ._+_
-*-distribʳ-+ = comm+distrˡ⇒distrʳ *-comm *-distribˡ-+
+*-distribʳ-+ = comm∧distrˡ⇒distrʳ *-comm *-distribˡ-+
 
 *-distrib-+ : ℙ._*_ DistributesOver ℙ._+_
 *-distrib-+ = *-distribˡ-+ , *-distribʳ-+

From 86f43fe13e07daeb9dfa9ed77e3fd192fbd11cad Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Sat, 25 Nov 2023 19:42:02 +0800
Subject: [PATCH 191/316] Add biased versions of Function structures (#2210)

---
 CHANGELOG.md                                  |   6 +-
 src/Algebra/Structures/Biased.agda            |   2 +-
 .../Function/Dependent/Propositional.agda     |   2 +-
 src/Function.agda                             |   1 +
 src/Function/Construct/Symmetry.agda          |   2 +-
 src/Function/Structures.agda                  |   2 +-
 src/Function/Structures/Biased.agda           | 127 ++++++++++++++++++
 .../Binary/PropositionalEquality.agda         |   1 -
 8 files changed, 136 insertions(+), 7 deletions(-)
 create mode 100644 src/Function/Structures/Biased.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 25065935ce..8706638585 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -755,8 +755,10 @@ Non-backwards compatible changes
    - The records in  `Function.Structures` and `Function.Bundles` export proofs
          of these under the names `strictlySurjective`, `strictlyInverseˡ` and
          `strictlyInverseʳ`,
-   - Conversion functions have been added in both directions to
-         `Function.Consequences(.Propositional/Setoid)`.
+   - Conversion functions for the definitions have been added in both directions 
+		 to `Function.Consequences(.Propositional/Setoid)`.
+   - Conversion functions for structures have been added in 
+		 `Function.Structures.Biased`.
 
 ### New `Function.Strict`
 
diff --git a/src/Algebra/Structures/Biased.agda b/src/Algebra/Structures/Biased.agda
index cb730e9ba6..a2511ed843 100644
--- a/src/Algebra/Structures/Biased.agda
+++ b/src/Algebra/Structures/Biased.agda
@@ -2,7 +2,7 @@
 -- The Agda standard library
 --
 -- Ways to give instances of certain structures where some fields can
--- be given in terms of others
+-- be given in terms of others. Re-exported via `Algebra`.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index 04cb6a7097..48fdff0af9 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -185,7 +185,7 @@ module _ where
        (∀ {x} → A x ↠ B (to I↠J x)) →
        Σ I A ↠ Σ J B
   Σ-↠ {I = I} {J = J} {A = A} {B = B} I↠J A↠B =
-    mk↠ (strictlySurjective⇒surjective strictlySurjective′)
+    mk↠ₛ strictlySurjective′
     where
     to′ : Σ I A → Σ J B
     to′ = map (to I↠J) (to A↠B)
diff --git a/src/Function.agda b/src/Function.agda
index 1d45b05901..b4155ef8ef 100644
--- a/src/Function.agda
+++ b/src/Function.agda
@@ -13,4 +13,5 @@ open import Function.Base public
 open import Function.Strict public
 open import Function.Definitions public
 open import Function.Structures public
+open import Function.Structures.Biased public
 open import Function.Bundles public
diff --git a/src/Function/Construct/Symmetry.agda b/src/Function/Construct/Symmetry.agda
index 834ee0d4fa..5506f93dd4 100644
--- a/src/Function/Construct/Symmetry.agda
+++ b/src/Function/Construct/Symmetry.agda
@@ -108,7 +108,7 @@ module _ {≈₁ : Rel A ℓ₁} {≈₂ : Rel B ℓ₂} {f : A → B} {f⁻¹ :
   isLeftInverse : IsRightInverse ≈₁ ≈₂ f f⁻¹ → IsLeftInverse ≈₂ ≈₁ f⁻¹ f
   isLeftInverse inv = record
     { isCongruent = isCongruent F.isCongruent F.from-cong
-    ; from-cong   = F.cong₁
+    ; from-cong   = F.to-cong
     ; inverseˡ    = inverseˡ ≈₁ ≈₂ F.inverseʳ
     } where module F = IsRightInverse inv
 
diff --git a/src/Function/Structures.agda b/src/Function/Structures.agda
index ae34203fe0..9533cfb179 100644
--- a/src/Function/Structures.agda
+++ b/src/Function/Structures.agda
@@ -120,7 +120,7 @@ record IsRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁
     inverseʳ    : Inverseʳ _≈₁_ _≈₂_ to from
 
   open IsCongruent isCongruent public
-    renaming (cong to cong₁)
+    renaming (cong to to-cong)
 
   strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from
   strictlyInverseʳ x = inverseʳ Eq₂.refl
diff --git a/src/Function/Structures/Biased.agda b/src/Function/Structures/Biased.agda
new file mode 100644
index 0000000000..e25c4970c9
--- /dev/null
+++ b/src/Function/Structures/Biased.agda
@@ -0,0 +1,127 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Ways to give instances of certain structures where some fields can
+-- be given in terms of others.
+-- The contents of this file should usually be accessed from `Function`.
+------------------------------------------------------------------------
+
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+
+module Function.Structures.Biased {a b ℓ₁ ℓ₂}
+  {A : Set a} (_≈₁_ : Rel A ℓ₁) -- Equality over the domain
+  {B : Set b} (_≈₂_ : Rel B ℓ₂) -- Equality over the codomain
+  where
+
+open import Data.Product.Base as Product using (∃; _×_; _,_)
+open import Function.Base
+open import Function.Definitions
+open import Function.Structures _≈₁_ _≈₂_
+open import Function.Consequences.Setoid
+open import Level using (_⊔_)
+
+------------------------------------------------------------------------
+-- Surjection
+------------------------------------------------------------------------
+
+record IsStrictSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+  field
+    isCongruent : IsCongruent f
+    strictlySurjective : StrictlySurjective _≈₂_ f
+
+  open IsCongruent isCongruent public
+
+  isSurjection : IsSurjection f
+  isSurjection = record
+    { isCongruent = isCongruent
+    ; surjective = strictlySurjective⇒surjective
+        Eq₁.setoid Eq₂.setoid cong strictlySurjective
+    }
+
+open IsStrictSurjection public
+  using () renaming (isSurjection to isStrictSurjection)
+
+------------------------------------------------------------------------
+-- Bijection
+------------------------------------------------------------------------
+
+record IsStrictBijection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+  field
+    isInjection : IsInjection f
+    strictlySurjective  : StrictlySurjective _≈₂_ f
+
+  isBijection : IsBijection f
+  isBijection = record
+    { isInjection = isInjection
+    ; surjective = strictlySurjective⇒surjective
+        Eq₁.setoid Eq₂.setoid cong strictlySurjective
+    } where open IsInjection isInjection
+
+open IsStrictBijection public
+  using () renaming (isBijection to isStrictBijection)
+
+------------------------------------------------------------------------
+-- Left inverse
+------------------------------------------------------------------------
+
+record IsStrictLeftInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+  field
+    isCongruent  : IsCongruent to
+    from-cong    : Congruent _≈₂_ _≈₁_ from
+    strictlyInverseˡ : StrictlyInverseˡ _≈₂_ to from
+
+  isLeftInverse : IsLeftInverse to from
+  isLeftInverse = record
+    { isCongruent = isCongruent
+    ; from-cong = from-cong
+    ; inverseˡ = strictlyInverseˡ⇒inverseˡ
+        Eq₁.setoid Eq₂.setoid cong strictlyInverseˡ
+    } where open IsCongruent isCongruent
+
+open IsStrictLeftInverse public
+  using () renaming (isLeftInverse to isStrictLeftInverse)
+
+------------------------------------------------------------------------
+-- Right inverse
+------------------------------------------------------------------------
+
+record IsStrictRightInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+  field
+    isCongruent : IsCongruent to
+    from-cong   : Congruent _≈₂_ _≈₁_ from
+    strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from
+
+  isRightInverse : IsRightInverse to from
+  isRightInverse = record
+    { isCongruent = isCongruent
+    ; from-cong = from-cong
+    ; inverseʳ = strictlyInverseʳ⇒inverseʳ
+        Eq₁.setoid Eq₂.setoid from-cong strictlyInverseʳ
+    } where open IsCongruent isCongruent
+
+open IsStrictRightInverse public
+  using () renaming (isRightInverse to isStrictRightInverse)
+
+------------------------------------------------------------------------
+-- Inverse
+------------------------------------------------------------------------
+
+record IsStrictInverse (to : A → B) (from : B → A) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+  field
+    isLeftInverse : IsLeftInverse to from
+    strictlyInverseʳ : StrictlyInverseʳ _≈₁_ to from
+
+  isInverse : IsInverse to from
+  isInverse = record
+    { isLeftInverse = isLeftInverse
+    ; inverseʳ      = strictlyInverseʳ⇒inverseʳ
+        Eq₁.setoid Eq₂.setoid from-cong strictlyInverseʳ
+    } where open IsLeftInverse isLeftInverse
+
+open IsStrictInverse public
+  using () renaming (isInverse to isStrictInverse)
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 7ba3825e31..6c2eb19bed 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -135,4 +135,3 @@ isPropositional = Irrelevant
 Please use Relation.Nullary.Irrelevant instead. "
 #-}
 
-

From 56ed8fedfa4841d18f00efc4af5755e3fec0a150 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Sun, 26 Nov 2023 10:07:51 +0800
Subject: [PATCH 192/316] Fixes #2166 by fixing names in `IsSemilattice`
 (#2211)

* Fix names in IsSemilattice

* Add reference to changes to Semilattice to CHANGELOG
---
 CHANGELOG.md                                    |  4 +++-
 src/Algebra/Lattice/Structures.agda             | 17 ++++++-----------
 .../Binary/Construct/NaturalOrder/Left.agda     |  6 +++---
 3 files changed, 12 insertions(+), 15 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8706638585..db31e5138b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -205,7 +205,9 @@ Non-backwards compatible changes
 * Added new `IsBoundedSemilattice`/`BoundedSemilattice` records.
 
 * Added new aliases `Is(Meet/Join)(Bounded)Semilattice` for `Is(Bounded)Semilattice`
-  which can be used to indicate meet/join-ness of the original structures.
+  which can be used to indicate meet/join-ness of the original structures, and
+  the field names in `IsSemilattice` and `Semilattice` have been renamed from 
+  `∧-cong` to `∙-cong`to indicate their undirected nature.
 
 * Finally, the following auxiliary files have been moved:
   ```agda
diff --git a/src/Algebra/Lattice/Structures.agda b/src/Algebra/Lattice/Structures.agda
index fc40c0c300..116ed85271 100644
--- a/src/Algebra/Lattice/Structures.agda
+++ b/src/Algebra/Lattice/Structures.agda
@@ -37,20 +37,15 @@ record IsSemilattice (∙ : Op₂ A) : Set (a ⊔ ℓ) where
     comm   : Commutative ∙
 
   open IsBand isBand public
-    renaming
-    ( ∙-cong  to ∧-cong
-    ; ∙-congˡ to ∧-congˡ
-    ; ∙-congʳ to ∧-congʳ
-    )
 
 -- Used to bring names appropriate for a meet semilattice into scope.
 IsMeetSemilattice = IsSemilattice
 module IsMeetSemilattice {∧} (L : IsMeetSemilattice ∧) where
   open IsSemilattice L public
     renaming
-    ( ∧-cong  to ∧-cong
-    ; ∧-congˡ to ∧-congˡ
-    ; ∧-congʳ to ∧-congʳ
+    ( ∙-cong  to ∧-cong
+    ; ∙-congˡ to ∧-congˡ
+    ; ∙-congʳ to ∧-congʳ
     )
 
 -- Used to bring names appropriate for a join semilattice into scope.
@@ -58,9 +53,9 @@ IsJoinSemilattice = IsSemilattice
 module IsJoinSemilattice {∨} (L : IsJoinSemilattice ∨) where
   open IsSemilattice L public
     renaming
-    ( ∧-cong  to ∨-cong
-    ; ∧-congˡ to ∨-congˡ
-    ; ∧-congʳ to ∨-congʳ
+    ( ∙-cong  to ∨-cong
+    ; ∙-congˡ to ∨-congˡ
+    ; ∙-congʳ to ∨-congʳ
     )
 
 ------------------------------------------------------------------------
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Left.agda b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
index 22a5031d27..c4846142ec 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Left.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
@@ -99,21 +99,21 @@ module _ (semi : IsSemilattice _∙_) where
 
   x∙y≤x : ∀ x y → (x ∙ y) ≤ x
   x∙y≤x x y = begin
-    x ∙ y       ≈⟨ ∧-cong (sym (idem x)) S.refl ⟩
+    x ∙ y       ≈⟨ ∙-cong (sym (idem x)) S.refl ⟩
     (x ∙ x) ∙ y ≈⟨ assoc x x y ⟩
     x ∙ (x ∙ y) ≈⟨ comm x (x ∙ y) ⟩
     (x ∙ y) ∙ x ∎
 
   x∙y≤y : ∀ x y → (x ∙ y) ≤ y
   x∙y≤y x y = begin
-    x ∙ y        ≈⟨ ∧-cong S.refl (sym (idem y)) ⟩
+    x ∙ y        ≈⟨ ∙-cong S.refl (sym (idem y)) ⟩
     x ∙ (y ∙ y)  ≈⟨ sym (assoc x y y) ⟩
     (x ∙ y) ∙ y  ∎
 
   ∙-presʳ-≤ : ∀ {x y} z → z ≤ x → z ≤ y → z ≤ (x ∙ y)
   ∙-presʳ-≤ {x} {y} z z≤x z≤y = begin
     z            ≈⟨ z≤y ⟩
-    z ∙ y        ≈⟨ ∧-cong z≤x S.refl ⟩
+    z ∙ y        ≈⟨ ∙-cong z≤x S.refl ⟩
     (z ∙ x) ∙ y  ≈⟨ assoc z x y ⟩
     z ∙ (x ∙ y)  ∎
 

From 67afe7c5bb31b8ee5cd90db3afd0a56d87732388 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 26 Nov 2023 02:08:45 +0000
Subject: [PATCH 193/316] remove final references to `Category.*` (#2214)

---
 .../Properties/BooleanAlgebra/Expression.agda        | 12 ++++++------
 .../Construct/Closure/ReflexiveTransitive.agda       |  4 ++--
 src/Text/Pretty.agda                                 |  4 ++--
 3 files changed, 10 insertions(+), 10 deletions(-)

diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
index 88a5cc5518..64ac6d7864 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
@@ -20,8 +20,8 @@ open import Data.Fin.Base using (Fin)
 open import Data.Nat.Base
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec)
-import Data.Vec.Effectful as VecCat
-import Function.Identity.Effectful as IdCat
+import Data.Vec.Effectful as Vec
+import Function.Identity.Effectful as Identity
 open import Data.Vec.Properties using (lookup-map)
 open import Data.Vec.Relation.Binary.Pointwise.Extensional as PW
   using (Pointwise; ext)
@@ -165,18 +165,18 @@ lift n = record
     }
   }
   where
-  open RawApplicative VecCat.applicative
+  open RawApplicative Vec.applicative
     using (pure; zipWith) renaming (_<$>_ to map)
 
   ⟦_⟧Id : ∀ {n} → Expr n → Vec Carrier n → Carrier
-  ⟦_⟧Id = Semantics.⟦_⟧ IdCat.applicative
+  ⟦_⟧Id = Semantics.⟦_⟧ Identity.applicative
 
   ⟦_⟧Vec : ∀ {m n} → Expr n → Vec (Vec Carrier m) n → Vec Carrier m
-  ⟦_⟧Vec = Semantics.⟦_⟧ VecCat.applicative
+  ⟦_⟧Vec = Semantics.⟦_⟧ Vec.applicative
 
   open module R {n} (i : Fin n) =
     Reflection setoid var
       (λ e ρ → Vec.lookup (⟦ e ⟧Vec ρ) i)
       (λ e ρ → ⟦ e ⟧Id (Vec.map (flip Vec.lookup i) ρ))
       (λ e ρ → sym $ reflexive $
-                 Naturality.natural (VecCat.lookup-morphism i) e ρ)
+                 Naturality.natural (Vec.lookup-morphism i) e ρ)
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda
index 71ab8535d8..43778816c3 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive.agda
@@ -130,11 +130,11 @@ _>>=_ : ∀ {i t u} {I : Set i} {T : Rel I t} {U : Rel I u} {j k} →
 m >>= f = (f ⋆) m
 
 -- Note that the monad-like structure above is not an indexed monad
--- (as defined in Category.Monad.Indexed). If it were, then _>>=_
+-- (as defined in Effect.Monad.Indexed). If it were, then _>>=_
 -- would have a type similar to
 --
 --   ∀ {I} {T U : Rel I t} {i j k} →
 --   Star T i j → (T i j → Star U j k) → Star U i k.
 --                  ^^^^^
 -- Note, however, that there is no scope for applying T to any indices
--- in the definition used in Category.Monad.Indexed.
+-- in the definition used in Effect.Monad.Indexed.
diff --git a/src/Text/Pretty.agda b/src/Text/Pretty.agda
index 0bdfe0cf2f..a4b3c1860d 100644
--- a/src/Text/Pretty.agda
+++ b/src/Text/Pretty.agda
@@ -23,8 +23,8 @@ open import Data.String.Base using (String; fromList; replicate)
 open import Function.Base using (_∘_; _∘′_; _$_)
 
 open import Effect.Monad using (RawMonad)
-import Data.List.Effectful as Cat
-open RawMonad (Cat.monad {Level.zero})
+import Data.List.Effectful as List
+open RawMonad (List.monad {Level.zero})
 
 import Data.Nat.Properties as ℕₚ
 open import Data.List.Extrema.Core ℕₚ.≤-totalOrder using (⊓ᴸ)

From b6ac7e0cd427eaa25f35d11e05c79eb5b4ae9fd1 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Sun, 26 Nov 2023 10:48:05 +0800
Subject: [PATCH 194/316] Fix #2195 by removing redundant zero from IsRing
 (#2209)

* Fix #2195 by removing redundant zero from IsRing

* Moved proofs eariler to IsSemiringWithoutOne

* Updated CHANGELOG

* Fix bug

* Refactored Properties.Ring

* Fix renaming
---
 CHANGELOG.md                                  |  6 ++
 src/Algebra/Bundles.agda                      |  3 +
 src/Algebra/Construct/DirectProduct.agda      |  3 -
 .../Lattice/Properties/BooleanAlgebra.agda    |  1 -
 src/Algebra/Morphism/RingMonomorphism.agda    |  1 -
 src/Algebra/Properties/Ring.agda              | 62 ++------------
 src/Algebra/Properties/RingWithoutOne.agda    | 31 +++++--
 src/Algebra/Structures.agda                   | 84 ++++++-------------
 src/Algebra/Structures/Biased.agda            | 55 +++++++-----
 src/Data/Integer/Properties.agda              |  1 -
 src/Data/Parity/Properties.agda               |  1 -
 .../Rational/Unnormalised/Properties.agda     |  1 -
 12 files changed, 98 insertions(+), 151 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index db31e5138b..6704471d12 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -33,6 +33,12 @@ Highlights
 Bug-fixes
 ---------
 
+* In `Algebra.Structures` the records `IsRing` and `IsRingWithoutOne` contained an unnecessary field
+  `zero : RightZero 0# *`, which could be derived from the other ring axioms.
+  Consequently this field has been removed from the record, and the record
+  `IsRingWithoutAnnihilatingZero` in `Algebra.Structures.Biased` has been
+  deprecated as it is now identical to is `IsRing`.
+  
 * In `Algebra.Definitions.RawSemiring` the record `Prime` did not
   enforce that the number was not divisible by `1#`. To fix this
   `p∤1 : p ∤ 1#` has been added as a field.
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index 9c1c14fec0..6b6384458a 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -944,6 +944,9 @@ record Ring c ℓ : Set (suc (c ⊔ ℓ)) where
   +-abelianGroup : AbelianGroup _ _
   +-abelianGroup = record { isAbelianGroup = +-isAbelianGroup }
 
+  ringWithoutOne : RingWithoutOne _ _
+  ringWithoutOne = record { isRingWithoutOne = isRingWithoutOne }
+
   semiring : Semiring _ _
   semiring = record { isSemiring = isSemiring }
 
diff --git a/src/Algebra/Construct/DirectProduct.agda b/src/Algebra/Construct/DirectProduct.agda
index ef9f377d29..3c0ec699ee 100644
--- a/src/Algebra/Construct/DirectProduct.agda
+++ b/src/Algebra/Construct/DirectProduct.agda
@@ -335,8 +335,6 @@ ringWithoutOne R S = record
       ; *-assoc          = Semigroup.assoc (semigroup R.*-semigroup S.*-semigroup)
       ; distrib          = (λ x y z → (R.distribˡ , S.distribˡ) <*> x <*> y <*> z)
                             , (λ x y z → (R.distribʳ , S.distribʳ) <*> x <*> y <*> z)
-      ; zero             = uncurry (λ x y → R.zeroˡ x , S.zeroˡ y)
-                            , uncurry (λ x y → R.zeroʳ x , S.zeroʳ y)
       }
 
   } where module R = RingWithoutOne R; module S = RingWithoutOne S
@@ -389,7 +387,6 @@ ring R S = record
       ; *-assoc          = Semiring.*-assoc Semi
       ; *-identity       = Semiring.*-identity Semi
       ; distrib          = Semiring.distrib Semi
-      ; zero             = Semiring.zero Semi
       }
   }
   where
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
index 702e9fa1c8..673434a48f 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra.agda
@@ -516,7 +516,6 @@ module XorRing
     ; *-assoc = ∧-assoc
     ; *-identity = ∧-identity
     ; distrib = ∧-distrib-⊕
-    ; zero = ∧-zero
     }
 
   ⊕-∧-isCommutativeRing : IsCommutativeRing _⊕_ _∧_ id ⊥ ⊤
diff --git a/src/Algebra/Morphism/RingMonomorphism.agda b/src/Algebra/Morphism/RingMonomorphism.agda
index 9230c802a9..35a191a0f8 100644
--- a/src/Algebra/Morphism/RingMonomorphism.agda
+++ b/src/Algebra/Morphism/RingMonomorphism.agda
@@ -150,7 +150,6 @@ isRing isRing = record
   ; *-assoc          = *-assoc R.*-isMagma R.*-assoc
   ; *-identity       = *-identity R.*-isMagma R.*-identity
   ; distrib          = distrib R.+-isGroup R.*-isMagma R.distrib
-  ; zero             = zero R.+-isGroup R.*-isMagma R.zero
   } where module R = IsRing isRing
 
 isCommutativeRing : IsCommutativeRing _≈₂_ _⊕_ _⊛_ ⊝_ 0#₂ 1#₂ →
diff --git a/src/Algebra/Properties/Ring.agda b/src/Algebra/Properties/Ring.agda
index 22dabbac29..461a9a040b 100644
--- a/src/Algebra/Properties/Ring.agda
+++ b/src/Algebra/Properties/Ring.agda
@@ -12,73 +12,21 @@ module Algebra.Properties.Ring {r₁ r₂} (R : Ring r₁ r₂) where
 
 open Ring R
 
-import Algebra.Properties.AbelianGroup as AbelianGroupProperties
+import Algebra.Properties.RingWithoutOne as RingWithoutOneProperties
 open import Function.Base using (_$_)
 open import Relation.Binary.Reasoning.Setoid setoid
 open import Algebra.Definitions _≈_
 
 ------------------------------------------------------------------------
--- Export properties of abelian groups
+-- Export properties of rings without a 1#.
 
-open AbelianGroupProperties +-abelianGroup public
-  renaming
-  ( ε⁻¹≈ε            to -0#≈0#
-  ; ∙-cancelˡ        to +-cancelˡ
-  ; ∙-cancelʳ        to +-cancelʳ
-  ; ∙-cancel         to +-cancel
-  ; ⁻¹-involutive    to -‿involutive
-  ; ⁻¹-injective     to -‿injective
-  ; ⁻¹-anti-homo-∙   to -‿anti-homo-+
-  ; identityˡ-unique to +-identityˡ-unique
-  ; identityʳ-unique to +-identityʳ-unique
-  ; identity-unique  to +-identity-unique
-  ; inverseˡ-unique  to +-inverseˡ-unique
-  ; inverseʳ-unique  to +-inverseʳ-unique
-  ; ⁻¹-∙-comm        to -‿+-comm
-  )
+open RingWithoutOneProperties ringWithoutOne public
 
 ------------------------------------------------------------------------
--- Properties of -_
-
--‿distribˡ-* : ∀ x y → - (x * y) ≈ - x * y
--‿distribˡ-* x y = sym $ begin
-  - x * y                        ≈⟨ sym $ +-identityʳ _ ⟩
-  - x * y + 0#                   ≈⟨ +-congˡ $ sym (-‿inverseʳ _) ⟩
-  - x * y + (x * y + - (x * y))  ≈⟨ sym $ +-assoc _ _ _  ⟩
-  - x * y + x * y + - (x * y)    ≈⟨ +-congʳ $ sym (distribʳ _ _ _) ⟩
-  (- x + x) * y + - (x * y)      ≈⟨ +-congʳ $ *-congʳ $ -‿inverseˡ _ ⟩
-  0# * y + - (x * y)             ≈⟨ +-congʳ $ zeroˡ _ ⟩
-  0# + - (x * y)                 ≈⟨ +-identityˡ _ ⟩
-  - (x * y)                      ∎
-
--‿distribʳ-* : ∀ x y → - (x * y) ≈ x * - y
--‿distribʳ-* x y = sym $ begin
-  x * - y                        ≈⟨ sym $ +-identityˡ _ ⟩
-  0# + x * - y                   ≈⟨ +-congʳ $ sym (-‿inverseˡ _) ⟩
-  - (x * y) + x * y + x * - y    ≈⟨ +-assoc _ _ _  ⟩
-  - (x * y) + (x * y + x * - y)  ≈⟨ +-congˡ $ sym (distribˡ _ _ _)  ⟩
-  - (x * y) + x * (y + - y)      ≈⟨ +-congˡ $ *-congˡ $ -‿inverseʳ _ ⟩
-  - (x * y) + x * 0#             ≈⟨ +-congˡ $ zeroʳ _ ⟩
-  - (x * y) + 0#                 ≈⟨ +-identityʳ _ ⟩
-  - (x * y)                      ∎
+-- Extra properties of 1#
 
 -1*x≈-x : ∀ x → - 1# * x ≈ - x
 -1*x≈-x x = begin
-  - 1# * x    ≈⟨ sym (-‿distribˡ-* 1# x ) ⟩
+  - 1# * x    ≈⟨ -‿distribˡ-* 1# x ⟨
   - (1# * x)  ≈⟨ -‿cong ( *-identityˡ x ) ⟩
   - x         ∎
-
-x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#
-x+x≈x⇒x≈0 x eq = +-identityˡ-unique x x eq
-
-x[y-z]≈xy-xz : ∀ x y z → x * (y - z) ≈ x * y - x * z
-x[y-z]≈xy-xz x y z = begin
-  x * (y - z)      ≈⟨ distribˡ x y (- z) ⟩
-  x * y + x * - z  ≈⟨ +-congˡ (sym (-‿distribʳ-* x z)) ⟩
-  x * y - x * z    ∎
-
-[y-z]x≈yx-zx : ∀ x y z → (y - z) * x ≈ (y * x) - (z * x)
-[y-z]x≈yx-zx x y z = begin
-  (y - z) * x      ≈⟨ distribʳ x y (- z) ⟩
-  y * x + - z * x  ≈⟨ +-congˡ (sym (-‿distribˡ-* z x)) ⟩
-  y * x - z * x    ∎
diff --git a/src/Algebra/Properties/RingWithoutOne.agda b/src/Algebra/Properties/RingWithoutOne.agda
index cab4d405cb..d81ef58477 100644
--- a/src/Algebra/Properties/RingWithoutOne.agda
+++ b/src/Algebra/Properties/RingWithoutOne.agda
@@ -38,10 +38,10 @@ open AbelianGroupProperties +-abelianGroup public
 
 -‿distribˡ-* : ∀ x y → - (x * y) ≈ - x * y
 -‿distribˡ-* x y = sym $ begin
-  - x * y                        ≈⟨ sym $ +-identityʳ (- x * y) ⟩
-  - x * y + 0#                   ≈⟨ +-congˡ $ sym ( -‿inverseʳ (x * y) ) ⟩
-  - x * y + (x * y + - (x * y))  ≈⟨ sym $ +-assoc (- x * y) (x * y) (- (x * y))  ⟩
-  - x * y + x * y + - (x * y)    ≈⟨ +-congʳ $ sym ( distribʳ y (- x) x ) ⟩
+  - x * y                        ≈⟨ +-identityʳ (- x * y) ⟨
+  - x * y + 0#                   ≈⟨ +-congˡ $ -‿inverseʳ (x * y) ⟨
+  - x * y + (x * y + - (x * y))  ≈⟨ +-assoc (- x * y) (x * y) (- (x * y)) ⟨
+  - x * y + x * y + - (x * y)    ≈⟨ +-congʳ $ distribʳ y (- x) x ⟨
   (- x + x) * y + - (x * y)      ≈⟨ +-congʳ $ *-congʳ $ -‿inverseˡ x ⟩
   0# * y + - (x * y)             ≈⟨ +-congʳ $ zeroˡ y ⟩
   0# + - (x * y)                 ≈⟨ +-identityˡ (- (x * y)) ⟩
@@ -49,11 +49,26 @@ open AbelianGroupProperties +-abelianGroup public
 
 -‿distribʳ-* : ∀ x y → - (x * y) ≈ x * - y
 -‿distribʳ-* x y = sym $ begin
-  x * - y                        ≈⟨ sym $ +-identityˡ (x * (- y)) ⟩
-  0# + x * - y                   ≈⟨ +-congʳ $ sym ( -‿inverseˡ (x * y) ) ⟩
-  - (x * y) + x * y + x * - y    ≈⟨ +-assoc (- (x * y)) (x * y) (x * (- y)) ⟩
-  - (x * y) + (x * y + x * - y)  ≈⟨ +-congˡ $ sym ( distribˡ x y ( - y) )  ⟩
+  x * - y                        ≈⟨ +-identityˡ (x * - y) ⟨
+  0# + x * - y                   ≈⟨ +-congʳ $ -‿inverseˡ (x * y) ⟨
+  - (x * y) + x * y + x * - y    ≈⟨ +-assoc (- (x * y)) (x * y) (x * - y) ⟩
+  - (x * y) + (x * y + x * - y)  ≈⟨ +-congˡ $ distribˡ x y (- y) ⟨
   - (x * y) + x * (y + - y)      ≈⟨ +-congˡ $ *-congˡ $ -‿inverseʳ y ⟩
   - (x * y) + x * 0#             ≈⟨ +-congˡ $ zeroʳ x ⟩
   - (x * y) + 0#                 ≈⟨ +-identityʳ (- (x * y)) ⟩
   - (x * y)                      ∎
+
+x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#
+x+x≈x⇒x≈0 x eq = +-identityˡ-unique x x eq
+
+x[y-z]≈xy-xz : ∀ x y z → x * (y - z) ≈ x * y - x * z
+x[y-z]≈xy-xz x y z = begin
+  x * (y - z)      ≈⟨ distribˡ x y (- z) ⟩
+  x * y + x * - z  ≈⟨ +-congˡ (sym (-‿distribʳ-* x z)) ⟩
+  x * y - x * z    ∎
+
+[y-z]x≈yx-zx : ∀ x y z → (y - z) * x ≈ (y * x) - (z * x)
+[y-z]x≈yx-zx x y z = begin
+  (y - z) * x      ≈⟨ distribʳ x y (- z) ⟩
+  y * x + - z * x  ≈⟨ +-congˡ (sym (-‿distribˡ-* z x)) ⟩
+  y * x - z * x    ∎
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index 237e50a876..44208367ea 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -651,7 +651,6 @@ record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ
     *-cong           : Congruent₂ *
     *-assoc          : Associative *
     distrib          : * DistributesOver +
-    zero             : Zero 0# *
 
   open IsAbelianGroup +-isAbelianGroup public
     renaming
@@ -679,23 +678,28 @@ record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ
     ; isGroup                 to +-isGroup
     )
 
-  *-isMagma : IsMagma *
-  *-isMagma = record
-    { isEquivalence = isEquivalence
-    ; ∙-cong        = *-cong
-    }
+  distribˡ : * DistributesOverˡ +
+  distribˡ = proj₁ distrib
+
+  distribʳ : * DistributesOverʳ +
+  distribʳ = proj₂ distrib
 
   zeroˡ : LeftZero 0# *
-  zeroˡ = proj₁ zero
+  zeroˡ = Consequences.assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ setoid
+    +-cong *-cong +-assoc distribʳ +-identityʳ -‿inverseʳ
 
   zeroʳ : RightZero 0# *
-  zeroʳ = proj₂ zero
+  zeroʳ = Consequences.assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ setoid
+    +-cong *-cong +-assoc distribˡ +-identityʳ -‿inverseʳ
 
-  distribˡ : * DistributesOverˡ +
-  distribˡ = proj₁ distrib
+  zero : Zero 0# *
+  zero = zeroˡ , zeroʳ
 
-  distribʳ : * DistributesOverʳ +
-  distribʳ = proj₂ distrib
+  *-isMagma : IsMagma *
+  *-isMagma = record
+    { isEquivalence = isEquivalence
+    ; ∙-cong        = *-cong
+    }
 
   *-isSemigroup : IsSemigroup *
   *-isSemigroup = record
@@ -806,45 +810,17 @@ record IsRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
     *-assoc          : Associative *
     *-identity       : Identity 1# *
     distrib          : * DistributesOver +
-    zero             : Zero 0# *
 
-  open IsAbelianGroup +-isAbelianGroup public
-    renaming
-    ( assoc                   to +-assoc
-    ; ∙-cong                  to +-cong
-    ; ∙-congˡ                 to +-congˡ
-    ; ∙-congʳ                 to +-congʳ
-    ; identity                to +-identity
-    ; identityˡ               to +-identityˡ
-    ; identityʳ               to +-identityʳ
-    ; inverse                 to -‿inverse
-    ; inverseˡ                to -‿inverseˡ
-    ; inverseʳ                to -‿inverseʳ
-    ; ⁻¹-cong                 to -‿cong
-    ; comm                    to +-comm
-    ; isMagma                 to +-isMagma
-    ; isSemigroup             to +-isSemigroup
-    ; isMonoid                to +-isMonoid
-    ; isUnitalMagma           to +-isUnitalMagma
-    ; isCommutativeMagma      to +-isCommutativeMagma
-    ; isCommutativeMonoid     to +-isCommutativeMonoid
-    ; isCommutativeSemigroup  to +-isCommutativeSemigroup
-    ; isInvertibleMagma       to +-isInvertibleMagma
-    ; isInvertibleUnitalMagma to +-isInvertibleUnitalMagma
-    ; isGroup                 to +-isGroup
-    )
-
-  *-isMagma : IsMagma *
-  *-isMagma = record
-    { isEquivalence = isEquivalence
-    ; ∙-cong        = *-cong
+  isRingWithoutOne : IsRingWithoutOne + * -_ 0#
+  isRingWithoutOne = record
+    { +-isAbelianGroup = +-isAbelianGroup
+    ; *-cong  = *-cong
+    ; *-assoc = *-assoc
+    ; distrib = distrib
     }
 
-  *-isSemigroup : IsSemigroup *
-  *-isSemigroup = record
-    { isMagma = *-isMagma
-    ; assoc   = *-assoc
-    }
+  open IsRingWithoutOne isRingWithoutOne public
+    hiding (+-isAbelianGroup; *-cong; *-assoc; distrib)
 
   *-isMonoid : IsMonoid * 1#
   *-isMonoid = record
@@ -855,18 +831,10 @@ record IsRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
   open IsMonoid *-isMonoid public
     using ()
     renaming
-    ( ∙-congˡ     to *-congˡ
-    ; ∙-congʳ     to *-congʳ
-    ; identityˡ   to *-identityˡ
+    ( identityˡ   to *-identityˡ
     ; identityʳ   to *-identityʳ
     )
 
-  zeroˡ : LeftZero 0# *
-  zeroˡ = proj₁ zero
-
-  zeroʳ : RightZero 0# *
-  zeroʳ = proj₂ zero
-
   isSemiringWithoutAnnihilatingZero
     : IsSemiringWithoutAnnihilatingZero + * 0# 1#
   isSemiringWithoutAnnihilatingZero = record
@@ -885,7 +853,7 @@ record IsRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
     }
 
   open IsSemiring isSemiring public
-    using (distribˡ; distribʳ; isNearSemiring; isSemiringWithoutOne)
+    using (isNearSemiring; isSemiringWithoutOne)
 
 
 record IsCommutativeRing
diff --git a/src/Algebra/Structures/Biased.agda b/src/Algebra/Structures/Biased.agda
index a2511ed843..8218a1ee1b 100644
--- a/src/Algebra/Structures/Biased.agda
+++ b/src/Algebra/Structures/Biased.agda
@@ -192,6 +192,33 @@ open IsCommutativeSemiringʳ public
 ------------------------------------------------------------------------
 -- IsRing
 
+record IsRing* (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
+  field
+    +-isAbelianGroup : IsAbelianGroup + 0# -_
+    *-isMonoid       : IsMonoid * 1#
+    distrib          : * DistributesOver +
+    zero             : Zero 0# *
+
+  isRing : IsRing + * -_ 0# 1#
+  isRing = record
+    { +-isAbelianGroup = +-isAbelianGroup
+    ; *-cong = ∙-cong
+    ; *-assoc = assoc
+    ; *-identity = identity
+    ; distrib = distrib
+    } where open IsMonoid *-isMonoid
+
+open IsRing* public
+  using () renaming (isRing to isRing*)
+
+
+
+------------------------------------------------------------------------
+-- Deprecated
+------------------------------------------------------------------------
+
+-- Version 2.0
+
 -- We can recover a ring without proving that 0# annihilates *.
 record IsRingWithoutAnnihilatingZero (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A)
                                      : Set (a ⊔ ℓ) where
@@ -224,28 +251,16 @@ record IsRingWithoutAnnihilatingZero (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A)
     ; *-assoc          = *.assoc
     ; *-identity       = *.identity
     ; distrib          = distrib
-    ; zero             = zero
     }
 
 open IsRingWithoutAnnihilatingZero public
   using () renaming (isRing to isRingWithoutAnnihilatingZero)
 
-record IsRing* (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
-  field
-    +-isAbelianGroup : IsAbelianGroup + 0# -_
-    *-isMonoid       : IsMonoid * 1#
-    distrib          : * DistributesOver +
-    zero             : Zero 0# *
-
-  isRing : IsRing + * -_ 0# 1#
-  isRing = record
-    { +-isAbelianGroup = +-isAbelianGroup
-    ; *-cong = ∙-cong
-    ; *-assoc = assoc
-    ; *-identity = identity
-    ; distrib = distrib
-    ; zero = zero
-    } where open IsMonoid *-isMonoid
-
-open IsRing* public
-  using () renaming (isRing to isRing*)
+{-# WARNING_ON_USAGE IsRingWithoutAnnihilatingZero
+"Warning: IsRingWithoutAnnihilatingZero was deprecated in v2.0.
+Please use the standard `IsRing` instead."
+#-}
+{-# WARNING_ON_USAGE isRingWithoutAnnihilatingZero
+"Warning: isRingWithoutAnnihilatingZero was deprecated in v2.0.
+Please use the standard `IsRing` instead."
+#-}
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 01bdfcfe0c..ca69682633 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -1532,7 +1532,6 @@ private
   ; *-assoc          = *-assoc
   ; *-identity       = *-identity
   ; distrib          = *-distrib-+
-  ; zero             = *-zero
   }
 
 +-*-isCommutativeRing : IsCommutativeRing _+_ _*_ -_ 0ℤ 1ℤ
diff --git a/src/Data/Parity/Properties.agda b/src/Data/Parity/Properties.agda
index 13fdb3ed57..079257482b 100644
--- a/src/Data/Parity/Properties.agda
+++ b/src/Data/Parity/Properties.agda
@@ -371,7 +371,6 @@ p+p≡0ℙ 1ℙ = refl
   ; *-assoc          = *-assoc
   ; *-identity       = *-identity
   ; distrib          = *-distrib-+
-  ; zero             = *-zero
   }
 
 +-*-ring : Ring 0ℓ 0ℓ
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index dce11c3f21..5a18639aae 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -1387,7 +1387,6 @@ nonNeg*nonNeg⇒nonNeg p q = nonNegative
   ; *-assoc          = *-assoc
   ; *-identity       = *-identity
   ; distrib          = *-distrib-+
-  ; zero             = *-zero
   }
 
 +-*-isCommutativeRing : IsCommutativeRing _≃_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ

From 47b458c3838a43b8fe734cc5ffc1896615b2df81 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 29 Nov 2023 16:35:49 +0800
Subject: [PATCH 195/316] Fix #2216 by making divisibility definitions records
 (#2217)

* Fix #2216 by making divisibility definitions records

* remove spurious/ambiguous `import`

---------

Co-authored-by: jamesmckinna <j.mckinna@hw.ac.uk>
---
 CHANGELOG.md                                  |  4 +++
 src/Algebra/Definitions/RawMagma.agda         | 25 ++++++++++++++-----
 .../Properties/Magma/Divisibility.agda        |  2 +-
 src/Data/Nat/Base.agda                        |  3 +--
 4 files changed, 25 insertions(+), 9 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6704471d12..f494eee972 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1048,6 +1048,10 @@ Non-backwards compatible changes
 
 * In `Algebra.Core` the operations `Opₗ` and `Opᵣ` have moved to `Algebra.Module.Core`.
 
+* In `Algebra.Definitions.RawMagma.Divisibility` the definitions for `_∣ˡ_` and `_∣ʳ_`
+  have been changed from being defined as raw products to being defined as records. However,
+  the record constructors are called `_,_` so the changes required are minimal.
+
 * In `Codata.Guarded.Stream` the following functions have been modified to have simpler definitions:
   * `cycle`
   * `interleave⁺`
diff --git a/src/Algebra/Definitions/RawMagma.agda b/src/Algebra/Definitions/RawMagma.agda
index dc96fedc76..f235f5d622 100644
--- a/src/Algebra/Definitions/RawMagma.agda
+++ b/src/Algebra/Definitions/RawMagma.agda
@@ -27,18 +27,31 @@ open RawMagma M renaming (Carrier to A)
 
 infix 5 _∣ˡ_ _∤ˡ_ _∣ʳ_ _∤ʳ_ _∣_ _∤_
 
--- Divisibility from the left
-
-_∣ˡ_ : Rel A (a ⊔ ℓ)
-x ∣ˡ y = ∃ λ q → (x ∙ q) ≈ y
+-- Divisibility from the left.
+--
+-- This and, the definition of right divisibility below, are defined as
+-- records rather than in terms of the base product type in order to
+-- make the use of pattern synonyms more ergonomic (see #2216 for
+-- further details). The record field names are not designed to be
+-- used explicitly and indeed aren't re-exported publicly by
+-- `Algebra.X.Properties.Divisibility` modules.
+
+record _∣ˡ_ (x y : A) : Set (a ⊔ ℓ) where
+  constructor _,_
+  field
+    quotient : A
+    equality : x ∙ quotient ≈ y
 
 _∤ˡ_ : Rel A (a ⊔ ℓ)
 x ∤ˡ y = ¬ x ∣ˡ y
 
 -- Divisibility from the right
 
-_∣ʳ_ : Rel A (a ⊔ ℓ)
-x ∣ʳ y = ∃ λ q → (q ∙ x) ≈ y
+record _∣ʳ_ (x y : A) : Set (a ⊔ ℓ) where
+  constructor _,_
+  field
+    quotient : A
+    equality : quotient ∙ x ≈ y
 
 _∤ʳ_ : Rel A (a ⊔ ℓ)
 x ∤ʳ y = ¬ x ∣ʳ y
diff --git a/src/Algebra/Properties/Magma/Divisibility.agda b/src/Algebra/Properties/Magma/Divisibility.agda
index d8ffc208f0..629406faa1 100644
--- a/src/Algebra/Properties/Magma/Divisibility.agda
+++ b/src/Algebra/Properties/Magma/Divisibility.agda
@@ -18,7 +18,7 @@ open Magma M
 -- Re-export divisibility relations publicly
 
 open import Algebra.Definitions.RawMagma rawMagma public
-  using (_∣_; _∤_; _∣∣_; _∤∤_; _∣ˡ_; _∤ˡ_; _∣ʳ_; _∤ʳ_)
+  using (_∣_; _∤_; _∣∣_; _∤∤_; _∣ˡ_; _∤ˡ_; _∣ʳ_; _∤ʳ_; _,_)
 
 ------------------------------------------------------------------------
 -- Properties of divisibility
diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index 6135cfe01d..478b71f783 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -12,10 +12,9 @@
 module Data.Nat.Base where
 
 open import Algebra.Bundles.Raw using (RawMagma; RawMonoid; RawNearSemiring; RawSemiring)
-open import Algebra.Definitions.RawMagma using (_∣ˡ_)
+open import Algebra.Definitions.RawMagma using (_∣ˡ_; _,_)
 open import Data.Bool.Base using (Bool; true; false; T; not)
 open import Data.Parity.Base using (Parity; 0ℙ; 1ℙ)
-open import Data.Product.Base using (_,_)
 open import Level using (0ℓ)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.PropositionalEquality.Core

From 39cfbcf0cfcd291c6d731050e249a9c76766d8b8 Mon Sep 17 00:00:00 2001
From: Alex Rice <alexrice999@hotmail.co.uk>
Date: Wed, 6 Dec 2023 15:17:15 +0000
Subject: [PATCH 196/316] Fix deprecated modules (#2224)

* Fix deprecated modules

* [ ci ] Also build deprecated modules

* [ ci ] ignore user warnings in test

* [ ci ] fix filtering logic

Deprecation and safety are not the same thing

---------

Co-authored-by: Guillaume Allais <guillaume.allais@ens-lyon.org>
---
 .github/workflows/ci-ubuntu.yml               | 10 +++++-
 GenerateEverything.hs                         | 32 +++++++++++--------
 src/Algebra/Operations/Semiring.agda          |  6 ++--
 src/Algebra/Properties/BooleanAlgebra.agda    |  9 +++---
 .../Properties/DistributiveLattice.agda       |  5 ++-
 src/Data/Fin/Substitution/Example.agda        |  2 +-
 6 files changed, 37 insertions(+), 27 deletions(-)

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index 9ac08748df..a40fe804ad 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -151,10 +151,19 @@ jobs:
 
       - name: Test stdlib
         run: |
+          # Including deprecated modules purely for testing
+          cabal run GenerateEverything -- --include-deprecated
+          ${{ env.AGDA }} -WnoUserWarning --safe EverythingSafe.agda
+          ${{ env.AGDA }} -WnoUserWarning Everything.agda
+
+      - name: Prepare HTML index
+        run: |
+          # Regenerating the Everything files without the deprecated modules
           cabal run GenerateEverything
           cp .github/tooling/* .
           ./index.sh
           ${{ env.AGDA }} --safe EverythingSafe.agda
+          ${{ env.AGDA }} Everything.agda
           ${{ env.AGDA }} index.agda
 
       - name: Golden testing
@@ -177,7 +186,6 @@ jobs:
           rm -f '${{ env.AGDA_HTML_DIR }}'/*.html
           rm -f '${{ env.AGDA_HTML_DIR }}'/*.css
           ${{ env.AGDA }} --html --html-dir ${{ env.AGDA_HTML_DIR }} index.agda
-
           cp .github/tooling/* .
           ./landing.sh
 
diff --git a/GenerateEverything.hs b/GenerateEverything.hs
index edea56a0f0..f6add73649 100644
--- a/GenerateEverything.hs
+++ b/GenerateEverything.hs
@@ -1,6 +1,7 @@
 {-# LANGUAGE PatternGuards #-}
 {-# LANGUAGE PatternSynonyms #-}
 {-# LANGUAGE RecordWildCards #-}
+{-# LANGUAGE MultiWayIf #-}
 
 import Control.Applicative
 import Control.Monad
@@ -227,10 +228,10 @@ extractHeader mod = extract
 
 -- | A crude classifier looking for lines containing options
 
-data Status = Deprecated | Unsafe | Safe
-  deriving (Eq)
+data Safety = Unsafe | Safe deriving (Eq)
+data Status = Deprecated | Active deriving (Eq)
 
-classify :: FilePath -> [String] -> [String] -> Exc Status
+classify :: FilePath -> [String] -> [String] -> Exc (Safety, Status)
 classify fp hd ls
   -- We start with sanity checks
   | isUnsafe && safe          = throwError $ fp ++ contradiction "unsafe" "safe"
@@ -239,11 +240,12 @@ classify fp hd ls
   | isWithK && not withK      = throwError $ fp ++ missingWithK
   | not (isWithK || cubicalC) = throwError $ fp ++ uncategorized "as relying on K" "cubical-compatible"
   -- And then perform the actual classification
-  | deprecated                = pure $ Deprecated
-  | isUnsafe                  = pure $ Unsafe
-  | safe                      = pure $ Safe
-  -- We know that @not (isUnsafe || safe)@, all cases are covered
-  | otherwise                 = error "IMPOSSIBLE"
+  | otherwise = do
+      let safety = if | safe -> Safe
+                      | isUnsafe -> Unsafe
+                      | otherwise -> error "IMPOSSIBLE"
+      let status = if deprecated then Deprecated else Active
+      pure (safety, status)
 
   where
 
@@ -280,18 +282,20 @@ classify fp hd ls
 data LibraryFile = LibraryFile
   { filepath   :: FilePath -- ^ FilePath of the source file
   , header     :: [String] -- ^ All lines in the headers are already prefixed with \"-- \".
-  , status     :: Status   -- ^ Safety options used by the module
+  , safety     :: Safety
+  , status     :: Status   -- ^ Deprecation status options used by the module
   }
 
 analyse :: FilePath -> IO LibraryFile
 analyse fp = do
   ls <- lines <$> readFileUTF8 fp
   hd <- runExc $ extractHeader fp ls
-  cl <- runExc $ classify fp hd ls
+  (sf, st) <- runExc $ classify fp hd ls
   return $ LibraryFile
-    { filepath   = fp
-    , header     = hd
-    , status     = cl
+    { filepath = fp
+    , header   = hd
+    , safety   = sf
+    , status   = st
     }
 
 checkFilePaths :: String -> [FilePath] -> IO ()
@@ -356,7 +360,7 @@ main = do
     unlines [ header
             , "{-# OPTIONS --safe --guardedness #-}\n"
             , mkModule safeOutputFile
-            , format $ filter ((Unsafe /=) . status) libraryfiles
+            , format $ filter ((Unsafe /=) . safety) libraryfiles
             ]
 
 -- | Usage info.
diff --git a/src/Algebra/Operations/Semiring.agda b/src/Algebra/Operations/Semiring.agda
index fbc6c254c2..08d55c884d 100644
--- a/src/Algebra/Operations/Semiring.agda
+++ b/src/Algebra/Operations/Semiring.agda
@@ -26,6 +26,6 @@ open Semiring S
 -- Re-exports
 
 open MonoidOperations +-commutativeMonoid public
-open import Algebra.Properties.Semiring.Exponentiation S public
-open import Algebra.Properties.Semiring.Multiplication S public
-  using (×1-homo-*; ×′1-homo-*)
+open import Algebra.Properties.Semiring.Exp S public
+open import Algebra.Properties.Semiring.Mult S public
+  using (×1-homo-*)
diff --git a/src/Algebra/Properties/BooleanAlgebra.agda b/src/Algebra/Properties/BooleanAlgebra.agda
index 2e1f148701..0445f20505 100644
--- a/src/Algebra/Properties/BooleanAlgebra.agda
+++ b/src/Algebra/Properties/BooleanAlgebra.agda
@@ -28,8 +28,7 @@ import Algebra.Properties.DistributiveLattice as DistribLatticeProperties
 
 open import Algebra.Structures _≈_
 open import Relation.Binary
-open import Function.Equality using (_⟨$⟩_)
-open import Function.Equivalence using (_⇔_; module Equivalence)
+open import Function.Bundles using (module Equivalence; _⇔_)
 open import Data.Product.Base using (_,_)
 
 ------------------------------------------------------------------------
@@ -47,9 +46,9 @@ replace-equality {_≈′_} ≈⇔≈′ = record
   { isBooleanAlgebra = record
     { isDistributiveLattice = DistributiveLattice.isDistributiveLattice
         (DistribLatticeProperties.replace-equality distributiveLattice ≈⇔≈′)
-    ; ∨-complement          = ((λ x → to ⟨$⟩ ∨-complementˡ x) , λ x → to ⟨$⟩ ∨-complementʳ x)
-    ; ∧-complement          = ((λ x → to ⟨$⟩ ∧-complementˡ x) , λ x → to ⟨$⟩ ∧-complementʳ x)
-    ; ¬-cong                = λ i≈j → to ⟨$⟩ ¬-cong (from ⟨$⟩ i≈j)
+    ; ∨-complement          = ((λ x → to (∨-complementˡ x)) , λ x → to (∨-complementʳ x))
+    ; ∧-complement          = ((λ x → to (∧-complementˡ x)) , λ x → to (∧-complementʳ x))
+    ; ¬-cong                = λ i≈j → to (¬-cong (from i≈j))
     }
   } where open module E {x y} = Equivalence (≈⇔≈′ {x} {y})
 {-# WARNING_ON_USAGE replace-equality
diff --git a/src/Algebra/Properties/DistributiveLattice.agda b/src/Algebra/Properties/DistributiveLattice.agda
index adb6266e62..df03dfa38a 100644
--- a/src/Algebra/Properties/DistributiveLattice.agda
+++ b/src/Algebra/Properties/DistributiveLattice.agda
@@ -12,8 +12,7 @@
 open import Algebra.Lattice.Bundles
 open import Algebra.Lattice.Structures.Biased
 open import Relation.Binary
-open import Function.Equality
-open import Function.Equivalence
+open import Function.Bundles using (module Equivalence; _⇔_)
 import Algebra.Construct.Subst.Equality as SubstEq
 
 module Algebra.Properties.DistributiveLattice
@@ -44,7 +43,7 @@ replace-equality {_≈′_} ≈⇔≈′ = record
   { isDistributiveLattice = isDistributiveLatticeʳʲᵐ (record
     { isLattice    = Lattice.isLattice
                        (LatticeProperties.replace-equality lattice ≈⇔≈′)
-    ; ∨-distribʳ-∧ = λ x y z → to ⟨$⟩ ∨-distribʳ-∧ x y z
+    ; ∨-distribʳ-∧ = λ x y z → to (∨-distribʳ-∧ x y z)
     })
   } where open module E {x y} = Equivalence (≈⇔≈′ {x} {y})
 {-# WARNING_ON_USAGE replace-equality
diff --git a/src/Data/Fin/Substitution/Example.agda b/src/Data/Fin/Substitution/Example.agda
index 02c98c539f..453c47d67f 100644
--- a/src/Data/Fin/Substitution/Example.agda
+++ b/src/Data/Fin/Substitution/Example.agda
@@ -20,7 +20,7 @@ open import Data.Fin.Substitution.Lemmas
 open import Data.Nat.Base hiding (_/_)
 open import Data.Fin.Base using (Fin)
 open import Data.Vec.Base
-open import Relation.Binary.PropositionalEquality.Core as PropEq
+open import Relation.Binary.PropositionalEquality as PropEq
   using (_≡_; refl; sym; cong; cong₂)
 open PropEq.≡-Reasoning
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive

From 24370a342660cab84b81d533a5e6dade7d5db3eb Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Mon, 11 Dec 2023 18:47:02 +0800
Subject: [PATCH 197/316] Final admin changes for v2.0 release (#2225)

* Final admin changes for v2.0 release

* Fix Agda versions
---
 .github/workflows/ci-ubuntu.yml | 2 +-
 CHANGELOG.md                    | 6 +++---
 CITATION.cff                    | 4 ++--
 LICENCE                         | 2 +-
 doc/installation-guide.md       | 2 +-
 5 files changed, 8 insertions(+), 8 deletions(-)

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index a40fe804ad..095f28b1f8 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -79,7 +79,7 @@ jobs:
             echo "AGDA_HTML_DIR=html/experimental" >> $GITHUB_ENV
           else
             # Pick Agda version for master
-            echo "AGDA_COMMIT=tags/v2.6.4" >> $GITHUB_ENV;
+            echo "AGDA_COMMIT=tags/v2.6.4.1" >> $GITHUB_ENV;
             echo "AGDA_HTML_DIR=html/master" >> $GITHUB_ENV
           fi
 
diff --git a/CHANGELOG.md b/CHANGELOG.md
index f494eee972..134309712c 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,7 +1,7 @@
-Version 2.0-rc1
-===============
+Version 2.0
+===========
 
-The library has been tested using Agda 2.6.4.
+The library has been tested using Agda 2.6.4 and 2.6.4.1.
 
 NOTE: Version `2.0` contains various breaking changes and is not backwards
 compatible with code written with version `1.X` of the library.
diff --git a/CITATION.cff b/CITATION.cff
index 416e595b98..5e965d00fa 100644
--- a/CITATION.cff
+++ b/CITATION.cff
@@ -3,6 +3,6 @@ message: "If you use this software, please cite it as below."
 authors:
 - name: "The Agda Community"
 title: "Agda Standard Library"
-version: 1.7.2
-date-released: 2023-02-01
+version: 2.0
+date-released: 2023-12-11
 url: "https://github.com/agda/agda-stdlib"
\ No newline at end of file
diff --git a/LICENCE b/LICENCE
index df31f24985..3a9d1c3624 100644
--- a/LICENCE
+++ b/LICENCE
@@ -1,4 +1,4 @@
-Copyright (c) 2007-2021 Nils Anders Danielsson, Ulf Norell, Shin-Cheng
+Copyright (c) 2007-2023 Nils Anders Danielsson, Ulf Norell, Shin-Cheng
 Mu, Bradley Hardy, Samuel Bronson, Dan Doel, Patrik Jansson,
 Liang-Ting Chen, Jean-Philippe Bernardy, Andrés Sicard-Ramírez,
 Nicolas Pouillard, Darin Morrison, Peter Berry, Daniel Brown,
diff --git a/doc/installation-guide.md b/doc/installation-guide.md
index 67d1685ad9..956fc317db 100644
--- a/doc/installation-guide.md
+++ b/doc/installation-guide.md
@@ -3,7 +3,7 @@ Installation instructions
 
 Note: the full story on installing Agda libraries can be found at [readthedocs](http://agda.readthedocs.io/en/latest/tools/package-system.html).
 
-Use version v2.0 of the standard library with Agda 2.6.4.
+Use version v2.0 of the standard library with Agda 2.6.4 or 2.6.4.1.
 
 1. Navigate to a suitable directory `$HERE` (replace appropriately) where
    you would like to install the library.

From 94abfa5fb3ec04df9a0c0c19c4ad39738d74433e Mon Sep 17 00:00:00 2001
From: Alex Rice <alexrice999@hotmail.co.uk>
Date: Tue, 12 Dec 2023 16:47:45 +0000
Subject: [PATCH 198/316] Fix typo in raise deprecation message (#2226)

---
 src/Data/Fin/Base.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Data/Fin/Base.agda b/src/Data/Fin/Base.agda
index 9b4c303f2f..d762e10007 100644
--- a/src/Data/Fin/Base.agda
+++ b/src/Data/Fin/Base.agda
@@ -317,7 +317,7 @@ compare (suc i) (suc j) with compare i j
 raise = _↑ʳ_
 {-# WARNING_ON_USAGE raise
 "Warning: raise was deprecated in v2.0.
-Please use _↑_ʳ instead."
+Please use _↑ʳ_ instead."
 #-}
 inject+ : ∀ {m} n → Fin m → Fin (m ℕ.+ n)
 inject+ n i = i ↑ˡ n

From 73ff4960b61791e1bcdbe31e38add499d20e2931 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 13 Dec 2023 23:33:05 +0800
Subject: [PATCH 199/316] Setup for v2.1 (#2227)

---
 CHANGELOG.md          | 3891 +---------------------------------------
 CHANGELOG/v2.0.md     | 3911 +++++++++++++++++++++++++++++++++++++++++
 doc/release-guide.txt |    4 +-
 3 files changed, 3916 insertions(+), 3890 deletions(-)
 create mode 100644 CHANGELOG/v2.0.md

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 134309712c..c32883f2b7 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,3911 +1,28 @@
-Version 2.0
-===========
+Version 2.1-dev
+===============
 
 The library has been tested using Agda 2.6.4 and 2.6.4.1.
 
-NOTE: Version `2.0` contains various breaking changes and is not backwards
-compatible with code written with version `1.X` of the library.
-
 Highlights
 ----------
 
-* A new tactic `cong!` available from `Tactic.Cong` which automatically
-  infers the argument to `cong` for you via anti-unification.
-
-* Improved the `solve` tactic in `Tactic.RingSolver` to work in a much
-  wider range of situations.
-
-* A massive refactoring of the unindexed `Functor`/`Applicative`/`Monad` hierarchy
-  and the `MonadReader` / `MonadState` type classes. These are now usable with
-  instance arguments as demonstrated in the tests/monad examples.
-
-* Significant tightening of `import` statements internally in the library,
-  drastically decreasing the dependencies and hence load time of many key
-  modules.
-
-* A golden testing library in `Test.Golden`. This allows you to run a set
-  of tests and make sure their output matches an expected `golden` value.
-  The test runner has many options: filtering tests by name, dumping the
-  list of failures to a file, timing the runs, coloured output, etc.
-  Cf. the comments in `Test.Golden` and the standard library's own tests
-  in `tests/` for documentation on how to use the library.
-
 Bug-fixes
 ---------
 
-* In `Algebra.Structures` the records `IsRing` and `IsRingWithoutOne` contained an unnecessary field
-  `zero : RightZero 0# *`, which could be derived from the other ring axioms.
-  Consequently this field has been removed from the record, and the record
-  `IsRingWithoutAnnihilatingZero` in `Algebra.Structures.Biased` has been
-  deprecated as it is now identical to is `IsRing`.
-  
-* In `Algebra.Definitions.RawSemiring` the record `Prime` did not
-  enforce that the number was not divisible by `1#`. To fix this
-  `p∤1 : p ∤ 1#` has been added as a field.
-
-* In `Data.Container.FreeMonad`, we give a direct definition of `_⋆_` as an inductive
-  type rather than encoding it as an instance of `μ`. This ensures Agda notices that
-  `C ⋆ X` is strictly positive in `X` which in turn allows us to use the free monad
-  when defining auxiliary (co)inductive types (cf. the `Tap` example in
-  `README.Data.Container.FreeMonad`).
-
-* In `Data.Fin.Properties` the `i` argument to `opposite-suc` was implicit
-  but could not be inferred in general. It has been made explicit.
-
-* In `Data.List.Membership.Setoid` the operations `_∷=_` and `_─_`
-  had an extraneous `{A : Set a}` parameter. This has been removed.
-
-* In `Data.List.Relation.Ternary.Appending.Setoid` the constructors
-  were re-exported in their full generality which lead to unsolved meta
-  variables at their use sites. Now versions of the constructors specialised
-  to use the setoid's carrier set are re-exported.
-
-* In `Data.Nat.DivMod` the parameter `o` in the proof `/-monoˡ-≤` was
-  implicit but not inferrable. It has been changed to be explicit.
-
-* In `Data.Nat.DivMod` the parameter `m` in the proof `+-distrib-/-∣ʳ` was
-  implicit but not inferrable, while `n` is explicit but inferrable.
-  They have been to explicit and implicit respectively.
-
-* In `Data.Nat.GeneralisedArithmetic` the `s` and `z` arguments to the
-  following functions were implicit but not inferrable:
-  `fold-+`, `fold-k`, `fold-*`, `fold-pull`. They have been made explicit.
-
-* In `Data.Rational(.Unnormalised).Properties` the module `≤-Reasoning`
-  exported equality combinators using the generic setoid symbol `_≈_`. They
-  have been renamed to use the same `_≃_` symbol used for non-propositional
-  equality over `Rational`s, i.e.
-  ```agda
-  step-≈  ↦  step-≃
-  step-≈˘ ↦  step-≃˘
-  ```
-  with corresponding associated syntax:
-  ```agda
-  _≈⟨_⟩_ ↦ _≃⟨_⟩_
-  _≈⟨_⟨_ ↦ _≃⟨_⟨_
-  ```
-
-* In `Function.Construct.Composition` the combinators
-  `_⟶-∘_`, `_↣-∘_`, `_↠-∘_`, `_⤖-∘_`, `_⇔-∘_`, `_↩-∘_`, `_↪-∘_`, `_↔-∘_`
-  had their arguments in the wrong order. They have been flipped so they can
-  actually be  used as a composition operator.
-
-* In `Function.Definitions` the definitions of `Surjection`, `Inverseˡ`,
-  `Inverseʳ` were not being re-exported correctly and therefore had an unsolved
-  meta-variable whenever this module was explicitly parameterised. This has
-  been fixed.
-
-* In `System.Exit` the `ExitFailure` constructor is now carrying an integer
-  rather than a natural. The previous binding was incorrectly assuming that
-  all exit codes where non-negative.
-
 Non-backwards compatible changes
 --------------------------------
 
-### Removed deprecated names
-
-* All modules and names that were deprecated in `v1.2` and before have
-  been removed.
-
-### Changes to `LeftCancellative` and `RightCancellative` in `Algebra.Definitions`
-
-* The definitions of the types for cancellativity in `Algebra.Definitions` previously
-  made some of their arguments implicit. This was under the assumption that the operators were
-  defined by pattern matching on the left argument so that Agda could always infer the
-  argument on the RHS.
-
-* Although many of the operators defined in the library follow this convention, this is not
-  always true and cannot be assumed in user's code.
-
-* Therefore the definitions have been changed as follows to make all their arguments explicit:
-  - `LeftCancellative _∙_`
-    - From: `∀ x {y z} → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
-    - To: `∀ x y z → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
-
-  - `RightCancellative _∙_`
-    - From: `∀ {x} y z → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
-    - To: `∀ x y z → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
-
-  - `AlmostLeftCancellative e _∙_`
-    - From: `∀ {x} y z → ¬ x ≈ e → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
-    - To: `∀ x y z → ¬ x ≈ e → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
-
-  - `AlmostRightCancellative e _∙_`
-    - From: `∀ {x} y z → ¬ x ≈ e → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
-    - To: `∀ x y z → ¬ x ≈ e → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
-
-* Correspondingly some proofs of the above types will need additional arguments passed explicitly.
-  Instances can easily be fixed by adding additional underscores, e.g.
-  - `∙-cancelˡ x` to `∙-cancelˡ x _ _`
-  - `∙-cancelʳ y z` to `∙-cancelʳ _ y z`
-
-### Changes to ring structures in `Algebra`
-
-* Several ring-like structures now have the multiplicative structure defined by
-  its laws rather than as a substructure, to avoid repeated proofs that the
-  underlying relation is an equivalence. These are:
-  * `IsNearSemiring`
-  * `IsSemiringWithoutOne`
-  * `IsSemiringWithoutAnnihilatingZero`
-  * `IsRing`
-
-* To aid with migration, structures matching the old style ones have been added
-  to `Algebra.Structures.Biased`, with conversion functions:
-  * `IsNearSemiring*` and `isNearSemiring*`
-  * `IsSemiringWithoutOne*` and `isSemiringWithoutOne*`
-  * `IsSemiringWithoutAnnihilatingZero*` and `isSemiringWithoutAnnihilatingZero*`
-  * `IsRing*` and `isRing*`
-
-### Refactoring of lattices in `Algebra.Structures/Bundles` hierarchy
-
-* In order to improve modularity and consistency with `Relation.Binary.Lattice`,
-  the structures & bundles for `Semilattice`, `Lattice`, `DistributiveLattice`
-  & `BooleanAlgebra` have been moved out of the `Algebra` modules and into their
-  own hierarchy in `Algebra.Lattice`.
-
-* All submodules, (e.g. `Algebra.Properties.Semilattice` or `Algebra.Morphism.Lattice`)
-  have been moved to the corresponding place under `Algebra.Lattice` (e.g.
-  `Algebra.Lattice.Properties.Semilattice` or `Algebra.Lattice.Morphism.Lattice`). See
-  the `Deprecated modules` section below for full details.
-
-* The definition of `IsDistributiveLattice` and `IsBooleanAlgebra` have changed so
-  that they are no longer right-biased which hindered compositionality.
-  More concretely, `IsDistributiveLattice` now has fields:
-  ```agda
-  ∨-distrib-∧ : ∨ DistributesOver ∧
-  ∧-distrib-∨ : ∧ DistributesOver ∨
-  ```
-  instead of
-  ```agda
-  ∨-distribʳ-∧ : ∨ DistributesOverʳ ∧
-  ```
-  and `IsBooleanAlgebra` now has fields:
-  ```agda
-  ∨-complement : Inverse ⊤ ¬ ∨
-  ∧-complement : Inverse ⊥ ¬ ∧
-  ```
-  instead of:
-  ```agda
-  ∨-complementʳ : RightInverse ⊤ ¬ ∨
-  ∧-complementʳ : RightInverse ⊥ ¬ ∧
-  ```
-
-* To allow construction of these structures via their old form, smart constructors
-  have been added to a new module `Algebra.Lattice.Structures.Biased`, which are by
-  re-exported automatically by `Algebra.Lattice`. For example, if before you wrote:
-  ```agda
-  ∧-∨-isDistributiveLattice = record
-    { isLattice    = ∧-∨-isLattice
-    ; ∨-distribʳ-∧ = ∨-distribʳ-∧
-    }
-  ```
-  you can use the smart constructor `isDistributiveLatticeʳʲᵐ` to write:
-  ```agda
-  ∧-∨-isDistributiveLattice = isDistributiveLatticeʳʲᵐ (record
-    { isLattice    = ∧-∨-isLattice
-    ; ∨-distribʳ-∧ = ∨-distribʳ-∧
-    })
-  ```
-  without having to prove full distributivity.
-
-* Added new `IsBoundedSemilattice`/`BoundedSemilattice` records.
-
-* Added new aliases `Is(Meet/Join)(Bounded)Semilattice` for `Is(Bounded)Semilattice`
-  which can be used to indicate meet/join-ness of the original structures, and
-  the field names in `IsSemilattice` and `Semilattice` have been renamed from 
-  `∧-cong` to `∙-cong`to indicate their undirected nature.
-
-* Finally, the following auxiliary files have been moved:
-  ```agda
-  Algebra.Properties.Semilattice               ↦ Algebra.Lattice.Properties.Semilattice
-  Algebra.Properties.Lattice                   ↦ Algebra.Lattice.Properties.Lattice
-  Algebra.Properties.DistributiveLattice       ↦ Algebra.Lattice.Properties.DistributiveLattice
-  Algebra.Properties.BooleanAlgebra            ↦ Algebra.Lattice.Properties.BooleanAlgebra
-  Algebra.Properties.BooleanAlgebra.Expression ↦ Algebra.Lattice.Properties.BooleanAlgebra.Expression
-  Algebra.Morphism.LatticeMonomorphism         ↦ Algebra.Lattice.Morphism.LatticeMonomorphism
-  ```
-
-#### Changes to `Algebra.Morphism.Structures`
-
-* Previously the record definitions:
-  ```
-  IsNearSemiringHomomorphism
-  IsSemiringHomomorphism
-  IsRingHomomorphism
-  ```
-  all had two separate proofs of `IsRelHomomorphism` within them.
-
-* To fix this they have all been redefined to build up from `IsMonoidHomomorphism`,
-  `IsNearSemiringHomomorphism`, and `IsSemiringHomomorphism` respectively,
-  adding a single property at each step.
-
-* Similarly, `IsLatticeHomomorphism` is now built as
-  `IsRelHomomorphism` along with proofs that `_∧_` and `_∨_` are homomorphic.
-
-* Finally `⁻¹-homo` in `IsRingHomomorphism` has been renamed to `-‿homo`.
-
-#### Renamed `Category` modules to `Effect`
-
-* As observed by Wen Kokke in issue #1636, it no longer really makes sense
-  to group the modules which correspond to the variety of concepts of
-  (effectful) type constructor arising in functional programming (esp. in Haskell)
-  such as `Monad`, `Applicative`, `Functor`, etc, under `Category.*`,
-  as this obstructs the importing of the `agda-categories` development into
-  the Standard Library, and moreover needlessly restricts the applicability of
-  categorical concepts to this (highly specific) mode of use.
-
-* Correspondingly, client modules grouped under  `*.Categorical.*` which
-  exploit such structure  for effectful programming have been  renamed
-  `*.Effectful`, with the originals being deprecated.
-
-* Full list of moved modules:
-  ```agda
-  Codata.Sized.Colist.Categorical            ↦ Codata.Sized.Colist.Effectful
-  Codata.Sized.Covec.Categorical             ↦ Codata.Sized.Covec.Effectful
-  Codata.Sized.Delay.Categorical             ↦ Codata.Sized.Delay.Effectful
-  Codata.Sized.Stream.Categorical            ↦ Codata.Sized.Stream.Effectful
-  Data.List.Categorical                      ↦ Data.List.Effectful
-  Data.List.Categorical.Transformer          ↦ Data.List.Effectful.Transformer
-  Data.List.NonEmpty.Categorical             ↦ Data.List.NonEmpty.Effectful
-  Data.List.NonEmpty.Categorical.Transformer ↦ Data.List.NonEmpty.Effectful.Transformer
-  Data.Maybe.Categorical                     ↦ Data.Maybe.Effectful
-  Data.Maybe.Categorical.Transformer         ↦ Data.Maybe.Effectful.Transformer
-  Data.Product.Categorical.Examples          ↦ Data.Product.Effectful.Examples
-  Data.Product.Categorical.Left              ↦ Data.Product.Effectful.Left
-  Data.Product.Categorical.Left.Base         ↦ Data.Product.Effectful.Left.Base
-  Data.Product.Categorical.Right             ↦ Data.Product.Effectful.Right
-  Data.Product.Categorical.Right.Base        ↦ Data.Product.Effectful.Right.Base
-  Data.Sum.Categorical.Examples              ↦ Data.Sum.Effectful.Examples
-  Data.Sum.Categorical.Left                  ↦ Data.Sum.Effectful.Left
-  Data.Sum.Categorical.Left.Transformer      ↦ Data.Sum.Effectful.Left.Transformer
-  Data.Sum.Categorical.Right                 ↦ Data.Sum.Effectful.Right
-  Data.Sum.Categorical.Right.Transformer     ↦ Data.Sum.Effectful.Right.Transformer
-  Data.These.Categorical.Examples            ↦ Data.These.Effectful.Examples
-  Data.These.Categorical.Left                ↦ Data.These.Effectful.Left
-  Data.These.Categorical.Left.Base           ↦ Data.These.Effectful.Left.Base
-  Data.These.Categorical.Right               ↦ Data.These.Effectful.Right
-  Data.These.Categorical.Right.Base          ↦ Data.These.Effectful.Right.Base
-  Data.Vec.Categorical                       ↦ Data.Vec.Effectful
-  Data.Vec.Categorical.Transformer           ↦ Data.Vec.Effectful.Transformer
-  Data.Vec.Recursive.Categorical             ↦ Data.Vec.Recursive.Effectful
-  Function.Identity.Categorical              ↦ Function.Identity.Effectful
-  IO.Categorical                             ↦ IO.Effectful
-  Reflection.TCM.Categorical                 ↦ Reflection.TCM.Effectful
-  ```
-
-* Full list of new modules:
-  ```
-  Algebra.Construct.Initial
-  Algebra.Construct.Terminal
-  Data.List.Effectful.Transformer
-  Data.List.NonEmpty.Effectful.Transformer
-  Data.Maybe.Effectful.Transformer
-  Data.Sum.Effectful.Left.Transformer
-  Data.Sum.Effectful.Right.Transformer
-  Data.Vec.Effectful.Transformer
-  Effect.Empty
-  Effect.Choice
-  Effect.Monad.Error.Transformer
-  Effect.Monad.Identity
-  Effect.Monad.IO
-  Effect.Monad.IO.Instances
-  Effect.Monad.Reader.Indexed
-  Effect.Monad.Reader.Instances
-  Effect.Monad.Reader.Transformer
-  Effect.Monad.Reader.Transformer.Base
-  Effect.Monad.State.Indexed
-  Effect.Monad.State.Instances
-  Effect.Monad.State.Transformer
-  Effect.Monad.State.Transformer.Base
-  Effect.Monad.Writer
-  Effect.Monad.Writer.Indexed
-  Effect.Monad.Writer.Instances
-  Effect.Monad.Writer.Transformer
-  Effect.Monad.Writer.Transformer.Base
-  IO.Effectful
-  IO.Instances
-  ```
-
-### Refactoring of the unindexed Functor/Applicative/Monad hierarchy in `Effect`
-
-* The unindexed versions are not defined in terms of the named versions anymore.
-
-* The `RawApplicative` and `RawMonad` type classes have been relaxed so that the underlying
-  functors do not need their domain and codomain to live at the same Set level.
-  This is needed for level-increasing functors like `IO : Set l → Set (suc l)`.
-
-* `RawApplicative` is now `RawFunctor + pure + _<*>_` and `RawMonad` is now
-  `RawApplicative` + `_>>=_`.
-  This reorganisation means in particular that the functor/applicative of a monad
-  are not computed using `_>>=_`. This may break proofs.
-
-* When `F : Set f → Set f` we moreover have a definable join/μ operator
-  `join : (M : RawMonad F) → F (F A) → F A`.
-
-* We now have `RawEmpty` and `RawChoice` respectively packing `empty : M A` and
-  `(<|>) : M A → M A → M A`. `RawApplicativeZero`, `RawAlternative`, `RawMonadZero`,
-  `RawMonadPlus` are all defined in terms of these.
-
-* `MonadT T` now returns a `MonadTd` record that packs both a proof that the
-  `Monad M` transformed by `T` is a monad and that we can `lift` a computation
-  `M A` to a transformed computation `T M A`.
-
-* The monad transformer are not mere aliases anymore, they are record-wrapped
-  which allows constraints such as `MonadIO (StateT S (ReaderT R IO))` to be
-  discharged by instance arguments.
-
-* The mtl-style type classes (`MonadState`, `MonadReader`) do not contain a proof
-  that the underlying functor is a `Monad` anymore. This ensures we do not have
-  conflicting `Monad M` instances from a pair of `MonadState S M` & `MonadReader R M`
-  constraints.
-
-* `MonadState S M` is now defined in terms of
-  ```agda
-  gets : (S → A) → M A
-  modify : (S → S) → M ⊤
-  ```
-  with `get` and `put` defined as derived notions.
-  This is needed because `MonadState S M` does not pack a `Monad M` instance anymore
-  and so we cannot define `modify f` as `get >>= λ s → put (f s)`.
-
-* `MonadWriter 𝕎 M` is defined similarly:
-   ```agda
-   writer : W × A → M A
-   listen : M A → M (W × A)
-   pass   : M ((W → W) × A) → M A
-   ```
-   with `tell` defined as a derived notion.
-   Note that `𝕎` is a `RawMonoid`, not a `Set` and `W` is the carrier of the monoid.
-
-
-#### Moved `Codata` modules to `Codata.Sized`
-
-* Due to the change in Agda 2.6.2 where sized types are no longer compatible
-  with the `--safe` flag, it has become clear that a third variant of codata
-  will be needed using coinductive records.
-
-* Therefore all existing modules in `Codata` which used sized types have been
-  moved inside a new folder named `Codata.Sized`,  e.g. `Codata.Stream`
-  has become `Codata.Sized.Stream`.
-
-### New proof-irrelevant for empty type in `Data.Empty`
-
-* The definition of `⊥` has been changed to
-  ```agda
-  private
-    data Empty : Set where
-
-  ⊥ : Set
-  ⊥ = Irrelevant Empty
-  ```
-  in order to make ⊥ proof irrelevant. Any two proofs of `⊥` or of a negated
-  statements are now *judgmentally* equal to each other.
-
-* Consequently the following two definitions have been modified:
-
-  + In `Relation.Nullary.Decidable.Core`, the type of `dec-no` has changed
-    ```agda
-    dec-no : (p? : Dec P) → ¬ P → ∃ λ ¬p′ → p? ≡ no ¬p′
-      ↦
-    dec-no : (p? : Dec P) (¬p : ¬ P) → p? ≡ no ¬p
-    ```
-
-  + In `Relation.Binary.PropositionalEquality`, the type of `≢-≟-identity` has changed
-    ```agda
-    ≢-≟-identity : x ≢ y → ∃ λ ¬eq → x ≟ y ≡ no ¬eq
-      ↦
-    ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y
-    ```
-
-### Deprecation of `_≺_` in `Data.Fin.Base`
-
-* In `Data.Fin.Base` the relation `_≺_` and its single constructor `_≻toℕ_`
-  have been deprecated in favour of their extensional equivalent `_<_`
-  but omitting the inversion principle which pattern matching on `_≻toℕ_`
-  would achieve; this instead is proxied by the property `Data.Fin.Properties.toℕ<`.
-
-* Consequently in `Data.Fin.Induction`:
-  ```
-  ≺-Rec
-  ≺-wellFounded
-  ≺-recBuilder
-  ≺-rec
-  ```
-  these functions are also deprecated.
-
-* Likewise in `Data.Fin.Properties` the proofs `≺⇒<′` and `<′⇒≺` have been deprecated
-  in favour of their proxy counterparts `<⇒<′` and `<′⇒<`.
-
-### Standardisation of `insertAt`/`updateAt`/`removeAt` in `Data.List`/`Data.Vec`
-
-* Previously, the names and argument order of index-based insertion, update and removal functions for
-  various types of lists and vectors were inconsistent.
-
-* To fix this the names have all been standardised to `insertAt`/`updateAt`/`removeAt`.
-
-* Correspondingly the following changes have occurred:
-
-* In `Data.List.Base` the following have been added:
-  ```agda
-  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
-  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
-  removeAt : (xs : List A) → Fin (length xs) → List A
-  ```
-  and the following has been deprecated
-  ```
-  _─_ ↦ removeAt
-  ```
-
-* In `Data.Vec.Base`:
-  ```agda
-  insert ↦ insertAt
-  remove ↦ removeAt
-
-  updateAt : Fin n → (A → A) → Vec A n → Vec A n
-    ↦
-  updateAt : Vec A n → Fin n → (A → A) → Vec A n
-  ```
-
-* In `Data.Vec.Functional`:
-  ```agda
-  remove : Fin (suc n) → Vector A (suc n) → Vector A n
-    ↦
-  removeAt : Vector A (suc n) → Fin (suc n) → Vector A n
-
-  updateAt : Fin n → (A → A) → Vector A n → Vector A n
-    ↦
-  updateAt : Vector A n → Fin n → (A → A) → Vector A n
-  ```
-
-* The old names (and the names of all proofs about these functions) have been deprecated appropriately.
-
-#### Standardisation of `lookup` in `Data.(List/Vec/...)`
-
-* All the types of `lookup` functions (and variants) in the following modules
-  have been changed to match the argument convention adopted in the `List` module (i.e.
-  `lookup` takes its container first and the index, whose type may depend on the
-  container value, second):
-  ```
-  Codata.Guarded.Stream
-  Codata.Guarded.Stream.Relation.Binary.Pointwise
-  Codata.Musical.Colist.Base
-  Codata.Musical.Colist.Relation.Unary.Any.Properties
-  Codata.Musical.Covec
-  Codata.Musical.Stream
-  Codata.Sized.Colist
-  Codata.Sized.Covec
-  Codata.Sized.Stream
-  Data.Vec.Relation.Unary.All
-  Data.Star.Environment
-  Data.Star.Pointer
-  Data.Star.Vec
-  Data.Trie
-  Data.Trie.NonEmpty
-  Data.Tree.AVL
-  Data.Tree.AVL.Indexed
-  Data.Tree.AVL.Map
-  Data.Tree.AVL.NonEmpty
-  Data.Vec.Recursive
-  ```
-
-* To accommodate this in in `Data.Vec.Relation.Unary.Linked.Properties`
-  and `Codata.Guarded.Stream.Relation.Binary.Pointwise`, the proofs
-  called `lookup` have been renamed `lookup⁺`.
-
-#### Changes to `Data.(Nat/Integer/Rational)` proofs of `NonZero`/`Positive`/`Negative` to use instance arguments
-
-* Many numeric operations in the library require their arguments to be non-zero,
-  and various proofs require their arguments to be non-zero/positive/negative etc.
-  As discussed on the [mailing list](https://lists.chalmers.se/pipermail/agda/2021/012693.html),
-  the previous way of constructing and passing round these proofs was extremely
-  clunky and lead to messy and difficult to read code.
-
-* We have therefore changed every occurrence where we need a proof of
-  non-zeroness/positivity/etc. to take it as an irrelevant
-  [instance argument](https://agda.readthedocs.io/en/latest/language/instance-arguments.html).
-  See the mailing list discussion for a fuller explanation of the motivation and implementation.
-
-* For example, whereas the type of division over `ℕ` used to be:
-  ```agda
-  _/_ : (dividend divisor : ℕ) {≢0 : False (divisor ≟ 0)} → ℕ
-  ```
-  it is now:
-  ```agda
-  _/_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ
-  ```
-
-* This means that as long as an instance of `NonZero n` is in scope then you can write
-  `m / n` without having to explicitly provide a proof, as instance search will fill it in
-  for you. The full list of such operations changed is as follows:
-    - In `Data.Nat.DivMod`: `_/_`, `_%_`, `_div_`, `_mod_`
-    - In `Data.Nat.Pseudorandom.LCG`: `Generator`
-    - In `Data.Integer.DivMod`: `_divℕ_`, `_div_`, `_modℕ_`, `_mod_`
-    - In `Data.Rational`: `mkℚ+`, `normalize`, `_/_`, `1/_`
-    - In `Data.Rational.Unnormalised`: `_/_`, `1/_`, `_÷_`
-
-* At the moment, there are 4 different ways such instance arguments can be provided,
-  listed in order of convenience and clarity:
-
-  1. *Automatic basic instances* - the standard library provides instances based on
-         the constructors of each numeric type in `Data.X.Base`. For example,
-         `Data.Nat.Base` constrains an instance of `NonZero (suc n)` for any `n`
-         and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`.
-         Consequently, if the argument is of the required form, these instances will always
-         be filled in by instance search automatically, e.g.
-        ```agda
-        0/n≡0 : 0 / suc n ≡ 0
-        ```
-
-  2. *Add an instance argument parameter* - You can provide the instance argument as
-         a parameter to your function and Agda's instance search will automatically use it
-         in the correct place without you having to explicitly pass it, e.g.
-    ```agda
-    0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
-    ```
-
-  3. *Define the instance locally* - You can define an instance argument in scope
-         (e.g. in a `where` clause) and Agda's instance search will again find it automatically,
-         e.g.
-        ```agda
-        instance
-          n≢0 : NonZero n
-          n≢0 = ...
-
-        0/n≡0 : 0 / n ≡ 0
-        ```
-
-  4. *Pass the instance argument explicitly* - Finally, if all else fails you can pass the
-         instance argument explicitly into the function using `{{ }}`, e.g.
-         ```
-         0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
-         ```
-
-* Suitable constructors for `NonZero`/`Positive` etc. can be found in `Data.X.Base`.
-
-* A full list of proofs that have changed to use instance arguments is available
-  at the end of this file. Notable changes to proofs are now discussed below.
-
-* Previously one of the hacks used in proofs was to explicitly refer to everything
-  in the correct form, e.g. if the argument `n` had to be non-zero then you would
-  refer to the argument as `suc n` everywhere instead of `n`, e.g.
-  ```
-  n/n≡1 : ∀ n → suc n / suc n ≡ 1
-  ```
-  This made the proofs extremely difficult to use if your term wasn't in the form `suc n`.
-  After being updated to use instance arguments instead, the proof above becomes:
-  ```
-  n/n≡1 : ∀ n {{_ : NonZero n}} → n / n ≡ 1
-  ```
-  However, note that this means that if you passed in the value `x` to these proofs
-  before, then you will now have to pass in `suc x`. The proofs for which the
-  arguments have changed form in this way are highlighted in the list at the bottom
-  of the file.
-
-* Finally, in `Data.Rational.Unnormalised.Base` the definition of `_≢0` is now
-  redundant and so has been removed. Additionally the following proofs about it have
-  also been removed from `Data.Rational.Unnormalised.Properties`:
-  ```
-  p≄0⇒∣↥p∣≢0 : ∀ p → p ≠ 0ℚᵘ → ℤ.∣ (↥ p) ∣ ≢0
-  ∣↥p∣≢0⇒p≄0 : ∀ p → ℤ.∣ (↥ p) ∣ ≢0 → p ≠ 0ℚᵘ
-  ```
-
-### Changes to the definition of `_≤″_` in `Data.Nat.Base` (issue #1919)
-
-* The definition of `_≤″_` was previously:
-  ```agda
-  record _≤″_ (m n : ℕ) : Set where
-    constructor less-than-or-equal
-    field
-      {k}   : ℕ
-      proof : m + k ≡ n
-  ```
-  which introduced a spurious additional definition, when this is in fact, modulo
-  field names and implicit/explicit qualifiers, equivalent to the definition of left-
-  divisibility, `_∣ˡ_` for the `RawMagma` structure of `_+_`.
-
-* Since the addition of raw bundles to `Data.X.Base`, this definition can now be
-  made directly. Accordingly, the definition has been changed to:
-  ```agda
-  _≤″_ : (m n : ℕ)  → Set
-  _≤″_ = _∣ˡ_ +-rawMagma
-
-  pattern less-than-or-equal {k} prf = k , prf
-  ```
-
-* Knock-on consequences include the need to retain the old constructor
-  name, now introduced as a pattern synonym, and introduction of (a function
-  equivalent to) the former field name/projection function `proof` as
-  `≤″-proof` in `Data.Nat.Properties`.
-
-### Changes to definition of `Prime` in `Data.Nat.Primality`
-
-* The definition of `Prime` was:
-  ```agda
-  Prime 0             = ⊥
-  Prime 1             = ⊥
-  Prime (suc (suc n)) = (i : Fin n) → 2 + toℕ i ∤ 2 + n
-  ```
-  which was very hard to reason about as, not only did it involve conversion
-  to and from the `Fin` type, it also required that the divisor was of the form
-  `2 + toℕ i`, which has exactly the same problem as the `suc n` hack described
-  above used for non-zeroness.
-
-* To make it easier to use, reason about and read, the definition has been
-  changed to:
-  ```agda
-  record Prime (p : ℕ) : Set where
-    constructor prime
-    field
-      .{{nontrivial}} : NonTrivial p
-      notComposite    : ¬ Composite p
-  ```
-  where `Composite` is now defined as the diagonal of the new relation
-  `_HasNonTrivialDivisorLessThan_` in `Data.Nat.Divisibility.Core`.
-
-### Changes to operation reduction behaviour in `Data.Rational(.Unnormalised)`
-
-* Currently arithmetic expressions involving rationals (both normalised and
-  unnormalised) undergo disastrous exponential normalisation. For example,
-  `p + q` would often be normalised by Agda to
-  `(↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)`. While the normalised form
-  of `p + q + r + s + t + u + v` would be ~700 lines long. This behaviour
-  often chokes both type-checking and the display of the expressions in the IDE.
-
-* To avoid this expansion and make non-trivial reasoning about rationals actually feasible:
-  1. the records `ℚᵘ` and `ℚ` have both had the `no-eta-equality` flag enabled
-  2. definition of arithmetic operations have trivial pattern matching added to
-     prevent them reducing, e.g.
-     ```agda
-     p + q = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)
-     ```
-     has been changed to
-     ```
-     p@record{} + q@record{} = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)
-     ```
-
-* As a consequence of this, some proofs that relied either on this reduction behaviour
-  or on eta-equality may no longer type-check.
-
-* There are several ways to fix this:
-
-  1. The principled way is to not rely on this reduction behaviour in the first place.
-     The `Properties` files for rational numbers have been greatly expanded in `v1.7`
-     and `v2.0`, and we believe most proofs should be able to be built up from existing
-     proofs contained within these files.
-
-  2. Alternatively, annotating any rational arguments to a proof with either
-     `@record{}` or `@(mkℚ _ _ _)` should restore the old reduction behaviour for any
-     terms involving those parameters.
-
-  3. Finally, if the above approaches are not viable then you may be forced to explicitly
-     use `cong` combined with a lemma that proves the old reduction behaviour.
-
-* Similarly, in order to prevent reduction, the equality `_≃_` in `Data.Rational.Base`
-  has been made into a data type with the single constructor `*≡*`. The destructor
-  `drop-*≡*` has been added to `Data.Rational.Properties`.
-
-#### Deprecation of old `Function` hierarchy
-
-* The new `Function` hierarchy was introduced in `v1.2` which follows
-  the same `Core`/`Definitions`/`Bundles`/`Structures` as all the other
-  hierarchies in the library.
-
-* At the time the old function hierarchy in:
-  ```
-  Function.Equality
-  Function.Injection
-  Function.Surjection
-  Function.Bijection
-  Function.LeftInverse
-  Function.Inverse
-  Function.HalfAdjointEquivalence
-  Function.Related
-  ```
-  was unofficially deprecated, but could not be officially deprecated because
-  of it's widespread use in the rest of the library.
-
-* Now, the old hierarchy modules have all been officially deprecated. All
-  uses of them in the rest of the library have been switched to use the
-  new hierarchy.
-
-* The latter is unfortunately a relatively big breaking change, but was judged
-  to be unavoidable given how widely used the old hierarchy was.
-
-#### Changes to the new `Function` hierarchy
-
-* In `Function.Bundles` the names of the fields in the records have been
-  changed from `f`, `g`, `cong₁` and `cong₂` to `to`, `from`, `to-cong`, `from-cong`.
-
-* In `Function.Definitions`, the module no longer has two equalities as
-  module arguments, as they did not interact as intended with the re-exports
-  from `Function.Definitions.(Core1/Core2)`. The latter two modules have
-  been removed and their definitions folded into `Function.Definitions`.
-
-* In `Function.Definitions` the following definitions have been changed from:
-  ```
-  Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
-  Inverseˡ f g = ∀ y → f (g y) ≈₂ y
-  Inverseʳ f g = ∀ x → g (f x) ≈₁ x
-  ```
-  to:
-  ```
-  Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y
-  Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
-  Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
-  ```
-  This is for several reasons:
-          i) the new definitions compose much more easily,
-          ii) Agda can better infer the equalities used.
-
-* To ease backwards compatibility:
-   - the old definitions have been moved to the new names  `StrictlySurjective`,
-         `StrictlyInverseˡ` and `StrictlyInverseʳ`.
-   - The records in  `Function.Structures` and `Function.Bundles` export proofs
-         of these under the names `strictlySurjective`, `strictlyInverseˡ` and
-         `strictlyInverseʳ`,
-   - Conversion functions for the definitions have been added in both directions 
-		 to `Function.Consequences(.Propositional/Setoid)`.
-   - Conversion functions for structures have been added in 
-		 `Function.Structures.Biased`.
-
-### New `Function.Strict`
-
-* The module `Strict` has been deprecated in favour of `Function.Strict`
-  and the definitions of strict application, `_$!_` and `_$!′_`, have been
-  moved from `Function.Base` to `Function.Strict`.
-
-* The contents of `Function.Strict` is now re-exported by `Function`.
-
-### Change to the definition of `WfRec` in `Induction.WellFounded` (issue #2083)
-
-* Previously, the definition of `WfRec` was:
-  ```agda
-  WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
-  WfRec _<_ P x = ∀ y → y < x → P y
-  ```
-  which meant that all arguments involving accessibility and wellfoundedness proofs
-  were polluted by almost-always-inferrable explicit arguments for the `y` position.
-
-* The definition has now been changed to make that argument *implicit*, as
-  ```agda
-  WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
-  WfRec _<_ P x = ∀ {y} → y < x → P y
-  ```
-
-### Reorganisation of `Reflection` modules
-
-* Under the `Reflection` module, there were various impending name clashes
-  between the core AST as exposed by Agda and the annotated AST defined in
-  the library.
-
-* While the content of the modules remain the same, the modules themselves
-  have therefore been renamed as follows:
-  ```
-  Reflection.Annotated              ↦ Reflection.AnnotatedAST
-  Reflection.Annotated.Free         ↦ Reflection.AnnotatedAST.Free
-
-  Reflection.Abstraction            ↦ Reflection.AST.Abstraction
-  Reflection.Argument               ↦ Reflection.AST.Argument
-  Reflection.Argument.Information   ↦ Reflection.AST.Argument.Information
-  Reflection.Argument.Quantity      ↦ Reflection.AST.Argument.Quantity
-  Reflection.Argument.Relevance     ↦ Reflection.AST.Argument.Relevance
-  Reflection.Argument.Modality      ↦ Reflection.AST.Argument.Modality
-  Reflection.Argument.Visibility    ↦ Reflection.AST.Argument.Visibility
-  Reflection.DeBruijn               ↦ Reflection.AST.DeBruijn
-  Reflection.Definition             ↦ Reflection.AST.Definition
-  Reflection.Instances              ↦ Reflection.AST.Instances
-  Reflection.Literal                ↦ Reflection.AST.Literal
-  Reflection.Meta                   ↦ Reflection.AST.Meta
-  Reflection.Name                   ↦ Reflection.AST.Name
-  Reflection.Pattern                ↦ Reflection.AST.Pattern
-  Reflection.Show                   ↦ Reflection.AST.Show
-  Reflection.Traversal              ↦ Reflection.AST.Traversal
-  Reflection.Universe               ↦ Reflection.AST.Universe
-
-  Reflection.TypeChecking.Monad             ↦ Reflection.TCM
-  Reflection.TypeChecking.Monad.Categorical ↦ Reflection.TCM.Categorical
-  Reflection.TypeChecking.Monad.Format      ↦ Reflection.TCM.Format
-  Reflection.TypeChecking.Monad.Syntax      ↦ Reflection.TCM.Instances
-  Reflection.TypeChecking.Monad.Instances   ↦ Reflection.TCM.Syntax
-  ```
-
-* A new module `Reflection.AST` that re-exports the contents of the
-  submodules has been added.
-
-
-### Reorganisation of the `Relation.Nullary` hierarchy
-
-* It was very difficult to use the `Relation.Nullary` modules, as
-  `Relation.Nullary` contained the basic definitions of negation, decidability etc.,
-  and the operations and proofs about these definitions were spread over
-  `Relation.Nullary.(Negation/Product/Sum/Implication etc.)`.
-
-* To fix this all the contents of the latter is now exported by `Relation.Nullary`.
-
-* In order to achieve this the following backwards compatible changes have been made:
-
-  1. the definition of `Dec` and `recompute` have been moved to `Relation.Nullary.Decidable.Core`
-
-  2. the definition of `Reflects` has been moved to `Relation.Nullary.Reflects`
-
-  3. the definition of `¬_` has been moved to `Relation.Nullary.Negation.Core`
-
-  4. The modules `Relation.Nullary.(Product/Sum/Implication)` have been deprecated
-         and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`.
-		 
-  5. The proof `T?` has been moved from `Data.Bool.Properties` to `Relation.Nullary.Decidable.Core`
-	 (but is still re-exported by the former).
-
-  as well as the following breaking changes:
-
-  1. `¬?` has been moved from `Relation.Nullary.Negation.Core` to
-         `Relation.Nullary.Decidable.Core`
-
-  2. `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to
-         `Relation.Nullary.Reflects`.
-
-  3. `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved
-         from `Relation.Nullary.Negation` to `Relation.Nullary.Decidable`.
-
-  4. `fromDec` and `toDec` have been moved from `Data.Sum.Base` to `Data.Sum`.
-
-
-### (Issue #2096) Introduction of flipped and negated relation symbols to bundles in `Relation.Binary.Bundles`
-
-* Previously, bundles such as `Preorder`, `Poset`, `TotalOrder` etc. did not have the flipped
-  and negated versions of the operators exposed. In some cases they could obtained by opening the
-  relevant `Relation.Binary.Properties.X` file but usually they had to be redefined every time.
-
-* To fix this, these bundles now all export all 4 versions of the operator: normal, converse, negated,
-  converse-negated. Accordingly they are no longer exported from the corresponding `Properties` file.
-
-* To make this work for `Preorder`, it was necessary to change the name of the relation symbol.
-  Previously, the symbol was `_∼_`  which is (notationally) symmetric, so that its
-  converse relation could only be discussed *semantically* in terms of `flip _∼_`.
-
-* Now, the `Preorder` record field `_∼_` has been renamed to `_≲_`, with `_≳_`
-  introduced as a definition in `Relation.Binary.Bundles.Preorder`.
-  Partial backwards compatible has been achieved by redeclaring a deprecated version
-  of the old symbol in the record. Therefore, only _declarations_ of `PartialOrder` records will
-  need their field names updating.
-
-
-### Changes to definition of `IsStrictTotalOrder` in `Relation.Binary.Structures`
-
-* The previous definition of the record `IsStrictTotalOrder` did not
-  build upon `IsStrictPartialOrder` as would be expected.
-  Instead it omitted several fields like irreflexivity as they were derivable from the
-  proof of trichotomy. However, this led to problems further up the hierarchy where
-  bundles such as `StrictTotalOrder` which contained multiple distinct proofs of
-  `IsStrictPartialOrder`.
-
-* To remedy this the definition of `IsStrictTotalOrder` has been changed to so
-  that it builds upon `IsStrictPartialOrder` as would be expected.
-
-* To aid migration, the old record definition has been moved to
-  `Relation.Binary.Structures.Biased` which contains the `isStrictTotalOrderᶜ`
-  smart constructor (which is re-exported by `Relation.Binary`) . Therefore the old code:
-  ```agda
-  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
-  <-isStrictTotalOrder = record
-        { isEquivalence = isEquivalence
-        ; trans         = <-trans
-        ; compare       = <-cmp
-        }
-  ```
-  can be migrated either by updating to the new record fields if you have a proof of `IsStrictPartialOrder`
-  available:
-  ```agda
-  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
-  <-isStrictTotalOrder = record
-        { isStrictPartialOrder = <-isStrictPartialOrder
-        ; compare              = <-cmp
-        }
-  ```
-  or simply applying the smart constructor to the record definition as follows:
-  ```agda
-  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
-  <-isStrictTotalOrder = isStrictTotalOrderᶜ record
-        { isEquivalence = isEquivalence
-        ; trans         = <-trans
-        ; compare       = <-cmp
-        }
-  ```
-
-### Changes to the interface of `Relation.Binary.Reasoning.Triple`
-
-* The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof
-  that the strict relation is irreflexive.
-
-* This allows the addition of the following new proof combinator:
-  ```agda
-  begin-contradiction : (r : x IsRelatedTo x) → {s : True (IsStrict? r)} → A
-  ```
-  that takes a proof that a value is strictly less than itself and then applies the
-  principle of explosion to derive anything.
-
-* Specialised versions of this combinator are available in the `Reasoning` modules
-  exported by the following modules:
-  ```
-  Data.Nat.Properties
-  Data.Nat.Binary.Properties
-  Data.Integer.Properties
-  Data.Rational.Unnormalised.Properties
-  Data.Rational.Properties
-  Data.Vec.Relation.Binary.Lex.Strict
-  Data.Vec.Relation.Binary.Lex.NonStrict
-  Relation.Binary.Reasoning.StrictPartialOrder
-  Relation.Binary.Reasoning.PartialOrder
-  ```
-
-### A more modular design for `Relation.Binary.Reasoning`
-
-* Previously, every `Reasoning` module in the library tended to roll it's own set
-  of syntax for the combinators. This hindered consistency and maintainability.
-
-* To improve the situation, a new module `Relation.Binary.Reasoning.Syntax`
-  has been introduced which exports a wide range of sub-modules containing
-  pre-existing reasoning combinator syntax.
-
-* This makes it possible to add new or rename existing reasoning combinators to a
-  pre-existing `Reasoning` module in just a couple of lines
-  (e.g. see `∣-Reasoning` in `Data.Nat.Divisibility`)
-
-* One pre-requisite for that is that `≡-Reasoning` has been moved from
-  `Relation.Binary.PropositionalEquality.Core` (which shouldn't be
-  imported anyway as it's a `Core` module) to
-  `Relation.Binary.PropositionalEquality.Properties`.
-  It is still exported by `Relation.Binary.PropositionalEquality`.
-
-### Renaming of symmetric combinators in `Reasoning` modules
-
-* We've had various complaints about the symmetric version of reasoning combinators
-  that use the syntax `_R˘⟨_⟩_` for some relation `R`, (e.g. `_≡˘⟨_⟩_` and `_≃˘⟨_⟩_`)
-  introduced in `v1.0`. In particular:
-  1. The symbol `˘` is hard to type.
-  2. The symbol `˘` doesn't convey the direction of the equality
-  3. The left brackets aren't vertically aligned with the left brackets of the non-symmetric version.
-
-* To address these problems we have renamed all the symmetric versions of the
-  combinators from `_R˘⟨_⟩_` to `_R⟨_⟨_` (the direction of the last bracket is flipped
-  to indicate the quality goes from right to left).
-
-* The old combinators still exist but have been deprecated. However due to
-  [Agda issue #5617](https://github.com/agda/agda/issues/5617), the deprecation warnings
-  don't fire correctly. We will not remove the old combinators before the above issue is
-  addressed. However, we still encourage migrating to the new combinators!
-
-* On a Linux-based system, the following command was used to globally migrate all uses of the
-  old combinators to the new ones in the standard library itself.
-  It *may* be useful when trying to migrate your own code:
-  ```bash
-  find . -type f -name "*.agda" -print0 | xargs -0 sed -i -e 's/˘⟨\(.*\)⟩/⟨\1⟨/g'
-  ```
-  USUAL CAVEATS: It has not been widely tested and the standard library developers are not
-  responsible for the results of this command. It is strongly recommended you back up your
-  work before you attempt to run it.
-
-* NOTE: this refactoring may require some extra bracketing around the operator `_⟨_⟩_` from
-  `Function.Base` if the `_⟨_⟩_` operator is used within the reasoning proof. The symptom
-  for this is a `Don't know how to parse` error.
-
-### Improvements to `Text.Pretty` and `Text.Regex`
-
-* In `Text.Pretty`, `Doc` is now a record rather than a type alias. This
-  helps Agda reconstruct the `width` parameter when the module is opened
-  without it being applied. In particular this allows users to write
-  width-generic pretty printers and only pick a width when calling the
-  renderer by using this import style:
-  ```
-  open import Text.Pretty using (Doc; render)
-  --                      ^-- no width parameter for Doc & render
-  open module Pretty {w} = Text.Pretty w hiding (Doc; render)
-  --                 ^-- implicit width parameter for the combinators
-
-  pretty : Doc w
-  pretty = ? -- you can use the combinators here and there won't be any
-             -- issues related to the fact that `w` cannot be reconstructed
-             -- anymore
-
-  main = do
-    -- you can now use the same pretty with different widths:
-    putStrLn $ render 40 pretty
-    putStrLn $ render 80 pretty
-  ```
-
-* In `Text.Regex.Search` the `searchND` function finding infix matches has
-  been tweaked to make sure the returned solution is a local maximum in terms
-  of length. It used to be that we would make the match start as late as
-  possible and finish as early as possible. It's now the reverse.
-
-  So `[a-zA-Z]+.agdai?` run on "the path _build/Main.agdai corresponds to"
-  will return "Main.agdai" when it used to be happy to just return "n.agda".
-
-### Other
-
-* In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
-  the `--without-K` flag now use the `--cubical-compatible` flag instead.
-
-* In `Algebra.Core` the operations `Opₗ` and `Opᵣ` have moved to `Algebra.Module.Core`.
-
-* In `Algebra.Definitions.RawMagma.Divisibility` the definitions for `_∣ˡ_` and `_∣ʳ_`
-  have been changed from being defined as raw products to being defined as records. However,
-  the record constructors are called `_,_` so the changes required are minimal.
-
-* In `Codata.Guarded.Stream` the following functions have been modified to have simpler definitions:
-  * `cycle`
-  * `interleave⁺`
-  * `cantor`
-  Furthermore, the direction of interleaving of `cantor` has changed. Precisely,
-  suppose `pair` is the cantor pairing function, then `lookup (pair i j) (cantor xss)`
-  according to the old definition corresponds to `lookup (pair j i) (cantor xss)`
-  according to the new definition.
-  For a concrete example see the one included at the end of the module.
-
-* In `Data.Fin.Base` the relations `_≤_` `_≥_` `_<_` `_>_`  have been
-  generalised so they can now relate `Fin` terms with different indices.
-  Should be mostly backwards compatible, but very occasionally when proving
-  properties about the orderings themselves the second index must be provided
-  explicitly.
-
-* In `Data.Fin.Properties` the proof `inj⇒≟` that an injection from a type
-  `A` into `Fin n` induces a decision procedure for `_≡_` on `A` has been
-  generalised to other equivalences over `A` (i.e. to arbitrary setoids), and
-  renamed from `eq?` to the more descriptive  and `inj⇒decSetoid`.
-
-* In `Data.Fin.Properties` the proof `pigeonhole` has been strengthened
-  so that the a proof that `i < j` rather than a mere `i ≢ j` is returned.
-
-* In `Data.Fin.Substitution.TermSubst` the fixity of `_/Var_` has been changed
-  from `infix 8` to `infixl 8`.
-
-* In `Data.Integer.DivMod` the previous implementations of
-  `_divℕ_`, `_div_`, `_modℕ_`, `_mod_` internally converted to the unary
-  `Fin` datatype resulting in poor performance. The implementation has been
-  updated to use the corresponding operations from `Data.Nat.DivMod` which are
-  efficiently implemented using the Haskell backend.
-
-* In `Data.Integer.Properties` the first two arguments of `m≡n⇒m-n≡0`
-  (now renamed `i≡j⇒i-j≡0`) have been made implicit.
-
-* In `Data.(List|Vec).Relation.Binary.Lex.Strict` the argument `xs` in
-  `xs≮[]` in introduced in PRs #1648 and #1672 has now been made implicit.
-
-* In `Data.List.NonEmpty` the functions `split`, `flatten` and `flatten-split`
-  have been removed from. In their place `groupSeqs` and `ungroupSeqs`
-  have been added to `Data.List.NonEmpty.Base` which morally perform the same
-  operations but without computing the accompanying proofs. The proofs can be
-  found in `Data.List.NonEmpty.Properties` under the names `groupSeqs-groups`
-  and `ungroupSeqs` and `groupSeqs`.
-
-* In `Data.List.Relation.Unary.Grouped.Properties` the proofs `map⁺` and `map⁻`
-  have had their preconditions weakened so the equivalences no longer require congruence
-  proofs.
-
-* In `Data.Nat.Divisibility` the proof `m/n/o≡m/[n*o]` has been removed. In it's
-  place a new more general proof `m/n/o≡m/[n*o]` has been added to `Data.Nat.DivMod`
-  that doesn't require the `n * o ∣ m` pre-condition.
-
-* In `Data.Product.Relation.Binary.Lex.Strict` the proof of wellfoundedness
-  of the lexicographic ordering on products, no longer
-  requires the assumption of symmetry for the first equality relation `_≈₁_`.
-
-* In `Data.Rational.Base` the constructors `+0` and `+[1+_]` from `Data.Integer.Base`
-  are no longer re-exported by `Data.Rational.Base`. You will have to open `Data.Integer(.Base)`
-  directly to use them.
-
-* In `Data.Rational(.Unnormalised).Properties` the types of the proofs
-  `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` have been switched,
-  as the previous naming scheme didn't correctly generalise to e.g. `pos+pos⇒pos`.
-  For example the types of `pos⇒1/pos`/`1/pos⇒pos` were:
-  ```agda
-  pos⇒1/pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
-  1/pos⇒pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
-  ```
-  but are now:
-  ```agda
-  pos⇒1/pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
-  1/pos⇒pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
-  ```
-
-* In `Data.Sum.Base` the definitions `fromDec` and `toDec` have been moved to `Data.Sum`.
-
-* In `Data.Vec.Base` the definition of `_>>=_` under `Data.Vec.Base` has been
-  moved to the submodule `CartesianBind` in order to avoid clashing with the
-  new, alternative definition of `_>>=_`, located in the second new submodule
-  `DiagonalBind`.
-
-* In `Data.Vec.Base` the definitions `init` and `last` have been changed from the `initLast`
-  view-derived implementation to direct recursive definitions.
-
-* In `Data.Vec.Properties` the type of the proof `zipWith-comm` has been generalised from:
-  ```agda
-  zipWith-comm : ∀ {f : A → A → B} (comm : ∀ x y → f x y ≡ f y x) (xs ys : Vec A n) →
-                 zipWith f xs ys ≡ zipWith f ys xs
-  ```
-  to
-  ```agda
-  zipWith-comm : ∀ {f : A → B → C} {g : B → A → C}  (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys →
-                 zipWith f xs ys ≡ zipWith g ys xs
-  ```
-
-* In `Data.Vec.Relation.Unary.All` the functions `lookup` and `tabulate` have
-  been moved to `Data.Vec.Relation.Unary.All.Properties` and renamed
-  `lookup⁺` and `lookup⁻` respectively.
-
-* In `Data.Vec.Base` and `Data.Vec.Functional` the functions `iterate` and `replicate`
-  now take the length of vector, `n`, as an explicit rather than an implicit argument, i.e.
-  the new types are:
-  ```agda
-  iterate : (A → A) → A → ∀ n → Vec A n
-  replicate : ∀ n → A → Vec A n
-  ```
-
-* In `Relation.Binary.Construct.Closure.Symmetric` the operation
-  `SymClosure` on relations in has been reimplemented
-  as a data type `SymClosure _⟶_ a b` that is parameterized by the
-  input relation `_⟶_` (as well as the elements `a` and `b` of the
-  domain) so that `_⟶_` can be inferred, which it could not from the
-  previous implementation using the sum type `a ⟶ b ⊎ b ⟶ a`.
-
-* In `Relation.Nullary.Decidable.Core` the name `excluded-middle` has been
-  renamed to `¬¬-excluded-middle`.
-
-
 Other major improvements
 ------------------------
 
-### Improvements to ring solver tactic
-
-* The ring solver tactic has been greatly improved. In particular:
-
-  1. When the solver is used for concrete ring types, e.g. ℤ, the equality can now use
-     all the ring operations defined natively for that type, rather than having
-     to use the operations defined in `AlmostCommutativeRing`. For example
-     previously you could not use `Data.Integer.Base._*_` but instead had to
-     use `AlmostCommutativeRing._*_`.
-
-  2. The solver now supports use of the subtraction operator `_-_` whenever
-     it is defined immediately in terms of `_+_` and `-_`. This is the case for
-     `Data.Integer` and `Data.Rational`.
-
-### Moved `_%_` and `_/_` operators to `Data.Nat.Base`
-
-* Previously the division and modulus operators were defined in `Data.Nat.DivMod`
-  which in turn meant that using them required importing `Data.Nat.Properties`
-  which is a very heavy dependency.
-
-* To fix this, these operators have been moved to `Data.Nat.Base`. The properties
-  for them still live in `Data.Nat.DivMod` (which also publicly re-exports them
-  to provide backwards compatibility).
-
-* Beneficiaries of this change include `Data.Rational.Unnormalised.Base` whose
-  dependencies are now significantly smaller.
-
-### Moved raw bundles from `Data.X.Properties` to `Data.X.Base`
-
-* Raw bundles by design don't contain any proofs so should in theory be able to live
-  in `Data.X.Base` rather than `Data.X.Bundles`.
-
-* To achieve this while keeping the dependency footprint small, the definitions of
-  raw bundles (`RawMagma`, `RawMonoid` etc.) have been moved from `Algebra(.Lattice)Bundles` to
-  a new module `Algebra(.Lattice).Bundles.Raw` which can be imported at much lower cost
-  from `Data.X.Base`.
-
-* We have then moved raw bundles defined in `Data.X.Properties` to `Data.X.Base` for
-  `X` = `Nat`/`Nat.Binary`/`Integer`/`Rational`/`Rational.Unnormalised`.
-  
-### Upgrades to `README` sub-library
-
-* The `README` sub-library has been moved to `doc/README` and a new `doc/standard-library-doc.agda-lib` has been added.
-
-* The first consequence is that `README` files now can be type-checked in Emacs
-  using an out-of-the-box standard Agda installation without altering the main
-  `standard-library.agda-lib` file.
-
-* The second is that the `README` files are now their own first-class library 
-  and can be imported like an other library.
-
 Deprecated modules
 ------------------
 
-### Moving `Data.Erased` to `Data.Irrelevant`
-
-* This fixes the fact we had picked the wrong name originally. The erased modality
-  corresponds to `@0` whereas the irrelevance one corresponds to `.`.
-
-### Deprecation of `Data.Fin.Substitution.Example`
-
-* The module `Data.Fin.Substitution.Example` has been deprecated, and moved to `README.Data.Fin.Substitution.UntypedLambda`
-
-### Deprecation of `Data.Nat.Properties.Core`
-
-* The module `Data.Nat.Properties.Core` has been deprecated, and its one lemma moved to `Data.Nat.Base`, renamed as `s≤s⁻¹`
-
-### Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`
-
-* This module has been deprecated, as none of its contents actually depended on axiom K.
-  The contents has been moved to `Data.Product.Function.Dependent.Setoid`.
-
-### Moving `Function.Related`
-
-* The module `Function.Related` has been deprecated in favour of `Function.Related.Propositional`
-  whose code uses the new function hierarchy. This also opens up the possibility of a more
-  general `Function.Related.Setoid` at a later date. Several of the names have been changed
-  in this process to bring them into line with the camelcase naming convention used
-  in the rest of the library:
-  ```agda
-  reverse-implication ↦ reverseImplication
-  reverse-injection   ↦ reverseInjection
-  left-inverse        ↦ leftInverse
-
-  Symmetric-kind      ↦ SymmetricKind
-  Forward-kind        ↦ ForwardKind
-  Backward-kind       ↦ BackwardKind
-  Equivalence-kind    ↦ EquivalenceKind
-  ```
-
-### Moving `Relation.Binary.Construct.(Converse/Flip)`
-
-* The following files have been moved:
-  ```agda
-  Relation.Binary.Construct.Converse ↦ Relation.Binary.Construct.Flip.EqAndOrd
-  Relation.Binary.Construct.Flip     ↦ Relation.Binary.Construct.Flip.Ord
-  ```
-
-### Moving `Relation.Binary.Properties.XLattice`
-
-* The following files have been moved:
-  ```agda
-  Relation.Binary.Properties.BoundedJoinSemilattice.agda  ↦ Relation.Binary.Lattice.Properties.BoundedJoinSemilattice.agda
-  Relation.Binary.Properties.BoundedLattice.agda          ↦ Relation.Binary.Lattice.Properties.BoundedLattice.agda
-  Relation.Binary.Properties.BoundedMeetSemilattice.agda  ↦ Relation.Binary.Lattice.Properties.BoundedMeetSemilattice.agda
-  Relation.Binary.Properties.DistributiveLattice.agda     ↦ Relation.Binary.Lattice.Properties.DistributiveLattice.agda
-  Relation.Binary.Properties.HeytingAlgebra.agda         ↦ Relation.Binary.Lattice.Properties.HeytingAlgebra.agda
-  Relation.Binary.Properties.JoinSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.JoinSemilattice.agda
-  Relation.Binary.Properties.Lattice.agda                 ↦ Relation.Binary.Lattice.Properties.Lattice.agda
-  Relation.Binary.Properties.MeetSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.MeetSemilattice.agda
-  ```
-
 Deprecated names
 ----------------
 
-* In `Algebra.Consequences.Propositional`:
-  ```agda
-  comm+assoc⇒middleFour             ↦  comm∧assoc⇒middleFour
-  identity+middleFour⇒assoc         ↦  identity∧middleFour⇒assoc
-  identity+middleFour⇒comm          ↦  identity∧middleFour⇒comm
-  comm+distrˡ⇒distrʳ                ↦  comm∧distrˡ⇒distrʳ
-  comm+distrʳ⇒distrˡ                ↦  comm∧distrʳ⇒distrˡ
-  assoc+distribʳ+idʳ+invʳ⇒zeˡ       ↦  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
-  assoc+distribˡ+idʳ+invʳ⇒zeʳ       ↦  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
-  assoc+id+invʳ⇒invˡ-unique         ↦  assoc∧id∧invʳ⇒invˡ-unique
-  assoc+id+invˡ⇒invʳ-unique         ↦  assoc∧id∧invˡ⇒invʳ-unique
-  subst+comm⇒sym                    ↦  subst∧comm⇒sym
-  ```
-
-* In `Algebra.Consequences.Setoid`:
-  ```agda
-  comm+assoc⇒middleFour             ↦  comm∧assoc⇒middleFour
-  identity+middleFour⇒assoc         ↦  identity∧middleFour⇒assoc
-  identity+middleFour⇒comm          ↦  identity∧middleFour⇒comm
-  comm+cancelˡ⇒cancelʳ              ↦  comm∧cancelˡ⇒cancelʳ
-  comm+cancelʳ⇒cancelˡ              ↦  comm∧cancelʳ⇒cancelˡ
-  comm+almostCancelˡ⇒almostCancelʳ  ↦  comm∧almostCancelˡ⇒almostCancelʳ
-  comm+almostCancelʳ⇒almostCancelˡ  ↦  comm∧almostCancelʳ⇒almostCancelˡ
-  comm+idˡ⇒idʳ                      ↦  comm∧idˡ⇒idʳ
-  comm+idʳ⇒idˡ                      ↦  comm∧idʳ⇒idˡ
-  comm+zeˡ⇒zeʳ                      ↦  comm∧zeˡ⇒zeʳ
-  comm+zeʳ⇒zeˡ                      ↦  comm∧zeʳ⇒zeˡ
-  comm+invˡ⇒invʳ                    ↦  comm∧invˡ⇒invʳ
-  comm+invʳ⇒invˡ                    ↦  comm∧invʳ⇒invˡ
-  comm+distrˡ⇒distrʳ                ↦  comm∧distrˡ⇒distrʳ
-  comm+distrʳ⇒distrˡ                ↦  comm∧distrʳ⇒distrˡ
-  assoc+distribʳ+idʳ+invʳ⇒zeˡ       ↦  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
-  assoc+distribˡ+idʳ+invʳ⇒zeʳ       ↦  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
-  assoc+id+invʳ⇒invˡ-unique         ↦  assoc∧id∧invʳ⇒invˡ-unique
-  assoc+id+invˡ⇒invʳ-unique         ↦  assoc∧id∧invˡ⇒invʳ-unique
-  subst+comm⇒sym                    ↦  subst∧comm⇒sym
-  ```
-
-* In `Algebra.Construct.Zero`:
-  ```agda
-  rawMagma   ↦  Algebra.Construct.Terminal.rawMagma
-  magma      ↦  Algebra.Construct.Terminal.magma
-  semigroup  ↦  Algebra.Construct.Terminal.semigroup
-  band       ↦  Algebra.Construct.Terminal.band
-  ```
-
-* In `Codata.Guarded.Stream.Properties`:
-  ```agda
-  map-identity  ↦  map-id
-  map-fusion    ↦  map-∘
-  drop-fusion   ↦  drop-drop
-  ```
-
-* In `Codata.Sized.Colist.Properties`:
-  ```agda
-  map-identity      ↦  map-id
-  map-map-fusion    ↦  map-∘
-  drop-drop-fusion  ↦  drop-drop
-  ```
-
-* In `Codata.Sized.Covec.Properties`:
-  ```agda
-  map-identity   ↦  map-id
-  map-map-fusion  ↦  map-∘
-  ```
-
-* In `Codata.Sized.Delay.Properties`:
-  ```agda
-  map-identity      ↦  map-id
-  map-map-fusion    ↦  map-∘
-  map-unfold-fusion  ↦  map-unfold
-  ```
-
-* In `Codata.Sized.M.Properties`:
-  ```agda
-  map-compose  ↦  map-∘
-  ```
-
-* In `Codata.Sized.Stream.Properties`:
-  ```agda
-  map-identity   ↦  map-id
-  map-map-fusion  ↦  map-∘
-  ```
-
-* In `Data.Container.Related`:
-  ```agda
-  _∼[_]_  ↦  _≲[_]_
-  ```
-
-* In `Data.Bool.Properties` (Issue #2046):
-  ```agda
-  push-function-into-if ↦ if-float
-  ```
-
-* In `Data.Fin.Base`: two new, hopefully more memorable, names `↑ˡ` `↑ʳ`
-  for the 'left', resp. 'right' injection of a Fin m into a 'larger' type,
-  `Fin (m + n)`, resp. `Fin (n + m)`, with argument order to reflect the
-  position of the `Fin m` argument.
-  ```
-  inject+  ↦  flip _↑ˡ_
-  raise    ↦  _↑ʳ_
-  ```
-
-* In `Data.Fin.Properties`:
-  ```
-  toℕ-raise        ↦ toℕ-↑ʳ
-  toℕ-inject+      ↦ toℕ-↑ˡ
-  splitAt-inject+  ↦ splitAt-↑ˡ m i n
-  splitAt-raise    ↦ splitAt-↑ʳ
-  Fin0↔⊥           ↦ 0↔⊥
-  eq?              ↦ inj⇒≟
-  ```
-
-* In `Data.Fin.Permutation.Components`:
-  ```
-  reverse            ↦ Data.Fin.Base.opposite
-  reverse-prop       ↦ Data.Fin.Properties.opposite-prop
-  reverse-involutive ↦ Data.Fin.Properties.opposite-involutive
-  reverse-suc        ↦ Data.Fin.Properties.opposite-suc
-  ```
-
-* In `Data.Integer.DivMod` the operator names have been renamed to
-  be consistent with those in `Data.Nat.DivMod`:
-  ```
-  _divℕ_  ↦ _/ℕ_
-  _div_   ↦ _/_
-  _modℕ_  ↦ _%ℕ_
-  _mod_   ↦ _%_
-  ```
-
-* In `Data.Integer.Properties` references to variables in names have
-  been made consistent so that `m`, `n` always refer to naturals and
-  `i` and `j` always refer to integers:
-  ```
-  ≤-steps        ↦  i≤j⇒i≤k+j
-  ≤-step         ↦  i≤j⇒i≤1+j
-
-  ≤-steps-neg    ↦  i≤j⇒i-k≤j
-  ≤-step-neg     ↦  i≤j⇒pred[i]≤j
-
-  n≮n            ↦  i≮i
-  ∣n∣≡0⇒n≡0       ↦  ∣i∣≡0⇒i≡0
-  ∣-n∣≡∣n∣         ↦  ∣-i∣≡∣i∣
-  0≤n⇒+∣n∣≡n      ↦  0≤i⇒+∣i∣≡i
-  +∣n∣≡n⇒0≤n      ↦  +∣i∣≡i⇒0≤i
-  +∣n∣≡n⊎+∣n∣≡-n   ↦  +∣i∣≡i⊎+∣i∣≡-i
-  ∣m+n∣≤∣m∣+∣n∣     ↦  ∣i+j∣≤∣i∣+∣j∣
-  ∣m-n∣≤∣m∣+∣n∣     ↦  ∣i-j∣≤∣i∣+∣j∣
-  signₙ◃∣n∣≡n     ↦  signᵢ◃∣i∣≡i
-  ◃-≡            ↦  ◃-cong
-  ∣m-n∣≡∣n-m∣      ↦  ∣i-j∣≡∣j-i∣
-  m≡n⇒m-n≡0      ↦  i≡j⇒i-j≡0
-  m-n≡0⇒m≡n      ↦  i-j≡0⇒i≡j
-  m≤n⇒m-n≤0      ↦  i≤j⇒i-j≤0
-  m-n≤0⇒m≤n      ↦  i-j≤0⇒i≤j
-  m≤n⇒0≤n-m      ↦  i≤j⇒0≤j-i
-  0≤n-m⇒m≤n      ↦  0≤i-j⇒j≤i
-  n≤1+n          ↦  i≤suc[i]
-  n≢1+n          ↦  i≢suc[i]
-  m≤pred[n]⇒m<n  ↦  i≤pred[j]⇒i<j
-  m<n⇒m≤pred[n]  ↦  i<j⇒i≤pred[j]
-  -1*n≡-n        ↦  -1*i≡-i
-  m*n≡0⇒m≡0∨n≡0  ↦  i*j≡0⇒i≡0∨j≡0
-  ∣m*n∣≡∣m∣*∣n∣  ↦  ∣i*j∣≡∣i∣*∣j∣
-  m≤m+n          ↦  i≤i+j
-  n≤m+n          ↦  i≤j+i
-  m-n≤m          ↦  i≤i-j
-
-  +-pos-monoʳ-≤    ↦  +-monoʳ-≤
-  +-neg-monoʳ-≤    ↦  +-monoʳ-≤
-  *-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
-  *-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
-  *-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
-  *-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
-  *-cancelˡ-<-neg  ↦  *-cancelˡ-<-nonPos
-  *-cancelʳ-<-neg  ↦  *-cancelʳ-<-nonPos
-
-  ^-semigroup-morphism ↦ ^-isMagmaHomomorphism
-  ^-monoid-morphism    ↦ ^-isMonoidHomomorphism
-
-  pos-distrib-* ↦ pos-*
-  pos-+-commute ↦ pos-+
-  abs-*-commute ↦ abs-*
-
-  +-isAbelianGroup ↦ +-0-isAbelianGroup
-  ```
-
-* In `Data.List.Base`:
-  ```agda
-  _─_  ↦  removeAt
-  ```
-
-* In `Data.List.Properties`:
-  ```agda
-  map-id₂         ↦  map-id-local
-  map-cong₂       ↦  map-cong-local
-  map-compose     ↦  map-∘
-
-  map-++-commute       ↦  map-++
-  sum-++-commute       ↦  sum-++
-  reverse-map-commute  ↦  reverse-map
-  reverse-++-commute   ↦  reverse-++
-
-  zipWith-identityˡ  ↦  zipWith-zeroˡ
-  zipWith-identityʳ  ↦  zipWith-zeroʳ
-
-  ʳ++-++  ↦  ++-ʳ++
-
-  take++drop  ↦  take++drop≡id
-
-  length-─  ↦  length-removeAt
-  map-─     ↦  map-removeAt
-  ```
-
-* In `Data.List.NonEmpty.Properties`:
-  ```agda
-  map-compose     ↦  map-∘
-
-  map-++⁺-commute ↦  map-++⁺
-  ```
-
-* In `Data.List.Relation.Unary.All.Properties`:
-  ```agda
-  gmap  ↦  gmap⁺
-
-  updateAt-id-relative      ↦  updateAt-id-local
-  updateAt-compose-relative ↦  updateAt-updateAt-local
-  updateAt-compose          ↦  updateAt-updateAt
-  updateAt-cong-relative    ↦  updateAt-cong-local
-  ```
-
-* In `Data.List.Relation.Unary.Any.Properties`:
-  ```agda
-  map-with-∈⁺  ↦  mapWith∈⁺
-  map-with-∈⁻  ↦  mapWith∈⁻
-  map-with-∈↔  ↦  mapWith∈↔
-  ```
-
-* In `Data.List.Relation.Binary.Subset.Propositional.Properties`:
-  ```agda
-  map-with-∈⁺  ↦  mapWith∈⁺
-  ```
-
-* In `Data.List.Zipper.Properties`:
-  ```agda
-  toList-reverse-commute ↦  toList-reverse
-  toList-ˡ++-commute     ↦  toList-ˡ++
-  toList-++ʳ-commute     ↦  toList-++ʳ
-  toList-map-commute    ↦  toList-map
-  toList-foldr-commute  ↦  toList-foldr
-  ```
-
-* In `Data.Maybe.Properties`:
-  ```agda
-  map-id₂     ↦  map-id-local
-  map-cong₂   ↦  map-cong-local
-
-  map-compose    ↦  map-∘
-
-  map-<∣>-commute ↦  map-<∣>
-  ```
-
-* In `Data.Nat.Properties`:
-  ```agda
-  suc[pred[n]]≡n  ↦  suc-pred
-  ≤-step          ↦  m≤n⇒m≤1+n
-  ≤-stepsˡ        ↦  m≤n⇒m≤o+n
-  ≤-stepsʳ        ↦  m≤n⇒m≤n+o
-  <-step          ↦  m<n⇒m<1+n
-  pred-mono       ↦  pred-mono-≤
-
-  <-transʳ        ↦  ≤-<-trans
-  <-transˡ        ↦  <-≤-trans
-  ```
-
-* In `Data.Rational.Unnormalised.Base`:
-  ```agda
-  _≠_  ↦  _≄_
-  +-rawMonoid ↦ +-0-rawMonoid
-  *-rawMonoid ↦ *-1-rawMonoid
-  ```
-
-* In `Data.Rational.Unnormalised.Properties`:
-  ```agda
-  ↥[p/q]≡p         ↦  ↥[n/d]≡n
-  ↧[p/q]≡q         ↦  ↧[n/d]≡d
-  *-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
-  *-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
-  ≤-steps          ↦  p≤q⇒p≤r+q
-  *-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
-  *-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
-  *-cancelˡ-<-pos  ↦  *-cancelˡ-<-nonNeg
-  *-cancelʳ-<-pos  ↦  *-cancelʳ-<-nonNeg
-
-  positive⇒nonNegative  ↦ pos⇒nonNeg
-  negative⇒nonPositive  ↦ neg⇒nonPos
-  negative<positive     ↦ neg<pos
-  ```
-
-* In `Data.Rational.Base`:
-  ```agda
-  +-rawMonoid ↦ +-0-rawMonoid
-  *-rawMonoid ↦ *-1-rawMonoid
-  ```
-
-* In `Data.Rational.Properties`:
-  ```agda
-  *-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
-  *-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
-  *-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
-  *-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
-  *-cancelˡ-<-pos  ↦  *-cancelˡ-<-nonNeg
-  *-cancelʳ-<-pos  ↦  *-cancelʳ-<-nonNeg
-  *-cancelˡ-<-neg  ↦  *-cancelˡ-<-nonPos
-  *-cancelʳ-<-neg  ↦  *-cancelʳ-<-nonPos
-
-  negative<positive     ↦ neg<pos
-  ```
-
-* In `Data.Rational.Unnormalised.Base`:
-  ```agda
-  +-rawMonoid ↦ +-0-rawMonoid
-  *-rawMonoid ↦ *-1-rawMonoid
-  ```
-
-* In `Data.Rational.Unnormalised.Properties`:
-  ```agda
-  ≤-steps  ↦  p≤q⇒p≤r+q
-  ```
-
-* In `Data.Sum.Properties`:
-  ```agda
-  [,]-∘-distr      ↦  [,]-∘
-  [,]-map-commute  ↦  [,]-map
-  map-commute      ↦  map-map
-  map₁₂-commute    ↦  map₁₂-map₂₁
-  ```
-
-* In `Data.Tree.AVL`:
-  ```agda
-  _∈?_ ↦ member
-  ```
-
-* In `Data.Tree.AVL.IndexedMap`:
-  ```agda
-  _∈?_ ↦ member
-  ```
-
-* In `Data.Tree.AVL.Map`:
-  ```agda
-  _∈?_ ↦ member
-  ```
-
-* In `Data.Tree.AVL.Sets`:
-  ```agda
-  _∈?_ ↦ member
-  ```
-
-* In `Data.Tree.Binary.Zipper.Properties`:
-  ```agda
-  toTree-#nodes-commute   ↦  toTree-#nodes
-  toTree-#leaves-commute  ↦  toTree-#leaves
-  toTree-map-commute      ↦  toTree-map
-  toTree-foldr-commute    ↦  toTree-foldr
-  toTree-⟪⟫ˡ-commute      ↦  toTree--⟪⟫ˡ
-  toTree-⟪⟫ʳ-commute      ↦  toTree-⟪⟫ʳ
-  ```
-
-* In `Data.Tree.Rose.Properties`:
-  ```agda
-  map-compose     ↦  map-∘
-  ```
-
-* In `Data.Vec.Base`:
-  ```agda
-  remove  ↦  removeAt
-  insert  ↦  insertAt
-  ```
-
-* In `Data.Vec.Properties`:
-  ```agda
-  take-distr-zipWith ↦  take-zipWith
-  take-distr-map     ↦  take-map
-  drop-distr-zipWith ↦  drop-zipWith
-  drop-distr-map     ↦  drop-map
-
-  updateAt-id-relative      ↦  updateAt-id-local
-  updateAt-compose-relative ↦  updateAt-updateAt-local
-  updateAt-compose          ↦  updateAt-updateAt
-  updateAt-cong-relative    ↦  updateAt-cong-local
-
-  []%=-compose    ↦  []%=-∘
-
-  []≔-++-inject+  ↦ []≔-++-↑ˡ
-  []≔-++-raise    ↦ []≔-++-↑ʳ
-  idIsFold        ↦ id-is-foldr
-  sum-++-commute  ↦ sum-++
-
-  take-drop-id  ↦  take++drop≡id
-
-  map-insert       ↦  map-insertAt
-
-  insert-lookup    ↦  insertAt-lookup
-  insert-punchIn   ↦  insertAt-punchIn
-  remove-PunchOut  ↦  removeAt-punchOut
-  remove-insert    ↦  removeAt-insertAt
-  insert-remove    ↦  insertAt-removeAt
-
-  lookup-inject≤-take ↦ lookup-take-inject≤
-  ```
-
-* In `Data.Vec.Functional.Properties`:
-  ```agda
-  updateAt-id-relative      ↦  updateAt-id-local
-  updateAt-compose-relative ↦  updateAt-updateAt-local
-  updateAt-compose          ↦  updateAt-updateAt
-  updateAt-cong-relative    ↦  updateAt-cong-local
-
-  map-updateAt              ↦  map-updateAt-local
-
-  insert-lookup             ↦  insertAt-lookup
-  insert-punchIn            ↦  insertAt-punchIn
-  remove-punchOut           ↦  removeAt-punchOut
-  remove-insert             ↦  removeAt-insertAt
-  insert-remove             ↦  insertAt-removeAt
-  ```
-  NB. This last one is complicated by the *addition* of a 'global' property `map-updateAt`
-
-* In `Data.Vec.Recursive.Properties`:
-  ```agda
-  append-cons-commute  ↦  append-cons
-  ```
-
-* In `Data.Vec.Relation.Binary.Equality.Setoid`:
-  ```agda
-  map-++-commute ↦  map-++
-  ```
-
-* In `Function.Base`:
-  ```agda
-  case_return_of_   ↦   case_returning_of_
-  ```
-
-* In `Function.Construct.Composition`:
-  ```agda
-  _∘-⟶_   ↦   _⟶-∘_
-  _∘-↣_   ↦   _↣-∘_
-  _∘-↠_   ↦   _↠-∘_
-  _∘-⤖_  ↦   _⤖-∘_
-  _∘-⇔_   ↦   _⇔-∘_
-  _∘-↩_   ↦   _↩-∘_
-  _∘-↪_   ↦   _↪-∘_
-  _∘-↔_   ↦   _↔-∘_
-  ```
-
-  * In `Function.Construct.Identity`:
-  ```agda
-  id-⟶   ↦   ⟶-id
-  id-↣   ↦   ↣-id
-  id-↠   ↦   ↠-id
-  id-⤖  ↦   ⤖-id
-  id-⇔   ↦   ⇔-id
-  id-↩   ↦   ↩-id
-  id-↪   ↦   ↪-id
-  id-↔   ↦   ↔-id
-  ```
-
-* In `Function.Construct.Symmetry`:
-  ```agda
-  sym-⤖   ↦   ⤖-sym
-  sym-⇔   ↦   ⇔-sym
-  sym-↩   ↦   ↩-sym
-  sym-↪   ↦   ↪-sym
-  sym-↔   ↦   ↔-sym
-  ```
-
-* In `Function.Related.Propositional.Reasoning`:
-  ```agda
-  _↔⟨⟩_  ↦  _≡⟨⟩_
-  ```
-
-* In `Foreign.Haskell.Either` and `Foreign.Haskell.Pair`:
-  ```agda
-  toForeign   ↦ Foreign.Haskell.Coerce.coerce
-  fromForeign ↦ Foreign.Haskell.Coerce.coerce
-  ```
-
-* In `Relation.Binary.Properties.Preorder`:
-  ```agda
-  invIsPreorder ↦ converse-isPreorder
-  invPreorder   ↦ converse-preorder
-  ```
-
-* In `Relation.Binary.PropositionalEquality`:
-  ```agda
-  isPropositional ↦ Relation.Nullary.Irrelevant
-  ```
-
-* In `Relation.Unary.Consequences`:
-  ```agda
-  dec⟶recomputable  ↦  dec⇒recomputable
-  ```
-
-## Missing fixity declarations added
-
-* An effort has been made to add sensible fixities for many declarations:
-  ```
-  infix   4 _≟H_ _≟N_                                        (Algebra.Solver.Ring)
-  infixr  5 _∷_                                              (Codata.Guarded.Stream)
-  infix   4 _[_]                                             (Codata.Guarded.Stream)
-  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Binary.Pointwise)
-  infix   4 _≈∞_                                             (Codata.Guarded.Stream.Relation.Binary.Pointwise)
-  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Unary.All)
-  infixr  5 _∷_                                              (Codata.Musical.Colist)
-  infix   4 _≈_                                              (Codata.Musical.Conat)
-  infixr  5 _∷_                                              (Codata.Musical.Colist.Bisimilarity)
-  infixr  5 _∷_                                              (Codata.Musical.Colist.Relation.Unary.All)
-  infixr  5 _∷_                                              (Codata.Sized.Colist)
-  infixr  5 _∷_                                              (Codata.Sized.Covec)
-  infixr  5 _∷_                                              (Codata.Sized.Cowriter)
-  infixl  1 _>>=_                                            (Codata.Sized.Cowriter)
-  infixr  5 _∷_                                              (Codata.Sized.Stream)
-  infixr  5 _∷_                                              (Codata.Sized.Colist.Bisimilarity)
-  infix   4 _ℕ≤?_                                            (Codata.Sized.Conat.Properties)
-  infixr  5 _∷_                                              (Codata.Sized.Covec.Bisimilarity)
-  infixr  5 _∷_                                              (Codata.Sized.Cowriter.Bisimilarity)
-  infixr  5 _∷_                                              (Codata.Sized.Stream.Bisimilarity)
-  infix   4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
-  infix   6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
-  infixr  8 _⇒_ _⊸_                                          (Data.Container.Core)
-  infixr -1 _<$>_ _<*>_                                      (Data.Container.FreeMonad)
-  infixl  1 _>>=_                                            (Data.Container.FreeMonad)
-  infix   5 _▷_                                              (Data.Container.Indexed)
-  infixr  4 _,_                                              (Data.Container.Relation.Binary.Pointwise)
-  infix   4 _≈_                                              (Data.Float.Base)
-  infixl  4 _+ _*                                            (Data.List.Kleene.Base)
-  infixr  4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
-  infix   4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
-  infix   4 _≢∈_                                             (Data.List.Membership.Propositional)
-  infixr  5 _`∷_                                             (Data.List.Reflection)
-  infix   4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
-  infixr  5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
-  infixr  5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
-  infixr  5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
-  infixr  5 _∷=_                                             (Data.List.Relation.Unary.Any)
-  infixr  5 _++_                                             (Data.List.Ternary.Appending)
-  infixl  7 _⊓′_                                             (Data.Nat.Base)
-  infixl  6 _⊔′_                                             (Data.Nat.Base)
-  infixr  8 _^_                                              (Data.Nat.Base)
-  infix   4 _!≢0 _!*_!≢0                                     (Data.Nat.Properties)
-  infixl  6.5 _P′_ _P_ _C′_ _C_                              (Data.Nat.Combinatorics.Base)
-  infix   8 _⁻¹                                              (Data.Parity.Base)
-  infixr  2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
-  infixr  2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
-  infixr  2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
-  infixr  2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
-  infix   4 _≃?_                                             (Data.Rational.Unnormalised.Properties)
-  infixr  4 _,_                                              (Data.Refinement)
-  infixr  1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
-  infixr  1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
-  infixr  1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
-  infixr  1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
-  infixr  4 _,_                                              (Data.Tree.AVL.Value)
-  infix   4 _≈ₖᵥ_                                            (Data.Tree.AVL.Map.Membership.Propositional)
-  infixr  5 _`∷_                                             (Data.Vec.Reflection)
-  infixr  5 _∷=_                                             (Data.Vec.Membership.Setoid)
-  infix  -1 _$ⁿ_                                             (Data.Vec.N-ary)
-  infix   4 _≋_                                              (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
-  infixl  1 _>>=-cong_ _≡->>=-cong_                          (Effect.Monad.Partiality)
-  infixl  1 _?>=′_                                           (Effect.Monad.Predicate)
-  infixl  1 _>>=-cong_ _>>=-congP_                           (Effect.Monad.Partiality.All)
-  infixr  5 _∷_                                              (Foreign.Haskell.List.NonEmpty)
-  infixr  4 _,_                                              (Foreign.Haskell.Pair)
-  infixr  8 _^_                                              (Function.Endomorphism.Propositional)
-  infixr  8 _^_                                              (Function.Endomorphism.Setoid)
-  infix   4 _≃_                                              (Function.HalfAdjointEquivalence)
-  infix   4 _≈_ _≈ᵢ_ _≤_                                     (Function.Metric.Bundles)
-  infixl  6 _∙_                                              (Function.Metric.Bundles)
-  infix   4 _≈_                                              (Function.Metric.Nat.Bundles)
-  infix   4 _≈_                                              (Function.Metric.Rational.Bundles)
-  infix   3 _←_ _↢_                                          (Function.Related)
-  infix   4 _<_                                              (Induction.WellFounded)
-  infixl  6 _ℕ+_                                             (Level.Literals)
-  infixr  4 _,_                                              (Reflection.AnnotatedAST)
-  infix   4 _≟_                                              (Reflection.AST.Definition)
-  infix   4 _≡ᵇ_                                             (Reflection.AST.Literal)
-  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Meta)
-  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Name)
-  infix   4 _≟-Telescope_                                    (Reflection.AST.Term)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Information)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Modality)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Quantity)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Relevance)
-  infix   4 _≟_                                              (Reflection.AST.Argument.Visibility)
-  infixr  4 _,_                                              (Reflection.AST.Traversal)
-  infix   4 _∈FV_                                            (Reflection.AST.DeBruijn)
-  infixr  9 _;_                                              (Relation.Binary.Construct.Composition)
-  infixl  6 _+²_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
-  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Heterogeneous.Construct.At)
-  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Homogeneous.Construct.At)
-  infix   4 _∈_ _∉_                                          (Relation.Unary.Indexed)
-  infix   4 _≈_                                              (Relation.Binary.Bundles)
-  infixl  6 _∩_                                              (Relation.Binary.Construct.Intersection)
-  infix   4 _<₋_                                             (Relation.Binary.Construct.Add.Infimum.Strict)
-  infix   4 _≈∙_                                             (Relation.Binary.Construct.Add.Point.Equality)
-  infix   4 _≤⁺_ _≤⊤⁺                                        (Relation.Binary.Construct.Add.Supremum.NonStrict)
-  infixr  5 _∷ʳ_                                             (Relation.Binary.Construct.Closure.Transitive)
-  infixr  6 _∪_                                              (Relation.Binary.Construct.Union)
-  infixl  6 _+ℤ_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
-  infix   4 _≉_ _≈ᵢ_ _≤ᵢ_                                    (Relation.Binary.Indexed.Homogeneous.Bundles)
-  infixr  9 _⍮_                                              (Relation.Unary.PredicateTransformer)
-  infix   8 ∼_                                               (Relation.Unary.PredicateTransformer)
-  infix   2 _×?_ _⊙?_                                        (Relation.Unary.Properties)
-  infix   10 _~?                                             (Relation.Unary.Properties)
-  infixr  1 _⊎?_                                             (Relation.Unary.Properties)
-  infixr  7 _∩?_                                             (Relation.Unary.Properties)
-  infixr  6 _∪?_                                             (Relation.Unary.Properties)
-  infixr  5 _∷ᴹ_ _∷⁻¹ᴹ_                                      (Text.Regex.Search)
-  infixl  6 _`⊜_                                             (Tactic.RingSolver)
-  infix   8 ⊝_                                               (Tactic.RingSolver.Core.Expression)
-  infix   4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]                  (Text.Regex.Properties)
-  infix   4 _∈?_ _∉?_                                        (Text.Regex.Derivative.Brzozowski)
-  infix   4 _∈_ _∉_ _∈?_ _∉?_                                (Text.Regex.String.Unsafe)
-  ```
-
-New modules
------------
-
-* Constructive algebraic structures with apartness relations:
-  ```
-  Algebra.Apartness
-  Algebra.Apartness.Bundles
-  Algebra.Apartness.Structures
-  Algebra.Apartness.Properties.CommutativeHeytingAlgebra
-  Relation.Binary.Properties.ApartnessRelation
-  ```
-
-* Algebraic structures obtained by flipping their binary operations:
-  ```
-  Algebra.Construct.Flip.Op
-  ```
-
-* Algebraic structures when freely adding an identity element:
-  ```
-  Algebra.Construct.Add.Identity
-  ```
-
-* Definitions for algebraic modules:
-  ```
-  Algebra.Module
-  Algebra.Module.Core
-  Algebra.Module.Definitions.Bi.Simultaneous
-  Algebra.Module.Morphism.Construct.Composition
-  Algebra.Module.Morphism.Construct.Identity
-  Algebra.Module.Morphism.Definitions
-  Algebra.Module.Morphism.ModuleHomomorphism
-  Algebra.Module.Morphism.Structures
-  Algebra.Module.Properties
-  ```
-
-* Identity morphisms and composition of morphisms between algebraic structures:
-  ```
-  Algebra.Morphism.Construct.Composition
-  Algebra.Morphism.Construct.Identity
-  ```
-
-* Properties of various new algebraic structures:
-  ```
-  Algebra.Properties.MoufangLoop
-  Algebra.Properties.Quasigroup
-  Algebra.Properties.MiddleBolLoop
-  Algebra.Properties.Loop
-  Algebra.Properties.KleeneAlgebra
-  ```
-
-* Properties of rings without a unit
-  ```
-  Algebra.Properties.RingWithoutOne`
-  ```
-
-* Proof of the Binomial Theorem for semirings
-  ```
-  Algebra.Properties.Semiring.Binomial
-  Algebra.Properties.CommutativeSemiring.Binomial
-  ```
-
-* 'Optimised' tail-recursive exponentiation properties:
-  ```
-  Algebra.Properties.Semiring.Exp.TailRecursiveOptimised
-  ```
-
-* An implementation of M-types with `--guardedness` flag:
-  ```
-  Codata.Guarded.M
-  ```
-
-* A definition of infinite streams using coinductive records:
-  ```
-  Codata.Guarded.Stream
-  Codata.Guarded.Stream.Properties
-  Codata.Guarded.Stream.Relation.Binary.Pointwise
-  Codata.Guarded.Stream.Relation.Unary.All
-  Codata.Guarded.Stream.Relation.Unary.Any
-  ```
-
-* A small library for function arguments with default values:
-  ```
-  Data.Default
-  ```
-
-* A small library defining a structurally recursive view of `Fin n`:
-  ```
-  Data.Fin.Relation.Unary.Top
-  ```
-
-* A small library for a non-empty fresh list:
-  ```
-  Data.List.Fresh.NonEmpty
-  ```
-
-* A small library defining a structurally inductive view of lists:
-  ```
-  Data.List.Relation.Unary.Sufficient
-  ```
-
-* Combinations and permutations for ℕ.
-  ```
-  Data.Nat.Combinatorics
-  Data.Nat.Combinatorics.Base
-  Data.Nat.Combinatorics.Spec
-  ```
-
-* A small library defining parity and its algebra:
-  ```
-  Data.Parity
-  Data.Parity.Base
-  Data.Parity.Instances
-  Data.Parity.Properties
-  ```
-
-* New base module for `Data.Product` containing only the basic definitions.
-  ```
-  Data.Product.Base
-  ```
-
-* Reflection utilities for some specific types:
-  ```
-  Data.List.Reflection
-  Data.Vec.Reflection
-  ```
-
-* The `All` predicate over non-empty lists:
-  ```
-  Data.List.NonEmpty.Relation.Unary.All
-  ```
-
-* Some n-ary functions manipulating lists
-  ```
-  Data.List.Nary.NonDependent
-  ```
-
-* Added Logarithm base 2 on natural numbers:
-  ```
-  Data.Nat.Logarithm.Core
-  Data.Nat.Logarithm
-  ```
-
-* Show module for unnormalised rationals:
-  ```
-  Data.Rational.Unnormalised.Show
-  ```
-
-* Membership relations for maps and sets
-  ```
-  Data.Tree.AVL.Map.Membership.Propositional
-  Data.Tree.AVL.Map.Membership.Propositional.Properties
-  Data.Tree.AVL.Sets.Membership
-  Data.Tree.AVL.Sets.Membership.Properties
-  ```
-
-* Port of `Linked` to `Vec`:
-  ```
-  Data.Vec.Relation.Unary.Linked
-  Data.Vec.Relation.Unary.Linked.Properties
-  ```
-
-* Combinators for propositional equational reasoning on vectors with different indices
-  ```
-  Data.Vec.Relation.Binary.Equality.Cast
-  ```
-
-* Relations on indexed sets
-  ```
-  Function.Indexed.Bundles
-  ```
-
-* Properties of various types of functions:
-  ```
-  Function.Consequences
-  Function.Consequences.Setoid
-  Function.Consequences.Propositional
-  Function.Properties.Bijection
-  Function.Properties.RightInverse
-  Function.Properties.Surjection
-  Function.Construct.Constant
-  ```
-
-* New interface for `NonEmpty` Haskell lists:
-  ```
-  Foreign.Haskell.List.NonEmpty
-  ```
-
-* In order to improve modularity, the contents of `Relation.Binary.Lattice` has been
-  split out into the standard:
-  ```
-  Relation.Binary.Lattice.Definitions
-  Relation.Binary.Lattice.Structures
-  Relation.Binary.Lattice.Bundles
-  ```
-  All contents is re-exported by `Relation.Binary.Lattice` as before.
-
-
-* Added relational reasoning over apartness relations:
-  ```
-  Relation.Binary.Reasoning.Base.Apartness`
-  ```
-
-* Algebraic properties of `_∩_` and `_∪_` for predicates
-  ```
-  Relation.Unary.Algebra
-  ```
-
-* Both versions of equality on predicates are equivalences
-  ```
-  Relation.Unary.Relation.Binary.Equality
-  ```
-
-* The subset relations on predicates define an order
-  ```
-  Relation.Unary.Relation.Binary.Subset
-  ```
-
-* Polymorphic versions of some unary relations and their properties
-  ```
-  Relation.Unary.Polymorphic
-  Relation.Unary.Polymorphic.Properties
-  ```
-
-* Alpha equality over reflected terms
-  ```
-  Reflection.AST.AlphaEquality
-  ```
-
-* Various system types and primitives:
-  ```
-  System.Clock.Primitive
-  System.Clock
-  System.Console.ANSI
-  System.Directory.Primitive
-  System.Directory
-  System.FilePath.Posix.Primitive
-  System.FilePath.Posix
-  System.Process.Primitive
-  System.Process
-  ```
-
-* A new `cong!` tactic for automatically deriving arguments to `cong`
-  ```
-  Tactic.Cong
-  ```
-
-* A golden testing library with test pools, an options parser, a runner:
-  ```
-  Test.Golden
-  ```
+New modules
+-----------
 
 Additions to existing modules
 -----------------------------
-
-* The module `Algebra` now publicly re-exports the contents of
-  `Algebra.Structures.Biased`.
-
-* Added new definitions to `Algebra.Bundles`:
-  ```agda
-  record UnitalMagma           c ℓ : Set (suc (c ⊔ ℓ))
-  record InvertibleMagma       c ℓ : Set (suc (c ⊔ ℓ))
-  record InvertibleUnitalMagma c ℓ : Set (suc (c ⊔ ℓ))
-  record Quasigroup            c ℓ : Set (suc (c ⊔ ℓ))
-  record Loop                  c ℓ : Set (suc (c ⊔ ℓ))
-  record RingWithoutOne        c ℓ : Set (suc (c ⊔ ℓ))
-  record IdempotentSemiring    c ℓ : Set (suc (c ⊔ ℓ))
-  record KleeneAlgebra         c ℓ : Set (suc (c ⊔ ℓ))
-  record Quasiring             c ℓ : Set (suc (c ⊔ ℓ))
-  record Nearring              c ℓ : Set (suc (c ⊔ ℓ))
-  record IdempotentMagma       c ℓ : Set (suc (c ⊔ ℓ))
-  record AlternateMagma        c ℓ : Set (suc (c ⊔ ℓ))
-  record FlexibleMagma         c ℓ : Set (suc (c ⊔ ℓ))
-  record MedialMagma           c ℓ : Set (suc (c ⊔ ℓ))
-  record SemimedialMagma       c ℓ : Set (suc (c ⊔ ℓ))
-  record LeftBolLoop           c ℓ : Set (suc (c ⊔ ℓ))
-  record RightBolLoop          c ℓ : Set (suc (c ⊔ ℓ))
-  record MoufangLoop           c ℓ : Set (suc (c ⊔ ℓ))
-  record NonAssociativeRing    c ℓ : Set (suc (c ⊔ ℓ))
-  record MiddleBolLoop         c ℓ : Set (suc (c ⊔ ℓ))
-  ```
-
-* Added new definitions to `Algebra.Bundles.Raw`:
-  ```agda
-  record RawLoop               c ℓ : Set (suc (c ⊔ ℓ))
-  record RawQuasiGroup         c ℓ : Set (suc (c ⊔ ℓ))
-  record RawRingWithoutOne     c ℓ : Set (suc (c ⊔ ℓ))
-  ```
-
-* Added new definitions to `Algebra.Structures`:
-  ```agda
-  record IsUnitalMagma (_∙_ : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsInvertibleMagma (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
-  record IsInvertibleUnitalMagma (_∙_ : Op₂ A) (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
-  record IsQuasigroup (∙ \\ // : Op₂ A) : Set (a ⊔ ℓ)
-  record IsLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ)
-  record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ)
-  record IsKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
-  record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
-  record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
-  record IsIdempotentMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsAlternativeMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsFlexibleMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsMedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsSemimedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
-  record IsLeftBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsRightBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsMoufangLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
-  record IsMiddleBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
-  ```
-
-* Added new proof to `Algebra.Consequences.Base`:
-  ```agda
-  reflexive∧selfInverse⇒involutive : Reflexive _≈_ → SelfInverse _≈_ f → Involutive _≈_ f
-  ```
-
-* Added new proofs to `Algebra.Consequences.Propositional`:
-  ```agda
-  comm∧assoc⇒middleFour     : Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
-  identity∧middleFour⇒assoc : Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
-  identity∧middleFour⇒comm  : Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
-  ```
-
-* Added new proofs to `Algebra.Consequences.Setoid`:
-  ```agda
-  comm∧assoc⇒middleFour     : Congruent₂ _∙_ → Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
-  identity∧middleFour⇒assoc : Congruent₂ _∙_ → Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
-  identity∧middleFour⇒comm  : Congruent₂ _∙_ → Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
-
-  involutive⇒surjective  : Involutive f  → Surjective f
-  selfInverse⇒involutive : SelfInverse f → Involutive f
-  selfInverse⇒congruent  : SelfInverse f → Congruent f
-  selfInverse⇒inverseᵇ   : SelfInverse f → Inverseᵇ f f
-  selfInverse⇒surjective : SelfInverse f → Surjective f
-  selfInverse⇒injective  : SelfInverse f → Injective f
-  selfInverse⇒bijective  : SelfInverse f → Bijective f
-
-  comm∧idˡ⇒id              : Commutative _∙_ → LeftIdentity  e _∙_ → Identity e _∙_
-  comm∧idʳ⇒id              : Commutative _∙_ → RightIdentity e _∙_ → Identity e _∙_
-  comm∧zeˡ⇒ze              : Commutative _∙_ → LeftZero      e _∙_ → Zero     e _∙_
-  comm∧zeʳ⇒ze              : Commutative _∙_ → RightZero     e _∙_ → Zero     e _∙_
-  comm∧invˡ⇒inv            : Commutative _∙_ → LeftInverse  e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
-  comm∧invʳ⇒inv            : Commutative _∙_ → RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
-  comm∧distrˡ⇒distr        : Commutative _∙_ → _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
-  comm∧distrʳ⇒distr        : Commutative _∙_ → _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
-  distrib∧absorbs⇒distribˡ : Associative _∙_ → Commutative _◦_ → _∙_ Absorbs _◦_ → _◦_ Absorbs _∙_ → _◦_ DistributesOver _∙_ → _∙_ DistributesOverˡ _◦_
-  ```
-
-* Added new functions to `Algebra.Construct.DirectProduct`:
-  ```agda
-  rawSemiring                     : RawSemiring a ℓ₁ → RawSemiring b ℓ₂ → RawSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawRing                         : RawRing a ℓ₁ → RawRing b ℓ₂ → RawRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero a ℓ₁ →
-                                    SemiringWithoutAnnihilatingZero b ℓ₂ →
-                                    SemiringWithoutAnnihilatingZero (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  semiring                        : Semiring a ℓ₁ → Semiring b ℓ₂ → Semiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  commutativeSemiring             : CommutativeSemiring a ℓ₁ →
-                                        CommutativeSemiring b ℓ₂ →
-                                    CommutativeSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  ring                            : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  commutativeRing                 : CommutativeRing a ℓ₁ → CommutativeRing b ℓ₂ → CommutativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawQuasigroup                   : RawQuasigroup a ℓ₁ → RawQuasigroup b ℓ₂ → RawQuasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawLoop                         : RawLoop a ℓ₁ → RawLoop b ℓ₂ → RawLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  unitalMagma                     : UnitalMagma a ℓ₁ → UnitalMagma b ℓ₂ → UnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  invertibleMagma                 : InvertibleMagma a ℓ₁ → InvertibleMagma b ℓ₂ → InvertibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  invertibleUnitalMagma           : InvertibleUnitalMagma a ℓ₁ →
-                                        InvertibleUnitalMagma b ℓ₂ →
-                                    InvertibleUnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  quasigroup                      : Quasigroup a ℓ₁ → Quasigroup b ℓ₂ → Quasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  loop                            : Loop a ℓ₁ → Loop b ℓ₂ → Loop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  idempotentSemiring              : IdempotentSemiring a ℓ₁ →
-                                    IdempotentSemiring b ℓ₂ →
-                                                                        IdempotentSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  idempotentMagma                 : IdempotentMagma a ℓ₁ → IdempotentMagma b ℓ₂ → IdempotentMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  alternativeMagma                : AlternativeMagma a ℓ₁ → AlternativeMagma b ℓ₂ → AlternativeMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  flexibleMagma                   : FlexibleMagma a ℓ₁ → FlexibleMagma b ℓ₂ → FlexibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  medialMagma                     : MedialMagma a ℓ₁ → MedialMagma b ℓ₂ → MedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  semimedialMagma                 : SemimedialMagma a ℓ₁ → SemimedialMagma b ℓ₂ → SemimedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  kleeneAlgebra                   : KleeneAlgebra a ℓ₁ → KleeneAlgebra b ℓ₂ → KleeneAlgebra (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  leftBolLoop                     : LeftBolLoop a ℓ₁ → LeftBolLoop b ℓ₂ → LeftBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rightBolLoop                    : RightBolLoop a ℓ₁ → RightBolLoop b ℓ₂ → RightBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  middleBolLoop                   : MiddleBolLoop a ℓ₁ → MiddleBolLoop b ℓ₂ → MiddleBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  moufangLoop                     : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  rawRingWithoutOne               : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  ringWithoutOne                  : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  nonAssociativeRing              : NonAssociativeRing a ℓ₁ →
-                                            NonAssociativeRing b ℓ₂ →
-                                                                        NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  quasiring                       : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  nearring                        : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
-  ```
-
-* Added new definition to `Algebra.Definitions`:
-  ```agda
-  _*_ MiddleFourExchange _+_ = ∀ w x y z → ((w + x) * (y + z)) ≈ ((w + y) * (x + z))
-
-  SelfInverse f = ∀ {x y} → f x ≈ y → f y ≈ x
-
-  LeftDividesˡ  _∙_ _\\_ = ∀ x y → (x ∙ (x \\ y)) ≈ y
-  LeftDividesʳ  _∙_ _\\_ = ∀ x y → (x \\ (x ∙ y)) ≈ y
-  RightDividesˡ _∙_ _//_ = ∀ x y → ((y // x) ∙ x) ≈ y
-  RightDividesʳ _∙_ _//_ = ∀ x y → ((y ∙ x) // x) ≈ y
-  LeftDivides    ∙   \\ = (LeftDividesˡ ∙ \\) × (LeftDividesʳ ∙ \\)
-  RightDivides   ∙   // = (RightDividesˡ ∙ //) × (RightDividesʳ ∙ //)
-
-  LeftInvertible  e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e
-  RightInvertible e _∙_ x = ∃[ x⁻¹ ] (x ∙ x⁻¹) ≈ e
-  Invertible      e _∙_ x = ∃[ x⁻¹ ] ((x⁻¹ ∙ x) ≈ e) × ((x ∙ x⁻¹) ≈ e)
-  StarRightExpansive e _+_ _∙_ _⁻* = ∀ x → (e + (x ∙ (x ⁻*))) ≈ (x ⁻*)
-  StarLeftExpansive e _+_ _∙_ _⁻* = ∀ x →  (e + ((x ⁻*) ∙ x)) ≈ (x ⁻*)
-  StarExpansive e _+_ _∙_ _* = (StarLeftExpansive e _+_ _∙_ _*) × (StarRightExpansive e _+_ _∙_ _*)
-  StarLeftDestructive _+_ _∙_ _* = ∀ a b x → (b + (a ∙ x)) ≈ x → ((a *) ∙ b) ≈ x
-  StarRightDestructive _+_ _∙_ _* = ∀ a b x → (b + (x ∙ a)) ≈ x → (b ∙ (a *)) ≈ x
-  StarDestructive _+_ _∙_ _* = (StarLeftDestructive _+_ _∙_ _*) × (StarRightDestructive _+_ _∙_ _*)
-  LeftAlternative _∙_ = ∀ x y  →  ((x ∙ x) ∙ y) ≈ (x ∙ (y ∙ y))
-  RightAlternative _∙_ = ∀ x y → (x ∙ (y ∙ y)) ≈ ((x ∙ y) ∙ y)
-  Alternative _∙_ = (LeftAlternative _∙_ ) × (RightAlternative _∙_)
-  Flexible _∙_ = ∀ x y → ((x ∙ y) ∙ x) ≈ (x ∙ (y ∙ x))
-  Medial _∙_ = ∀ x y u z → ((x ∙ y) ∙ (u ∙ z)) ≈ ((x ∙ u) ∙ (y ∙ z))
-  LeftSemimedial _∙_ = ∀ x y z → ((x ∙ x) ∙ (y ∙ z)) ≈ ((x ∙ y) ∙ (x ∙ z))
-  RightSemimedial _∙_ = ∀ x y z → ((y ∙ z) ∙ (x ∙ x)) ≈ ((y ∙ x) ∙ (z ∙ x))
-  Semimedial _∙_ = (LeftSemimedial _∙_) × (RightSemimedial _∙_)
-  LeftBol _∙_ = ∀ x y z → (x ∙ (y ∙ (x ∙ z))) ≈ ((x ∙ (y ∙ z)) ∙ z )
-  RightBol _∙_ = ∀ x y z → (((z ∙ x) ∙ y) ∙ x) ≈ (z ∙ ((x ∙ y) ∙ x))
-  MiddleBol _∙_ _\\_ _//_ = ∀ x y z → (x ∙ ((y ∙ z) \\ x)) ≈ ((x // z) ∙ (y \\ x))
-  ```
-
-* Added new functions to `Algebra.Definitions.RawSemiring`:
-  ```agda
-  _^[_]*_ : A → ℕ → A → A
-  _^ᵗ_    : A → ℕ → A
-  ```
-
-* In `Algebra.Bundles.Lattice` the existing record `Lattice` now provides
-  ```agda
-  ∨-commutativeSemigroup : CommutativeSemigroup c ℓ
-  ∧-commutativeSemigroup : CommutativeSemigroup c ℓ
-  ```
-  and their corresponding algebraic sub-bundles.
-
-* In `Algebra.Lattice.Structures` the record `IsLattice` now provides
-  ```
-  ∨-isCommutativeSemigroup : IsCommutativeSemigroup ∨
-  ∧-isCommutativeSemigroup : IsCommutativeSemigroup ∧
-  ```
-  and their corresponding algebraic substructures.
-
-* The module `Algebra.Properties.Magma.Divisibility` now re-exports operations
-  `_∣ˡ_`, `_∤ˡ_`, `_∣ʳ_`, `_∤ʳ_` from `Algebra.Definitions.Magma`.
-
-* Added new records to `Algebra.Morphism.Structures`:
-  ```agda
-  record IsQuasigroupHomomorphism     (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsQuasigroupMonomorphism     (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsQuasigroupIsomorphism      (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsLoopHomomorphism           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsLoopMonomorphism           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsLoopIsomorphism            (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsRingWithoutOneIsoMorphism  (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsKleeneAlgebraHomomorphism  (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsKleeneAlgebraMonomorphism  (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
-  record IsKleeneAlgebraIsomorphism   (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  ```
-
-* Added new proofs to `Algebra.Properties.CommutativeSemigroup`:
-  ```agda
-  xy∙xx≈x∙yxx : (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))
-
-  interchange : Interchangable _∙_ _∙_
-  leftSemimedial : LeftSemimedial _∙_
-  rightSemimedial : RightSemimedial _∙_
-  middleSemimedial : (x ∙ y) ∙ (z ∙ x) ≈ (x ∙ z) ∙ (y ∙ x)
-  semimedial : Semimedial _∙_
-  ```
-
-* Added new functions to `Algebra.Properties.CommutativeMonoid`
-  ```agda
-  invertibleˡ⇒invertibleʳ : LeftInvertible  _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x
-  invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible  _≈_ 0# _+_ x
-  invertibleˡ⇒invertible  : LeftInvertible  _≈_ 0# _+_ x → Invertible      _≈_ 0# _+_ x
-  invertibleʳ⇒invertible  : RightInvertible _≈_ 0# _+_ x → Invertible      _≈_ 0# _+_ x
-  ```
-
-* Added new proof to `Algebra.Properties.Monoid.Mult`:
-  ```agda
-  ×-congˡ : (_× x) Preserves _≡_ ⟶ _≈_
-  ```
-
-* Added new proof to `Algebra.Properties.Monoid.Sum`:
-  ```agda
-  sum-init-last : (t : Vector _ (suc n)) → sum t ≈ sum (init t) + last t
-  ```
-
-* Added new proofs to `Algebra.Properties.Semigroup`:
-  ```agda
-  leftAlternative  : LeftAlternative _∙_
-  rightAlternative : RightAlternative _∙_
-  alternative      : Alternative _∙_
-  flexible         : Flexible _∙_
-  ```
-
-* Added new proofs to `Algebra.Properties.Semiring.Exp`:
-  ```agda
-  ^-congʳ               : (x ^_) Preserves _≡_ ⟶ _≈_
-  y*x^m*y^n≈x^m*y^[n+1] : x * y ≈ y * x → y * (x ^ m * y ^ n) ≈ x ^ m * y ^ suc n
-  ```
-
-* Added new proofs to `Algebra.Properties.Semiring.Mult`:
-  ```agda
-  1×-identityʳ : 1 × x ≈ x
-  ×-comm-*    : x * (n × y) ≈ n × (x * y)
-  ×-assoc-*   : (n × x) * y ≈ n × (x * y)
-  ```
-
-* Added new proofs to `Algebra.Properties.Ring`:
-  ```agda
-  -1*x≈-x      : - 1# * x ≈ - x
-  x+x≈x⇒x≈0    : x + x ≈ x → x ≈ 0#
-  x[y-z]≈xy-xz : x * (y - z) ≈ x * y - x * z
-  [y-z]x≈yx-zx : (y - z) * x ≈ (y * x) - (z * x)
-  ```
-
-* Added new functions in `Codata.Guarded.Stream`:
-  ```
-  transpose  : List (Stream A) → Stream (List A)
-  transpose⁺ : List⁺ (Stream A) → Stream (List⁺ A)
-  concat     : Stream (List⁺ A) → Stream A
-  ```
-
-* Added new proofs in `Codata.Guarded.Stream.Properties`:
-  ```
-  cong-concat       : ass ≈ bss → concat ass ≈ concat bss
-  map-concat        : map f (concat ass) ≈ concat (map (List⁺.map f) ass)
-  lookup-transpose  : lookup n (transpose ass) ≡ List.map (lookup n) ass
-  lookup-transpose⁺ : lookup n (transpose⁺ ass) ≡ List⁺.map (lookup n) ass
-  ```
-
-* Added new proofs in `Data.Bool.Properties`:
-  ```agda
-  <-wellFounded : WellFounded _<_
-  ∨-conicalˡ : LeftConical false _∨_
-  ∨-conicalʳ : RightConical false _∨_
-  ∨-conical  : Conical false _∨_
-  ∧-conicalˡ : LeftConical true _∧_
-  ∧-conicalʳ : RightConical true _∧_
-  ∧-conical  : Conical true _∧_
-
-  true-xor            : true xor x ≡ not x
-  xor-same            : x xor x ≡ false
-  not-distribˡ-xor    : not (x xor y) ≡ (not x) xor y
-  not-distribʳ-xor    : not (x xor y) ≡ x xor (not y)
-  xor-assoc           : Associative _xor_
-  xor-comm            : Commutative _xor_
-  xor-identityˡ       : LeftIdentity false _xor_
-  xor-identityʳ       : RightIdentity false _xor_
-  xor-identity        : Identity false _xor_
-  xor-inverseˡ        : LeftInverse true not _xor_
-  xor-inverseʳ        : RightInverse true not _xor_
-  xor-inverse         : Inverse true not _xor_
-  ∧-distribˡ-xor      : _∧_ DistributesOverˡ _xor_
-  ∧-distribʳ-xor      : _∧_ DistributesOverʳ _xor_
-  ∧-distrib-xor       : _∧_ DistributesOver _xor_
-  xor-annihilates-not : (not x) xor (not y) ≡ x xor y
-  ```
-
-* Exposed container combinator conversion functions from `Data.Container.Combinator.Properties`
-  separately from their correctness proofs in `Data.Container.Combinator`:
-  ```
-  to-id      : F.id A → ⟦ id ⟧ A
-  from-id    : ⟦ id ⟧ A → F.id A
-  to-const   : A → ⟦ const A ⟧ B
-  from-const : ⟦ const A ⟧ B → A
-  to-∘       : ⟦ C₁ ⟧ (⟦ C₂ ⟧ A) → ⟦ C₁ ∘ C₂ ⟧ A
-  from-∘     : ⟦ C₁ ∘ C₂ ⟧ A → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A)
-  to-×       : ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A → ⟦ C₁ × C₂ ⟧ A
-  from-×     : ⟦ C₁ × C₂ ⟧ A → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A
-  to-Π       : (∀ i → ⟦ Cᵢ i ⟧ A) → ⟦ Π I Cᵢ ⟧ A
-  from-Π     : ⟦ Π I Cᵢ ⟧ A → ∀ i → ⟦ Cᵢ i ⟧ A
-  to-⊎       : ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A → ⟦ C₁ ⊎ C₂ ⟧ A
-  from-⊎     : ⟦ C₁ ⊎ C₂ ⟧ A → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A
-  to-Σ       : (∃ λ i → ⟦ C i ⟧ A) → ⟦ Σ I C ⟧ A
-  from-Σ     : ⟦ Σ I C ⟧ A → ∃ λ i → ⟦ C i ⟧ A
-  ```
-
-* Added new functions in `Data.Fin.Base`:
-  ```
-  finToFun  : Fin (m ^ n) → (Fin n → Fin m)
-  funToFin  : (Fin m → Fin n) → Fin (n ^ m)
-  quotient  : Fin (m * n) → Fin m
-  remainder : Fin (m * n) → Fin n
-  ```
-
-* Added new proofs in `Data.Fin.Induction`:
-  ```agda
-  spo-wellFounded : IsStrictPartialOrder _≈_ _⊏_ → WellFounded _⊏_
-  spo-noetherian  : IsStrictPartialOrder _≈_ _⊏_ → WellFounded (flip _⊏_)
-
-  <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
-  ```
-
-* Added new definitions and proofs in `Data.Fin.Permutation`:
-  ```agda
-  insert         : Fin (suc m) → Fin (suc n) → Permutation m n → Permutation (suc m) (suc n)
-  insert-punchIn : insert i j π ⟨$⟩ʳ punchIn i k ≡ punchIn j (π ⟨$⟩ʳ k)
-  insert-remove  : insert i (π ⟨$⟩ʳ i) (remove i π) ≈ π
-  remove-insert  : remove i (insert i j π) ≈ π
-  ```
-
-* Added new proofs in `Data.Fin.Properties`:
-  ```
-  1↔⊤                : Fin 1 ↔ ⊤
-  2↔Bool             : Fin 2 ↔ Bool
-
-  0≢1+n              : zero ≢ suc i
-
-  ↑ˡ-injective       : i ↑ˡ n ≡ j ↑ˡ n → i ≡ j
-  ↑ʳ-injective       : n ↑ʳ i ≡ n ↑ʳ j → i ≡ j
-  finTofun-funToFin  : funToFin ∘ finToFun ≗ id
-  funTofin-funToFun  : finToFun (funToFin f) ≗ f
-  ^↔→                : Extensionality _ _ → Fin (m ^ n) ↔ (Fin n → Fin m)
-
-  toℕ-mono-<         : i < j → toℕ i ℕ.< toℕ j
-  toℕ-mono-≤         : i ≤ j → toℕ i ℕ.≤ toℕ j
-  toℕ-cancel-≤       : toℕ i ℕ.≤ toℕ j → i ≤ j
-  toℕ-cancel-<       : toℕ i ℕ.< toℕ j → i < j
-
-  splitAt⁻¹-↑ˡ       : splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i
-  splitAt⁻¹-↑ʳ       : splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i
-
-  toℕ-combine        : toℕ (combine i j) ≡ k ℕ.* toℕ i ℕ.+ toℕ j
-  combine-injectiveˡ : combine i j ≡ combine k l → i ≡ k
-  combine-injectiveʳ : combine i j ≡ combine k l → j ≡ l
-  combine-injective  : combine i j ≡ combine k l → i ≡ k × j ≡ l
-  combine-surjective : ∀ i → ∃₂ λ j k → combine j k ≡ i
-  combine-monoˡ-<    : i < j → combine i k < combine j l
-
-  ℕ-ℕ≡toℕ‿ℕ-         : n ℕ-ℕ i ≡ toℕ (n ℕ- i)
-
-  lower₁-injective   : lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
-  pinch-injective    : suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k
-
-  i<1+i              : i < suc i
-
-  injective⇒≤               : ∀ {f : Fin m → Fin n} → Injective f → m ℕ.≤ n
-  <⇒notInjective            : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective f)
-  ℕ→Fin-notInjective        : ∀ (f : ℕ → Fin n) → ¬ (Injective f)
-  cantor-schröder-bernstein : ∀ {f : Fin m → Fin n} {g : Fin n → Fin m} → Injective f → Injective g → m ≡ n
-
-  cast-is-id    : cast eq k ≡ k
-  subst-is-cast : subst Fin eq k ≡ cast eq k
-  cast-trans    : cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k
-
-  fromℕ≢inject₁      : {i : Fin n} → fromℕ n ≢ inject₁ i
-
-  inject≤-trans      : inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (≤-trans m≤n n≤o)
-  inject≤-irrelevant : inject≤ i m≤n ≡ inject≤ i m≤n′
-  i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j
-  ```
-
-* Added new lemmas in `Data.Fin.Substitution.Lemmas.TermLemmas`:
-  ```
-  map-var≡ : (∀ x → lookup ρ₁ x ≡ f x) → (∀ x → lookup ρ₂ x ≡ T.var (f x)) → map var ρ₁ ≡ ρ₂
-  wk≡wk    : map var VarSubst.wk ≡ T.wk {n = n}
-  id≡id    : map var VarSubst.id ≡ T.id {n = n}
-  sub≡sub  : map var (VarSubst.sub x) ≡ T.sub (T.var x)
-  ↑≡↑      : map var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
-  /Var≡/   : t /Var ρ ≡ t T./ map T.var ρ
-
-  sub-renaming-commutes : t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
-  sub-commutes-with-renaming : t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)
-  ```
-
-* Added new functions and definitions in `Data.Integer.Base`:
-  ```agda
-  _^_ : ℤ → ℕ → ℤ
-
-  +-0-rawGroup  : Rawgroup 0ℓ 0ℓ
-
-  *-rawMagma    : RawMagma 0ℓ 0ℓ
-  *-1-rawMonoid : RawMonoid 0ℓ 0ℓ
- ```
-
-* Added new proofs in `Data.Integer.Properties`:
-  ```agda
-  sign-cong′ : s₁ ◃ n₁ ≡ s₂ ◃ n₂ → s₁ ≡ s₂ ⊎ (n₁ ≡ 0 × n₂ ≡ 0)
-  ≤-⊖        : m ℕ.≤ n → n ⊖ m ≡ + (n ∸ m)
-  ∣⊖∣-≤      : m ℕ.≤ n → ∣ m ⊖ n ∣ ≡ n ∸ m
-  ∣-∣-≤      : i ≤ j → + ∣ i - j ∣ ≡ j - i
-
-  i^n≡0⇒i≡0      : i ^ n ≡ 0ℤ → i ≡ 0ℤ
-  ^-identityʳ    : i ^ 1 ≡ i
-  ^-zeroˡ        : 1 ^ n ≡ 1
-  ^-*-assoc      : (i ^ m) ^ n ≡ i ^ (m ℕ.* n)
-  ^-distribˡ-+-* : i ^ (m ℕ.+ n) ≡ i ^ m * i ^ n
-
-  ^-isMagmaHomomorphism  : IsMagmaHomomorphism  ℕ.+-rawMagma    *-rawMagma    (i ^_)
-  ^-isMonoidHomomorphism : IsMonoidHomomorphism ℕ.+-0-rawMonoid *-1-rawMonoid (i ^_)
-  ```
-
-* Added new proofs in `Data.Integer.GCD`:
-  ```agda
-  gcd-assoc : Associative gcd
-  gcd-zeroˡ : LeftZero 1ℤ gcd
-  gcd-zeroʳ : RightZero 1ℤ gcd
-  gcd-zero  : Zero 1ℤ gcd
-  ```
-
-* Added new functions and definitions to `Data.List.Base`:
-  ```agda
-  takeWhileᵇ   : (A → Bool) → List A → List A
-  dropWhileᵇ   : (A → Bool) → List A → List A
-  filterᵇ      : (A → Bool) → List A → List A
-  partitionᵇ   : (A → Bool) → List A → List A × List A
-  spanᵇ        : (A → Bool) → List A → List A × List A
-  breakᵇ       : (A → Bool) → List A → List A × List A
-  linesByᵇ     : (A → Bool) → List A → List (List A)
-  wordsByᵇ     : (A → Bool) → List A → List (List A)
-  derunᵇ       : (A → A → Bool) → List A → List A
-  deduplicateᵇ : (A → A → Bool) → List A → List A
-
-  findᵇ        : (A → Bool) → List A -> Maybe A
-  findIndexᵇ   : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)
-  findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)
-  find         : Decidable P → List A → Maybe A
-  findIndex    : Decidable P → (xs : List A) → Maybe $ Fin (length xs)
-  findIndices  : Decidable P → (xs : List A) → List $ Fin (length xs)
-
-  catMaybes       : List (Maybe A) → List A
-  ap              : List (A → B) → List A → List B
-  ++-rawMagma     : Set a → RawMagma a _
-  ++-[]-rawMonoid : Set a → RawMonoid a _
-
-  iterate : (A → A) → A → ℕ → List A
-  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
-  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
-  ```
-
-* Added new proofs in `Data.List.Relation.Binary.Lex.Strict`:
-  ```agda
-  xs≮[] : ¬ xs < []
-  ```
-
-* Added new proofs to `Data.List.Relation.Binary.Permutation.Propositional.Properties`:
-  ```agda
-  Any-resp-[σ⁻¹∘σ] : (σ : xs ↭ ys) → (ix : Any P xs) → Any-resp-↭ (trans σ (↭-sym σ)) ix ≡ ix
-  ∈-resp-[σ⁻¹∘σ]   : (σ : xs ↭ ys) → (ix : x ∈ xs)   → ∈-resp-↭   (trans σ (↭-sym σ)) ix ≡ ix
-  ```
-
-* In `Data.List.Relation.Binary.Permutation.Setoid.Properties`:
-  ```agda
-  foldr-commMonoid : xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys
-  ```
-
-* Added new function to `Data.List.Relation.Binary.Permutation.Propositional.Properties`
-  ```agda
-  ↭-reverse : reverse xs ↭ xs
-  ```
-
-* Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
-  ```
-  ⊆-mergeˡ : xs ⊆ merge _≤?_ xs ys
-  ⊆-mergeʳ : ys ⊆ merge _≤?_ xs ys
-  ```
-
-* Added new functions in `Data.List.Relation.Unary.All`:
-  ```
-  decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
-  ```
-
-* Added new proof to `Data.List.Relation.Unary.All.Properties`:
-  ```agda
-  gmap⁻ : Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P
-  ```
-
-* Added new functions in `Data.List.Fresh.Relation.Unary.All`:
-  ```
-  decide :  Π[ P ∪ Q ] → Π[ All {R = R} P ∪ Any Q ]
-  ```
-
-* Added new proofs to `Data.List.Membership.Propositional.Properties`:
-  ```agda
-  mapWith∈-id  : mapWith∈ xs (λ {x} _ → x) ≡ xs
-  map-mapWith∈ : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
-  ```
-
-* Added new proofs to `Data.List.Membership.Setoid.Properties`:
-  ```agda
-  mapWith∈-id     : mapWith∈ xs (λ {x} _ → x) ≡ xs
-  map-mapWith∈    : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
-  index-injective : index x₁∈xs ≡ index x₂∈xs → x₁ ≈ x₂
-
-  ∈-++⁺∘++⁻         : (p : v ∈ xs ++ ys) → [ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ (∈-++⁻ xs p) ≡ p
-  ∈-++⁻∘++⁺         : (p : v ∈ xs ⊎ v ∈ ys) → ∈-++⁻ xs ([ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ p) ≡ p
-  ∈-++↔             : (v ∈ xs ⊎ v ∈ ys) ↔ v ∈ xs ++ ys
-  ∈-++-comm         : v ∈ xs ++ ys → v ∈ ys ++ xs
-  ∈-++-comm∘++-comm : (p : v ∈ xs ++ ys) → ∈-++-comm ys xs (∈-++-comm xs ys p) ≡ p
-  ∈-++↔++           : v ∈ xs ++ ys ↔ v ∈ ys ++ xs
-  ```
-
-* Add new proofs in `Data.List.Properties`:
-  ```agda
-  ∈⇒∣product : n ∈ ns → n ∣ product ns
-  ∷ʳ-++      : xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
-
-  concatMap-cong : f ≗ g → concatMap f ≗ concatMap g
-  concatMap-pure : concatMap [_] ≗ id
-  concatMap-map  : concatMap g (map f xs) ≡ concatMap (g ∘′ f) xs
-  map-concatMap  : map f ∘′ concatMap g ≗ concatMap (map f ∘′ g)
-
-  length-isMagmaHomomorphism  : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
-  length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
-
-  take-map : take n (map f xs) ≡ map f (take n xs)
-  drop-map : drop n (map f xs) ≡ map f (drop n xs)
-  head-map : head (map f xs) ≡ Maybe.map f (head xs)
-
-  take-suc               : take (suc m) xs ≡ take m xs ∷ʳ lookup xs i
-  take-suc-tabulate      : take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f i
-  drop-take-suc          : drop m (take (suc m) xs) ≡ [ lookup xs i ]
-  drop-take-suc-tabulate : drop m (take (suc m) (tabulate f)) ≡ [ f i ]
-
-  take-all : n ≥ length xs → take n xs ≡ xs
-  drop-all : n ≥ length xs → drop n xs ≡ []
-
-  take-[] : take m [] ≡ []
-  drop-[] : drop m [] ≡ []
-
-  drop-drop : drop n (drop m xs) ≡ drop (m + n) xs
-
-  lookup-replicate  : lookup (replicate n x) i ≡ x
-  map-replicate     : map f (replicate n x) ≡ replicate n (f x)
-  zipWith-replicate : zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)
-
-  length-iterate : length (iterate f x n) ≡ n
-  iterate-id     : iterate id x n ≡ replicate n x
-  lookup-iterate : lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)
-
-  length-insertAt   : length (insertAt xs i v) ≡ suc (length xs)
-  length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
-  removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
-  insertAt-removeAt : insertAt (removeAt xs i) (cast (sym (lengthAt-removeAt xs i)) i) (lookup xs i) ≡ xs
-
-  cartesianProductWith-zeroˡ       : cartesianProductWith f [] ys ≡ []
-  cartesianProductWith-zeroʳ       : cartesianProductWith f xs [] ≡ []
-  cartesianProductWith-distribʳ-++ : cartesianProductWith f (xs ++ ys) zs ≡
-                                     cartesianProductWith f xs zs ++ cartesianProductWith f ys zs
-
-  foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
-  foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
-  ```
-
-* In `Data.List.NonEmpty.Base`:
-  ```agda
-  drop+ : ℕ → List⁺ A → List⁺ A
-  ```
-
-* Added new proofs in `Data.List.NonEmpty.Properties`:
-  ```agda
-  length-++⁺      : length (xs ++⁺ ys) ≡ length xs + length ys
-  length-++⁺-tail : length (xs ++⁺ ys) ≡ suc (length xs + length (tail ys))
-  ++-++⁺          : (xs ++ ys) ++⁺ zs ≡ xs ++⁺ ys ++⁺ zs
-  ++⁺-cancelˡ′    : xs ++⁺ zs ≡ ys ++⁺ zs′ → List.length xs ≡ List.length ys → zs ≡ zs′
-  ++⁺-cancelˡ     : xs ++⁺ ys ≡ xs ++⁺ zs → ys ≡ zs
-  drop+-++⁺       : drop+ (length xs) (xs ++⁺ ys) ≡ ys
-  map-++⁺-commute : map f (xs ++⁺ ys) ≡ map f xs ++⁺ map f ys
-  length-map      : length (map f xs) ≡ length xs
-  map-cong        : f ≗ g → map f ≗ map g
-  map-compose     : map (g ∘ f) ≗ map g ∘ map f
-  ```
-
-* Added new proof to `Data.Maybe.Properties`
-  ```agda
-  <∣>-idem : Idempotent _<∣>_
-  ```
-
-* Added new patterns and definitions to `Data.Nat.Base`:
-  ```agda
-  pattern z<s {n}         = s≤s (z≤n {n})
-  pattern s<s {m} {n} m<n = s≤s {m} {n} m<n
-
-  s≤s⁻¹ : suc m ≤ suc n → m ≤ n
-  s<s⁻¹ : suc m < suc n → m < n
-
-  pattern <′-base          = ≤′-refl
-  pattern <′-step {n} m<′n = ≤′-step {n} m<′n
-
-  pattern ≤″-offset k = less-than-or-equal {k} refl
-  pattern <″-offset k = ≤″-offset k
-
-  s≤″s⁻¹ : suc m ≤″ suc n → m ≤″ n
-  s<″s⁻¹ : suc m <″ suc n → m <″ n
-
-  pattern 2+ n = suc (suc n)
-  pattern 1<2+n {n} = s<s (z<s {n})
-
-  NonTrivial            : Pred ℕ 0ℓ
-  instance nonTrivial   : NonTrivial (2+ n)
-  n>1⇒nonTrivial        : 1 < n → NonTrivial n
-  nonZero⇒≢1⇒nonTrivial : .{{NonZero n}} → n ≢ 1 → NonTrivial n
-  recompute-nonTrivial  : .{{NonTrivial n}} → NonTrivial n
-  nonTrivial⇒nonZero    : .{{NonTrivial n}} → NonZero n
-  nonTrivial⇒n>1        : .{{NonTrivial n}} → 1 < n
-  nonTrivial⇒≢1         : .{{NonTrivial n}} → n ≢ 1
-
-  _⊔′_ : ℕ → ℕ → ℕ
-  _⊓′_ : ℕ → ℕ → ℕ
-  ∣_-_∣′ : ℕ → ℕ → ℕ
-  _! : ℕ → ℕ
-
-  parity : ℕ → Parity
-
-  +-rawMagma          : RawMagma 0ℓ 0ℓ
-  +-0-rawMonoid       : RawMonoid 0ℓ 0ℓ
-  *-rawMagma          : RawMagma 0ℓ 0ℓ
-  *-1-rawMonoid       : RawMonoid 0ℓ 0ℓ
-  +-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
-  +-*-rawSemiring     : RawSemiring 0ℓ 0ℓ
-  ```
-
-* Added a new proof to `Data.Nat.Binary.Properties`:
-  ```agda
-  suc-injective : Injective _≡_ _≡_ suc
-  toℕ-inverseˡ  : Inverseˡ _≡_ _≡_ toℕ fromℕ
-  toℕ-inverseʳ  : Inverseʳ _≡_ _≡_ toℕ fromℕ
-  toℕ-inverseᵇ  : Inverseᵇ _≡_ _≡_ toℕ fromℕ
-
-  <-asym : Asymmetric _<_
-  ```
-
-* Added a new pattern synonym and a new definition to `Data.Nat.Divisibility.Core`:
-  ```agda
-  pattern divides-refl q = divides q refl
-  record _HasNonTrivialDivisorLessThan_ (m n : ℕ) : Set where
-  ```
-
-* Added new proofs to `Data.Nat.Divisibility`:
-  ```agda
-  hasNonTrivialDivisor-≢ : .{{NonTrivial d}} → .{{NonZero n}} → d ≢ n → d ∣ n → n HasNonTrivialDivisorLessThan n
-  hasNonTrivialDivisor-∣ : m HasNonTrivialDivisorLessThan n → m ∣ o → o HasNonTrivialDivisorLessThan n
-  hasNonTrivialDivisor-≤ : m HasNonTrivialDivisorLessThan n → n ≤ o → m HasNonTrivialDivisorLessThan o
-  ```
-
-* Added new definitions, smart constructors and proofs to `Data.Nat.Primality`:
-  ```agda
-  infix 10 _Rough_
-  _Rough_           : ℕ → Pred ℕ _
-  0-rough           : 0 Rough n
-  1-rough           : 1 Rough n
-  2-rough           : 2 Rough n
-  rough⇒≤           : .{{NonTrivial n}} → m Rough n → m ≤ n
-  ∤⇒rough-suc        : m ∤ n → m Rough n → (suc m) Rough n
-  rough∧∣⇒rough     : m Rough o → n ∣ o → m Rough n
-  Composite         : ℕ → Set
-  composite-≢       : .{{NonTrivial d}} → .{{NonZero n}} → d ≢ n → d ∣ n → Composite n
-  composite-∣       : .{{NonZero n}} → Composite m → m ∣ n → Composite n
-  composite?        : Decidable Composite
-  Irreducible       : ℕ → Set
-  irreducible?      : Decidable Irreducible
-  composite⇒¬prime  : Composite n → ¬ Prime n
-  ¬composite⇒prime  : .{{NonTrivial n} → ¬ Composite n → Prime n
-  prime⇒¬composite  : Prime n → ¬ Composite n
-  ¬prime⇒composite  : .{{NonTrivial n} → ¬ Prime n → Composite n
-  prime⇒irreducible : Prime p → Irreducible p
-  irreducible⇒prime : .{{NonTrivial p}} → Irreducible p → Prime p
-  euclidsLemma      : Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n
-  ```
-
-* Added new proofs in `Data.Nat.Properties`:
-  ```agda
-  nonZero?  : Decidable NonZero
-  n≮0       : n ≮ 0
-  n+1+m≢m   : n + suc m ≢ m
-  m*n≡0⇒m≡0 : .{{_ : NonZero n}} → m * n ≡ 0 → m ≡ 0
-  n>0⇒n≢0   : n > 0 → n ≢ 0
-  m*n≢0     : .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
-  m*n≢0⇒m≢0 : .{{NonZero (m * n)}} → NonZero m
-  m*n≢0⇒n≢0 : .{{NonZero (m * n)}} → NonZero n
-  m≢0∧n>1⇒m*n>1 : .{{_ : NonZero m}} .{{_ : NonTrivial n}} → NonTrivial (m * n)
-  n≢0∧m>1⇒m*n>1 : .{{_ : NonZero n}} .{{_ : NonTrivial m}} → NonTrivial (m * n)
-  m^n≢0     : .{{_ : NonZero m}} → NonZero (m ^ n)
-  m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n
-
-  s<s-injective : s<s p ≡ s<s q → p ≡ q
-  <-step        : m < n → m < 1 + n
-  m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
-
-  pred-mono-≤   : m ≤ n → pred m ≤ pred n
-  pred-mono-<   : .{{_ : NonZero m}} → m < n → pred m < pred n
-
-  z<′s : zero <′ suc n
-  s<′s : m <′ n → suc m <′ suc n
-  <⇒<′ : m < n → m <′ n
-  <′⇒< : m <′ n → m < n
-
-  ≤″-proof : (le : m ≤″ n) → let less-than-or-equal {k} _ = le in m + k ≡ n
-
-  m+n≤p⇒m≤p∸n         : m + n ≤ p → m ≤ p ∸ n
-  m≤p∸n⇒m+n≤p         : n ≤ p → m ≤ p ∸ n → m + n ≤ p
-
-  1≤n!    : 1 ≤ n !
-  _!≢0    : NonZero (n !)
-  _!*_!≢0 : NonZero (m ! * n !)
-
-  anyUpTo? : ∀ (P? : U.Decidable P) (v : ℕ) → Dec (∃ λ n → n < v × P n)
-  allUpTo? : ∀ (P? : U.Decidable P) (v : ℕ) → Dec (∀ {n} → n < v → P n)
-
-  n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
-
-  m^n>0 : .{{_ : NonZero m}} n → m ^ n > 0
-
-  ^-monoˡ-≤ : (_^ n) Preserves _≤_ ⟶ _≤_
-  ^-monoʳ-≤ : .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
-  ^-monoˡ-< : .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
-  ^-monoʳ-< : 1 < m → (m ^_) Preserves _<_ ⟶ _<_
-
-  n≡⌊n+n/2⌋ : n ≡ ⌊ n + n /2⌋
-  n≡⌈n+n/2⌉ : n ≡ ⌈ n + n /2⌉
-
-  m<n⇒m<n*o : .{{_ : NonZero o}} → m < n → m < n * o
-  m<n⇒m<o*n : .{{_ : NonZero o}} → m < n → m < o * n
-  ∸-monoˡ-< : m < o → n ≤ m → m ∸ n < o ∸ n
-
-  m≤n⇒∣m-n∣≡n∸m : m ≤ n → ∣ m - n ∣ ≡ n ∸ m
-
-  ⊔≡⊔′ : m ⊔ n ≡ m ⊔′ n
-  ⊓≡⊓′ : m ⊓ n ≡ m ⊓′ n
-  ∣-∣≡∣-∣′ : ∣ m - n ∣ ≡ ∣ m - n ∣′
-
-  nonTrivial? : Decidable NonTrivial
-  eq? : A ↣ ℕ → DecidableEquality A
-  ≤-<-connex : Connex _≤_ _<_
-  ≥->-connex : Connex _≥_ _>_
-  <-≤-connex : Connex _<_ _≤_
-  >-≥-connex : Connex _>_ _≥_
-  <-cmp : Trichotomous _≡_ _<_
-  anyUpTo? : (P? : Decidable P) → ∀ v → Dec (∃ λ n → n < v × P n)
-  allUpTo? : (P? : Decidable P) → ∀ v → Dec (∀ {n} → n < v → P n)
-  ```
-
-* Added new proofs in `Data.Nat.Combinatorics`:
-  ```agda
-  [n-k]*[n-k-1]!≡[n-k]!   : k < n → (n ∸ k) * (n ∸ suc k) ! ≡ (n ∸ k) !
-  [n-k]*d[k+1]≡[k+1]*d[k] : k < n → (n ∸ k) * ((suc k) ! * (n ∸ suc k) !) ≡ (suc k) * (k ! * (n ∸ k) !)
-  k![n∸k]!∣n!             : k ≤ n → k ! * (n ∸ k) ! ∣ n !
-  nP1≡n                   : n P 1 ≡ n
-  nC1≡n                   : n C 1 ≡ n
-  nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
-  ```
-
-* Added new proofs in `Data.Nat.DivMod`:
-  ```agda
-  m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n
-  m*n/m!≡n/[m∸1]! : .{{_ : NonZero m}} → m * n / m ! ≡ n / (pred m) !
-
-  %-congˡ             : .{{_ : NonZero o}} → m ≡ n → m % o ≡ n % o
-  %-congʳ             : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ≡ n → o % m ≡ o % n
-  m≤n⇒[n∸m]%m≡n%m     : .{{_ : NonZero m}} → m ≤ n → (n ∸ m) % m ≡ n % m
-  m*n≤o⇒[o∸m*n]%n≡o%n : .{{_ : NonZero n}} → m * n ≤ o → (o ∸ m * n) % n ≡ o % n
-  m∣n⇒o%n%m≡o%m       : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ∣ n → o % n % m ≡ o % m
-  m<n⇒m%n≡m           : .{{_ : NonZero n}} → m < n → m % n ≡ m
-  m*n/o*n≡m/o         : .{{_ : NonZero o}} {{_ : NonZero (o * n)}} → m * n / (o * n) ≡ m / o
-  m<n*o⇒m/o<n         : .{{_ : NonZero o}} → m < n * o → m / o < n
-  [m∸n]/n≡m/n∸1       : .{{_ : NonZero n}} → (m ∸ n) / n ≡ pred (m / n)
-  [m∸n*o]/o≡m/o∸n     : .{{_ : NonZero o}} → (m ∸ n * o) / o ≡ m / o ∸ n
-  m/n/o≡m/[n*o]       : .{{_ : NonZero n}} .{{_ : NonZero o}} .{{_ : NonZero (n * o)}} → m / n / o ≡ m / (n * o)
-  m%[n*o]/o≡m/o%n     : .{{_ : NonZero n}} .{{_ : NonZero o}} {{_ : NonZero (n * o)}} → m % (n * o) / o ≡ m / o % n
-  m%n*o≡m*o%[n*o]     : .{{_ : NonZero n}} {{_ : NonZero (n * o)}} → m % n * o ≡ m * o % (n * o)
-
-  [m*n+o]%[p*n]≡[m*n]%[p*n]+o : {{_ : NonZero (p * n)}} → o < n → (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o
-  ```
-
-* Added new proofs in `Data.Nat.Divisibility`:
-  ```agda
-  n∣m*n*o       : n ∣ m * n * o
-  m*n∣⇒m∣       : m * n ∣ i → m ∣ i
-  m*n∣⇒n∣       : m * n ∣ i → n ∣ i
-  m≤n⇒m!∣n!     : m ≤ n → m ! ∣ n !
-  m/n/o≡m/[n*o] : .{{NonZero n}} .{{NonZero o}} → n * o ∣ m → (m / n) / o ≡ m / (n * o)
-  ```
-
-* Added new proofs in `Data.Nat.GCD`:
-  ```agda
-  gcd-assoc     : Associative gcd
-  gcd-identityˡ : LeftIdentity 0 gcd
-  gcd-identityʳ : RightIdentity 0 gcd
-  gcd-identity  : Identity 0 gcd
-  gcd-zeroˡ     : LeftZero 1 gcd
-  gcd-zeroʳ     : RightZero 1 gcd
-  gcd-zero      : Zero 1 gcd
-  ```
-
-* Added new patterns in `Data.Nat.Reflection`:
-  ```agda
-  pattern `ℕ     = def (quote ℕ) []
-  pattern `zero  = con (quote ℕ.zero) []
-  pattern `suc x = con (quote ℕ.suc) (x ⟨∷⟩ [])
-  ```
-
-* Added new functions and proofs in `Data.Nat.GeneralisedArithmetic`:
-  ```agda
-  iterate         : (A → A) → A → ℕ → A
-  iterate-is-fold : fold z s m ≡ iterate s z m
-  ```
-
-* Added new proofs in `Data.Parity.Properties`:
-  ```agda
-  suc-homo-⁻¹ : (parity (suc n)) ⁻¹ ≡ parity n
-  +-homo-+    : parity (m ℕ.+ n) ≡ parity m ℙ.+ parity n
-  *-homo-*    : parity (m ℕ.* n) ≡ parity m ℙ.* parity n
-  parity-isMagmaHomomorphism : IsMagmaHomomorphism ℕ.+-rawMagma ℙ.+-rawMagma parity
-  parity-isMonoidHomomorphism : IsMonoidHomomorphism ℕ.+-0-rawMonoid ℙ.+-0-rawMonoid parity
-  parity-isNearSemiringHomomorphism : IsNearSemiringHomomorphism ℕ.+-*-rawNearSemiring ℙ.+-*-rawNearSemiring parity
-  parity-isSemiringHomomorphism : IsSemiringHomomorphism ℕ.+-*-rawSemiring ℙ.+-*-rawSemiring parity
-  ```
-
-* Added new rounding functions in `Data.Rational.Base`:
-  ```agda
-  floor ceiling truncate round ⌊_⌋ ⌈_⌉ [_] : ℚ → ℤ
-  fracPart : ℚ → ℚ
-  ```
-
-* Added new definitions and proofs in `Data.Rational.Properties`:
-  ```agda
-  ↥ᵘ-toℚᵘ     : ↥ᵘ (toℚᵘ p) ≡ ↥ p
-  ↧ᵘ-toℚᵘ     : ↧ᵘ (toℚᵘ p) ≡ ↧ p
-  ↥p≡↥q≡0⇒p≡q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≡ q
-
-  _≥?_ : Decidable _≥_
-  _>?_ : Decidable _>_
-
-  +-*-rawNearSemiring                 : RawNearSemiring 0ℓ 0ℓ
-  +-*-rawSemiring                     : RawSemiring 0ℓ 0ℓ
-  toℚᵘ-isNearSemiringHomomorphism-+-* : IsNearSemiringHomomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ
-  toℚᵘ-isNearSemiringMonomorphism-+-* : IsNearSemiringMonomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ
-  toℚᵘ-isSemiringHomomorphism-+-*     : IsSemiringHomomorphism     +-*-rawSemiring     ℚᵘ.+-*-rawSemiring     toℚᵘ
-  toℚᵘ-isSemiringMonomorphism-+-*     : IsSemiringMonomorphism     +-*-rawSemiring     ℚᵘ.+-*-rawSemiring     toℚᵘ
-
-  pos⇒nonZero       : .{{Positive p}} → NonZero p
-  neg⇒nonZero       : .{{Negative p}} → NonZero p
-  nonZero⇒1/nonZero : .{{_ : NonZero p}} → NonZero (1/ p)
-
-  <-dense              : Dense _<_
-  <-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
-  <-denseLinearOrder   : DenseLinearOrder 0ℓ 0ℓ 0ℓ
-  ```
-
-* Added new rounding functions in `Data.Rational.Unnormalised.Base`:
-  ```agda
-  floor ceiling truncate round ⌊_⌋ ⌈_⌉ [_] : ℚᵘ → ℤ
-  fracPart : ℚᵘ → ℚᵘ
-  ```
-
-* Added new definitions in `Data.Rational.Unnormalised.Properties`:
-  ```agda
-  ↥p≡0⇒p≃0    : ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ
-  p≃0⇒↥p≡0    : p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ
-  ↥p≡↥q≡0⇒p≃q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≃ q
-
-  +-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
-  +-*-rawSemiring     : RawSemiring 0ℓ 0ℓ
-
-  ≰⇒≥ : _≰_ ⇒ _≥_
-
-  _≥?_ : Decidable _≥_
-  _>?_ : Decidable _>_
-
-  *-mono-≤-nonNeg   : .{{_ : NonNegative p}} .{{_ : NonNegative r}} → p ≤ q → r ≤ s → p * r ≤ q * s
-  *-mono-<-nonNeg   : .{{_ : NonNegative p}} .{{_ : NonNegative r}} → p < q → r < s → p * r < q * s
-  1/-antimono-≤-pos : .{{_ : Positive p}}    .{{_ : Positive q}}    → p ≤ q → 1/ q ≤ 1/ p
-  ⊓-mono-<          : _⊓_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
-  ⊔-mono-<          : _⊔_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
-
-  pos⇒nonZero          : .{{_ : Positive p}} → NonZero p
-  neg⇒nonZero          : .{{_ : Negative p}} → NonZero p
-  pos+pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p + q)
-  nonNeg+nonNeg⇒nonNeg : .{{_ : NonNegative p}} .{{_ : NonNegative q}} → NonNegative (p + q)
-  pos*pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p * q)
-  nonNeg*nonNeg⇒nonNeg : .{{_ : NonNegative p}} .{{_ : NonNegative q}} → NonNegative (p * q)
-  pos⊓pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊓ q)
-  pos⊔pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊔ q)
-  1/nonZero⇒nonZero    : .{{_ : NonZero p}} → NonZero (1/ p)
-
-  0≄1             : 0ℚᵘ ≄ 1ℚᵘ
-  ≃-≄-irreflexive : Irreflexive _≃_ _≄_
-  ≄-symmetric     : Symmetric _≄_
-  ≄-cotransitive  : Cotransitive _≄_
-  ≄⇒invertible    : p ≄ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
-
-  <-dense         : Dense _<_
-
-  <-isDenseLinearOrder         : IsDenseLinearOrder _≃_ _<_
-  +-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
-  +-*-isHeytingField           : IsHeytingField _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
-
-  <-denseLinearOrder           : DenseLinearOrder 0ℓ 0ℓ 0ℓ
-  +-*-heytingCommutativeRing   : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
-  +-*-heytingField             : HeytingField 0ℓ 0ℓ 0ℓ
-
-  module ≃-Reasoning = SetoidReasoning ≃-setoid
-  ```
-
-* Added new functions to `Data.Product.Nary.NonDependent`:
-  ```agda
-  zipWith : (∀ k → Projₙ as k → Projₙ bs k → Projₙ cs k) → Product n as → Product n bs → Product n cs
-  ```
-
-* Added new proof to `Data.Product.Properties`:
-  ```agda
-  map-cong : f ≗ g → h ≗ i → map f h ≗ map g i
-  ```
-
-* Added new definitions to `Data.Product.Properties`:
-  ```
-  Σ-≡,≡→≡ : (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂) → p₁ ≡ p₂
-  Σ-≡,≡←≡ : p₁ ≡ p₂ → (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂)
-  ×-≡,≡→≡ : (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂) → p₁ ≡ p₂
-  ×-≡,≡←≡ : p₁ ≡ p₂ → (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂)
-  ```
-
-* Added new proofs to `Data.Product.Relation.Binary.Lex.Strict`
-  ```agda
-  ×-respectsʳ : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_
-  ×-respectsˡ : Symmetric _≈₁_ → Transitive _≈₁_ → _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_
-  ×-wellFounded' : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ → WellFounded _<₁_ → WellFounded _<₂_ → WellFounded _<ₗₑₓ_
-  ```
-
-* Added new definitions to `Data.Sign.Base`:
-  ```agda
-  *-rawMagma : RawMagma 0ℓ 0ℓ
-  *-1-rawMonoid : RawMonoid 0ℓ 0ℓ
-  *-1-rawGroup : RawGroup 0ℓ 0ℓ
-  ```
-
-* Added new proofs to `Data.Sign.Properties`:
-  ```agda
-  *-inverse : Inverse + id _*_
-  *-isCommutativeSemigroup : IsCommutativeSemigroup _*_
-  *-isCommutativeMonoid : IsCommutativeMonoid _*_ +
-  *-isGroup : IsGroup _*_ + id
-  *-isAbelianGroup : IsAbelianGroup _*_ + id
-  *-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ
-  *-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
-  *-group : Group 0ℓ 0ℓ
-  *-abelianGroup : AbelianGroup 0ℓ 0ℓ
-  ≡-isDecEquivalence : IsDecEquivalence _≡_
-  ```
-
-* Added new functions in `Data.String.Base`:
-  ```agda
-  wordsByᵇ : (Char → Bool) → String → List String
-  linesByᵇ : (Char → Bool) → String → List String
-  ```
-
-* Added new proofs in `Data.String.Properties`:
-  ```agda
-  ≤-isDecTotalOrder-≈ : IsDecTotalOrder _≈_ _≤_
-  ≤-decTotalOrder-≈   : DecTotalOrder _ _ _
-  ```
-* Added new definitions in `Data.Sum.Properties`:
-  ```agda
-  swap-↔ : (A ⊎ B) ↔ (B ⊎ A)
-  ```
-
-* Added new proofs in `Data.Sum.Properties`:
-  ```agda
-  map-assocˡ : map (map f g) h ∘ assocˡ ≗ assocˡ ∘ map f (map g h)
-  map-assocʳ : map f (map g h) ∘ assocʳ ≗ assocʳ ∘ map (map f g) h
-  ```
-
-* Made `Map` public in `Data.Tree.AVL.IndexedMap`
-
-* Added new definitions in `Data.Vec.Base`:
-  ```agda
-  truncate : m ≤ n → Vec A n → Vec A m
-  pad      : m ≤ n → A → Vec A m → Vec A n
-
-  FoldrOp A B = A → B n → B (suc n)
-  FoldlOp A B = B n → A → B (suc n)
-
-  foldr′ : (A → B → B) → B → Vec A n → B
-  foldl′ : (B → A → B) → B → Vec A n → B
-  countᵇ : (A → Bool) → Vec A n → ℕ
-
-  iterate : (A → A) → A → Vec A n
-
-  diagonal           : Vec (Vec A n) n → Vec A n
-  DiagonalBind._>>=_ : Vec A n → (A → Vec B n) → Vec B n
-
-  _ʳ++_              : Vec A m → Vec A n → Vec A (m + n)
-
-  cast : .(eq : m ≡ n) → Vec A m → Vec A n
-  ```
-
-* Added new instance in `Data.Vec.Effectful`:
-  ```agda
-  monad : RawMonad (λ (A : Set a) → Vec A n)
-  ```
-
-* Added new proofs in `Data.Vec.Properties`:
-  ```agda
-  padRight-refl      : padRight ≤-refl a xs ≡ xs
-  padRight-replicate : replicate a ≡ padRight le a (replicate a)
-  padRight-trans     : padRight (≤-trans m≤n n≤p) a xs ≡ padRight n≤p a (padRight m≤n a xs)
-
-  truncate-refl     : truncate ≤-refl xs ≡ xs
-  truncate-trans    : truncate (≤-trans m≤n n≤p) xs ≡ truncate m≤n (truncate n≤p xs)
-  truncate-padRight : truncate m≤n (padRight m≤n a xs) ≡ xs
-
-  map-const    : map (const x) xs ≡ replicate x
-  map-⊛        : map f xs ⊛ map g xs ≡ map (f ˢ g) xs
-  map-++       : map f (xs ++ ys) ≡ map f xs ++ map f ys
-  map-is-foldr : map f ≗ foldr (Vec B) (λ x ys → f x ∷ ys) []
-  map-∷ʳ       : map f (xs ∷ʳ x) ≡ (map f xs) ∷ʳ (f x)
-  map-reverse  : map f (reverse xs) ≡ reverse (map f xs)
-  map-ʳ++      : map f (xs ʳ++ ys) ≡ map f xs ʳ++ map f ys
-  map-insert   : map f (insert xs i x) ≡ insert (map f xs) i (f x)
-  toList-map   : toList (map f xs) ≡ List.map f (toList xs)
-
-  lookup-concat : lookup (concat xss) (combine i j) ≡ lookup (lookup xss i) j
-
-  ⊛-is->>=    : fs ⊛ xs ≡ fs >>= flip map xs
-  lookup-⊛*   : lookup (fs ⊛* xs) (combine i j) ≡ (lookup fs i $ lookup xs j)
-  ++-is-foldr : xs ++ ys ≡ foldr ((Vec A) ∘ (_+ n)) _∷_ ys xs
-  []≔-++-↑ʳ   : (xs ++ ys) [ m ↑ʳ i ]≔ y ≡ xs ++ (ys [ i ]≔ y)
-  unfold-ʳ++  : xs ʳ++ ys ≡ reverse xs ++ ys
-
-  foldl-universal : (e : C zero) → ∀ {n} → Vec A n → C n) →
-                    (∀ ... → h C g e [] ≡ e) →
-                    (∀ ... → h C g e ∘ (x ∷_) ≗ h (C ∘ suc) g (g e x)) →
-                    h B f e ≗ foldl B f e
-  foldl-fusion  : h d ≡ e → (∀ ... → h (f b x) ≡ g (h b) x) → h ∘ foldl B f d ≗ foldl C g e
-  foldl-∷ʳ      : foldl B f e (ys ∷ʳ y) ≡ f (foldl B f e ys) y
-  foldl-[]      : foldl B f e [] ≡ e
-  foldl-reverse : foldl B {n} f e ∘ reverse ≗ foldr B (flip f) e
-
-  foldr-[]      : foldr B f e [] ≡ e
-  foldr-++      : foldr B f e (xs ++ ys) ≡ foldr (B ∘ (_+ n)) f (foldr B f e ys) xs
-  foldr-∷ʳ      : foldr B f e (ys ∷ʳ y) ≡ foldr (B ∘ suc) f (f y e) ys
-  foldr-ʳ++     : foldr B f e (xs ʳ++ ys) ≡ foldl (B ∘ (_+ n)) (flip f) (foldr B f e ys) xs
-  foldr-reverse : foldr B f e ∘ reverse ≗ foldl B (flip f) e
-
-  ∷ʳ-injective  : xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
-  ∷ʳ-injectiveˡ : xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys
-  ∷ʳ-injectiveʳ : xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y
-
-  unfold-∷ʳ : cast eq (xs ∷ʳ x) ≡ xs ++ [ x ]
-  init-∷ʳ   : init (xs ∷ʳ x) ≡ xs
-  last-∷ʳ   : last (xs ∷ʳ x) ≡ x
-  cast-∷ʳ   : cast eq (xs ∷ʳ x) ≡ (cast (cong pred eq) xs) ∷ʳ x
-  ++-∷ʳ     : cast eq ((xs ++ ys) ∷ʳ z) ≡ xs ++ (ys ∷ʳ z)
-  ∷ʳ-++     : cast eq ((xs ∷ʳ a) ++ ys) ≡ xs ++ (a ∷ ys)
-
-  reverse-∷          : reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
-  reverse-involutive : Involutive _≡_ reverse
-  reverse-reverse    : reverse xs ≡ ys → reverse ys ≡ xs
-  reverse-injective  : reverse xs ≡ reverse ys → xs ≡ ys
-
-  transpose-replicate : transpose (replicate xs) ≡ map replicate xs
-  toList-replicate    : toList (replicate {n = n} a) ≡ List.replicate n a
-
-  toList-++ : toList (xs ++ ys) ≡ toList xs List.++ toList ys
-
-  cast-is-id    : cast eq xs ≡ xs
-  subst-is-cast : subst (Vec A) eq xs ≡ cast eq xs
-  cast-sym      : cast eq xs ≡ ys → cast (sym eq) ys ≡ xs
-  cast-trans    : cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
-  map-cast      : map f (cast eq xs) ≡ cast eq (map f xs)
-  lookup-cast   : lookup (cast eq xs) (Fin.cast eq i) ≡ lookup xs i
-  lookup-cast₁  : lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)
-  lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
-  cast-reverse  : cast eq ∘ reverse ≗ reverse ∘ cast eq
-  cast-++ˡ      : cast (cong (_+ n) eq) (xs ++ ys) ≡ cast eq xs ++ ys
-  cast-++ʳ      : cast (cong (m +_) eq) (xs ++ ys) ≡ xs ++ cast eq ys
-
-  iterate-id     : iterate id x n ≡ replicate x
-  take-iterate   : take n (iterate f x (n + m)) ≡ iterate f x n
-  drop-iterate   : drop n (iterate f x n) ≡ []
-  lookup-iterate : lookup (iterate f x n) i ≡ ℕ.iterate f x (toℕ i)
-  toList-iterate : toList (iterate f x n) ≡ List.iterate f x n
-
-  zipwith-++ : zipWith f (xs ++ ys) (xs' ++ ys') ≡ zipWith f xs xs' ++ zipWith f ys ys'
-
-  ++-assoc     : cast eq ((xs ++ ys) ++ zs) ≡ xs ++ (ys ++ zs)
-  ++-identityʳ : cast eq (xs ++ []) ≡ xs
-  init-reverse : init ∘ reverse ≗ reverse ∘ tail
-  last-reverse : last ∘ reverse ≗ head
-  reverse-++   : cast eq (reverse (xs ++ ys)) ≡ reverse ys ++ reverse xs
-
-  toList-cast   : toList (cast eq xs) ≡ toList xs
-  cast-fromList : cast _ (fromList xs) ≡ fromList ys
-  fromList-map  : cast _ (fromList (List.map f xs)) ≡ map f (fromList xs)
-  fromList-++   : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
-  fromList-reverse : cast (Listₚ.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)
-
-  ∷-ʳ++   : cast eq ((a ∷ xs) ʳ++ ys) ≡ xs ʳ++ (a ∷ ys)
-  ++-ʳ++  : cast eq ((xs ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ʳ++ zs)
-  ʳ++-ʳ++ : cast eq ((xs ʳ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ++ zs)
-
-  length-toList   : List.length (toList xs) ≡ length xs
-  toList-insertAt : toList (insertAt xs i v) ≡ List.insertAt (toList xs) (Fin.cast (cong suc (sym (length-toList xs))) i) v
-
-  truncate≡take       : .(eq : n ≡ m + o) → truncate m≤n xs ≡ take m (cast eq xs)
-  take≡truncate       : take m xs ≡ truncate (m≤m+n m n) xs
-  lookup-truncate     : lookup (truncate m≤n xs) i ≡ lookup xs (Fin.inject≤ i m≤n)
-  lookup-take-inject≤ : lookup (take m xs) i ≡ lookup xs (Fin.inject≤ i (m≤m+n m n))
-  ```
-
-* Added new proofs in `Data.Vec.Membership.Propositional.Properties`:
-  ```agda
-  index-∈-fromList⁺ : Any.index (∈-fromList⁺ v∈xs) ≡ indexₗ v∈xs
-  ```
-
-* Added new isomorphisms to `Data.Unit.Polymorphic.Properties`:
-  ```agda
-  ⊤↔⊤* : ⊤ ↔ ⊤*
-  ```
-
-* Added new proofs in `Data.Vec.Functional.Properties`:
-  ```
-  map-updateAt : f ∘ g ≗ h ∘ f → map f (updateAt i g xs) ≗ updateAt i h (map f xs)
-  ```
-
-* Added new proofs in `Data.Vec.Relation.Binary.Lex.Strict`:
-  ```agda
-  xs≮[] : ¬ xs < []
-
-  <-respectsˡ : IsPartialEquivalence _≈_ → _≺_ Respectsˡ _≈_ → _<_ Respectsˡ _≋_
-  <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ → _<_ _Respectsʳ _≋_
-  <-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ → WellFounded _<_
-  ```
-
-* Added new function to `Data.Vec.Relation.Binary.Pointwise.Inductive`
-  ```agda
-  cong-[_]≔ : Pointwise _∼_ xs ys → Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)
-  ```
-
-* Added new function to `Data.Vec.Relation.Binary.Equality.Setoid`
-  ```agda
-  map-[]≔ : map f (xs [ i ]≔ p) ≋ map f xs [ i ]≔ f p
-  ```
-
-* Added new functions in `Data.Vec.Relation.Unary.Any`:
-  ```
-  lookup : Any P xs → A
-  ```
-
-* Added new functions in `Data.Vec.Relation.Unary.All`:
-  ```
-  decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
-
-  lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
-  lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
-  lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
-  lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
-  ```
-
-* Added vector associativity proof to  `Data.Vec.Relation.Binary.Equality.Setoid`:
-  ```
-  ++-assoc : (xs ++ ys) ++ zs ≋ xs ++ (ys ++ zs)
-  ```
-
-* Added new isomorphisms to `Data.Vec.N-ary`:
-  ```agda
-  Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B
-  ```
-
-* Added new isomorphisms to `Data.Vec.Recursive`:
-  ```agda
-  lift↔             : A ↔ B → A ^ n ↔ B ^ n
-  Fin[m^n]↔Fin[m]^n : Fin (m ^ n) ↔ Fin m Vec.^ n
-  ```
-
-* Added new functions in `Effect.Monad.State`:
-  ```
-  runState  : State s a → s → a × s
-  evalState : State s a → s → a
-  execState : State s a → s → s
-  ```
-
-* Added a non-dependent version of `flip` in `Function.Base`:
-  ```agda
-  flip′ : (A → B → C) → (B → A → C)
-  ```
-
-* Added new proofs and definitions in `Function.Bundles`:
-  ```agda
-  LeftInverse.isSplitSurjection : LeftInverse → IsSplitSurjection to
-  LeftInverse.surjection        : LeftInverse → Surjection
-  SplitSurjection               = LeftInverse
-  _⟨$⟩_ = Func.to
-  ```
-
-* Added new proofs to `Function.Properties.Inverse`:
-  ```agda
-  ↔-refl  : Reflexive _↔_
-  ↔-sym   : Symmetric _↔_
-  ↔-trans : Transitive _↔_
-
-  ↔⇒↣   : A ↔ B → A ↣ B
-  ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)
-
-  Inverse⇒Injection : Inverse S T → Injection S T
-  ```
-
-* Added new proofs in `Function.Construct.Symmetry`:
-  ```
-  bijective     : Bijective ≈₁ ≈₂ f → Symmetric ≈₂ → Transitive ≈₂ → Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹
-  isBijection   : IsBijection ≈₁ ≈₂ f → Congruent ≈₂ ≈₁ f⁻¹ → IsBijection ≈₂ ≈₁ f⁻¹
-  isBijection-≡ : IsBijection ≈₁ _≡_ f → IsBijection _≡_ ≈₁ f⁻¹
-  bijection     : Bijection R S → Congruent IB.Eq₂._≈_ IB.Eq₁._≈_ f⁻¹ → Bijection S R
-  bijection-≡   : Bijection R (setoid B) → Bijection (setoid B) R
-  sym-⤖        : A ⤖ B → B ⤖ A
-  ```
-
-* Added new operations in `Function.Strict`:
-  ```
-  _!|>_  : (a : A) → (∀ a → B a) → B a
-  _!|>′_ : A → (A → B) → B
-  ```
-
-* Added new proof and record in `Function.Structures`:
-  ```agda
-  record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
-  ```
-
-* Added new definition to the `Surjection` module in `Function.Related.Surjection`:
-  ```
-  f⁻ = proj₁ ∘ surjective
-  ```
-
-* Added new proof to `Induction.WellFounded`
-  ```agda
-  Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
-  acc⇒asym        : Acc _<_ x → x < y → ¬ (y < x)
-  wf⇒asym         : WellFounded _<_ → Asymmetric _<_
-  wf⇒irrefl       : _<_ Respects₂ _≈_ → Symmetric _≈_ → WellFounded _<_ → Irreflexive _≈_ _<_
-  ```
-
-* Added new operations in `IO`:
-  ```
-  Colist.forM  : Colist A → (A → IO B) → IO (Colist B)
-  Colist.forM′ : Colist A → (A → IO B) → IO ⊤
-
-  List.forM  : List A → (A → IO B) → IO (List B)
-  List.forM′ : List A → (A → IO B) → IO ⊤
-  ```
-
-* Added new operations in `IO.Base`:
-  ```
-  lift! : IO A → IO (Lift b A)
-  _<$_  : B → IO A → IO B
-  _=<<_ : (A → IO B) → IO A → IO B
-  _<<_  : IO B → IO A → IO B
-  lift′ : Prim.IO ⊤ → IO {a} ⊤
-
-  when   : Bool → IO ⊤ → IO ⊤
-  unless : Bool → IO ⊤ → IO ⊤
-
-  whenJust  : Maybe A → (A → IO ⊤) → IO ⊤
-  untilJust : IO (Maybe A) → IO A
-  untilRight : (A → IO (A ⊎ B)) → A → IO B
-  ```
-
-* Added new functions in `Reflection.AST.Term`:
-  ```
-  stripPis     : Term → List (String × Arg Type) × Term
-  prependLams  : List (String × Visibility) → Term → Term
-  prependHLams : List String → Term → Term
-  prependVLams : List String → Term → Term
-  ```
-
-* Added new types and operations to `Reflection.TCM`:
-  ```
-  Blocker : Set
-  blockerMeta : Meta → Blocker
-  blockerAny : List Blocker → Blocker
-  blockerAll : List Blocker → Blocker
-  blockTC : Blocker → TC A
-  ```
-
-* Added new operations in `Relation.Binary.Construct.Closure.Equivalence`:
-  ```
-  fold   : IsEquivalence _∼_ → _⟶_ ⇒ _∼_ → EqClosure _⟶_ ⇒ _∼_
-  gfold  : IsEquivalence _∼_ → _⟶_ =[ f ]⇒ _∼_ → EqClosure _⟶_ =[ f ]⇒ _∼_
-  return : _⟶_ ⇒ EqClosure _⟶_
-  join   : EqClosure (EqClosure _⟶_) ⇒ EqClosure _⟶_
-  _⋆     : _⟶₁_ ⇒ EqClosure _⟶₂_ → EqClosure _⟶₁_ ⇒ EqClosure _⟶₂_
-  ```
-
-* Added new operations in `Relation.Binary.Construct.Closure.Symmetric`:
-  ```
-  fold   : Symmetric _∼_ → _⟶_ ⇒ _∼_ → SymClosure _⟶_ ⇒ _∼_
-  gfold  : Symmetric _∼_ → _⟶_ =[ f ]⇒ _∼_ → SymClosure _⟶_ =[ f ]⇒ _∼_
-  return : _⟶_ ⇒ SymClosure _⟶_
-  join   : SymClosure (SymClosure _⟶_) ⇒ SymClosure _⟶_
-  _⋆     : _⟶₁_ ⇒ SymClosure _⟶₂_ → SymClosure _⟶₁_ ⇒ SymClosure _⟶₂_
-  ```
-
-* Added new proofs to `Relation.Binary.Lattice.Properties.{Join,Meet}Semilattice`:
-  ```agda
-  isPosemigroup           : IsPosemigroup _≈_ _≤_ _∨_
-  posemigroup             : Posemigroup c ℓ₁ ℓ₂
-  ≈-dec⇒≤-dec             : Decidable _≈_ → Decidable _≤_
-  ≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_
-  ```
-
-* Added new proofs to `Relation.Binary.Lattice.Properties.Bounded{Join,Meet}Semilattice`:
-  ```agda
-  isCommutativePomonoid : IsCommutativePomonoid _≈_ _≤_ _∨_ ⊥
-  commutativePomonoid   : CommutativePomonoid c ℓ₁ ℓ₂
-  ```
-
-* Added new proofs to `Relation.Binary.Properties.Poset`:
-  ```agda
-  ≤-dec⇒≈-dec             : Decidable _≤_ → Decidable _≈_
-  ≤-dec⇒isDecPartialOrder : Decidable _≤_ → IsDecPartialOrder _≈_ _≤_
-  ```
-
-* Added new proofs in `Relation.Binary.Properties.StrictPartialOrder`:
-  ```agda
-  >-strictPartialOrder : StrictPartialOrder s₁ s₂ s₃
-  ```
-
-* Added new proofs in `Relation.Binary.PropositionalEquality.Properties`:
-  ```
-  subst-application′ : subst Q eq (f x p) ≡ f y (subst P eq p)
-  sym-cong           : sym (cong f p) ≡ cong f (sym p)
-  ```
-
-* Added new proofs in `Relation.Binary.HeterogeneousEquality`:
-  ```
-  subst₂-removable : subst₂ _∼_ eq₁ eq₂ p ≅ p
-  ```
-
-* Added new definitions in `Relation.Unary`:
-  ```
-  _≐_  : Pred A ℓ₁ → Pred A ℓ₂ → Set _
-  _≐′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _
-  _∖_  : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _
-  ```
-
-* Added new proofs in `Relation.Unary.Properties`:
-  ```
-  ⊆-reflexive : _≐_ ⇒ _⊆_
-  ⊆-antisym   : Antisymmetric _≐_ _⊆_
-  ⊆-min       : Min _⊆_ ∅
-  ⊆-max       : Max _⊆_ U
-  ⊂⇒⊆         : _⊂_ ⇒ _⊆_
-  ⊂-trans     : Trans _⊂_ _⊂_ _⊂_
-  ⊂-⊆-trans   : Trans _⊂_ _⊆_ _⊂_
-  ⊆-⊂-trans   : Trans _⊆_ _⊂_ _⊂_
-  ⊂-respʳ-≐   : _⊂_ Respectsʳ _≐_
-  ⊂-respˡ-≐   : _⊂_ Respectsˡ _≐_
-  ⊂-resp-≐    : _⊂_ Respects₂ _≐_
-  ⊂-irrefl    : Irreflexive _≐_ _⊂_
-  ⊂-antisym   : Antisymmetric _≐_ _⊂_
-
-  ∅-⊆′         : (P : Pred A ℓ) → ∅ ⊆′ P
-  ⊆′-U         : (P : Pred A ℓ) → P ⊆′ U
-  ⊆′-refl      : Reflexive {A = Pred A ℓ} _⊆′_
-  ⊆′-reflexive : _≐′_ ⇒ _⊆′_
-  ⊆′-trans     : Trans _⊆′_ _⊆′_ _⊆′_
-  ⊆′-antisym   : Antisymmetric _≐′_ _⊆′_
-  ⊆′-min       : Min _⊆′_ ∅
-  ⊆′-max       : Max _⊆′_ U
-  ⊂′-trans     : Trans _⊂′_ _⊂′_ _⊂′_
-  ⊂′-⊆′-trans  : Trans _⊂′_ _⊆′_ _⊂′_
-  ⊆′-⊂′-trans  : Trans _⊆′_ _⊂′_ _⊂′_
-  ⊂′-respʳ-≐′  : _⊂′_ Respectsʳ _≐′_
-  ⊂′-respˡ-≐′  : _⊂′_ Respectsˡ _≐′_
-  ⊂′-resp-≐′   : _⊂′_ Respects₂ _≐′_
-  ⊂′-irrefl    : Irreflexive _≐′_ _⊂′_
-  ⊂′-antisym   : Antisymmetric _≐′_ _⊂′_
-  ⊂′⇒⊆′        : _⊂′_ ⇒ _⊆′_
-  ⊆⇒⊆′         : _⊆_ ⇒ _⊆′_
-  ⊆′⇒⊆         : _⊆′_ ⇒ _⊆_
-  ⊂⇒⊂′         : _⊂_ ⇒ _⊂′_
-  ⊂′⇒⊂         : _⊂′_ ⇒ _⊂_
-
-  ≐-refl   : Reflexive _≐_
-  ≐-sym    : Sym _≐_ _≐_
-  ≐-trans  : Trans _≐_ _≐_ _≐_
-  ≐′-refl  : Reflexive _≐′_
-  ≐′-sym   : Sym _≐′_ _≐′_
-  ≐′-trans : Trans _≐′_ _≐′_ _≐′_
-  ≐⇒≐′     : _≐_ ⇒ _≐′_
-  ≐′⇒≐     : _≐′_ ⇒ _≐_
-
-  U-irrelevant : Irrelevant U
-  ∁-irrelevant : (P : Pred A ℓ) → Irrelevant (∁ P)
-  ```
-
-* Generalised proofs in `Relation.Unary.Properties`:
-  ```
-  ⊆-trans : Trans _⊆_ _⊆_ _⊆_
-  ```
-
-* Added new proofs in `Relation.Binary.Properties.Setoid`:
-  ```
-  ≈-isPreorder     : IsPreorder _≈_ _≈_
-  ≈-isPartialOrder : IsPartialOrder _≈_ _≈_
-
-  ≈-preorder : Preorder a ℓ ℓ
-  ≈-poset    : Poset a ℓ ℓ
-  ```
-
-* Added new definitions in `Relation.Binary.Definitions`:
-  ```
-  RightTrans R S = Trans R S R
-  LeftTrans  S R = Trans S R R
-
-  Dense        _<_ = x < y → ∃[ z ] x < z × z < y
-  Cotransitive _#_ = x # y → ∀ z → (x # z) ⊎ (z # y)
-  Tight    _≈_ _#_ = (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)
-
-  Monotonic₁         _≤_ _⊑_ f     = f Preserves _≤_ ⟶ _⊑_
-  Antitonic₁         _≤_ _⊑_ f     = f Preserves (flip _≤_) ⟶ _⊑_
-  Monotonic₂         _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ _⊑_ ⟶ _≼_
-  MonotonicAntitonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ (flip _⊑_) ⟶ _≼_
-  AntitonicMonotonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ _⊑_ ⟶ _≼_
-  Antitonic₂         _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ (flip _⊑_) ⟶ _≼_
-  Adjoint            _≤_ _⊑_ f g   = (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)
-  ```
-
-* Added new definitions in `Relation.Binary.Bundles`:
-  ```
-  record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  record ApartnessRelation c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
-  ```
-
-* Added new definitions in `Relation.Binary.Structures`:
-  ```
-  record IsDenseLinearOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
-  record IsApartnessRelation (_#_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
-  ```
-
-* Added new proofs to `Relation.Binary.Consequences`:
-  ```
-  sym⇒¬-sym       : Symmetric _∼_ → Symmetric (¬_ ∘₂ _∼_)
-  cotrans⇒¬-trans : Cotransitive _∼_ → Transitive (¬_ ∘₂ _∼_)
-  irrefl⇒¬-refl   : Reflexive _≈_ → Irreflexive _≈_ _∼_ →  Reflexive (¬_ ∘₂ _∼_)
-  mono₂⇒cong₂     : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ → ∀ {f} →
-                    f Preserves₂ ≤₁ ⟶ ≤₁ ⟶ ≤₂ →
-                    f Preserves₂ ≈₁ ⟶ ≈₁ ⟶ ≈₂
-  ```
-
-* Added new proofs to `Relation.Binary.Construct.Closure.Transitive`:
-  ```
-  accessible⁻  : Acc _∼⁺_ x → Acc _∼_ x
-  wellFounded⁻ : WellFounded _∼⁺_ → WellFounded _∼_
-  accessible   : Acc _∼_ x → Acc _∼⁺_ x
-  ```
-
-* Added new operations in `Relation.Binary.PropositionalEquality.Properties`:
-  ```
-  J       : (B : (y : A) → x ≡ y → Set b) (p : x ≡ y) → B x refl → B y p
-  dcong   : (p : x ≡ y) → subst B p (f x) ≡ f y
-  dcong₂  : (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → f x₁ y₁ ≡ f x₂ y₂
-  dsubst₂ : (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → C x₁ y₁ → C x₂ y₂
-  ddcong₂ : (p : x₁ ≡ x₂) (q : subst B p y₁ ≡ y₂) → dsubst₂ C p q (f x₁ y₁) ≡ f x₂ y₂
-
-  isPartialOrder : IsPartialOrder _≡_ _≡_
-  poset          : Set a → Poset _ _ _
-  ```
-
-* Added new proof in `Relation.Nullary.Reflects`:
-  ```agda
-  T-reflects : Reflects (T b) b
-  T-reflects-elim : Reflects (T a) b → b ≡ a
-  ```
-
-* Added new operations in `System.Exit`:
-  ```
-  isSuccess : ExitCode → Bool
-  isFailure : ExitCode → Bool
-  ```
-
-NonZero/Positive/Negative changes
----------------------------------
-
-This is a full list of proofs that have changed form to use irrelevant instance arguments:
-
-* In `Data.Nat.Base`:
-  ```
-  ≢-nonZero⁻¹ : ∀ {n} → .(NonZero n) → n ≢ 0
-  >-nonZero⁻¹ : ∀ {n} → .(NonZero n) → n > 0
-  ```
-
-* In `Data.Nat.Properties`:
-  ```
-  *-cancelʳ-≡ : ∀ m n {o} → m * suc o ≡ n * suc o → m ≡ n
-  *-cancelˡ-≡ : ∀ {m n} o → suc o * m ≡ suc o * n → m ≡ n
-  *-cancelʳ-≤ : ∀ m n o → m * suc o ≤ n * suc o → m ≤ n
-  *-cancelˡ-≤ : ∀ {m n} o → suc o * m ≤ suc o * n → m ≤ n
-  *-monoˡ-<   : ∀ n → (_* suc n) Preserves _<_ ⟶ _<_
-  *-monoʳ-<   : ∀ n → (suc n *_) Preserves _<_ ⟶ _<_
-
-  m≤m*n          : ∀ m {n} → 0 < n → m ≤ m * n
-  m≤n*m          : ∀ m {n} → 0 < n → m ≤ n * m
-  m<m*n          : ∀ {m n} → 0 < m → 1 < n → m < m * n
-  suc[pred[n]]≡n : ∀ {n} → n ≢ 0 → suc (pred n) ≡ n
-  ```
-
-* In `Data.Nat.Coprimality`:
-  ```
-  ¬0-coprimeTo-2+ : ∀ {n} → ¬ Coprime 0 (2 + n)
-  Bézout-coprime  : ∀ {i j d} → Bézout.Identity (suc d) (i * suc d) (j * suc d) → Coprime i j
-  prime⇒coprime   : ∀ m → Prime m → ∀ n → 0 < n → n < m → Coprime m n
-  ```
-
-* In `Data.Nat.Divisibility`
-  ```agda
-  m%n≡0⇒n∣m : ∀ m n → m % suc n ≡ 0 → suc n ∣ m
-  n∣m⇒m%n≡0 : ∀ m n → suc n ∣ m → m % suc n ≡ 0
-  m%n≡0⇔n∣m : ∀ m n → m % suc n ≡ 0 ⇔ suc n ∣ m
-  ∣⇒≤       : ∀ {m n} → m ∣ suc n → m ≤ suc n
-  >⇒∤        : ∀ {m n} → m > suc n → m ∤ suc n
-  *-cancelˡ-∣ : ∀ {i j} k → suc k * i ∣ suc k * j → i ∣ j
-  ```
-
-* In `Data.Nat.DivMod`:
-  ```
-  m≡m%n+[m/n]*n : ∀ m n → m ≡ m % suc n + (m / suc n) * suc n
-  m%n≡m∸m/n*n   : ∀ m n → m % suc n ≡ m ∸ (m / suc n) * suc n
-  n%n≡0         : ∀ n → suc n % suc n ≡ 0
-  m%n%n≡m%n     : ∀ m n → m % suc n % suc n ≡ m % suc n
-  [m+n]%n≡m%n   : ∀ m n → (m + suc n) % suc n ≡ m % suc n
-  [m+kn]%n≡m%n  : ∀ m k n → (m + k * (suc n)) % suc n ≡ m % suc n
-  m*n%n≡0       : ∀ m n → (m * suc n) % suc n ≡ 0
-  m%n<n         : ∀ m n → m % suc n < suc n
-  m%n≤m         : ∀ m n → m % suc n ≤ m
-  m≤n⇒m%n≡m     : ∀ {m n} → m ≤ n → m % suc n ≡ m
-
-  m/n<m         : ∀ m n {≢0} → m ≥ 1 → n ≥ 2 → (m / n) {≢0} < m
-  ```
-
-* In `Data.Nat.GCD`
-  ```
-  GCD-* : ∀ {m n d c} → GCD (m * suc c) (n * suc c) (d * suc c) → GCD m n d
-  gcd[m,n]≤n : ∀ m n → gcd m (suc n) ≤ suc n
-  ```
-
-* In `Data.Integer.Properties`:
-  ```
-  positive⁻¹        : ∀ {i} → Positive i → i > 0ℤ
-  negative⁻¹        : ∀ {i} → Negative i → i < 0ℤ
-  nonPositive⁻¹     : ∀ {i} → NonPositive i → i ≤ 0ℤ
-  nonNegative⁻¹     : ∀ {i} → NonNegative i → i ≥ 0ℤ
-  negative<positive : ∀ {i j} → Negative i → Positive j → i < j
-
-  sign-◃    : ∀ s n → sign (s ◃ suc n) ≡ s
-  sign-cong : ∀ {s₁ s₂ n₁ n₂} → s₁ ◃ suc n₁ ≡ s₂ ◃ suc n₂ → s₁ ≡ s₂
-  -◃<+◃     : ∀ m n → Sign.- ◃ (suc m) < Sign.+ ◃ n
-  m⊖1+n<m   : ∀ m n → m ⊖ suc n < + m
-
-  *-cancelʳ-≡     : ∀ i j k → k ≢ + 0 → i * k ≡ j * k → i ≡ j
-  *-cancelˡ-≡     : ∀ i j k → i ≢ + 0 → i * j ≡ i * k → j ≡ k
-  *-cancelʳ-≤-pos : ∀ m n o → m * + suc o ≤ n * + suc o → m ≤ n
-
-  ≤-steps     : ∀ n → i ≤ j → i ≤ + n + j
-  ≤-steps-neg : ∀ n → i ≤ j → i - + n ≤ j
-  n≤m+n       : ∀ n → i ≤ + n + i
-  m≤m+n       : ∀ n → i ≤ i + + n
-  m-n≤m       : ∀ i n → i - + n ≤ i
-
-  *-cancelʳ-≤-pos    : ∀ m n o → m * + suc o ≤ n * + suc o → m ≤ n
-  *-cancelˡ-≤-pos    : ∀ m j k → + suc m * j ≤ + suc m * k → j ≤ k
-  *-cancelˡ-≤-neg    : ∀ m {j k} → -[1+ m ] * j ≤ -[1+ m ] * k → j ≥ k
-  *-cancelʳ-≤-neg    : ∀ {n o} m → n * -[1+ m ] ≤ o * -[1+ m ] → n ≥ o
-  *-cancelˡ-<-nonNeg : ∀ n → + n * i < + n * j → i < j
-  *-cancelʳ-<-nonNeg : ∀ n → i * + n < j * + n → i < j
-  *-monoʳ-≤-nonNeg   : ∀ n → (_* + n) Preserves _≤_ ⟶ _≤_
-  *-monoˡ-≤-nonNeg   : ∀ n → (+ n *_) Preserves _≤_ ⟶ _≤_
-  *-monoˡ-≤-nonPos   : ∀ i → NonPositive i → (i *_) Preserves _≤_ ⟶ _≥_
-  *-monoʳ-≤-nonPos   : ∀ i → NonPositive i → (_* i) Preserves _≤_ ⟶ _≥_
-  *-monoˡ-<-pos      : ∀ n → (+[1+ n ] *_) Preserves _<_ ⟶ _<_
-  *-monoʳ-<-pos      : ∀ n → (_* +[1+ n ]) Preserves _<_ ⟶ _<_
-  *-monoˡ-<-neg      : ∀ n → (-[1+ n ] *_) Preserves _<_ ⟶ _>_
-  *-monoʳ-<-neg      : ∀ n → (_* -[1+ n ]) Preserves _<_ ⟶ _>_
-  *-cancelˡ-<-nonPos : ∀ n → NonPositive n → n * i < n * j → i > j
-  *-cancelʳ-<-nonPos : ∀ n → NonPositive n → i * n < j * n → i > j
-
-  *-distribˡ-⊓-nonNeg : ∀ m j k → + m * (j ⊓ k) ≡ (+ m * j) ⊓ (+ m * k)
-  *-distribʳ-⊓-nonNeg : ∀ m j k → (j ⊓ k) * + m ≡ (j * + m) ⊓ (k * + m)
-  *-distribˡ-⊓-nonPos : ∀ i → NonPositive i → ∀ j k → i * (j ⊓ k) ≡ (i * j) ⊔ (i * k)
-  *-distribʳ-⊓-nonPos : ∀ i → NonPositive i → ∀ j k → (j ⊓ k) * i ≡ (j * i) ⊔ (k * i)
-  *-distribˡ-⊔-nonNeg : ∀ m j k → + m * (j ⊔ k) ≡ (+ m * j) ⊔ (+ m * k)
-  *-distribʳ-⊔-nonNeg : ∀ m j k → (j ⊔ k) * + m ≡ (j * + m) ⊔ (k * + m)
-  *-distribˡ-⊔-nonPos : ∀ i → NonPositive i → ∀ j k → i * (j ⊔ k) ≡ (i * j) ⊓ (i * k)
-  *-distribʳ-⊔-nonPos : ∀ i → NonPositive i → ∀ j k → (j ⊔ k) * i ≡ (j * i) ⊓ (k * i)
-  ```
-
-* In `Data.Integer.Divisibility`:
-  ```
-  *-cancelˡ-∣ : ∀ k {i j} → k ≢ + 0 → k * i ∣ k * j → i ∣ j
-  *-cancelʳ-∣ : ∀ k {i j} → k ≢ + 0 → i * k ∣ j * k → i ∣ j
-  ```
-
-* In `Data.Integer.Divisibility.Signed`:
-  ```
-  *-cancelˡ-∣ : ∀ k {i j} → k ≢ + 0 → k * i ∣ k * j → i ∣ j
-  *-cancelʳ-∣ : ∀ k {i j} → k ≢ + 0 → i * k ∣ j * k → i ∣ j
-  ```
-
-* In `Data.Rational.Unnormalised.Properties`:
-  ```agda
-  positive⁻¹           : ∀ {q} → .(Positive q) → q > 0ℚᵘ
-  nonNegative⁻¹        : ∀ {q} → .(NonNegative q) → q ≥ 0ℚᵘ
-  negative⁻¹           : ∀ {q} → .(Negative q) → q < 0ℚᵘ
-  nonPositive⁻¹        : ∀ {q} → .(NonPositive q) → q ≤ 0ℚᵘ
-  positive⇒nonNegative : ∀ {p} → Positive p → NonNegative p
-  negative⇒nonPositive : ∀ {p} → Negative p → NonPositive p
-  negative<positive    : ∀ {p q} → .(Negative p) → .(Positive q) → p < q
-  nonNeg∧nonPos⇒0      : ∀ {p} → .(NonNegative p) → .(NonPositive p) → p ≃ 0ℚᵘ
-
-  p≤p+q   : ∀ {p q} → NonNegative q → p ≤ p + q
-  p≤q+p   : ∀ {p} → NonNegative p → ∀ {q} → q ≤ p + q
-
-  *-cancelʳ-≤-pos    : ∀ {r} → Positive r → ∀ {p q} → p * r ≤ q * r → p ≤ q
-  *-cancelˡ-≤-pos    : ∀ {r} → Positive r → ∀ {p q} → r * p ≤ r * q → p ≤ q
-  *-cancelʳ-≤-neg    : ∀ r → Negative r → ∀ {p q} → p * r ≤ q * r → q ≤ p
-  *-cancelˡ-≤-neg    : ∀ r → Negative r → ∀ {p q} → r * p ≤ r * q → q ≤ p
-  *-cancelˡ-<-nonNeg : ∀ {r} → NonNegative r → ∀ {p q} → r * p < r * q → p < q
-  *-cancelʳ-<-nonNeg : ∀ {r} → NonNegative r → ∀ {p q} → p * r < q * r → p < q
-  *-cancelˡ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → r * p < r * q → q < p
-  *-cancelʳ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → p * r < q * r → q < p
-  *-monoˡ-≤-nonNeg   : ∀ {r} → NonNegative r → (_* r) Preserves _≤_ ⟶ _≤_
-  *-monoʳ-≤-nonNeg   : ∀ {r} → NonNegative r → (r *_) Preserves _≤_ ⟶ _≤_
-  *-monoˡ-≤-nonPos   : ∀ r → NonPositive r → (_* r) Preserves _≤_ ⟶ _≥_
-  *-monoʳ-≤-nonPos   : ∀ r → NonPositive r → (r *_) Preserves _≤_ ⟶ _≥_
-  *-monoˡ-<-pos      : ∀ {r} → Positive r → (_* r) Preserves _<_ ⟶ _<_
-  *-monoʳ-<-pos      : ∀ {r} → Positive r → (r *_) Preserves _<_ ⟶ _<_
-  *-monoˡ-<-neg      : ∀ r → Negative r → (_* r) Preserves _<_ ⟶ _>_
-  *-monoʳ-<-neg      : ∀ r → Negative r → (r *_) Preserves _<_ ⟶ _>_
-
-  pos⇒1/pos : ∀ p (p>0 : Positive p) → Positive ((1/ p) {{pos⇒≢0 p p>0}})
-  neg⇒1/neg : ∀ p (p<0 : Negative p) → Negative ((1/ p) {{neg⇒≢0 p p<0}})
-
-  *-distribʳ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊓ (r * p)
-  *-distribˡ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊓ (p * r)
-  *-distribˡ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊔ (p * r)
-  *-distribʳ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊔ (r * p)
-  *-distribˡ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊓ (p * r)
-  *-distribʳ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊓ (r * p)
-  *-distribˡ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊔ (p * r)
-  *-distribʳ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊔ (r * p)
-  ```
-
-* In `Data.Rational.Properties`:
-  ```
-  positive⁻¹ : Positive p → p > 0ℚ
-  nonNegative⁻¹ : NonNegative p → p ≥ 0ℚ
-  negative⁻¹ : Negative p → p < 0ℚ
-  nonPositive⁻¹ : NonPositive p → p ≤ 0ℚ
-  negative<positive : Negative p → Positive q → p < q
-  nonNeg≢neg : ∀ p q → NonNegative p → Negative q → p ≢ q
-  pos⇒nonNeg : ∀ p → Positive p → NonNegative p
-  neg⇒nonPos : ∀ p → Negative p → NonPositive p
-  nonNeg∧nonZero⇒pos : ∀ p → NonNegative p → NonZero p → Positive p
-
-  *-cancelʳ-≤-pos    : ∀ r → Positive r → ∀ {p q} → p * r ≤ q * r → p ≤ q
-  *-cancelˡ-≤-pos    : ∀ r → Positive r → ∀ {p q} → r * p ≤ r * q → p ≤ q
-  *-cancelʳ-≤-neg    : ∀ r → Negative r → ∀ {p q} → p * r ≤ q * r → p ≥ q
-  *-cancelˡ-≤-neg    : ∀ r → Negative r → ∀ {p q} → r * p ≤ r * q → p ≥ q
-  *-cancelˡ-<-nonNeg : ∀ r → NonNegative r → ∀ {p q} → r * p < r * q → p < q
-  *-cancelʳ-<-nonNeg : ∀ r → NonNegative r → ∀ {p q} → p * r < q * r → p < q
-  *-cancelˡ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → r * p < r * q → p > q
-  *-cancelʳ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → p * r < q * r → p > q
-  *-monoʳ-≤-nonNeg   : ∀ r → NonNegative r → (_* r) Preserves _≤_ ⟶ _≤_
-  *-monoˡ-≤-nonNeg   : ∀ r → NonNegative r → (r *_) Preserves _≤_ ⟶ _≤_
-  *-monoʳ-≤-nonPos   : ∀ r → NonPositive r → (_* r) Preserves _≤_ ⟶ _≥_
-  *-monoˡ-≤-nonPos   : ∀ r → NonPositive r → (r *_) Preserves _≤_ ⟶ _≥_
-  *-monoˡ-<-pos      : ∀ r → Positive r → (_* r) Preserves _<_ ⟶ _<_
-  *-monoʳ-<-pos      : ∀ r → Positive r → (r *_) Preserves _<_ ⟶ _<_
-  *-monoˡ-<-neg      : ∀ r → Negative r → (_* r) Preserves _<_ ⟶ _>_
-  *-monoʳ-<-neg      : ∀ r → Negative r → (r *_) Preserves _<_ ⟶ _>_
-
-  *-distribˡ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊓ (p * r)
-  *-distribʳ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊓ (r * p)
-  *-distribˡ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊔ (p * r)
-  *-distribʳ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊔ (r * p)
-  *-distribˡ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊓ (p * r)
-  *-distribʳ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊓ (r * p)
-  *-distribˡ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊔ (p * r)
-  *-distribʳ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊔ (r * p)
-
-  pos⇒1/pos : ∀ p (p>0 : Positive p) → Positive ((1/ p) {{pos⇒≢0 p p>0}})
-  neg⇒1/neg : ∀ p (p<0 : Negative p) → Negative ((1/ p) {{neg⇒≢0 p p<0}})
-  1/pos⇒pos : ∀ p .{{_ : NonZero p}} → (1/p : Positive (1/ p)) → Positive p
-  1/neg⇒neg : ∀ p .{{_ : NonZero p}} → (1/p : Negative (1/ p)) → Negative p
-  ```
diff --git a/CHANGELOG/v2.0.md b/CHANGELOG/v2.0.md
new file mode 100644
index 0000000000..134309712c
--- /dev/null
+++ b/CHANGELOG/v2.0.md
@@ -0,0 +1,3911 @@
+Version 2.0
+===========
+
+The library has been tested using Agda 2.6.4 and 2.6.4.1.
+
+NOTE: Version `2.0` contains various breaking changes and is not backwards
+compatible with code written with version `1.X` of the library.
+
+Highlights
+----------
+
+* A new tactic `cong!` available from `Tactic.Cong` which automatically
+  infers the argument to `cong` for you via anti-unification.
+
+* Improved the `solve` tactic in `Tactic.RingSolver` to work in a much
+  wider range of situations.
+
+* A massive refactoring of the unindexed `Functor`/`Applicative`/`Monad` hierarchy
+  and the `MonadReader` / `MonadState` type classes. These are now usable with
+  instance arguments as demonstrated in the tests/monad examples.
+
+* Significant tightening of `import` statements internally in the library,
+  drastically decreasing the dependencies and hence load time of many key
+  modules.
+
+* A golden testing library in `Test.Golden`. This allows you to run a set
+  of tests and make sure their output matches an expected `golden` value.
+  The test runner has many options: filtering tests by name, dumping the
+  list of failures to a file, timing the runs, coloured output, etc.
+  Cf. the comments in `Test.Golden` and the standard library's own tests
+  in `tests/` for documentation on how to use the library.
+
+Bug-fixes
+---------
+
+* In `Algebra.Structures` the records `IsRing` and `IsRingWithoutOne` contained an unnecessary field
+  `zero : RightZero 0# *`, which could be derived from the other ring axioms.
+  Consequently this field has been removed from the record, and the record
+  `IsRingWithoutAnnihilatingZero` in `Algebra.Structures.Biased` has been
+  deprecated as it is now identical to is `IsRing`.
+  
+* In `Algebra.Definitions.RawSemiring` the record `Prime` did not
+  enforce that the number was not divisible by `1#`. To fix this
+  `p∤1 : p ∤ 1#` has been added as a field.
+
+* In `Data.Container.FreeMonad`, we give a direct definition of `_⋆_` as an inductive
+  type rather than encoding it as an instance of `μ`. This ensures Agda notices that
+  `C ⋆ X` is strictly positive in `X` which in turn allows us to use the free monad
+  when defining auxiliary (co)inductive types (cf. the `Tap` example in
+  `README.Data.Container.FreeMonad`).
+
+* In `Data.Fin.Properties` the `i` argument to `opposite-suc` was implicit
+  but could not be inferred in general. It has been made explicit.
+
+* In `Data.List.Membership.Setoid` the operations `_∷=_` and `_─_`
+  had an extraneous `{A : Set a}` parameter. This has been removed.
+
+* In `Data.List.Relation.Ternary.Appending.Setoid` the constructors
+  were re-exported in their full generality which lead to unsolved meta
+  variables at their use sites. Now versions of the constructors specialised
+  to use the setoid's carrier set are re-exported.
+
+* In `Data.Nat.DivMod` the parameter `o` in the proof `/-monoˡ-≤` was
+  implicit but not inferrable. It has been changed to be explicit.
+
+* In `Data.Nat.DivMod` the parameter `m` in the proof `+-distrib-/-∣ʳ` was
+  implicit but not inferrable, while `n` is explicit but inferrable.
+  They have been to explicit and implicit respectively.
+
+* In `Data.Nat.GeneralisedArithmetic` the `s` and `z` arguments to the
+  following functions were implicit but not inferrable:
+  `fold-+`, `fold-k`, `fold-*`, `fold-pull`. They have been made explicit.
+
+* In `Data.Rational(.Unnormalised).Properties` the module `≤-Reasoning`
+  exported equality combinators using the generic setoid symbol `_≈_`. They
+  have been renamed to use the same `_≃_` symbol used for non-propositional
+  equality over `Rational`s, i.e.
+  ```agda
+  step-≈  ↦  step-≃
+  step-≈˘ ↦  step-≃˘
+  ```
+  with corresponding associated syntax:
+  ```agda
+  _≈⟨_⟩_ ↦ _≃⟨_⟩_
+  _≈⟨_⟨_ ↦ _≃⟨_⟨_
+  ```
+
+* In `Function.Construct.Composition` the combinators
+  `_⟶-∘_`, `_↣-∘_`, `_↠-∘_`, `_⤖-∘_`, `_⇔-∘_`, `_↩-∘_`, `_↪-∘_`, `_↔-∘_`
+  had their arguments in the wrong order. They have been flipped so they can
+  actually be  used as a composition operator.
+
+* In `Function.Definitions` the definitions of `Surjection`, `Inverseˡ`,
+  `Inverseʳ` were not being re-exported correctly and therefore had an unsolved
+  meta-variable whenever this module was explicitly parameterised. This has
+  been fixed.
+
+* In `System.Exit` the `ExitFailure` constructor is now carrying an integer
+  rather than a natural. The previous binding was incorrectly assuming that
+  all exit codes where non-negative.
+
+Non-backwards compatible changes
+--------------------------------
+
+### Removed deprecated names
+
+* All modules and names that were deprecated in `v1.2` and before have
+  been removed.
+
+### Changes to `LeftCancellative` and `RightCancellative` in `Algebra.Definitions`
+
+* The definitions of the types for cancellativity in `Algebra.Definitions` previously
+  made some of their arguments implicit. This was under the assumption that the operators were
+  defined by pattern matching on the left argument so that Agda could always infer the
+  argument on the RHS.
+
+* Although many of the operators defined in the library follow this convention, this is not
+  always true and cannot be assumed in user's code.
+
+* Therefore the definitions have been changed as follows to make all their arguments explicit:
+  - `LeftCancellative _∙_`
+    - From: `∀ x {y z} → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
+    - To: `∀ x y z → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
+
+  - `RightCancellative _∙_`
+    - From: `∀ {x} y z → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
+    - To: `∀ x y z → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
+
+  - `AlmostLeftCancellative e _∙_`
+    - From: `∀ {x} y z → ¬ x ≈ e → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (x ∙ y) ≈ (x ∙ z) → y ≈ z`
+
+  - `AlmostRightCancellative e _∙_`
+    - From: `∀ {x} y z → ¬ x ≈ e → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
+    - To: `∀ x y z → ¬ x ≈ e → (y ∙ x) ≈ (z ∙ x) → y ≈ z`
+
+* Correspondingly some proofs of the above types will need additional arguments passed explicitly.
+  Instances can easily be fixed by adding additional underscores, e.g.
+  - `∙-cancelˡ x` to `∙-cancelˡ x _ _`
+  - `∙-cancelʳ y z` to `∙-cancelʳ _ y z`
+
+### Changes to ring structures in `Algebra`
+
+* Several ring-like structures now have the multiplicative structure defined by
+  its laws rather than as a substructure, to avoid repeated proofs that the
+  underlying relation is an equivalence. These are:
+  * `IsNearSemiring`
+  * `IsSemiringWithoutOne`
+  * `IsSemiringWithoutAnnihilatingZero`
+  * `IsRing`
+
+* To aid with migration, structures matching the old style ones have been added
+  to `Algebra.Structures.Biased`, with conversion functions:
+  * `IsNearSemiring*` and `isNearSemiring*`
+  * `IsSemiringWithoutOne*` and `isSemiringWithoutOne*`
+  * `IsSemiringWithoutAnnihilatingZero*` and `isSemiringWithoutAnnihilatingZero*`
+  * `IsRing*` and `isRing*`
+
+### Refactoring of lattices in `Algebra.Structures/Bundles` hierarchy
+
+* In order to improve modularity and consistency with `Relation.Binary.Lattice`,
+  the structures & bundles for `Semilattice`, `Lattice`, `DistributiveLattice`
+  & `BooleanAlgebra` have been moved out of the `Algebra` modules and into their
+  own hierarchy in `Algebra.Lattice`.
+
+* All submodules, (e.g. `Algebra.Properties.Semilattice` or `Algebra.Morphism.Lattice`)
+  have been moved to the corresponding place under `Algebra.Lattice` (e.g.
+  `Algebra.Lattice.Properties.Semilattice` or `Algebra.Lattice.Morphism.Lattice`). See
+  the `Deprecated modules` section below for full details.
+
+* The definition of `IsDistributiveLattice` and `IsBooleanAlgebra` have changed so
+  that they are no longer right-biased which hindered compositionality.
+  More concretely, `IsDistributiveLattice` now has fields:
+  ```agda
+  ∨-distrib-∧ : ∨ DistributesOver ∧
+  ∧-distrib-∨ : ∧ DistributesOver ∨
+  ```
+  instead of
+  ```agda
+  ∨-distribʳ-∧ : ∨ DistributesOverʳ ∧
+  ```
+  and `IsBooleanAlgebra` now has fields:
+  ```agda
+  ∨-complement : Inverse ⊤ ¬ ∨
+  ∧-complement : Inverse ⊥ ¬ ∧
+  ```
+  instead of:
+  ```agda
+  ∨-complementʳ : RightInverse ⊤ ¬ ∨
+  ∧-complementʳ : RightInverse ⊥ ¬ ∧
+  ```
+
+* To allow construction of these structures via their old form, smart constructors
+  have been added to a new module `Algebra.Lattice.Structures.Biased`, which are by
+  re-exported automatically by `Algebra.Lattice`. For example, if before you wrote:
+  ```agda
+  ∧-∨-isDistributiveLattice = record
+    { isLattice    = ∧-∨-isLattice
+    ; ∨-distribʳ-∧ = ∨-distribʳ-∧
+    }
+  ```
+  you can use the smart constructor `isDistributiveLatticeʳʲᵐ` to write:
+  ```agda
+  ∧-∨-isDistributiveLattice = isDistributiveLatticeʳʲᵐ (record
+    { isLattice    = ∧-∨-isLattice
+    ; ∨-distribʳ-∧ = ∨-distribʳ-∧
+    })
+  ```
+  without having to prove full distributivity.
+
+* Added new `IsBoundedSemilattice`/`BoundedSemilattice` records.
+
+* Added new aliases `Is(Meet/Join)(Bounded)Semilattice` for `Is(Bounded)Semilattice`
+  which can be used to indicate meet/join-ness of the original structures, and
+  the field names in `IsSemilattice` and `Semilattice` have been renamed from 
+  `∧-cong` to `∙-cong`to indicate their undirected nature.
+
+* Finally, the following auxiliary files have been moved:
+  ```agda
+  Algebra.Properties.Semilattice               ↦ Algebra.Lattice.Properties.Semilattice
+  Algebra.Properties.Lattice                   ↦ Algebra.Lattice.Properties.Lattice
+  Algebra.Properties.DistributiveLattice       ↦ Algebra.Lattice.Properties.DistributiveLattice
+  Algebra.Properties.BooleanAlgebra            ↦ Algebra.Lattice.Properties.BooleanAlgebra
+  Algebra.Properties.BooleanAlgebra.Expression ↦ Algebra.Lattice.Properties.BooleanAlgebra.Expression
+  Algebra.Morphism.LatticeMonomorphism         ↦ Algebra.Lattice.Morphism.LatticeMonomorphism
+  ```
+
+#### Changes to `Algebra.Morphism.Structures`
+
+* Previously the record definitions:
+  ```
+  IsNearSemiringHomomorphism
+  IsSemiringHomomorphism
+  IsRingHomomorphism
+  ```
+  all had two separate proofs of `IsRelHomomorphism` within them.
+
+* To fix this they have all been redefined to build up from `IsMonoidHomomorphism`,
+  `IsNearSemiringHomomorphism`, and `IsSemiringHomomorphism` respectively,
+  adding a single property at each step.
+
+* Similarly, `IsLatticeHomomorphism` is now built as
+  `IsRelHomomorphism` along with proofs that `_∧_` and `_∨_` are homomorphic.
+
+* Finally `⁻¹-homo` in `IsRingHomomorphism` has been renamed to `-‿homo`.
+
+#### Renamed `Category` modules to `Effect`
+
+* As observed by Wen Kokke in issue #1636, it no longer really makes sense
+  to group the modules which correspond to the variety of concepts of
+  (effectful) type constructor arising in functional programming (esp. in Haskell)
+  such as `Monad`, `Applicative`, `Functor`, etc, under `Category.*`,
+  as this obstructs the importing of the `agda-categories` development into
+  the Standard Library, and moreover needlessly restricts the applicability of
+  categorical concepts to this (highly specific) mode of use.
+
+* Correspondingly, client modules grouped under  `*.Categorical.*` which
+  exploit such structure  for effectful programming have been  renamed
+  `*.Effectful`, with the originals being deprecated.
+
+* Full list of moved modules:
+  ```agda
+  Codata.Sized.Colist.Categorical            ↦ Codata.Sized.Colist.Effectful
+  Codata.Sized.Covec.Categorical             ↦ Codata.Sized.Covec.Effectful
+  Codata.Sized.Delay.Categorical             ↦ Codata.Sized.Delay.Effectful
+  Codata.Sized.Stream.Categorical            ↦ Codata.Sized.Stream.Effectful
+  Data.List.Categorical                      ↦ Data.List.Effectful
+  Data.List.Categorical.Transformer          ↦ Data.List.Effectful.Transformer
+  Data.List.NonEmpty.Categorical             ↦ Data.List.NonEmpty.Effectful
+  Data.List.NonEmpty.Categorical.Transformer ↦ Data.List.NonEmpty.Effectful.Transformer
+  Data.Maybe.Categorical                     ↦ Data.Maybe.Effectful
+  Data.Maybe.Categorical.Transformer         ↦ Data.Maybe.Effectful.Transformer
+  Data.Product.Categorical.Examples          ↦ Data.Product.Effectful.Examples
+  Data.Product.Categorical.Left              ↦ Data.Product.Effectful.Left
+  Data.Product.Categorical.Left.Base         ↦ Data.Product.Effectful.Left.Base
+  Data.Product.Categorical.Right             ↦ Data.Product.Effectful.Right
+  Data.Product.Categorical.Right.Base        ↦ Data.Product.Effectful.Right.Base
+  Data.Sum.Categorical.Examples              ↦ Data.Sum.Effectful.Examples
+  Data.Sum.Categorical.Left                  ↦ Data.Sum.Effectful.Left
+  Data.Sum.Categorical.Left.Transformer      ↦ Data.Sum.Effectful.Left.Transformer
+  Data.Sum.Categorical.Right                 ↦ Data.Sum.Effectful.Right
+  Data.Sum.Categorical.Right.Transformer     ↦ Data.Sum.Effectful.Right.Transformer
+  Data.These.Categorical.Examples            ↦ Data.These.Effectful.Examples
+  Data.These.Categorical.Left                ↦ Data.These.Effectful.Left
+  Data.These.Categorical.Left.Base           ↦ Data.These.Effectful.Left.Base
+  Data.These.Categorical.Right               ↦ Data.These.Effectful.Right
+  Data.These.Categorical.Right.Base          ↦ Data.These.Effectful.Right.Base
+  Data.Vec.Categorical                       ↦ Data.Vec.Effectful
+  Data.Vec.Categorical.Transformer           ↦ Data.Vec.Effectful.Transformer
+  Data.Vec.Recursive.Categorical             ↦ Data.Vec.Recursive.Effectful
+  Function.Identity.Categorical              ↦ Function.Identity.Effectful
+  IO.Categorical                             ↦ IO.Effectful
+  Reflection.TCM.Categorical                 ↦ Reflection.TCM.Effectful
+  ```
+
+* Full list of new modules:
+  ```
+  Algebra.Construct.Initial
+  Algebra.Construct.Terminal
+  Data.List.Effectful.Transformer
+  Data.List.NonEmpty.Effectful.Transformer
+  Data.Maybe.Effectful.Transformer
+  Data.Sum.Effectful.Left.Transformer
+  Data.Sum.Effectful.Right.Transformer
+  Data.Vec.Effectful.Transformer
+  Effect.Empty
+  Effect.Choice
+  Effect.Monad.Error.Transformer
+  Effect.Monad.Identity
+  Effect.Monad.IO
+  Effect.Monad.IO.Instances
+  Effect.Monad.Reader.Indexed
+  Effect.Monad.Reader.Instances
+  Effect.Monad.Reader.Transformer
+  Effect.Monad.Reader.Transformer.Base
+  Effect.Monad.State.Indexed
+  Effect.Monad.State.Instances
+  Effect.Monad.State.Transformer
+  Effect.Monad.State.Transformer.Base
+  Effect.Monad.Writer
+  Effect.Monad.Writer.Indexed
+  Effect.Monad.Writer.Instances
+  Effect.Monad.Writer.Transformer
+  Effect.Monad.Writer.Transformer.Base
+  IO.Effectful
+  IO.Instances
+  ```
+
+### Refactoring of the unindexed Functor/Applicative/Monad hierarchy in `Effect`
+
+* The unindexed versions are not defined in terms of the named versions anymore.
+
+* The `RawApplicative` and `RawMonad` type classes have been relaxed so that the underlying
+  functors do not need their domain and codomain to live at the same Set level.
+  This is needed for level-increasing functors like `IO : Set l → Set (suc l)`.
+
+* `RawApplicative` is now `RawFunctor + pure + _<*>_` and `RawMonad` is now
+  `RawApplicative` + `_>>=_`.
+  This reorganisation means in particular that the functor/applicative of a monad
+  are not computed using `_>>=_`. This may break proofs.
+
+* When `F : Set f → Set f` we moreover have a definable join/μ operator
+  `join : (M : RawMonad F) → F (F A) → F A`.
+
+* We now have `RawEmpty` and `RawChoice` respectively packing `empty : M A` and
+  `(<|>) : M A → M A → M A`. `RawApplicativeZero`, `RawAlternative`, `RawMonadZero`,
+  `RawMonadPlus` are all defined in terms of these.
+
+* `MonadT T` now returns a `MonadTd` record that packs both a proof that the
+  `Monad M` transformed by `T` is a monad and that we can `lift` a computation
+  `M A` to a transformed computation `T M A`.
+
+* The monad transformer are not mere aliases anymore, they are record-wrapped
+  which allows constraints such as `MonadIO (StateT S (ReaderT R IO))` to be
+  discharged by instance arguments.
+
+* The mtl-style type classes (`MonadState`, `MonadReader`) do not contain a proof
+  that the underlying functor is a `Monad` anymore. This ensures we do not have
+  conflicting `Monad M` instances from a pair of `MonadState S M` & `MonadReader R M`
+  constraints.
+
+* `MonadState S M` is now defined in terms of
+  ```agda
+  gets : (S → A) → M A
+  modify : (S → S) → M ⊤
+  ```
+  with `get` and `put` defined as derived notions.
+  This is needed because `MonadState S M` does not pack a `Monad M` instance anymore
+  and so we cannot define `modify f` as `get >>= λ s → put (f s)`.
+
+* `MonadWriter 𝕎 M` is defined similarly:
+   ```agda
+   writer : W × A → M A
+   listen : M A → M (W × A)
+   pass   : M ((W → W) × A) → M A
+   ```
+   with `tell` defined as a derived notion.
+   Note that `𝕎` is a `RawMonoid`, not a `Set` and `W` is the carrier of the monoid.
+
+
+#### Moved `Codata` modules to `Codata.Sized`
+
+* Due to the change in Agda 2.6.2 where sized types are no longer compatible
+  with the `--safe` flag, it has become clear that a third variant of codata
+  will be needed using coinductive records.
+
+* Therefore all existing modules in `Codata` which used sized types have been
+  moved inside a new folder named `Codata.Sized`,  e.g. `Codata.Stream`
+  has become `Codata.Sized.Stream`.
+
+### New proof-irrelevant for empty type in `Data.Empty`
+
+* The definition of `⊥` has been changed to
+  ```agda
+  private
+    data Empty : Set where
+
+  ⊥ : Set
+  ⊥ = Irrelevant Empty
+  ```
+  in order to make ⊥ proof irrelevant. Any two proofs of `⊥` or of a negated
+  statements are now *judgmentally* equal to each other.
+
+* Consequently the following two definitions have been modified:
+
+  + In `Relation.Nullary.Decidable.Core`, the type of `dec-no` has changed
+    ```agda
+    dec-no : (p? : Dec P) → ¬ P → ∃ λ ¬p′ → p? ≡ no ¬p′
+      ↦
+    dec-no : (p? : Dec P) (¬p : ¬ P) → p? ≡ no ¬p
+    ```
+
+  + In `Relation.Binary.PropositionalEquality`, the type of `≢-≟-identity` has changed
+    ```agda
+    ≢-≟-identity : x ≢ y → ∃ λ ¬eq → x ≟ y ≡ no ¬eq
+      ↦
+    ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y
+    ```
+
+### Deprecation of `_≺_` in `Data.Fin.Base`
+
+* In `Data.Fin.Base` the relation `_≺_` and its single constructor `_≻toℕ_`
+  have been deprecated in favour of their extensional equivalent `_<_`
+  but omitting the inversion principle which pattern matching on `_≻toℕ_`
+  would achieve; this instead is proxied by the property `Data.Fin.Properties.toℕ<`.
+
+* Consequently in `Data.Fin.Induction`:
+  ```
+  ≺-Rec
+  ≺-wellFounded
+  ≺-recBuilder
+  ≺-rec
+  ```
+  these functions are also deprecated.
+
+* Likewise in `Data.Fin.Properties` the proofs `≺⇒<′` and `<′⇒≺` have been deprecated
+  in favour of their proxy counterparts `<⇒<′` and `<′⇒<`.
+
+### Standardisation of `insertAt`/`updateAt`/`removeAt` in `Data.List`/`Data.Vec`
+
+* Previously, the names and argument order of index-based insertion, update and removal functions for
+  various types of lists and vectors were inconsistent.
+
+* To fix this the names have all been standardised to `insertAt`/`updateAt`/`removeAt`.
+
+* Correspondingly the following changes have occurred:
+
+* In `Data.List.Base` the following have been added:
+  ```agda
+  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
+  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
+  removeAt : (xs : List A) → Fin (length xs) → List A
+  ```
+  and the following has been deprecated
+  ```
+  _─_ ↦ removeAt
+  ```
+
+* In `Data.Vec.Base`:
+  ```agda
+  insert ↦ insertAt
+  remove ↦ removeAt
+
+  updateAt : Fin n → (A → A) → Vec A n → Vec A n
+    ↦
+  updateAt : Vec A n → Fin n → (A → A) → Vec A n
+  ```
+
+* In `Data.Vec.Functional`:
+  ```agda
+  remove : Fin (suc n) → Vector A (suc n) → Vector A n
+    ↦
+  removeAt : Vector A (suc n) → Fin (suc n) → Vector A n
+
+  updateAt : Fin n → (A → A) → Vector A n → Vector A n
+    ↦
+  updateAt : Vector A n → Fin n → (A → A) → Vector A n
+  ```
+
+* The old names (and the names of all proofs about these functions) have been deprecated appropriately.
+
+#### Standardisation of `lookup` in `Data.(List/Vec/...)`
+
+* All the types of `lookup` functions (and variants) in the following modules
+  have been changed to match the argument convention adopted in the `List` module (i.e.
+  `lookup` takes its container first and the index, whose type may depend on the
+  container value, second):
+  ```
+  Codata.Guarded.Stream
+  Codata.Guarded.Stream.Relation.Binary.Pointwise
+  Codata.Musical.Colist.Base
+  Codata.Musical.Colist.Relation.Unary.Any.Properties
+  Codata.Musical.Covec
+  Codata.Musical.Stream
+  Codata.Sized.Colist
+  Codata.Sized.Covec
+  Codata.Sized.Stream
+  Data.Vec.Relation.Unary.All
+  Data.Star.Environment
+  Data.Star.Pointer
+  Data.Star.Vec
+  Data.Trie
+  Data.Trie.NonEmpty
+  Data.Tree.AVL
+  Data.Tree.AVL.Indexed
+  Data.Tree.AVL.Map
+  Data.Tree.AVL.NonEmpty
+  Data.Vec.Recursive
+  ```
+
+* To accommodate this in in `Data.Vec.Relation.Unary.Linked.Properties`
+  and `Codata.Guarded.Stream.Relation.Binary.Pointwise`, the proofs
+  called `lookup` have been renamed `lookup⁺`.
+
+#### Changes to `Data.(Nat/Integer/Rational)` proofs of `NonZero`/`Positive`/`Negative` to use instance arguments
+
+* Many numeric operations in the library require their arguments to be non-zero,
+  and various proofs require their arguments to be non-zero/positive/negative etc.
+  As discussed on the [mailing list](https://lists.chalmers.se/pipermail/agda/2021/012693.html),
+  the previous way of constructing and passing round these proofs was extremely
+  clunky and lead to messy and difficult to read code.
+
+* We have therefore changed every occurrence where we need a proof of
+  non-zeroness/positivity/etc. to take it as an irrelevant
+  [instance argument](https://agda.readthedocs.io/en/latest/language/instance-arguments.html).
+  See the mailing list discussion for a fuller explanation of the motivation and implementation.
+
+* For example, whereas the type of division over `ℕ` used to be:
+  ```agda
+  _/_ : (dividend divisor : ℕ) {≢0 : False (divisor ≟ 0)} → ℕ
+  ```
+  it is now:
+  ```agda
+  _/_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ
+  ```
+
+* This means that as long as an instance of `NonZero n` is in scope then you can write
+  `m / n` without having to explicitly provide a proof, as instance search will fill it in
+  for you. The full list of such operations changed is as follows:
+    - In `Data.Nat.DivMod`: `_/_`, `_%_`, `_div_`, `_mod_`
+    - In `Data.Nat.Pseudorandom.LCG`: `Generator`
+    - In `Data.Integer.DivMod`: `_divℕ_`, `_div_`, `_modℕ_`, `_mod_`
+    - In `Data.Rational`: `mkℚ+`, `normalize`, `_/_`, `1/_`
+    - In `Data.Rational.Unnormalised`: `_/_`, `1/_`, `_÷_`
+
+* At the moment, there are 4 different ways such instance arguments can be provided,
+  listed in order of convenience and clarity:
+
+  1. *Automatic basic instances* - the standard library provides instances based on
+         the constructors of each numeric type in `Data.X.Base`. For example,
+         `Data.Nat.Base` constrains an instance of `NonZero (suc n)` for any `n`
+         and `Data.Integer.Base` contains an instance of `NonNegative (+ n)` for any `n`.
+         Consequently, if the argument is of the required form, these instances will always
+         be filled in by instance search automatically, e.g.
+        ```agda
+        0/n≡0 : 0 / suc n ≡ 0
+        ```
+
+  2. *Add an instance argument parameter* - You can provide the instance argument as
+         a parameter to your function and Agda's instance search will automatically use it
+         in the correct place without you having to explicitly pass it, e.g.
+    ```agda
+    0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
+    ```
+
+  3. *Define the instance locally* - You can define an instance argument in scope
+         (e.g. in a `where` clause) and Agda's instance search will again find it automatically,
+         e.g.
+        ```agda
+        instance
+          n≢0 : NonZero n
+          n≢0 = ...
+
+        0/n≡0 : 0 / n ≡ 0
+        ```
+
+  4. *Pass the instance argument explicitly* - Finally, if all else fails you can pass the
+         instance argument explicitly into the function using `{{ }}`, e.g.
+         ```
+         0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
+         ```
+
+* Suitable constructors for `NonZero`/`Positive` etc. can be found in `Data.X.Base`.
+
+* A full list of proofs that have changed to use instance arguments is available
+  at the end of this file. Notable changes to proofs are now discussed below.
+
+* Previously one of the hacks used in proofs was to explicitly refer to everything
+  in the correct form, e.g. if the argument `n` had to be non-zero then you would
+  refer to the argument as `suc n` everywhere instead of `n`, e.g.
+  ```
+  n/n≡1 : ∀ n → suc n / suc n ≡ 1
+  ```
+  This made the proofs extremely difficult to use if your term wasn't in the form `suc n`.
+  After being updated to use instance arguments instead, the proof above becomes:
+  ```
+  n/n≡1 : ∀ n {{_ : NonZero n}} → n / n ≡ 1
+  ```
+  However, note that this means that if you passed in the value `x` to these proofs
+  before, then you will now have to pass in `suc x`. The proofs for which the
+  arguments have changed form in this way are highlighted in the list at the bottom
+  of the file.
+
+* Finally, in `Data.Rational.Unnormalised.Base` the definition of `_≢0` is now
+  redundant and so has been removed. Additionally the following proofs about it have
+  also been removed from `Data.Rational.Unnormalised.Properties`:
+  ```
+  p≄0⇒∣↥p∣≢0 : ∀ p → p ≠ 0ℚᵘ → ℤ.∣ (↥ p) ∣ ≢0
+  ∣↥p∣≢0⇒p≄0 : ∀ p → ℤ.∣ (↥ p) ∣ ≢0 → p ≠ 0ℚᵘ
+  ```
+
+### Changes to the definition of `_≤″_` in `Data.Nat.Base` (issue #1919)
+
+* The definition of `_≤″_` was previously:
+  ```agda
+  record _≤″_ (m n : ℕ) : Set where
+    constructor less-than-or-equal
+    field
+      {k}   : ℕ
+      proof : m + k ≡ n
+  ```
+  which introduced a spurious additional definition, when this is in fact, modulo
+  field names and implicit/explicit qualifiers, equivalent to the definition of left-
+  divisibility, `_∣ˡ_` for the `RawMagma` structure of `_+_`.
+
+* Since the addition of raw bundles to `Data.X.Base`, this definition can now be
+  made directly. Accordingly, the definition has been changed to:
+  ```agda
+  _≤″_ : (m n : ℕ)  → Set
+  _≤″_ = _∣ˡ_ +-rawMagma
+
+  pattern less-than-or-equal {k} prf = k , prf
+  ```
+
+* Knock-on consequences include the need to retain the old constructor
+  name, now introduced as a pattern synonym, and introduction of (a function
+  equivalent to) the former field name/projection function `proof` as
+  `≤″-proof` in `Data.Nat.Properties`.
+
+### Changes to definition of `Prime` in `Data.Nat.Primality`
+
+* The definition of `Prime` was:
+  ```agda
+  Prime 0             = ⊥
+  Prime 1             = ⊥
+  Prime (suc (suc n)) = (i : Fin n) → 2 + toℕ i ∤ 2 + n
+  ```
+  which was very hard to reason about as, not only did it involve conversion
+  to and from the `Fin` type, it also required that the divisor was of the form
+  `2 + toℕ i`, which has exactly the same problem as the `suc n` hack described
+  above used for non-zeroness.
+
+* To make it easier to use, reason about and read, the definition has been
+  changed to:
+  ```agda
+  record Prime (p : ℕ) : Set where
+    constructor prime
+    field
+      .{{nontrivial}} : NonTrivial p
+      notComposite    : ¬ Composite p
+  ```
+  where `Composite` is now defined as the diagonal of the new relation
+  `_HasNonTrivialDivisorLessThan_` in `Data.Nat.Divisibility.Core`.
+
+### Changes to operation reduction behaviour in `Data.Rational(.Unnormalised)`
+
+* Currently arithmetic expressions involving rationals (both normalised and
+  unnormalised) undergo disastrous exponential normalisation. For example,
+  `p + q` would often be normalised by Agda to
+  `(↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)`. While the normalised form
+  of `p + q + r + s + t + u + v` would be ~700 lines long. This behaviour
+  often chokes both type-checking and the display of the expressions in the IDE.
+
+* To avoid this expansion and make non-trivial reasoning about rationals actually feasible:
+  1. the records `ℚᵘ` and `ℚ` have both had the `no-eta-equality` flag enabled
+  2. definition of arithmetic operations have trivial pattern matching added to
+     prevent them reducing, e.g.
+     ```agda
+     p + q = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)
+     ```
+     has been changed to
+     ```
+     p@record{} + q@record{} = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)
+     ```
+
+* As a consequence of this, some proofs that relied either on this reduction behaviour
+  or on eta-equality may no longer type-check.
+
+* There are several ways to fix this:
+
+  1. The principled way is to not rely on this reduction behaviour in the first place.
+     The `Properties` files for rational numbers have been greatly expanded in `v1.7`
+     and `v2.0`, and we believe most proofs should be able to be built up from existing
+     proofs contained within these files.
+
+  2. Alternatively, annotating any rational arguments to a proof with either
+     `@record{}` or `@(mkℚ _ _ _)` should restore the old reduction behaviour for any
+     terms involving those parameters.
+
+  3. Finally, if the above approaches are not viable then you may be forced to explicitly
+     use `cong` combined with a lemma that proves the old reduction behaviour.
+
+* Similarly, in order to prevent reduction, the equality `_≃_` in `Data.Rational.Base`
+  has been made into a data type with the single constructor `*≡*`. The destructor
+  `drop-*≡*` has been added to `Data.Rational.Properties`.
+
+#### Deprecation of old `Function` hierarchy
+
+* The new `Function` hierarchy was introduced in `v1.2` which follows
+  the same `Core`/`Definitions`/`Bundles`/`Structures` as all the other
+  hierarchies in the library.
+
+* At the time the old function hierarchy in:
+  ```
+  Function.Equality
+  Function.Injection
+  Function.Surjection
+  Function.Bijection
+  Function.LeftInverse
+  Function.Inverse
+  Function.HalfAdjointEquivalence
+  Function.Related
+  ```
+  was unofficially deprecated, but could not be officially deprecated because
+  of it's widespread use in the rest of the library.
+
+* Now, the old hierarchy modules have all been officially deprecated. All
+  uses of them in the rest of the library have been switched to use the
+  new hierarchy.
+
+* The latter is unfortunately a relatively big breaking change, but was judged
+  to be unavoidable given how widely used the old hierarchy was.
+
+#### Changes to the new `Function` hierarchy
+
+* In `Function.Bundles` the names of the fields in the records have been
+  changed from `f`, `g`, `cong₁` and `cong₂` to `to`, `from`, `to-cong`, `from-cong`.
+
+* In `Function.Definitions`, the module no longer has two equalities as
+  module arguments, as they did not interact as intended with the re-exports
+  from `Function.Definitions.(Core1/Core2)`. The latter two modules have
+  been removed and their definitions folded into `Function.Definitions`.
+
+* In `Function.Definitions` the following definitions have been changed from:
+  ```
+  Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
+  Inverseˡ f g = ∀ y → f (g y) ≈₂ y
+  Inverseʳ f g = ∀ x → g (f x) ≈₁ x
+  ```
+  to:
+  ```
+  Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y
+  Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
+  Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
+  ```
+  This is for several reasons:
+          i) the new definitions compose much more easily,
+          ii) Agda can better infer the equalities used.
+
+* To ease backwards compatibility:
+   - the old definitions have been moved to the new names  `StrictlySurjective`,
+         `StrictlyInverseˡ` and `StrictlyInverseʳ`.
+   - The records in  `Function.Structures` and `Function.Bundles` export proofs
+         of these under the names `strictlySurjective`, `strictlyInverseˡ` and
+         `strictlyInverseʳ`,
+   - Conversion functions for the definitions have been added in both directions 
+		 to `Function.Consequences(.Propositional/Setoid)`.
+   - Conversion functions for structures have been added in 
+		 `Function.Structures.Biased`.
+
+### New `Function.Strict`
+
+* The module `Strict` has been deprecated in favour of `Function.Strict`
+  and the definitions of strict application, `_$!_` and `_$!′_`, have been
+  moved from `Function.Base` to `Function.Strict`.
+
+* The contents of `Function.Strict` is now re-exported by `Function`.
+
+### Change to the definition of `WfRec` in `Induction.WellFounded` (issue #2083)
+
+* Previously, the definition of `WfRec` was:
+  ```agda
+  WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
+  WfRec _<_ P x = ∀ y → y < x → P y
+  ```
+  which meant that all arguments involving accessibility and wellfoundedness proofs
+  were polluted by almost-always-inferrable explicit arguments for the `y` position.
+
+* The definition has now been changed to make that argument *implicit*, as
+  ```agda
+  WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
+  WfRec _<_ P x = ∀ {y} → y < x → P y
+  ```
+
+### Reorganisation of `Reflection` modules
+
+* Under the `Reflection` module, there were various impending name clashes
+  between the core AST as exposed by Agda and the annotated AST defined in
+  the library.
+
+* While the content of the modules remain the same, the modules themselves
+  have therefore been renamed as follows:
+  ```
+  Reflection.Annotated              ↦ Reflection.AnnotatedAST
+  Reflection.Annotated.Free         ↦ Reflection.AnnotatedAST.Free
+
+  Reflection.Abstraction            ↦ Reflection.AST.Abstraction
+  Reflection.Argument               ↦ Reflection.AST.Argument
+  Reflection.Argument.Information   ↦ Reflection.AST.Argument.Information
+  Reflection.Argument.Quantity      ↦ Reflection.AST.Argument.Quantity
+  Reflection.Argument.Relevance     ↦ Reflection.AST.Argument.Relevance
+  Reflection.Argument.Modality      ↦ Reflection.AST.Argument.Modality
+  Reflection.Argument.Visibility    ↦ Reflection.AST.Argument.Visibility
+  Reflection.DeBruijn               ↦ Reflection.AST.DeBruijn
+  Reflection.Definition             ↦ Reflection.AST.Definition
+  Reflection.Instances              ↦ Reflection.AST.Instances
+  Reflection.Literal                ↦ Reflection.AST.Literal
+  Reflection.Meta                   ↦ Reflection.AST.Meta
+  Reflection.Name                   ↦ Reflection.AST.Name
+  Reflection.Pattern                ↦ Reflection.AST.Pattern
+  Reflection.Show                   ↦ Reflection.AST.Show
+  Reflection.Traversal              ↦ Reflection.AST.Traversal
+  Reflection.Universe               ↦ Reflection.AST.Universe
+
+  Reflection.TypeChecking.Monad             ↦ Reflection.TCM
+  Reflection.TypeChecking.Monad.Categorical ↦ Reflection.TCM.Categorical
+  Reflection.TypeChecking.Monad.Format      ↦ Reflection.TCM.Format
+  Reflection.TypeChecking.Monad.Syntax      ↦ Reflection.TCM.Instances
+  Reflection.TypeChecking.Monad.Instances   ↦ Reflection.TCM.Syntax
+  ```
+
+* A new module `Reflection.AST` that re-exports the contents of the
+  submodules has been added.
+
+
+### Reorganisation of the `Relation.Nullary` hierarchy
+
+* It was very difficult to use the `Relation.Nullary` modules, as
+  `Relation.Nullary` contained the basic definitions of negation, decidability etc.,
+  and the operations and proofs about these definitions were spread over
+  `Relation.Nullary.(Negation/Product/Sum/Implication etc.)`.
+
+* To fix this all the contents of the latter is now exported by `Relation.Nullary`.
+
+* In order to achieve this the following backwards compatible changes have been made:
+
+  1. the definition of `Dec` and `recompute` have been moved to `Relation.Nullary.Decidable.Core`
+
+  2. the definition of `Reflects` has been moved to `Relation.Nullary.Reflects`
+
+  3. the definition of `¬_` has been moved to `Relation.Nullary.Negation.Core`
+
+  4. The modules `Relation.Nullary.(Product/Sum/Implication)` have been deprecated
+         and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`.
+		 
+  5. The proof `T?` has been moved from `Data.Bool.Properties` to `Relation.Nullary.Decidable.Core`
+	 (but is still re-exported by the former).
+
+  as well as the following breaking changes:
+
+  1. `¬?` has been moved from `Relation.Nullary.Negation.Core` to
+         `Relation.Nullary.Decidable.Core`
+
+  2. `¬-reflects` has been moved from `Relation.Nullary.Negation.Core` to
+         `Relation.Nullary.Reflects`.
+
+  3. `decidable-stable`, `excluded-middle` and `¬-drop-Dec` have been moved
+         from `Relation.Nullary.Negation` to `Relation.Nullary.Decidable`.
+
+  4. `fromDec` and `toDec` have been moved from `Data.Sum.Base` to `Data.Sum`.
+
+
+### (Issue #2096) Introduction of flipped and negated relation symbols to bundles in `Relation.Binary.Bundles`
+
+* Previously, bundles such as `Preorder`, `Poset`, `TotalOrder` etc. did not have the flipped
+  and negated versions of the operators exposed. In some cases they could obtained by opening the
+  relevant `Relation.Binary.Properties.X` file but usually they had to be redefined every time.
+
+* To fix this, these bundles now all export all 4 versions of the operator: normal, converse, negated,
+  converse-negated. Accordingly they are no longer exported from the corresponding `Properties` file.
+
+* To make this work for `Preorder`, it was necessary to change the name of the relation symbol.
+  Previously, the symbol was `_∼_`  which is (notationally) symmetric, so that its
+  converse relation could only be discussed *semantically* in terms of `flip _∼_`.
+
+* Now, the `Preorder` record field `_∼_` has been renamed to `_≲_`, with `_≳_`
+  introduced as a definition in `Relation.Binary.Bundles.Preorder`.
+  Partial backwards compatible has been achieved by redeclaring a deprecated version
+  of the old symbol in the record. Therefore, only _declarations_ of `PartialOrder` records will
+  need their field names updating.
+
+
+### Changes to definition of `IsStrictTotalOrder` in `Relation.Binary.Structures`
+
+* The previous definition of the record `IsStrictTotalOrder` did not
+  build upon `IsStrictPartialOrder` as would be expected.
+  Instead it omitted several fields like irreflexivity as they were derivable from the
+  proof of trichotomy. However, this led to problems further up the hierarchy where
+  bundles such as `StrictTotalOrder` which contained multiple distinct proofs of
+  `IsStrictPartialOrder`.
+
+* To remedy this the definition of `IsStrictTotalOrder` has been changed to so
+  that it builds upon `IsStrictPartialOrder` as would be expected.
+
+* To aid migration, the old record definition has been moved to
+  `Relation.Binary.Structures.Biased` which contains the `isStrictTotalOrderᶜ`
+  smart constructor (which is re-exported by `Relation.Binary`) . Therefore the old code:
+  ```agda
+  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
+  <-isStrictTotalOrder = record
+        { isEquivalence = isEquivalence
+        ; trans         = <-trans
+        ; compare       = <-cmp
+        }
+  ```
+  can be migrated either by updating to the new record fields if you have a proof of `IsStrictPartialOrder`
+  available:
+  ```agda
+  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
+  <-isStrictTotalOrder = record
+        { isStrictPartialOrder = <-isStrictPartialOrder
+        ; compare              = <-cmp
+        }
+  ```
+  or simply applying the smart constructor to the record definition as follows:
+  ```agda
+  <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
+  <-isStrictTotalOrder = isStrictTotalOrderᶜ record
+        { isEquivalence = isEquivalence
+        ; trans         = <-trans
+        ; compare       = <-cmp
+        }
+  ```
+
+### Changes to the interface of `Relation.Binary.Reasoning.Triple`
+
+* The module `Relation.Binary.Reasoning.Base.Triple` now takes an extra proof
+  that the strict relation is irreflexive.
+
+* This allows the addition of the following new proof combinator:
+  ```agda
+  begin-contradiction : (r : x IsRelatedTo x) → {s : True (IsStrict? r)} → A
+  ```
+  that takes a proof that a value is strictly less than itself and then applies the
+  principle of explosion to derive anything.
+
+* Specialised versions of this combinator are available in the `Reasoning` modules
+  exported by the following modules:
+  ```
+  Data.Nat.Properties
+  Data.Nat.Binary.Properties
+  Data.Integer.Properties
+  Data.Rational.Unnormalised.Properties
+  Data.Rational.Properties
+  Data.Vec.Relation.Binary.Lex.Strict
+  Data.Vec.Relation.Binary.Lex.NonStrict
+  Relation.Binary.Reasoning.StrictPartialOrder
+  Relation.Binary.Reasoning.PartialOrder
+  ```
+
+### A more modular design for `Relation.Binary.Reasoning`
+
+* Previously, every `Reasoning` module in the library tended to roll it's own set
+  of syntax for the combinators. This hindered consistency and maintainability.
+
+* To improve the situation, a new module `Relation.Binary.Reasoning.Syntax`
+  has been introduced which exports a wide range of sub-modules containing
+  pre-existing reasoning combinator syntax.
+
+* This makes it possible to add new or rename existing reasoning combinators to a
+  pre-existing `Reasoning` module in just a couple of lines
+  (e.g. see `∣-Reasoning` in `Data.Nat.Divisibility`)
+
+* One pre-requisite for that is that `≡-Reasoning` has been moved from
+  `Relation.Binary.PropositionalEquality.Core` (which shouldn't be
+  imported anyway as it's a `Core` module) to
+  `Relation.Binary.PropositionalEquality.Properties`.
+  It is still exported by `Relation.Binary.PropositionalEquality`.
+
+### Renaming of symmetric combinators in `Reasoning` modules
+
+* We've had various complaints about the symmetric version of reasoning combinators
+  that use the syntax `_R˘⟨_⟩_` for some relation `R`, (e.g. `_≡˘⟨_⟩_` and `_≃˘⟨_⟩_`)
+  introduced in `v1.0`. In particular:
+  1. The symbol `˘` is hard to type.
+  2. The symbol `˘` doesn't convey the direction of the equality
+  3. The left brackets aren't vertically aligned with the left brackets of the non-symmetric version.
+
+* To address these problems we have renamed all the symmetric versions of the
+  combinators from `_R˘⟨_⟩_` to `_R⟨_⟨_` (the direction of the last bracket is flipped
+  to indicate the quality goes from right to left).
+
+* The old combinators still exist but have been deprecated. However due to
+  [Agda issue #5617](https://github.com/agda/agda/issues/5617), the deprecation warnings
+  don't fire correctly. We will not remove the old combinators before the above issue is
+  addressed. However, we still encourage migrating to the new combinators!
+
+* On a Linux-based system, the following command was used to globally migrate all uses of the
+  old combinators to the new ones in the standard library itself.
+  It *may* be useful when trying to migrate your own code:
+  ```bash
+  find . -type f -name "*.agda" -print0 | xargs -0 sed -i -e 's/˘⟨\(.*\)⟩/⟨\1⟨/g'
+  ```
+  USUAL CAVEATS: It has not been widely tested and the standard library developers are not
+  responsible for the results of this command. It is strongly recommended you back up your
+  work before you attempt to run it.
+
+* NOTE: this refactoring may require some extra bracketing around the operator `_⟨_⟩_` from
+  `Function.Base` if the `_⟨_⟩_` operator is used within the reasoning proof. The symptom
+  for this is a `Don't know how to parse` error.
+
+### Improvements to `Text.Pretty` and `Text.Regex`
+
+* In `Text.Pretty`, `Doc` is now a record rather than a type alias. This
+  helps Agda reconstruct the `width` parameter when the module is opened
+  without it being applied. In particular this allows users to write
+  width-generic pretty printers and only pick a width when calling the
+  renderer by using this import style:
+  ```
+  open import Text.Pretty using (Doc; render)
+  --                      ^-- no width parameter for Doc & render
+  open module Pretty {w} = Text.Pretty w hiding (Doc; render)
+  --                 ^-- implicit width parameter for the combinators
+
+  pretty : Doc w
+  pretty = ? -- you can use the combinators here and there won't be any
+             -- issues related to the fact that `w` cannot be reconstructed
+             -- anymore
+
+  main = do
+    -- you can now use the same pretty with different widths:
+    putStrLn $ render 40 pretty
+    putStrLn $ render 80 pretty
+  ```
+
+* In `Text.Regex.Search` the `searchND` function finding infix matches has
+  been tweaked to make sure the returned solution is a local maximum in terms
+  of length. It used to be that we would make the match start as late as
+  possible and finish as early as possible. It's now the reverse.
+
+  So `[a-zA-Z]+.agdai?` run on "the path _build/Main.agdai corresponds to"
+  will return "Main.agdai" when it used to be happy to just return "n.agda".
+
+### Other
+
+* In accordance with changes to the flags in Agda 2.6.3, all modules that previously used
+  the `--without-K` flag now use the `--cubical-compatible` flag instead.
+
+* In `Algebra.Core` the operations `Opₗ` and `Opᵣ` have moved to `Algebra.Module.Core`.
+
+* In `Algebra.Definitions.RawMagma.Divisibility` the definitions for `_∣ˡ_` and `_∣ʳ_`
+  have been changed from being defined as raw products to being defined as records. However,
+  the record constructors are called `_,_` so the changes required are minimal.
+
+* In `Codata.Guarded.Stream` the following functions have been modified to have simpler definitions:
+  * `cycle`
+  * `interleave⁺`
+  * `cantor`
+  Furthermore, the direction of interleaving of `cantor` has changed. Precisely,
+  suppose `pair` is the cantor pairing function, then `lookup (pair i j) (cantor xss)`
+  according to the old definition corresponds to `lookup (pair j i) (cantor xss)`
+  according to the new definition.
+  For a concrete example see the one included at the end of the module.
+
+* In `Data.Fin.Base` the relations `_≤_` `_≥_` `_<_` `_>_`  have been
+  generalised so they can now relate `Fin` terms with different indices.
+  Should be mostly backwards compatible, but very occasionally when proving
+  properties about the orderings themselves the second index must be provided
+  explicitly.
+
+* In `Data.Fin.Properties` the proof `inj⇒≟` that an injection from a type
+  `A` into `Fin n` induces a decision procedure for `_≡_` on `A` has been
+  generalised to other equivalences over `A` (i.e. to arbitrary setoids), and
+  renamed from `eq?` to the more descriptive  and `inj⇒decSetoid`.
+
+* In `Data.Fin.Properties` the proof `pigeonhole` has been strengthened
+  so that the a proof that `i < j` rather than a mere `i ≢ j` is returned.
+
+* In `Data.Fin.Substitution.TermSubst` the fixity of `_/Var_` has been changed
+  from `infix 8` to `infixl 8`.
+
+* In `Data.Integer.DivMod` the previous implementations of
+  `_divℕ_`, `_div_`, `_modℕ_`, `_mod_` internally converted to the unary
+  `Fin` datatype resulting in poor performance. The implementation has been
+  updated to use the corresponding operations from `Data.Nat.DivMod` which are
+  efficiently implemented using the Haskell backend.
+
+* In `Data.Integer.Properties` the first two arguments of `m≡n⇒m-n≡0`
+  (now renamed `i≡j⇒i-j≡0`) have been made implicit.
+
+* In `Data.(List|Vec).Relation.Binary.Lex.Strict` the argument `xs` in
+  `xs≮[]` in introduced in PRs #1648 and #1672 has now been made implicit.
+
+* In `Data.List.NonEmpty` the functions `split`, `flatten` and `flatten-split`
+  have been removed from. In their place `groupSeqs` and `ungroupSeqs`
+  have been added to `Data.List.NonEmpty.Base` which morally perform the same
+  operations but without computing the accompanying proofs. The proofs can be
+  found in `Data.List.NonEmpty.Properties` under the names `groupSeqs-groups`
+  and `ungroupSeqs` and `groupSeqs`.
+
+* In `Data.List.Relation.Unary.Grouped.Properties` the proofs `map⁺` and `map⁻`
+  have had their preconditions weakened so the equivalences no longer require congruence
+  proofs.
+
+* In `Data.Nat.Divisibility` the proof `m/n/o≡m/[n*o]` has been removed. In it's
+  place a new more general proof `m/n/o≡m/[n*o]` has been added to `Data.Nat.DivMod`
+  that doesn't require the `n * o ∣ m` pre-condition.
+
+* In `Data.Product.Relation.Binary.Lex.Strict` the proof of wellfoundedness
+  of the lexicographic ordering on products, no longer
+  requires the assumption of symmetry for the first equality relation `_≈₁_`.
+
+* In `Data.Rational.Base` the constructors `+0` and `+[1+_]` from `Data.Integer.Base`
+  are no longer re-exported by `Data.Rational.Base`. You will have to open `Data.Integer(.Base)`
+  directly to use them.
+
+* In `Data.Rational(.Unnormalised).Properties` the types of the proofs
+  `pos⇒1/pos`/`1/pos⇒pos` and `neg⇒1/neg`/`1/neg⇒neg` have been switched,
+  as the previous naming scheme didn't correctly generalise to e.g. `pos+pos⇒pos`.
+  For example the types of `pos⇒1/pos`/`1/pos⇒pos` were:
+  ```agda
+  pos⇒1/pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
+  1/pos⇒pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
+  ```
+  but are now:
+  ```agda
+  pos⇒1/pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
+  1/pos⇒pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
+  ```
+
+* In `Data.Sum.Base` the definitions `fromDec` and `toDec` have been moved to `Data.Sum`.
+
+* In `Data.Vec.Base` the definition of `_>>=_` under `Data.Vec.Base` has been
+  moved to the submodule `CartesianBind` in order to avoid clashing with the
+  new, alternative definition of `_>>=_`, located in the second new submodule
+  `DiagonalBind`.
+
+* In `Data.Vec.Base` the definitions `init` and `last` have been changed from the `initLast`
+  view-derived implementation to direct recursive definitions.
+
+* In `Data.Vec.Properties` the type of the proof `zipWith-comm` has been generalised from:
+  ```agda
+  zipWith-comm : ∀ {f : A → A → B} (comm : ∀ x y → f x y ≡ f y x) (xs ys : Vec A n) →
+                 zipWith f xs ys ≡ zipWith f ys xs
+  ```
+  to
+  ```agda
+  zipWith-comm : ∀ {f : A → B → C} {g : B → A → C}  (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys →
+                 zipWith f xs ys ≡ zipWith g ys xs
+  ```
+
+* In `Data.Vec.Relation.Unary.All` the functions `lookup` and `tabulate` have
+  been moved to `Data.Vec.Relation.Unary.All.Properties` and renamed
+  `lookup⁺` and `lookup⁻` respectively.
+
+* In `Data.Vec.Base` and `Data.Vec.Functional` the functions `iterate` and `replicate`
+  now take the length of vector, `n`, as an explicit rather than an implicit argument, i.e.
+  the new types are:
+  ```agda
+  iterate : (A → A) → A → ∀ n → Vec A n
+  replicate : ∀ n → A → Vec A n
+  ```
+
+* In `Relation.Binary.Construct.Closure.Symmetric` the operation
+  `SymClosure` on relations in has been reimplemented
+  as a data type `SymClosure _⟶_ a b` that is parameterized by the
+  input relation `_⟶_` (as well as the elements `a` and `b` of the
+  domain) so that `_⟶_` can be inferred, which it could not from the
+  previous implementation using the sum type `a ⟶ b ⊎ b ⟶ a`.
+
+* In `Relation.Nullary.Decidable.Core` the name `excluded-middle` has been
+  renamed to `¬¬-excluded-middle`.
+
+
+Other major improvements
+------------------------
+
+### Improvements to ring solver tactic
+
+* The ring solver tactic has been greatly improved. In particular:
+
+  1. When the solver is used for concrete ring types, e.g. ℤ, the equality can now use
+     all the ring operations defined natively for that type, rather than having
+     to use the operations defined in `AlmostCommutativeRing`. For example
+     previously you could not use `Data.Integer.Base._*_` but instead had to
+     use `AlmostCommutativeRing._*_`.
+
+  2. The solver now supports use of the subtraction operator `_-_` whenever
+     it is defined immediately in terms of `_+_` and `-_`. This is the case for
+     `Data.Integer` and `Data.Rational`.
+
+### Moved `_%_` and `_/_` operators to `Data.Nat.Base`
+
+* Previously the division and modulus operators were defined in `Data.Nat.DivMod`
+  which in turn meant that using them required importing `Data.Nat.Properties`
+  which is a very heavy dependency.
+
+* To fix this, these operators have been moved to `Data.Nat.Base`. The properties
+  for them still live in `Data.Nat.DivMod` (which also publicly re-exports them
+  to provide backwards compatibility).
+
+* Beneficiaries of this change include `Data.Rational.Unnormalised.Base` whose
+  dependencies are now significantly smaller.
+
+### Moved raw bundles from `Data.X.Properties` to `Data.X.Base`
+
+* Raw bundles by design don't contain any proofs so should in theory be able to live
+  in `Data.X.Base` rather than `Data.X.Bundles`.
+
+* To achieve this while keeping the dependency footprint small, the definitions of
+  raw bundles (`RawMagma`, `RawMonoid` etc.) have been moved from `Algebra(.Lattice)Bundles` to
+  a new module `Algebra(.Lattice).Bundles.Raw` which can be imported at much lower cost
+  from `Data.X.Base`.
+
+* We have then moved raw bundles defined in `Data.X.Properties` to `Data.X.Base` for
+  `X` = `Nat`/`Nat.Binary`/`Integer`/`Rational`/`Rational.Unnormalised`.
+  
+### Upgrades to `README` sub-library
+
+* The `README` sub-library has been moved to `doc/README` and a new `doc/standard-library-doc.agda-lib` has been added.
+
+* The first consequence is that `README` files now can be type-checked in Emacs
+  using an out-of-the-box standard Agda installation without altering the main
+  `standard-library.agda-lib` file.
+
+* The second is that the `README` files are now their own first-class library 
+  and can be imported like an other library.
+
+Deprecated modules
+------------------
+
+### Moving `Data.Erased` to `Data.Irrelevant`
+
+* This fixes the fact we had picked the wrong name originally. The erased modality
+  corresponds to `@0` whereas the irrelevance one corresponds to `.`.
+
+### Deprecation of `Data.Fin.Substitution.Example`
+
+* The module `Data.Fin.Substitution.Example` has been deprecated, and moved to `README.Data.Fin.Substitution.UntypedLambda`
+
+### Deprecation of `Data.Nat.Properties.Core`
+
+* The module `Data.Nat.Properties.Core` has been deprecated, and its one lemma moved to `Data.Nat.Base`, renamed as `s≤s⁻¹`
+
+### Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`
+
+* This module has been deprecated, as none of its contents actually depended on axiom K.
+  The contents has been moved to `Data.Product.Function.Dependent.Setoid`.
+
+### Moving `Function.Related`
+
+* The module `Function.Related` has been deprecated in favour of `Function.Related.Propositional`
+  whose code uses the new function hierarchy. This also opens up the possibility of a more
+  general `Function.Related.Setoid` at a later date. Several of the names have been changed
+  in this process to bring them into line with the camelcase naming convention used
+  in the rest of the library:
+  ```agda
+  reverse-implication ↦ reverseImplication
+  reverse-injection   ↦ reverseInjection
+  left-inverse        ↦ leftInverse
+
+  Symmetric-kind      ↦ SymmetricKind
+  Forward-kind        ↦ ForwardKind
+  Backward-kind       ↦ BackwardKind
+  Equivalence-kind    ↦ EquivalenceKind
+  ```
+
+### Moving `Relation.Binary.Construct.(Converse/Flip)`
+
+* The following files have been moved:
+  ```agda
+  Relation.Binary.Construct.Converse ↦ Relation.Binary.Construct.Flip.EqAndOrd
+  Relation.Binary.Construct.Flip     ↦ Relation.Binary.Construct.Flip.Ord
+  ```
+
+### Moving `Relation.Binary.Properties.XLattice`
+
+* The following files have been moved:
+  ```agda
+  Relation.Binary.Properties.BoundedJoinSemilattice.agda  ↦ Relation.Binary.Lattice.Properties.BoundedJoinSemilattice.agda
+  Relation.Binary.Properties.BoundedLattice.agda          ↦ Relation.Binary.Lattice.Properties.BoundedLattice.agda
+  Relation.Binary.Properties.BoundedMeetSemilattice.agda  ↦ Relation.Binary.Lattice.Properties.BoundedMeetSemilattice.agda
+  Relation.Binary.Properties.DistributiveLattice.agda     ↦ Relation.Binary.Lattice.Properties.DistributiveLattice.agda
+  Relation.Binary.Properties.HeytingAlgebra.agda         ↦ Relation.Binary.Lattice.Properties.HeytingAlgebra.agda
+  Relation.Binary.Properties.JoinSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.JoinSemilattice.agda
+  Relation.Binary.Properties.Lattice.agda                 ↦ Relation.Binary.Lattice.Properties.Lattice.agda
+  Relation.Binary.Properties.MeetSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.MeetSemilattice.agda
+  ```
+
+Deprecated names
+----------------
+
+* In `Algebra.Consequences.Propositional`:
+  ```agda
+  comm+assoc⇒middleFour             ↦  comm∧assoc⇒middleFour
+  identity+middleFour⇒assoc         ↦  identity∧middleFour⇒assoc
+  identity+middleFour⇒comm          ↦  identity∧middleFour⇒comm
+  comm+distrˡ⇒distrʳ                ↦  comm∧distrˡ⇒distrʳ
+  comm+distrʳ⇒distrˡ                ↦  comm∧distrʳ⇒distrˡ
+  assoc+distribʳ+idʳ+invʳ⇒zeˡ       ↦  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
+  assoc+distribˡ+idʳ+invʳ⇒zeʳ       ↦  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
+  assoc+id+invʳ⇒invˡ-unique         ↦  assoc∧id∧invʳ⇒invˡ-unique
+  assoc+id+invˡ⇒invʳ-unique         ↦  assoc∧id∧invˡ⇒invʳ-unique
+  subst+comm⇒sym                    ↦  subst∧comm⇒sym
+  ```
+
+* In `Algebra.Consequences.Setoid`:
+  ```agda
+  comm+assoc⇒middleFour             ↦  comm∧assoc⇒middleFour
+  identity+middleFour⇒assoc         ↦  identity∧middleFour⇒assoc
+  identity+middleFour⇒comm          ↦  identity∧middleFour⇒comm
+  comm+cancelˡ⇒cancelʳ              ↦  comm∧cancelˡ⇒cancelʳ
+  comm+cancelʳ⇒cancelˡ              ↦  comm∧cancelʳ⇒cancelˡ
+  comm+almostCancelˡ⇒almostCancelʳ  ↦  comm∧almostCancelˡ⇒almostCancelʳ
+  comm+almostCancelʳ⇒almostCancelˡ  ↦  comm∧almostCancelʳ⇒almostCancelˡ
+  comm+idˡ⇒idʳ                      ↦  comm∧idˡ⇒idʳ
+  comm+idʳ⇒idˡ                      ↦  comm∧idʳ⇒idˡ
+  comm+zeˡ⇒zeʳ                      ↦  comm∧zeˡ⇒zeʳ
+  comm+zeʳ⇒zeˡ                      ↦  comm∧zeʳ⇒zeˡ
+  comm+invˡ⇒invʳ                    ↦  comm∧invˡ⇒invʳ
+  comm+invʳ⇒invˡ                    ↦  comm∧invʳ⇒invˡ
+  comm+distrˡ⇒distrʳ                ↦  comm∧distrˡ⇒distrʳ
+  comm+distrʳ⇒distrˡ                ↦  comm∧distrʳ⇒distrˡ
+  assoc+distribʳ+idʳ+invʳ⇒zeˡ       ↦  assoc∧distribʳ∧idʳ∧invʳ⇒zeˡ
+  assoc+distribˡ+idʳ+invʳ⇒zeʳ       ↦  assoc∧distribˡ∧idʳ∧invʳ⇒zeʳ
+  assoc+id+invʳ⇒invˡ-unique         ↦  assoc∧id∧invʳ⇒invˡ-unique
+  assoc+id+invˡ⇒invʳ-unique         ↦  assoc∧id∧invˡ⇒invʳ-unique
+  subst+comm⇒sym                    ↦  subst∧comm⇒sym
+  ```
+
+* In `Algebra.Construct.Zero`:
+  ```agda
+  rawMagma   ↦  Algebra.Construct.Terminal.rawMagma
+  magma      ↦  Algebra.Construct.Terminal.magma
+  semigroup  ↦  Algebra.Construct.Terminal.semigroup
+  band       ↦  Algebra.Construct.Terminal.band
+  ```
+
+* In `Codata.Guarded.Stream.Properties`:
+  ```agda
+  map-identity  ↦  map-id
+  map-fusion    ↦  map-∘
+  drop-fusion   ↦  drop-drop
+  ```
+
+* In `Codata.Sized.Colist.Properties`:
+  ```agda
+  map-identity      ↦  map-id
+  map-map-fusion    ↦  map-∘
+  drop-drop-fusion  ↦  drop-drop
+  ```
+
+* In `Codata.Sized.Covec.Properties`:
+  ```agda
+  map-identity   ↦  map-id
+  map-map-fusion  ↦  map-∘
+  ```
+
+* In `Codata.Sized.Delay.Properties`:
+  ```agda
+  map-identity      ↦  map-id
+  map-map-fusion    ↦  map-∘
+  map-unfold-fusion  ↦  map-unfold
+  ```
+
+* In `Codata.Sized.M.Properties`:
+  ```agda
+  map-compose  ↦  map-∘
+  ```
+
+* In `Codata.Sized.Stream.Properties`:
+  ```agda
+  map-identity   ↦  map-id
+  map-map-fusion  ↦  map-∘
+  ```
+
+* In `Data.Container.Related`:
+  ```agda
+  _∼[_]_  ↦  _≲[_]_
+  ```
+
+* In `Data.Bool.Properties` (Issue #2046):
+  ```agda
+  push-function-into-if ↦ if-float
+  ```
+
+* In `Data.Fin.Base`: two new, hopefully more memorable, names `↑ˡ` `↑ʳ`
+  for the 'left', resp. 'right' injection of a Fin m into a 'larger' type,
+  `Fin (m + n)`, resp. `Fin (n + m)`, with argument order to reflect the
+  position of the `Fin m` argument.
+  ```
+  inject+  ↦  flip _↑ˡ_
+  raise    ↦  _↑ʳ_
+  ```
+
+* In `Data.Fin.Properties`:
+  ```
+  toℕ-raise        ↦ toℕ-↑ʳ
+  toℕ-inject+      ↦ toℕ-↑ˡ
+  splitAt-inject+  ↦ splitAt-↑ˡ m i n
+  splitAt-raise    ↦ splitAt-↑ʳ
+  Fin0↔⊥           ↦ 0↔⊥
+  eq?              ↦ inj⇒≟
+  ```
+
+* In `Data.Fin.Permutation.Components`:
+  ```
+  reverse            ↦ Data.Fin.Base.opposite
+  reverse-prop       ↦ Data.Fin.Properties.opposite-prop
+  reverse-involutive ↦ Data.Fin.Properties.opposite-involutive
+  reverse-suc        ↦ Data.Fin.Properties.opposite-suc
+  ```
+
+* In `Data.Integer.DivMod` the operator names have been renamed to
+  be consistent with those in `Data.Nat.DivMod`:
+  ```
+  _divℕ_  ↦ _/ℕ_
+  _div_   ↦ _/_
+  _modℕ_  ↦ _%ℕ_
+  _mod_   ↦ _%_
+  ```
+
+* In `Data.Integer.Properties` references to variables in names have
+  been made consistent so that `m`, `n` always refer to naturals and
+  `i` and `j` always refer to integers:
+  ```
+  ≤-steps        ↦  i≤j⇒i≤k+j
+  ≤-step         ↦  i≤j⇒i≤1+j
+
+  ≤-steps-neg    ↦  i≤j⇒i-k≤j
+  ≤-step-neg     ↦  i≤j⇒pred[i]≤j
+
+  n≮n            ↦  i≮i
+  ∣n∣≡0⇒n≡0       ↦  ∣i∣≡0⇒i≡0
+  ∣-n∣≡∣n∣         ↦  ∣-i∣≡∣i∣
+  0≤n⇒+∣n∣≡n      ↦  0≤i⇒+∣i∣≡i
+  +∣n∣≡n⇒0≤n      ↦  +∣i∣≡i⇒0≤i
+  +∣n∣≡n⊎+∣n∣≡-n   ↦  +∣i∣≡i⊎+∣i∣≡-i
+  ∣m+n∣≤∣m∣+∣n∣     ↦  ∣i+j∣≤∣i∣+∣j∣
+  ∣m-n∣≤∣m∣+∣n∣     ↦  ∣i-j∣≤∣i∣+∣j∣
+  signₙ◃∣n∣≡n     ↦  signᵢ◃∣i∣≡i
+  ◃-≡            ↦  ◃-cong
+  ∣m-n∣≡∣n-m∣      ↦  ∣i-j∣≡∣j-i∣
+  m≡n⇒m-n≡0      ↦  i≡j⇒i-j≡0
+  m-n≡0⇒m≡n      ↦  i-j≡0⇒i≡j
+  m≤n⇒m-n≤0      ↦  i≤j⇒i-j≤0
+  m-n≤0⇒m≤n      ↦  i-j≤0⇒i≤j
+  m≤n⇒0≤n-m      ↦  i≤j⇒0≤j-i
+  0≤n-m⇒m≤n      ↦  0≤i-j⇒j≤i
+  n≤1+n          ↦  i≤suc[i]
+  n≢1+n          ↦  i≢suc[i]
+  m≤pred[n]⇒m<n  ↦  i≤pred[j]⇒i<j
+  m<n⇒m≤pred[n]  ↦  i<j⇒i≤pred[j]
+  -1*n≡-n        ↦  -1*i≡-i
+  m*n≡0⇒m≡0∨n≡0  ↦  i*j≡0⇒i≡0∨j≡0
+  ∣m*n∣≡∣m∣*∣n∣  ↦  ∣i*j∣≡∣i∣*∣j∣
+  m≤m+n          ↦  i≤i+j
+  n≤m+n          ↦  i≤j+i
+  m-n≤m          ↦  i≤i-j
+
+  +-pos-monoʳ-≤    ↦  +-monoʳ-≤
+  +-neg-monoʳ-≤    ↦  +-monoʳ-≤
+  *-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
+  *-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
+  *-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
+  *-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
+  *-cancelˡ-<-neg  ↦  *-cancelˡ-<-nonPos
+  *-cancelʳ-<-neg  ↦  *-cancelʳ-<-nonPos
+
+  ^-semigroup-morphism ↦ ^-isMagmaHomomorphism
+  ^-monoid-morphism    ↦ ^-isMonoidHomomorphism
+
+  pos-distrib-* ↦ pos-*
+  pos-+-commute ↦ pos-+
+  abs-*-commute ↦ abs-*
+
+  +-isAbelianGroup ↦ +-0-isAbelianGroup
+  ```
+
+* In `Data.List.Base`:
+  ```agda
+  _─_  ↦  removeAt
+  ```
+
+* In `Data.List.Properties`:
+  ```agda
+  map-id₂         ↦  map-id-local
+  map-cong₂       ↦  map-cong-local
+  map-compose     ↦  map-∘
+
+  map-++-commute       ↦  map-++
+  sum-++-commute       ↦  sum-++
+  reverse-map-commute  ↦  reverse-map
+  reverse-++-commute   ↦  reverse-++
+
+  zipWith-identityˡ  ↦  zipWith-zeroˡ
+  zipWith-identityʳ  ↦  zipWith-zeroʳ
+
+  ʳ++-++  ↦  ++-ʳ++
+
+  take++drop  ↦  take++drop≡id
+
+  length-─  ↦  length-removeAt
+  map-─     ↦  map-removeAt
+  ```
+
+* In `Data.List.NonEmpty.Properties`:
+  ```agda
+  map-compose     ↦  map-∘
+
+  map-++⁺-commute ↦  map-++⁺
+  ```
+
+* In `Data.List.Relation.Unary.All.Properties`:
+  ```agda
+  gmap  ↦  gmap⁺
+
+  updateAt-id-relative      ↦  updateAt-id-local
+  updateAt-compose-relative ↦  updateAt-updateAt-local
+  updateAt-compose          ↦  updateAt-updateAt
+  updateAt-cong-relative    ↦  updateAt-cong-local
+  ```
+
+* In `Data.List.Relation.Unary.Any.Properties`:
+  ```agda
+  map-with-∈⁺  ↦  mapWith∈⁺
+  map-with-∈⁻  ↦  mapWith∈⁻
+  map-with-∈↔  ↦  mapWith∈↔
+  ```
+
+* In `Data.List.Relation.Binary.Subset.Propositional.Properties`:
+  ```agda
+  map-with-∈⁺  ↦  mapWith∈⁺
+  ```
+
+* In `Data.List.Zipper.Properties`:
+  ```agda
+  toList-reverse-commute ↦  toList-reverse
+  toList-ˡ++-commute     ↦  toList-ˡ++
+  toList-++ʳ-commute     ↦  toList-++ʳ
+  toList-map-commute    ↦  toList-map
+  toList-foldr-commute  ↦  toList-foldr
+  ```
+
+* In `Data.Maybe.Properties`:
+  ```agda
+  map-id₂     ↦  map-id-local
+  map-cong₂   ↦  map-cong-local
+
+  map-compose    ↦  map-∘
+
+  map-<∣>-commute ↦  map-<∣>
+  ```
+
+* In `Data.Nat.Properties`:
+  ```agda
+  suc[pred[n]]≡n  ↦  suc-pred
+  ≤-step          ↦  m≤n⇒m≤1+n
+  ≤-stepsˡ        ↦  m≤n⇒m≤o+n
+  ≤-stepsʳ        ↦  m≤n⇒m≤n+o
+  <-step          ↦  m<n⇒m<1+n
+  pred-mono       ↦  pred-mono-≤
+
+  <-transʳ        ↦  ≤-<-trans
+  <-transˡ        ↦  <-≤-trans
+  ```
+
+* In `Data.Rational.Unnormalised.Base`:
+  ```agda
+  _≠_  ↦  _≄_
+  +-rawMonoid ↦ +-0-rawMonoid
+  *-rawMonoid ↦ *-1-rawMonoid
+  ```
+
+* In `Data.Rational.Unnormalised.Properties`:
+  ```agda
+  ↥[p/q]≡p         ↦  ↥[n/d]≡n
+  ↧[p/q]≡q         ↦  ↧[n/d]≡d
+  *-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
+  *-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
+  ≤-steps          ↦  p≤q⇒p≤r+q
+  *-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
+  *-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
+  *-cancelˡ-<-pos  ↦  *-cancelˡ-<-nonNeg
+  *-cancelʳ-<-pos  ↦  *-cancelʳ-<-nonNeg
+
+  positive⇒nonNegative  ↦ pos⇒nonNeg
+  negative⇒nonPositive  ↦ neg⇒nonPos
+  negative<positive     ↦ neg<pos
+  ```
+
+* In `Data.Rational.Base`:
+  ```agda
+  +-rawMonoid ↦ +-0-rawMonoid
+  *-rawMonoid ↦ *-1-rawMonoid
+  ```
+
+* In `Data.Rational.Properties`:
+  ```agda
+  *-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
+  *-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
+  *-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
+  *-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
+  *-cancelˡ-<-pos  ↦  *-cancelˡ-<-nonNeg
+  *-cancelʳ-<-pos  ↦  *-cancelʳ-<-nonNeg
+  *-cancelˡ-<-neg  ↦  *-cancelˡ-<-nonPos
+  *-cancelʳ-<-neg  ↦  *-cancelʳ-<-nonPos
+
+  negative<positive     ↦ neg<pos
+  ```
+
+* In `Data.Rational.Unnormalised.Base`:
+  ```agda
+  +-rawMonoid ↦ +-0-rawMonoid
+  *-rawMonoid ↦ *-1-rawMonoid
+  ```
+
+* In `Data.Rational.Unnormalised.Properties`:
+  ```agda
+  ≤-steps  ↦  p≤q⇒p≤r+q
+  ```
+
+* In `Data.Sum.Properties`:
+  ```agda
+  [,]-∘-distr      ↦  [,]-∘
+  [,]-map-commute  ↦  [,]-map
+  map-commute      ↦  map-map
+  map₁₂-commute    ↦  map₁₂-map₂₁
+  ```
+
+* In `Data.Tree.AVL`:
+  ```agda
+  _∈?_ ↦ member
+  ```
+
+* In `Data.Tree.AVL.IndexedMap`:
+  ```agda
+  _∈?_ ↦ member
+  ```
+
+* In `Data.Tree.AVL.Map`:
+  ```agda
+  _∈?_ ↦ member
+  ```
+
+* In `Data.Tree.AVL.Sets`:
+  ```agda
+  _∈?_ ↦ member
+  ```
+
+* In `Data.Tree.Binary.Zipper.Properties`:
+  ```agda
+  toTree-#nodes-commute   ↦  toTree-#nodes
+  toTree-#leaves-commute  ↦  toTree-#leaves
+  toTree-map-commute      ↦  toTree-map
+  toTree-foldr-commute    ↦  toTree-foldr
+  toTree-⟪⟫ˡ-commute      ↦  toTree--⟪⟫ˡ
+  toTree-⟪⟫ʳ-commute      ↦  toTree-⟪⟫ʳ
+  ```
+
+* In `Data.Tree.Rose.Properties`:
+  ```agda
+  map-compose     ↦  map-∘
+  ```
+
+* In `Data.Vec.Base`:
+  ```agda
+  remove  ↦  removeAt
+  insert  ↦  insertAt
+  ```
+
+* In `Data.Vec.Properties`:
+  ```agda
+  take-distr-zipWith ↦  take-zipWith
+  take-distr-map     ↦  take-map
+  drop-distr-zipWith ↦  drop-zipWith
+  drop-distr-map     ↦  drop-map
+
+  updateAt-id-relative      ↦  updateAt-id-local
+  updateAt-compose-relative ↦  updateAt-updateAt-local
+  updateAt-compose          ↦  updateAt-updateAt
+  updateAt-cong-relative    ↦  updateAt-cong-local
+
+  []%=-compose    ↦  []%=-∘
+
+  []≔-++-inject+  ↦ []≔-++-↑ˡ
+  []≔-++-raise    ↦ []≔-++-↑ʳ
+  idIsFold        ↦ id-is-foldr
+  sum-++-commute  ↦ sum-++
+
+  take-drop-id  ↦  take++drop≡id
+
+  map-insert       ↦  map-insertAt
+
+  insert-lookup    ↦  insertAt-lookup
+  insert-punchIn   ↦  insertAt-punchIn
+  remove-PunchOut  ↦  removeAt-punchOut
+  remove-insert    ↦  removeAt-insertAt
+  insert-remove    ↦  insertAt-removeAt
+
+  lookup-inject≤-take ↦ lookup-take-inject≤
+  ```
+
+* In `Data.Vec.Functional.Properties`:
+  ```agda
+  updateAt-id-relative      ↦  updateAt-id-local
+  updateAt-compose-relative ↦  updateAt-updateAt-local
+  updateAt-compose          ↦  updateAt-updateAt
+  updateAt-cong-relative    ↦  updateAt-cong-local
+
+  map-updateAt              ↦  map-updateAt-local
+
+  insert-lookup             ↦  insertAt-lookup
+  insert-punchIn            ↦  insertAt-punchIn
+  remove-punchOut           ↦  removeAt-punchOut
+  remove-insert             ↦  removeAt-insertAt
+  insert-remove             ↦  insertAt-removeAt
+  ```
+  NB. This last one is complicated by the *addition* of a 'global' property `map-updateAt`
+
+* In `Data.Vec.Recursive.Properties`:
+  ```agda
+  append-cons-commute  ↦  append-cons
+  ```
+
+* In `Data.Vec.Relation.Binary.Equality.Setoid`:
+  ```agda
+  map-++-commute ↦  map-++
+  ```
+
+* In `Function.Base`:
+  ```agda
+  case_return_of_   ↦   case_returning_of_
+  ```
+
+* In `Function.Construct.Composition`:
+  ```agda
+  _∘-⟶_   ↦   _⟶-∘_
+  _∘-↣_   ↦   _↣-∘_
+  _∘-↠_   ↦   _↠-∘_
+  _∘-⤖_  ↦   _⤖-∘_
+  _∘-⇔_   ↦   _⇔-∘_
+  _∘-↩_   ↦   _↩-∘_
+  _∘-↪_   ↦   _↪-∘_
+  _∘-↔_   ↦   _↔-∘_
+  ```
+
+  * In `Function.Construct.Identity`:
+  ```agda
+  id-⟶   ↦   ⟶-id
+  id-↣   ↦   ↣-id
+  id-↠   ↦   ↠-id
+  id-⤖  ↦   ⤖-id
+  id-⇔   ↦   ⇔-id
+  id-↩   ↦   ↩-id
+  id-↪   ↦   ↪-id
+  id-↔   ↦   ↔-id
+  ```
+
+* In `Function.Construct.Symmetry`:
+  ```agda
+  sym-⤖   ↦   ⤖-sym
+  sym-⇔   ↦   ⇔-sym
+  sym-↩   ↦   ↩-sym
+  sym-↪   ↦   ↪-sym
+  sym-↔   ↦   ↔-sym
+  ```
+
+* In `Function.Related.Propositional.Reasoning`:
+  ```agda
+  _↔⟨⟩_  ↦  _≡⟨⟩_
+  ```
+
+* In `Foreign.Haskell.Either` and `Foreign.Haskell.Pair`:
+  ```agda
+  toForeign   ↦ Foreign.Haskell.Coerce.coerce
+  fromForeign ↦ Foreign.Haskell.Coerce.coerce
+  ```
+
+* In `Relation.Binary.Properties.Preorder`:
+  ```agda
+  invIsPreorder ↦ converse-isPreorder
+  invPreorder   ↦ converse-preorder
+  ```
+
+* In `Relation.Binary.PropositionalEquality`:
+  ```agda
+  isPropositional ↦ Relation.Nullary.Irrelevant
+  ```
+
+* In `Relation.Unary.Consequences`:
+  ```agda
+  dec⟶recomputable  ↦  dec⇒recomputable
+  ```
+
+## Missing fixity declarations added
+
+* An effort has been made to add sensible fixities for many declarations:
+  ```
+  infix   4 _≟H_ _≟N_                                        (Algebra.Solver.Ring)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream)
+  infix   4 _[_]                                             (Codata.Guarded.Stream)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Binary.Pointwise)
+  infix   4 _≈∞_                                             (Codata.Guarded.Stream.Relation.Binary.Pointwise)
+  infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Unary.All)
+  infixr  5 _∷_                                              (Codata.Musical.Colist)
+  infix   4 _≈_                                              (Codata.Musical.Conat)
+  infixr  5 _∷_                                              (Codata.Musical.Colist.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Musical.Colist.Relation.Unary.All)
+  infixr  5 _∷_                                              (Codata.Sized.Colist)
+  infixr  5 _∷_                                              (Codata.Sized.Covec)
+  infixr  5 _∷_                                              (Codata.Sized.Cowriter)
+  infixl  1 _>>=_                                            (Codata.Sized.Cowriter)
+  infixr  5 _∷_                                              (Codata.Sized.Stream)
+  infixr  5 _∷_                                              (Codata.Sized.Colist.Bisimilarity)
+  infix   4 _ℕ≤?_                                            (Codata.Sized.Conat.Properties)
+  infixr  5 _∷_                                              (Codata.Sized.Covec.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Sized.Cowriter.Bisimilarity)
+  infixr  5 _∷_                                              (Codata.Sized.Stream.Bisimilarity)
+  infix   4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
+  infix   6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
+  infixr  8 _⇒_ _⊸_                                          (Data.Container.Core)
+  infixr -1 _<$>_ _<*>_                                      (Data.Container.FreeMonad)
+  infixl  1 _>>=_                                            (Data.Container.FreeMonad)
+  infix   5 _▷_                                              (Data.Container.Indexed)
+  infixr  4 _,_                                              (Data.Container.Relation.Binary.Pointwise)
+  infix   4 _≈_                                              (Data.Float.Base)
+  infixl  4 _+ _*                                            (Data.List.Kleene.Base)
+  infixr  4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
+  infix   4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
+  infix   4 _≢∈_                                             (Data.List.Membership.Propositional)
+  infixr  5 _`∷_                                             (Data.List.Reflection)
+  infix   4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
+  infixr  5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
+  infixr  5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
+  infixr  5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
+  infixr  5 _∷=_                                             (Data.List.Relation.Unary.Any)
+  infixr  5 _++_                                             (Data.List.Ternary.Appending)
+  infixl  7 _⊓′_                                             (Data.Nat.Base)
+  infixl  6 _⊔′_                                             (Data.Nat.Base)
+  infixr  8 _^_                                              (Data.Nat.Base)
+  infix   4 _!≢0 _!*_!≢0                                     (Data.Nat.Properties)
+  infixl  6.5 _P′_ _P_ _C′_ _C_                              (Data.Nat.Combinatorics.Base)
+  infix   8 _⁻¹                                              (Data.Parity.Base)
+  infixr  2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
+  infixr  2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
+  infixr  2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
+  infixr  2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
+  infix   4 _≃?_                                             (Data.Rational.Unnormalised.Properties)
+  infixr  4 _,_                                              (Data.Refinement)
+  infixr  1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
+  infixr  1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
+  infixr  1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
+  infixr  1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
+  infixr  4 _,_                                              (Data.Tree.AVL.Value)
+  infix   4 _≈ₖᵥ_                                            (Data.Tree.AVL.Map.Membership.Propositional)
+  infixr  5 _`∷_                                             (Data.Vec.Reflection)
+  infixr  5 _∷=_                                             (Data.Vec.Membership.Setoid)
+  infix  -1 _$ⁿ_                                             (Data.Vec.N-ary)
+  infix   4 _≋_                                              (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
+  infixl  1 _>>=-cong_ _≡->>=-cong_                          (Effect.Monad.Partiality)
+  infixl  1 _?>=′_                                           (Effect.Monad.Predicate)
+  infixl  1 _>>=-cong_ _>>=-congP_                           (Effect.Monad.Partiality.All)
+  infixr  5 _∷_                                              (Foreign.Haskell.List.NonEmpty)
+  infixr  4 _,_                                              (Foreign.Haskell.Pair)
+  infixr  8 _^_                                              (Function.Endomorphism.Propositional)
+  infixr  8 _^_                                              (Function.Endomorphism.Setoid)
+  infix   4 _≃_                                              (Function.HalfAdjointEquivalence)
+  infix   4 _≈_ _≈ᵢ_ _≤_                                     (Function.Metric.Bundles)
+  infixl  6 _∙_                                              (Function.Metric.Bundles)
+  infix   4 _≈_                                              (Function.Metric.Nat.Bundles)
+  infix   4 _≈_                                              (Function.Metric.Rational.Bundles)
+  infix   3 _←_ _↢_                                          (Function.Related)
+  infix   4 _<_                                              (Induction.WellFounded)
+  infixl  6 _ℕ+_                                             (Level.Literals)
+  infixr  4 _,_                                              (Reflection.AnnotatedAST)
+  infix   4 _≟_                                              (Reflection.AST.Definition)
+  infix   4 _≡ᵇ_                                             (Reflection.AST.Literal)
+  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Meta)
+  infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Name)
+  infix   4 _≟-Telescope_                                    (Reflection.AST.Term)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Information)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Modality)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Quantity)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Relevance)
+  infix   4 _≟_                                              (Reflection.AST.Argument.Visibility)
+  infixr  4 _,_                                              (Reflection.AST.Traversal)
+  infix   4 _∈FV_                                            (Reflection.AST.DeBruijn)
+  infixr  9 _;_                                              (Relation.Binary.Construct.Composition)
+  infixl  6 _+²_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
+  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Heterogeneous.Construct.At)
+  infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Homogeneous.Construct.At)
+  infix   4 _∈_ _∉_                                          (Relation.Unary.Indexed)
+  infix   4 _≈_                                              (Relation.Binary.Bundles)
+  infixl  6 _∩_                                              (Relation.Binary.Construct.Intersection)
+  infix   4 _<₋_                                             (Relation.Binary.Construct.Add.Infimum.Strict)
+  infix   4 _≈∙_                                             (Relation.Binary.Construct.Add.Point.Equality)
+  infix   4 _≤⁺_ _≤⊤⁺                                        (Relation.Binary.Construct.Add.Supremum.NonStrict)
+  infixr  5 _∷ʳ_                                             (Relation.Binary.Construct.Closure.Transitive)
+  infixr  6 _∪_                                              (Relation.Binary.Construct.Union)
+  infixl  6 _+ℤ_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
+  infix   4 _≉_ _≈ᵢ_ _≤ᵢ_                                    (Relation.Binary.Indexed.Homogeneous.Bundles)
+  infixr  9 _⍮_                                              (Relation.Unary.PredicateTransformer)
+  infix   8 ∼_                                               (Relation.Unary.PredicateTransformer)
+  infix   2 _×?_ _⊙?_                                        (Relation.Unary.Properties)
+  infix   10 _~?                                             (Relation.Unary.Properties)
+  infixr  1 _⊎?_                                             (Relation.Unary.Properties)
+  infixr  7 _∩?_                                             (Relation.Unary.Properties)
+  infixr  6 _∪?_                                             (Relation.Unary.Properties)
+  infixr  5 _∷ᴹ_ _∷⁻¹ᴹ_                                      (Text.Regex.Search)
+  infixl  6 _`⊜_                                             (Tactic.RingSolver)
+  infix   8 ⊝_                                               (Tactic.RingSolver.Core.Expression)
+  infix   4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]                  (Text.Regex.Properties)
+  infix   4 _∈?_ _∉?_                                        (Text.Regex.Derivative.Brzozowski)
+  infix   4 _∈_ _∉_ _∈?_ _∉?_                                (Text.Regex.String.Unsafe)
+  ```
+
+New modules
+-----------
+
+* Constructive algebraic structures with apartness relations:
+  ```
+  Algebra.Apartness
+  Algebra.Apartness.Bundles
+  Algebra.Apartness.Structures
+  Algebra.Apartness.Properties.CommutativeHeytingAlgebra
+  Relation.Binary.Properties.ApartnessRelation
+  ```
+
+* Algebraic structures obtained by flipping their binary operations:
+  ```
+  Algebra.Construct.Flip.Op
+  ```
+
+* Algebraic structures when freely adding an identity element:
+  ```
+  Algebra.Construct.Add.Identity
+  ```
+
+* Definitions for algebraic modules:
+  ```
+  Algebra.Module
+  Algebra.Module.Core
+  Algebra.Module.Definitions.Bi.Simultaneous
+  Algebra.Module.Morphism.Construct.Composition
+  Algebra.Module.Morphism.Construct.Identity
+  Algebra.Module.Morphism.Definitions
+  Algebra.Module.Morphism.ModuleHomomorphism
+  Algebra.Module.Morphism.Structures
+  Algebra.Module.Properties
+  ```
+
+* Identity morphisms and composition of morphisms between algebraic structures:
+  ```
+  Algebra.Morphism.Construct.Composition
+  Algebra.Morphism.Construct.Identity
+  ```
+
+* Properties of various new algebraic structures:
+  ```
+  Algebra.Properties.MoufangLoop
+  Algebra.Properties.Quasigroup
+  Algebra.Properties.MiddleBolLoop
+  Algebra.Properties.Loop
+  Algebra.Properties.KleeneAlgebra
+  ```
+
+* Properties of rings without a unit
+  ```
+  Algebra.Properties.RingWithoutOne`
+  ```
+
+* Proof of the Binomial Theorem for semirings
+  ```
+  Algebra.Properties.Semiring.Binomial
+  Algebra.Properties.CommutativeSemiring.Binomial
+  ```
+
+* 'Optimised' tail-recursive exponentiation properties:
+  ```
+  Algebra.Properties.Semiring.Exp.TailRecursiveOptimised
+  ```
+
+* An implementation of M-types with `--guardedness` flag:
+  ```
+  Codata.Guarded.M
+  ```
+
+* A definition of infinite streams using coinductive records:
+  ```
+  Codata.Guarded.Stream
+  Codata.Guarded.Stream.Properties
+  Codata.Guarded.Stream.Relation.Binary.Pointwise
+  Codata.Guarded.Stream.Relation.Unary.All
+  Codata.Guarded.Stream.Relation.Unary.Any
+  ```
+
+* A small library for function arguments with default values:
+  ```
+  Data.Default
+  ```
+
+* A small library defining a structurally recursive view of `Fin n`:
+  ```
+  Data.Fin.Relation.Unary.Top
+  ```
+
+* A small library for a non-empty fresh list:
+  ```
+  Data.List.Fresh.NonEmpty
+  ```
+
+* A small library defining a structurally inductive view of lists:
+  ```
+  Data.List.Relation.Unary.Sufficient
+  ```
+
+* Combinations and permutations for ℕ.
+  ```
+  Data.Nat.Combinatorics
+  Data.Nat.Combinatorics.Base
+  Data.Nat.Combinatorics.Spec
+  ```
+
+* A small library defining parity and its algebra:
+  ```
+  Data.Parity
+  Data.Parity.Base
+  Data.Parity.Instances
+  Data.Parity.Properties
+  ```
+
+* New base module for `Data.Product` containing only the basic definitions.
+  ```
+  Data.Product.Base
+  ```
+
+* Reflection utilities for some specific types:
+  ```
+  Data.List.Reflection
+  Data.Vec.Reflection
+  ```
+
+* The `All` predicate over non-empty lists:
+  ```
+  Data.List.NonEmpty.Relation.Unary.All
+  ```
+
+* Some n-ary functions manipulating lists
+  ```
+  Data.List.Nary.NonDependent
+  ```
+
+* Added Logarithm base 2 on natural numbers:
+  ```
+  Data.Nat.Logarithm.Core
+  Data.Nat.Logarithm
+  ```
+
+* Show module for unnormalised rationals:
+  ```
+  Data.Rational.Unnormalised.Show
+  ```
+
+* Membership relations for maps and sets
+  ```
+  Data.Tree.AVL.Map.Membership.Propositional
+  Data.Tree.AVL.Map.Membership.Propositional.Properties
+  Data.Tree.AVL.Sets.Membership
+  Data.Tree.AVL.Sets.Membership.Properties
+  ```
+
+* Port of `Linked` to `Vec`:
+  ```
+  Data.Vec.Relation.Unary.Linked
+  Data.Vec.Relation.Unary.Linked.Properties
+  ```
+
+* Combinators for propositional equational reasoning on vectors with different indices
+  ```
+  Data.Vec.Relation.Binary.Equality.Cast
+  ```
+
+* Relations on indexed sets
+  ```
+  Function.Indexed.Bundles
+  ```
+
+* Properties of various types of functions:
+  ```
+  Function.Consequences
+  Function.Consequences.Setoid
+  Function.Consequences.Propositional
+  Function.Properties.Bijection
+  Function.Properties.RightInverse
+  Function.Properties.Surjection
+  Function.Construct.Constant
+  ```
+
+* New interface for `NonEmpty` Haskell lists:
+  ```
+  Foreign.Haskell.List.NonEmpty
+  ```
+
+* In order to improve modularity, the contents of `Relation.Binary.Lattice` has been
+  split out into the standard:
+  ```
+  Relation.Binary.Lattice.Definitions
+  Relation.Binary.Lattice.Structures
+  Relation.Binary.Lattice.Bundles
+  ```
+  All contents is re-exported by `Relation.Binary.Lattice` as before.
+
+
+* Added relational reasoning over apartness relations:
+  ```
+  Relation.Binary.Reasoning.Base.Apartness`
+  ```
+
+* Algebraic properties of `_∩_` and `_∪_` for predicates
+  ```
+  Relation.Unary.Algebra
+  ```
+
+* Both versions of equality on predicates are equivalences
+  ```
+  Relation.Unary.Relation.Binary.Equality
+  ```
+
+* The subset relations on predicates define an order
+  ```
+  Relation.Unary.Relation.Binary.Subset
+  ```
+
+* Polymorphic versions of some unary relations and their properties
+  ```
+  Relation.Unary.Polymorphic
+  Relation.Unary.Polymorphic.Properties
+  ```
+
+* Alpha equality over reflected terms
+  ```
+  Reflection.AST.AlphaEquality
+  ```
+
+* Various system types and primitives:
+  ```
+  System.Clock.Primitive
+  System.Clock
+  System.Console.ANSI
+  System.Directory.Primitive
+  System.Directory
+  System.FilePath.Posix.Primitive
+  System.FilePath.Posix
+  System.Process.Primitive
+  System.Process
+  ```
+
+* A new `cong!` tactic for automatically deriving arguments to `cong`
+  ```
+  Tactic.Cong
+  ```
+
+* A golden testing library with test pools, an options parser, a runner:
+  ```
+  Test.Golden
+  ```
+
+Additions to existing modules
+-----------------------------
+
+* The module `Algebra` now publicly re-exports the contents of
+  `Algebra.Structures.Biased`.
+
+* Added new definitions to `Algebra.Bundles`:
+  ```agda
+  record UnitalMagma           c ℓ : Set (suc (c ⊔ ℓ))
+  record InvertibleMagma       c ℓ : Set (suc (c ⊔ ℓ))
+  record InvertibleUnitalMagma c ℓ : Set (suc (c ⊔ ℓ))
+  record Quasigroup            c ℓ : Set (suc (c ⊔ ℓ))
+  record Loop                  c ℓ : Set (suc (c ⊔ ℓ))
+  record RingWithoutOne        c ℓ : Set (suc (c ⊔ ℓ))
+  record IdempotentSemiring    c ℓ : Set (suc (c ⊔ ℓ))
+  record KleeneAlgebra         c ℓ : Set (suc (c ⊔ ℓ))
+  record Quasiring             c ℓ : Set (suc (c ⊔ ℓ))
+  record Nearring              c ℓ : Set (suc (c ⊔ ℓ))
+  record IdempotentMagma       c ℓ : Set (suc (c ⊔ ℓ))
+  record AlternateMagma        c ℓ : Set (suc (c ⊔ ℓ))
+  record FlexibleMagma         c ℓ : Set (suc (c ⊔ ℓ))
+  record MedialMagma           c ℓ : Set (suc (c ⊔ ℓ))
+  record SemimedialMagma       c ℓ : Set (suc (c ⊔ ℓ))
+  record LeftBolLoop           c ℓ : Set (suc (c ⊔ ℓ))
+  record RightBolLoop          c ℓ : Set (suc (c ⊔ ℓ))
+  record MoufangLoop           c ℓ : Set (suc (c ⊔ ℓ))
+  record NonAssociativeRing    c ℓ : Set (suc (c ⊔ ℓ))
+  record MiddleBolLoop         c ℓ : Set (suc (c ⊔ ℓ))
+  ```
+
+* Added new definitions to `Algebra.Bundles.Raw`:
+  ```agda
+  record RawLoop               c ℓ : Set (suc (c ⊔ ℓ))
+  record RawQuasiGroup         c ℓ : Set (suc (c ⊔ ℓ))
+  record RawRingWithoutOne     c ℓ : Set (suc (c ⊔ ℓ))
+  ```
+
+* Added new definitions to `Algebra.Structures`:
+  ```agda
+  record IsUnitalMagma (_∙_ : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsInvertibleMagma (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
+  record IsInvertibleUnitalMagma (_∙_ : Op₂ A) (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
+  record IsQuasigroup (∙ \\ // : Op₂ A) : Set (a ⊔ ℓ)
+  record IsLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ)
+  record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ)
+  record IsKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
+  record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
+  record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
+  record IsIdempotentMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsAlternativeMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsFlexibleMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsMedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsSemimedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
+  record IsLeftBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsRightBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsMoufangLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
+  record IsMiddleBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
+  ```
+
+* Added new proof to `Algebra.Consequences.Base`:
+  ```agda
+  reflexive∧selfInverse⇒involutive : Reflexive _≈_ → SelfInverse _≈_ f → Involutive _≈_ f
+  ```
+
+* Added new proofs to `Algebra.Consequences.Propositional`:
+  ```agda
+  comm∧assoc⇒middleFour     : Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
+  identity∧middleFour⇒assoc : Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
+  identity∧middleFour⇒comm  : Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
+  ```
+
+* Added new proofs to `Algebra.Consequences.Setoid`:
+  ```agda
+  comm∧assoc⇒middleFour     : Congruent₂ _∙_ → Commutative _∙_ → Associative _∙_ → _∙_ MiddleFourExchange _∙_
+  identity∧middleFour⇒assoc : Congruent₂ _∙_ → Identity e _∙_ → _∙_ MiddleFourExchange _∙_ → Associative _∙_
+  identity∧middleFour⇒comm  : Congruent₂ _∙_ → Identity e _+_ → _∙_ MiddleFourExchange _+_ → Commutative _∙_
+
+  involutive⇒surjective  : Involutive f  → Surjective f
+  selfInverse⇒involutive : SelfInverse f → Involutive f
+  selfInverse⇒congruent  : SelfInverse f → Congruent f
+  selfInverse⇒inverseᵇ   : SelfInverse f → Inverseᵇ f f
+  selfInverse⇒surjective : SelfInverse f → Surjective f
+  selfInverse⇒injective  : SelfInverse f → Injective f
+  selfInverse⇒bijective  : SelfInverse f → Bijective f
+
+  comm∧idˡ⇒id              : Commutative _∙_ → LeftIdentity  e _∙_ → Identity e _∙_
+  comm∧idʳ⇒id              : Commutative _∙_ → RightIdentity e _∙_ → Identity e _∙_
+  comm∧zeˡ⇒ze              : Commutative _∙_ → LeftZero      e _∙_ → Zero     e _∙_
+  comm∧zeʳ⇒ze              : Commutative _∙_ → RightZero     e _∙_ → Zero     e _∙_
+  comm∧invˡ⇒inv            : Commutative _∙_ → LeftInverse  e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm∧invʳ⇒inv            : Commutative _∙_ → RightInverse e _⁻¹ _∙_ → Inverse e _⁻¹ _∙_
+  comm∧distrˡ⇒distr        : Commutative _∙_ → _∙_ DistributesOverˡ _◦_ → _∙_ DistributesOver _◦_
+  comm∧distrʳ⇒distr        : Commutative _∙_ → _∙_ DistributesOverʳ _◦_ → _∙_ DistributesOver _◦_
+  distrib∧absorbs⇒distribˡ : Associative _∙_ → Commutative _◦_ → _∙_ Absorbs _◦_ → _◦_ Absorbs _∙_ → _◦_ DistributesOver _∙_ → _∙_ DistributesOverˡ _◦_
+  ```
+
+* Added new functions to `Algebra.Construct.DirectProduct`:
+  ```agda
+  rawSemiring                     : RawSemiring a ℓ₁ → RawSemiring b ℓ₂ → RawSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawRing                         : RawRing a ℓ₁ → RawRing b ℓ₂ → RawRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero a ℓ₁ →
+                                    SemiringWithoutAnnihilatingZero b ℓ₂ →
+                                    SemiringWithoutAnnihilatingZero (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  semiring                        : Semiring a ℓ₁ → Semiring b ℓ₂ → Semiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  commutativeSemiring             : CommutativeSemiring a ℓ₁ →
+                                        CommutativeSemiring b ℓ₂ →
+                                    CommutativeSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  ring                            : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  commutativeRing                 : CommutativeRing a ℓ₁ → CommutativeRing b ℓ₂ → CommutativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawQuasigroup                   : RawQuasigroup a ℓ₁ → RawQuasigroup b ℓ₂ → RawQuasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawLoop                         : RawLoop a ℓ₁ → RawLoop b ℓ₂ → RawLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  unitalMagma                     : UnitalMagma a ℓ₁ → UnitalMagma b ℓ₂ → UnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  invertibleMagma                 : InvertibleMagma a ℓ₁ → InvertibleMagma b ℓ₂ → InvertibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  invertibleUnitalMagma           : InvertibleUnitalMagma a ℓ₁ →
+                                        InvertibleUnitalMagma b ℓ₂ →
+                                    InvertibleUnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  quasigroup                      : Quasigroup a ℓ₁ → Quasigroup b ℓ₂ → Quasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  loop                            : Loop a ℓ₁ → Loop b ℓ₂ → Loop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  idempotentSemiring              : IdempotentSemiring a ℓ₁ →
+                                    IdempotentSemiring b ℓ₂ →
+                                                                        IdempotentSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  idempotentMagma                 : IdempotentMagma a ℓ₁ → IdempotentMagma b ℓ₂ → IdempotentMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  alternativeMagma                : AlternativeMagma a ℓ₁ → AlternativeMagma b ℓ₂ → AlternativeMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  flexibleMagma                   : FlexibleMagma a ℓ₁ → FlexibleMagma b ℓ₂ → FlexibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  medialMagma                     : MedialMagma a ℓ₁ → MedialMagma b ℓ₂ → MedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  semimedialMagma                 : SemimedialMagma a ℓ₁ → SemimedialMagma b ℓ₂ → SemimedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  kleeneAlgebra                   : KleeneAlgebra a ℓ₁ → KleeneAlgebra b ℓ₂ → KleeneAlgebra (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  leftBolLoop                     : LeftBolLoop a ℓ₁ → LeftBolLoop b ℓ₂ → LeftBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rightBolLoop                    : RightBolLoop a ℓ₁ → RightBolLoop b ℓ₂ → RightBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  middleBolLoop                   : MiddleBolLoop a ℓ₁ → MiddleBolLoop b ℓ₂ → MiddleBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  moufangLoop                     : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  rawRingWithoutOne               : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  ringWithoutOne                  : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  nonAssociativeRing              : NonAssociativeRing a ℓ₁ →
+                                            NonAssociativeRing b ℓ₂ →
+                                                                        NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  quasiring                       : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  nearring                        : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
+  ```
+
+* Added new definition to `Algebra.Definitions`:
+  ```agda
+  _*_ MiddleFourExchange _+_ = ∀ w x y z → ((w + x) * (y + z)) ≈ ((w + y) * (x + z))
+
+  SelfInverse f = ∀ {x y} → f x ≈ y → f y ≈ x
+
+  LeftDividesˡ  _∙_ _\\_ = ∀ x y → (x ∙ (x \\ y)) ≈ y
+  LeftDividesʳ  _∙_ _\\_ = ∀ x y → (x \\ (x ∙ y)) ≈ y
+  RightDividesˡ _∙_ _//_ = ∀ x y → ((y // x) ∙ x) ≈ y
+  RightDividesʳ _∙_ _//_ = ∀ x y → ((y ∙ x) // x) ≈ y
+  LeftDivides    ∙   \\ = (LeftDividesˡ ∙ \\) × (LeftDividesʳ ∙ \\)
+  RightDivides   ∙   // = (RightDividesˡ ∙ //) × (RightDividesʳ ∙ //)
+
+  LeftInvertible  e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e
+  RightInvertible e _∙_ x = ∃[ x⁻¹ ] (x ∙ x⁻¹) ≈ e
+  Invertible      e _∙_ x = ∃[ x⁻¹ ] ((x⁻¹ ∙ x) ≈ e) × ((x ∙ x⁻¹) ≈ e)
+  StarRightExpansive e _+_ _∙_ _⁻* = ∀ x → (e + (x ∙ (x ⁻*))) ≈ (x ⁻*)
+  StarLeftExpansive e _+_ _∙_ _⁻* = ∀ x →  (e + ((x ⁻*) ∙ x)) ≈ (x ⁻*)
+  StarExpansive e _+_ _∙_ _* = (StarLeftExpansive e _+_ _∙_ _*) × (StarRightExpansive e _+_ _∙_ _*)
+  StarLeftDestructive _+_ _∙_ _* = ∀ a b x → (b + (a ∙ x)) ≈ x → ((a *) ∙ b) ≈ x
+  StarRightDestructive _+_ _∙_ _* = ∀ a b x → (b + (x ∙ a)) ≈ x → (b ∙ (a *)) ≈ x
+  StarDestructive _+_ _∙_ _* = (StarLeftDestructive _+_ _∙_ _*) × (StarRightDestructive _+_ _∙_ _*)
+  LeftAlternative _∙_ = ∀ x y  →  ((x ∙ x) ∙ y) ≈ (x ∙ (y ∙ y))
+  RightAlternative _∙_ = ∀ x y → (x ∙ (y ∙ y)) ≈ ((x ∙ y) ∙ y)
+  Alternative _∙_ = (LeftAlternative _∙_ ) × (RightAlternative _∙_)
+  Flexible _∙_ = ∀ x y → ((x ∙ y) ∙ x) ≈ (x ∙ (y ∙ x))
+  Medial _∙_ = ∀ x y u z → ((x ∙ y) ∙ (u ∙ z)) ≈ ((x ∙ u) ∙ (y ∙ z))
+  LeftSemimedial _∙_ = ∀ x y z → ((x ∙ x) ∙ (y ∙ z)) ≈ ((x ∙ y) ∙ (x ∙ z))
+  RightSemimedial _∙_ = ∀ x y z → ((y ∙ z) ∙ (x ∙ x)) ≈ ((y ∙ x) ∙ (z ∙ x))
+  Semimedial _∙_ = (LeftSemimedial _∙_) × (RightSemimedial _∙_)
+  LeftBol _∙_ = ∀ x y z → (x ∙ (y ∙ (x ∙ z))) ≈ ((x ∙ (y ∙ z)) ∙ z )
+  RightBol _∙_ = ∀ x y z → (((z ∙ x) ∙ y) ∙ x) ≈ (z ∙ ((x ∙ y) ∙ x))
+  MiddleBol _∙_ _\\_ _//_ = ∀ x y z → (x ∙ ((y ∙ z) \\ x)) ≈ ((x // z) ∙ (y \\ x))
+  ```
+
+* Added new functions to `Algebra.Definitions.RawSemiring`:
+  ```agda
+  _^[_]*_ : A → ℕ → A → A
+  _^ᵗ_    : A → ℕ → A
+  ```
+
+* In `Algebra.Bundles.Lattice` the existing record `Lattice` now provides
+  ```agda
+  ∨-commutativeSemigroup : CommutativeSemigroup c ℓ
+  ∧-commutativeSemigroup : CommutativeSemigroup c ℓ
+  ```
+  and their corresponding algebraic sub-bundles.
+
+* In `Algebra.Lattice.Structures` the record `IsLattice` now provides
+  ```
+  ∨-isCommutativeSemigroup : IsCommutativeSemigroup ∨
+  ∧-isCommutativeSemigroup : IsCommutativeSemigroup ∧
+  ```
+  and their corresponding algebraic substructures.
+
+* The module `Algebra.Properties.Magma.Divisibility` now re-exports operations
+  `_∣ˡ_`, `_∤ˡ_`, `_∣ʳ_`, `_∤ʳ_` from `Algebra.Definitions.Magma`.
+
+* Added new records to `Algebra.Morphism.Structures`:
+  ```agda
+  record IsQuasigroupHomomorphism     (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsQuasigroupMonomorphism     (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsQuasigroupIsomorphism      (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsLoopHomomorphism           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsLoopMonomorphism           (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsLoopIsomorphism            (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsRingWithoutOneIsoMorphism  (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraHomomorphism  (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraMonomorphism  (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
+  record IsKleeneAlgebraIsomorphism   (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  ```
+
+* Added new proofs to `Algebra.Properties.CommutativeSemigroup`:
+  ```agda
+  xy∙xx≈x∙yxx : (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))
+
+  interchange : Interchangable _∙_ _∙_
+  leftSemimedial : LeftSemimedial _∙_
+  rightSemimedial : RightSemimedial _∙_
+  middleSemimedial : (x ∙ y) ∙ (z ∙ x) ≈ (x ∙ z) ∙ (y ∙ x)
+  semimedial : Semimedial _∙_
+  ```
+
+* Added new functions to `Algebra.Properties.CommutativeMonoid`
+  ```agda
+  invertibleˡ⇒invertibleʳ : LeftInvertible  _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x
+  invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible  _≈_ 0# _+_ x
+  invertibleˡ⇒invertible  : LeftInvertible  _≈_ 0# _+_ x → Invertible      _≈_ 0# _+_ x
+  invertibleʳ⇒invertible  : RightInvertible _≈_ 0# _+_ x → Invertible      _≈_ 0# _+_ x
+  ```
+
+* Added new proof to `Algebra.Properties.Monoid.Mult`:
+  ```agda
+  ×-congˡ : (_× x) Preserves _≡_ ⟶ _≈_
+  ```
+
+* Added new proof to `Algebra.Properties.Monoid.Sum`:
+  ```agda
+  sum-init-last : (t : Vector _ (suc n)) → sum t ≈ sum (init t) + last t
+  ```
+
+* Added new proofs to `Algebra.Properties.Semigroup`:
+  ```agda
+  leftAlternative  : LeftAlternative _∙_
+  rightAlternative : RightAlternative _∙_
+  alternative      : Alternative _∙_
+  flexible         : Flexible _∙_
+  ```
+
+* Added new proofs to `Algebra.Properties.Semiring.Exp`:
+  ```agda
+  ^-congʳ               : (x ^_) Preserves _≡_ ⟶ _≈_
+  y*x^m*y^n≈x^m*y^[n+1] : x * y ≈ y * x → y * (x ^ m * y ^ n) ≈ x ^ m * y ^ suc n
+  ```
+
+* Added new proofs to `Algebra.Properties.Semiring.Mult`:
+  ```agda
+  1×-identityʳ : 1 × x ≈ x
+  ×-comm-*    : x * (n × y) ≈ n × (x * y)
+  ×-assoc-*   : (n × x) * y ≈ n × (x * y)
+  ```
+
+* Added new proofs to `Algebra.Properties.Ring`:
+  ```agda
+  -1*x≈-x      : - 1# * x ≈ - x
+  x+x≈x⇒x≈0    : x + x ≈ x → x ≈ 0#
+  x[y-z]≈xy-xz : x * (y - z) ≈ x * y - x * z
+  [y-z]x≈yx-zx : (y - z) * x ≈ (y * x) - (z * x)
+  ```
+
+* Added new functions in `Codata.Guarded.Stream`:
+  ```
+  transpose  : List (Stream A) → Stream (List A)
+  transpose⁺ : List⁺ (Stream A) → Stream (List⁺ A)
+  concat     : Stream (List⁺ A) → Stream A
+  ```
+
+* Added new proofs in `Codata.Guarded.Stream.Properties`:
+  ```
+  cong-concat       : ass ≈ bss → concat ass ≈ concat bss
+  map-concat        : map f (concat ass) ≈ concat (map (List⁺.map f) ass)
+  lookup-transpose  : lookup n (transpose ass) ≡ List.map (lookup n) ass
+  lookup-transpose⁺ : lookup n (transpose⁺ ass) ≡ List⁺.map (lookup n) ass
+  ```
+
+* Added new proofs in `Data.Bool.Properties`:
+  ```agda
+  <-wellFounded : WellFounded _<_
+  ∨-conicalˡ : LeftConical false _∨_
+  ∨-conicalʳ : RightConical false _∨_
+  ∨-conical  : Conical false _∨_
+  ∧-conicalˡ : LeftConical true _∧_
+  ∧-conicalʳ : RightConical true _∧_
+  ∧-conical  : Conical true _∧_
+
+  true-xor            : true xor x ≡ not x
+  xor-same            : x xor x ≡ false
+  not-distribˡ-xor    : not (x xor y) ≡ (not x) xor y
+  not-distribʳ-xor    : not (x xor y) ≡ x xor (not y)
+  xor-assoc           : Associative _xor_
+  xor-comm            : Commutative _xor_
+  xor-identityˡ       : LeftIdentity false _xor_
+  xor-identityʳ       : RightIdentity false _xor_
+  xor-identity        : Identity false _xor_
+  xor-inverseˡ        : LeftInverse true not _xor_
+  xor-inverseʳ        : RightInverse true not _xor_
+  xor-inverse         : Inverse true not _xor_
+  ∧-distribˡ-xor      : _∧_ DistributesOverˡ _xor_
+  ∧-distribʳ-xor      : _∧_ DistributesOverʳ _xor_
+  ∧-distrib-xor       : _∧_ DistributesOver _xor_
+  xor-annihilates-not : (not x) xor (not y) ≡ x xor y
+  ```
+
+* Exposed container combinator conversion functions from `Data.Container.Combinator.Properties`
+  separately from their correctness proofs in `Data.Container.Combinator`:
+  ```
+  to-id      : F.id A → ⟦ id ⟧ A
+  from-id    : ⟦ id ⟧ A → F.id A
+  to-const   : A → ⟦ const A ⟧ B
+  from-const : ⟦ const A ⟧ B → A
+  to-∘       : ⟦ C₁ ⟧ (⟦ C₂ ⟧ A) → ⟦ C₁ ∘ C₂ ⟧ A
+  from-∘     : ⟦ C₁ ∘ C₂ ⟧ A → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A)
+  to-×       : ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A → ⟦ C₁ × C₂ ⟧ A
+  from-×     : ⟦ C₁ × C₂ ⟧ A → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A
+  to-Π       : (∀ i → ⟦ Cᵢ i ⟧ A) → ⟦ Π I Cᵢ ⟧ A
+  from-Π     : ⟦ Π I Cᵢ ⟧ A → ∀ i → ⟦ Cᵢ i ⟧ A
+  to-⊎       : ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A → ⟦ C₁ ⊎ C₂ ⟧ A
+  from-⊎     : ⟦ C₁ ⊎ C₂ ⟧ A → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A
+  to-Σ       : (∃ λ i → ⟦ C i ⟧ A) → ⟦ Σ I C ⟧ A
+  from-Σ     : ⟦ Σ I C ⟧ A → ∃ λ i → ⟦ C i ⟧ A
+  ```
+
+* Added new functions in `Data.Fin.Base`:
+  ```
+  finToFun  : Fin (m ^ n) → (Fin n → Fin m)
+  funToFin  : (Fin m → Fin n) → Fin (n ^ m)
+  quotient  : Fin (m * n) → Fin m
+  remainder : Fin (m * n) → Fin n
+  ```
+
+* Added new proofs in `Data.Fin.Induction`:
+  ```agda
+  spo-wellFounded : IsStrictPartialOrder _≈_ _⊏_ → WellFounded _⊏_
+  spo-noetherian  : IsStrictPartialOrder _≈_ _⊏_ → WellFounded (flip _⊏_)
+
+  <-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j
+  ```
+
+* Added new definitions and proofs in `Data.Fin.Permutation`:
+  ```agda
+  insert         : Fin (suc m) → Fin (suc n) → Permutation m n → Permutation (suc m) (suc n)
+  insert-punchIn : insert i j π ⟨$⟩ʳ punchIn i k ≡ punchIn j (π ⟨$⟩ʳ k)
+  insert-remove  : insert i (π ⟨$⟩ʳ i) (remove i π) ≈ π
+  remove-insert  : remove i (insert i j π) ≈ π
+  ```
+
+* Added new proofs in `Data.Fin.Properties`:
+  ```
+  1↔⊤                : Fin 1 ↔ ⊤
+  2↔Bool             : Fin 2 ↔ Bool
+
+  0≢1+n              : zero ≢ suc i
+
+  ↑ˡ-injective       : i ↑ˡ n ≡ j ↑ˡ n → i ≡ j
+  ↑ʳ-injective       : n ↑ʳ i ≡ n ↑ʳ j → i ≡ j
+  finTofun-funToFin  : funToFin ∘ finToFun ≗ id
+  funTofin-funToFun  : finToFun (funToFin f) ≗ f
+  ^↔→                : Extensionality _ _ → Fin (m ^ n) ↔ (Fin n → Fin m)
+
+  toℕ-mono-<         : i < j → toℕ i ℕ.< toℕ j
+  toℕ-mono-≤         : i ≤ j → toℕ i ℕ.≤ toℕ j
+  toℕ-cancel-≤       : toℕ i ℕ.≤ toℕ j → i ≤ j
+  toℕ-cancel-<       : toℕ i ℕ.< toℕ j → i < j
+
+  splitAt⁻¹-↑ˡ       : splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i
+  splitAt⁻¹-↑ʳ       : splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i
+
+  toℕ-combine        : toℕ (combine i j) ≡ k ℕ.* toℕ i ℕ.+ toℕ j
+  combine-injectiveˡ : combine i j ≡ combine k l → i ≡ k
+  combine-injectiveʳ : combine i j ≡ combine k l → j ≡ l
+  combine-injective  : combine i j ≡ combine k l → i ≡ k × j ≡ l
+  combine-surjective : ∀ i → ∃₂ λ j k → combine j k ≡ i
+  combine-monoˡ-<    : i < j → combine i k < combine j l
+
+  ℕ-ℕ≡toℕ‿ℕ-         : n ℕ-ℕ i ≡ toℕ (n ℕ- i)
+
+  lower₁-injective   : lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
+  pinch-injective    : suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k
+
+  i<1+i              : i < suc i
+
+  injective⇒≤               : ∀ {f : Fin m → Fin n} → Injective f → m ℕ.≤ n
+  <⇒notInjective            : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective f)
+  ℕ→Fin-notInjective        : ∀ (f : ℕ → Fin n) → ¬ (Injective f)
+  cantor-schröder-bernstein : ∀ {f : Fin m → Fin n} {g : Fin n → Fin m} → Injective f → Injective g → m ≡ n
+
+  cast-is-id    : cast eq k ≡ k
+  subst-is-cast : subst Fin eq k ≡ cast eq k
+  cast-trans    : cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k
+
+  fromℕ≢inject₁      : {i : Fin n} → fromℕ n ≢ inject₁ i
+
+  inject≤-trans      : inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (≤-trans m≤n n≤o)
+  inject≤-irrelevant : inject≤ i m≤n ≡ inject≤ i m≤n′
+  i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j
+  ```
+
+* Added new lemmas in `Data.Fin.Substitution.Lemmas.TermLemmas`:
+  ```
+  map-var≡ : (∀ x → lookup ρ₁ x ≡ f x) → (∀ x → lookup ρ₂ x ≡ T.var (f x)) → map var ρ₁ ≡ ρ₂
+  wk≡wk    : map var VarSubst.wk ≡ T.wk {n = n}
+  id≡id    : map var VarSubst.id ≡ T.id {n = n}
+  sub≡sub  : map var (VarSubst.sub x) ≡ T.sub (T.var x)
+  ↑≡↑      : map var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
+  /Var≡/   : t /Var ρ ≡ t T./ map T.var ρ
+
+  sub-renaming-commutes : t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
+  sub-commutes-with-renaming : t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)
+  ```
+
+* Added new functions and definitions in `Data.Integer.Base`:
+  ```agda
+  _^_ : ℤ → ℕ → ℤ
+
+  +-0-rawGroup  : Rawgroup 0ℓ 0ℓ
+
+  *-rawMagma    : RawMagma 0ℓ 0ℓ
+  *-1-rawMonoid : RawMonoid 0ℓ 0ℓ
+ ```
+
+* Added new proofs in `Data.Integer.Properties`:
+  ```agda
+  sign-cong′ : s₁ ◃ n₁ ≡ s₂ ◃ n₂ → s₁ ≡ s₂ ⊎ (n₁ ≡ 0 × n₂ ≡ 0)
+  ≤-⊖        : m ℕ.≤ n → n ⊖ m ≡ + (n ∸ m)
+  ∣⊖∣-≤      : m ℕ.≤ n → ∣ m ⊖ n ∣ ≡ n ∸ m
+  ∣-∣-≤      : i ≤ j → + ∣ i - j ∣ ≡ j - i
+
+  i^n≡0⇒i≡0      : i ^ n ≡ 0ℤ → i ≡ 0ℤ
+  ^-identityʳ    : i ^ 1 ≡ i
+  ^-zeroˡ        : 1 ^ n ≡ 1
+  ^-*-assoc      : (i ^ m) ^ n ≡ i ^ (m ℕ.* n)
+  ^-distribˡ-+-* : i ^ (m ℕ.+ n) ≡ i ^ m * i ^ n
+
+  ^-isMagmaHomomorphism  : IsMagmaHomomorphism  ℕ.+-rawMagma    *-rawMagma    (i ^_)
+  ^-isMonoidHomomorphism : IsMonoidHomomorphism ℕ.+-0-rawMonoid *-1-rawMonoid (i ^_)
+  ```
+
+* Added new proofs in `Data.Integer.GCD`:
+  ```agda
+  gcd-assoc : Associative gcd
+  gcd-zeroˡ : LeftZero 1ℤ gcd
+  gcd-zeroʳ : RightZero 1ℤ gcd
+  gcd-zero  : Zero 1ℤ gcd
+  ```
+
+* Added new functions and definitions to `Data.List.Base`:
+  ```agda
+  takeWhileᵇ   : (A → Bool) → List A → List A
+  dropWhileᵇ   : (A → Bool) → List A → List A
+  filterᵇ      : (A → Bool) → List A → List A
+  partitionᵇ   : (A → Bool) → List A → List A × List A
+  spanᵇ        : (A → Bool) → List A → List A × List A
+  breakᵇ       : (A → Bool) → List A → List A × List A
+  linesByᵇ     : (A → Bool) → List A → List (List A)
+  wordsByᵇ     : (A → Bool) → List A → List (List A)
+  derunᵇ       : (A → A → Bool) → List A → List A
+  deduplicateᵇ : (A → A → Bool) → List A → List A
+
+  findᵇ        : (A → Bool) → List A -> Maybe A
+  findIndexᵇ   : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)
+  findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)
+  find         : Decidable P → List A → Maybe A
+  findIndex    : Decidable P → (xs : List A) → Maybe $ Fin (length xs)
+  findIndices  : Decidable P → (xs : List A) → List $ Fin (length xs)
+
+  catMaybes       : List (Maybe A) → List A
+  ap              : List (A → B) → List A → List B
+  ++-rawMagma     : Set a → RawMagma a _
+  ++-[]-rawMonoid : Set a → RawMonoid a _
+
+  iterate : (A → A) → A → ℕ → List A
+  insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
+  updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
+  ```
+
+* Added new proofs in `Data.List.Relation.Binary.Lex.Strict`:
+  ```agda
+  xs≮[] : ¬ xs < []
+  ```
+
+* Added new proofs to `Data.List.Relation.Binary.Permutation.Propositional.Properties`:
+  ```agda
+  Any-resp-[σ⁻¹∘σ] : (σ : xs ↭ ys) → (ix : Any P xs) → Any-resp-↭ (trans σ (↭-sym σ)) ix ≡ ix
+  ∈-resp-[σ⁻¹∘σ]   : (σ : xs ↭ ys) → (ix : x ∈ xs)   → ∈-resp-↭   (trans σ (↭-sym σ)) ix ≡ ix
+  ```
+
+* In `Data.List.Relation.Binary.Permutation.Setoid.Properties`:
+  ```agda
+  foldr-commMonoid : xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys
+  ```
+
+* Added new function to `Data.List.Relation.Binary.Permutation.Propositional.Properties`
+  ```agda
+  ↭-reverse : reverse xs ↭ xs
+  ```
+
+* Added new proofs to `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
+  ```
+  ⊆-mergeˡ : xs ⊆ merge _≤?_ xs ys
+  ⊆-mergeʳ : ys ⊆ merge _≤?_ xs ys
+  ```
+
+* Added new functions in `Data.List.Relation.Unary.All`:
+  ```
+  decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
+  ```
+
+* Added new proof to `Data.List.Relation.Unary.All.Properties`:
+  ```agda
+  gmap⁻ : Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P
+  ```
+
+* Added new functions in `Data.List.Fresh.Relation.Unary.All`:
+  ```
+  decide :  Π[ P ∪ Q ] → Π[ All {R = R} P ∪ Any Q ]
+  ```
+
+* Added new proofs to `Data.List.Membership.Propositional.Properties`:
+  ```agda
+  mapWith∈-id  : mapWith∈ xs (λ {x} _ → x) ≡ xs
+  map-mapWith∈ : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
+  ```
+
+* Added new proofs to `Data.List.Membership.Setoid.Properties`:
+  ```agda
+  mapWith∈-id     : mapWith∈ xs (λ {x} _ → x) ≡ xs
+  map-mapWith∈    : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
+  index-injective : index x₁∈xs ≡ index x₂∈xs → x₁ ≈ x₂
+
+  ∈-++⁺∘++⁻         : (p : v ∈ xs ++ ys) → [ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ (∈-++⁻ xs p) ≡ p
+  ∈-++⁻∘++⁺         : (p : v ∈ xs ⊎ v ∈ ys) → ∈-++⁻ xs ([ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ p) ≡ p
+  ∈-++↔             : (v ∈ xs ⊎ v ∈ ys) ↔ v ∈ xs ++ ys
+  ∈-++-comm         : v ∈ xs ++ ys → v ∈ ys ++ xs
+  ∈-++-comm∘++-comm : (p : v ∈ xs ++ ys) → ∈-++-comm ys xs (∈-++-comm xs ys p) ≡ p
+  ∈-++↔++           : v ∈ xs ++ ys ↔ v ∈ ys ++ xs
+  ```
+
+* Add new proofs in `Data.List.Properties`:
+  ```agda
+  ∈⇒∣product : n ∈ ns → n ∣ product ns
+  ∷ʳ-++      : xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
+
+  concatMap-cong : f ≗ g → concatMap f ≗ concatMap g
+  concatMap-pure : concatMap [_] ≗ id
+  concatMap-map  : concatMap g (map f xs) ≡ concatMap (g ∘′ f) xs
+  map-concatMap  : map f ∘′ concatMap g ≗ concatMap (map f ∘′ g)
+
+  length-isMagmaHomomorphism  : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
+  length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length
+
+  take-map : take n (map f xs) ≡ map f (take n xs)
+  drop-map : drop n (map f xs) ≡ map f (drop n xs)
+  head-map : head (map f xs) ≡ Maybe.map f (head xs)
+
+  take-suc               : take (suc m) xs ≡ take m xs ∷ʳ lookup xs i
+  take-suc-tabulate      : take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f i
+  drop-take-suc          : drop m (take (suc m) xs) ≡ [ lookup xs i ]
+  drop-take-suc-tabulate : drop m (take (suc m) (tabulate f)) ≡ [ f i ]
+
+  take-all : n ≥ length xs → take n xs ≡ xs
+  drop-all : n ≥ length xs → drop n xs ≡ []
+
+  take-[] : take m [] ≡ []
+  drop-[] : drop m [] ≡ []
+
+  drop-drop : drop n (drop m xs) ≡ drop (m + n) xs
+
+  lookup-replicate  : lookup (replicate n x) i ≡ x
+  map-replicate     : map f (replicate n x) ≡ replicate n (f x)
+  zipWith-replicate : zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)
+
+  length-iterate : length (iterate f x n) ≡ n
+  iterate-id     : iterate id x n ≡ replicate n x
+  lookup-iterate : lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)
+
+  length-insertAt   : length (insertAt xs i v) ≡ suc (length xs)
+  length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
+  removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
+  insertAt-removeAt : insertAt (removeAt xs i) (cast (sym (lengthAt-removeAt xs i)) i) (lookup xs i) ≡ xs
+
+  cartesianProductWith-zeroˡ       : cartesianProductWith f [] ys ≡ []
+  cartesianProductWith-zeroʳ       : cartesianProductWith f xs [] ≡ []
+  cartesianProductWith-distribʳ-++ : cartesianProductWith f (xs ++ ys) zs ≡
+                                     cartesianProductWith f xs zs ++ cartesianProductWith f ys zs
+
+  foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
+  foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs
+  ```
+
+* In `Data.List.NonEmpty.Base`:
+  ```agda
+  drop+ : ℕ → List⁺ A → List⁺ A
+  ```
+
+* Added new proofs in `Data.List.NonEmpty.Properties`:
+  ```agda
+  length-++⁺      : length (xs ++⁺ ys) ≡ length xs + length ys
+  length-++⁺-tail : length (xs ++⁺ ys) ≡ suc (length xs + length (tail ys))
+  ++-++⁺          : (xs ++ ys) ++⁺ zs ≡ xs ++⁺ ys ++⁺ zs
+  ++⁺-cancelˡ′    : xs ++⁺ zs ≡ ys ++⁺ zs′ → List.length xs ≡ List.length ys → zs ≡ zs′
+  ++⁺-cancelˡ     : xs ++⁺ ys ≡ xs ++⁺ zs → ys ≡ zs
+  drop+-++⁺       : drop+ (length xs) (xs ++⁺ ys) ≡ ys
+  map-++⁺-commute : map f (xs ++⁺ ys) ≡ map f xs ++⁺ map f ys
+  length-map      : length (map f xs) ≡ length xs
+  map-cong        : f ≗ g → map f ≗ map g
+  map-compose     : map (g ∘ f) ≗ map g ∘ map f
+  ```
+
+* Added new proof to `Data.Maybe.Properties`
+  ```agda
+  <∣>-idem : Idempotent _<∣>_
+  ```
+
+* Added new patterns and definitions to `Data.Nat.Base`:
+  ```agda
+  pattern z<s {n}         = s≤s (z≤n {n})
+  pattern s<s {m} {n} m<n = s≤s {m} {n} m<n
+
+  s≤s⁻¹ : suc m ≤ suc n → m ≤ n
+  s<s⁻¹ : suc m < suc n → m < n
+
+  pattern <′-base          = ≤′-refl
+  pattern <′-step {n} m<′n = ≤′-step {n} m<′n
+
+  pattern ≤″-offset k = less-than-or-equal {k} refl
+  pattern <″-offset k = ≤″-offset k
+
+  s≤″s⁻¹ : suc m ≤″ suc n → m ≤″ n
+  s<″s⁻¹ : suc m <″ suc n → m <″ n
+
+  pattern 2+ n = suc (suc n)
+  pattern 1<2+n {n} = s<s (z<s {n})
+
+  NonTrivial            : Pred ℕ 0ℓ
+  instance nonTrivial   : NonTrivial (2+ n)
+  n>1⇒nonTrivial        : 1 < n → NonTrivial n
+  nonZero⇒≢1⇒nonTrivial : .{{NonZero n}} → n ≢ 1 → NonTrivial n
+  recompute-nonTrivial  : .{{NonTrivial n}} → NonTrivial n
+  nonTrivial⇒nonZero    : .{{NonTrivial n}} → NonZero n
+  nonTrivial⇒n>1        : .{{NonTrivial n}} → 1 < n
+  nonTrivial⇒≢1         : .{{NonTrivial n}} → n ≢ 1
+
+  _⊔′_ : ℕ → ℕ → ℕ
+  _⊓′_ : ℕ → ℕ → ℕ
+  ∣_-_∣′ : ℕ → ℕ → ℕ
+  _! : ℕ → ℕ
+
+  parity : ℕ → Parity
+
+  +-rawMagma          : RawMagma 0ℓ 0ℓ
+  +-0-rawMonoid       : RawMonoid 0ℓ 0ℓ
+  *-rawMagma          : RawMagma 0ℓ 0ℓ
+  *-1-rawMonoid       : RawMonoid 0ℓ 0ℓ
+  +-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
+  +-*-rawSemiring     : RawSemiring 0ℓ 0ℓ
+  ```
+
+* Added a new proof to `Data.Nat.Binary.Properties`:
+  ```agda
+  suc-injective : Injective _≡_ _≡_ suc
+  toℕ-inverseˡ  : Inverseˡ _≡_ _≡_ toℕ fromℕ
+  toℕ-inverseʳ  : Inverseʳ _≡_ _≡_ toℕ fromℕ
+  toℕ-inverseᵇ  : Inverseᵇ _≡_ _≡_ toℕ fromℕ
+
+  <-asym : Asymmetric _<_
+  ```
+
+* Added a new pattern synonym and a new definition to `Data.Nat.Divisibility.Core`:
+  ```agda
+  pattern divides-refl q = divides q refl
+  record _HasNonTrivialDivisorLessThan_ (m n : ℕ) : Set where
+  ```
+
+* Added new proofs to `Data.Nat.Divisibility`:
+  ```agda
+  hasNonTrivialDivisor-≢ : .{{NonTrivial d}} → .{{NonZero n}} → d ≢ n → d ∣ n → n HasNonTrivialDivisorLessThan n
+  hasNonTrivialDivisor-∣ : m HasNonTrivialDivisorLessThan n → m ∣ o → o HasNonTrivialDivisorLessThan n
+  hasNonTrivialDivisor-≤ : m HasNonTrivialDivisorLessThan n → n ≤ o → m HasNonTrivialDivisorLessThan o
+  ```
+
+* Added new definitions, smart constructors and proofs to `Data.Nat.Primality`:
+  ```agda
+  infix 10 _Rough_
+  _Rough_           : ℕ → Pred ℕ _
+  0-rough           : 0 Rough n
+  1-rough           : 1 Rough n
+  2-rough           : 2 Rough n
+  rough⇒≤           : .{{NonTrivial n}} → m Rough n → m ≤ n
+  ∤⇒rough-suc        : m ∤ n → m Rough n → (suc m) Rough n
+  rough∧∣⇒rough     : m Rough o → n ∣ o → m Rough n
+  Composite         : ℕ → Set
+  composite-≢       : .{{NonTrivial d}} → .{{NonZero n}} → d ≢ n → d ∣ n → Composite n
+  composite-∣       : .{{NonZero n}} → Composite m → m ∣ n → Composite n
+  composite?        : Decidable Composite
+  Irreducible       : ℕ → Set
+  irreducible?      : Decidable Irreducible
+  composite⇒¬prime  : Composite n → ¬ Prime n
+  ¬composite⇒prime  : .{{NonTrivial n} → ¬ Composite n → Prime n
+  prime⇒¬composite  : Prime n → ¬ Composite n
+  ¬prime⇒composite  : .{{NonTrivial n} → ¬ Prime n → Composite n
+  prime⇒irreducible : Prime p → Irreducible p
+  irreducible⇒prime : .{{NonTrivial p}} → Irreducible p → Prime p
+  euclidsLemma      : Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n
+  ```
+
+* Added new proofs in `Data.Nat.Properties`:
+  ```agda
+  nonZero?  : Decidable NonZero
+  n≮0       : n ≮ 0
+  n+1+m≢m   : n + suc m ≢ m
+  m*n≡0⇒m≡0 : .{{_ : NonZero n}} → m * n ≡ 0 → m ≡ 0
+  n>0⇒n≢0   : n > 0 → n ≢ 0
+  m*n≢0     : .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
+  m*n≢0⇒m≢0 : .{{NonZero (m * n)}} → NonZero m
+  m*n≢0⇒n≢0 : .{{NonZero (m * n)}} → NonZero n
+  m≢0∧n>1⇒m*n>1 : .{{_ : NonZero m}} .{{_ : NonTrivial n}} → NonTrivial (m * n)
+  n≢0∧m>1⇒m*n>1 : .{{_ : NonZero n}} .{{_ : NonTrivial m}} → NonTrivial (m * n)
+  m^n≢0     : .{{_ : NonZero m}} → NonZero (m ^ n)
+  m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n
+
+  s<s-injective : s<s p ≡ s<s q → p ≡ q
+  <-step        : m < n → m < 1 + n
+  m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n
+
+  pred-mono-≤   : m ≤ n → pred m ≤ pred n
+  pred-mono-<   : .{{_ : NonZero m}} → m < n → pred m < pred n
+
+  z<′s : zero <′ suc n
+  s<′s : m <′ n → suc m <′ suc n
+  <⇒<′ : m < n → m <′ n
+  <′⇒< : m <′ n → m < n
+
+  ≤″-proof : (le : m ≤″ n) → let less-than-or-equal {k} _ = le in m + k ≡ n
+
+  m+n≤p⇒m≤p∸n         : m + n ≤ p → m ≤ p ∸ n
+  m≤p∸n⇒m+n≤p         : n ≤ p → m ≤ p ∸ n → m + n ≤ p
+
+  1≤n!    : 1 ≤ n !
+  _!≢0    : NonZero (n !)
+  _!*_!≢0 : NonZero (m ! * n !)
+
+  anyUpTo? : ∀ (P? : U.Decidable P) (v : ℕ) → Dec (∃ λ n → n < v × P n)
+  allUpTo? : ∀ (P? : U.Decidable P) (v : ℕ) → Dec (∀ {n} → n < v → P n)
+
+  n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1
+
+  m^n>0 : .{{_ : NonZero m}} n → m ^ n > 0
+
+  ^-monoˡ-≤ : (_^ n) Preserves _≤_ ⟶ _≤_
+  ^-monoʳ-≤ : .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
+  ^-monoˡ-< : .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
+  ^-monoʳ-< : 1 < m → (m ^_) Preserves _<_ ⟶ _<_
+
+  n≡⌊n+n/2⌋ : n ≡ ⌊ n + n /2⌋
+  n≡⌈n+n/2⌉ : n ≡ ⌈ n + n /2⌉
+
+  m<n⇒m<n*o : .{{_ : NonZero o}} → m < n → m < n * o
+  m<n⇒m<o*n : .{{_ : NonZero o}} → m < n → m < o * n
+  ∸-monoˡ-< : m < o → n ≤ m → m ∸ n < o ∸ n
+
+  m≤n⇒∣m-n∣≡n∸m : m ≤ n → ∣ m - n ∣ ≡ n ∸ m
+
+  ⊔≡⊔′ : m ⊔ n ≡ m ⊔′ n
+  ⊓≡⊓′ : m ⊓ n ≡ m ⊓′ n
+  ∣-∣≡∣-∣′ : ∣ m - n ∣ ≡ ∣ m - n ∣′
+
+  nonTrivial? : Decidable NonTrivial
+  eq? : A ↣ ℕ → DecidableEquality A
+  ≤-<-connex : Connex _≤_ _<_
+  ≥->-connex : Connex _≥_ _>_
+  <-≤-connex : Connex _<_ _≤_
+  >-≥-connex : Connex _>_ _≥_
+  <-cmp : Trichotomous _≡_ _<_
+  anyUpTo? : (P? : Decidable P) → ∀ v → Dec (∃ λ n → n < v × P n)
+  allUpTo? : (P? : Decidable P) → ∀ v → Dec (∀ {n} → n < v → P n)
+  ```
+
+* Added new proofs in `Data.Nat.Combinatorics`:
+  ```agda
+  [n-k]*[n-k-1]!≡[n-k]!   : k < n → (n ∸ k) * (n ∸ suc k) ! ≡ (n ∸ k) !
+  [n-k]*d[k+1]≡[k+1]*d[k] : k < n → (n ∸ k) * ((suc k) ! * (n ∸ suc k) !) ≡ (suc k) * (k ! * (n ∸ k) !)
+  k![n∸k]!∣n!             : k ≤ n → k ! * (n ∸ k) ! ∣ n !
+  nP1≡n                   : n P 1 ≡ n
+  nC1≡n                   : n C 1 ≡ n
+  nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k
+  ```
+
+* Added new proofs in `Data.Nat.DivMod`:
+  ```agda
+  m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n
+  m*n/m!≡n/[m∸1]! : .{{_ : NonZero m}} → m * n / m ! ≡ n / (pred m) !
+
+  %-congˡ             : .{{_ : NonZero o}} → m ≡ n → m % o ≡ n % o
+  %-congʳ             : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ≡ n → o % m ≡ o % n
+  m≤n⇒[n∸m]%m≡n%m     : .{{_ : NonZero m}} → m ≤ n → (n ∸ m) % m ≡ n % m
+  m*n≤o⇒[o∸m*n]%n≡o%n : .{{_ : NonZero n}} → m * n ≤ o → (o ∸ m * n) % n ≡ o % n
+  m∣n⇒o%n%m≡o%m       : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ∣ n → o % n % m ≡ o % m
+  m<n⇒m%n≡m           : .{{_ : NonZero n}} → m < n → m % n ≡ m
+  m*n/o*n≡m/o         : .{{_ : NonZero o}} {{_ : NonZero (o * n)}} → m * n / (o * n) ≡ m / o
+  m<n*o⇒m/o<n         : .{{_ : NonZero o}} → m < n * o → m / o < n
+  [m∸n]/n≡m/n∸1       : .{{_ : NonZero n}} → (m ∸ n) / n ≡ pred (m / n)
+  [m∸n*o]/o≡m/o∸n     : .{{_ : NonZero o}} → (m ∸ n * o) / o ≡ m / o ∸ n
+  m/n/o≡m/[n*o]       : .{{_ : NonZero n}} .{{_ : NonZero o}} .{{_ : NonZero (n * o)}} → m / n / o ≡ m / (n * o)
+  m%[n*o]/o≡m/o%n     : .{{_ : NonZero n}} .{{_ : NonZero o}} {{_ : NonZero (n * o)}} → m % (n * o) / o ≡ m / o % n
+  m%n*o≡m*o%[n*o]     : .{{_ : NonZero n}} {{_ : NonZero (n * o)}} → m % n * o ≡ m * o % (n * o)
+
+  [m*n+o]%[p*n]≡[m*n]%[p*n]+o : {{_ : NonZero (p * n)}} → o < n → (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o
+  ```
+
+* Added new proofs in `Data.Nat.Divisibility`:
+  ```agda
+  n∣m*n*o       : n ∣ m * n * o
+  m*n∣⇒m∣       : m * n ∣ i → m ∣ i
+  m*n∣⇒n∣       : m * n ∣ i → n ∣ i
+  m≤n⇒m!∣n!     : m ≤ n → m ! ∣ n !
+  m/n/o≡m/[n*o] : .{{NonZero n}} .{{NonZero o}} → n * o ∣ m → (m / n) / o ≡ m / (n * o)
+  ```
+
+* Added new proofs in `Data.Nat.GCD`:
+  ```agda
+  gcd-assoc     : Associative gcd
+  gcd-identityˡ : LeftIdentity 0 gcd
+  gcd-identityʳ : RightIdentity 0 gcd
+  gcd-identity  : Identity 0 gcd
+  gcd-zeroˡ     : LeftZero 1 gcd
+  gcd-zeroʳ     : RightZero 1 gcd
+  gcd-zero      : Zero 1 gcd
+  ```
+
+* Added new patterns in `Data.Nat.Reflection`:
+  ```agda
+  pattern `ℕ     = def (quote ℕ) []
+  pattern `zero  = con (quote ℕ.zero) []
+  pattern `suc x = con (quote ℕ.suc) (x ⟨∷⟩ [])
+  ```
+
+* Added new functions and proofs in `Data.Nat.GeneralisedArithmetic`:
+  ```agda
+  iterate         : (A → A) → A → ℕ → A
+  iterate-is-fold : fold z s m ≡ iterate s z m
+  ```
+
+* Added new proofs in `Data.Parity.Properties`:
+  ```agda
+  suc-homo-⁻¹ : (parity (suc n)) ⁻¹ ≡ parity n
+  +-homo-+    : parity (m ℕ.+ n) ≡ parity m ℙ.+ parity n
+  *-homo-*    : parity (m ℕ.* n) ≡ parity m ℙ.* parity n
+  parity-isMagmaHomomorphism : IsMagmaHomomorphism ℕ.+-rawMagma ℙ.+-rawMagma parity
+  parity-isMonoidHomomorphism : IsMonoidHomomorphism ℕ.+-0-rawMonoid ℙ.+-0-rawMonoid parity
+  parity-isNearSemiringHomomorphism : IsNearSemiringHomomorphism ℕ.+-*-rawNearSemiring ℙ.+-*-rawNearSemiring parity
+  parity-isSemiringHomomorphism : IsSemiringHomomorphism ℕ.+-*-rawSemiring ℙ.+-*-rawSemiring parity
+  ```
+
+* Added new rounding functions in `Data.Rational.Base`:
+  ```agda
+  floor ceiling truncate round ⌊_⌋ ⌈_⌉ [_] : ℚ → ℤ
+  fracPart : ℚ → ℚ
+  ```
+
+* Added new definitions and proofs in `Data.Rational.Properties`:
+  ```agda
+  ↥ᵘ-toℚᵘ     : ↥ᵘ (toℚᵘ p) ≡ ↥ p
+  ↧ᵘ-toℚᵘ     : ↧ᵘ (toℚᵘ p) ≡ ↧ p
+  ↥p≡↥q≡0⇒p≡q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≡ q
+
+  _≥?_ : Decidable _≥_
+  _>?_ : Decidable _>_
+
+  +-*-rawNearSemiring                 : RawNearSemiring 0ℓ 0ℓ
+  +-*-rawSemiring                     : RawSemiring 0ℓ 0ℓ
+  toℚᵘ-isNearSemiringHomomorphism-+-* : IsNearSemiringHomomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ
+  toℚᵘ-isNearSemiringMonomorphism-+-* : IsNearSemiringMonomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ
+  toℚᵘ-isSemiringHomomorphism-+-*     : IsSemiringHomomorphism     +-*-rawSemiring     ℚᵘ.+-*-rawSemiring     toℚᵘ
+  toℚᵘ-isSemiringMonomorphism-+-*     : IsSemiringMonomorphism     +-*-rawSemiring     ℚᵘ.+-*-rawSemiring     toℚᵘ
+
+  pos⇒nonZero       : .{{Positive p}} → NonZero p
+  neg⇒nonZero       : .{{Negative p}} → NonZero p
+  nonZero⇒1/nonZero : .{{_ : NonZero p}} → NonZero (1/ p)
+
+  <-dense              : Dense _<_
+  <-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
+  <-denseLinearOrder   : DenseLinearOrder 0ℓ 0ℓ 0ℓ
+  ```
+
+* Added new rounding functions in `Data.Rational.Unnormalised.Base`:
+  ```agda
+  floor ceiling truncate round ⌊_⌋ ⌈_⌉ [_] : ℚᵘ → ℤ
+  fracPart : ℚᵘ → ℚᵘ
+  ```
+
+* Added new definitions in `Data.Rational.Unnormalised.Properties`:
+  ```agda
+  ↥p≡0⇒p≃0    : ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ
+  p≃0⇒↥p≡0    : p ≃ 0ℚᵘ → ↥ p ≡ 0ℤ
+  ↥p≡↥q≡0⇒p≃q : ↥ p ≡ 0ℤ → ↥ q ≡ 0ℤ → p ≃ q
+
+  +-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
+  +-*-rawSemiring     : RawSemiring 0ℓ 0ℓ
+
+  ≰⇒≥ : _≰_ ⇒ _≥_
+
+  _≥?_ : Decidable _≥_
+  _>?_ : Decidable _>_
+
+  *-mono-≤-nonNeg   : .{{_ : NonNegative p}} .{{_ : NonNegative r}} → p ≤ q → r ≤ s → p * r ≤ q * s
+  *-mono-<-nonNeg   : .{{_ : NonNegative p}} .{{_ : NonNegative r}} → p < q → r < s → p * r < q * s
+  1/-antimono-≤-pos : .{{_ : Positive p}}    .{{_ : Positive q}}    → p ≤ q → 1/ q ≤ 1/ p
+  ⊓-mono-<          : _⊓_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
+  ⊔-mono-<          : _⊔_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
+
+  pos⇒nonZero          : .{{_ : Positive p}} → NonZero p
+  neg⇒nonZero          : .{{_ : Negative p}} → NonZero p
+  pos+pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p + q)
+  nonNeg+nonNeg⇒nonNeg : .{{_ : NonNegative p}} .{{_ : NonNegative q}} → NonNegative (p + q)
+  pos*pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p * q)
+  nonNeg*nonNeg⇒nonNeg : .{{_ : NonNegative p}} .{{_ : NonNegative q}} → NonNegative (p * q)
+  pos⊓pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊓ q)
+  pos⊔pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊔ q)
+  1/nonZero⇒nonZero    : .{{_ : NonZero p}} → NonZero (1/ p)
+
+  0≄1             : 0ℚᵘ ≄ 1ℚᵘ
+  ≃-≄-irreflexive : Irreflexive _≃_ _≄_
+  ≄-symmetric     : Symmetric _≄_
+  ≄-cotransitive  : Cotransitive _≄_
+  ≄⇒invertible    : p ≄ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)
+
+  <-dense         : Dense _<_
+
+  <-isDenseLinearOrder         : IsDenseLinearOrder _≃_ _<_
+  +-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
+  +-*-isHeytingField           : IsHeytingField _≃_ _≄_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
+
+  <-denseLinearOrder           : DenseLinearOrder 0ℓ 0ℓ 0ℓ
+  +-*-heytingCommutativeRing   : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
+  +-*-heytingField             : HeytingField 0ℓ 0ℓ 0ℓ
+
+  module ≃-Reasoning = SetoidReasoning ≃-setoid
+  ```
+
+* Added new functions to `Data.Product.Nary.NonDependent`:
+  ```agda
+  zipWith : (∀ k → Projₙ as k → Projₙ bs k → Projₙ cs k) → Product n as → Product n bs → Product n cs
+  ```
+
+* Added new proof to `Data.Product.Properties`:
+  ```agda
+  map-cong : f ≗ g → h ≗ i → map f h ≗ map g i
+  ```
+
+* Added new definitions to `Data.Product.Properties`:
+  ```
+  Σ-≡,≡→≡ : (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂) → p₁ ≡ p₂
+  Σ-≡,≡←≡ : p₁ ≡ p₂ → (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂)
+  ×-≡,≡→≡ : (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂) → p₁ ≡ p₂
+  ×-≡,≡←≡ : p₁ ≡ p₂ → (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂)
+  ```
+
+* Added new proofs to `Data.Product.Relation.Binary.Lex.Strict`
+  ```agda
+  ×-respectsʳ : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_
+  ×-respectsˡ : Symmetric _≈₁_ → Transitive _≈₁_ → _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_
+  ×-wellFounded' : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ → WellFounded _<₁_ → WellFounded _<₂_ → WellFounded _<ₗₑₓ_
+  ```
+
+* Added new definitions to `Data.Sign.Base`:
+  ```agda
+  *-rawMagma : RawMagma 0ℓ 0ℓ
+  *-1-rawMonoid : RawMonoid 0ℓ 0ℓ
+  *-1-rawGroup : RawGroup 0ℓ 0ℓ
+  ```
+
+* Added new proofs to `Data.Sign.Properties`:
+  ```agda
+  *-inverse : Inverse + id _*_
+  *-isCommutativeSemigroup : IsCommutativeSemigroup _*_
+  *-isCommutativeMonoid : IsCommutativeMonoid _*_ +
+  *-isGroup : IsGroup _*_ + id
+  *-isAbelianGroup : IsAbelianGroup _*_ + id
+  *-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ
+  *-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
+  *-group : Group 0ℓ 0ℓ
+  *-abelianGroup : AbelianGroup 0ℓ 0ℓ
+  ≡-isDecEquivalence : IsDecEquivalence _≡_
+  ```
+
+* Added new functions in `Data.String.Base`:
+  ```agda
+  wordsByᵇ : (Char → Bool) → String → List String
+  linesByᵇ : (Char → Bool) → String → List String
+  ```
+
+* Added new proofs in `Data.String.Properties`:
+  ```agda
+  ≤-isDecTotalOrder-≈ : IsDecTotalOrder _≈_ _≤_
+  ≤-decTotalOrder-≈   : DecTotalOrder _ _ _
+  ```
+* Added new definitions in `Data.Sum.Properties`:
+  ```agda
+  swap-↔ : (A ⊎ B) ↔ (B ⊎ A)
+  ```
+
+* Added new proofs in `Data.Sum.Properties`:
+  ```agda
+  map-assocˡ : map (map f g) h ∘ assocˡ ≗ assocˡ ∘ map f (map g h)
+  map-assocʳ : map f (map g h) ∘ assocʳ ≗ assocʳ ∘ map (map f g) h
+  ```
+
+* Made `Map` public in `Data.Tree.AVL.IndexedMap`
+
+* Added new definitions in `Data.Vec.Base`:
+  ```agda
+  truncate : m ≤ n → Vec A n → Vec A m
+  pad      : m ≤ n → A → Vec A m → Vec A n
+
+  FoldrOp A B = A → B n → B (suc n)
+  FoldlOp A B = B n → A → B (suc n)
+
+  foldr′ : (A → B → B) → B → Vec A n → B
+  foldl′ : (B → A → B) → B → Vec A n → B
+  countᵇ : (A → Bool) → Vec A n → ℕ
+
+  iterate : (A → A) → A → Vec A n
+
+  diagonal           : Vec (Vec A n) n → Vec A n
+  DiagonalBind._>>=_ : Vec A n → (A → Vec B n) → Vec B n
+
+  _ʳ++_              : Vec A m → Vec A n → Vec A (m + n)
+
+  cast : .(eq : m ≡ n) → Vec A m → Vec A n
+  ```
+
+* Added new instance in `Data.Vec.Effectful`:
+  ```agda
+  monad : RawMonad (λ (A : Set a) → Vec A n)
+  ```
+
+* Added new proofs in `Data.Vec.Properties`:
+  ```agda
+  padRight-refl      : padRight ≤-refl a xs ≡ xs
+  padRight-replicate : replicate a ≡ padRight le a (replicate a)
+  padRight-trans     : padRight (≤-trans m≤n n≤p) a xs ≡ padRight n≤p a (padRight m≤n a xs)
+
+  truncate-refl     : truncate ≤-refl xs ≡ xs
+  truncate-trans    : truncate (≤-trans m≤n n≤p) xs ≡ truncate m≤n (truncate n≤p xs)
+  truncate-padRight : truncate m≤n (padRight m≤n a xs) ≡ xs
+
+  map-const    : map (const x) xs ≡ replicate x
+  map-⊛        : map f xs ⊛ map g xs ≡ map (f ˢ g) xs
+  map-++       : map f (xs ++ ys) ≡ map f xs ++ map f ys
+  map-is-foldr : map f ≗ foldr (Vec B) (λ x ys → f x ∷ ys) []
+  map-∷ʳ       : map f (xs ∷ʳ x) ≡ (map f xs) ∷ʳ (f x)
+  map-reverse  : map f (reverse xs) ≡ reverse (map f xs)
+  map-ʳ++      : map f (xs ʳ++ ys) ≡ map f xs ʳ++ map f ys
+  map-insert   : map f (insert xs i x) ≡ insert (map f xs) i (f x)
+  toList-map   : toList (map f xs) ≡ List.map f (toList xs)
+
+  lookup-concat : lookup (concat xss) (combine i j) ≡ lookup (lookup xss i) j
+
+  ⊛-is->>=    : fs ⊛ xs ≡ fs >>= flip map xs
+  lookup-⊛*   : lookup (fs ⊛* xs) (combine i j) ≡ (lookup fs i $ lookup xs j)
+  ++-is-foldr : xs ++ ys ≡ foldr ((Vec A) ∘ (_+ n)) _∷_ ys xs
+  []≔-++-↑ʳ   : (xs ++ ys) [ m ↑ʳ i ]≔ y ≡ xs ++ (ys [ i ]≔ y)
+  unfold-ʳ++  : xs ʳ++ ys ≡ reverse xs ++ ys
+
+  foldl-universal : (e : C zero) → ∀ {n} → Vec A n → C n) →
+                    (∀ ... → h C g e [] ≡ e) →
+                    (∀ ... → h C g e ∘ (x ∷_) ≗ h (C ∘ suc) g (g e x)) →
+                    h B f e ≗ foldl B f e
+  foldl-fusion  : h d ≡ e → (∀ ... → h (f b x) ≡ g (h b) x) → h ∘ foldl B f d ≗ foldl C g e
+  foldl-∷ʳ      : foldl B f e (ys ∷ʳ y) ≡ f (foldl B f e ys) y
+  foldl-[]      : foldl B f e [] ≡ e
+  foldl-reverse : foldl B {n} f e ∘ reverse ≗ foldr B (flip f) e
+
+  foldr-[]      : foldr B f e [] ≡ e
+  foldr-++      : foldr B f e (xs ++ ys) ≡ foldr (B ∘ (_+ n)) f (foldr B f e ys) xs
+  foldr-∷ʳ      : foldr B f e (ys ∷ʳ y) ≡ foldr (B ∘ suc) f (f y e) ys
+  foldr-ʳ++     : foldr B f e (xs ʳ++ ys) ≡ foldl (B ∘ (_+ n)) (flip f) (foldr B f e ys) xs
+  foldr-reverse : foldr B f e ∘ reverse ≗ foldl B (flip f) e
+
+  ∷ʳ-injective  : xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
+  ∷ʳ-injectiveˡ : xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys
+  ∷ʳ-injectiveʳ : xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y
+
+  unfold-∷ʳ : cast eq (xs ∷ʳ x) ≡ xs ++ [ x ]
+  init-∷ʳ   : init (xs ∷ʳ x) ≡ xs
+  last-∷ʳ   : last (xs ∷ʳ x) ≡ x
+  cast-∷ʳ   : cast eq (xs ∷ʳ x) ≡ (cast (cong pred eq) xs) ∷ʳ x
+  ++-∷ʳ     : cast eq ((xs ++ ys) ∷ʳ z) ≡ xs ++ (ys ∷ʳ z)
+  ∷ʳ-++     : cast eq ((xs ∷ʳ a) ++ ys) ≡ xs ++ (a ∷ ys)
+
+  reverse-∷          : reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
+  reverse-involutive : Involutive _≡_ reverse
+  reverse-reverse    : reverse xs ≡ ys → reverse ys ≡ xs
+  reverse-injective  : reverse xs ≡ reverse ys → xs ≡ ys
+
+  transpose-replicate : transpose (replicate xs) ≡ map replicate xs
+  toList-replicate    : toList (replicate {n = n} a) ≡ List.replicate n a
+
+  toList-++ : toList (xs ++ ys) ≡ toList xs List.++ toList ys
+
+  cast-is-id    : cast eq xs ≡ xs
+  subst-is-cast : subst (Vec A) eq xs ≡ cast eq xs
+  cast-sym      : cast eq xs ≡ ys → cast (sym eq) ys ≡ xs
+  cast-trans    : cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
+  map-cast      : map f (cast eq xs) ≡ cast eq (map f xs)
+  lookup-cast   : lookup (cast eq xs) (Fin.cast eq i) ≡ lookup xs i
+  lookup-cast₁  : lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)
+  lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
+  cast-reverse  : cast eq ∘ reverse ≗ reverse ∘ cast eq
+  cast-++ˡ      : cast (cong (_+ n) eq) (xs ++ ys) ≡ cast eq xs ++ ys
+  cast-++ʳ      : cast (cong (m +_) eq) (xs ++ ys) ≡ xs ++ cast eq ys
+
+  iterate-id     : iterate id x n ≡ replicate x
+  take-iterate   : take n (iterate f x (n + m)) ≡ iterate f x n
+  drop-iterate   : drop n (iterate f x n) ≡ []
+  lookup-iterate : lookup (iterate f x n) i ≡ ℕ.iterate f x (toℕ i)
+  toList-iterate : toList (iterate f x n) ≡ List.iterate f x n
+
+  zipwith-++ : zipWith f (xs ++ ys) (xs' ++ ys') ≡ zipWith f xs xs' ++ zipWith f ys ys'
+
+  ++-assoc     : cast eq ((xs ++ ys) ++ zs) ≡ xs ++ (ys ++ zs)
+  ++-identityʳ : cast eq (xs ++ []) ≡ xs
+  init-reverse : init ∘ reverse ≗ reverse ∘ tail
+  last-reverse : last ∘ reverse ≗ head
+  reverse-++   : cast eq (reverse (xs ++ ys)) ≡ reverse ys ++ reverse xs
+
+  toList-cast   : toList (cast eq xs) ≡ toList xs
+  cast-fromList : cast _ (fromList xs) ≡ fromList ys
+  fromList-map  : cast _ (fromList (List.map f xs)) ≡ map f (fromList xs)
+  fromList-++   : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
+  fromList-reverse : cast (Listₚ.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)
+
+  ∷-ʳ++   : cast eq ((a ∷ xs) ʳ++ ys) ≡ xs ʳ++ (a ∷ ys)
+  ++-ʳ++  : cast eq ((xs ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ʳ++ zs)
+  ʳ++-ʳ++ : cast eq ((xs ʳ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ++ zs)
+
+  length-toList   : List.length (toList xs) ≡ length xs
+  toList-insertAt : toList (insertAt xs i v) ≡ List.insertAt (toList xs) (Fin.cast (cong suc (sym (length-toList xs))) i) v
+
+  truncate≡take       : .(eq : n ≡ m + o) → truncate m≤n xs ≡ take m (cast eq xs)
+  take≡truncate       : take m xs ≡ truncate (m≤m+n m n) xs
+  lookup-truncate     : lookup (truncate m≤n xs) i ≡ lookup xs (Fin.inject≤ i m≤n)
+  lookup-take-inject≤ : lookup (take m xs) i ≡ lookup xs (Fin.inject≤ i (m≤m+n m n))
+  ```
+
+* Added new proofs in `Data.Vec.Membership.Propositional.Properties`:
+  ```agda
+  index-∈-fromList⁺ : Any.index (∈-fromList⁺ v∈xs) ≡ indexₗ v∈xs
+  ```
+
+* Added new isomorphisms to `Data.Unit.Polymorphic.Properties`:
+  ```agda
+  ⊤↔⊤* : ⊤ ↔ ⊤*
+  ```
+
+* Added new proofs in `Data.Vec.Functional.Properties`:
+  ```
+  map-updateAt : f ∘ g ≗ h ∘ f → map f (updateAt i g xs) ≗ updateAt i h (map f xs)
+  ```
+
+* Added new proofs in `Data.Vec.Relation.Binary.Lex.Strict`:
+  ```agda
+  xs≮[] : ¬ xs < []
+
+  <-respectsˡ : IsPartialEquivalence _≈_ → _≺_ Respectsˡ _≈_ → _<_ Respectsˡ _≋_
+  <-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ → _<_ _Respectsʳ _≋_
+  <-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ → WellFounded _<_
+  ```
+
+* Added new function to `Data.Vec.Relation.Binary.Pointwise.Inductive`
+  ```agda
+  cong-[_]≔ : Pointwise _∼_ xs ys → Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)
+  ```
+
+* Added new function to `Data.Vec.Relation.Binary.Equality.Setoid`
+  ```agda
+  map-[]≔ : map f (xs [ i ]≔ p) ≋ map f xs [ i ]≔ f p
+  ```
+
+* Added new functions in `Data.Vec.Relation.Unary.Any`:
+  ```
+  lookup : Any P xs → A
+  ```
+
+* Added new functions in `Data.Vec.Relation.Unary.All`:
+  ```
+  decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]
+
+  lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
+  lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
+  lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
+  lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
+  ```
+
+* Added vector associativity proof to  `Data.Vec.Relation.Binary.Equality.Setoid`:
+  ```
+  ++-assoc : (xs ++ ys) ++ zs ≋ xs ++ (ys ++ zs)
+  ```
+
+* Added new isomorphisms to `Data.Vec.N-ary`:
+  ```agda
+  Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B
+  ```
+
+* Added new isomorphisms to `Data.Vec.Recursive`:
+  ```agda
+  lift↔             : A ↔ B → A ^ n ↔ B ^ n
+  Fin[m^n]↔Fin[m]^n : Fin (m ^ n) ↔ Fin m Vec.^ n
+  ```
+
+* Added new functions in `Effect.Monad.State`:
+  ```
+  runState  : State s a → s → a × s
+  evalState : State s a → s → a
+  execState : State s a → s → s
+  ```
+
+* Added a non-dependent version of `flip` in `Function.Base`:
+  ```agda
+  flip′ : (A → B → C) → (B → A → C)
+  ```
+
+* Added new proofs and definitions in `Function.Bundles`:
+  ```agda
+  LeftInverse.isSplitSurjection : LeftInverse → IsSplitSurjection to
+  LeftInverse.surjection        : LeftInverse → Surjection
+  SplitSurjection               = LeftInverse
+  _⟨$⟩_ = Func.to
+  ```
+
+* Added new proofs to `Function.Properties.Inverse`:
+  ```agda
+  ↔-refl  : Reflexive _↔_
+  ↔-sym   : Symmetric _↔_
+  ↔-trans : Transitive _↔_
+
+  ↔⇒↣   : A ↔ B → A ↣ B
+  ↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)
+
+  Inverse⇒Injection : Inverse S T → Injection S T
+  ```
+
+* Added new proofs in `Function.Construct.Symmetry`:
+  ```
+  bijective     : Bijective ≈₁ ≈₂ f → Symmetric ≈₂ → Transitive ≈₂ → Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹
+  isBijection   : IsBijection ≈₁ ≈₂ f → Congruent ≈₂ ≈₁ f⁻¹ → IsBijection ≈₂ ≈₁ f⁻¹
+  isBijection-≡ : IsBijection ≈₁ _≡_ f → IsBijection _≡_ ≈₁ f⁻¹
+  bijection     : Bijection R S → Congruent IB.Eq₂._≈_ IB.Eq₁._≈_ f⁻¹ → Bijection S R
+  bijection-≡   : Bijection R (setoid B) → Bijection (setoid B) R
+  sym-⤖        : A ⤖ B → B ⤖ A
+  ```
+
+* Added new operations in `Function.Strict`:
+  ```
+  _!|>_  : (a : A) → (∀ a → B a) → B a
+  _!|>′_ : A → (A → B) → B
+  ```
+
+* Added new proof and record in `Function.Structures`:
+  ```agda
+  record IsSplitSurjection (f : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
+  ```
+
+* Added new definition to the `Surjection` module in `Function.Related.Surjection`:
+  ```
+  f⁻ = proj₁ ∘ surjective
+  ```
+
+* Added new proof to `Induction.WellFounded`
+  ```agda
+  Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_
+  acc⇒asym        : Acc _<_ x → x < y → ¬ (y < x)
+  wf⇒asym         : WellFounded _<_ → Asymmetric _<_
+  wf⇒irrefl       : _<_ Respects₂ _≈_ → Symmetric _≈_ → WellFounded _<_ → Irreflexive _≈_ _<_
+  ```
+
+* Added new operations in `IO`:
+  ```
+  Colist.forM  : Colist A → (A → IO B) → IO (Colist B)
+  Colist.forM′ : Colist A → (A → IO B) → IO ⊤
+
+  List.forM  : List A → (A → IO B) → IO (List B)
+  List.forM′ : List A → (A → IO B) → IO ⊤
+  ```
+
+* Added new operations in `IO.Base`:
+  ```
+  lift! : IO A → IO (Lift b A)
+  _<$_  : B → IO A → IO B
+  _=<<_ : (A → IO B) → IO A → IO B
+  _<<_  : IO B → IO A → IO B
+  lift′ : Prim.IO ⊤ → IO {a} ⊤
+
+  when   : Bool → IO ⊤ → IO ⊤
+  unless : Bool → IO ⊤ → IO ⊤
+
+  whenJust  : Maybe A → (A → IO ⊤) → IO ⊤
+  untilJust : IO (Maybe A) → IO A
+  untilRight : (A → IO (A ⊎ B)) → A → IO B
+  ```
+
+* Added new functions in `Reflection.AST.Term`:
+  ```
+  stripPis     : Term → List (String × Arg Type) × Term
+  prependLams  : List (String × Visibility) → Term → Term
+  prependHLams : List String → Term → Term
+  prependVLams : List String → Term → Term
+  ```
+
+* Added new types and operations to `Reflection.TCM`:
+  ```
+  Blocker : Set
+  blockerMeta : Meta → Blocker
+  blockerAny : List Blocker → Blocker
+  blockerAll : List Blocker → Blocker
+  blockTC : Blocker → TC A
+  ```
+
+* Added new operations in `Relation.Binary.Construct.Closure.Equivalence`:
+  ```
+  fold   : IsEquivalence _∼_ → _⟶_ ⇒ _∼_ → EqClosure _⟶_ ⇒ _∼_
+  gfold  : IsEquivalence _∼_ → _⟶_ =[ f ]⇒ _∼_ → EqClosure _⟶_ =[ f ]⇒ _∼_
+  return : _⟶_ ⇒ EqClosure _⟶_
+  join   : EqClosure (EqClosure _⟶_) ⇒ EqClosure _⟶_
+  _⋆     : _⟶₁_ ⇒ EqClosure _⟶₂_ → EqClosure _⟶₁_ ⇒ EqClosure _⟶₂_
+  ```
+
+* Added new operations in `Relation.Binary.Construct.Closure.Symmetric`:
+  ```
+  fold   : Symmetric _∼_ → _⟶_ ⇒ _∼_ → SymClosure _⟶_ ⇒ _∼_
+  gfold  : Symmetric _∼_ → _⟶_ =[ f ]⇒ _∼_ → SymClosure _⟶_ =[ f ]⇒ _∼_
+  return : _⟶_ ⇒ SymClosure _⟶_
+  join   : SymClosure (SymClosure _⟶_) ⇒ SymClosure _⟶_
+  _⋆     : _⟶₁_ ⇒ SymClosure _⟶₂_ → SymClosure _⟶₁_ ⇒ SymClosure _⟶₂_
+  ```
+
+* Added new proofs to `Relation.Binary.Lattice.Properties.{Join,Meet}Semilattice`:
+  ```agda
+  isPosemigroup           : IsPosemigroup _≈_ _≤_ _∨_
+  posemigroup             : Posemigroup c ℓ₁ ℓ₂
+  ≈-dec⇒≤-dec             : Decidable _≈_ → Decidable _≤_
+  ≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_
+  ```
+
+* Added new proofs to `Relation.Binary.Lattice.Properties.Bounded{Join,Meet}Semilattice`:
+  ```agda
+  isCommutativePomonoid : IsCommutativePomonoid _≈_ _≤_ _∨_ ⊥
+  commutativePomonoid   : CommutativePomonoid c ℓ₁ ℓ₂
+  ```
+
+* Added new proofs to `Relation.Binary.Properties.Poset`:
+  ```agda
+  ≤-dec⇒≈-dec             : Decidable _≤_ → Decidable _≈_
+  ≤-dec⇒isDecPartialOrder : Decidable _≤_ → IsDecPartialOrder _≈_ _≤_
+  ```
+
+* Added new proofs in `Relation.Binary.Properties.StrictPartialOrder`:
+  ```agda
+  >-strictPartialOrder : StrictPartialOrder s₁ s₂ s₃
+  ```
+
+* Added new proofs in `Relation.Binary.PropositionalEquality.Properties`:
+  ```
+  subst-application′ : subst Q eq (f x p) ≡ f y (subst P eq p)
+  sym-cong           : sym (cong f p) ≡ cong f (sym p)
+  ```
+
+* Added new proofs in `Relation.Binary.HeterogeneousEquality`:
+  ```
+  subst₂-removable : subst₂ _∼_ eq₁ eq₂ p ≅ p
+  ```
+
+* Added new definitions in `Relation.Unary`:
+  ```
+  _≐_  : Pred A ℓ₁ → Pred A ℓ₂ → Set _
+  _≐′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _
+  _∖_  : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _
+  ```
+
+* Added new proofs in `Relation.Unary.Properties`:
+  ```
+  ⊆-reflexive : _≐_ ⇒ _⊆_
+  ⊆-antisym   : Antisymmetric _≐_ _⊆_
+  ⊆-min       : Min _⊆_ ∅
+  ⊆-max       : Max _⊆_ U
+  ⊂⇒⊆         : _⊂_ ⇒ _⊆_
+  ⊂-trans     : Trans _⊂_ _⊂_ _⊂_
+  ⊂-⊆-trans   : Trans _⊂_ _⊆_ _⊂_
+  ⊆-⊂-trans   : Trans _⊆_ _⊂_ _⊂_
+  ⊂-respʳ-≐   : _⊂_ Respectsʳ _≐_
+  ⊂-respˡ-≐   : _⊂_ Respectsˡ _≐_
+  ⊂-resp-≐    : _⊂_ Respects₂ _≐_
+  ⊂-irrefl    : Irreflexive _≐_ _⊂_
+  ⊂-antisym   : Antisymmetric _≐_ _⊂_
+
+  ∅-⊆′         : (P : Pred A ℓ) → ∅ ⊆′ P
+  ⊆′-U         : (P : Pred A ℓ) → P ⊆′ U
+  ⊆′-refl      : Reflexive {A = Pred A ℓ} _⊆′_
+  ⊆′-reflexive : _≐′_ ⇒ _⊆′_
+  ⊆′-trans     : Trans _⊆′_ _⊆′_ _⊆′_
+  ⊆′-antisym   : Antisymmetric _≐′_ _⊆′_
+  ⊆′-min       : Min _⊆′_ ∅
+  ⊆′-max       : Max _⊆′_ U
+  ⊂′-trans     : Trans _⊂′_ _⊂′_ _⊂′_
+  ⊂′-⊆′-trans  : Trans _⊂′_ _⊆′_ _⊂′_
+  ⊆′-⊂′-trans  : Trans _⊆′_ _⊂′_ _⊂′_
+  ⊂′-respʳ-≐′  : _⊂′_ Respectsʳ _≐′_
+  ⊂′-respˡ-≐′  : _⊂′_ Respectsˡ _≐′_
+  ⊂′-resp-≐′   : _⊂′_ Respects₂ _≐′_
+  ⊂′-irrefl    : Irreflexive _≐′_ _⊂′_
+  ⊂′-antisym   : Antisymmetric _≐′_ _⊂′_
+  ⊂′⇒⊆′        : _⊂′_ ⇒ _⊆′_
+  ⊆⇒⊆′         : _⊆_ ⇒ _⊆′_
+  ⊆′⇒⊆         : _⊆′_ ⇒ _⊆_
+  ⊂⇒⊂′         : _⊂_ ⇒ _⊂′_
+  ⊂′⇒⊂         : _⊂′_ ⇒ _⊂_
+
+  ≐-refl   : Reflexive _≐_
+  ≐-sym    : Sym _≐_ _≐_
+  ≐-trans  : Trans _≐_ _≐_ _≐_
+  ≐′-refl  : Reflexive _≐′_
+  ≐′-sym   : Sym _≐′_ _≐′_
+  ≐′-trans : Trans _≐′_ _≐′_ _≐′_
+  ≐⇒≐′     : _≐_ ⇒ _≐′_
+  ≐′⇒≐     : _≐′_ ⇒ _≐_
+
+  U-irrelevant : Irrelevant U
+  ∁-irrelevant : (P : Pred A ℓ) → Irrelevant (∁ P)
+  ```
+
+* Generalised proofs in `Relation.Unary.Properties`:
+  ```
+  ⊆-trans : Trans _⊆_ _⊆_ _⊆_
+  ```
+
+* Added new proofs in `Relation.Binary.Properties.Setoid`:
+  ```
+  ≈-isPreorder     : IsPreorder _≈_ _≈_
+  ≈-isPartialOrder : IsPartialOrder _≈_ _≈_
+
+  ≈-preorder : Preorder a ℓ ℓ
+  ≈-poset    : Poset a ℓ ℓ
+  ```
+
+* Added new definitions in `Relation.Binary.Definitions`:
+  ```
+  RightTrans R S = Trans R S R
+  LeftTrans  S R = Trans S R R
+
+  Dense        _<_ = x < y → ∃[ z ] x < z × z < y
+  Cotransitive _#_ = x # y → ∀ z → (x # z) ⊎ (z # y)
+  Tight    _≈_ _#_ = (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)
+
+  Monotonic₁         _≤_ _⊑_ f     = f Preserves _≤_ ⟶ _⊑_
+  Antitonic₁         _≤_ _⊑_ f     = f Preserves (flip _≤_) ⟶ _⊑_
+  Monotonic₂         _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ _⊑_ ⟶ _≼_
+  MonotonicAntitonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ (flip _⊑_) ⟶ _≼_
+  AntitonicMonotonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ _⊑_ ⟶ _≼_
+  Antitonic₂         _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ (flip _⊑_) ⟶ _≼_
+  Adjoint            _≤_ _⊑_ f g   = (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)
+  ```
+
+* Added new definitions in `Relation.Binary.Bundles`:
+  ```
+  record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
+  record ApartnessRelation c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
+  ```
+
+* Added new definitions in `Relation.Binary.Structures`:
+  ```
+  record IsDenseLinearOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
+  record IsApartnessRelation (_#_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
+  ```
+
+* Added new proofs to `Relation.Binary.Consequences`:
+  ```
+  sym⇒¬-sym       : Symmetric _∼_ → Symmetric (¬_ ∘₂ _∼_)
+  cotrans⇒¬-trans : Cotransitive _∼_ → Transitive (¬_ ∘₂ _∼_)
+  irrefl⇒¬-refl   : Reflexive _≈_ → Irreflexive _≈_ _∼_ →  Reflexive (¬_ ∘₂ _∼_)
+  mono₂⇒cong₂     : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ → ∀ {f} →
+                    f Preserves₂ ≤₁ ⟶ ≤₁ ⟶ ≤₂ →
+                    f Preserves₂ ≈₁ ⟶ ≈₁ ⟶ ≈₂
+  ```
+
+* Added new proofs to `Relation.Binary.Construct.Closure.Transitive`:
+  ```
+  accessible⁻  : Acc _∼⁺_ x → Acc _∼_ x
+  wellFounded⁻ : WellFounded _∼⁺_ → WellFounded _∼_
+  accessible   : Acc _∼_ x → Acc _∼⁺_ x
+  ```
+
+* Added new operations in `Relation.Binary.PropositionalEquality.Properties`:
+  ```
+  J       : (B : (y : A) → x ≡ y → Set b) (p : x ≡ y) → B x refl → B y p
+  dcong   : (p : x ≡ y) → subst B p (f x) ≡ f y
+  dcong₂  : (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → f x₁ y₁ ≡ f x₂ y₂
+  dsubst₂ : (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → C x₁ y₁ → C x₂ y₂
+  ddcong₂ : (p : x₁ ≡ x₂) (q : subst B p y₁ ≡ y₂) → dsubst₂ C p q (f x₁ y₁) ≡ f x₂ y₂
+
+  isPartialOrder : IsPartialOrder _≡_ _≡_
+  poset          : Set a → Poset _ _ _
+  ```
+
+* Added new proof in `Relation.Nullary.Reflects`:
+  ```agda
+  T-reflects : Reflects (T b) b
+  T-reflects-elim : Reflects (T a) b → b ≡ a
+  ```
+
+* Added new operations in `System.Exit`:
+  ```
+  isSuccess : ExitCode → Bool
+  isFailure : ExitCode → Bool
+  ```
+
+NonZero/Positive/Negative changes
+---------------------------------
+
+This is a full list of proofs that have changed form to use irrelevant instance arguments:
+
+* In `Data.Nat.Base`:
+  ```
+  ≢-nonZero⁻¹ : ∀ {n} → .(NonZero n) → n ≢ 0
+  >-nonZero⁻¹ : ∀ {n} → .(NonZero n) → n > 0
+  ```
+
+* In `Data.Nat.Properties`:
+  ```
+  *-cancelʳ-≡ : ∀ m n {o} → m * suc o ≡ n * suc o → m ≡ n
+  *-cancelˡ-≡ : ∀ {m n} o → suc o * m ≡ suc o * n → m ≡ n
+  *-cancelʳ-≤ : ∀ m n o → m * suc o ≤ n * suc o → m ≤ n
+  *-cancelˡ-≤ : ∀ {m n} o → suc o * m ≤ suc o * n → m ≤ n
+  *-monoˡ-<   : ∀ n → (_* suc n) Preserves _<_ ⟶ _<_
+  *-monoʳ-<   : ∀ n → (suc n *_) Preserves _<_ ⟶ _<_
+
+  m≤m*n          : ∀ m {n} → 0 < n → m ≤ m * n
+  m≤n*m          : ∀ m {n} → 0 < n → m ≤ n * m
+  m<m*n          : ∀ {m n} → 0 < m → 1 < n → m < m * n
+  suc[pred[n]]≡n : ∀ {n} → n ≢ 0 → suc (pred n) ≡ n
+  ```
+
+* In `Data.Nat.Coprimality`:
+  ```
+  ¬0-coprimeTo-2+ : ∀ {n} → ¬ Coprime 0 (2 + n)
+  Bézout-coprime  : ∀ {i j d} → Bézout.Identity (suc d) (i * suc d) (j * suc d) → Coprime i j
+  prime⇒coprime   : ∀ m → Prime m → ∀ n → 0 < n → n < m → Coprime m n
+  ```
+
+* In `Data.Nat.Divisibility`
+  ```agda
+  m%n≡0⇒n∣m : ∀ m n → m % suc n ≡ 0 → suc n ∣ m
+  n∣m⇒m%n≡0 : ∀ m n → suc n ∣ m → m % suc n ≡ 0
+  m%n≡0⇔n∣m : ∀ m n → m % suc n ≡ 0 ⇔ suc n ∣ m
+  ∣⇒≤       : ∀ {m n} → m ∣ suc n → m ≤ suc n
+  >⇒∤        : ∀ {m n} → m > suc n → m ∤ suc n
+  *-cancelˡ-∣ : ∀ {i j} k → suc k * i ∣ suc k * j → i ∣ j
+  ```
+
+* In `Data.Nat.DivMod`:
+  ```
+  m≡m%n+[m/n]*n : ∀ m n → m ≡ m % suc n + (m / suc n) * suc n
+  m%n≡m∸m/n*n   : ∀ m n → m % suc n ≡ m ∸ (m / suc n) * suc n
+  n%n≡0         : ∀ n → suc n % suc n ≡ 0
+  m%n%n≡m%n     : ∀ m n → m % suc n % suc n ≡ m % suc n
+  [m+n]%n≡m%n   : ∀ m n → (m + suc n) % suc n ≡ m % suc n
+  [m+kn]%n≡m%n  : ∀ m k n → (m + k * (suc n)) % suc n ≡ m % suc n
+  m*n%n≡0       : ∀ m n → (m * suc n) % suc n ≡ 0
+  m%n<n         : ∀ m n → m % suc n < suc n
+  m%n≤m         : ∀ m n → m % suc n ≤ m
+  m≤n⇒m%n≡m     : ∀ {m n} → m ≤ n → m % suc n ≡ m
+
+  m/n<m         : ∀ m n {≢0} → m ≥ 1 → n ≥ 2 → (m / n) {≢0} < m
+  ```
+
+* In `Data.Nat.GCD`
+  ```
+  GCD-* : ∀ {m n d c} → GCD (m * suc c) (n * suc c) (d * suc c) → GCD m n d
+  gcd[m,n]≤n : ∀ m n → gcd m (suc n) ≤ suc n
+  ```
+
+* In `Data.Integer.Properties`:
+  ```
+  positive⁻¹        : ∀ {i} → Positive i → i > 0ℤ
+  negative⁻¹        : ∀ {i} → Negative i → i < 0ℤ
+  nonPositive⁻¹     : ∀ {i} → NonPositive i → i ≤ 0ℤ
+  nonNegative⁻¹     : ∀ {i} → NonNegative i → i ≥ 0ℤ
+  negative<positive : ∀ {i j} → Negative i → Positive j → i < j
+
+  sign-◃    : ∀ s n → sign (s ◃ suc n) ≡ s
+  sign-cong : ∀ {s₁ s₂ n₁ n₂} → s₁ ◃ suc n₁ ≡ s₂ ◃ suc n₂ → s₁ ≡ s₂
+  -◃<+◃     : ∀ m n → Sign.- ◃ (suc m) < Sign.+ ◃ n
+  m⊖1+n<m   : ∀ m n → m ⊖ suc n < + m
+
+  *-cancelʳ-≡     : ∀ i j k → k ≢ + 0 → i * k ≡ j * k → i ≡ j
+  *-cancelˡ-≡     : ∀ i j k → i ≢ + 0 → i * j ≡ i * k → j ≡ k
+  *-cancelʳ-≤-pos : ∀ m n o → m * + suc o ≤ n * + suc o → m ≤ n
+
+  ≤-steps     : ∀ n → i ≤ j → i ≤ + n + j
+  ≤-steps-neg : ∀ n → i ≤ j → i - + n ≤ j
+  n≤m+n       : ∀ n → i ≤ + n + i
+  m≤m+n       : ∀ n → i ≤ i + + n
+  m-n≤m       : ∀ i n → i - + n ≤ i
+
+  *-cancelʳ-≤-pos    : ∀ m n o → m * + suc o ≤ n * + suc o → m ≤ n
+  *-cancelˡ-≤-pos    : ∀ m j k → + suc m * j ≤ + suc m * k → j ≤ k
+  *-cancelˡ-≤-neg    : ∀ m {j k} → -[1+ m ] * j ≤ -[1+ m ] * k → j ≥ k
+  *-cancelʳ-≤-neg    : ∀ {n o} m → n * -[1+ m ] ≤ o * -[1+ m ] → n ≥ o
+  *-cancelˡ-<-nonNeg : ∀ n → + n * i < + n * j → i < j
+  *-cancelʳ-<-nonNeg : ∀ n → i * + n < j * + n → i < j
+  *-monoʳ-≤-nonNeg   : ∀ n → (_* + n) Preserves _≤_ ⟶ _≤_
+  *-monoˡ-≤-nonNeg   : ∀ n → (+ n *_) Preserves _≤_ ⟶ _≤_
+  *-monoˡ-≤-nonPos   : ∀ i → NonPositive i → (i *_) Preserves _≤_ ⟶ _≥_
+  *-monoʳ-≤-nonPos   : ∀ i → NonPositive i → (_* i) Preserves _≤_ ⟶ _≥_
+  *-monoˡ-<-pos      : ∀ n → (+[1+ n ] *_) Preserves _<_ ⟶ _<_
+  *-monoʳ-<-pos      : ∀ n → (_* +[1+ n ]) Preserves _<_ ⟶ _<_
+  *-monoˡ-<-neg      : ∀ n → (-[1+ n ] *_) Preserves _<_ ⟶ _>_
+  *-monoʳ-<-neg      : ∀ n → (_* -[1+ n ]) Preserves _<_ ⟶ _>_
+  *-cancelˡ-<-nonPos : ∀ n → NonPositive n → n * i < n * j → i > j
+  *-cancelʳ-<-nonPos : ∀ n → NonPositive n → i * n < j * n → i > j
+
+  *-distribˡ-⊓-nonNeg : ∀ m j k → + m * (j ⊓ k) ≡ (+ m * j) ⊓ (+ m * k)
+  *-distribʳ-⊓-nonNeg : ∀ m j k → (j ⊓ k) * + m ≡ (j * + m) ⊓ (k * + m)
+  *-distribˡ-⊓-nonPos : ∀ i → NonPositive i → ∀ j k → i * (j ⊓ k) ≡ (i * j) ⊔ (i * k)
+  *-distribʳ-⊓-nonPos : ∀ i → NonPositive i → ∀ j k → (j ⊓ k) * i ≡ (j * i) ⊔ (k * i)
+  *-distribˡ-⊔-nonNeg : ∀ m j k → + m * (j ⊔ k) ≡ (+ m * j) ⊔ (+ m * k)
+  *-distribʳ-⊔-nonNeg : ∀ m j k → (j ⊔ k) * + m ≡ (j * + m) ⊔ (k * + m)
+  *-distribˡ-⊔-nonPos : ∀ i → NonPositive i → ∀ j k → i * (j ⊔ k) ≡ (i * j) ⊓ (i * k)
+  *-distribʳ-⊔-nonPos : ∀ i → NonPositive i → ∀ j k → (j ⊔ k) * i ≡ (j * i) ⊓ (k * i)
+  ```
+
+* In `Data.Integer.Divisibility`:
+  ```
+  *-cancelˡ-∣ : ∀ k {i j} → k ≢ + 0 → k * i ∣ k * j → i ∣ j
+  *-cancelʳ-∣ : ∀ k {i j} → k ≢ + 0 → i * k ∣ j * k → i ∣ j
+  ```
+
+* In `Data.Integer.Divisibility.Signed`:
+  ```
+  *-cancelˡ-∣ : ∀ k {i j} → k ≢ + 0 → k * i ∣ k * j → i ∣ j
+  *-cancelʳ-∣ : ∀ k {i j} → k ≢ + 0 → i * k ∣ j * k → i ∣ j
+  ```
+
+* In `Data.Rational.Unnormalised.Properties`:
+  ```agda
+  positive⁻¹           : ∀ {q} → .(Positive q) → q > 0ℚᵘ
+  nonNegative⁻¹        : ∀ {q} → .(NonNegative q) → q ≥ 0ℚᵘ
+  negative⁻¹           : ∀ {q} → .(Negative q) → q < 0ℚᵘ
+  nonPositive⁻¹        : ∀ {q} → .(NonPositive q) → q ≤ 0ℚᵘ
+  positive⇒nonNegative : ∀ {p} → Positive p → NonNegative p
+  negative⇒nonPositive : ∀ {p} → Negative p → NonPositive p
+  negative<positive    : ∀ {p q} → .(Negative p) → .(Positive q) → p < q
+  nonNeg∧nonPos⇒0      : ∀ {p} → .(NonNegative p) → .(NonPositive p) → p ≃ 0ℚᵘ
+
+  p≤p+q   : ∀ {p q} → NonNegative q → p ≤ p + q
+  p≤q+p   : ∀ {p} → NonNegative p → ∀ {q} → q ≤ p + q
+
+  *-cancelʳ-≤-pos    : ∀ {r} → Positive r → ∀ {p q} → p * r ≤ q * r → p ≤ q
+  *-cancelˡ-≤-pos    : ∀ {r} → Positive r → ∀ {p q} → r * p ≤ r * q → p ≤ q
+  *-cancelʳ-≤-neg    : ∀ r → Negative r → ∀ {p q} → p * r ≤ q * r → q ≤ p
+  *-cancelˡ-≤-neg    : ∀ r → Negative r → ∀ {p q} → r * p ≤ r * q → q ≤ p
+  *-cancelˡ-<-nonNeg : ∀ {r} → NonNegative r → ∀ {p q} → r * p < r * q → p < q
+  *-cancelʳ-<-nonNeg : ∀ {r} → NonNegative r → ∀ {p q} → p * r < q * r → p < q
+  *-cancelˡ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → r * p < r * q → q < p
+  *-cancelʳ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → p * r < q * r → q < p
+  *-monoˡ-≤-nonNeg   : ∀ {r} → NonNegative r → (_* r) Preserves _≤_ ⟶ _≤_
+  *-monoʳ-≤-nonNeg   : ∀ {r} → NonNegative r → (r *_) Preserves _≤_ ⟶ _≤_
+  *-monoˡ-≤-nonPos   : ∀ r → NonPositive r → (_* r) Preserves _≤_ ⟶ _≥_
+  *-monoʳ-≤-nonPos   : ∀ r → NonPositive r → (r *_) Preserves _≤_ ⟶ _≥_
+  *-monoˡ-<-pos      : ∀ {r} → Positive r → (_* r) Preserves _<_ ⟶ _<_
+  *-monoʳ-<-pos      : ∀ {r} → Positive r → (r *_) Preserves _<_ ⟶ _<_
+  *-monoˡ-<-neg      : ∀ r → Negative r → (_* r) Preserves _<_ ⟶ _>_
+  *-monoʳ-<-neg      : ∀ r → Negative r → (r *_) Preserves _<_ ⟶ _>_
+
+  pos⇒1/pos : ∀ p (p>0 : Positive p) → Positive ((1/ p) {{pos⇒≢0 p p>0}})
+  neg⇒1/neg : ∀ p (p<0 : Negative p) → Negative ((1/ p) {{neg⇒≢0 p p<0}})
+
+  *-distribʳ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊓ (r * p)
+  *-distribˡ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊓ (p * r)
+  *-distribˡ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊔ (p * r)
+  *-distribʳ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊔ (r * p)
+  *-distribˡ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊓ (p * r)
+  *-distribʳ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊓ (r * p)
+  *-distribˡ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊔ (p * r)
+  *-distribʳ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊔ (r * p)
+  ```
+
+* In `Data.Rational.Properties`:
+  ```
+  positive⁻¹ : Positive p → p > 0ℚ
+  nonNegative⁻¹ : NonNegative p → p ≥ 0ℚ
+  negative⁻¹ : Negative p → p < 0ℚ
+  nonPositive⁻¹ : NonPositive p → p ≤ 0ℚ
+  negative<positive : Negative p → Positive q → p < q
+  nonNeg≢neg : ∀ p q → NonNegative p → Negative q → p ≢ q
+  pos⇒nonNeg : ∀ p → Positive p → NonNegative p
+  neg⇒nonPos : ∀ p → Negative p → NonPositive p
+  nonNeg∧nonZero⇒pos : ∀ p → NonNegative p → NonZero p → Positive p
+
+  *-cancelʳ-≤-pos    : ∀ r → Positive r → ∀ {p q} → p * r ≤ q * r → p ≤ q
+  *-cancelˡ-≤-pos    : ∀ r → Positive r → ∀ {p q} → r * p ≤ r * q → p ≤ q
+  *-cancelʳ-≤-neg    : ∀ r → Negative r → ∀ {p q} → p * r ≤ q * r → p ≥ q
+  *-cancelˡ-≤-neg    : ∀ r → Negative r → ∀ {p q} → r * p ≤ r * q → p ≥ q
+  *-cancelˡ-<-nonNeg : ∀ r → NonNegative r → ∀ {p q} → r * p < r * q → p < q
+  *-cancelʳ-<-nonNeg : ∀ r → NonNegative r → ∀ {p q} → p * r < q * r → p < q
+  *-cancelˡ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → r * p < r * q → p > q
+  *-cancelʳ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → p * r < q * r → p > q
+  *-monoʳ-≤-nonNeg   : ∀ r → NonNegative r → (_* r) Preserves _≤_ ⟶ _≤_
+  *-monoˡ-≤-nonNeg   : ∀ r → NonNegative r → (r *_) Preserves _≤_ ⟶ _≤_
+  *-monoʳ-≤-nonPos   : ∀ r → NonPositive r → (_* r) Preserves _≤_ ⟶ _≥_
+  *-monoˡ-≤-nonPos   : ∀ r → NonPositive r → (r *_) Preserves _≤_ ⟶ _≥_
+  *-monoˡ-<-pos      : ∀ r → Positive r → (_* r) Preserves _<_ ⟶ _<_
+  *-monoʳ-<-pos      : ∀ r → Positive r → (r *_) Preserves _<_ ⟶ _<_
+  *-monoˡ-<-neg      : ∀ r → Negative r → (_* r) Preserves _<_ ⟶ _>_
+  *-monoʳ-<-neg      : ∀ r → Negative r → (r *_) Preserves _<_ ⟶ _>_
+
+  *-distribˡ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊓ (p * r)
+  *-distribʳ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊓ (r * p)
+  *-distribˡ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊔ (p * r)
+  *-distribʳ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊔ (r * p)
+  *-distribˡ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊓ (p * r)
+  *-distribʳ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊓ (r * p)
+  *-distribˡ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊔ (p * r)
+  *-distribʳ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊔ (r * p)
+
+  pos⇒1/pos : ∀ p (p>0 : Positive p) → Positive ((1/ p) {{pos⇒≢0 p p>0}})
+  neg⇒1/neg : ∀ p (p<0 : Negative p) → Negative ((1/ p) {{neg⇒≢0 p p<0}})
+  1/pos⇒pos : ∀ p .{{_ : NonZero p}} → (1/p : Positive (1/ p)) → Positive p
+  1/neg⇒neg : ∀ p .{{_ : NonZero p}} → (1/p : Negative (1/ p)) → Negative p
+  ```
diff --git a/doc/release-guide.txt b/doc/release-guide.txt
index 0c60b5d4b3..1fb5565b9b 100644
--- a/doc/release-guide.txt
+++ b/doc/release-guide.txt
@@ -64,7 +64,7 @@ procedure should be followed:
 * Generate and upload documentation for the released version:
 
     cp .github/tooling/* .
-    runhaskell GenerateEverything.hs
+    cabal run GenerateEverything.hs
     ./index.sh
     agda -i. -idoc -isrc --html index.agda
     mv html v$VERSION
@@ -81,5 +81,3 @@ procedure should be followed:
 * Move the CHANGELOG.md into the old CHANGELOG folders
 
 * Create new CHANGELOG.md file
-
-* Revert changes in README.md to reference development version

From 40175aadd094318bf877285bc6b4cf3caf8055da Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Thu, 14 Dec 2023 03:38:48 +0100
Subject: [PATCH 200/316] Added tabulate+< (#2190)

* added tabulate+<

* renamed tabulate function

---------

Co-authored-by: Guilherme <alvares@chalmers.se>
Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                                   |  5 +++++
 .../Relation/Unary/AllPairs/Properties.agda    | 18 +++++++++++-------
 2 files changed, 16 insertions(+), 7 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index c32883f2b7..e10ca58108 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -26,3 +26,8 @@ New modules
 
 Additions to existing modules
 -----------------------------
+
+* In `Data.List.Relation.Unary.AllPairs.Properties`:
+  ```
+  tabulate⁺-< : (i < j → R (f i) (f j)) → AllPairs R (tabulate f)
+  ```
diff --git a/src/Data/List/Relation/Unary/AllPairs/Properties.agda b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
index c116fcb00f..027d9bb4cc 100644
--- a/src/Data/List/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
@@ -13,9 +13,9 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
 open import Data.Bool.Base using (true; false)
-open import Data.Fin.Base using (Fin)
-open import Data.Fin.Properties using (suc-injective)
-open import Data.Nat.Base using (zero; suc; _<_; z≤n; s≤s)
+open import Data.Fin.Base as F using (Fin)
+open import Data.Fin.Properties using (suc-injective; <⇒≢)
+open import Data.Nat.Base using (zero; suc; _<_; z≤n; s≤s; z<s; s<s)
 open import Data.Nat.Properties using (≤-refl; m<n⇒m<1+n)
 open import Function.Base using (_∘_; flip)
 open import Level using (Level)
@@ -115,12 +115,16 @@ module _ {R : Rel A ℓ} where
 
 module _ {R : Rel A ℓ} where
 
+  tabulate⁺-< : ∀ {n} {f : Fin n → A} → (∀ {i j} → i F.< j → R (f i) (f j)) →
+              AllPairs R (tabulate f)
+  tabulate⁺-< {zero}  fᵢ~fⱼ = []
+  tabulate⁺-< {suc n} fᵢ~fⱼ =
+    All.tabulate⁺ (λ _ → fᵢ~fⱼ z<s) ∷
+    tabulate⁺-< (fᵢ~fⱼ ∘ s<s)
+
   tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → i ≢ j → R (f i) (f j)) →
               AllPairs R (tabulate f)
-  tabulate⁺ {zero}  fᵢ~fⱼ = []
-  tabulate⁺ {suc n} fᵢ~fⱼ =
-    All.tabulate⁺ (λ j → fᵢ~fⱼ λ()) ∷
-    tabulate⁺ (fᵢ~fⱼ ∘ (_∘ suc-injective))
+  tabulate⁺ fᵢ~fⱼ = tabulate⁺-< (fᵢ~fⱼ ∘ <⇒≢)
 
 ------------------------------------------------------------------------
 -- filter

From f5f9727050a2683b6ee63c4ad619c67d983426aa Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Thu, 14 Dec 2023 05:06:36 +0100
Subject: [PATCH 201/316] Added pointwise and catmaybe in Lists (#2222)

* added pointwise and catmaybe

* added pointwise to any

* added cat maybe all

* changed pointwise definition

* changed files name

* fixed changelogs and properties

* changed Any solution

* changed pointwise to catMaybe

---------

Co-authored-by: Guilherme <alvares@chalmers.se>
Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                                     | 12 ++++++++++++
 src/Data/List/Relation/Unary/All/Properties.agda | 14 +++++++++++++-
 .../List/Relation/Unary/AllPairs/Properties.agda | 16 ++++++++++++++--
 src/Data/Maybe/Relation/Binary/Pointwise.agda    |  5 +++++
 4 files changed, 44 insertions(+), 3 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e10ca58108..2701cdde9b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -27,7 +27,19 @@ New modules
 Additions to existing modules
 -----------------------------
 
+* In `Data.Maybe.Relation.Binary.Pointwise`:
+  ```agda
+  pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
+  ```
+
+* In `Data.List.Relation.Unary.All.Properties`:
+  ```agda
+  All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)
+  Any-catMaybes⁺ : All (Maybe.Any P) xs → All P (catMaybes xs)
+  ```
+
 * In `Data.List.Relation.Unary.AllPairs.Properties`:
   ```
+  catMaybes⁺ : AllPairs (Pointwise R) xs → AllPairs R (catMaybes xs)
   tabulate⁺-< : (i < j → R (f i) (f j)) → AllPairs R (tabulate f)
   ```
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index 1b55bcd2b8..d2a57b7143 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -28,7 +28,8 @@ import Data.List.Relation.Binary.Equality.Setoid as ListEq using (_≋_; []; _
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
 open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_)
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
-open import Data.Maybe.Relation.Unary.All as Maybe using (just; nothing)
+open import Data.Maybe.Relation.Unary.All as Maybe using (just; nothing; fromAny)
+open import Data.Maybe.Relation.Unary.Any as Maybe using (just)
 open import Data.Nat.Base using (zero; suc; s≤s; _<_; z<s; s<s)
 open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)
 open import Data.Product.Base as Prod using (_×_; _,_; uncurry; uncurry′)
@@ -388,6 +389,17 @@ mapMaybe⁺ {xs = x ∷ xs} {f = f} (px ∷ pxs) with f x
 ... | just v with px
 ...   | just pv = pv ∷ mapMaybe⁺ pxs
 
+------------------------------------------------------------------------
+-- catMaybes
+
+All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)
+All-catMaybes⁺ [] = []
+All-catMaybes⁺ (just px ∷ pxs) = px ∷ All-catMaybes⁺ pxs
+All-catMaybes⁺ (nothing ∷ pxs) = All-catMaybes⁺ pxs
+
+Any-catMaybes⁺ : All (Maybe.Any P) xs → All P (catMaybes xs)
+Any-catMaybes⁺ = All-catMaybes⁺ ∘ All.map fromAny
+
 ------------------------------------------------------------------------
 -- _++_
 
diff --git a/src/Data/List/Relation/Unary/AllPairs/Properties.agda b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
index 027d9bb4cc..9b145c3834 100644
--- a/src/Data/List/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
@@ -10,9 +10,11 @@ module Data.List.Relation.Unary.AllPairs.Properties where
 
 open import Data.List.Base hiding (any)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
-import Data.List.Relation.Unary.All.Properties as All
+open import Data.List.Relation.Unary.All.Properties as All using (Any-catMaybes⁺)
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
 open import Data.Bool.Base using (true; false)
+open import Data.Maybe using (Maybe; nothing; just)
+open import Data.Maybe.Relation.Binary.Pointwise using (pointwise⊆any; Pointwise)
 open import Data.Fin.Base as F using (Fin)
 open import Data.Fin.Properties using (suc-injective; <⇒≢)
 open import Data.Nat.Base using (zero; suc; _<_; z≤n; s≤s; z<s; s<s)
@@ -22,7 +24,7 @@ open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (DecSetoid)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_)
-open import Relation.Unary using (Pred; Decidable)
+open import Relation.Unary using (Pred; Decidable; _⊆_)
 open import Relation.Nullary.Decidable using (does)
 
 private
@@ -136,3 +138,13 @@ module _ {R : Rel A ℓ} {P : Pred A p} (P? : Decidable P) where
   filter⁺ {x ∷ xs} (x∉xs ∷ xs!) with does (P? x)
   ... | false = filter⁺ xs!
   ... | true  = All.filter⁺ P? x∉xs ∷ filter⁺ xs!
+
+------------------------------------------------------------------------
+-- catMaybes
+
+module _ {R : Rel A ℓ} where
+
+  catMaybes⁺ : {xs : List (Maybe A)} → AllPairs (Pointwise R) xs → AllPairs R (catMaybes xs)
+  catMaybes⁺ {xs = []} [] = []
+  catMaybes⁺ {xs = nothing ∷  _} (x∼xs ∷ pxs) = catMaybes⁺ pxs
+  catMaybes⁺ {xs = just x  ∷ xs} (x∼xs ∷ pxs) = Any-catMaybes⁺ (All.map pointwise⊆any x∼xs) ∷ catMaybes⁺ pxs
diff --git a/src/Data/Maybe/Relation/Binary/Pointwise.agda b/src/Data/Maybe/Relation/Binary/Pointwise.agda
index 48fa144a2e..3da2abd9f1 100644
--- a/src/Data/Maybe/Relation/Binary/Pointwise.agda
+++ b/src/Data/Maybe/Relation/Binary/Pointwise.agda
@@ -11,6 +11,7 @@ module Data.Maybe.Relation.Binary.Pointwise where
 open import Level
 open import Data.Product.Base using (∃; _×_; -,_; _,_)
 open import Data.Maybe.Base using (Maybe; just; nothing)
+open import Data.Maybe.Relation.Unary.Any using (Any; just)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Binary.Core using (REL; Rel; _⇒_)
 open import Relation.Binary.Bundles using (Setoid; DecSetoid)
@@ -18,6 +19,7 @@ open import Relation.Binary.Definitions using (Reflexive; Sym; Trans; Decidable)
 open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary
+open import Relation.Unary using (_⊆_)
 import Relation.Nullary.Decidable as Dec
 
 ------------------------------------------------------------------------
@@ -93,6 +95,9 @@ module _ {a r} {A : Set a} {R : Rel A r} where
     ; _≟_           = dec R._≟_
     } where module R = IsDecEquivalence R-isDecEquivalence
 
+  pointwise⊆any : ∀ {x} → Pointwise R (just x) ⊆ Any (R x)
+  pointwise⊆any (just Rxy) = just Rxy
+
 module _ {c ℓ} where
 
   setoid : Setoid c ℓ → Setoid c (c ⊔ ℓ)

From c649694289adc712498ea60468ebcb5ace447cb2 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 14 Dec 2023 06:05:35 +0000
Subject: [PATCH 202/316] [fixes #1711] Refactoring `Data.Nat.Divisibility` and
 `Data.Nat.DivMod` (#2182)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* added new definitions to `_∣_`

* `CHANGELOG`

* don't declare `quotient≢0` as an `instance`

* replace use of `subst` with one of `trans`

* what's sauce for the goose...

* switch to a `rewrite`-based solution...

* tightened `import`s

* simplified dependenciess

* simplified dependencies; `CHANGELOG`

* removed `module` abstractions

* delegated proof of `quotient≢0` to `Data.Nat.Properties`

* removed redundant property

* cosmetic review changes; others to follow

* better proof of `quotient>1`

* `where` clause layout

* leaving in the flipped equality; moved everything else

* new lemmas moved from `Core`; knock-on consequences; lots of tidying up

* tidying up; `CHANGELOG`

* cosmetic tweaks

* reverted to simple version

* problems with exporting `quotient`

* added last lemma: defining equation for `_/_`

* improved `CHANGELOG`

* revert: simplified imports

* improved `CHANGELOG`

* endpoint of simplifying the proof of `*-pres-∣`

* simplified the proof of `n/m≡quotient`

* simplified the proof of `∣m+n∣m⇒∣n`

* simplified the proof of `∣m∸n∣n⇒∣m`

* simplified `import`s

* simplified a lot of proofs, esp. wrt `divides-refl` and `NonZero` reasoning

* simplified more proofs, esp. wrt `divides-refl` reasoning

* simplified `import`s

* moved `equalityᵒ` proof out of `Core`

* `CHANGELOG`

* temporary fix: further `NonZero` refactoring advised!

* regularised use of instance brackets

* further instance simplification

* further streamlining

* tidied up `CHANGELOG`

* simplified `NonZero` pattern matching

* regularised use of instance brackets

* simplified proof of `/-*-interchange`

* simplified proof of `/-*-interchange`

* ... permitting the migration of `*-pres-∣` to `Data.Nat.Divisibility`

* tweaked proof of `/-*-interchange`

* narrowed `import`s

* simplified proof; renamed new proofs

* Capitalisation

* streamlined `import`s; streamlined proof of decidability

* spurious duplication after merge

* missing symbol import

* replaced one use of `1 < m` with `{{NonTrivial m}}`

* fixed `CHANGELOG`

* removed use of identifier `k`

* refactoring: more use of `NonTrivial` instances

* knock-on consequences: simplified function

* two new lemmas

* refactoring: use of `connex` in proofs

* new lemmas about `pred`

* new lemmas about monus

* refactoring: use of the new properties, simplifying pattern analysis

* whitespace

* questionable? revision after comments on #2221

* silly argument name typo; remove parens

* tidied up imports of `Relation.Nullary`

* removed spurious `instance`

* localised appeals to `Reasoning`

* further use of `variable`s

* tidied up `record` name in comment

* cosmetic

* reconciled implicit/explicit arguments in various monus lemmas

* fixed knock-on change re monus; reverted change to `m/n<m`

* reverted change to `m/n≢0⇒n≤m`

* reverted breaking changes involving `NonZero` inference

* revised `CHANGELOG`

* restored deleted proof

* fix whitespace

* renaming: `DivMod.nonZeroDivisor`

* localised use of `≤-Reasoning`

* reverted export; removed anonymous module

* revert commit re `yes/no`

* renamed flipped equality

* tweaked comment

* added alias for `equality`
---
 CHANGELOG.md                        |  43 ++++-
 src/Data/Digit.agda                 |   7 +-
 src/Data/Fin/Properties.agda        |   3 +
 src/Data/Nat/Coprimality.agda       |  14 +-
 src/Data/Nat/DivMod.agda            | 253 ++++++++++++------------
 src/Data/Nat/Divisibility.agda      | 285 +++++++++++++++-------------
 src/Data/Nat/Divisibility/Core.agda |  38 ++--
 src/Data/Nat/Properties.agda        |  53 ++++--
 src/Data/Rational/Properties.agda   |  11 +-
 9 files changed, 407 insertions(+), 300 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2701cdde9b..bf2d01d6a3 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -21,15 +21,20 @@ Deprecated modules
 Deprecated names
 ----------------
 
+* In `Data.Nat.Divisibility.Core`:
+  ```agda
+  *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
+  ```
+
 New modules
 -----------
 
 Additions to existing modules
 -----------------------------
 
-* In `Data.Maybe.Relation.Binary.Pointwise`:
+* In `Data.Fin.Properties`:
   ```agda
-  pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
+  nonZeroIndex : Fin n → ℕ.NonZero n
   ```
 
 * In `Data.List.Relation.Unary.All.Properties`:
@@ -39,7 +44,39 @@ Additions to existing modules
   ```
 
 * In `Data.List.Relation.Unary.AllPairs.Properties`:
-  ```
+  ```agda
   catMaybes⁺ : AllPairs (Pointwise R) xs → AllPairs R (catMaybes xs)
   tabulate⁺-< : (i < j → R (f i) (f j)) → AllPairs R (tabulate f)
   ```
+
+* In `Data.Maybe.Relation.Binary.Pointwise`:
+  ```agda
+  pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
+  ```
+
+* In `Data.Nat.Divisibility`:
+  ```agda
+  quotient≢0       : m ∣ n → .{{NonZero n}} → NonZero quotient
+
+  m|n⇒n≡quotient*m : m ∣ n → n ≡ quotient * m
+  m|n⇒n≡m*quotient : m ∣ n → n ≡ m * quotient
+  quotient-∣       : m ∣ n → quotient ∣ n
+  quotient>1       : m ∣ n → m < n → 1 < quotient
+  quotient-<       : m ∣ n → .{{NonTrivial m}} → .{{NonZero n}} → quotient < n
+  n/m≡quotient     : m ∣ n → .{{_ : NonZero m}} → n / m ≡ quotient
+
+  m/n≡0⇒m<n    : .{{_ : NonZero n}} → m / n ≡ 0 → m < n
+  m/n≢0⇒n≤m    : .{{_ : NonZero n}} → m / n ≢ 0 → n ≤ m
+
+  nonZeroDivisor : DivMod dividend divisor → NonZero divisor
+  ```
+
+* Added new proofs in `Data.Nat.Properties`:
+  ```agda
+  m≤n+o⇒m∸n≤o : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o
+  m<n+o⇒m∸n<o : ∀ m n {o} → .{{NonZero o}} → m < n + o → m ∸ n < o
+  pred-cancel-≤ : pred m ≤ pred n → (m ≡ 1 × n ≡ 0) ⊎ m ≤ n
+  pred-cancel-< : pred m < pred n → m < n
+  pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n
+  pred-cancel-≡ : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n
+  ```
diff --git a/src/Data/Digit.agda b/src/Data/Digit.agda
index 3f11c1c3af..207b0280b7 100644
--- a/src/Data/Digit.agda
+++ b/src/Data/Digit.agda
@@ -55,11 +55,8 @@ toNatDigits base@(suc (suc _)) n = aux (<-wellFounded-fast n) []
   aux : {n : ℕ} → Acc _<_ n → List ℕ → List ℕ
   aux {zero}        _      xs =  (0 ∷ xs)
   aux {n@(suc _)} (acc wf) xs with does (0 <? n / base)
-  ... | false = (n % base) ∷ xs
-  ... | true  = aux (wf q<n) ((n % base) ∷ xs)
-    where
-    q<n : n / base < n
-    q<n = m/n<m n base (s<s z<s)
+  ... | false = (n % base) ∷ xs -- Could this more simply be n ∷ xs here?
+  ... | true  = aux (wf (m/n<m n base sz<ss)) ((n % base) ∷ xs)
 
 ------------------------------------------------------------------------
 -- Converting between `ℕ` and expansions of `Digit base`
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index cdcbeb4e85..303cf64905 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -65,6 +65,9 @@ private
 ¬Fin0 : ¬ Fin 0
 ¬Fin0 ()
 
+nonZeroIndex : Fin n → ℕ.NonZero n
+nonZeroIndex {n = suc _} _ = _
+
 ------------------------------------------------------------------------
 -- Bundles
 
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index f385944f50..f513e82fa4 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -21,7 +21,7 @@ open import Function.Base using (_∘_)
 open import Level using (0ℓ)
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; trans; cong; subst)
-open import Relation.Nullary as Nullary using (¬_; contradiction; yes ; no)
+open import Relation.Nullary as Nullary using (¬_; contradiction; map′)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Symmetric; Decidable)
 
@@ -65,18 +65,8 @@ coprime-/gcd m n = GCD≡1⇒coprime (GCD-/gcd m n)
 sym : Symmetric Coprime
 sym c = c ∘ swap
 
-private
-  0≢1 : 0 ≢ 1
-  0≢1 ()
-
-  2+≢1 : ∀ {n} → 2+ n ≢ 1
-  2+≢1 ()
-
 coprime? : Decidable Coprime
-coprime? m n with mkGCD m n
-... | (0    , g) = no  (0≢1  ∘ GCD.unique g ∘ coprime⇒GCD≡1)
-... | (1    , g) = yes (GCD≡1⇒coprime g)
-... | (2+ _ , g) = no  (2+≢1 ∘ GCD.unique g ∘ coprime⇒GCD≡1)
+coprime? m n = map′ gcd≡1⇒coprime coprime⇒gcd≡1 (gcd m n ≟ 1)
 
 ------------------------------------------------------------------------
 -- Other basic properties
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index bbb8288838..5ff23e4c72 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -11,15 +11,16 @@ module Data.Nat.DivMod where
 open import Agda.Builtin.Nat using (div-helper; mod-helper)
 
 open import Data.Fin.Base using (Fin; toℕ; fromℕ<)
-open import Data.Fin.Properties using (toℕ-fromℕ<)
-open import Data.Nat.Base as Nat
+open import Data.Fin.Properties using (nonZeroIndex; toℕ-fromℕ<)
+open import Data.Nat.Base
 open import Data.Nat.DivMod.Core
 open import Data.Nat.Divisibility.Core
 open import Data.Nat.Induction
 open import Data.Nat.Properties
-open import Function.Base using (_$_)
+open import Data.Sum.Base using (inj₁; inj₂)
+open import Function.Base using (_$_; _∘_)
 open import Relation.Binary.PropositionalEquality
-open import Relation.Nullary.Decidable using (yes; no)
+open import Relation.Nullary.Negation using (contradiction)
 
 open ≤-Reasoning
 
@@ -34,7 +35,7 @@ open import Data.Nat.Base public
   using (_%_; _/_)
 
 ------------------------------------------------------------------------
--- Relationship between _%_ and _div_
+-- Relationship between _%_ and _/_
 
 m≡m%n+[m/n]*n : ∀ m n .{{_ : NonZero n}} → m ≡ m % n + (m / n) * n
 m≡m%n+[m/n]*n m (suc n) = div-mod-lemma 0 0 m n
@@ -49,10 +50,10 @@ m%n≡m∸m/n*n m n = begin-equality
 ------------------------------------------------------------------------
 -- Properties of _%_
 
-%-congˡ : .⦃ _ : NonZero o ⦄ → m ≡ n → m % o ≡ n % o
+%-congˡ : .{{_ : NonZero o}} → m ≡ n → m % o ≡ n % o
 %-congˡ refl = refl
 
-%-congʳ : .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ≡ n →
+%-congʳ : .{{_ : NonZero m}} .{{_ : NonZero n}} → m ≡ n →
           o % m ≡ o % n
 %-congʳ refl = refl
 
@@ -71,45 +72,25 @@ m%n%n≡m%n m (suc n-1) = modₕ-idem 0 m n-1
 [m+kn]%n≡m%n : ∀ m k n .{{_ : NonZero n}} → (m + k * n) % n ≡ m % n
 [m+kn]%n≡m%n m zero    n = cong (_% n) (+-identityʳ m)
 [m+kn]%n≡m%n m (suc k) n = begin-equality
-  (m + (n + k * n)) % n ≡⟨ cong (_% n) (sym (+-assoc m n (k * n))) ⟩
+  (m + (n + k * n)) % n ≡⟨ cong (_% n) (+-assoc m n (k * n)) ⟨
   (m + n + k * n)   % n ≡⟨ [m+kn]%n≡m%n (m + n) k n ⟩
   (m + n)           % n ≡⟨ [m+n]%n≡m%n m n ⟩
   m                 % n ∎
 
-m≤n⇒[n∸m]%m≡n%m : .⦃ _ : NonZero m ⦄ → m ≤ n →
+m≤n⇒[n∸m]%m≡n%m : .{{_ : NonZero m}} → m ≤ n →
                   (n ∸ m) % m ≡ n % m
 m≤n⇒[n∸m]%m≡n%m {m} {n} m≤n = begin-equality
   (n ∸ m) % m     ≡⟨ [m+n]%n≡m%n (n ∸ m) m ⟨
   (n ∸ m + m) % m ≡⟨ cong (_% m) (m∸n+n≡m m≤n) ⟩
   n % m           ∎
 
-m*n≤o⇒[o∸m*n]%n≡o%n : ∀ m {n o} .⦃ _ : NonZero n ⦄ → m * n ≤ o →
+m*n≤o⇒[o∸m*n]%n≡o%n : ∀ m {n o} .{{_ : NonZero n}} → m * n ≤ o →
                       (o ∸ m * n) % n ≡ o % n
 m*n≤o⇒[o∸m*n]%n≡o%n m {n} {o} m*n≤o = begin-equality
   (o ∸ m * n) % n         ≡⟨ [m+kn]%n≡m%n (o ∸ m * n) m n ⟨
   (o ∸ m * n + m * n) % n ≡⟨ cong (_% n) (m∸n+n≡m m*n≤o) ⟩
   o % n                   ∎
 
-m∣n⇒o%n%m≡o%m : ∀ m n o .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ∣ n →
-                o % n % m ≡ o % m
-m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
-  o % n % m                ≡⟨⟩
-  o % pm % m               ≡⟨ %-congˡ (m%n≡m∸m/n*n o pm) ⟩
-  (o ∸ o / pm * pm) % m    ≡⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟨
-  (o ∸ o / pm * p * m) % m ≡⟨ m*n≤o⇒[o∸m*n]%n≡o%n (o / pm * p) lem ⟩
-  o % m                    ∎
-  where
-  pm = p * m
-
-  lem : o / pm * p * m ≤ o
-  lem = begin
-    o / pm * p * m       ≡⟨ *-assoc (o / pm) p m ⟩
-    -- Sort out dependencies in this file, then use m/n*n≤m instead.
-    o / pm * pm          ≤⟨ m≤m+n (o / pm * pm) (o % pm) ⟩
-    o / pm * pm + o % pm ≡⟨ +-comm _ (o % pm) ⟩
-    o % pm + o / pm * pm ≡⟨ m≡m%n+[m/n]*n o pm ⟨
-    o                    ∎
-
 m*n%n≡0 : ∀ m n .{{_ : NonZero n}} → (m * n) % n ≡ 0
 m*n%n≡0 m n@(suc _) = [m+kn]%n≡m%n 0 m n
 
@@ -123,10 +104,10 @@ m%n≤m : ∀ m n .{{_ : NonZero n}} → m % n ≤ m
 m%n≤m m (suc n-1) = a[modₕ]n≤a 0 m n-1
 
 m≤n⇒m%n≡m : m ≤ n → m % suc n ≡ m
-m≤n⇒m%n≡m {m = m} m≤n with less-than-or-equal {k} refl ← ≤⇒≤″ m≤n
+m≤n⇒m%n≡m {m = m} m≤n with ≤″-offset k ← ≤⇒≤″ m≤n
   = a≤n⇒a[modₕ]n≡a 0 (m + k) m k
 
-m<n⇒m%n≡m : .⦃ _ : NonZero n ⦄ → m < n → m % n ≡ m
+m<n⇒m%n≡m : .{{_ : NonZero n}} → m < n → m % n ≡ m
 m<n⇒m%n≡m {n = suc _} m<n = m≤n⇒m%n≡m (<⇒≤pred m<n)
 
 %-pred-≡0 : ∀ {m n} .{{_ : NonZero n}} → (suc m % n) ≡ 0 → (m % n) ≡ n ∸ 1
@@ -143,15 +124,15 @@ m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-
   (m + n)                         % d ≡⟨ cong (λ v → (v + n) % d) (m≡m%n+[m/n]*n m d) ⟩
   (m % d +  m / d * d + n)        % d ≡⟨ cong (_% d) (+-assoc (m % d) _ n) ⟩
   (m % d + (m / d * d + n))       % d ≡⟨ cong (λ v → (m % d + v) % d) (+-comm _ n) ⟩
-  (m % d + (n + m / d * d))       % d ≡⟨ cong (_% d) (sym (+-assoc (m % d) n _)) ⟩
+  (m % d + (n + m / d * d))       % d ≡⟨ cong (_% d) (+-assoc (m % d) n _) ⟨
   (m % d +  n + m / d * d)        % d ≡⟨ [m+kn]%n≡m%n (m % d + n) (m / d) d ⟩
   (m % d +  n)                    % d ≡⟨ cong (λ v → (m % d + v) % d) (m≡m%n+[m/n]*n n d) ⟩
-  (m % d + (n % d + (n / d) * d)) % d ≡⟨ sym (cong (_% d) (+-assoc (m % d) (n % d) _)) ⟩
+  (m % d + (n % d + (n / d) * d)) % d ≡⟨ cong (_% d) (+-assoc (m % d) (n % d) _) ⟨
   (m % d +  n % d + (n / d) * d)  % d ≡⟨ [m+kn]%n≡m%n (m % d + n % d) (n / d) d ⟩
   (m % d +  n % d)                % d ∎
 
 %-distribˡ-* : ∀ m n d .{{_ : NonZero d}} → (m * n) % d ≡ ((m % d) * (n % d)) % d
-%-distribˡ-* m n d@(suc d-1) = begin-equality
+%-distribˡ-* m n d = begin-equality
   (m * n)                                             % d ≡⟨ cong (λ h → (h * n) % d) (m≡m%n+[m/n]*n m d) ⟩
   ((m′ + k * d) * n)                                  % d ≡⟨ cong (λ h → ((m′ + k * d) * h) % d) (m≡m%n+[m/n]*n n d) ⟩
   ((m′ + k * d) * (n′ + j * d))                       % d ≡⟨ cong (_% d) lemma ⟩
@@ -173,14 +154,14 @@ m<[1+n%d]⇒m≤[n%d] {m} n (suc d-1) = k<1+a[modₕ]n⇒k≤a[modₕ]n 0 m n d-
     m′ * n′ + (m′ * j + (n′ + j * d) * k) * d         ∎
 
 %-remove-+ˡ : ∀ {m} n {d} .{{_ : NonZero d}} → d ∣ m → (m + n) % d ≡ n % d
-%-remove-+ˡ {m@.(p * d)} n {d@(suc _)} (divides-refl p) = begin-equality
+%-remove-+ˡ {m@.(p * d)} n {d} (divides-refl p) = begin-equality
   (m + n)     % d ≡⟨⟩
   (p * d + n) % d ≡⟨ cong (_% d) (+-comm (p * d) n) ⟩
   (n + p * d) % d ≡⟨ [m+kn]%n≡m%n n p d ⟩
   n           % d ∎
 
 %-remove-+ʳ : ∀ m {n d} .{{_ : NonZero d}} → d ∣ n → (m + n) % d ≡ m % d
-%-remove-+ʳ m {n} {suc _} eq rewrite +-comm m n = %-remove-+ˡ {n} m eq
+%-remove-+ʳ m {n} rewrite +-comm m n = %-remove-+ˡ {n} m
 
 ------------------------------------------------------------------------
 -- Properties of _/_
@@ -204,16 +185,16 @@ m*n/n≡m : ∀ m n .{{_ : NonZero n}} → m * n / n ≡ m
 m*n/n≡m m (suc n-1) = a*n[divₕ]n≡a 0 m n-1
 
 m/n*n≡m : ∀ {m n} .{{_ : NonZero n}} → n ∣ m → m / n * n ≡ m
-m/n*n≡m {_} {n@(suc _)} (divides-refl q) = cong (_* n) (m*n/n≡m q n)
+m/n*n≡m {n = n} (divides-refl q) = cong (_* n) (m*n/n≡m q n)
 
 m*[n/m]≡n : .{{_ : NonZero m}} → m ∣ n → m * (n / m) ≡ n
 m*[n/m]≡n {m} m∣n = trans (*-comm m (_ / m)) (m/n*n≡m m∣n)
 
 m/n*n≤m : ∀ m n .{{_ : NonZero n}} → (m / n) * n ≤ m
-m/n*n≤m m n@(suc n-1) = begin
+m/n*n≤m m n = begin
   (m / n) * n          ≤⟨ m≤m+n ((m / n) * n) (m % n) ⟩
   (m / n) * n + m % n  ≡⟨ +-comm _ (m % n) ⟩
-  m % n + (m / n) * n  ≡⟨ sym (m≡m%n+[m/n]*n m n) ⟩
+  m % n + (m / n) * n  ≡⟨ m≡m%n+[m/n]*n m n ⟨
   m                    ∎
 
 m/n≤m : ∀ m n .{{_ : NonZero n}} → (m / n) ≤ m
@@ -222,11 +203,12 @@ m/n≤m m n = *-cancelʳ-≤ (m / n) m n (begin
   m           ≤⟨ m≤m*n m n ⟩
   m * n       ∎)
 
-m/n<m : ∀ m n .{{_ : NonZero m}} .{{_ : NonZero n}} → n ≥ 2 → m / n < m
-m/n<m m n n≥2 = *-cancelʳ-< _ (m / n) m (begin-strict
-  (m / n) * n ≤⟨ m/n*n≤m m n ⟩
-  m           <⟨ m<m*n m n n≥2 ⟩
-  m * n       ∎)
+m/n<m : ∀ m n .{{_ : NonZero m}} .{{_ : NonZero n}} →
+        1 < n → m / n < m
+m/n<m m n 1<n = *-cancelʳ-< _ (m / n) m $ begin-strict
+  m / n * n ≤⟨ m/n*n≤m m n ⟩
+  m         <⟨ m<m*n m n 1<n ⟩
+  m * n     ∎
 
 /-mono-≤ : .{{_ : NonZero o}} .{{_ : NonZero p}} →
            m ≤ n → o ≥ p → m / o ≤ n / p
@@ -243,8 +225,8 @@ m/n<m m n n≥2 = *-cancelʳ-< _ (m / n) m (begin-strict
               o ∣ m → o ∣ n → m / o ≡ n / o → m ≡ n
 /-cancelʳ-≡ {m} {n} {o} o∣m o∣n m/o≡n/o = begin-equality
   m           ≡⟨ m*[n/m]≡n {o} {m} o∣m ⟨
-  o * (m / o) ≡⟨  cong (o *_) m/o≡n/o ⟩
-  o * (n / o) ≡⟨  m*[n/m]≡n {o} {n} o∣n ⟩
+  o * (m / o) ≡⟨ cong (o *_) m/o≡n/o ⟩
+  o * (n / o) ≡⟨ m*[n/m]≡n {o} {n} o∣n ⟩
   n           ∎
 
 m<n⇒m/n≡0 : ∀ {m n} .{{_ : NonZero n}} → m < n → m / n ≡ 0
@@ -252,30 +234,41 @@ m<n⇒m/n≡0 {m} {suc n-1} (s≤s m≤n) = divₕ-finish n-1 m n-1 m≤n
 
 m≥n⇒m/n>0 : ∀ {m n} .{{_ : NonZero n}} → m ≥ n → m / n > 0
 m≥n⇒m/n>0 {m@(suc _)} {n@(suc _)} m≥n = begin
-  1     ≡⟨ sym (n/n≡1 m) ⟩
+  1     ≡⟨ n/n≡1 m ⟨
   m / m ≤⟨ /-monoʳ-≤ m m≥n ⟩
   m / n ∎
 
+m/n≡0⇒m<n : ∀ {m n} .{{_ : NonZero n}} → m / n ≡ 0 → m < n
+m/n≡0⇒m<n {m} {n@(suc _)} m/n≡0  with <-≤-connex m n
+... | inj₁ m<n = m<n
+... | inj₂ n≤m = contradiction m/n≡0 (≢-nonZero⁻¹ _)
+  where instance _ =  >-nonZero (m≥n⇒m/n>0 n≤m)
+
+m/n≢0⇒n≤m : ∀ {m n} .{{_ : NonZero n}} → m / n ≢ 0 → n ≤ m
+m/n≢0⇒n≤m {m} {n@(suc _)} m/n≢0 with <-≤-connex m n
+... | inj₁ m<n = contradiction (m<n⇒m/n≡0 m<n) m/n≢0
+... | inj₂ n≤m = n≤m
+
 +-distrib-/ : ∀ m n {d} .{{_ : NonZero d}} → m % d + n % d < d →
               (m + n) / d ≡ m / d + n / d
 +-distrib-/ m n {suc d-1} leq = +-distrib-divₕ 0 0 m n d-1 leq
 
 +-distrib-/-∣ˡ : ∀ {m} n {d} .{{_ : NonZero d}} →
                  d ∣ m → (m + n) / d ≡ m / d + n / d
-+-distrib-/-∣ˡ {m@.(p * d)} n {d} (divides-refl p) = +-distrib-/ m n (begin-strict
++-distrib-/-∣ˡ {m@.(p * d)} n {d} (divides-refl p) = +-distrib-/ m n $ begin-strict
   m % d + n % d     ≡⟨⟩
   p * d % d + n % d ≡⟨ cong (_+ n % d) (m*n%n≡0 p d) ⟩
   n % d             <⟨ m%n<n n d ⟩
-  d                 ∎)
+  d                 ∎
 
 +-distrib-/-∣ʳ : ∀ m {n} {d} .{{_ : NonZero d}} →
                  d ∣ n → (m + n) / d ≡ m / d + n / d
-+-distrib-/-∣ʳ m {n@.(p * d)} {d} (divides-refl p) = +-distrib-/ m n (begin-strict
++-distrib-/-∣ʳ m {n@.(p * d)} {d} (divides-refl p) = +-distrib-/ m n $ begin-strict
   m % d + n % d     ≡⟨⟩
   m % d + p * d % d ≡⟨ cong (m % d +_) (m*n%n≡0 p d) ⟩
   m % d + 0         ≡⟨ +-identityʳ _ ⟩
   m % d             <⟨ m%n<n m d ⟩
-  d                 ∎)
+  d                 ∎
 
 m/n≡1+[m∸n]/n : ∀ {m n} .{{_ : NonZero n}} → m ≥ n → m / n ≡ 1 + ((m ∸ n) / n)
 m/n≡1+[m∸n]/n {m@(suc m-1)} {n@(suc n-1)} m≥n = begin-equality
@@ -285,64 +278,77 @@ m/n≡1+[m∸n]/n {m@(suc m-1)} {n@(suc n-1)} m≥n = begin-equality
   1 + (div-helper 0 n-1 (m ∸ n) n-1) ≡⟨⟩
   1 + (m ∸ n) / n                    ∎
 
+[m∸n]/n≡m/n∸1 : ∀ m n .{{_ : NonZero n}} → (m ∸ n) / n ≡ pred (m / n)
+[m∸n]/n≡m/n∸1 m n with <-≤-connex m n
+... | inj₁ m<n = begin-equality
+  (m ∸ n) / n  ≡⟨ m<n⇒m/n≡0 (≤-<-trans (m∸n≤m m n) m<n) ⟩
+  0            ≡⟨⟩
+  pred 0       ≡⟨ cong pred (m<n⇒m/n≡0 m<n) ⟨
+  pred (m / n) ∎
+... | inj₂ n≥m = begin-equality
+  (m ∸ n) / n            ≡⟨⟩
+  pred (1 + (m ∸ n) / n) ≡⟨ cong pred (m/n≡1+[m∸n]/n n≥m) ⟨
+  pred (m / n)           ∎
+
+m∣n⇒o%n%m≡o%m : ∀ m n o .{{_ : NonZero m}} .{{_ : NonZero n}} → m ∣ n →
+                o % n % m ≡ o % m
+m∣n⇒o%n%m≡o%m m n@.(p * m) o (divides-refl p) = begin-equality
+  o % n % m                ≡⟨⟩
+  o % pm % m               ≡⟨ %-congˡ (m%n≡m∸m/n*n o pm) ⟩
+  (o ∸ o / pm * pm) % m    ≡⟨ cong (λ # → (o ∸ #) % m) (*-assoc (o / pm) p m) ⟨
+  (o ∸ o / pm * p * m) % m ≡⟨ m*n≤o⇒[o∸m*n]%n≡o%n (o / pm * p) lem ⟩
+  o % m                    ∎
+  where
+  pm = p * m
+
+  lem : o / pm * p * m ≤ o
+  lem = begin
+    o / pm * p * m       ≡⟨ *-assoc (o / pm) p m ⟩
+    o / pm * pm          ≤⟨ m/n*n≤m o pm ⟩
+    o                    ∎
+
 m*n/m*o≡n/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (m * o)}} →
               (m * n) / (m * o) ≡ n / o
-m*n/m*o≡n/o m@(suc _) n o = helper (<-wellFounded n)
+m*n/m*o≡n/o m n o = helper (<-wellFounded n)
   where
+  instance _ = m*n≢0 m o
   helper : ∀ {n} → Acc _<_ n → (m * n) / (m * o) ≡ n / o
-  helper {n} (acc rec) with n <? o
-  ... | yes n<o = trans (m<n⇒m/n≡0 (*-monoʳ-< m n<o)) (sym (m<n⇒m/n≡0 n<o))
-  ... | no  n≮o = begin-equality
-    (m * n) / (m * o)             ≡⟨  m/n≡1+[m∸n]/n (*-monoʳ-≤ m (≮⇒≥ n≮o)) ⟩
-    1 + (m * n ∸ m * o) / (m * o) ≡⟨ cong (λ v → 1 + v / (m * o)) (*-distribˡ-∸ m n o) ⟨
-    1 + (m * (n ∸ o)) / (m * o)   ≡⟨  cong suc (helper (rec n∸o<n)) ⟩
-    1 + (n ∸ o) / o               ≡⟨ cong₂ _+_ (n/n≡1 o) refl ⟨
-    o / o + (n ∸ o) / o           ≡⟨ +-distrib-/-∣ˡ (n ∸ o) (divides 1 ((sym (*-identityˡ o)))) ⟨
-    (o + (n ∸ o)) / o             ≡⟨  cong (_/ o) (m+[n∸m]≡n (≮⇒≥ n≮o)) ⟩
+  helper {n} (acc rec) with <-≤-connex n o
+  ... | inj₁ n<o = trans (m<n⇒m/n≡0 (*-monoʳ-< m n<o)) (sym (m<n⇒m/n≡0 n<o))
+    where instance _ = m*n≢0⇒m≢0 m
+  ... | inj₂ n≥o = begin-equality
+    (m * n) / (m * o)             ≡⟨ m/n≡1+[m∸n]/n (*-monoʳ-≤ m n≥o) ⟩
+    1 + (m * n ∸ m * o) / (m * o) ≡⟨ cong (suc ∘ (_/ (m * o))) (*-distribˡ-∸ m n o) ⟨
+    1 + (m * (n ∸ o)) / (m * o)   ≡⟨ cong suc (helper (rec n∸o<n)) ⟩
+    1 + (n ∸ o) / o               ≡⟨ m/n≡1+[m∸n]/n n≥o ⟨
     n / o                         ∎
-    where n∸o<n = ∸-monoʳ-< (n≢0⇒n>0 (≢-nonZero⁻¹ o)) (≮⇒≥ n≮o)
+    where n∸o<n = ∸-monoʳ-< (n≢0⇒n>0 (≢-nonZero⁻¹ o)) n≥o
 
-m*n/o*n≡m/o : ∀ m n o .⦃ _ : NonZero o ⦄ ⦃ _ : NonZero (o * n) ⦄ →
+m*n/o*n≡m/o : ∀ m n o .{{_ : NonZero o}} .{{_ : NonZero (o * n)}} →
               m * n / (o * n) ≡ m / o
-m*n/o*n≡m/o m n o ⦃ _ ⦄ ⦃ o*n≢0 ⦄ = begin-equality
+m*n/o*n≡m/o m n o = begin-equality
   m * n / (o * n) ≡⟨ /-congˡ (*-comm m n) ⟩
   n * m / (o * n) ≡⟨ /-congʳ (*-comm o n) ⟩
   n * m / (n * o) ≡⟨ m*n/m*o≡n/o n m o ⟩
   m / o           ∎
-  where instance n*o≢0 = subst NonZero (*-comm o n) o*n≢0
-
-m<n*o⇒m/o<n : ∀ {m n o} .⦃ _ : NonZero o ⦄ → m < n * o → m / o < n
-m<n*o⇒m/o<n {m} {suc n} {o} m<n*o with m <? o
-... | yes m<o = begin-strict
+  where instance
+    _ : NonZero n
+    _ = m*n≢0⇒n≢0 o
+    _ : NonZero (n * o)
+    _ = m*n≢0 n o
+
+m<n*o⇒m/o<n : ∀ {m n o} .{{_ : NonZero o}} → m < n * o → m / o < n
+m<n*o⇒m/o<n {m} {1} {o} m<o rewrite *-identityˡ o = begin-strict
   m / o ≡⟨ m<n⇒m/n≡0 m<o ⟩
   0     <⟨ z<s ⟩
-  suc n ∎
-... | no m≮o = begin-strict
-  m / o             ≡⟨ m/n≡1+[m∸n]/n (≮⇒≥ m≮o) ⟩
-  suc ((m ∸ o) / o) <⟨ s≤s (m<n*o⇒m/o<n lem) ⟩
-  suc n             ∎
-  where
-  lem : m ∸ o < n * o
-  lem = begin-strict
-    m ∸ o         <⟨ ∸-monoˡ-< m<n*o (≮⇒≥ m≮o) ⟩
-    o + n * o ∸ o ≡⟨ m+n∸m≡n o (n * o) ⟩
-    n * o         ∎
-
-[m∸n]/n≡m/n∸1 : ∀ m n .⦃ _ : NonZero n ⦄ → (m ∸ n) / n ≡ pred (m / n)
-[m∸n]/n≡m/n∸1 m n with m <? n
-... | yes m<n = begin-equality
-  (m ∸ n) / n  ≡⟨ m<n⇒m/n≡0 (≤-<-trans (m∸n≤m m n) m<n) ⟩
-  0            ≡⟨⟩
-  0 ∸ 1        ≡⟨ cong (_∸ 1) (m<n⇒m/n≡0 m<n) ⟨
-  m / n ∸ 1    ≡⟨⟩
-  pred (m / n) ∎
-... | no m≮n = begin-equality
-  (m ∸ n) / n           ≡⟨⟩
-  suc ((m ∸ n) / n) ∸ 1 ≡⟨ cong (_∸ 1) (m/n≡1+[m∸n]/n (≮⇒≥ m≮n)) ⟨
-  m / n ∸ 1             ≡⟨⟩
-  pred (m / n)          ∎
-
-[m∸n*o]/o≡m/o∸n : ∀ m n o .⦃ _ : NonZero o ⦄ → (m ∸ n * o) / o ≡ m / o ∸ n
+  1 ∎
+m<n*o⇒m/o<n {m} {suc n@(suc _)} {o} m<n*o = pred-cancel-< $ begin-strict
+  pred (m / o) ≡⟨ [m∸n]/n≡m/n∸1 m o ⟨
+  (m ∸ o) / o  <⟨ m<n*o⇒m/o<n (m<n+o⇒m∸n<o m o m<n*o) ⟩
+  n ∎
+  where instance _ = m*n≢0 n o
+
+[m∸n*o]/o≡m/o∸n : ∀ m n o .{{_ : NonZero o}} → (m ∸ n * o) / o ≡ m / o ∸ n
 [m∸n*o]/o≡m/o∸n m zero    o = refl
 [m∸n*o]/o≡m/o∸n m (suc n) o = begin-equality
   (m ∸ (o + n * o)) / o ≡⟨ /-congˡ (∸-+-assoc m o (n * o)) ⟨
@@ -351,8 +357,8 @@ m<n*o⇒m/o<n {m} {suc n} {o} m<n*o with m <? o
   m / o ∸ 1 ∸ n         ≡⟨ ∸-+-assoc (m / o) 1 n ⟩
   m / o ∸ suc n         ∎
 
-m/n/o≡m/[n*o] : ∀ m n o .⦃ _ : NonZero n ⦄ .⦃ _ : NonZero o ⦄
-                .⦃ _ : NonZero (n * o) ⦄ → m / n / o ≡ m / (n * o)
+m/n/o≡m/[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}}
+                .{{_ : NonZero (n * o)}} → m / n / o ≡ m / (n * o)
 m/n/o≡m/[n*o] m n o = begin-equality
   m / n / o                             ≡⟨ /-congˡ {o = o} (/-congˡ (m≡m%n+[m/n]*n m n*o)) ⟩
   (m % n*o + m / n*o * n*o) / n / o     ≡⟨ /-congˡ (+-distrib-/-∣ʳ (m % n*o) lem₁) ⟩
@@ -385,27 +391,34 @@ m/n/o≡m/[n*o] m n o = begin-equality
     o*n     ∎
 
 *-/-assoc : ∀ m {n d} .{{_ : NonZero d}} → d ∣ n → m * n / d ≡ m * (n / d)
-*-/-assoc zero    {_} {d@(suc _)} d∣n = 0/n≡0 (suc d)
-*-/-assoc (suc m) {n} {d@(suc _)} d∣n = begin-equality
+*-/-assoc zero    {_} {d} d∣n = 0/n≡0 d
+*-/-assoc (suc m) {n} {d} d∣n = begin-equality
   (n + m * n) / d     ≡⟨ +-distrib-/-∣ˡ _ d∣n ⟩
   n / d + (m * n) / d ≡⟨ cong (n / d +_) (*-/-assoc m d∣n) ⟩
   n / d + m * (n / d) ∎
 
-/-*-interchange : ∀ {m n o p} .{{_ : NonZero o}} .{{_ : NonZero p}} .{{_ : NonZero (o * p)}} →
+/-*-interchange : .{{_ : NonZero o}} .{{_ : NonZero p}} .{{_ : NonZero (o * p)}} →
                   o ∣ m → p ∣ n → (m * n) / (o * p) ≡ (m / o) * (n / p)
-/-*-interchange {m} {n} {o@(suc _)} {p@(suc _)} o∣m p∣n = *-cancelˡ-≡ _ _ (o * p) (begin-equality
-  (o * p) * ((m * n) / (o * p)) ≡⟨  m*[n/m]≡n (*-pres-∣ o∣m p∣n) ⟩
-  m * n                         ≡⟨ cong₂ _*_ (m*[n/m]≡n o∣m) (m*[n/m]≡n p∣n) ⟨
-  (o * (m / o)) * (p * (n / p)) ≡⟨ [m*n]*[o*p]≡[m*o]*[n*p] o (m / o) p (n / p) ⟩
-  (o * p) * ((m / o) * (n / p)) ∎)
+/-*-interchange {o} {p} {m@.(q * o)} {n@.(r * p)} (divides-refl q) (divides-refl r)
+  = begin-equality
+  (m * n) / (o * p) ≡⟨⟩
+  q * o * (r * p) / (o * p) ≡⟨ /-congˡ ([m*n]*[o*p]≡[m*o]*[n*p] q o r p) ⟩
+  q * r * (o * p) / (o * p) ≡⟨ m*n/n≡m (q * r) (o * p) ⟩
+  q * r                     ≡⟨ cong₂ _*_ (m*n/n≡m q o) (m*n/n≡m r p) ⟨
+  (q * o / o) * (r * p / p) ≡⟨⟩
+  (m / o) * (n / p)         ∎
 
 m*n/m!≡n/[m∸1]! : ∀ m n .{{_ : NonZero m}} →
-                 (m * n / m !) {{m !≢0}}  ≡ (n / (pred m) !) {{pred m !≢0}}
-m*n/m!≡n/[m∸1]! (suc m) n = m*n/m*o≡n/o (suc m) n (m !) {{m !≢0}} {{suc m !≢0}}
-
-m%[n*o]/o≡m/o%n : ∀ m n o .⦃ _ : NonZero n ⦄ .⦃ _ : NonZero o ⦄
-                  ⦃ _ : NonZero (n * o) ⦄ → m % (n * o) / o ≡ m / o % n
-m%[n*o]/o≡m/o%n m n o ⦃ _ ⦄ ⦃ _ ⦄ ⦃ n*o≢0 ⦄ = begin-equality
+                  let instance _ = m !≢0 ; instance _ = (pred m) !≢0 in
+                  (m * n / m !) ≡ (n / (pred m) !)
+m*n/m!≡n/[m∸1]! m′@(suc m) n = m*n/m*o≡n/o m′ n (m !)
+  where instance
+    _ = m !≢0
+    _ = m′ !≢0
+
+m%[n*o]/o≡m/o%n : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero o}} →
+                  {{_ : NonZero (n * o)}} → m % (n * o) / o ≡ m / o % n
+m%[n*o]/o≡m/o%n m n o = begin-equality
   m % (n * o) / o                   ≡⟨ /-congˡ (m%n≡m∸m/n*n m (n * o)) ⟩
   (m ∸ (m / (n * o) * (n * o))) / o ≡⟨ cong (λ # → (m ∸ #) / o) (*-assoc (m / (n * o)) n o) ⟨
   (m ∸ (m / (n * o) * n * o)) / o   ≡⟨ [m∸n*o]/o≡m/o∸n m (m / (n * o) * n) o ⟩
@@ -413,11 +426,11 @@ m%[n*o]/o≡m/o%n m n o ⦃ _ ⦄ ⦃ _ ⦄ ⦃ n*o≢0 ⦄ = begin-equality
   m / o ∸ m / (o * n) * n           ≡⟨ cong (λ # → m / o ∸ # * n) (m/n/o≡m/[n*o] m o n ) ⟨
   m / o ∸ m / o / n * n             ≡⟨ m%n≡m∸m/n*n (m / o) n ⟨
   m / o % n                         ∎
-  where instance o*n≢0 = subst NonZero (*-comm n o) n*o≢0
+  where instance _ = m*n≢0 o n
 
-m%n*o≡m*o%[n*o] : ∀ m n o .⦃ _ : NonZero n ⦄ ⦃ _ : NonZero (n * o) ⦄ →
+m%n*o≡m*o%[n*o] : ∀ m n o .{{_ : NonZero n}} .{{_ : NonZero (n * o)}} →
                   m % n * o ≡ m * o % (n * o)
-m%n*o≡m*o%[n*o] m n o ⦃ _ ⦄ ⦃ n*o≢0 ⦄ = begin-equality
+m%n*o≡m*o%[n*o] m n o = begin-equality
   m % n * o                         ≡⟨ cong (_* o) (m%n≡m∸m/n*n m n) ⟩
   (m ∸ m / n * n) * o               ≡⟨ *-distribʳ-∸ o m (m / n * n) ⟩
   m * o ∸ m / n * n * o             ≡⟨ cong (λ # → m * o ∸ # * n * o) (m*n/o*n≡m/o m o n) ⟨
@@ -425,9 +438,9 @@ m%n*o≡m*o%[n*o] m n o ⦃ _ ⦄ ⦃ n*o≢0 ⦄ = begin-equality
   m * o ∸ m * o / (n * o) * (n * o) ≡⟨ m%n≡m∸m/n*n (m * o) (n * o) ⟨
   m * o % (n * o)                   ∎
 
-[m*n+o]%[p*n]≡[m*n]%[p*n]+o : ∀ m {n o} p ⦃ _ : NonZero (p * n) ⦄ → o < n →
+[m*n+o]%[p*n]≡[m*n]%[p*n]+o : ∀ m {n o} p .{{_ : NonZero (p * n)}} → o < n →
                               (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o
-[m*n+o]%[p*n]≡[m*n]%[p*n]+o m {n} {o} p@(suc p-1) ⦃ p*n≢0 ⦄ o<n = begin-equality
+[m*n+o]%[p*n]≡[m*n]%[p*n]+o m {n} {o} p@(suc p-1) o<n = begin-equality
   (mn + o) % pn           ≡⟨ %-distribˡ-+ mn o pn ⟩
   (mn % pn + o % pn) % pn ≡⟨ cong (λ # → (mn % pn + #) % pn) (m<n⇒m%n≡m (m<n⇒m<o*n p o<n)) ⟩
   (mn % pn + o) % pn      ≡⟨ m<n⇒m%n≡m lem₂ ⟩
@@ -458,6 +471,10 @@ record DivMod (dividend divisor : ℕ) : Set where
     remainder : Fin divisor
     property  : dividend ≡ toℕ remainder + quotient * divisor
 
+  nonZeroDivisor : NonZero divisor
+  nonZeroDivisor = nonZeroIndex remainder
+
+
 infixl 7 _div_ _mod_ _divMod_
 
 _div_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index 9083642f80..abac2915a7 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -8,11 +8,11 @@
 
 module Data.Nat.Divisibility where
 
-open import Algebra
 open import Data.Nat.Base
 open import Data.Nat.DivMod
+  using (m≡m%n+[m/n]*n; m%n≡m∸m/n*n; m*n/n≡m; m*n%n≡0; *-/-assoc)
 open import Data.Nat.Properties
-open import Function.Base using (_∘′_; _$_)
+open import Function.Base using (_∘′_; _$_; flip)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (0ℓ)
 open import Relation.Nullary.Decidable as Dec using (yes; no)
@@ -25,24 +25,62 @@ open import Relation.Binary.Definitions
   using (Reflexive; Transitive; Antisymmetric; Decidable)
 import Relation.Binary.Reasoning.Preorder as PreorderReasoning
 open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst)
+  using (_≡_; _≢_; refl; sym; cong; subst)
 open import Relation.Binary.Reasoning.Syntax
-import Relation.Binary.PropositionalEquality.Properties as PropEq
+open import Relation.Binary.PropositionalEquality.Properties
+  using (isEquivalence; module ≡-Reasoning)
+
+private
+  variable d m n o p : ℕ
+
 
 ------------------------------------------------------------------------
--- Definition
+-- Definition and derived properties
+
+open import Data.Nat.Divisibility.Core public hiding (*-pres-∣)
+
+quotient≢0 : (m∣n : m ∣ n) → .{{NonZero n}} → NonZero (quotient m∣n)
+quotient≢0 m∣n rewrite _∣_.equality m∣n = m*n≢0⇒m≢0 (quotient m∣n)
+
+m|n⇒n≡quotient*m : (m∣n : m ∣ n) → n ≡ (quotient m∣n) * m
+m|n⇒n≡quotient*m m∣n = _∣_.equality m∣n
+
+m|n⇒n≡m*quotient : (m∣n : m ∣ n) → n ≡ m * (quotient m∣n)
+m|n⇒n≡m*quotient {m = m} m∣n rewrite _∣_.equality m∣n = *-comm (quotient m∣n) m
+
+quotient-∣ : (m∣n : m ∣ n) → (quotient m∣n) ∣ n
+quotient-∣ {m = m} m∣n = divides m (m|n⇒n≡m*quotient m∣n)
+
+quotient>1 : (m∣n : m ∣ n) → m < n → 1 < quotient m∣n
+quotient>1 {m} {n} m∣n m<n = *-cancelˡ-< m 1 (quotient m∣n) $ begin-strict
+    m * 1        ≡⟨ *-identityʳ m ⟩
+    m            <⟨ m<n ⟩
+    n            ≡⟨ m|n⇒n≡m*quotient m∣n ⟩
+    m * quotient m∣n ∎
+  where open ≤-Reasoning
 
-open import Data.Nat.Divisibility.Core public
+quotient-< : (m∣n : m ∣ n) → .{{NonTrivial m}} → .{{NonZero n}} → quotient m∣n < n
+quotient-< {m} {n} m∣n = begin-strict
+  quotient m∣n     <⟨ m<m*n (quotient m∣n) m (nonTrivial⇒n>1 m) ⟩
+  quotient m∣n * m ≡⟨ _∣_.equality m∣n ⟨
+  n                ∎
+  where open ≤-Reasoning; instance _ = quotient≢0 m∣n
+
+------------------------------------------------------------------------
+-- Relating _/_ and quotient
+
+n/m≡quotient : (m∣n : m ∣ n) .{{_ : NonZero m}} → n / m ≡ quotient m∣n
+n/m≡quotient {m = m} (divides-refl q) = m*n/n≡m q m
 
 ------------------------------------------------------------------------
 -- Relationship with _%_
 
 m%n≡0⇒n∣m : ∀ m n .{{_ : NonZero n}} → m % n ≡ 0 → n ∣ m
-m%n≡0⇒n∣m m n eq = divides (m / n) (begin-equality
-  m                  ≡⟨ m≡m%n+[m/n]*n m n ⟩
-  m % n + m / n * n  ≡⟨ cong₂ _+_ eq refl ⟩
-  m / n * n          ∎)
-  where open ≤-Reasoning
+m%n≡0⇒n∣m m n eq = divides (m / n) $ begin
+  m                ≡⟨ m≡m%n+[m/n]*n m n ⟩
+  m % n + [m/n]*n  ≡⟨ cong (_+ [m/n]*n) eq ⟩
+  [m/n]*n          ∎
+  where open ≡-Reasoning; [m/n]*n = m / n * n
 
 n∣m⇒m%n≡0 : ∀ m n .{{_ : NonZero n}} → n ∣ m → m % n ≡ 0
 n∣m⇒m%n≡0 .(q * n) n (divides-refl q) = m*n%n≡0 q n
@@ -53,15 +91,11 @@ m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)
 ------------------------------------------------------------------------
 -- Properties of _∣_ and _≤_
 
-∣⇒≤ : ∀ {m n} .{{_ : NonZero n}} → m ∣ n → m ≤ n
-∣⇒≤ {m} {n@(suc _)} (divides (suc q) eq) = begin
-  m          ≤⟨ m≤m+n m (q * m) ⟩
-  suc q * m  ≡⟨ sym eq ⟩
-  n          ∎
-  where open ≤-Reasoning
+∣⇒≤ : .{{_ : NonZero n}} → m ∣ n → m ≤ n
+∣⇒≤ {n@.(q * m)} {m} (divides-refl q@(suc p)) = m≤m+n m (p * m)
 
->⇒∤ : ∀ {m n} .{{_ : NonZero n}} → m > n → m ∤ n
->⇒∤ (s≤s m>n) m∣n = contradiction (∣⇒≤ m∣n) (≤⇒≯ m>n)
+>⇒∤ : .{{_ : NonZero n}} → m > n → m ∤ n
+>⇒∤ {n@(suc _)} n<m@(s<s _) m∣n = contradiction (∣⇒≤ m∣n) (<⇒≱ n<m)
 
 ------------------------------------------------------------------------
 -- _∣_ is a partial order
@@ -79,20 +113,20 @@ m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)
   divides (q * p) (sym (*-assoc q p _))
 
 ∣-antisym : Antisymmetric _≡_ _∣_
-∣-antisym {m}     {zero}  _ (divides-refl q) = *-zeroʳ q
-∣-antisym {zero}  {n}     (divides p eq) _   = sym (trans eq (*-comm p 0))
-∣-antisym {suc m} {suc n} p∣q           q∣p  = ≤-antisym (∣⇒≤ p∣q) (∣⇒≤ q∣p)
+∣-antisym {m}     {zero}   _  q∣p = m|n⇒n≡m*quotient q∣p
+∣-antisym {zero}  {n}     p∣q  _  = sym (m|n⇒n≡m*quotient p∣q)
+∣-antisym {suc m} {suc n} p∣q q∣p = ≤-antisym (∣⇒≤ p∣q) (∣⇒≤ q∣p)
 
 infix 4 _∣?_
 
 _∣?_ : Decidable _∣_
 zero  ∣? zero   = yes (divides-refl 0)
 zero  ∣? suc m  = no ((λ()) ∘′ ∣-antisym (divides-refl 0))
-suc n ∣? m      = Dec.map (m%n≡0⇔n∣m m (suc n)) (m % suc n ≟ 0)
+n@(suc _) ∣? m  = Dec.map (m%n≡0⇔n∣m m n) (m % n ≟ 0)
 
 ∣-isPreorder : IsPreorder _≡_ _∣_
 ∣-isPreorder = record
-  { isEquivalence = PropEq.isEquivalence
+  { isEquivalence = isEquivalence
   ; reflexive     = ∣-reflexive
   ; trans         = ∣-trans
   }
@@ -125,42 +159,43 @@ module ∣-Reasoning where
 
   open ∣-syntax _IsRelatedTo_ _IsRelatedTo_ ∣-go public
 
+
 ------------------------------------------------------------------------
 -- Simple properties of _∣_
 
-infix 10 1∣_ _∣0
-
-1∣_ : ∀ n → 1 ∣ n
-1∣ n = divides n (sym (*-identityʳ n))
+infix 10 _∣0 1∣_
 
 _∣0 : ∀ n → n ∣ 0
 n ∣0 = divides-refl 0
 
-0∣⇒≡0 : ∀ {n} → 0 ∣ n → n ≡ 0
+0∣⇒≡0 : 0 ∣ n → n ≡ 0
 0∣⇒≡0 {n} 0∣n = ∣-antisym (n ∣0) 0∣n
 
-∣1⇒≡1 : ∀ {n} → n ∣ 1 → n ≡ 1
+1∣_ : ∀ n → 1 ∣ n
+1∣ n = divides n (sym (*-identityʳ n))
+
+∣1⇒≡1 : n ∣ 1 → n ≡ 1
 ∣1⇒≡1 {n} n∣1 = ∣-antisym n∣1 (1∣ n)
 
-n∣n : ∀ {n} → n ∣ n
-n∣n {n} = ∣-refl
+n∣n : n ∣ n
+n∣n = ∣-refl
 
 ------------------------------------------------------------------------
 -- Properties of _∣_ and _+_
 
-∣m∣n⇒∣m+n : ∀ {i m n} → i ∣ m → i ∣ n → i ∣ m + n
+∣m∣n⇒∣m+n : d ∣ m → d ∣ n → d ∣ m + n
 ∣m∣n⇒∣m+n (divides-refl p) (divides-refl q) =
   divides (p + q) (sym (*-distribʳ-+ _ p q))
 
-∣m+n∣m⇒∣n : ∀ {i m n} → i ∣ m + n → i ∣ m → i ∣ n
-∣m+n∣m⇒∣n {i} {m} {n} (divides p m+n≡p*i) (divides q m≡q*i) =
+∣m+n∣m⇒∣n : d ∣ m + n → d ∣ m → d ∣ n
+∣m+n∣m⇒∣n {d} {m} {n} (divides p m+n≡p*d) (divides-refl q) =
   divides (p ∸ q) $ begin-equality
-    n             ≡⟨ sym (m+n∸n≡m n m) ⟩
+    n             ≡⟨ m+n∸n≡m n m ⟨
     n + m ∸ m     ≡⟨ cong (_∸ m) (+-comm n m) ⟩
-    m + n ∸ m     ≡⟨ cong₂ _∸_ m+n≡p*i m≡q*i ⟩
-    p * i ∸ q * i ≡⟨ sym (*-distribʳ-∸ i p q) ⟩
-    (p ∸ q) * i   ∎
-  where open ∣-Reasoning
+    m + n ∸ m     ≡⟨ cong (_∸ m) m+n≡p*d ⟩
+    p * d ∸ q * d ≡⟨ *-distribʳ-∸ d p q ⟨
+    (p ∸ q) * d   ∎
+    where open ∣-Reasoning
 
 ------------------------------------------------------------------------
 -- Properties of _∣_ and _*_
@@ -174,99 +209,91 @@ m∣m*n n = divides n (*-comm _ n)
 n∣m*n*o : ∀ m {n} o → n ∣ m * n * o
 n∣m*n*o m o = ∣-trans (n∣m*n m) (m∣m*n o)
 
-∣m⇒∣m*n : ∀ {i m} n → i ∣ m → i ∣ m * n
-∣m⇒∣m*n {i} {m} n (divides-refl q) = ∣-trans (n∣m*n q) (m∣m*n n)
+∣m⇒∣m*n : ∀ n → d ∣ m → d ∣ m * n
+∣m⇒∣m*n n (divides-refl q) = ∣-trans (n∣m*n q) (m∣m*n n)
 
-∣n⇒∣m*n : ∀ {i} m {n} → i ∣ n → i ∣ m * n
+∣n⇒∣m*n : ∀ m {n} → d ∣ n → d ∣ m * n
 ∣n⇒∣m*n m {n} rewrite *-comm m n = ∣m⇒∣m*n m
 
-m*n∣⇒m∣ : ∀ {i} m n → m * n ∣ i → m ∣ i
+m*n∣⇒m∣ : ∀ m n → m * n ∣ d → m ∣ d
 m*n∣⇒m∣ m n (divides-refl q) = ∣n⇒∣m*n q (m∣m*n n)
 
-m*n∣⇒n∣ : ∀ {i} m n → m * n ∣ i → n ∣ i
+m*n∣⇒n∣ : ∀ m n → m * n ∣ d → n ∣ d
 m*n∣⇒n∣ m n rewrite *-comm m n = m*n∣⇒m∣ n m
 
-*-monoʳ-∣ : ∀ {i j} k → i ∣ j → k * i ∣ k * j
-*-monoʳ-∣ {i} {j@.(q * i)} k (divides-refl q) = divides q $ begin-equality
-  k * j        ≡⟨⟩
-  k * (q * i)  ≡⟨ sym (*-assoc k q i) ⟩
-  (k * q) * i  ≡⟨ cong (_* i) (*-comm k q) ⟩
-  (q * k) * i  ≡⟨ *-assoc q k i ⟩
-  q * (k * i)  ∎
-  where open ≤-Reasoning
+*-pres-∣ : o ∣ m → p ∣ n → o * p ∣ m * n
+*-pres-∣ {o} {m@.(q * o)} {p} {n@.(r * p)} (divides-refl q) (divides-refl r) =
+  divides (q * r) ([m*n]*[o*p]≡[m*o]*[n*p] q o r p)
 
-*-monoˡ-∣ : ∀ {i j} k → i ∣ j → i * k ∣ j * k
-*-monoˡ-∣ {i} {j} k rewrite *-comm i k | *-comm j k = *-monoʳ-∣ k
+*-monoʳ-∣ : ∀ o → m ∣ n → o * m ∣ o * n
+*-monoʳ-∣ o = *-pres-∣ (∣-refl {o})
 
-*-cancelˡ-∣ : ∀ {i j} k .{{_ : NonZero k}} → k * i ∣ k * j → i ∣ j
-*-cancelˡ-∣ {i} {j} k@(suc _) (divides q eq) =
-  divides q $ *-cancelʳ-≡ j (q * i) _ $ begin-equality
-    j * k        ≡⟨ *-comm j k ⟩
-    k * j        ≡⟨ eq ⟩
-    q * (k * i)  ≡⟨ cong (q *_) (*-comm k i) ⟩
-    q * (i * k)  ≡⟨ sym (*-assoc q i k) ⟩
-    (q * i) * k  ∎
-    where open ≤-Reasoning
+*-monoˡ-∣ : ∀ o → m ∣ n → m * o ∣ n * o
+*-monoˡ-∣ o = flip *-pres-∣ (∣-refl {o})
+
+*-cancelˡ-∣ : ∀ o .{{_ : NonZero o}} → o * m ∣ o * n → m ∣ n
+*-cancelˡ-∣ {m} {n} o o*m∣o*n = divides q $ *-cancelˡ-≡ n (q * m) o $ begin-equality
+  o * n       ≡⟨ m|n⇒n≡m*quotient o*m∣o*n ⟩
+  o * m * q   ≡⟨ *-assoc o m q ⟩
+  o * (m * q) ≡⟨ cong (o *_) (*-comm q m) ⟨
+  o * (q * m) ∎
+  where
+  open ∣-Reasoning
+  q = quotient o*m∣o*n
 
-*-cancelʳ-∣ : ∀ {i j} k .{{_ : NonZero k}} → i * k ∣ j * k → i ∣ j
-*-cancelʳ-∣ {i} {j} k rewrite *-comm i k | *-comm j k = *-cancelˡ-∣ k
+*-cancelʳ-∣ : ∀ o .{{_ : NonZero o}} → m * o ∣ n * o → m ∣ n
+*-cancelʳ-∣ {m} {n} o rewrite *-comm m o | *-comm n o = *-cancelˡ-∣ o
 
 ------------------------------------------------------------------------
 -- Properties of _∣_ and _∸_
 
-∣m∸n∣n⇒∣m : ∀ i {m n} → n ≤ m → i ∣ m ∸ n → i ∣ n → i ∣ m
-∣m∸n∣n⇒∣m i {m} {n} n≤m (divides p m∸n≡p*i) (divides q n≡q*o) =
+∣m∸n∣n⇒∣m : ∀ d → n ≤ m → d ∣ m ∸ n → d ∣ n → d ∣ m
+∣m∸n∣n⇒∣m {n} {m} d n≤m (divides p m∸n≡p*d) (divides-refl q) =
   divides (p + q) $ begin-equality
-    m             ≡⟨ sym (m+[n∸m]≡n n≤m) ⟩
+    m             ≡⟨ m+[n∸m]≡n n≤m ⟨
     n + (m ∸ n)   ≡⟨ +-comm n (m ∸ n) ⟩
-    m ∸ n + n     ≡⟨ cong₂ _+_ m∸n≡p*i n≡q*o ⟩
-    p * i + q * i ≡⟨ sym (*-distribʳ-+ i p q)  ⟩
-    (p + q) * i   ∎
-  where open ≤-Reasoning
+    m ∸ n + n     ≡⟨ cong (_+ n) m∸n≡p*d ⟩
+    p * d + q * d ≡⟨ *-distribʳ-+ d p q ⟨
+    (p + q) * d   ∎
+  where open ∣-Reasoning
 
 ------------------------------------------------------------------------
 -- Properties of _∣_ and _/_
 
-m/n∣m : ∀ {m n} .{{_ : NonZero n}} → n ∣ m → m / n ∣ m
-m/n∣m {m@.(p * n)} {n} (divides-refl p) = begin
-  m / n     ≡⟨⟩
-  p * n / n ≡⟨ m*n/n≡m p n ⟩
-  p         ∣⟨ m∣m*n n ⟩
-  p * n     ≡⟨⟩
-  m         ∎
+m/n∣m : .{{_ : NonZero n}} → n ∣ m → m / n ∣ m
+m/n∣m {n} {m} n∣m = begin
+  m / n        ≡⟨ n/m≡quotient n∣m ⟩
+  quotient n∣m ∣⟨ quotient-∣ n∣m ⟩
+  m            ∎
   where open ∣-Reasoning
 
-m*n∣o⇒m∣o/n : ∀ m n {o} .{{_ : NonZero n}} → m * n ∣ o → m ∣ o / n
-m*n∣o⇒m∣o/n m n {o@.(p * (m * n))} (divides-refl p) = begin
-  m               ∣⟨ n∣m*n p ⟩
-  p * m           ≡⟨ sym (*-identityʳ (p * m)) ⟩
-  p * m * 1       ≡⟨ sym (cong (p * m *_) (n/n≡1 n)) ⟩
-  p * m * (n / n) ≡⟨ sym (*-/-assoc (p * m) (n∣n {n})) ⟩
-  p * m * n / n   ≡⟨ cong (_/ n) (*-assoc p m n) ⟩
-  p * (m * n) / n ≡⟨⟩
-  o / n           ∎
+m*n∣o⇒m∣o/n : ∀ m n .{{_ : NonZero n}} → m * n ∣ o → m ∣ o / n
+m*n∣o⇒m∣o/n m n (divides-refl p) = divides p $ begin-equality
+  p * (m * n) / n   ≡⟨ *-/-assoc p (n∣m*n m) ⟩
+  p * ((m * n) / n) ≡⟨ cong (p *_) (m*n/n≡m m n) ⟩
+  p * m ∎
   where open ∣-Reasoning
 
-m*n∣o⇒n∣o/m : ∀ m n {o} .{{_ : NonZero m}} → m * n ∣ o → n ∣ (o / m)
+m*n∣o⇒n∣o/m : ∀ m n .{{_ : NonZero m}} → m * n ∣ o → n ∣ (o / m)
 m*n∣o⇒n∣o/m m n rewrite *-comm m n = m*n∣o⇒m∣o/n n m
 
-m∣n/o⇒m*o∣n : ∀ {m n o} .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → m * o ∣ n
-m∣n/o⇒m*o∣n {m} {n} {o} (divides-refl p) m∣p*o/o = begin
+m∣n/o⇒m*o∣n : .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → m * o ∣ n
+m∣n/o⇒m*o∣n {o} {n@.(p * o)} {m} (divides-refl p) m∣p*o/o = begin
   m * o ∣⟨ *-monoˡ-∣ o (subst (m ∣_) (m*n/n≡m p o) m∣p*o/o) ⟩
   p * o ∎
   where open ∣-Reasoning
 
-m∣n/o⇒o*m∣n : ∀ {m n o} .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → o * m ∣ n
-m∣n/o⇒o*m∣n {m} {_} {o} rewrite *-comm o m = m∣n/o⇒m*o∣n
+m∣n/o⇒o*m∣n : .{{_ : NonZero o}} → o ∣ n → m ∣ n / o → o * m ∣ n
+m∣n/o⇒o*m∣n {o} {_} {m} rewrite *-comm o m = m∣n/o⇒m*o∣n
 
-m/n∣o⇒m∣o*n : ∀ {m n o} .{{_ : NonZero n}} → n ∣ m → m / n ∣ o → m ∣ o * n
-m/n∣o⇒m∣o*n {_} {n} {o} (divides-refl p) p*n/n∣o = begin
+m/n∣o⇒m∣o*n : .{{_ : NonZero n}} → n ∣ m → m / n ∣ o → m ∣ o * n
+m/n∣o⇒m∣o*n {n} {m@.(p * n)} {o} (divides-refl p) p*n/n∣o = begin
   p * n ∣⟨ *-monoˡ-∣ n (subst (_∣ o) (m*n/n≡m p n) p*n/n∣o) ⟩
   o * n ∎
   where open ∣-Reasoning
 
-m∣n*o⇒m/n∣o : ∀ {m n o} .{{_ : NonZero n}} → n ∣ m → m ∣ o * n → m / n ∣ o
-m∣n*o⇒m/n∣o {m@.(p * n)} {n@(suc _)} {o} (divides-refl p) pn∣on = begin
+m∣n*o⇒m/n∣o : .{{_ : NonZero n}} → n ∣ m → m ∣ o * n → m / n ∣ o
+m∣n*o⇒m/n∣o {n} {m@.(p * n)} {o} (divides-refl p) pn∣on = begin
   m / n     ≡⟨⟩
   p * n / n ≡⟨ m*n/n≡m p n ⟩
   p         ∣⟨ *-cancelʳ-∣ n pn∣on ⟩
@@ -276,58 +303,58 @@ m∣n*o⇒m/n∣o {m@.(p * n)} {n@(suc _)} {o} (divides-refl p) pn∣on = begin
 ------------------------------------------------------------------------
 -- Properties of _∣_ and _%_
 
-∣n∣m%n⇒∣m : ∀ {m n d} .{{_ : NonZero n}} → d ∣ n → d ∣ m % n → d ∣ m
-∣n∣m%n⇒∣m {m} {n@.(a * d)} {d} (divides-refl a) (divides b m%n≡bd) =
-  divides (b + (m / n) * a) (begin-equality
+∣n∣m%n⇒∣m : .{{_ : NonZero n}} → d ∣ n → d ∣ m % n → d ∣ m
+∣n∣m%n⇒∣m {n@.(p * d)} {d} {m} (divides-refl p) (divides q m%n≡qd) =
+  divides (q + (m / n) * p) $ begin-equality
     m                         ≡⟨ m≡m%n+[m/n]*n m n ⟩
-    m % n + (m / n) * n       ≡⟨ cong (_+ (m / n) * n) m%n≡bd ⟩
-    b * d + (m / n) * n       ≡⟨⟩
-    b * d + (m / n) * (a * d) ≡⟨ sym (cong (b * d +_) (*-assoc (m / n) a d)) ⟩
-    b * d + ((m / n) * a) * d ≡⟨ sym (*-distribʳ-+ d b _) ⟩
-    (b + (m / n) * a) * d     ∎)
-    where open ≤-Reasoning
-
-%-presˡ-∣ : ∀ {m n d} .{{_ : NonZero n}} → d ∣ m → d ∣ n → d ∣ m % n
-%-presˡ-∣ {m@.(a * d)} {n} {d} (divides-refl a) (divides b 1+n≡bd) =
-  divides (a ∸ m / n * b) $ begin-equality
-    m % n                   ≡⟨  m%n≡m∸m/n*n m n ⟩
-    m ∸ m / n * n           ≡⟨  cong (λ v → m ∸ m / n * v) 1+n≡bd ⟩
-    m ∸ m / n * (b * d)     ≡⟨ cong (m ∸_) (*-assoc (m / n) b d) ⟨
-    m  ∸ (m / n * b) * d    ≡⟨⟩
-    a * d ∸ (m / n * b) * d ≡⟨ *-distribʳ-∸ d a (m / n * b) ⟨
-    (a ∸ m / n * b) * d     ∎
-  where open ≤-Reasoning
+    m % n + (m / n) * n       ≡⟨ cong (_+ (m / n) * n) m%n≡qd ⟩
+    q * d + (m / n) * n       ≡⟨⟩
+    q * d + (m / n) * (p * d) ≡⟨ cong (q * d +_) (*-assoc (m / n) p d) ⟨
+    q * d + ((m / n) * p) * d ≡⟨ *-distribʳ-+ d q _ ⟨
+    (q + (m / n) * p) * d     ∎
+  where open ∣-Reasoning
+
+%-presˡ-∣ : .{{_ : NonZero n}} → d ∣ m → d ∣ n → d ∣ m % n
+%-presˡ-∣ {n} {d} {m@.(p * d)} (divides-refl p) (divides q 1+n≡qd) =
+  divides (p ∸ m / n * q) $ begin-equality
+    m % n                   ≡⟨ m%n≡m∸m/n*n m n ⟩
+    m ∸ m / n * n           ≡⟨ cong (λ v → m ∸ m / n * v) 1+n≡qd ⟩
+    m ∸ m / n * (q * d)     ≡⟨ cong (m ∸_) (*-assoc (m / n) q d) ⟨
+    m  ∸ (m / n * q) * d    ≡⟨⟩
+    p * d ∸ (m / n * q) * d ≡⟨ *-distribʳ-∸ d p (m / n * q) ⟨
+    (p ∸ m / n * q) * d     ∎
+  where open ∣-Reasoning
 
 ------------------------------------------------------------------------
 -- Properties of _∣_ and !_
 
-m≤n⇒m!∣n! : ∀ {m n} → m ≤ n → m ! ∣ n !
+m≤n⇒m!∣n! : m ≤ n → m ! ∣ n !
 m≤n⇒m!∣n! m≤n = help (≤⇒≤′ m≤n)
   where
-  help : ∀ {m n} → m ≤′ n → m ! ∣ n !
-  help {m} {n}     ≤′-refl        = ∣-refl
-  help {m} {suc n} (≤′-step m≤′n) = ∣n⇒∣m*n (suc n) (help m≤′n)
+  help : m ≤′ n → m ! ∣ n !
+  help         ≤′-refl       = ∣-refl
+  help {n = n} (≤′-step m≤n) = ∣n⇒∣m*n n (help m≤n)
 
 ------------------------------------------------------------------------
--- Properties of _BoundedNonTrivialDivisor_
+-- Properties of _HasNonTrivialDivisorLessThan_
 
 -- Smart constructor
 
-hasNonTrivialDivisor-≢ : ∀ {d n} → .{{NonTrivial d}} → .{{NonZero n}} →
+hasNonTrivialDivisor-≢ : .{{NonTrivial d}} → .{{NonZero n}} →
                          d ≢ n → d ∣ n → n HasNonTrivialDivisorLessThan n
 hasNonTrivialDivisor-≢ d≢n d∣n
   = hasNonTrivialDivisor (≤∧≢⇒< (∣⇒≤ d∣n) d≢n) d∣n
 
 -- Monotonicity wrt ∣
 
-hasNonTrivialDivisor-∣ : ∀ {m n o} → m HasNonTrivialDivisorLessThan n → m ∣ o →
+hasNonTrivialDivisor-∣ : m HasNonTrivialDivisorLessThan n → m ∣ o →
                          o HasNonTrivialDivisorLessThan n
 hasNonTrivialDivisor-∣ (hasNonTrivialDivisor d<n d∣m) n∣o
   = hasNonTrivialDivisor d<n (∣-trans d∣m n∣o)
 
 -- Monotonicity wrt ≤
 
-hasNonTrivialDivisor-≤ : ∀ {m n o} → m HasNonTrivialDivisorLessThan n → n ≤ o →
-                             m HasNonTrivialDivisorLessThan o
-hasNonTrivialDivisor-≤ (hasNonTrivialDivisor d<n d∣m) m≤o =
-  hasNonTrivialDivisor (<-≤-trans d<n m≤o) d∣m
+hasNonTrivialDivisor-≤ : m HasNonTrivialDivisorLessThan n → n ≤ o →
+                         m HasNonTrivialDivisorLessThan o
+hasNonTrivialDivisor-≤ (hasNonTrivialDivisor d<n d∣m) m≤o
+  = hasNonTrivialDivisor (<-≤-trans d<n m≤o) d∣m
diff --git a/src/Data/Nat/Divisibility/Core.agda b/src/Data/Nat/Divisibility/Core.agda
index 58ef913a5f..1b43ab22d6 100644
--- a/src/Data/Nat/Divisibility/Core.agda
+++ b/src/Data/Nat/Divisibility/Core.agda
@@ -14,11 +14,14 @@ module Data.Nat.Divisibility.Core where
 
 open import Data.Nat.Base using (ℕ; _*_; _<_; NonTrivial)
 open import Data.Nat.Properties
-open import Level using (0ℓ)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; sym; cong₂; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl)
+
+
+private
+  variable m n o p : ℕ
 
 ------------------------------------------------------------------------
 -- Main definition
@@ -35,15 +38,16 @@ record _∣_ (m n : ℕ) : Set where
   constructor divides
   field quotient : ℕ
         equality : n ≡ quotient * m
-open _∣_ using (quotient) public
 
-_∤_ : Rel ℕ 0ℓ
+_∤_ : Rel ℕ _
 m ∤ n = ¬ (m ∣ n)
 
 -- Smart constructor
 
 pattern divides-refl q = divides q refl
 
+open _∣_ using (quotient) public
+
 ------------------------------------------------------------------------
 -- Restricted divisor relation
 
@@ -60,12 +64,18 @@ record _HasNonTrivialDivisorLessThan_ (m n : ℕ) : Set where
     divisor-∣       : divisor ∣ m
 
 ------------------------------------------------------------------------
--- Basic properties
-
-*-pres-∣ : ∀ {m n o p} → o ∣ m → p ∣ n → o * p ∣ m * n
-*-pres-∣ {m} {n} {o} {p} (divides c m≡c*o) (divides d n≡d*p) =
-  divides (c * d) (begin
-    m * n             ≡⟨ cong₂ _*_ m≡c*o n≡d*p ⟩
-    (c * o) * (d * p) ≡⟨ [m*n]*[o*p]≡[m*o]*[n*p] c o d p ⟩
-    (c * d) * (o * p) ∎)
-  where open ≡-Reasoning
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.1
+
+*-pres-∣ : o ∣ m → p ∣ n → o * p ∣ m * n
+*-pres-∣ {o} {m@.(q * o)} {p} {n@.(r * p)} (divides-refl q) (divides-refl r) =
+  divides (q * r) ([m*n]*[o*p]≡[m*o]*[n*p] q o r p)
+
+{-# WARNING_ON_USAGE *-pres-∣
+"Warning: *-pres-∣ was deprecated in v2.1.
+ Please use Data.Nat.Divisibility.*-pres-∣ instead."
+#-}
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index ab1678ac9b..b09b46db7d 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -908,7 +908,7 @@ m*n≡0⇒m≡0∨n≡0 : ∀ m {n} → m * n ≡ 0 → m ≡ 0 ⊎ n ≡ 0
 m*n≡0⇒m≡0∨n≡0 zero    {n}     eq = inj₁ refl
 m*n≡0⇒m≡0∨n≡0 (suc m) {zero}  eq = inj₂ refl
 
-m*n≢0 : ∀ m n → .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
+m*n≢0 : ∀ m n .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
 m*n≢0 (suc m) (suc n) = _
 
 m*n≢0⇒m≢0 : ∀ m {n} → .{{NonZero (m * n)}} → NonZero m
@@ -947,9 +947,9 @@ n≢0∧m>1⇒m*n>1 m n rewrite *-comm m n = m≢0∧n>1⇒m*n>1 n m
 -- Other properties of _*_ and _≤_/_<_
 
 *-cancelʳ-≤ : ∀ m n o .{{_ : NonZero o}} → m * o ≤ n * o → m ≤ n
-*-cancelʳ-≤ zero    _       (suc o) _  = z≤n
-*-cancelʳ-≤ (suc m) (suc n) (suc o) le =
-  s≤s (*-cancelʳ-≤ m n (suc o) (+-cancelˡ-≤ _ _ _ le))
+*-cancelʳ-≤ zero    _       _         _  = z≤n
+*-cancelʳ-≤ (suc m) (suc n) o@(suc _) le =
+  s≤s (*-cancelʳ-≤ m n o (+-cancelˡ-≤ _ _ _ le))
 
 *-cancelˡ-≤ : ∀ o .{{_ : NonZero o}} → o * m ≤ o * n → m ≤ n
 *-cancelˡ-≤ {m} {n} o rewrite *-comm o m | *-comm o n = *-cancelʳ-≤ m n o
@@ -969,14 +969,12 @@ n≢0∧m>1⇒m*n>1 m n rewrite *-comm m n = m≢0∧n>1⇒m*n>1 n m
 *-mono-< (s<s m<n@(s≤s _)) u<v@(s≤s _) = +-mono-< u<v (*-mono-< m<n u<v)
 
 *-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_* n) Preserves _<_ ⟶ _<_
-*-monoˡ-< (suc n) z<s       = 0<1+n
-*-monoˡ-< (suc n) (s<s m<o@(s≤s _)) =
-  +-mono-≤-< (≤-refl {suc n}) (*-monoˡ-< (suc n) m<o)
+*-monoˡ-< n@(suc _) z<s               = 0<1+n
+*-monoˡ-< n@(suc _) (s<s m<o@(s≤s _)) = +-mono-≤-< ≤-refl (*-monoˡ-< n m<o)
 
 *-monoʳ-< : ∀ n .{{_ : NonZero n}} → (n *_) Preserves _<_ ⟶ _<_
 *-monoʳ-< (suc zero)      m<o@(s≤s _) = +-mono-≤ m<o z≤n
-*-monoʳ-< (suc n@(suc _)) m<o@(s≤s _) =
-  +-mono-≤ m<o (<⇒≤ (*-monoʳ-< n m<o))
+*-monoʳ-< (suc n@(suc _)) m<o@(s≤s _) = +-mono-≤ m<o (<⇒≤ (*-monoʳ-< n m<o))
 
 m≤m*n : ∀ m n .{{_ : NonZero n}} → m ≤ m * n
 m≤m*n m n@(suc _) = begin
@@ -1598,10 +1596,20 @@ m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
   (m + n) ∸ o          ≡⟨ +-∸-assoc m o≤n ⟩
   m + (n ∸ o)          ∎
 
-m+n≤o⇒m≤o∸n : ∀ m n o → m + n ≤ o → m ≤ o ∸ n
-m+n≤o⇒m≤o∸n zero    n o       le       = z≤n
-m+n≤o⇒m≤o∸n (suc m) n (suc o) (s≤s le)
-  rewrite +-∸-assoc 1 (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m n o le)
+m≤n+o⇒m∸n≤o : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o
+m≤n+o⇒m∸n≤o      m  zero    le = le
+m≤n+o⇒m∸n≤o zero    (suc n)  _ = z≤n
+m≤n+o⇒m∸n≤o (suc m) (suc n) le = m≤n+o⇒m∸n≤o m n (s≤s⁻¹ le)
+
+m<n+o⇒m∸n<o : ∀ m n {o} → .{{NonZero o}} → m < n + o → m ∸ n < o
+m<n+o⇒m∸n<o      m  zero                lt = lt
+m<n+o⇒m∸n<o zero    (suc n) {o@(suc _)} lt = z<s
+m<n+o⇒m∸n<o (suc m) (suc n)             lt = m<n+o⇒m∸n<o m n  (s<s⁻¹ lt)
+
+m+n≤o⇒m≤o∸n : ∀ m {n o} → m + n ≤ o → m ≤ o ∸ n
+m+n≤o⇒m≤o∸n zero    le       = z≤n
+m+n≤o⇒m≤o∸n (suc m) (s≤s le)
+  rewrite +-∸-assoc 1 (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m le)
 
 m≤o∸n⇒m+n≤o : ∀ m {n o} (n≤o : n ≤ o) → m ≤ o ∸ n → m + n ≤ o
 m≤o∸n⇒m+n≤o m         z≤n       le rewrite +-identityʳ m = le
@@ -1728,6 +1736,25 @@ pred-mono-≤ {suc _} {suc _} m≤n = s≤s⁻¹ m≤n
 pred-mono-< : .{{NonZero m}} → m < n → pred m < pred n
 pred-mono-< {m = suc _} {n = suc _} = s<s⁻¹
 
+pred-cancel-≤ : pred m ≤ pred n → (m ≡ 1 × n ≡ 0) ⊎ m ≤ n
+pred-cancel-≤ {m = zero}  {n = zero}  _  = inj₂ z≤n
+pred-cancel-≤ {m = zero}  {n = suc _} _  = inj₂ z≤n
+pred-cancel-≤ {m = suc _} {n = zero} z≤n = inj₁ (refl , refl)
+pred-cancel-≤ {m = suc _} {n = suc _} le = inj₂ (s≤s le)
+
+pred-cancel-< : pred m < pred n → m < n
+pred-cancel-< {m = zero}  {n = suc _} _ = z<s
+pred-cancel-< {m = suc _} {n = suc _}   = s<s
+
+pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n
+pred-injective {suc m} {suc n} = cong suc
+
+pred-cancel-≡ : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n
+pred-cancel-≡ {m = zero}  {n = zero}  _    = inj₂ refl
+pred-cancel-≡ {m = zero}  {n = suc _} refl = inj₁ (inj₁ (refl , refl))
+pred-cancel-≡ {m = suc _} {n = zero}  refl = inj₁ (inj₂ (refl , refl))
+pred-cancel-≡ {m = suc _} {n = suc _}      = inj₂ ∘ pred-injective
+
 ------------------------------------------------------------------------
 -- Properties of ∣_-_∣
 ------------------------------------------------------------------------
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index f0a7d11c96..6ff6a8bc6d 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -311,12 +311,12 @@ normalize-injective-≃ : ∀ m n c d {{_ : ℕ.NonZero c}} {{_ : ℕ.NonZero d}
 normalize-injective-≃ m n c d eq = ℕ./-cancelʳ-≡
   md∣gcd[m,c]gcd[n,d]
   nc∣gcd[m,c]gcd[n,d]
-  (begin
+  $ begin
     (m ℕ.* d) ℕ./ (gcd[m,c] ℕ.* gcd[n,d]) ≡⟨  ℕ./-*-interchange gcd[m,c]∣m gcd[n,d]∣d ⟩
     (m ℕ./ gcd[m,c]) ℕ.* (d ℕ./ gcd[n,d]) ≡⟨  cong₂ ℕ._*_ m/gcd[m,c]≡n/gcd[n,d] (sym c/gcd[m,c]≡d/gcd[n,d]) ⟩
     (n ℕ./ gcd[n,d]) ℕ.* (c ℕ./ gcd[m,c]) ≡⟨ ℕ./-*-interchange gcd[n,d]∣n gcd[m,c]∣c ⟨
-    (n ℕ.* c) ℕ./ (gcd[n,d] ℕ.* gcd[m,c]) ≡⟨  ℕ./-congʳ (ℕ.*-comm gcd[n,d] gcd[m,c]) ⟩
-    (n ℕ.* c) ℕ./ (gcd[m,c] ℕ.* gcd[n,d]) ∎)
+    (n ℕ.* c) ℕ./ (gcd[n,d] ℕ.* gcd[m,c]) ≡⟨ ℕ./-congʳ (ℕ.*-comm gcd[n,d] gcd[m,c]) ⟩
+    (n ℕ.* c) ℕ./ (gcd[m,c] ℕ.* gcd[n,d]) ∎
   where
   open ≡-Reasoning
   gcd[m,c]   = ℕ.gcd m c
@@ -331,14 +331,13 @@ normalize-injective-≃ m n c d eq = ℕ./-cancelʳ-≡
 
   gcd[m,c]≢0′          = ℕ.gcd[m,n]≢0 m c (inj₂ (ℕ.≢-nonZero⁻¹ c))
   gcd[n,d]≢0′          = ℕ.gcd[m,n]≢0 n d (inj₂ (ℕ.≢-nonZero⁻¹ d))
-  gcd[m,c]*gcd[n,d]≢0′ = Sum.[ gcd[m,c]≢0′ , gcd[n,d]≢0′ ] ∘ ℕ.m*n≡0⇒m≡0∨n≡0 _
   instance
     gcd[m,c]≢0   = ℕ.≢-nonZero gcd[m,c]≢0′
     gcd[n,d]≢0   = ℕ.≢-nonZero gcd[n,d]≢0′
     c/gcd[m,c]≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 m c {{gcd≢0 = gcd[m,c]≢0}})
     d/gcd[n,d]≢0 = ℕ.≢-nonZero (ℕ.n/gcd[m,n]≢0 n d {{gcd≢0 = gcd[n,d]≢0}})
-    gcd[m,c]*gcd[n,d]≢0 = ℕ.≢-nonZero gcd[m,c]*gcd[n,d]≢0′
-    gcd[n,d]*gcd[m,c]≢0 = ℕ.≢-nonZero (subst (_≢ 0) (ℕ.*-comm gcd[m,c] gcd[n,d]) gcd[m,c]*gcd[n,d]≢0′)
+    gcd[m,c]*gcd[n,d]≢0 = ℕ.m*n≢0 gcd[m,c] gcd[n,d]
+    gcd[n,d]*gcd[m,c]≢0 = ℕ.m*n≢0 gcd[n,d] gcd[m,c]
 
   div = mkℚ+-injective eq
   m/gcd[m,c]≡n/gcd[n,d] = proj₁ div

From 52926d4673e485517a51c609d86eefcc5c2cf057 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 16 Dec 2023 06:40:03 +0000
Subject: [PATCH 203/316] [fixes #2232] (#2233)

* fixes #2232

* corrected major version number
---
 doc/release-guide.txt     | 2 ++
 standard-library.agda-lib | 2 +-
 2 files changed, 3 insertions(+), 1 deletion(-)

diff --git a/doc/release-guide.txt b/doc/release-guide.txt
index 1fb5565b9b..77a5e77ba5 100644
--- a/doc/release-guide.txt
+++ b/doc/release-guide.txt
@@ -81,3 +81,5 @@ procedure should be followed:
 * Move the CHANGELOG.md into the old CHANGELOG folders
 
 * Create new CHANGELOG.md file
+
+* Update `standard-library.agda-lib` to the new version/milestone on `master`
diff --git a/standard-library.agda-lib b/standard-library.agda-lib
index 30ce05e7dc..ba7ae571b9 100644
--- a/standard-library.agda-lib
+++ b/standard-library.agda-lib
@@ -1,4 +1,4 @@
-name: standard-library-2.0
+name: standard-library-2.1
 include: src
 flags:
   --warning=noUnsupportedIndexedMatch

From d80dfdbf1ba0e5953b532b313360113dabc33278 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Piotr=20Paradzi=C5=84ski?= <piotr.paradzinski@gmail.com>
Date: Wed, 27 Dec 2023 05:12:23 +0100
Subject: [PATCH 204/316] Add `map` to `Data.String.Base` (#2208)

* add map to Data.String

* Add test for Data.String map

* CHANGELOG.md add map to list of new functions in Data.String.Base

* precise import of Data.String to avoid conflict on map

* Fix import statements

---------

Co-authored-by: lemastero <piotr.paradzinski@iohk.io>
Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                 | 5 +++++
 src/Data/String/Base.agda    | 6 ++++++
 src/Data/Tree/Rose/Show.agda | 2 +-
 src/IO.agda                  | 2 +-
 src/Text/Printf/Generic.agda | 2 +-
 5 files changed, 14 insertions(+), 3 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index bf2d01d6a3..8a3ccbdae8 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -80,3 +80,8 @@ Additions to existing modules
   pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n
   pred-cancel-≡ : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n
   ```
+
+* Added new functions in `Data.String.Base`:
+  ```agda
+  map : (Char → Char) → String → String
+  ```
diff --git a/src/Data/String/Base.agda b/src/Data/String/Base.agda
index d583da042c..fb0ebc1526 100644
--- a/src/Data/String/Base.agda
+++ b/src/Data/String/Base.agda
@@ -185,3 +185,9 @@ lines = linesByᵇ ('\n' Char.≈ᵇ_)
 -- `lines` preserves empty lines
 _ : lines "\nabc\n\nb\n\n\n" ≡ "" ∷ "abc" ∷ "" ∷ "b" ∷ "" ∷ "" ∷ []
 _ = refl
+
+map : (Char → Char) → String → String
+map f = fromList ∘ List.map f ∘ toList
+
+_ : map Char.toUpper "abc" ≡ "ABC"
+_ = refl
diff --git a/src/Data/Tree/Rose/Show.agda b/src/Data/Tree/Rose/Show.agda
index 5b293aeb70..1aa0509aae 100644
--- a/src/Data/Tree/Rose/Show.agda
+++ b/src/Data/Tree/Rose/Show.agda
@@ -15,7 +15,7 @@ open import Data.DifferenceList as DList renaming (DiffList to DList) using ()
 open import Data.List.Base as List using (List; []; [_]; _∷_; _∷ʳ_)
 open import Data.Nat.Base using (ℕ; _∸_)
 open import Data.Product.Base using (_×_; _,_)
-open import Data.String.Base hiding (show)
+open import Data.String.Base using (String; _++_)
 open import Data.Tree.Rose using (Rose; node; map; fromBinary)
 open import Function.Base using (flip; _∘′_; id)
 
diff --git a/src/IO.agda b/src/IO.agda
index eebb5d7d92..96f7db2760 100644
--- a/src/IO.agda
+++ b/src/IO.agda
@@ -11,7 +11,7 @@ module IO where
 open import Codata.Musical.Notation
 open import Codata.Musical.Costring
 open import Data.Unit.Polymorphic.Base
-open import Data.String.Base
+open import Data.String.Base using (String)
 import Data.Unit.Base as Unit0
 open import Function.Base using (_∘_; flip)
 import IO.Primitive as Prim
diff --git a/src/Text/Printf/Generic.agda b/src/Text/Printf/Generic.agda
index 07c8856443..21aa36133a 100644
--- a/src/Text/Printf/Generic.agda
+++ b/src/Text/Printf/Generic.agda
@@ -15,7 +15,7 @@ open import Data.Maybe.Base hiding (map)
 open import Data.Nat.Base using (ℕ)
 open import Data.Product.Base hiding (map)
 open import Data.Product.Nary.NonDependent
-open import Data.String.Base
+open import Data.String.Base using (String)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Unit.Base using (⊤)
 open import Function.Nary.NonDependent

From f940caec3ee2d774809bd531045f7f6f0d0e6da5 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 30 Dec 2023 05:36:26 +0000
Subject: [PATCH 205/316] fixes issue #2237 (#2238)

* fixes issue #2237

* leftover from #2182: subtle naming 'bug'/anomaly
---
 CHANGELOG.md                   |  4 ++--
 src/Data/Nat/Divisibility.agda | 26 +++++++++++++-------------
 src/Data/Nat/Primality.agda    |  2 +-
 3 files changed, 16 insertions(+), 16 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8a3ccbdae8..d454783b0b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -58,8 +58,8 @@ Additions to existing modules
   ```agda
   quotient≢0       : m ∣ n → .{{NonZero n}} → NonZero quotient
 
-  m|n⇒n≡quotient*m : m ∣ n → n ≡ quotient * m
-  m|n⇒n≡m*quotient : m ∣ n → n ≡ m * quotient
+  m∣n⇒n≡quotient*m : m ∣ n → n ≡ quotient * m
+  m∣n⇒n≡m*quotient : m ∣ n → n ≡ m * quotient
   quotient-∣       : m ∣ n → quotient ∣ n
   quotient>1       : m ∣ n → m < n → 1 < quotient
   quotient-<       : m ∣ n → .{{NonTrivial m}} → .{{NonZero n}} → quotient < n
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index abac2915a7..f980291cc4 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -42,20 +42,20 @@ open import Data.Nat.Divisibility.Core public hiding (*-pres-∣)
 quotient≢0 : (m∣n : m ∣ n) → .{{NonZero n}} → NonZero (quotient m∣n)
 quotient≢0 m∣n rewrite _∣_.equality m∣n = m*n≢0⇒m≢0 (quotient m∣n)
 
-m|n⇒n≡quotient*m : (m∣n : m ∣ n) → n ≡ (quotient m∣n) * m
-m|n⇒n≡quotient*m m∣n = _∣_.equality m∣n
+m∣n⇒n≡quotient*m : (m∣n : m ∣ n) → n ≡ (quotient m∣n) * m
+m∣n⇒n≡quotient*m m∣n = _∣_.equality m∣n
 
-m|n⇒n≡m*quotient : (m∣n : m ∣ n) → n ≡ m * (quotient m∣n)
-m|n⇒n≡m*quotient {m = m} m∣n rewrite _∣_.equality m∣n = *-comm (quotient m∣n) m
+m∣n⇒n≡m*quotient : (m∣n : m ∣ n) → n ≡ m * (quotient m∣n)
+m∣n⇒n≡m*quotient {m = m} m∣n rewrite _∣_.equality m∣n = *-comm (quotient m∣n) m
 
 quotient-∣ : (m∣n : m ∣ n) → (quotient m∣n) ∣ n
-quotient-∣ {m = m} m∣n = divides m (m|n⇒n≡m*quotient m∣n)
+quotient-∣ {m = m} m∣n = divides m (m∣n⇒n≡m*quotient m∣n)
 
 quotient>1 : (m∣n : m ∣ n) → m < n → 1 < quotient m∣n
 quotient>1 {m} {n} m∣n m<n = *-cancelˡ-< m 1 (quotient m∣n) $ begin-strict
     m * 1        ≡⟨ *-identityʳ m ⟩
     m            <⟨ m<n ⟩
-    n            ≡⟨ m|n⇒n≡m*quotient m∣n ⟩
+    n            ≡⟨ m∣n⇒n≡m*quotient m∣n ⟩
     m * quotient m∣n ∎
   where open ≤-Reasoning
 
@@ -113,8 +113,8 @@ m%n≡0⇔n∣m m n = mk⇔ (m%n≡0⇒n∣m m n) (n∣m⇒m%n≡0 m n)
   divides (q * p) (sym (*-assoc q p _))
 
 ∣-antisym : Antisymmetric _≡_ _∣_
-∣-antisym {m}     {zero}   _  q∣p = m|n⇒n≡m*quotient q∣p
-∣-antisym {zero}  {n}     p∣q  _  = sym (m|n⇒n≡m*quotient p∣q)
+∣-antisym {m}     {zero}   _  q∣p = m∣n⇒n≡m*quotient q∣p
+∣-antisym {zero}  {n}     p∣q  _  = sym (m∣n⇒n≡m*quotient p∣q)
 ∣-antisym {suc m} {suc n} p∣q q∣p = ≤-antisym (∣⇒≤ p∣q) (∣⇒≤ q∣p)
 
 infix 4 _∣?_
@@ -233,7 +233,7 @@ m*n∣⇒n∣ m n rewrite *-comm m n = m*n∣⇒m∣ n m
 
 *-cancelˡ-∣ : ∀ o .{{_ : NonZero o}} → o * m ∣ o * n → m ∣ n
 *-cancelˡ-∣ {m} {n} o o*m∣o*n = divides q $ *-cancelˡ-≡ n (q * m) o $ begin-equality
-  o * n       ≡⟨ m|n⇒n≡m*quotient o*m∣o*n ⟩
+  o * n       ≡⟨ m∣n⇒n≡m*quotient o*m∣o*n ⟩
   o * m * q   ≡⟨ *-assoc o m q ⟩
   o * (m * q) ≡⟨ cong (o *_) (*-comm q m) ⟨
   o * (q * m) ∎
@@ -349,12 +349,12 @@ hasNonTrivialDivisor-≢ d≢n d∣n
 
 hasNonTrivialDivisor-∣ : m HasNonTrivialDivisorLessThan n → m ∣ o →
                          o HasNonTrivialDivisorLessThan n
-hasNonTrivialDivisor-∣ (hasNonTrivialDivisor d<n d∣m) n∣o
-  = hasNonTrivialDivisor d<n (∣-trans d∣m n∣o)
+hasNonTrivialDivisor-∣ (hasNonTrivialDivisor d<n d∣m) m∣o
+  = hasNonTrivialDivisor d<n (∣-trans d∣m m∣o)
 
 -- Monotonicity wrt ≤
 
 hasNonTrivialDivisor-≤ : m HasNonTrivialDivisorLessThan n → n ≤ o →
                          m HasNonTrivialDivisorLessThan o
-hasNonTrivialDivisor-≤ (hasNonTrivialDivisor d<n d∣m) m≤o
-  = hasNonTrivialDivisor (<-≤-trans d<n m≤o) d∣m
+hasNonTrivialDivisor-≤ (hasNonTrivialDivisor d<n d∣m) n≤o
+  = hasNonTrivialDivisor (<-≤-trans d<n n≤o) d∣m
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index 620f0c87b0..683c7d8a87 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -317,7 +317,7 @@ prime⇒¬composite (prime p) = p
   where 2≡1⊎2≡0 = irr[0] {2} (divides-refl 0)
 
 irreducible[1] : Irreducible 1
-irreducible[1] m|1 = inj₁ (∣1⇒≡1 m|1)
+irreducible[1] m∣1 = inj₁ (∣1⇒≡1 m∣1)
 
 irreducible[2] : Irreducible 2
 irreducible[2] {zero}  0∣2 with () ← 0∣⇒≡0 0∣2

From e7d71a300f9c24716992b89df11c834ddb93929f Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Mon, 1 Jan 2024 15:12:26 +0100
Subject: [PATCH 206/316] Bring back a convenient short-cut for infix Func
 (#2239)

* Bring back a convenient short-cut for infix Func

* change name as per suggestion from Matthew

* propagate use of shortcut to where it increases readability

* Revert "propagate use of shortcut to where it increases readability"

This reverts commit debfec19a0b6ad93cadce4e88f559790b287a316.

* Move definitions up. Fix comments.
---
 CHANGELOG.md              |  3 +++
 src/Function/Bundles.agda | 11 +++++++++++
 2 files changed, 14 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d454783b0b..8be73afc66 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -85,3 +85,6 @@ Additions to existing modules
   ```agda
   map : (Char → Char) → String → String
   ```
+
+* In `Function.Bundles`, added `_➙_` as a synonym for `Func` that can
+  be used infix.
diff --git a/src/Function/Bundles.agda b/src/Function/Bundles.agda
index 98aac7a48c..6532715a21 100644
--- a/src/Function/Bundles.agda
+++ b/src/Function/Bundles.agda
@@ -400,6 +400,17 @@ module _ (From : Setoid a ℓ₁) (To : Setoid b ℓ₂) where
   module SplitSurjection (splitSurjection : SplitSurjection) =
     LeftInverse splitSurjection
 
+------------------------------------------------------------------------
+-- Infix abbreviations for oft-used items
+------------------------------------------------------------------------
+
+-- Same naming convention as used for propositional equality below, with
+-- appended ₛ (for 'S'etoid).
+
+infixr 0 _⟶ₛ_
+_⟶ₛ_ : Setoid a ℓ₁ → Setoid b ℓ₂ → Set _
+_⟶ₛ_ = Func
+
 ------------------------------------------------------------------------
 -- Bundles specialised for propositional equality
 ------------------------------------------------------------------------

From d3521b6b0816cd7a54be09b2c42e812cdcc54eac Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 1 Jan 2024 15:00:10 +0000
Subject: [PATCH 207/316] fixes #2234 (#2241)

---
 README.md | 15 ++++++++-------
 1 file changed, 8 insertions(+), 7 deletions(-)

diff --git a/README.md b/README.md
index 3c66d46ee0..4eba8bf8cd 100644
--- a/README.md
+++ b/README.md
@@ -37,13 +37,14 @@ See the [installation instructions](https://github.com/agda/agda-stdlib/blob/mas
 If you're using an old version of Agda, you can download the corresponding version
 of the standard library on the [Agda wiki](http://wiki.portal.chalmers.se/agda/pmwiki.php?n=Libraries.StandardLibrary).
 The module index for older versions of the library is also available. For example,
-version 0.17 can be found at https://agda.github.io/agda-stdlib/v0.17/, just
-replace in the URL 0.17 with the version that you need.
+version 1.7 can be found at https://agda.github.io/agda-stdlib/v1.7/, just
+replace in the URL 1.7 with the version that you need.
 
 #### Development version of Agda
 
 If you're using a development version of Agda rather than the latest official release,
 you should use the `experimental` branch of the standard library rather than `master`.
+[Instructions for updating the `experimental` branch](https://github.com/agda/agda-stdlib/blob/master/doc/updating-experimental.txt).
 The `experimental` branch contains non-backward compatible patches for upcoming
 changes to the language.
 
@@ -52,14 +53,14 @@ changes to the language.
 #### The `--safe` flag
 
 Most of the library can be type-checked using the `--safe` flag. Please consult
-[GenerateEverything.hs](https://github.com/agda/agda-stdlib/blob/master/GenerateEverything.hs#L23)
+[GenerateEverything.hs](https://github.com/agda/agda-stdlib/blob/master/GenerateEverything.hs#L32-L82)
 for a full list of modules that use unsafe features.
 
-#### The `--without-k` flag
+#### The `--cubical-compatible` flag
 
-Most of the library can be type-checked using the `--without-k` flag. Please consult
-[GenerateEverything.hs](https://github.com/agda/agda-stdlib/blob/master/GenerateEverything.hs#L74)
-for a full list of modules that use axiom K.
+Most of the library can be type-checked using the `--cubical-compatible` flag, which since Agda v2.6.3 supersedes the former `--without-K` flag. Please consult
+[GenerateEverything.hs](https://github.com/agda/agda-stdlib/blob/master/GenerateEverything.hs#L91-L111)
+for a full list of modules that use axiom K, requiring the `--with-K` flag.
 
 ## Contributing to the library
 

From e362f5c9006dccb464c24a233b123b36c30a811c Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 1 Jan 2024 15:00:26 +0000
Subject: [PATCH 208/316] Update `HACKING` (#2242)

* added paragraph on coupled contributions

* typo
---
 HACKING.md | 16 ++++++++++++++--
 1 file changed, 14 insertions(+), 2 deletions(-)

diff --git a/HACKING.md b/HACKING.md
index 9b0e7454ac..ff859c0507 100644
--- a/HACKING.md
+++ b/HACKING.md
@@ -27,6 +27,18 @@ What is an acceptable contribution?
 In general, if something is in a general undergraduate Computer Science or Mathematics
 textbook it is probably (!) contributable.
 
+Note on contributions to related/'coupled' modules
+==================================================
+
+Before making changes to a `Data` module please have a look at related modules
+and see if they have any content along similar lines. If so, then please 
+follow those conventions (e.g. naming, argument order).
+For example, if working on `Data.Rational`, please check `Data.Rational.Unnormalised`
+or if working on `Data.Vec` please check `Data.List` and vice versa.
+Likewise, if adding to such modules, please try to make companion additions
+to the related ones, or at least to make a task-list in the comments on your PR,
+indicating what outstanding work may be left for subsequent contributions/contributors.
+
 Setup
 =====
 
@@ -110,9 +122,9 @@ git checkout master
 git pull
 ```
 
-The second step is to create a branch for that feature based off of `master`.
+The second step is to create a branch for that feature based off `master`.
 Make sure to pick a fresh name in place of `new_feature`. We promptly push
-this new branch to our fork using the `-u` option fo `push`.
+this new branch to our fork using the `-u` option for `push`.
 
 ```shell
 git checkout -b new_feature

From cc10e33c29dbfcdb10fb7abf5d87be103560127a Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Wed, 3 Jan 2024 19:03:50 +0100
Subject: [PATCH 209/316] Bring back shortcut [fix CHANGELOG] (#2246)

* Bring back a convenient short-cut for infix Func

* change name as per suggestion from Matthew

* propagate use of shortcut to where it increases readability

* Revert "propagate use of shortcut to where it increases readability"

This reverts commit debfec19a0b6ad93cadce4e88f559790b287a316.

* Move definitions up. Fix comments.

* fix CHANGELOG to report the correct syntax.
---
 CHANGELOG.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8be73afc66..5b3079dee4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -86,5 +86,5 @@ Additions to existing modules
   map : (Char → Char) → String → String
   ```
 
-* In `Function.Bundles`, added `_➙_` as a synonym for `Func` that can
+* In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
   be used infix.

From a05ca4aabd97723052b8df32a6dc8c07733de962 Mon Sep 17 00:00:00 2001
From: Ambroise <chaster_killer@hotmail.fr>
Date: Sat, 20 Jan 2024 02:24:21 +0100
Subject: [PATCH 210/316] fix toList-replicate's statement about vectors
 (#2261)

---
 CHANGELOG.md                 | 4 ++++
 src/Data/Vec/Properties.agda | 2 +-
 2 files changed, 5 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5b3079dee4..c690740a1e 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -9,6 +9,10 @@ Highlights
 Bug-fixes
 ---------
 
+* Fix statement of `Data.Vec.Properties.toList-replicate`, where `replicate`
+  was mistakenly applied to the level of the type `A` instead of the
+  variable `x` of type `A`.
+
 Non-backwards compatible changes
 --------------------------------
 
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 4c5801fd7b..4d469a205c 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -1144,7 +1144,7 @@ zipWith-replicate₂ _⊕_ (x ∷ xs) y =
   cong (x ⊕ y ∷_) (zipWith-replicate₂ _⊕_ xs y)
 
 toList-replicate : ∀ (n : ℕ) (x : A) →
-                   toList (replicate n a) ≡ List.replicate n a
+                   toList (replicate n x) ≡ List.replicate n x
 toList-replicate zero    x = refl
 toList-replicate (suc n) x = cong (_ List.∷_) (toList-replicate n x)
 

From 6c44d200ae728a777b2cbe68be10a5cd4ac44cc9 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 20 Jan 2024 13:31:40 +0000
Subject: [PATCH 211/316] Remove all external use of `less-than-or-equal`
 beyond `Data.Nat.*` (#2256)

* removed all external use of `less-than-or-equal` beyond `Data.Nat.*`

* use of `variable`s
---
 src/Data/Vec/Bounded/Base.agda | 97 +++++++++++++++++-----------------
 1 file changed, 48 insertions(+), 49 deletions(-)

diff --git a/src/Data/Vec/Bounded/Base.agda b/src/Data/Vec/Bounded/Base.agda
index c926c9dfb1..7476afb2d4 100644
--- a/src/Data/Vec/Bounded/Base.agda
+++ b/src/Data/Vec/Bounded/Base.agda
@@ -9,11 +9,11 @@
 module Data.Vec.Bounded.Base where
 
 open import Data.Nat.Base
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.List.Base as List using (List)
-open import Data.List.Extrema ℕₚ.≤-totalOrder
+open import Data.List.Extrema ℕ.≤-totalOrder
 open import Data.List.Relation.Unary.All as All using (All)
-import Data.List.Relation.Unary.All.Properties as Allₚ
+import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Membership.Propositional using (mapWith∈)
 open import Data.Product.Base using (∃; _×_; _,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec)
@@ -21,7 +21,7 @@ open import Data.These.Base as These using (These)
 open import Function.Base using (_∘_; id; _$_)
 open import Level using (Level)
 open import Relation.Nullary.Decidable.Core using (recompute)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 
@@ -31,6 +31,7 @@ private
     A : Set a
     B : Set b
     C : Set c
+    m n : ℕ
 
 ------------------------------------------------------------------------
 -- Types
@@ -45,37 +46,35 @@ record Vec≤ (A : Set a) (n : ℕ) : Set a where
 ------------------------------------------------------------------------
 -- Conversion functions
 
-fromVec : ∀ {n} → Vec A n → Vec≤ A n
-fromVec v = v , ℕₚ.≤-refl
+fromVec : Vec A n → Vec≤ A n
+fromVec v = v , ℕ.≤-refl
 
-padRight : ∀ {n} → A → Vec≤ A n → Vec A n
+padRight : A → Vec≤ A n → Vec A n
 padRight a (vs , m≤n)
-  with recompute (_ ℕₚ.≤″? _) (ℕₚ.≤⇒≤″ m≤n)
-... | less-than-or-equal refl = vs Vec.++ Vec.replicate  _ a
+  with ≤″-offset k ← recompute (_ ℕ.≤″? _) (ℕ.≤⇒≤″ m≤n)
+  = vs Vec.++ Vec.replicate k a
 
-padLeft : ∀ {n} → A → Vec≤ A n → Vec A n
-padLeft a as@(vs , m≤n)
-  with recompute (_ ℕₚ.≤″? _) (ℕₚ.≤⇒≤″ m≤n)
-... | less-than-or-equal {k} ∣as∣+k≡n
-  with P.trans (ℕₚ.+-comm k (Vec≤.length as)) ∣as∣+k≡n
-...   | refl = Vec.replicate k a Vec.++ vs
+padLeft : A → Vec≤ A n → Vec A n
+padLeft a record { length = m ; vec = vs ; bound = m≤n }
+  with ≤″-offset k ← recompute (_ ℕ.≤″? _) (ℕ.≤⇒≤″ m≤n)
+  rewrite ℕ.+-comm m k
+  = Vec.replicate k a Vec.++ vs
 
 private
-  split : ∀ {m n} k → m + k ≡ n → ⌊ k /2⌋ + (m + ⌈ k /2⌉) ≡ n
-  split {m} {n} k eq = begin
-    ⌊ k /2⌋ + (m + ⌈ k /2⌉) ≡⟨ ℕₚ.+-comm ⌊ k /2⌋ _ ⟩
-    m + ⌈ k /2⌉ + ⌊ k /2⌋   ≡⟨ ℕₚ.+-assoc m ⌈ k /2⌉ ⌊ k /2⌋ ⟩
-    m + (⌈ k /2⌉ + ⌊ k /2⌋) ≡⟨ P.cong (m +_) (ℕₚ.+-comm ⌈ k /2⌉ ⌊ k /2⌋) ⟩
-    m + (⌊ k /2⌋ + ⌈ k /2⌉) ≡⟨ P.cong (m +_) (ℕₚ.⌊n/2⌋+⌈n/2⌉≡n k) ⟩
-    m + k                   ≡⟨ eq ⟩
-    n                       ∎ where open ≡-Reasoning
-
-padBoth : ∀ {n} → A → A → Vec≤ A n → Vec A n
-padBoth aₗ aᵣ as@(vs , m≤n)
-  with recompute (_ ℕₚ.≤″? _) (ℕₚ.≤⇒≤″ m≤n)
-... | less-than-or-equal {k} ∣as∣+k≡n
-    with split k ∣as∣+k≡n
-... | refl = Vec.replicate ⌊ k /2⌋ aₗ
+  split : ∀ m k → m + k ≡ ⌊ k /2⌋ + (m + ⌈ k /2⌉)
+  split m k = begin
+    m + k                   ≡⟨ ≡.cong (m +_) (ℕ.⌊n/2⌋+⌈n/2⌉≡n k) ⟨
+    m + (⌊ k /2⌋ + ⌈ k /2⌉) ≡⟨ ≡.cong (m +_) (ℕ.+-comm ⌊ k /2⌋ ⌈ k /2⌉) ⟩
+    m + (⌈ k /2⌉ + ⌊ k /2⌋) ≡⟨ ℕ.+-assoc m ⌈ k /2⌉ ⌊ k /2⌋ ⟨
+    m + ⌈ k /2⌉ + ⌊ k /2⌋   ≡⟨ ℕ.+-comm _ ⌊ k /2⌋ ⟩
+    ⌊ k /2⌋ + (m + ⌈ k /2⌉) ∎
+    where open ≡-Reasoning
+
+padBoth : A → A → Vec≤ A n → Vec A n
+padBoth aₗ aᵣ record { length = m ; vec = vs ; bound = m≤n }
+  with ≤″-offset k ← recompute (_ ℕ.≤″? _) (ℕ.≤⇒≤″ m≤n)
+  rewrite split m k
+  = Vec.replicate ⌊ k /2⌋ aₗ
       Vec.++ vs
       Vec.++ Vec.replicate ⌈ k /2⌉ aᵣ
 
@@ -83,59 +82,59 @@ padBoth aₗ aᵣ as@(vs , m≤n)
 fromList : (as : List A) → Vec≤ A (List.length as)
 fromList = fromVec ∘ Vec.fromList
 
-toList : ∀ {n} → Vec≤ A n → List A
+toList : Vec≤ A n → List A
 toList = Vec.toList ∘ Vec≤.vec
 
 ------------------------------------------------------------------------
 -- Creating new Vec≤ vectors
 
-replicate : ∀ {m n} .(m≤n : m ≤ n) → A → Vec≤ A n
+replicate : .(m≤n : m ≤ n) → A → Vec≤ A n
 replicate m≤n a = Vec.replicate _ a , m≤n
 
-[] : ∀ {n} → Vec≤ A n
+[] : Vec≤ A n
 [] = Vec.[] , z≤n
 
 infixr 5 _∷_
-_∷_ : ∀ {n} → A → Vec≤ A n → Vec≤ A (suc n)
+_∷_ : A → Vec≤ A n → Vec≤ A (suc n)
 a ∷ (as , p) = a Vec.∷ as , s≤s p
 
 ------------------------------------------------------------------------
 -- Modifying Vec≤ vectors
 
-≤-cast : ∀ {m n} → .(m≤n : m ≤ n) → Vec≤ A m → Vec≤ A n
-≤-cast m≤n (v , p) = v , ℕₚ.≤-trans p m≤n
+≤-cast : .(m≤n : m ≤ n) → Vec≤ A m → Vec≤ A n
+≤-cast m≤n (v , p) = v , ℕ.≤-trans p m≤n
 
-≡-cast : ∀ {m n} → .(eq : m ≡ n) → Vec≤ A m → Vec≤ A n
-≡-cast m≡n = ≤-cast (ℕₚ.≤-reflexive m≡n)
+≡-cast : .(eq : m ≡ n) → Vec≤ A m → Vec≤ A n
+≡-cast m≡n = ≤-cast (ℕ.≤-reflexive m≡n)
 
-map : (A → B) → ∀ {n} → Vec≤ A n → Vec≤ B n
+map : (A → B) → Vec≤ A n → Vec≤ B n
 map f (v , p) = Vec.map f v , p
 
-reverse : ∀ {n} → Vec≤ A n → Vec≤ A n
+reverse : Vec≤ A n → Vec≤ A n
 reverse (v , p) = Vec.reverse v , p
 
 -- Align and Zip.
 
-alignWith : (These A B → C) → ∀ {n} → Vec≤ A n → Vec≤ B n → Vec≤ C n
-alignWith f (as , p) (bs , q) = Vec.alignWith f as bs , ℕₚ.⊔-lub p q
+alignWith : (These A B → C) → Vec≤ A n → Vec≤ B n → Vec≤ C n
+alignWith f (as , p) (bs , q) = Vec.alignWith f as bs , ℕ.⊔-lub p q
 
-zipWith : (A → B → C) → ∀ {n} → Vec≤ A n → Vec≤ B n → Vec≤ C n
-zipWith f (as , p) (bs , q) = Vec.restrictWith f as bs , ℕₚ.m≤n⇒m⊓o≤n _ p
+zipWith : (A → B → C) → Vec≤ A n → Vec≤ B n → Vec≤ C n
+zipWith f (as , p) (bs , q) = Vec.restrictWith f as bs , ℕ.m≤n⇒m⊓o≤n _ p
 
-zip : ∀ {n} → Vec≤ A n → Vec≤ B n → Vec≤ (A × B) n
+zip : Vec≤ A n → Vec≤ B n → Vec≤ (A × B) n
 zip = zipWith _,_
 
-align : ∀ {n} → Vec≤ A n → Vec≤ B n → Vec≤ (These A B) n
+align : Vec≤ A n → Vec≤ B n → Vec≤ (These A B) n
 align = alignWith id
 
 -- take and drop
 
-take : ∀ {m} n → Vec≤ A m → Vec≤ A (n ⊓ m)
+take : ∀ n → Vec≤ A m → Vec≤ A (n ⊓ m)
 take             zero    _                = []
 take             (suc n) (Vec.[] , p)     = []
 take {m = suc m} (suc n) (a Vec.∷ as , p) = a ∷ take n (as , s≤s⁻¹ p)
 
-drop : ∀ {m} n → Vec≤ A m → Vec≤ A (m ∸ n)
+drop : ∀ n → Vec≤ A m → Vec≤ A (m ∸ n)
 drop             zero    v                = v
 drop             (suc n) (Vec.[] , p)     = []
 drop {m = suc m} (suc n) (a Vec.∷ as , p) = drop n (as , s≤s⁻¹ p)
@@ -150,7 +149,7 @@ rectangle {A = A} rows = width , padded where
   width = max 0 sizes
 
   all≤ : All (λ v → proj₁ v ≤ width) rows
-  all≤ = Allₚ.map⁻ (xs≤max 0 sizes)
+  all≤ = All.map⁻ (xs≤max 0 sizes)
 
   padded : List (Vec≤ A width)
   padded = mapWith∈ rows $ λ {x} x∈rows →

From 63b61c6c590dcc1b00a934ce4e9724b305ad6f9d Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Sat, 27 Jan 2024 19:25:55 +0100
Subject: [PATCH 212/316] Raw modules bundles (#2263)

* Raw bundles for modules

* Export raw bundles from module bundles

* Update chnagelog

* Remove redundant field

* Reexport raw bundles, add raw bundles for zero

* Add missing reexports, raw bundles for direct product

* Raw bundles for tensor unit

* Update changelog again

* Remove unused open

* Fix typo

* Put a few more definitions in the fake record modules for RawModule and RawSemimodule
---
 CHANGELOG.md                                  |  40 +++
 src/Algebra/Module/Bundles.agda               | 105 ++++++--
 src/Algebra/Module/Bundles/Raw.agda           | 251 ++++++++++++++++++
 .../Module/Construct/DirectProduct.agda       |  79 +++++-
 src/Algebra/Module/Construct/TensorUnit.agda  |  56 ++++
 src/Algebra/Module/Construct/Zero.agda        |  31 ++-
 6 files changed, 539 insertions(+), 23 deletions(-)
 create mode 100644 src/Algebra/Module/Bundles/Raw.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index c690740a1e..97eb05989b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -33,9 +33,49 @@ Deprecated names
 New modules
 -----------
 
+* `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
+
 Additions to existing modules
 -----------------------------
 
+* In `Algebra.Module.Bundles`, raw bundles are re-exported and the bundles expose their raw counterparts.
+
+* In `Algebra.Module.Construct.DirectProduct`:
+  ```agda
+  rawLeftSemimodule  : RawLeftSemimodule R m ℓm → RawLeftSemimodule m′ ℓm′ → RawLeftSemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  rawLeftModule      : RawLeftModule R m ℓm → RawLeftModule m′ ℓm′ → RawLeftModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  rawRightSemimodule : RawRightSemimodule R m ℓm → RawRightSemimodule m′ ℓm′ → RawRightSemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  rawRightModule     : RawRightModule R m ℓm → RawRightModule m′ ℓm′ → RawRightModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  rawBisemimodule    : RawBisemimodule R m ℓm → RawBisemimodule m′ ℓm′ → RawBisemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  rawBimodule        : RawBimodule R m ℓm → RawBimodule m′ ℓm′ → RawBimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  rawSemimodule      : RawSemimodule R m ℓm → RawSemimodule m′ ℓm′ → RawSemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  rawModule          : RawModule R m ℓm → RawModule m′ ℓm′ → RawModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+  ```
+
+* In `Algebra.Module.Construct.TensorUnit`:
+  ```agda
+  rawLeftSemimodule  : RawLeftSemimodule _ c ℓ
+  rawLeftModule      : RawLeftModule _ c ℓ
+  rawRightSemimodule : RawRightSemimodule _ c ℓ
+  rawRightModule     : RawRightModule _ c ℓ
+  rawBisemimodule    : RawBisemimodule _ _ c ℓ
+  rawBimodule        : RawBimodule _ _ c ℓ
+  rawSemimodule      : RawSemimodule _ c ℓ
+  rawModule          : RawModule _ c ℓ
+  ```
+
+* In `Algebra.Module.Construct.Zero`:
+  ```agda
+  rawLeftSemimodule  : RawLeftSemimodule R c ℓ
+  rawLeftModule      : RawLeftModule R c ℓ
+  rawRightSemimodule : RawRightSemimodule R c ℓ
+  rawRightModule     : RawRightModule R c ℓ
+  rawBisemimodule    : RawBisemimodule R c ℓ
+  rawBimodule        : RawBimodule R c ℓ
+  rawSemimodule      : RawSemimodule R c ℓ
+  rawModule          : RawModule R c ℓ
+  ```
+
 * In `Data.Fin.Properties`:
   ```agda
   nonZeroIndex : Fin n → ℕ.NonZero n
diff --git a/src/Algebra/Module/Bundles.agda b/src/Algebra/Module/Bundles.agda
index f36e5bccd0..56b01fcf97 100644
--- a/src/Algebra/Module/Bundles.agda
+++ b/src/Algebra/Module/Bundles.agda
@@ -28,6 +28,7 @@ module Algebra.Module.Bundles where
 open import Algebra.Bundles
 open import Algebra.Core
 open import Algebra.Definitions using (Involutive)
+import Algebra.Module.Bundles.Raw as Raw
 open import Algebra.Module.Core
 open import Algebra.Module.Structures
 open import Algebra.Module.Definitions
@@ -42,6 +43,16 @@ private
   variable
     r ℓr s ℓs : Level
 
+------------------------------------------------------------------------
+-- Re-export definitions of 'raw' bundles
+
+open Raw public
+  using ( RawLeftSemimodule; RawLeftModule
+        ; RawRightSemimodule; RawRightModule
+        ; RawBisemimodule; RawBimodule
+        ; RawSemimodule; RawModule
+        )
+
 ------------------------------------------------------------------------
 -- Left modules
 ------------------------------------------------------------------------
@@ -62,10 +73,6 @@ record LeftSemimodule (semiring : Semiring r ℓr) m ℓm
     0ᴹ : Carrierᴹ
     isLeftSemimodule : IsLeftSemimodule semiring _≈ᴹ_ _+ᴹ_ 0ᴹ _*ₗ_
 
-  infix 4 _≉ᴹ_
-  _≉ᴹ_ : Rel Carrierᴹ _
-  a ≉ᴹ b = ¬ (a ≈ᴹ b)
-
   open IsLeftSemimodule isLeftSemimodule public
 
   +ᴹ-commutativeMonoid : CommutativeMonoid m ℓm
@@ -80,8 +87,17 @@ record LeftSemimodule (semiring : Semiring r ℓr) m ℓm
     ; magma     to +ᴹ-magma
     ; rawMagma  to +ᴹ-rawMagma
     ; rawMonoid to +ᴹ-rawMonoid
+    ; _≉_ to _≉ᴹ_
     )
 
+  rawLeftSemimodule : RawLeftSemimodule Carrier m ℓm
+  rawLeftSemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; 0ᴹ = 0ᴹ
+    }
+
 record LeftModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
   open Ring ring
 
@@ -106,7 +122,7 @@ record LeftModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔ 
 
   open LeftSemimodule leftSemimodule public
     using ( +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma
-          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; _≉ᴹ_)
+          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; rawLeftSemimodule; _≉ᴹ_)
 
   +ᴹ-abelianGroup : AbelianGroup m ℓm
   +ᴹ-abelianGroup = record { isAbelianGroup = +ᴹ-isAbelianGroup }
@@ -114,6 +130,15 @@ record LeftModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔ 
   open AbelianGroup +ᴹ-abelianGroup public
     using () renaming (group to +ᴹ-group; rawGroup to +ᴹ-rawGroup)
 
+  rawLeftModule : RawLeftModule Carrier m ℓm
+  rawLeftModule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; 0ᴹ = 0ᴹ
+    ; -ᴹ_ = -ᴹ_
+    }
+
 ------------------------------------------------------------------------
 -- Right modules
 ------------------------------------------------------------------------
@@ -134,10 +159,6 @@ record RightSemimodule (semiring : Semiring r ℓr) m ℓm
     0ᴹ : Carrierᴹ
     isRightSemimodule : IsRightSemimodule semiring _≈ᴹ_ _+ᴹ_ 0ᴹ _*ᵣ_
 
-  infix 4 _≉ᴹ_
-  _≉ᴹ_ : Rel Carrierᴹ _
-  a ≉ᴹ b = ¬ (a ≈ᴹ b)
-
   open IsRightSemimodule isRightSemimodule public
 
   +ᴹ-commutativeMonoid : CommutativeMonoid m ℓm
@@ -152,8 +173,17 @@ record RightSemimodule (semiring : Semiring r ℓr) m ℓm
     ; magma     to +ᴹ-magma
     ; rawMagma  to +ᴹ-rawMagma
     ; rawMonoid to +ᴹ-rawMonoid
+    ; _≉_ to _≉ᴹ_
     )
 
+  rawRightSemimodule : RawRightSemimodule Carrier m ℓm
+  rawRightSemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    }
+
 record RightModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
   open Ring ring
 
@@ -178,7 +208,7 @@ record RightModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔
 
   open RightSemimodule rightSemimodule public
     using ( +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma
-          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; _≉ᴹ_)
+          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; rawRightSemimodule; _≉ᴹ_)
 
   +ᴹ-abelianGroup : AbelianGroup m ℓm
   +ᴹ-abelianGroup = record { isAbelianGroup = +ᴹ-isAbelianGroup }
@@ -186,6 +216,15 @@ record RightModule (ring : Ring r ℓr) m ℓm : Set (r ⊔ ℓr ⊔ suc (m ⊔
   open AbelianGroup +ᴹ-abelianGroup public
     using () renaming (group to +ᴹ-group; rawGroup to +ᴹ-rawGroup)
 
+  rawRightModule : RawRightModule Carrier m ℓm
+  rawRightModule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    ; -ᴹ_ = -ᴹ_
+    }
+
 ------------------------------------------------------------------------
 -- Bimodules
 ------------------------------------------------------------------------
@@ -220,7 +259,19 @@ record Bisemimodule (R-semiring : Semiring r ℓr) (S-semiring : Semiring s ℓs
 
   open LeftSemimodule leftSemimodule public
     using ( +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma; +ᴹ-rawMagma
-          ; +ᴹ-rawMonoid; _≉ᴹ_)
+          ; +ᴹ-rawMonoid; rawLeftSemimodule; _≉ᴹ_)
+
+  open RightSemimodule rightSemimodule public
+    using ( rawRightSemimodule )
+
+  rawBisemimodule : RawBisemimodule R.Carrier S.Carrier m ℓm
+  rawBisemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    }
 
 record Bimodule (R-ring : Ring r ℓr) (S-ring : Ring s ℓs) m ℓm
                 : Set (r ⊔ s ⊔ ℓr ⊔ ℓs ⊔ suc (m ⊔ ℓm)) where
@@ -254,13 +305,26 @@ record Bimodule (R-ring : Ring r ℓr) (S-ring : Ring s ℓs) m ℓm
   open LeftModule leftModule public
     using ( +ᴹ-abelianGroup; +ᴹ-commutativeMonoid; +ᴹ-group; +ᴹ-monoid
           ; +ᴹ-semigroup; +ᴹ-magma; +ᴹ-rawMagma; +ᴹ-rawMonoid; +ᴹ-rawGroup
-          ; _≉ᴹ_)
+          ; rawLeftSemimodule; rawLeftModule; _≉ᴹ_)
+
+  open RightModule rightModule public
+    using ( rawRightSemimodule; rawRightModule )
 
   bisemimodule : Bisemimodule R.semiring S.semiring m ℓm
   bisemimodule = record { isBisemimodule = isBisemimodule }
 
   open Bisemimodule bisemimodule public
-    using (leftSemimodule; rightSemimodule)
+    using (leftSemimodule; rightSemimodule; rawBisemimodule)
+
+  rawBimodule : RawBimodule R.Carrier S.Carrier m ℓm
+  rawBimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    ; -ᴹ_ = -ᴹ_
+    }
 
 ------------------------------------------------------------------------
 -- Modules over commutative structures
@@ -296,7 +360,9 @@ record Semimodule (commutativeSemiring : CommutativeSemiring r ℓr) m ℓm
   open Bisemimodule bisemimodule public
     using ( leftSemimodule; rightSemimodule
           ; +ᴹ-commutativeMonoid; +ᴹ-monoid; +ᴹ-semigroup; +ᴹ-magma
-          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; _≉ᴹ_)
+          ; +ᴹ-rawMagma; +ᴹ-rawMonoid; rawLeftSemimodule; rawRightSemimodule
+          ; rawBisemimodule; _≉ᴹ_
+          )
 
   open SetR ≈ᴹ-setoid
 
@@ -314,6 +380,9 @@ record Semimodule (commutativeSemiring : CommutativeSemiring r ℓr) m ℓm
     m *ᵣ (y * x)  ≈⟨ ≈ᴹ-sym (*ᵣ-assoc m y x) ⟩
     m *ᵣ y *ᵣ x   ∎
 
+  rawSemimodule : RawSemimodule Carrier m ℓm
+  rawSemimodule = rawBisemimodule
+
 record Module (commutativeRing : CommutativeRing r ℓr) m ℓm
               : Set (r ⊔ ℓr ⊔ suc (m ⊔ ℓm)) where
   open CommutativeRing commutativeRing
@@ -343,9 +412,13 @@ record Module (commutativeRing : CommutativeRing r ℓr) m ℓm
     using ( leftModule; rightModule; leftSemimodule; rightSemimodule
           ; +ᴹ-abelianGroup; +ᴹ-group; +ᴹ-commutativeMonoid; +ᴹ-monoid
           ; +ᴹ-semigroup; +ᴹ-magma ; +ᴹ-rawMonoid; +ᴹ-rawMagma
-          ; +ᴹ-rawGroup; _≉ᴹ_)
+          ; +ᴹ-rawGroup; rawLeftSemimodule; rawLeftModule; rawRightSemimodule
+          ; rawRightModule; rawBisemimodule; rawBimodule; _≉ᴹ_)
 
   semimodule : Semimodule commutativeSemiring m ℓm
   semimodule = record { isSemimodule = isSemimodule }
 
-  open Semimodule semimodule public using (*ₗ-comm; *ᵣ-comm)
+  open Semimodule semimodule public using (*ₗ-comm; *ᵣ-comm; rawSemimodule)
+
+  rawModule : RawModule Carrier m ℓm
+  rawModule = rawBimodule
diff --git a/src/Algebra/Module/Bundles/Raw.agda b/src/Algebra/Module/Bundles/Raw.agda
new file mode 100644
index 0000000000..b9cc108915
--- /dev/null
+++ b/src/Algebra/Module/Bundles/Raw.agda
@@ -0,0 +1,251 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Definitions of 'raw' bundles for module-like algebraic structures
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Algebra.Module.Bundles.Raw where
+
+open import Algebra.Bundles.Raw
+open import Algebra.Core
+open import Algebra.Module.Core
+open import Level
+open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Binary.Core using (Rel)
+
+private
+  variable
+    r ℓr s ℓs : Level
+
+------------------------------------------------------------------------
+-- Raw left modules
+------------------------------------------------------------------------
+
+record RawLeftSemimodule (R : Set r) m ℓm : Set (r ⊔ suc (m ⊔ ℓm)) where
+  infixr 7 _*ₗ_
+  infixl 6 _+ᴹ_
+  infix 4 _≈ᴹ_
+
+  field
+    Carrierᴹ : Set m
+    _≈ᴹ_ : Rel Carrierᴹ ℓm
+    _+ᴹ_ : Op₂ Carrierᴹ
+    _*ₗ_ : Opₗ R Carrierᴹ
+    0ᴹ : Carrierᴹ
+
+  +ᴹ-rawMonoid : RawMonoid m ℓm
+  +ᴹ-rawMonoid = record
+    { _≈_ = _≈ᴹ_
+    ; _∙_ = _+ᴹ_
+    ; ε = 0ᴹ
+    }
+
+  open RawMonoid +ᴹ-rawMonoid public
+    using ()
+    renaming (rawMagma to +ᴹ-rawMagma; _≉_ to _≉ᴹ_)
+
+record RawLeftModule (R : Set r) m ℓm : Set (r ⊔ suc (m ⊔ ℓm)) where
+  infix 8 -ᴹ_
+  infixr 7 _*ₗ_
+  infixl 6 _+ᴹ_
+  infix 4 _≈ᴹ_
+
+  field
+    Carrierᴹ : Set m
+    _≈ᴹ_ : Rel Carrierᴹ ℓm
+    _+ᴹ_ : Op₂ Carrierᴹ
+    _*ₗ_ : Opₗ R Carrierᴹ
+    0ᴹ : Carrierᴹ
+    -ᴹ_ : Op₁ Carrierᴹ
+
+  rawLeftSemimodule : RawLeftSemimodule R m ℓm
+  rawLeftSemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; 0ᴹ = 0ᴹ
+    }
+
+  open RawLeftSemimodule rawLeftSemimodule public
+    using (+ᴹ-rawMagma; +ᴹ-rawMonoid; _≉ᴹ_)
+
+  +ᴹ-rawGroup : RawGroup m ℓm
+  +ᴹ-rawGroup = record
+    { _≈_ = _≈ᴹ_
+    ; _∙_ = _+ᴹ_
+    ; ε = 0ᴹ
+    ; _⁻¹ = -ᴹ_
+    }
+
+------------------------------------------------------------------------
+-- Raw right modules
+------------------------------------------------------------------------
+
+record RawRightSemimodule (R : Set r) m ℓm : Set (r ⊔ suc (m ⊔ ℓm)) where
+  infixl 7 _*ᵣ_
+  infixl 6 _+ᴹ_
+  infix 4 _≈ᴹ_
+
+  field
+    Carrierᴹ : Set m
+    _≈ᴹ_ : Rel Carrierᴹ ℓm
+    _+ᴹ_ : Op₂ Carrierᴹ
+    _*ᵣ_ : Opᵣ R Carrierᴹ
+    0ᴹ : Carrierᴹ
+
+  +ᴹ-rawMonoid : RawMonoid m ℓm
+  +ᴹ-rawMonoid = record
+    { _≈_ = _≈ᴹ_
+    ; _∙_ = _+ᴹ_
+    ; ε = 0ᴹ
+    }
+
+  open RawMonoid +ᴹ-rawMonoid public
+    using ()
+    renaming (rawMagma to +ᴹ-rawMagma; _≉_ to _≉ᴹ_)
+
+record RawRightModule (R : Set r) m ℓm : Set (r ⊔ suc (m ⊔ ℓm)) where
+  infix 8 -ᴹ_
+  infixl 7 _*ᵣ_
+  infixl 6 _+ᴹ_
+  infix 4 _≈ᴹ_
+
+  field
+    Carrierᴹ : Set m
+    _≈ᴹ_ : Rel Carrierᴹ ℓm
+    _+ᴹ_ : Op₂ Carrierᴹ
+    _*ᵣ_ : Opᵣ R Carrierᴹ
+    0ᴹ : Carrierᴹ
+    -ᴹ_ : Op₁ Carrierᴹ
+
+  rawRightSemimodule : RawRightSemimodule R m ℓm
+  rawRightSemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    }
+
+  open RawRightSemimodule rawRightSemimodule public
+    using (+ᴹ-rawMagma; +ᴹ-rawMonoid; _≉ᴹ_)
+
+  +ᴹ-rawGroup : RawGroup m ℓm
+  +ᴹ-rawGroup = record
+    { _≈_ = _≈ᴹ_
+    ; _∙_ = _+ᴹ_
+    ; ε = 0ᴹ
+    ; _⁻¹ = -ᴹ_
+    }
+
+------------------------------------------------------------------------
+-- Bimodules
+------------------------------------------------------------------------
+
+record RawBisemimodule (R : Set r) (S : Set s) m ℓm : Set (r ⊔ s ⊔ suc (m ⊔ ℓm)) where
+  infixr 7 _*ₗ_
+  infixl 7 _*ᵣ_
+  infixl 6 _+ᴹ_
+  infix 4 _≈ᴹ_
+
+  field
+    Carrierᴹ : Set m
+    _≈ᴹ_ : Rel Carrierᴹ ℓm
+    _+ᴹ_ : Op₂ Carrierᴹ
+    _*ₗ_ : Opₗ R Carrierᴹ
+    _*ᵣ_ : Opᵣ S Carrierᴹ
+    0ᴹ : Carrierᴹ
+
+  rawLeftSemimodule : RawLeftSemimodule R m ℓm
+  rawLeftSemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; 0ᴹ = 0ᴹ
+    }
+
+  rawRightSemimodule : RawRightSemimodule S m ℓm
+  rawRightSemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    }
+
+  open RawLeftSemimodule rawLeftSemimodule public
+    using (+ᴹ-rawMagma; +ᴹ-rawMonoid; _≉ᴹ_)
+
+record RawBimodule (R : Set r) (S : Set s) m ℓm : Set (r ⊔ s ⊔ suc (m ⊔ ℓm)) where
+  infix 8 -ᴹ_
+  infixr 7 _*ₗ_
+  infixl 7 _*ᵣ_
+  infixl 6 _+ᴹ_
+  infix 4 _≈ᴹ_
+
+  field
+    Carrierᴹ : Set m
+    _≈ᴹ_ : Rel Carrierᴹ ℓm
+    _+ᴹ_ : Op₂ Carrierᴹ
+    _*ₗ_ : Opₗ R Carrierᴹ
+    _*ᵣ_ : Opᵣ S Carrierᴹ
+    0ᴹ : Carrierᴹ
+    -ᴹ_ : Op₁ Carrierᴹ
+
+  rawLeftModule : RawLeftModule R m ℓm
+  rawLeftModule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; 0ᴹ = 0ᴹ
+    ; -ᴹ_ = -ᴹ_
+    }
+ 
+  rawRightModule : RawRightModule S m ℓm
+  rawRightModule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    ; -ᴹ_ = -ᴹ_
+    }
+
+  rawBisemimodule : RawBisemimodule R S m ℓm
+  rawBisemimodule = record
+    { _≈ᴹ_ = _≈ᴹ_
+    ; _+ᴹ_ = _+ᴹ_
+    ; _*ₗ_ = _*ₗ_
+    ; _*ᵣ_ = _*ᵣ_
+    ; 0ᴹ = 0ᴹ
+    }
+
+  open RawBisemimodule rawBisemimodule public
+    using (+ᴹ-rawMagma; +ᴹ-rawMonoid; rawLeftSemimodule; rawRightSemimodule; _≉ᴹ_)
+
+  open RawLeftModule rawLeftModule public
+    using (+ᴹ-rawGroup)
+
+------------------------------------------------------------------------
+-- Modules over commutative structures
+------------------------------------------------------------------------
+
+RawSemimodule : ∀ (R : Set r) m ℓm → Set (r ⊔ suc (m ⊔ ℓm))
+RawSemimodule R = RawBisemimodule R R
+
+module RawSemimodule {R : Set r} {m ℓm} (M : RawSemimodule R m ℓm) where
+  open RawBisemimodule M public
+
+  rawBisemimodule : RawBisemimodule R R m ℓm
+  rawBisemimodule = M
+
+RawModule : ∀ (R : Set r) m ℓm → Set (r ⊔ suc(m ⊔ ℓm))
+RawModule R = RawBimodule R R
+
+module RawModule {R : Set r} {m ℓm} (M : RawModule R m ℓm) where
+  open RawBimodule M public
+
+  rawBimodule : RawBimodule R R m ℓm
+  rawBimodule = M
+
+  rawSemimodule : RawSemimodule R m ℓm
+  rawSemimodule = rawBisemimodule
diff --git a/src/Algebra/Module/Construct/DirectProduct.agda b/src/Algebra/Module/Construct/DirectProduct.agda
index 429a2d11ca..473e240801 100644
--- a/src/Algebra/Module/Construct/DirectProduct.agda
+++ b/src/Algebra/Module/Construct/DirectProduct.agda
@@ -14,7 +14,7 @@ module Algebra.Module.Construct.DirectProduct where
 open import Algebra.Bundles
 open import Algebra.Construct.DirectProduct
 open import Algebra.Module.Bundles
-open import Data.Product.Base using (map; _,_; proj₁; proj₂)
+open import Data.Product.Base using (map; zip; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Level
 
@@ -22,6 +22,83 @@ private
   variable
     r s ℓr ℓs m m′ ℓm ℓm′ : Level
 
+------------------------------------------------------------------------
+-- Raw bundles
+
+rawLeftSemimodule : {R : Set r} →
+                    RawLeftSemimodule R m ℓm →
+                    RawLeftSemimodule R m′ ℓm′ →
+                    RawLeftSemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawLeftSemimodule M N = record
+  { _≈ᴹ_ = Pointwise M._≈ᴹ_ N._≈ᴹ_
+  ; _+ᴹ_ = zip M._+ᴹ_ N._+ᴹ_
+  ; _*ₗ_ = λ r → map (r M.*ₗ_) (r N.*ₗ_)
+  ; 0ᴹ = M.0ᴹ , N.0ᴹ
+  } where module M = RawLeftSemimodule M; module N = RawLeftSemimodule N
+
+rawLeftModule : {R : Set r} →
+                RawLeftModule R m ℓm →
+                RawLeftModule R m′ ℓm′ →
+                RawLeftModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawLeftModule M N = record
+  { RawLeftSemimodule (rawLeftSemimodule M.rawLeftSemimodule N.rawLeftSemimodule)
+  ; -ᴹ_ = map M.-ᴹ_ N.-ᴹ_
+  } where module M = RawLeftModule M; module N = RawLeftModule N
+
+
+rawRightSemimodule : {R : Set r} →
+                    RawRightSemimodule R m ℓm →
+                    RawRightSemimodule R m′ ℓm′ →
+                    RawRightSemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawRightSemimodule M N = record
+  { _≈ᴹ_ = Pointwise M._≈ᴹ_ N._≈ᴹ_
+  ; _+ᴹ_ = zip M._+ᴹ_ N._+ᴹ_
+  ; _*ᵣ_ = λ mn r → map (M._*ᵣ r) (N._*ᵣ r) mn
+  ; 0ᴹ = M.0ᴹ , N.0ᴹ
+  } where module M = RawRightSemimodule M; module N = RawRightSemimodule N
+
+rawRightModule : {R : Set r} →
+                RawRightModule R m ℓm →
+                RawRightModule R m′ ℓm′ →
+                RawRightModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawRightModule M N = record
+  { RawRightSemimodule (rawRightSemimodule M.rawRightSemimodule N.rawRightSemimodule)
+  ; -ᴹ_ = map M.-ᴹ_ N.-ᴹ_
+  } where module M = RawRightModule M; module N = RawRightModule N
+
+rawBisemimodule : {R : Set r} {S : Set s} →
+                    RawBisemimodule R S m ℓm →
+                    RawBisemimodule R S m′ ℓm′ →
+                    RawBisemimodule R S (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawBisemimodule M N = record
+  { _≈ᴹ_ = Pointwise M._≈ᴹ_ N._≈ᴹ_
+  ; _+ᴹ_ = zip M._+ᴹ_ N._+ᴹ_
+  ; _*ₗ_ = λ r → map (r M.*ₗ_) (r N.*ₗ_)
+  ; _*ᵣ_ = λ mn r → map (M._*ᵣ r) (N._*ᵣ r) mn
+  ; 0ᴹ = M.0ᴹ , N.0ᴹ
+  } where module M = RawBisemimodule M; module N = RawBisemimodule N
+
+rawBimodule : {R : Set r} {S : Set s} →
+                RawBimodule R S m ℓm →
+                RawBimodule R S m′ ℓm′ →
+                RawBimodule R S (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawBimodule M N = record
+  { RawBisemimodule (rawBisemimodule M.rawBisemimodule N.rawBisemimodule)
+  ; -ᴹ_ = map M.-ᴹ_ N.-ᴹ_
+  } where module M = RawBimodule M; module N = RawBimodule N
+
+rawSemimodule : {R : Set r} →
+                RawSemimodule R m ℓm →
+                RawSemimodule R m′ ℓm′ →
+                RawSemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawSemimodule M N = rawBisemimodule M N
+
+rawModule : {R : Set r} →
+            RawModule R m ℓm →
+            RawModule R m′ ℓm′ →
+            RawModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
+rawModule M N = rawBimodule M N
+
 ------------------------------------------------------------------------
 -- Bundles
 
diff --git a/src/Algebra/Module/Construct/TensorUnit.agda b/src/Algebra/Module/Construct/TensorUnit.agda
index c76f2a39ba..6c97b67de0 100644
--- a/src/Algebra/Module/Construct/TensorUnit.agda
+++ b/src/Algebra/Module/Construct/TensorUnit.agda
@@ -19,6 +19,62 @@ private
   variable
     c ℓ : Level
 
+------------------------------------------------------------------------
+-- Raw bundles
+
+rawLeftSemimodule : {R : RawSemiring c ℓ} → RawLeftSemimodule _ c ℓ
+rawLeftSemimodule {R = R} = record
+  { _≈ᴹ_ = _≈_
+  ; _+ᴹ_ = _+_
+  ; _*ₗ_ = _*_
+  ; 0ᴹ = 0#
+  } where open RawSemiring R
+
+rawLeftModule : {R : RawRing c ℓ} → RawLeftModule _ c ℓ
+rawLeftModule {R = R} = record
+  { RawLeftSemimodule (rawLeftSemimodule {R = rawSemiring})
+  ; -ᴹ_ = -_
+  } where open RawRing R
+
+rawRightSemimodule : {R : RawSemiring c ℓ} → RawRightSemimodule _ c ℓ
+rawRightSemimodule {R = R} = record
+  { _≈ᴹ_ = _≈_
+  ; _+ᴹ_ = _+_
+  ; _*ᵣ_ = _*_
+  ; 0ᴹ = 0#
+  } where open RawSemiring R
+
+rawRightModule : {R : RawRing c ℓ} → RawRightModule _ c ℓ
+rawRightModule {R = R} = record
+  { RawRightSemimodule (rawRightSemimodule {R = rawSemiring})
+  ; -ᴹ_ = -_
+  } where open RawRing R
+
+rawBisemimodule : {R : RawSemiring c ℓ} → RawBisemimodule _ _ c ℓ
+rawBisemimodule {R = R} = record
+  { _≈ᴹ_ = _≈_
+  ; _+ᴹ_ = _+_
+  ; _*ₗ_ = _*_
+  ; _*ᵣ_ = _*_
+  ; 0ᴹ = 0#
+  } where open RawSemiring R
+
+rawBimodule : {R : RawRing c ℓ} → RawBimodule _ _ c ℓ
+rawBimodule {R = R} = record
+  { RawBisemimodule (rawBisemimodule {R = rawSemiring})
+  ; -ᴹ_ = -_
+  } where open RawRing R
+
+
+rawSemimodule : {R : RawSemiring c ℓ} → RawSemimodule _ c ℓ
+rawSemimodule {R = R} = rawBisemimodule {R = R}
+
+rawModule : {R : RawRing c ℓ} → RawModule _ c ℓ
+rawModule {R = R} = rawBimodule {R = R}
+
+------------------------------------------------------------------------
+-- Bundles
+
 leftSemimodule : {R : Semiring c ℓ} → LeftSemimodule R c ℓ
 leftSemimodule {R = semiring} = record
   { Carrierᴹ = Carrier
diff --git a/src/Algebra/Module/Construct/Zero.agda b/src/Algebra/Module/Construct/Zero.agda
index 624cd04598..256f70e8c0 100644
--- a/src/Algebra/Module/Construct/Zero.agda
+++ b/src/Algebra/Module/Construct/Zero.agda
@@ -35,14 +35,33 @@ module ℤero where
   _≈ᴹ_     : Rel Carrierᴹ ℓ
   _ ≈ᴹ _ = ⊤
 
-open ℤero
-
 ------------------------------------------------------------------------
--- Raw bundles NOT YET IMPLEMENTED issue #1828
-{-
-rawModule : RawModule c ℓ
+-- Raw bundles
+
+rawLeftSemimodule : {R : Set r} → RawLeftSemimodule R c ℓ
+rawLeftSemimodule = record { ℤero }
+
+rawLeftModule : {R : Set r} → RawLeftModule R c ℓ
+rawLeftModule = record { ℤero }
+
+rawRightSemimodule : {R : Set r} → RawRightSemimodule R c ℓ
+rawRightSemimodule = record { ℤero }
+
+rawRightModule : {R : Set r} → RawRightModule R c ℓ
+rawRightModule = record { ℤero }
+
+rawBisemimodule : {R : Set r} {S : Set s} → RawBisemimodule R S c ℓ
+rawBisemimodule = record { ℤero }
+
+rawBimodule : {R : Set r} {S : Set s} → RawBimodule R S c ℓ
+rawBimodule = record { ℤero }
+
+rawSemimodule : {R : Set r} → RawSemimodule R c ℓ
+rawSemimodule = record { ℤero }
+
+rawModule : {R : Set r} → RawModule R c ℓ
 rawModule = record { ℤero }
--}
+
 ------------------------------------------------------------------------
 -- Bundles
 

From 64e2ba0cffc865a5a91821cda3be02276d9104d7 Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Wed, 31 Jan 2024 01:15:41 +0100
Subject: [PATCH 213/316] Parametrize module morphisms by raw bundles (#2265)

* Index module morphisms by raw bundles

* Use synonyms for RawModule's RawBimodule etc

* Update changelog

* Update constructed morphisms

* Fix Algebra.Module.Morphism.ModuleHomomorphism
---
 CHANGELOG.md                                  |   8 ++
 .../Morphism/Construct/Composition.agda       | 130 ++++++++++--------
 .../Module/Morphism/Construct/Identity.agda   |  34 ++---
 .../Module/Morphism/ModuleHomomorphism.agda   |   6 +-
 src/Algebra/Module/Morphism/Structures.agda   | 129 ++++++++---------
 5 files changed, 153 insertions(+), 154 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 97eb05989b..e8f0a568f2 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -16,6 +16,14 @@ Bug-fixes
 Non-backwards compatible changes
 --------------------------------
 
+* The modules and morphisms in `Algebra.Module.Morphism.Structures` are now
+  parametrized by _raw_ bundles rather than lawful bundles, in line with other
+  modules defining morphism structures.
+* The definitions in `Algebra.Module.Morphism.Construct.Composition` are now
+  parametrized by _raw_ bundles, and as such take a proof of transitivity.
+* The definitions in `Algebra.Module.Morphism.Construct.Identity` are now
+  parametrized by _raw_ bundles, and as such take a proof of reflexivity.
+
 Other major improvements
 ------------------------
 
diff --git a/src/Algebra/Module/Morphism/Construct/Composition.agda b/src/Algebra/Module/Morphism/Construct/Composition.agda
index 8c39496dd4..ddeeef6893 100644
--- a/src/Algebra/Module/Morphism/Construct/Composition.agda
+++ b/src/Algebra/Module/Morphism/Construct/Composition.agda
@@ -8,24 +8,25 @@
 
 module Algebra.Module.Morphism.Construct.Composition where
 
-open import Algebra.Bundles
-open import Algebra.Module.Bundles
+open import Algebra.Module.Bundles.Raw
 open import Algebra.Module.Morphism.Structures
 open import Algebra.Morphism.Construct.Composition
 open import Function.Base using (_∘_)
 import Function.Construct.Composition as Func
 open import Level using (Level)
+open import Relation.Binary.Definitions using (Transitive)
 
 private
   variable
-    r ℓr s ℓs m₁ ℓm₁ m₂ ℓm₂ m₃ ℓm₃ : Level
+    r s m₁ ℓm₁ m₂ ℓm₂ m₃ ℓm₃ : Level
 
 module _
-  {semiring : Semiring r ℓr}
-  {M₁ : LeftSemimodule semiring m₁ ℓm₁}
-  {M₂ : LeftSemimodule semiring m₂ ℓm₂}
-  {M₃ : LeftSemimodule semiring m₃ ℓm₃}
-  (open LeftSemimodule)
+  {R : Set r}
+  {M₁ : RawLeftSemimodule R m₁ ℓm₁}
+  {M₂ : RawLeftSemimodule R m₂ ℓm₂}
+  {M₃ : RawLeftSemimodule R m₃ ℓm₃}
+  (open RawLeftSemimodule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -34,8 +35,8 @@ module _
                                  IsLeftSemimoduleHomomorphism M₂ M₃ g →
                                  IsLeftSemimoduleHomomorphism M₁ M₃ (g ∘ f)
   isLeftSemimoduleHomomorphism f-homo g-homo = record
-    { +ᴹ-isMonoidHomomorphism = isMonoidHomomorphism (≈ᴹ-trans M₃) F.+ᴹ-isMonoidHomomorphism G.+ᴹ-isMonoidHomomorphism
-    ; *ₗ-homo                 = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
+    { +ᴹ-isMonoidHomomorphism = isMonoidHomomorphism ≈ᴹ₃-trans F.+ᴹ-isMonoidHomomorphism G.+ᴹ-isMonoidHomomorphism
+    ; *ₗ-homo                 = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
     } where module F = IsLeftSemimoduleHomomorphism f-homo; module G = IsLeftSemimoduleHomomorphism g-homo
 
   isLeftSemimoduleMonomorphism : IsLeftSemimoduleMonomorphism M₁ M₂ f →
@@ -55,11 +56,12 @@ module _
     } where module F = IsLeftSemimoduleIsomorphism f-iso; module G = IsLeftSemimoduleIsomorphism g-iso
 
 module _
-  {ring : Ring r ℓr}
-  {M₁ : LeftModule ring m₁ ℓm₁}
-  {M₂ : LeftModule ring m₂ ℓm₂}
-  {M₃ : LeftModule ring m₃ ℓm₃}
-  (open LeftModule)
+  {R : Set r}
+  {M₁ : RawLeftModule R m₁ ℓm₁}
+  {M₂ : RawLeftModule R m₂ ℓm₂}
+  {M₃ : RawLeftModule R m₃ ℓm₃}
+  (open RawLeftModule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -68,8 +70,8 @@ module _
                              IsLeftModuleHomomorphism M₂ M₃ g →
                              IsLeftModuleHomomorphism M₁ M₃ (g ∘ f)
   isLeftModuleHomomorphism f-homo g-homo = record
-    { +ᴹ-isGroupHomomorphism = isGroupHomomorphism (≈ᴹ-trans M₃) F.+ᴹ-isGroupHomomorphism G.+ᴹ-isGroupHomomorphism
-    ; *ₗ-homo = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
+    { +ᴹ-isGroupHomomorphism = isGroupHomomorphism ≈ᴹ₃-trans F.+ᴹ-isGroupHomomorphism G.+ᴹ-isGroupHomomorphism
+    ; *ₗ-homo = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
     } where module F = IsLeftModuleHomomorphism f-homo; module G = IsLeftModuleHomomorphism g-homo
 
   isLeftModuleMonomorphism : IsLeftModuleMonomorphism M₁ M₂ f →
@@ -89,11 +91,12 @@ module _
     } where module F = IsLeftModuleIsomorphism f-iso; module G = IsLeftModuleIsomorphism g-iso
 
 module _
-  {semiring : Semiring r ℓr}
-  {M₁ : RightSemimodule semiring m₁ ℓm₁}
-  {M₂ : RightSemimodule semiring m₂ ℓm₂}
-  {M₃ : RightSemimodule semiring m₃ ℓm₃}
-  (open RightSemimodule)
+  {R : Set r}
+  {M₁ : RawRightSemimodule R m₁ ℓm₁}
+  {M₂ : RawRightSemimodule R m₂ ℓm₂}
+  {M₃ : RawRightSemimodule R m₃ ℓm₃}
+  (open RawRightSemimodule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -102,8 +105,8 @@ module _
                                   IsRightSemimoduleHomomorphism M₂ M₃ g →
                                   IsRightSemimoduleHomomorphism M₁ M₃ (g ∘ f)
   isRightSemimoduleHomomorphism f-homo g-homo = record
-    { +ᴹ-isMonoidHomomorphism = isMonoidHomomorphism (≈ᴹ-trans M₃) F.+ᴹ-isMonoidHomomorphism G.+ᴹ-isMonoidHomomorphism
-    ; *ᵣ-homo                 = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
+    { +ᴹ-isMonoidHomomorphism = isMonoidHomomorphism ≈ᴹ₃-trans F.+ᴹ-isMonoidHomomorphism G.+ᴹ-isMonoidHomomorphism
+    ; *ᵣ-homo                 = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
     } where module F = IsRightSemimoduleHomomorphism f-homo; module G = IsRightSemimoduleHomomorphism g-homo
 
   isRightSemimoduleMonomorphism : IsRightSemimoduleMonomorphism M₁ M₂ f →
@@ -123,11 +126,12 @@ module _
     } where module F = IsRightSemimoduleIsomorphism f-iso; module G = IsRightSemimoduleIsomorphism g-iso
 
 module _
-  {ring : Ring r ℓr}
-  {M₁ : RightModule ring m₁ ℓm₁}
-  {M₂ : RightModule ring m₂ ℓm₂}
-  {M₃ : RightModule ring m₃ ℓm₃}
-  (open RightModule)
+  {R : Set r}
+  {M₁ : RawRightModule R m₁ ℓm₁}
+  {M₂ : RawRightModule R m₂ ℓm₂}
+  {M₃ : RawRightModule R m₃ ℓm₃}
+  (open RawRightModule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -136,8 +140,8 @@ module _
                               IsRightModuleHomomorphism M₂ M₃ g →
                               IsRightModuleHomomorphism M₁ M₃ (g ∘ f)
   isRightModuleHomomorphism f-homo g-homo = record
-    { +ᴹ-isGroupHomomorphism = isGroupHomomorphism (≈ᴹ-trans M₃) F.+ᴹ-isGroupHomomorphism G.+ᴹ-isGroupHomomorphism
-    ; *ᵣ-homo                = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
+    { +ᴹ-isGroupHomomorphism = isGroupHomomorphism ≈ᴹ₃-trans F.+ᴹ-isGroupHomomorphism G.+ᴹ-isGroupHomomorphism
+    ; *ᵣ-homo                = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
     } where module F = IsRightModuleHomomorphism f-homo; module G = IsRightModuleHomomorphism g-homo
 
   isRightModuleMonomorphism : IsRightModuleMonomorphism M₁ M₂ f →
@@ -157,12 +161,13 @@ module _
     } where module F = IsRightModuleIsomorphism f-iso; module G = IsRightModuleIsomorphism g-iso
 
 module _
-  {R-semiring : Semiring r ℓr}
-  {S-semiring : Semiring s ℓs}
-  {M₁ : Bisemimodule R-semiring S-semiring m₁ ℓm₁}
-  {M₂ : Bisemimodule R-semiring S-semiring m₂ ℓm₂}
-  {M₃ : Bisemimodule R-semiring S-semiring m₃ ℓm₃}
-  (open Bisemimodule)
+  {R : Set r}
+  {S : Set s}
+  {M₁ : RawBisemimodule R S m₁ ℓm₁}
+  {M₂ : RawBisemimodule R S m₂ ℓm₂}
+  {M₃ : RawBisemimodule R S m₃ ℓm₃}
+  (open RawBisemimodule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -171,9 +176,9 @@ module _
                                IsBisemimoduleHomomorphism M₂ M₃ g →
                                IsBisemimoduleHomomorphism M₁ M₃ (g ∘ f)
   isBisemimoduleHomomorphism f-homo g-homo = record
-    { +ᴹ-isMonoidHomomorphism = isMonoidHomomorphism (≈ᴹ-trans M₃) F.+ᴹ-isMonoidHomomorphism G.+ᴹ-isMonoidHomomorphism
-    ; *ₗ-homo                 = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
-    ; *ᵣ-homo                 = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
+    { +ᴹ-isMonoidHomomorphism = isMonoidHomomorphism ≈ᴹ₃-trans F.+ᴹ-isMonoidHomomorphism G.+ᴹ-isMonoidHomomorphism
+    ; *ₗ-homo                 = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
+    ; *ᵣ-homo                 = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
     } where module F = IsBisemimoduleHomomorphism f-homo; module G = IsBisemimoduleHomomorphism g-homo
 
   isBisemimoduleMonomorphism : IsBisemimoduleMonomorphism M₁ M₂ f →
@@ -193,12 +198,13 @@ module _
     } where module F = IsBisemimoduleIsomorphism f-iso; module G = IsBisemimoduleIsomorphism g-iso
 
 module _
-  {R-ring : Ring r ℓr}
-  {S-ring : Ring s ℓs}
-  {M₁ : Bimodule R-ring S-ring m₁ ℓm₁}
-  {M₂ : Bimodule R-ring S-ring m₂ ℓm₂}
-  {M₃ : Bimodule R-ring S-ring m₃ ℓm₃}
-  (open Bimodule)
+  {R : Set r}
+  {S : Set s}
+  {M₁ : RawBimodule R S m₁ ℓm₁}
+  {M₂ : RawBimodule R S m₂ ℓm₂}
+  {M₃ : RawBimodule R S m₃ ℓm₃}
+  (open RawBimodule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -207,9 +213,9 @@ module _
                            IsBimoduleHomomorphism M₂ M₃ g →
                            IsBimoduleHomomorphism M₁ M₃ (g ∘ f)
   isBimoduleHomomorphism f-homo g-homo = record
-    { +ᴹ-isGroupHomomorphism = isGroupHomomorphism (≈ᴹ-trans M₃) F.+ᴹ-isGroupHomomorphism G.+ᴹ-isGroupHomomorphism
-    ; *ₗ-homo                = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
-    ; *ᵣ-homo                = λ r x → ≈ᴹ-trans M₃ (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
+    { +ᴹ-isGroupHomomorphism = isGroupHomomorphism ≈ᴹ₃-trans F.+ᴹ-isGroupHomomorphism G.+ᴹ-isGroupHomomorphism
+    ; *ₗ-homo                = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ₗ-homo r x)) (G.*ₗ-homo r (f x))
+    ; *ᵣ-homo                = λ r x → ≈ᴹ₃-trans (G.⟦⟧-cong (F.*ᵣ-homo r x)) (G.*ᵣ-homo r (f x))
     } where module F = IsBimoduleHomomorphism f-homo; module G = IsBimoduleHomomorphism g-homo
 
   isBimoduleMonomorphism : IsBimoduleMonomorphism M₁ M₂ f →
@@ -229,11 +235,12 @@ module _
     } where module F = IsBimoduleIsomorphism f-iso; module G = IsBimoduleIsomorphism g-iso
 
 module _
-  {commutativeSemiring : CommutativeSemiring r ℓr}
-  {M₁ : Semimodule commutativeSemiring m₁ ℓm₁}
-  {M₂ : Semimodule commutativeSemiring m₂ ℓm₂}
-  {M₃ : Semimodule commutativeSemiring m₃ ℓm₃}
-  (open Semimodule)
+  {R : Set r}
+  {M₁ : RawSemimodule R m₁ ℓm₁}
+  {M₂ : RawSemimodule R m₂ ℓm₂}
+  {M₃ : RawSemimodule R m₃ ℓm₃}
+  (open RawSemimodule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -242,7 +249,7 @@ module _
                              IsSemimoduleHomomorphism M₂ M₃ g →
                              IsSemimoduleHomomorphism M₁ M₃ (g ∘ f)
   isSemimoduleHomomorphism f-homo g-homo = record
-    { isBisemimoduleHomomorphism = isBisemimoduleHomomorphism F.isBisemimoduleHomomorphism G.isBisemimoduleHomomorphism
+    { isBisemimoduleHomomorphism = isBisemimoduleHomomorphism ≈ᴹ₃-trans F.isBisemimoduleHomomorphism G.isBisemimoduleHomomorphism
     } where module F = IsSemimoduleHomomorphism f-homo; module G = IsSemimoduleHomomorphism g-homo
 
   isSemimoduleMonomorphism : IsSemimoduleMonomorphism M₁ M₂ f →
@@ -262,11 +269,12 @@ module _
     } where module F = IsSemimoduleIsomorphism f-iso; module G = IsSemimoduleIsomorphism g-iso
 
 module _
-  {commutativeRing : CommutativeRing r ℓr}
-  {M₁ : Module commutativeRing m₁ ℓm₁}
-  {M₂ : Module commutativeRing m₂ ℓm₂}
-  {M₃ : Module commutativeRing m₃ ℓm₃}
-  (open Module)
+  {R : Set r}
+  {M₁ : RawModule R m₁ ℓm₁}
+  {M₂ : RawModule R m₂ ℓm₂}
+  {M₃ : RawModule R m₃ ℓm₃}
+  (open RawModule)
+  (≈ᴹ₃-trans : Transitive (_≈ᴹ_ M₃))
   {f : Carrierᴹ M₁ → Carrierᴹ M₂}
   {g : Carrierᴹ M₂ → Carrierᴹ M₃}
   where
@@ -275,7 +283,7 @@ module _
                          IsModuleHomomorphism M₂ M₃ g →
                          IsModuleHomomorphism M₁ M₃ (g ∘ f)
   isModuleHomomorphism f-homo g-homo = record
-    { isBimoduleHomomorphism = isBimoduleHomomorphism F.isBimoduleHomomorphism G.isBimoduleHomomorphism
+    { isBimoduleHomomorphism = isBimoduleHomomorphism ≈ᴹ₃-trans F.isBimoduleHomomorphism G.isBimoduleHomomorphism
     } where module F = IsModuleHomomorphism f-homo; module G = IsModuleHomomorphism g-homo
 
   isModuleMonomorphism : IsModuleMonomorphism M₁ M₂ f →
diff --git a/src/Algebra/Module/Morphism/Construct/Identity.agda b/src/Algebra/Module/Morphism/Construct/Identity.agda
index 55735c2b24..54a08af98c 100644
--- a/src/Algebra/Module/Morphism/Construct/Identity.agda
+++ b/src/Algebra/Module/Morphism/Construct/Identity.agda
@@ -8,8 +8,7 @@
 
 module Algebra.Module.Morphism.Construct.Identity where
 
-open import Algebra.Bundles
-open import Algebra.Module.Bundles
+open import Algebra.Module.Bundles.Raw
 open import Algebra.Module.Morphism.Structures
   using ( module LeftSemimoduleMorphisms
         ; module LeftModuleMorphisms
@@ -25,13 +24,13 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (id)
 import Function.Construct.Identity as Id
 open import Level using (Level)
+open import Relation.Binary.Definitions using (Reflexive)
 
 private
   variable
-    r ℓr s ℓs m ℓm : Level
+    r s m ℓm : Level
 
-module _ {semiring : Semiring r ℓr} (M : LeftSemimodule semiring m ℓm) where
-  open LeftSemimodule M using (≈ᴹ-refl)
+module _ {R : Set r} (M : RawLeftSemimodule R m ℓm) (open RawLeftSemimodule M) (≈ᴹ-refl : Reflexive _≈ᴹ_) where
   open LeftSemimoduleMorphisms M M
 
   isLeftSemimoduleHomomorphism : IsLeftSemimoduleHomomorphism id
@@ -52,8 +51,7 @@ module _ {semiring : Semiring r ℓr} (M : LeftSemimodule semiring m ℓm) where
     ; surjective                   = Id.surjective _
     }
 
-module _ {ring : Ring r ℓr} (M : LeftModule ring m ℓm) where
-  open LeftModule M using (≈ᴹ-refl)
+module _ {R : Set r} (M : RawLeftModule R m ℓm) (open RawLeftModule M) (≈ᴹ-refl : Reflexive _≈ᴹ_)  where
   open LeftModuleMorphisms M M
 
   isLeftModuleHomomorphism : IsLeftModuleHomomorphism id
@@ -74,8 +72,7 @@ module _ {ring : Ring r ℓr} (M : LeftModule ring m ℓm) where
     ; surjective               = Id.surjective _
     }
 
-module _ {semiring : Semiring r ℓr} (M : RightSemimodule semiring m ℓm) where
-  open RightSemimodule M using (≈ᴹ-refl)
+module _ {R : Set r} (M : RawRightSemimodule R m ℓm) (open RawRightSemimodule M) (≈ᴹ-refl : Reflexive _≈ᴹ_) where
   open RightSemimoduleMorphisms M M
 
   isRightSemimoduleHomomorphism : IsRightSemimoduleHomomorphism id
@@ -96,8 +93,7 @@ module _ {semiring : Semiring r ℓr} (M : RightSemimodule semiring m ℓm) wher
     ; surjective                    = Id.surjective _
     }
 
-module _ {ring : Ring r ℓr} (M : RightModule ring m ℓm) where
-  open RightModule M using (≈ᴹ-refl)
+module _ {R : Set r} (M : RawRightModule R m ℓm) (open RawRightModule M) (≈ᴹ-refl : Reflexive _≈ᴹ_) where
   open RightModuleMorphisms M M
 
   isRightModuleHomomorphism : IsRightModuleHomomorphism id
@@ -118,8 +114,7 @@ module _ {ring : Ring r ℓr} (M : RightModule ring m ℓm) where
     ; surjective                = Id.surjective _
     }
 
-module _ {R-semiring : Semiring r ℓr} {S-semiring : Semiring s ℓs} (M : Bisemimodule R-semiring S-semiring m ℓm) where
-  open Bisemimodule M using (≈ᴹ-refl)
+module _ {R : Set r} {S : Set s} (M : RawBisemimodule R S m ℓm) (open RawBisemimodule M) (≈ᴹ-refl : Reflexive _≈ᴹ_) where
   open BisemimoduleMorphisms M M
 
   isBisemimoduleHomomorphism : IsBisemimoduleHomomorphism id
@@ -141,8 +136,7 @@ module _ {R-semiring : Semiring r ℓr} {S-semiring : Semiring s ℓs} (M : Bise
     ; surjective                 = Id.surjective _
     }
 
-module _ {R-ring : Ring r ℓr} {S-ring : Ring r ℓr} (M : Bimodule R-ring S-ring m ℓm) where
-  open Bimodule M using (≈ᴹ-refl)
+module _ {R : Set r} {S : Set s} (M : RawBimodule R S m ℓm) (open RawBimodule M) (≈ᴹ-refl : Reflexive _≈ᴹ_) where
   open BimoduleMorphisms M M
 
   isBimoduleHomomorphism : IsBimoduleHomomorphism id
@@ -164,13 +158,12 @@ module _ {R-ring : Ring r ℓr} {S-ring : Ring r ℓr} (M : Bimodule R-ring S-ri
     ; surjective             = Id.surjective _
     }
 
-module _ {commutativeSemiring : CommutativeSemiring r ℓr} (M : Semimodule commutativeSemiring m ℓm) where
-  open Semimodule M using (≈ᴹ-refl)
+module _ {R : Set r} (M : RawSemimodule R m ℓm) (open RawSemimodule M) (≈ᴹ-refl : Reflexive _≈ᴹ_)  where
   open SemimoduleMorphisms M M
 
   isSemimoduleHomomorphism : IsSemimoduleHomomorphism id
   isSemimoduleHomomorphism = record
-    { isBisemimoduleHomomorphism = isBisemimoduleHomomorphism _
+    { isBisemimoduleHomomorphism = isBisemimoduleHomomorphism _ ≈ᴹ-refl
     }
 
   isSemimoduleMonomorphism : IsSemimoduleMonomorphism id
@@ -185,13 +178,12 @@ module _ {commutativeSemiring : CommutativeSemiring r ℓr} (M : Semimodule comm
     ; surjective               = Id.surjective _
     }
 
-module _ {commutativeRing : CommutativeRing r ℓr} (M : Module commutativeRing m ℓm) where
-  open Module M using (≈ᴹ-refl)
+module _ {R : Set r} (M : RawModule R m ℓm) (open RawModule M) (≈ᴹ-refl : Reflexive _≈ᴹ_) where
   open ModuleMorphisms M M
 
   isModuleHomomorphism : IsModuleHomomorphism id
   isModuleHomomorphism = record
-    { isBimoduleHomomorphism = isBimoduleHomomorphism _
+    { isBimoduleHomomorphism = isBimoduleHomomorphism _ ≈ᴹ-refl
     }
 
   isModuleMonomorphism : IsModuleMonomorphism id
diff --git a/src/Algebra/Module/Morphism/ModuleHomomorphism.agda b/src/Algebra/Module/Morphism/ModuleHomomorphism.agda
index 704ec3863c..6bd7f2b4e6 100644
--- a/src/Algebra/Module/Morphism/ModuleHomomorphism.agda
+++ b/src/Algebra/Module/Morphism/ModuleHomomorphism.agda
@@ -17,10 +17,10 @@ module Algebra.Module.Morphism.ModuleHomomorphism
   {r ℓr m ℓm : Level}
   {ring      : CommutativeRing r ℓr}
   (modA modB  : Module ring m ℓm)
-  (open Module modA using () renaming (Carrierᴹ to A))
-  (open Module modB using () renaming (Carrierᴹ to B))
+  (open Module modA using () renaming (Carrierᴹ to A; rawModule to rawModA))
+  (open Module modB using () renaming (Carrierᴹ to B; rawModule to rawModB))
   {f : A → B}
-  (open MorphismStructures.ModuleMorphisms modA modB)
+  (open MorphismStructures.ModuleMorphisms rawModA rawModB)
   (isModuleHomomorphism : IsModuleHomomorphism f)
   where
 
diff --git a/src/Algebra/Module/Morphism/Structures.agda b/src/Algebra/Module/Morphism/Structures.agda
index 96fb5ffa0a..e874f4d341 100644
--- a/src/Algebra/Module/Morphism/Structures.agda
+++ b/src/Algebra/Module/Morphism/Structures.agda
@@ -7,8 +7,7 @@
 
 module Algebra.Module.Morphism.Structures where
 
-open import Algebra.Bundles
-open import Algebra.Module.Bundles
+open import Algebra.Module.Bundles.Raw
 import Algebra.Module.Morphism.Definitions as MorphismDefinitions
 import Algebra.Morphism.Structures as MorphismStructures
 open import Function.Definitions
@@ -16,19 +15,18 @@ open import Level
 
 private
   variable
-    r ℓr s ℓs m₁ m₂ ℓm₁ ℓm₂ : Level
+    r s m₁ m₂ ℓm₁ ℓm₂ : Level
 
 module LeftSemimoduleMorphisms
-  {semiring : Semiring r ℓr}
-  (M₁ : LeftSemimodule semiring m₁ ℓm₁)
-  (M₂ : LeftSemimodule semiring m₂ ℓm₂)
+  {R : Set r}
+  (M₁ : RawLeftSemimodule R m₁ ℓm₁)
+  (M₂ : RawLeftSemimodule R m₂ ℓm₂)
   where
 
-  open Semiring semiring renaming (Carrier to R)
-  open LeftSemimodule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _≈ᴹ_ to _≈ᴹ₁_)
-  open LeftSemimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _≈ᴹ_ to _≈ᴹ₂_)
+  open RawLeftSemimodule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawLeftSemimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open MorphismStructures.MonoidMorphisms (LeftSemimodule.+ᴹ-rawMonoid M₁) (LeftSemimodule.+ᴹ-rawMonoid M₂)
+  open MorphismStructures.MonoidMorphisms (RawLeftSemimodule.+ᴹ-rawMonoid M₁) (RawLeftSemimodule.+ᴹ-rawMonoid M₂)
 
   record IsLeftSemimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
@@ -74,17 +72,16 @@ module LeftSemimoduleMorphisms
       renaming (isMagmaIsomorphism to +ᴹ-isMagmaIsomorphism)
 
 module LeftModuleMorphisms
-  {ring : Ring r ℓr}
-  (M₁ : LeftModule ring m₁ ℓm₁)
-  (M₂ : LeftModule ring m₂ ℓm₂)
+  {R : Set r}
+  (M₁ : RawLeftModule R m₁ ℓm₁)
+  (M₂ : RawLeftModule R m₂ ℓm₂)
   where
 
-  open Ring ring renaming (Carrier to R)
-  open LeftModule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _≈ᴹ_ to _≈ᴹ₁_)
-  open LeftModule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _≈ᴹ_ to _≈ᴹ₂_)
+  open RawLeftModule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawLeftModule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open MorphismStructures.GroupMorphisms (LeftModule.+ᴹ-rawGroup M₁) (LeftModule.+ᴹ-rawGroup M₂)
-  open LeftSemimoduleMorphisms (LeftModule.leftSemimodule M₁) (LeftModule.leftSemimodule M₂)
+  open MorphismStructures.GroupMorphisms (RawLeftModule.+ᴹ-rawGroup M₁) (RawLeftModule.+ᴹ-rawGroup M₂)
+  open LeftSemimoduleMorphisms (RawLeftModule.rawLeftSemimodule M₁) (RawLeftModule.rawLeftSemimodule M₂)
 
   record IsLeftModuleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
@@ -148,16 +145,15 @@ module LeftModuleMorphisms
       }
 
 module RightSemimoduleMorphisms
-  {semiring : Semiring r ℓr}
-  (M₁ : RightSemimodule semiring m₁ ℓm₁)
-  (M₂ : RightSemimodule semiring m₂ ℓm₂)
+  {R : Set r}
+  (M₁ : RawRightSemimodule R m₁ ℓm₁)
+  (M₂ : RawRightSemimodule R m₂ ℓm₂)
   where
 
-  open Semiring semiring renaming (Carrier to R)
-  open RightSemimodule M₁ renaming (Carrierᴹ to A; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
-  open RightSemimodule M₂ renaming (Carrierᴹ to B; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
+  open RawRightSemimodule M₁ renaming (Carrierᴹ to A; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawRightSemimodule M₂ renaming (Carrierᴹ to B; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open MorphismStructures.MonoidMorphisms (RightSemimodule.+ᴹ-rawMonoid M₁) (RightSemimodule.+ᴹ-rawMonoid M₂)
+  open MorphismStructures.MonoidMorphisms (RawRightSemimodule.+ᴹ-rawMonoid M₁) (RawRightSemimodule.+ᴹ-rawMonoid M₂)
 
   record IsRightSemimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
@@ -203,17 +199,16 @@ module RightSemimoduleMorphisms
       renaming (isMagmaIsomorphism to +ᴹ-isMagmaIsomorphism)
 
 module RightModuleMorphisms
-  {ring : Ring r ℓr}
-  (M₁ : RightModule ring m₁ ℓm₁)
-  (M₂ : RightModule ring m₂ ℓm₂)
+  {R : Set r}
+  (M₁ : RawRightModule R m₁ ℓm₁)
+  (M₂ : RawRightModule R m₂ ℓm₂)
   where
 
-  open Ring ring renaming (Carrier to R)
-  open RightModule M₁ renaming (Carrierᴹ to A; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
-  open RightModule M₂ renaming (Carrierᴹ to B; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
+  open RawRightModule M₁ renaming (Carrierᴹ to A; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawRightModule M₂ renaming (Carrierᴹ to B; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_
-  open MorphismStructures.GroupMorphisms (RightModule.+ᴹ-rawGroup M₁) (RightModule.+ᴹ-rawGroup M₂)
-  open RightSemimoduleMorphisms (RightModule.rightSemimodule M₁) (RightModule.rightSemimodule M₂)
+  open MorphismStructures.GroupMorphisms (RawRightModule.+ᴹ-rawGroup M₁) (RawRightModule.+ᴹ-rawGroup M₂)
+  open RightSemimoduleMorphisms (RawRightModule.rawRightSemimodule M₁) (RawRightModule.rawRightSemimodule M₂)
 
   record IsRightModuleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
@@ -276,21 +271,19 @@ module RightModuleMorphisms
       }
 
 module BisemimoduleMorphisms
-  {R-semiring : Semiring r ℓr}
-  {S-semiring : Semiring s ℓs}
-  (M₁ : Bisemimodule R-semiring S-semiring m₁ ℓm₁)
-  (M₂ : Bisemimodule R-semiring S-semiring m₂ ℓm₂)
+  {R : Set r}
+  {S : Set s}
+  (M₁ : RawBisemimodule R S m₁ ℓm₁)
+  (M₂ : RawBisemimodule R S m₂ ℓm₂)
   where
 
-  open Semiring R-semiring renaming (Carrier to R)
-  open Semiring S-semiring renaming (Carrier to S)
-  open Bisemimodule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
-  open Bisemimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
+  open RawBisemimodule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawBisemimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_ using (Homomorphicₗ)
   open MorphismDefinitions S A B _≈ᴹ₂_ using (Homomorphicᵣ)
-  open MorphismStructures.MonoidMorphisms (Bisemimodule.+ᴹ-rawMonoid M₁) (Bisemimodule.+ᴹ-rawMonoid M₂)
-  open LeftSemimoduleMorphisms (Bisemimodule.leftSemimodule M₁) (Bisemimodule.leftSemimodule M₂)
-  open RightSemimoduleMorphisms (Bisemimodule.rightSemimodule M₁) (Bisemimodule.rightSemimodule M₂)
+  open MorphismStructures.MonoidMorphisms (RawBisemimodule.+ᴹ-rawMonoid M₁) (RawBisemimodule.+ᴹ-rawMonoid M₂)
+  open LeftSemimoduleMorphisms (RawBisemimodule.rawLeftSemimodule M₁) (RawBisemimodule.rawLeftSemimodule M₂)
+  open RightSemimoduleMorphisms (RawBisemimodule.rawRightSemimodule M₁) (RawBisemimodule.rawRightSemimodule M₂)
 
   record IsBisemimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ s ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
@@ -358,22 +351,20 @@ module BisemimoduleMorphisms
       }
 
 module BimoduleMorphisms
-  {R-ring : Ring r ℓr}
-  {S-ring : Ring s ℓs}
-  (M₁ : Bimodule R-ring S-ring m₁ ℓm₁)
-  (M₂ : Bimodule R-ring S-ring m₂ ℓm₂)
+  {R : Set r}
+  {S : Set s}
+  (M₁ : RawBimodule R S m₁ ℓm₁)
+  (M₂ : RawBimodule R S m₂ ℓm₂)
   where
 
-  open Ring R-ring renaming (Carrier to R)
-  open Ring S-ring renaming (Carrier to S)
-  open Bimodule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
-  open Bimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
+  open RawBimodule M₁ renaming (Carrierᴹ to A; _*ₗ_ to _*ₗ₁_; _*ᵣ_ to _*ᵣ₁_; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawBimodule M₂ renaming (Carrierᴹ to B; _*ₗ_ to _*ₗ₂_; _*ᵣ_ to _*ᵣ₂_; _≈ᴹ_ to _≈ᴹ₂_)
   open MorphismDefinitions R A B _≈ᴹ₂_ using (Homomorphicₗ)
   open MorphismDefinitions S A B _≈ᴹ₂_ using (Homomorphicᵣ)
-  open MorphismStructures.GroupMorphisms (Bimodule.+ᴹ-rawGroup M₁) (Bimodule.+ᴹ-rawGroup M₂)
-  open LeftModuleMorphisms (Bimodule.leftModule M₁) (Bimodule.leftModule M₂)
-  open RightModuleMorphisms (Bimodule.rightModule M₁) (Bimodule.rightModule M₂)
-  open BisemimoduleMorphisms (Bimodule.bisemimodule M₁) (Bimodule.bisemimodule M₂)
+  open MorphismStructures.GroupMorphisms (RawBimodule.+ᴹ-rawGroup M₁) (RawBimodule.+ᴹ-rawGroup M₂)
+  open LeftModuleMorphisms (RawBimodule.rawLeftModule M₁) (RawBimodule.rawLeftModule M₂)
+  open RightModuleMorphisms (RawBimodule.rawRightModule M₁) (RawBimodule.rawRightModule M₂)
+  open BisemimoduleMorphisms (RawBimodule.rawBisemimodule M₁) (RawBimodule.rawBisemimodule M₂)
 
   record IsBimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ s ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
@@ -492,14 +483,14 @@ module BimoduleMorphisms
       }
 
 module SemimoduleMorphisms
-  {commutativeSemiring : CommutativeSemiring r ℓr}
-  (M₁ : Semimodule commutativeSemiring m₁ ℓm₁)
-  (M₂ : Semimodule commutativeSemiring m₂ ℓm₂)
+  {R : Set r}
+  (M₁ : RawSemimodule R m₁ ℓm₁)
+  (M₂ : RawSemimodule R m₂ ℓm₂)
   where
 
-  open Semimodule M₁ renaming (Carrierᴹ to A; _≈ᴹ_ to _≈ᴹ₁_)
-  open Semimodule M₂ renaming (Carrierᴹ to B; _≈ᴹ_ to _≈ᴹ₂_)
-  open BisemimoduleMorphisms (Semimodule.bisemimodule M₁) (Semimodule.bisemimodule M₂)
+  open RawSemimodule M₁ renaming (Carrierᴹ to A; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawSemimodule M₂ renaming (Carrierᴹ to B; _≈ᴹ_ to _≈ᴹ₂_)
+  open BisemimoduleMorphisms (RawSemimodule.rawBisemimodule M₁) (RawSemimodule.rawBisemimodule M₂)
 
   record IsSemimoduleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field
@@ -544,15 +535,15 @@ module SemimoduleMorphisms
             )
 
 module ModuleMorphisms
-  {commutativeRing : CommutativeRing r ℓr}
-  (M₁ : Module commutativeRing m₁ ℓm₁)
-  (M₂ : Module commutativeRing m₂ ℓm₂)
+  {R : Set r}
+  (M₁ : RawModule R m₁ ℓm₁)
+  (M₂ : RawModule R m₂ ℓm₂)
   where
 
-  open Module M₁ renaming (Carrierᴹ to A; _≈ᴹ_ to _≈ᴹ₁_)
-  open Module M₂ renaming (Carrierᴹ to B; _≈ᴹ_ to _≈ᴹ₂_)
-  open BimoduleMorphisms (Module.bimodule M₁) (Module.bimodule M₂)
-  open SemimoduleMorphisms (Module.semimodule M₁) (Module.semimodule M₂)
+  open RawModule M₁ renaming (Carrierᴹ to A; _≈ᴹ_ to _≈ᴹ₁_)
+  open RawModule M₂ renaming (Carrierᴹ to B; _≈ᴹ_ to _≈ᴹ₂_)
+  open BimoduleMorphisms (RawModule.rawBimodule M₁) (RawModule.rawBimodule M₂)
+  open SemimoduleMorphisms (RawModule.rawBisemimodule M₁) (RawModule.rawBisemimodule M₂)
 
   record IsModuleHomomorphism (⟦_⟧ : A → B) : Set (r ⊔ m₁ ⊔ ℓm₁ ⊔ ℓm₂) where
     field

From e76e2135eaf3833691108eb453306178e5c44766 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 5 Feb 2024 01:37:54 +0000
Subject: [PATCH 214/316] add lemma (#2271)

---
 CHANGELOG.md                        | 5 +++++
 src/Relation/Nullary/Decidable.agda | 6 +++++-
 2 files changed, 10 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e8f0a568f2..8dd7be0361 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -140,3 +140,8 @@ Additions to existing modules
 
 * In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
   be used infix.
+
+* Added new proof in `Relation.Nullary.Decidable`:
+  ```agda
+  ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
+  ```
diff --git a/src/Relation/Nullary/Decidable.agda b/src/Relation/Nullary/Decidable.agda
index cb5f72b6e0..4335a5df5a 100644
--- a/src/Relation/Nullary/Decidable.agda
+++ b/src/Relation/Nullary/Decidable.agda
@@ -19,7 +19,8 @@ open import Relation.Binary.Bundles using (Setoid; module Setoid)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary
 open import Relation.Nullary.Reflects using (invert)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong′)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; trans; cong′)
 
 private
   variable
@@ -80,3 +81,6 @@ dec-no (no _) _ | refl = refl
 dec-yes-irr : (a? : Dec A) → Irrelevant A → (a : A) → a? ≡ yes a
 dec-yes-irr a? irr a with dec-yes a? a
 ... | a′ , eq rewrite irr a a′ = eq
+
+⌊⌋-map′ : ∀ t f (a? : Dec A) → ⌊ map′ {B = B} t f a? ⌋ ≡ ⌊ a? ⌋
+⌊⌋-map′ t f a? = trans (isYes≗does (map′ t f a?)) (sym (isYes≗does a?))

From 98a6a8cb602942860a960ff0cc5e6d51b319f5ff Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 5 Feb 2024 01:40:48 +0000
Subject: [PATCH 215/316] additions to `style-guide` (#2270)

* added stub content on programming idioms

* added para on qualified import names
---
 doc/style-guide.md | 39 +++++++++++++++++++++++++++++++++++++--
 1 file changed, 37 insertions(+), 2 deletions(-)

diff --git a/doc/style-guide.md b/doc/style-guide.md
index 5732747cda..0e3a54841a 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -121,8 +121,9 @@ automate most of this.
 
 * If it is important that certain names only come into scope later in
   the file then the module should still be imported at the top of the
-  file but it can be given a shorter name using the keyword `as` and then
-  opened later on in the file when needed, e.g.
+  file but it can be imported *qualified*, i.e. given a shorter name
+  using the keyword `as` and then opened later on in the file when needed,
+  e.g.
   ```agda
   import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
   ...
@@ -130,6 +131,21 @@ automate most of this.
   open SetoidEquality S
   ```
 
+* Naming conventions for qualified `import`s: if importing a module under
+  a root of the form `Data.X` (e.g. the `Base` module for basic operations,
+  or `Properties` for lemmas about them etc.) then conventionally, the
+  qualified name(s) for the import(s) should (all) share as qualfied name
+  that of the name of the `X` datatype defined: i.e. `Data.Nat.Base`
+  should be imported as `ℕ`, `Data.List.Properties` as `List`,  etc.
+  In this spirit, the convention applies also to (the datatype defined by)
+  `Relation.Binary.PropositionalEquality.*` which should be imported qualified
+  with the name `≡`.
+  Other modules should be given a 'suitable' qualified name based on its 'long'
+  path-derived name (such as `SetoidEquality` in the example above); commonly
+  occcurring examples such as `Algebra.Structures` should be imported qualified
+  as `Structures` etc.
+  NB. Historical legacy means that these conventions have not always been observed!
+
 * When using only a few items (i.e. < 5) from a module, it is a good practice to
   enumerate the items that will be used by declaring the import statement
   with the directive `using`. This makes the dependencies clearer, e.g.
@@ -532,3 +548,22 @@ word within a compound word is capitalized except for the first word.
 
 * The names of patterns for reflected syntax are also *appended* with an
   additional backtick.
+
+#### Specific pragmatics/idiomatic patterns
+
+## Use of `with` notation
+
+Thinking on this has changed since the early days of the library, with
+a desire to avoid 'unnecessary' uses of `with`: see Issues
+[#1937](https://github.com/agda/agda-stdlib/issues/1937) and
+[#2123](https://github.com/agda/agda-stdlib/issues/2123).
+
+## Proving instances of `Decidable` for sets, predicates, relations, ...
+
+Issue [#803](https://github.com/agda/agda-stdlib/issues/803)
+articulates a programming pattern for writing proofs of decidability,
+used successfully in PR
+[#799](https://github.com/agda/agda-stdlib/pull/799) and made
+systematic for `Nary` relations in PR
+[#811](https://github.com/agda/agda-stdlib/pull/811)
+

From 654a7a9aa11aba060fb2d8b71fb4382c3d12588e Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 5 Feb 2024 01:45:59 +0000
Subject: [PATCH 216/316] fixes issue #1688 (#2254)

---
 CHANGELOG.md             | 6 ++++++
 src/Algebra/Bundles.agda | 6 ++++++
 2 files changed, 12 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 8dd7be0361..6e4a51e5c3 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -46,6 +46,12 @@ New modules
 Additions to existing modules
 -----------------------------
 
+* Exporting more `Raw` substructures from `Algebra.Bundles.Ring`:
+  ```agda
+  rawNearSemiring   : RawNearSemiring _ _
+  rawRingWithoutOne : RawRingWithoutOne _ _
+  +-rawGroup        : RawGroup _ _
+
 * In `Algebra.Module.Bundles`, raw bundles are re-exported and the bundles expose their raw counterparts.
 
 * In `Algebra.Module.Construct.DirectProduct`:
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index 6b6384458a..1a31d16496 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -961,6 +961,9 @@ record Ring c ℓ : Set (suc (c ⊔ ℓ)) where
     ; semiringWithoutAnnihilatingZero
     )
 
+  open NearSemiring nearSemiring public
+    using (rawNearSemiring)
+
   open AbelianGroup +-abelianGroup public
     using () renaming (group to +-group; invertibleMagma to +-invertibleMagma; invertibleUnitalMagma to +-invertibleUnitalMagma)
 
@@ -974,6 +977,9 @@ record Ring c ℓ : Set (suc (c ⊔ ℓ)) where
     ; 1#  = 1#
     }
 
+  open RawRing rawRing public
+    using (rawRingWithoutOne; +-rawGroup)
+
 
 record CommutativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
   infix  8 -_

From 7c7c030812212a42c773a4b43ed147f9c9e4737a Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 5 Feb 2024 03:03:37 +0000
Subject: [PATCH 217/316] Systematise relational definitions at all arities
 (#2259)

* fixes #2091

* revised along the lines of @MatthewDaggitt's suggestions

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                         | 17 +++++++++++
 src/Relation/Binary/Definitions.agda | 43 +++++++++++++++-------------
 src/Relation/Nullary.agda            | 26 +++++++++++++++--
 src/Relation/Unary.agda              | 37 +++++++++++++++---------
 4 files changed, 87 insertions(+), 36 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6e4a51e5c3..6de6fbc159 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -147,6 +147,23 @@ Additions to existing modules
 * In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
   be used infix.
 
+* Added new definitions in `Relation.Binary`
+  ```
+  Stable          : Pred A ℓ → Set _
+  ```
+
+* Added new definitions in `Relation.Nullary`
+  ```
+  Recomputable    : Set _
+  WeaklyDecidable : Set _
+  ```
+
+* Added new definitions in `Relation.Unary`
+  ```
+  Stable          : Pred A ℓ → Set _
+  WeaklyDecidable : Pred A ℓ → Set _
+  ```
+
 * Added new proof in `Relation.Nullary.Decidable`:
   ```agda
   ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
diff --git a/src/Relation/Binary/Definitions.agda b/src/Relation/Binary/Definitions.agda
index 887cb7c4a6..dd0e016c12 100644
--- a/src/Relation/Binary/Definitions.agda
+++ b/src/Relation/Binary/Definitions.agda
@@ -12,14 +12,12 @@ module Relation.Binary.Definitions where
 
 open import Agda.Builtin.Equality using (_≡_)
 
-open import Data.Maybe.Base using (Maybe)
 open import Data.Product.Base using (_×_; ∃-syntax)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_on_; flip)
 open import Level
 open import Relation.Binary.Core
-open import Relation.Nullary.Decidable.Core using (Dec)
-open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Nullary as Nullary using (¬_; Dec)
 
 private
   variable
@@ -206,35 +204,40 @@ P Respects₂ _∼_ = (P Respectsʳ _∼_) × (P Respectsˡ _∼_)
 Substitutive : Rel A ℓ₁ → (ℓ₂ : Level) → Set _
 Substitutive {A = A} _∼_ p = (P : A → Set p) → P Respects _∼_
 
--- Decidability - it is possible to determine whether a given pair of
--- elements are related.
+-- Irrelevancy - all proofs that a given pair of elements are related
+-- are indistinguishable.
 
-Decidable : REL A B ℓ → Set _
-Decidable _∼_ = ∀ x y → Dec (x ∼ y)
+Irrelevant : REL A B ℓ → Set _
+Irrelevant _∼_ = ∀ {x y} → Nullary.Irrelevant (x ∼ y)
+
+-- Recomputability - we can rebuild a relevant proof given an
+-- irrelevant one.
+
+Recomputable : REL A B ℓ → Set _
+Recomputable _∼_ = ∀ {x y} → Nullary.Recomputable (x ∼ y)
+
+-- Stability
+
+Stable : REL A B ℓ → Set _
+Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)
 
 -- Weak decidability - it is sometimes possible to determine if a given
 -- pair of elements are related.
 
 WeaklyDecidable : REL A B ℓ → Set _
-WeaklyDecidable _∼_ = ∀ x y → Maybe (x ∼ y)
+WeaklyDecidable _∼_ = ∀ x y → Nullary.WeaklyDecidable (x ∼ y)
+
+-- Decidability - it is possible to determine whether a given pair of
+-- elements are related.
+
+Decidable : REL A B ℓ → Set _
+Decidable _∼_ = ∀ x y → Dec (x ∼ y)
 
 -- Propositional equality is decidable for the type.
 
 DecidableEquality : (A : Set a) → Set _
 DecidableEquality A = Decidable {A = A} _≡_
 
--- Irrelevancy - all proofs that a given pair of elements are related
--- are indistinguishable.
-
-Irrelevant : REL A B ℓ → Set _
-Irrelevant _∼_ = ∀ {x y} (a b : x ∼ y) → a ≡ b
-
--- Recomputability - we can rebuild a relevant proof given an
--- irrelevant one.
-
-Recomputable : REL A B ℓ → Set _
-Recomputable _∼_ = ∀ {x y} → .(x ∼ y) → x ∼ y
-
 -- Universal - all pairs of elements are related
 
 Universal : REL A B ℓ → Set _
diff --git a/src/Relation/Nullary.agda b/src/Relation/Nullary.agda
index 0b0f83f77b..92e0607f4d 100644
--- a/src/Relation/Nullary.agda
+++ b/src/Relation/Nullary.agda
@@ -10,7 +10,15 @@
 
 module Relation.Nullary where
 
-open import Agda.Builtin.Equality
+open import Agda.Builtin.Equality using (_≡_)
+open import Agda.Builtin.Maybe using (Maybe)
+open import Level using (Level)
+
+private
+  variable
+    p : Level
+    P : Set p
+
 
 ------------------------------------------------------------------------
 -- Re-exports
@@ -22,5 +30,19 @@ open import Relation.Nullary.Decidable.Core public
 ------------------------------------------------------------------------
 -- Irrelevant types
 
-Irrelevant : ∀ {p} → Set p → Set p
+Irrelevant : Set p → Set p
 Irrelevant P = ∀ (p₁ p₂ : P) → p₁ ≡ p₂
+
+------------------------------------------------------------------------
+-- Recomputability - we can rebuild a relevant proof given an
+-- irrelevant one.
+
+Recomputable : Set p → Set p
+Recomputable P = .P → P
+
+------------------------------------------------------------------------
+-- Weak decidability
+-- `nothing` is 'don't know'/'give up'; `just` is `yes`/`definitely`
+
+WeaklyDecidable : Set p → Set p
+WeaklyDecidable = Maybe
diff --git a/src/Relation/Unary.agda b/src/Relation/Unary.agda
index f6d7cb13b0..357237a2be 100644
--- a/src/Relation/Unary.agda
+++ b/src/Relation/Unary.agda
@@ -14,8 +14,7 @@ open import Data.Product.Base using (_×_; _,_; Σ-syntax; ∃; uncurry; swap)
 open import Data.Sum.Base using (_⊎_; [_,_])
 open import Function.Base using (_∘_; _|>_)
 open import Level using (Level; _⊔_; 0ℓ; suc; Lift)
-open import Relation.Nullary.Decidable.Core using (Dec; True)
-open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Nullary as Nullary using (¬_; Dec; True)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 private
@@ -162,6 +161,28 @@ IUniversal P = ∀ {x} → x ∈ P
 
 syntax IUniversal P = ∀[ P ]
 
+-- Irrelevance - any two proofs that an element satifies P are
+-- indistinguishable.
+
+Irrelevant : Pred A ℓ → Set _
+Irrelevant P = ∀ {x} → Nullary.Irrelevant (P x)
+
+-- Recomputability - we can rebuild a relevant proof given an
+-- irrelevant one.
+
+Recomputable : Pred A ℓ → Set _
+Recomputable P = ∀ {x} → Nullary.Recomputable (P x)
+
+-- Weak Decidability
+
+Stable : Pred A ℓ → Set _
+Stable P = ∀ x → Nullary.Stable (P x)
+
+-- Weak Decidability
+
+WeaklyDecidable : Pred A ℓ → Set _
+WeaklyDecidable P = ∀ x → Nullary.WeaklyDecidable (P x)
+
 -- Decidability - it is possible to determine if an arbitrary element
 -- satisfies P.
 
@@ -174,18 +195,6 @@ Decidable P = ∀ x → Dec (P x)
 ⌊_⌋ : {P : Pred A ℓ} → Decidable P → Pred A ℓ
 ⌊ P? ⌋ a = Lift _ (True (P? a))
 
--- Irrelevance - any two proofs that an element satifies P are
--- indistinguishable.
-
-Irrelevant : Pred A ℓ → Set _
-Irrelevant P = ∀ {x} (a : P x) (b : P x) → a ≡ b
-
--- Recomputability - we can rebuild a relevant proof given an
--- irrelevant one.
-
-Recomputable : Pred A ℓ → Set _
-Recomputable P = ∀ {x} → .(P x) → P x
-
 ------------------------------------------------------------------------
 -- Operations on sets
 

From 709da18e7bd8b1c6fbda7b0906e00c1b983b43f4 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 5 Feb 2024 06:38:12 +0000
Subject: [PATCH 218/316] =?UTF-8?q?lemmas=20about=20semiring=20structure?=
 =?UTF-8?q?=20induced=20by=20`=5F=C3=97=20x`=20(#2272)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* tidying up proofs of, and using, semiring structure induced by `_× x`

* reinstated lemma about `0#`

* fixed name clash

* added companion lemma for `1#`

* new lemma, plus refactoring

* removed anonymous `module`

* don't inline use  of the lemma

* revised deprecation warning
---
 CHANGELOG.md                                  | 18 +++++++
 src/Algebra/Properties/Monoid/Mult.agda       |  6 +++
 src/Algebra/Properties/Semiring/Binomial.agda |  2 +-
 src/Algebra/Properties/Semiring/Mult.agda     | 50 ++++++++++++++-----
 4 files changed, 62 insertions(+), 14 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6de6fbc159..62fed1f0f4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -33,6 +33,11 @@ Deprecated modules
 Deprecated names
 ----------------
 
+* In `Algebra.Properties.Semiring.Mult`:
+  ```agda
+  1×-identityʳ  ↦  ×-homo-1
+  ```
+
 * In `Data.Nat.Divisibility.Core`:
   ```agda
   *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
@@ -90,6 +95,19 @@ Additions to existing modules
   rawModule          : RawModule R c ℓ
   ```
 
+* In `Algebra.Properties.Monoid.Mult`:
+  ```agda
+  ×-homo-0 : ∀ x → 0 × x ≈ 0#
+  ×-homo-1 : ∀ x → 1 × x ≈ x
+  ```
+
+* In `Algebra.Properties.Semiring.Mult`:
+  ```agda
+  ×-homo-0#     : ∀ x → 0 × x ≈ 0# * x
+  ×-homo-1#     : ∀ x → 1 × x ≈ 1# * x
+  idem-×-homo-* : (_*_ IdempotentOn x) → (m × x) * (n × x) ≈ (m ℕ.* n) × x
+  ```
+
 * In `Data.Fin.Properties`:
   ```agda
   nonZeroIndex : Fin n → ℕ.NonZero n
diff --git a/src/Algebra/Properties/Monoid/Mult.agda b/src/Algebra/Properties/Monoid/Mult.agda
index 88c4258fcb..2963523b77 100644
--- a/src/Algebra/Properties/Monoid/Mult.agda
+++ b/src/Algebra/Properties/Monoid/Mult.agda
@@ -51,6 +51,12 @@ open import Algebra.Definitions.RawMonoid rawMonoid public
 
 -- _×_ is homomorphic with respect to _ℕ+_/_+_.
 
+×-homo-0 : ∀ x → 0 × x ≈ 0#
+×-homo-0 x = refl
+
+×-homo-1 : ∀ x → 1 × x ≈ x
+×-homo-1 = +-identityʳ
+
 ×-homo-+ : ∀ x m n → (m ℕ.+ n) × x ≈ m × x + n × x
 ×-homo-+ x 0       n = sym (+-identityˡ (n × x))
 ×-homo-+ x (suc m) n = begin
diff --git a/src/Algebra/Properties/Semiring/Binomial.agda b/src/Algebra/Properties/Semiring/Binomial.agda
index 6731018af6..28ea35a8e8 100644
--- a/src/Algebra/Properties/Semiring/Binomial.agda
+++ b/src/Algebra/Properties/Semiring/Binomial.agda
@@ -216,7 +216,7 @@ term₁+term₂≈term x*y≈y*x n (suc i) with view i
 theorem : x * y ≈ y * x → ∀ n → (x + y) ^ n ≈ binomialExpansion n
 theorem x*y≈y*x zero    = begin
   (x + y) ^ 0                     ≡⟨⟩
-  1#                              ≈⟨ 1×-identityʳ 1# ⟨
+  1#                              ≈⟨ ×-homo-1 1# ⟨
   1 × 1#                          ≈⟨ *-identityʳ (1 × 1#) ⟨
   (1 × 1#) * 1#                   ≈⟨ ×-assoc-* 1 1# 1# ⟩
   1 × (1# * 1#)                   ≡⟨⟩
diff --git a/src/Algebra/Properties/Semiring/Mult.agda b/src/Algebra/Properties/Semiring/Mult.agda
index 801e4f1966..6301d274c6 100644
--- a/src/Algebra/Properties/Semiring/Mult.agda
+++ b/src/Algebra/Properties/Semiring/Mult.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra
+open import Algebra.Bundles using (Semiring)
 open import Data.Nat.Base as ℕ using (zero; suc)
 
 module Algebra.Properties.Semiring.Mult
@@ -14,6 +14,7 @@ module Algebra.Properties.Semiring.Mult
 
 open Semiring S renaming (zero to *-zero)
 open import Relation.Binary.Reasoning.Setoid setoid
+open import Algebra.Definitions _≈_ using (_IdempotentOn_)
 
 ------------------------------------------------------------------------
 -- Re-export definition from the monoid
@@ -23,21 +24,15 @@ open import Algebra.Properties.Monoid.Mult +-monoid public
 ------------------------------------------------------------------------
 -- Properties of _×_
 
--- (_× 1#) is homomorphic with respect to _ℕ.*_/_*_.
+-- (0 ×_) is (0# *_)
 
-×1-homo-* : ∀ m n → (m ℕ.* n) × 1# ≈ (m × 1#) * (n × 1#)
-×1-homo-* 0       n = sym (zeroˡ (n × 1#))
-×1-homo-* (suc m) n = begin
-  (n ℕ.+ m ℕ.* n) × 1#                ≈⟨  ×-homo-+ 1# n (m ℕ.* n) ⟩
-  n × 1# + (m ℕ.* n) × 1#             ≈⟨  +-congˡ (×1-homo-* m n) ⟩
-  n × 1# + (m × 1#) * (n × 1#)        ≈⟨ +-congʳ (*-identityˡ _) ⟨
-  1# * (n × 1#) + (m × 1#) * (n × 1#) ≈⟨ distribʳ (n × 1#) 1# (m × 1#) ⟨
-  (1# + m × 1#) * (n × 1#)            ∎
+×-homo-0# : ∀ x → 0 × x ≈ 0# * x
+×-homo-0# x = sym (zeroˡ x)
 
--- (1 ×_) is the identity
+-- (1 ×_) is (1# *_)
 
-1×-identityʳ : ∀ x → 1 × x ≈ x
-1×-identityʳ = +-identityʳ
+×-homo-1# : ∀ x → 1 × x ≈ 1# * x
+×-homo-1# x = trans (×-homo-1 x) (sym (*-identityˡ x))
 
 -- (n ×_) commutes with _*_
 
@@ -60,3 +55,32 @@ open import Algebra.Properties.Monoid.Mult +-monoid public
   x * y + (n × x) * y   ≈⟨ +-congˡ (×-assoc-* n _ _) ⟩
   x * y + n × (x * y)   ≡⟨⟩
   suc n × (x * y)       ∎
+
+-- (_× x) is homomorphic with respect to _ℕ.*_/_*_ for idempotent x.
+
+idem-×-homo-* : ∀ m n {x} → (_*_ IdempotentOn x) → (m × x) * (n × x) ≈ (m ℕ.* n) × x
+idem-×-homo-* m n {x} idem = begin
+  (m × x) * (n × x)   ≈⟨ ×-assoc-* m x (n × x) ⟩
+  m × (x * (n × x))   ≈⟨ ×-congʳ m (×-comm-* n x x) ⟩
+  m × (n × (x * x))   ≈⟨ ×-assocˡ _ m n ⟩
+  (m ℕ.* n) × (x * x) ≈⟨ ×-congʳ (m ℕ.* n) idem ⟩
+  (m ℕ.* n) × x       ∎
+
+-- (_× 1#) is homomorphic with respect to _ℕ.*_/_*_.
+
+×1-homo-* : ∀ m n → (m ℕ.* n) × 1# ≈ (m × 1#) * (n × 1#)
+×1-homo-* m n = sym (idem-×-homo-* m n (*-identityʳ 1#))
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.1
+
+1×-identityʳ = ×-homo-1
+{-# WARNING_ON_USAGE 1×-identityʳ
+"Warning: 1×-identityʳ was deprecated in v2.1.
+Please use ×-homo-1 instead. "
+#-}

From 07988c07febd0f7605e470bc66a08fa760a7f2b9 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 6 Feb 2024 21:45:51 +0000
Subject: [PATCH 219/316] Qualified import of `Data.Nat` fixing #2280 (#2281)

* `Algebra.Properties.Semiring.Binomial`

* `Codata.Sized.Cowriter`

* `Data.Char.Properties`

* `Data.Difference*`

* `Data.Fin.*`

* `Data.Float.Properties`

* `Data.Graph.Acyclic`

* `Data.Integer.Divisibility`: still need to disambiguate `Divisibility`?

* `Data.List.Extrema.Nat`

* `Data.List.Relation.Binary.*`

* `Data.Nat.Binary.*`

* `Data.Rational.Base`

* `Data.String`

* `Data.Vec.*`

* `Data.Word`

* `Induction.Lexicographic`

* `Reflection.AST`

* `Tactic.*`

* `Text.*`
---
 src/Algebra/Properties/Semiring/Binomial.agda |  10 +-
 src/Codata/Sized/Cowriter.agda                |   2 +-
 src/Data/Char/Properties.agda                 |  26 +--
 src/Data/DifferenceNat.agda                   |   6 +-
 src/Data/DifferenceVec.agda                   |  22 +--
 src/Data/Fin.agda                             |   5 +-
 src/Data/Fin/Permutation/Components.agda      |   1 -
 src/Data/Fin/Properties.agda                  |  90 +++++------
 src/Data/Fin/Subset/Properties.agda           |  18 +--
 src/Data/Float/Properties.agda                |  14 +-
 src/Data/Graph/Acyclic.agda                   |  11 +-
 src/Data/Integer/Divisibility.agda            |   4 +-
 src/Data/List/Extrema/Nat.agda                |   4 +-
 .../Infix/Heterogeneous/Properties.agda       |  16 +-
 .../Sublist/Heterogeneous/Properties.agda     |  24 +--
 .../Binary/Sublist/Heterogeneous/Solver.agda  |  12 +-
 .../Binary/Sublist/Setoid/Properties.agda     |   4 +-
 src/Data/Nat/Binary/Properties.agda           | 150 +++++++++---------
 src/Data/Nat/Binary/Subtraction.agda          |  42 ++---
 src/Data/Rational/Base.agda                   |   8 +-
 src/Data/String.agda                          |   4 +-
 src/Data/Vec.agda                             |   6 +-
 src/Data/Vec/Functional/Properties.agda       |  24 +--
 src/Data/Word/Properties.agda                 |   8 +-
 src/Induction/Lexicographic.agda              |   4 +-
 src/Reflection/AST/Definition.agda            |   8 +-
 src/Reflection/AST/Meta.agda                  |   4 +-
 src/Tactic/Cong.agda                          |  12 +-
 .../Polynomial/Homomorphism/Addition.agda     |  12 +-
 .../Homomorphism/Exponentiation.agda          |   6 +-
 .../Core/Polynomial/Homomorphism/Lemmas.agda  |   4 +-
 src/Text/Pretty.agda                          |   4 +-
 src/Text/Printf.agda                          |  13 +-
 33 files changed, 286 insertions(+), 292 deletions(-)

diff --git a/src/Algebra/Properties/Semiring/Binomial.agda b/src/Algebra/Properties/Semiring/Binomial.agda
index 28ea35a8e8..bbd9adce21 100644
--- a/src/Algebra/Properties/Semiring/Binomial.agda
+++ b/src/Algebra/Properties/Semiring/Binomial.agda
@@ -10,10 +10,10 @@
 
 open import Algebra.Bundles using (Semiring)
 open import Data.Bool.Base using (true)
-open import Data.Nat.Base as Nat hiding (_+_; _*_; _^_)
+open import Data.Nat.Base as ℕ hiding (_+_; _*_; _^_)
 open import Data.Nat.Combinatorics
   using (_C_; nCn≡1; nC1≡n; nCk+nC[k+1]≡[n+1]C[k+1])
-open import Data.Nat.Properties as Nat
+open import Data.Nat.Properties as ℕ
   using (<⇒<ᵇ; n<1+n; n∸n≡0; +-∸-assoc)
 open import Data.Fin.Base as Fin
   using (Fin; zero; suc; toℕ; fromℕ; inject₁)
@@ -154,8 +154,8 @@ y*lemma x*y≈y*x {n} j = begin
 -- Now, a lemma characterising the sum of the term₁ and term₂ expressions
 
 private
-  n<ᵇ1+n : ∀ n → (n Nat.<ᵇ suc n) ≡ true
-  n<ᵇ1+n n with true ← n Nat.<ᵇ suc n | _ ← <⇒<ᵇ (n<1+n n) = ≡.refl
+  n<ᵇ1+n : ∀ n → (n ℕ.<ᵇ suc n) ≡ true
+  n<ᵇ1+n n with true ← n ℕ.<ᵇ suc n | _ ← <⇒<ᵇ (n<1+n n) = ≡.refl
 
 term₁+term₂≈term : x * y ≈ y * x → ∀ n i → term₁ n i + term₂ n i ≈ binomialTerm (suc n) i
 
@@ -193,7 +193,7 @@ term₁+term₂≈term x*y≈y*x n (suc i) with view i
     ≈⟨ +-congˡ (×-congˡ nC[k+1]≡nC[j+1]) ⟨
   (nCk × [x^k+1]*[y^n-k]) + (nC[k+1] × [x^k+1]*[y^n-k])
     ≈⟨ ×-homo-+ [x^k+1]*[y^n-k] nCk nC[k+1] ⟨
-  (nCk Nat.+ nC[k+1]) × [x^k+1]*[y^n-k]
+  (nCk ℕ.+ nC[k+1]) × [x^k+1]*[y^n-k]
     ≡⟨ cong (_× [x^k+1]*[y^n-k]) (nCk+nC[k+1]≡[n+1]C[k+1] n k) ⟩
   ((suc n) C (suc k)) × [x^k+1]*[y^n-k]
     ≡⟨⟩
diff --git a/src/Codata/Sized/Cowriter.agda b/src/Codata/Sized/Cowriter.agda
index 7206c83c86..1cd3d2ba9c 100644
--- a/src/Codata/Sized/Cowriter.agda
+++ b/src/Codata/Sized/Cowriter.agda
@@ -17,7 +17,7 @@ open import Codata.Sized.Stream as Stream using (Stream; _∷_)
 open import Data.Unit.Base
 open import Data.List.Base using (List; []; _∷_)
 open import Data.List.NonEmpty.Base using (List⁺; _∷_)
-open import Data.Nat.Base as Nat using (ℕ; zero; suc)
+open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Data.Product.Base as Prod using (_×_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.Vec.Base using (Vec; []; _∷_)
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index c399bf30e3..cd4ccdce29 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -11,7 +11,7 @@ module Data.Char.Properties where
 open import Data.Bool.Base using (Bool)
 open import Data.Char.Base
 import Data.Nat.Base as ℕ
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (_,_)
 
 open import Function.Base
@@ -56,7 +56,7 @@ open import Agda.Builtin.Char.Properties
 
 infix 4 _≟_
 _≟_ : Decidable {A = Char} _≡_
-x ≟ y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕₚ.≟ toℕ y)
+x ≟ y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕ.≟ toℕ y)
 
 setoid : Setoid _ _
 setoid = PropEq.setoid Char
@@ -95,22 +95,22 @@ private
 
 infix 4 _<?_
 _<?_ : Decidable _<_
-_<?_ = On.decidable toℕ ℕ._<_ ℕₚ._<?_
+_<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
 
 <-cmp : Trichotomous _≡_ _<_
-<-cmp c d with ℕₚ.<-cmp (toℕ c) (toℕ d)
+<-cmp c d with ℕ.<-cmp (toℕ c) (toℕ d)
 ... | tri< lt ¬eq ¬gt = tri< lt (≉⇒≢ ¬eq) ¬gt
 ... | tri≈ ¬lt eq ¬gt = tri≈ ¬lt (≈⇒≡ eq) ¬gt
 ... | tri> ¬lt ¬eq gt = tri> ¬lt (≉⇒≢ ¬eq) gt
 
 <-irrefl : Irreflexive _≡_ _<_
-<-irrefl = ℕₚ.<-irrefl ∘′ cong toℕ
+<-irrefl = ℕ.<-irrefl ∘′ cong toℕ
 
 <-trans : Transitive _<_
-<-trans {c} {d} {e} = On.transitive toℕ ℕ._<_ ℕₚ.<-trans {c} {d} {e}
+<-trans {c} {d} {e} = On.transitive toℕ ℕ._<_ ℕ.<-trans {c} {d} {e}
 
 <-asym : Asymmetric _<_
-<-asym {c} {d} = On.asymmetric toℕ ℕ._<_ ℕₚ.<-asym {c} {d}
+<-asym {c} {d} = On.asymmetric toℕ ℕ._<_ ℕ.<-asym {c} {d}
 
 <-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_
 <-isStrictPartialOrder = record
@@ -151,7 +151,7 @@ _≤?_ = Reflₚ.decidable <-cmp
 ≤-trans = Reflₚ.trans (λ {a} {b} {c} → <-trans {a} {b} {c})
 
 ≤-antisym : Antisymmetric _≡_ _≤_
-≤-antisym = Reflₚ.antisym _≡_ refl ℕₚ.<-asym
+≤-antisym = Reflₚ.antisym _≡_ refl ℕ.<-asym
 
 ≤-isPreorder : IsPreorder _≡_ _≤_
 ≤-isPreorder = record
@@ -220,7 +220,7 @@ Please use Propositional Equality's subst instead."
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-x ≈? y = toℕ x ℕₚ.≟ toℕ y
+x ≈? y = toℕ x ℕ.≟ toℕ y
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -277,28 +277,28 @@ Please use decSetoid instead."
 #-}
 
 <-isStrictPartialOrder-≈ : IsStrictPartialOrder _≈_ _<_
-<-isStrictPartialOrder-≈ = On.isStrictPartialOrder toℕ ℕₚ.<-isStrictPartialOrder
+<-isStrictPartialOrder-≈ = On.isStrictPartialOrder toℕ ℕ.<-isStrictPartialOrder
 {-# WARNING_ON_USAGE <-isStrictPartialOrder-≈
 "Warning: <-isStrictPartialOrder-≈ was deprecated in v1.5.
 Please use <-isStrictPartialOrder instead."
 #-}
 
 <-isStrictTotalOrder-≈ : IsStrictTotalOrder _≈_ _<_
-<-isStrictTotalOrder-≈ = On.isStrictTotalOrder toℕ ℕₚ.<-isStrictTotalOrder
+<-isStrictTotalOrder-≈ = On.isStrictTotalOrder toℕ ℕ.<-isStrictTotalOrder
 {-# WARNING_ON_USAGE <-isStrictTotalOrder-≈
 "Warning: <-isStrictTotalOrder-≈ was deprecated in v1.5.
 Please use <-isStrictTotalOrder instead."
 #-}
 
 <-strictPartialOrder-≈ : StrictPartialOrder _ _ _
-<-strictPartialOrder-≈ = On.strictPartialOrder ℕₚ.<-strictPartialOrder toℕ
+<-strictPartialOrder-≈ = On.strictPartialOrder ℕ.<-strictPartialOrder toℕ
 {-# WARNING_ON_USAGE <-strictPartialOrder-≈
 "Warning: <-strictPartialOrder-≈ was deprecated in v1.5.
 Please use <-strictPartialOrder instead."
 #-}
 
 <-strictTotalOrder-≈ : StrictTotalOrder _ _ _
-<-strictTotalOrder-≈ = On.strictTotalOrder ℕₚ.<-strictTotalOrder toℕ
+<-strictTotalOrder-≈ = On.strictTotalOrder ℕ.<-strictTotalOrder toℕ
 {-# WARNING_ON_USAGE <-strictTotalOrder-≈
 "Warning: <-strictTotalOrder-≈ was deprecated in v1.5.
 Please use <-strictTotalOrder instead."
diff --git a/src/Data/DifferenceNat.agda b/src/Data/DifferenceNat.agda
index c0f33a1fea..5e23cc5030 100644
--- a/src/Data/DifferenceNat.agda
+++ b/src/Data/DifferenceNat.agda
@@ -9,7 +9,7 @@
 
 module Data.DifferenceNat where
 
-open import Data.Nat.Base as N using (ℕ)
+open import Data.Nat.Base as ℕ using (ℕ)
 open import Function.Base using (_⟨_⟩_)
 
 infixl 6 _+_
@@ -21,7 +21,7 @@ Diffℕ = ℕ → ℕ
 0# = λ k → k
 
 suc : Diffℕ → Diffℕ
-suc n = λ k → N.suc (n k)
+suc n = λ k → ℕ.suc (n k)
 
 1# : Diffℕ
 1# = suc 0#
@@ -35,4 +35,4 @@ toℕ n = n 0
 -- fromℕ n is linear in the size of n.
 
 fromℕ : ℕ → Diffℕ
-fromℕ n = λ k → n ⟨ N._+_ ⟩ k
+fromℕ n = λ k → n ⟨ ℕ._+_ ⟩ k
diff --git a/src/Data/DifferenceVec.agda b/src/Data/DifferenceVec.agda
index c34382a051..22af2a6247 100644
--- a/src/Data/DifferenceVec.agda
+++ b/src/Data/DifferenceVec.agda
@@ -9,9 +9,9 @@
 module Data.DifferenceVec where
 
 open import Data.DifferenceNat
-open import Data.Vec.Base as V using (Vec)
+open import Data.Vec.Base as Vec using (Vec)
 open import Function.Base using (_⟨_⟩_)
-import Data.Nat.Base as N
+import Data.Nat.Base as ℕ
 
 infixr 5 _∷_ _++_
 
@@ -22,7 +22,7 @@ DiffVec A m = ∀ {n} → Vec A n → Vec A (m n)
 [] = λ k → k
 
 _∷_ : ∀ {a} {A : Set a} {n} → A → DiffVec A n → DiffVec A (suc n)
-x ∷ xs = λ k → V._∷_ x (xs k)
+x ∷ xs = λ k → Vec._∷_ x (xs k)
 
 [_] : ∀ {a} {A : Set a} → A → DiffVec A 1#
 [ x ] = x ∷ []
@@ -32,25 +32,25 @@ _++_ : ∀ {a} {A : Set a} {m n} →
 xs ++ ys = λ k → xs (ys k)
 
 toVec : ∀ {a} {A : Set a} {n} → DiffVec A n → Vec A (toℕ n)
-toVec xs = xs V.[]
+toVec xs = xs Vec.[]
 
 -- fromVec xs is linear in the length of xs.
 
 fromVec : ∀ {a} {A : Set a} {n} → Vec A n → DiffVec A (fromℕ n)
-fromVec xs = λ k → xs ⟨ V._++_ ⟩ k
+fromVec xs = λ k → xs ⟨ Vec._++_ ⟩ k
 
 head : ∀ {a} {A : Set a} {n} → DiffVec A (suc n) → A
-head xs = V.head (toVec xs)
+head xs = Vec.head (toVec xs)
 
 tail : ∀ {a} {A : Set a} {n} → DiffVec A (suc n) → DiffVec A n
-tail xs = λ k → V.tail (xs k)
+tail xs = λ k → Vec.tail (xs k)
 
 take : ∀ {a} {A : Set a} m {n} →
        DiffVec A (fromℕ m + n) → DiffVec A (fromℕ m)
-take N.zero    xs = []
-take (N.suc m) xs = head xs ∷ take m (tail xs)
+take ℕ.zero    xs = []
+take (ℕ.suc m) xs = head xs ∷ take m (tail xs)
 
 drop : ∀ {a} {A : Set a} m {n} →
        DiffVec A (fromℕ m + n) → DiffVec A n
-drop N.zero    xs = xs
-drop (N.suc m) xs = drop m (tail xs)
+drop ℕ.zero    xs = xs
+drop (ℕ.suc m) xs = drop m (tail xs)
diff --git a/src/Data/Fin.agda b/src/Data/Fin.agda
index e9b6a865ba..d0c0e12c5c 100644
--- a/src/Data/Fin.agda
+++ b/src/Data/Fin.agda
@@ -9,8 +9,7 @@
 module Data.Fin where
 
 open import Relation.Nullary.Decidable.Core
-open import Data.Nat.Base using (suc)
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 
 ------------------------------------------------------------------------
 -- Publicly re-export the contents of the base module
@@ -27,5 +26,5 @@ open import Data.Fin.Properties public
 
 infix 10 #_
 
-#_ : ∀ m {n} {m<n : True (suc m ℕₚ.≤? n)} → Fin n
+#_ : ∀ m {n} {m<n : True (m ℕ.<? n)} → Fin n
 #_ _ {m<n = m<n} = fromℕ< (toWitness m<n)
diff --git a/src/Data/Fin/Permutation/Components.agda b/src/Data/Fin/Permutation/Components.agda
index 5a2f27f7f8..d345018778 100644
--- a/src/Data/Fin/Permutation/Components.agda
+++ b/src/Data/Fin/Permutation/Components.agda
@@ -12,7 +12,6 @@ open import Data.Bool.Base using (Bool; true; false)
 open import Data.Fin.Base
 open import Data.Fin.Properties
 open import Data.Nat.Base as ℕ using (zero; suc; _∸_)
-import Data.Nat.Properties as ℕₚ
 open import Data.Product.Base using (proj₂)
 open import Function.Base using (_∘_)
 open import Relation.Nullary.Reflects using (invert)
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 303cf64905..11dbddea63 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Fin.Base
 open import Data.Fin.Patterns
 open import Data.Nat.Base as ℕ
   using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; s<s⁻¹; _∸_; _^_)
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
 open import Data.Unit using (⊤; tt)
 open import Data.Product.Base as Prod
@@ -174,14 +174,14 @@ toℕ≤pred[n] zero                 = z≤n
 toℕ≤pred[n] (suc {n = suc n} i)  = s≤s (toℕ≤pred[n] i)
 
 toℕ≤n : ∀ (i : Fin n) → toℕ i ℕ.≤ n
-toℕ≤n {suc n} i = ℕₚ.m≤n⇒m≤1+n (toℕ≤pred[n] i)
+toℕ≤n {suc n} i = ℕ.m≤n⇒m≤1+n (toℕ≤pred[n] i)
 
 -- A simpler implementation of toℕ≤pred[n],
 -- however, with a different reduction behavior.
 -- If no one needs the reduction behavior of toℕ≤pred[n],
 -- it can be removed in favor of toℕ≤pred[n]′.
 toℕ≤pred[n]′ : ∀ (i : Fin n) → toℕ i ℕ.≤ ℕ.pred n
-toℕ≤pred[n]′ i = ℕₚ.<⇒≤pred (toℕ<n i)
+toℕ≤pred[n]′ i = ℕ.<⇒≤pred (toℕ<n i)
 
 toℕ-mono-< : i < j → toℕ i ℕ.< toℕ j
 toℕ-mono-< i<j = i<j
@@ -223,7 +223,7 @@ toℕ-fromℕ< {m = zero}  {n = suc _} _   = refl
 toℕ-fromℕ< {m = suc m} {n = suc _} m<n = cong suc (toℕ-fromℕ< (ℕ.s<s⁻¹ m<n))
 
 -- fromℕ is a special case of fromℕ<.
-fromℕ-def : ∀ n → fromℕ n ≡ fromℕ< ℕₚ.≤-refl
+fromℕ-def : ∀ n → fromℕ n ≡ fromℕ< ℕ.≤-refl
 fromℕ-def zero    = refl
 fromℕ-def (suc n) = cong suc (fromℕ-def n)
 
@@ -231,7 +231,7 @@ fromℕ<-cong : ∀ m n {o} → m ≡ n → .(m<o : m ℕ.< o) .(n<o : n ℕ.< o
               fromℕ< m<o ≡ fromℕ< n<o
 fromℕ<-cong 0       0                   _ _   _   = refl
 fromℕ<-cong (suc _) (suc _) {o = suc _} r m<n n<o
-  = cong suc (fromℕ<-cong _ _ (ℕₚ.suc-injective r) (ℕ.s<s⁻¹ m<n) (ℕ.s<s⁻¹ n<o))
+  = cong suc (fromℕ<-cong _ _ (ℕ.suc-injective r) (ℕ.s<s⁻¹ m<n) (ℕ.s<s⁻¹ n<o))
 
 fromℕ<-injective : ∀ m n {o} → .(m<o : m ℕ.< o) .(n<o : n ℕ.< o) →
                    fromℕ< m<o ≡ fromℕ< n<o → m ≡ n
@@ -252,8 +252,8 @@ fromℕ<≡fromℕ<″ {m = suc m} m<n (ℕ.<″-offset _)
 
 toℕ-fromℕ<″ : ∀ (m<n : m ℕ.<″ n) → toℕ (fromℕ<″ m m<n) ≡ m
 toℕ-fromℕ<″ {m} {n} m<n = begin
-  toℕ (fromℕ<″ m m<n)  ≡⟨ cong toℕ (sym (fromℕ<≡fromℕ<″ (ℕₚ.≤″⇒≤ m<n) m<n)) ⟩
-  toℕ (fromℕ< _)       ≡⟨ toℕ-fromℕ< (ℕₚ.≤″⇒≤ m<n) ⟩
+  toℕ (fromℕ<″ m m<n)  ≡⟨ cong toℕ (sym (fromℕ<≡fromℕ<″ (ℕ.≤″⇒≤ m<n) m<n)) ⟩
+  toℕ (fromℕ< _)       ≡⟨ toℕ-fromℕ< (ℕ.≤″⇒≤ m<n) ⟩
   m                    ∎
   where open ≡-Reasoning
 
@@ -267,7 +267,7 @@ toℕ-cast {n = suc n} eq (suc k) = cong suc (toℕ-cast (cong ℕ.pred eq) k)
 
 cast-is-id : .(eq : m ≡ m) (k : Fin m) → cast eq k ≡ k
 cast-is-id eq zero    = refl
-cast-is-id eq (suc k) = cong suc (cast-is-id (ℕₚ.suc-injective eq) k)
+cast-is-id eq (suc k) = cong suc (cast-is-id (ℕ.suc-injective eq) k)
 
 subst-is-cast : (eq : m ≡ n) (k : Fin m) → subst Fin eq k ≡ cast eq k
 subst-is-cast refl k = sym (cast-is-id refl k)
@@ -276,7 +276,7 @@ cast-trans : .(eq₁ : m ≡ n) .(eq₂ : n ≡ o) (k : Fin m) →
              cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k
 cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ zero = refl
 cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (suc k) =
-  cong suc (cast-trans (ℕₚ.suc-injective eq₁) (ℕₚ.suc-injective eq₂) k)
+  cong suc (cast-trans (ℕ.suc-injective eq₁) (ℕ.suc-injective eq₂) k)
 
 ------------------------------------------------------------------------
 -- Properties of _≤_
@@ -284,30 +284,30 @@ cast-trans {m = suc _} {n = suc _} {o = suc _} eq₁ eq₂ (suc k) =
 -- Relational properties
 
 ≤-reflexive : _≡_ ⇒ (_≤_ {n})
-≤-reflexive refl = ℕₚ.≤-refl
+≤-reflexive refl = ℕ.≤-refl
 
 ≤-refl : Reflexive (_≤_ {n})
 ≤-refl = ≤-reflexive refl
 
 ≤-trans : Transitive (_≤_ {n})
-≤-trans = ℕₚ.≤-trans
+≤-trans = ℕ.≤-trans
 
 ≤-antisym : Antisymmetric _≡_ (_≤_ {n})
-≤-antisym x≤y y≤x = toℕ-injective (ℕₚ.≤-antisym x≤y y≤x)
+≤-antisym x≤y y≤x = toℕ-injective (ℕ.≤-antisym x≤y y≤x)
 
 ≤-total : Total (_≤_ {n})
-≤-total x y = ℕₚ.≤-total (toℕ x) (toℕ y)
+≤-total x y = ℕ.≤-total (toℕ x) (toℕ y)
 
 ≤-irrelevant : Irrelevant (_≤_ {m} {n})
-≤-irrelevant = ℕₚ.≤-irrelevant
+≤-irrelevant = ℕ.≤-irrelevant
 
 infix 4 _≤?_ _<?_
 
 _≤?_ : B.Decidable (_≤_ {m} {n})
-a ≤? b = toℕ a ℕₚ.≤? toℕ b
+a ≤? b = toℕ a ℕ.≤? toℕ b
 
 _<?_ : B.Decidable (_<_ {m} {n})
-m <? n = suc (toℕ m) ℕₚ.≤? toℕ n
+m <? n = suc (toℕ m) ℕ.≤? toℕ n
 
 ------------------------------------------------------------------------
 -- Structures
@@ -367,13 +367,13 @@ m <? n = suc (toℕ m) ℕₚ.≤? toℕ n
 -- Relational properties
 
 <-irrefl : Irreflexive _≡_ (_<_ {n})
-<-irrefl refl = ℕₚ.<-irrefl refl
+<-irrefl refl = ℕ.<-irrefl refl
 
 <-asym : Asymmetric (_<_ {n})
-<-asym = ℕₚ.<-asym
+<-asym = ℕ.<-asym
 
 <-trans : Transitive (_<_ {n})
-<-trans = ℕₚ.<-trans
+<-trans = ℕ.<-trans
 
 <-cmp : Trichotomous _≡_ (_<_ {n})
 <-cmp zero    zero    = tri≈ (λ()) refl  (λ())
@@ -394,7 +394,7 @@ m <? n = suc (toℕ m) ℕₚ.≤? toℕ n
 <-resp₂-≡ = <-respʳ-≡ , <-respˡ-≡
 
 <-irrelevant : Irrelevant (_<_ {m} {n})
-<-irrelevant = ℕₚ.<-irrelevant
+<-irrelevant = ℕ.<-irrelevant
 
 ------------------------------------------------------------------------
 -- Structures
@@ -430,10 +430,10 @@ m <? n = suc (toℕ m) ℕₚ.≤? toℕ n
 -- Other properties
 
 i<1+i : ∀ (i : Fin n) → i < suc i
-i<1+i = ℕₚ.n<1+n ∘ toℕ
+i<1+i = ℕ.n<1+n ∘ toℕ
 
 <⇒≢ : i < j → i ≢ j
-<⇒≢ i<i refl = ℕₚ.n≮n _ i<i
+<⇒≢ i<i refl = ℕ.n≮n _ i<i
 
 ≤∧≢⇒< : i ≤ j → i ≢ j → i < j
 ≤∧≢⇒< {i = zero}  {zero}  _         0≢0   = contradiction refl 0≢0
@@ -466,13 +466,13 @@ toℕ-inject₁ zero    = refl
 toℕ-inject₁ (suc i) = cong suc (toℕ-inject₁ i)
 
 toℕ-inject₁-≢ : ∀ (i : Fin n) → n ≢ toℕ (inject₁ i)
-toℕ-inject₁-≢ (suc i) = toℕ-inject₁-≢ i ∘ ℕₚ.suc-injective
+toℕ-inject₁-≢ (suc i) = toℕ-inject₁-≢ i ∘ ℕ.suc-injective
 
 inject₁ℕ< : ∀ (i : Fin n) → toℕ (inject₁ i) ℕ.< n
 inject₁ℕ< i rewrite toℕ-inject₁ i = toℕ<n i
 
 inject₁ℕ≤ : ∀ (i : Fin n) → toℕ (inject₁ i) ℕ.≤ n
-inject₁ℕ≤ = ℕₚ.<⇒≤ ∘ inject₁ℕ<
+inject₁ℕ≤ = ℕ.<⇒≤ ∘ inject₁ℕ<
 
 ≤̄⇒inject₁< : i ≤ j → inject₁ i < suc j
 ≤̄⇒inject₁< {i = i} i≤j rewrite sym (toℕ-inject₁ i) = s<s i≤j
@@ -482,7 +482,7 @@ inject₁ℕ≤ = ℕₚ.<⇒≤ ∘ inject₁ℕ<
 
 i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j
 i≤inject₁[j]⇒i≤1+j {i = zero}              _   = z≤n
-i≤inject₁[j]⇒i≤1+j {i = suc i} {j = suc j} i≤j = s≤s (ℕₚ.m≤n⇒m≤1+n (subst (toℕ i ℕ.≤_) (toℕ-inject₁ j) (ℕ.s≤s⁻¹ i≤j)))
+i≤inject₁[j]⇒i≤1+j {i = suc i} {j = suc j} i≤j = s≤s (ℕ.m≤n⇒m≤1+n (subst (toℕ i ℕ.≤_) (toℕ-inject₁ j) (ℕ.s≤s⁻¹ i≤j)))
 
 ------------------------------------------------------------------------
 -- lower₁
@@ -552,7 +552,7 @@ inject≤-idempotent {_} {suc n} {suc o} (suc i) _ _ _ =
   cong suc (inject≤-idempotent i _ _ _)
 
 inject≤-trans : ∀ (i : Fin m) .(m≤n : m ℕ.≤ n) .(n≤o : n ℕ.≤ o) →
-                inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (ℕₚ.≤-trans m≤n n≤o)
+                inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (ℕ.≤-trans m≤n n≤o)
 inject≤-trans i _ _ = inject≤-idempotent i _ _ _
 
 inject≤-injective : ∀ .(m≤n m≤n′ : m ℕ.≤ n) i j →
@@ -571,7 +571,7 @@ inject≤-irrelevant _ _ i = refl
 
 pred< : ∀ (i : Fin (suc n)) → i ≢ zero → pred i < i
 pred< zero    i≢0 = contradiction refl i≢0
-pred< (suc i) _   = ≤̄⇒inject₁< ℕₚ.≤-refl
+pred< (suc i) _   = ≤̄⇒inject₁< ℕ.≤-refl
 
 ------------------------------------------------------------------------
 -- splitAt
@@ -665,7 +665,7 @@ toℕ-combine {suc m} {n} i@0F j = begin
   toℕ (combine i j)          ≡⟨⟩
   toℕ (j ↑ˡ (m ℕ.* n))       ≡⟨ toℕ-↑ˡ j (m ℕ.* n) ⟩
   toℕ j                      ≡⟨⟩
-  0 ℕ.+ toℕ j                ≡⟨ cong (ℕ._+ toℕ j) (ℕₚ.*-zeroʳ n) ⟨
+  0 ℕ.+ toℕ j                ≡⟨ cong (ℕ._+ toℕ j) (ℕ.*-zeroʳ n) ⟨
   n ℕ.* toℕ i ℕ.+ toℕ j      ∎
   where open ≡-Reasoning
 toℕ-combine {suc m} {n} (suc i) j = begin
@@ -680,15 +680,15 @@ combine-monoˡ-< : ∀ {i j : Fin m} (k l : Fin n) →
                   i < j → combine i k < combine j l
 combine-monoˡ-< {m} {n} {i} {j} k l i<j = begin-strict
   toℕ (combine i k)      ≡⟨ toℕ-combine i k ⟩
-  n ℕ.* toℕ i ℕ.+ toℕ k  <⟨ ℕₚ.+-monoʳ-< (n ℕ.* toℕ i) (toℕ<n k) ⟩
-  n ℕ.* toℕ i ℕ.+ n      ≡⟨ ℕₚ.+-comm _ n ⟩
-  n ℕ.+ n ℕ.* toℕ i      ≡⟨ cong (n ℕ.+_) (ℕₚ.*-comm n _) ⟩
-  n ℕ.+ toℕ i ℕ.* n      ≡⟨ ℕₚ.*-comm (suc (toℕ i)) n ⟩
-  n ℕ.* suc (toℕ i)      ≤⟨ ℕₚ.*-monoʳ-≤ n (toℕ-mono-< i<j) ⟩
-  n ℕ.* toℕ j            ≤⟨ ℕₚ.m≤m+n (n ℕ.* toℕ j) (toℕ l) ⟩
+  n ℕ.* toℕ i ℕ.+ toℕ k  <⟨ ℕ.+-monoʳ-< (n ℕ.* toℕ i) (toℕ<n k) ⟩
+  n ℕ.* toℕ i ℕ.+ n      ≡⟨ ℕ.+-comm _ n ⟩
+  n ℕ.+ n ℕ.* toℕ i      ≡⟨ cong (n ℕ.+_) (ℕ.*-comm n _) ⟩
+  n ℕ.+ toℕ i ℕ.* n      ≡⟨ ℕ.*-comm (suc (toℕ i)) n ⟩
+  n ℕ.* suc (toℕ i)      ≤⟨ ℕ.*-monoʳ-≤ n (toℕ-mono-< i<j) ⟩
+  n ℕ.* toℕ j            ≤⟨ ℕ.m≤m+n (n ℕ.* toℕ j) (toℕ l) ⟩
   n ℕ.* toℕ j ℕ.+ toℕ l  ≡⟨ toℕ-combine j l ⟨
   toℕ (combine j l)      ∎
-  where open ℕₚ.≤-Reasoning; open +-*-Solver
+  where open ℕ.≤-Reasoning; open +-*-Solver
 
 combine-injectiveˡ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
                      combine i j ≡ combine k l → i ≡ k
@@ -701,7 +701,7 @@ combine-injectiveʳ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
                      combine i j ≡ combine k l → j ≡ l
 combine-injectiveʳ {m} {n} i j k l cᵢⱼ≡cₖₗ
   with refl ← combine-injectiveˡ i j k l cᵢⱼ≡cₖₗ
-  = toℕ-injective (ℕₚ.+-cancelˡ-≡ (n ℕ.* toℕ i) _ _ (begin
+  = toℕ-injective (ℕ.+-cancelˡ-≡ (n ℕ.* toℕ i) _ _ (begin
   n ℕ.* toℕ i ℕ.+ toℕ j ≡⟨ toℕ-combine i j ⟨
   toℕ (combine i j)     ≡⟨ cong toℕ cᵢⱼ≡cₖₗ ⟩
   toℕ (combine i l)     ≡⟨ toℕ-combine i l ⟩
@@ -817,12 +817,12 @@ toℕ‿ℕ- (suc n) (suc i)  = toℕ‿ℕ- n i
 ℕ-ℕ≡toℕ‿ℕ- (suc n) (suc i) = ℕ-ℕ≡toℕ‿ℕ- n i
 
 nℕ-ℕi≤n : ∀ n i → n ℕ-ℕ i ℕ.≤ n
-nℕ-ℕi≤n n       zero     = ℕₚ.≤-refl
+nℕ-ℕi≤n n       zero     = ℕ.≤-refl
 nℕ-ℕi≤n (suc n) (suc i)  = begin
   n ℕ-ℕ i  ≤⟨ nℕ-ℕi≤n n i ⟩
-  n        ≤⟨ ℕₚ.n≤1+n n ⟩
+  n        ≤⟨ ℕ.n≤1+n n ⟩
   suc n    ∎
-  where open ℕₚ.≤-Reasoning
+  where open ℕ.≤-Reasoning
 
 ------------------------------------------------------------------------
 -- punchIn
@@ -1017,10 +1017,10 @@ injective⇒≤ {suc _} {suc _} {f} inj = s≤s (injective⇒≤ (λ eq →
     (contraInjective inj 0≢1+n) eq))))
 
 <⇒notInjective : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective _≡_ _≡_ f)
-<⇒notInjective n<m inj = ℕₚ.≤⇒≯ (injective⇒≤ inj) n<m
+<⇒notInjective n<m inj = ℕ.≤⇒≯ (injective⇒≤ inj) n<m
 
 ℕ→Fin-notInjective : ∀ (f : ℕ → Fin n) → ¬ (Injective _≡_ _≡_ f)
-ℕ→Fin-notInjective f inj = ℕₚ.<-irrefl refl
+ℕ→Fin-notInjective f inj = ℕ.<-irrefl refl
   (injective⇒≤ (Comp.injective _≡_ _≡_ _≡_ toℕ-injective inj))
 
 -- Cantor-Schröder-Bernstein for finite sets
@@ -1028,7 +1028,7 @@ injective⇒≤ {suc _} {suc _} {f} inj = s≤s (injective⇒≤ (λ eq →
 cantor-schröder-bernstein : ∀ {f : Fin m → Fin n} {g : Fin n → Fin m} →
                             Injective _≡_ _≡_ f → Injective _≡_ _≡_ g →
                             m ≡ n
-cantor-schröder-bernstein f-inj g-inj = ℕₚ.≤-antisym
+cantor-schröder-bernstein f-inj g-inj = ℕ.≤-antisym
   (injective⇒≤ f-inj) (injective⇒≤ g-inj)
 
 ------------------------------------------------------------------------
@@ -1086,7 +1086,7 @@ opposite-involutive : Involutive {A = Fin n} _≡_ opposite
 opposite-involutive {suc n} i = toℕ-injective (begin
   toℕ (opposite (opposite i)) ≡⟨ opposite-prop (opposite i) ⟩
   n ∸ (toℕ (opposite i))      ≡⟨ cong (n ∸_) (opposite-prop i) ⟩
-  n ∸ (n ∸ (toℕ i))           ≡⟨ ℕₚ.m∸[m∸n]≡n (toℕ≤pred[n] i) ⟩
+  n ∸ (n ∸ (toℕ i))           ≡⟨ ℕ.m∸[m∸n]≡n (toℕ≤pred[n] i) ⟩
   toℕ i                       ∎)
   where open ≡-Reasoning
 
@@ -1172,14 +1172,14 @@ private
   ≺⇒< (n ≻toℕ i) = toℕ<n i
 
 ≺⇒<′ : _≺_ ⇒ ℕ._<′_
-≺⇒<′ lt = ℕₚ.<⇒<′ (≺⇒< lt)
+≺⇒<′ lt = ℕ.<⇒<′ (≺⇒< lt)
 {-# WARNING_ON_USAGE ≺⇒<′
 "Warning: ≺⇒<′ was deprecated in v2.0.
 Please use <⇒<′ instead."
 #-}
 
 <′⇒≺ : ℕ._<′_ ⇒ _≺_
-<′⇒≺ lt = <⇒≺ (ℕₚ.<′⇒< lt)
+<′⇒≺ lt = <⇒≺ (ℕ.<′⇒< lt)
 {-# WARNING_ON_USAGE <′⇒≺
 "Warning: <′⇒≺ was deprecated in v2.0.
 Please use <′⇒< instead."
diff --git a/src/Data/Fin/Subset/Properties.agda b/src/Data/Fin/Subset/Properties.agda
index 6029b9a78c..fb230ea468 100644
--- a/src/Data/Fin/Subset/Properties.agda
+++ b/src/Data/Fin/Subset/Properties.agda
@@ -22,7 +22,7 @@ open import Data.Fin.Base using (Fin; suc; zero)
 open import Data.Fin.Subset
 open import Data.Fin.Properties using (any?; decFinSubset)
 open import Data.Nat.Base hiding (∣_-_∣)
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Product as Product using (∃; ∄; _×_; _,_; proj₁)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Vec.Base
@@ -134,8 +134,8 @@ nonempty? p = any? (_∈? p)
 ∣p∣≤n = count≤n (_≟ inside)
 
 ∣p∣≤∣x∷p∣ : ∀ x (p : Subset n)  → ∣ p ∣ ≤ ∣ x ∷ p ∣
-∣p∣≤∣x∷p∣ outside p = ℕₚ.≤-refl
-∣p∣≤∣x∷p∣ inside  p = ℕₚ.n≤1+n ∣ p ∣
+∣p∣≤∣x∷p∣ outside p = ℕ.≤-refl
+∣p∣≤∣x∷p∣ inside  p = ℕ.n≤1+n ∣ p ∣
 
 ------------------------------------------------------------------------
 -- ⊥
@@ -166,8 +166,8 @@ nonempty? p = any? (_∈? p)
 
 ∣p∣≡n⇒p≡⊤ : ∣ p ∣ ≡ n → p ≡ ⊤ {n}
 ∣p∣≡n⇒p≡⊤ {p = []}          _     = refl
-∣p∣≡n⇒p≡⊤ {p = outside ∷ p} |p|≡n = contradiction |p|≡n (ℕₚ.<⇒≢ (s≤s (∣p∣≤n p)))
-∣p∣≡n⇒p≡⊤ {p = inside  ∷ p} |p|≡n = cong (inside ∷_) (∣p∣≡n⇒p≡⊤ (ℕₚ.suc-injective |p|≡n))
+∣p∣≡n⇒p≡⊤ {p = outside ∷ p} |p|≡n = contradiction |p|≡n (ℕ.<⇒≢ (s≤s (∣p∣≤n p)))
+∣p∣≡n⇒p≡⊤ {p = inside  ∷ p} |p|≡n = cong (inside ∷_) (∣p∣≡n⇒p≡⊤ (ℕ.suc-injective |p|≡n))
 
 ------------------------------------------------------------------------
 -- ⁅_⁆
@@ -257,7 +257,7 @@ module _ (n : ℕ) where
 p⊆q⇒∣p∣≤∣q∣ : p ⊆ q → ∣ p ∣ ≤ ∣ q ∣
 p⊆q⇒∣p∣≤∣q∣ {p = []}          {[]}          p⊆q = z≤n
 p⊆q⇒∣p∣≤∣q∣ {p = outside ∷ p} {outside ∷ q} p⊆q = p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ p⊆q)
-p⊆q⇒∣p∣≤∣q∣ {p = outside ∷ p} {inside  ∷ q} p⊆q = ℕₚ.m≤n⇒m≤1+n (p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ p⊆q))
+p⊆q⇒∣p∣≤∣q∣ {p = outside ∷ p} {inside  ∷ q} p⊆q = ℕ.m≤n⇒m≤1+n (p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ p⊆q))
 p⊆q⇒∣p∣≤∣q∣ {p = inside  ∷ p} {outside ∷ q} p⊆q = contradiction (p⊆q here) λ()
 p⊆q⇒∣p∣≤∣q∣ {p = inside  ∷ p} {inside  ∷ q} p⊆q = s≤s (p⊆q⇒∣p∣≤∣q∣ (drop-∷-⊆ p⊆q))
 
@@ -358,7 +358,7 @@ p∪∁p≡⊤ (inside  ∷ p) = cong (inside ∷_) (p∪∁p≡⊤ p)
 ∣∁p∣≡n∸∣p∣ (inside  ∷ p) = ∣∁p∣≡n∸∣p∣ p
 ∣∁p∣≡n∸∣p∣ (outside ∷ p) = begin
   suc ∣ ∁ p ∣     ≡⟨ cong suc (∣∁p∣≡n∸∣p∣ p) ⟩
-  suc (_ ∸ ∣ p ∣) ≡⟨ sym (ℕₚ.+-∸-assoc 1 (∣p∣≤n p)) ⟩
+  suc (_ ∸ ∣ p ∣) ≡⟨ sym (ℕ.+-∸-assoc 1 (∣p∣≤n p)) ⟩
   suc  _ ∸ ∣ p ∣  ∎
   where open ≡-Reasoning
 
@@ -515,7 +515,7 @@ x∈p∩q⁻ (s      ∷ p) (t      ∷ q) (there x∈p∩q) =
 ∣p∩q∣≤∣q∣ p q = p⊆q⇒∣p∣≤∣q∣ (p∩q⊆q p q)
 
 ∣p∩q∣≤∣p∣⊓∣q∣ : ∀ (p q : Subset n) → ∣ p ∩ q ∣ ≤ ∣ p ∣ ⊓ ∣ q ∣
-∣p∩q∣≤∣p∣⊓∣q∣ p q = ℕₚ.⊓-glb (∣p∩q∣≤∣p∣ p q) (∣p∩q∣≤∣q∣ p q)
+∣p∩q∣≤∣p∣⊓∣q∣ p q = ℕ.⊓-glb (∣p∩q∣≤∣p∣ p q) (∣p∩q∣≤∣q∣ p q)
 
 ------------------------------------------------------------------------
 -- _∪_
@@ -753,7 +753,7 @@ x∈p∪q⁺ (inj₂ x∈q) = q⊆p∪q _ _ x∈q
 ∣q∣≤∣p∪q∣ p q = p⊆q⇒∣p∣≤∣q∣ (q⊆p∪q p q)
 
 ∣p∣⊔∣q∣≤∣p∪q∣ : ∀ (p q : Subset n) → ∣ p ∣ ⊔ ∣ q ∣ ≤ ∣ p ∪ q ∣
-∣p∣⊔∣q∣≤∣p∪q∣ p q = ℕₚ.⊔-lub (∣p∣≤∣p∪q∣ p q) (∣q∣≤∣p∪q∣ p q)
+∣p∣⊔∣q∣≤∣p∪q∣ p q = ℕ.⊔-lub (∣p∣≤∣p∪q∣ p q) (∣q∣≤∣p∪q∣ p q)
 
 ------------------------------------------------------------------------
 -- Properties of _─_
diff --git a/src/Data/Float/Properties.agda b/src/Data/Float/Properties.agda
index a397dd53ec..526f71f6c4 100644
--- a/src/Data/Float/Properties.agda
+++ b/src/Data/Float/Properties.agda
@@ -10,11 +10,11 @@ module Data.Float.Properties where
 
 open import Data.Bool.Base as Bool using (Bool)
 open import Data.Float.Base
-import Data.Maybe.Base as M
-import Data.Maybe.Properties as Mₚ
-import Data.Nat.Properties as Nₚ
+import Data.Maybe.Base as Maybe
+import Data.Maybe.Properties as Maybe
+import Data.Nat.Properties as ℕ
 import Data.Word.Base as Word
-import Data.Word.Properties as Wₚ
+import Data.Word.Properties as Word
 open import Function.Base using (_∘_)
 open import Relation.Nullary.Decidable as RN using (map′)
 open import Relation.Binary.Core using (_⇒_)
@@ -37,10 +37,10 @@ open import Agda.Builtin.Float.Properties
 -- Properties of _≈_
 
 ≈⇒≡ : _≈_ ⇒ _≡_
-≈⇒≡ eq = toWord-injective _ _ (Mₚ.map-injective Wₚ.≈⇒≡ eq)
+≈⇒≡ eq = toWord-injective _ _ (Maybe.map-injective Word.≈⇒≡ eq)
 
 ≈-reflexive : _≡_ ⇒ _≈_
-≈-reflexive eq = cong (M.map Word.toℕ ∘ toWord) eq
+≈-reflexive eq = cong (Maybe.map Word.toℕ ∘ toWord) eq
 
 ≈-refl : Reflexive _≈_
 ≈-refl = refl
@@ -56,7 +56,7 @@ open import Agda.Builtin.Float.Properties
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-_≈?_ = On.decidable (M.map Word.toℕ ∘ toWord) _≡_ (Mₚ.≡-dec Nₚ._≟_)
+_≈?_ = On.decidable (Maybe.map Word.toℕ ∘ toWord) _≡_ (Maybe.≡-dec ℕ._≟_)
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index db9c7286fa..c842cf3d30 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -13,13 +13,12 @@
 module Data.Graph.Acyclic where
 
 open import Level using (_⊔_)
-open import Data.Nat.Base as Nat using (ℕ; zero; suc; _<′_)
+open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _<′_)
 open import Data.Nat.Induction using (<′-rec; <′-Rec)
-import Data.Nat.Properties as Nat
+import Data.Nat.Properties as ℕ
 open import Data.Fin as Fin
   using (Fin; Fin′; zero; suc; #_; toℕ; _≟_; opposite) renaming (_ℕ-ℕ_ to _-_)
-import Data.Fin.Properties as FP
-import Data.Fin.Permutation.Components as PC
+import Data.Fin.Properties as Fin
 open import Data.Product.Base as Prod using (∃; _×_; _,_)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; decToMaybe)
 open import Data.Empty
@@ -36,7 +35,7 @@ open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 private
 
   lemma : ∀ n (i : Fin n) → n - suc i <′ n
-  lemma (suc n) i  = Nat.≤⇒≤′ $ Nat.s≤s $ FP.nℕ-ℕi≤n n i
+  lemma (suc n) i  = ℕ.≤⇒≤′ $ ℕ.s≤s $ Fin.nℕ-ℕi≤n n i
 
 ------------------------------------------------------------------------
 -- Node contexts
@@ -253,7 +252,7 @@ private
 -- Weakens a node label.
 
 weaken : ∀ {n} {i : Fin n} → Fin (n - suc i) → Fin n
-weaken {n} {i} j = Fin.inject≤ j (FP.nℕ-ℕi≤n n (suc i))
+weaken {n} {i} j = Fin.inject≤ j (Fin.nℕ-ℕi≤n n (suc i))
 
 -- Labels each node with its node number.
 
diff --git a/src/Data/Integer/Divisibility.agda b/src/Data/Integer/Divisibility.agda
index a8c55a1e2a..0ad36ad1f4 100644
--- a/src/Data/Integer/Divisibility.agda
+++ b/src/Data/Integer/Divisibility.agda
@@ -1,4 +1,4 @@
-------------------------------------------------------------------------
+-----------------------------------------------------------------------
 -- The Agda standard library
 --
 -- Unsigned divisibility
@@ -14,9 +14,7 @@ open import Function.Base using (_on_; _$_)
 open import Data.Integer.Base
 open import Data.Integer.Properties
 import Data.Nat.Base as ℕ
-import Data.Nat.Properties as ℕᵖ
 import Data.Nat.Divisibility as ℕᵈ
-import Data.Nat.Coprimality as ℕᶜ
 open import Level
 open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/List/Extrema/Nat.agda b/src/Data/List/Extrema/Nat.agda
index 89f513e2bf..28b75e922c 100644
--- a/src/Data/List/Extrema/Nat.agda
+++ b/src/Data/List/Extrema/Nat.agda
@@ -12,7 +12,7 @@
 module Data.List.Extrema.Nat where
 
 open import Data.Nat.Base using (ℕ; _≤_; _<_)
-open import Data.Nat.Properties as ℕₚ using (≤∧≢⇒<; <⇒≤; <⇒≢)
+open import Data.Nat.Properties as ℕ using (≤∧≢⇒<; <⇒≤; <⇒≢)
 open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.List.Base using (List)
 import Data.List.Extrema
@@ -33,7 +33,7 @@ private
   <×⇒< : ∀ {x y} → x ≤ y × x ≢ y → x < y
   <×⇒< (x≤y , x≢y) = ≤∧≢⇒< x≤y x≢y
 
-  module Extrema = Data.List.Extrema ℕₚ.≤-totalOrder
+  module Extrema = Data.List.Extrema ℕ.≤-totalOrder
 
 ------------------------------------------------------------------------
 -- Re-export the contents of Extrema
diff --git a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
index 9af0162ded..e833ade444 100644
--- a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.Empty using (⊥-elim)
 open import Data.List.Base as List using (List; []; _∷_; length; map; filter; replicate)
 open import Data.Nat.Base using (zero; suc; _≤_)
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (case_of_; _$′_)
 
@@ -68,7 +68,7 @@ module _ {c t} {C : Set c} {T : REL A C t} where
 
 length-mono : ∀ {as bs} → Infix R as bs → length as ≤ length bs
 length-mono (here pref) = Prefixₚ.length-mono pref
-length-mono (there p)   = ℕₚ.m≤n⇒m≤1+n (length-mono p)
+length-mono (there p)   = ℕ.m≤n⇒m≤1+n (length-mono p)
 
 ------------------------------------------------------------------------
 -- As an order
@@ -98,20 +98,20 @@ module _ {c t} {C : Set c} {T : REL A C t} where
   antisym : Antisym R S T → Antisym (Infix R) (Infix S) (Pointwise T)
   antisym asym (here p) (here q) = Prefixₚ.antisym asym p q
   antisym asym {i = a ∷ as} {j = bs} p@(here _) (there q)
-    = ⊥-elim $′ ℕₚ.<-irrefl refl $′ begin-strict
+    = ⊥-elim $′ ℕ.<-irrefl refl $′ begin-strict
       length as <⟨ length-mono p ⟩
       length bs ≤⟨ length-mono q ⟩
-      length as ∎ where open ℕₚ.≤-Reasoning
+      length as ∎ where open ℕ.≤-Reasoning
   antisym asym {i = as} {j = b ∷ bs} (there p) q@(here _)
-    = ⊥-elim $′ ℕₚ.<-irrefl refl $′ begin-strict
+    = ⊥-elim $′ ℕ.<-irrefl refl $′ begin-strict
       length bs <⟨ length-mono q ⟩
       length as ≤⟨ length-mono p ⟩
-      length bs ∎ where open ℕₚ.≤-Reasoning
+      length bs ∎ where open ℕ.≤-Reasoning
   antisym asym {i = a ∷ as} {j = b ∷ bs} (there p) (there q)
-    = ⊥-elim $′ ℕₚ.<-irrefl refl $′ begin-strict
+    = ⊥-elim $′ ℕ.<-irrefl refl $′ begin-strict
       length as <⟨ length-mono p ⟩
       length bs <⟨ length-mono q ⟩
-      length as ∎ where open ℕₚ.≤-Reasoning
+      length as ∎ where open ℕ.≤-Reasoning
 
 ------------------------------------------------------------------------
 -- map
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index b11e011d76..5e44480bc3 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -23,7 +23,7 @@ open import Data.List.Relation.Binary.Sublist.Heterogeneous
 
 open import Data.Maybe.Relation.Unary.All as MAll using (nothing; just)
 open import Data.Nat.Base using (ℕ; _≤_; _≥_); open ℕ; open _≤_
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (∃₂; _×_; _,_; <_,_>; proj₂; uncurry)
 
 open import Function.Base
@@ -62,7 +62,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   length-mono-≤ : ∀ {as bs} → Sublist R as bs → length as ≤ length bs
   length-mono-≤ []        = z≤n
-  length-mono-≤ (y ∷ʳ rs) = ℕₚ.m≤n⇒m≤1+n (length-mono-≤ rs)
+  length-mono-≤ (y ∷ʳ rs) = ℕ.m≤n⇒m≤1+n (length-mono-≤ rs)
   length-mono-≤ (r ∷ rs)  = s≤s (length-mono-≤ rs)
 
 ------------------------------------------------------------------------
@@ -75,9 +75,9 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   toPointwise : ∀ {as bs} → length as ≡ length bs →
                 Sublist R as bs → Pointwise R as bs
   toPointwise {bs = []}     eq []         = []
-  toPointwise {bs = b ∷ bs} eq (r ∷ rs)   = r ∷ toPointwise (ℕₚ.suc-injective eq) rs
+  toPointwise {bs = b ∷ bs} eq (r ∷ rs)   = r ∷ toPointwise (ℕ.suc-injective eq) rs
   toPointwise {bs = b ∷ bs} eq (b ∷ʳ rs) =
-    ⊥-elim $ ℕₚ.<-irrefl eq (s≤s (length-mono-≤ rs))
+    ⊥-elim $ ℕ.<-irrefl eq (s≤s (length-mono-≤ rs))
 
 ------------------------------------------------------------------------
 -- Various functions' outputs are sublists
@@ -190,7 +190,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   ++⁻ : ∀ {as bs cs ds} → length as ≡ length bs →
         Sublist R (as ++ cs) (bs ++ ds) → Sublist R cs ds
   ++⁻ {[]}     {[]}     eq rs = rs
-  ++⁻ {a ∷ as} {b ∷ bs} eq rs = ++⁻ (ℕₚ.suc-injective eq) (∷⁻ rs)
+  ++⁻ {a ∷ as} {b ∷ bs} eq rs = ++⁻ (ℕ.suc-injective eq) (∷⁻ rs)
 
   ++ˡ : ∀ {as bs} (cs : List B) → Sublist R as bs → Sublist R as (cs ++ bs)
   ++ˡ zs = ++⁺ (minimum zs)
@@ -223,7 +223,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
           Sublist R (drop m as) (drop n bs)
   drop⁺ {m} z≤n       rs        = drop-Sublist m rs
   drop⁺     (s≤s m≥n) []        = []
-  drop⁺     (s≤s m≥n) (y ∷ʳ rs) = drop⁺ (ℕₚ.m≤n⇒m≤1+n m≥n) rs
+  drop⁺     (s≤s m≥n) (y ∷ʳ rs) = drop⁺ (ℕ.m≤n⇒m≤1+n m≥n) rs
   drop⁺     (s≤s m≥n) (r ∷ rs)  = drop⁺ m≥n rs
 
   drop⁺-≥ : ∀ {m n as bs} → m ≥ n → Pointwise R as bs →
@@ -232,7 +232,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   drop⁺-⊆ : ∀ {as bs} m → Sublist R as bs →
             Sublist R (drop m as) (drop m bs)
-  drop⁺-⊆ m = drop⁺ (ℕₚ.≤-refl {m})
+  drop⁺-⊆ m = drop⁺ (ℕ.≤-refl {m})
 
 module _ {a b r p q} {A : Set a} {B : Set b}
          {R : REL A B r} {P : Pred A p} {Q : Pred B q}
@@ -385,25 +385,25 @@ module Antisymmetry
     {R : REL A B r} {S : REL B A s} {E : REL A B e}
     (rs⇒e : Antisym R S E) where
 
-  open ℕₚ.≤-Reasoning
+  open ℕ.≤-Reasoning
 
   antisym : Antisym (Sublist R) (Sublist S) (Pointwise E)
   antisym []        []        = []
   antisym (r ∷ rs)  (s ∷ ss)  = rs⇒e r s ∷ antisym rs ss
   -- impossible cases
   antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷ʳ_ {ys₂} {zs} z ss) =
-    ⊥-elim $ ℕₚ.<-irrefl P.refl $ begin
+    ⊥-elim $ ℕ.<-irrefl P.refl $ begin
     length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩
-    length zs        ≤⟨ ℕₚ.n≤1+n (length zs) ⟩
+    length zs        ≤⟨ ℕ.n≤1+n (length zs) ⟩
     length (z ∷ zs)  ≤⟨ length-mono-≤ rs ⟩
     length ys₁       ∎
   antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷_ {y} {ys₂} {z} {zs} s ss)  =
-    ⊥-elim $ ℕₚ.<-irrefl P.refl $ begin
+    ⊥-elim $ ℕ.<-irrefl P.refl $ begin
     length (z ∷ zs) ≤⟨ length-mono-≤ rs ⟩
     length ys₁      ≤⟨ length-mono-≤ ss ⟩
     length zs       ∎
   antisym (_∷_ {x} {xs} {y} {ys₁} r rs)  (_∷ʳ_ {ys₂} {zs} z ss) =
-    ⊥-elim $ ℕₚ.<-irrefl P.refl $ begin
+    ⊥-elim $ ℕ.<-irrefl P.refl $ begin
     length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩
     length xs        ≤⟨ length-mono-≤ rs ⟩
     length ys₁       ∎
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index e7b95805be..1813568d5b 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -21,8 +21,8 @@ module Data.List.Relation.Binary.Sublist.Heterogeneous.Solver
 
 open import Level using (_⊔_)
 open import Data.Fin as Fin
-open import Data.Maybe.Base as M
-open import Data.Nat.Base as Nat using (ℕ)
+open import Data.Maybe.Base as Maybe
+open import Data.Nat.Base as ℕ using (ℕ)
 open import Data.Product.Base using (Σ-syntax; _,_)
 open import Data.Vec.Base as Vec using (Vec ; lookup)
 open import Data.List.Base hiding (lookup)
@@ -124,8 +124,8 @@ private
 
 -- Solver for items
 solveI : ∀ {n} (a b : Item n) → Maybe (a ⊆I b)
-solveI (var k) (var l) = M.map var $ decToMaybe (k Fin.≟ l)
-solveI (val a) (val b) = M.map val $ decToMaybe (R? a b)
+solveI (var k) (var l) = Maybe.map var $ decToMaybe (k Fin.≟ l)
+solveI (val a) (val b) = Maybe.map val $ decToMaybe (R? a b)
 solveI _ _ = nothing
 
 -- Solver for linearised expressions
@@ -135,8 +135,8 @@ solveR [] e  = just (λ ρ → minimum _)
 solveR d  [] = nothing
 -- actual work
 solveR (a ∷ d) (b ∷ e) with solveI a b
-... | just it = M.map (keep-it it d e) (solveR d e)
-... | nothing = M.map (skip-it b (a ∷ d) e) (solveR (a ∷ d) e)
+... | just it = Maybe.map (keep-it it d e) (solveR d e)
+... | nothing = Maybe.map (skip-it b (a ∷ d) e) (solveR (a ∷ d) e)
 
 -- Coming back to ASTs thanks to flatten
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
index 19fe765456..cf12328323 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
@@ -16,7 +16,7 @@ open import Data.List.Base hiding (_∷ʳ_)
 open import Data.List.Relation.Unary.Any using (Any)
 import Data.Maybe.Relation.Unary.All as Maybe
 open import Data.Nat.Base using (_≤_; _≥_)
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (∃; _,_; proj₂)
 open import Function.Base
 open import Function.Bundles using (_⇔_; _⤖_)
@@ -159,7 +159,7 @@ module _ {m n} {xs : List A} where
 module _ {xs ys : List A} where
 
   drop⁺-⊆ : ∀ n → xs ⊆ ys → drop n xs ⊆ drop n ys
-  drop⁺-⊆ n xs⊆ys = drop⁺ {n} ℕₚ.≤-refl xs⊆ys
+  drop⁺-⊆ n xs⊆ys = drop⁺ {n} ℕ.≤-refl xs⊆ys
 
 ------------------------------------------------------------------------
 -- takeWhile / dropWhile
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 4f0dc58e73..625634af1d 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -18,8 +18,8 @@ open import Data.Nat.Binary.Base
 open import Data.Nat as ℕ using (ℕ; z≤n; s≤s; s<s⁻¹)
 open import Data.Nat.DivMod using (_%_; _/_; m/n≤m; +-distrib-/-∣ˡ)
 open import Data.Nat.Divisibility using (∣-refl)
-import Data.Nat.Base as ℕᵇ
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Base as ℕ
+import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
@@ -39,7 +39,7 @@ open import Relation.Nullary.Negation using (contradiction)
 open import Algebra.Definitions {A = ℕᵇ} _≡_
 open import Algebra.Structures {A = ℕᵇ} _≡_
 import Algebra.Properties.CommutativeSemigroup as CommSemigProp
-import Algebra.Properties.CommutativeSemigroup ℕₚ.+-commutativeSemigroup
+import Algebra.Properties.CommutativeSemigroup ℕ.+-commutativeSemigroup
   as ℕ-+-semigroupProperties
 import Relation.Binary.Construct.StrictToNonStrict _≡_ _<_
   as StrictToNonStrict
@@ -94,16 +94,16 @@ zero     ≟ 1+[2 _ ] =  no λ()
 toℕ-double : ∀ x → toℕ (double x) ≡ 2 ℕ.* (toℕ x)
 toℕ-double zero     =  refl
 toℕ-double 1+[2 x ] =  cong ((2 ℕ.*_) ∘ ℕ.suc) (toℕ-double x)
-toℕ-double 2[1+ x ] =  cong (2 ℕ.*_) (sym (ℕₚ.*-distribˡ-+ 2 1 (toℕ x)))
+toℕ-double 2[1+ x ] =  cong (2 ℕ.*_) (sym (ℕ.*-distribˡ-+ 2 1 (toℕ x)))
 
 toℕ-suc : ∀ x → toℕ (suc x) ≡ ℕ.suc (toℕ x)
 toℕ-suc zero     =  refl
 toℕ-suc 2[1+ x ] =  cong (ℕ.suc ∘ (2 ℕ.*_)) (toℕ-suc x)
-toℕ-suc 1+[2 x ] =  ℕₚ.*-distribˡ-+ 2 1 (toℕ x)
+toℕ-suc 1+[2 x ] =  ℕ.*-distribˡ-+ 2 1 (toℕ x)
 
 toℕ-pred : ∀ x → toℕ (pred x) ≡ ℕ.pred (toℕ x)
 toℕ-pred zero     =  refl
-toℕ-pred 2[1+ x ] =  cong ℕ.pred $ sym $ ℕₚ.*-distribˡ-+ 2 1 (toℕ x)
+toℕ-pred 2[1+ x ] =  cong ℕ.pred $ sym $ ℕ.*-distribˡ-+ 2 1 (toℕ x)
 toℕ-pred 1+[2 x ] =  toℕ-double x
 
 toℕ-fromℕ' : toℕ ∘ fromℕ' ≗ id
@@ -116,7 +116,7 @@ toℕ-fromℕ' (ℕ.suc n) = begin
   where open ≡-Reasoning
 
 fromℕ≡fromℕ' : fromℕ ≗ fromℕ'
-fromℕ≡fromℕ' n = fromℕ-helper≡fromℕ' n n ℕₚ.≤-refl
+fromℕ≡fromℕ' n = fromℕ-helper≡fromℕ' n n ℕ.≤-refl
   where
   split : ℕᵇ → Maybe Bool × ℕᵇ
   split zero     = nothing , zero
@@ -170,7 +170,7 @@ fromℕ≡fromℕ' n = fromℕ-helper≡fromℕ' n n ℕₚ.≤-refl
       just (n % 2 ℕ.≡ᵇ 0) , fromℕ' (n / 2)               ≡⟨ cong₂ _,_ (head-homo n) (tail-homo n) ⟨
       head (fromℕ' (ℕ.suc n)) , tail (fromℕ' (ℕ.suc n))  ≡⟨⟩
       split (fromℕ' (ℕ.suc n))                           ∎)
-    where rec-n/2 = fromℕ-helper≡fromℕ' (n / 2) w (ℕₚ.≤-trans (m/n≤m n 2) n≤w)
+    where rec-n/2 = fromℕ-helper≡fromℕ' (n / 2) w (ℕ.≤-trans (m/n≤m n 2) n≤w)
 
 toℕ-fromℕ : toℕ ∘ fromℕ ≗ id
 toℕ-fromℕ n rewrite fromℕ≡fromℕ' n = toℕ-fromℕ' n
@@ -179,20 +179,20 @@ toℕ-injective : Injective _≡_ _≡_ toℕ
 toℕ-injective {zero}     {zero}     _               =  refl
 toℕ-injective {2[1+ x ]} {2[1+ y ]} 2[1+xN]≡2[1+yN] =  cong 2[1+_] x≡y
   where
-  1+xN≡1+yN = ℕₚ.*-cancelˡ-≡ (ℕ.suc _) (ℕ.suc _) 2 2[1+xN]≡2[1+yN]
+  1+xN≡1+yN = ℕ.*-cancelˡ-≡ (ℕ.suc _) (ℕ.suc _) 2 2[1+xN]≡2[1+yN]
   xN≡yN     = cong ℕ.pred 1+xN≡1+yN
   x≡y       = toℕ-injective xN≡yN
 
 toℕ-injective {2[1+ x ]} {1+[2 y ]} 2[1+xN]≡1+2yN =
-  contradiction 2[1+xN]≡1+2yN (ℕₚ.even≢odd (ℕ.suc (toℕ x)) (toℕ y))
+  contradiction 2[1+xN]≡1+2yN (ℕ.even≢odd (ℕ.suc (toℕ x)) (toℕ y))
 
 toℕ-injective {1+[2 x ]} {2[1+ y ]} 1+2xN≡2[1+yN] =
-  contradiction (sym 1+2xN≡2[1+yN]) (ℕₚ.even≢odd (ℕ.suc (toℕ y)) (toℕ x))
+  contradiction (sym 1+2xN≡2[1+yN]) (ℕ.even≢odd (ℕ.suc (toℕ y)) (toℕ x))
 
 toℕ-injective {1+[2 x ]} {1+[2 y ]} 1+2xN≡1+2yN =  cong 1+[2_] x≡y
   where
   2xN≡2yN = cong ℕ.pred 1+2xN≡1+2yN
-  xN≡yN   = ℕₚ.*-cancelˡ-≡ _ _ 2 2xN≡2yN
+  xN≡yN   = ℕ.*-cancelˡ-≡ _ _ 2 2xN≡2yN
   x≡y     = toℕ-injective xN≡yN
 
 toℕ-surjective : Surjective _≡_ _≡_ toℕ
@@ -285,43 +285,43 @@ x≢0⇒x>0 {1+[2 _ ]} _   =  0<odd
 -- Properties of _<_ and toℕ & fromℕ.
 
 toℕ-mono-< : toℕ Preserves _<_ ⟶ ℕ._<_
-toℕ-mono-< {zero}     {2[1+ _ ]} _               =  ℕₚ.0<1+n
-toℕ-mono-< {zero}     {1+[2 _ ]} _               =  ℕₚ.0<1+n
+toℕ-mono-< {zero}     {2[1+ _ ]} _               =  ℕ.0<1+n
+toℕ-mono-< {zero}     {1+[2 _ ]} _               =  ℕ.0<1+n
 toℕ-mono-< {2[1+ x ]} {2[1+ y ]} (even<even x<y) =  begin
-  ℕ.suc (2 ℕ.* (ℕ.suc xN))    ≤⟨ ℕₚ.+-monoʳ-≤ 1 (ℕₚ.*-monoʳ-≤ 2 xN<yN) ⟩
-  ℕ.suc (2 ℕ.* yN)            ≤⟨ ℕₚ.n≤1+n _ ⟩
-  2 ℕ.+ (2 ℕ.* yN)            ≡⟨ sym (ℕₚ.*-distribˡ-+ 2 1 yN) ⟩
+  ℕ.suc (2 ℕ.* (ℕ.suc xN))    ≤⟨ ℕ.+-monoʳ-≤ 1 (ℕ.*-monoʳ-≤ 2 xN<yN) ⟩
+  ℕ.suc (2 ℕ.* yN)            ≤⟨ ℕ.n≤1+n _ ⟩
+  2 ℕ.+ (2 ℕ.* yN)            ≡⟨ sym (ℕ.*-distribˡ-+ 2 1 yN) ⟩
   2 ℕ.* (ℕ.suc yN)            ∎
-  where open ℕₚ.≤-Reasoning;  xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y
+  where open ℕ.≤-Reasoning;  xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y
 toℕ-mono-< {2[1+ x ]} {1+[2 y ]} (even<odd x<y) =
-  ℕₚ.+-monoʳ-≤ 1 (ℕₚ.*-monoʳ-≤ 2 (toℕ-mono-< x<y))
+  ℕ.+-monoʳ-≤ 1 (ℕ.*-monoʳ-≤ 2 (toℕ-mono-< x<y))
 toℕ-mono-< {1+[2 x ]} {2[1+ y ]} (odd<even (inj₁ x<y)) =   begin
   ℕ.suc (ℕ.suc (2 ℕ.* xN))    ≡⟨⟩
-  2 ℕ.+ (2 ℕ.* xN)            ≡⟨ sym (ℕₚ.*-distribˡ-+ 2 1 xN) ⟩
-  2 ℕ.* (ℕ.suc xN)            ≤⟨ ℕₚ.*-monoʳ-≤ 2 xN<yN ⟩
-  2 ℕ.* yN                    ≤⟨ ℕₚ.*-monoʳ-≤ 2 (ℕₚ.n≤1+n _) ⟩
+  2 ℕ.+ (2 ℕ.* xN)            ≡⟨ sym (ℕ.*-distribˡ-+ 2 1 xN) ⟩
+  2 ℕ.* (ℕ.suc xN)            ≤⟨ ℕ.*-monoʳ-≤ 2 xN<yN ⟩
+  2 ℕ.* yN                    ≤⟨ ℕ.*-monoʳ-≤ 2 (ℕ.n≤1+n _) ⟩
   2 ℕ.* (ℕ.suc yN)            ∎
-  where open ℕₚ.≤-Reasoning;  xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y
+  where open ℕ.≤-Reasoning;  xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y
 toℕ-mono-< {1+[2 x ]} {2[1+ .x ]} (odd<even (inj₂ refl)) =
-  ℕₚ.≤-reflexive (sym (ℕₚ.*-distribˡ-+ 2 1 (toℕ x)))
-toℕ-mono-< {1+[2 x ]} {1+[2 y ]} (odd<odd x<y) =  ℕₚ.+-monoʳ-< 1 (ℕₚ.*-monoʳ-< 2 xN<yN)
+  ℕ.≤-reflexive (sym (ℕ.*-distribˡ-+ 2 1 (toℕ x)))
+toℕ-mono-< {1+[2 x ]} {1+[2 y ]} (odd<odd x<y) =  ℕ.+-monoʳ-< 1 (ℕ.*-monoʳ-< 2 xN<yN)
   where xN = toℕ x;  yN = toℕ y;  xN<yN = toℕ-mono-< x<y
 
 toℕ-cancel-< : ∀ {x y} → toℕ x ℕ.< toℕ y → x < y
 toℕ-cancel-< {zero}     {2[1+ y ]} x<y = 0<even
 toℕ-cancel-< {zero}     {1+[2 y ]} x<y = 0<odd
 toℕ-cancel-< {2[1+ x ]} {2[1+ y ]} x<y =
-  even<even (toℕ-cancel-< (s<s⁻¹ (ℕₚ.*-cancelˡ-< 2 _ _ x<y)))
+  even<even (toℕ-cancel-< (s<s⁻¹ (ℕ.*-cancelˡ-< 2 _ _ x<y)))
 toℕ-cancel-< {2[1+ x ]} {1+[2 y ]} x<y
-  rewrite ℕₚ.*-distribˡ-+ 2 1 (toℕ x) =
-  even<odd (toℕ-cancel-< (ℕₚ.*-cancelˡ-< 2 _ _ (ℕₚ.≤-trans (s≤s (ℕₚ.n≤1+n _)) (s<s⁻¹ x<y))))
-toℕ-cancel-< {1+[2 x ]} {2[1+ y ]} x<y with toℕ x ℕₚ.≟ toℕ y
+  rewrite ℕ.*-distribˡ-+ 2 1 (toℕ x) =
+  even<odd (toℕ-cancel-< (ℕ.*-cancelˡ-< 2 _ _ (ℕ.≤-trans (s≤s (ℕ.n≤1+n _)) (s<s⁻¹ x<y))))
+toℕ-cancel-< {1+[2 x ]} {2[1+ y ]} x<y with toℕ x ℕ.≟ toℕ y
 ... | yes x≡y = odd<even (inj₂ (toℕ-injective x≡y))
 ... | no  x≢y
-  rewrite ℕₚ.+-suc (toℕ y) (toℕ y ℕ.+ 0) =
-  odd<even (inj₁ (toℕ-cancel-< (ℕₚ.≤∧≢⇒< (ℕₚ.*-cancelˡ-≤ 2 (ℕₚ.+-cancelˡ-≤ 2 _ _ x<y)) x≢y)))
+  rewrite ℕ.+-suc (toℕ y) (toℕ y ℕ.+ 0) =
+  odd<even (inj₁ (toℕ-cancel-< (ℕ.≤∧≢⇒< (ℕ.*-cancelˡ-≤ 2 (ℕ.+-cancelˡ-≤ 2 _ _ x<y)) x≢y)))
 toℕ-cancel-< {1+[2 x ]} {1+[2 y ]} x<y =
-  odd<odd (toℕ-cancel-< (ℕₚ.*-cancelˡ-< 2 _ _ (s<s⁻¹ x<y)))
+  odd<odd (toℕ-cancel-< (ℕ.*-cancelˡ-< 2 _ _ (s<s⁻¹ x<y)))
 
 fromℕ-cancel-< : ∀ {x y} → fromℕ x < fromℕ y → x ℕ.< y
 fromℕ-cancel-< = subst₂ ℕ._<_ (toℕ-fromℕ _) (toℕ-fromℕ _) ∘ toℕ-mono-<
@@ -482,13 +482,13 @@ x≤0⇒x≡0 (inj₂ x≡0) = x≡0
 -- Properties of _<_ and toℕ & fromℕ.
 
 fromℕ-mono-≤ : fromℕ Preserves ℕ._≤_ ⟶ _≤_
-fromℕ-mono-≤ m≤n  with ℕₚ.m≤n⇒m<n∨m≡n m≤n
+fromℕ-mono-≤ m≤n  with ℕ.m≤n⇒m<n∨m≡n m≤n
 ... | inj₁ m<n =  inj₁ (fromℕ-mono-< m<n)
 ... | inj₂ m≡n =  inj₂ (cong fromℕ m≡n)
 
 toℕ-mono-≤ : toℕ Preserves _≤_ ⟶ ℕ._≤_
-toℕ-mono-≤ (inj₁ x<y)  =  ℕₚ.<⇒≤ (toℕ-mono-< x<y)
-toℕ-mono-≤ (inj₂ refl) =  ℕₚ.≤-reflexive refl
+toℕ-mono-≤ (inj₁ x<y)  =  ℕ.<⇒≤ (toℕ-mono-< x<y)
+toℕ-mono-≤ (inj₂ refl) =  ℕ.≤-reflexive refl
 
 toℕ-cancel-≤ : ∀ {x y} → toℕ x ℕ.≤ toℕ y → x ≤ y
 toℕ-cancel-≤ = subst₂ _≤_ (fromℕ-toℕ _) (fromℕ-toℕ _) ∘ fromℕ-mono-≤
@@ -637,8 +637,8 @@ module ≤-Reasoning where
 
 toℕ-homo-+ : ∀ x y → toℕ (x + y) ≡ toℕ x ℕ.+ toℕ y
 toℕ-homo-+ zero     _        = refl
-toℕ-homo-+ 2[1+ x ] zero     = cong ℕ.suc (sym (ℕₚ.+-identityʳ _))
-toℕ-homo-+ 1+[2 x ] zero     = cong ℕ.suc (sym (ℕₚ.+-identityʳ _))
+toℕ-homo-+ 2[1+ x ] zero     = cong ℕ.suc (sym (ℕ.+-identityʳ _))
+toℕ-homo-+ 1+[2 x ] zero     = cong ℕ.suc (sym (ℕ.+-identityʳ _))
 toℕ-homo-+ 2[1+ x ] 2[1+ y ] = begin
   toℕ (2[1+ x ] + 2[1+ y ])          ≡⟨⟩
   toℕ 2[1+ (suc (x + y)) ]           ≡⟨⟩
@@ -686,19 +686,19 @@ toℕ-homo-+ 1+[2 x ] 1+[2 y ] = begin
   toℕ 1+[2 x ] ℕ.+ toℕ 1+[2 y ]           ∎
   where open ≡-Reasoning;  m = toℕ x;  n = toℕ y
 
-toℕ-isMagmaHomomorphism-+ : IsMagmaHomomorphism +-rawMagma ℕᵇ.+-rawMagma toℕ
+toℕ-isMagmaHomomorphism-+ : IsMagmaHomomorphism +-rawMagma ℕ.+-rawMagma toℕ
 toℕ-isMagmaHomomorphism-+ = record
   { isRelHomomorphism = toℕ-isRelHomomorphism
   ; homo              = toℕ-homo-+
   }
 
-toℕ-isMonoidHomomorphism-+ : IsMonoidHomomorphism +-0-rawMonoid ℕᵇ.+-0-rawMonoid toℕ
+toℕ-isMonoidHomomorphism-+ : IsMonoidHomomorphism +-0-rawMonoid ℕ.+-0-rawMonoid toℕ
 toℕ-isMonoidHomomorphism-+ = record
   { isMagmaHomomorphism = toℕ-isMagmaHomomorphism-+
   ; ε-homo              = refl
   }
 
-toℕ-isMonoidMonomorphism-+ : IsMonoidMonomorphism +-0-rawMonoid ℕᵇ.+-0-rawMonoid toℕ
+toℕ-isMonoidMonomorphism-+ : IsMonoidMonomorphism +-0-rawMonoid ℕ.+-0-rawMonoid toℕ
 toℕ-isMonoidMonomorphism-+ = record
   { isMonoidHomomorphism = toℕ-isMonoidHomomorphism-+
   ; injective            = toℕ-injective
@@ -745,25 +745,25 @@ private
   module +-Monomorphism = MonoidMonomorphism toℕ-isMonoidMonomorphism-+
 
 +-assoc : Associative _+_
-+-assoc = +-Monomorphism.assoc ℕₚ.+-isMagma ℕₚ.+-assoc
++-assoc = +-Monomorphism.assoc ℕ.+-isMagma ℕ.+-assoc
 
 +-comm : Commutative _+_
-+-comm = +-Monomorphism.comm ℕₚ.+-isMagma ℕₚ.+-comm
++-comm = +-Monomorphism.comm ℕ.+-isMagma ℕ.+-comm
 
 +-identityˡ : LeftIdentity zero _+_
 +-identityˡ _ = refl
 
 +-identityʳ : RightIdentity zero _+_
-+-identityʳ = +-Monomorphism.identityʳ ℕₚ.+-isMagma ℕₚ.+-identityʳ
++-identityʳ = +-Monomorphism.identityʳ ℕ.+-isMagma ℕ.+-identityʳ
 
 +-identity : Identity zero _+_
 +-identity = +-identityˡ , +-identityʳ
 
 +-cancelˡ-≡ : LeftCancellative _+_
-+-cancelˡ-≡ = +-Monomorphism.cancelˡ ℕₚ.+-isMagma ℕₚ.+-cancelˡ-≡
++-cancelˡ-≡ = +-Monomorphism.cancelˡ ℕ.+-isMagma ℕ.+-cancelˡ-≡
 
 +-cancelʳ-≡ : RightCancellative _+_
-+-cancelʳ-≡ = +-Monomorphism.cancelʳ ℕₚ.+-isMagma ℕₚ.+-cancelʳ-≡
++-cancelʳ-≡ = +-Monomorphism.cancelʳ ℕ.+-isMagma ℕ.+-cancelʳ-≡
 
 ------------------------------------------------------------------------
 -- Structures for _+_
@@ -772,7 +772,7 @@ private
 +-isMagma = isMagma _+_
 
 +-isSemigroup : IsSemigroup _+_
-+-isSemigroup = +-Monomorphism.isSemigroup ℕₚ.+-isSemigroup
++-isSemigroup = +-Monomorphism.isSemigroup ℕ.+-isSemigroup
 
 +-isCommutativeSemigroup : IsCommutativeSemigroup _+_
 +-isCommutativeSemigroup = record
@@ -781,10 +781,10 @@ private
   }
 
 +-0-isMonoid : IsMonoid _+_ 0ᵇ
-+-0-isMonoid = +-Monomorphism.isMonoid ℕₚ.+-0-isMonoid
++-0-isMonoid = +-Monomorphism.isMonoid ℕ.+-0-isMonoid
 
 +-0-isCommutativeMonoid : IsCommutativeMonoid _+_ 0ᵇ
-+-0-isCommutativeMonoid = +-Monomorphism.isCommutativeMonoid ℕₚ.+-0-isCommutativeMonoid
++-0-isCommutativeMonoid = +-Monomorphism.isCommutativeMonoid ℕ.+-0-isCommutativeMonoid
 
 ------------------------------------------------------------------------
 -- Bundles for _+_
@@ -822,7 +822,7 @@ module Bin+CSemigroup = CommSemigProp +-commutativeSemigroup
 +-mono-≤ {x} {x′} {y} {y′} x≤x′ y≤y′ =  begin
   x + y                 ≡⟨ sym $ cong₂ _+_ (fromℕ-toℕ x) (fromℕ-toℕ y) ⟩
   fromℕ m + fromℕ n     ≡⟨ sym (fromℕ-homo-+ m n) ⟩
-  fromℕ (m ℕ.+ n)       ≤⟨ fromℕ-mono-≤ (ℕₚ.+-mono-≤ m≤m′ n≤n′) ⟩
+  fromℕ (m ℕ.+ n)       ≤⟨ fromℕ-mono-≤ (ℕ.+-mono-≤ m≤m′ n≤n′) ⟩
   fromℕ (m′ ℕ.+ n′)     ≡⟨ fromℕ-homo-+ m′ n′ ⟩
   fromℕ m′ + fromℕ n′   ≡⟨ cong₂ _+_ (fromℕ-toℕ x′) (fromℕ-toℕ y′) ⟩
   x′ + y′               ∎
@@ -842,7 +842,7 @@ module Bin+CSemigroup = CommSemigProp +-commutativeSemigroup
 +-mono-<-≤ {x} {x′} {y} {y′} x<x′ y≤y′ =  begin-strict
   x + y                  ≡⟨ sym $ cong₂ _+_ (fromℕ-toℕ x) (fromℕ-toℕ y) ⟩
   fromℕ m + fromℕ n      ≡⟨ sym (fromℕ-homo-+ m n) ⟩
-  fromℕ (m ℕ.+ n)        <⟨ fromℕ-mono-< (ℕₚ.+-mono-<-≤ m<m′ n≤n′) ⟩
+  fromℕ (m ℕ.+ n)        <⟨ fromℕ-mono-< (ℕ.+-mono-<-≤ m<m′ n≤n′) ⟩
   fromℕ (m′ ℕ.+ n′)      ≡⟨ fromℕ-homo-+ m′ n′ ⟩
   fromℕ m′ + fromℕ n′    ≡⟨ cong₂ _+_ (fromℕ-toℕ x′) (fromℕ-toℕ y′) ⟩
   x′ + y′                ∎
@@ -880,7 +880,7 @@ x≤x+y x y =  begin
 x<x+y : ∀ x {y} → y > 0ᵇ → x < x + y
 x<x+y x {y} y>0 = begin-strict
   x                             ≡⟨ sym (fromℕ-toℕ x) ⟩
-  fromℕ (toℕ x)                 <⟨ fromℕ-mono-< (ℕₚ.m<m+n (toℕ x) (toℕ-mono-< y>0)) ⟩
+  fromℕ (toℕ x)                 <⟨ fromℕ-mono-< (ℕ.m<m+n (toℕ x) (toℕ-mono-< y>0)) ⟩
   fromℕ (toℕ x ℕ.+ toℕ y)       ≡⟨ fromℕ-homo-+ (toℕ x) (toℕ y) ⟩
   fromℕ (toℕ x) + fromℕ (toℕ y) ≡⟨ cong₂ _+_ (fromℕ-toℕ x) (fromℕ-toℕ y) ⟩
   x + y                         ∎
@@ -910,12 +910,12 @@ x≢0⇒x+y≢0 {zero}     _    0≢0 =  contradiction refl 0≢0
 private  2*ₙ2*ₙ =  (2 ℕ.*_) ∘ (2 ℕ.*_)
 
 toℕ-homo-* : ∀ x y → toℕ (x * y) ≡ toℕ x ℕ.* toℕ y
-toℕ-homo-* x y =  aux x y (size x ℕ.+ size y) ℕₚ.≤-refl
+toℕ-homo-* x y =  aux x y (size x ℕ.+ size y) ℕ.≤-refl
   where
   aux : (x y : ℕᵇ) → (cnt : ℕ) → (size x ℕ.+ size y ℕ.≤ cnt) →  toℕ (x * y) ≡ toℕ x ℕ.* toℕ y
   aux zero     _        _ _ = refl
-  aux 2[1+ x ] zero     _ _ = sym (ℕₚ.*-zeroʳ (toℕ x ℕ.+ (ℕ.suc (toℕ x ℕ.+ 0))))
-  aux 1+[2 x ] zero     _ _ = sym (ℕₚ.*-zeroʳ (toℕ x ℕ.+ (toℕ x ℕ.+ 0)))
+  aux 2[1+ x ] zero     _ _ = sym (ℕ.*-zeroʳ (toℕ x ℕ.+ (ℕ.suc (toℕ x ℕ.+ 0))))
+  aux 1+[2 x ] zero     _ _ = sym (ℕ.*-zeroʳ (toℕ x ℕ.+ (toℕ x ℕ.+ 0)))
   aux 2[1+ x ] 2[1+ y ] (ℕ.suc cnt) (s≤s |x|+1+|y|≤cnt) = begin
     toℕ (2[1+ x ] * 2[1+ y ])                ≡⟨⟩
     toℕ (double 2[1+ (x + (y + xy)) ])       ≡⟨ toℕ-double 2[1+ (x + (y + xy)) ] ⟩
@@ -932,7 +932,7 @@ toℕ-homo-* x y =  aux x y (size x ℕ.+ size y) ℕₚ.≤-refl
     where
     open ≡-Reasoning;  m = toℕ x;  n = toℕ y;  xy = x * y
 
-    |x|+|y|≤cnt = ℕₚ.≤-trans (ℕₚ.+-monoʳ-≤ (size x) (ℕₚ.n≤1+n (size y))) |x|+1+|y|≤cnt
+    |x|+|y|≤cnt = ℕ.≤-trans (ℕ.+-monoʳ-≤ (size x) (ℕ.n≤1+n (size y))) |x|+1+|y|≤cnt
 
   aux 2[1+ x ] 1+[2 y ] (ℕ.suc cnt) (s≤s |x|+1+|y|≤cnt) = begin
     toℕ (2[1+ x ] * 1+[2 y ])                  ≡⟨⟩
@@ -980,13 +980,13 @@ toℕ-homo-* x y =  aux x y (size x ℕ.+ size y) ℕₚ.≤-refl
     ℕ.suc (2 ℕ.* (toℕ (x + y * 1+2x)))      ≡⟨ cong (ℕ.suc ∘ (2 ℕ.*_)) (toℕ-homo-+ x (y * 1+2x)) ⟩
     ℕ.suc (2 ℕ.* (m ℕ.+ (toℕ (y * 1+2x))))  ≡⟨ cong (ℕ.suc ∘ (2 ℕ.*_) ∘ (m ℕ.+_))
                                                  (aux y 1+2x cnt |y|+1+|x|≤cnt) ⟩
-    ℕ.suc (2 ℕ.* (m ℕ.+ (n ℕ.* [1+2x]′)))   ≡⟨ cong ℕ.suc $ ℕₚ.*-distribˡ-+ 2 m (n ℕ.* [1+2x]′) ⟩
-    ℕ.suc (2m ℕ.+ (2 ℕ.* (n ℕ.* [1+2x]′)))  ≡⟨ cong (ℕ.suc ∘ (2m ℕ.+_)) (sym (ℕₚ.*-assoc 2 n _)) ⟩
+    ℕ.suc (2 ℕ.* (m ℕ.+ (n ℕ.* [1+2x]′)))   ≡⟨ cong ℕ.suc $ ℕ.*-distribˡ-+ 2 m (n ℕ.* [1+2x]′) ⟩
+    ℕ.suc (2m ℕ.+ (2 ℕ.* (n ℕ.* [1+2x]′)))  ≡⟨ cong (ℕ.suc ∘ (2m ℕ.+_)) (sym (ℕ.*-assoc 2 n _)) ⟩
     (ℕ.suc 2m) ℕ.+ 2n ℕ.* [1+2x]′           ≡⟨⟩
     [1+2x]′ ℕ.+ 2n ℕ.* [1+2x]′              ≡⟨ cong (ℕ._+ (2n ℕ.* [1+2x]′)) $
-                                                    sym (ℕₚ.*-identityˡ [1+2x]′) ⟩
-    1 ℕ.* [1+2x]′ ℕ.+ 2n ℕ.* [1+2x]′        ≡⟨ sym (ℕₚ.*-distribʳ-+ [1+2x]′ 1 2n) ⟩
-    (ℕ.suc 2n) ℕ.* [1+2x]′                  ≡⟨ ℕₚ.*-comm (ℕ.suc 2n) [1+2x]′ ⟩
+                                                    sym (ℕ.*-identityˡ [1+2x]′) ⟩
+    1 ℕ.* [1+2x]′ ℕ.+ 2n ℕ.* [1+2x]′        ≡⟨ sym (ℕ.*-distribʳ-+ [1+2x]′ 1 2n) ⟩
+    (ℕ.suc 2n) ℕ.* [1+2x]′                  ≡⟨ ℕ.*-comm (ℕ.suc 2n) [1+2x]′ ⟩
     toℕ 1+[2 x ] ℕ.* toℕ 1+[2 y ]           ∎
     where
     open ≡-Reasoning
@@ -999,19 +999,19 @@ toℕ-homo-* x y =  aux x y (size x ℕ.+ size y) ℕₚ.≤-refl
     |y|+1+|x|≤cnt = subst (ℕ._≤ cnt) eq |x|+1+|y|≤cnt
 
 
-toℕ-isMagmaHomomorphism-* : IsMagmaHomomorphism *-rawMagma ℕᵇ.*-rawMagma toℕ
+toℕ-isMagmaHomomorphism-* : IsMagmaHomomorphism *-rawMagma ℕ.*-rawMagma toℕ
 toℕ-isMagmaHomomorphism-* = record
   { isRelHomomorphism = toℕ-isRelHomomorphism
   ; homo              = toℕ-homo-*
   }
 
-toℕ-isMonoidHomomorphism-* : IsMonoidHomomorphism *-1-rawMonoid ℕᵇ.*-1-rawMonoid toℕ
+toℕ-isMonoidHomomorphism-* : IsMonoidHomomorphism *-1-rawMonoid ℕ.*-1-rawMonoid toℕ
 toℕ-isMonoidHomomorphism-* = record
   { isMagmaHomomorphism = toℕ-isMagmaHomomorphism-*
   ; ε-homo              = refl
   }
 
-toℕ-isMonoidMonomorphism-* : IsMonoidMonomorphism *-1-rawMonoid ℕᵇ.*-1-rawMonoid toℕ
+toℕ-isMonoidMonomorphism-* : IsMonoidMonomorphism *-1-rawMonoid ℕ.*-1-rawMonoid toℕ
 toℕ-isMonoidMonomorphism-* = record
   { isMonoidHomomorphism = toℕ-isMonoidHomomorphism-*
   ; injective            = toℕ-injective
@@ -1038,13 +1038,13 @@ private
 -- by `toℕ`/`fromℕ`.
 
 *-assoc : Associative _*_
-*-assoc = *-Monomorphism.assoc ℕₚ.*-isMagma ℕₚ.*-assoc
+*-assoc = *-Monomorphism.assoc ℕ.*-isMagma ℕ.*-assoc
 
 *-comm : Commutative _*_
-*-comm = *-Monomorphism.comm ℕₚ.*-isMagma ℕₚ.*-comm
+*-comm = *-Monomorphism.comm ℕ.*-isMagma ℕ.*-comm
 
 *-identityˡ : LeftIdentity 1ᵇ _*_
-*-identityˡ = *-Monomorphism.identityˡ ℕₚ.*-isMagma ℕₚ.*-identityˡ
+*-identityˡ = *-Monomorphism.identityˡ ℕ.*-isMagma ℕ.*-identityˡ
 
 *-identityʳ : RightIdentity 1ᵇ _*_
 *-identityʳ x =  trans (*-comm x 1ᵇ) (*-identityˡ x)
@@ -1068,7 +1068,7 @@ private
   a * (b + c)                          ≡⟨ sym (fromℕ-toℕ (a * (b + c))) ⟩
   fromℕ (toℕ (a * (b + c)))            ≡⟨ cong fromℕ (toℕ-homo-* a (b + c)) ⟩
   fromℕ (k ℕ.* (toℕ (b + c)))          ≡⟨ cong (fromℕ ∘ (k ℕ.*_)) (toℕ-homo-+ b c) ⟩
-  fromℕ (k ℕ.* (m ℕ.+ n))              ≡⟨ cong fromℕ (ℕₚ.*-distribˡ-+ k m n) ⟩
+  fromℕ (k ℕ.* (m ℕ.+ n))              ≡⟨ cong fromℕ (ℕ.*-distribˡ-+ k m n) ⟩
   fromℕ (k ℕ.* m ℕ.+ k ℕ.* n)          ≡⟨ cong fromℕ $ sym $
                                             cong₂ ℕ._+_ (toℕ-homo-* a b) (toℕ-homo-* a c) ⟩
   fromℕ (toℕ (a * b) ℕ.+ toℕ (a * c))  ≡⟨ cong fromℕ (sym (toℕ-homo-+ (a * b) (a * c))) ⟩
@@ -1089,13 +1089,13 @@ private
 *-isMagma = isMagma _*_
 
 *-isSemigroup : IsSemigroup _*_
-*-isSemigroup = *-Monomorphism.isSemigroup ℕₚ.*-isSemigroup
+*-isSemigroup = *-Monomorphism.isSemigroup ℕ.*-isSemigroup
 
 *-1-isMonoid : IsMonoid _*_ 1ᵇ
-*-1-isMonoid = *-Monomorphism.isMonoid ℕₚ.*-1-isMonoid
+*-1-isMonoid = *-Monomorphism.isMonoid ℕ.*-1-isMonoid
 
 *-1-isCommutativeMonoid : IsCommutativeMonoid _*_ 1ᵇ
-*-1-isCommutativeMonoid = *-Monomorphism.isCommutativeMonoid ℕₚ.*-1-isCommutativeMonoid
+*-1-isCommutativeMonoid = *-Monomorphism.isCommutativeMonoid ℕ.*-1-isCommutativeMonoid
 
 +-*-isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _+_ _*_ zero 1ᵇ
 +-*-isSemiringWithoutAnnihilatingZero = record
@@ -1157,10 +1157,10 @@ private
 *-mono-≤ : _*_ Preserves₂ _≤_ ⟶ _≤_ ⟶ _≤_
 *-mono-≤ {x} {u} {y} {v} x≤u y≤v = toℕ-cancel-≤ (begin
   toℕ (x * y)      ≡⟨ toℕ-homo-* x y ⟩
-  toℕ x ℕ.* toℕ y  ≤⟨ ℕₚ.*-mono-≤ (toℕ-mono-≤ x≤u) (toℕ-mono-≤ y≤v) ⟩
+  toℕ x ℕ.* toℕ y  ≤⟨ ℕ.*-mono-≤ (toℕ-mono-≤ x≤u) (toℕ-mono-≤ y≤v) ⟩
   toℕ u ℕ.* toℕ v  ≡⟨ sym (toℕ-homo-* u v) ⟩
   toℕ (u * v)      ∎)
-  where open ℕₚ.≤-Reasoning
+  where open ℕ.≤-Reasoning
 
 *-monoʳ-≤ : ∀ x → (x *_) Preserves _≤_ ⟶ _≤_
 *-monoʳ-≤ x y≤y′ = *-mono-≤ (≤-refl {x}) y≤y′
@@ -1171,10 +1171,10 @@ private
 *-mono-< : _*_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
 *-mono-< {x} {u} {y} {v} x<u y<v = toℕ-cancel-< (begin-strict
   toℕ (x * y)      ≡⟨ toℕ-homo-* x y ⟩
-  toℕ x ℕ.* toℕ y  <⟨ ℕₚ.*-mono-< (toℕ-mono-< x<u) (toℕ-mono-< y<v) ⟩
+  toℕ x ℕ.* toℕ y  <⟨ ℕ.*-mono-< (toℕ-mono-< x<u) (toℕ-mono-< y<v) ⟩
   toℕ u ℕ.* toℕ v  ≡⟨ sym (toℕ-homo-* u v) ⟩
   toℕ (u * v)      ∎)
-  where open ℕₚ.≤-Reasoning
+  where open ℕ.≤-Reasoning
 
 *-monoʳ-< : ∀ x → ((1ᵇ + x) *_) Preserves _<_ ⟶ _<_
 *-monoʳ-< x {y} {z} y<z = begin-strict
@@ -1455,7 +1455,7 @@ pred-mono-≤ : pred Preserves _≤_ ⟶ _≤_
 pred-mono-≤ {x} {y} x≤y = begin
   pred x             ≡⟨ cong pred (sym (fromℕ-toℕ x)) ⟩
   pred (fromℕ m)     ≡⟨ sym (fromℕ-pred m) ⟩
-  fromℕ (ℕ.pred m)   ≤⟨ fromℕ-mono-≤ (ℕₚ.pred-mono-≤ (toℕ-mono-≤ x≤y)) ⟩
+  fromℕ (ℕ.pred m)   ≤⟨ fromℕ-mono-≤ (ℕ.pred-mono-≤ (toℕ-mono-≤ x≤y)) ⟩
   fromℕ (ℕ.pred n)   ≡⟨ fromℕ-pred n ⟩
   pred (fromℕ n)     ≡⟨ cong pred (fromℕ-toℕ y) ⟩
   pred y             ∎
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index a4d0ba125d..079f138ffa 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -15,7 +15,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.Nat as ℕ using (ℕ)
 open import Data.Nat.Binary.Base
 open import Data.Nat.Binary.Properties
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Vec.Base using ([]; _∷_)
@@ -77,39 +77,39 @@ toℕ-homo-∸ 2[1+ x ] zero     = refl
 toℕ-homo-∸ 2[1+ x ] 2[1+ y ] = begin
   toℕ (double (x ∸ y))          ≡⟨ toℕ-double (x ∸ y) ⟩
   2 ℕ.* toℕ (x ∸ y)             ≡⟨ cong (2 ℕ.*_) (toℕ-homo-∸ x y) ⟩
-  2 ℕ.* (toℕ x ℕ.∸ toℕ y)       ≡⟨ ℕₚ.*-distribˡ-∸ 2 (ℕ.suc (toℕ x)) (ℕ.suc (toℕ y)) ⟩
+  2 ℕ.* (toℕ x ℕ.∸ toℕ y)       ≡⟨ ℕ.*-distribˡ-∸ 2 (ℕ.suc (toℕ x)) (ℕ.suc (toℕ y)) ⟩
   toℕ 2[1+ x ] ℕ.∸ toℕ 2[1+ y ] ∎
   where open ≡-Reasoning
 toℕ-homo-∸ 2[1+ x ] 1+[2 y ] with x <? y
-... | yes x<y  = sym (ℕₚ.m≤n⇒m∸n≡0 (toℕ-mono-≤ (inj₁ (even<odd x<y))))
+... | yes x<y  = sym (ℕ.m≤n⇒m∸n≡0 (toℕ-mono-≤ (inj₁ (even<odd x<y))))
 ... | no  x≮y  = begin
   ℕ.suc (2 ℕ.* toℕ (x ∸ y))                     ≡⟨ cong (ℕ.suc ∘ (2 ℕ.*_)) (toℕ-homo-∸ x y) ⟩
-  ℕ.suc (2 ℕ.* (toℕ x ℕ.∸ toℕ y))               ≡⟨ cong ℕ.suc (ℕₚ.*-distribˡ-∸ 2 (toℕ x) (toℕ y)) ⟩
-  ℕ.suc (2 ℕ.* toℕ x ℕ.∸ 2 ℕ.* toℕ y)           ≡⟨ sym (ℕₚ.+-∸-assoc 1 (ℕₚ.*-monoʳ-≤ 2 (toℕ-mono-≤ (≮⇒≥ x≮y)))) ⟩
-  ℕ.suc (2 ℕ.* toℕ x) ℕ.∸ 2 ℕ.* toℕ y           ≡⟨ sym (cong (ℕ._∸ 2 ℕ.* toℕ y) (ℕₚ.+-suc (toℕ x) (1 ℕ.* toℕ x))) ⟩
+  ℕ.suc (2 ℕ.* (toℕ x ℕ.∸ toℕ y))               ≡⟨ cong ℕ.suc (ℕ.*-distribˡ-∸ 2 (toℕ x) (toℕ y)) ⟩
+  ℕ.suc (2 ℕ.* toℕ x ℕ.∸ 2 ℕ.* toℕ y)           ≡⟨ sym (ℕ.+-∸-assoc 1 (ℕ.*-monoʳ-≤ 2 (toℕ-mono-≤ (≮⇒≥ x≮y)))) ⟩
+  ℕ.suc (2 ℕ.* toℕ x) ℕ.∸ 2 ℕ.* toℕ y           ≡⟨ sym (cong (ℕ._∸ 2 ℕ.* toℕ y) (ℕ.+-suc (toℕ x) (1 ℕ.* toℕ x))) ⟩
   2 ℕ.* (ℕ.suc (toℕ x)) ℕ.∸ ℕ.suc (2 ℕ.* toℕ y) ∎
   where open ≡-Reasoning
 toℕ-homo-∸ 1+[2 x ] zero     = refl
 toℕ-homo-∸ 1+[2 x ] 2[1+ y ] with x ≤? y
-... | yes x≤y = sym (ℕₚ.m≤n⇒m∸n≡0 (toℕ-mono-≤ (inj₁ (odd<even x≤y))))
+... | yes x≤y = sym (ℕ.m≤n⇒m∸n≡0 (toℕ-mono-≤ (inj₁ (odd<even x≤y))))
 ... | no  _   = begin
   toℕ (pred (double (x ∸ y)))                   ≡⟨ toℕ-pred (double (x ∸ y)) ⟩
   ℕ.pred (toℕ (double (x ∸ y)))                 ≡⟨ cong ℕ.pred (toℕ-double (x ∸ y)) ⟩
   ℕ.pred (2 ℕ.* toℕ (x ∸ y))                    ≡⟨ cong (ℕ.pred ∘ (2 ℕ.*_)) (toℕ-homo-∸ x y) ⟩
-  ℕ.pred (2 ℕ.* (toℕ x ℕ.∸ toℕ y))              ≡⟨ cong ℕ.pred (ℕₚ.*-distribˡ-∸ 2 (toℕ x) (toℕ y)) ⟩
-  ℕ.pred (2 ℕ.* toℕ x ℕ.∸ 2 ℕ.* toℕ y)          ≡⟨ ℕₚ.pred[m∸n]≡m∸[1+n] (2 ℕ.* toℕ x) (2 ℕ.* toℕ y) ⟩
-  2 ℕ.* toℕ x ℕ.∸ ℕ.suc (2 ℕ.* toℕ y)           ≡⟨ sym (cong (2 ℕ.* toℕ x ℕ.∸_) (ℕₚ.+-suc (toℕ y) (1 ℕ.* toℕ y))) ⟩
+  ℕ.pred (2 ℕ.* (toℕ x ℕ.∸ toℕ y))              ≡⟨ cong ℕ.pred (ℕ.*-distribˡ-∸ 2 (toℕ x) (toℕ y)) ⟩
+  ℕ.pred (2 ℕ.* toℕ x ℕ.∸ 2 ℕ.* toℕ y)          ≡⟨ ℕ.pred[m∸n]≡m∸[1+n] (2 ℕ.* toℕ x) (2 ℕ.* toℕ y) ⟩
+  2 ℕ.* toℕ x ℕ.∸ ℕ.suc (2 ℕ.* toℕ y)           ≡⟨ sym (cong (2 ℕ.* toℕ x ℕ.∸_) (ℕ.+-suc (toℕ y) (1 ℕ.* toℕ y))) ⟩
   ℕ.suc (2 ℕ.* toℕ x) ℕ.∸ 2 ℕ.* (ℕ.suc (toℕ y)) ∎
   where open ≡-Reasoning
 toℕ-homo-∸ 1+[2 x ] 1+[2 y ] = begin
   toℕ (double (x ∸ y))        ≡⟨ toℕ-double (x ∸ y) ⟩
   2 ℕ.* toℕ (x ∸ y)           ≡⟨ cong (2 ℕ.*_) (toℕ-homo-∸ x y) ⟩
-  2 ℕ.* (toℕ x ℕ.∸ toℕ y)     ≡⟨ ℕₚ.*-distribˡ-∸ 2 (toℕ x) (toℕ y) ⟩
+  2 ℕ.* (toℕ x ℕ.∸ toℕ y)     ≡⟨ ℕ.*-distribˡ-∸ 2 (toℕ x) (toℕ y) ⟩
   2 ℕ.* toℕ x ℕ.∸ 2 ℕ.* toℕ y ∎
   where open ≡-Reasoning
 
 fromℕ-homo-∸ : ∀ m n → fromℕ (m ℕ.∸ n) ≡ (fromℕ m) ∸ (fromℕ n)
-fromℕ-homo-∸ = homomorphic₂-inv ∸-magma ℕₚ.∸-magma
+fromℕ-homo-∸ = homomorphic₂-inv ∸-magma ℕ.∸-magma
   (cong fromℕ) toℕ-inverseᵇ toℕ-homo-∸
 
 ------------------------------------------------------------------------
@@ -126,13 +126,13 @@ odd∸even-for> {x} {y} x>y with x ≤? y
 ... | yes x≤y = contradiction x>y (≤⇒≯ x≤y)
 
 x≤y⇒x∸y≡0 : x ≤ y → x ∸ y ≡ 0ᵇ
-x≤y⇒x∸y≡0 {x} {y} = toℕ-injective ∘ trans (toℕ-homo-∸ x y) ∘ ℕₚ.m≤n⇒m∸n≡0 ∘ toℕ-mono-≤
+x≤y⇒x∸y≡0 {x} {y} = toℕ-injective ∘ trans (toℕ-homo-∸ x y) ∘ ℕ.m≤n⇒m∸n≡0 ∘ toℕ-mono-≤
 
 x∸y≡0⇒x≤y : x ∸ y ≡ 0ᵇ → x ≤ y
-x∸y≡0⇒x≤y {x} {y} = toℕ-cancel-≤ ∘ ℕₚ.m∸n≡0⇒m≤n ∘ trans (sym (toℕ-homo-∸ x y)) ∘ cong toℕ
+x∸y≡0⇒x≤y {x} {y} = toℕ-cancel-≤ ∘ ℕ.m∸n≡0⇒m≤n ∘ trans (sym (toℕ-homo-∸ x y)) ∘ cong toℕ
 
 x<y⇒y∸x>0 : x < y → y ∸ x > 0ᵇ
-x<y⇒y∸x>0 {x} {y} = toℕ-cancel-< ∘ subst (ℕ._> 0) (sym (toℕ-homo-∸ y x)) ∘ ℕₚ.m<n⇒0<n∸m ∘ toℕ-mono-<
+x<y⇒y∸x>0 {x} {y} = toℕ-cancel-< ∘ subst (ℕ._> 0) (sym (toℕ-homo-∸ y x)) ∘ ℕ.m<n⇒0<n∸m ∘ toℕ-mono-<
 
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _+_
@@ -141,7 +141,7 @@ x<y⇒y∸x>0 {x} {y} = toℕ-cancel-< ∘ subst (ℕ._> 0) (sym (toℕ-homo-∸
 [x∸y]+y≡x {x} {y} x≥y = toℕ-injective (begin
   toℕ (x ∸ y + y)             ≡⟨ toℕ-homo-+ (x ∸ y) y ⟩
   toℕ (x ∸ y) ℕ.+ toℕ y       ≡⟨ cong (ℕ._+ toℕ y) (toℕ-homo-∸ x y) ⟩
-  (toℕ x ℕ.∸ toℕ y) ℕ.+ toℕ y ≡⟨ ℕₚ.m∸n+n≡m (toℕ-mono-≤ x≥y) ⟩
+  (toℕ x ℕ.∸ toℕ y) ℕ.+ toℕ y ≡⟨ ℕ.m∸n+n≡m (toℕ-mono-≤ x≥y) ⟩
   toℕ x                       ∎)
   where open ≡-Reasoning
 
@@ -162,7 +162,7 @@ x+[y∸x]≡y {x} {y} x≤y = begin-equality
 ∸-+-assoc x y z = toℕ-injective $ begin
   toℕ ((x ∸ y) ∸ z)       ≡⟨ toℕ-homo-∸ (x ∸ y) z ⟩
   toℕ (x ∸ y) ℕ.∸ n       ≡⟨ cong (ℕ._∸ n) (toℕ-homo-∸ x y) ⟩
-  (k ℕ.∸ m) ℕ.∸ n         ≡⟨ ℕₚ.∸-+-assoc k m n ⟩
+  (k ℕ.∸ m) ℕ.∸ n         ≡⟨ ℕ.∸-+-assoc k m n ⟩
   k ℕ.∸ (m ℕ.+ n)         ≡⟨ cong (k ℕ.∸_) (sym (toℕ-homo-+ y z)) ⟩
   k ℕ.∸ (toℕ (y + z))     ≡⟨ sym (toℕ-homo-∸ x (y + z)) ⟩
   toℕ (x ∸ (y + z))       ∎
@@ -172,7 +172,7 @@ x+[y∸x]≡y {x} {y} x≤y = begin-equality
 +-∸-assoc x {y} {z} z≤y = toℕ-injective $ begin
   toℕ ((x + y) ∸ z)     ≡⟨ toℕ-homo-∸ (x + y) z ⟩
   (toℕ (x + y)) ℕ.∸ n   ≡⟨ cong (ℕ._∸ n) (toℕ-homo-+ x y) ⟩
-  (k ℕ.+ m) ℕ.∸ n       ≡⟨ ℕₚ.+-∸-assoc k n≤m ⟩
+  (k ℕ.+ m) ℕ.∸ n       ≡⟨ ℕ.+-∸-assoc k n≤m ⟩
   k ℕ.+ (m ℕ.∸ n)       ≡⟨ cong (k ℕ.+_) (sym (toℕ-homo-∸ y z)) ⟩
   k ℕ.+ toℕ (y ∸ z)     ≡⟨ sym (toℕ-homo-+ x (y ∸ z)) ⟩
   toℕ (x + (y ∸ z))     ∎
@@ -196,10 +196,10 @@ suc[x]∸suc[y] x y = begin-equality
 ∸-mono-≤ : _∸_ Preserves₂ _≤_ ⟶ _≥_ ⟶ _≤_
 ∸-mono-≤ {x} {y} {u} {v} x≤y v≤u = toℕ-cancel-≤ (begin
   toℕ (x ∸ u)      ≡⟨ toℕ-homo-∸ x u ⟩
-  toℕ x ℕ.∸ toℕ u  ≤⟨ ℕₚ.∸-mono (toℕ-mono-≤ x≤y) (toℕ-mono-≤ v≤u) ⟩
+  toℕ x ℕ.∸ toℕ u  ≤⟨ ℕ.∸-mono (toℕ-mono-≤ x≤y) (toℕ-mono-≤ v≤u) ⟩
   toℕ y ℕ.∸ toℕ v  ≡⟨ sym (toℕ-homo-∸ y v) ⟩
   toℕ (y ∸ v)      ∎)
-  where open ℕₚ.≤-Reasoning
+  where open ℕ.≤-Reasoning
 
 ∸-monoˡ-≤ : (x : ℕᵇ) → (_∸ x) Preserves _≤_ ⟶ _≤_
 ∸-monoˡ-≤ x y≤z = ∸-mono-≤ y≤z (≤-refl {x})
@@ -230,7 +230,7 @@ x∸y<x {x} {y} x≢0 y≢0 = begin-strict
 *-distribˡ-∸ x y z = toℕ-injective $ begin
   toℕ (x * (y ∸ z))              ≡⟨ toℕ-homo-* x (y ∸ z) ⟩
   k ℕ.* (toℕ (y ∸ z))            ≡⟨ cong (k ℕ.*_) (toℕ-homo-∸ y z) ⟩
-  k ℕ.* (m ℕ.∸ n)                ≡⟨ ℕₚ.*-distribˡ-∸ k m n ⟩
+  k ℕ.* (m ℕ.∸ n)                ≡⟨ ℕ.*-distribˡ-∸ k m n ⟩
   (k ℕ.* m) ℕ.∸ (k ℕ.* n)        ≡⟨ cong₂ ℕ._∸_ (sym (toℕ-homo-* x y)) (sym (toℕ-homo-* x z)) ⟩
   toℕ (x * y) ℕ.∸ toℕ (x * z)    ≡⟨ sym (toℕ-homo-∸ (x * y) (x * z)) ⟩
   toℕ ((x * y) ∸ (x * z))        ∎
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 0948447163..32fdfc3ae0 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -14,7 +14,7 @@ open import Data.Integer.Base as ℤ
   using (ℤ; +_; +0; +[1+_]; -[1+_])
   hiding (module ℤ)
 open import Data.Nat.GCD
-open import Data.Nat.Coprimality as C
+open import Data.Nat.Coprimality as ℕ
   using (Coprime; Bézout-coprime; coprime-/gcd; coprime?; ¬0-coprimeTo-2+)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; 2+) hiding (module ℕ)
 open import Data.Rational.Unnormalised.Base as ℚᵘ using (ℚᵘ; mkℚᵘ)
@@ -193,7 +193,7 @@ open ℤ public
 ≢-nonZero {mkℚ +[1+ _ ] _         _} _   = _
 ≢-nonZero {mkℚ +0       zero      _} p≢0 = contradiction refl p≢0
 ≢-nonZero {mkℚ +0       d@(suc m) c} p≢0 =
-  contradiction (λ {d} → C.recompute c {d}) (¬0-coprimeTo-2+ {{ℕ.nonTrivial {m}}})
+  contradiction (λ {d} → ℕ.recompute c {d}) (¬0-coprimeTo-2+ {{ℕ.nonTrivial {m}}})
 
 >-nonZero : ∀ {p} → p > 0ℚ → NonZero p
 >-nonZero {p@(mkℚ _ _ _)} (*<* p<q) = ℚᵘ.>-nonZero {toℚᵘ p} (ℚᵘ.*<* p<q)
@@ -237,8 +237,8 @@ p - q = p + (- q)
 
 -- reciprocal: requires a proof that the numerator is not zero
 1/_ : (p : ℚ) → .{{_ : NonZero p}} → ℚ
-1/ mkℚ +[1+ n ] d prf = mkℚ +[1+ d ] n (C.sym prf)
-1/ mkℚ -[1+ n ] d prf = mkℚ -[1+ d ] n (C.sym prf)
+1/ mkℚ +[1+ n ] d prf = mkℚ +[1+ d ] n (ℕ.sym prf)
+1/ mkℚ -[1+ n ] d prf = mkℚ -[1+ d ] n (ℕ.sym prf)
 
 -- division: requires a proof that the denominator is not zero
 _÷_ : (p q : ℚ) → .{{_ : NonZero q}} → ℚ
diff --git a/src/Data/String.agda b/src/Data/String.agda
index f8ca8385e4..e92054a673 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -12,9 +12,9 @@ open import Data.Bool.Base using (if_then_else_)
 open import Data.Char as Char using (Char)
 open import Function.Base using (_∘_; _$_)
 open import Data.Nat.Base as ℕ using (ℕ)
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.List.Base as List using (List)
-open import Data.List.Extrema ℕₚ.≤-totalOrder
+open import Data.List.Extrema ℕ.≤-totalOrder
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Char.Base as Char using (Char)
 import Data.Char.Properties as Char using (_≟_)
diff --git a/src/Data/Vec.agda b/src/Data/Vec.agda
index 62b16e0cc6..2e293acdf9 100644
--- a/src/Data/Vec.agda
+++ b/src/Data/Vec.agda
@@ -18,7 +18,7 @@ module Data.Vec where
 
 open import Level
 open import Data.Bool.Base
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Vec.Bounded.Base as Vec≤
   using (Vec≤; ≤-cast; fromVec)
 open import Relation.Nullary
@@ -43,7 +43,7 @@ module _ {P : A → Set p} (P? : Decidable P) where
   filter []       = Vec≤.[]
   filter (a ∷ as) with does (P? a)
   ... | true  = a Vec≤.∷ filter as
-  ... | false = ≤-cast (ℕₚ.n≤1+n _) (filter as)
+  ... | false = ≤-cast (ℕ.n≤1+n _) (filter as)
 
   takeWhile : ∀ {n} → Vec A n → Vec≤ A n
   takeWhile []       = Vec≤.[]
@@ -54,5 +54,5 @@ module _ {P : A → Set p} (P? : Decidable P) where
   dropWhile : ∀ {n} → Vec A n → Vec≤ A n
   dropWhile Vec.[]       = Vec≤.[]
   dropWhile (a Vec.∷ as) with does (P? a)
-  ... | true  = ≤-cast (ℕₚ.n≤1+n _) (dropWhile as)
+  ... | true  = ≤-cast (ℕ.n≤1+n _) (dropWhile as)
   ... | false = fromVec (a Vec.∷ as)
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index d1d5f8c628..39d9e25abb 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -11,13 +11,13 @@ module Data.Vec.Functional.Properties where
 open import Data.Empty using (⊥-elim)
 open import Data.Fin.Base
 open import Data.Nat as ℕ
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
-open import Data.List.Base as L using (List)
-import Data.List.Properties as Lₚ
+open import Data.List.Base as List using (List)
+import Data.List.Properties as List
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
-open import Data.Vec.Base as V using (Vec)
-import Data.Vec.Properties as Vₚ
+open import Data.Vec.Base as Vec using (Vec)
+import Data.Vec.Properties as Vec
 open import Data.Vec.Functional
 open import Function.Base
 open import Level using (Level)
@@ -177,7 +177,7 @@ module _ {ys ys′ : Vector A m} where
 
   ++-cong : ∀ (xs xs′ : Vector A n) →
             xs ≗ xs′ → ys ≗ ys′ → xs ++ ys ≗ xs′ ++ ys′
-  ++-cong {n} xs xs′ eq₁ eq₂ i with toℕ i ℕₚ.<? n
+  ++-cong {n} xs xs′ eq₁ eq₂ i with toℕ i ℕ.<? n
   ... | yes i<n = begin
     (xs ++ ys) i      ≡⟨ lookup-++-< xs ys i i<n ⟩
     xs (fromℕ< i<n)   ≡⟨ eq₁ (fromℕ< i<n) ⟩
@@ -185,9 +185,9 @@ module _ {ys ys′ : Vector A m} where
     (xs′ ++ ys′) i    ∎
     where open ≡-Reasoning
   ... | no i≮n = begin
-    (xs ++ ys) i               ≡⟨ lookup-++-≥ xs ys i (ℕₚ.≮⇒≥ i≮n) ⟩
-    ys (reduce≥ i (ℕₚ.≮⇒≥ i≮n))   ≡⟨ eq₂ (reduce≥ i (ℕₚ.≮⇒≥ i≮n)) ⟩
-    ys′ (reduce≥ i (ℕₚ.≮⇒≥ i≮n))  ≡⟨ lookup-++-≥ xs′ ys′ i (ℕₚ.≮⇒≥ i≮n) ⟨
+    (xs ++ ys) i               ≡⟨ lookup-++-≥ xs ys i (ℕ.≮⇒≥ i≮n) ⟩
+    ys (reduce≥ i (ℕ.≮⇒≥ i≮n))   ≡⟨ eq₂ (reduce≥ i (ℕ.≮⇒≥ i≮n)) ⟩
+    ys′ (reduce≥ i (ℕ.≮⇒≥ i≮n))  ≡⟨ lookup-++-≥ xs′ ys′ i (ℕ.≮⇒≥ i≮n) ⟨
     (xs′ ++ ys′) i             ∎
     where open ≡-Reasoning
 
@@ -255,13 +255,13 @@ insertAt-removeAt {n = suc n} xs (suc i) (suc j) = insertAt-removeAt (tail xs) i
 -- Conversion functions
 
 toVec∘fromVec : (xs : Vec A n) → toVec (fromVec xs) ≡ xs
-toVec∘fromVec = Vₚ.tabulate∘lookup
+toVec∘fromVec = Vec.tabulate∘lookup
 
 fromVec∘toVec : (xs : Vector A n) → fromVec (toVec xs) ≗ xs
-fromVec∘toVec = Vₚ.lookup∘tabulate
+fromVec∘toVec = Vec.lookup∘tabulate
 
 toList∘fromList : (xs : List A) → toList (fromList xs) ≡ xs
-toList∘fromList = Lₚ.tabulate-lookup
+toList∘fromList = List.tabulate-lookup
 
 ------------------------------------------------------------------------
 -- DEPRECATED NAMES
diff --git a/src/Data/Word/Properties.agda b/src/Data/Word/Properties.agda
index 3f9ad94ef1..e0c2308909 100644
--- a/src/Data/Word/Properties.agda
+++ b/src/Data/Word/Properties.agda
@@ -11,7 +11,7 @@ module Data.Word.Properties where
 import Data.Nat.Base as ℕ
 open import Data.Bool.Base using (Bool)
 open import Data.Word.Base
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Function.Base
 open import Relation.Nullary.Decidable using (map′; ⌊_⌋)
 open import Relation.Binary
@@ -51,7 +51,7 @@ open import Agda.Builtin.Word.Properties
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-x ≈? y = toℕ x ℕₚ.≟ toℕ y
+x ≈? y = toℕ x ℕ.≟ toℕ y
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -100,7 +100,7 @@ w₁ == w₂ = ⌊ w₁ ≟ w₂ ⌋
 
 infix 4 _<?_
 _<?_ : Decidable _<_
-_<?_ = On.decidable toℕ ℕ._<_ ℕₚ._<?_
+_<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
 
 <-strictTotalOrder-≈ : StrictTotalOrder _ _ _
-<-strictTotalOrder-≈ = On.strictTotalOrder ℕₚ.<-strictTotalOrder toℕ
+<-strictTotalOrder-≈ = On.strictTotalOrder ℕ.<-strictTotalOrder toℕ
diff --git a/src/Induction/Lexicographic.agda b/src/Induction/Lexicographic.agda
index 6acd96b7f1..d70b989acd 100644
--- a/src/Induction/Lexicographic.agda
+++ b/src/Induction/Lexicographic.agda
@@ -68,13 +68,13 @@ RecA ⊗ RecB = Σ-Rec RecA (λ _ → RecB)
 private
 
   open import Data.Nat.Base
-  open import Data.Nat.Induction as N
+  open import Data.Nat.Induction as ℕ
 
   -- The Ackermann function à la Rózsa Péter.
 
   ackermann : ℕ → ℕ → ℕ
   ackermann m n =
-    build [ N.recBuilder ⊗ N.recBuilder ]
+    build [ ℕ.recBuilder ⊗ ℕ.recBuilder ]
           (λ _ → ℕ)
           (λ { (zero  , n)     _                   → 1 + n
              ; (suc m , zero)  (_         , ackm•) → ackm• 1
diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index b97fe47cae..3f06fef070 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -8,8 +8,8 @@
 
 module Reflection.AST.Definition where
 
-import Data.List.Properties as Listₚ                   using (≡-dec)
-import Data.Nat.Properties as ℕₚ                       using (_≟_)
+import Data.List.Properties as List                    using (≡-dec)
+import Data.Nat.Properties as ℕ                        using (_≟_)
 open import Data.Product.Base                          using (_×_; <_,_>; uncurry)
 open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_; yes; no)
 open import Relation.Binary.Definitions                using (DecidableEquality)
@@ -66,10 +66,10 @@ function cs       ≟ function cs′        =
   map′ (cong function) function-injective (cs Term.≟-Clauses cs′)
 data-type pars cs ≟ data-type pars′ cs′ =
   map′ (uncurry (cong₂ data-type)) data-type-injective
-           (pars ℕₚ.≟ pars′ ×-dec Listₚ.≡-dec Name._≟_ cs cs′)
+           (pars ℕ.≟ pars′ ×-dec List.≡-dec Name._≟_ cs cs′)
 record′ c fs      ≟ record′ c′ fs′      =
   map′ (uncurry (cong₂ record′)) record′-injective
-           (c Name.≟ c′ ×-dec Listₚ.≡-dec (Arg.≡-dec Name._≟_) fs fs′)
+           (c Name.≟ c′ ×-dec List.≡-dec (Arg.≡-dec Name._≟_) fs fs′)
 constructor′ d    ≟ constructor′ d′     =
   map′ (cong constructor′) constructor′-injective (d Name.≟ d′)
 axiom             ≟ axiom               = yes refl
diff --git a/src/Reflection/AST/Meta.agda b/src/Reflection/AST/Meta.agda
index babcd23670..a3cac80c9a 100644
--- a/src/Reflection/AST/Meta.agda
+++ b/src/Reflection/AST/Meta.agda
@@ -8,7 +8,7 @@
 
 module Reflection.AST.Meta where
 
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Function.Base                              using (_on_)
 open import Relation.Nullary.Decidable.Core            using (map′)
 open import Relation.Binary.Core                       using (Rel)
@@ -30,7 +30,7 @@ _≈_ : Rel Meta _
 _≈_ = _≡_ on toℕ
 
 _≈?_ : Decidable _≈_
-_≈?_ = On.decidable toℕ _≡_ ℕₚ._≟_
+_≈?_ = On.decidable toℕ _≡_ ℕ._≟_
 
 _≟_ : DecidableEquality Meta
 m ≟ n = map′ (toℕ-injective _ _) (cong toℕ) (m ≈? n)
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 2c1b4196b1..846fde2484 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -31,7 +31,7 @@ open import Data.Char.Base   as Char  using (toℕ)
 open import Data.Float.Base  as Float using (_≡ᵇ_)
 open import Data.List.Base   as List  using ([]; _∷_)
 open import Data.Maybe.Base  as Maybe using (Maybe; just; nothing)
-open import Data.Nat.Base    as Nat   using (ℕ; zero; suc; _≡ᵇ_; _+_)
+open import Data.Nat.Base    as ℕ     using (ℕ; zero; suc; _≡ᵇ_; _+_)
 open import Data.Unit.Base            using (⊤)
 open import Data.Word.Base   as Word  using (toℕ)
 open import Data.Product.Base         using (_×_; map₁; _,_)
@@ -62,7 +62,7 @@ open import Reflection.TCM.Syntax
 private
   -- Descend past a variable.
   varDescend : ℕ → ℕ → ℕ
-  varDescend ϕ x = if ϕ Nat.≤ᵇ x then suc x else x
+  varDescend ϕ x = if ϕ ℕ.≤ᵇ x then suc x else x
 
   -- Descend a variable underneath pattern variables.
   patternDescend : ℕ → Pattern → Pattern × ℕ
@@ -136,7 +136,7 @@ private
   antiUnifyClauses : ℕ → Clauses → Clauses → Maybe Clauses
   antiUnifyClause  : ℕ → Clause → Clause → Maybe Clause
 
-  antiUnify ϕ (var x args) (var y args') with x Nat.≡ᵇ y | antiUnifyArgs ϕ args args'
+  antiUnify ϕ (var x args) (var y args') with x ℕ.≡ᵇ y | antiUnifyArgs ϕ args args'
   ... | _     | nothing    = var ϕ []
   ... | false | just uargs = var ϕ uargs
   ... | true  | just uargs = var (varDescend ϕ x) uargs
@@ -158,13 +158,13 @@ private
     Π[ s ∶ arg i (antiUnify ϕ a a') ] antiUnify (suc ϕ) b b'
   antiUnify ϕ (sort (set t)) (sort (set t')) =
     sort (set (antiUnify ϕ t t'))
-  antiUnify ϕ (sort (lit n)) (sort (lit n')) with n Nat.≡ᵇ n'
+  antiUnify ϕ (sort (lit n)) (sort (lit n')) with n ℕ.≡ᵇ n'
   ... | true  = sort (lit n)
   ... | false = var ϕ []
-  antiUnify ϕ (sort (propLit n)) (sort (propLit n')) with n Nat.≡ᵇ n'
+  antiUnify ϕ (sort (propLit n)) (sort (propLit n')) with n ℕ.≡ᵇ n'
   ... | true  = sort (propLit n)
   ... | false = var ϕ []
-  antiUnify ϕ (sort (inf n)) (sort (inf n')) with n Nat.≡ᵇ n'
+  antiUnify ϕ (sort (inf n)) (sort (inf n')) with n ℕ.≡ᵇ n'
   ... | true  = sort (inf n)
   ... | false = var ϕ []
   antiUnify ϕ (sort unknown) (sort unknown) =
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
index bad78c4da2..e71ad07419 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
@@ -13,12 +13,12 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Addition
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Nat            as ℕ  using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
-open import Data.Nat.Properties as ℕₚ using (≤′-trans)
-open import Data.Product.Base         using (_,_; _×_; proj₂)
-open import Data.List.Base            using (_∷_; [])
+open import Data.Nat            as ℕ using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
+open import Data.Nat.Properties as ℕ using (≤′-trans)
+open import Data.Product.Base        using (_,_; _×_; proj₂)
+open import Data.List.Base           using (_∷_; [])
 open import Data.List.Kleene
-open import Data.Vec.Base             using (Vec)
+open import Data.Vec.Base            using (Vec)
 open import Function.Base using (_⟨_⟩_; flip)
 open import Relation.Unary
 
@@ -206,7 +206,7 @@ mutual
             ρ ^ suc j * ((ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ k)
           ≈⟨ pow-add _ _ k j ⟩
             (ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ (k ℕ.+ suc j)
-            ≡⟨ ≡.cong (λ i → (ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ i) (ℕₚ.+-comm k (suc j)) ⟩
+            ≡⟨ ≡.cong (λ i → (ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ i) (ℕ.+-comm k (suc j)) ⟩
             (ρ * ⅀?⟦ xs ⟧ (ρ , Ρ) + ⟦ poly x ⟧ Ρ) *⟨ ρ ⟩^ (suc j ℕ.+ k)
           ∎)
     ⟩
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
index 4f6607d663..58086538ed 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Exponentiation.agda
@@ -20,7 +20,7 @@ open import Data.Product.Base  using (_,_; _×_; proj₁; proj₂)
 open import Data.List.Kleene
 open import Data.Vec.Base      using (Vec)
 
-import Data.Nat.Properties as ℕ-Prop
+import Data.Nat.Properties as ℕ
 import Relation.Binary.PropositionalEquality.Core as ≡
 
 open Homomorphism homo
@@ -64,7 +64,7 @@ pow-eval-hom x (suc i) = (*-homo _ x) ⟨ trans ⟩ (≪* pow-eval-hom x i)
   rearrange zero =
     begin
       (ρ′ RawPow.^ (i ℕ.* 0)) * (⟦ x ⟧ Ρ RawPow.^ suc i)
-    ≡⟨ ≡.cong (λ k →  (ρ′ RawPow.^ k) * (⟦ x ⟧ Ρ RawPow.^ suc i)) (ℕ-Prop.*-zeroʳ i) ⟩
+    ≡⟨ ≡.cong (λ k →  (ρ′ RawPow.^ k) * (⟦ x ⟧ Ρ RawPow.^ suc i)) (ℕ.*-zeroʳ i) ⟩
       1# * (⟦ x ⟧ Ρ RawPow.^ suc i)
     ≈⟨ *-identityˡ _ ⟩
       ⟦ x ⟧ Ρ RawPow.^ suc i
@@ -72,7 +72,7 @@ pow-eval-hom x (suc i) = (*-homo _ x) ⟨ trans ⟩ (≪* pow-eval-hom x i)
   rearrange j@(suc j′) =
     begin
       (ρ′ RawPow.^ (suc i ℕ.* j))           * (⟦ x ⟧ Ρ RawPow.^ suc i)
-    ≡⟨ ≡.cong (λ v → (ρ′ RawPow.^ v) * (⟦ x ⟧ Ρ RawPow.^ suc i)) (ℕ-Prop.*-comm (suc i) j) ⟩
+    ≡⟨ ≡.cong (λ v → (ρ′ RawPow.^ v) * (⟦ x ⟧ Ρ RawPow.^ suc i)) (ℕ.*-comm (suc i) j) ⟩
       (ρ′ RawPow.^ (j ℕ.* suc i))           * (⟦ x ⟧ Ρ RawPow.^ suc i)
     ≈⟨ ≪* sym (RawPow.^-assocʳ ρ′ j (suc i)) ⟩
       ((ρ′ RawPow.^ suc j′) RawPow.^ suc i) * (⟦ x ⟧ Ρ RawPow.^ suc i)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index f014d54c9f..c879bdef38 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -15,7 +15,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas
 
 open import Data.Bool                                       using (Bool;true;false)
 open import Data.Nat.Base as ℕ                              using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
-open import Data.Nat.Properties as ℕₚ                       using (≤′-trans)
+open import Data.Nat.Properties as ℕ                        using (≤′-trans)
 open import Data.Vec.Base as Vec                            using (Vec; _∷_)
 open import Data.Fin                                        using (Fin; zero; suc)
 open import Data.List.Base                                  using (_∷_; [])
@@ -65,7 +65,7 @@ pow-hom : ∀ {n} i
         → (xs : Coeff n +)
         → ∀ ρ ρs
         → ⅀⟦ xs ⟧ (ρ , ρs) *⟨ ρ ⟩^ i ≈ ⅀⟦ xs ⍓+ i ⟧ (ρ , ρs)
-pow-hom zero (x Δ j & xs) ρ ρs rewrite ℕₚ.+-identityʳ j = refl
+pow-hom zero (x Δ j & xs) ρ ρs rewrite ℕ.+-identityʳ j = refl
 pow-hom (suc i) (x ≠0 Δ j & xs) ρ ρs =
   begin
     ρ ^ (suc i) * (((x , xs) ⟦∷⟧ (ρ , ρs)) *⟨ ρ ⟩^ j)
diff --git a/src/Text/Pretty.agda b/src/Text/Pretty.agda
index a4b3c1860d..a9202b1f82 100644
--- a/src/Text/Pretty.agda
+++ b/src/Text/Pretty.agda
@@ -26,8 +26,8 @@ open import Effect.Monad using (RawMonad)
 import Data.List.Effectful as List
 open RawMonad (List.monad {Level.zero})
 
-import Data.Nat.Properties as ℕₚ
-open import Data.List.Extrema.Core ℕₚ.≤-totalOrder using (⊓ᴸ)
+import Data.Nat.Properties as ℕ
+open import Data.List.Extrema.Core ℕ.≤-totalOrder using (⊓ᴸ)
 
 ------------------------------------------------------------------------
 -- Internal representation of documents and rendering function
diff --git a/src/Text/Printf.agda b/src/Text/Printf.agda
index dbb5a51d28..b7f1330c89 100644
--- a/src/Text/Printf.agda
+++ b/src/Text/Printf.agda
@@ -11,18 +11,17 @@ module Text.Printf where
 open import Data.String.Base using (String; fromChar; concat)
 open import Function.Base using (id)
 
-import Data.Char.Base    as Cₛ
-import Data.Integer.Show as ℤₛ
-import Data.Float        as Fₛ
-import Data.Nat.Show     as ℕₛ  using (show)
+import Data.Integer.Show as ℤ
+import Data.Float        as Float
+import Data.Nat.Show     as ℕ
 
 open import Text.Format as Format hiding (Error)
 open import Text.Printf.Generic
 
 printfSpec : PrintfSpec formatSpec String
-printfSpec .PrintfSpec.renderArg ℕArg      = ℕₛ.show
-printfSpec .PrintfSpec.renderArg ℤArg      = ℤₛ.show
-printfSpec .PrintfSpec.renderArg FloatArg  = Fₛ.show
+printfSpec .PrintfSpec.renderArg ℕArg      = ℕ.show
+printfSpec .PrintfSpec.renderArg ℤArg      = ℤ.show
+printfSpec .PrintfSpec.renderArg FloatArg  = Float.show
 printfSpec .PrintfSpec.renderArg CharArg   = fromChar
 printfSpec .PrintfSpec.renderArg StringArg = id
 printfSpec .PrintfSpec.renderStr           = id

From 04cc05cf8fb844929672da8462cb74c69ab5af4a Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Wed, 7 Feb 2024 20:31:13 +0100
Subject: [PATCH 220/316] Fix import in
 README.Data.Fin.Substitution.UntypedLambda (#2279)

Also import this module in doc/README.agda so that it is covered by CI.

Closes #2278.
---
 doc/README.agda                                     | 4 ++++
 doc/README/Data/Fin/Substitution/UntypedLambda.agda | 2 +-
 2 files changed, 5 insertions(+), 1 deletion(-)

diff --git a/doc/README.agda b/doc/README.agda
index 41b536a0d0..6a9a3e398c 100644
--- a/doc/README.agda
+++ b/doc/README.agda
@@ -222,6 +222,10 @@ import Text.Printf
 
 import README.Case
 
+-- Showcasing the framework for well-scoped substitutions
+
+import README.Data.Fin.Substitution.UntypedLambda
+
 -- Some examples showing how combinators can be used to emulate
 -- "functional reasoning"
 
diff --git a/doc/README/Data/Fin/Substitution/UntypedLambda.agda b/doc/README/Data/Fin/Substitution/UntypedLambda.agda
index 126959958f..036f7ed5ca 100644
--- a/doc/README/Data/Fin/Substitution/UntypedLambda.agda
+++ b/doc/README/Data/Fin/Substitution/UntypedLambda.agda
@@ -14,7 +14,7 @@ open import Data.Fin.Substitution.Lemmas
 open import Data.Nat.Base hiding (_/_)
 open import Data.Fin.Base using (Fin)
 open import Data.Vec.Base
-open import Relation.Binary.PropositionalEquality.Core
+open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong; cong₂; module ≡-Reasoning)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
   using (Star; ε; _◅_)

From 8e35057497cba3b931c85bb197e68e9f27d71530 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 8 Feb 2024 20:17:05 +0000
Subject: [PATCH 221/316] Qualified import of reasoning modules fixing #2280
 (#2282)

* `Data.List.Relation.Relation.Binary.BagAndSetEquality`

* `Relation.Binary.Reasoning.*`

* preorder reasoning

* setoid reasoning

* alignment
---
 .../Properties/HeytingCommutativeRing.agda    |  8 +--
 src/Algebra/Construct/LexProduct.agda         |  2 +-
 src/Algebra/Construct/LiftedChoice.agda       |  4 +-
 .../Lattice/Morphism/LatticeMonomorphism.agda |  4 +-
 src/Algebra/Module/Bundles.agda               |  4 +-
 src/Algebra/Module/Consequences.agda          |  4 +-
 src/Algebra/Morphism.agda                     |  4 +-
 src/Algebra/Morphism/Consequences.agda        |  6 +-
 src/Algebra/Morphism/GroupMonomorphism.agda   |  6 +-
 src/Algebra/Morphism/MagmaMonomorphism.agda   |  4 +-
 src/Algebra/Morphism/MonoidMonomorphism.agda  |  4 +-
 src/Algebra/Morphism/RingMonomorphism.agda    |  4 +-
 .../CommutativeSemigroup/Divisibility.agda    |  4 +-
 src/Algebra/Solver/CommutativeMonoid.agda     |  8 +--
 .../Solver/IdempotentCommutativeMonoid.agda   |  8 +--
 src/Codata/Musical/Colist.agda                |  8 +--
 src/Data/Integer/Divisibility/Signed.agda     | 35 +++++-----
 src/Data/List/Properties.agda                 |  4 +-
 .../Relation/Binary/BagAndSetEquality.agda    | 70 +++++++++----------
 .../Relation/Binary/Permutation/Setoid.agda   |  4 +-
 .../Subset/Propositional/Properties.agda      |  2 +-
 .../Binary/Subset/Setoid/Properties.agda      |  4 +-
 src/Data/Nat/Divisibility.agda                |  4 +-
 .../Rational/Unnormalised/Properties.agda     |  4 +-
 src/Function/Properties/Bijection.agda        |  4 +-
 src/Function/Properties/Equivalence.agda      |  4 +-
 src/Function/Properties/Injection.agda        |  2 -
 src/Function/Properties/Inverse.agda          | 10 +--
 src/Function/Properties/Surjection.agda       |  8 +--
 .../ReflexiveTransitive/Properties.agda       | 12 ++--
 .../Binary/Construct/NaturalOrder/Left.agda   | 14 ++--
 .../Binary/Construct/NaturalOrder/Right.agda  | 12 ++--
 .../Lattice/Properties/HeytingAlgebra.agda    | 14 ++--
 .../Binary/Lattice/Properties/Lattice.agda    | 14 ++--
 .../Binary/Reasoning/Base/Apartness.agda      |  6 +-
 .../Binary/Reasoning/Base/Double.agda         |  6 +-
 .../Binary/Reasoning/Base/Partial.agda        |  2 +-
 .../Binary/Reasoning/Base/Single.agda         |  4 +-
 .../Binary/Reasoning/Base/Triple.agda         |  8 +--
 .../Binary/Reasoning/MultiSetoid.agda         |  4 +-
 .../Binary/Reasoning/PartialSetoid.agda       |  6 +-
 src/Relation/Binary/Reasoning/Syntax.agda     |  6 +-
 src/Relation/Binary/Reflection.agda           |  8 +--
 src/Tactic/MonoidSolver.agda                  |  4 +-
 44 files changed, 177 insertions(+), 180 deletions(-)

diff --git a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
index 868224bf0c..d8b3ffcdfd 100644
--- a/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
+++ b/src/Algebra/Apartness/Properties/HeytingCommutativeRing.agda
@@ -21,7 +21,7 @@ open CommutativeRing commutativeRing using (ring; *-commutativeMonoid)
 open import Algebra.Properties.Ring ring
   using (-0#≈0#; -‿distribˡ-*; -‿distribʳ-*; -‿anti-homo-+; -‿involutive)
 open import Relation.Binary.Definitions using (Symmetric)
-import Relation.Binary.Reasoning.Setoid as ReasonSetoid
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 open import Algebra.Properties.CommutativeMonoid *-commutativeMonoid
 
 private variable
@@ -50,7 +50,7 @@ x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
   helper (x⁻¹ , x⁻¹*x≈1 , x*x⁻¹≈1) (y⁻¹ , y⁻¹*y≈1 , y*y⁻¹≈1)
     = invertibleˡ⇒# (y⁻¹ * x⁻¹ , y⁻¹*x⁻¹*x*y≈1)
     where
-    open ReasonSetoid setoid
+    open ≈-Reasoning setoid
 
     y⁻¹*x⁻¹*x*y≈1 : y⁻¹ * x⁻¹ * (x * y - 0#) ≈ 1#
     y⁻¹*x⁻¹*x*y≈1 = begin
@@ -67,7 +67,7 @@ x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
 #-sym : Symmetric _#_
 #-sym {x} {y} x#y = invertibleˡ⇒# (- x-y⁻¹ , x-y⁻¹*y-x≈1)
   where
-  open ReasonSetoid setoid
+  open ≈-Reasoning setoid
   InvX-Y : Invertible _≈_ 1# _*_ (x - y)
   InvX-Y = #⇒invertible x#y
 
@@ -96,7 +96,7 @@ x#0y#0→xy#0 {x} {y} x#0 y#0 = helper (#⇒invertible x#0) (#⇒invertible y#0)
   helper (x-z⁻¹ , x-z⁻¹*x-z≈1# , x-z*x-z⁻¹≈1#)
     = invertibleˡ⇒# (x-z⁻¹ , x-z⁻¹*y-z≈1)
     where
-    open ReasonSetoid setoid
+    open ≈-Reasoning setoid
 
     x-z⁻¹*y-z≈1 : x-z⁻¹ * (y - z) ≈ 1#
     x-z⁻¹*y-z≈1 = begin
diff --git a/src/Algebra/Construct/LexProduct.agda b/src/Algebra/Construct/LexProduct.agda
index e4d51ff9ed..e2f5825824 100644
--- a/src/Algebra/Construct/LexProduct.agda
+++ b/src/Algebra/Construct/LexProduct.agda
@@ -16,7 +16,7 @@ open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Nullary using (¬_; does; yes; no)
 open import Relation.Nullary.Negation using (contradiction; contradiction₂)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 module Algebra.Construct.LexProduct
   {ℓ₁ ℓ₂ ℓ₃ ℓ₄} (M : Magma ℓ₁ ℓ₂) (N : Magma ℓ₃ ℓ₄)
diff --git a/src/Algebra/Construct/LiftedChoice.agda b/src/Algebra/Construct/LiftedChoice.agda
index f53eecc7f1..9b141f8f7e 100644
--- a/src/Algebra/Construct/LiftedChoice.agda
+++ b/src/Algebra/Construct/LiftedChoice.agda
@@ -20,7 +20,7 @@ open import Level using (Level; _⊔_)
 open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Unary using (Pred)
 
-import Relation.Binary.Reasoning.Setoid as EqReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 private
   variable
@@ -46,7 +46,7 @@ module _ {_≈_ : Rel B ℓ} {_∙_ : Op₂ B}
 
   private module M = IsSelectiveMagma ∙-isSelectiveMagma
   open M hiding (sel; isMagma)
-  open EqReasoning setoid
+  open ≈-Reasoning setoid
 
   module _ (f : A → B) where
 
diff --git a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
index 9eb1d4da24..9e16a46c4a 100644
--- a/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
+++ b/src/Algebra/Lattice/Morphism/LatticeMonomorphism.agda
@@ -18,7 +18,7 @@ import Algebra.Lattice.Properties.Lattice as LatticeProperties
 open import Data.Product.Base using (_,_; map)
 open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.Morphism.RelMonomorphism as RelMonomorphisms
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 module Algebra.Lattice.Morphism.LatticeMonomorphism
   {a b ℓ₁ ℓ₂} {L₁ : RawLattice a ℓ₁} {L₂ : RawLattice b ℓ₂} {⟦_⟧}
@@ -73,7 +73,7 @@ module _ (⊔-⊓-isLattice : IsLattice _≈₂_ _⊔_ _⊓_) where
   setoid : Setoid b ℓ₂
   setoid = record { isEquivalence = isEquivalence }
 
-  open SetoidReasoning setoid
+  open ≈-Reasoning setoid
 
   ∨-absorbs-∧ : _Absorbs_ _≈₁_ _∨_ _∧_
   ∨-absorbs-∧ x y = injective (begin
diff --git a/src/Algebra/Module/Bundles.agda b/src/Algebra/Module/Bundles.agda
index 56b01fcf97..33e6bfa731 100644
--- a/src/Algebra/Module/Bundles.agda
+++ b/src/Algebra/Module/Bundles.agda
@@ -37,7 +37,7 @@ open import Function.Base
 open import Level
 open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary    using (¬_)
-import Relation.Binary.Reasoning.Setoid as SetR
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 private
   variable
@@ -364,7 +364,7 @@ record Semimodule (commutativeSemiring : CommutativeSemiring r ℓr) m ℓm
           ; rawBisemimodule; _≉ᴹ_
           )
 
-  open SetR ≈ᴹ-setoid
+  open ≈-Reasoning ≈ᴹ-setoid
 
   *ₗ-comm : L.Commutative _*ₗ_
   *ₗ-comm x y m = begin
diff --git a/src/Algebra/Module/Consequences.agda b/src/Algebra/Module/Consequences.agda
index 9acd6d6413..2ee2c98d7e 100644
--- a/src/Algebra/Module/Consequences.agda
+++ b/src/Algebra/Module/Consequences.agda
@@ -16,7 +16,7 @@ open import Function.Base using (flip)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (Setoid)
-import Relation.Binary.Reasoning.Setoid as Rea
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 private
   variable
@@ -27,7 +27,7 @@ private
 module _ (_≈ᴬ_ : Rel {a} A ℓa) (S : Setoid c ℓ) where
 
   open Setoid S
-  open Rea S
+  open ≈-Reasoning S
   open Defs _≈ᴬ_
 
   private
diff --git a/src/Algebra/Morphism.agda b/src/Algebra/Morphism.agda
index 1bba00bab2..45b7a46a1d 100644
--- a/src/Algebra/Morphism.agda
+++ b/src/Algebra/Morphism.agda
@@ -14,7 +14,7 @@ import Algebra.Properties.Group as GroupP
 open import Function.Base
 open import Level
 open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
-import Relation.Binary.Reasoning.Setoid as EqR
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 private
   variable
@@ -137,7 +137,7 @@ module _ {c₁ ℓ₁ c₂ ℓ₂}
     open IsMonoidMorphism mn-homo public
 
     ⁻¹-homo : Homomorphic₁ ⟦_⟧ F._⁻¹ T._⁻¹
-    ⁻¹-homo x = let open EqR T.setoid in T.uniqueˡ-⁻¹ ⟦ x F.⁻¹ ⟧ ⟦ x ⟧ $ begin
+    ⁻¹-homo x = let open ≈-Reasoning T.setoid in T.uniqueˡ-⁻¹ ⟦ x F.⁻¹ ⟧ ⟦ x ⟧ $ begin
       ⟦ x F.⁻¹ ⟧ T.∙ ⟦ x ⟧ ≈⟨ T.sym (∙-homo (x F.⁻¹) x) ⟩
       ⟦ x F.⁻¹ F.∙ x ⟧     ≈⟨ ⟦⟧-cong (F.inverseˡ x) ⟩
       ⟦ F.ε ⟧              ≈⟨ ε-homo ⟩
diff --git a/src/Algebra/Morphism/Consequences.agda b/src/Algebra/Morphism/Consequences.agda
index e297a8dbc9..b451a87088 100644
--- a/src/Algebra/Morphism/Consequences.agda
+++ b/src/Algebra/Morphism/Consequences.agda
@@ -13,7 +13,7 @@ open import Algebra.Morphism.Definitions
 open import Data.Product.Base using (_,_)
 open import Function.Base using (id; _∘_)
 open import Function.Definitions
-import Relation.Binary.Reasoning.Setoid as EqR
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- If f and g are mutually inverse maps between A and B, g is congruent,
@@ -34,7 +34,7 @@ module _ {α α= β β=} (M₁ : Magma α α=) (M₂ : Magma β β=) where
     g (f (g x) ∙₂ f (g y)) ≈⟨ g-cong (homo (g x) (g y)) ⟨
     g (f (g x ∙₁ g y))     ≈⟨ invʳ M₂.refl ⟩
     g x ∙₁ g y             ∎
-    where open EqR M₁.setoid
+    where open ≈-Reasoning M₁.setoid
 
   homomorphic₂-inj  : ∀ {f g} → Injective _≈₁_ _≈₂_ f →
                       Inverseˡ _≈₁_ _≈₂_ f g →
@@ -45,4 +45,4 @@ module _ {α α= β β=} (M₁ : Magma α α=) (M₂ : Magma β β=) where
     x ∙₂ y              ≈⟨ M₂.∙-cong (invˡ M₁.refl) (invˡ M₁.refl) ⟨
     f (g x) ∙₂ f (g y)  ≈⟨ homo (g x) (g y) ⟨
     f (g x ∙₁ g y)      ∎)
-    where open EqR M₂.setoid
+    where open ≈-Reasoning M₂.setoid
diff --git a/src/Algebra/Morphism/GroupMonomorphism.agda b/src/Algebra/Morphism/GroupMonomorphism.agda
index 51cdcbafe7..4b3349d141 100644
--- a/src/Algebra/Morphism/GroupMonomorphism.agda
+++ b/src/Algebra/Morphism/GroupMonomorphism.agda
@@ -27,7 +27,7 @@ open RawGroup G₂ renaming
 open import Algebra.Definitions
 open import Algebra.Structures
 open import Data.Product.Base using (_,_)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- Re-export all properties of monoid monomorphisms
@@ -41,7 +41,7 @@ open import Algebra.Morphism.MonoidMonomorphism
 module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
 
   open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)
-  open SetoidReasoning setoid
+  open ≈-Reasoning setoid
 
   inverseˡ : LeftInverse _≈₂_ ε₂ _⁻¹₂ _◦_ → LeftInverse _≈₁_ ε₁ _⁻¹₁ _∙_
   inverseˡ invˡ x = injective (begin
@@ -72,7 +72,7 @@ module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
 module _ (◦-isAbelianGroup : IsAbelianGroup _≈₂_ _◦_ ε₂ _⁻¹₂) where
 
   open IsAbelianGroup ◦-isAbelianGroup renaming (∙-cong to ◦-cong; ⁻¹-cong to ⁻¹₂-cong)
-  open SetoidReasoning setoid
+  open ≈-Reasoning setoid
 
   ⁻¹-distrib-∙ : (∀ x y → (x ◦ y) ⁻¹₂ ≈₂ (x ⁻¹₂) ◦ (y ⁻¹₂)) → (∀ x y → (x ∙ y) ⁻¹₁ ≈₁ (x ⁻¹₁) ∙ (y ⁻¹₁))
   ⁻¹-distrib-∙ ⁻¹-distrib-∙ x y = injective (begin
diff --git a/src/Algebra/Morphism/MagmaMonomorphism.agda b/src/Algebra/Morphism/MagmaMonomorphism.agda
index 87027ba5e1..fbe08a5861 100644
--- a/src/Algebra/Morphism/MagmaMonomorphism.agda
+++ b/src/Algebra/Morphism/MagmaMonomorphism.agda
@@ -27,7 +27,7 @@ open import Algebra.Structures
 open import Algebra.Definitions
 open import Data.Product.Base using (map)
 open import Data.Sum.Base using (inj₁; inj₂)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 import Relation.Binary.Morphism.RelMonomorphism isRelMonomorphism as RelMorphism
 
 ------------------------------------------------------------------------
@@ -36,7 +36,7 @@ import Relation.Binary.Morphism.RelMonomorphism isRelMonomorphism as RelMorphism
 module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
 
   open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)
-  open SetoidReasoning setoid
+  open ≈-Reasoning setoid
 
   cong : Congruent₂ _≈₁_ _∙_
   cong {x} {y} {u} {v} x≈y u≈v = injective (begin
diff --git a/src/Algebra/Morphism/MonoidMonomorphism.agda b/src/Algebra/Morphism/MonoidMonomorphism.agda
index bfb412302e..1b918295af 100644
--- a/src/Algebra/Morphism/MonoidMonomorphism.agda
+++ b/src/Algebra/Morphism/MonoidMonomorphism.agda
@@ -25,7 +25,7 @@ open RawMonoid M₂ renaming (Carrier to B; _≈_ to _≈₂_; _∙_ to _◦_; 
 open import Algebra.Definitions
 open import Algebra.Structures
 open import Data.Product.Base using (map)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- Re-export all properties of magma monomorphisms
@@ -39,7 +39,7 @@ open import Algebra.Morphism.MagmaMonomorphism
 module _ (◦-isMagma : IsMagma _≈₂_ _◦_) where
 
   open IsMagma ◦-isMagma renaming (∙-cong to ◦-cong)
-  open SetoidReasoning setoid
+  open ≈-Reasoning setoid
 
   identityˡ : LeftIdentity _≈₂_ ε₂ _◦_ → LeftIdentity _≈₁_ ε₁ _∙_
   identityˡ idˡ x = injective (begin
diff --git a/src/Algebra/Morphism/RingMonomorphism.agda b/src/Algebra/Morphism/RingMonomorphism.agda
index 35a191a0f8..40e7a80d83 100644
--- a/src/Algebra/Morphism/RingMonomorphism.agda
+++ b/src/Algebra/Morphism/RingMonomorphism.agda
@@ -29,7 +29,7 @@ open RawRing R₂ renaming
 open import Algebra.Definitions
 open import Algebra.Structures
 open import Data.Product.Base using (proj₁; proj₂; _,_)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- Re-export all properties of group and monoid monomorphisms
@@ -83,7 +83,7 @@ module _ (+-isGroup : IsGroup _≈₂_ _⊕_ 0#₂ ⊝_)
 
   open IsGroup +-isGroup hiding (setoid; refl; sym)
   open IsMagma *-isMagma renaming (∙-cong to ◦-cong)
-  open SetoidReasoning setoid
+  open ≈-Reasoning setoid
 
   distribˡ : _DistributesOverˡ_ _≈₂_ _⊛_ _⊕_ → _DistributesOverˡ_ _≈₁_ _*_ _+_
   distribˡ distribˡ x y z = injective (begin
diff --git a/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda b/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
index da8d3abf45..782b094fbe 100644
--- a/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
+++ b/src/Algebra/Properties/CommutativeSemigroup/Divisibility.agda
@@ -8,7 +8,7 @@
 
 open import Algebra using (CommutativeSemigroup)
 open import Data.Product.Base using (_,_)
-import Relation.Binary.Reasoning.Setoid as EqReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 module Algebra.Properties.CommutativeSemigroup.Divisibility
   {a ℓ} (CS : CommutativeSemigroup a ℓ)
@@ -16,7 +16,7 @@ module Algebra.Properties.CommutativeSemigroup.Divisibility
 
 open CommutativeSemigroup CS
 open import Algebra.Properties.CommutativeSemigroup CS using (x∙yz≈xz∙y; x∙yz≈y∙xz)
-open EqReasoning setoid
+open ≈-Reasoning setoid
 
 ------------------------------------------------------------------------
 -- Re-export the contents of divisibility over semigroups
diff --git a/src/Algebra/Solver/CommutativeMonoid.agda b/src/Algebra/Solver/CommutativeMonoid.agda
index 8062ff34df..f15f83824f 100644
--- a/src/Algebra/Solver/CommutativeMonoid.agda
+++ b/src/Algebra/Solver/CommutativeMonoid.agda
@@ -22,16 +22,16 @@ open import Data.Vec.Base using (Vec; []; _∷_; lookup; replicate)
 
 open import Function.Base using (_∘_)
 
-import Relation.Binary.Reasoning.Setoid as EqReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 import Relation.Binary.Reflection as Reflection
 import Relation.Nullary.Decidable as Dec
 import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Nullary.Decidable using (Dec)
 
 open CommutativeMonoid M
-open EqReasoning setoid
+open ≈-Reasoning setoid
 
 private
   variable
@@ -189,7 +189,7 @@ prove′ e₁ e₂ =
   lemma : normalise e₁ ≡ normalise e₂ → ∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ
   lemma eq ρ =
     R.prove ρ e₁ e₂ (begin
-      ⟦ normalise e₁ ⟧⇓ ρ  ≡⟨ P.cong (λ e → ⟦ e ⟧⇓ ρ) eq ⟩
+      ⟦ normalise e₁ ⟧⇓ ρ  ≡⟨ ≡.cong (λ e → ⟦ e ⟧⇓ ρ) eq ⟩
       ⟦ normalise e₂ ⟧⇓ ρ  ∎)
 
 -- This procedure can be combined with from-just.
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
index fc65652787..35cfcc5ea6 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
@@ -20,19 +20,19 @@ open import Data.Vec.Base using (Vec; []; _∷_; lookup; replicate)
 
 open import Function.Base using (_∘_)
 
-import Relation.Binary.Reasoning.Setoid  as EqReasoning
+import Relation.Binary.Reasoning.Setoid  as ≈-Reasoning
 import Relation.Binary.Reflection            as Reflection
 import Relation.Nullary.Decidable            as Dec
 import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 
-open import Relation.Binary.PropositionalEquality as P using (_≡_; decSetoid)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; decSetoid)
 open import Relation.Nullary.Decidable using (Dec)
 
 module Algebra.Solver.IdempotentCommutativeMonoid
   {m₁ m₂} (M : IdempotentCommutativeMonoid m₁ m₂) where
 
 open IdempotentCommutativeMonoid M
-open EqReasoning setoid
+open ≈-Reasoning setoid
 
 private
   variable
@@ -202,7 +202,7 @@ prove′ e₁ e₂ =
   lemma : normalise e₁ ≡ normalise e₂ → ∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ
   lemma eq ρ =
     R.prove ρ e₁ e₂ (begin
-      ⟦ normalise e₁ ⟧⇓ ρ  ≡⟨ P.cong (λ e → ⟦ e ⟧⇓ ρ) eq ⟩
+      ⟦ normalise e₁ ⟧⇓ ρ  ≡⟨ ≡.cong (λ e → ⟦ e ⟧⇓ ρ) eq ⟩
       ⟦ normalise e₂ ⟧⇓ ρ  ∎)
 
 -- This procedure can be combined with from-just.
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index 3699ef64be..18d250f566 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -31,8 +31,8 @@ open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Bundles using (Poset; Setoid; Preorder)
 open import Relation.Binary.Definitions using (Transitive; Antisymmetric)
 import Relation.Binary.Construct.FromRel as Ind
-import Relation.Binary.Reasoning.Preorder as PreR
-import Relation.Binary.Reasoning.PartialOrder as POR
+import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
+import Relation.Binary.Reasoning.PartialOrder as ≤-Reasoning
 open import Relation.Binary.PropositionalEquality as P using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
 open import Relation.Nullary.Reflects using (invert)
@@ -164,7 +164,7 @@ Any-∈ {P = P} = mk↔ₛ′
   antisym (x ∷ p₁) p₂ = x ∷ ♯ antisym (♭ p₁) (tail p₂)
 
 module ⊑-Reasoning {a} {A : Set a} where
-  private module Base = POR (⊑-Poset A)
+  private module Base = ≤-Reasoning (⊑-Poset A)
 
   open Base public hiding (step-<; step-≤)
 
@@ -188,7 +188,7 @@ module ⊑-Reasoning {a} {A : Set a} where
 --      ys ≡⟨ ys≡zs ⟩
 --      zs  ∎
 module ⊆-Reasoning {A : Set a} where
-  private module Base = PreR (⊆-Preorder A)
+  private module Base = ≲-Reasoning (⊆-Preorder A)
 
   open Base public
     hiding (step-≲; step-∼)
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index 5149871534..d171b7015e 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -18,8 +18,8 @@ import Data.Nat.Base as ℕ
 import Data.Nat.Divisibility as ℕ
 import Data.Nat.Coprimality as ℕ
 import Data.Nat.Properties as ℕ
-import Data.Sign as S
-import Data.Sign.Properties as SProp
+import Data.Sign as Sign
+import Data.Sign.Properties as Sign
 open import Level
 open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
 open import Relation.Binary.Bundles using (Preorder)
@@ -27,9 +27,8 @@ open import Relation.Binary.Structures using (IsPreorder)
 open import Relation.Binary.Definitions
   using (Reflexive; Transitive; Decidable)
 open import Relation.Binary.PropositionalEquality
-import Relation.Binary.Reasoning.Preorder as PreorderReasoning
-open import Relation.Nullary.Decidable using (yes; no)
-import Relation.Nullary.Decidable as DEC
+import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
+open import Relation.Nullary.Decidable as Dec using (yes; no)
 open import Relation.Binary.Reasoning.Syntax
 
 ------------------------------------------------------------------------
@@ -50,19 +49,19 @@ open _∣_ using (quotient) public
 ∣ᵤ⇒∣ {k} {i} (divides 0           eq) = divides (+ 0) (∣i∣≡0⇒i≡0 eq)
 ∣ᵤ⇒∣ {k} {i} (divides q@(ℕ.suc _) eq) with k ≟ +0
 ... | yes refl = divides +0 (∣i∣≡0⇒i≡0 (trans eq (ℕ.*-zeroʳ q)))
-... | no  neq  = divides (sign i S.* sign k ◃ q) (◃-cong sign-eq abs-eq)
+... | no  neq  = divides (sign i Sign.* sign k ◃ q) (◃-cong sign-eq abs-eq)
   where
-  ikq = sign i S.* sign k ◃ q
+  ikq = sign i Sign.* sign k ◃ q
 
   *-nonZero : ∀ m n .{{_ : ℕ.NonZero m}} .{{_ : ℕ.NonZero n}} → ℕ.NonZero (m ℕ.* n)
   *-nonZero (ℕ.suc _) (ℕ.suc _) = _
 
   ◃-nonZero : ∀ s n .{{_ : ℕ.NonZero n}} → NonZero (s ◃ n)
-  ◃-nonZero S.- (ℕ.suc _) = _
-  ◃-nonZero S.+ (ℕ.suc _) = _
+  ◃-nonZero Sign.- (ℕ.suc _) = _
+  ◃-nonZero Sign.+ (ℕ.suc _) = _
 
   ikq≢0 : NonZero ikq
-  ikq≢0 = ◃-nonZero (sign i S.* sign k) q
+  ikq≢0 = ◃-nonZero (sign i Sign.* sign k) q
 
   instance
     ikq*∣k∣≢0 : ℕ.NonZero (∣ ikq ∣ ℕ.* ∣ k ∣)
@@ -70,18 +69,18 @@ open _∣_ using (quotient) public
 
   sign-eq : sign i ≡ sign (ikq * k)
   sign-eq = sym $ begin
-    sign (ikq * k)                  ≡⟨ sign-◃ (sign ikq S.* sign k) (∣ ikq ∣ ℕ.* ∣ k ∣) ⟩
-    sign ikq S.* sign k             ≡⟨ cong (S._* sign k) (sign-◃ (sign i S.* sign k) q) ⟩
-    (sign i S.* sign k) S.* sign k  ≡⟨ SProp.*-assoc (sign i) (sign k) (sign k) ⟩
-    sign i S.* (sign k S.* sign k)  ≡⟨ cong (sign i S.*_) (SProp.s*s≡+ (sign k)) ⟩
-    sign i S.* S.+                  ≡⟨ SProp.*-identityʳ (sign i) ⟩
+    sign (ikq * k)                        ≡⟨ sign-◃ (sign ikq Sign.* sign k) (∣ ikq ∣ ℕ.* ∣ k ∣) ⟩
+    sign ikq Sign.* sign k                ≡⟨ cong (Sign._* sign k) (sign-◃ (sign i Sign.* sign k) q) ⟩
+    (sign i Sign.* sign k) Sign.* sign k  ≡⟨ Sign.*-assoc (sign i) (sign k) (sign k) ⟩
+    sign i Sign.* (sign k Sign.* sign k)  ≡⟨ cong (sign i Sign.*_) (Sign.s*s≡+ (sign k)) ⟩
+    sign i Sign.* Sign.+                  ≡⟨ Sign.*-identityʳ (sign i) ⟩
     sign i                          ∎
     where open ≡-Reasoning
 
   abs-eq : ∣ i ∣ ≡ ∣ ikq * k ∣
   abs-eq = sym $ begin
     ∣ ikq * k ∣        ≡⟨ ∣i*j∣≡∣i∣*∣j∣ ikq k ⟩
-    ∣ ikq ∣ ℕ.* ∣ k ∣  ≡⟨ cong (ℕ._* ∣ k ∣) (abs-◃ (sign i S.* sign k) q) ⟩
+    ∣ ikq ∣ ℕ.* ∣ k ∣  ≡⟨ cong (ℕ._* ∣ k ∣) (abs-◃ (sign i Sign.* sign k) q) ⟩
     q ℕ.* ∣ k ∣        ≡⟨ sym eq ⟩
     ∣ i ∣              ∎
     where open ≡-Reasoning
@@ -119,7 +118,7 @@ open _∣_ using (quotient) public
 -- Divisibility reasoning
 
 module ∣-Reasoning where
-  private module Base = PreorderReasoning ∣-preorder
+  private module Base = ≲-Reasoning ∣-preorder
 
   open Base public
     hiding (step-≲; step-∼; step-≈; step-≈˘)
@@ -133,7 +132,7 @@ module ∣-Reasoning where
 infix 4 _∣?_
 
 _∣?_ : Decidable _∣_
-k ∣? m = DEC.map′ ∣ᵤ⇒∣ ∣⇒∣ᵤ (∣ k ∣ ℕ.∣? ∣ m ∣)
+k ∣? m = Dec.map′ ∣ᵤ⇒∣ ∣⇒∣ᵤ (∣ k ∣ ℕ.∣? ∣ m ∣)
 
 0∣⇒≡0 : ∀ {m} → 0ℤ ∣ m → m ≡ 0ℤ
 0∣⇒≡0 0|m = ∣i∣≡0⇒i≡0 (ℕ.0∣⇒≡0 (∣⇒∣ᵤ 0|m))
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 5bd47dc763..4188bab92a 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -35,8 +35,8 @@ open import Function.Base using (id; _∘_; _∘′_; _∋_; _-⟨_∣; ∣_⟩-
 open import Function.Definitions using (Injective)
 open import Level using (Level)
 open import Relation.Binary.Definitions as B using (DecidableEquality)
-import Relation.Binary.Reasoning.Setoid as EqR
-open import Relation.Binary.PropositionalEquality as P hiding ([_])
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
+open import Relation.Binary.PropositionalEquality as ≡ hiding ([_])
 open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (¬_; Dec; does; _because_; yes; no; contradiction)
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 17af3d8955..389721d4c5 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -15,10 +15,11 @@ open import Data.Empty
 open import Data.Fin.Base
 open import Data.List.Base
 open import Data.List.Effectful using (monad; module Applicative; module MonadProperties)
-import Data.List.Properties as LP
+import Data.List.Properties as List
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Data.List.Relation.Unary.Any.Properties hiding (++-comm)
 open import Data.List.Membership.Propositional using (_∈_)
+open import Data.List.Membership.Propositional.Properties
 open import Data.List.Relation.Binary.Subset.Propositional.Properties
   using (⊆-preorder)
 open import Data.List.Relation.Binary.Permutation.Propositional
@@ -39,13 +40,12 @@ open import Level using (Level)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Definitions using (Trans)
 open import Relation.Binary.Bundles using (Preorder; Setoid)
-import Relation.Binary.Reasoning.Setoid as EqR
-import Relation.Binary.Reasoning.Preorder as PreorderReasoning
-open import Relation.Binary.PropositionalEquality as P
-  using (_≡_; _≢_; _≗_; refl)
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
+import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
+open import Relation.Binary.PropositionalEquality as ≡
+  using (_≡_; _≢_; _≗_; refl; module ≡-Reasoning)
 open import Relation.Binary.Reasoning.Syntax
 open import Relation.Nullary.Negation using (¬_)
-open import Data.List.Membership.Propositional.Properties
 
 private
   variable
@@ -93,7 +93,7 @@ bag-=⇒ xs≈ys = ↔⇒ xs≈ys
 -- "Equational" reasoning for _⊆_ along with an additional relatedness
 
 module ⊆-Reasoning {A : Set a} where
-  private module Base = PreorderReasoning (⊆-preorder A)
+  private module Base = ≲-Reasoning (⊆-preorder A)
 
   open Base public
     hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)
@@ -140,10 +140,10 @@ module _ {k} {f g : A → B} {xs ys} where
 
     helper : ∀ y → x ≡ f y ↔ x ≡ g y
     helper y = mk↔ₛ′
-      (λ x≡fy → P.trans x≡fy (        f≗g y))
-      (λ x≡gy → P.trans x≡gy (P.sym $ f≗g y))
-      (λ { P.refl → P.trans-symˡ (f≗g y) })
-      λ { P.refl → P.trans-symʳ (f≗g y) }
+      (λ x≡fy → ≡.trans x≡fy (        f≗g y))
+      (λ x≡gy → ≡.trans x≡gy (≡.sym $ f≗g y))
+      (λ { ≡.refl → ≡.trans-symˡ (f≗g y) })
+      λ { ≡.refl → ≡.trans-symʳ (f≗g y) }
 
 ------------------------------------------------------------------------
 -- _++_
@@ -195,11 +195,11 @@ module _ {k} {A B : Set a} {fs gs : List (A → B)} {xs ys} where
   ⊛-cong : fs ∼[ k ] gs → xs ∼[ k ] ys → (fs ⊛ xs) ∼[ k ] (gs ⊛ ys)
   ⊛-cong fs≈gs xs≈ys {x} = begin
     x ∈ (fs ⊛ xs)
-      ≡⟨ P.cong (x ∈_) (Applicative.unfold-⊛ fs xs) ⟩
+      ≡⟨ ≡.cong (x ∈_) (Applicative.unfold-⊛ fs xs) ⟩
     x ∈ (fs >>= λ f → xs >>= λ x → pure (f x))
       ∼⟨ >>=-cong fs≈gs (λ f → >>=-cong xs≈ys λ x → K-refl) ⟩
     x ∈ (gs >>= λ g → ys >>= λ y → pure (g y))
-      ≡⟨ P.cong (x ∈_) (Applicative.unfold-⊛ gs ys) ⟨
+      ≡⟨ ≡.cong (x ∈_) (Applicative.unfold-⊛ gs ys) ⟨
     x ∈ (gs ⊛ ys) ∎
     where open Related.EquationalReasoning
 
@@ -232,7 +232,7 @@ commutativeMonoid {a} k A = record
         ; ∙-cong        = ++-cong
         }
       ; assoc         = λ xs ys zs →
-                          Eq.reflexive (LP.++-assoc xs ys zs)
+                          Eq.reflexive (List.++-assoc xs ys zs)
       }
     ; identityˡ = λ xs → K-refl
     ; comm      = λ xs ys → ↔⇒ (++↔++ xs ys)
@@ -284,10 +284,10 @@ empty-unique {xs = _ ∷ _} ∷∼[] with ⇒→ ∷∼[] (here refl)
                 (xs₂ >>= pure ∘ f))       ≈⟨ >>=-left-distributive fs ⟩
 
   (fs >>= λ f → xs₁ >>= pure ∘ f) ++
-  (fs >>= λ f → xs₂ >>= pure ∘ f)         ≡⟨ P.cong₂ _++_ (Applicative.unfold-⊛ fs xs₁) (Applicative.unfold-⊛ fs xs₂) ⟨
+  (fs >>= λ f → xs₂ >>= pure ∘ f)         ≡⟨ ≡.cong₂ _++_ (Applicative.unfold-⊛ fs xs₁) (Applicative.unfold-⊛ fs xs₂) ⟨
 
   (fs ⊛ xs₁) ++ (fs ⊛ xs₂)                ∎
-  where open EqR ([ bag ]-Equality B)
+  where open ≈-Reasoning ([ bag ]-Equality B)
 
 private
 
@@ -397,7 +397,7 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
 
     to (Fin-length-cong xs≈ys) (index-of p)                        ∎
     where
-    open P.≡-Reasoning
+    open ≡-Reasoning
     open Inverse
 
     lemma :
@@ -423,11 +423,11 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
     index-of (Inverse.to xs≈ys q)
   index-equality-preserved {p = p} {q} xs≈ys eq =
     index-of (Inverse.to xs≈ys p)                  ≡⟨ index-of-commutes xs≈ys p ⟩
-    Inverse.to (Fin-length-cong xs≈ys) (index-of p)  ≡⟨ P.cong (Inverse.to (Fin-length-cong xs≈ys)) eq ⟩
-    Inverse.to (Fin-length-cong xs≈ys) (index-of q)  ≡⟨ P.sym $ index-of-commutes xs≈ys q ⟩
+    Inverse.to (Fin-length-cong xs≈ys) (index-of p)  ≡⟨ ≡.cong (Inverse.to (Fin-length-cong xs≈ys)) eq ⟩
+    Inverse.to (Fin-length-cong xs≈ys) (index-of q)  ≡⟨ ≡.sym $ index-of-commutes xs≈ys q ⟩
     index-of (Inverse.to xs≈ys q)                  ∎
     where
-    open P.≡-Reasoning
+    open ≡-Reasoning
 
   -- The old inspect idiom.
 
@@ -487,17 +487,17 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
           ∀ b → g (from f) from-hyp (g (to f) to-hyp b) ≡ b
     g∘g f to-hyp from-hyp b = g∘g′
       where
-      open P.≡-Reasoning
+      open ≡-Reasoning
 
       g∘g′ : g (from f) from-hyp (g (to f) to-hyp b) ≡ b
       g∘g′ with inspect (to f (inj₂ b))
       g∘g′ | inj₂ c , eq₁ with inspect (from f (inj₂ c))
       ...   | inj₂ b′ , eq₂ = inj₂-injective (
-        inj₂ b′            ≡⟨ P.sym eq₂ ⟩
+        inj₂ b′            ≡⟨ ≡.sym eq₂ ⟩
         from f (inj₂ c)   ≡⟨ to-from f eq₁ ⟩
         inj₂ b            ∎)
       ...   | inj₁ a  , eq₂ with
-        inj₁ a             ≡⟨ P.sym eq₂ ⟩
+        inj₁ a             ≡⟨ ≡.sym eq₂ ⟩
         from f (inj₂ c)    ≡⟨ to-from f eq₁ ⟩
         inj₂ b             ∎
       ... | ()
@@ -505,7 +505,7 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
       g∘g′ | inj₁ a , eq₁ | inj₁ a′ , eq₂ = ⊥-elim $ to-hyp eq₁ eq₂
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ with inspect (from f (inj₂ c))
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₂ b′ , eq₃ with
-        inj₁ a             ≡⟨ P.sym (to-from f eq₂) ⟩
+        inj₁ a             ≡⟨ ≡.sym (to-from f eq₂) ⟩
         from f (inj₂ c)    ≡⟨ eq₃ ⟩
         inj₂ b′            ∎
       ... | ()
@@ -513,12 +513,12 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₁ a″ , eq₄ = ⊥-elim $ from-hyp eq₃ eq₄
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₂ b′ , eq₄ = inj₂-injective (
         let lemma =
-              inj₁ a′             ≡⟨ P.sym eq₃ ⟩
+              inj₁ a′             ≡⟨ ≡.sym eq₃ ⟩
               from f (inj₂ c)    ≡⟨ to-from f eq₂ ⟩
               inj₁ a             ∎
         in
-        inj₂ b′             ≡⟨ P.sym eq₄ ⟩
-        from f (inj₁ a′)    ≡⟨ P.cong (from f ∘ inj₁) $ inj₁-injective lemma ⟩
+        inj₂ b′             ≡⟨ ≡.sym eq₄ ⟩
+        from f (inj₁ a′)    ≡⟨ ≡.cong (from f ∘ inj₁) $ inj₁-injective lemma ⟩
         from f (inj₁ a)    ≡⟨ to-from f eq₁ ⟩
         inj₂ b              ∎)
 
@@ -541,21 +541,21 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
   lemma {xs} inv {b = z∈xs} {a = p} {a′ = q} hyp₁ hyp₂ with
     zero                                                                  ≡⟨⟩
     index-of {xs = x ∷ xs} (here p)                                       ≡⟨⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₁ p)                         ≡⟨ P.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ P.sym $
+    index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₁ p)                         ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ ≡.sym $
                                                                                to-from (∼→⊎↔⊎ inv) {x = inj₁ p} hyp₂ ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) $ (from (∼→⊎↔⊎ inv) $ inj₁ q))  ≡⟨ P.cong index-of $
+    index-of {xs = x ∷ xs} (to (∷↔ _) $ (from (∼→⊎↔⊎ inv) $ inj₁ q))  ≡⟨ ≡.cong index-of $
                                                                                strictlyInverseˡ (∷↔ _) (from inv (here q)) ⟩
     index-of {xs = x ∷ xs} (to (SK-sym inv) $ here q)                   ≡⟨ index-equality-preserved (SK-sym inv) refl ⟩
-    index-of {xs = x ∷ xs} (to (SK-sym inv) $ here p)                   ≡⟨ P.cong index-of $ P.sym $
+    index-of {xs = x ∷ xs} (to (SK-sym inv) $ here p)                   ≡⟨ ≡.cong index-of $ ≡.sym $
                                                                                strictlyInverseˡ (∷↔ _) (from inv (here p)) ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) (from (∼→⊎↔⊎ inv) $ inj₁ p))  ≡⟨ P.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $
+    index-of {xs = x ∷ xs} (to (∷↔ _) (from (∼→⊎↔⊎ inv) $ inj₁ p))  ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $
                                                                                to-from (∼→⊎↔⊎ inv) {x = inj₂ z∈xs} hyp₁ ⟩
     index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₂ z∈xs)                      ≡⟨⟩
     index-of {xs = x ∷ xs} (there z∈xs)                                   ≡⟨⟩
     suc (index-of {xs = xs} z∈xs)                                         ∎
     where
     open Inverse
-    open P.≡-Reasoning
+    open ≡-Reasoning
   ... | ()
 
 ------------------------------------------------------------------------
@@ -573,10 +573,10 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
   from∘to : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ xs) → from p (to p q) ≡ q
   from∘to refl          v∈xs                 = refl
   from∘to (prep _ _)    (here refl)          = refl
-  from∘to (prep _ p)    (there v∈xs)         = P.cong there (from∘to p v∈xs)
+  from∘to (prep _ p)    (there v∈xs)         = ≡.cong there (from∘to p v∈xs)
   from∘to (swap x y p)  (here refl)          = refl
   from∘to (swap x y p)  (there (here refl))  = refl
-  from∘to (swap x y p)  (there (there v∈xs)) = P.cong (there ∘ there) (from∘to p v∈xs)
+  from∘to (swap x y p)  (there (there v∈xs)) = ≡.cong (there ∘ there) (from∘to p v∈xs)
   from∘to (trans p₁ p₂) v∈xs
     rewrite from∘to p₂ (Any-resp-↭ p₁ v∈xs)
           | from∘to p₁ v∈xs                  = refl
@@ -588,7 +588,7 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
 ∼bag⇒↭ : _∼[ bag ]_ ⇒ _↭_ {A = A}
 ∼bag⇒↭ {A = A} {[]} eq with empty-unique (↔-sym eq)
 ... | refl = refl
-∼bag⇒↭ {A = A} {x ∷ xs} eq with ∈-∃++ (Inverse.to (eq {x}) (here P.refl))
+∼bag⇒↭ {A = A} {x ∷ xs} eq with ∈-∃++ (Inverse.to (eq {x}) (here ≡.refl))
 ... | zs₁ , zs₂ , p rewrite p = begin
   x ∷ xs           <⟨ ∼bag⇒↭ (drop-cons (↔-trans eq (comm zs₁ (x ∷ zs₂)))) ⟩
   x ∷ (zs₂ ++ zs₁) <⟨ ++-comm zs₂ zs₁ ⟩
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid.agda b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
index edcf62b6da..4409e78cc4 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid.agda
@@ -24,7 +24,7 @@ open import Data.List.Relation.Binary.Equality.Setoid S
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Level using (_⊔_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 private
   module Eq = Setoid S
@@ -88,7 +88,7 @@ steps (trans xs↭ys ys↭zs) = steps xs↭ys + steps ys↭zs
 
 module PermutationReasoning where
 
-  private module Base = SetoidReasoning ↭-setoid
+  private module Base = ≈-Reasoning ↭-setoid
 
   open Base public
     hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨)
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index 71313d087e..7b498f2a45 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -38,7 +38,7 @@ open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Bundles using (Preorder)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; _≗_; isEquivalence; subst; resp; refl; setoid; module ≡-Reasoning)
-import Relation.Binary.Reasoning.Preorder as PreorderReasoning
+import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 
 private
   open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index 0c344758b7..bc1a693a2f 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -31,7 +31,7 @@ open import Relation.Binary.Definitions
   using (Reflexive; Transitive; _Respectsʳ_; _Respectsˡ_; _Respects_)
 open import Relation.Binary.Bundles using (Setoid; Preorder)
 open import Relation.Binary.Structures using (IsPreorder)
-import Relation.Binary.Reasoning.Preorder as PreorderReasoning
+import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 open import Relation.Binary.Reasoning.Syntax
 
 open Setoid using (Carrier)
@@ -115,7 +115,7 @@ module _ (S : Setoid a ℓ) where
 module ⊆-Reasoning (S : Setoid a ℓ) where
   open Membership S using (_∈_)
 
-  private module Base = PreorderReasoning (⊆-preorder S)
+  private module Base = ≲-Reasoning (⊆-preorder S)
 
   open Base public
     hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-≲; step-∼)
diff --git a/src/Data/Nat/Divisibility.agda b/src/Data/Nat/Divisibility.agda
index f980291cc4..c5d42a2c90 100644
--- a/src/Data/Nat/Divisibility.agda
+++ b/src/Data/Nat/Divisibility.agda
@@ -23,7 +23,7 @@ open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder)
 open import Relation.Binary.Definitions
   using (Reflexive; Transitive; Antisymmetric; Decidable)
-import Relation.Binary.Reasoning.Preorder as PreorderReasoning
+import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; _≢_; refl; sym; cong; subst)
 open import Relation.Binary.Reasoning.Syntax
@@ -151,7 +151,7 @@ n@(suc _) ∣? m  = Dec.map (m%n≡0⇔n∣m m n) (m % n ≟ 0)
 -- A reasoning module for the _∣_ relation
 
 module ∣-Reasoning where
-  private module Base = PreorderReasoning ∣-preorder
+  private module Base = ≲-Reasoning ∣-preorder
 
   open Base public
     hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 5a18639aae..2174d29c74 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -42,7 +42,7 @@ open import Relation.Binary.Definitions
 import Relation.Binary.Consequences as BC
 open import Relation.Binary.PropositionalEquality
 import Relation.Binary.Properties.Poset as PosetProperties
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 open import Relation.Binary.Reasoning.Syntax
 
 open import Algebra.Properties.CommutativeSemigroup ℤ.*-commutativeSemigroup
@@ -153,7 +153,7 @@ proj₂ (≄-tight p q) p≃q p≄q = p≄q p≃q
   { isDecEquivalence = ≃-isDecEquivalence
   }
 
-module ≃-Reasoning = SetoidReasoning ≃-setoid
+module ≃-Reasoning = ≈-Reasoning ≃-setoid
 
 ↥p≡0⇒p≃0 : ∀ p → ↥ p ≡ 0ℤ → p ≃ 0ℚᵘ
 ↥p≡0⇒p≃0 p ↥p≡0 = *≡* (cong (ℤ._* (↧ 0ℚᵘ)) ↥p≡0)
diff --git a/src/Function/Properties/Bijection.agda b/src/Function/Properties/Bijection.agda
index 1028e694e8..50c84ce3c9 100644
--- a/src/Function/Properties/Bijection.agda
+++ b/src/Function/Properties/Bijection.agda
@@ -13,7 +13,7 @@ open import Level using (Level)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions using (Reflexive; Trans)
-open import Relation.Binary.PropositionalEquality as P using (setoid)
+open import Relation.Binary.PropositionalEquality as ≡ using (setoid)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
 open import Function.Properties.Surjection using (injective⇒to⁻-cong)
@@ -37,7 +37,7 @@ refl = Identity.bijection _
 
 -- Can't prove full symmetry as we have no proof that the witness
 -- produced by the surjection proof preserves equality
-sym-≡ : Bijection S (P.setoid B) → Bijection (P.setoid B) S
+sym-≡ : Bijection S (≡.setoid B) → Bijection (≡.setoid B) S
 sym-≡ = Symmetry.bijection-≡
 
 trans : Trans (Bijection {a} {ℓ₁} {b} {ℓ₂}) (Bijection {b} {ℓ₂} {c} {ℓ₃}) Bijection
diff --git a/src/Function/Properties/Equivalence.agda b/src/Function/Properties/Equivalence.agda
index ad1264ec7c..5872569945 100644
--- a/src/Function/Properties/Equivalence.agda
+++ b/src/Function/Properties/Equivalence.agda
@@ -14,7 +14,7 @@ open import Level
 open import Relation.Binary.Definitions
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
-import Relation.Binary.PropositionalEquality.Properties as Eq
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 import Function.Construct.Identity as Identity
 import Function.Construct.Symmetry as Symmetry
@@ -83,7 +83,7 @@ setoid s₁ s₂ = record
 
 ⇔-isEquivalence : IsEquivalence {ℓ = ℓ} _⇔_
 ⇔-isEquivalence = record
-  { refl = λ {x} → Identity.equivalence (Eq.setoid x)
+  { refl = λ {x} → Identity.equivalence (≡.setoid x)
   ; sym = Symmetry.equivalence
   ; trans = Composition.equivalence
   }
diff --git a/src/Function/Properties/Injection.agda b/src/Function/Properties/Injection.agda
index 0e773cf9b8..b191513549 100644
--- a/src/Function/Properties/Injection.agda
+++ b/src/Function/Properties/Injection.agda
@@ -16,9 +16,7 @@ import Function.Construct.Composition as Compose
 open import Level using (Level)
 open import Data.Product.Base using (proj₁; proj₂)
 open import Relation.Binary.Definitions
-open import Relation.Binary.PropositionalEquality using ()
 open import Relation.Binary using (Setoid)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
 
 private
   variable
diff --git a/src/Function/Properties/Inverse.agda b/src/Function/Properties/Inverse.agda
index 5ade85d843..0579d70a62 100644
--- a/src/Function/Properties/Inverse.agda
+++ b/src/Function/Properties/Inverse.agda
@@ -16,9 +16,9 @@ open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (REL)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 import Function.Consequences.Setoid as Consequences
 
 import Function.Construct.Identity as Identity
@@ -148,8 +148,8 @@ module _ (ext : ∀ {a b} → Extensionality a b) where
   ↔-fun A↔B C↔D = mk↔ₛ′
     (λ a→c b → to C↔D (a→c (from A↔B b)))
     (λ b→d a → from C↔D (b→d (to A↔B a)))
-    (λ b→d → ext λ _ → P.trans (strictlyInverseˡ C↔D _ ) (P.cong b→d (strictlyInverseˡ A↔B _)))
-    (λ a→c → ext λ _ → P.trans (strictlyInverseʳ C↔D _ ) (P.cong a→c (strictlyInverseʳ A↔B _)))
+    (λ b→d → ext λ _ → ≡.trans (strictlyInverseˡ C↔D _ ) (≡.cong b→d (strictlyInverseˡ A↔B _)))
+    (λ a→c → ext λ _ → ≡.trans (strictlyInverseʳ C↔D _ ) (≡.cong a→c (strictlyInverseʳ A↔B _)))
     where open Inverse
 
 module _ (I : Inverse S T) where
diff --git a/src/Function/Properties/Surjection.agda b/src/Function/Properties/Surjection.agda
index 6f0bc11e6b..99e13c218d 100644
--- a/src/Function/Properties/Surjection.agda
+++ b/src/Function/Properties/Surjection.agda
@@ -15,10 +15,10 @@ import Function.Construct.Identity as Identity
 import Function.Construct.Composition as Compose
 open import Level using (Level)
 open import Data.Product.Base using (proj₁; proj₂)
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality as ≡
 open import Relation.Binary.Definitions
 open import Relation.Binary.Bundles using (Setoid)
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 private
   variable
@@ -44,7 +44,7 @@ mkSurjection f surjective = record
 ↠⇒⟶ = Surjection.function
 
 ↠⇒↪ : A ↠ B → B ↪ A
-↠⇒↪ s = mk↪ {from = to} λ { P.refl → proj₂ (strictlySurjective _)}
+↠⇒↪ s = mk↪ {from = to} λ { ≡.refl → proj₂ (strictlySurjective _)}
   where open Surjection s
 
 ↠⇒⇔ : A ↠ B → A ⇔ B
@@ -75,6 +75,6 @@ injective⇒to⁻-cong {T = T} surj injective {x} {y} x≈y = injective $ begin
   y          ≈⟨ to∘to⁻ y ⟨
   to (to⁻ y) ∎
   where
-  open SetoidReasoning T
+  open ≈-Reasoning T
   open Surjection surj
 
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
index a78d14938b..a68071d25e 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties.agda
@@ -14,10 +14,10 @@ open import Relation.Binary.Bundles using (Preorder)
 open import Relation.Binary.Structures using (IsPreorder)
 open import Relation.Binary.Definitions using (Transitive; Reflexive)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
-open import Relation.Binary.PropositionalEquality.Core as PropEq
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; refl; sym; cong; cong₂)
-import Relation.Binary.PropositionalEquality.Properties as PropEq
-import Relation.Binary.Reasoning.Preorder as PreorderReasoning
+import Relation.Binary.PropositionalEquality.Properties as ≡
+import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 open import Relation.Binary.Reasoning.Syntax
 
 ------------------------------------------------------------------------
@@ -82,7 +82,7 @@ fold-◅◅ P _⊕_ ∅ left-unit assoc (x ◅ xs) ys = begin
                                                    fold-◅◅ P _⊕_ ∅ left-unit assoc xs ys ⟩
   (x ⊕ (fold P _⊕_ ∅ xs  ⊕ fold P _⊕_ ∅ ys))  ≡⟨ sym (assoc x _ _) ⟩
   ((x ⊕ fold P _⊕_ ∅ xs) ⊕ fold P _⊕_ ∅ ys)   ∎
-  where open PropEq.≡-Reasoning
+  where open ≡.≡-Reasoning
 
 ------------------------------------------------------------------------
 -- Relational properties
@@ -97,7 +97,7 @@ module _ {i t} {I : Set i} (T : Rel I t) where
 
   isPreorder : IsPreorder _≡_ (Star T)
   isPreorder = record
-    { isEquivalence = PropEq.isEquivalence
+    { isEquivalence = ≡.isEquivalence
     ; reflexive     = reflexive
     ; trans         = trans
     }
@@ -113,7 +113,7 @@ module _ {i t} {I : Set i} (T : Rel I t) where
 -- Preorder reasoning for Star
 
 module StarReasoning {i t} {I : Set i} (T : Rel I t) where
-  private module Base = PreorderReasoning (preorder T)
+  private module Base = ≲-Reasoning (preorder T)
 
   open Base public
     hiding (step-≈; step-≈˘; step-≈-⟩; step-≈-⟨; step-∼; step-≲)
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Left.agda b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
index c4846142ec..47465e8926 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Left.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
@@ -18,7 +18,7 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Definitions
   using (Symmetric; Transitive; Reflexive; Antisymmetric; Total; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Decidable)
 open import Relation.Nullary.Negation using (¬_)
-import Relation.Binary.Reasoning.Setoid as EqReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 open import Relation.Binary.Lattice using (Infimum)
 
 module Relation.Binary.Construct.NaturalOrder.Left
@@ -44,7 +44,7 @@ reflexive magma idem {x} {y} x≈y = begin
   x     ≈⟨ sym (idem x) ⟩
   x ∙ x ≈⟨ ∙-cong refl x≈y ⟩
   x ∙ y ∎
-  where open IsMagma magma; open EqReasoning setoid
+  where open IsMagma magma; open ≈-Reasoning setoid
 
 refl : Symmetric _≈_ → Idempotent _∙_ → Reflexive _≤_
 refl sym idem {x} = sym (idem x)
@@ -55,7 +55,7 @@ antisym isEq comm {x} {y} x≤y y≤x = begin
   x ∙ y ≈⟨ comm x y ⟩
   y ∙ x ≈⟨ sym y≤x ⟩
   y     ∎
-  where open IsEquivalence isEq; open EqReasoning (record { isEquivalence = isEq })
+  where open IsEquivalence isEq; open ≈-Reasoning (record { isEquivalence = isEq })
 
 total : Symmetric _≈_ → Transitive _≈_ → Selective _∙_ → Commutative _∙_ → Total _≤_
 total sym trans sel comm x y with sel x y
@@ -69,14 +69,14 @@ trans semi {x} {y} {z} x≤y y≤z = begin
   x ∙ (y ∙ z) ≈⟨ sym (assoc x y z) ⟩
   (x ∙ y) ∙ z ≈⟨ ∙-cong (sym x≤y) S.refl ⟩
   x ∙ z       ∎
-  where open module S = IsSemigroup semi; open EqReasoning S.setoid
+  where open module S = IsSemigroup semi; open ≈-Reasoning S.setoid
 
 respʳ : IsMagma _∙_ → _≤_ Respectsʳ _≈_
 respʳ magma {x} {y} {z} y≈z x≤y = begin
   x     ≈⟨ x≤y ⟩
   x ∙ y ≈⟨ ∙-cong M.refl y≈z ⟩
   x ∙ z ∎
-  where open module M = IsMagma magma; open EqReasoning M.setoid
+  where open module M = IsMagma magma; open ≈-Reasoning M.setoid
 
 respˡ : IsMagma _∙_ → _≤_ Respectsˡ _≈_
 respˡ magma {x} {y} {z} y≈z y≤x = begin
@@ -84,7 +84,7 @@ respˡ magma {x} {y} {z} y≈z y≤x = begin
   y     ≈⟨ y≤x ⟩
   y ∙ x ≈⟨ ∙-cong y≈z M.refl ⟩
   z ∙ x ∎
-  where open module M = IsMagma magma; open EqReasoning M.setoid
+  where open module M = IsMagma magma; open ≈-Reasoning M.setoid
 
 resp₂ : IsMagma _∙_ →  _≤_ Respects₂ _≈_
 resp₂ magma = respʳ magma , respˡ magma
@@ -95,7 +95,7 @@ dec _≟_ x y = x ≟ (x ∙ y)
 module _ (semi : IsSemilattice _∙_) where
 
   private open module S = IsSemilattice semi
-  open EqReasoning setoid
+  open ≈-Reasoning setoid
 
   x∙y≤x : ∀ x y → (x ∙ y) ≤ x
   x∙y≤x x y = begin
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Right.agda b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
index 3c2aa348d6..76529350e2 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Right.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
@@ -19,7 +19,7 @@ open import Relation.Binary.Definitions
   using (Symmetric; Transitive; Reflexive; Antisymmetric; Total; _Respectsʳ_; _Respectsˡ_; _Respects₂_; Decidable)
 
 open import Relation.Nullary.Negation using (¬_)
-import Relation.Binary.Reasoning.Setoid as EqReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 module Relation.Binary.Construct.NaturalOrder.Right
   {a ℓ} {A : Set a} (_≈_ : Rel A ℓ) (_∙_ : Op₂ A) where
@@ -44,7 +44,7 @@ reflexive magma idem {x} {y} x≈y = begin
   x     ≈⟨ sym (idem x) ⟩
   x ∙ x ≈⟨ ∙-cong x≈y refl ⟩
   y ∙ x ∎
-  where open IsMagma magma; open EqReasoning setoid
+  where open IsMagma magma; open ≈-Reasoning setoid
 
 refl : Symmetric _≈_ → Idempotent _∙_ → Reflexive _≤_
 refl sym idem {x} = sym (idem x)
@@ -55,7 +55,7 @@ antisym isEq comm {x} {y} x≤y y≤x = begin
   y ∙ x ≈⟨  comm y x ⟩
   x ∙ y ≈⟨ y≤x ⟨
   y     ∎
-  where open EqReasoning (record { isEquivalence = isEq })
+  where open ≈-Reasoning (record { isEquivalence = isEq })
 
 total : Symmetric _≈_ → Transitive _≈_ → Selective _∙_ → Commutative _∙_ → Total _≤_
 total sym trans sel comm x y with sel x y
@@ -69,14 +69,14 @@ trans semi {x} {y} {z} x≤y y≤z = begin
   (z ∙ y) ∙ x ≈⟨ assoc z y x ⟩
   z ∙ (y ∙ x) ≈⟨ ∙-cong S.refl (sym x≤y) ⟩
   z ∙ x       ∎
-  where open module S = IsSemigroup semi; open EqReasoning S.setoid
+  where open module S = IsSemigroup semi; open ≈-Reasoning S.setoid
 
 respʳ : IsMagma _∙_ → _≤_ Respectsʳ _≈_
 respʳ magma {x} {y} {z} y≈z x≤y = begin
   x     ≈⟨ x≤y ⟩
   y ∙ x ≈⟨ ∙-cong y≈z M.refl ⟩
   z ∙ x ∎
-  where open module M = IsMagma magma; open EqReasoning M.setoid
+  where open module M = IsMagma magma; open ≈-Reasoning M.setoid
 
 respˡ : IsMagma _∙_ → _≤_ Respectsˡ _≈_
 respˡ magma {x} {y} {z} y≈z y≤x = begin
@@ -84,7 +84,7 @@ respˡ magma {x} {y} {z} y≈z y≤x = begin
   y     ≈⟨ y≤x ⟩
   x ∙ y ≈⟨ ∙-cong M.refl y≈z ⟩
   x ∙ z ∎
-  where open module M = IsMagma magma; open EqReasoning M.setoid
+  where open module M = IsMagma magma; open ≈-Reasoning M.setoid
 
 resp₂ : IsMagma _∙_ →  _≤_ Respects₂ _≈_
 resp₂ magma = respʳ magma , respˡ magma
diff --git a/src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda b/src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda
index e7090dc6c5..4727088718 100644
--- a/src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda
+++ b/src/Relation/Binary/Lattice/Properties/HeytingAlgebra.agda
@@ -19,13 +19,13 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (_$_; flip; _∘_)
 open import Level using (_⊔_)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-import Relation.Binary.Reasoning.PartialOrder as POR
+import Relation.Binary.Reasoning.PartialOrder as ≤-Reasoning
 open import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice
 open import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice
 import Relation.Binary.Lattice.Properties.BoundedMeetSemilattice boundedMeetSemilattice as BM
 open import Relation.Binary.Lattice.Properties.Lattice lattice
 open import Relation.Binary.Lattice.Properties.BoundedLattice boundedLattice
-import Relation.Binary.Reasoning.Setoid as EqReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- Useful lemmas
@@ -62,7 +62,7 @@ y≤x⇨y = transpose-⇨ (x∧y≤x _ _)
   x ⇨ u ≤⟨ ⇨ʳ-covariant u≤v ⟩
   x ⇨ v ≤⟨ ⇨ˡ-contravariant y≤x ⟩
   y ⇨ v ∎
-  where open POR poset
+  where open ≤-Reasoning poset
 
 ⇨-cong : _⇨_ Preserves₂ _≈_ ⟶ _≈_ ⟶ _≈_
 ⇨-cong x≈y u≈v = antisym (⇨-relax (reflexive $ Eq.sym x≈y) (reflexive u≈v))
@@ -113,7 +113,7 @@ y≤x⇨y = transpose-⇨ (x∧y≤x _ _)
   (((x ⇨ y) ∧ x) ∧ (x ⇨ z)) ∧ x  ≈⟨ ∧-assoc _ _ _ ⟩
   (((x ⇨ y) ∧ x) ∧ (x ⇨ z)  ∧ x) ≤⟨ ∧-monotonic ⇨-eval ⇨-eval ⟩
   y ∧ z                          ∎)
-  where open POR poset
+  where open ≤-Reasoning poset
 
 ⇨-distribˡ-∧ : _⇨_ DistributesOverˡ _∧_
 ⇨-distribˡ-∧ x y z = antisym (⇨-distribˡ-∧-≤ x y z) (⇨-distribˡ-∧-≥ x y z)
@@ -175,7 +175,7 @@ de-morgan₂-≤ x y = transpose-⇨ $ begin
       ¬ ¬ y ∧ ¬ y               ≤⟨ ⇨-eval ⟩
       ⊥                         ∎ ⟩
   ⊥                             ∎
-  where open POR poset
+  where open ≤-Reasoning poset
 
 de-morgan₂-≥ : ∀ x y → ¬ ¬ (¬ x ∨ ¬ y) ≤ ¬ (x ∧ y)
 de-morgan₂-≥ x y = transpose-⇨ $ ⇨-applyˡ $ transpose-⇨ $ begin
@@ -184,7 +184,7 @@ de-morgan₂-≥ x y = transpose-⇨ $ ⇨-applyˡ $ transpose-⇨ $ begin
                                                (⇨-applyʳ (x∧y≤y _ _)) ⟩
   ⊥ ∨ ⊥                         ≈⟨ ∨-idempotent _ ⟩
   ⊥                             ∎
-  where open POR poset
+  where open ≤-Reasoning poset
 
 de-morgan₂ : ∀ x y → ¬ (x ∧ y) ≈ ¬ ¬ (¬ x ∨ ¬ y)
 de-morgan₂ x y = antisym (de-morgan₂-≤ x y) (de-morgan₂-≥ x y)
@@ -196,4 +196,4 @@ weak-lem {x} = begin
   ⊥ ∧ (x ⇨ ⊥) ⇨ ⊥ ≈⟨ ⇨-cong (∧-zeroˡ _) Eq.refl ⟩
   ⊥ ⇨ ⊥           ≈⟨ ⇨-unit ⟩
   ⊤               ∎
-  where open EqReasoning setoid
+  where open ≈-Reasoning setoid
diff --git a/src/Relation/Binary/Lattice/Properties/Lattice.agda b/src/Relation/Binary/Lattice/Properties/Lattice.agda
index fd95effaaf..261371a3f0 100644
--- a/src/Relation/Binary/Lattice/Properties/Lattice.agda
+++ b/src/Relation/Binary/Lattice/Properties/Lattice.agda
@@ -21,8 +21,8 @@ open import Function.Base using (flip)
 open import Relation.Binary.Properties.Poset poset
 import Relation.Binary.Lattice.Properties.JoinSemilattice joinSemilattice as J
 import Relation.Binary.Lattice.Properties.MeetSemilattice meetSemilattice as M
-import Relation.Binary.Reasoning.Setoid as EqReasoning
-import Relation.Binary.Reasoning.PartialOrder as POR
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
+import Relation.Binary.Reasoning.PartialOrder as ≤-Reasoning
 
 ∨-absorbs-∧ : _∨_ Absorbs _∧_
 ∨-absorbs-∧ x y =
@@ -44,7 +44,7 @@ absorptive = ∨-absorbs-∧ , ∧-absorbs-∨
   x ∧ y ≤⟨ x∧y≤x x y ⟩
   x     ≤⟨ x≤x∨y x y ⟩
   x ∨ y ∎
-  where open POR poset
+  where open ≤-Reasoning poset
 
 -- two quadrilateral arguments
 
@@ -55,14 +55,14 @@ quadrilateral₁ {x} {y} x∨y≈x = begin
   y ∧ (x ∨ y) ≈⟨ M.∧-cong Eq.refl (J.∨-comm _ _) ⟩
   y ∧ (y ∨ x) ≈⟨ ∧-absorbs-∨ _ _ ⟩
   y           ∎
-  where open EqReasoning setoid
+  where open ≈-Reasoning setoid
 
 quadrilateral₂ : ∀ {x y} → x ∧ y ≈ y → x ∨ y ≈ x
 quadrilateral₂ {x} {y} x∧y≈y = begin
   x ∨ y       ≈⟨ J.∨-cong Eq.refl (Eq.sym x∧y≈y) ⟩
   x ∨ (x ∧ y) ≈⟨ ∨-absorbs-∧ _ _ ⟩
   x           ∎
-  where open EqReasoning setoid
+  where open ≈-Reasoning setoid
 
 -- collapsing sublattice
 
@@ -75,7 +75,7 @@ collapse₁ {x} {y} x≈y = begin
   x ∨ y ∎
   where
   y≤x = reflexive (Eq.sym x≈y)
-  open EqReasoning setoid
+  open ≈-Reasoning setoid
 
 -- this can also be proved by quadrilateral argument, but it's much less symmetric.
 collapse₂ : ∀ {x y} → x ∨ y ≤ x ∧ y → x ≈ y
@@ -88,7 +88,7 @@ collapse₂ {x} {y} ∨≤∧ = antisym
          x ∨ y ≤⟨ ∨≤∧ ⟩
          x ∧ y ≤⟨ x∧y≤x _ _ ⟩
          x     ∎)
-  where open POR poset
+  where open ≤-Reasoning poset
 
 ------------------------------------------------------------------------
 -- The dual construction is also a lattice.
diff --git a/src/Relation/Binary/Reasoning/Base/Apartness.agda b/src/Relation/Binary/Reasoning/Base/Apartness.agda
index 3d4d72f2c8..22eee2d505 100644
--- a/src/Relation/Binary/Reasoning/Base/Apartness.agda
+++ b/src/Relation/Binary/Reasoning/Base/Apartness.agda
@@ -13,7 +13,7 @@ open import Relation.Nullary.Decidable using (Dec; yes; no)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions using (Reflexive; Transitive; Symmetric; Trans)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
 
 
@@ -38,8 +38,8 @@ data _IsRelatedTo_ (x y : A) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
 
 ≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
 ≡-go x≡y nothing = nothing
-≡-go x≡y (apartness y#z) = apartness (case x≡y of λ where P.refl → y#z)
-≡-go x≡y (equals y≈z) = equals (case x≡y of λ where P.refl → y≈z)
+≡-go x≡y (apartness y#z) = apartness (case x≡y of λ where ≡.refl → y#z)
+≡-go x≡y (equals y≈z) = equals (case x≡y of λ where ≡.refl → y≈z)
 
 ≈-go  : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_
 ≈-go x≈y nothing         = nothing
diff --git a/src/Relation/Binary/Reasoning/Base/Double.agda b/src/Relation/Binary/Reasoning/Base/Double.agda
index 6fa901d573..b5fd9f66e4 100644
--- a/src/Relation/Binary/Reasoning/Base/Double.agda
+++ b/src/Relation/Binary/Reasoning/Base/Double.agda
@@ -16,7 +16,7 @@ open import Relation.Nullary.Decidable.Core using (Dec; yes; no)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Definitions using (Reflexive; Trans)
 open import Relation.Binary.Structures using (IsPreorder)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
 
 
@@ -40,8 +40,8 @@ start (equals x≈y) = reflexive x≈y
 start (nonstrict x≲y) = x≲y
 
 ≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
-≡-go x≡y (equals y≈z) = equals (case x≡y of λ where P.refl → y≈z)
-≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where P.refl → y≤z)
+≡-go x≡y (equals y≈z) = equals (case x≡y of λ where ≡.refl → y≈z)
+≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where ≡.refl → y≤z)
 
 ≲-go : Trans _≲_ _IsRelatedTo_ _IsRelatedTo_
 ≲-go x≲y (equals y≈z) = nonstrict (∼-respʳ-≈ y≈z x≲y)
diff --git a/src/Relation/Binary/Reasoning/Base/Partial.agda b/src/Relation/Binary/Reasoning/Base/Partial.agda
index bd1fcd8693..3d24edf501 100644
--- a/src/Relation/Binary/Reasoning/Base/Partial.agda
+++ b/src/Relation/Binary/Reasoning/Base/Partial.agda
@@ -11,7 +11,7 @@ open import Level using (_⊔_)
 open import Relation.Binary.Core
 open import Relation.Binary.Definitions
 open import Relation.Nullary.Decidable using (Dec; yes; no)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
 
 
diff --git a/src/Relation/Binary/Reasoning/Base/Single.agda b/src/Relation/Binary/Reasoning/Base/Single.agda
index ccbf63dda3..2f34ee8525 100644
--- a/src/Relation/Binary/Reasoning/Base/Single.agda
+++ b/src/Relation/Binary/Reasoning/Base/Single.agda
@@ -10,7 +10,7 @@ open import Level using (_⊔_)
 open import Function.Base using (case_of_)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Definitions using (Reflexive; Transitive; Trans)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
 
 module Relation.Binary.Reasoning.Base.Single
@@ -36,7 +36,7 @@ start (relTo x∼y) = x∼y
 ∼-go x∼y (relTo y∼z) = relTo (trans x∼y y∼z)
 
 ≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
-≡-go x≡y (relTo y∼z) = relTo (case x≡y of λ where P.refl → y∼z)
+≡-go x≡y (relTo y∼z) = relTo (case x≡y of λ where ≡.refl → y∼z)
 
 stop : Reflexive _IsRelatedTo_
 stop = relTo refl
diff --git a/src/Relation/Binary/Reasoning/Base/Triple.agda b/src/Relation/Binary/Reasoning/Base/Triple.agda
index 460115eff9..6afb937497 100644
--- a/src/Relation/Binary/Reasoning/Base/Triple.agda
+++ b/src/Relation/Binary/Reasoning/Base/Triple.agda
@@ -19,7 +19,7 @@ open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Structures using (IsPreorder)
 open import Relation.Binary.Definitions
   using (Transitive; _Respects₂_; Reflexive; Trans; Irreflexive; Asymmetric)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
 
 module Relation.Binary.Reasoning.Base.Triple {a ℓ₁ ℓ₂ ℓ₃} {A : Set a}
@@ -48,9 +48,9 @@ start (nonstrict x≤y) = x≤y
 start (strict x<y) = <⇒≤ x<y
 
 ≡-go : Trans _≡_ _IsRelatedTo_ _IsRelatedTo_
-≡-go x≡y (equals y≈z) = equals (case x≡y of λ where P.refl → y≈z)
-≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where P.refl → y≤z)
-≡-go x≡y (strict y<z) = strict (case x≡y of λ where P.refl → y<z)
+≡-go x≡y (equals y≈z) = equals (case x≡y of λ where ≡.refl → y≈z)
+≡-go x≡y (nonstrict y≤z) = nonstrict (case x≡y of λ where ≡.refl → y≤z)
+≡-go x≡y (strict y<z) = strict (case x≡y of λ where ≡.refl → y<z)
 
 ≈-go : Trans _≈_ _IsRelatedTo_ _IsRelatedTo_
 ≈-go x≈y (equals y≈z) = equals (Eq.trans x≈y y≈z)
diff --git a/src/Relation/Binary/Reasoning/MultiSetoid.agda b/src/Relation/Binary/Reasoning/MultiSetoid.agda
index aff53dd984..5665472508 100644
--- a/src/Relation/Binary/Reasoning/MultiSetoid.agda
+++ b/src/Relation/Binary/Reasoning/MultiSetoid.agda
@@ -31,7 +31,7 @@ open import Function.Base using (case_of_)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Definitions using (Trans; Reflexive)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
 
 private
@@ -51,7 +51,7 @@ module _ (S : Setoid a ℓ) where
   start (relTo x≈y) = x≈y
 
   ≡-go : Trans _≡_ IsRelatedTo IsRelatedTo
-  ≡-go x≡y (relTo y∼z) = relTo (case x≡y of λ where P.refl → y∼z)
+  ≡-go x≡y (relTo y∼z) = relTo (case x≡y of λ where ≡.refl → y∼z)
 
   ≈-go : Trans _≈_ IsRelatedTo IsRelatedTo
   ≈-go x≈y (relTo y≈z) = relTo (trans x≈y y≈z)
diff --git a/src/Relation/Binary/Reasoning/PartialSetoid.agda b/src/Relation/Binary/Reasoning/PartialSetoid.agda
index 1149f1a298..5163c99947 100644
--- a/src/Relation/Binary/Reasoning/PartialSetoid.agda
+++ b/src/Relation/Binary/Reasoning/PartialSetoid.agda
@@ -15,14 +15,14 @@ module Relation.Binary.Reasoning.PartialSetoid
 open PartialSetoid S
 
 import Relation.Binary.Reasoning.Base.Partial _≈_ trans
-  as PartialReasoning
+  as ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- Reasoning combinators
 
 -- Export the combinators for partial relation reasoning, hiding the
 -- single misnamed combinator.
-open PartialReasoning public hiding (step-∼)
+open ≈-Reasoning public hiding (step-∼)
 
 -- Re-export the equality-based combinators instead
-open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ PartialReasoning.∼-go sym public
+open ≈-syntax _IsRelatedTo_ _IsRelatedTo_ ≈-Reasoning.∼-go sym public
diff --git a/src/Relation/Binary/Reasoning/Syntax.agda b/src/Relation/Binary/Reasoning/Syntax.agda
index 4806512296..809d238b39 100644
--- a/src/Relation/Binary/Reasoning/Syntax.agda
+++ b/src/Relation/Binary/Reasoning/Syntax.agda
@@ -12,7 +12,7 @@ open import Relation.Nullary.Decidable.Core
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Binary.Core using (Rel; REL; _⇒_)
 open import Relation.Binary.Definitions
-open import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_)
 
 -- List of `Reasoning` modules that do not use this framework and so
@@ -395,7 +395,7 @@ module ≡-syntax
   step-≡-∣ : ∀ x {y} → R x y → R x y
   step-≡-∣ x xRy = xRy
 
-  step-≡-⟨ = backward R R step P.sym
+  step-≡-⟨ = backward R R step ≡.sym
 
   syntax step-≡-⟩ x yRz x≡y = x ≡⟨ x≡y ⟩ yRz
   syntax step-≡-∣ x xRy     = x ≡⟨⟩ xRy
@@ -424,7 +424,7 @@ module ≡-noncomputing-syntax (R : REL A B ℓ₁) where
 
   private
     step : Trans _≡_ R R
-    step P.refl xRy = xRy
+    step ≡.refl xRy = xRy
 
   open ≡-syntax R step public
 
diff --git a/src/Relation/Binary/Reflection.agda b/src/Relation/Binary/Reflection.agda
index a21288100a..12665df8cc 100644
--- a/src/Relation/Binary/Reflection.agda
+++ b/src/Relation/Binary/Reflection.agda
@@ -14,7 +14,7 @@ open import Function.Base using (id; _⟨_⟩_)
 open import Function.Bundles using (module Equivalence)
 open import Level using (Level)
 open import Relation.Binary.Bundles using (Setoid)
-import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Core as ≡
 
 -- Think of the parameters as follows:
 --
@@ -42,10 +42,10 @@ module Relation.Binary.Reflection
 
 open import Data.Vec.N-ary
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
-import Relation.Binary.Reasoning.Setoid as Eq
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 open Setoid Sem
-open Eq Sem
+open ≈-Reasoning Sem
 
 -- If two normalised expressions are semantically equal, then their
 -- non-normalised forms are also equal.
@@ -91,7 +91,7 @@ solve₁ : ∀ n (f : N-ary n (Expr n) (Expr n × Expr n)) →
                  ⟦ proj₁ (close n f)  ⟧ ρ ≈ ⟦ proj₂ (close n f)  ⟧ ρ)
 solve₁ n f =
   Equivalence.from (uncurry-∀ⁿ n) λ ρ →
-    P.subst id (P.sym (left-inverse (λ _ → _ ≈ _ → _ ≈ _) ρ))
+    ≡.subst id (≡.sym (left-inverse (λ _ → _ ≈ _ → _ ≈ _) ρ))
       (prove ρ (proj₁ (close n f)) (proj₂ (close n f)))
 
 -- A variant of _,_ which is intended to make uses of solve and solve₁
diff --git a/src/Tactic/MonoidSolver.agda b/src/Tactic/MonoidSolver.agda
index c62ea9dccf..4e370bea25 100644
--- a/src/Tactic/MonoidSolver.agda
+++ b/src/Tactic/MonoidSolver.agda
@@ -88,7 +88,7 @@ import Reflection.AST.Name as Name
 open import Reflection.TCM
 open import Reflection.TCM.Syntax
 
-import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- The Expr type with homomorphism proofs
@@ -103,7 +103,7 @@ data Expr {a} (A : Set a) : Set a where
 module _ {m₁ m₂} (monoid : Monoid m₁ m₂) where
 
   open Monoid monoid
-  open SetoidReasoning setoid
+  open ≈-Reasoning setoid
 
   -- Convert the AST to an expression (i.e. evaluate it) without
   -- normalising.

From 95023d561b4b4ac3929d3e6aa4b6840880077030 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 8 Feb 2024 20:21:17 +0000
Subject: [PATCH 222/316] Qualified import of `Data.Product.Base` fixing #2280
 (#2284)

* Qualified import of `Data.Product.Base as Product`

* more modules affected
---
 src/Algebra/Construct/Flip/Op.agda            |  6 +-
 src/Algebra/Construct/Subst/Equality.agda     | 16 ++---
 .../Lattice/Construct/Subst/Equality.agda     |  2 +-
 src/Codata/Musical/Colist.agda                |  6 +-
 src/Codata/Musical/Colist/Infinite-merge.agda |  4 +-
 src/Codata/Sized/Colist/Properties.agda       |  4 +-
 src/Codata/Sized/Cowriter.agda                |  6 +-
 src/Codata/Sized/M/Properties.agda            |  2 +-
 src/Codata/Sized/Stream/Properties.agda       |  8 +--
 src/Data/Container/Core.agda                  |  6 +-
 src/Data/Fin/Properties.agda                  |  4 +-
 src/Data/List/Base.agda                       | 14 ++---
 .../Propositional/Properties/Core.agda        |  4 +-
 src/Data/List/Membership/Setoid.agda          |  4 +-
 .../List/Membership/Setoid/Properties.agda    |  6 +-
 src/Data/List/Nary/NonDependent.agda          |  4 +-
 src/Data/List/Properties.agda                 | 44 +++++++-------
 .../Prefix/Heterogeneous/Properties.agda      |  2 +-
 .../Subset/Propositional/Properties.agda      | 14 ++---
 .../List/Relation/Ternary/Interleaving.agda   | 10 ++--
 src/Data/List/Relation/Unary/All.agda         |  8 +--
 .../List/Relation/Unary/All/Properties.agda   |  8 +--
 src/Data/List/Relation/Unary/Any.agda         |  2 +-
 .../List/Relation/Unary/Any/Properties.agda   | 10 ++--
 src/Data/List/Sort/MergeSort.agda             |  2 +-
 src/Data/Maybe/Relation/Unary/All.agda        |  4 +-
 src/Data/Maybe/Relation/Unary/Any.agda        |  6 +-
 .../Function/Dependent/Propositional.agda     |  6 +-
 .../Product/Function/NonDependent/Setoid.agda | 58 +++++++++----------
 .../Relation/Binary/Pointwise/Dependent.agda  |  2 +-
 .../Binary/Pointwise/NonDependent.agda        | 16 ++---
 src/Data/Sum/Function/Setoid.agda             | 10 ++--
 .../Membership/Propositional/Properties.agda  |  6 +-
 src/Data/Tree/AVL/Map/Relation/Unary/Any.agda | 18 +++---
 src/Data/Tree/AVL/Relation/Unary/Any.agda     |  4 +-
 src/Data/Tree/AVL/Sets.agda                   |  6 +-
 src/Data/Tree/AVL/Sets/Membership.agda        |  2 +-
 .../Tree/AVL/Sets/Membership/Properties.agda  |  2 +-
 src/Data/Trie/NonEmpty.agda                   |  4 +-
 src/Data/Vec/Base.agda                        |  6 +-
 src/Data/Vec/N-ary.agda                       |  4 +-
 src/Data/Vec/Properties.agda                  | 12 ++--
 src/Data/Vec/Recursive.agda                   |  4 +-
 src/Data/Vec/Relation/Unary/All.agda          |  8 +--
 .../Vec/Relation/Unary/All/Properties.agda    |  4 +-
 src/Data/Vec/Relation/Unary/Any.agda          |  4 +-
 .../Vec/Relation/Unary/Any/Properties.agda    |  6 +-
 src/Data/Vec/Relation/Unary/Linked.agda       |  4 +-
 src/Function/Consequences.agda                |  6 +-
 src/Function/Related/TypeIsomorphisms.agda    | 10 ++--
 .../Closure/Reflexive/Properties.agda         |  4 +-
 .../Binary/Construct/Subst/Equality.agda      |  2 +-
 src/Relation/Binary/Properties/Preorder.agda  |  4 +-
 src/Relation/Nary.agda                        |  2 +-
 src/Relation/Nullary/Negation.agda            |  4 +-
 src/Relation/Nullary/Universe.agda            |  4 +-
 56 files changed, 214 insertions(+), 214 deletions(-)

diff --git a/src/Algebra/Construct/Flip/Op.agda b/src/Algebra/Construct/Flip/Op.agda
index 4dd5849a6f..dd139e4ba1 100644
--- a/src/Algebra/Construct/Flip/Op.agda
+++ b/src/Algebra/Construct/Flip/Op.agda
@@ -10,7 +10,7 @@
 module Algebra.Construct.Flip.Op where
 
 open import Algebra
-import Data.Product.Base as Prod
+import Data.Product.Base as Product
 import Data.Sum.Base as Sum
 open import Function.Base using (flip)
 open import Level using (Level)
@@ -39,7 +39,7 @@ module _ (≈ : Rel A ℓ) (∙ : Op₂ A) where
   associative sym assoc x y z = sym (assoc z y x)
 
   identity : Identity ≈ ε ∙ → Identity ≈ ε (flip ∙)
-  identity id = Prod.swap id
+  identity id = Product.swap id
 
   commutative : Commutative ≈ ∙ → Commutative ≈ (flip ∙)
   commutative comm = flip comm
@@ -51,7 +51,7 @@ module _ (≈ : Rel A ℓ) (∙ : Op₂ A) where
   idempotent idem = idem
 
   inverse : Inverse ≈ ε ⁻¹ ∙ → Inverse ≈ ε ⁻¹ (flip ∙)
-  inverse inv = Prod.swap inv
+  inverse inv = Product.swap inv
 
 ------------------------------------------------------------------------
 -- Structures
diff --git a/src/Algebra/Construct/Subst/Equality.agda b/src/Algebra/Construct/Subst/Equality.agda
index 05d0277398..6dc1ced9e2 100644
--- a/src/Algebra/Construct/Subst/Equality.agda
+++ b/src/Algebra/Construct/Subst/Equality.agda
@@ -9,7 +9,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
 open import Relation.Binary.Core
 
 module Algebra.Construct.Subst.Equality
@@ -45,13 +45,13 @@ sel : ∀ {∙} → Selective ≈₁ ∙ → Selective ≈₂ ∙
 sel sel x y = Sum.map to to (sel x y)
 
 identity : ∀ {∙ e} → Identity ≈₁ e ∙ → Identity ≈₂ e ∙
-identity = Prod.map (to ∘_) (to ∘_)
+identity = Product.map (to ∘_) (to ∘_)
 
 inverse : ∀ {∙ e ⁻¹} → Inverse ≈₁ ⁻¹ ∙ e → Inverse ≈₂ ⁻¹ ∙ e
-inverse = Prod.map (to ∘_) (to ∘_)
+inverse = Product.map (to ∘_) (to ∘_)
 
 absorptive : ∀ {∙ ◦} → Absorptive ≈₁ ∙ ◦ → Absorptive ≈₂ ∙ ◦
-absorptive = Prod.map (λ f x y → to (f x y)) (λ f x y → to (f x y))
+absorptive = Product.map (λ f x y → to (f x y)) (λ f x y → to (f x y))
 
 distribˡ : ∀ {∙ ◦} → _DistributesOverˡ_ ≈₁ ∙ ◦ → _DistributesOverˡ_ ≈₂ ∙ ◦
 distribˡ distribˡ x y z = to (distribˡ x y z)
@@ -60,7 +60,7 @@ distribʳ : ∀ {∙ ◦} → _DistributesOverʳ_ ≈₁ ∙ ◦ → _Distribute
 distribʳ distribʳ x y z = to (distribʳ x y z)
 
 distrib : ∀ {∙ ◦} → _DistributesOver_ ≈₁ ∙ ◦ → _DistributesOver_ ≈₂ ∙ ◦
-distrib {∙} {◦} = Prod.map (distribˡ {∙} {◦}) (distribʳ {∙} {◦})
+distrib {∙} {◦} = Product.map (distribˡ {∙} {◦}) (distribʳ {∙} {◦})
 
 ------------------------------------------------------------------------
 -- Structures
@@ -92,7 +92,7 @@ isSelectiveMagma S = record
 isMonoid : ∀ {∙ ε} → IsMonoid ≈₁ ∙ ε → IsMonoid ≈₂ ∙ ε
 isMonoid S = record
   { isSemigroup = isSemigroup S.isSemigroup
-  ; identity    = Prod.map (to ∘_) (to ∘_) S.identity
+  ; identity    = Product.map (to ∘_) (to ∘_) S.identity
   } where module S = IsMonoid S
 
 isCommutativeMonoid : ∀ {∙ ε} →
@@ -113,7 +113,7 @@ isIdempotentCommutativeMonoid {∙} S = record
 isGroup : ∀ {∙ ε ⁻¹} → IsGroup ≈₁ ∙ ε ⁻¹ → IsGroup ≈₂ ∙ ε ⁻¹
 isGroup S = record
   { isMonoid = isMonoid S.isMonoid
-  ; inverse  = Prod.map (to ∘_) (to ∘_) S.inverse
+  ; inverse  = Product.map (to ∘_) (to ∘_) S.inverse
   ; ⁻¹-cong  = cong₁ S.⁻¹-cong
   } where module S = IsGroup S
 
@@ -141,7 +141,7 @@ isSemiringWithoutOne {+} {*} S = record
   ; *-cong                = cong₂ S.*-cong
   ; *-assoc               = assoc {*} S.*-assoc
   ; distrib               = distrib {*} {+} S.distrib
-  ; zero                  = Prod.map (to ∘_) (to ∘_) S.zero
+  ; zero                  = Product.map (to ∘_) (to ∘_) S.zero
   } where module S = IsSemiringWithoutOne S
 
 isCommutativeSemiringWithoutOne : ∀ {+ * 0#} →
diff --git a/src/Algebra/Lattice/Construct/Subst/Equality.agda b/src/Algebra/Lattice/Construct/Subst/Equality.agda
index e756319aaf..8a498afbb2 100644
--- a/src/Algebra/Lattice/Construct/Subst/Equality.agda
+++ b/src/Algebra/Lattice/Construct/Subst/Equality.agda
@@ -12,7 +12,7 @@
 open import Algebra.Core using (Op₂)
 open import Algebra.Definitions
 open import Algebra.Lattice.Structures
-open import Data.Product.Base as Prod
+open import Data.Product.Base using (_,_)
 open import Function.Base
 open import Relation.Binary.Core
 
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index 18d250f566..c65d33e8bf 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -21,7 +21,7 @@ open import Data.Maybe.Relation.Unary.Any using (just)
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.List.Base using (List; []; _∷_)
 open import Data.List.NonEmpty using (List⁺; _∷_)
-open import Data.Product.Base as Prod using (∃; _×_; _,_)
+open import Data.Product.Base as Product using (∃; _×_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Vec.Bounded as Vec≤ using (Vec≤)
 open import Function.Base
@@ -108,7 +108,7 @@ Any-∈ {P = P} = mk↔ₛ′
   where
   to : ∀ {xs} → Any P xs → ∃ λ x → x ∈ xs × P x
   to (here  p) = _ , here P.refl , p
-  to (there p) = Prod.map id (Prod.map there id) (to p)
+  to (there p) = Product.map id (Product.map there id) (to p)
 
   from : ∀ {x xs} → x ∈ xs → P x → Any P xs
   from (here P.refl) p = here p
@@ -118,7 +118,7 @@ Any-∈ {P = P} = mk↔ₛ′
             to (from x∈xs p) ≡ (x , x∈xs , p)
   to∘from (here P.refl) p = P.refl
   to∘from (there x∈xs)  p =
-    P.cong (Prod.map id (Prod.map there id)) (to∘from x∈xs p)
+    P.cong (Product.map id (Product.map there id)) (to∘from x∈xs p)
 
   from∘to : ∀ {xs} (p : Any P xs) →
             let (x , x∈xs , px) = to p in from x∈xs px ≡ p
diff --git a/src/Codata/Musical/Colist/Infinite-merge.agda b/src/Codata/Musical/Colist/Infinite-merge.agda
index 584880a768..ecf393ca83 100644
--- a/src/Codata/Musical/Colist/Infinite-merge.agda
+++ b/src/Codata/Musical/Colist/Infinite-merge.agda
@@ -13,7 +13,7 @@ open import Codata.Musical.Colist as Colist hiding (_⋎_)
 open import Data.Nat.Base
 open import Data.Nat.Induction using (<′-wellFounded)
 open import Data.Nat.Properties
-open import Data.Product.Base as Prod using (_×_; _,_; ∃; ∃₂; proj₁; proj₂)
+open import Data.Product.Base as Product using (_×_; _,_; ∃; ∃₂; proj₁; proj₂)
 open import Data.Sum.Base
 open import Data.Sum.Properties
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
@@ -195,7 +195,7 @@ Any-merge {P = P} xss = mk↔ₛ′ (proj₁ ∘ to xss) from to∘from (proj₂
          | index-Any-⋎P xs p
     ... | inj₁ q | P.refl | _   = here (inj₂ q) , P.refl
     ... | inj₂ q | P.refl | q≤p =
-      Prod.map there
+      Product.map there
                (P.cong (there ∘ (Inverse.from (Any-⋎P xs)) ∘ inj₂))
                (rec (s≤′s q≤p))
 
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index a4196bb952..b5852c7055 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -27,7 +27,7 @@ open import Data.Maybe.Base as Maybe using (Maybe; nothing; just)
 import Data.Maybe.Properties as Maybeₚ
 open import Data.Maybe.Relation.Unary.All using (All; nothing; just)
 open import Data.Nat.Base as ℕ using (zero; suc; z≤n; s≤s)
-open import Data.Product.Base as Prod using (_×_; _,_; uncurry)
+open import Data.Product.Base as Product using (_×_; _,_; uncurry)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Function.Base
@@ -106,7 +106,7 @@ lookup-replicate (suc k) (suc n) a = lookup-replicate k (n .force) a
 -- Properties of unfold
 
 map-unfold : ∀ (f : B → C) (alg : A → Maybe (A × B)) a →
-             i ⊢ map f (unfold alg a) ≈ unfold (Maybe.map (Prod.map₂ f) ∘ alg) a
+             i ⊢ map f (unfold alg a) ≈ unfold (Maybe.map (Product.map₂ f) ∘ alg) a
 map-unfold f alg a with alg a
 ... | nothing       = []
 ... | just (a′ , b) = Eq.refl ∷ λ where .force → map-unfold f alg a′
diff --git a/src/Codata/Sized/Cowriter.agda b/src/Codata/Sized/Cowriter.agda
index 1cd3d2ba9c..03d24d3d3c 100644
--- a/src/Codata/Sized/Cowriter.agda
+++ b/src/Codata/Sized/Cowriter.agda
@@ -18,7 +18,7 @@ open import Data.Unit.Base
 open import Data.List.Base using (List; []; _∷_)
 open import Data.List.NonEmpty.Base using (List⁺; _∷_)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
-open import Data.Product.Base as Prod using (_×_; _,_)
+open import Data.Product.Base as Product using (_×_; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.Vec.Base using (Vec; []; _∷_)
 open import Data.Vec.Bounded.Base as Vec≤ using (Vec≤; _,_)
@@ -68,11 +68,11 @@ length (w ∷ cw) = suc λ where .force → length (cw .force)
 splitAt : ∀ (n : ℕ) → Cowriter W A ∞ → (Vec W n × Cowriter W A ∞) ⊎ (Vec≤ W n × A)
 splitAt zero    cw       = inj₁ ([] , cw)
 splitAt (suc n) [ a ]    = inj₂ (Vec≤.[] , a)
-splitAt (suc n) (w ∷ cw) = Sum.map (Prod.map₁ (w ∷_)) (Prod.map₁ (w Vec≤.∷_))
+splitAt (suc n) (w ∷ cw) = Sum.map (Product.map₁ (w ∷_)) (Product.map₁ (w Vec≤.∷_))
                          $ splitAt n (cw .force)
 
 take : ∀ (n : ℕ) → Cowriter W A ∞ → Vec W n ⊎ (Vec≤ W n × A)
-take n = Sum.map₁ Prod.proj₁ ∘′ splitAt n
+take n = Sum.map₁ Product.proj₁ ∘′ splitAt n
 
 infixr 5 _++_ _⁺++_
 _++_ : ∀ {i} → List W → Cowriter W A i → Cowriter W A i
diff --git a/src/Codata/Sized/M/Properties.agda b/src/Codata/Sized/M/Properties.agda
index f9801448f5..a02410d43e 100644
--- a/src/Codata/Sized/M/Properties.agda
+++ b/src/Codata/Sized/M/Properties.agda
@@ -15,7 +15,7 @@ open import Codata.Sized.M
 open import Codata.Sized.M.Bisimilarity
 open import Data.Container.Core as C hiding (map)
 import Data.Container.Morphism as Mp
-open import Data.Product.Base as Prod using (_,_)
+open import Data.Product.Base as Product using (_,_)
 open import Data.Product.Properties hiding (map-cong)
 open import Function.Base using (_$′_; _∘′_)
 import Relation.Binary.PropositionalEquality.Core as P
diff --git a/src/Codata/Sized/Stream/Properties.agda b/src/Codata/Sized/Stream/Properties.agda
index 40cb3fc961..0fd436f1bd 100644
--- a/src/Codata/Sized/Stream/Properties.agda
+++ b/src/Codata/Sized/Stream/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Nat.GeneralisedArithmetic using (fold; fold-pull)
 open import Data.List.Base as List using ([]; _∷_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)
 import Data.List.Relation.Binary.Equality.Propositional as Eq
-open import Data.Product.Base as Prod using (_,_)
+open import Data.Product.Base as Product using (_,_)
 open import Data.Vec.Base as Vec using (_∷_)
 
 open import Function.Base using (id; _$_; _∘′_; const)
@@ -49,7 +49,7 @@ take-repeat-identity (suc n) a = P.cong (a Vec.∷_) (take-repeat-identity n a)
 
 splitAt-repeat-identity : (n : ℕ) (a : A) → splitAt n (repeat a) ≡ (Vec.replicate n a , repeat a)
 splitAt-repeat-identity zero    a = P.refl
-splitAt-repeat-identity (suc n) a = P.cong (Prod.map₁ (a ∷_)) (splitAt-repeat-identity n a)
+splitAt-repeat-identity (suc n) a = P.cong (Product.map₁ (a ∷_)) (splitAt-repeat-identity n a)
 
 replicate-repeat : ∀ {i} (n : ℕ) (a : A) → i ⊢ List.replicate n a ++ repeat a ≈ repeat a
 replicate-repeat zero    a = refl
@@ -103,10 +103,10 @@ map-∘ f g (a ∷ as) = P.refl ∷ λ where .force → map-∘ f g (as .force)
 -- splitAt
 
 splitAt-map : ∀ n (f : A → B) xs →
-  splitAt n (map f xs) ≡ Prod.map (Vec.map f) (map f) (splitAt n xs)
+  splitAt n (map f xs) ≡ Product.map (Vec.map f) (map f) (splitAt n xs)
 splitAt-map zero    f xs       = P.refl
 splitAt-map (suc n) f (x ∷ xs) =
-  P.cong (Prod.map₁ (f x Vec.∷_)) (splitAt-map n f (xs .force))
+  P.cong (Product.map₁ (f x Vec.∷_)) (splitAt-map n f (xs .force))
 
 ------------------------------------------------------------------------
 -- iterate
diff --git a/src/Data/Container/Core.agda b/src/Data/Container/Core.agda
index f03f757c7e..006f433d2e 100644
--- a/src/Data/Container/Core.agda
+++ b/src/Data/Container/Core.agda
@@ -9,7 +9,7 @@
 module Data.Container.Core where
 
 open import Level
-open import Data.Product.Base as Prod using (Σ-syntax)
+open import Data.Product.Base as Product using (Σ-syntax)
 open import Function.Base
 open import Function using (Inverse; _↔_)
 open import Relation.Unary using (Pred; _⊆_)
@@ -33,7 +33,7 @@ open Container public
 
 map : ∀ {s p x y} {C : Container s p} {X : Set x} {Y : Set y} →
       (X → Y) → ⟦ C ⟧ X → ⟦ C ⟧ Y
-map f = Prod.map₂ (f ∘_)
+map f = Product.map₂ (f ∘_)
 
 -- Representation of container morphisms.
 
@@ -47,7 +47,7 @@ record _⇒_ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Containe
     position : ∀ {s} → Position C₂ (shape s) → Position C₁ s
 
   ⟪_⟫ : ∀ {x} {X : Set x} → ⟦ C₁ ⟧ X → ⟦ C₂ ⟧ X
-  ⟪_⟫ = Prod.map shape (_∘′ position)
+  ⟪_⟫ = Product.map shape (_∘′ position)
 
 open _⇒_ public
 
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 11dbddea63..d105d9e8d7 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -23,7 +23,7 @@ open import Data.Nat.Base as ℕ
 import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
 open import Data.Unit using (⊤; tt)
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
   using (∃; ∃₂; _×_; _,_; map; proj₁; proj₂; uncurry; <_,_>)
 open import Data.Product.Properties using (,-injective)
 open import Data.Product.Algebra using (×-cong)
@@ -644,7 +644,7 @@ splitAt-≥ (suc m) (suc i) i≥m = cong (Sum.map suc id) (splitAt-≥ m i (ℕ.
 remQuot-combine : ∀ {n k} (i : Fin n) j → remQuot k (combine i j) ≡ (i , j)
 remQuot-combine {suc n} {k} zero    j rewrite splitAt-↑ˡ k j (n ℕ.* k) = refl
 remQuot-combine {suc n} {k} (suc i) j rewrite splitAt-↑ʳ k   (n ℕ.* k) (combine i j) =
-  cong (Prod.map₁ suc) (remQuot-combine i j)
+  cong (Product.map₁ suc) (remQuot-combine i j)
 
 combine-remQuot : ∀ {n} k (i : Fin (n ℕ.* k)) → uncurry combine (remQuot {n} k i) ≡ i
 combine-remQuot {suc n} k i with splitAt k i in eq
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 16546110b6..8b988c646d 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -17,7 +17,7 @@ open import Data.Bool.Base as Bool
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _*_ ; _≤_ ; s≤s)
-open import Data.Product.Base as Prod using (_×_; _,_; map₁; map₂′)
+open import Data.Product.Base as Product using (_×_; _,_; map₁; map₂′)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Function.Base
@@ -96,13 +96,13 @@ zipWith f _        _        = []
 unalignWith : (A → These B C) → List A → List B × List C
 unalignWith f []       = [] , []
 unalignWith f (a ∷ as) with f a
-... | this b    = Prod.map₁ (b ∷_) (unalignWith f as)
-... | that c    = Prod.map₂ (c ∷_) (unalignWith f as)
-... | these b c = Prod.map (b ∷_) (c ∷_) (unalignWith f as)
+... | this b    = Product.map₁ (b ∷_) (unalignWith f as)
+... | that c    = Product.map₂ (c ∷_) (unalignWith f as)
+... | these b c = Product.map (b ∷_) (c ∷_) (unalignWith f as)
 
 unzipWith : (A → B × C) → List A → List B × List C
 unzipWith f []         = [] , []
-unzipWith f (xy ∷ xys) = Prod.zip _∷_ _∷_ (f xy) (unzipWith f xys)
+unzipWith f (xy ∷ xys) = Product.zip _∷_ _∷_ (f xy) (unzipWith f xys)
 
 partitionSumsWith : (A → B ⊎ C) → List A → List B × List C
 partitionSumsWith f = unalignWith (These.fromSum ∘′ f)
@@ -340,7 +340,7 @@ drop (suc n) (x ∷ xs) = drop n xs
 splitAt : ℕ → List A → List A × List A
 splitAt zero    xs       = ([] , xs)
 splitAt (suc n) []       = ([] , [])
-splitAt (suc n) (x ∷ xs) = Prod.map₁ (x ∷_) (splitAt n xs)
+splitAt (suc n) (x ∷ xs) = Product.map₁ (x ∷_) (splitAt n xs)
 
 removeAt : (xs : List A) → Fin (length xs) → List A
 removeAt (x ∷ xs) zero     = xs
@@ -406,7 +406,7 @@ partitionᵇ p = partition (T? ∘ p)
 span : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
 span P? []       = ([] , [])
 span P? ys@(x ∷ xs) with does (P? x)
-... | true  = Prod.map (x ∷_) id (span P? xs)
+... | true  = Product.map (x ∷_) id (span P? xs)
 ... | false = ([] , ys)
 
 
diff --git a/src/Data/List/Membership/Propositional/Properties/Core.agda b/src/Data/List/Membership/Propositional/Properties/Core.agda
index fadd49bb4b..f7543d05f8 100644
--- a/src/Data/List/Membership/Propositional/Properties/Core.agda
+++ b/src/Data/List/Membership/Propositional/Properties/Core.agda
@@ -17,7 +17,7 @@ open import Function.Bundles
 open import Data.List.Base using (List)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
   using (_,_; proj₁; proj₂; uncurry′; ∃; _×_)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core as P
@@ -42,7 +42,7 @@ map∘find (there p) hyp = P.cong there (map∘find p hyp)
 
 find∘map : ∀ {P : Pred A p} {Q : Pred A q}
            {xs : List A} (p : Any P xs) (f : P ⊆ Q) →
-           find (Any.map f p) ≡ Prod.map id (Prod.map id f) (find p)
+           find (Any.map f p) ≡ Product.map id (Product.map id f) (find p)
 find∘map (here  p) f = refl
 find∘map (there p) f rewrite find∘map p f = refl
 
diff --git a/src/Data/List/Membership/Setoid.agda b/src/Data/List/Membership/Setoid.agda
index 1a19169992..cadc600646 100644
--- a/src/Data/List/Membership/Setoid.agda
+++ b/src/Data/List/Membership/Setoid.agda
@@ -15,7 +15,7 @@ open import Function.Base using (_∘_; id; flip)
 open import Data.List.Base as List using (List; []; _∷_; length; lookup)
 open import Data.List.Relation.Unary.Any as Any
   using (Any; index; map; here; there)
-open import Data.Product.Base as Prod using (∃; _×_; _,_)
+open import Data.Product.Base as Product using (∃; _×_; _,_)
 open import Relation.Unary using (Pred)
 open import Relation.Nullary.Negation using (¬_)
 
@@ -50,7 +50,7 @@ module _ {p} {P : Pred A p} where
 
   find : ∀ {xs} → Any P xs → ∃ λ x → x ∈ xs × P x
   find (here px)   = (_ , here refl , px)
-  find (there pxs) = Prod.map id (Prod.map there id) (find pxs)
+  find (there pxs) = Product.map id (Product.map there id) (find pxs)
 
   lose : P Respects _≈_ →  ∀ {x xs} → x ∈ xs → P x → Any P xs
   lose resp x∈xs px = map (flip resp px) x∈xs
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index 1a74ace45f..f63391f882 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -21,7 +21,7 @@ import Data.List.Relation.Binary.Equality.Setoid as Equality
 import Data.List.Relation.Unary.Unique.Setoid as Unique
 open import Data.Nat.Base using (suc; z≤n; s≤s; _≤_; _<_)
 open import Data.Nat.Properties using (≤-trans; n≤1+n)
-open import Data.Product.Base as Prod using (∃; _×_; _,_ ; ∃₂; proj₁; proj₂)
+open import Data.Product.Base as Product using (∃; _×_; _,_ ; ∃₂; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (_$_; flip; _∘_; _∘′_; id)
@@ -342,10 +342,10 @@ module _ (S : Setoid c ℓ) {P : Pred (Carrier S) p}
 
   ∈-filter⁻ : ∀ {v xs} → v ∈ filter P? xs → v ∈ xs × P v
   ∈-filter⁻ {xs = x ∷ xs} v∈f[x∷xs] with P? x
-  ... | false because  _   = Prod.map there id (∈-filter⁻ v∈f[x∷xs])
+  ... | false because  _   = Product.map there id (∈-filter⁻ v∈f[x∷xs])
   ... |  true because [Px] with v∈f[x∷xs]
   ...   | here  v≈x   = here v≈x , resp (sym v≈x) (invert [Px])
-  ...   | there v∈fxs = Prod.map there id (∈-filter⁻ v∈fxs)
+  ...   | there v∈fxs = Product.map there id (∈-filter⁻ v∈fxs)
 
 ------------------------------------------------------------------------
 -- derun and deduplicate
diff --git a/src/Data/List/Nary/NonDependent.agda b/src/Data/List/Nary/NonDependent.agda
index f95da6345c..c008efa6c3 100644
--- a/src/Data/List/Nary/NonDependent.agda
+++ b/src/Data/List/Nary/NonDependent.agda
@@ -10,7 +10,7 @@ module Data.List.Nary.NonDependent where
 
 open import Data.Nat.Base using (zero; suc)
 open import Data.List.Base as List using (List; []; _∷_)
-open import Data.Product.Base as Prod using (_,_)
+open import Data.Product.Base as Product using (_,_)
 open import Data.Product.Nary.NonDependent using (Product)
 open import Function.Base using ()
 open import Function.Nary.NonDependent.Base
@@ -37,7 +37,7 @@ zipWith : ∀ n {ls} {as : Sets n ls} {r} {R : Set r} →
 zipWith 0               f       = []
 zipWith 1               f xs    = List.map f xs
 zipWith (suc n@(suc _)) f xs ys =
-  zipWith n (Prod.uncurry f) (List.zipWith _,_ xs ys)
+  zipWith n (Product.uncurry f) (List.zipWith _,_ xs ys)
 
 unzipWith : ∀ n {ls} {as : Sets n ls} {a} {A : Set a} →
             (A → Product n as) → (List A → Product n (List <$> as))
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 4188bab92a..0dd6574711 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -25,9 +25,9 @@ open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.Properties
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
   using (_×_; _,_; uncurry; uncurry′; proj₁; proj₂; <_,_>)
-import Data.Product.Relation.Unary.All as Prod using (All)
+import Data.Product.Relation.Unary.All as Product using (All)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Fin.Properties using (toℕ-cast)
@@ -370,20 +370,20 @@ module _ (f : A → B → C) where
 
 unalignWith-this : unalignWith ((A → These A B) ∋ this) ≗ (_, [])
 unalignWith-this []       = refl
-unalignWith-this (a ∷ as) = cong (Prod.map₁ (a ∷_)) (unalignWith-this as)
+unalignWith-this (a ∷ as) = cong (Product.map₁ (a ∷_)) (unalignWith-this as)
 
 unalignWith-that : unalignWith ((B → These A B) ∋ that) ≗ ([] ,_)
 unalignWith-that []       = refl
-unalignWith-that (b ∷ bs) = cong (Prod.map₂ (b ∷_)) (unalignWith-that bs)
+unalignWith-that (b ∷ bs) = cong (Product.map₂ (b ∷_)) (unalignWith-that bs)
 
 module _ {f g : C → These A B} where
 
   unalignWith-cong : f ≗ g → unalignWith f ≗ unalignWith g
   unalignWith-cong f≗g []       = refl
   unalignWith-cong f≗g (c ∷ cs) with f c | g c | f≗g c
-  ... | this a    | ._ | refl = cong (Prod.map₁ (a ∷_)) (unalignWith-cong f≗g cs)
-  ... | that b    | ._ | refl = cong (Prod.map₂ (b ∷_)) (unalignWith-cong f≗g cs)
-  ... | these a b | ._ | refl = cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-cong f≗g cs)
+  ... | this a    | ._ | refl = cong (Product.map₁ (a ∷_)) (unalignWith-cong f≗g cs)
+  ... | that b    | ._ | refl = cong (Product.map₂ (b ∷_)) (unalignWith-cong f≗g cs)
+  ... | these a b | ._ | refl = cong (Product.map (a ∷_) (b ∷_)) (unalignWith-cong f≗g cs)
 
 module _ (f : C → These A B) where
 
@@ -391,17 +391,17 @@ module _ (f : C → These A B) where
                     unalignWith f (map g ds) ≡ unalignWith (f ∘′ g) ds
   unalignWith-map g []       = refl
   unalignWith-map g (d ∷ ds) with f (g d)
-  ... | this a    = cong (Prod.map₁ (a ∷_)) (unalignWith-map g ds)
-  ... | that b    = cong (Prod.map₂ (b ∷_)) (unalignWith-map g ds)
-  ... | these a b = cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-map g ds)
+  ... | this a    = cong (Product.map₁ (a ∷_)) (unalignWith-map g ds)
+  ... | that b    = cong (Product.map₂ (b ∷_)) (unalignWith-map g ds)
+  ... | these a b = cong (Product.map (a ∷_) (b ∷_)) (unalignWith-map g ds)
 
   map-unalignWith : (g : A → D) (h : B → E) →
-    Prod.map (map g) (map h) ∘′ unalignWith f ≗ unalignWith (These.map g h ∘′ f)
+    Product.map (map g) (map h) ∘′ unalignWith f ≗ unalignWith (These.map g h ∘′ f)
   map-unalignWith g h []       = refl
   map-unalignWith g h (c ∷ cs) with f c
-  ... | this a    = cong (Prod.map₁ (g a ∷_)) (map-unalignWith g h cs)
-  ... | that b    = cong (Prod.map₂ (h b ∷_)) (map-unalignWith g h cs)
-  ... | these a b = cong (Prod.map (g a ∷_) (h b ∷_)) (map-unalignWith g h cs)
+  ... | this a    = cong (Product.map₁ (g a ∷_)) (map-unalignWith g h cs)
+  ... | that b    = cong (Product.map₂ (h b ∷_)) (map-unalignWith g h cs)
+  ... | these a b = cong (Product.map (g a ∷_) (h b ∷_)) (map-unalignWith g h cs)
 
   unalignWith-alignWith : (g : These A B → C) → f ∘′ g ≗ id → ∀ as bs →
                           unalignWith f (alignWith g as bs) ≡ (as , bs)
@@ -417,7 +417,7 @@ module _ (f : C → These A B) where
     as , []                            ∎
   unalignWith-alignWith g g∘f≗id (a ∷ as)   (b ∷ bs)
     rewrite g∘f≗id (these a b) =
-    cong (Prod.map (a ∷_) (b ∷_)) (unalignWith-alignWith g g∘f≗id as bs)
+    cong (Product.map (a ∷_) (b ∷_)) (unalignWith-alignWith g g∘f≗id as bs)
 
 ------------------------------------------------------------------------
 -- unzipWith
@@ -896,7 +896,7 @@ lookup-iterate f x (suc n) (suc i) = lookup-iterate f (f x) n i
 splitAt-defn : ∀ n → splitAt {A = A} n ≗ < take n , drop n >
 splitAt-defn zero    xs       = refl
 splitAt-defn (suc n) []       = refl
-splitAt-defn (suc n) (x ∷ xs) = cong (Prod.map (x ∷_) id) (splitAt-defn n xs)
+splitAt-defn (suc n) (x ∷ xs) = cong (Product.map (x ∷_) id) (splitAt-defn n xs)
 
 ------------------------------------------------------------------------
 -- takeWhile, dropWhile, and span
@@ -912,7 +912,7 @@ module _ {P : Pred A p} (P? : Decidable P) where
   span-defn : span P? ≗ < takeWhile P? , dropWhile P? >
   span-defn []       = refl
   span-defn (x ∷ xs) with does (P? x)
-  ... | true  = cong (Prod.map (x ∷_) id) (span-defn xs)
+  ... | true  = cong (Product.map (x ∷_) id) (span-defn xs)
   ... | false = refl
 
 ------------------------------------------------------------------------
@@ -1023,15 +1023,15 @@ module _ {P : Pred A p} (P? : Decidable P) where
   partition-defn : partition P? ≗ < filter P? , filter (∁? P?) >
   partition-defn []       = refl
   partition-defn (x ∷ xs) with ih ← partition-defn xs | does (P? x)
-  ...  | true  = cong (Prod.map (x ∷_) id) ih
-  ...  | false = cong (Prod.map id (x ∷_)) ih
+  ...  | true  = cong (Product.map (x ∷_) id) ih
+  ...  | false = cong (Product.map id (x ∷_)) ih
 
   length-partition : ∀ xs → (let (ys , zs) = partition P? xs) →
                      length ys ≤ length xs × length zs ≤ length xs
   length-partition []       = z≤n , z≤n
   length-partition (x ∷ xs) with ih ← length-partition xs | does (P? x)
-  ...  | true  = Prod.map s≤s m≤n⇒m≤1+n ih
-  ...  | false = Prod.map m≤n⇒m≤1+n s≤s ih
+  ...  | true  = Product.map s≤s m≤n⇒m≤1+n ih
+  ...  | false = Product.map m≤n⇒m≤1+n s≤s ih
 
 ------------------------------------------------------------------------
 -- _ʳ++_
@@ -1181,7 +1181,7 @@ module _ {x y : A} where
   ∷ʳ-injective : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
   ∷ʳ-injective []          []          refl = (refl , refl)
   ∷ʳ-injective (x ∷ xs)    (y  ∷ ys)   eq   with refl , eq′  ← ∷-injective eq
-    = Prod.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
+    = Product.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
   ∷ʳ-injective []          (_ ∷ _ ∷ _) ()
   ∷ʳ-injective (_ ∷ _ ∷ _) []          ()
 
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
index 202bff89b0..776d065423 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
@@ -19,7 +19,7 @@ open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_
 open import Data.List.Relation.Binary.Prefix.Heterogeneous as Prefix hiding (PrefixView; _++_)
 open import Data.Nat.Base using (ℕ; zero; suc; _≤_; z≤n; s≤s)
 open import Data.Nat.Properties using (suc-injective)
-open import Data.Product.Base as Prod using (_×_; _,_; proj₁; proj₂; uncurry)
+open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Function.Base
 
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index 7b498f2a45..f50a29e82c 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -26,7 +26,7 @@ open import Data.List.Relation.Binary.Subset.Propositional
 open import Data.List.Relation.Binary.Permutation.Propositional
 import Data.List.Relation.Binary.Permutation.Propositional.Properties as Permutation
 open import Data.Nat using (ℕ; _≤_)
-import Data.Product.Base as Prod
+import Data.Product.Base as Product
 import Data.Sum.Base as Sum
 open import Effect.Monad
 open import Function.Base using (_∘_; _∘′_; id; _$_)
@@ -130,7 +130,7 @@ All-resp-⊇ = Setoidₚ.All-resp-⊇ (setoid _) (subst _)
 map⁺ : ∀ (f : A → B) → xs ⊆ ys → map f xs ⊆ map f ys
 map⁺ f xs⊆ys =
   Inverse.to (map-∈↔ f) ∘
-  Prod.map₂ (Prod.map₁ xs⊆ys) ∘
+  Product.map₂ (Product.map₁ xs⊆ys) ∘
   Inverse.from (map-∈↔ f)
 
 ------------------------------------------------------------------------
@@ -171,7 +171,7 @@ module _ {xss yss : List (List A)} where
   concat⁺ : xss ⊆ yss → concat xss ⊆ concat yss
   concat⁺ xss⊆yss =
     Inverse.to concat-∈↔ ∘
-    Prod.map₂ (Prod.map₂ xss⊆yss) ∘
+    Product.map₂ (Product.map₂ xss⊆yss) ∘
     Inverse.from concat-∈↔
 
 ------------------------------------------------------------------------
@@ -188,7 +188,7 @@ module _ {A B : Set a} (f g : A → List B) where
   >>=⁺ : xs ⊆ ys → (∀ {x} → f x ⊆ g x) → (xs >>= f) ⊆ (ys >>= g)
   >>=⁺ xs⊆ys f⊆g =
     Inverse.to >>=-∈↔ ∘
-    Prod.map₂ (Prod.map xs⊆ys f⊆g) ∘
+    Product.map₂ (Product.map xs⊆ys f⊆g) ∘
     Inverse.from >>=-∈↔
 
 ------------------------------------------------------------------------
@@ -199,7 +199,7 @@ module _ {A B : Set a} {fs gs : List (A → B)} where
   ⊛⁺ : fs ⊆ gs → xs ⊆ ys → (fs ⊛ xs) ⊆ (gs ⊛ ys)
   ⊛⁺ fs⊆gs xs⊆ys =
     (Inverse.to $ ⊛-∈↔ gs) ∘
-    Prod.map₂ (Prod.map₂ (Prod.map fs⊆gs (Prod.map₁ xs⊆ys))) ∘
+    Product.map₂ (Product.map₂ (Product.map fs⊆gs (Product.map₁ xs⊆ys))) ∘
     (Inverse.from $ ⊛-∈↔ fs)
 
 ------------------------------------------------------------------------
@@ -210,7 +210,7 @@ module _ {A B : Set a} {ws xs : List A} {ys zs : List B} where
   ⊗⁺ : ws ⊆ xs → ys ⊆ zs → (ws ⊗ ys) ⊆ (xs ⊗ zs)
   ⊗⁺ ws⊆xs ys⊆zs =
     Inverse.to ⊗-∈↔ ∘
-    Prod.map ws⊆xs ys⊆zs ∘
+    Product.map ws⊆xs ys⊆zs ∘
     Inverse.from ⊗-∈↔
 
 ------------------------------------------------------------------------
@@ -235,7 +235,7 @@ module _ {xs : List A} {f : ∀ {x} → x ∈ xs → B}
                 mapWith∈ xs f ⊆ mapWith∈ ys g
   mapWith∈⁺ xs⊆ys f≈g {x} =
     Inverse.to Any.mapWith∈↔ ∘
-    Prod.map₂ (Prod.map xs⊆ys (λ {x∈xs} x≡fx∈xs → begin
+    Product.map₂ (Product.map xs⊆ys (λ {x∈xs} x≡fx∈xs → begin
       x               ≡⟨ x≡fx∈xs ⟩
       f x∈xs          ≡⟨ f≈g x∈xs ⟩
       g (xs⊆ys x∈xs)  ∎)) ∘
diff --git a/src/Data/List/Relation/Ternary/Interleaving.agda b/src/Data/List/Relation/Ternary/Interleaving.agda
index 08d9f728e7..17055652fb 100644
--- a/src/Data/List/Relation/Ternary/Interleaving.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving.agda
@@ -12,7 +12,7 @@ module Data.List.Relation.Ternary.Interleaving where
 open import Level
 open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
-open import Data.Product.Base as Prod using (∃; ∃₂; _×_; uncurry; _,_; -,_; proj₂)
+open import Data.Product.Base as Product using (∃; ∃₂; _×_; uncurry; _,_; -,_; proj₂)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
 open import Relation.Binary.Core using (REL; _⇒_)
@@ -95,11 +95,11 @@ module _ {a b l r} {A : Set a} {B : Set b} {L : REL A B l} {R : REL A B r} where
   split : ∀ {as bs} → Pointwise (λ a b → L a b ⊎ R a b) as bs →
           ∃₂ λ asr asl → Interleaving L R asl asr bs
   split []            = [] , [] , []
-  split (inj₁ l ∷ pw) = Prod.map _ (Prod.map _ (l ∷ˡ_)) (split pw)
-  split (inj₂ r ∷ pw) = Prod.map _ (Prod.map _ (r ∷ʳ_)) (split pw)
+  split (inj₁ l ∷ pw) = Product.map _ (Product.map _ (l ∷ˡ_)) (split pw)
+  split (inj₂ r ∷ pw) = Product.map _ (Product.map _ (r ∷ʳ_)) (split pw)
 
   unsplit : ∀ {l r as} → Interleaving L R l r as →
             ∃ λ bs → Pointwise (λ a b → L a b ⊎ R a b) bs as
   unsplit []        = -, []
-  unsplit (l ∷ˡ sp) = Prod.map _ (inj₁ l ∷_) (unsplit sp)
-  unsplit (r ∷ʳ sp) = Prod.map _ (inj₂ r ∷_) (unsplit sp)
+  unsplit (l ∷ˡ sp) = Product.map _ (inj₁ l ∷_) (unsplit sp)
+  unsplit (r ∷ʳ sp) = Product.map _ (inj₂ r ∷_) (unsplit sp)
diff --git a/src/Data/List/Relation/Unary/All.agda b/src/Data/List/Relation/Unary/All.agda
index 97da349699..d9f358df04 100644
--- a/src/Data/List/Relation/Unary/All.agda
+++ b/src/Data/List/Relation/Unary/All.agda
@@ -15,7 +15,7 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional renaming (_∈_ to _∈ₚ_)
 import Data.List.Membership.Setoid as SetoidMembership
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
   using (∃; -,_; _×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_; id; const)
@@ -88,7 +88,7 @@ reduce f (px ∷ pxs) = f px ∷ reduce f pxs
 
 construct : (f : B → ∃ P) (xs : List B) → ∃ (All P)
 construct f []       = [] , []
-construct f (x ∷ xs) = Prod.zip _∷_ _∷_ (f x) (construct f xs)
+construct f (x ∷ xs) = Product.zip _∷_ _∷_ (f x) (construct f xs)
 
 fromList : (xs : List (∃ P)) → All P (List.map proj₁ xs)
 fromList []              = []
@@ -107,7 +107,7 @@ zipWith f (px ∷ pxs , qx ∷ qxs) = f (px , qx) ∷ zipWith f (pxs , qxs)
 
 unzipWith : R ⊆ P ∩ Q → All R ⊆ All P ∩ All Q
 unzipWith f []         = [] , []
-unzipWith f (rx ∷ rxs) = Prod.zip _∷_ _∷_ (f rx) (unzipWith f rxs)
+unzipWith f (rx ∷ rxs) = Product.zip _∷_ _∷_ (f rx) (unzipWith f rxs)
 
 zip : All P ∩ All Q ⊆ All (P ∩ Q)
 zip = zipWith id
@@ -189,7 +189,7 @@ lookupAny (px ∷ pxs) (here qx) = px , qx
 lookupAny (px ∷ pxs) (there i) = lookupAny pxs i
 
 lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
-lookupWith f pxs i = Prod.uncurry f (lookupAny pxs i)
+lookupWith f pxs i = Product.uncurry f (lookupAny pxs i)
 
 lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
 lookup pxs = lookupWith (λ { px refl → px }) pxs
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index d2a57b7143..a36a07f3b1 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -32,7 +32,7 @@ open import Data.Maybe.Relation.Unary.All as Maybe using (just; nothing; fromAny
 open import Data.Maybe.Relation.Unary.Any as Maybe using (just)
 open import Data.Nat.Base using (zero; suc; s≤s; _<_; z<s; s<s)
 open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)
-open import Data.Product.Base as Prod using (_×_; _,_; uncurry; uncurry′)
+open import Data.Product.Base as Product using (_×_; _,_; uncurry; uncurry′)
 open import Function.Base
 open import Function.Bundles
 open import Level using (Level)
@@ -417,7 +417,7 @@ Any-catMaybes⁺ = All-catMaybes⁺ ∘ All.map fromAny
 
 ++⁻ : ∀ xs {ys} → All P (xs ++ ys) → All P xs × All P ys
 ++⁻ []       p          = [] , p
-++⁻ (x ∷ xs) (px ∷ pxs) = Prod.map (px ∷_) id (++⁻ _ pxs)
+++⁻ (x ∷ xs) (px ∷ pxs) = Product.map (px ∷_) id (++⁻ _ pxs)
 
 ++↔ : (All P xs × All P ys) ↔ All P (xs ++ ys)
 ++↔ {xs = zs} = mk↔ₛ′ (uncurry ++⁺) (++⁻ zs) (++⁺∘++⁻ zs) ++⁻∘++⁺
@@ -448,7 +448,7 @@ concat⁻ {xss = xs ∷ xss} pxs = ++⁻ˡ xs pxs ∷ concat⁻ (++⁻ʳ xs pxs)
 ∷ʳ⁺ pxs px = ++⁺ pxs (px ∷ [])
 
 ∷ʳ⁻ : All P (xs ∷ʳ x) → All P xs × P x
-∷ʳ⁻ pxs = Prod.map₂ singleton⁻ $ ++⁻ _ pxs
+∷ʳ⁻ pxs = Product.map₂ singleton⁻ $ ++⁻ _ pxs
 
 -- unsnoc
 
@@ -656,7 +656,7 @@ module _ {R : A → A → Set q} (R? : B.Decidable R) where
     where
     aux : ∀ {z} → z ∈ filter (¬? ∘ ¬? ∘ R? x) (deduplicate R? xs) → P z
     aux {z = z} z∈filter = resp (decidable-stable (R? x z)
-      (Prod.proj₂ (∈-filter⁻ (¬? ∘ ¬? ∘ R? x) {z} {deduplicate R? xs} z∈filter))) px
+      (Product.proj₂ (∈-filter⁻ (¬? ∘ ¬? ∘ R? x) {z} {deduplicate R? xs} z∈filter))) px
 
 ------------------------------------------------------------------------
 -- zipWith
diff --git a/src/Data/List/Relation/Unary/Any.agda b/src/Data/List/Relation/Unary/Any.agda
index 169f6020aa..1b6093cadc 100644
--- a/src/Data/List/Relation/Unary/Any.agda
+++ b/src/Data/List/Relation/Unary/Any.agda
@@ -11,7 +11,7 @@ module Data.List.Relation.Unary.Any where
 open import Data.Empty
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List; []; [_]; _∷_; removeAt)
-open import Data.Product.Base as Prod using (∃; _,_)
+open import Data.Product.Base as Product using (∃; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary using (¬_; yes; no; _⊎-dec_)
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index e356e9eca1..949f6d9fdb 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -25,7 +25,7 @@ open import Data.Nat using (zero; suc; _<_; z<s; s<s; s≤s)
 open import Data.Nat.Properties using (_≟_; ≤∧≢⇒<; ≤-refl; m<n⇒m<1+n)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.Any as MAny using (just)
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
   using (_×_; _,_; ∃; ∃₂; proj₁; proj₂; uncurry′)
 open import Data.Product.Properties
 open import Data.Product.Function.NonDependent.Propositional
@@ -214,7 +214,7 @@ Any-×⁺ (p , q) = Any.map (λ p → Any.map (λ q → (p , q)) q) p
 
 Any-×⁻ : Any (λ x → Any (λ y → P x × Q y) ys) xs →
          Any P xs × Any Q ys
-Any-×⁻ pq with Prod.map₂ (Prod.map₂ find) (find pq)
+Any-×⁻ pq with Product.map₂ (Product.map₂ find) (find pq)
 ... | (x , x∈xs , y , y∈ys , p , q) = lose x∈xs p , lose y∈ys q
 
 ×↔ : ∀ {xs ys} →
@@ -286,7 +286,7 @@ module _ {_~_ : REL A B r} where
 
   Any-Σ⁻ʳ : Any (∃ ∘ _~_) xs → ∃ λ x → Any (_~ x) xs
   Any-Σ⁻ʳ (here (b , x)) = b , here x
-  Any-Σ⁻ʳ (there xs) = Prod.map₂ there $ Any-Σ⁻ʳ xs
+  Any-Σ⁻ʳ (there xs) = Product.map₂ there $ Any-Σ⁻ʳ xs
 
 ------------------------------------------------------------------------
 -- Invertible introduction (⁺) and elimination (⁻) rules for various
@@ -514,7 +514,7 @@ tabulate⁺ (suc i) p = there (tabulate⁺ i p)
 
 tabulate⁻ : ∀ {n} {f : Fin n → A} → Any P (tabulate f) → ∃ λ i → P (f i)
 tabulate⁻ {n = suc _} (here p)  = zero , p
-tabulate⁻ {n = suc _} (there p) = Prod.map suc id (tabulate⁻ p)
+tabulate⁻ {n = suc _} (there p) = Product.map suc id (tabulate⁻ p)
 
 ------------------------------------------------------------------------
 -- filter
@@ -593,7 +593,7 @@ module _ {P : B → Set p} where
                 ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)
   mapWith∈⁻ (y ∷ xs) f (here  p) = (y , here refl , p)
   mapWith∈⁻ (y ∷ xs) f (there p) =
-    Prod.map₂ (Prod.map there id) $ mapWith∈⁻ xs (f ∘ there) p
+    Product.map₂ (Product.map there id) $ mapWith∈⁻ xs (f ∘ there) p
 
   mapWith∈↔ : ∀ {xs : List A} {f : ∀ {x} → x ∈ xs → B} →
                 (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) ↔ Any P (mapWith∈ xs f)
diff --git a/src/Data/List/Sort/MergeSort.agda b/src/Data/List/Sort/MergeSort.agda
index bded259428..f476d83b96 100644
--- a/src/Data/List/Sort/MergeSort.agda
+++ b/src/Data/List/Sort/MergeSort.agda
@@ -24,7 +24,7 @@ open import Relation.Nullary.Decidable using (does)
 open import Data.Nat.Base using (_<_; _>_; z<s; s<s)
 open import Data.Nat.Induction
 open import Data.Nat.Properties using (m<n⇒m<1+n)
-open import Data.Product.Base as Prod using (_,_)
+open import Data.Product.Base as Product using (_,_)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Bundles using (DecTotalOrder)
 open import Relation.Nullary.Negation using (¬_)
diff --git a/src/Data/Maybe/Relation/Unary/All.agda b/src/Data/Maybe/Relation/Unary/All.agda
index c070bd279d..9a57aa62f5 100644
--- a/src/Data/Maybe/Relation/Unary/All.agda
+++ b/src/Data/Maybe/Relation/Unary/All.agda
@@ -12,7 +12,7 @@ open import Effect.Applicative
 open import Effect.Monad
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.Any using (Any; just)
-open import Data.Product.Base as Prod using (_,_)
+open import Data.Product.Base as Product using (_,_)
 open import Function.Base using (id; _∘′_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
@@ -56,7 +56,7 @@ module _ {a p q r} {A : Set a} {P : Pred A p} {Q : Pred A q} {R : Pred A r} wher
   zipWith f (nothing , nothing) = nothing
 
   unzipWith : P ⊆ Q ∩ R → All P ⊆ All Q ∩ All R
-  unzipWith f (just px) = Prod.map just just (f px)
+  unzipWith f (just px) = Product.map just just (f px)
   unzipWith f nothing   = nothing , nothing
 
 module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
diff --git a/src/Data/Maybe/Relation/Unary/Any.agda b/src/Data/Maybe/Relation/Unary/Any.agda
index 48cd402ce0..767a5624cc 100644
--- a/src/Data/Maybe/Relation/Unary/Any.agda
+++ b/src/Data/Maybe/Relation/Unary/Any.agda
@@ -9,7 +9,7 @@
 module Data.Maybe.Relation.Unary.Any where
 
 open import Data.Maybe.Base using (Maybe; just; nothing)
-open import Data.Product.Base as Prod using (∃; _,_; -,_)
+open import Data.Product.Base as Product using (∃; _,_; -,_)
 open import Function.Base using (id)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
@@ -50,7 +50,7 @@ module _ {a p q r} {A : Set a} {P : Pred A p} {Q : Pred A q} {R : Pred A r} wher
   zipWith f (just px , just qx) = just (f (px , qx))
 
   unzipWith : P ⊆ Q ∩ R → Any P ⊆ Any Q ∩ Any R
-  unzipWith f (just px) = Prod.map just just (f px)
+  unzipWith f (just px) = Product.map just just (f px)
 
 module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
 
@@ -73,4 +73,4 @@ module _ {a p} {A : Set a} {P : Pred A p} where
   irrelevant P-irrelevant (just p) (just q) = cong just (P-irrelevant p q)
 
   satisfiable : Satisfiable P → Satisfiable (Any P)
-  satisfiable P-satisfiable = Prod.map just just P-satisfiable
+  satisfiable P-satisfiable = Product.map just just P-satisfiable
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index 48fdff0af9..b99e1934f8 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.Dependent.Propositional where
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
 open import Data.Product.Function.NonDependent.Setoid using ()
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using ()
 open import Data.Product.Properties using (Σ-≡,≡→≡; Σ-≡,≡↔≡; Σ-≡,≡←≡)
@@ -41,7 +41,7 @@ module _ where
   Σ-⟶ : (I⟶J : I ⟶ J) →
          (∀ {i} → A i ⟶ B (to I⟶J i)) →
          Σ I A ⟶ Σ J B
-  Σ-⟶ I⟶J A⟶B = mk⟶ $ Prod.map (to I⟶J) (to A⟶B)
+  Σ-⟶ I⟶J A⟶B = mk⟶ $ Product.map (to I⟶J) (to A⟶B)
 
 ------------------------------------------------------------------------
 -- Equivalences
@@ -117,7 +117,7 @@ module _ where
     open Injection
 
     to′ : Σ I A → Σ J B
-    to′ = Prod.map (_≃_.to I≃J) (to A↣B)
+    to′ = Product.map (_≃_.to I≃J) (to A↣B)
 
     to-injective : Injective _≡_ _≡_ to′
     to-injective {(x₁ , x₂)} {(y₁ , y₂)} =
diff --git a/src/Data/Product/Function/NonDependent/Setoid.agda b/src/Data/Product/Function/NonDependent/Setoid.agda
index 653f122bb5..95951951b4 100644
--- a/src/Data/Product/Function/NonDependent/Setoid.agda
+++ b/src/Data/Product/Function/NonDependent/Setoid.agda
@@ -9,7 +9,7 @@
 
 module Data.Product.Function.NonDependent.Setoid where
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
 open import Level using (Level)
 open import Relation.Binary
@@ -44,8 +44,8 @@ swapₛ = < proj₂ₛ , proj₁ₛ >ₛ
 
 _×-function_ : Func A B → Func C D → Func (A ×ₛ C) (B ×ₛ D)
 f ×-function g = record
-  { to    = Prod.map (to f) (to g)
-  ; cong  = Prod.map (cong f) (cong g)
+  { to    = Product.map (to f) (to g)
+  ; cong  = Product.map (cong f) (cong g)
   } where open Func
 
 infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_
@@ -53,54 +53,54 @@ infixr 2 _×-equivalence_ _×-injection_ _×-left-inverse_
 _×-equivalence_ : Equivalence A B → Equivalence C D →
                   Equivalence (A ×ₛ C) (B ×ₛ D)
 _×-equivalence_ f g = record
-  { to        = Prod.map (to f) (to g)
-  ; from      = Prod.map (from f) (from g)
-  ; to-cong   = Prod.map (to-cong f) (to-cong g)
-  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  { to        = Product.map (to f) (to g)
+  ; from      = Product.map (from f) (from g)
+  ; to-cong   = Product.map (to-cong f) (to-cong g)
+  ; from-cong = Product.map (from-cong f) (from-cong g)
   } where open Equivalence
 
 _×-injection_ : Injection A B → Injection C D →
                 Injection (A ×ₛ C) (B ×ₛ D)
 f ×-injection g = record
-  { to        = Prod.map (to f) (to g)
-  ; cong      = Prod.map (cong f) (cong g)
-  ; injective = Prod.map (injective f) (injective g)
+  { to        = Product.map (to f) (to g)
+  ; cong      = Product.map (cong f) (cong g)
+  ; injective = Product.map (injective f) (injective g)
   } where open Injection
 
 _×-surjection_ : Surjection A B → Surjection C D →
                  Surjection (A ×ₛ C) (B ×ₛ D)
 f ×-surjection g = record
-  { to         = Prod.map (to f) (to g)
-  ; cong       = Prod.map (cong f) (cong g)
-  ; surjective = λ y → Prod.zip _,_ (λ ff gg x₂ → (ff (proj₁ x₂)) , (gg (proj₂ x₂))) (surjective f (proj₁ y)) (surjective g (proj₂ y))
+  { to         = Product.map (to f) (to g)
+  ; cong       = Product.map (cong f) (cong g)
+  ; surjective = λ y → Product.zip _,_ (λ ff gg x₂ → (ff (proj₁ x₂)) , (gg (proj₂ x₂))) (surjective f (proj₁ y)) (surjective g (proj₂ y))
   } where open Surjection
 
 _×-bijection_ : Bijection A B → Bijection C D →
                 Bijection (A ×ₛ C) (B ×ₛ D)
 f ×-bijection g = record
-  { to         = Prod.map (to f) (to g)
-  ; cong       = Prod.map (cong f) (cong g)
-  ; bijective  = Prod.map (injective f) (injective g) ,
-                 λ { (y₀ , y₁) → Prod.zip _,_ (λ {ff gg (x₀ , x₁) → ff x₀ , gg x₁}) (surjective f y₀) (surjective g y₁)}
+  { to         = Product.map (to f) (to g)
+  ; cong       = Product.map (cong f) (cong g)
+  ; bijective  = Product.map (injective f) (injective g) ,
+                 λ { (y₀ , y₁) → Product.zip _,_ (λ {ff gg (x₀ , x₁) → ff x₀ , gg x₁}) (surjective f y₀) (surjective g y₁)}
   } where open Bijection
 
 _×-leftInverse_ : LeftInverse A B → LeftInverse C D →
                   LeftInverse (A ×ₛ C) (B ×ₛ D)
 f ×-leftInverse g = record
-  { to        = Prod.map (to f) (to g)
-  ; from      = Prod.map (from f) (from g)
-  ; to-cong   = Prod.map (to-cong f) (to-cong g)
-  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  { to        = Product.map (to f) (to g)
+  ; from      = Product.map (from f) (from g)
+  ; to-cong   = Product.map (to-cong f) (to-cong g)
+  ; from-cong = Product.map (from-cong f) (from-cong g)
   ; inverseˡ   = λ x → inverseˡ f (proj₁ x) , inverseˡ g (proj₂ x)
   } where open LeftInverse
 
 _×-rightInverse_ : RightInverse A B → RightInverse C D →
                    RightInverse (A ×ₛ C) (B ×ₛ D)
 f ×-rightInverse g = record
-  { to        = Prod.map (to f) (to g)
-  ; from      = Prod.map (from f) (from g)
-  ; to-cong   = Prod.map (to-cong f) (to-cong g)
-  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  { to        = Product.map (to f) (to g)
+  ; from      = Product.map (from f) (from g)
+  ; to-cong   = Product.map (to-cong f) (to-cong g)
+  ; from-cong = Product.map (from-cong f) (from-cong g)
   ; inverseʳ   = λ x → inverseʳ f (proj₁ x) , inverseʳ g (proj₂ x)
   } where open RightInverse
 
@@ -109,10 +109,10 @@ infixr 2 _×-surjection_ _×-inverse_
 _×-inverse_ : Inverse A B → Inverse C D →
               Inverse (A ×ₛ C) (B ×ₛ D)
 f ×-inverse g = record
-  { to        = Prod.map (to f) (to g)
-  ; from      = Prod.map (from f) (from g)
-  ; to-cong   = Prod.map (to-cong f) (to-cong g)
-  ; from-cong = Prod.map (from-cong f) (from-cong g)
+  { to        = Product.map (to f) (to g)
+  ; from      = Product.map (from f) (from g)
+  ; to-cong   = Product.map (to-cong f) (to-cong g)
+  ; from-cong = Product.map (from-cong f) (from-cong g)
   ; inverse   = (λ x → inverseˡ f (proj₁ x) , inverseˡ g (proj₂ x)) ,
                 (λ x → inverseʳ f (proj₁ x) , inverseʳ g (proj₂ x))
   } where open Inverse
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
index fd22ff6cce..ac0f48df7b 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
@@ -8,7 +8,7 @@
 
 module Data.Product.Relation.Binary.Pointwise.Dependent where
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
 open import Level
 open import Function.Base
 open import Relation.Binary.Core using (Rel; REL; _⇒_)
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index efa64a8605..95a8978cf5 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -8,7 +8,7 @@
 
 module Data.Product.Relation.Binary.Pointwise.NonDependent where
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
 open import Data.Product.Properties using (≡-dec)
 open import Data.Sum.Base
 open import Data.Unit.Base using (⊤)
@@ -35,7 +35,7 @@ Pointwise R S = (R on proj₁) -×- (S on proj₂)
 -- Pointwise preserves many relational properties
 
 ×-reflexive : ≈₁ ⇒ R → ≈₂ ⇒ S → Pointwise ≈₁ ≈₂ ⇒ Pointwise R S
-×-reflexive refl₁ refl₂ = Prod.map refl₁ refl₂
+×-reflexive refl₁ refl₂ = Product.map refl₁ refl₂
 
 ×-refl : Reflexive R → Reflexive S → Reflexive (Pointwise R S)
 ×-refl refl₁ refl₂ = refl₁ , refl₂
@@ -49,14 +49,14 @@ Pointwise R S = (R on proj₁) -×- (S on proj₂)
 ×-irreflexive₂ ir x≈y x<y = ir (proj₂ x≈y) (proj₂ x<y)
 
 ×-symmetric : Symmetric R → Symmetric S → Symmetric (Pointwise R S)
-×-symmetric sym₁ sym₂ = Prod.map sym₁ sym₂
+×-symmetric sym₁ sym₂ = Product.map sym₁ sym₂
 
 ×-transitive : Transitive R → Transitive S → Transitive (Pointwise R S)
-×-transitive trans₁ trans₂ = Prod.zip trans₁ trans₂
+×-transitive trans₁ trans₂ = Product.zip trans₁ trans₂
 
 ×-antisymmetric : Antisymmetric ≈₁ R → Antisymmetric ≈₂ S →
                   Antisymmetric (Pointwise ≈₁ ≈₂) (Pointwise R S)
-×-antisymmetric antisym₁ antisym₂ = Prod.zip antisym₁ antisym₂
+×-antisymmetric antisym₁ antisym₂ = Product.zip antisym₁ antisym₂
 
 ×-asymmetric₁ : Asymmetric R → Asymmetric (Pointwise R S)
 ×-asymmetric₁ asym₁ x<y y<x = asym₁ (proj₁ x<y) (proj₁ y<x)
@@ -66,15 +66,15 @@ Pointwise R S = (R on proj₁) -×- (S on proj₂)
 
 ×-respectsʳ : R Respectsʳ ≈₁ → S Respectsʳ ≈₂ →
              (Pointwise R S) Respectsʳ (Pointwise ≈₁ ≈₂)
-×-respectsʳ resp₁ resp₂ = Prod.zip resp₁ resp₂
+×-respectsʳ resp₁ resp₂ = Product.zip resp₁ resp₂
 
 ×-respectsˡ : R Respectsˡ ≈₁ → S Respectsˡ ≈₂ →
              (Pointwise R S) Respectsˡ (Pointwise ≈₁ ≈₂)
-×-respectsˡ resp₁ resp₂ = Prod.zip resp₁ resp₂
+×-respectsˡ resp₁ resp₂ = Product.zip resp₁ resp₂
 
 ×-respects₂ : R Respects₂ ≈₁ → S Respects₂ ≈₂ →
               (Pointwise R S) Respects₂ (Pointwise ≈₁ ≈₂)
-×-respects₂ = Prod.zip ×-respectsʳ ×-respectsˡ
+×-respects₂ = Product.zip ×-respectsʳ ×-respectsˡ
 
 ×-total : Symmetric R → Total R → Total S → Total (Pointwise R S)
 ×-total sym₁ total₁ total₂ (x₁ , x₂) (y₁ , y₂)
diff --git a/src/Data/Sum/Function/Setoid.agda b/src/Data/Sum/Function/Setoid.agda
index d0b28634fa..659bad1ef3 100644
--- a/src/Data/Sum/Function/Setoid.agda
+++ b/src/Data/Sum/Function/Setoid.agda
@@ -8,7 +8,7 @@
 
 module Data.Sum.Function.Setoid where
 
-open import Data.Product.Base as Prod using (_,_)
+open import Data.Product.Base as Product using (_,_)
 open import Data.Sum.Base as Sum
 open import Data.Sum.Relation.Binary.Pointwise as Pointwise
 open import Relation.Binary
@@ -60,8 +60,8 @@ swapₛ = [ inj₂ₛ , inj₁ₛ ]ₛ
               StrictlySurjective ≈₂ g →
               StrictlySurjective (Pointwise ≈₁ ≈₂) (Sum.map f g)
 ⊎-strictlySurjective f-sur g-sur =
-  [ Prod.map inj₁ inj₁ ∘ f-sur
-  , Prod.map inj₂ inj₂ ∘ g-sur
+  [ Product.map inj₁ inj₁ ∘ f-sur
+  , Product.map inj₂ inj₂ ∘ g-sur
   ]
 
 ⊎-surjective : ∀ {f : A → B} {g : C → D} →
@@ -69,8 +69,8 @@ swapₛ = [ inj₂ₛ , inj₁ₛ ]ₛ
               Surjective ≈₃ ≈₄ g →
               Surjective (Pointwise ≈₁ ≈₃) (Pointwise ≈₂ ≈₄) (Sum.map f g)
 ⊎-surjective f-sur g-sur =
-  [ Prod.map inj₁ (λ { fwd (inj₁ x) → inj₁ (fwd x)}) ∘ f-sur
-  , Prod.map inj₂ (λ { fwd (inj₂ y) → inj₂ (fwd y)}) ∘ g-sur
+  [ Product.map inj₁ (λ { fwd (inj₁ x) → inj₁ (fwd x)}) ∘ f-sur
+  , Product.map inj₂ (λ { fwd (inj₂ y) → inj₂ (fwd y)}) ∘ g-sur
   ]
 
 
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
index 64eb574b04..3c98800c66 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
@@ -15,7 +15,7 @@ module Data.Tree.AVL.Map.Membership.Propositional.Properties
 
 open import Data.Bool.Base using (true; false)
 open import Data.Maybe.Base using (just; nothing; is-just)
-open import Data.Product.Base as Prod using (_×_; ∃-syntax; _,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (_×_; ∃-syntax; _,_; proj₁; proj₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent renaming (Pointwise to _×ᴿ_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _∘′_)
@@ -37,7 +37,7 @@ import Data.Tree.AVL.Indexed.Relation.Unary.Any strictTotalOrder as IAny
 import Data.Tree.AVL.Indexed.Relation.Unary.Any.Properties strictTotalOrder as IAnyₚ
 open import Data.Tree.AVL.Key strictTotalOrder using (⊥⁺<[_]<⊤⁺)
 open import Data.Tree.AVL.Map strictTotalOrder
-open import Data.Tree.AVL.Map.Relation.Unary.Any strictTotalOrder as Mapₚ using (Any)
+open import Data.Tree.AVL.Map.Relation.Unary.Any strictTotalOrder as Map using (Any)
 open import Data.Tree.AVL.Map.Membership.Propositional strictTotalOrder
 open import Data.Tree.AVL.Relation.Unary.Any strictTotalOrder as Any using (tree)
 
@@ -113,7 +113,7 @@ private
 
 ∈ₖᵥ-lookup⁻ : lookup m k ≡ just x → (k , x) ∈ₖᵥ m
 ∈ₖᵥ-lookup⁻ {m = tree t} {k = k} {x = x} eq
-  = tree (IAny.map (Prod.map sym ≡-sym) (IAnyₚ.lookup⁻ t k x (⊥⁺<[ _ ]<⊤⁺) eq))
+  = tree (IAny.map (Product.map sym ≡-sym) (IAnyₚ.lookup⁻ t k x (⊥⁺<[ _ ]<⊤⁺) eq))
 
 ∈ₖᵥ-lookup-nothing⁺ : (∀ x → (k , x) ∉ₖᵥ m) → lookup m k ≡ nothing
 ∈ₖᵥ-lookup-nothing⁺ {k = k} {m = m@(tree t)} k∉m with lookup m k in eq
diff --git a/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
index 5ad457d54f..a6e16267c4 100644
--- a/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Map/Relation/Unary/Any.agda
@@ -12,7 +12,7 @@ module Data.Tree.AVL.Map.Relation.Unary.Any
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
   where
 
-open import Data.Product.Base as Prod using (_×_; ∃; _,_)
+open import Data.Product.Base as Product using (_×_; ∃; _,_)
 open import Function.Base using (_∘_; _∘′_; id)
 open import Level using (Level; _⊔_)
 
@@ -21,7 +21,7 @@ open import Relation.Unary
 open StrictTotalOrder strictTotalOrder renaming (Carrier to Key)
 open import Data.Tree.AVL.Map strictTotalOrder using (Map)
 open import Data.Tree.AVL.Indexed strictTotalOrder using (toPair)
-import Data.Tree.AVL.Relation.Unary.Any strictTotalOrder as Mapₚ
+import Data.Tree.AVL.Relation.Unary.Any strictTotalOrder as Map
 
 private
   variable
@@ -39,30 +39,30 @@ private
 -- See `Relation.Unary` for an explanation of predicates.
 
 Any : {V : Set v} → Pred (Key × V) p → Pred (Map V) (a ⊔ ℓ₂ ⊔ v ⊔ p)
-Any P = Mapₚ.Any (P ∘′ toPair)
+Any P = Map.Any (P ∘′ toPair)
 
 ------------------------------------------------------------------------
 -- Operations on Any
 
 map : P ⊆ Q → Any P ⊆ Any Q
-map f = Mapₚ.map f
+map f = Map.map f
 
 lookup : Any {V = V} P m → Key × V
-lookup = toPair ∘′ Mapₚ.lookup
+lookup = toPair ∘′ Map.lookup
 
 lookupKey : Any P m → Key
-lookupKey = Mapₚ.lookupKey
+lookupKey = Map.lookupKey
 
 -- If any element satisfies P, then P is satisfied.
 
 satisfied : Any P m → ∃ P
-satisfied = Prod.map toPair id ∘′ Mapₚ.satisfied
+satisfied = Product.map toPair id ∘′ Map.satisfied
 
 ------------------------------------------------------------------------
 -- Properties of predicates preserved by Any
 
 any? : Decidable P → Decidable (Any P)
-any? P? = Mapₚ.any? (P? ∘ toPair)
+any? P? = Map.any? (P? ∘ toPair)
 
 satisfiable : Satisfiable P → Satisfiable (Any P)
-satisfiable ((k , v) , p) = Mapₚ.satisfiable k (v , p)
+satisfiable ((k , v) , p) = Map.satisfiable k (v , p)
diff --git a/src/Data/Tree/AVL/Relation/Unary/Any.agda b/src/Data/Tree/AVL/Relation/Unary/Any.agda
index dfa96935ac..5efc9363a3 100644
--- a/src/Data/Tree/AVL/Relation/Unary/Any.agda
+++ b/src/Data/Tree/AVL/Relation/Unary/Any.agda
@@ -12,7 +12,7 @@ module Data.Tree.AVL.Relation.Unary.Any
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂)
   where
 
-open import Data.Product.Base as Prod using (∃)
+open import Data.Product.Base as Product using (∃)
 open import Function.Base using (_∘_; _$_)
 open import Level using (Level; _⊔_)
 
@@ -68,5 +68,5 @@ any? : Decidable P → Decidable (Any {V = V} P)
 any? P? (tree p) = map′ tree (λ where (tree p) → p) (AVLₚ.any? P? p)
 
 satisfiable : (k : Key) → Satisfiable (P ∘ (k ,_)) → Satisfiable (Any {V = V} P)
-satisfiable k sat = Prod.map tree tree
+satisfiable k sat = Product.map tree tree
                   $ AVLₚ.satisfiable Indexed.⊥⁺<[ k ] Indexed.[ k ]<⊤⁺ sat
diff --git a/src/Data/Tree/AVL/Sets.agda b/src/Data/Tree/AVL/Sets.agda
index f20fded531..6065671a41 100644
--- a/src/Data/Tree/AVL/Sets.agda
+++ b/src/Data/Tree/AVL/Sets.agda
@@ -16,7 +16,7 @@ open import Data.Bool.Base using (Bool)
 open import Data.List.Base as List using (List)
 open import Data.Maybe.Base as Maybe
 open import Data.Nat.Base using (ℕ)
-open import Data.Product.Base as Prod using (_×_; _,_; proj₁)
+open import Data.Product.Base as Product using (_×_; _,_; proj₁)
 open import Data.Unit.Base
 open import Function.Base
 open import Level using (Level; _⊔_)
@@ -54,10 +54,10 @@ member : A → ⟨Set⟩ → Bool
 member = AVL.member
 
 headTail : ⟨Set⟩ → Maybe (A × ⟨Set⟩)
-headTail s = Maybe.map (Prod.map₁ proj₁) (AVL.headTail s)
+headTail s = Maybe.map (Product.map₁ proj₁) (AVL.headTail s)
 
 initLast : ⟨Set⟩ → Maybe (⟨Set⟩ × A)
-initLast s = Maybe.map (Prod.map₂ proj₁) (AVL.initLast s)
+initLast s = Maybe.map (Product.map₂ proj₁) (AVL.initLast s)
 
 fromList : List A → ⟨Set⟩
 fromList = AVL.fromList ∘ List.map (_, _)
diff --git a/src/Data/Tree/AVL/Sets/Membership.agda b/src/Data/Tree/AVL/Sets/Membership.agda
index f65022ac92..ba6a231227 100644
--- a/src/Data/Tree/AVL/Sets/Membership.agda
+++ b/src/Data/Tree/AVL/Sets/Membership.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Sets.Membership
   where
 
 open import Data.Bool.Base using (true; false)
-open import Data.Product.Base as Prod using (_,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (_,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
 open import Function.Base using (_∘_; _∘′_; const)
diff --git a/src/Data/Tree/AVL/Sets/Membership/Properties.agda b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
index db3f0d38cc..8b46933af5 100644
--- a/src/Data/Tree/AVL/Sets/Membership/Properties.agda
+++ b/src/Data/Tree/AVL/Sets/Membership/Properties.agda
@@ -13,7 +13,7 @@ module Data.Tree.AVL.Sets.Membership.Properties
   where
 
 open import Data.Bool.Base using (true; false)
-open import Data.Product.Base as Prod using (_,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (_,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
 open import Function.Base using (_∘_; _∘′_; const)
diff --git a/src/Data/Trie/NonEmpty.agda b/src/Data/Trie/NonEmpty.agda
index f32d20d92f..3bdc550a68 100644
--- a/src/Data/Trie/NonEmpty.agda
+++ b/src/Data/Trie/NonEmpty.agda
@@ -13,7 +13,7 @@ module Data.Trie.NonEmpty {k e r} (S : StrictTotalOrder k e r) where
 open import Level
 open import Size
 open import Effect.Monad
-open import Data.Product.Base as Prod using (∃; uncurry; -,_)
+open import Data.Product.Base as Product using (∃; uncurry; -,_)
 open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; [_]; concatMap)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′) hiding (module Maybe)
@@ -125,7 +125,7 @@ toList⁺ (node nd) = These.mergeThese List⁺._⁺++⁺_
   where
 
   fromVal    = [_] ∘′ -,_
-  fromTrie⁺  = λ (k , v) → List⁺.map (Prod.map (k ∷_) id) (toList⁺ v)
+  fromTrie⁺  = λ (k , v) → List⁺.map (Product.map (k ∷_) id) (toList⁺ v)
   fromTries⁺ = concatMap fromTrie⁺ ∘′ Tree⁺.toList⁺
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 10684dee30..5f57c22b2f 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -12,7 +12,7 @@ open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Nat.Base
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List)
-open import Data.Product.Base as Prod using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Function.Base using (const; _∘′_; id; _∘_)
 open import Level using (Level)
@@ -131,7 +131,7 @@ zipWith f (x ∷ xs) (y ∷ ys) = f x y ∷ zipWith f xs ys
 
 unzipWith : (A → B × C) → Vec A n → Vec B n × Vec C n
 unzipWith f []       = [] , []
-unzipWith f (a ∷ as) = Prod.zip _∷_ _∷_ (f a) (unzipWith f as)
+unzipWith f (a ∷ as) = Product.zip _∷_ _∷_ (f a) (unzipWith f as)
 
 align : Vec A m → Vec B n → Vec (These A B) (m ⊔ n)
 align = alignWith id
@@ -282,7 +282,7 @@ group (suc n) k xs  =
 split : Vec A n → Vec A ⌈ n /2⌉ × Vec A ⌊ n /2⌋
 split []           = ([]     , [])
 split (x ∷ [])     = (x ∷ [] , [])
-split (x ∷ y ∷ xs) = Prod.map (x ∷_) (y ∷_) (split xs)
+split (x ∷ y ∷ xs) = Product.map (x ∷_) (y ∷_) (split xs)
 
 uncons : Vec A (suc n) → A × Vec A n
 uncons (x ∷ xs) = x , xs
diff --git a/src/Data/Vec/N-ary.agda b/src/Data/Vec/N-ary.agda
index 13c7cb29ae..fa8fc7589c 100644
--- a/src/Data/Vec/N-ary.agda
+++ b/src/Data/Vec/N-ary.agda
@@ -11,7 +11,7 @@ module Data.Vec.N-ary where
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)
 open import Data.Nat.Base hiding (_⊔_)
-open import Data.Product.Base as Prod using (∃; _,_)
+open import Data.Product.Base as Product using (∃; _,_)
 open import Data.Vec.Base using (Vec; []; _∷_; head; tail)
 open import Function.Base using (_∘_; id; flip; constᵣ)
 open import Function.Bundles using (_⇔_; mk⇔)
@@ -126,7 +126,7 @@ uncurry-∃ⁿ {a} {A} {ℓ} n = mk⇔ (⇒ n) (⇐ n)
   ⇒ : ∀ n {P : N-ary n A (Set ℓ)} →
       ∃ⁿ n P → (∃ λ (xs : Vec A n) → P $ⁿ xs)
   ⇒ zero    p       = ([] , p)
-  ⇒ (suc n) (x , p) = Prod.map (_∷_ x) id (⇒ n p)
+  ⇒ (suc n) (x , p) = Product.map (_∷_ x) id (⇒ n p)
 
   ⇐ : ∀ n {P : N-ary n A (Set ℓ)} →
       (∃ λ (xs : Vec A n) → P $ⁿ xs) → ∃ⁿ n P
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 4d469a205c..a860a2d925 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -17,7 +17,7 @@ import Data.List.Properties as Listₚ
 open import Data.Nat.Base
 open import Data.Nat.Properties
   using (+-assoc; m≤n⇒m≤1+n; m≤m+n; ≤-refl; ≤-trans; ≤-irrelevant; ≤⇒≤″; suc-injective; +-comm; +-suc)
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
   using (_×_; _,_; proj₁; proj₂; <_,_>; uncurry)
 open import Data.Sum.Base using ([_,_]′)
 open import Data.Sum.Properties using ([,]-map)
@@ -689,7 +689,7 @@ map-<,>-zip f g []       = refl
 map-<,>-zip f g (x ∷ xs) = cong (_ ∷_) (map-<,>-zip f g xs)
 
 map-zip : ∀ (f : A → B) (g : C → D) (xs : Vec A n) (ys : Vec C n) →
-          map (Prod.map f g) (zip xs ys) ≡ zip (map f xs) (map g ys)
+          map (Product.map f g) (zip xs ys) ≡ zip (map f xs) (map g ys)
 map-zip f g []       []       = refl
 map-zip f g (x ∷ xs) (y ∷ ys) = cong (_ ∷_) (map-zip f g xs ys)
 
@@ -705,10 +705,10 @@ lookup-unzip (suc i) ((x , y) ∷ xys) = lookup-unzip i xys
 
 map-unzip : ∀ (f : A → B) (g : C → D) (xys : Vec (A × C) n) →
             let xs , ys = unzip xys
-            in (map f xs , map g ys) ≡ unzip (map (Prod.map f g) xys)
+            in (map f xs , map g ys) ≡ unzip (map (Product.map f g) xys)
 map-unzip f g []              = refl
 map-unzip f g ((x , y) ∷ xys) =
-  cong (Prod.map (f x ∷_) (g y ∷_)) (map-unzip f g xys)
+  cong (Product.map (f x ∷_) (g y ∷_)) (map-unzip f g xys)
 
 -- Products of vectors are isomorphic to vectors of products.
 
@@ -716,7 +716,7 @@ unzip∘zip : ∀ (xs : Vec A n) (ys : Vec B n) →
             unzip (zip xs ys) ≡ (xs , ys)
 unzip∘zip [] []             = refl
 unzip∘zip (x ∷ xs) (y ∷ ys) =
-  cong (Prod.map (x ∷_) (y ∷_)) (unzip∘zip xs ys)
+  cong (Product.map (x ∷_) (y ∷_)) (unzip∘zip xs ys)
 
 zip∘unzip : ∀ (xys : Vec (A × B) n) → uncurry zip (unzip xys) ≡ xys
 zip∘unzip []         = refl
@@ -869,7 +869,7 @@ unfold-∷ʳ eq x (y ∷ xs) = cong (y ∷_) (unfold-∷ʳ (cong pred eq) x xs)
 ∷ʳ-injective : ∀ (xs ys : Vec A n) → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
 ∷ʳ-injective []       []        refl = (refl , refl)
 ∷ʳ-injective (x ∷ xs) (y  ∷ ys) eq   with ∷-injective eq
-... | refl , eq′ = Prod.map₁ (cong (x ∷_)) (∷ʳ-injective xs ys eq′)
+... | refl , eq′ = Product.map₁ (cong (x ∷_)) (∷ʳ-injective xs ys eq′)
 
 ∷ʳ-injectiveˡ : ∀ (xs ys : Vec A n) → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys
 ∷ʳ-injectiveˡ xs ys eq = proj₁ (∷ʳ-injective xs ys eq)
diff --git a/src/Data/Vec/Recursive.agda b/src/Data/Vec/Recursive.agda
index ea7d04285f..0eb04b264b 100644
--- a/src/Data/Vec/Recursive.agda
+++ b/src/Data/Vec/Recursive.agda
@@ -21,7 +21,7 @@ open import Data.Nat.Properties using (+-comm; *-comm)
 open import Data.Empty.Polymorphic
 open import Data.Fin.Base as Fin using (Fin; zero; suc)
 open import Data.Fin.Properties using (1↔⊤; *↔×)
-open import Data.Product.Base as Prod using (_×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.Product.Algebra using (×-cong)
 open import Data.Sum.Base as Sum using (_⊎_)
 open import Data.Unit.Base using (tt)
@@ -146,7 +146,7 @@ zipWith f ((suc n@(suc _))) (a , as) (b , bs) = f a b , zipWith f n as bs
 unzipWith : (A → B × C) → ∀ n → A ^ n → B ^ n × C ^ n
 unzipWith f 0               as       = [] , []
 unzipWith f 1               a        = f a
-unzipWith f (suc n@(suc _)) (a , as) = Prod.zip _,_ _,_ (f a) (unzipWith f n as)
+unzipWith f (suc n@(suc _)) (a , as) = Product.zip _,_ _,_ (f a) (unzipWith f n as)
 
 zip : ∀ n → A ^ n → B ^ n → (A × B) ^ n
 zip = zipWith _,_
diff --git a/src/Data/Vec/Relation/Unary/All.agda b/src/Data/Vec/Relation/Unary/All.agda
index e8d24fbd65..4cc1805c25 100644
--- a/src/Data/Vec/Relation/Unary/All.agda
+++ b/src/Data/Vec/Relation/Unary/All.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Relation.Unary.All where
 
 open import Data.Nat.Base using (ℕ; zero; suc; NonZero)
-open import Data.Product.Base as Prod using (_×_; _,_; uncurry; <_,_>)
+open import Data.Product.Base as Product using (_×_; _,_; uncurry; <_,_>)
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)
@@ -71,7 +71,7 @@ zip (px ∷ pxs , qx ∷ qxs) = (px , qx) ∷ zip (pxs , qxs)
 
 unzip : All (P ∩ Q) {n} ⊆ All P ∩ All Q
 unzip []           = [] , []
-unzip (pqx ∷ pqxs) = Prod.zip _∷_ _∷_ pqx (unzip pqxs)
+unzip (pqx ∷ pqxs) = Product.zip _∷_ _∷_ pqx (unzip pqxs)
 
 module _ {P : Pred A p} {Q : Pred B q} {R : Pred C r} where
 
@@ -91,7 +91,7 @@ lookupAny (px ∷ pxs) (here qx) = px , qx
 lookupAny (px ∷ pxs) (there i) = lookupAny pxs i
 
 lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
-lookupWith f pxs i = Prod.uncurry f (lookupAny pxs i)
+lookupWith f pxs i = Product.uncurry f (lookupAny pxs i)
 
 lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
 lookup pxs = lookupWith (λ { px P.refl → px }) pxs
@@ -121,7 +121,7 @@ irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =
 
 satisfiable : Satisfiable P → ∀ {n} → Satisfiable (All P {n})
 satisfiable (x , p) {zero}  = [] , []
-satisfiable (x , p) {suc n} = Prod.map (x ∷_) (p ∷_) (satisfiable (x , p))
+satisfiable (x , p) {suc n} = Product.map (x ∷_) (p ∷_) (satisfiable (x , p))
 
 ------------------------------------------------------------------------
 -- Generalised decidability procedure
diff --git a/src/Data/Vec/Relation/Unary/All/Properties.agda b/src/Data/Vec/Relation/Unary/All/Properties.agda
index 4b8951ff9d..57f06a7087 100644
--- a/src/Data/Vec/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/All/Properties.agda
@@ -12,7 +12,7 @@ open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base using ([]; _∷_)
 open import Data.List.Relation.Unary.All as List using ([]; _∷_)
-open import Data.Product.Base as Prod using (_×_; _,_; uncurry; uncurry′)
+open import Data.Product.Base as Product using (_×_; _,_; uncurry; uncurry′)
 open import Data.Vec.Base as Vec
 import Data.Vec.Properties as Vecₚ
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
@@ -82,7 +82,7 @@ gmap g = map⁺ ∘ All.map g
 ++⁻ : (xs : Vec A m) {ys : Vec A n} →
       All P (xs ++ ys) → All P xs × All P ys
 ++⁻ []       p          = [] , p
-++⁻ (x ∷ xs) (px ∷ pxs) = Prod.map₁ (px ∷_) (++⁻ _ pxs)
+++⁻ (x ∷ xs) (px ∷ pxs) = Product.map₁ (px ∷_) (++⁻ _ pxs)
 
 ++⁺∘++⁻ : (xs : Vec A m) {ys : Vec A n} →
           (p : All P (xs ++ ys)) →
diff --git a/src/Data/Vec/Relation/Unary/Any.agda b/src/Data/Vec/Relation/Unary/Any.agda
index de901fe571..b4d857a69c 100644
--- a/src/Data/Vec/Relation/Unary/Any.agda
+++ b/src/Data/Vec/Relation/Unary/Any.agda
@@ -13,7 +13,7 @@ open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Nat.Base using (ℕ; zero; suc; NonZero)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Vec.Base as Vec using (Vec; []; [_]; _∷_)
-open import Data.Product.Base as Prod using (∃; _,_)
+open import Data.Product.Base as Product using (∃; _,_)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Nullary.Decidable as Dec using (yes; no; _⊎-dec_)
@@ -80,7 +80,7 @@ any? P? (x ∷ xs) = Dec.map′ fromSum toSum (P? x ⊎-dec any? P? xs)
 
 satisfiable : Satisfiable P → ∀ {n} → Satisfiable (Any P {suc n})
 satisfiable (x , p) {zero}  = x ∷ [] , here p
-satisfiable (x , p) {suc n} = Prod.map (x ∷_) there (satisfiable (x , p))
+satisfiable (x , p) {suc n} = Product.map (x ∷_) there (satisfiable (x , p))
 
 any = any?
 {-# WARNING_ON_USAGE any
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index 437e7933e7..d330ea5cef 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -15,7 +15,7 @@ open import Data.List.Base using ([]; _∷_)
 import Data.List.Relation.Unary.Any as List
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
-open import Data.Product.Base as Prod using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
+open import Data.Product.Base as Product using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
 open import Data.Vec.Base hiding (here; there)
 open import Data.Vec.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.Vec.Membership.Propositional
@@ -365,7 +365,7 @@ module _ {P : Pred A p} where
   tabulate⁻ : ∀ {n} {f : Fin n → A} →
               Any P (tabulate f) → ∃ λ i → P (f i)
   tabulate⁻ (here p)  = zero , p
-  tabulate⁻ (there p) = Prod.map suc id (tabulate⁻ p)
+  tabulate⁻ (there p) = Product.map suc id (tabulate⁻ p)
 
 ------------------------------------------------------------------------
 -- mapWith∈
@@ -384,7 +384,7 @@ module _ {P : Pred B p} where
               ∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)
   mapWith∈⁻ (y ∷ xs) f (here  p) = (y , here refl , p)
   mapWith∈⁻ (y ∷ xs) f (there p) =
-    Prod.map id (Prod.map there id) $ mapWith∈⁻ xs (f ∘ there) p
+    Product.map id (Product.map there id) $ mapWith∈⁻ xs (f ∘ there) p
 
   mapWith∈↔ : ∀ {n} {xs : Vec A n} {f : ∀ {x} → x ∈ xs → B} →
     (∃₂ λ x (x∈xs : x ∈ xs) → P (f x∈xs)) ↔ Any P (mapWith∈ xs f)
diff --git a/src/Data/Vec/Relation/Unary/Linked.agda b/src/Data/Vec/Relation/Unary/Linked.agda
index 368c22ddc1..c0f6b5fe89 100644
--- a/src/Data/Vec/Relation/Unary/Linked.agda
+++ b/src/Data/Vec/Relation/Unary/Linked.agda
@@ -11,7 +11,7 @@ module Data.Vec.Relation.Unary.Linked {a} {A : Set a} where
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
-open import Data.Product.Base as Prod using (_,_; _×_; uncurry; <_,_>)
+open import Data.Product.Base as Product using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (Rel; _⇒_)
@@ -66,7 +66,7 @@ zipWith f (px ∷ pxs , qx ∷ qxs) = f (px , qx) ∷ zipWith f (pxs , qxs)
 unzipWith : R ⇒ P ∩ᵇ Q → Linked R {n} ⊆ Linked P ∩ᵘ Linked Q
 unzipWith f []         = [] , []
 unzipWith f [-]        = [-] , [-]
-unzipWith f (rx ∷ rxs) = Prod.zip _∷_ _∷_ (f rx) (unzipWith f rxs)
+unzipWith f (rx ∷ rxs) = Product.zip _∷_ _∷_ (f rx) (unzipWith f rxs)
 
 zip : Linked P {n} ∩ᵘ Linked Q ⊆ Linked (P ∩ᵇ Q)
 zip = zipWith id
diff --git a/src/Function/Consequences.agda b/src/Function/Consequences.agda
index 2fda92224e..2a13d452f6 100644
--- a/src/Function/Consequences.agda
+++ b/src/Function/Consequences.agda
@@ -10,7 +10,7 @@
 
 module Function.Consequences where
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
 open import Function.Definitions
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
@@ -72,14 +72,14 @@ surjective⇒strictlySurjective : ∀ (≈₂ : Rel B ℓ₂) →
                                  Surjective ≈₁ ≈₂ f →
                                  StrictlySurjective ≈₂ f
 surjective⇒strictlySurjective _ refl surj x =
-  Prod.map₂ (λ v → v refl) (surj x)
+  Product.map₂ (λ v → v refl) (surj x)
 
 strictlySurjective⇒surjective : Transitive ≈₂ →
                                  Congruent ≈₁ ≈₂ f →
                                  StrictlySurjective ≈₂ f →
                                  Surjective ≈₁ ≈₂ f
 strictlySurjective⇒surjective trans cong surj x =
-  Prod.map₂ (λ fy≈x z≈y → trans (cong z≈y) fy≈x) (surj x)
+  Product.map₂ (λ fy≈x z≈y → trans (cong z≈y) fy≈x) (surj x)
 
 ------------------------------------------------------------------------
 -- StrictlyInverseˡ
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index 9fb1fded28..d8a78e052c 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -15,7 +15,7 @@ open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Bool.Base using (true; false)
 open import Data.Empty using (⊥-elim)
 open import Data.Empty.Polymorphic using (⊥) renaming (⊥-elim to ⊥ₚ-elim)
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
   using (_×_; Σ; curry; uncurry; _,_; -,_; <_,_>; proj₁; proj₂; ∃₂; ∃)
 open import Data.Product.Function.NonDependent.Propositional
 open import Data.Sum.Base as Sum
@@ -47,12 +47,12 @@ private
 -- Σ is associative
 Σ-assoc : ∀ {A : Set a} {B : A → Set b} {C : (a : A) → B a → Set c} →
           Σ (Σ A B) (uncurry C) ↔ Σ A (λ a → Σ (B a) (C a))
-Σ-assoc = mk↔ₛ′ Prod.assocʳ Prod.assocˡ (λ _ → P.refl) (λ _ → P.refl)
+Σ-assoc = mk↔ₛ′ Product.assocʳ Product.assocˡ (λ _ → P.refl) (λ _ → P.refl)
 
 -- × is commutative
 
 ×-comm : ∀ (A : Set a) (B : Set b) → (A × B) ↔ (B × A)
-×-comm _ _ = mk↔ₛ′ Prod.swap Prod.swap (λ _ → P.refl) λ _ → P.refl
+×-comm _ _ = mk↔ₛ′ Product.swap Product.swap (λ _ → P.refl) λ _ → P.refl
 
 -- × has ⊤ as its identity
 
@@ -114,14 +114,14 @@ private
 ×-distribˡ-⊎ : ∀ ℓ → _DistributesOverˡ_ {ℓ = ℓ} _↔_ _×_ _⊎_
 ×-distribˡ-⊎ ℓ _ _ _ = mk↔ₛ′
   (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)
-  [ Prod.map₂ inj₁ , Prod.map₂ inj₂ ]′
+  [ Product.map₂ inj₁ , Product.map₂ inj₂ ]′
   [ (λ _ → P.refl) , (λ _ → P.refl) ]
   (uncurry λ _ → [ (λ _ → P.refl) , (λ _ → P.refl) ])
 
 ×-distribʳ-⊎ : ∀ ℓ → _DistributesOverʳ_ {ℓ = ℓ} _↔_ _×_ _⊎_
 ×-distribʳ-⊎ ℓ _ _ _ = mk↔ₛ′
   (uncurry [ curry inj₁ , curry inj₂ ]′)
-  [ Prod.map₁ inj₁ , Prod.map₁ inj₂ ]′
+  [ Product.map₁ inj₁ , Product.map₁ inj₂ ]′
   [ (λ _ → P.refl) , (λ _ → P.refl) ]
   (uncurry [ (λ _ _ → P.refl) , (λ _ _ → P.refl) ])
 
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index 29defea059..ff04ed09a1 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -8,7 +8,7 @@
 
 module Relation.Binary.Construct.Closure.Reflexive.Properties where
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base as Product
 open import Data.Sum.Base as Sum
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Function.Base using (id)
@@ -103,7 +103,7 @@ module _ {_~_ : Rel A ℓ} {P : Pred A p} where
 module _ {_~_ : Rel A ℓ} {P : Rel A p} where
 
   resp₂ : P Respects₂ _~_ → P Respects₂ (ReflClosure _~_)
-  resp₂ = Prod.map respˡ respʳ
+  resp₂ = Product.map respˡ respʳ
 
 ------------------------------------------------------------------------
 -- Structures
diff --git a/src/Relation/Binary/Construct/Subst/Equality.agda b/src/Relation/Binary/Construct/Subst/Equality.agda
index 95172b57ee..8cde3b1a66 100644
--- a/src/Relation/Binary/Construct/Subst/Equality.agda
+++ b/src/Relation/Binary/Construct/Subst/Equality.agda
@@ -9,7 +9,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Product.Base as Prod
+open import Data.Product.Base using (_,_)
 open import Relation.Binary.Core using (Rel; _⇔_)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
diff --git a/src/Relation/Binary/Properties/Preorder.agda b/src/Relation/Binary/Properties/Preorder.agda
index b2be296828..93ff3ba71e 100644
--- a/src/Relation/Binary/Properties/Preorder.agda
+++ b/src/Relation/Binary/Properties/Preorder.agda
@@ -13,7 +13,7 @@ module Relation.Binary.Properties.Preorder
   {p₁ p₂ p₃} (P : Preorder p₁ p₂ p₃) where
 
 open import Function.Base using (flip)
-open import Data.Product.Base as Prod using (_×_; _,_; swap)
+open import Data.Product.Base as Product using (_×_; _,_; swap)
 import Relation.Binary.Construct.Flip.EqAndOrd as EqAndOrd
 
 open Preorder P
@@ -37,7 +37,7 @@ InducedEquivalence = record
   ; isEquivalence = record
     { refl  = (refl , refl)
     ; sym   = swap
-    ; trans = Prod.zip trans (flip trans)
+    ; trans = Product.zip trans (flip trans)
     }
   }
 
diff --git a/src/Relation/Nary.agda b/src/Relation/Nary.agda
index 6610b16439..64308696fe 100644
--- a/src/Relation/Nary.agda
+++ b/src/Relation/Nary.agda
@@ -19,7 +19,7 @@ open import Data.Unit.Base
 open import Data.Bool.Base using (true; false)
 open import Data.Empty
 open import Data.Nat.Base using (zero; suc)
-open import Data.Product.Base as Prod using (_×_; _,_)
+open import Data.Product.Base as Product using (_×_; _,_)
 open import Data.Product.Nary.NonDependent
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_$_; _∘′_)
diff --git a/src/Relation/Nullary/Negation.agda b/src/Relation/Nullary/Negation.agda
index 4d9ee54208..b9116504ec 100644
--- a/src/Relation/Nullary/Negation.agda
+++ b/src/Relation/Nullary/Negation.agda
@@ -11,7 +11,7 @@ module Relation.Nullary.Negation where
 open import Effect.Monad using (RawMonad; mkRawMonad)
 open import Data.Bool.Base using (Bool; false; true; if_then_else_; not)
 open import Data.Empty using (⊥-elim)
-open import Data.Product.Base as Prod using (_,_; Σ; Σ-syntax; ∃; curry; uncurry)
+open import Data.Product.Base as Product using (_,_; Σ; Σ-syntax; ∃; curry; uncurry)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
 open import Function.Base using (flip; _∘_; const; _∘′_)
 open import Level using (Level)
@@ -82,7 +82,7 @@ independence-of-premise : A → (B → Σ A P) → DoubleNegation (Σ[ x ∈ A ]
 independence-of-premise {A = A} {B = B} {P = P} q f = ¬¬-map helper ¬¬-excluded-middle
   where
   helper : Dec B → Σ[ x ∈ A ] (B → P x)
-  helper (yes p) = Prod.map₂ const (f p)
+  helper (yes p) = Product.map₂ const (f p)
   helper (no ¬p) = (q , ⊥-elim ∘′ ¬p)
 
 -- The independence of premise rule for binary sums.
diff --git a/src/Relation/Nullary/Universe.agda b/src/Relation/Nullary/Universe.agda
index 34e087ec69..56b67682fc 100644
--- a/src/Relation/Nullary/Universe.agda
+++ b/src/Relation/Nullary/Universe.agda
@@ -19,7 +19,7 @@ import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   as Trivial
 open import Data.Sum.Base as Sum  hiding (map)
 open import Data.Sum.Relation.Binary.Pointwise using (_⊎ₛ_; inj₁; inj₂)
-open import Data.Product.Base as Prod hiding (map)
+open import Data.Product.Base as Product hiding (map)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 open import Function.Base using (_∘_; id)
 open import Function.Indexed.Relation.Binary.Equality using (≡-setoid)
@@ -69,7 +69,7 @@ map : ∀ {p} (F : PropF p) {P Q} → (P → Q) → ⟦ F ⟧ P → ⟦ F ⟧ Q
 map Id        f  p = f p
 map (K P)     f  p = p
 map (F₁ ∨ F₂) f FP = Sum.map  (map F₁ f) (map F₂ f) FP
-map (F₁ ∧ F₂) f FP = Prod.map (map F₁ f) (map F₂ f) FP
+map (F₁ ∧ F₂) f FP = Product.map (map F₁ f) (map F₂ f) FP
 map (P₁ ⇒ F₂) f FP = map F₂ f ∘ FP
 map (¬¬ F)    f FP = ¬¬-map (map F f) FP
 

From 95184f12383b7ce8a8ad09faab06dc86e95d685a Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 8 Feb 2024 20:28:28 +0000
Subject: [PATCH 223/316] standardise use of `Properties` modules (#2283)

---
 .../Stream/Relation/Binary/Pointwise.agda     |  22 ++--
 src/Codata/Sized/Colist/Properties.agda       | 108 +++++++++---------
 src/Data/Char/Properties.agda                 |  26 ++---
 src/Data/Digit/Properties.agda                |   8 +-
 src/Data/Fin/Induction.agda                   |   4 +-
 src/Data/Fin/Substitution/Lemmas.agda         |  34 +++---
 src/Data/Integer/Properties.agda              |   6 +-
 src/Data/List/Countdown.agda                  |  12 +-
 .../Fresh/Membership/Setoid/Properties.agda   |  21 ++--
 .../Membership/Propositional/Properties.agda  |  88 +++++++-------
 .../Propositional/Properties/WithK.agda       |   6 +-
 .../List/Membership/Setoid/Properties.agda    |  30 ++---
 .../Infix/Heterogeneous/Properties.agda       |  34 +++---
 .../Sublist/Heterogeneous/Properties.agda     |  10 +-
 .../Subset/Propositional/Properties.agda      |  32 +++---
 .../Suffix/Heterogeneous/Properties.agda      |  20 ++--
 .../Ternary/Appending/Propositional.agda      |   6 +-
 .../Appending/Propositional/Properties.agda   |   9 +-
 src/Data/String/Properties.agda               |  24 ++--
 src/Data/Tree/Rose/Properties.agda            |  14 +--
 src/Data/Vec/Properties.agda                  |  14 +--
 .../Vec/Relation/Unary/All/Properties.agda    |   1 -
 .../Relation/Unary/AllPairs/Properties.agda   |  14 +--
 .../Vec/Relation/Unary/Linked/Properties.agda |   4 +-
 .../Unary/Unique/Setoid/Properties.agda       |  16 +--
 src/System/Clock.agda                         |   4 +-
 src/Text/Pretty/Core.agda                     |  12 +-
 src/Text/Regex/String.agda                    |   4 +-
 src/Text/Tabular/List.agda                    |   4 +-
 29 files changed, 292 insertions(+), 295 deletions(-)

diff --git a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
index a4908e851f..0ba8d9539c 100644
--- a/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
+++ b/src/Codata/Guarded/Stream/Relation/Binary/Pointwise.agda
@@ -17,8 +17,8 @@ open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions
   using (Reflexive; Sym; Trans; Antisym; Symmetric; Transitive)
 open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -114,16 +114,16 @@ module _ {A : Set a} where
  _≈_ = Pointwise _≡_
 
  refl : Reflexive _≈_
- refl = reflexive P.refl
+ refl = reflexive ≡.refl
 
  sym : Symmetric _≈_
- sym = symmetric P.sym
+ sym = symmetric ≡.sym
 
  trans : Transitive _≈_
- trans = transitive P.trans
+ trans = transitive ≡.trans
 
  ≈-setoid : Setoid _ _
- ≈-setoid = setoid (P.setoid A)
+ ≈-setoid = setoid (≡.setoid A)
 
 ------------------------------------------------------------------------
 -- Pointwise DSL
@@ -161,7 +161,7 @@ module pw-Reasoning (S : Setoid a ℓ) where
 
   `head : ∀ {as bs} → `Pointwise as bs → as .head ∼ bs .head
   `head (`lift rs)         = rs .head
-  `head (`refl eq)         = S.reflexive (P.cong head eq)
+  `head (`refl eq)         = S.reflexive (≡.cong head eq)
   `head (`bisim rs)        = S.reflexive (rs .head)
   `head (`step `rs)        = `rs .head
   `head (`sym `rs)         = S.sym (`head `rs)
@@ -169,7 +169,7 @@ module pw-Reasoning (S : Setoid a ℓ) where
 
   `tail : ∀ {as bs} → `Pointwise as bs → `Pointwise (as .tail)  (bs .tail)
   `tail (`lift rs)         = `lift (rs .tail)
-  `tail (`refl eq)         = `refl (P.cong tail eq)
+  `tail (`refl eq)         = `refl (≡.cong tail eq)
   `tail (`bisim rs)        = `bisim (rs .tail)
   `tail (`step `rs)        = `rs .tail
   `tail (`sym `rs)         = `sym (`tail `rs)
@@ -196,8 +196,8 @@ module pw-Reasoning (S : Setoid a ℓ) where
   pattern _≈⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`bisim bs∼as)) bs∼cs
   pattern _≡⟨_⟩_  as as∼bs bs∼cs = `trans {as = as} (`refl as∼bs) bs∼cs
   pattern _≡⟨_⟨_ as bs∼as bs∼cs = `trans {as = as} (`sym (`refl bs∼as)) bs∼cs
-  pattern _≡⟨⟩_   as as∼bs       = `trans {as = as} (`refl P.refl) as∼bs
-  pattern _∎      as             = `refl  {as = as} P.refl
+  pattern _≡⟨⟩_   as as∼bs       = `trans {as = as} (`refl ≡.refl) as∼bs
+  pattern _∎      as             = `refl  {as = as} ≡.refl
 
 
   -- Deprecated v2.0
@@ -226,7 +226,7 @@ module pw-Reasoning (S : Setoid a ℓ) where
 
 module ≈-Reasoning {a} {A : Set a} where
 
-  open pw-Reasoning (P.setoid A) public
+  open pw-Reasoning (≡.setoid A) public
 
   infix 4 _≈∞_
   _≈∞_ = `Pointwise∞
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index b5852c7055..c03b3c2096 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -14,24 +14,24 @@ open import Codata.Sized.Thunk as Thunk using (Thunk; force)
 open import Codata.Sized.Colist
 open import Codata.Sized.Colist.Bisimilarity
 open import Codata.Sized.Conat
-open import Codata.Sized.Conat.Bisimilarity as coℕᵇ using (zero; suc)
-import Codata.Sized.Conat.Properties as coℕₚ
+open import Codata.Sized.Conat.Bisimilarity as Conat using (zero; suc)
+import Codata.Sized.Conat.Properties as Conat
 open import Codata.Sized.Cowriter as Cowriter using ([_]; _∷_)
-open import Codata.Sized.Cowriter.Bisimilarity as coWriterᵇ using ([_]; _∷_)
+open import Codata.Sized.Cowriter.Bisimilarity as Cowriter using ([_]; _∷_)
 open import Codata.Sized.Stream as Stream using (Stream; _∷_)
 open import Data.Vec.Bounded as Vec≤ using (Vec≤)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)
 open import Data.List.Relation.Binary.Equality.Propositional using (≋-refl)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just)
-import Data.Maybe.Properties as Maybeₚ
+import Data.Maybe.Properties as Maybe
 open import Data.Maybe.Relation.Unary.All using (All; nothing; just)
 open import Data.Nat.Base as ℕ using (zero; suc; z≤n; s≤s)
 open import Data.Product.Base as Product using (_×_; _,_; uncurry)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Function.Base
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 private
   variable
@@ -47,13 +47,13 @@ private
 
 map-id : ∀ (as : Colist A ∞) → i ⊢ map id as ≈ as
 map-id []       = []
-map-id (a ∷ as) = Eq.refl ∷ λ where .force → map-id (as .force)
+map-id (a ∷ as) = ≡.refl ∷ λ where .force → map-id (as .force)
 
 map-∘ : ∀ (f : A → B) (g : B → C) as {i} →
                  i ⊢ map g (map f as) ≈ map (g ∘ f) as
 map-∘ f g []       = []
 map-∘ f g (a ∷ as) =
-  Eq.refl ∷ λ where .force → map-∘ f g (as .force)
+  ≡.refl ∷ λ where .force → map-∘ f g (as .force)
 
 ------------------------------------------------------------------------
 -- Relation to Cowriter
@@ -62,24 +62,24 @@ fromCowriter∘toCowriter≗id : ∀ (as : Colist A ∞) →
   i ⊢ fromCowriter (toCowriter as) ≈ as
 fromCowriter∘toCowriter≗id []       = []
 fromCowriter∘toCowriter≗id (a ∷ as) =
-  Eq.refl ∷ λ where .force → fromCowriter∘toCowriter≗id (as .force)
+  ≡.refl ∷ λ where .force → fromCowriter∘toCowriter≗id (as .force)
 
 ------------------------------------------------------------------------
 -- Properties of length
 
-length-∷ : ∀ (a : A) as → i coℕᵇ.⊢ length (a ∷ as) ≈ 1 ℕ+ length (as .force)
-length-∷ a as = suc (λ where .force → coℕᵇ.refl)
+length-∷ : ∀ (a : A) as → i Conat.⊢ length (a ∷ as) ≈ 1 ℕ+ length (as .force)
+length-∷ a as = suc (λ where .force → Conat.refl)
 
-length-replicate : ∀ n (a : A) → i coℕᵇ.⊢ length (replicate n a) ≈ n
+length-replicate : ∀ n (a : A) → i Conat.⊢ length (replicate n a) ≈ n
 length-replicate zero    a = zero
 length-replicate (suc n) a = suc λ where .force → length-replicate (n .force) a
 
 length-++ : (as bs : Colist A ∞) →
-            i coℕᵇ.⊢ length (as ++ bs) ≈ length as + length bs
-length-++ []       bs = coℕᵇ.refl
+            i Conat.⊢ length (as ++ bs) ≈ length as + length bs
+length-++ []       bs = Conat.refl
 length-++ (a ∷ as) bs = suc λ where .force → length-++ (as .force) bs
 
-length-map : ∀ (f : A → B) as → i coℕᵇ.⊢ length (map f as) ≈ length as
+length-map : ∀ (f : A → B) as → i Conat.⊢ length (map f as) ≈ length as
 length-map f []       = zero
 length-map f (a ∷ as) = suc λ where .force → length-map f (as .force)
 
@@ -89,17 +89,17 @@ length-map f (a ∷ as) = suc λ where .force → length-map f (as .force)
 replicate-+ : ∀ m n (a : A) →
               i ⊢ replicate (m + n) a ≈ replicate m a ++ replicate n a
 replicate-+ zero    n a = refl
-replicate-+ (suc m) n a = Eq.refl ∷ λ where .force → replicate-+ (m .force) n a
+replicate-+ (suc m) n a = ≡.refl ∷ λ where .force → replicate-+ (m .force) n a
 
 map-replicate : ∀ (f : A → B) n a →
                 i ⊢ map f (replicate n a) ≈ replicate n (f a)
 map-replicate f zero    a = []
 map-replicate f (suc n) a =
-  Eq.refl ∷ λ where .force → map-replicate f (n .force) a
+  ≡.refl ∷ λ where .force → map-replicate f (n .force) a
 
 lookup-replicate : ∀ k n (a : A) → All (a ≡_) (lookup (replicate n a) k)
 lookup-replicate k zero          a = nothing
-lookup-replicate zero    (suc n) a = just Eq.refl
+lookup-replicate zero    (suc n) a = just ≡.refl
 lookup-replicate (suc k) (suc n) a = lookup-replicate k (n .force) a
 
 ------------------------------------------------------------------------
@@ -109,31 +109,31 @@ map-unfold : ∀ (f : B → C) (alg : A → Maybe (A × B)) a →
              i ⊢ map f (unfold alg a) ≈ unfold (Maybe.map (Product.map₂ f) ∘ alg) a
 map-unfold f alg a with alg a
 ... | nothing       = []
-... | just (a′ , b) = Eq.refl ∷ λ where .force → map-unfold f alg a′
+... | just (a′ , b) = ≡.refl ∷ λ where .force → map-unfold f alg a′
 
 module _ {alg : A → Maybe (A × B)} {a} where
 
   unfold-nothing : alg a ≡ nothing → unfold alg a ≡ []
   unfold-nothing eq with alg a
-  ... | nothing = Eq.refl
+  ... | nothing = ≡.refl
 
   unfold-just : ∀ {a′ b} → alg a ≡ just (a′ , b) →
                 i ⊢ unfold alg a ≈ b ∷ λ where .force → unfold alg a′
   unfold-just eq with alg a
-  unfold-just Eq.refl | just (a′ , b) = Eq.refl ∷ λ where .force → refl
+  unfold-just ≡.refl | just (a′ , b) = ≡.refl ∷ λ where .force → refl
 
 ------------------------------------------------------------------------
 -- Properties of scanl
 
 length-scanl : ∀ (c : B → A → B) n as →
-               i coℕᵇ.⊢ length (scanl c n as) ≈ 1 ℕ+ length as
+               i Conat.⊢ length (scanl c n as) ≈ 1 ℕ+ length as
 length-scanl c n []       = suc λ where .force → zero
 length-scanl c n (a ∷ as) = suc λ { .force → begin
   length (scanl c (c n a) (as .force))
     ≈⟨ length-scanl c (c n a) (as .force) ⟩
   1 ℕ+ length (as .force)
     ≈⟨ length-∷ a as ⟨
-  length (a ∷ as) ∎ } where open coℕᵇ.≈-Reasoning
+  length (a ∷ as) ∎ } where open Conat.≈-Reasoning
 
 module _ (cons : C → B → C) (alg : A → Maybe (A × B)) where
 
@@ -145,15 +145,15 @@ module _ (cons : C → B → C) (alg : A → Maybe (A × B)) where
   scanl-unfold : ∀ nil a → i ⊢ scanl cons nil (unfold alg a)
                              ≈ nil ∷ (λ where .force → unfold alg′ (a , nil))
   scanl-unfold nil a with alg a in eq
-  ... | nothing      = Eq.refl ∷ λ { .force →
-    sym (fromEq (unfold-nothing (Maybeₚ.map-nothing eq))) }
-  ... | just (a′ , b) = Eq.refl ∷ λ { .force → begin
+  ... | nothing      = ≡.refl ∷ λ { .force →
+    sym (fromEq (unfold-nothing (Maybe.map-nothing eq))) }
+  ... | just (a′ , b) = ≡.refl ∷ λ { .force → begin
     scanl cons (cons nil b) (unfold alg a′)
      ≈⟨ scanl-unfold (cons nil b) a′ ⟩
     (cons nil b ∷ _)
-     ≈⟨ Eq.refl ∷ (λ where .force → refl) ⟩
+     ≈⟨ ≡.refl ∷ (λ where .force → refl) ⟩
     (cons nil b ∷ _)
-     ≈⟨ unfold-just (Maybeₚ.map-just eq) ⟨
+     ≈⟨ unfold-just (Maybe.map-just eq) ⟨
     unfold alg′ (a , nil) ∎ } where open ≈-Reasoning
 
 ------------------------------------------------------------------------
@@ -164,10 +164,10 @@ map-alignWith : ∀ (f : C → D) (al : These A B → C) as bs →
 map-alignWith f al []         bs       = map-∘ (al ∘′ that) f bs
 map-alignWith f al as@(_ ∷ _) []       = map-∘ (al ∘′ this) f as
 map-alignWith f al (a ∷ as)   (b ∷ bs) =
-  Eq.refl ∷ λ where .force → map-alignWith f al (as .force) (bs .force)
+  ≡.refl ∷ λ where .force → map-alignWith f al (as .force) (bs .force)
 
 length-alignWith : ∀ (al : These A B → C) as bs →
-                   i coℕᵇ.⊢ length (alignWith al as bs) ≈ length as ⊔ length bs
+                   i Conat.⊢ length (alignWith al as bs) ≈ length as ⊔ length bs
 length-alignWith al []         bs       = length-map (al ∘ that) bs
 length-alignWith al as@(_ ∷ _) []       = length-map (al ∘ this) as
 length-alignWith al (a ∷ as)   (b ∷ bs) =
@@ -181,10 +181,10 @@ map-zipWith : ∀ (f : C → D) (zp : A → B → C) as bs →
 map-zipWith f zp []       _        = []
 map-zipWith f zp (_ ∷ _)  []       = []
 map-zipWith f zp (a ∷ as) (b ∷ bs) =
-  Eq.refl ∷ λ where .force → map-zipWith f zp (as .force) (bs .force)
+  ≡.refl ∷ λ where .force → map-zipWith f zp (as .force) (bs .force)
 
 length-zipWith : ∀ (zp : A → B → C) as bs →
-                 i coℕᵇ.⊢ length (zipWith zp as bs) ≈ length as ⊓ length bs
+                 i Conat.⊢ length (zipWith zp as bs) ≈ length as ⊓ length bs
 length-zipWith zp []         bs       = zero
 length-zipWith zp as@(_ ∷ _) []       = zero
 length-zipWith zp (a ∷ as)   (b ∷ bs) =
@@ -208,25 +208,25 @@ map-drop f zero    as       = refl
 map-drop f (suc m) []       = []
 map-drop f (suc m) (a ∷ as) = map-drop f m (as .force)
 
-length-drop : ∀ m (as : Colist A ∞) → i coℕᵇ.⊢ length (drop m as) ≈ length as ∸ m
-length-drop zero    as       = coℕᵇ.refl
-length-drop (suc m) []       = coℕᵇ.sym (coℕₚ.0∸m≈0 m)
+length-drop : ∀ m (as : Colist A ∞) → i Conat.⊢ length (drop m as) ≈ length as ∸ m
+length-drop zero    as       = Conat.refl
+length-drop (suc m) []       = Conat.sym (Conat.0∸m≈0 m)
 length-drop (suc m) (a ∷ as) = length-drop m (as .force)
 
 drop-fromList-++-identity : ∀ (as : List A) bs →
                             drop (List.length as) (fromList as ++ bs) ≡ bs
-drop-fromList-++-identity []       bs = Eq.refl
+drop-fromList-++-identity []       bs = ≡.refl
 drop-fromList-++-identity (a ∷ as) bs = drop-fromList-++-identity as bs
 
 drop-fromList-++-≤ : ∀ (as : List A) bs {m} → m ℕ.≤ List.length as →
                      drop m (fromList as ++ bs) ≡ fromList (List.drop m as) ++ bs
-drop-fromList-++-≤ []       bs z≤n     = Eq.refl
-drop-fromList-++-≤ (a ∷ as) bs z≤n     = Eq.refl
+drop-fromList-++-≤ []       bs z≤n     = ≡.refl
+drop-fromList-++-≤ (a ∷ as) bs z≤n     = ≡.refl
 drop-fromList-++-≤ (a ∷ as) bs (s≤s p) = drop-fromList-++-≤ as bs p
 
 drop-fromList-++-≥ : ∀ (as : List A) bs {m} → m ℕ.≥ List.length as →
                      drop m (fromList as ++ bs) ≡ drop (m ℕ.∸ List.length as) bs
-drop-fromList-++-≥ []       bs z≤n     = Eq.refl
+drop-fromList-++-≥ []       bs z≤n     = ≡.refl
 drop-fromList-++-≥ (a ∷ as) bs (s≤s p) = drop-fromList-++-≥ as bs p
 
 drop-⁺++-identity : ∀ (as : List⁺ A) bs →
@@ -236,7 +236,7 @@ drop-⁺++-identity (a ∷ as) bs = drop-fromList-++-identity as (bs .force)
 ------------------------------------------------------------------------
 -- Properties of cotake
 
-length-cotake : ∀ n (as : Stream A ∞) → i coℕᵇ.⊢ length (cotake n as) ≈ n
+length-cotake : ∀ n (as : Stream A ∞) → i Conat.⊢ length (cotake n as) ≈ n
 length-cotake zero    as       = zero
 length-cotake (suc n) (a ∷ as) =
   suc λ where .force → length-cotake (n .force) (as .force)
@@ -245,7 +245,7 @@ map-cotake : ∀ (f : A → B) n as →
              i ⊢ map f (cotake n as) ≈ cotake n (Stream.map f as)
 map-cotake f zero    as       = []
 map-cotake f (suc n) (a ∷ as) =
-  Eq.refl ∷ λ where .force → map-cotake f (n .force) (as .force)
+  ≡.refl ∷ λ where .force → map-cotake f (n .force) (as .force)
 
 ------------------------------------------------------------------------
 -- Properties of chunksOf
@@ -255,20 +255,20 @@ module Map-ChunksOf (f : A → B) n where
   open ChunksOf n using (chunksOfAcc)
 
   map-chunksOf : ∀ as →
-    i coWriterᵇ.⊢ Cowriter.map (Vec.map f) (Vec≤.map f) (chunksOf n as)
+    i Cowriter.⊢ Cowriter.map (Vec.map f) (Vec≤.map f) (chunksOf n as)
                 ≈ chunksOf n (map f as)
   map-chunksOfAcc : ∀ m as {k≤ k≡ k≤′ k≡′} →
                     (∀ vs → Vec≤.map f (k≤ vs) ≡ k≤′ (Vec≤.map f vs)) →
                     (∀ vs → Vec.map f (k≡ vs) ≡ k≡′ (Vec.map f vs)) →
-                    i coWriterᵇ.⊢ Cowriter.map (Vec.map f) (Vec≤.map f)
+                    i Cowriter.⊢ Cowriter.map (Vec.map f) (Vec≤.map f)
                                         (chunksOfAcc m k≤ k≡ as)
                                 ≈ chunksOfAcc m k≤′ k≡′ (map f as)
 
-  map-chunksOf as = map-chunksOfAcc n as (λ vs → Eq.refl) (λ vs → Eq.refl)
+  map-chunksOf as = map-chunksOfAcc n as (λ vs → ≡.refl) (λ vs → ≡.refl)
 
   map-chunksOfAcc zero    as       eq-≤ eq-≡ =
       eq-≡ [] ∷ λ where .force → map-chunksOf as
-  map-chunksOfAcc (suc m) []       eq-≤ eq-≡ = coWriterᵇ.[ eq-≤ Vec≤.[] ]
+  map-chunksOfAcc (suc m) []       eq-≤ eq-≡ = Cowriter.[ eq-≤ Vec≤.[] ]
   map-chunksOfAcc (suc m) (a ∷ as) eq-≤ eq-≡ =
     map-chunksOfAcc m (as .force) (eq-≤ ∘ (a Vec≤.∷_)) (eq-≡ ∘ (a Vec.∷_))
 
@@ -280,21 +280,21 @@ open Map-ChunksOf using (map-chunksOf) public
 fromList-++ : (as bs : List A) →
               i ⊢ fromList (as List.++ bs) ≈ fromList as ++ fromList bs
 fromList-++ []       bs = refl
-fromList-++ (a ∷ as) bs = Eq.refl ∷ λ where .force → fromList-++ as bs
+fromList-++ (a ∷ as) bs = ≡.refl ∷ λ where .force → fromList-++ as bs
 
 fromList-scanl : ∀ (c : B → A → B) n as →
                  i ⊢ fromList (List.scanl c n as) ≈ scanl c n (fromList as)
-fromList-scanl c n []       = Eq.refl ∷ λ where .force → refl
+fromList-scanl c n []       = ≡.refl ∷ λ where .force → refl
 fromList-scanl c n (a ∷ as) =
-  Eq.refl ∷ λ where .force → fromList-scanl c (c n a) as
+  ≡.refl ∷ λ where .force → fromList-scanl c (c n a) as
 
 map-fromList : ∀ (f : A → B) as →
                i ⊢ map f (fromList as) ≈ fromList (List.map f as)
 map-fromList f []       = []
-map-fromList f (a ∷ as) = Eq.refl ∷ λ where .force → map-fromList f as
+map-fromList f (a ∷ as) = ≡.refl ∷ λ where .force → map-fromList f as
 
 length-fromList : (as : List A) →
-                  i coℕᵇ.⊢ length (fromList as) ≈ fromℕ (List.length as)
+                  i Conat.⊢ length (fromList as) ≈ fromℕ (List.length as)
 length-fromList []       = zero
 length-fromList (a ∷ as) = suc (λ where .force → length-fromList as)
 
@@ -304,19 +304,19 @@ length-fromList (a ∷ as) = suc (λ where .force → length-fromList as)
 fromStream-++ : ∀ (as : List A) bs →
                 i ⊢ fromStream (as Stream.++ bs) ≈ fromList as ++ fromStream bs
 fromStream-++ []       bs = refl
-fromStream-++ (a ∷ as) bs = Eq.refl ∷ λ where .force → fromStream-++ as bs
+fromStream-++ (a ∷ as) bs = ≡.refl ∷ λ where .force → fromStream-++ as bs
 
 fromStream-⁺++ : ∀ (as : List⁺ A) bs →
                  i ⊢ fromStream (as Stream.⁺++ bs)
                    ≈ fromList⁺ as ++ fromStream (bs .force)
 fromStream-⁺++ (a ∷ as) bs =
-  Eq.refl ∷ λ where .force → fromStream-++ as (bs .force)
+  ≡.refl ∷ λ where .force → fromStream-++ as (bs .force)
 
 fromStream-concat : (ass : Stream (List⁺ A) ∞) →
                     i ⊢ concat (fromStream ass) ≈ fromStream (Stream.concat ass)
 fromStream-concat (as@(a ∷ _) ∷ ass) = begin
   concat (fromStream (as ∷ ass))
-    ≈⟨ Eq.refl ∷ (λ { .force → ++⁺ ≋-refl (fromStream-concat (ass .force))}) ⟩
+    ≈⟨ ≡.refl ∷ (λ { .force → ++⁺ ≋-refl (fromStream-concat (ass .force))}) ⟩
   a ∷ _
     ≈⟨ sym (fromStream-⁺++ as _) ⟩
   fromStream (Stream.concat (as ∷ ass)) ∎ where open ≈-Reasoning
@@ -325,12 +325,12 @@ fromStream-scanl : ∀ (c : B → A → B) n as →
                    i ⊢ scanl c n (fromStream as)
                      ≈ fromStream (Stream.scanl c n as)
 fromStream-scanl c n (a ∷ as) =
-  Eq.refl ∷ λ where .force → fromStream-scanl c (c n a) (as .force)
+  ≡.refl ∷ λ where .force → fromStream-scanl c (c n a) (as .force)
 
 map-fromStream : ∀ (f : A → B) as →
                  i ⊢ map f (fromStream as) ≈ fromStream (Stream.map f as)
 map-fromStream f (a ∷ as) =
-  Eq.refl ∷ λ where .force → map-fromStream f (as .force)
+  ≡.refl ∷ λ where .force → map-fromStream f (as .force)
 
 ------------------------------------------------------------------------
 -- DEPRECATED
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index cd4ccdce29..b9c7270881 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -27,10 +27,10 @@ open import Relation.Binary.Definitions
 import Relation.Binary.Construct.On as On
 import Relation.Binary.Construct.Subst.Equality as Subst
 import Relation.Binary.Construct.Closure.Reflexive as Refl
-import Relation.Binary.Construct.Closure.Reflexive.Properties as Reflₚ
-open import Relation.Binary.PropositionalEquality.Core as PropEq
+import Relation.Binary.Construct.Closure.Reflexive.Properties as Refl
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≢_; refl; cong; sym; trans; subst)
-import Relation.Binary.PropositionalEquality.Properties as PropEq
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 ------------------------------------------------------------------------
 -- Primitive properties
@@ -59,13 +59,13 @@ _≟_ : Decidable {A = Char} _≡_
 x ≟ y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕ.≟ toℕ y)
 
 setoid : Setoid _ _
-setoid = PropEq.setoid Char
+setoid = ≡.setoid Char
 
 decSetoid : DecSetoid _ _
-decSetoid = PropEq.decSetoid _≟_
+decSetoid = ≡.decSetoid _≟_
 
 isDecEquivalence : IsDecEquivalence _≡_
-isDecEquivalence = PropEq.isDecEquivalence _≟_
+isDecEquivalence = ≡.isDecEquivalence _≟_
 
 ------------------------------------------------------------------------
 -- Boolean equality test.
@@ -114,11 +114,11 @@ _<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
 
 <-isStrictPartialOrder : IsStrictPartialOrder _≡_ _<_
 <-isStrictPartialOrder = record
-  { isEquivalence = PropEq.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; irrefl        = <-irrefl
   ; trans         = λ {a} {b} {c} → <-trans {a} {b} {c}
-  ; <-resp-≈      = (λ {c} → PropEq.subst (c <_))
-                  , (λ {c} → PropEq.subst (_< c))
+  ; <-resp-≈      = (λ {c} → ≡.subst (c <_))
+                  , (λ {c} → ≡.subst (_< c))
   }
 
 <-isStrictTotalOrder : IsStrictTotalOrder _≡_ _<_
@@ -142,20 +142,20 @@ _<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
 
 infix 4 _≤?_
 _≤?_ : Decidable _≤_
-_≤?_ = Reflₚ.decidable <-cmp
+_≤?_ = Refl.decidable <-cmp
 
 ≤-reflexive : _≡_ ⇒ _≤_
 ≤-reflexive = Refl.reflexive
 
 ≤-trans : Transitive _≤_
-≤-trans = Reflₚ.trans (λ {a} {b} {c} → <-trans {a} {b} {c})
+≤-trans = Refl.trans (λ {a} {b} {c} → <-trans {a} {b} {c})
 
 ≤-antisym : Antisymmetric _≡_ _≤_
-≤-antisym = Reflₚ.antisym _≡_ refl ℕ.<-asym
+≤-antisym = Refl.antisym _≡_ refl ℕ.<-asym
 
 ≤-isPreorder : IsPreorder _≡_ _≤_
 ≤-isPreorder = record
-  { isEquivalence = PropEq.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; reflexive     = ≤-reflexive
   ; trans         = ≤-trans
   }
diff --git a/src/Data/Digit/Properties.agda b/src/Data/Digit/Properties.agda
index fb52d2af72..98810d4e27 100644
--- a/src/Data/Digit/Properties.agda
+++ b/src/Data/Digit/Properties.agda
@@ -7,13 +7,13 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.Digit
-import Data.Char.Properties as Charₚ
+import Data.Char.Properties as Char
 open import Data.Nat.Base using (ℕ)
 open import Data.Nat.Properties using (_≤?_)
 open import Data.Fin.Properties using (inject≤-injective)
 open import Data.Product.Base using (_,_; proj₁)
 open import Data.Vec.Relation.Unary.Unique.Propositional using (Unique)
-import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Uniqueₚ
+import Data.Vec.Relation.Unary.Unique.Propositional.Properties as Unique
 open import Data.Vec.Relation.Unary.AllPairs using (allPairs?)
 open import Relation.Nullary.Decidable.Core using (True; from-yes; ¬?)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
@@ -22,7 +22,7 @@ open import Function.Base using (_∘_)
 module Data.Digit.Properties where
 
 digitCharsUnique : Unique digitChars
-digitCharsUnique = from-yes (allPairs? (λ x y → ¬? (x Charₚ.≟ y)) digitChars)
+digitCharsUnique = from-yes (allPairs? (λ x y → ¬? (x Char.≟ y)) digitChars)
 
 module _ (base : ℕ) where
   module _ {base≥2 base≥2′ : True (2 ≤? base)} where
@@ -32,4 +32,4 @@ module _ (base : ℕ) where
 
   module _ {base≤16 base≤16′ : True (base ≤? 16)} where
     showDigit-injective : (n m : Digit base) → showDigit {base} {base≤16} n ≡ showDigit {base} {base≤16′} m → n ≡ m
-    showDigit-injective n m = inject≤-injective _ _ n m ∘ Uniqueₚ.lookup-injective digitCharsUnique _ _
+    showDigit-injective n m = inject≤-injective _ _ n m ∘ Unique.lookup-injective digitCharsUnique _ _
diff --git a/src/Data/Fin/Induction.agda b/src/Data/Fin/Induction.agda
index 47e37446bf..97ca14015b 100644
--- a/src/Data/Fin/Induction.agda
+++ b/src/Data/Fin/Induction.agda
@@ -16,7 +16,7 @@ import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (_,_)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.Linked as Linked using (Linked; [-]; _∷_)
-import Data.Vec.Relation.Unary.Linked.Properties as Linkedₚ
+import Data.Vec.Relation.Unary.Linked.Properties as Linked
 open import Function.Base using (flip; _$_)
 open import Induction
 open import Induction.WellFounded as WF
@@ -124,7 +124,7 @@ module _ {_≈_ : Rel (Fin n) ℓ} where
     pigeon : {xs : Vec (Fin n) n} → Linked (flip _⊏_) (i ∷ xs) → WellFounded _⊏_
     pigeon {xs} i∷xs↑ =
       let (i₁ , i₂ , i₁<i₂ , xs[i₁]≡xs[i₂]) = pigeonhole (n<1+n n) (Vec.lookup (i ∷ xs)) in
-      let xs[i₁]⊏xs[i₂] = Linkedₚ.lookup⁺ (Ord.transitive _⊏_ ⊏.trans) i∷xs↑ i₁<i₂ in
+      let xs[i₁]⊏xs[i₂] = Linked.lookup⁺ (Ord.transitive _⊏_ ⊏.trans) i∷xs↑ i₁<i₂ in
       let xs[i₁]⊏xs[i₁] = ⊏.<-respʳ-≈ (⊏.Eq.reflexive xs[i₁]≡xs[i₂]) xs[i₁]⊏xs[i₂] in
       contradiction xs[i₁]⊏xs[i₁] (⊏.irrefl ⊏.Eq.refl)
 
diff --git a/src/Data/Fin/Substitution/Lemmas.agda b/src/Data/Fin/Substitution/Lemmas.agda
index 8b4d0b176a..05e177acf7 100644
--- a/src/Data/Fin/Substitution/Lemmas.agda
+++ b/src/Data/Fin/Substitution/Lemmas.agda
@@ -12,9 +12,9 @@ open import Data.Fin.Substitution
 open import Data.Nat hiding (_⊔_; _/_)
 open import Data.Fin.Base using (Fin; zero; suc; lift)
 open import Data.Vec.Base
-import Data.Vec.Properties as VecProp
+import Data.Vec.Properties as Vec
 open import Function.Base as Fun using (_∘_; _$_; flip)
-open import Relation.Binary.PropositionalEquality.Core as PropEq
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; refl; sym; cong; cong₂)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
@@ -68,9 +68,9 @@ record Lemmas₀ (T : Pred ℕ ℓ) : Set ℓ where
   lookup-map-weaken-↑⋆ zero    x           = refl
   lookup-map-weaken-↑⋆ (suc k) zero        = refl
   lookup-map-weaken-↑⋆ (suc k) (suc x) {ρ} = begin
-    lookup (map weaken (map weaken ρ ↑⋆ k)) x        ≡⟨ VecProp.lookup-map x weaken (map weaken ρ ↑⋆ k) ⟩
+    lookup (map weaken (map weaken ρ ↑⋆ k)) x        ≡⟨ Vec.lookup-map x weaken (map weaken ρ ↑⋆ k) ⟩
     weaken (lookup (map weaken ρ ↑⋆ k) x)            ≡⟨ cong weaken (lookup-map-weaken-↑⋆ k x) ⟩
-    weaken (lookup ((ρ ↑) ↑⋆ k) (lift k suc x))      ≡⟨ sym $ VecProp.lookup-map (lift k suc x) weaken ((ρ ↑) ↑⋆ k) ⟩
+    weaken (lookup ((ρ ↑) ↑⋆ k) (lift k suc x))      ≡⟨ sym $ Vec.lookup-map (lift k suc x) weaken ((ρ ↑) ↑⋆ k) ⟩
     lookup (map weaken ((ρ ↑) ↑⋆ k)) (lift k suc x)  ∎
 
 record Lemmas₁ (T : Pred ℕ ℓ) : Set ℓ where
@@ -85,7 +85,7 @@ record Lemmas₁ (T : Pred ℕ ℓ) : Set ℓ where
                       lookup             ρ  x ≡ var      y →
                       lookup (map weaken ρ) x ≡ var (suc y)
   lookup-map-weaken x {y} {ρ} hyp = begin
-    lookup (map weaken ρ) x  ≡⟨ VecProp.lookup-map x weaken ρ ⟩
+    lookup (map weaken ρ) x  ≡⟨ Vec.lookup-map x weaken ρ ⟩
     weaken (lookup ρ x)      ≡⟨ cong weaken hyp ⟩
     weaken (var y)           ≡⟨ weaken-var ⟩
     var (suc y)              ∎
@@ -153,7 +153,7 @@ record Lemmas₂ (T : Pred ℕ ℓ) : Set ℓ where
 
   lookup-⊙ : ∀ x {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
              lookup (ρ₁ ⊙ ρ₂) x ≡ lookup ρ₁ x / ρ₂
-  lookup-⊙ x {ρ₁} {ρ₂} = VecProp.lookup-map x (λ t → t / ρ₂) ρ₁
+  lookup-⊙ x {ρ₁} {ρ₂} = Vec.lookup-map x (λ t → t / ρ₂) ρ₁
 
   lookup-⨀ : ∀ x (ρs : Subs T m n) →
              lookup (⨀ ρs) x ≡ var x /✶ ρs
@@ -239,8 +239,8 @@ record Lemmas₃ (T : Pred ℕ ℓ) : Set ℓ where
 
   ⊙-id : {ρ : Sub T m n} → ρ ⊙ id ≡ ρ
   ⊙-id {ρ = ρ} = begin
-    map (λ t → t / id) ρ  ≡⟨ VecProp.map-cong id-vanishes ρ ⟩
-    map Fun.id         ρ  ≡⟨ VecProp.map-id ρ ⟩
+    map (λ t → t / id) ρ  ≡⟨ Vec.map-cong id-vanishes ρ ⟩
+    map Fun.id         ρ  ≡⟨ Vec.map-id ρ ⟩
     ρ                     ∎
 
   open Lemmas₂ lemmas₂ public hiding (wk-⊙-sub′)
@@ -264,13 +264,13 @@ record Lemmas₄ (T : Pred ℕ ℓ) : Set ℓ where
       ρ₁ ↑ ⊙ ρ₂ ↑                             ∎
       where
       lemma = begin
-        map weaken (map (λ t → t / ρ₂) ρ₁)    ≡⟨ sym (VecProp.map-∘ _ _ _) ⟩
-        map (λ t → weaken (t / ρ₂)) ρ₁        ≡⟨ VecProp.map-cong (λ t → begin
+        map weaken (map (λ t → t / ρ₂) ρ₁)    ≡⟨ sym (Vec.map-∘ _ _ _) ⟩
+        map (λ t → weaken (t / ρ₂)) ρ₁        ≡⟨ Vec.map-cong (λ t → begin
                                                    weaken (t / ρ₂)  ≡⟨ sym /-wk ⟩
                                                    t / ρ₂ / wk      ≡⟨ hyp t ⟩
                                                    t / wk / ρ₂ ↑    ≡⟨ cong₂ _/_ /-wk refl ⟩
                                                    weaken t / ρ₂ ↑  ∎) ρ₁ ⟩
-        map (λ t → weaken t / ρ₂ ↑) ρ₁        ≡⟨ VecProp.map-∘ _ _ _ ⟩
+        map (λ t → weaken t / ρ₂ ↑) ρ₁        ≡⟨ Vec.map-∘ _ _ _ ⟩
         map (λ t → t / ρ₂ ↑) (map weaken ρ₁)  ∎
 
     ↑⋆-distrib′ : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} →
@@ -284,7 +284,7 @@ record Lemmas₄ (T : Pred ℕ ℓ) : Set ℓ where
 
   map-weaken : {ρ : Sub T m n} → map weaken ρ ≡ ρ ⊙ wk
   map-weaken {ρ = ρ} = begin
-    map weaken ρ          ≡⟨ VecProp.map-cong (λ _ → sym /-wk) ρ ⟩
+    map weaken ρ          ≡⟨ Vec.map-cong (λ _ → sym /-wk) ρ ⟩
     map (λ t → t / wk) ρ  ≡⟨ refl ⟩
     ρ ⊙ wk                ∎
 
@@ -324,8 +324,8 @@ record Lemmas₄ (T : Pred ℕ ℓ) : Set ℓ where
   ⊙-assoc : {ρ₁ : Sub T m n} {ρ₂ : Sub T n o} {ρ₃ : Sub T o p} →
             ρ₁ ⊙ (ρ₂ ⊙ ρ₃) ≡ (ρ₁ ⊙ ρ₂) ⊙ ρ₃
   ⊙-assoc {ρ₁ = ρ₁} {ρ₂} {ρ₃} = begin
-    map (λ t → t / ρ₂ ⊙ ρ₃) ρ₁                  ≡⟨ VecProp.map-cong /-⊙ ρ₁ ⟩
-    map (λ t → t / ρ₂ / ρ₃) ρ₁                  ≡⟨ VecProp.map-∘ _ _ _ ⟩
+    map (λ t → t / ρ₂ ⊙ ρ₃) ρ₁                  ≡⟨ Vec.map-cong /-⊙ ρ₁ ⟩
+    map (λ t → t / ρ₂ / ρ₃) ρ₁                  ≡⟨ Vec.map-∘ _ _ _ ⟩
     map (λ t → t / ρ₃) (map (λ t → t / ρ₂) ρ₁)  ∎
 
   map-weaken-⊙-sub : ∀ {ρ : Sub T m n} {t} → map weaken ρ ⊙ sub t ≡ ρ
@@ -560,7 +560,7 @@ record TermLemmas (T : ℕ → Set) : Set₁ where
              (∀ x → lookup ρ₂ x ≡ T.var (f x)) →
              map T.var ρ₁ ≡ ρ₂
   map-var≡ {ρ₁ = ρ₁} {ρ₂ = ρ₂} {f = f} hyp₁ hyp₂ = extensionality λ x →
-    lookup (map T.var ρ₁) x  ≡⟨ VecProp.lookup-map x _ ρ₁ ⟩
+    lookup (map T.var ρ₁) x  ≡⟨ Vec.lookup-map x _ ρ₁ ⟩
     T.var (lookup ρ₁ x)      ≡⟨ cong T.var $ hyp₁ x ⟩
     T.var (f x)              ≡⟨ sym $ hyp₂ x ⟩
     lookup ρ₂ x              ∎
@@ -577,7 +577,7 @@ record TermLemmas (T : ℕ → Set) : Set₁ where
   ↑≡↑ : {ρ : Sub Fin m n} → map T.var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
   ↑≡↑ {ρ = ρ} = map-var≡
     (VarLemmas.lookup-↑⋆ (lookup ρ) (λ _ → refl) 1)
-    (lookup-↑⋆ (lookup ρ) (λ _ → VecProp.lookup-map _ _ ρ) 1)
+    (lookup-↑⋆ (lookup ρ) (λ _ → Vec.lookup-map _ _ ρ) 1)
 
   /Var≡/ : ∀ {ρ : Sub Fin m n} {t} → t /Var ρ ≡ t T./ map T.var ρ
   /Var≡/ {ρ = ρ} {t = t} =
@@ -585,7 +585,7 @@ record TermLemmas (T : ℕ → Set) : Set₁ where
       (λ k x →
          T.var x /Var ρ VarSubst.↑⋆ k        ≡⟨ app-var ⟩
          T.var (lookup (ρ VarSubst.↑⋆ k) x)  ≡⟨ cong T.var $ VarLemmas.lookup-↑⋆ _ (λ _ → refl) k _ ⟩
-         T.var (lift k (VarSubst._/ ρ) x)    ≡⟨ sym $ lookup-↑⋆ _ (λ _ → VecProp.lookup-map _ _ ρ) k _ ⟩
+         T.var (lift k (VarSubst._/ ρ) x)    ≡⟨ sym $ lookup-↑⋆ _ (λ _ → Vec.lookup-map _ _ ρ) k _ ⟩
          lookup (map T.var ρ T.↑⋆ k) x       ≡⟨ sym app-var ⟩
          T.var x T./ map T.var ρ T.↑⋆ k      ∎)
       zero t
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index ca69682633..840b4302b5 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -24,7 +24,7 @@ open import Data.Nat.Solver
 open import Data.Product.Base using (proj₁; proj₂; _,_; _×_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sign as Sign using (Sign) renaming (_*_ to _𝕊*_)
-import Data.Sign.Properties as 𝕊ₚ
+import Data.Sign.Properties as Sign
 open import Function.Base using (_∘_; _$_; id)
 open import Level using (0ℓ)
 open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
@@ -1597,7 +1597,7 @@ abs-* i j = abs-◃ _ _
 *-cancelʳ-≡ : ∀ i j k .{{_ : NonZero k}} → i * k ≡ j * k → i ≡ j
 *-cancelʳ-≡ i j k eq with sign-cong′ eq
 ... | inj₁ s[ik]≡s[jk] = ◃-cong
-  (𝕊ₚ.*-cancelʳ-≡ (sign k) (sign i) (sign j) s[ik]≡s[jk])
+  (Sign.*-cancelʳ-≡ (sign k) (sign i) (sign j) s[ik]≡s[jk])
   (ℕ.*-cancelʳ-≡ ∣ i ∣ ∣ j ∣ _ (abs-cong eq))
 ... | inj₂ (∣ik∣≡0 , ∣jk∣≡0) = trans
   (∣i∣≡0⇒i≡0 (ℕ.m*n≡0⇒m≡0 _ _ ∣ik∣≡0))
@@ -1709,7 +1709,7 @@ neg-distribʳ-* i j = begin
 ◃-distrib-* s t zero    (suc n) = refl
 ◃-distrib-* s t (suc m) zero    =
   trans
-    (cong₂ _◃_ (𝕊ₚ.*-comm s t) (ℕ.*-comm m 0))
+    (cong₂ _◃_ (Sign.*-comm s t) (ℕ.*-comm m 0))
     (*-comm (t ◃ zero) (s ◃ suc m))
 ◃-distrib-* s t (suc m) (suc n) =
   sym (cong₂ _◃_
diff --git a/src/Data/List/Countdown.agda b/src/Data/List/Countdown.agda
index dbfd75b332..dea756e108 100644
--- a/src/Data/List/Countdown.agda
+++ b/src/Data/List/Countdown.agda
@@ -29,10 +29,10 @@ open import Data.Sum.Properties
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary
 open import Relation.Nullary.Decidable using (dec-true; dec-false)
-open import Relation.Binary.PropositionalEquality.Core as PropEq
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≢_; refl; cong)
-import Relation.Binary.PropositionalEquality.Properties as PropEq
-open PropEq.≡-Reasoning
+import Relation.Binary.PropositionalEquality.Properties as ≡
+open ≡.≡-Reasoning
 
 private
   open module D = DecSetoid D
@@ -124,12 +124,12 @@ record _⊕_ (counted : List Elem) (n : ℕ) : Set where
 
 -- A countdown can be initialised by proving that Elem is finite.
 
-empty : ∀ {n} → Injection D.setoid (PropEq.setoid (Fin n)) → [] ⊕ n
+empty : ∀ {n} → Injection D.setoid (≡.setoid (Fin n)) → [] ⊕ n
 empty inj =
   record { kind      = inj₂ ∘ to
          ; injective = λ {x} {y} {i} eq₁ eq₂ → injective (begin
              to x ≡⟨ inj₂-injective eq₁ ⟩
-             i    ≡⟨ PropEq.sym $ inj₂-injective eq₂ ⟩
+             i    ≡⟨ ≡.sym $ inj₂-injective eq₂ ⟩
              to y ∎)
          }
   where open Injection inj
@@ -199,7 +199,7 @@ insert {counted} {n} counted⊕1+n x x∉counted =
   inj eq₁ eq₂ | no  _ | no  _ | inj₂ i         | inj₂ _ | inj₂ _ | _ | _ | hlp =
     hlp _ refl refl $
       punchOut-injective {i = i} _ _ $
-        (PropEq.trans (inj₂-injective eq₁) (PropEq.sym (inj₂-injective eq₂)))
+        (≡.trans (inj₂-injective eq₁) (≡.sym (inj₂-injective eq₂)))
 
 -- Counts an element if it has not already been counted.
 
diff --git a/src/Data/List/Fresh/Membership/Setoid/Properties.agda b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
index ce3f7df88b..72bdc6c0ba 100644
--- a/src/Data/List/Fresh/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Fresh/Membership/Setoid/Properties.agda
@@ -20,17 +20,16 @@ open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; fromInj₂)
 
 open import Function.Base using (id; _∘′_; _$_)
 open import Relation.Nullary
-open import Relation.Unary as U using (Pred)
-import Relation.Binary.Definitions as B
-import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Unary as Unary using (Pred)
+import Relation.Binary.Definitions as Binary
+import Relation.Binary.PropositionalEquality.Core as ≡
 open import Relation.Nary
 
 open import Data.List.Fresh
 open import Data.List.Fresh.Properties
 open import Data.List.Fresh.Membership.Setoid S
 open import Data.List.Fresh.Relation.Unary.Any using (Any; here; there; _─_)
-import Data.List.Fresh.Relation.Unary.Any.Properties as Anyₚ
-import Data.List.Fresh.Relation.Unary.All.Properties as Allₚ
+import Data.List.Fresh.Relation.Unary.Any.Properties as List#
 
 open Setoid S renaming (Carrier to A)
 
@@ -82,7 +81,7 @@ module _ {R : Rel A r} where
 module _ {R : Rel A r} (R⇒≉ : ∀[ R ⇒ _≉_ ]) (≉⇒R : ∀[ _≉_ ⇒ R ]) where
 
   private
-    R≈ : R B.Respectsˡ _≈_
+    R≈ : R Binary.Respectsˡ _≈_
     R≈ x≈y Rxz = ≉⇒R (R⇒≉ Rxz ∘′ trans x≈y)
 
   fresh-remove : ∀ {x} {xs : List# A R} (x∈xs : x ∈ xs) → x # (xs ─ x∈xs)
@@ -103,7 +102,7 @@ module _ {R : Rel A r} (R⇒≉ : ∀[ R ⇒ _≉_ ]) where
   injection {[]}                 {ys} inj = z≤n
   injection {xxs@(cons x xs pr)} {ys} inj = begin
     length xxs               ≤⟨ s≤s (injection step) ⟩
-    suc (length (ys ─ x∈ys)) ≡⟨ P.sym (Anyₚ.length-remove x∈ys) ⟩
+    suc (length (ys ─ x∈ys)) ≡⟨ ≡.sym (List#.length-remove x∈ys) ⟩
     length ys                ∎
 
     where
@@ -123,7 +122,7 @@ module _ {R : Rel A r} (R⇒≉ : ∀[ R ⇒ _≉_ ]) where
    (∃ λ x → x ∈ ys × x ∉ xs) → length xs < length ys
   strict-injection {xs} {ys} inj (x , x∈ys , x∉xs) = begin
     suc (length xs)          ≤⟨ s≤s (injection step) ⟩
-    suc (length (ys ─ x∈ys)) ≡⟨ P.sym (Anyₚ.length-remove x∈ys) ⟩
+    suc (length (ys ─ x∈ys)) ≡⟨ ≡.sym (List#.length-remove x∈ys) ⟩
     length ys                ∎
 
     where
@@ -138,12 +137,12 @@ module _ {R : Rel A r} (R⇒≉ : ∀[ R ⇒ _≉_ ]) where
 ------------------------------------------------------------------------
 -- proof irrelevance
 
-module _ {R : Rel A r} (R⇒≉ : ∀[ R ⇒ _≉_ ]) (≈-irrelevant : B.Irrelevant _≈_) where
+module _ {R : Rel A r} (R⇒≉ : ∀[ R ⇒ _≉_ ]) (≈-irrelevant : Binary.Irrelevant _≈_) where
 
   ∈-irrelevant : ∀ {x} {xs : List# A R} → Irrelevant (x ∈ xs)
   -- positive cases
-  ∈-irrelevant (here x≈y₁)   (here x≈y₂)   = P.cong here (≈-irrelevant x≈y₁ x≈y₂)
-  ∈-irrelevant (there x∈xs₁) (there x∈xs₂) = P.cong there (∈-irrelevant x∈xs₁ x∈xs₂)
+  ∈-irrelevant (here x≈y₁)   (here x≈y₂)   = ≡.cong here (≈-irrelevant x≈y₁ x≈y₂)
+  ∈-irrelevant (there x∈xs₁) (there x∈xs₂) = ≡.cong there (∈-irrelevant x∈xs₁ x∈xs₂)
   -- absurd cases
   ∈-irrelevant {xs = cons x xs pr} (here x≈y)    (there x∈xs₂) =
     ⊥-elim (distinct x∈xs₂ (fresh⇒∉ R⇒≉ pr) x≈y)
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index 06240c6be5..c5ab013b61 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -16,7 +16,7 @@ open import Data.List.Base as List
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Relation.Unary.Any.Properties
 open import Data.List.Membership.Propositional
-import Data.List.Membership.Setoid.Properties as Membershipₛ
+import Data.List.Membership.Setoid.Properties as Membership
 open import Data.List.Relation.Binary.Equality.Propositional
   using (_≋_; ≡⇒≋; ≋⇒≡)
 open import Data.List.Effectful using (monad)
@@ -35,8 +35,8 @@ open import Function.Related.TypeIsomorphisms
 open import Function.Construct.Identity using (↔-id)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.Definitions as B hiding (Decidable)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.Definitions as Binary hiding (Decidable)
+open import Relation.Binary.PropositionalEquality as ≡
   using (_≡_; _≢_; refl; sym; trans; cong; subst; →-to-⟶; _≗_)
 import Relation.Binary.Properties.DecTotalOrder as DTOProperties
 open import Relation.Unary using (_⟨×⟩_; Decidable)
@@ -62,10 +62,10 @@ open import Data.List.Membership.Propositional.Properties.Core public
 -- Equality
 
 ∈-resp-≋ : ∀ {x : A} → (x ∈_) Respects _≋_
-∈-resp-≋ = Membershipₛ.∈-resp-≋ (P.setoid _)
+∈-resp-≋ = Membership.∈-resp-≋ (≡.setoid _)
 
 ∉-resp-≋ : ∀ {x : A} → (x ∉_) Respects _≋_
-∉-resp-≋ = Membershipₛ.∉-resp-≋ (P.setoid _)
+∉-resp-≋ = Membership.∉-resp-≋ (≡.setoid _)
 
 ------------------------------------------------------------------------
 -- mapWith∈
@@ -74,19 +74,19 @@ mapWith∈-cong : ∀ (xs : List A) → (f g : ∀ {x} → x ∈ xs → B) →
                 (∀ {x} → (x∈xs : x ∈ xs) → f x∈xs ≡ g x∈xs) →
                 mapWith∈ xs f ≡ mapWith∈ xs g
 mapWith∈-cong []       f g cong = refl
-mapWith∈-cong (x ∷ xs) f g cong = P.cong₂ _∷_ (cong (here refl))
+mapWith∈-cong (x ∷ xs) f g cong = ≡.cong₂ _∷_ (cong (here refl))
   (mapWith∈-cong xs (f ∘ there) (g ∘ there) (cong ∘ there))
 
 mapWith∈≗map : ∀ (f : A → B) xs → mapWith∈ xs (λ {x} _ → f x) ≡ map f xs
 mapWith∈≗map f xs =
-  ≋⇒≡ (Membershipₛ.mapWith∈≗map (P.setoid _) (P.setoid _) f xs)
+  ≋⇒≡ (Membership.mapWith∈≗map (≡.setoid _) (≡.setoid _) f xs)
 
 mapWith∈-id : (xs : List A) → mapWith∈ xs (λ {x} _ → x) ≡ xs
-mapWith∈-id = Membershipₛ.mapWith∈-id (P.setoid _)
+mapWith∈-id = Membership.mapWith∈-id (≡.setoid _)
 
 map-mapWith∈ : (xs : List A) (f : ∀ {x} → x ∈ xs → B) (g : B → C) →
                map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
-map-mapWith∈ = Membershipₛ.map-mapWith∈ (P.setoid _)
+map-mapWith∈ = Membership.map-mapWith∈ (≡.setoid _)
 
 ------------------------------------------------------------------------
 -- map
@@ -94,10 +94,10 @@ map-mapWith∈ = Membershipₛ.map-mapWith∈ (P.setoid _)
 module _ (f : A → B) where
 
   ∈-map⁺ : ∀ {x xs} → x ∈ xs → f x ∈ map f xs
-  ∈-map⁺ = Membershipₛ.∈-map⁺ (P.setoid A) (P.setoid B) (P.cong f)
+  ∈-map⁺ = Membership.∈-map⁺ (≡.setoid A) (≡.setoid B) (≡.cong f)
 
   ∈-map⁻ : ∀ {y xs} → y ∈ map f xs → ∃ λ x → x ∈ xs × y ≡ f x
-  ∈-map⁻ = Membershipₛ.∈-map⁻ (P.setoid A) (P.setoid B)
+  ∈-map⁻ = Membership.∈-map⁻ (≡.setoid A) (≡.setoid B)
 
   map-∈↔ : ∀ {y xs} → (∃ λ x → x ∈ xs × y ≡ f x) ↔ y ∈ map f xs
   map-∈↔ {y} {xs} =
@@ -112,19 +112,19 @@ module _ (f : A → B) where
 module _ {v : A} where
 
   ∈-++⁺ˡ : ∀ {xs ys} → v ∈ xs → v ∈ xs ++ ys
-  ∈-++⁺ˡ = Membershipₛ.∈-++⁺ˡ (P.setoid A)
+  ∈-++⁺ˡ = Membership.∈-++⁺ˡ (≡.setoid A)
 
   ∈-++⁺ʳ : ∀ xs {ys} → v ∈ ys → v ∈ xs ++ ys
-  ∈-++⁺ʳ = Membershipₛ.∈-++⁺ʳ (P.setoid A)
+  ∈-++⁺ʳ = Membership.∈-++⁺ʳ (≡.setoid A)
 
   ∈-++⁻ : ∀ xs {ys} → v ∈ xs ++ ys → (v ∈ xs) ⊎ (v ∈ ys)
-  ∈-++⁻ = Membershipₛ.∈-++⁻ (P.setoid A)
+  ∈-++⁻ = Membership.∈-++⁻ (≡.setoid A)
 
   ∈-insert : ∀ xs {ys} → v ∈ xs ++ [ v ] ++ ys
-  ∈-insert xs = Membershipₛ.∈-insert (P.setoid A) xs refl
+  ∈-insert xs = Membership.∈-insert (≡.setoid A) xs refl
 
   ∈-∃++ : ∀ {xs} → v ∈ xs → ∃₂ λ ys zs → xs ≡ ys ++ [ v ] ++ zs
-  ∈-∃++ v∈xs with Membershipₛ.∈-∃++ (P.setoid A) v∈xs
+  ∈-∃++ v∈xs with Membership.∈-∃++ (≡.setoid A) v∈xs
   ... | ys , zs , _ , refl , eq = ys , zs , ≋⇒≡ eq
 
 ------------------------------------------------------------------------
@@ -133,17 +133,17 @@ module _ {v : A} where
 module _ {v : A} where
 
   ∈-concat⁺ : ∀ {xss} → Any (v ∈_) xss → v ∈ concat xss
-  ∈-concat⁺ = Membershipₛ.∈-concat⁺ (P.setoid A)
+  ∈-concat⁺ = Membership.∈-concat⁺ (≡.setoid A)
 
   ∈-concat⁻ : ∀ xss → v ∈ concat xss → Any (v ∈_) xss
-  ∈-concat⁻ = Membershipₛ.∈-concat⁻ (P.setoid A)
+  ∈-concat⁻ = Membership.∈-concat⁻ (≡.setoid A)
 
   ∈-concat⁺′ : ∀ {vs xss} → v ∈ vs → vs ∈ xss → v ∈ concat xss
   ∈-concat⁺′ v∈vs vs∈xss =
-    Membershipₛ.∈-concat⁺′ (P.setoid A) v∈vs (Any.map ≡⇒≋ vs∈xss)
+    Membership.∈-concat⁺′ (≡.setoid A) v∈vs (Any.map ≡⇒≋ vs∈xss)
 
   ∈-concat⁻′ : ∀ xss → v ∈ concat xss → ∃ λ xs → v ∈ xs × xs ∈ xss
-  ∈-concat⁻′ xss v∈c with Membershipₛ.∈-concat⁻′ (P.setoid A) xss v∈c
+  ∈-concat⁻′ xss v∈c with Membership.∈-concat⁻′ (≡.setoid A) xss v∈c
   ... | xs , v∈xs , xs∈xss = xs , v∈xs , Any.map ≋⇒≡ xs∈xss
 
   concat-∈↔ : ∀ {xss : List (List A)} →
@@ -162,13 +162,13 @@ module _ (f : A → B → C) where
 
   ∈-cartesianProductWith⁺ : ∀ {xs ys a b} → a ∈ xs → b ∈ ys →
                             f a b ∈ cartesianProductWith f xs ys
-  ∈-cartesianProductWith⁺ = Membershipₛ.∈-cartesianProductWith⁺
-    (P.setoid A) (P.setoid B) (P.setoid C) (P.cong₂ f)
+  ∈-cartesianProductWith⁺ = Membership.∈-cartesianProductWith⁺
+    (≡.setoid A) (≡.setoid B) (≡.setoid C) (≡.cong₂ f)
 
   ∈-cartesianProductWith⁻ : ∀ xs ys {v} → v ∈ cartesianProductWith f xs ys →
                             ∃₂ λ a b → a ∈ xs × b ∈ ys × v ≡ f a b
-  ∈-cartesianProductWith⁻ = Membershipₛ.∈-cartesianProductWith⁻
-    (P.setoid A) (P.setoid B) (P.setoid C) f
+  ∈-cartesianProductWith⁻ = Membership.∈-cartesianProductWith⁻
+    (≡.setoid A) (≡.setoid B) (≡.setoid C) f
 
 ------------------------------------------------------------------------
 -- cartesianProduct
@@ -188,11 +188,11 @@ module _ (f : A → B → C) where
 module _ (f : ℕ → A) where
 
   ∈-applyUpTo⁺ : ∀ {i n} → i < n → f i ∈ applyUpTo f n
-  ∈-applyUpTo⁺ = Membershipₛ.∈-applyUpTo⁺ (P.setoid _) f
+  ∈-applyUpTo⁺ = Membership.∈-applyUpTo⁺ (≡.setoid _) f
 
   ∈-applyUpTo⁻ : ∀ {v n} → v ∈ applyUpTo f n →
                  ∃ λ i → i < n × v ≡ f i
-  ∈-applyUpTo⁻ = Membershipₛ.∈-applyUpTo⁻ (P.setoid _) f
+  ∈-applyUpTo⁻ = Membership.∈-applyUpTo⁻ (≡.setoid _) f
 
 ------------------------------------------------------------------------
 -- upTo
@@ -210,11 +210,11 @@ module _ (f : ℕ → A) where
 module _ (f : ℕ → A) where
 
   ∈-applyDownFrom⁺ : ∀ {i n} → i < n → f i ∈ applyDownFrom f n
-  ∈-applyDownFrom⁺ = Membershipₛ.∈-applyDownFrom⁺ (P.setoid _) f
+  ∈-applyDownFrom⁺ = Membership.∈-applyDownFrom⁺ (≡.setoid _) f
 
   ∈-applyDownFrom⁻ : ∀ {v n} → v ∈ applyDownFrom f n →
                      ∃ λ i → i < n × v ≡ f i
-  ∈-applyDownFrom⁻ = Membershipₛ.∈-applyDownFrom⁻ (P.setoid _) f
+  ∈-applyDownFrom⁻ = Membership.∈-applyDownFrom⁻ (≡.setoid _) f
 
 ------------------------------------------------------------------------
 -- downFrom
@@ -232,10 +232,10 @@ module _ (f : ℕ → A) where
 module _ {n} {f : Fin n → A} where
 
   ∈-tabulate⁺ : ∀ i → f i ∈ tabulate f
-  ∈-tabulate⁺ = Membershipₛ.∈-tabulate⁺ (P.setoid _)
+  ∈-tabulate⁺ = Membership.∈-tabulate⁺ (≡.setoid _)
 
   ∈-tabulate⁻ : ∀ {v} → v ∈ tabulate f → ∃ λ i → v ≡ f i
-  ∈-tabulate⁻ = Membershipₛ.∈-tabulate⁻ (P.setoid _)
+  ∈-tabulate⁻ = Membership.∈-tabulate⁻ (≡.setoid _)
 
 ------------------------------------------------------------------------
 -- filter
@@ -243,29 +243,29 @@ module _ {n} {f : Fin n → A} where
 module _ {p} {P : A → Set p} (P? : Decidable P) where
 
   ∈-filter⁺ : ∀ {x xs} → x ∈ xs → P x → x ∈ filter P? xs
-  ∈-filter⁺ = Membershipₛ.∈-filter⁺ (P.setoid A) P? (P.subst P)
+  ∈-filter⁺ = Membership.∈-filter⁺ (≡.setoid A) P? (≡.subst P)
 
   ∈-filter⁻ : ∀ {v xs} → v ∈ filter P? xs → v ∈ xs × P v
-  ∈-filter⁻ = Membershipₛ.∈-filter⁻ (P.setoid A) P? (P.subst P)
+  ∈-filter⁻ = Membership.∈-filter⁻ (≡.setoid A) P? (≡.subst P)
 
 ------------------------------------------------------------------------
 -- derun and deduplicate
 
-module _ {r} {R : Rel A r} (R? : B.Decidable R) where
+module _ {r} {R : Rel A r} (R? : Binary.Decidable R) where
 
   ∈-derun⁻ : ∀ xs {z} → z ∈ derun R? xs → z ∈ xs
-  ∈-derun⁻ xs z∈derun[R,xs] = Membershipₛ.∈-derun⁻ (P.setoid A) R? xs z∈derun[R,xs]
+  ∈-derun⁻ xs z∈derun[R,xs] = Membership.∈-derun⁻ (≡.setoid A) R? xs z∈derun[R,xs]
 
   ∈-deduplicate⁻ : ∀ xs {z} → z ∈ deduplicate R? xs → z ∈ xs
-  ∈-deduplicate⁻ xs z∈dedup[R,xs] = Membershipₛ.∈-deduplicate⁻ (P.setoid A) R? xs z∈dedup[R,xs]
+  ∈-deduplicate⁻ xs z∈dedup[R,xs] = Membership.∈-deduplicate⁻ (≡.setoid A) R? xs z∈dedup[R,xs]
 
-module _ (_≈?_ : B.Decidable {A = A} _≡_) where
+module _ (_≈?_ : Binary.Decidable {A = A} _≡_) where
 
   ∈-derun⁺ : ∀ {xs z} → z ∈ xs → z ∈ derun _≈?_ xs
-  ∈-derun⁺ z∈xs = Membershipₛ.∈-derun⁺ (P.setoid A) _≈?_ (flip trans) z∈xs
+  ∈-derun⁺ z∈xs = Membership.∈-derun⁺ (≡.setoid A) _≈?_ (flip trans) z∈xs
 
   ∈-deduplicate⁺ : ∀ {xs z} → z ∈ xs → z ∈ deduplicate _≈?_ xs
-  ∈-deduplicate⁺ z∈xs = Membershipₛ.∈-deduplicate⁺ (P.setoid A) _≈?_ (λ c≡b a≡b → trans a≡b (sym c≡b)) z∈xs
+  ∈-deduplicate⁺ z∈xs = Membership.∈-deduplicate⁺ (≡.setoid A) _≈?_ (λ c≡b a≡b → trans a≡b (sym c≡b)) z∈xs
 
 ------------------------------------------------------------------------
 -- _>>=_
@@ -307,13 +307,13 @@ module _ (_≈?_ : B.Decidable {A = A} _≡_) where
 -- length
 
 ∈-length : ∀ {x : A} {xs} → x ∈ xs → 1 ≤ length xs
-∈-length = Membershipₛ.∈-length (P.setoid _)
+∈-length = Membership.∈-length (≡.setoid _)
 
 ------------------------------------------------------------------------
 -- lookup
 
 ∈-lookup : ∀ {xs : List A} i → lookup xs i ∈ xs
-∈-lookup {xs = xs} i = Membershipₛ.∈-lookup (P.setoid _) xs i
+∈-lookup {xs = xs} i = Membership.∈-lookup (≡.setoid _) xs i
 
 ------------------------------------------------------------------------
 -- foldr
@@ -322,7 +322,7 @@ module _ {_•_ : Op₂ A} where
 
   foldr-selective : Selective _≡_ _•_ → ∀ e xs →
                     (foldr _•_ e xs ≡ e) ⊎ (foldr _•_ e xs ∈ xs)
-  foldr-selective = Membershipₛ.foldr-selective (P.setoid A)
+  foldr-selective = Membership.foldr-selective (≡.setoid A)
 
 ------------------------------------------------------------------------
 -- allFin
@@ -380,7 +380,7 @@ finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
     f′-injective′ : Injective _≡_ _≡_ f′
     f′-injective′ {j} {k} eq with i ≤ᵇ j | Reflects.invert (≤ᵇ-reflects-≤ i j)
                                 | i ≤ᵇ k | Reflects.invert (≤ᵇ-reflects-≤ i k)
-    ... | true  | p | true  | q = P.cong pred (f-inj eq)
+    ... | true  | p | true  | q = ≡.cong pred (f-inj eq)
     ... | true  | p | false | q = contradiction (f-inj eq) (lemma p q)
     ... | false | p | true  | q = contradiction (f-inj eq) (lemma q p ∘ sym)
     ... | false | p | false | q = f-inj eq
@@ -388,7 +388,7 @@ finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
     f′-inj : ℕ ↣ _
     f′-inj = record
       { to        = f′
-      ; cong      = P.cong f′
+      ; cong      = ≡.cong f′
       ; injective = f′-injective′
       }
 
@@ -398,4 +398,4 @@ finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
 there-injective-≢∈ : ∀ {xs} {x y z : A} {x∈xs : x ∈ xs} {y∈xs : y ∈ xs} →
                      there {x = z} x∈xs ≢∈ there y∈xs →
                      x∈xs ≢∈ y∈xs
-there-injective-≢∈ neq refl eq = neq refl (P.cong there eq)
+there-injective-≢∈ neq refl eq = neq refl (≡.cong there eq)
diff --git a/src/Data/List/Membership/Propositional/Properties/WithK.agda b/src/Data/List/Membership/Propositional/Properties/WithK.agda
index 60a86ee5e3..a86f23a8ea 100644
--- a/src/Data/List/Membership/Propositional/Properties/WithK.agda
+++ b/src/Data/List/Membership/Propositional/Properties/WithK.agda
@@ -12,9 +12,9 @@ module Data.List.Membership.Propositional.Properties.WithK where
 open import Data.List.Base
 open import Data.List.Relation.Unary.Unique.Propositional
 open import Data.List.Membership.Propositional
-import Data.List.Membership.Setoid.Properties as Membershipₛ
+import Data.List.Membership.Setoid.Properties as Membership
 open import Relation.Unary using (Irrelevant)
-open import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Relation.Binary.PropositionalEquality.WithK
 
 ------------------------------------------------------------------------
@@ -22,4 +22,4 @@ open import Relation.Binary.PropositionalEquality.WithK
 
 unique⇒irrelevant : ∀ {a} {A : Set a} {xs : List A} →
                     Unique xs → Irrelevant (_∈ xs)
-unique⇒irrelevant = Membershipₛ.unique⇒irrelevant (P.setoid _) ≡-irrelevant
+unique⇒irrelevant = Membership.unique⇒irrelevant (≡.setoid _) ≡-irrelevant
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index f63391f882..c06897367b 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -28,10 +28,10 @@ open import Function.Base using (_$_; flip; _∘_; _∘′_; id)
 open import Function.Bundles using (_↔_)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_; _Preserves_⟶_)
-open import Relation.Binary.Definitions as B hiding (Decidable)
+open import Relation.Binary.Definitions as Binary hiding (Decidable)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-open import Relation.Unary as U using (Decidable; Pred)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+open import Relation.Unary as Unary using (Decidable; Pred)
 open import Relation.Nullary using (¬_; does; _because_; yes; no)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary.Negation using (contradiction)
@@ -85,11 +85,11 @@ module _ (S : Setoid c ℓ) where
     ∉×∈⇒≉ : ∀ {x y xs} → All (y ≉_) xs → x ∈ xs → x ≉ y
     ∉×∈⇒≉ = All.lookupWith λ y≉z x≈z x≈y → y≉z (trans (sym x≈y) x≈z)
 
-  unique⇒irrelevant : B.Irrelevant _≈_ → ∀ {xs} → Unique xs → U.Irrelevant (_∈ xs)
+  unique⇒irrelevant : Binary.Irrelevant _≈_ → ∀ {xs} → Unique xs → Unary.Irrelevant (_∈ xs)
   unique⇒irrelevant ≈-irr _        (here p)  (here q)  =
-    P.cong here (≈-irr p q)
+    ≡.cong here (≈-irr p q)
   unique⇒irrelevant ≈-irr (_  ∷ u) (there p) (there q) =
-    P.cong there (unique⇒irrelevant ≈-irr u p q)
+    ≡.cong there (unique⇒irrelevant ≈-irr u p q)
   unique⇒irrelevant ≈-irr (≉s ∷ _) (here p)  (there q) =
     contradiction p (∉×∈⇒≉ ≉s q)
   unique⇒irrelevant ≈-irr (≉s ∷ _) (there p) (here q)  =
@@ -130,18 +130,18 @@ module _ (S : Setoid c ℓ) where
 
   length-mapWith∈ : ∀ {a} {A : Set a} xs {f : ∀ {x} → x ∈ xs → A} →
                     length (mapWith∈ xs f) ≡ length xs
-  length-mapWith∈ []       = P.refl
-  length-mapWith∈ (x ∷ xs) = P.cong suc (length-mapWith∈ xs)
+  length-mapWith∈ []       = ≡.refl
+  length-mapWith∈ (x ∷ xs) = ≡.cong suc (length-mapWith∈ xs)
 
   mapWith∈-id : ∀ xs → mapWith∈ xs (λ {x} _ → x) ≡ xs
-  mapWith∈-id []       = P.refl
-  mapWith∈-id (x ∷ xs) = P.cong (x ∷_) (mapWith∈-id xs)
+  mapWith∈-id []       = ≡.refl
+  mapWith∈-id (x ∷ xs) = ≡.cong (x ∷_) (mapWith∈-id xs)
 
   map-mapWith∈ : ∀ {a b} {A : Set a} {B : Set b} →
                  ∀ xs (f : ∀ {x} → x ∈ xs → A) (g : A → B) →
                  map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
-  map-mapWith∈ []       f g = P.refl
-  map-mapWith∈ (x ∷ xs) f g = P.cong (_ ∷_) (map-mapWith∈ xs (f ∘ there) g)
+  map-mapWith∈ []       f g = ≡.refl
+  map-mapWith∈ (x ∷ xs) f g = ≡.cong (_ ∷_) (map-mapWith∈ xs (f ∘ there) g)
 
 ------------------------------------------------------------------------
 -- map
@@ -164,8 +164,8 @@ module _ (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) where
   map-∷= : ∀ {f} (f≈ : f Preserves _≈₁_ ⟶ _≈₂_)
            {xs x v} → (x∈xs : x ∈₁ xs) →
            map f (x∈xs M₁.∷= v) ≡ ∈-map⁺ f≈ x∈xs M₂.∷= f v
-  map-∷= f≈ (here x≈y)   = P.refl
-  map-∷= f≈ (there x∈xs) = P.cong (_ ∷_) (map-∷= f≈ x∈xs)
+  map-∷= f≈ (here x≈y)   = ≡.refl
+  map-∷= f≈ (there x∈xs) = ≡.cong (_ ∷_) (map-∷= f≈ x∈xs)
 
 ------------------------------------------------------------------------
 -- _++_
@@ -350,7 +350,7 @@ module _ (S : Setoid c ℓ) {P : Pred (Carrier S) p}
 ------------------------------------------------------------------------
 -- derun and deduplicate
 
-module _ (S : Setoid c ℓ) {R : Rel (Carrier S) ℓ₂} (R? : B.Decidable R) where
+module _ (S : Setoid c ℓ) {R : Rel (Carrier S) ℓ₂} (R? : Binary.Decidable R) where
 
   open Setoid S using (_≈_)
   open Membership S using (_∈_)
diff --git a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
index e833ade444..7e85ae8661 100644
--- a/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Infix/Heterogeneous/Properties.agda
@@ -27,7 +27,7 @@ open import Data.List.Relation.Binary.Pointwise.Base as Pointwise using (Pointwi
 open import Data.List.Relation.Binary.Infix.Heterogeneous
 open import Data.List.Relation.Binary.Prefix.Heterogeneous
   as Prefix using (Prefix; []; _∷_)
-import Data.List.Relation.Binary.Prefix.Heterogeneous.Properties as Prefixₚ
+import Data.List.Relation.Binary.Prefix.Heterogeneous.Properties as Prefix
 open import Data.List.Relation.Binary.Suffix.Heterogeneous
   as Suffix using (Suffix; here; there)
 
@@ -43,7 +43,7 @@ private
 -- Conversion functions
 
 fromPointwise : ∀ {as bs} → Pointwise R as bs → Infix R as bs
-fromPointwise pw = here (Prefixₚ.fromPointwise pw)
+fromPointwise pw = here (Prefix.fromPointwise pw)
 
 fromSuffix : ∀ {as bs} → Suffix R as bs → Infix R as bs
 fromSuffix (here pw) = fromPointwise pw
@@ -52,11 +52,11 @@ fromSuffix (there p) = there (fromSuffix p)
 module _ {c t} {C : Set c} {T : REL A C t} where
 
   fromPrefixSuffix : Trans R S T → Trans (Prefix R) (Suffix S) (Infix T)
-  fromPrefixSuffix tr p (here q)  = here (Prefixₚ.trans tr p (Prefixₚ.fromPointwise q))
+  fromPrefixSuffix tr p (here q)  = here (Prefix.trans tr p (Prefix.fromPointwise q))
   fromPrefixSuffix tr p (there q) = there (fromPrefixSuffix tr p q)
 
   fromSuffixPrefix : Trans R S T → Trans (Suffix R) (Prefix S) (Infix T)
-  fromSuffixPrefix tr (here p)  q       = here (Prefixₚ.trans tr (Prefixₚ.fromPointwise p) q)
+  fromSuffixPrefix tr (here p)  q       = here (Prefix.trans tr (Prefix.fromPointwise p) q)
   fromSuffixPrefix tr (there p) (_ ∷ q) = there (fromSuffixPrefix tr p q)
 
 ∷⁻ : ∀ {as b bs} → Infix R as (b ∷ bs) → Prefix R as (b ∷ bs) ⊎ Infix R as bs
@@ -67,7 +67,7 @@ module _ {c t} {C : Set c} {T : REL A C t} where
 -- length
 
 length-mono : ∀ {as bs} → Infix R as bs → length as ≤ length bs
-length-mono (here pref) = Prefixₚ.length-mono pref
+length-mono (here pref) = Prefix.length-mono pref
 length-mono (there p)   = ℕ.m≤n⇒m≤1+n (length-mono p)
 
 ------------------------------------------------------------------------
@@ -76,11 +76,11 @@ length-mono (there p)   = ℕ.m≤n⇒m≤1+n (length-mono p)
 module _ {c t} {C : Set c} {T : REL A C t} where
 
   Prefix-Infix-trans : Trans R S T → Trans (Prefix R) (Infix S) (Infix T)
-  Prefix-Infix-trans tr p (here q)  = here (Prefixₚ.trans tr p q)
+  Prefix-Infix-trans tr p (here q)  = here (Prefix.trans tr p q)
   Prefix-Infix-trans tr p (there q) = there (Prefix-Infix-trans tr p q)
 
   Infix-Prefix-trans : Trans R S T → Trans (Infix R) (Prefix S) (Infix T)
-  Infix-Prefix-trans tr (here p)  q       = here (Prefixₚ.trans tr p q)
+  Infix-Prefix-trans tr (here p)  q       = here (Prefix.trans tr p q)
   Infix-Prefix-trans tr (there p) (_ ∷ q) = there (Infix-Prefix-trans tr p q)
 
   Suffix-Infix-trans : Trans R S T → Trans (Suffix R) (Infix S) (Infix T)
@@ -88,7 +88,7 @@ module _ {c t} {C : Set c} {T : REL A C t} where
   Suffix-Infix-trans tr p (there q) = there (Suffix-Infix-trans tr p q)
 
   Infix-Suffix-trans : Trans R S T → Trans (Infix R) (Suffix S) (Infix T)
-  Infix-Suffix-trans tr p (here q)  = Infix-Prefix-trans tr p (Prefixₚ.fromPointwise q)
+  Infix-Suffix-trans tr p (here q)  = Infix-Prefix-trans tr p (Prefix.fromPointwise q)
   Infix-Suffix-trans tr p (there q) = there (Infix-Suffix-trans tr p q)
 
   trans : Trans R S T → Trans (Infix R) (Infix S) (Infix T)
@@ -96,7 +96,7 @@ module _ {c t} {C : Set c} {T : REL A C t} where
   trans tr p (there q) = there (trans tr p q)
 
   antisym : Antisym R S T → Antisym (Infix R) (Infix S) (Pointwise T)
-  antisym asym (here p) (here q) = Prefixₚ.antisym asym p q
+  antisym asym (here p) (here q) = Prefix.antisym asym p q
   antisym asym {i = a ∷ as} {j = bs} p@(here _) (there q)
     = ⊥-elim $′ ℕ.<-irrefl refl $′ begin-strict
       length as <⟨ length-mono p ⟩
@@ -121,14 +121,14 @@ module _ {c d r} {C : Set c} {D : Set d} {R : REL C D r} where
   map⁺ : ∀ {as bs} (f : A → C) (g : B → D) →
          Infix (λ a b → R (f a) (g b)) as bs →
          Infix R (List.map f as) (List.map g bs)
-  map⁺ f g (here p)  = here (Prefixₚ.map⁺ f g p)
+  map⁺ f g (here p)  = here (Prefix.map⁺ f g p)
   map⁺ f g (there p) = there (map⁺ f g p)
 
   map⁻ : ∀ {as bs} (f : A → C) (g : B → D) →
          Infix R (List.map f as) (List.map g bs) →
          Infix (λ a b → R (f a) (g b)) as bs
-  map⁻ {bs = []}     f g (here p)  = here (Prefixₚ.map⁻ f g p)
-  map⁻ {bs = b ∷ bs} f g (here p)  = here (Prefixₚ.map⁻ f g p)
+  map⁻ {bs = []}     f g (here p)  = here (Prefix.map⁻ f g p)
+  map⁻ {bs = b ∷ bs} f g (here p)  = here (Prefix.map⁻ f g p)
   map⁻ {bs = b ∷ bs} f g (there p) = there (map⁻ f g p)
 
 ------------------------------------------------------------------------
@@ -139,7 +139,7 @@ module _ {p q} {P : Pred A p} {Q : Pred B q} (P? : U.Decidable P) (Q? : U.Decida
          where
 
   filter⁺ : ∀ {as bs} → Infix R as bs → Infix R (filter P? as) (filter Q? bs)
-  filter⁺ (here p) = here (Prefixₚ.filter⁺ P? Q? (λ _ → P⇒Q) (λ _ → Q⇒P) p)
+  filter⁺ (here p) = here (Prefix.filter⁺ P? Q? (λ _ → P⇒Q) (λ _ → Q⇒P) p)
   filter⁺ {bs = b ∷ bs} (there p) with does (Q? b)
   ... | true = there (filter⁺ p)
   ... | false = filter⁺ p
@@ -149,12 +149,12 @@ module _ {p q} {P : Pred A p} {Q : Pred B q} (P? : U.Decidable P) (Q? : U.Decida
 
 replicate⁺ : ∀ {m n a b} → m ≤ n → R a b →
              Infix R (replicate m a) (replicate n b)
-replicate⁺ m≤n r = here (Prefixₚ.replicate⁺ m≤n r)
+replicate⁺ m≤n r = here (Prefix.replicate⁺ m≤n r)
 
 replicate⁻ : ∀ {m n a b} → m ≢ 0 →
              Infix R (replicate m a) (replicate n b) → R a b
-replicate⁻ {m = m} {n = zero}  m≢0 (here p)  = Prefixₚ.replicate⁻ m≢0 p
-replicate⁻ {m = m} {n = suc n} m≢0 (here p)  = Prefixₚ.replicate⁻ m≢0 p
+replicate⁻ {m = m} {n = zero}  m≢0 (here p)  = Prefix.replicate⁻ m≢0 p
+replicate⁻ {m = m} {n = suc n} m≢0 (here p)  = Prefix.replicate⁻ m≢0 p
 replicate⁻ {m = m} {n = suc n} m≢0 (there p) = replicate⁻ m≢0 p
 
 ------------------------------------------------------------------------
@@ -165,4 +165,4 @@ infix? R? [] [] = yes (here [])
 infix? R? (a ∷ as) [] = no (λ where (here ()))
 infix? R? as bbs@(_ ∷ bs) =
   map′ [ here , there ]′ ∷⁻
-  (Prefixₚ.prefix? R? as bbs ⊎-dec infix? R? as bs)
+  (Prefix.prefix? R? as bbs ⊎-dec infix? R? as bs)
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index 5e44480bc3..3184ca4420 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -15,13 +15,13 @@ open import Data.Empty
 open import Data.List.Relation.Unary.All using (Null; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Data.List.Base as List hiding (map; _∷ʳ_)
-import Data.List.Properties as Lₚ
+import Data.List.Properties as List
 open import Data.List.Relation.Unary.Any.Properties
   using (here-injective; there-injective)
 open import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise; []; _∷_)
 open import Data.List.Relation.Binary.Sublist.Heterogeneous
 
-open import Data.Maybe.Relation.Unary.All as MAll using (nothing; just)
+open import Data.Maybe.Relation.Unary.All as Maybe using (nothing; just)
 open import Data.Nat.Base using (ℕ; _≤_; _≥_); open ℕ; open _≤_
 import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (∃₂; _×_; _,_; <_,_>; proj₂; uncurry)
@@ -90,9 +90,9 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   tail-Sublist : ∀ {as bs} → Sublist R as bs →
-                 MAll.All (λ as → Sublist R as bs) (tail as)
+                 Maybe.All (λ as → Sublist R as bs) (tail as)
   tail-Sublist []        = nothing
-  tail-Sublist (b ∷ʳ ps) = MAll.map (b ∷ʳ_) (tail-Sublist ps)
+  tail-Sublist (b ∷ʳ ps) = Maybe.map (b ∷ʳ_) (tail-Sublist ps)
   tail-Sublist (p ∷ ps)  = just (_ ∷ʳ ps)
 
   take-Sublist : ∀ {as bs} n → Sublist R as bs → Sublist R (take n as) bs
@@ -308,7 +308,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   reverse⁻ : ∀ {as bs} → Sublist R (reverse as) (reverse bs) → Sublist R as bs
   reverse⁻ {as} {bs} p = cast (reverse⁺ p) where
-    cast = P.subst₂ (Sublist R) (Lₚ.reverse-involutive as) (Lₚ.reverse-involutive bs)
+    cast = P.subst₂ (Sublist R) (List.reverse-involutive as) (List.reverse-involutive bs)
 
 ------------------------------------------------------------------------
 -- Inversion lemmas
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index f50a29e82c..3e91679bb2 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -21,7 +21,7 @@ open import Data.List.Effectful
 open import Data.List.Relation.Unary.Any using (Any)
 open import Data.List.Membership.Propositional
 open import Data.List.Membership.Propositional.Properties
-import Data.List.Relation.Binary.Subset.Setoid.Properties as Setoidₚ
+import Data.List.Relation.Binary.Subset.Setoid.Properties as Subset
 open import Data.List.Relation.Binary.Subset.Propositional
 open import Data.List.Relation.Binary.Permutation.Propositional
 import Data.List.Relation.Binary.Permutation.Propositional.Properties as Permutation
@@ -79,7 +79,7 @@ module _ (A : Set a) where
 ------------------------------------------------------------------------
 -- Relational properties with _↭_ (permutation)
 ------------------------------------------------------------------------
--- See issue #1354 for why these proofs can't be taken from `Setoidₚ`
+-- See issue #1354 for why these proofs can't be taken from `Subset`
 
 ⊆-reflexive-↭ : _↭_ {A = A} ⇒ _⊆_
 ⊆-reflexive-↭ xs↭ys = Permutation.∈-resp-↭ xs↭ys
@@ -109,7 +109,7 @@ module _ (A : Set a) where
 ------------------------------------------------------------------------
 
 module ⊆-Reasoning (A : Set a) where
-  open Setoidₚ.⊆-Reasoning (setoid A) public
+  open Subset.⊆-Reasoning (setoid A) public
     hiding (step-≋; step-≋˘)
 
 ------------------------------------------------------------------------
@@ -117,10 +117,10 @@ module ⊆-Reasoning (A : Set a) where
 ------------------------------------------------------------------------
 
 Any-resp-⊆ : ∀ {P : Pred A p} → (Any P) Respects _⊆_
-Any-resp-⊆ = Setoidₚ.Any-resp-⊆ (setoid _) (subst _)
+Any-resp-⊆ = Subset.Any-resp-⊆ (setoid _) (subst _)
 
 All-resp-⊇ : ∀ {P : Pred A p} → (All P) Respects _⊇_
-All-resp-⊇ = Setoidₚ.All-resp-⊇ (setoid _) (subst _)
+All-resp-⊇ = Subset.All-resp-⊇ (setoid _) (subst _)
 
 ------------------------------------------------------------------------
 -- Properties relating _⊆_ to various list functions
@@ -137,31 +137,31 @@ map⁺ f xs⊆ys =
 -- ∷
 
 xs⊆x∷xs : ∀ (xs : List A) x → xs ⊆ x ∷ xs
-xs⊆x∷xs = Setoidₚ.xs⊆x∷xs (setoid _)
+xs⊆x∷xs = Subset.xs⊆x∷xs (setoid _)
 
 ∷⁺ʳ : ∀ x → xs ⊆ ys → x ∷ xs ⊆ x ∷ ys
-∷⁺ʳ = Setoidₚ.∷⁺ʳ (setoid _)
+∷⁺ʳ = Subset.∷⁺ʳ (setoid _)
 
 ∈-∷⁺ʳ : ∀ {x} → x ∈ ys → xs ⊆ ys → x ∷ xs ⊆ ys
-∈-∷⁺ʳ = Setoidₚ.∈-∷⁺ʳ (setoid _)
+∈-∷⁺ʳ = Subset.∈-∷⁺ʳ (setoid _)
 
 ------------------------------------------------------------------------
 -- _++_
 
 xs⊆xs++ys : ∀ (xs ys : List A) → xs ⊆ xs ++ ys
-xs⊆xs++ys = Setoidₚ.xs⊆xs++ys (setoid _)
+xs⊆xs++ys = Subset.xs⊆xs++ys (setoid _)
 
 xs⊆ys++xs : ∀ (xs ys : List A) → xs ⊆ ys ++ xs
-xs⊆ys++xs = Setoidₚ.xs⊆ys++xs (setoid _)
+xs⊆ys++xs = Subset.xs⊆ys++xs (setoid _)
 
 ++⁺ʳ : ∀ zs → xs ⊆ ys → zs ++ xs ⊆ zs ++ ys
-++⁺ʳ = Setoidₚ.++⁺ʳ (setoid _)
+++⁺ʳ = Subset.++⁺ʳ (setoid _)
 
 ++⁺ˡ : ∀ zs → xs ⊆ ys → xs ++ zs ⊆ ys ++ zs
-++⁺ˡ = Setoidₚ.++⁺ˡ (setoid _)
+++⁺ˡ = Subset.++⁺ˡ (setoid _)
 
 ++⁺ : ws ⊆ xs → ys ⊆ zs → ws ++ ys ⊆ xs ++ zs
-++⁺ = Setoidₚ.++⁺ (setoid _)
+++⁺ = Subset.++⁺ (setoid _)
 
 ------------------------------------------------------------------------
 -- concat
@@ -178,7 +178,7 @@ module _ {xss yss : List (List A)} where
 -- applyUpTo
 
 applyUpTo⁺ : ∀ (f : ℕ → A) {m n} → m ≤ n → applyUpTo f m ⊆ applyUpTo f n
-applyUpTo⁺ = Setoidₚ.applyUpTo⁺ (setoid _)
+applyUpTo⁺ = Subset.applyUpTo⁺ (setoid _)
 
 ------------------------------------------------------------------------
 -- _>>=_
@@ -248,12 +248,12 @@ module _ {xs : List A} {f : ∀ {x} → x ∈ xs → B}
 module _ {P : Pred A p} (P? : Decidable P) where
 
   filter-⊆ : ∀ xs → filter P? xs ⊆ xs
-  filter-⊆ = Setoidₚ.filter-⊆ (setoid A) P?
+  filter-⊆ = Subset.filter-⊆ (setoid A) P?
 
   module _ {Q : Pred A q} (Q? : Decidable Q) where
 
     filter⁺′ : P ⋐ Q → ∀ {xs ys} → xs ⊆ ys → filter P? xs ⊆ filter Q? ys
-    filter⁺′ = Setoidₚ.filter⁺′ (setoid A) P? (resp P) Q? (resp Q)
+    filter⁺′ = Subset.filter⁺′ (setoid A) P? (resp P) Q? (resp Q)
 
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
index 3ea10495e6..17a0c25c77 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
@@ -30,8 +30,8 @@ open import Relation.Binary.Definitions as B
 open import Relation.Binary.PropositionalEquality.Core as P
   using (_≡_; _≢_; refl; sym; subst; subst₂)
 
-import Data.List.Properties as Listₚ
-import Data.List.Relation.Binary.Prefix.Heterogeneous.Properties as Prefixₚ
+import Data.List.Properties as List
+import Data.List.Relation.Binary.Prefix.Heterogeneous.Properties as Prefix
 
 ------------------------------------------------------------------------
 -- Suffix and Prefix are linked via reverse
@@ -43,15 +43,15 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   fromPrefix {as} {bs} p with Prefix.toView p
   ... | Prefix._++_ {cs} rs ds =
     subst (Suffix R (reverse as))
-      (sym (Listₚ.reverse-++ cs ds))
+      (sym (List.reverse-++ cs ds))
       (Suffix.fromView (reverse ds Suffix.++ Pw.reverse⁺ rs))
 
   fromPrefix-rev : ∀ {as bs} → Prefix R (reverse as) (reverse bs) →
                    Suffix R as bs
   fromPrefix-rev pre =
     subst₂ (Suffix R)
-      (Listₚ.reverse-involutive _)
-      (Listₚ.reverse-involutive _)
+      (List.reverse-involutive _)
+      (List.reverse-involutive _)
       (fromPrefix pre)
 
   toPrefix-rev : ∀ {as bs} → Suffix R as bs →
@@ -59,15 +59,15 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   toPrefix-rev {as} {bs} s with Suffix.toView s
   ... | Suffix._++_ cs {ds} rs =
     subst (Prefix R (reverse as))
-      (sym (Listₚ.reverse-++ cs ds))
+      (sym (List.reverse-++ cs ds))
       (Prefix.fromView (Pw.reverse⁺ rs Prefix.++ reverse cs))
 
   toPrefix : ∀ {as bs} → Suffix R (reverse as) (reverse bs) →
              Prefix R as bs
   toPrefix suf =
     subst₂ (Prefix R)
-      (Listₚ.reverse-involutive _)
-      (Listₚ.reverse-involutive _)
+      (List.reverse-involutive _)
+      (List.reverse-involutive _)
       (toPrefix-rev suf)
 
 ------------------------------------------------------------------------
@@ -139,7 +139,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
     ds<cs = begin-strict
       length ds                   ≤⟨ m≤n+m (length ds) (length bs) ⟩
       length bs + length ds       <⟨ ≤-refl ⟩
-      suc (length bs + length ds) ≡⟨ sym $ Listₚ.length-++ (b ∷ bs) ⟩
+      suc (length bs + length ds) ≡⟨ sym $ List.length-++ (b ∷ bs) ⟩
       length (b ∷ bs ++ ds)       ≡⟨ sym $ Pointwise-length rs ⟩
       length cs                   ∎
 
@@ -209,4 +209,4 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   suffix? : B.Decidable R → B.Decidable (Suffix R)
   suffix? R? as bs = Dec.map′ fromPrefix-rev toPrefix-rev
-                   $ Prefixₚ.prefix? R? (reverse as) (reverse bs)
+                   $ Prefix.prefix? R? (reverse as) (reverse bs)
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional.agda b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
index f85ff2f480..f807a0d649 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
@@ -13,14 +13,14 @@ module Data.List.Relation.Ternary.Appending.Propositional
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.Product.Base using (_,_)
 
-import Data.List.Properties as Listₚ
+import Data.List.Properties as List
 import Data.List.Relation.Binary.Pointwise as Pw using (≡⇒Pointwise-≡; Pointwise-≡⇒≡)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; setoid; refl; trans; cong₂; module ≡-Reasoning)
 
 import Data.List.Relation.Ternary.Appending.Setoid (setoid A) as General
 import Data.List.Relation.Ternary.Appending.Setoid.Properties (setoid A)
-  as Appendingₚ
+  as Appending
 
 ------------------------------------------------------------------------
 -- Re-export the basic combinators
@@ -37,7 +37,7 @@ _++_ : (as bs : List A) → Appending as bs (as List.++ bs)
 as ++ bs = Pw.≡⇒Pointwise-≡ refl General.++ Pw.≡⇒Pointwise-≡ refl
 
 _++[] : (as : List A) → Appending as [] as
-as ++[] = Appendingₚ.respʳ-≋ (as ++ []) (Pw.≡⇒Pointwise-≡ (Listₚ.++-identityʳ as))
+as ++[] = Appending.respʳ-≋ (as ++ []) (Pw.≡⇒Pointwise-≡ (List.++-identityʳ as))
 
 break : ∀ {as bs cs} → Appending as bs cs → as List.++ bs ≡ cs
 break {as} {bs} {cs} lrs = let (cs₁ , cs₂ , eq , acs , bcs) = General.break lrs in begin
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
index 54e19f59d2..46cf757ed0 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional/Properties.agda
@@ -9,7 +9,6 @@
 module Data.List.Relation.Ternary.Appending.Propositional.Properties {a} {A : Set a} where
 
 open import Data.List.Base as List using (List; [])
-import Data.List.Properties as Listₚ
 import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise-≡⇒≡)
 open import Data.List.Relation.Binary.Equality.Propositional using (_≋_)
 open import Data.List.Relation.Ternary.Appending.Propositional {A = A}
@@ -18,7 +17,7 @@ open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 import Data.List.Relation.Ternary.Appending.Setoid.Properties (setoid A)
-  as Appendingₚ
+  as Appending
 
 private
   variable
@@ -27,14 +26,14 @@ private
 ------------------------------------------------------------------------
 -- Re-export existing properties
 
-open Appendingₚ public
+open Appending public
   hiding ([]++⁻¹; ++[]⁻¹)
 
 ------------------------------------------------------------------------
 -- Prove propositional-specific ones
 
 []++⁻¹ : Appending [] bs cs → bs ≡ cs
-[]++⁻¹ = Pw.Pointwise-≡⇒≡ ∘′ Appendingₚ.[]++⁻¹
+[]++⁻¹ = Pw.Pointwise-≡⇒≡ ∘′ Appending.[]++⁻¹
 
 ++[]⁻¹ : Appending as [] cs → as ≡ cs
-++[]⁻¹ = Pw.Pointwise-≡⇒≡ ∘′ Appendingₚ.++[]⁻¹
+++[]⁻¹ = Pw.Pointwise-≡⇒≡ ∘′ Appending.++[]⁻¹
diff --git a/src/Data/String/Properties.agda b/src/Data/String/Properties.agda
index 9c5ae194e2..1d2e9a9a1c 100644
--- a/src/Data/String/Properties.agda
+++ b/src/Data/String/Properties.agda
@@ -9,8 +9,8 @@
 module Data.String.Properties where
 
 open import Data.Bool.Base using (Bool)
-import Data.Char.Properties as Charₚ
-import Data.List.Properties as Listₚ
+import Data.Char.Properties as Char
+import Data.List.Properties as List
 import Data.List.Relation.Binary.Pointwise as Pointwise
 import Data.List.Relation.Binary.Lex.Strict as StrictLex
 open import Data.String.Base
@@ -59,7 +59,7 @@ open import Agda.Builtin.String.Properties public
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-x ≈? y = Pointwise.decidable Charₚ._≟_ (toList x) (toList y)
+x ≈? y = Pointwise.decidable Char._≟_ (toList x) (toList y)
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -103,54 +103,54 @@ x ≟ y = map′ ≈⇒≡ ≈-reflexive $ x ≈? y
 
 infix 4 _<?_
 _<?_ : Decidable _<_
-x <? y = StrictLex.<-decidable Charₚ._≟_ Charₚ._<?_ (toList x) (toList y)
+x <? y = StrictLex.<-decidable Char._≟_ Char._<?_ (toList x) (toList y)
 
 <-isStrictPartialOrder-≈ : IsStrictPartialOrder _≈_ _<_
 <-isStrictPartialOrder-≈ =
   On.isStrictPartialOrder
     toList
-    (StrictLex.<-isStrictPartialOrder Charₚ.<-isStrictPartialOrder)
+    (StrictLex.<-isStrictPartialOrder Char.<-isStrictPartialOrder)
 
 <-isStrictTotalOrder-≈ : IsStrictTotalOrder _≈_ _<_
 <-isStrictTotalOrder-≈ =
   On.isStrictTotalOrder
     toList
-    (StrictLex.<-isStrictTotalOrder Charₚ.<-isStrictTotalOrder)
+    (StrictLex.<-isStrictTotalOrder Char.<-isStrictTotalOrder)
 
 <-strictPartialOrder-≈ : StrictPartialOrder _ _ _
 <-strictPartialOrder-≈ =
   On.strictPartialOrder
-    (StrictLex.<-strictPartialOrder Charₚ.<-strictPartialOrder)
+    (StrictLex.<-strictPartialOrder Char.<-strictPartialOrder)
     toList
 
 <-strictTotalOrder-≈ : StrictTotalOrder _ _ _
 <-strictTotalOrder-≈ =
   On.strictTotalOrder
-    (StrictLex.<-strictTotalOrder Charₚ.<-strictTotalOrder)
+    (StrictLex.<-strictTotalOrder Char.<-strictTotalOrder)
     toList
 
 ≤-isDecPartialOrder-≈ : IsDecPartialOrder _≈_ _≤_
 ≤-isDecPartialOrder-≈ =
   On.isDecPartialOrder
     toList
-    (StrictLex.≤-isDecPartialOrder Charₚ.<-isStrictTotalOrder)
+    (StrictLex.≤-isDecPartialOrder Char.<-isStrictTotalOrder)
 
 ≤-isDecTotalOrder-≈ : IsDecTotalOrder _≈_ _≤_
 ≤-isDecTotalOrder-≈ =
   On.isDecTotalOrder
     toList
-    (StrictLex.≤-isDecTotalOrder Charₚ.<-isStrictTotalOrder)
+    (StrictLex.≤-isDecTotalOrder Char.<-isStrictTotalOrder)
 
 ≤-decTotalOrder-≈ :  DecTotalOrder _ _ _
 ≤-decTotalOrder-≈ =
   On.decTotalOrder
-    (StrictLex.≤-decTotalOrder Charₚ.<-strictTotalOrder)
+    (StrictLex.≤-decTotalOrder Char.<-strictTotalOrder)
     toList
 
 ≤-decPoset-≈ : DecPoset _ _ _
 ≤-decPoset-≈ =
   On.decPoset
-    (StrictLex.≤-decPoset Charₚ.<-strictTotalOrder)
+    (StrictLex.≤-decPoset Char.<-strictTotalOrder)
     toList
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Tree/Rose/Properties.agda b/src/Data/Tree/Rose/Properties.agda
index 054103fdd1..91600e68c8 100644
--- a/src/Data/Tree/Rose/Properties.agda
+++ b/src/Data/Tree/Rose/Properties.agda
@@ -12,7 +12,7 @@ open import Level using (Level)
 open import Size
 open import Data.List.Base as List using (List)
 open import Data.List.Extrema.Nat
-import Data.List.Properties as Listₚ
+import Data.List.Properties as List
 open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.Tree.Rose
 open import Function.Base
@@ -33,14 +33,14 @@ private
 map-∘ : (f : A → B) (g : B → C) →
               map {i = i} (g ∘′ f) ≗ map {i = i} g ∘′ map {i = i} f
 map-∘ f g (node a ts) = cong (node (g (f a))) $ begin
-  List.map (map (g ∘′ f)) ts             ≡⟨ Listₚ.map-cong (map-∘ f g) ts ⟩
-  List.map (map g ∘ map f) ts            ≡⟨ Listₚ.map-∘ ts ⟩
+  List.map (map (g ∘′ f)) ts             ≡⟨ List.map-cong (map-∘ f g) ts ⟩
+  List.map (map g ∘ map f) ts            ≡⟨ List.map-∘ ts ⟩
   List.map (map g) (List.map (map f) ts) ∎
 
 depth-map : (f : A → B) (t : Rose A i) → depth {i = i} (map {i = i} f t) ≡ depth {i = i} t
 depth-map f (node a ts) = cong (suc ∘′ max 0) $ begin
-  List.map depth (List.map (map f) ts) ≡⟨ Listₚ.map-∘ ts ⟨
-  List.map (depth ∘′ map f) ts         ≡⟨ Listₚ.map-cong (depth-map f) ts ⟩
+  List.map depth (List.map (map f) ts) ≡⟨ List.map-∘ ts ⟨
+  List.map (depth ∘′ map f) ts         ≡⟨ List.map-cong (depth-map f) ts ⟩
   List.map depth ts ∎
 
 ------------------------------------------------------------------------
@@ -49,8 +49,8 @@ depth-map f (node a ts) = cong (suc ∘′ max 0) $ begin
 foldr-map : (f : A → B) (n : B → List C → C) (ts : Rose A i) →
             foldr {i = i} n (map {i = i} f ts) ≡ foldr {i = i} (n ∘′ f) ts
 foldr-map f n (node a ts) = cong (n (f a)) $ begin
-  List.map (foldr n) (List.map (map f) ts) ≡⟨ Listₚ.map-∘ ts ⟨
-  List.map (foldr n ∘′ map f) ts           ≡⟨ Listₚ.map-cong (foldr-map f n) ts ⟩
+  List.map (foldr n) (List.map (map f) ts) ≡⟨ List.map-∘ ts ⟨
+  List.map (foldr n ∘′ map f) ts           ≡⟨ List.map-cong (foldr-map f n) ts ⟩
   List.map (foldr (n ∘′ f)) ts             ∎
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index a860a2d925..66752c65e7 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base as Fin
   using (Fin; zero; suc; toℕ; fromℕ<; _↑ˡ_; _↑ʳ_)
 open import Data.List.Base as List using (List)
-import Data.List.Properties as Listₚ
+import Data.List.Properties as List
 open import Data.Nat.Base
 open import Data.Nat.Properties
   using (+-assoc; m≤n⇒m≤1+n; m≤m+n; ≤-refl; ≤-trans; ≤-irrelevant; ≤⇒≤″; suc-injective; +-comm; +-suc)
@@ -1294,29 +1294,29 @@ cast-fromList : ∀ {xs ys : List A} (eq : xs ≡ ys) →
                 cast (cong List.length eq) (fromList xs) ≡ fromList ys
 cast-fromList {xs = List.[]}     {ys = List.[]}     eq = refl
 cast-fromList {xs = x List.∷ xs} {ys = y List.∷ ys} eq =
-  let x≡y , xs≡ys = Listₚ.∷-injective eq in begin
+  let x≡y , xs≡ys = List.∷-injective eq in begin
   x ∷ cast (cong (pred ∘ List.length) eq) (fromList xs) ≡⟨ cong (_ ∷_) (cast-fromList xs≡ys) ⟩
   x ∷ fromList ys                                       ≡⟨ cong (_∷ _) x≡y ⟩
   y ∷ fromList ys                                       ∎
   where open ≡-Reasoning
 
 fromList-map : ∀ (f : A → B) (xs : List A) →
-               cast (Listₚ.length-map f xs) (fromList (List.map f xs)) ≡ map f (fromList xs)
+               cast (List.length-map f xs) (fromList (List.map f xs)) ≡ map f (fromList xs)
 fromList-map f List.[]       = refl
 fromList-map f (x List.∷ xs) = cong (f x ∷_) (fromList-map f xs)
 
 fromList-++ : ∀ (xs : List A) {ys : List A} →
-              cast (Listₚ.length-++ xs) (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
+              cast (List.length-++ xs) (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
 fromList-++ List.[]       {ys} = cast-is-id refl (fromList ys)
 fromList-++ (x List.∷ xs) {ys} = cong (x ∷_) (fromList-++ xs)
 
-fromList-reverse : (xs : List A) → cast (Listₚ.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)
+fromList-reverse : (xs : List A) → cast (List.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)
 fromList-reverse List.[] = refl
 fromList-reverse (x List.∷ xs) = begin
-  fromList (List.reverse (x List.∷ xs))         ≈⟨ cast-fromList (Listₚ.ʳ++-defn xs) ⟩
+  fromList (List.reverse (x List.∷ xs))         ≈⟨ cast-fromList (List.ʳ++-defn xs) ⟩
   fromList (List.reverse xs List.++ List.[ x ]) ≈⟨ fromList-++ (List.reverse xs) ⟩
   fromList (List.reverse xs) ++ [ x ]           ≈⟨ unfold-∷ʳ (+-comm 1 _) x (fromList (List.reverse xs)) ⟨
-  fromList (List.reverse xs) ∷ʳ x               ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (Listₚ.length-reverse xs)) _ _)
+  fromList (List.reverse xs) ∷ʳ x               ≈⟨ ≈-cong (_∷ʳ x) (cast-∷ʳ (cong suc (List.length-reverse xs)) _ _)
                                                           (fromList-reverse xs) ⟩
   reverse (fromList xs) ∷ʳ x                    ≂⟨ reverse-∷ x (fromList xs) ⟨
   reverse (x ∷ fromList xs)                     ≈⟨⟩
diff --git a/src/Data/Vec/Relation/Unary/All/Properties.agda b/src/Data/Vec/Relation/Unary/All/Properties.agda
index 57f06a7087..3c2a126230 100644
--- a/src/Data/Vec/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/All/Properties.agda
@@ -14,7 +14,6 @@ open import Data.List.Base using ([]; _∷_)
 open import Data.List.Relation.Unary.All as List using ([]; _∷_)
 open import Data.Product.Base as Product using (_×_; _,_; uncurry; uncurry′)
 open import Data.Vec.Base as Vec
-import Data.Vec.Properties as Vecₚ
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 open import Level using (Level)
 open import Function.Base using (_∘_; id)
diff --git a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
index ffc0ec6c4c..b13962b750 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
@@ -9,9 +9,9 @@
 module Data.Vec.Relation.Unary.AllPairs.Properties where
 
 open import Data.Vec
-import Data.Vec.Properties as Vecₚ
+import Data.Vec.Properties as Vec
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
-import Data.Vec.Relation.Unary.All.Properties as Allₚ
+import Data.Vec.Relation.Unary.All.Properties as All
 open import Data.Vec.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
 open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base using (Fin)
@@ -39,7 +39,7 @@ module _ {R : Rel A ℓ} {f : B → A} where
   map⁺ : ∀ {n xs} → AllPairs (λ x y → R (f x) (f y)) {n} xs →
          AllPairs R {n} (map f xs)
   map⁺ []           = []
-  map⁺ (x∉xs ∷ xs!) = Allₚ.map⁺ x∉xs ∷ map⁺ xs!
+  map⁺ (x∉xs ∷ xs!) = All.map⁺ x∉xs ∷ map⁺ xs!
 
 ------------------------------------------------------------------------
 -- ++
@@ -49,7 +49,7 @@ module _ {R : Rel A ℓ} where
   ++⁺ : ∀ {n m xs ys} → AllPairs R {n} xs → AllPairs R {m} ys →
         All (λ x → All (R x) ys) xs → AllPairs R (xs ++ ys)
   ++⁺ []         Rys _              = Rys
-  ++⁺ (px ∷ Rxs) Rys (Rxys ∷ Rxsys) = Allₚ.++⁺ px Rxys ∷ ++⁺ Rxs Rys Rxsys
+  ++⁺ (px ∷ Rxs) Rys (Rxys ∷ Rxsys) = All.++⁺ px Rxys ∷ ++⁺ Rxs Rys Rxsys
 
 ------------------------------------------------------------------------
 -- concat
@@ -61,7 +61,7 @@ module _ {R : Rel A ℓ} where
             AllPairs R (concat xss)
   concat⁺ []           []              = []
   concat⁺ (pxs ∷ pxss) (Rxsxss ∷ Rxss) = ++⁺ pxs (concat⁺ pxss Rxss)
-    (All.map Allₚ.concat⁺ (Allₚ.All-swap Rxsxss))
+    (All.map All.concat⁺ (All.All-swap Rxsxss))
 
 ------------------------------------------------------------------------
 -- take and drop
@@ -70,7 +70,7 @@ module _ {R : Rel A ℓ} where
 
   take⁺ : ∀ {n} m {xs} → AllPairs R {m + n} xs → AllPairs R {m} (take m xs)
   take⁺ zero pxs = []
-  take⁺ (suc m) {x ∷ xs} (px ∷ pxs) = Allₚ.take⁺ m px ∷ take⁺ m pxs
+  take⁺ (suc m) {x ∷ xs} (px ∷ pxs) = All.take⁺ m px ∷ take⁺ m pxs
 
   drop⁺ : ∀ {n} m {xs} → AllPairs R {m + n} xs → AllPairs R {n} (drop m xs)
   drop⁺ zero pxs = pxs
@@ -85,5 +85,5 @@ module _ {R : Rel A ℓ} where
               AllPairs R (tabulate f)
   tabulate⁺ {zero}  fᵢ~fⱼ = []
   tabulate⁺ {suc n} fᵢ~fⱼ =
-    Allₚ.tabulate⁺ (λ j → fᵢ~fⱼ λ()) ∷
+    All.tabulate⁺ (λ j → fᵢ~fⱼ λ()) ∷
     tabulate⁺ (fᵢ~fⱼ ∘ (_∘ suc-injective))
diff --git a/src/Data/Vec/Relation/Unary/Linked/Properties.agda b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
index 8ff5a4d2e3..e49cda76f6 100644
--- a/src/Data/Vec/Relation/Unary/Linked/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Linked/Properties.agda
@@ -10,7 +10,7 @@ module Data.Vec.Relation.Unary.Linked.Properties where
 
 open import Data.Vec.Base as Vec
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
-import Data.Vec.Relation.Unary.All.Properties as Allₚ
+import Data.Vec.Relation.Unary.All.Properties as All
 open import Data.Vec.Relation.Unary.Linked as Linked
   using (Linked; []; [-]; _∷_)
 open import Data.Fin.Base using (zero; suc; _<_)
@@ -43,7 +43,7 @@ module _ (trans : Transitive R) where
   lookup⁺ : ∀ {i j} {xs : Vec _ n} →
            Linked R xs → i < j →
            R (lookup xs i) (lookup xs j)
-  lookup⁺ {i = zero}  {j = suc j} (rx ∷ rxs) i<j = Allₚ.lookup⁺ (Linked⇒All rx rxs) j
+  lookup⁺ {i = zero}  {j = suc j} (rx ∷ rxs) i<j = All.lookup⁺ (Linked⇒All rx rxs) j
   lookup⁺ {i = suc i} {j = suc j} (_  ∷ rxs) i<j = lookup⁺ rxs (s<s⁻¹ i<j)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
index cb276cae9a..692f2530bf 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
@@ -11,10 +11,10 @@ module Data.Vec.Relation.Unary.Unique.Setoid.Properties where
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Vec.Base as Vec
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
-import Data.Vec.Relation.Unary.All.Properties as Allₚ
+import Data.Vec.Relation.Unary.All.Properties as All
 open import Data.Vec.Relation.Unary.AllPairs as AllPairs using (AllPairs)
 open import Data.Vec.Relation.Unary.Unique.Setoid
-import Data.Vec.Relation.Unary.AllPairs.Properties as AllPairsₚ
+import Data.Vec.Relation.Unary.AllPairs.Properties as AllPairs
 open import Data.Nat.Base using (ℕ; _+_)
 open import Function.Base using (_∘_; id)
 open import Level using (Level)
@@ -39,7 +39,7 @@ module _ (S : Setoid a ℓ₁) (R : Setoid b ℓ₂) where
 
   map⁺ : ∀ {f} → (∀ {x y} → f x ≈₂ f y → x ≈₁ y) →
          ∀ {n xs} → Unique S {n} xs → Unique R {n} (map f xs)
-  map⁺ inj xs! = AllPairsₚ.map⁺ (AllPairs.map (contraposition inj) xs!)
+  map⁺ inj xs! = AllPairs.map⁺ (AllPairs.map (contraposition inj) xs!)
 
 ------------------------------------------------------------------------
 -- take & drop
@@ -47,10 +47,10 @@ module _ (S : Setoid a ℓ₁) (R : Setoid b ℓ₂) where
 module _ (S : Setoid a ℓ) where
 
   drop⁺ : ∀ {n} m {xs} → Unique S {m + n} xs → Unique S {n} (drop m xs)
-  drop⁺ = AllPairsₚ.drop⁺
+  drop⁺ = AllPairs.drop⁺
 
   take⁺ : ∀ {n} m {xs} → Unique S {m + n} xs → Unique S {m} (take m xs)
-  take⁺ = AllPairsₚ.take⁺
+  take⁺ = AllPairs.take⁺
 
 ------------------------------------------------------------------------
 -- tabulate
@@ -61,7 +61,7 @@ module _ (S : Setoid a ℓ) where
 
   tabulate⁺ : ∀ {n} {f : Fin n → A} → (∀ {i j} → f i ≈ f j → i ≡ j) →
               Unique S (tabulate f)
-  tabulate⁺ f-inj = AllPairsₚ.tabulate⁺ (_∘ f-inj)
+  tabulate⁺ f-inj = AllPairs.tabulate⁺ (_∘ f-inj)
 
 ------------------------------------------------------------------------
 -- lookup
@@ -72,6 +72,6 @@ module _ (S : Setoid a ℓ) where
 
   lookup-injective : ∀ {n xs} → Unique S {n} xs → ∀ i j → lookup xs i ≈ lookup xs j → i ≡ j
   lookup-injective (px ∷ pxs) zero zero eq = P.refl
-  lookup-injective (px ∷ pxs) zero (suc j) eq = contradiction eq (Allₚ.lookup⁺ px j)
-  lookup-injective (px ∷ pxs) (suc i) zero eq = contradiction (sym eq) (Allₚ.lookup⁺ px i)
+  lookup-injective (px ∷ pxs) zero (suc j) eq = contradiction eq (All.lookup⁺ px j)
+  lookup-injective (px ∷ pxs) (suc i) zero eq = contradiction (sym eq) (All.lookup⁺ px i)
   lookup-injective (px ∷ pxs) (suc i) (suc j) eq = P.cong suc (lookup-injective pxs i j eq)
diff --git a/src/System/Clock.agda b/src/System/Clock.agda
index d8b8745b17..7ee4200207 100644
--- a/src/System/Clock.agda
+++ b/src/System/Clock.agda
@@ -14,7 +14,7 @@ open import Data.Fin.Base using (Fin; toℕ)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_; _∸_; _^_; _<ᵇ_)
 import Data.Nat.Show as ℕ using (show)
 open import Data.Nat.DivMod using (_/_)
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.String.Base using (String; _++_; padLeft)
 
 open import IO.Base
@@ -104,5 +104,5 @@ show : Time →   -- Time in seconds and nanoseconds
 show (mkTime s ns) prec = secs ++ "s" ++ padLeft '0' decimals nsecs where
   decimals = toℕ prec
   secs     = ℕ.show s
-  prf      = ℕₚ.m^n≢0 10 (9 ∸ decimals)
+  prf      = ℕ.m^n≢0 10 (9 ∸ decimals)
   nsecs    = ℕ.show ((ns / (10 ^ (9 ∸ decimals))) {{prf}})
diff --git a/src/Text/Pretty/Core.agda b/src/Text/Pretty/Core.agda
index 96415feb7f..39910f4dd5 100644
--- a/src/Text/Pretty/Core.agda
+++ b/src/Text/Pretty/Core.agda
@@ -23,15 +23,15 @@ import Data.Product.Relation.Unary.All as Allᴾ
 open import Data.Tree.Binary as Tree using (Tree; leaf; node; #nodes; mapₙ)
 open import Data.Tree.Binary.Relation.Unary.All as Allᵀ using (leaf; node)
 open import Data.Unit using (⊤; tt)
-import Data.Tree.Binary.Relation.Unary.All.Properties as Allᵀₚ
-import Data.Tree.Binary.Properties as Treeₚ
+import Data.Tree.Binary.Relation.Unary.All.Properties as Allᵀ
+import Data.Tree.Binary.Properties as Tree
 
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; maybe′)
 open import Data.Maybe.Relation.Unary.All as Allᴹ using (nothing; just)
 
 open import Data.String.Base as String
   using (String; length; replicate; _++_; unlines)
-open import Data.String.Unsafe as Stringₚ
+open import Data.String.Unsafe as String
 open import Function.Base
 open import Relation.Nullary.Decidable using (Dec)
 open import Relation.Unary using (IUniversal; _⇒_; U)
@@ -162,10 +162,10 @@ private
 
     size-indents : ∀ ma t → #nodes (indents ma t) ≡ #nodes t
     size-indents nothing    t = refl
-    size-indents (just pad) t = Treeₚ.#nodes-mapₙ (pad ++_) t
+    size-indents (just pad) t = Tree.#nodes-mapₙ (pad ++_) t
 
     unfold-indents : ∀ ma t → indents ma t ≡ mapₙ (indent ma) t
-    unfold-indents nothing    t = sym (Treeₚ.map-id t)
+    unfold-indents nothing    t = sym (Tree.map-id t)
     unfold-indents (just pad) t = refl
 
     vContent : Content × String
@@ -250,7 +250,7 @@ private
       All≤-node? (≤-Content (m≤m⊔n _ _) ∣xs∣)
                  middle
                  (subst (Allᵀ.All _ U) (sym $ unfold-indents pad tl)
-                 $ Allᵀₚ.mapₙ⁺ (indent pad) (Allᵀ.mapₙ (indented _) ∣tl∣))
+                 $ Allᵀ.mapₙ⁺ (indent pad) (Allᵀ.mapₙ (indented _) ∣tl∣))
       where
 
       middle : length (lastx ++ hd) ≤ vMaxWidth
diff --git a/src/Text/Regex/String.agda b/src/Text/Regex/String.agda
index 911c130dc0..b526c77827 100644
--- a/src/Text/Regex/String.agda
+++ b/src/Text/Regex/String.agda
@@ -8,9 +8,9 @@
 
 module Text.Regex.String where
 
-import Data.Char.Properties as Charₚ
+import Data.Char.Properties as Char
 
 ------------------------------------------------------------------------
 -- Re-exporting definitions
 
-open import Text.Regex Charₚ.≤-decPoset public
+open import Text.Regex Char.≤-decPoset public
diff --git a/src/Text/Tabular/List.agda b/src/Text/Tabular/List.agda
index 1f1603424b..ca8cce5b70 100644
--- a/src/Text/Tabular/List.agda
+++ b/src/Text/Tabular/List.agda
@@ -10,7 +10,7 @@ module Text.Tabular.List where
 
 open import Data.String.Base using (String)
 open import Data.List.Base
-import Data.Nat.Properties as ℕₚ
+import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (-,_; proj₂)
 open import Data.Vec.Base as Vec using (Vec)
 open import Data.Vec.Bounded.Base as Vec≤ using (Vec≤)
@@ -24,7 +24,7 @@ display c a rows = Show.display c alignment rectangle
   where
   alignment : Vec Alignment _
   alignment = Vec≤.padRight Left
-            $ Vec≤.≤-cast (ℕₚ.m⊓n≤m _ _)
+            $ Vec≤.≤-cast (ℕ.m⊓n≤m _ _)
             $ Vec≤.take _ (Vec≤.fromList a)
 
   rectangle : Vec (Vec String _) _

From f5cfcc7fec00abde2232f35e5597bd198c79aa9c Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 10 Feb 2024 20:08:33 +0000
Subject: [PATCH 224/316] missing code endquote (#2286)

---
 CHANGELOG.md | 1 +
 1 file changed, 1 insertion(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 62fed1f0f4..7fa21bf90e 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -56,6 +56,7 @@ Additions to existing modules
   rawNearSemiring   : RawNearSemiring _ _
   rawRingWithoutOne : RawRingWithoutOne _ _
   +-rawGroup        : RawGroup _ _
+  ```
 
 * In `Algebra.Module.Bundles`, raw bundles are re-exported and the bundles expose their raw counterparts.
 

From 8931b966b29261f55ec886812578807682b06b9a Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 14 Feb 2024 08:28:17 +0000
Subject: [PATCH 225/316] manual fix for #1380 (#2285)

---
 src/Data/Bool/Solver.agda               | 2 +-
 src/Data/Integer/Solver.agda            | 2 +-
 src/Data/Integer/Tactic/RingSolver.agda | 2 +-
 src/Data/Nat/Solver.agda                | 2 +-
 4 files changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/Data/Bool/Solver.agda b/src/Data/Bool/Solver.agda
index 8de769b42f..6a1cc83b03 100644
--- a/src/Data/Bool/Solver.agda
+++ b/src/Data/Bool/Solver.agda
@@ -4,7 +4,7 @@
 -- Automatic solvers for equations over booleans
 ------------------------------------------------------------------------
 
--- See README.Nat for examples of how to use similar solvers
+-- See README.Data.Nat for examples of how to use similar solvers
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
diff --git a/src/Data/Integer/Solver.agda b/src/Data/Integer/Solver.agda
index fcb3e411dc..46e9398677 100644
--- a/src/Data/Integer/Solver.agda
+++ b/src/Data/Integer/Solver.agda
@@ -4,7 +4,7 @@
 -- Automatic solvers for equations over integers
 ------------------------------------------------------------------------
 
--- See README.Integer for examples of how to use this solver
+-- See README.Data.Integer for examples of how to use this solver
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
diff --git a/src/Data/Integer/Tactic/RingSolver.agda b/src/Data/Integer/Tactic/RingSolver.agda
index ec812e3f0a..affe9ecd2e 100644
--- a/src/Data/Integer/Tactic/RingSolver.agda
+++ b/src/Data/Integer/Tactic/RingSolver.agda
@@ -4,7 +4,7 @@
 -- Automatic solvers for equations over integers
 ------------------------------------------------------------------------
 
--- See README.Integer for examples of how to use this solver
+-- See README.Tactic.RingSolver for examples of how to use this solver
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
diff --git a/src/Data/Nat/Solver.agda b/src/Data/Nat/Solver.agda
index 4c851c0e4b..efd14902b4 100644
--- a/src/Data/Nat/Solver.agda
+++ b/src/Data/Nat/Solver.agda
@@ -4,7 +4,7 @@
 -- Automatic solvers for equations over naturals
 ------------------------------------------------------------------------
 
--- See README.Nat for examples of how to use this solver
+-- See README.Data.Nat for examples of how to use this solver
 
 {-# OPTIONS --cubical-compatible --safe #-}
 

From 685c213d57e6e2f7b46389bc246f47cdd80f8311 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 15 Feb 2024 01:38:23 +0000
Subject: [PATCH 226/316] fixed `sized-types` error in orphan module (#2295)

---
 doc/README/Data/Tree/Binary.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/doc/README/Data/Tree/Binary.agda b/doc/README/Data/Tree/Binary.agda
index cf7e2de90a..0f03ad7b79 100644
--- a/doc/README/Data/Tree/Binary.agda
+++ b/doc/README/Data/Tree/Binary.agda
@@ -4,7 +4,7 @@
 -- Some examples showing how the Binary tree module can be used
 ------------------------------------------------------------------------
 
-{-# OPTIONS --cubical-compatible --safe --sized-types #-}
+{-# OPTIONS --cubical-compatible --sized-types #-}
 
 module README.Data.Tree.Binary where
 

From 68b17e02857537c8ecca60c1c1f95a54790a8325 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 25 Feb 2024 08:45:55 +0000
Subject: [PATCH 227/316] Qualified import of `PropositionalEquality` etc.
 fixing #2280 (#2293)

* Qualified import of `PropositionalEquality` fixing #2280

* Qualified import of `PropositionalEquality.Core` fixing #2280

* Qualified import of `HeterogeneousEquality.Core` fixing #2280

* simplified imports; fixed `README` link

* simplified imports
---
 src/Algebra/Construct/LiftedChoice.agda       |   6 +-
 .../Properties/BooleanAlgebra/Expression.agda |  16 +-
 src/Algebra/Operations/CommutativeMonoid.agda |   6 +-
 .../Properties/CommutativeMonoid/Sum.agda     |   6 +-
 src/Algebra/Properties/Monoid/Mult.agda       |   4 +-
 .../Properties/Monoid/Mult/TCOptimised.agda   |   4 +-
 src/Algebra/Properties/Monoid/Sum.agda        |   6 +-
 src/Algebra/Properties/Semiring/Exp.agda      |   2 +-
 src/Algebra/Solver/Monoid.agda                |   4 +-
 src/Algebra/Solver/Ring.agda                  |   4 +-
 .../Ring/NaturalCoefficients/Default.agda     |   4 +-
 src/Codata/Guarded/Stream/Properties.agda     |  84 +++++-----
 src/Codata/Musical/Colist.agda                |  22 +--
 src/Codata/Musical/Colist/Infinite-merge.agda |  16 +-
 .../Colist/Relation/Unary/Any/Properties.agda |  12 +-
 src/Codata/Musical/Conat.agda                 |  10 +-
 src/Codata/Musical/Covec.agda                 |   6 +-
 src/Codata/Musical/Stream.agda                |  12 +-
 src/Codata/Sized/Colist/Bisimilarity.agda     |  14 +-
 src/Codata/Sized/Covec/Bisimilarity.agda      |   8 +-
 src/Codata/Sized/Covec/Properties.agda        |   6 +-
 src/Codata/Sized/Cowriter/Bisimilarity.agda   |  14 +-
 src/Codata/Sized/Delay/Bisimilarity.agda      |   8 +-
 src/Codata/Sized/Delay/Properties.agda        |  20 +--
 src/Codata/Sized/M/Bisimilarity.agda          |  10 +-
 src/Codata/Sized/M/Properties.agda            |  14 +-
 src/Codata/Sized/Stream/Bisimilarity.agda     |  12 +-
 src/Codata/Sized/Stream/Properties.agda       |  46 +++---
 src/Data/Container/Combinator/Properties.agda |  24 +--
 src/Data/Container/Indexed.agda               |  10 +-
 src/Data/Container/Indexed/Combinator.agda    |  12 +-
 src/Data/Container/Indexed/WithK.agda         |  14 +-
 src/Data/Container/Morphism/Properties.agda   |   6 +-
 .../Container/Relation/Binary/Pointwise.agda  |   3 +-
 .../Relation/Binary/Pointwise/Properties.agda |  10 +-
 .../Relation/Unary/Any/Properties.agda        |  90 +++++------
 src/Data/Digit.agda                           |   6 +-
 src/Data/Fin/Permutation.agda                 |  12 +-
 src/Data/Fin/Properties.agda                  |  12 +-
 src/Data/Fin/Substitution/Example.agda        |   8 +-
 src/Data/Graph/Acyclic.agda                   |  18 +--
 src/Data/List/Effectful.agda                  |  32 ++--
 .../Propositional/Properties/Core.agda        |  12 +-
 .../Binary/Equality/Propositional.agda        |  12 +-
 .../List/Relation/Binary/Equality/Setoid.agda |   4 +-
 src/Data/List/Relation/Binary/Pointwise.agda  |  26 ++--
 .../Relation/Binary/Pointwise/Properties.agda |   6 +-
 .../Prefix/Heterogeneous/Properties.agda      |   8 +-
 .../Binary/Sublist/Heterogeneous.agda         |   1 -
 .../Sublist/Heterogeneous/Properties.agda     |  46 +++---
 .../Binary/Sublist/Heterogeneous/Solver.agda  |   8 +-
 .../List/Relation/Binary/Sublist/Setoid.agda  |   2 +-
 .../Suffix/Heterogeneous/Properties.agda      |   8 +-
 .../List/Relation/Ternary/Interleaving.agda   |   6 +-
 src/Data/List/Relation/Unary/All.agda         |  10 +-
 .../List/Relation/Unary/Any/Properties.agda   |  68 ++++-----
 .../Unary/Enumerates/Setoid/Properties.agda   |   4 +-
 .../List/Relation/Unary/First/Properties.agda |  12 +-
 src/Data/Maybe/Relation/Binary/Connected.agda |   4 +-
 src/Data/Maybe/Relation/Binary/Pointwise.agda |   8 +-
 src/Data/Maybe/Relation/Unary/All.agda        |   4 +-
 src/Data/Maybe/Relation/Unary/Any.agda        |   2 +-
 src/Data/Nat/Coprimality.agda                 |   4 +-
 src/Data/Nat/GCD.agda                         |  28 ++--
 src/Data/Nat/LCM.agda                         |  12 +-
 .../Function/Dependent/Propositional.agda     | 144 +++++++++---------
 .../Dependent/Propositional/WithK.agda        |   1 -
 .../Product/Function/Dependent/Setoid.agda    |  22 +--
 .../Relation/Binary/Pointwise/Dependent.agda  |   2 +-
 .../Binary/Pointwise/Dependent/WithK.agda     |  14 +-
 .../Binary/Pointwise/NonDependent.agda        |  10 +-
 src/Data/Sum/Relation/Binary/LeftOrder.agda   |   2 +-
 src/Data/Sum/Relation/Binary/Pointwise.agda   |  12 +-
 .../Relation/Binary/Equality/Setoid.agda      |   6 +-
 src/Data/Vec/Properties/WithK.agda            |   8 +-
 src/Data/Vec/Recursive/Properties.agda        |  33 ++--
 src/Data/Vec/Relation/Binary/Lex/Core.agda    |   8 +-
 src/Data/Vec/Relation/Binary/Lex/Strict.agda  |  10 +-
 .../Binary/Pointwise/Extensional.agda         |   8 +-
 .../Relation/Binary/Pointwise/Inductive.agda  |  12 +-
 src/Data/Vec/Relation/Unary/All.agda          |   8 +-
 .../Vec/Relation/Unary/Any/Properties.agda    |  31 ++--
 .../Unary/Unique/Setoid/Properties.agda       |   6 +-
 src/Effect/Applicative.agda                   |   2 +-
 src/Effect/Applicative/Indexed.agda           |   4 +-
 src/Effect/Monad/Partiality.agda              |  42 ++---
 src/Effect/Monad/Partiality/All.agda          |  10 +-
 src/Function/Bijection.agda                   |   4 +-
 src/Function/Endomorphism/Propositional.agda  |  16 +-
 src/Function/Equality.agda                    |   8 +-
 src/Function/Equivalence.agda                 |   4 +-
 src/Function/Injection.agda                   |   6 +-
 src/Function/Inverse.agda                     |   4 +-
 src/Function/LeftInverse.agda                 |   4 +-
 src/Function/Related.agda                     |  30 ++--
 src/Function/Related/Propositional.agda       |  16 +-
 src/Function/Related/TypeIsomorphisms.agda    |  52 +++----
 src/Function/Surjection.agda                  |   4 +-
 .../Construct/Add/Infimum/NonStrict.agda      |  13 +-
 .../Binary/Construct/Add/Infimum/Strict.agda  |  15 +-
 .../Binary/Construct/Add/Point/Equality.agda  |   6 +-
 .../Construct/Add/Supremum/NonStrict.agda     |  14 +-
 .../Binary/Construct/Add/Supremum/Strict.agda |  15 +-
 .../Closure/Reflexive/Properties.agda         |   6 +-
 .../Binary/Construct/Closure/Transitive.agda  |   4 +-
 .../Binary/HeterogeneousEquality.agda         |  16 +-
 .../Binary/Indexed/Heterogeneous/Bundles.agda |   3 -
 .../Binary/Indexed/Heterogeneous/Core.agda    |   1 -
 .../Indexed/Heterogeneous/Definitions.agda    |   2 -
 .../Indexed/Heterogeneous/Structures.agda     |   4 +-
 .../Binary/Indexed/Homogeneous/Bundles.agda   |   7 +-
 .../Indexed/Homogeneous/Structures.agda       |   6 +-
 src/Relation/Binary/Properties/Setoid.agda    |   8 +-
 src/Relation/Binary/Structures.agda           |   4 +-
 src/Tactic/Cong.agda                          |   2 +-
 115 files changed, 792 insertions(+), 805 deletions(-)

diff --git a/src/Algebra/Construct/LiftedChoice.agda b/src/Algebra/Construct/LiftedChoice.agda
index 9b141f8f7e..2377650cf8 100644
--- a/src/Algebra/Construct/LiftedChoice.agda
+++ b/src/Algebra/Construct/LiftedChoice.agda
@@ -17,7 +17,7 @@ open import Relation.Nullary using (¬_; yes; no)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
 open import Data.Product.Base using (_×_; _,_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Unary using (Pred)
 
 import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
@@ -55,8 +55,8 @@ module _ {_≈_ : Rel B ℓ} {_∙_ : Op₂ B}
 
     sel-≡ : Selective _≡_ _◦_
     sel-≡ x y with M.sel (f x) (f y)
-    ... | inj₁ _ = inj₁ P.refl
-    ... | inj₂ _ = inj₂ P.refl
+    ... | inj₁ _ = inj₁ ≡.refl
+    ... | inj₂ _ = inj₂ ≡.refl
 
     distrib : ∀ x y → ((f x) ∙ (f y)) ≈ f (x ◦ y)
     distrib x y with M.sel (f x) (f y)
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
index 64ac6d7864..5583f22a5b 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
@@ -26,7 +26,7 @@ open import Data.Vec.Properties using (lookup-map)
 open import Data.Vec.Relation.Binary.Pointwise.Extensional as PW
   using (Pointwise; ext)
 open import Function.Base using (_∘_; _$_; flip)
-open import Relation.Binary.PropositionalEquality as P using (_≗_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
 import Relation.Binary.Reflection as Reflection
 
 -- Expressions made up of variables and the operations of a boolean
@@ -68,7 +68,7 @@ module Naturality
   (f : Applicative.Morphism A₁ A₂)
   where
 
-  open P.≡-Reasoning
+  open ≡.≡-Reasoning
   open Applicative.Morphism f
   open Semantics A₁ renaming (⟦_⟧ to ⟦_⟧₁)
   open Semantics A₂ renaming (⟦_⟧ to ⟦_⟧₂)
@@ -77,21 +77,21 @@ module Naturality
 
   natural : ∀ {n} (e : Expr n) → op ∘ ⟦ e ⟧₁ ≗ ⟦ e ⟧₂ ∘ Vec.map op
   natural (var x) ρ = begin
-    op (Vec.lookup ρ x)                                            ≡⟨ P.sym $ lookup-map x op ρ ⟩
+    op (Vec.lookup ρ x)                                            ≡⟨ ≡.sym $ lookup-map x op ρ ⟩
     Vec.lookup (Vec.map op ρ) x                                    ∎
   natural (e₁ or e₂) ρ = begin
     op (_∨_ <$>₁ ⟦ e₁ ⟧₁ ρ ⊛₁ ⟦ e₂ ⟧₁ ρ)                       ≡⟨ op-⊛ _ _ ⟩
-    op (_∨_ <$>₁ ⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ P.cong₂ _⊛₂_ (op-<$> _ _) P.refl ⟩
-    _∨_ <$>₂ op (⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ P.cong₂ (λ e₁ e₂ → _∨_ <$>₂ e₁ ⊛₂ e₂) (natural e₁ ρ) (natural e₂ ρ) ⟩
+    op (_∨_ <$>₁ ⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ ≡.cong₂ _⊛₂_ (op-<$> _ _) ≡.refl ⟩
+    _∨_ <$>₂ op (⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ ≡.cong₂ (λ e₁ e₂ → _∨_ <$>₂ e₁ ⊛₂ e₂) (natural e₁ ρ) (natural e₂ ρ) ⟩
     _∨_ <$>₂ ⟦ e₁ ⟧₂ (Vec.map op ρ) ⊛₂ ⟦ e₂ ⟧₂ (Vec.map op ρ)  ∎
   natural (e₁ and e₂) ρ = begin
     op (_∧_ <$>₁ ⟦ e₁ ⟧₁ ρ ⊛₁ ⟦ e₂ ⟧₁ ρ)                       ≡⟨ op-⊛ _ _ ⟩
-    op (_∧_ <$>₁ ⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ P.cong₂ _⊛₂_ (op-<$> _ _) P.refl ⟩
-    _∧_ <$>₂ op (⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ P.cong₂ (λ e₁ e₂ → _∧_ <$>₂ e₁ ⊛₂ e₂) (natural e₁ ρ) (natural e₂ ρ) ⟩
+    op (_∧_ <$>₁ ⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ ≡.cong₂ _⊛₂_ (op-<$> _ _) ≡.refl ⟩
+    _∧_ <$>₂ op (⟦ e₁ ⟧₁ ρ) ⊛₂ op (⟦ e₂ ⟧₁ ρ)                  ≡⟨ ≡.cong₂ (λ e₁ e₂ → _∧_ <$>₂ e₁ ⊛₂ e₂) (natural e₁ ρ) (natural e₂ ρ) ⟩
     _∧_ <$>₂ ⟦ e₁ ⟧₂ (Vec.map op ρ) ⊛₂ ⟦ e₂ ⟧₂ (Vec.map op ρ)  ∎
   natural (not e) ρ = begin
     op (¬_ <$>₁ ⟦ e ⟧₁ ρ)                                      ≡⟨ op-<$> _ _ ⟩
-    ¬_ <$>₂ op (⟦ e ⟧₁ ρ)                                      ≡⟨ P.cong (¬_ <$>₂_) (natural e ρ) ⟩
+    ¬_ <$>₂ op (⟦ e ⟧₁ ρ)                                      ≡⟨ ≡.cong (¬_ <$>₂_) (natural e ρ) ⟩
     ¬_ <$>₂ ⟦ e ⟧₂ (Vec.map op ρ)                              ∎
   natural top ρ = begin
     op (pure₁ ⊤)                                                   ≡⟨ op-pure _ ⟩
diff --git a/src/Algebra/Operations/CommutativeMonoid.agda b/src/Algebra/Operations/CommutativeMonoid.agda
index 52d72af156..626d7570fb 100644
--- a/src/Algebra/Operations/CommutativeMonoid.agda
+++ b/src/Algebra/Operations/CommutativeMonoid.agda
@@ -12,7 +12,7 @@ open import Data.Fin.Base using (Fin; zero)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 module Algebra.Operations.CommutativeMonoid
   {s₁ s₂} (CM : CommutativeMonoid s₁ s₂)
@@ -58,7 +58,7 @@ suc n ×′ x = x + n ×′ x
 ×-congʳ (suc n) x≈x′ = +-cong x≈x′ (×-congʳ n x≈x′)
 
 ×-cong : _×_ Preserves₂ _≡_ ⟶ _≈_ ⟶ _≈_
-×-cong {u} P.refl x≈x′ = ×-congʳ u x≈x′
+×-cong {u} ≡.refl x≈x′ = ×-congʳ u x≈x′
 
 -- _×_ is homomorphic with respect to _ℕ+_/_+_.
 
@@ -98,7 +98,7 @@ suc n ×′ x = x + n ×′ x
 -- _×′_ preserves equality.
 
 ×′-cong : _×′_ Preserves₂ _≡_ ⟶ _≈_ ⟶ _≈_
-×′-cong {n} {_} {x} {y} P.refl x≈y = begin
+×′-cong {n} {_} {x} {y} ≡.refl x≈y = begin
   n  ×′ x ≈⟨ sym (×≈×′ n x) ⟩
   n  ×  x ≈⟨ ×-congʳ n x≈y ⟩
   n  ×  y ≈⟨ ×≈×′ n y ⟩
diff --git a/src/Algebra/Properties/CommutativeMonoid/Sum.agda b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
index eed54de4ed..351363c0d6 100644
--- a/src/Algebra/Properties/CommutativeMonoid/Sum.agda
+++ b/src/Algebra/Properties/CommutativeMonoid/Sum.agda
@@ -14,7 +14,7 @@ open import Data.Fin.Permutation as Perm using (Permutation; _⟨$⟩ˡ_; _⟨$
 open import Data.Fin.Patterns using (0F)
 open import Data.Vec.Functional
 open import Function.Base using (_∘_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Nullary.Negation using (contradiction)
 
 module Algebra.Properties.CommutativeMonoid.Sum
@@ -90,9 +90,9 @@ sum-permute {zero}  {suc n} f π = contradiction π (Perm.refute λ())
 sum-permute {suc m} {zero}  f π = contradiction π (Perm.refute λ())
 sum-permute {suc m} {suc n} f π = begin
   sum f                                    ≡⟨⟩
-  f 0F  + sum f/0                          ≡⟨ P.cong (_+ sum f/0) (P.cong f (Perm.inverseʳ π)) ⟨
+  f 0F  + sum f/0                          ≡⟨ ≡.cong (_+ sum f/0) (≡.cong f (Perm.inverseʳ π)) ⟨
   πf π₀ + sum f/0                          ≈⟨ +-congˡ (sum-permute f/0 (Perm.remove π₀ π)) ⟩
-  πf π₀ + sum (rearrange (π/0 ⟨$⟩ʳ_) f/0)  ≡⟨ P.cong (πf π₀ +_) (sum-cong-≗ (P.cong f ∘ Perm.punchIn-permute′ π 0F)) ⟨
+  πf π₀ + sum (rearrange (π/0 ⟨$⟩ʳ_) f/0)  ≡⟨ ≡.cong (πf π₀ +_) (sum-cong-≗ (≡.cong f ∘ Perm.punchIn-permute′ π 0F)) ⟨
   πf π₀ + sum (removeAt πf π₀)             ≈⟨ sym (sum-remove πf) ⟩
   sum πf                                   ∎
   where
diff --git a/src/Algebra/Properties/Monoid/Mult.agda b/src/Algebra/Properties/Monoid/Mult.agda
index 2963523b77..ce837c8134 100644
--- a/src/Algebra/Properties/Monoid/Mult.agda
+++ b/src/Algebra/Properties/Monoid/Mult.agda
@@ -9,7 +9,7 @@
 open import Algebra.Bundles using (Monoid)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; NonZero)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 module Algebra.Properties.Monoid.Mult {a ℓ} (M : Monoid a ℓ) where
 
@@ -44,7 +44,7 @@ open import Algebra.Definitions.RawMonoid rawMonoid public
 ×-congʳ (suc n) x≈x′ = +-cong x≈x′ (×-congʳ n x≈x′)
 
 ×-cong : _×_ Preserves₂ _≡_ ⟶ _≈_ ⟶ _≈_
-×-cong {n} P.refl x≈x′ = ×-congʳ n x≈x′
+×-cong {n} ≡.refl x≈x′ = ×-congʳ n x≈x′
 
 ×-congˡ : ∀ {x} → (_× x) Preserves _≡_ ⟶ _≈_
 ×-congˡ m≡n = ×-cong m≡n refl
diff --git a/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda b/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
index c862c5e3f2..a147954cb0 100644
--- a/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
+++ b/src/Algebra/Properties/Monoid/Mult/TCOptimised.agda
@@ -10,7 +10,7 @@
 open import Algebra.Bundles using (Monoid)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 module Algebra.Properties.Monoid.Mult.TCOptimised
   {a ℓ} (M : Monoid a ℓ) where
@@ -75,7 +75,7 @@ open import Algebra.Definitions.RawMonoid rawMonoid public
 ×-congʳ (suc n@(suc _)) x≈y = +-cong (×-congʳ n x≈y) x≈y
 
 ×-cong : _×_ Preserves₂ _≡_ ⟶ _≈_ ⟶ _≈_
-×-cong {n} P.refl x≈y = ×-congʳ n x≈y
+×-cong {n} ≡.refl x≈y = ×-congʳ n x≈y
 
 ×-assocˡ : ∀ x m n → m × (n × x) ≈ (m ℕ.* n) × x
 ×-assocˡ x m n = begin
diff --git a/src/Algebra/Properties/Monoid/Sum.agda b/src/Algebra/Properties/Monoid/Sum.agda
index 6ba7d12fc8..fb383f3768 100644
--- a/src/Algebra/Properties/Monoid/Sum.agda
+++ b/src/Algebra/Properties/Monoid/Sum.agda
@@ -13,7 +13,7 @@ open import Data.Fin.Base using (zero; suc)
 open import Data.Unit using (tt)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Core using (_Preserves_⟶_)
-open import Relation.Binary.PropositionalEquality as P using (_≗_; _≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≗_; _≡_)
 
 module Algebra.Properties.Monoid.Sum {a ℓ} (M : Monoid a ℓ) where
 
@@ -61,8 +61,8 @@ sum-cong-≋ {zero}  xs≋ys = refl
 sum-cong-≋ {suc n} xs≋ys = +-cong (xs≋ys zero) (sum-cong-≋ (xs≋ys ∘ suc))
 
 sum-cong-≗ : ∀ {n} → sum {n} Preserves _≗_ ⟶ _≡_
-sum-cong-≗ {zero}  xs≗ys = P.refl
-sum-cong-≗ {suc n} xs≗ys = P.cong₂ _+_ (xs≗ys zero) (sum-cong-≗ (xs≗ys ∘ suc))
+sum-cong-≗ {zero}  xs≗ys = ≡.refl
+sum-cong-≗ {suc n} xs≗ys = ≡.cong₂ _+_ (xs≗ys zero) (sum-cong-≗ (xs≗ys ∘ suc))
 
 sum-replicate : ∀ n {x} → sum (replicate n x) ≈ n × x
 sum-replicate zero    = refl
diff --git a/src/Algebra/Properties/Semiring/Exp.agda b/src/Algebra/Properties/Semiring/Exp.agda
index ed8b094520..f4adeee842 100644
--- a/src/Algebra/Properties/Semiring/Exp.agda
+++ b/src/Algebra/Properties/Semiring/Exp.agda
@@ -9,7 +9,7 @@
 open import Algebra
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Relation.Binary.Core using (_Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 import Data.Nat.Properties as ℕ
 
 module Algebra.Properties.Semiring.Exp
diff --git a/src/Algebra/Solver/Monoid.agda b/src/Algebra/Solver/Monoid.agda
index f80a2847f6..ba6aabee06 100644
--- a/src/Algebra/Solver/Monoid.agda
+++ b/src/Algebra/Solver/Monoid.agda
@@ -22,7 +22,7 @@ open import Data.Vec.Base using (Vec; lookup)
 open import Function.Base using (_∘_; _$_)
 open import Relation.Binary.Definitions using (Decidable)
 
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 import Relation.Binary.Reflection
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
@@ -128,7 +128,7 @@ prove′ e₁ e₂ =
   lemma : normalise e₁ ≡ normalise e₂ → ∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ
   lemma eq ρ =
     R.prove ρ e₁ e₂ (begin
-      ⟦ normalise e₁ ⟧⇓ ρ  ≡⟨ P.cong (λ e → ⟦ e ⟧⇓ ρ) eq ⟩
+      ⟦ normalise e₁ ⟧⇓ ρ  ≡⟨ cong (λ e → ⟦ e ⟧⇓ ρ) eq ⟩
       ⟦ normalise e₂ ⟧⇓ ρ  ∎)
 
 -- This procedure can be combined with from-just.
diff --git a/src/Algebra/Solver/Ring.agda b/src/Algebra/Solver/Ring.agda
index 50c191903b..a2a982d5c2 100644
--- a/src/Algebra/Solver/Ring.agda
+++ b/src/Algebra/Solver/Ring.agda
@@ -41,7 +41,7 @@ open import Algebra.Properties.Semiring.Exp semiring
 
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Binary.Reasoning.Setoid setoid
-import Relation.Binary.PropositionalEquality.Core as PropEq
+import Relation.Binary.PropositionalEquality.Core as ≡
 import Relation.Binary.Reflection as Reflection
 
 open import Data.Nat.Base using (ℕ; suc; zero)
@@ -534,7 +534,7 @@ correct (con c)  ρ = correct-con c ρ
 correct (var i)  ρ = correct-var i ρ
 correct (p :^ k) ρ = begin
   ⟦ normalise p ^N k ⟧N ρ  ≈⟨ ^N-homo (normalise p) k ρ ⟩
-  ⟦ p ⟧↓ ρ ^ k             ≈⟨ correct p ρ ⟨ ^-cong ⟩ PropEq.refl {x = k} ⟩
+  ⟦ p ⟧↓ ρ ^ k             ≈⟨ correct p ρ ⟨ ^-cong ⟩ ≡.refl {x = k} ⟩
   ⟦ p ⟧ ρ ^ k              ∎
 correct (:- p) ρ = begin
   ⟦ -N normalise p ⟧N ρ  ≈⟨ -N‿-homo (normalise p) ρ ⟩
diff --git a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
index 9b9671c527..da1b56b93f 100644
--- a/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
+++ b/src/Algebra/Solver/Ring/NaturalCoefficients/Default.agda
@@ -20,13 +20,13 @@ import Algebra.Properties.Semiring.Mult as SemiringMultiplication
 open import Data.Maybe.Base using (Maybe; map)
 open import Data.Nat using (_≟_)
 open import Relation.Binary.Consequences using (dec⇒weaklyDec)
-import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Core as ≡
 
 open CommutativeSemiring R
 open SemiringMultiplication semiring
 
 private
   dec : ∀ m n → Maybe (m × 1# ≈ n × 1#)
-  dec m n = map (λ { P.refl → refl }) (dec⇒weaklyDec _≟_ m n)
+  dec m n = map (λ { ≡.refl → refl }) (dec⇒weaklyDec _≟_ m n)
 
 open import Algebra.Solver.Ring.NaturalCoefficients R dec public
diff --git a/src/Codata/Guarded/Stream/Properties.agda b/src/Codata/Guarded/Stream/Properties.agda
index 3742aab856..c5f8e6b0cc 100644
--- a/src/Codata/Guarded/Stream/Properties.agda
+++ b/src/Codata/Guarded/Stream/Properties.agda
@@ -20,7 +20,7 @@ open import Data.Product.Base as Prod using (_×_; _,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec; _∷_)
 open import Function.Base using (const; flip; id; _∘′_; _$′_; _⟨_⟩_; _∘₂′_)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 
@@ -39,7 +39,7 @@ cong-lookup : ∀ {as bs : Stream A} → as ≈ bs → ∀ n → lookup as n ≡
 cong-lookup = B.lookup⁺
 
 cong-take : ∀ n {as bs : Stream A} → as ≈ bs → take n as ≡ take n bs
-cong-take zero    as≈bs = P.refl
+cong-take zero    as≈bs = refl
 cong-take (suc n) as≈bs = cong₂ _∷_ (as≈bs .head) (cong-take n (as≈bs .tail))
 
 cong-drop : ∀ n {as bs : Stream A} → as ≈ bs → drop n as ≈ drop n bs
@@ -63,7 +63,7 @@ cong-concat ass≈bss = cong-++-concat [] ass≈bss
     cong-++-concat : ∀ (as : List A) {ass bss} → ass ≈ bss → ++-concat as ass ≈ ++-concat as bss
     cong-++-concat [] ass≈bss .head = cong List⁺.head (ass≈bss .head)
     cong-++-concat [] ass≈bss .tail rewrite ass≈bss .head = cong-++-concat _ (ass≈bss .tail)
-    cong-++-concat (a ∷ as) ass≈bss .head = P.refl
+    cong-++-concat (a ∷ as) ass≈bss .head = refl
     cong-++-concat (a ∷ as) ass≈bss .tail = cong-++-concat as ass≈bss
 
 cong-interleave : {as bs cs ds : Stream A} → as ≈ bs → cs ≈ ds →
@@ -79,11 +79,11 @@ cong-chunksOf n as≈bs .tail = cong-chunksOf n (cong-drop n as≈bs)
 -- Properties of repeat
 
 lookup-repeat : ∀ n (a : A) → lookup (repeat a) n ≡ a
-lookup-repeat zero    a = P.refl
+lookup-repeat zero    a = refl
 lookup-repeat (suc n) a = lookup-repeat n a
 
 splitAt-repeat : ∀ n (a : A) → splitAt n (repeat a) ≡ (Vec.replicate n a , repeat a)
-splitAt-repeat zero    a = P.refl
+splitAt-repeat zero    a = refl
 splitAt-repeat (suc n) a = cong (Prod.map₁ (a ∷_)) (splitAt-repeat n a)
 
 take-repeat : ∀ n (a : A) → take n (repeat a) ≡ Vec.replicate n a
@@ -93,28 +93,28 @@ drop-repeat : ∀ n (a : A) → drop n (repeat a) ≡ repeat a
 drop-repeat n a = cong proj₂ (splitAt-repeat n a)
 
 map-repeat : ∀ (f : A → B) a → map f (repeat a) ≈ repeat (f a)
-map-repeat f a .head = P.refl
+map-repeat f a .head = refl
 map-repeat f a .tail = map-repeat f a
 
 ap-repeat : ∀ (f : A → B) a → ap (repeat f) (repeat a) ≈ repeat (f a)
-ap-repeat f a .head = P.refl
+ap-repeat f a .head = refl
 ap-repeat f a .tail = ap-repeat f a
 
 ap-repeatˡ : ∀ (f : A → B) as → ap (repeat f) as ≈ map f as
-ap-repeatˡ f as .head = P.refl
+ap-repeatˡ f as .head = refl
 ap-repeatˡ f as .tail = ap-repeatˡ f (as .tail)
 
 ap-repeatʳ : ∀ (fs : Stream (A → B)) a → ap fs (repeat a) ≈ map (_$′ a) fs
-ap-repeatʳ fs a .head = P.refl
+ap-repeatʳ fs a .head = refl
 ap-repeatʳ fs a .tail = ap-repeatʳ (fs .tail) a
 
 interleave-repeat : (a : A) → interleave (repeat a) (repeat a) ≈ repeat a
-interleave-repeat a .head = P.refl
+interleave-repeat a .head = refl
 interleave-repeat a .tail = interleave-repeat a
 
 zipWith-repeat : ∀ (f : A → B → C) a b →
                  zipWith f (repeat a) (repeat b) ≈ repeat (f a b)
-zipWith-repeat f a b .head = P.refl
+zipWith-repeat f a b .head = refl
 zipWith-repeat f a b .tail = zipWith-repeat f a b
 
 chunksOf-repeat : ∀ n (a : A) → chunksOf n (repeat a) ≈ repeat (Vec.replicate n a)
@@ -125,7 +125,7 @@ chunksOf-repeat n a = begin go where
   go : chunksOf n (repeat a) ≈∞ repeat (Vec.replicate n a)
   go .head = take-repeat n a
   go .tail =
-    chunksOf n (drop n (repeat a)) ≡⟨ P.cong (chunksOf n) (drop-repeat n a) ⟩
+    chunksOf n (drop n (repeat a)) ≡⟨ cong (chunksOf n) (drop-repeat n a) ⟩
     chunksOf n (repeat a)          ↺⟨ go ⟩
     repeat (Vec.replicate n a)     ∎
 
@@ -133,34 +133,34 @@ chunksOf-repeat n a = begin go where
 -- Properties of map
 
 map-const : (a : A) (bs : Stream B) → map (const a) bs ≈ repeat a
-map-const a bs .head = P.refl
+map-const a bs .head = refl
 map-const a bs .tail = map-const a (bs .tail)
 
 map-id : (as : Stream A) → map id as ≈ as
-map-id as .head = P.refl
+map-id as .head = refl
 map-id as .tail = map-id (as .tail)
 
 map-∘ : ∀ (g : B → C) (f : A → B) as → map g (map f as) ≈ map (g ∘′ f) as
-map-∘ g f as .head = P.refl
+map-∘ g f as .head = refl
 map-∘ g f as .tail = map-∘ g f (as .tail)
 
 map-unfold : ∀ (g : B → C) (f : A → A × B) a →
              map g (unfold f a) ≈ unfold (Prod.map₂ g ∘′ f) a
-map-unfold g f a .head = P.refl
+map-unfold g f a .head = refl
 map-unfold g f a .tail = map-unfold g f (proj₁ (f a))
 
 map-drop : ∀ (f : A → B) n as → map f (drop n as) ≡ drop n (map f as)
-map-drop f zero    as = P.refl
+map-drop f zero    as = refl
 map-drop f (suc n) as = map-drop f n (as .tail)
 
 map-zipWith : ∀ (g : C → D) (f : A → B → C) as bs →
               map g (zipWith f as bs) ≈ zipWith (g ∘₂′ f) as bs
-map-zipWith g f as bs .head = P.refl
+map-zipWith g f as bs .head = refl
 map-zipWith g f as bs .tail = map-zipWith g f (as .tail) (bs .tail)
 
 map-interleave : ∀ (f : A → B) as bs →
                  map f (interleave as bs) ≈ interleave (map f as) (map f bs)
-map-interleave f as bs .head = P.refl
+map-interleave f as bs .head = refl
 map-interleave f as bs .tail = map-interleave f bs (as .tail)
 
 map-concat : ∀ (f : A → B) ass → map f (concat ass) ≈ concat (map (List⁺.map f) ass)
@@ -168,9 +168,9 @@ map-concat f ass = map-++-concat [] ass
   where
     open Concat
     map-++-concat : ∀ acc ass → map f (++-concat acc ass) ≈ ++-concat (List.map f acc) (map (List⁺.map f) ass)
-    map-++-concat [] ass .head = P.refl
+    map-++-concat [] ass .head = refl
     map-++-concat [] ass .tail = map-++-concat (ass .head .List⁺.tail) (ass .tail)
-    map-++-concat (a ∷ as) ass .head = P.refl
+    map-++-concat (a ∷ as) ass .head = refl
     map-++-concat (a ∷ as) ass .tail = map-++-concat as ass
 
 map-cycle : ∀ (f : A → B) as → map f (cycle as) ≈ cycle (List⁺.map f as)
@@ -186,29 +186,29 @@ map-cycle f as = run
 -- Properties of lookup
 
 lookup-drop : ∀ m (as : Stream A) n → lookup (drop m as) n ≡ lookup as (m + n)
-lookup-drop zero    as n = P.refl
+lookup-drop zero    as n = refl
 lookup-drop (suc m) as n = lookup-drop m (as .tail) n
 
 lookup-map : ∀ n (f : A → B) as → lookup (map f as) n ≡ f (lookup as n)
-lookup-map zero    f as = P.refl
+lookup-map zero    f as = refl
 lookup-map (suc n) f as = lookup-map n f (as . tail)
 
 lookup-iterate : ∀ n f (x : A) → lookup (iterate f x) n ≡ ℕ.iterate f x n
-lookup-iterate zero    f x = P.refl
+lookup-iterate zero    f x = refl
 lookup-iterate (suc n) f x = lookup-iterate n f (f x)
 
 lookup-zipWith : ∀ n (f : A → B → C) as bs →
                  lookup (zipWith f as bs) n ≡ f (lookup as n) (lookup bs n)
-lookup-zipWith zero f as bs = P.refl
+lookup-zipWith zero f as bs = refl
 lookup-zipWith (suc n) f as bs = lookup-zipWith n f (as .tail) (bs .tail)
 
 lookup-unfold : ∀ n (f : A → A × B) a →
                 lookup (unfold f a) n ≡ proj₂ (f (ℕ.iterate (proj₁ ∘′ f) a n))
-lookup-unfold zero    f a = P.refl
+lookup-unfold zero    f a = refl
 lookup-unfold (suc n) f a = lookup-unfold n f (proj₁ (f a))
 
 lookup-tabulate : ∀ n (f : ℕ → A) → lookup (tabulate f) n ≡ f n
-lookup-tabulate zero f = P.refl
+lookup-tabulate zero f = refl
 lookup-tabulate (suc n) f = lookup-tabulate n (f ∘′ suc)
 
 lookup-transpose : ∀ n (ass : List (Stream A)) →
@@ -237,33 +237,33 @@ lookup-tails zero    as = B.refl
 lookup-tails (suc n) as = lookup-tails n (as .tail)
 
 lookup-evens : ∀ n (as : Stream A) → lookup (evens as) n ≡ lookup as (n * 2)
-lookup-evens zero    as = P.refl
+lookup-evens zero    as = refl
 lookup-evens (suc n) as = lookup-evens n (as .tail .tail)
 
 lookup-odds : ∀ n (as : Stream A) → lookup (odds as) n ≡ lookup as (suc (n * 2))
-lookup-odds zero    as = P.refl
+lookup-odds zero    as = refl
 lookup-odds (suc n) as = lookup-odds n (as .tail .tail)
 
 lookup-interleave-even : ∀ n (as bs : Stream A) →
                          lookup (interleave as bs) (n * 2) ≡ lookup as n
-lookup-interleave-even zero    as bs = P.refl
+lookup-interleave-even zero    as bs = refl
 lookup-interleave-even (suc n) as bs = lookup-interleave-even n (as .tail) (bs .tail)
 
 lookup-interleave-odd : ∀ n (as bs : Stream A) →
                         lookup (interleave as bs) (suc (n * 2)) ≡ lookup bs n
-lookup-interleave-odd zero    as bs = P.refl
+lookup-interleave-odd zero    as bs = refl
 lookup-interleave-odd (suc n) as bs = lookup-interleave-odd n (as .tail) (bs .tail)
 
 ------------------------------------------------------------------------
 -- Properties of take
 
 take-iterate : ∀ n f (x : A) → take n (iterate f x) ≡ Vec.iterate f x n
-take-iterate zero    f x = P.refl
+take-iterate zero    f x = refl
 take-iterate (suc n) f x = cong (x ∷_) (take-iterate n f (f x))
 
 take-zipWith : ∀ n (f : A → B → C) as bs →
                take n (zipWith f as bs) ≡ Vec.zipWith f (take n as) (take n bs)
-take-zipWith zero    f as bs = P.refl
+take-zipWith zero    f as bs = refl
 take-zipWith (suc n) f as bs =
   cong (f (as .head) (bs .head) ∷_) (take-zipWith n f (as .tail) (bs . tail))
 
@@ -271,21 +271,21 @@ take-zipWith (suc n) f as bs =
 -- Properties of drop
 
 drop-drop : ∀ m n (as : Stream A) → drop n (drop m as) ≡ drop (m + n) as
-drop-drop zero    n as = P.refl
+drop-drop zero    n as = refl
 drop-drop (suc m) n as = drop-drop m n (as .tail)
 
 drop-zipWith : ∀ n (f : A → B → C) as bs →
                drop n (zipWith f as bs) ≡ zipWith f (drop n as) (drop n bs)
-drop-zipWith zero    f as bs = P.refl
+drop-zipWith zero    f as bs = refl
 drop-zipWith (suc n) f as bs = drop-zipWith n f (as .tail) (bs .tail)
 
 drop-ap : ∀ n (fs : Stream (A → B)) as →
           drop n (ap fs as) ≡ ap (drop n fs) (drop n as)
-drop-ap zero    fs as = P.refl
+drop-ap zero    fs as = refl
 drop-ap (suc n) fs as = drop-ap n (fs .tail) (as .tail)
 
 drop-iterate : ∀ n f (x : A) → drop n (iterate f x) ≡ iterate f (ℕ.iterate f x n)
-drop-iterate zero    f x = P.refl
+drop-iterate zero    f x = refl
 drop-iterate (suc n) f x = drop-iterate n f (f x)
 
 ------------------------------------------------------------------------
@@ -293,25 +293,25 @@ drop-iterate (suc n) f x = drop-iterate n f (f x)
 
 zipWith-defn : ∀ (f : A → B → C) as bs →
                zipWith f as bs ≈ (repeat f ⟨ ap ⟩ as ⟨ ap ⟩ bs)
-zipWith-defn f as bs .head = P.refl
+zipWith-defn f as bs .head = refl
 zipWith-defn f as bs .tail = zipWith-defn f (as .tail) (bs .tail)
 
 zipWith-const : (as : Stream A) (bs : Stream B) →
                 zipWith const as bs ≈ as
-zipWith-const as bs .head = P.refl
+zipWith-const as bs .head = refl
 zipWith-const as bs .tail = zipWith-const (as .tail) (bs .tail)
 
 zipWith-flip : ∀ (f : A → B → C) as bs →
                zipWith (flip f) as bs ≈ zipWith f bs as
-zipWith-flip f as bs .head = P.refl
+zipWith-flip f as bs .head = refl
 zipWith-flip f as bs .tail = zipWith-flip f (as .tail) (bs. tail)
 
 ------------------------------------------------------------------------
 -- Properties of interleave
 
 interleave-evens-odds : (as : Stream A) → interleave (evens as) (odds as) ≈ as
-interleave-evens-odds as .head       = P.refl
-interleave-evens-odds as .tail .head = P.refl
+interleave-evens-odds as .head       = refl
+interleave-evens-odds as .tail .head = refl
 interleave-evens-odds as .tail .tail = interleave-evens-odds (as .tail .tail)
 
 ------------------------------------------------------------------------
diff --git a/src/Codata/Musical/Colist.agda b/src/Codata/Musical/Colist.agda
index c65d33e8bf..74bd2a2aa4 100644
--- a/src/Codata/Musical/Colist.agda
+++ b/src/Codata/Musical/Colist.agda
@@ -33,7 +33,7 @@ open import Relation.Binary.Definitions using (Transitive; Antisymmetric)
 import Relation.Binary.Construct.FromRel as Ind
 import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 import Relation.Binary.Reasoning.PartialOrder as ≤-Reasoning
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Binary.Reasoning.Syntax
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary
@@ -107,23 +107,23 @@ Any-∈ {P = P} = mk↔ₛ′
   from∘to
   where
   to : ∀ {xs} → Any P xs → ∃ λ x → x ∈ xs × P x
-  to (here  p) = _ , here P.refl , p
+  to (here  p) = _ , here refl , p
   to (there p) = Product.map id (Product.map there id) (to p)
 
   from : ∀ {x xs} → x ∈ xs → P x → Any P xs
-  from (here P.refl) p = here p
+  from (here refl) p = here p
   from (there x∈xs)  p = there (from x∈xs p)
 
   to∘from : ∀ {x xs} (x∈xs : x ∈ xs) (p : P x) →
             to (from x∈xs p) ≡ (x , x∈xs , p)
-  to∘from (here P.refl) p = P.refl
+  to∘from (here refl) p = refl
   to∘from (there x∈xs)  p =
-    P.cong (Product.map id (Product.map there id)) (to∘from x∈xs p)
+    cong (Product.map id (Product.map there id)) (to∘from x∈xs p)
 
   from∘to : ∀ {xs} (p : Any P xs) →
             let (x , x∈xs , px) = to p in from x∈xs px ≡ p
-  from∘to (here _)  = P.refl
-  from∘to (there p) = P.cong there (from∘to p)
+  from∘to (here _)  = refl
+  from∘to (there p) = cong there (from∘to p)
 
 -- Prefixes are subsets.
 
@@ -176,8 +176,8 @@ module ⊑-Reasoning {a} {A : Set a} where
 
 ⊆-Preorder : ∀ {ℓ} → Set ℓ → Preorder _ _ _
 ⊆-Preorder A = Ind.preorder (setoid A) _∈_
-                 (λ xs≈ys → ⊑⇒⊆ (⊑P.reflexive xs≈ys))
-  where module ⊑P = Poset (⊑-Poset A)
+                 (λ xs≈ys → ⊑⇒⊆ (⊑A.reflexive xs≈ys))
+  where module ⊑A = Poset (⊑-Poset A)
 
 -- Example uses:
 --
@@ -220,7 +220,7 @@ infixr 5 _∷_
 module Finite-injective where
 
  ∷-injective : ∀ {x : A} {xs p q} → (Finite (x ∷ xs) ∋ x ∷ p) ≡ x ∷ q → p ≡ q
- ∷-injective P.refl = P.refl
+ ∷-injective refl = refl
 
 -- Infinite xs means that xs has infinite length.
 
@@ -230,7 +230,7 @@ data Infinite {A : Set a} : Colist A → Set a where
 module Infinite-injective where
 
  ∷-injective : ∀ {x : A} {xs p q} → (Infinite (x ∷ xs) ∋ x ∷ p) ≡ x ∷ q → p ≡ q
- ∷-injective P.refl = P.refl
+ ∷-injective refl = refl
 
 -- Colists which are not finite are infinite.
 
diff --git a/src/Codata/Musical/Colist/Infinite-merge.agda b/src/Codata/Musical/Colist/Infinite-merge.agda
index ecf393ca83..c4343bd9e5 100644
--- a/src/Codata/Musical/Colist/Infinite-merge.agda
+++ b/src/Codata/Musical/Colist/Infinite-merge.agda
@@ -25,7 +25,7 @@ open import Function.Related.TypeIsomorphisms
 open import Level
 open import Relation.Unary using (Pred)
 import Induction.WellFounded as WF
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 import Relation.Binary.Construct.On as On
@@ -149,9 +149,9 @@ Any-merge {P = P} xss = mk↔ₛ′ (proj₁ ∘ to xss) from to∘from (proj₂
 
   from-injective : ∀ {xss} (p₁ p₂ : Any Q xss) →
                    from p₁ ≡ from p₂ → p₁ ≡ p₂
-  from-injective (here (inj₁ p))  (here (inj₁ .p)) P.refl = P.refl
+  from-injective (here (inj₁ p))  (here (inj₁ .p)) refl = refl
   from-injective (here (inj₂ p₁)) (here (inj₂ p₂)) eq     =
-    P.cong (here ∘ inj₂) $
+    cong (here ∘ inj₂) $
     inj₁-injective $
     Injection.injective (↔⇒↣ (↔-sym (Any-⋎P _))) $
     there-injective eq
@@ -166,7 +166,7 @@ Any-merge {P = P} xss = mk↔ₛ′ (proj₁ ∘ to xss) from to∘from (proj₂
                            (there-injective eq)
   ... | ()
   from-injective (there {x = _ , xs} p₁) (there p₂) eq =
-    P.cong there $
+    cong there $
     from-injective p₁ p₂ $
     inj₂-injective $
     Injection.injective (↔⇒↣ (↔-sym (Any-⋎P xs))) $
@@ -188,15 +188,15 @@ Any-merge {P = P} xss = mk↔ₛ′ (proj₁ ∘ to xss) from to∘from (proj₂
 
     step : ∀ p → WF.WfRec (_<′_ on size) InputPred p → InputPred p
     step ([]             , ())      rec
-    step ((x , xs) ∷ xss , here  p) rec = here (inj₁ p) , P.refl
+    step ((x , xs) ∷ xss , here  p) rec = here (inj₁ p) , refl
     step ((x , xs) ∷ xss , there p) rec
       with Inverse.to (Any-⋎P xs) p
          | Inverse.strictlyInverseʳ (Any-⋎P xs) p
          | index-Any-⋎P xs p
-    ... | inj₁ q | P.refl | _   = here (inj₂ q) , P.refl
-    ... | inj₂ q | P.refl | q≤p =
+    ... | inj₁ q | refl | _   = here (inj₂ q) , refl
+    ... | inj₂ q | refl | q≤p =
       Product.map there
-               (P.cong (there ∘ (Inverse.from (Any-⋎P xs)) ∘ inj₂))
+               (cong (there ∘ (Inverse.from (Any-⋎P xs)) ∘ inj₂))
                (rec (s≤′s q≤p))
 
   to∘from = λ p → from-injective _ _ (proj₂ (to xss (from p)))
diff --git a/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda b/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
index f7173c7e91..438922dc42 100644
--- a/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
+++ b/src/Codata/Musical/Colist/Relation/Unary/Any/Properties.agda
@@ -21,7 +21,7 @@ open import Function.Base using (_∋_; _∘_)
 open import Function.Bundles
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; cong)
 open import Relation.Unary using (Pred)
 
@@ -66,15 +66,15 @@ Any-cong {A = A} {P} {Q} {xs} {ys} P↔Q xs≈ys =
 
   to∘from : ∀ {xs ys} (xs≈ys : xs ≈ ys) (q : Any Q ys) →
             to xs≈ys (from xs≈ys q) ≡ q
-  to∘from (x ∷ xs≈) (there q) = P.cong there (to∘from (♭ xs≈) q)
+  to∘from (x ∷ xs≈) (there q) = cong there (to∘from (♭ xs≈) q)
   to∘from (x ∷ xs≈) (here qx) =
-    P.cong here (Inverse.strictlyInverseˡ P↔Q qx)
+    cong here (Inverse.strictlyInverseˡ P↔Q qx)
 
   from∘to : ∀ {xs ys} (xs≈ys : xs ≈ ys) (p : Any P xs) →
             from xs≈ys (to xs≈ys p) ≡ p
-  from∘to (x ∷ xs≈) (there p) = P.cong there (from∘to (♭ xs≈) p)
+  from∘to (x ∷ xs≈) (there p) = cong there (from∘to (♭ xs≈) p)
   from∘to (x ∷ xs≈) (here px) =
-    P.cong here (Inverse.strictlyInverseʳ P↔Q px)
+    cong here (Inverse.strictlyInverseʳ P↔Q px)
 
 ------------------------------------------------------------------------
 -- map
@@ -164,7 +164,7 @@ lookup-index (there p) = lookup-index p
 index-Any-resp : ∀ {f : ∀ {x} → P x → Q x} {xs ys}
                  (xs≈ys : xs ≈ ys) (p : Any P xs) →
                  index (Any-resp f xs≈ys p) ≡ index p
-index-Any-resp (x ∷ xs≈) (here px) = P.refl
+index-Any-resp (x ∷ xs≈) (here px) = refl
 index-Any-resp (x ∷ xs≈) (there p) =
   cong suc (index-Any-resp (♭ xs≈) p)
 
diff --git a/src/Codata/Musical/Conat.agda b/src/Codata/Musical/Conat.agda
index f4dab55a6f..760d733a20 100644
--- a/src/Codata/Musical/Conat.agda
+++ b/src/Codata/Musical/Conat.agda
@@ -14,7 +14,7 @@ open import Function.Base using (_∋_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 ------------------------------------------------------------------------
 -- Re-exporting the type and basic operations
@@ -27,11 +27,11 @@ open import Codata.Musical.Conat.Base public
 module Coℕ-injective where
 
  suc-injective : ∀ {m n} → (Coℕ ∋ suc m) ≡ suc n → m ≡ n
- suc-injective P.refl = P.refl
+ suc-injective ≡.refl = ≡.refl
 
 fromℕ-injective : ∀ {m n} → fromℕ m ≡ fromℕ n → m ≡ n
-fromℕ-injective {zero}  {zero}  eq = P.refl
-fromℕ-injective {suc m} {suc n} eq = P.cong suc (fromℕ-injective (P.cong pred eq))
+fromℕ-injective {zero}  {zero}  eq = ≡.refl
+fromℕ-injective {suc m} {suc n} eq = ≡.cong suc (fromℕ-injective (≡.cong pred eq))
 
 ------------------------------------------------------------------------
 -- Equality
@@ -45,7 +45,7 @@ data _≈_ : Coℕ → Coℕ → Set where
 module ≈-injective where
 
  suc-injective : ∀ {m n p q} → (suc m ≈ suc n ∋ suc p) ≡ suc q → p ≡ q
- suc-injective P.refl = P.refl
+ suc-injective ≡.refl = ≡.refl
 
 setoid : Setoid _ _
 setoid = record
diff --git a/src/Codata/Musical/Covec.agda b/src/Codata/Musical/Covec.agda
index 2658fd0311..64156344d9 100644
--- a/src/Codata/Musical/Covec.agda
+++ b/src/Codata/Musical/Covec.agda
@@ -21,7 +21,7 @@ open import Relation.Binary.Core using (_⇒_; _=[_]⇒_)
 open import Relation.Binary.Bundles using (Setoid; Poset)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive; Antisymmetric)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 private
   variable
@@ -38,10 +38,10 @@ data Covec (A : Set a) : Coℕ → Set a where
   _∷_ : ∀ {n} (x : A) (xs : ∞ (Covec A (♭ n))) → Covec A (suc n)
 
 ∷-injectiveˡ : ∀ {a b} {n} {as bs} → (Covec A (suc n) ∋ a ∷ as) ≡ b ∷ bs → a ≡ b
-∷-injectiveˡ P.refl = P.refl
+∷-injectiveˡ ≡.refl = ≡.refl
 
 ∷-injectiveʳ : ∀ {a b} {n} {as bs} → (Covec A (suc n) ∋ a ∷ as) ≡ b ∷ bs → as ≡ bs
-∷-injectiveʳ P.refl = P.refl
+∷-injectiveʳ ≡.refl = ≡.refl
 
 ------------------------------------------------------------------------
 -- Some operations
diff --git a/src/Codata/Musical/Stream.agda b/src/Codata/Musical/Stream.agda
index d0d655af65..12bf3b5fb7 100644
--- a/src/Codata/Musical/Stream.agda
+++ b/src/Codata/Musical/Stream.agda
@@ -16,7 +16,7 @@ open import Data.Nat.Base using (ℕ; zero; suc)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 private
   variable
@@ -144,13 +144,13 @@ setoid A = record
   }
   where
   refl : Reflexive _≈_
-  refl {_ ∷ _} = P.refl ∷ ♯ refl
+  refl {_ ∷ _} = ≡.refl ∷ ♯ refl
 
   sym : Symmetric _≈_
-  sym (x≡ ∷ xs≈) = P.sym x≡ ∷ ♯ sym (♭ xs≈)
+  sym (x≡ ∷ xs≈) = ≡.sym x≡ ∷ ♯ sym (♭ xs≈)
 
   trans : Transitive _≈_
-  trans (x≡ ∷ xs≈) (y≡ ∷ ys≈) = P.trans x≡ y≡ ∷ ♯ trans (♭ xs≈) (♭ ys≈)
+  trans (x≡ ∷ xs≈) (y≡ ∷ ys≈) = ≡.trans x≡ y≡ ∷ ♯ trans (♭ xs≈) (♭ ys≈)
 
 head-cong : {xs ys : Stream A} → xs ≈ ys → head xs ≡ head ys
 head-cong (x≡ ∷ _) = x≡
@@ -160,13 +160,13 @@ tail-cong (_ ∷ xs≈) = ♭ xs≈
 
 map-cong : ∀ (f : A → B) {xs ys} →
            xs ≈ ys → map f xs ≈ map f ys
-map-cong f (x≡ ∷ xs≈) = P.cong f x≡ ∷ ♯ map-cong f (♭ xs≈)
+map-cong f (x≡ ∷ xs≈) = ≡.cong f x≡ ∷ ♯ map-cong f (♭ xs≈)
 
 zipWith-cong : ∀ (_∙_ : A → B → C) {xs xs′ ys ys′} →
                xs ≈ xs′ → ys ≈ ys′ →
                zipWith _∙_ xs ys ≈ zipWith _∙_ xs′ ys′
 zipWith-cong _∙_ (x≡ ∷ xs≈) (y≡ ∷ ys≈) =
-  P.cong₂ _∙_ x≡ y≡ ∷ ♯ zipWith-cong _∙_ (♭ xs≈) (♭ ys≈)
+  ≡.cong₂ _∙_ x≡ y≡ ∷ ♯ zipWith-cong _∙_ (♭ xs≈) (♭ ys≈)
 
 infixr 5 _⋎-cong_
 
diff --git a/src/Codata/Sized/Colist/Bisimilarity.agda b/src/Codata/Sized/Colist/Bisimilarity.agda
index 4ea3a18047..94370a7fc2 100644
--- a/src/Codata/Sized/Colist/Bisimilarity.agda
+++ b/src/Codata/Sized/Colist/Bisimilarity.agda
@@ -20,8 +20,8 @@ open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive; Sym; Trans)
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as Eq
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -82,16 +82,16 @@ module _ {A : Set a} where
   _⊢_≈_ = Bisim _≡_
 
   refl : Reflexive (i ⊢_≈_)
-  refl = reflexive Eq.refl
+  refl = reflexive ≡.refl
 
   fromEq : ∀ {as bs} → as ≡ bs → i ⊢ as ≈ bs
-  fromEq Eq.refl = refl
+  fromEq ≡.refl = refl
 
   sym : Symmetric (i ⊢_≈_)
-  sym = symmetric Eq.sym
+  sym = symmetric ≡.sym
 
   trans : Transitive (i ⊢_≈_)
-  trans = transitive Eq.trans
+  trans = transitive ≡.trans
 
 isEquivalence : {R : Rel A r} → IsEquivalence R → IsEquivalence (Bisim R i)
 isEquivalence equiv^R = record
@@ -107,4 +107,4 @@ setoid S i = record
 
 module ≈-Reasoning {a} {A : Set a} {i} where
 
-  open import Relation.Binary.Reasoning.Setoid (setoid (Eq.setoid A) i) public
+  open import Relation.Binary.Reasoning.Setoid (setoid (≡.setoid A) i) public
diff --git a/src/Codata/Sized/Covec/Bisimilarity.agda b/src/Codata/Sized/Covec/Bisimilarity.agda
index 685b7f180f..f50e249bfa 100644
--- a/src/Codata/Sized/Covec/Bisimilarity.agda
+++ b/src/Codata/Sized/Covec/Bisimilarity.agda
@@ -15,7 +15,7 @@ open import Codata.Sized.Conat hiding (_⊔_)
 open import Codata.Sized.Covec
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive; Sym; Trans)
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) (i : Size) :
            ∀ m n (xs : Covec A ∞ m) (ys : Covec B ∞ n) → Set (r ⊔ a ⊔ b) where
@@ -56,10 +56,10 @@ module _ {ℓ} {A : Set ℓ} where
  _,_⊢_≈_ i m = Bisim _≡_ i m m
 
  refl : ∀ {i m} → Reflexive (i , m ⊢_≈_)
- refl = reflexive Eq.refl
+ refl = reflexive ≡.refl
 
  sym : ∀ {i m} → Symmetric (i , m ⊢_≈_)
- sym = symmetric Eq.sym
+ sym = symmetric ≡.sym
 
  trans : ∀ {i m} → Transitive (i , m ⊢_≈_)
- trans = transitive Eq.trans
+ trans = transitive ≡.trans
diff --git a/src/Codata/Sized/Covec/Properties.agda b/src/Codata/Sized/Covec/Properties.agda
index 08122efe0b..7d45bad63c 100644
--- a/src/Codata/Sized/Covec/Properties.agda
+++ b/src/Codata/Sized/Covec/Properties.agda
@@ -14,7 +14,7 @@ open import Codata.Sized.Conat
 open import Codata.Sized.Covec
 open import Codata.Sized.Covec.Bisimilarity
 open import Function.Base using (id; _∘_)
-open import Relation.Binary.PropositionalEquality.Core as Eq
+open import Relation.Binary.PropositionalEquality.Core as ≡
 
 -- Functor laws
 
@@ -22,13 +22,13 @@ module _ {a} {A : Set a} where
 
  map-id : ∀ {m} (as : Covec A ∞ m) {i} → i , m ⊢ map id as ≈ as
  map-id []       = []
- map-id (a ∷ as) = Eq.refl ∷ λ where .force → map-id (as .force)
+ map-id (a ∷ as) = ≡.refl ∷ λ where .force → map-id (as .force)
 
 module _ {a b c} {A : Set a} {B : Set b} {C : Set c} where
 
  map-∘ : ∀ (f : A → B) (g : B → C) {m} as {i} → i , m ⊢ map g (map f as) ≈ map (g ∘ f) as
  map-∘ f g []       = []
- map-∘ f g (a ∷ as) = Eq.refl ∷ λ where .force → map-∘ f g (as .force)
+ map-∘ f g (a ∷ as) = ≡.refl ∷ λ where .force → map-∘ f g (as .force)
 
 ------------------------------------------------------------------------
 -- DEPRECATED
diff --git a/src/Codata/Sized/Cowriter/Bisimilarity.agda b/src/Codata/Sized/Cowriter/Bisimilarity.agda
index d9ce3e2d6f..c38ef8180b 100644
--- a/src/Codata/Sized/Cowriter/Bisimilarity.agda
+++ b/src/Codata/Sized/Cowriter/Bisimilarity.agda
@@ -17,8 +17,8 @@ open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive; Sym; Trans)
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as Eq
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -74,16 +74,16 @@ module _ {W : Set w} {A : Set a} where
   _⊢_≈_ = Bisim _≡_ _≡_
 
   refl : Reflexive (i ⊢_≈_)
-  refl = reflexive Eq.refl Eq.refl
+  refl = reflexive ≡.refl ≡.refl
 
   fromEq : ∀ {as bs} → as ≡ bs → i ⊢ as ≈ bs
-  fromEq Eq.refl = refl
+  fromEq ≡.refl = refl
 
   sym : Symmetric (i ⊢_≈_)
-  sym = symmetric Eq.sym Eq.sym
+  sym = symmetric ≡.sym ≡.sym
 
   trans : Transitive (i ⊢_≈_)
-  trans = transitive Eq.trans Eq.trans
+  trans = transitive ≡.trans ≡.trans
 
 module _ {R : Rel W r} {S : Rel A s}
          (equiv^R : IsEquivalence R) (equiv^S : IsEquivalence S) where
@@ -108,4 +108,4 @@ setoid R S i = record
 module ≈-Reasoning {W : Set w} {A : Set a} {i} where
 
   open import Relation.Binary.Reasoning.Setoid
-              (setoid (Eq.setoid W) (Eq.setoid A) i) public
+              (setoid (≡.setoid W) (≡.setoid A) i) public
diff --git a/src/Codata/Sized/Delay/Bisimilarity.agda b/src/Codata/Sized/Delay/Bisimilarity.agda
index f4845669ba..fb01d94ebc 100644
--- a/src/Codata/Sized/Delay/Bisimilarity.agda
+++ b/src/Codata/Sized/Delay/Bisimilarity.agda
@@ -14,7 +14,7 @@ open import Codata.Sized.Delay
 open import Level
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive; Sym; Trans)
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 data Bisim {a b r} {A : Set a} {B : Set b} (R : A → B → Set r) i :
            (xs : Delay A ∞) (ys : Delay B ∞) → Set (a ⊔ b ⊔ r) where
@@ -53,10 +53,10 @@ module _ {ℓ} {A : Set ℓ} where
  _⊢_≈_ = Bisim _≡_
 
  refl : ∀ {i} → Reflexive (i ⊢_≈_)
- refl = reflexive Eq.refl
+ refl = reflexive ≡.refl
 
  sym : ∀ {i} → Symmetric (i ⊢_≈_)
- sym = symmetric Eq.sym
+ sym = symmetric ≡.sym
 
  trans : ∀ {i} → Transitive (i ⊢_≈_)
- trans = transitive Eq.trans
+ trans = transitive ≡.trans
diff --git a/src/Codata/Sized/Delay/Properties.agda b/src/Codata/Sized/Delay/Properties.agda
index 39a56d1ef9..de2b3c95b4 100644
--- a/src/Codata/Sized/Delay/Properties.agda
+++ b/src/Codata/Sized/Delay/Properties.agda
@@ -17,7 +17,7 @@ open import Codata.Sized.Conat.Bisimilarity as Coℕ using (zero ; suc)
 open import Codata.Sized.Delay
 open import Codata.Sized.Delay.Bisimilarity
 open import Function.Base using (id; _∘′_)
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 module _ {a} {A : Set a} where
 
@@ -40,19 +40,19 @@ module _ {a b c} {A : Set a} {B : Set b} {C : Set c} where
    suc λ where .force →  length-zipWith f (da .force) (db .force)
 
  map-id : ∀ da {i} → i ⊢ map (id {A = A}) da ≈ da
- map-id (now a)    = now Eq.refl
+ map-id (now a)    = now ≡.refl
  map-id (later da) = later λ where .force → map-id (da .force)
 
  map-∘ : ∀ (f : A → B) (g : B → C) da {i}  →
    i ⊢ map g (map f da) ≈ map (g ∘′ f) da
- map-∘ f g (now a)    = now Eq.refl
+ map-∘ f g (now a)    = now ≡.refl
  map-∘ f g (later da) = later λ where .force → map-∘ f g (da .force)
 
  map-unfold : ∀ (f : B → C) n (s : A) {i} →
    i ⊢ map f (unfold n s) ≈ unfold (Sum.map id f ∘′ n) s
  map-unfold f n s with n s
  ... | Sum.inj₁ s′ = later λ where .force → map-unfold f n s′
- ... | Sum.inj₂ b = now Eq.refl
+ ... | Sum.inj₂ b = now ≡.refl
 
 
 ------------------------------------------------------------------------
@@ -62,9 +62,9 @@ module _ {a b c} {A : Set a} {B : Set b} {C : Set c} where
            {d : Delay A ∞} →
            (d⇓₁ : d ⇓) → (d⇓₂ : d ⇓) →
            d⇓₁ ≡ d⇓₂
-⇓-unique {d = now s} (now s) (now s) = Eq.refl
+⇓-unique {d = now s} (now s) (now s) = ≡.refl
 ⇓-unique {d = later d'} (later l) (later r) =
-  Eq.cong later (⇓-unique {d = force d'} l r)
+  ≡.cong later (⇓-unique {d = force d'} l r)
 
 module _ {a} {A B : Set a} where
 
@@ -85,7 +85,7 @@ module _ {a} {A B : Set a} where
   extract-bind-⇓ : {d : Delay A Size.∞} → {f : A → Delay B Size.∞} →
                    (d⇓ : d ⇓) → (f⇓ : f (extract d⇓) ⇓) →
                    extract (bind-⇓ d⇓ {f} f⇓) ≡ extract f⇓
-  extract-bind-⇓ (now a) f⇓ = Eq.refl
+  extract-bind-⇓ (now a) f⇓ = ≡.refl
   extract-bind-⇓ (later t) f⇓ = extract-bind-⇓ t f⇓
 
   -- If the right element of a bind returns a certain value so does the
@@ -94,7 +94,7 @@ module _ {a} {A B : Set a} where
     (d : Delay A ∞) {f : A → Delay B ∞} →
     (bind⇓ : bind d f ⇓) →
     extract (bind̅₂ d bind⇓) ≡ extract bind⇓
-  extract-bind̅₂-bind⇓ (now s) bind⇓ = Eq.refl
+  extract-bind̅₂-bind⇓ (now s) bind⇓ = ≡.refl
   extract-bind̅₂-bind⇓ (later s) (later bind⇓) =
     extract-bind̅₂-bind⇓ (force s) bind⇓
 
@@ -106,9 +106,9 @@ module _ {a} {A B : Set a} where
       (d⇓ : d ⇓) → (f⇓ : f (extract d⇓) ⇓) →
       toℕ (length-⇓ bind⇓) ≡ toℕ (length-⇓ d⇓) ℕ.+ toℕ (length-⇓ f⇓)
   bind⇓-length {f = f} bind⇓ d⇓@(now s') f⇓ =
-    Eq.cong (toℕ ∘′ length-⇓) (⇓-unique bind⇓ f⇓)
+    ≡.cong (toℕ ∘′ length-⇓) (⇓-unique bind⇓ f⇓)
   bind⇓-length {d = d@(later dt)} {f = f} bind⇓@(later bind'⇓) d⇓@(later r) f⇓ =
-    Eq.cong ℕ.suc (bind⇓-length bind'⇓ r f⇓)
+    ≡.cong ℕ.suc (bind⇓-length bind'⇓ r f⇓)
 
 ------------------------------------------------------------------------
 -- DEPRECATED
diff --git a/src/Codata/Sized/M/Bisimilarity.agda b/src/Codata/Sized/M/Bisimilarity.agda
index 9c641ec834..cc0b934990 100644
--- a/src/Codata/Sized/M/Bisimilarity.agda
+++ b/src/Codata/Sized/M/Bisimilarity.agda
@@ -21,7 +21,7 @@ open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
-import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Core as ≡
 
 data Bisim {s p} (C : Container s p) (i : Size) : Rel (M C ∞) (s ⊔ p) where
   inf : ∀ {t u} → Pointwise C (Thunk^R (Bisim C) i) t u → Bisim C i (inf t) (inf u)
@@ -32,14 +32,14 @@ module _ {s p} {C : Container s p} where
   -- combinators C.refl, C.sym and C.trans
 
   refl : ∀ {i} → Reflexive (Bisim C i)
-  refl {x = inf t} = inf (P.refl , λ where p .force → refl)
+  refl {x = inf t} = inf (≡.refl , λ where p .force → refl)
 
   sym : ∀ {i} → Symmetric (Bisim C i)
-  sym  (inf (P.refl , f)) = inf (P.refl , λ where p .force → sym (f p .force))
+  sym  (inf (≡.refl , f)) = inf (≡.refl , λ where p .force → sym (f p .force))
 
   trans : ∀ {i} → Transitive (Bisim C i)
-  trans (inf (P.refl , f)) (inf (P.refl , g)) =
-    inf (P.refl , λ where p .force → trans (f p .force) (g p .force))
+  trans (inf (≡.refl , f)) (inf (≡.refl , g)) =
+    inf (≡.refl , λ where p .force → trans (f p .force) (g p .force))
 
   isEquivalence : ∀ {i} → IsEquivalence (Bisim C i)
   isEquivalence = record
diff --git a/src/Codata/Sized/M/Properties.agda b/src/Codata/Sized/M/Properties.agda
index a02410d43e..8d9724f9a7 100644
--- a/src/Codata/Sized/M/Properties.agda
+++ b/src/Codata/Sized/M/Properties.agda
@@ -18,19 +18,19 @@ import Data.Container.Morphism as Mp
 open import Data.Product.Base as Product using (_,_)
 open import Data.Product.Properties hiding (map-cong)
 open import Function.Base using (_$′_; _∘′_)
-import Relation.Binary.PropositionalEquality.Core as P
-import Relation.Binary.PropositionalEquality.Properties as P
+import Relation.Binary.PropositionalEquality.Core as ≡
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 open import Data.Container.Relation.Binary.Pointwise using (_,_)
 import Data.Container.Relation.Binary.Equality.Setoid as EqSetoid
 
-private module Eq {a} (A : Set a) = EqSetoid (P.setoid A)
+private module Eq {a} (A : Set a) = EqSetoid (≡.setoid A)
 open Eq using (Eq)
 
 module _ {s p} {C : Container s p} where
 
   map-id : ∀ {i} c → Bisim C i (map (Mp.id C) c) c
-  map-id (inf (s , f)) = inf (P.refl , λ where p .force → map-id (f p .force))
+  map-id (inf (s , f)) = inf (≡.refl , λ where p .force → map-id (f p .force))
 
 module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂} where
 
@@ -39,7 +39,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
              ∀ c₁ → Bisim C₂ i (map f c₁) (map g c₁)
   map-cong {f = f} {g} f≗g (inf t@(s , n)) with f≗g t
   ... | eqs , eqf = inf (eqs , λ where
-     p .force {j} → P.subst (λ t → Bisim C₂ j (map f (n (position f p) .force))
+     p .force {j} → ≡.subst (λ t → Bisim C₂ j (map f (n (position f p) .force))
                                               (map g (t .force)))
                     (eqf p)
                     (map-cong f≗g (n (position f p) .force)))
@@ -49,7 +49,7 @@ module _ {s₁ s₂ s₃ p₁ p₂ p₃} {C₁ : Container s₁ p₁}
 
   map-∘ : ∀ {i} {g : C₂ ⇒ C₃} {f : C₁ ⇒ C₂} c₁ →
                 Bisim C₃ i (map (g Mp.∘ f) c₁) (map g $′ map f c₁)
-  map-∘ (inf (s , f)) = inf (P.refl , λ where p .force → map-∘ (f _ .force))
+  map-∘ (inf (s , f)) = inf (≡.refl , λ where p .force → map-∘ (f _ .force))
 
 
 module _ {s₁ s₂ p₁ p₂ s} {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂}
@@ -57,7 +57,7 @@ module _ {s₁ s₂ p₁ p₂ s} {C₁ : Container s₁ p₁} {C₂ : Container
 
   map-unfold : ∀ {i} s → Bisim C₂ i (map f (unfold alg s))
                                     (unfold (⟪ f ⟫ ∘′ alg) s)
-  map-unfold s = inf (P.refl , λ where p .force → map-unfold _)
+  map-unfold s = inf (≡.refl , λ where p .force → map-unfold _)
 
 ------------------------------------------------------------------------
 -- DEPRECATED
diff --git a/src/Codata/Sized/Stream/Bisimilarity.agda b/src/Codata/Sized/Stream/Bisimilarity.agda
index f08d0daeb8..57516709df 100644
--- a/src/Codata/Sized/Stream/Bisimilarity.agda
+++ b/src/Codata/Sized/Stream/Bisimilarity.agda
@@ -19,8 +19,8 @@ open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive; Sym; Trans)
 open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as Eq
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -87,14 +87,14 @@ module _ {A : Set a} where
  _⊢_≈_ = Bisim _≡_
 
  refl : ∀ {i} → Reflexive (i ⊢_≈_)
- refl = reflexive Eq.refl
+ refl = reflexive ≡.refl
 
  sym : ∀ {i} → Symmetric (i ⊢_≈_)
- sym = symmetric Eq.sym
+ sym = symmetric ≡.sym
 
  trans : ∀ {i} → Transitive (i ⊢_≈_)
- trans = transitive Eq.trans
+ trans = transitive ≡.trans
 
 module ≈-Reasoning {a} {A : Set a} {i} where
 
-  open import Relation.Binary.Reasoning.Setoid (setoid (Eq.setoid A) i) public
+  open import Relation.Binary.Reasoning.Setoid (setoid (≡.setoid A) i) public
diff --git a/src/Codata/Sized/Stream/Properties.agda b/src/Codata/Sized/Stream/Properties.agda
index 0fd436f1bd..913b84edae 100644
--- a/src/Codata/Sized/Stream/Properties.agda
+++ b/src/Codata/Sized/Stream/Properties.agda
@@ -19,12 +19,12 @@ open import Data.Nat.GeneralisedArithmetic using (fold; fold-pull)
 
 open import Data.List.Base as List using ([]; _∷_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)
-import Data.List.Relation.Binary.Equality.Propositional as Eq
+import Data.List.Relation.Binary.Equality.Propositional as ≋
 open import Data.Product.Base as Product using (_,_)
 open import Data.Vec.Base as Vec using (_∷_)
 
 open import Function.Base using (id; _$_; _∘′_; const)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; _≢_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; _≢_)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 
@@ -40,63 +40,63 @@ private
 -- repeat
 
 lookup-repeat-identity : (n : ℕ) (a : A) → lookup (repeat a) n ≡ a
-lookup-repeat-identity zero    a = P.refl
+lookup-repeat-identity zero    a = ≡.refl
 lookup-repeat-identity (suc n) a = lookup-repeat-identity n a
 
 take-repeat-identity : (n : ℕ) (a : A) → take n (repeat a) ≡ Vec.replicate n a
-take-repeat-identity zero    a = P.refl
-take-repeat-identity (suc n) a = P.cong (a Vec.∷_) (take-repeat-identity n a)
+take-repeat-identity zero    a = ≡.refl
+take-repeat-identity (suc n) a = ≡.cong (a Vec.∷_) (take-repeat-identity n a)
 
 splitAt-repeat-identity : (n : ℕ) (a : A) → splitAt n (repeat a) ≡ (Vec.replicate n a , repeat a)
-splitAt-repeat-identity zero    a = P.refl
-splitAt-repeat-identity (suc n) a = P.cong (Product.map₁ (a ∷_)) (splitAt-repeat-identity n a)
+splitAt-repeat-identity zero    a = ≡.refl
+splitAt-repeat-identity (suc n) a = ≡.cong (Product.map₁ (a ∷_)) (splitAt-repeat-identity n a)
 
 replicate-repeat : ∀ {i} (n : ℕ) (a : A) → i ⊢ List.replicate n a ++ repeat a ≈ repeat a
 replicate-repeat zero    a = refl
-replicate-repeat (suc n) a = P.refl ∷ λ where .force → replicate-repeat n a
+replicate-repeat (suc n) a = ≡.refl ∷ λ where .force → replicate-repeat n a
 
 cycle-replicate : ∀ {i} (n : ℕ) (n≢0 : n ≢ 0) (a : A) → i ⊢ cycle (List⁺.replicate n n≢0 a) ≈ repeat a
 cycle-replicate {i} n n≢0 a = let as = List⁺.replicate n n≢0 a in begin
   cycle as                           ≡⟨⟩
-  as ⁺++ _                           ≈⟨ ⁺++⁺ Eq.≋-refl (λ where .force → cycle-replicate n n≢0 a) ⟩
-  as ⁺++ (λ where .force → repeat a) ≈⟨ P.refl ∷ (λ where .force → replicate-repeat (pred n) a) ⟩
+  as ⁺++ _                           ≈⟨ ⁺++⁺ ≋.≋-refl (λ where .force → cycle-replicate n n≢0 a) ⟩
+  as ⁺++ (λ where .force → repeat a) ≈⟨ ≡.refl ∷ (λ where .force → replicate-repeat (pred n) a) ⟩
   repeat a                           ∎ where open ≈-Reasoning
 
 module _ {a b} {A : Set a} {B : Set b} where
 
   map-repeat : ∀ (f : A → B) a {i} → i ⊢ map f (repeat a) ≈ repeat (f a)
-  map-repeat f a = P.refl ∷ λ where .force → map-repeat f a
+  map-repeat f a = ≡.refl ∷ λ where .force → map-repeat f a
 
   ap-repeat : ∀ (f : A → B) a {i} → i ⊢ ap (repeat f) (repeat a) ≈ repeat (f a)
-  ap-repeat f a = P.refl ∷ λ where .force → ap-repeat f a
+  ap-repeat f a = ≡.refl ∷ λ where .force → ap-repeat f a
 
   ap-repeatˡ : ∀ (f : A → B) as {i} → i ⊢ ap (repeat f) as ≈ map f as
-  ap-repeatˡ f (a ∷ as) = P.refl ∷ λ where .force → ap-repeatˡ f (as .force)
+  ap-repeatˡ f (a ∷ as) = ≡.refl ∷ λ where .force → ap-repeatˡ f (as .force)
 
   ap-repeatʳ : ∀ (fs : Stream (A → B) ∞) (a : A) {i} → i ⊢ ap fs (repeat a) ≈ map (_$ a) fs
-  ap-repeatʳ (f ∷ fs) a = P.refl ∷ λ where .force → ap-repeatʳ (fs .force) a
+  ap-repeatʳ (f ∷ fs) a = ≡.refl ∷ λ where .force → ap-repeatʳ (fs .force) a
 
   map-++ : ∀ {i} (f : A → B) as xs → i ⊢ map f (as ++ xs) ≈ List.map f as ++ map f xs
   map-++ f []       xs = refl
-  map-++ f (a ∷ as) xs = P.refl ∷ λ where .force → map-++ f as xs
+  map-++ f (a ∷ as) xs = ≡.refl ∷ λ where .force → map-++ f as xs
 
   map-⁺++ : ∀ {i} (f : A → B) as xs → i ⊢ map f (as ⁺++ xs) ≈ List⁺.map f as ⁺++ Thunk.map (map f) xs
-  map-⁺++ f (a ∷ as) xs = P.refl ∷ (λ where .force → map-++ f as (xs .force))
+  map-⁺++ f (a ∷ as) xs = ≡.refl ∷ (λ where .force → map-++ f as (xs .force))
 
   map-cycle : ∀ {i} (f : A → B) as → i ⊢ map f (cycle as) ≈ cycle (List⁺.map f as)
   map-cycle f as = begin
     map f (cycle as)       ≈⟨ map-⁺++ f as _ ⟩
-    List⁺.map f as ⁺++ _   ≈⟨ ⁺++⁺ Eq.≋-refl (λ where .force → map-cycle f as) ⟩
+    List⁺.map f as ⁺++ _   ≈⟨ ⁺++⁺ ≋.≋-refl (λ where .force → map-cycle f as) ⟩
     cycle (List⁺.map f as) ∎ where open ≈-Reasoning
 
 ------------------------------------------------------------------------
 -- Functor laws
 
 map-id : ∀ (as : Stream A ∞) → i ⊢ map id as ≈ as
-map-id (a ∷ as) = P.refl ∷ λ where .force → map-id (as .force)
+map-id (a ∷ as) = ≡.refl ∷ λ where .force → map-id (as .force)
 
 map-∘ : ∀ (f : A → B) (g : B → C) as → i ⊢ map g (map f as) ≈ map (g ∘′ f) as
-map-∘ f g (a ∷ as) = P.refl ∷ λ where .force → map-∘ f g (as .force)
+map-∘ f g (a ∷ as) = ≡.refl ∷ λ where .force → map-∘ f g (as .force)
 
 
 ------------------------------------------------------------------------
@@ -104,19 +104,19 @@ map-∘ f g (a ∷ as) = P.refl ∷ λ where .force → map-∘ f g (as .force)
 
 splitAt-map : ∀ n (f : A → B) xs →
   splitAt n (map f xs) ≡ Product.map (Vec.map f) (map f) (splitAt n xs)
-splitAt-map zero    f xs       = P.refl
+splitAt-map zero    f xs       = ≡.refl
 splitAt-map (suc n) f (x ∷ xs) =
-  P.cong (Product.map₁ (f x Vec.∷_)) (splitAt-map n f (xs .force))
+  ≡.cong (Product.map₁ (f x Vec.∷_)) (splitAt-map n f (xs .force))
 
 ------------------------------------------------------------------------
 -- iterate
 
 lookup-iterate-identity : ∀ n f (a : A) → lookup (iterate f a) n ≡ fold a f n
-lookup-iterate-identity zero     f a = P.refl
+lookup-iterate-identity zero     f a = ≡.refl
 lookup-iterate-identity (suc n)  f a = begin
   lookup (iterate f a) (suc n) ≡⟨⟩
   lookup (iterate f (f a)) n   ≡⟨ lookup-iterate-identity n f (f a) ⟩
-  fold (f a) f n               ≡⟨ fold-pull a f (const ∘′ f) (f a) P.refl (λ _ → P.refl) n ⟩
+  fold (f a) f n               ≡⟨ fold-pull a f (const ∘′ f) (f a) ≡.refl (λ _ → ≡.refl) n ⟩
   f (fold a f n)               ≡⟨⟩
   fold a f (suc n)             ∎ where open ≡-Reasoning
 
diff --git a/src/Data/Container/Combinator/Properties.agda b/src/Data/Container/Combinator/Properties.agda
index 6f7c83e993..f4b9568eb2 100644
--- a/src/Data/Container/Combinator/Properties.agda
+++ b/src/Data/Container/Combinator/Properties.agda
@@ -18,7 +18,7 @@ open import Data.Sum.Base as S using (inj₁; inj₂; [_,_]′; [_,_])
 open import Function.Base as F using (_∘′_)
 open import Function.Bundles
 open import Level using (_⊔_; lower)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
+open import Relation.Binary.PropositionalEquality using (_≡_; _≗_; refl; cong)
 
 -- I have proved some of the correctness statements under the
 -- assumption of functional extensionality. I could have reformulated
@@ -27,35 +27,35 @@ open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
 module Identity where
 
   correct : ∀ {s p x} {X : Set x} → ⟦ id {s} {p} ⟧ X ↔ F.id X
-  correct {X = X} = mk↔ₛ′ from-id to-id (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ from-id to-id (λ _ → refl) (λ _ → refl)
 
 module Constant (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′) where
 
   correct : ∀ {x p y} (X : Set x) {Y : Set y} → ⟦ const {x} {p ⊔ y} X ⟧ Y ↔ F.const X Y
-  correct {x} {y} X {Y} = mk↔ₛ′ (from-const X) (to-const X) (λ _ → P.refl) from∘to
+  correct {x} {y} X {Y} = mk↔ₛ′ (from-const X) (to-const X) (λ _ → refl) from∘to
     where
     from∘to : (x : ⟦ const X ⟧ Y) → to-const X (proj₁ x) ≡ x
-    from∘to xs = P.cong (proj₁ xs ,_) (ext (λ x → ⊥-elim (lower x)))
+    from∘to xs = cong (proj₁ xs ,_) (ext (λ x → ⊥-elim (lower x)))
 
 module Composition {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) where
 
   correct : ∀ {x} {X : Set x} → ⟦ C₁ ∘ C₂ ⟧ X ↔ (⟦ C₁ ⟧ F.∘ ⟦ C₂ ⟧) X
-  correct {X = X} = mk↔ₛ′ (from-∘ C₁ C₂) (to-∘ C₁ C₂) (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ (from-∘ C₁ C₂) (to-∘ C₁ C₂) (λ _ → refl) (λ _ → refl)
 
 module Product (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′)
        {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) where
 
   correct : ∀ {x} {X : Set x} →  ⟦ C₁ × C₂ ⟧ X ↔ (⟦ C₁ ⟧ X Prod.× ⟦ C₂ ⟧ X)
-  correct {X = X} = mk↔ₛ′ (from-× C₁ C₂) (to-× C₁ C₂) (λ _ → P.refl) from∘to
+  correct {X = X} = mk↔ₛ′ (from-× C₁ C₂) (to-× C₁ C₂) (λ _ → refl) from∘to
     where
     from∘to : (to-× C₁ C₂) F.∘ (from-× C₁ C₂) ≗ F.id
     from∘to (s , f) =
-      P.cong (s ,_) (ext [ (λ _ → P.refl) , (λ _ → P.refl) ])
+      cong (s ,_) (ext [ (λ _ → refl) , (λ _ → refl) ])
 
 module IndexedProduct {i s p} {I : Set i} (Cᵢ : I → Container s p) where
 
   correct : ∀ {x} {X : Set x} → ⟦ Π I Cᵢ ⟧ X ↔ (∀ i → ⟦ Cᵢ i ⟧ X)
-  correct {X = X} = mk↔ₛ′ (from-Π I Cᵢ) (to-Π I Cᵢ) (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ (from-Π I Cᵢ) (to-Π I Cᵢ) (λ _ → refl) (λ _ → refl)
 
 module Sum {s₁ s₂ p} (C₁ : Container s₁ p) (C₂ : Container s₂ p) where
 
@@ -63,16 +63,16 @@ module Sum {s₁ s₂ p} (C₁ : Container s₁ p) (C₂ : Container s₂ p) whe
   correct {X = X} = mk↔ₛ′ (from-⊎ C₁ C₂) (to-⊎ C₁ C₂) to∘from from∘to
     where
     from∘to : (to-⊎ C₁ C₂) F.∘ (from-⊎ C₁ C₂) ≗ F.id
-    from∘to (inj₁ s₁ , f) = P.refl
-    from∘to (inj₂ s₂ , f) = P.refl
+    from∘to (inj₁ s₁ , f) = refl
+    from∘to (inj₂ s₂ , f) = refl
 
     to∘from : (from-⊎ C₁ C₂) F.∘ (to-⊎ C₁ C₂) ≗ F.id
-    to∘from = [ (λ _ → P.refl) , (λ _ → P.refl) ]
+    to∘from = [ (λ _ → refl) , (λ _ → refl) ]
 
 module IndexedSum {i s p} {I : Set i} (C : I → Container s p) where
 
   correct : ∀ {x} {X : Set x} → ⟦ Σ I C ⟧ X ↔ (∃ λ i → ⟦ C i ⟧ X)
-  correct {X = X} = mk↔ₛ′ (from-Σ I C) (to-Σ I C) (λ _ → P.refl) (λ _ → P.refl)
+  correct {X = X} = mk↔ₛ′ (from-Σ I C) (to-Σ I C) (λ _ → refl) (λ _ → refl)
 
 module ConstantExponentiation {i s p} {I : Set i} (C : Container s p) where
 
diff --git a/src/Data/Container/Indexed.agda b/src/Data/Container/Indexed.agda
index 65b36f0527..b34657a4be 100644
--- a/src/Data/Container/Indexed.agda
+++ b/src/Data/Container/Indexed.agda
@@ -18,7 +18,7 @@ open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
 open import Function using (_↔_; Inverse)
 open import Relation.Unary using (Pred; _⊆_)
 open import Relation.Binary.Core using (Rel; REL)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_; refl)
+open import Relation.Binary.PropositionalEquality using (_≡_; _≗_; refl; trans; subst)
 
 ------------------------------------------------------------------------
 
@@ -98,7 +98,7 @@ module _ {i₁ i₂ o₁ o₂}
               Container I₁ O₁ c₁ r → (I₁ → I₂) → (O₁ → O₂) →
               Container I₂ O₂ c₂ r → Set _
   C₁ ⇒C[ f / g ] C₂ = ContainerMorphism C₁ C₂ f g _≡_ (λ R₂ R₁ → R₂ ≡ R₁)
-                                        (λ r₂≡r₁ r₂ → P.subst ⟨id⟩ r₂≡r₁ r₂)
+                                        (λ r₂≡r₁ r₂ → subst ⟨id⟩ r₂≡r₁ r₂)
 
 -- Degenerate cases where no reindexing is performed.
 
@@ -123,7 +123,7 @@ module _ {i o c r} {I : Set i} {O : Set o} where
 ⟪_⟫ : ∀ {i o c r ℓ} {I : Set i} {O : Set o} {C₁ C₂ : Container I O c r} →
       C₁ ⇒ C₂ → (X : Pred I ℓ) → ⟦ C₁ ⟧ X ⊆ ⟦ C₂ ⟧ X
 ⟪ m ⟫ X (c , k) = command m c , λ r₂ →
-  P.subst X (coherent m) (k (response m r₂))
+  subst X (coherent m) (k (response m r₂))
 
 module PlainMorphism {i o c r} {I : Set i} {O : Set o} where
 
@@ -145,7 +145,7 @@ module PlainMorphism {i o c r} {I : Set i} {O : Set o} where
   f ∘ g = record
     { command  = command  f ⟨∘⟩ command g
     ; response = response g ⟨∘⟩ response f
-    ; coherent = coherent g ⟨ P.trans ⟩ coherent f
+    ; coherent = coherent g ⟨ trans ⟩ coherent f
     }
 
   -- Identity commutes with ⟪_⟫.
@@ -187,7 +187,7 @@ module CartesianMorphism
   morphism : C₁ ⇒ C₂
   morphism = record
     { command  = command m
-    ; response = P.subst ⟨id⟩ (response m)
+    ; response = subst ⟨id⟩ (response m)
     ; coherent = coherent m
     }
 
diff --git a/src/Data/Container/Indexed/Combinator.agda b/src/Data/Container/Indexed/Combinator.agda
index b35a1ec887..825d173db7 100644
--- a/src/Data/Container/Indexed/Combinator.agda
+++ b/src/Data/Container/Indexed/Combinator.agda
@@ -21,8 +21,8 @@ open import Function.Indexed.Bundles using (_↔ᵢ_)
 open import Level
 open import Relation.Unary using (Pred; _⊆_; _∪_; _∩_; ⋃; ⋂)
   renaming (_⟨×⟩_ to _⟪×⟫_; _⟨⊙⟩_ to _⟪⊙⟫_; _⟨⊎⟩_ to _⟪⊎⟫_)
-open import Relation.Binary.PropositionalEquality as P
-  using (_≗_; refl)
+open import Relation.Binary.PropositionalEquality
+  using (_≗_; refl; cong)
 
 private
   variable
@@ -167,7 +167,7 @@ module Constant (ext : ∀ {ℓ} → Extensionality ℓ ℓ) where
     from = < F.id , F.const ⊥-elim >
 
     to∘from : _
-    to∘from xs = P.cong (proj₁ xs ,_) (ext ⊥-elim)
+    to∘from xs = cong (proj₁ xs ,_) (ext ⊥-elim)
 
 module Duality where
 
@@ -202,7 +202,7 @@ module Product (ext : ∀ {ℓ} → Extensionality ℓ ℓ) where
 
     from∘to : from ⟨∘⟩ to ≗ F.id
     from∘to (c , _) =
-      P.cong (c ,_) (ext [ (λ _ → refl) , (λ _ → refl) ])
+      cong (c ,_) (ext [ (λ _ → refl) , (λ _ → refl) ])
 
 module IndexedProduct where
 
@@ -231,8 +231,8 @@ module Sum (ext : ∀ {ℓ₁ ℓ₂} → Extensionality ℓ₁ ℓ₂) where
     from (inj₂ (c , f)) = inj₂ c , λ{ (All.inj₂ r) → f r}
 
     from∘to : from ⟨∘⟩ to ≗ F.id
-    from∘to (inj₁ _ , _) = P.cong (inj₁ _ ,_) (ext λ{ (All.inj₁ r) → refl})
-    from∘to (inj₂ _ , _) = P.cong (inj₂ _ ,_) (ext λ{ (All.inj₂ r) → refl})
+    from∘to (inj₁ _ , _) = cong (inj₁ _ ,_) (ext λ{ (All.inj₁ r) → refl})
+    from∘to (inj₂ _ , _) = cong (inj₂ _ ,_) (ext λ{ (All.inj₂ r) → refl})
 
     to∘from : to ⟨∘⟩ from ≗ F.id
     to∘from =  [ (λ _ → refl) , (λ _ → refl) ]
diff --git a/src/Data/Container/Indexed/WithK.agda b/src/Data/Container/Indexed/WithK.agda
index 9897402264..bda5a15e34 100644
--- a/src/Data/Container/Indexed/WithK.agda
+++ b/src/Data/Container/Indexed/WithK.agda
@@ -20,8 +20,8 @@ open import Data.Product.Base
 open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
 open import Level
 open import Relation.Unary using (Pred; _⊆_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
-open import Relation.Binary.HeterogeneousEquality as H using (_≅_; refl)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)
+open import Relation.Binary.HeterogeneousEquality as ≅ using (_≅_; refl)
 open import Relation.Binary.Indexed.Heterogeneous
 
 ------------------------------------------------------------------------
@@ -43,7 +43,7 @@ private
          {xs : ⟦ C ⟧ X o₁} {ys : ⟦ C ⟧ X o₂} → Extensionality r ℓ →
          Eq C X X (λ x₁ x₂ → x₁ ≅ x₂) xs ys → xs ≅ ys
   Eq⇒≅ {xs = c , k} {.c , k′} ext (refl , refl , k≈k′) =
-    H.cong (_,_ c) (ext (λ _ → refl) (λ r → k≈k′ r r refl))
+    ≅.cong (_,_ c) (ext (λ _ → refl) (λ r → k≈k′ r r refl))
 
 setoid : ∀ {i o c r s} {I : Set i} {O : Set o} →
          Container I O c r → IndexedSetoid I s _ → IndexedSetoid O _ _
@@ -122,7 +122,7 @@ module PlainMorphism {i o c r} {I : Set i} {O : Set o} where
     module Y = IndexedSetoid Y
 
     lemma : ∀ {i j} (eq : i ≡ j) {x} →
-            P.subst Y.Carrier eq (f x) Y.≈ f (P.subst X eq x)
+            ≡.subst Y.Carrier eq (f x) Y.≈ f (≡.subst X eq x)
     lemma refl = Y.refl
 
   -- In fact, all natural functions of the right type are container
@@ -135,7 +135,7 @@ module PlainMorphism {i o c r} {I : Set i} {O : Set o} where
                      Eq C₂ X.Carrier X.Carrier X._≈_
                        (proj₁ nt X.Carrier xs) (⟪ m ⟫ X.Carrier {o} xs)
   complete {C₁} {C₂} (nt , nat) = m , (λ X xs → nat X
-    (λ { (r , eq) → P.subst (IndexedSetoid.Carrier X) eq (proj₂ xs r) })
+    (λ { (r , eq) → ≡.subst (IndexedSetoid.Carrier X) eq (proj₂ xs r) })
     (proj₁ xs , (λ r → r , refl)))
     where
 
@@ -167,9 +167,9 @@ module PlainMorphism {i o c r} {I : Set i} {O : Set o} where
     module X = IndexedSetoid X
 
     lemma : ∀ {i j k} (eq₁ : i ≡ j) (eq₂ : j ≡ k) {x} →
-      P.subst X.Carrier (P.trans eq₁ eq₂) x
+      ≡.subst X.Carrier (≡.trans eq₁ eq₂) x
       X.≈
-      P.subst X.Carrier eq₂ (P.subst X.Carrier eq₁ x)
+      ≡.subst X.Carrier eq₂ (≡.subst X.Carrier eq₁ x)
     lemma refl refl = X.refl
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Container/Morphism/Properties.agda b/src/Data/Container/Morphism/Properties.agda
index c0d84007b4..39471b5e48 100644
--- a/src/Data/Container/Morphism/Properties.agda
+++ b/src/Data/Container/Morphism/Properties.agda
@@ -12,7 +12,7 @@ open import Level using (_⊔_; suc)
 open import Function.Base as F using (_$_)
 open import Data.Product.Base using (∃; proj₁; proj₂; _,_)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; _≗_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_)
 
 open import Data.Container.Core
 open import Data.Container.Morphism
@@ -23,7 +23,7 @@ open import Data.Container.Relation.Binary.Equality.Setoid
 module _ {s p} (C : Container s p) where
 
   id-correct : ∀ {x} {X : Set x} → ⟪ id C ⟫ {X = X} ≗ F.id
-  id-correct x = P.refl
+  id-correct x = ≡.refl
 
 -- Composition.
 
@@ -33,7 +33,7 @@ module _ {s₁ s₂ s₃ p₁ p₂ p₃}
 
   ∘-correct : (f : C₂ ⇒ C₃) (g : C₁ ⇒ C₂) → ∀ {x} {X : Set x} →
               ⟪ f ∘ g ⟫ {X = X} ≗ (⟪ f ⟫ F.∘ ⟪ g ⟫)
-  ∘-correct f g xs = P.refl
+  ∘-correct f g xs = ≡.refl
 
 module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s₂ p₂} where
 
diff --git a/src/Data/Container/Relation/Binary/Pointwise.agda b/src/Data/Container/Relation/Binary/Pointwise.agda
index 2bd20bc1c5..0e2df66f78 100644
--- a/src/Data/Container/Relation/Binary/Pointwise.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise.agda
@@ -12,8 +12,7 @@ open import Data.Product.Base using (_,_; Σ-syntax; -,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
 open import Level using (_⊔_)
 open import Relation.Binary.Core using (REL; _⇒_)
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; subst; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; subst)
 
 open import Data.Container.Core using (Container; ⟦_⟧)
 
diff --git a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
index 45b824cc56..978bfc55e4 100644
--- a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
@@ -17,19 +17,19 @@ open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; subst; cong)
 
 module _ {s p x r} {X : Set x} (C : Container s p) (R : Rel X r) where
 
   refl : Reflexive R → Reflexive (Pointwise C R)
-  refl R-refl = P.refl , λ p → R-refl
+  refl R-refl = ≡.refl , λ p → R-refl
 
   sym : Symmetric R → Symmetric (Pointwise C R)
-  sym R-sym (P.refl , f) = P.refl , λ p → R-sym (f p)
+  sym R-sym (≡.refl , f) = ≡.refl , λ p → R-sym (f p)
 
   trans : Transitive R → Transitive (Pointwise C R)
-  trans R-trans (P.refl , f) (P.refl , g) = P.refl , λ p → R-trans (f p) (g p)
+  trans R-trans (≡.refl , f) (≡.refl , g) = ≡.refl , λ p → R-trans (f p) (g p)
 
 private
 
@@ -38,4 +38,4 @@ private
 
   Eq⇒≡ : ∀ {s p x} {C : Container s p} {X : Set x} {xs ys : ⟦ C ⟧ X} →
          Extensionality p x → Pointwise C _≡_ xs ys → xs ≡ ys
-  Eq⇒≡ ext (P.refl , f≈f′) = cong -,_ (ext f≈f′)
+  Eq⇒≡ ext (≡.refl , f≈f′) = cong -,_ (ext f≈f′)
diff --git a/src/Data/Container/Relation/Unary/Any/Properties.agda b/src/Data/Container/Relation/Unary/Any/Properties.agda
index 0a4340fa7b..601d922a56 100644
--- a/src/Data/Container/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Container/Relation/Unary/Any/Properties.agda
@@ -23,7 +23,7 @@ open import Function.Related.Propositional as Related using (Related; SK-sym)
 open import Function.Related.TypeIsomorphisms
 open import Relation.Unary using (Pred ; _∪_ ; _∩_)
 open import Relation.Binary.Core using (REL)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality as ≡
   using (_≡_; _≗_; refl)
 
 private
@@ -42,21 +42,21 @@ module _ {s p} (C : Container s p) {x} {X : Set x} {ℓ} {P : Pred X ℓ} where
 -- ◇ can be unwrapped to reveal the Σ type
 
   ↔Σ : ∀ {xs : ⟦ C ⟧ X} → ◇ C P xs ↔ ∃ λ p → P (proj₂ xs p)
-  ↔Σ {xs} = mk↔ₛ′ ◇.proof any (λ _ → P.refl) (λ _ → P.refl)
+  ↔Σ {xs} = mk↔ₛ′ ◇.proof any (λ _ → refl) (λ _ → refl)
 
 -- ◇ can be expressed using _∈_.
 
   ↔∈ : ∀ {xs : ⟦ C ⟧ X} → ◇ C P xs ↔ (∃ λ x → x ∈ xs × P x)
-  ↔∈ {xs} = mk↔ₛ′ to from to∘from (λ _ → P.refl) where
+  ↔∈ {xs} = mk↔ₛ′ to from to∘from (λ _ → refl) where
 
     to : ◇ C P xs → ∃ λ x → x ∈ xs × P x
-    to (any (p , Px)) = (proj₂ xs p , (any (p , P.refl)) , Px)
+    to (any (p , Px)) = (proj₂ xs p , (any (p , refl)) , Px)
 
     from : (∃ λ x → x ∈ xs × P x) → ◇ C P xs
     from (.(proj₂ xs p) , (any (p , refl)) , Px) = any (p , Px)
 
     to∘from : to ∘ from ≗ id
-    to∘from (.(proj₂ xs p) , any (p , refl) , Px) = P.refl
+    to∘from (.(proj₂ xs p) , any (p , refl) , Px) = refl
 
 module _ {s p} {C : Container s p} {x} {X : Set x}
          {ℓ₁ ℓ₂} {P₁ : Pred X ℓ₁} {P₂ : Pred X ℓ₂} where
@@ -105,7 +105,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
   flatten : ∀ (xss : ⟦ C₁ ⟧ (⟦ C₂ ⟧ X)) →
             ◇ C₁ (◇ C₂ P) xss ↔
             ◇ (C₁ C.∘ C₂) P (Inverse.from (Composition.correct C₁ C₂) xss)
-  flatten xss = mk↔ₛ′ t f (λ _ → P.refl) (λ _ → P.refl) where
+  flatten xss = mk↔ₛ′ t f (λ _ → refl) (λ _ → refl) where
 
     ◇₁ = ◇ C₁; ◇₂ = ◇ C₂; ◇₁₂ = ◇ (C₁ C.∘ C₂)
     open Inverse
@@ -132,11 +132,11 @@ module _ {s p} {C : Container s p} {x} {X : Set x}
     from = [ Any.map₂ inj₁ , Any.map₂ inj₂ ]
 
     from∘to : from ∘ to ≗ id
-    from∘to (any (pos , inj₁ p)) = P.refl
-    from∘to (any (pos , inj₂ q)) = P.refl
+    from∘to (any (pos , inj₁ p)) = refl
+    from∘to (any (pos , inj₂ q)) = refl
 
     to∘from : to ∘ from ≗ id
-    to∘from = [ (λ _ → P.refl) , (λ _ → P.refl) ]
+    to∘from = [ (λ _ → refl) , (λ _ → refl) ]
 
 -- Products "commute" with ◇.
 
@@ -145,7 +145,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
 
   ×◇↔◇◇× : ∀ {xs : ⟦ C₁ ⟧ X} {ys : ⟦ C₂ ⟧ Y} →
            ◇ C₁ (λ x → ◇ C₂ (λ y → P x × Q y) ys) xs ↔ (◇ C₁ P xs × ◇ C₂ Q ys)
-  ×◇↔◇◇× {xs} {ys} = mk↔ₛ′ to from (λ _ → P.refl) (λ _ → P.refl)
+  ×◇↔◇◇× {xs} {ys} = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)
     where
     ◇₁ = ◇ C₁; ◇₂ = ◇ C₂
 
@@ -209,7 +209,7 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
   remove-linear {xs} m = mk↔ₛ′ t f t∘f f∘t
     where
     open _≃_
-    open P.≡-Reasoning
+    open ≡.≡-Reasoning
 
     position⊸m : ∀ {s} → Position C₂ (shape⊸ m s) ≃ Position C₁ s
     position⊸m = ↔⇒≃ (position⊸ m)
@@ -222,66 +222,66 @@ module _ {s₁ s₂ p₁ p₂} {C₁ : Container s₁ p₁} {C₂ : Container s
     f : ◇₁ P xs → ◇₂ P (⟪ m ⟫⊸ xs)
     f (any (x , p)) =
       any $ from position⊸m x
-          , P.subst (P ∘′ proj₂ xs) (P.sym (right-inverse-of position⊸m _)) p
+          , ≡.subst (P ∘′ proj₂ xs) (≡.sym (right-inverse-of position⊸m _)) p
 
     f∘t : f ∘ t ≗ id
-    f∘t (any (p₂ , p)) = P.cong any $ Σ-≡,≡→≡
+    f∘t (any (p₂ , p)) = ≡.cong any $ Σ-≡,≡→≡
       ( left-inverse-of position⊸m p₂
-      , (P.subst (P ∘ proj₂ xs ∘ to position⊸m)
+      , (≡.subst (P ∘ proj₂ xs ∘ to position⊸m)
            (left-inverse-of position⊸m p₂)
-           (P.subst (P ∘ proj₂ xs)
-              (P.sym (right-inverse-of position⊸m
+           (≡.subst (P ∘ proj₂ xs)
+              (≡.sym (right-inverse-of position⊸m
                         (to position⊸m p₂)))
-              p)                                                ≡⟨ P.subst-∘ (left-inverse-of position⊸m _) ⟩
+              p)                                                ≡⟨ ≡.subst-∘ (left-inverse-of position⊸m _) ⟩
 
-         P.subst (P ∘ proj₂ xs)
-           (P.cong (to position⊸m)
+         ≡.subst (P ∘ proj₂ xs)
+           (≡.cong (to position⊸m)
               (left-inverse-of position⊸m p₂))
-           (P.subst (P ∘ proj₂ xs)
-              (P.sym (right-inverse-of position⊸m
+           (≡.subst (P ∘ proj₂ xs)
+              (≡.sym (right-inverse-of position⊸m
                         (to position⊸m p₂)))
-              p)                                                ≡⟨ P.cong (λ eq → P.subst (P ∘ proj₂ xs) eq
-                                                                                    (P.subst (P ∘ proj₂ xs)
-                                                                                       (P.sym (right-inverse-of position⊸m _)) _))
+              p)                                                ≡⟨ ≡.cong (λ eq → ≡.subst (P ∘ proj₂ xs) eq
+                                                                                    (≡.subst (P ∘ proj₂ xs)
+                                                                                       (≡.sym (right-inverse-of position⊸m _)) _))
                                                                      (_≃_.left-right position⊸m _) ⟩
-         P.subst (P ∘ proj₂ xs)
+         ≡.subst (P ∘ proj₂ xs)
            (right-inverse-of position⊸m
               (to position⊸m p₂))
-           (P.subst (P ∘ proj₂ xs)
-              (P.sym (right-inverse-of position⊸m
+           (≡.subst (P ∘ proj₂ xs)
+              (≡.sym (right-inverse-of position⊸m
                         (to position⊸m p₂)))
-              p)                                                ≡⟨ P.subst-subst (P.sym (right-inverse-of position⊸m _)) ⟩
+              p)                                                ≡⟨ ≡.subst-subst (≡.sym (right-inverse-of position⊸m _)) ⟩
 
-         P.subst (P ∘ proj₂ xs)
-           (P.trans
-              (P.sym (right-inverse-of position⊸m
+         ≡.subst (P ∘ proj₂ xs)
+           (≡.trans
+              (≡.sym (right-inverse-of position⊸m
                         (to position⊸m p₂)))
               (right-inverse-of position⊸m
                  (to position⊸m p₂)))
-           p                                                    ≡⟨ P.cong (λ eq → P.subst (P ∘ proj₂ xs) eq p)
-                                                                     (P.trans-symˡ (right-inverse-of position⊸m _)) ⟩
+           p                                                    ≡⟨ ≡.cong (λ eq → ≡.subst (P ∘ proj₂ xs) eq p)
+                                                                     (≡.trans-symˡ (right-inverse-of position⊸m _)) ⟩
 
-         P.subst (P ∘ proj₂ xs) P.refl p                        ≡⟨⟩
+         ≡.subst (P ∘ proj₂ xs) ≡.refl p                        ≡⟨⟩
 
         p                                                       ∎)
       )
 
     t∘f : t ∘ f ≗ id
-    t∘f (any (p₁ , p)) = P.cong any $ Σ-≡,≡→≡
+    t∘f (any (p₁ , p)) = ≡.cong any $ Σ-≡,≡→≡
       ( right-inverse-of position⊸m p₁
-      , (P.subst (P ∘ proj₂ xs)
+      , (≡.subst (P ∘ proj₂ xs)
            (right-inverse-of position⊸m p₁)
-           (P.subst (P ∘ proj₂ xs)
-              (P.sym (right-inverse-of position⊸m p₁))
-              p)                                                ≡⟨ P.subst-subst (P.sym (right-inverse-of position⊸m _)) ⟩
+           (≡.subst (P ∘ proj₂ xs)
+              (≡.sym (right-inverse-of position⊸m p₁))
+              p)                                                ≡⟨ ≡.subst-subst (≡.sym (right-inverse-of position⊸m _)) ⟩
 
-         P.subst (P ∘ proj₂ xs)
-           (P.trans
-              (P.sym (right-inverse-of position⊸m p₁))
+         ≡.subst (P ∘ proj₂ xs)
+           (≡.trans
+              (≡.sym (right-inverse-of position⊸m p₁))
               (right-inverse-of position⊸m p₁))
-           p                                                    ≡⟨ P.cong (λ eq → P.subst (P ∘ proj₂ xs) eq p)
-                                                                     (P.trans-symˡ (right-inverse-of position⊸m _)) ⟩
-         P.subst (P ∘ proj₂ xs) P.refl p                        ≡⟨⟩
+           p                                                    ≡⟨ ≡.cong (λ eq → ≡.subst (P ∘ proj₂ xs) eq p)
+                                                                     (≡.trans-symˡ (right-inverse-of position⊸m _)) ⟩
+         ≡.subst (P ∘ proj₂ xs) ≡.refl p                        ≡⟨⟩
 
         p                                                       ∎)
       )
diff --git a/src/Data/Digit.agda b/src/Data/Digit.agda
index 207b0280b7..49370776da 100644
--- a/src/Data/Digit.agda
+++ b/src/Data/Digit.agda
@@ -21,7 +21,7 @@ open import Data.Nat.DivMod
 open import Data.Nat.Induction
 open import Relation.Nullary.Decidable using (True; does; toWitness)
 open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; sym; cong)
 open import Function.Base using (_$_)
 
 ------------------------------------------------------------------------
@@ -85,7 +85,7 @@ toDigits base@(suc (suc k)) n = <′-rec Pred helper n
   Pred = λ n → ∃ λ ds → fromDigits ds ≡ n
 
   cons : ∀ {m} (r : Digit base) → Pred m → Pred (toℕ r + m * base)
-  cons r (ds , eq) = (r ∷ ds , P.cong (λ i → toℕ r + i * base) eq)
+  cons r (ds , eq) = (r ∷ ds , cong (λ i → toℕ r + i * base) eq)
 
   open ≤-Reasoning
   open +-*-Solver
@@ -104,7 +104,7 @@ toDigits base@(suc (suc k)) n = <′-rec Pred helper n
 
   helper : ∀ n → <′-Rec Pred n → Pred n
   helper n                       rec with n divMod base
-  ... | result zero    r eq = ([ r ] , P.sym eq)
+  ... | result zero    r eq = ([ r ] , sym eq)
   ... | result (suc x) r refl = cons r (rec (lem x k (toℕ r)))
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 59a5c6abdf..97412c094d 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -28,9 +28,9 @@ open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary using (does; ¬_; yes; no)
 open import Relation.Nullary.Decidable using (dec-yes; dec-no)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary.PropositionalEquality as P
-  using (_≡_; _≢_; refl; trans; sym; →-to-⟶; cong; cong₂)
-open P.≡-Reasoning
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; _≢_; refl; sym; trans; subst; →-to-⟶; cong; cong₂; module ≡-Reasoning)
+open ≡-Reasoning
 
 private
   variable
@@ -67,10 +67,10 @@ _⟨$⟩ˡ_ : Permutation m n → Fin n → Fin m
 _⟨$⟩ˡ_ = Inverse.from
 
 inverseˡ : ∀ (π : Permutation m n) {i} → π ⟨$⟩ˡ (π ⟨$⟩ʳ i) ≡ i
-inverseˡ π = Inverse.inverseʳ π P.refl
+inverseˡ π = Inverse.inverseʳ π refl
 
 inverseʳ : ∀ (π : Permutation m n) {i} → π ⟨$⟩ʳ (π ⟨$⟩ˡ i) ≡ i
-inverseʳ π = Inverse.inverseˡ π P.refl
+inverseʳ π = Inverse.inverseˡ π refl
 
 ------------------------------------------------------------------------
 -- Equality
@@ -251,7 +251,7 @@ module _ (π : Permutation (suc m) (suc n)) where
 ↔⇒≡ {suc m} {suc n} π = cong suc (↔⇒≡ (remove 0F π))
 
 fromPermutation : Permutation m n → Permutation′ m
-fromPermutation π = P.subst (Permutation _) (sym (↔⇒≡ π)) π
+fromPermutation π = subst (Permutation _) (sym (↔⇒≡ π)) π
 
 refute : m ≢ n → ¬ Permutation m n
 refute m≢n π = contradiction (↔⇒≡ π) m≢n
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index d105d9e8d7..ff9239359c 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -41,7 +41,7 @@ open import Relation.Binary.Bundles
   using (Preorder; Setoid; DecSetoid; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
 open import Relation.Binary.Structures
   using (IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality as ≡
   using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_; module ≡-Reasoning)
 open import Relation.Nullary.Decidable as Dec
   using (Dec; _because_; yes; no; _×-dec_; _⊎-dec_; map′)
@@ -104,7 +104,7 @@ suc x ≟ suc y = map′ (cong suc) suc-injective (x ≟ y)
 
 ≡-isDecEquivalence : IsDecEquivalence {A = Fin n} _≡_
 ≡-isDecEquivalence = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; _≟_           = _≟_
   }
 
@@ -112,10 +112,10 @@ suc x ≟ suc y = map′ (cong suc) suc-injective (x ≟ y)
 -- Bundles
 
 ≡-preorder : ℕ → Preorder _ _ _
-≡-preorder n = P.preorder (Fin n)
+≡-preorder n = ≡.preorder (Fin n)
 
 ≡-setoid : ℕ → Setoid _ _
-≡-setoid n = P.setoid (Fin n)
+≡-setoid n = ≡.setoid (Fin n)
 
 ≡-decSetoid : ℕ → DecSetoid _ _
 ≡-decSetoid n = record
@@ -314,7 +314,7 @@ m <? n = suc (toℕ m) ℕ.≤? toℕ n
 
 ≤-isPreorder : IsPreorder {A = Fin n} _≡_ _≤_
 ≤-isPreorder = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; reflexive     = ≤-reflexive
   ; trans         = ≤-trans
   }
@@ -401,7 +401,7 @@ m <? n = suc (toℕ m) ℕ.≤? toℕ n
 
 <-isStrictPartialOrder : IsStrictPartialOrder {A = Fin n} _≡_ _<_
 <-isStrictPartialOrder = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; irrefl        = <-irrefl
   ; trans         = <-trans
   ; <-resp-≈      = <-resp₂-≡
diff --git a/src/Data/Fin/Substitution/Example.agda b/src/Data/Fin/Substitution/Example.agda
index 453c47d67f..7ac3061b11 100644
--- a/src/Data/Fin/Substitution/Example.agda
+++ b/src/Data/Fin/Substitution/Example.agda
@@ -12,7 +12,7 @@ module Data.Fin.Substitution.Example where
 
 {-# WARNING_ON_IMPORT
 "Data.Fin.Substitution.Example was deprecated in v2.0.
-Please see README.Data.Nat.Fin.Substitution.UntypedLambda instead."
+Please see README.Data.Fin.Substitution.UntypedLambda instead."
 #-}
 
 open import Data.Fin.Substitution
@@ -20,9 +20,9 @@ open import Data.Fin.Substitution.Lemmas
 open import Data.Nat.Base hiding (_/_)
 open import Data.Fin.Base using (Fin)
 open import Data.Vec.Base
-open import Relation.Binary.PropositionalEquality as PropEq
-  using (_≡_; refl; sym; cong; cong₂)
-open PropEq.≡-Reasoning
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; sym; cong; cong₂; module ≡-Reasoning)
+open ≡-Reasoning
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
   using (Star; ε; _◅_)
 
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index c842cf3d30..bf4c7267b7 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -27,7 +27,7 @@ open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Function.Base using (_$_; _∘′_; _∘_; id)
 open import Relation.Nullary
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- A lemma
@@ -186,7 +186,7 @@ private
 
   test-nodes : nodes example ≡ (# 0 , 0) ∷ (# 1 , 1) ∷ (# 2 , 2) ∷
                                (# 3 , 3) ∷ (# 4 , 4) ∷ []
-  test-nodes = P.refl
+  test-nodes = refl
 
 
 module _ {ℓ e} {N : Set ℓ} {E : Set e} where
@@ -213,7 +213,7 @@ private
 
   test-edges : edges example ≡ (# 1 , 10 , # 1) ∷ (# 1 , 11 , # 1) ∷
                                (# 2 , 12 , # 0) ∷ []
-  test-edges = P.refl
+  test-edges = refl
 
 -- The successors of a given node i (edge label × node number relative
 -- to i).
@@ -225,7 +225,7 @@ sucs g i = successors $ head (g [ i ])
 private
 
   test-sucs : sucs example (# 1) ≡ (10 , # 1) ∷ (11 , # 1) ∷ []
-  test-sucs = P.refl
+  test-sucs = refl
 
 -- The predecessors of a given node i (node number relative to i ×
 -- edge label).
@@ -238,13 +238,13 @@ preds (c & g) (suc i) =
             (List.map (Prod.map suc id) $ preds g i)
   where
   p : ∀ {e} {E : Set e} {n} (i : Fin n) → E × Fin n → Maybe (Fin′ (suc i) × E)
-  p i (e , j) = Maybe.map (λ{ P.refl → zero , e }) (decToMaybe (i ≟ j))
+  p i (e , j) = Maybe.map (λ{ refl → zero , e }) (decToMaybe (i ≟ j))
 
 private
 
   test-preds : preds example (# 3) ≡
                (# 1 , 10) ∷ (# 1 , 11) ∷ (# 2 , 12) ∷ []
-  test-preds = P.refl
+  test-preds = refl
 
 ------------------------------------------------------------------------
 -- Operations
@@ -272,7 +272,7 @@ private
                  context (# 3 , 3) [] &
                  context (# 4 , 4) [] &
                  ∅)
-  test-number = P.refl
+  test-number = refl
 
 -- Reverses all the edges in the graph.
 
@@ -290,7 +290,7 @@ reverse {N = N} {E} g =
 private
 
   test-reverse : reverse (reverse example) ≡ example
-  test-reverse = P.refl
+  test-reverse = refl
 
 ------------------------------------------------------------------------
 -- Views
@@ -330,4 +330,4 @@ private
                     node 3 [] ∷
                     node 4 [] ∷
                     []
-  test-toForest = P.refl
+  test-toForest = refl
diff --git a/src/Data/List/Effectful.agda b/src/Data/List/Effectful.agda
index b21ddf7616..b6259d7e61 100644
--- a/src/Data/List/Effectful.agda
+++ b/src/Data/List/Effectful.agda
@@ -18,9 +18,9 @@ open import Effect.Applicative
 open import Effect.Monad
 open import Function.Base
 open import Level
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality as ≡
   using (_≡_; _≢_; _≗_; refl)
-open P.≡-Reasoning
+open ≡.≡-Reasoning
 
 private
   variable
@@ -125,7 +125,7 @@ module MonadProperties where
   right-identity : ∀ {ℓ} {A : Set ℓ} (xs : List A) →
                    (xs >>= pure) ≡ xs
   right-identity []       = refl
-  right-identity (x ∷ xs) = P.cong (x ∷_) (right-identity xs)
+  right-identity (x ∷ xs) = ≡.cong (x ∷_) (right-identity xs)
 
   left-zero : ∀ {ℓ} {A B : Set ℓ} (f : A → List B) → (∅ >>= f) ≡ ∅
   left-zero f = refl
@@ -147,8 +147,8 @@ module MonadProperties where
                        (xs ∣ ys >>= f) ≡ ((xs >>= f) ∣ (ys >>= f))
   right-distributive []       ys f = refl
   right-distributive (x ∷ xs) ys f = begin
-    f x ∣ (xs ∣ ys >>= f)              ≡⟨ P.cong (f x ∣_) $ right-distributive xs ys f ⟩
-    f x ∣ ((xs >>= f) ∣ (ys >>= f))    ≡⟨ P.sym $ ++-assoc (f x) _ _ ⟩
+    f x ∣ (xs ∣ ys >>= f)              ≡⟨ ≡.cong (f x ∣_) $ right-distributive xs ys f ⟩
+    f x ∣ ((xs >>= f) ∣ (ys >>= f))    ≡⟨ ≡.sym $ ++-assoc (f x) _ _ ⟩
     ((f x ∣ (xs >>= f)) ∣ (ys >>= f))  ∎
 
   associative : ∀ {ℓ} {A B C : Set ℓ}
@@ -156,13 +156,13 @@ module MonadProperties where
                 (xs >>= λ x → f x >>= g) ≡ (xs >>= f >>= g)
   associative []       f g = refl
   associative (x ∷ xs) f g = begin
-    (f x >>= g) ∣ (xs >>= λ x → f x >>= g)  ≡⟨ P.cong ((f x >>= g) ∣_) $ associative xs f g ⟩
-    (f x >>= g) ∣ (xs >>= f >>= g)          ≡⟨ P.sym $ right-distributive (f x) (xs >>= f) g ⟩
+    (f x >>= g) ∣ (xs >>= λ x → f x >>= g)  ≡⟨ ≡.cong ((f x >>= g) ∣_) $ associative xs f g ⟩
+    (f x >>= g) ∣ (xs >>= f >>= g)          ≡⟨ ≡.sym $ right-distributive (f x) (xs >>= f) g ⟩
     (f x ∣ (xs >>= f) >>= g)                ∎
 
   cong : ∀ {ℓ} {A B : Set ℓ} {xs₁ xs₂} {f₁ f₂ : A → List B} →
          xs₁ ≡ xs₂ → f₁ ≗ f₂ → (xs₁ >>= f₁) ≡ (xs₂ >>= f₂)
-  cong {xs₁ = xs} refl f₁≗f₂ = P.cong concat (map-cong f₁≗f₂ xs)
+  cong {xs₁ = xs} refl f₁≗f₂ = ≡.cong concat (map-cong f₁≗f₂ xs)
 
 ------------------------------------------------------------------------
 -- The applicative functor derived from the list monad.
@@ -202,14 +202,14 @@ module Applicative where
 
   unfold-<$> : ∀ {ℓ} {A B : Set ℓ} → (f : A → B) (as : List A) →
                (f <$> as) ≡ (pure f ⊛ as)
-  unfold-<$> f as = P.sym (++-identityʳ (f <$> as))
+  unfold-<$> f as = ≡.sym (++-identityʳ (f <$> as))
 
   -- _⊛_ unfolds to binds.
   unfold-⊛ : ∀ {ℓ} {A B : Set ℓ} → (fs : List (A → B)) (as : List A) →
              (fs ⊛ as) ≡ (fs >>= pam as)
   unfold-⊛ fs as = begin
     fs ⊛ as
-      ≡⟨ concatMap-cong (λ f → P.cong (map f) (concatMap-pure as)) fs ⟨
+      ≡⟨ concatMap-cong (λ f → ≡.cong (map f) (concatMap-pure as)) fs ⟨
     concatMap (λ f → map f (concatMap pure as)) fs
       ≡⟨ concatMap-cong (λ f → map-concatMap f pure as) fs ⟩
     concatMap (λ f → concatMap (λ x → pure (f x)) as) fs
@@ -224,7 +224,7 @@ module Applicative where
   right-distributive fs₁ fs₂ xs = begin
     (fs₁ ∣ fs₂) ⊛ xs                     ≡⟨ unfold-⊛ (fs₁ ∣ fs₂) xs ⟩
     (fs₁ ∣ fs₂ >>= pam xs)               ≡⟨ MonadProperties.right-distributive fs₁ fs₂ (pam xs) ⟩
-    (fs₁ >>= pam xs) ∣ (fs₂ >>= pam xs)  ≡⟨ P.cong₂ _∣_ (unfold-⊛ fs₁ xs) (unfold-⊛ fs₂ xs) ⟨
+    (fs₁ >>= pam xs) ∣ (fs₂ >>= pam xs)  ≡⟨ ≡.cong₂ _∣_ (unfold-⊛ fs₁ xs) (unfold-⊛ fs₂ xs) ⟨
     (fs₁ ⊛ xs ∣ fs₂ ⊛ xs)                ∎
 
   -- _⊛_ does not distribute over _∣_ from the left.
@@ -262,9 +262,9 @@ module Applicative where
     pure _∘′_ ⊛ fs ⊛ gs ⊛ xs
       ≡⟨ unfold-⊛ (pure _∘′_ ⊛ fs ⊛ gs) xs ⟩
     (pure _∘′_ ⊛ fs ⊛ gs >>= pam xs)
-      ≡⟨ P.cong (_>>= pam xs) (unfold-⊛ (pure _∘′_ ⊛ fs) gs) ⟩
+      ≡⟨ ≡.cong (_>>= pam xs) (unfold-⊛ (pure _∘′_ ⊛ fs) gs) ⟩
     (pure _∘′_ ⊛ fs >>= pam gs >>= pam xs)
-      ≡⟨ P.cong (λ h → h >>= pam gs >>= pam xs) (unfold-⊛ (pure _∘′_) fs) ⟩
+      ≡⟨ ≡.cong (λ h → h >>= pam gs >>= pam xs) (unfold-⊛ (pure _∘′_) fs) ⟩
     (pure _∘′_ >>= pam fs >>= pam gs >>= pam xs)
       ≡⟨ MP.cong (MP.cong (MP.left-identity _∘′_ (pam fs))
                  (λ f → refl {x = pam gs f}))
@@ -278,13 +278,13 @@ module Applicative where
     (fs >>= λ f → gs >>= λ g → pam xs (f ∘′ g))
       ≡⟨ (MP.cong (refl {x = fs}) λ f →
          MP.cong (refl {x = gs}) λ g →
-         P.sym $ pam-lemma xs g (pure ∘ f)) ⟩
+         ≡.sym $ pam-lemma xs g (pure ∘ f)) ⟩
     (fs >>= λ f → gs >>= λ g → pam (pam xs g) f)
       ≡⟨ MP.cong (refl {x = fs}) (λ f → MP.associative gs (pam xs) (pure ∘ f)) ⟩
     (fs >>= pam (gs >>= pam xs))
       ≡⟨ unfold-⊛ fs (gs >>= pam xs) ⟨
     fs ⊛ (gs >>= pam xs)
-      ≡⟨ P.cong (fs ⊛_) (unfold-⊛ gs xs) ⟨
+      ≡⟨ ≡.cong (fs ⊛_) (unfold-⊛ gs xs) ⟨
     fs ⊛ (gs ⊛ xs)
       ∎
 
@@ -303,6 +303,6 @@ module Applicative where
     (fs >>= pam (pure x))      ≡⟨ (MP.cong (refl {x = fs}) λ f →
                                       MP.left-identity x (pure ∘ f)) ⟩
     (fs >>= λ f → pure (f x))  ≡⟨⟩
-    (pam fs (_$′ x))           ≡⟨ P.sym $ MP.left-identity (_$′ x) (pam fs) ⟩
+    (pam fs (_$′ x))           ≡⟨ ≡.sym $ MP.left-identity (_$′ x) (pam fs) ⟩
     (pure (_$′ x) >>= pam fs)  ≡⟨ unfold-⊛ (pure (_$′ x)) fs ⟨
     pure (_$′ x) ⊛ fs          ∎
diff --git a/src/Data/List/Membership/Propositional/Properties/Core.agda b/src/Data/List/Membership/Propositional/Properties/Core.agda
index f7543d05f8..7ad4f94573 100644
--- a/src/Data/List/Membership/Propositional/Properties/Core.agda
+++ b/src/Data/List/Membership/Propositional/Properties/Core.agda
@@ -20,8 +20,8 @@ open import Data.List.Membership.Propositional
 open import Data.Product.Base as Product
   using (_,_; proj₁; proj₂; uncurry′; ∃; _×_)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; subst)
 open import Relation.Unary using (Pred; _⊆_)
 
 private
@@ -37,8 +37,8 @@ map∘find : ∀ {P : Pred A p} {xs}
            {f : _≡_ (proj₁ p′) ⊆ P} →
            f refl ≡ proj₂ (proj₂ p′) →
            Any.map f (proj₁ (proj₂ p′)) ≡ p
-map∘find (here  p) hyp = P.cong here  hyp
-map∘find (there p) hyp = P.cong there (map∘find p hyp)
+map∘find (here  p) hyp = cong here  hyp
+map∘find (there p) hyp = cong there (map∘find p hyp)
 
 find∘map : ∀ {P : Pred A p} {Q : Pred A q}
            {xs : List A} (p : Any P xs) (f : P ⊆ Q) →
@@ -61,13 +61,13 @@ find-∈ (there x∈xs) rewrite find-∈ x∈xs = refl
 lose∘find : ∀ {P : Pred A p} {xs : List A}
             (p : Any P xs) →
             uncurry′ lose (proj₂ (find p)) ≡ p
-lose∘find p = map∘find p P.refl
+lose∘find p = map∘find p refl
 
 find∘lose : ∀ (P : Pred A p) {x xs}
             (x∈xs : x ∈ xs) (pp : P x) →
             find {P = P} (lose x∈xs pp) ≡ (x , x∈xs , pp)
 find∘lose P x∈xs p
-  rewrite find∘map x∈xs (flip (P.subst P) p)
+  rewrite find∘map x∈xs (flip (subst P) p)
         | find-∈ x∈xs
         = refl
 
diff --git a/src/Data/List/Relation/Binary/Equality/Propositional.agda b/src/Data/List/Relation/Binary/Equality/Propositional.agda
index 8b45be7d27..ffae515086 100644
--- a/src/Data/List/Relation/Binary/Equality/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Equality/Propositional.agda
@@ -16,20 +16,20 @@ module Data.List.Relation.Binary.Equality.Propositional {a} {A : Set a} where
 
 open import Data.List.Base
 import Data.List.Relation.Binary.Equality.Setoid as SetoidEquality
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 ------------------------------------------------------------------------
 -- Re-export everything from setoid equality
 
-open SetoidEquality (P.setoid A) public
+open SetoidEquality (≡.setoid A) public
 
 ------------------------------------------------------------------------
 -- ≋ is propositional
 
 ≋⇒≡ : _≋_ ⇒ _≡_
-≋⇒≡ []               = P.refl
-≋⇒≡ (P.refl ∷ xs≈ys) = P.cong (_ ∷_) (≋⇒≡ xs≈ys)
+≋⇒≡ []             = refl
+≋⇒≡ (refl ∷ xs≈ys) = cong (_ ∷_) (≋⇒≡ xs≈ys)
 
 ≡⇒≋ : _≡_ ⇒ _≋_
-≡⇒≋ P.refl = ≋-refl
+≡⇒≋ refl = ≋-refl
diff --git a/src/Data/List/Relation/Binary/Equality/Setoid.agda b/src/Data/List/Relation/Binary/Equality/Setoid.agda
index fc6842c38d..af78f43c2c 100644
--- a/src/Data/List/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Equality/Setoid.agda
@@ -20,7 +20,7 @@ open import Data.List.Relation.Unary.Unique.Setoid S using (Unique)
 open import Function.Base using (_∘_)
 open import Level
 open import Relation.Binary.Core renaming (Rel to Rel₂)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
 open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
 open import Relation.Unary as U using (Pred)
 
@@ -50,7 +50,7 @@ open PW public
 ≋-refl = PW.refl refl
 
 ≋-reflexive : _≡_ ⇒ _≋_
-≋-reflexive P.refl = ≋-refl
+≋-reflexive ≡.refl = ≋-refl
 
 ≋-sym : Symmetric _≋_
 ≋-sym = PW.symmetric sym
diff --git a/src/Data/List/Relation/Binary/Pointwise.agda b/src/Data/List/Relation/Binary/Pointwise.agda
index 2728418ddc..76e9d81f54 100644
--- a/src/Data/List/Relation/Binary/Pointwise.agda
+++ b/src/Data/List/Relation/Binary/Pointwise.agda
@@ -29,8 +29,8 @@ open import Relation.Binary.Core renaming (Rel to Rel₂)
 open import Relation.Binary.Definitions using (_Respects_; _Respects₂_)
 open import Relation.Binary.Bundles using (Setoid; DecSetoid; Preorder; Poset)
 open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence; IsPartialOrder; IsPreorder)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -129,8 +129,8 @@ AllPairs-resp-Pointwise resp@(respₗ , respᵣ) (x∼y ∷ xs) (px ∷ pxs) =
 -- length
 
 Pointwise-length : Pointwise R xs ys → length xs ≡ length ys
-Pointwise-length []            = P.refl
-Pointwise-length (x∼y ∷ xs∼ys) = P.cong ℕ.suc (Pointwise-length xs∼ys)
+Pointwise-length []            = ≡.refl
+Pointwise-length (x∼y ∷ xs∼ys) = ≡.cong ℕ.suc (Pointwise-length xs∼ys)
 
 ------------------------------------------------------------------------
 -- tabulate
@@ -162,9 +162,9 @@ tabulate⁻ {n = suc n} (x∼y ∷ xs∼ys) (fsuc i) = tabulate⁻ xs∼ys i
 ++-cancelʳ (y ∷ ys) (z ∷ zs) (y∼z ∷ ys∼zs) = y∼z ∷ (++-cancelʳ ys zs ys∼zs)
 -- Impossible cases
 ++-cancelʳ {xs = xs}     []       (z ∷ zs) eq   =
-  contradiction (P.trans (Pointwise-length eq) (length-++ (z ∷ zs))) (m≢1+n+m (length xs))
+  contradiction (≡.trans (Pointwise-length eq) (length-++ (z ∷ zs))) (m≢1+n+m (length xs))
 ++-cancelʳ {xs = xs}     (y ∷ ys) []       eq   =
-  contradiction (P.trans (P.sym (length-++ (y ∷ ys))) (Pointwise-length eq)) (m≢1+n+m (length xs) ∘ P.sym)
+  contradiction (≡.trans (≡.sym (length-++ (y ∷ ys))) (Pointwise-length eq)) (m≢1+n+m (length xs) ∘ ≡.sym)
 
 ------------------------------------------------------------------------
 -- concat
@@ -246,8 +246,8 @@ lookup⁻ : length xs ≡ length ys →
           (∀ {i j} → toℕ i ≡ toℕ j → R (lookup xs i) (lookup ys j)) →
           Pointwise R xs ys
 lookup⁻ {xs = []}    {ys = []}    _             _  = []
-lookup⁻ {xs = _ ∷ _} {ys = _ ∷ _} |xs|≡|ys| eq = eq {fzero} P.refl ∷
-  lookup⁻ (suc-injective |xs|≡|ys|) (eq ∘ P.cong ℕ.suc)
+lookup⁻ {xs = _ ∷ _} {ys = _ ∷ _} |xs|≡|ys| eq = eq {fzero} ≡.refl ∷
+  lookup⁻ (suc-injective |xs|≡|ys|) (eq ∘ ≡.cong ℕ.suc)
 
 lookup⁺ : ∀ (Rxys : Pointwise R xs ys) →
           ∀ i → (let j = cast (Pointwise-length Rxys) i) →
@@ -260,14 +260,14 @@ lookup⁺ (_   ∷ Rxys) (fsuc i) = lookup⁺ Rxys i
 ------------------------------------------------------------------------
 
 Pointwise-≡⇒≡ : Pointwise {A = A} _≡_ ⇒ _≡_
-Pointwise-≡⇒≡ []               = P.refl
-Pointwise-≡⇒≡ (P.refl ∷ xs∼ys) with Pointwise-≡⇒≡ xs∼ys
-... | P.refl = P.refl
+Pointwise-≡⇒≡ []               = ≡.refl
+Pointwise-≡⇒≡ (≡.refl ∷ xs∼ys) with Pointwise-≡⇒≡ xs∼ys
+... | ≡.refl = ≡.refl
 
 ≡⇒Pointwise-≡ :  _≡_ ⇒ Pointwise {A = A} _≡_
-≡⇒Pointwise-≡ P.refl = refl P.refl
+≡⇒Pointwise-≡ ≡.refl = refl ≡.refl
 
-Pointwise-≡↔≡ : Inverse (setoid (P.setoid A)) (P.setoid (List A))
+Pointwise-≡↔≡ : Inverse (setoid (≡.setoid A)) (≡.setoid (List A))
 Pointwise-≡↔≡ = record
   { to = id
   ; from = id
diff --git a/src/Data/List/Relation/Binary/Pointwise/Properties.agda b/src/Data/List/Relation/Binary/Pointwise/Properties.agda
index 7dfba09296..1761dd3cda 100644
--- a/src/Data/List/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/List/Relation/Binary/Pointwise/Properties.agda
@@ -13,7 +13,7 @@ open import Data.List.Base using (List; []; _∷_)
 open import Level
 open import Relation.Binary.Core using (REL; _⇒_)
 open import Relation.Binary.Definitions
-import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Core as ≡
 open import Relation.Nullary using (yes; no; _×-dec_)
 import Relation.Nullary.Decidable as Dec
 
@@ -72,6 +72,6 @@ decidable R? (x ∷ xs) (y ∷ ys) = Dec.map′ (uncurry _∷_) uncons
   (R? x y ×-dec decidable R? xs ys)
 
 irrelevant : Irrelevant R → Irrelevant (Pointwise R)
-irrelevant irr []       []         = P.refl
+irrelevant irr []       []         = ≡.refl
 irrelevant irr (r ∷ rs) (r₁ ∷ rs₁) =
-  P.cong₂ _∷_ (irr r r₁) (irrelevant irr rs rs₁)
+  ≡.cong₂ _∷_ (irr r r₁) (irrelevant irr rs rs₁)
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
index 776d065423..18f9871f1b 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous/Properties.agda
@@ -28,7 +28,7 @@ open import Relation.Unary as U using (Pred)
 open import Relation.Binary.Core using (Rel; REL; _⇒_)
 open import Relation.Binary.Definitions
   using (Trans; Antisym; Irrelevant; Decidable)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; _≢_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; refl; cong₂)
 
 private
   variable
@@ -158,7 +158,7 @@ replicate⁺ (s≤s m≤n) r = r ∷ replicate⁺ m≤n r
 
 replicate⁻ : ∀ {m n a b} → m ≢ 0 →
              Prefix R (replicate m a) (replicate n b) → R a b
-replicate⁻ {m = zero}  {n}     m≢0 r  = ⊥-elim (m≢0 P.refl)
+replicate⁻ {m = zero}  {n}     m≢0 r  = ⊥-elim (m≢0 refl)
 replicate⁻ {m = suc m} {suc n} m≢0 rs = Prefix.head rs
 
 ------------------------------------------------------------------------
@@ -208,9 +208,9 @@ module _ {a b c d} {A : Set a} {B : Set b} {C : Set c} {D : Set d}
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   irrelevant : Irrelevant R → Irrelevant (Prefix R)
-  irrelevant R-irr []       []         = P.refl
+  irrelevant R-irr []       []         = refl
   irrelevant R-irr (r ∷ rs) (r′ ∷ rs′) =
-    P.cong₂ _∷_ (R-irr r r′) (irrelevant R-irr rs rs′)
+    cong₂ _∷_ (R-irr r r′) (irrelevant R-irr rs rs′)
 
 ------------------------------------------------------------------------
 -- Decidability
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
index 7aa3f4dd5d..a0fe6b150c 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous.agda
@@ -13,7 +13,6 @@ open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Level using (_⊔_)
 open import Relation.Binary.Core using (REL; _⇒_)
 open import Relation.Binary.Definitions using (_⟶_Respects_; Min)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Unary using (Pred)
 
 module Data.List.Relation.Binary.Sublist.Heterogeneous
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index 3184ca4420..b760e0287f 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -39,7 +39,7 @@ open import Relation.Binary.Definitions
   using (Reflexive; Trans; Antisym; Decidable; Irrelevant; Irreflexive)
 open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsDecPartialOrder)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 ------------------------------------------------------------------------
 -- Injectivity of constructors
@@ -48,15 +48,15 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   ∷-injectiveˡ : ∀ {x y xs ys} {px qx : R x y} {pxs qxs : Sublist R xs ys} →
                  (Sublist R (x ∷ xs) (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → px ≡ qx
-  ∷-injectiveˡ P.refl = P.refl
+  ∷-injectiveˡ ≡.refl = ≡.refl
 
   ∷-injectiveʳ : ∀ {x y xs ys} {px qx : R x y} {pxs qxs : Sublist R xs ys} →
                  (Sublist R (x ∷ xs) (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → pxs ≡ qxs
-  ∷-injectiveʳ P.refl = P.refl
+  ∷-injectiveʳ ≡.refl = ≡.refl
 
   ∷ʳ-injective : ∀ {y xs ys} {pxs qxs : Sublist R xs ys} →
                  (Sublist R xs (y ∷ ys) ∋ y ∷ʳ pxs) ≡ (y ∷ʳ qxs) → pxs ≡ qxs
-  ∷ʳ-injective P.refl = P.refl
+  ∷ʳ-injective ≡.refl = ≡.refl
 
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
@@ -308,7 +308,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   reverse⁻ : ∀ {as bs} → Sublist R (reverse as) (reverse bs) → Sublist R as bs
   reverse⁻ {as} {bs} p = cast (reverse⁺ p) where
-    cast = P.subst₂ (Sublist R) (List.reverse-involutive as) (List.reverse-involutive bs)
+    cast = ≡.subst₂ (Sublist R) (List.reverse-involutive as) (List.reverse-involutive bs)
 
 ------------------------------------------------------------------------
 -- Inversion lemmas
@@ -327,27 +327,27 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} {a as b bs} where
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   Sublist-[]-irrelevant : U.Irrelevant (Sublist R [])
-  Sublist-[]-irrelevant []       []        = P.refl
-  Sublist-[]-irrelevant (y ∷ʳ p) (.y ∷ʳ q) = P.cong (y ∷ʳ_) (Sublist-[]-irrelevant p q)
+  Sublist-[]-irrelevant []       []        = ≡.refl
+  Sublist-[]-irrelevant (y ∷ʳ p) (.y ∷ʳ q) = ≡.cong (y ∷ʳ_) (Sublist-[]-irrelevant p q)
 
 ------------------------------------------------------------------------
 -- (to/from)Any is a bijection
 
   toAny-injective : ∀ {xs x} {p q : Sublist R [ x ] xs} → toAny p ≡ toAny q → p ≡ q
   toAny-injective {p = y ∷ʳ p} {y ∷ʳ q} =
-    P.cong (y ∷ʳ_) ∘′ toAny-injective ∘′ there-injective
+    ≡.cong (y ∷ʳ_) ∘′ toAny-injective ∘′ there-injective
   toAny-injective {p = _ ∷ p}  {_ ∷ q}  =
-    P.cong₂ (flip _∷_) (Sublist-[]-irrelevant p q) ∘′ here-injective
+    ≡.cong₂ (flip _∷_) (Sublist-[]-irrelevant p q) ∘′ here-injective
 
   fromAny-injective : ∀ {xs x} {p q : Any (R x) xs} →
                       fromAny {R = R} p ≡ fromAny q → p ≡ q
-  fromAny-injective {p = here px} {here qx} = P.cong here ∘′ ∷-injectiveˡ
+  fromAny-injective {p = here px} {here qx} = ≡.cong here ∘′ ∷-injectiveˡ
   fromAny-injective {p = there p} {there q} =
-    P.cong there ∘′ fromAny-injective ∘′ ∷ʳ-injective
+    ≡.cong there ∘′ fromAny-injective ∘′ ∷ʳ-injective
 
   toAny∘fromAny≗id : ∀ {xs x} (p : Any (R x) xs) → toAny (fromAny {R = R} p) ≡ p
-  toAny∘fromAny≗id (here px) = P.refl
-  toAny∘fromAny≗id (there p) = P.cong there (toAny∘fromAny≗id p)
+  toAny∘fromAny≗id (here px) = ≡.refl
+  toAny∘fromAny≗id (there p) = ≡.cong there (toAny∘fromAny≗id p)
 
   Sublist-[x]-bijection : ∀ {x xs} → (Sublist R [ x ] xs) ⤖ (Any (R x) xs)
   Sublist-[x]-bijection = mk⤖ (toAny-injective , strictlySurjective⇒surjective < fromAny , toAny∘fromAny≗id >)
@@ -360,10 +360,10 @@ module Reflexivity
     (R-refl : Reflexive R) where
 
   reflexive : _≡_ ⇒ Sublist R
-  reflexive P.refl = fromPointwise (Pw.refl R-refl)
+  reflexive ≡.refl = fromPointwise (Pw.refl R-refl)
 
   refl : Reflexive (Sublist R)
-  refl = reflexive P.refl
+  refl = reflexive ≡.refl
 
 open Reflexivity public
 
@@ -392,18 +392,18 @@ module Antisymmetry
   antisym (r ∷ rs)  (s ∷ ss)  = rs⇒e r s ∷ antisym rs ss
   -- impossible cases
   antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷ʳ_ {ys₂} {zs} z ss) =
-    ⊥-elim $ ℕ.<-irrefl P.refl $ begin
+    ⊥-elim $ ℕ.<-irrefl ≡.refl $ begin
     length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩
     length zs        ≤⟨ ℕ.n≤1+n (length zs) ⟩
     length (z ∷ zs)  ≤⟨ length-mono-≤ rs ⟩
     length ys₁       ∎
   antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷_ {y} {ys₂} {z} {zs} s ss)  =
-    ⊥-elim $ ℕ.<-irrefl P.refl $ begin
+    ⊥-elim $ ℕ.<-irrefl ≡.refl $ begin
     length (z ∷ zs) ≤⟨ length-mono-≤ rs ⟩
     length ys₁      ≤⟨ length-mono-≤ ss ⟩
     length zs       ∎
   antisym (_∷_ {x} {xs} {y} {ys₁} r rs)  (_∷ʳ_ {ys₂} {zs} z ss) =
-    ⊥-elim $ ℕ.<-irrefl P.refl $ begin
+    ⊥-elim $ ℕ.<-irrefl ≡.refl $ begin
     length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩
     length xs        ≤⟨ length-mono-≤ rs ⟩
     length ys₁       ∎
@@ -506,10 +506,10 @@ module Disjointness {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   -- Disjoint is proof-irrelevant
 
   Disjoint-irrelevant : ∀{xs ys zs} → Irrelevant (Disjoint {R = R} {xs} {ys} {zs})
-  Disjoint-irrelevant [] [] = P.refl
-  Disjoint-irrelevant (y   ∷ₙ d₁) (.y   ∷ₙ d₂) = P.cong (y ∷ₙ_) (Disjoint-irrelevant d₁ d₂)
-  Disjoint-irrelevant (x≈y ∷ₗ d₁) (.x≈y ∷ₗ d₂) = P.cong (x≈y ∷ₗ_) (Disjoint-irrelevant d₁ d₂)
-  Disjoint-irrelevant (x≈y ∷ᵣ d₁) (.x≈y ∷ᵣ d₂) = P.cong (x≈y ∷ᵣ_) (Disjoint-irrelevant d₁ d₂)
+  Disjoint-irrelevant [] [] = ≡.refl
+  Disjoint-irrelevant (y   ∷ₙ d₁) (.y   ∷ₙ d₂) = ≡.cong (y ∷ₙ_) (Disjoint-irrelevant d₁ d₂)
+  Disjoint-irrelevant (x≈y ∷ₗ d₁) (.x≈y ∷ₗ d₂) = ≡.cong (x≈y ∷ₗ_) (Disjoint-irrelevant d₁ d₂)
+  Disjoint-irrelevant (x≈y ∷ᵣ d₁) (.x≈y ∷ᵣ d₂) = ≡.cong (x≈y ∷ᵣ_) (Disjoint-irrelevant d₁ d₂)
 
   -- Note: DisjointUnion is not proof-irrelevant unless the underlying relation R is.
   -- The proof is not entirely trivial, thus, we leave it for future work:
@@ -525,7 +525,7 @@ module Disjointness {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   Disjoint-irrefl′ (y ∷ₙ d) = Disjoint-irrefl′ d
 
   Disjoint-irrefl : ∀{x xs ys} → Irreflexive {A = x ∷ xs ⊆ ys } _≡_ Disjoint
-  Disjoint-irrefl P.refl x with Disjoint-irrefl′ x
+  Disjoint-irrefl ≡.refl x with Disjoint-irrefl′ x
   ... | () ∷ _
 
   -- Symmetry
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index 1813568d5b..86194916f8 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -32,11 +32,11 @@ open import Data.List.Relation.Binary.Sublist.Heterogeneous
 open import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
 open import Function
 
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality as ≡
   using (_≡_; _≗_; sym; cong; cong₂; subst₂)
 open import Relation.Nullary
 
-open P.≡-Reasoning
+open ≡.≡-Reasoning
 
 infix 4  _⊆I_ _⊆R_ _⊆T_
 
@@ -87,7 +87,7 @@ d ⊆R e = ∀ ρ → Sublist R (⟦ d ⟧R ρ) (⟦ e ⟧R ρ)
 -- Flattening in a semantics-respecting manner
 
 ⟦++⟧R : ∀ {n} xs ys (ρ : Vec (List A) n) → ⟦ xs ++ ys ⟧R ρ ≡ ⟦ xs ⟧R ρ ++ ⟦ ys ⟧R ρ
-⟦++⟧R []       ys ρ = P.refl
+⟦++⟧R []       ys ρ = ≡.refl
 ⟦++⟧R (x ∷ xs) ys ρ = begin
   ⟦ x ⟧I ρ ++ ⟦ xs ++ ys ⟧R ρ
     ≡⟨ cong (⟦ x ⟧I ρ ++_) (⟦++⟧R xs ys ρ) ⟩
@@ -97,7 +97,7 @@ d ⊆R e = ∀ ρ → Sublist R (⟦ d ⟧R ρ) (⟦ e ⟧R ρ)
     ∎
 
 flatten : ∀ {n} (t : TList n) → Σ[ r ∈ RList n ] ⟦ r ⟧R ≗ ⟦ t ⟧T
-flatten []       = [] , λ _ → P.refl
+flatten []       = [] , λ _ → ≡.refl
 flatten (It it)  = it ∷ [] , λ ρ → ++-identityʳ (⟦ It it ⟧T ρ)
 flatten (t <> u) =
   let (rt , eqt) = flatten t
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid.agda b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
index 86e16ab80c..83a16444d0 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
@@ -29,7 +29,7 @@ open import Data.Product.Base using (∃; ∃₂; _×_; _,_; proj₂)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Bundles using (Preorder; Poset)
 open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Nullary using (¬_; Dec; yes; no)
 
 open Setoid S renaming (Carrier to A)
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
index 17a0c25c77..abacb301d0 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
@@ -27,8 +27,8 @@ open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Binary.Core using (REL; Rel; _⇒_)
 open import Relation.Binary.Definitions as B
   using (Trans; Antisym; Irrelevant)
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; _≢_; refl; sym; subst; subst₂)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≢_; refl; sym; cong; subst; subst₂)
 
 import Data.List.Properties as List
 import Data.List.Relation.Binary.Prefix.Heterogeneous.Properties as Prefix
@@ -199,10 +199,10 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   irrelevant : Irrelevant R → Irrelevant (Suffix R)
-  irrelevant irr (here  rs)   (here  rs₁)   = P.cong here $ Pw.irrelevant irr rs rs₁
+  irrelevant irr (here  rs)   (here  rs₁)   = cong here $ Pw.irrelevant irr rs rs₁
   irrelevant irr (here  rs)   (there rsuf)  = contradiction (Pointwise-length rs) (S[as][bs]⇒∣as∣≢1+∣bs∣ rsuf)
   irrelevant irr (there rsuf) (here  rs)    = contradiction (Pointwise-length rs) (S[as][bs]⇒∣as∣≢1+∣bs∣ rsuf)
-  irrelevant irr (there rsuf) (there rsuf₁) = P.cong there $ irrelevant irr rsuf rsuf₁
+  irrelevant irr (there rsuf) (there rsuf₁) = cong there $ irrelevant irr rsuf rsuf₁
 
 ------------------------------------------------------------------------
 -- Decidability
diff --git a/src/Data/List/Relation/Ternary/Interleaving.agda b/src/Data/List/Relation/Ternary/Interleaving.agda
index 17055652fb..38ecbf5579 100644
--- a/src/Data/List/Relation/Ternary/Interleaving.agda
+++ b/src/Data/List/Relation/Ternary/Interleaving.agda
@@ -16,7 +16,7 @@ open import Data.Product.Base as Product using (∃; ∃₂; _×_; uncurry; _,_;
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base
 open import Relation.Binary.Core using (REL; _⇒_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- Definition
@@ -60,9 +60,9 @@ module _ {a b c l r} {A : Set a} {B : Set b} {C : Set c}
           Interleaving _≡_ _≡_ csl csr cs × Pointwise L l csl × Pointwise R r csr
   break []        = -, [] , [] , []
   break (l ∷ˡ sp) = let (_ , eq , pwl , pwr) = break sp in
-                    -, P.refl ∷ˡ eq , l ∷ pwl , pwr
+                    -, refl ∷ˡ eq , l ∷ pwl , pwr
   break (r ∷ʳ sp) = let (_ , eq , pwl , pwr) = break sp in
-                    -, P.refl ∷ʳ eq , pwl , r ∷ pwr
+                    -, refl ∷ʳ eq , pwl , r ∷ pwr
 
 -- map
 
diff --git a/src/Data/List/Relation/Unary/All.agda b/src/Data/List/Relation/Unary/All.agda
index d9f358df04..bd8677630b 100644
--- a/src/Data/List/Relation/Unary/All.agda
+++ b/src/Data/List/Relation/Unary/All.agda
@@ -25,8 +25,8 @@ import Relation.Nullary.Decidable as Dec
 open import Relation.Unary hiding (_∈_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (_Respects_)
-open import Relation.Binary.PropositionalEquality.Core as P
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -124,7 +124,7 @@ module _(S : Setoid a ℓ) {P : Pred (Setoid.Carrier S) p} where
   tabulateₛ {x ∷ xs} hyp = hyp (here refl₁) ∷ tabulateₛ (hyp ∘ there)
 
 tabulate : (∀ {x} → x ∈ₚ xs → P x) → All P xs
-tabulate = tabulateₛ (P.setoid _)
+tabulate = tabulateₛ (≡.setoid _)
 
 self : ∀ {xs : List A} → All (const A) xs
 self = tabulate (λ {x} _ → x)
@@ -213,9 +213,9 @@ universal u []       = []
 universal u (x ∷ xs) = u x ∷ universal u xs
 
 irrelevant : Irrelevant P → Irrelevant (All P)
-irrelevant irr []           []           = P.refl
+irrelevant irr []           []           = ≡.refl
 irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =
-  P.cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)
+  ≡.cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)
 
 satisfiable : Satisfiable (All P)
 satisfiable = [] , []
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index 949f6d9fdb..a6eaa9d480 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -41,8 +41,8 @@ open import Function.Related.Propositional as Related using (Kind; Related)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel; REL)
 open import Relation.Binary.Definitions as B
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; trans; cong; cong₂; subst)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 open import Relation.Unary as U
@@ -100,13 +100,13 @@ Any-cong {P = P} {Q = Q} {xs = xs} {ys} P↔Q xs≈ys =
 
 map-id : ∀ (f : P ⋐ P) → (∀ {x} (p : P x) → f p ≡ p) →
          (p : Any P xs) → Any.map f p ≡ p
-map-id f hyp (here  p) = P.cong here (hyp p)
-map-id f hyp (there p) = P.cong there $ map-id f hyp p
+map-id f hyp (here  p) = cong here (hyp p)
+map-id f hyp (there p) = cong there $ map-id f hyp p
 
 map-∘ : ∀ (f : Q ⋐ R) (g : P ⋐ Q) (p : Any P xs) →
         Any.map (f ∘ g) p ≡ Any.map f (Any.map g p)
 map-∘ f g (here  p) = refl
-map-∘ f g (there p) = P.cong there $ map-∘ f g p
+map-∘ f g (there p) = cong there $ map-∘ f g p
 
 ------------------------------------------------------------------------
 -- Any.lookup
@@ -133,16 +133,16 @@ swap-there : ∀ {P : A → B → Set ℓ} →
              (any : Any (λ x → Any (P x) ys) xs) →
              swap (Any.map (there {x = x}) any) ≡ there (swap any)
 swap-there (here  pys)  = refl
-swap-there (there pxys) = P.cong (Any.map there) (swap-there pxys)
+swap-there (there pxys) = cong (Any.map there) (swap-there pxys)
 
 swap-invol : ∀ {P : A → B → Set ℓ} →
              (any : Any (λ x → Any (P x) ys) xs) →
              swap (swap any) ≡ any
 swap-invol (here (here px))   = refl
 swap-invol (here (there pys)) =
-  P.cong (Any.map there) (swap-invol (here pys))
+  cong (Any.map there) (swap-invol (here pys))
 swap-invol (there pxys)       =
-  P.trans (swap-there (swap pxys)) (P.cong there (swap-invol pxys))
+  trans (swap-there (swap pxys)) (cong there (swap-invol pxys))
 
 swap↔ : ∀ {P : A → B → Set ℓ} →
        Any (λ x → Any (P x) ys) xs ↔ Any (λ y → Any (flip P y) xs) ys
@@ -228,31 +228,31 @@ Any-×⁻ pq with Product.map₂ (Product.map₂ find) (find pq)
 
     Any-×⁻ (Any-×⁺ (p , q))
 
-      ≡⟨⟩
+     ≡⟨⟩
 
     (let (x , x∈xs , pq)    = find (Any-×⁺ (p , q))
          (y , y∈ys , p , q) = find pq
      in  lose x∈xs p , lose y∈ys q)
 
-     ≡⟨ P.cong (λ • → let (x , x∈xs , pq)    = •
-                          (y , y∈ys , p , q) = find pq
-                      in  lose x∈xs p , lose y∈ys q)
-               (find∘map p (λ p → Any.map (p ,_) q)) ⟩
+     ≡⟨ cong (λ • → let (x , x∈xs , pq)    = •
+                        (y , y∈ys , p , q) = find pq
+                    in  lose x∈xs p , lose y∈ys q)
+             (find∘map p (λ p → Any.map (p ,_) q)) ⟩
 
     (let (x , x∈xs , p)     = find p
          (y , y∈ys , p , q) = find (Any.map (p ,_) q)
      in  lose x∈xs p , lose y∈ys q)
 
-     ≡⟨ P.cong (λ • → let (x , x∈xs , p)     = find p
-                          (y , y∈ys , p , q) = •
-                      in  lose x∈xs p , lose y∈ys q)
-               (find∘map q (proj₂ (proj₂ (find p)) ,_)) ⟩
+     ≡⟨ cong (λ • → let (x , x∈xs , p)     = find p
+                        (y , y∈ys , p , q) = •
+                    in  lose x∈xs p , lose y∈ys q)
+             (find∘map q (proj₂ (proj₂ (find p)) ,_)) ⟩
 
     (let (x , x∈xs , p) = find p
          (y , y∈ys , q) = find q
      in  lose x∈xs p , lose y∈ys q)
 
-     ≡⟨ P.cong₂ _,_ (lose∘find p) (lose∘find q) ⟩
+     ≡⟨ cong₂ _,_ (lose∘find p) (lose∘find q) ⟩
 
     (p , q) ∎
 
@@ -263,16 +263,16 @@ Any-×⁻ pq with Product.map₂ (Product.map₂ find) (find pq)
     with find pq′
       | (λ (f : (proj₁ (find pq′) ≡_) ⋐ _) → map∘find pq′ {f})
   ... | (y , y∈ys , p , q) | lem₂
-    rewrite P.sym $ map-∘ {R = λ x → Any (λ y → P x × Q y) ys}
-                          (λ p → Any.map (λ q → p , q) (lose y∈ys q))
-                          (λ y → P.subst P y p)
-                          x∈xs
+    rewrite sym $ map-∘ {R = λ x → Any (λ y → P x × Q y) ys}
+                        (λ p → Any.map (λ q → p , q) (lose y∈ys q))
+                        (λ y → subst P y p)
+                        x∈xs
             = lem₁ _ helper
     where
     helper : Any.map (λ q → p , q) (lose y∈ys q) ≡ pq′
-    helper rewrite P.sym $ map-∘ (λ q → p , q)
-                                 (λ y → P.subst Q y q)
-                                 y∈ys
+    helper rewrite sym $ map-∘ (λ q → p , q)
+                               (λ y → subst Q y q)
+                               y∈ys
            = lem₂ _ refl
 
 ------------------------------------------------------------------------
@@ -317,12 +317,12 @@ module _ {f : A → B} where
 
   map⁺∘map⁻ : (p : Any P (List.map f xs)) → map⁺ (map⁻ p) ≡ p
   map⁺∘map⁻ {xs = x ∷ xs} (here  p) = refl
-  map⁺∘map⁻ {xs = x ∷ xs} (there p) = P.cong there (map⁺∘map⁻ p)
+  map⁺∘map⁻ {xs = x ∷ xs} (there p) = cong there (map⁺∘map⁻ p)
 
   map⁻∘map⁺ : ∀ (P : Pred B p) →
               (p : Any (P ∘ f) xs) → map⁻ {P = P} (map⁺ p) ≡ p
   map⁻∘map⁺ P (here  p) = refl
-  map⁻∘map⁺ P (there p) = P.cong there (map⁻∘map⁺ P p)
+  map⁻∘map⁺ P (there p) = cong there (map⁻∘map⁺ P p)
 
   map↔ : Any (P ∘ f) xs ↔ Any P (List.map f xs)
   map↔ = mk↔ₛ′ map⁺ map⁻ map⁺∘map⁻ (map⁻∘map⁺ _)
@@ -363,8 +363,8 @@ module _ {P : A → Set p} where
   ++⁺∘++⁻ []       p         = refl
   ++⁺∘++⁻ (x ∷ xs) (here  p) = refl
   ++⁺∘++⁻ (x ∷ xs) (there p) with ++⁻ xs p | ++⁺∘++⁻ xs p
-  ... | inj₁ p′ | ih = P.cong there ih
-  ... | inj₂ p′ | ih = P.cong there ih
+  ... | inj₁ p′ | ih = cong there ih
+  ... | inj₂ p′ | ih = cong there ih
 
   ++⁻∘++⁺ : ∀ xs {ys} (p : Any P xs ⊎ Any P ys) →
             ++⁻ xs ([ ++⁺ˡ , ++⁺ʳ xs ]′ p) ≡ p
@@ -441,7 +441,7 @@ module _ {P : A → Set p} where
   concat⁻∘concat⁺ (here                      p) = concat⁻∘++⁺ˡ _ p
   concat⁻∘concat⁺ (there {x = xs} {xs = xss} p)
     rewrite concat⁻∘++⁺ʳ xs xss (concat⁺ p) =
-      P.cong there $ concat⁻∘concat⁺ p
+      cong there $ concat⁻∘concat⁺ p
 
   concat↔ : ∀ {xss} → Any (Any P) xss ↔ Any P (concat xss)
   concat↔ {xss} = mk↔ₛ′ concat⁺ (concat⁻ xss) (concat⁺∘concat⁻ xss) concat⁻∘concat⁺
@@ -496,7 +496,7 @@ applyUpTo⁻ f {suc n} (there p) with applyUpTo⁻ (f ∘ suc) p
 
 applyDownFrom⁺ : ∀ f {i n} → P (f i) → i < n → Any P (applyDownFrom f n)
 applyDownFrom⁺ f {i} {suc n} p (s≤s i≤n) with i ≟ n
-... | yes P.refl = here p
+... | yes refl = here p
 ... | no  i≢n    = there (applyDownFrom⁺ f p (≤∧≢⇒< i≤n i≢n))
 
 applyDownFrom⁻ : ∀ f {n} → Any P (applyDownFrom f n) →
@@ -611,7 +611,7 @@ module _ {P : B → Set p} where
               mapWith∈⁺ f (mapWith∈⁻ xs f p) ≡ p
     to∘from (y ∷ xs) f (here  p) = refl
     to∘from (y ∷ xs) f (there p) =
-      P.cong there $ to∘from xs (f ∘ there) p
+      cong there $ to∘from xs (f ∘ there) p
 
 ------------------------------------------------------------------------
 -- reverse
@@ -686,7 +686,7 @@ module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where
   Any (λ f → Any (P ∘ f) xs) fs                ↔⟨ Any-cong (λ _ → Any-cong (λ _ → pure↔) (_ ∎)) (_ ∎) ⟩
   Any (λ f → Any (Any P ∘ pure ∘ f) xs) fs     ↔⟨ Any-cong (λ _ → >>=↔ ) (_ ∎) ⟩
   Any (λ f → Any P (xs >>= pure ∘ f)) fs       ↔⟨ >>=↔ ⟩
-  Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡⟨ P.cong (Any P) (Listₑ.Applicative.unfold-⊛ fs xs) ⟨
+  Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡⟨ cong (Any P) (Listₑ.Applicative.unfold-⊛ fs xs) ⟨
   Any P (fs ⊛ xs)                               ∎
   where open Related.EquationalReasoning
 
@@ -706,7 +706,7 @@ module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where
   Any (λ x → Any (λ y → P (x , y)) ys) xs                           ↔⟨ pure↔ ⟩
   Any (λ _,_ → Any (λ x → Any (λ y → P (x , y)) ys) xs) (pure _,_)  ↔⟨ ⊛↔ ⟩
   Any (λ x, → Any (P ∘ x,) ys) (pure _,_ ⊛ xs)                      ↔⟨ ⊛↔ ⟩
-  Any P (pure _,_ ⊛ xs ⊛ ys)                                        ≡⟨ P.cong (Any P ∘′ (_⊛ ys)) (Listₑ.Applicative.unfold-<$> _,_ xs) ⟨
+  Any P (pure _,_ ⊛ xs ⊛ ys)                                        ≡⟨ cong (Any P ∘′ (_⊛ ys)) (Listₑ.Applicative.unfold-<$> _,_ xs) ⟨
   Any P (xs ⊗ ys)                                                   ∎
   where open Related.EquationalReasoning
 
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index f5e0a53c5c..f1a95b1e14 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -23,7 +23,7 @@ open import Function.Definitions using (Surjective)
 open import Function.Consequences using (strictlySurjective⇒surjective)
 open import Level
 open import Relation.Binary.Bundles using (Setoid; DecSetoid)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Properties.Setoid using (respʳ-flip)
 
 module Data.List.Relation.Unary.Enumerates.Setoid.Properties where
@@ -89,4 +89,4 @@ module _ (S : Setoid a ℓ₁) where
   lookup-surjective : ∀ {xs} → IsEnumeration S xs →
                       Surjective _≡_ _≈_ (lookup xs)
   lookup-surjective _∈xs = strictlySurjective⇒surjective
-    trans (λ { P.refl → refl}) (λ y → index (y ∈xs) , sym (lookup-index (y ∈xs)))
+    trans (λ { ≡.refl → refl}) (λ y → index (y ∈xs) , sym (lookup-index (y ∈xs)))
diff --git a/src/Data/List/Relation/Unary/First/Properties.agda b/src/Data/List/Relation/Unary/First/Properties.agda
index f30e037022..d598f14469 100644
--- a/src/Data/List/Relation/Unary/First/Properties.agda
+++ b/src/Data/List/Relation/Unary/First/Properties.agda
@@ -16,7 +16,7 @@ open import Data.List.Relation.Unary.Any as Any using (here; there)
 open import Data.List.Relation.Unary.First
 import Data.Sum as Sum
 open import Function.Base using (_∘′_; _$_; _∘_; id)
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl; _≗_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; refl; _≗_)
 open import Relation.Unary
 open import Relation.Nullary.Negation
 
@@ -66,14 +66,14 @@ module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
   unique-index p⇒¬q [ _ ]    [ _ ]    = refl
   unique-index p⇒¬q [ qx ]   (px ∷ _) = ⊥-elim (p⇒¬q px qx)
   unique-index p⇒¬q (px ∷ _) [ qx ]   = ⊥-elim (p⇒¬q px qx)
-  unique-index p⇒¬q (_ ∷ f₁) (_ ∷ f₂) = P.cong suc (unique-index p⇒¬q f₁ f₂)
+  unique-index p⇒¬q (_ ∷ f₁) (_ ∷ f₂) = ≡.cong suc (unique-index p⇒¬q f₁ f₂)
 
   irrelevant : P ⊆ ∁ Q → Irrelevant P → Irrelevant Q → Irrelevant (First P Q)
-  irrelevant p⇒¬q p-irr q-irr [ qx₁ ]    [ qx₂ ]    = P.cong [_] (q-irr qx₁ qx₂)
+  irrelevant p⇒¬q p-irr q-irr [ qx₁ ]    [ qx₂ ]    = ≡.cong [_] (q-irr qx₁ qx₂)
   irrelevant p⇒¬q p-irr q-irr [ qx₁ ]    (px₂ ∷ f₂) = ⊥-elim (p⇒¬q px₂ qx₁)
   irrelevant p⇒¬q p-irr q-irr (px₁ ∷ f₁) [ qx₂ ]    = ⊥-elim (p⇒¬q px₁ qx₂)
   irrelevant p⇒¬q p-irr q-irr (px₁ ∷ f₁) (px₂ ∷ f₂) =
-    P.cong₂ _∷_ (p-irr px₁ px₂) (irrelevant p⇒¬q p-irr q-irr f₁ f₂)
+    ≡.cong₂ _∷_ (p-irr px₁ px₂) (irrelevant p⇒¬q p-irr q-irr f₁ f₂)
 
 ------------------------------------------------------------------------
 -- Decidability
@@ -97,11 +97,11 @@ module _ {a p} {A : Set a} {P : Pred A p} where
 
   fromAny∘toAny≗id : ∀ {xs} → fromAny {Q = P} {x = xs} ∘′ toAny ≗ id
   fromAny∘toAny≗id [ qx ]      = refl
-  fromAny∘toAny≗id (px ∷ pqxs) = P.cong (_ ∷_) (fromAny∘toAny≗id pqxs)
+  fromAny∘toAny≗id (px ∷ pqxs) = ≡.cong (_ ∷_) (fromAny∘toAny≗id pqxs)
 
   toAny∘fromAny≗id : ∀ {xs} → toAny {Q = P} ∘′ fromAny {x = xs} ≗ id
   toAny∘fromAny≗id (here px) = refl
-  toAny∘fromAny≗id (there v) = P.cong there (toAny∘fromAny≗id v)
+  toAny∘fromAny≗id (there v) = ≡.cong there (toAny∘fromAny≗id v)
 
 ------------------------------------------------------------------------
 -- Equivalence between the inductive definition and the view
diff --git a/src/Data/Maybe/Relation/Binary/Connected.agda b/src/Data/Maybe/Relation/Binary/Connected.agda
index 8c174c9ea1..7bf8f33310 100644
--- a/src/Data/Maybe/Relation/Binary/Connected.agda
+++ b/src/Data/Maybe/Relation/Binary/Connected.agda
@@ -13,7 +13,7 @@ open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Relation.Binary.Core using (REL; _⇒_)
 open import Relation.Binary.Definitions using (Reflexive; Sym; Decidable)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 
@@ -52,7 +52,7 @@ refl R-refl {just _}  = just R-refl
 refl R-refl {nothing} = nothing
 
 reflexive : _≡_ ⇒ R → _≡_ ⇒ Connected R
-reflexive reflexive P.refl = refl (reflexive P.refl)
+reflexive reflexive ≡.refl = refl (reflexive ≡.refl)
 
 sym : Sym R S → Sym (Connected R) (Connected S)
 sym R-sym (just p)     = just (R-sym p)
diff --git a/src/Data/Maybe/Relation/Binary/Pointwise.agda b/src/Data/Maybe/Relation/Binary/Pointwise.agda
index 3da2abd9f1..073ac74a81 100644
--- a/src/Data/Maybe/Relation/Binary/Pointwise.agda
+++ b/src/Data/Maybe/Relation/Binary/Pointwise.agda
@@ -17,7 +17,7 @@ open import Relation.Binary.Core using (REL; Rel; _⇒_)
 open import Relation.Binary.Bundles using (Setoid; DecSetoid)
 open import Relation.Binary.Definitions using (Reflexive; Sym; Trans; Decidable)
 open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Nullary
 open import Relation.Unary using (_⊆_)
 import Relation.Nullary.Decidable as Dec
@@ -43,10 +43,10 @@ module _ {a b ℓ} {A : Set a} {B : Set b} {R : REL A B ℓ} where
   just-equivalence = mk⇔ just drop-just
 
   nothing-inv : ∀ {x} → Pointwise R nothing x → x ≡ nothing
-  nothing-inv nothing = P.refl
+  nothing-inv nothing = ≡.refl
 
   just-inv : ∀ {x y} → Pointwise R (just x) y → ∃ λ z → y ≡ just z × R x z
-  just-inv (just r) = -, P.refl , r
+  just-inv (just r) = -, ≡.refl , r
 
 ------------------------------------------------------------------------
 -- Relational properties
@@ -58,7 +58,7 @@ module _ {a r} {A : Set a} {R : Rel A r} where
   refl R-refl {nothing} = nothing
 
   reflexive : _≡_ ⇒ R → _≡_ ⇒ Pointwise R
-  reflexive reflexive P.refl = refl (reflexive P.refl)
+  reflexive reflexive ≡.refl = refl (reflexive ≡.refl)
 
 module _ {a b r₁ r₂} {A : Set a} {B : Set b}
          {R : REL A B r₁} {S : REL B A r₂} where
diff --git a/src/Data/Maybe/Relation/Unary/All.agda b/src/Data/Maybe/Relation/Unary/All.agda
index 9a57aa62f5..f26a261f42 100644
--- a/src/Data/Maybe/Relation/Unary/All.agda
+++ b/src/Data/Maybe/Relation/Unary/All.agda
@@ -16,7 +16,7 @@ open import Data.Product.Base as Product using (_,_)
 open import Function.Base using (id; _∘′_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Unary
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
@@ -114,7 +114,7 @@ module _ {a p} {A : Set a} {P : Pred A p} where
 
   irrelevant : Irrelevant P → Irrelevant (All P)
   irrelevant P-irrelevant (just p) (just q) = cong just (P-irrelevant p q)
-  irrelevant P-irrelevant nothing  nothing  = P.refl
+  irrelevant P-irrelevant nothing  nothing  = refl
 
   satisfiable : Satisfiable (All P)
   satisfiable = nothing , nothing
diff --git a/src/Data/Maybe/Relation/Unary/Any.agda b/src/Data/Maybe/Relation/Unary/Any.agda
index 767a5624cc..d379ce2998 100644
--- a/src/Data/Maybe/Relation/Unary/Any.agda
+++ b/src/Data/Maybe/Relation/Unary/Any.agda
@@ -13,7 +13,7 @@ open import Data.Product.Base as Product using (∃; _,_; -,_)
 open import Function.Base using (id)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 open import Relation.Unary
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
diff --git a/src/Data/Nat/Coprimality.agda b/src/Data/Nat/Coprimality.agda
index f513e82fa4..16a9cf9897 100644
--- a/src/Data/Nat/Coprimality.agda
+++ b/src/Data/Nat/Coprimality.agda
@@ -19,7 +19,7 @@ open import Data.Product.Base as Prod
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
 open import Function.Base using (_∘_)
 open import Level using (0ℓ)
-open import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≢_; refl; trans; cong; subst)
 open import Relation.Nullary as Nullary using (¬_; contradiction; map′)
 open import Relation.Binary.Core using (Rel)
@@ -47,7 +47,7 @@ coprime⇒GCD≡1 {m} {n} coprime = GCD.is (1∣ m , 1∣ n) (∣-reflexive ∘
 
 GCD≡1⇒coprime : GCD m n 1 → Coprime m n
 GCD≡1⇒coprime g cd with divides q eq ← GCD.greatest g cd
-  = m*n≡1⇒n≡1 q _ (P.sym eq)
+  = m*n≡1⇒n≡1 q _ (≡.sym eq)
 
 coprime⇒gcd≡1 : Coprime m n → gcd m n ≡ 1
 coprime⇒gcd≡1 coprime = GCD.unique (gcd-GCD _ _) (coprime⇒GCD≡1 coprime)
diff --git a/src/Data/Nat/GCD.agda b/src/Data/Nat/GCD.agda
index b88947c707..26d44549fb 100644
--- a/src/Data/Nat/GCD.agda
+++ b/src/Data/Nat/GCD.agda
@@ -22,7 +22,7 @@ open import Function.Base using (_$_; _∘_)
 open import Induction using (build)
 open import Induction.Lexicographic using (_⊗_; [_⊗_])
 open import Relation.Binary.Definitions using (tri<; tri>; tri≈; Symmetric)
-open import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≢_; subst; cong)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
@@ -79,7 +79,7 @@ gcd[m,n]∣m m n with <-cmp m n
 gcd[m,n]∣n : ∀ m n → gcd m n ∣ n
 gcd[m,n]∣n m n with <-cmp m n
 ... | tri< n<m _    _ = proj₁ (gcd′[m,n]∣m,n {n} {m} _ _)
-... | tri≈ _ P.refl _ = ∣-refl
+... | tri≈ _ ≡.refl _ = ∣-refl
 ... | tri> _ _    m<n = proj₂ (gcd′[m,n]∣m,n {m} {n} _ _)
 
 gcd-greatest : ∀ {m n c} → c ∣ m → c ∣ n → c ∣ gcd m n
@@ -102,11 +102,11 @@ gcd[m,n]≢0 m n (inj₁ m≢0) eq = m≢0 (0∣⇒≡0 (subst (_∣ m) eq (gcd[
 gcd[m,n]≢0 m n (inj₂ n≢0) eq = n≢0 (0∣⇒≡0 (subst (_∣ n) eq (gcd[m,n]∣n m n)))
 
 gcd[m,n]≡0⇒m≡0 : ∀ {m n} → gcd m n ≡ 0 → m ≡ 0
-gcd[m,n]≡0⇒m≡0 {zero}  {n} eq = P.refl
+gcd[m,n]≡0⇒m≡0 {zero}  {n} eq = ≡.refl
 gcd[m,n]≡0⇒m≡0 {suc m} {n} eq = contradiction eq (gcd[m,n]≢0 (suc m) n (inj₁ λ()))
 
 gcd[m,n]≡0⇒n≡0 : ∀ m {n} → gcd m n ≡ 0 → n ≡ 0
-gcd[m,n]≡0⇒n≡0 m {zero}  eq = P.refl
+gcd[m,n]≡0⇒n≡0 m {zero}  eq = ≡.refl
 gcd[m,n]≡0⇒n≡0 m {suc n} eq = contradiction eq (gcd[m,n]≢0 m (suc n) (inj₂ λ()))
 
 gcd-comm : Commutative gcd
@@ -140,12 +140,12 @@ gcd-assoc m n p = ∣-antisym
       p               ∎
 
 gcd-identityˡ : LeftIdentity 0 gcd
-gcd-identityˡ zero = P.refl
-gcd-identityˡ (suc _) = P.refl
+gcd-identityˡ zero = ≡.refl
+gcd-identityˡ (suc _) = ≡.refl
 
 gcd-identityʳ : RightIdentity 0 gcd
-gcd-identityʳ zero = P.refl
-gcd-identityʳ (suc _) = P.refl
+gcd-identityʳ zero = ≡.refl
+gcd-identityʳ (suc _) = ≡.refl
 
 gcd-identity : Algebra.Identity 0 gcd
 gcd-identity = gcd-identityˡ , gcd-identityʳ
@@ -187,9 +187,9 @@ gcd[cm,cn]/c≡gcd[m,n] c m n = gcd-universality forwards backwards
       *-cancelˡ-∣ c (∣-trans cd∣gcd[cm,n] (gcd[m,n]∣n (c * m) _))
 
 c*gcd[m,n]≡gcd[cm,cn] : ∀ c m n → c * gcd m n ≡ gcd (c * m) (c * n)
-c*gcd[m,n]≡gcd[cm,cn] zero      m n = P.sym gcd[0,0]≡0
+c*gcd[m,n]≡gcd[cm,cn] zero      m n = ≡.sym gcd[0,0]≡0
 c*gcd[m,n]≡gcd[cm,cn] c@(suc _) m n = begin
-  c * gcd m n                   ≡⟨ cong (c *_) (P.sym (gcd[cm,cn]/c≡gcd[m,n] c m n)) ⟩
+  c * gcd m n                   ≡⟨ cong (c *_) (≡.sym (gcd[cm,cn]/c≡gcd[m,n] c m n)) ⟩
   c * (gcd (c * m) (c * n) / c) ≡⟨ m*[n/m]≡n (gcd-greatest (m∣m*n m) (m∣m*n n)) ⟩
   gcd (c * m) (c * n)           ∎
   where open ≡-Reasoning
@@ -282,7 +282,7 @@ mkGCD m n = gcd m n , gcd-GCD m n
 
 gcd? : (m n d : ℕ) → Dec (GCD m n d)
 gcd? m n d =
-  Dec.map′ (λ { P.refl → gcd-GCD m n }) (GCD.unique (gcd-GCD m n))
+  Dec.map′ (λ { ≡.refl → gcd-GCD m n }) (GCD.unique (gcd-GCD m n))
            (gcd m n ≟ d)
 
 GCD-* : ∀ {m n d c} .{{_ : NonZero c}} → GCD (m * c) (n * c) (d * c) → GCD m n d
@@ -297,7 +297,7 @@ GCD-/ {m} {n} {d} {c} {{x}}
   rewrite m*n/n≡m p c {{x}} | m*n/n≡m q c {{x}} | m*n/n≡m r c {{x}} = GCD-* gcd
 
 GCD-/gcd : ∀ m n .{{_ : NonZero (gcd m n)}} → GCD (m / gcd m n) (n / gcd m n) 1
-GCD-/gcd m n rewrite P.sym (n/n≡1 (gcd m n)) =
+GCD-/gcd m n rewrite ≡.sym (n/n≡1 (gcd m n)) =
   GCD-/ (gcd[m,n]∣m m n) (gcd[m,n]∣n m n) ∣-refl (gcd-GCD m n)
 
 ------------------------------------------------------------------------
@@ -327,10 +327,10 @@ module Bézout where
     sym (-+ x y eq) = +- y x eq
 
     refl : ∀ {d} → Identity d d d
-    refl = -+ 0 1 P.refl
+    refl = -+ 0 1 ≡.refl
 
     base : ∀ {d} → Identity d 0 d
-    base = -+ 0 1 P.refl
+    base = -+ 0 1 ≡.refl
 
     private
       infixl 7 _⊕_
diff --git a/src/Data/Nat/LCM.agda b/src/Data/Nat/LCM.agda
index aead0fbf01..fc8937577b 100644
--- a/src/Data/Nat/LCM.agda
+++ b/src/Data/Nat/LCM.agda
@@ -17,8 +17,8 @@ open import Data.Nat.Properties
 open import Data.Nat.GCD
 open import Data.Product.Base using (_×_; _,_; uncurry′; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl; sym; trans; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; trans; cong; cong₂; subst)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 open import Relation.Nullary.Decidable using (False; fromWitnessFalse)
@@ -32,8 +32,8 @@ private
 -- Definition
 
 lcm : ℕ → ℕ → ℕ
-lcm zero        n = zero
-lcm m@(suc m-1) n = m * (n / gcd m n)
+lcm zero      n = zero
+lcm m@(suc _) n = m * (n / gcd m n)
   where instance _ = gcd≢0ˡ {m} {n}
 
 ------------------------------------------------------------------------
@@ -62,7 +62,7 @@ n∣lcm[m,n] m@(suc m-1) n = begin
 
 lcm-least : ∀ {m n c} → m ∣ c → n ∣ c → lcm m n ∣ c
 lcm-least {zero}      {n} {c} 0∣c _   = 0∣c
-lcm-least {m@(suc _)} {n} {c} m∣c n∣c = P.subst (_∣ c) (sym (rearrange m n))
+lcm-least {m@(suc _)} {n} {c} m∣c n∣c = subst (_∣ c) (sym (rearrange m n))
   (m∣n*o⇒m/n∣o gcd[m,n]∣m*n mn∣c*gcd)
   where
   instance _ = gcd≢0ˡ {m} {n}
@@ -73,7 +73,7 @@ lcm-least {m@(suc _)} {n} {c} m∣c n∣c = P.subst (_∣ c) (sym (rearrange m n
 
   mn∣c*gcd : m * n ∣ c * gcd m n
   mn∣c*gcd = begin
-    m * n               ∣⟨  gcd-greatest (P.subst (_∣ c * m) (*-comm n m) (*-monoˡ-∣ m n∣c)) (*-monoˡ-∣ n m∣c) ⟩
+    m * n               ∣⟨  gcd-greatest (subst (_∣ c * m) (*-comm n m) (*-monoˡ-∣ m n∣c)) (*-monoˡ-∣ n m∣c) ⟩
     gcd (c * m) (c * n) ≡⟨ c*gcd[m,n]≡gcd[cm,cn] c m n ⟨
     c * gcd m n         ∎
 
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index b99e1934f8..2063e1cfbb 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -24,7 +24,7 @@ open import Function.Consequences.Propositional
   using (inverseʳ⇒injective; strictlySurjective⇒surjective)
 open import Function.Definitions using (Inverseˡ; Inverseʳ; Injective; StrictlySurjective)
 open import Function.Bundles
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
 
 private
   variable
@@ -54,7 +54,7 @@ module _ where
          Σ I A ⇔ Σ J B
   Σ-⇔ {B = B} I↠J A⇔B = mk⇔
     (map (to  I↠J) (Equivalence.to A⇔B))
-    (map (to⁻ I↠J) (Equivalence.from A⇔B ∘ P.subst B (P.sym (proj₂ (surjective I↠J _) P.refl))))
+    (map (to⁻ I↠J) (Equivalence.from A⇔B ∘ ≡.subst B (≡.sym (proj₂ (surjective I↠J _) ≡.refl))))
 
   -- See also Data.Product.Relation.Binary.Pointwise.Dependent.WithK.↣.
 
@@ -68,7 +68,7 @@ module _ where
          Σ I A ↣ Σ J B
   Σ-↣ {I = I} {J = J} {A = A} {B = B} I↔J A↣B = mk↣ to-injective
     where
-    open P.≡-Reasoning
+    open ≡.≡-Reasoning
 
     I≃J = ↔⇒≃ I↔J
 
@@ -77,42 +77,42 @@ module _ where
       {x₁ x₂ : I} {y : A (from (to x₁))}
       (g : ∀ x → A (from (to x)) → B (to x))
       (eq : to x₁ ≡ to x₂) →
-      P.subst B eq (g x₁ y) ≡ g x₂ (P.subst A (P.cong from eq) y)
+      ≡.subst B eq (g x₁ y) ≡ g x₂ (≡.subst A (≡.cong from eq) y)
     subst-application′ {x₁} {x₂} {y} g eq =
-      P.subst B eq (g x₁ y)                      ≡⟨ P.cong (P.subst B eq) (P.sym (g′-lemma _ _)) ⟩
-      P.subst B eq (g′ (to x₁) y)                ≡⟨ P.subst-application A g′ eq ⟩
-      g′ (to x₂) (P.subst A (P.cong from eq) y)  ≡⟨ g′-lemma _ _ ⟩
-      g x₂ (P.subst A (P.cong from eq) y)        ∎
+      ≡.subst B eq (g x₁ y)                      ≡⟨ ≡.cong (≡.subst B eq) (≡.sym (g′-lemma _ _)) ⟩
+      ≡.subst B eq (g′ (to x₁) y)                ≡⟨ ≡.subst-application A g′ eq ⟩
+      g′ (to x₂) (≡.subst A (≡.cong from eq) y)  ≡⟨ g′-lemma _ _ ⟩
+      g x₂ (≡.subst A (≡.cong from eq) y)        ∎
       where
       open _≃_ I≃J
 
       g′ : ∀ x → A (from x) → B x
       g′ x =
-        P.subst B (right-inverse-of x) ∘
+        ≡.subst B (right-inverse-of x) ∘
         g (from x) ∘
-        P.subst A (P.sym (P.cong from (right-inverse-of x)))
+        ≡.subst A (≡.sym (≡.cong from (right-inverse-of x)))
 
       g′-lemma : ∀ x y → g′ (to x) y ≡ g x y
       g′-lemma x y =
-        P.subst B (right-inverse-of (to x))
+        ≡.subst B (right-inverse-of (to x))
           (g (from (to x)) $
-           P.subst A (P.sym (P.cong from (right-inverse-of (to x)))) y)  ≡⟨ P.cong (λ p → P.subst B p (g (from (to x))
-                                                                                                           (P.subst A (P.sym (P.cong from p)) y)))
-                                                                               (P.sym (left-right x)) ⟩
-        P.subst B (P.cong to (left-inverse-of x))
+           ≡.subst A (≡.sym (≡.cong from (right-inverse-of (to x)))) y)  ≡⟨ ≡.cong (λ p → ≡.subst B p (g (from (to x))
+                                                                                                           (≡.subst A (≡.sym (≡.cong from p)) y)))
+                                                                               (≡.sym (left-right x)) ⟩
+        ≡.subst B (≡.cong to (left-inverse-of x))
           (g (from (to x)) $
-           P.subst A
-             (P.sym (P.cong from (P.cong to (left-inverse-of x))))
+           ≡.subst A
+             (≡.sym (≡.cong from (≡.cong to (left-inverse-of x))))
              y)                                                           ≡⟨ lemma _ ⟩
 
         g x y                                                             ∎
         where
         lemma : ∀ {x′} eq {y : A (from (to x′))} →
-                  P.subst B (P.cong to eq)
+                  ≡.subst B (≡.cong to eq)
                     (g (from (to x))
-                      (P.subst A (P.sym (P.cong from (P.cong to eq))) y)) ≡
+                      (≡.subst A (≡.sym (≡.cong from (≡.cong to eq))) y)) ≡
                   g x′ y
-        lemma P.refl = P.refl
+        lemma ≡.refl = ≡.refl
 
     open Injection
 
@@ -125,49 +125,49 @@ module _ where
       Σ-≡,≡→≡ ∘′
 
       map (_≃_.injective I≃J) (λ {eq₁} eq₂ → injective A↣B (
-              to A↣B (P.subst A (_≃_.injective I≃J eq₁) x₂)             ≡⟨⟩
+              to A↣B (≡.subst A (_≃_.injective I≃J eq₁) x₂)             ≡⟨⟩
 
               (let eq =
-                      P.trans (P.sym (_≃_.left-inverse-of I≃J x₁))
-                        (P.trans (P.cong (_≃_.from I≃J) eq₁)
-                          (P.trans (_≃_.left-inverse-of I≃J y₁)
-                            P.refl)) in
-              to A↣B (P.subst A eq x₂))                                   ≡⟨ P.cong (λ p → to A↣B
-                                                                                             (P.subst A
-                                                                                               (P.trans (P.sym (_≃_.left-inverse-of I≃J _))
-                                                                                                  (P.trans (P.cong (_≃_.from I≃J) eq₁) p))
+                      ≡.trans (≡.sym (_≃_.left-inverse-of I≃J x₁))
+                        (≡.trans (≡.cong (_≃_.from I≃J) eq₁)
+                          (≡.trans (_≃_.left-inverse-of I≃J y₁)
+                            ≡.refl)) in
+              to A↣B (≡.subst A eq x₂))                                   ≡⟨ ≡.cong (λ p → to A↣B
+                                                                                             (≡.subst A
+                                                                                               (≡.trans (≡.sym (_≃_.left-inverse-of I≃J _))
+                                                                                                  (≡.trans (≡.cong (_≃_.from I≃J) eq₁) p))
                                                                                                x₂))
-                                                                               (P.trans-reflʳ _) ⟩
+                                                                               (≡.trans-reflʳ _) ⟩
 
-              (let eq = P.trans (P.sym (_≃_.left-inverse-of I≃J x₁))
-                          (P.trans (P.cong (_≃_.from I≃J) eq₁)
+              (let eq = ≡.trans (≡.sym (_≃_.left-inverse-of I≃J x₁))
+                          (≡.trans (≡.cong (_≃_.from I≃J) eq₁)
                             (_≃_.left-inverse-of I≃J y₁)) in
-              to A↣B (P.subst A eq x₂))                                  ≡⟨ P.cong (to A↣B)
-                                                                               (P.sym (P.subst-subst (P.sym (_≃_.left-inverse-of I≃J _)))) ⟩
+              to A↣B (≡.subst A eq x₂))                                  ≡⟨ ≡.cong (to A↣B)
+                                                                               (≡.sym (≡.subst-subst (≡.sym (_≃_.left-inverse-of I≃J _)))) ⟩
 
-              to A↣B ((P.subst A (P.trans (P.cong (_≃_.from I≃J) eq₁)
+              to A↣B ((≡.subst A (≡.trans (≡.cong (_≃_.from I≃J) eq₁)
                              (_≃_.left-inverse-of I≃J y₁)) $
-               P.subst A (P.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ P.cong (to A↣B)
-                                                                               (P.sym (P.subst-subst (P.cong (_≃_.from I≃J) eq₁))) ⟩
+               ≡.subst A (≡.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ ≡.cong (to A↣B)
+                                                                               (≡.sym (≡.subst-subst (≡.cong (_≃_.from I≃J) eq₁))) ⟩
               to A↣B (
-              (P.subst A (_≃_.left-inverse-of I≃J y₁) $
-               P.subst A (P.cong (_≃_.from I≃J) eq₁) $
-               P.subst A (P.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ P.sym (subst-application′
+              (≡.subst A (_≃_.left-inverse-of I≃J y₁) $
+               ≡.subst A (≡.cong (_≃_.from I≃J) eq₁) $
+               ≡.subst A (≡.sym (_≃_.left-inverse-of I≃J x₁)) x₂))      ≡⟨ ≡.sym (subst-application′
                                                                                       (λ x y → to A↣B
-                                                                                                 (P.subst A (_≃_.left-inverse-of I≃J x) y))
+                                                                                                 (≡.subst A (_≃_.left-inverse-of I≃J x) y))
                                                                                       eq₁) ⟩
-              P.subst B eq₁ (to A↣B $
-                 (P.subst A (_≃_.left-inverse-of I≃J x₁) $
-                  P.subst A (P.sym (_≃_.left-inverse-of I≃J x₁)) x₂))  ≡⟨ P.cong (P.subst B eq₁ ∘ to A↣B)
-                                                                               (P.subst-subst (P.sym (_≃_.left-inverse-of I≃J _))) ⟩
+              ≡.subst B eq₁ (to A↣B $
+                 (≡.subst A (_≃_.left-inverse-of I≃J x₁) $
+                  ≡.subst A (≡.sym (_≃_.left-inverse-of I≃J x₁)) x₂))  ≡⟨ ≡.cong (≡.subst B eq₁ ∘ to A↣B)
+                                                                               (≡.subst-subst (≡.sym (_≃_.left-inverse-of I≃J _))) ⟩
 
-              (let eq = P.trans (P.sym (_≃_.left-inverse-of I≃J x₁))
+              (let eq = ≡.trans (≡.sym (_≃_.left-inverse-of I≃J x₁))
                           (_≃_.left-inverse-of I≃J x₁) in
-              P.subst B eq₁ (to A↣B (P.subst A eq x₂)))                  ≡⟨ P.cong (λ p → P.subst B eq₁ (to A↣B (P.subst A p x₂)))
-                                                                              (P.trans-symˡ (_≃_.left-inverse-of I≃J _)) ⟩
-              P.subst B eq₁ (to A↣B (P.subst A P.refl x₂))               ≡⟨⟩
+              ≡.subst B eq₁ (to A↣B (≡.subst A eq x₂)))                  ≡⟨ ≡.cong (λ p → ≡.subst B eq₁ (to A↣B (≡.subst A p x₂)))
+                                                                              (≡.trans-symˡ (_≃_.left-inverse-of I≃J _)) ⟩
+              ≡.subst B eq₁ (to A↣B (≡.subst A ≡.refl x₂))               ≡⟨⟩
 
-              P.subst B eq₁ (to A↣B x₂)                                  ≡⟨ eq₂ ⟩
+              ≡.subst B eq₁ (to A↣B x₂)                                  ≡⟨ eq₂ ⟩
 
               to A↣B y₂                                                  ∎
 
@@ -191,7 +191,7 @@ module _ where
     to′ = map (to I↠J) (to A↠B)
 
     backcast : ∀ {i} → B i → B (to I↠J (to⁻ I↠J i))
-    backcast = P.subst B (P.sym (to∘to⁻ I↠J _))
+    backcast = ≡.subst B (≡.sym (to∘to⁻ I↠J _))
 
     to⁻′ : Σ J B → Σ I A
     to⁻′ = map (to⁻ I↠J) (Surjection.to⁻ A↠B ∘ backcast)
@@ -199,10 +199,10 @@ module _ where
     strictlySurjective′ : StrictlySurjective _≡_ to′
     strictlySurjective′ (x , y) = to⁻′ (x , y) , Σ-≡,≡→≡
       ( to∘to⁻ I↠J x
-      , (P.subst B (to∘to⁻ I↠J x) (to A↠B (to⁻ A↠B (backcast y))) ≡⟨ P.cong (P.subst B _) (to∘to⁻ A↠B _) ⟩
-         P.subst B (to∘to⁻ I↠J x) (backcast y)                      ≡⟨ P.subst-subst-sym (to∘to⁻ I↠J x) ⟩
+      , (≡.subst B (to∘to⁻ I↠J x) (to A↠B (to⁻ A↠B (backcast y))) ≡⟨ ≡.cong (≡.subst B _) (to∘to⁻ A↠B _) ⟩
+         ≡.subst B (to∘to⁻ I↠J x) (backcast y)                      ≡⟨ ≡.subst-subst-sym (to∘to⁻ I↠J x) ⟩
          y                                                          ∎)
-      ) where open P.≡-Reasoning
+      ) where open ≡.≡-Reasoning
 
 
 ------------------------------------------------------------------------
@@ -220,17 +220,17 @@ module _ where
     to′ = map (to I↩J) (to A↩B)
 
     backcast : ∀ {j} → B j → B (to I↩J (from I↩J j))
-    backcast = P.subst B (P.sym (inverseˡ I↩J P.refl))
+    backcast = ≡.subst B (≡.sym (inverseˡ I↩J ≡.refl))
 
     from′ : Σ J B → Σ I A
     from′ = map (from I↩J) (from A↩B ∘ backcast)
 
     inv : Inverseˡ _≡_ _≡_ to′ from′
-    inv {j , b} P.refl = Σ-≡,≡→≡ (strictlyInverseˡ I↩J j  , (
+    inv {j , b} ≡.refl = Σ-≡,≡→≡ (strictlyInverseˡ I↩J j  , (
       begin
-        P.subst B (inverseˡ I↩J P.refl) (to A↩B (from A↩B (backcast b))) ≡⟨ P.cong (P.subst B _) (inverseˡ A↩B P.refl) ⟩
-        P.subst B (inverseˡ I↩J P.refl) (backcast b)                       ≡⟨ P.subst-subst-sym (inverseˡ I↩J _) ⟩
-        b                                                                  ∎)) where open P.≡-Reasoning
+        ≡.subst B (inverseˡ I↩J ≡.refl) (to A↩B (from A↩B (backcast b))) ≡⟨ ≡.cong (≡.subst B _) (inverseˡ A↩B ≡.refl) ⟩
+        ≡.subst B (inverseˡ I↩J ≡.refl) (backcast b)                       ≡⟨ ≡.subst-subst-sym (inverseˡ I↩J _) ⟩
+        b                                                                  ∎)) where open ≡.≡-Reasoning
 
 ------------------------------------------------------------------------
 -- Right inverses
@@ -250,7 +250,7 @@ module _ where
     (Surjection.to∘to⁻ surjection′)
     left-inverse-of
     where
-    open P.≡-Reasoning
+    open ≡.≡-Reasoning
 
     I≃J = ↔⇒≃ I↔J
 
@@ -260,27 +260,27 @@ module _ where
     left-inverse-of : ∀ p → Surjection.to⁻ surjection′ (Surjection.to surjection′ p) ≡ p
     left-inverse-of (x , y) = to Σ-≡,≡↔≡
       ( _≃_.left-inverse-of I≃J x
-      , (P.subst A (_≃_.left-inverse-of I≃J x)
+      , (≡.subst A (_≃_.left-inverse-of I≃J x)
            (from A↔B
-              (P.subst B (P.sym (_≃_.right-inverse-of I≃J
+              (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J
                                     (_≃_.to I≃J x)))
-                 (to A↔B y)))                   ≡⟨ P.subst-application B (λ _ → from A↔B) _ ⟩
+                 (to A↔B y)))                   ≡⟨ ≡.subst-application B (λ _ → from A↔B) _ ⟩
 
          from A↔B
-           (P.subst B (P.cong (_≃_.to I≃J)
+           (≡.subst B (≡.cong (_≃_.to I≃J)
                           (_≃_.left-inverse-of I≃J x))
-              (P.subst B (P.sym (_≃_.right-inverse-of I≃J
+              (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J
                                     (_≃_.to I≃J x)))
-                 (to A↔B y)))                   ≡⟨ P.cong (λ eq → from A↔B (P.subst B eq
-                                                                                  (P.subst B (P.sym (_≃_.right-inverse-of I≃J _)) _)))
+                 (to A↔B y)))                   ≡⟨ ≡.cong (λ eq → from A↔B (≡.subst B eq
+                                                                                  (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J _)) _)))
                                                                    (_≃_.left-right I≃J _) ⟩
          from A↔B
-           (P.subst B (_≃_.right-inverse-of I≃J
+           (≡.subst B (_≃_.right-inverse-of I≃J
                           (_≃_.to I≃J x))
-              (P.subst B (P.sym (_≃_.right-inverse-of I≃J
+              (≡.subst B (≡.sym (_≃_.right-inverse-of I≃J
                                     (_≃_.to I≃J x)))
-                 (to A↔B y)))                   ≡⟨ P.cong (from A↔B)
-                                                                   (P.subst-subst-sym (_≃_.right-inverse-of I≃J _)) ⟩
+                 (to A↔B y)))                   ≡⟨ ≡.cong (from A↔B)
+                                                                   (≡.subst-subst-sym (_≃_.right-inverse-of I≃J _)) ⟩
 
          from A↔B (to A↔B y)      ≡⟨ Inverse.strictlyInverseʳ A↔B _ ⟩
 
@@ -297,7 +297,7 @@ private module _ where
     ∀ {x} → A (from I↔J x) ∼[ k ] B x
   swap-coercions {A = A} B I↔J eq {x} =
     A (from I↔J x)           ∼⟨ eq ⟩
-    B (to I↔J (from I↔J x)) ↔⟨ K-reflexive (P.cong B $ strictlyInverseˡ I↔J x) ⟩
+    B (to I↔J (from I↔J x)) ↔⟨ K-reflexive (≡.cong B $ strictlyInverseˡ I↔J x) ⟩
     B x                       ∎
     where open EquationalReasoning
 
diff --git a/src/Data/Product/Function/Dependent/Propositional/WithK.agda b/src/Data/Product/Function/Dependent/Propositional/WithK.agda
index 83aa4a9baa..0e0f63268b 100644
--- a/src/Data/Product/Function/Dependent/Propositional/WithK.agda
+++ b/src/Data/Product/Function/Dependent/Propositional/WithK.agda
@@ -20,7 +20,6 @@ open import Level using (Level)
 open import Function
 open import Function.Properties.Injection
 open import Function.Properties.Inverse as Inverse
-open import Relation.Binary.PropositionalEquality as P using (_≡_; refl)
 
 private
   variable
diff --git a/src/Data/Product/Function/Dependent/Setoid.agda b/src/Data/Product/Function/Dependent/Setoid.agda
index 22cb950626..0ad3785ed7 100644
--- a/src/Data/Product/Function/Dependent/Setoid.agda
+++ b/src/Data/Product/Function/Dependent/Setoid.agda
@@ -26,8 +26,8 @@ open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
   using (_atₛ_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -43,20 +43,20 @@ private module _ (A : IndexedSetoid I a ℓ₁) where
   open IndexedSetoid A
 
   cast : ∀ {i j} → j ≡ i → Carrier i → Carrier j
-  cast j≡i = P.subst Carrier (P.sym $ j≡i)
+  cast j≡i = ≡.subst Carrier (≡.sym $ j≡i)
 
   cast-cong : ∀ {i j} {x y : Carrier i}
                (j≡i : j ≡ i) →
                x ≈ y →
                cast j≡i x ≈ cast j≡i y
-  cast-cong P.refl p = p
+  cast-cong ≡.refl p = p
 
   cast-eq : ∀ {i j x} (eq : i ≡ j) → cast eq x ≈ x
-  cast-eq P.refl = IndexedSetoid.refl A
+  cast-eq ≡.refl = IndexedSetoid.refl A
 
 private
   _×ₛ_ : (I : Set i) → IndexedSetoid I a ℓ₁ → Setoid _ _
-  I ×ₛ A = Σ.setoid (P.setoid I) A
+  I ×ₛ A = Σ.setoid (≡.setoid I) A
 
 ------------------------------------------------------------------------
 -- Functions
@@ -77,7 +77,7 @@ module _ where
     to′ = map (to I⟶J) (to A⟶B)
 
     cong′ : Congruent (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′
-    cong′ (P.refl , ∼) = (P.refl , cong A⟶B ∼)
+    cong′ (≡.refl , ∼) = (≡.refl , cong A⟶B ∼)
 
 ------------------------------------------------------------------------
 -- Equivalences
@@ -149,7 +149,7 @@ module _ where
           i ≡ j →
           (_≈_ B (to A↣B x) (to A↣B y)) →
           _≈_ A x y
-      lemma P.refl = Injection.injective A↣B
+      lemma ≡.refl = Injection.injective A↣B
 
 ------------------------------------------------------------------------
 -- Surjections
@@ -224,13 +224,13 @@ module _ where
     to′ (i , x) = to I↔J i , to A↔B x
 
     to′-cong : Congruent (_≈_ (I ×ₛ A)) (_≈_ (J ×ₛ B)) to′
-    to′-cong (P.refl , x≈y) = to-cong I↔J P.refl , to-cong A↔B x≈y
+    to′-cong (≡.refl , x≈y) = to-cong I↔J ≡.refl , to-cong A↔B x≈y
 
     from′ : Carrier (J ×ₛ B) → Carrier (I ×ₛ A)
     from′ (j , y) = from I↔J j , from A↔B (cast B (strictlyInverseˡ I↔J _) y)
 
     from′-cong : Congruent (_≈_ (J ×ₛ B)) (_≈_ (I ×ₛ A)) from′
-    from′-cong (P.refl , x≈y) = from-cong I↔J P.refl , from-cong A↔B (cast-cong B (strictlyInverseˡ I↔J _) x≈y)
+    from′-cong (≡.refl , x≈y) = from-cong I↔J ≡.refl , from-cong A↔B (cast-cong B (strictlyInverseˡ I↔J _) x≈y)
 
     strictlyInvˡ : StrictlyInverseˡ (_≈_ (J ×ₛ B)) to′ from′
     strictlyInvˡ (i , x) = strictlyInverseˡ I↔J i ,
@@ -243,7 +243,7 @@ module _ where
     lem : ∀ {i j} → i ≡ j → ∀ {x : IndexedSetoid.Carrier B (to I↔J i)} {y : IndexedSetoid.Carrier B (to I↔J j)} →
           IndexedSetoid._≈_ B x y →
           IndexedSetoid._≈_ A (from A↔B x) (from A↔B y)
-    lem P.refl x≈y = from-cong A↔B x≈y
+    lem ≡.refl x≈y = from-cong A↔B x≈y
 
     strictlyInvʳ : StrictlyInverseʳ (_≈_ (I ×ₛ A)) to′ from′
     strictlyInvʳ (i , x) = strictlyInverseʳ I↔J i ,
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
index ac0f48df7b..7050e8bd3a 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent.agda
@@ -17,7 +17,7 @@ open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions as B
 open import Relation.Binary.Indexed.Heterogeneous as I
   using (IREL; IRel; IndexedSetoid; IsIndexedEquivalence)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 ------------------------------------------------------------------------
 -- Pointwise lifting
diff --git a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
index decb0c779f..cc6f36372f 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/Dependent/WithK.agda
@@ -15,10 +15,10 @@ open import Function.Base
 open import Function.Bundles using (Inverse)
 open import Level using (Level)
 open import Relation.Binary.Core using (_⇒_)
-open import Relation.Binary.HeterogeneousEquality as H using (_≅_)
+open import Relation.Binary.HeterogeneousEquality as ≅ using (_≅_)
 open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -31,16 +31,16 @@ private
 -- decomposed using Pointwise
 
 Pointwise-≡⇒≡ : Pointwise A _≡_ (λ x y → x ≅ y) ⇒ _≡_
-Pointwise-≡⇒≡ (P.refl , H.refl) = P.refl
+Pointwise-≡⇒≡ (≡.refl , ≅.refl) = ≡.refl
 
 ≡⇒Pointwise-≡ : _≡_ ⇒ Pointwise A _≡_ (λ x y → x ≅ y)
-≡⇒Pointwise-≡ P.refl = (P.refl , H.refl)
+≡⇒Pointwise-≡ ≡.refl = (≡.refl , ≅.refl)
 
-Pointwise-≡↔≡ : Inverse (setoid (P.setoid I) (H.indexedSetoid A)) (P.setoid (Σ I A))
+Pointwise-≡↔≡ : Inverse (setoid (≡.setoid I) (≅.indexedSetoid A)) (≡.setoid (Σ I A))
 Pointwise-≡↔≡ = record
   { to         = id
   ; to-cong    = Pointwise-≡⇒≡
   ; from       = id
   ; from-cong  = ≡⇒Pointwise-≡
-  ; inverse    = (λ {(P.refl , H.refl) → P.refl}) , λ {P.refl → (P.refl , H.refl)}
+  ; inverse    = (λ {(≡.refl , ≅.refl) → ≡.refl}) , λ {≡.refl → (≡.refl , ≅.refl)}
   }
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index 95a8978cf5..6c8004b559 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -16,8 +16,8 @@ open import Level using (Level; _⊔_; 0ℓ)
 open import Function
 open import Relation.Nullary.Decidable using (_×-dec_)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -188,12 +188,12 @@ _×ₛ_ = ×-setoid
 -- decomposed using ×-Rel
 
 ≡×≡⇒≡ : Pointwise _≡_ _≡_ ⇒ _≡_ {A = A × B}
-≡×≡⇒≡ (P.refl , P.refl) = P.refl
+≡×≡⇒≡ (≡.refl , ≡.refl) = ≡.refl
 
 ≡⇒≡×≡ : _≡_ {A = A × B} ⇒ Pointwise _≡_ _≡_
-≡⇒≡×≡ P.refl = (P.refl , P.refl)
+≡⇒≡×≡ ≡.refl = (≡.refl , ≡.refl)
 
-Pointwise-≡↔≡ : Inverse (P.setoid A ×ₛ P.setoid B) (P.setoid (A × B))
+Pointwise-≡↔≡ : Inverse (≡.setoid A ×ₛ ≡.setoid B) (≡.setoid (A × B))
 Pointwise-≡↔≡ = record
   { to         = id
   ; from       = id
diff --git a/src/Data/Sum/Relation/Binary/LeftOrder.agda b/src/Data/Sum/Relation/Binary/LeftOrder.agda
index 416d71f15a..d29a6a96ed 100644
--- a/src/Data/Sum/Relation/Binary/LeftOrder.agda
+++ b/src/Data/Sum/Relation/Binary/LeftOrder.agda
@@ -24,7 +24,7 @@ open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsStrictPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictTotalOrder)
 open import Relation.Binary.Definitions
   using (Reflexive; Transitive; Asymmetric; Total; Decidable; Irreflexive; Antisymmetric; Trichotomous; _Respectsʳ_; _Respectsˡ_; _Respects₂_; tri<; tri>; tri≈)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 ------------------------------------------------------------------------
 -- Definition
diff --git a/src/Data/Sum/Relation/Binary/Pointwise.agda b/src/Data/Sum/Relation/Binary/Pointwise.agda
index 943b6f5e49..6d6c84a253 100644
--- a/src/Data/Sum/Relation/Binary/Pointwise.agda
+++ b/src/Data/Sum/Relation/Binary/Pointwise.agda
@@ -17,8 +17,8 @@ open import Function.Bundles using (Inverse; mk↔)
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   variable
@@ -201,14 +201,14 @@ _⊎ₛ_ = ⊎-setoid
 -- decomposed using Pointwise
 
 Pointwise-≡⇒≡ : (Pointwise _≡_ _≡_) ⇒ _≡_ {A = A ⊎ B}
-Pointwise-≡⇒≡ (inj₁ x) = P.cong inj₁ x
-Pointwise-≡⇒≡ (inj₂ x) = P.cong inj₂ x
+Pointwise-≡⇒≡ (inj₁ x) = ≡.cong inj₁ x
+Pointwise-≡⇒≡ (inj₂ x) = ≡.cong inj₂ x
 
 ≡⇒Pointwise-≡ : _≡_ {A = A ⊎ B} ⇒ (Pointwise _≡_ _≡_)
-≡⇒Pointwise-≡ P.refl = ⊎-refl P.refl P.refl
+≡⇒Pointwise-≡ ≡.refl = ⊎-refl ≡.refl ≡.refl
 
 Pointwise-≡↔≡ : (A : Set a) (B : Set b) →
-                 Inverse (P.setoid A ⊎ₛ P.setoid B) (P.setoid (A ⊎ B))
+                 Inverse (≡.setoid A ⊎ₛ ≡.setoid B) (≡.setoid (A ⊎ B))
 Pointwise-≡↔≡ _ _ = record
   { to        = id
   ; from      = id
diff --git a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
index 5853bd1f68..1102985013 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Equality/Setoid.agda
@@ -7,7 +7,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.Nat.Base using (ℕ)
-open import Data.Vec.Functional as VF hiding (map)
+open import Data.Vec.Functional hiding (map)
 open import Data.Vec.Functional.Relation.Binary.Pointwise
   using (Pointwise)
 import Data.Vec.Functional.Relation.Binary.Pointwise.Properties as PW
@@ -17,7 +17,7 @@ open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 module Data.Vec.Functional.Relation.Binary.Equality.Setoid
   {a ℓ} (S : Setoid a ℓ) where
@@ -41,7 +41,7 @@ _≋_ = Pointwise _≈_
 ≋-refl {n} = PW.refl {R = _≈_} refl
 
 ≋-reflexive : ∀ {n} → _≡_ ⇒ (_≋_ {n = n})
-≋-reflexive P.refl = ≋-refl
+≋-reflexive ≡.refl = ≋-refl
 
 ≋-sym : ∀ {n} → Symmetric (_≋_ {n = n})
 ≋-sym = PW.sym {R = _≈_} sym
diff --git a/src/Data/Vec/Properties/WithK.agda b/src/Data/Vec/Properties/WithK.agda
index 7ca8cc2c91..f587d16b64 100644
--- a/src/Data/Vec/Properties/WithK.agda
+++ b/src/Data/Vec/Properties/WithK.agda
@@ -12,8 +12,8 @@ module Data.Vec.Properties.WithK where
 open import Data.Nat.Base
 open import Data.Nat.Properties using (+-assoc)
 open import Data.Vec.Base
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
-open import Relation.Binary.HeterogeneousEquality as H using (_≅_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
+open import Relation.Binary.HeterogeneousEquality as ≅ using (_≅_; refl)
 
 ------------------------------------------------------------------------
 -- _[_]=_
@@ -24,7 +24,7 @@ module _ {a} {A : Set a} where
                     (p q : xs [ i ]= x) → p ≡ q
   []=-irrelevant here            here             = refl
   []=-irrelevant (there xs[i]=x) (there xs[i]=x′) =
-    P.cong there ([]=-irrelevant xs[i]=x xs[i]=x′)
+    cong there ([]=-irrelevant xs[i]=x xs[i]=x′)
 
 ------------------------------------------------------------------------
 -- _++_
@@ -35,7 +35,7 @@ module _ {a} {A : Set a} where
              (xs ++ ys) ++ zs ≅ xs ++ (ys ++ zs)
   ++-assoc         []       ys zs = refl
   ++-assoc {suc m} (x ∷ xs) ys zs =
-    H.icong (Vec A) (+-assoc m _ _) (x ∷_) (++-assoc xs ys zs)
+    ≅.icong (Vec A) (+-assoc m _ _) (x ∷_) (++-assoc xs ys zs)
 
 ------------------------------------------------------------------------
 -- foldr
diff --git a/src/Data/Vec/Recursive/Properties.agda b/src/Data/Vec/Recursive/Properties.agda
index d036134a45..3f25aa19d1 100644
--- a/src/Data/Vec/Recursive/Properties.agda
+++ b/src/Data/Vec/Recursive/Properties.agda
@@ -14,7 +14,8 @@ open import Data.Product.Base
 open import Data.Vec.Recursive
 open import Data.Vec.Base using (Vec; _∷_)
 open import Function.Bundles using (_↔_; mk↔ₛ′)
-open import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 open ≡-Reasoning
@@ -28,31 +29,31 @@ private
 -- Basic proofs
 
 cons-head-tail-identity : ∀ n (as : A ^ suc n) → cons n (head n as) (tail n as) ≡ as
-cons-head-tail-identity 0       as = P.refl
-cons-head-tail-identity (suc n) as = P.refl
+cons-head-tail-identity 0       as = refl
+cons-head-tail-identity (suc n) as = refl
 
 head-cons-identity : ∀ n a (as : A ^ n) → head n (cons n a as) ≡ a
-head-cons-identity 0       a as = P.refl
-head-cons-identity (suc n) a as = P.refl
+head-cons-identity 0       a as = refl
+head-cons-identity (suc n) a as = refl
 
 tail-cons-identity : ∀ n a (as : A ^ n) → tail n (cons n a as) ≡ as
-tail-cons-identity 0       a as = P.refl
-tail-cons-identity (suc n) a as = P.refl
+tail-cons-identity 0       a as = refl
+tail-cons-identity (suc n) a as = refl
 
 append-cons : ∀ m n a (xs : A ^ m) ys →
   append (suc m) n (cons m a xs) ys ≡ cons (m + n) a (append m n xs ys)
-append-cons 0       n a xs ys = P.refl
-append-cons (suc m) n a xs ys = P.refl
+append-cons 0       n a xs ys = refl
+append-cons (suc m) n a xs ys = refl
 
 append-splitAt-identity : ∀ m n (as : A ^ (m + n)) → uncurry (append m n) (splitAt m n as) ≡ as
-append-splitAt-identity 0       n as = P.refl
+append-splitAt-identity 0       n as = refl
 append-splitAt-identity (suc m) n as = begin
   let x         = head (m + n) as in
   let (xs , ys) = splitAt m n (tail (m + n) as) in
   append (suc m) n (cons m (head (m + n) as) xs) ys
     ≡⟨ append-cons m n x xs ys ⟩
   cons (m + n) x (append m n xs ys)
-    ≡⟨ P.cong (cons (m + n) x) (append-splitAt-identity m n (tail (m + n) as)) ⟩
+    ≡⟨ cong (cons (m + n) x) (append-splitAt-identity m n (tail (m + n) as)) ⟩
   cons (m + n) x (tail (m + n) as)
     ≡⟨ cons-head-tail-identity (m + n) as ⟩
   as
@@ -62,21 +63,21 @@ append-splitAt-identity (suc m) n as = begin
 -- Conversion to and from Vec
 
 fromVec∘toVec : ∀ n (xs : A ^ n) → fromVec (toVec n xs) ≡ xs
-fromVec∘toVec 0       _  = P.refl
+fromVec∘toVec 0       _  = refl
 fromVec∘toVec (suc n) xs = begin
   cons n (head n xs) (fromVec (toVec n (tail n xs)))
-    ≡⟨ P.cong (cons n (head n xs)) (fromVec∘toVec n (tail n xs)) ⟩
+    ≡⟨ cong (cons n (head n xs)) (fromVec∘toVec n (tail n xs)) ⟩
   cons n (head n xs) (tail n xs)
     ≡⟨ cons-head-tail-identity n xs ⟩
   xs ∎
 
 toVec∘fromVec : ∀ {n} (xs : Vec A n) → toVec n (fromVec xs) ≡ xs
-toVec∘fromVec             Vec.[]       = P.refl
+toVec∘fromVec             Vec.[]       = refl
 toVec∘fromVec {n = suc n} (x Vec.∷ xs) = begin
   head n (cons n x (fromVec xs)) Vec.∷ toVec n (tail n (cons n x (fromVec xs)))
-    ≡⟨ P.cong₂ (λ x xs → x Vec.∷ toVec n xs) hd-prf tl-prf ⟩
+    ≡⟨ cong₂ (λ x xs → x Vec.∷ toVec n xs) hd-prf tl-prf ⟩
   x Vec.∷ toVec n (fromVec xs)
-    ≡⟨ P.cong (x Vec.∷_) (toVec∘fromVec xs) ⟩
+    ≡⟨ cong (x Vec.∷_) (toVec∘fromVec xs) ⟩
   x Vec.∷ xs
     ∎ where
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index 7b35081911..ed23c7cf6a 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -22,7 +22,7 @@ open import Relation.Binary.Core using (Rel; REL)
 open import Relation.Binary.Definitions
   using (Transitive; Symmetric; Asymmetric; Antisymmetric; Irreflexive; Trans; _Respects₂_; _Respectsˡ_; _Respectsʳ_; Decidable; Irrelevant)
 open import Relation.Binary.Structures using (IsPartialEquivalence)
-open import Relation.Binary.PropositionalEquality.Core as P
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; refl; cong)
 import Relation.Nullary as Nullary
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _×-dec_; _⊎-dec_)
@@ -101,9 +101,9 @@ module _ {P : Set} {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
 
     transitive′ : ∀ {m n o P₂} → Trans (Lex P _≈_ _≺_ {m} {n}) (Lex P₂ _≈_ _≺_ {n} {o}) (Lex (P × P₂) _≈_ _≺_)
     transitive′ (base p₁)        (base p₂)        = base (p₁ , p₂)
-    transitive′ (this x≺y m≡n)   (this y≺z n≡o)   = this (≺-trans x≺y y≺z) (P.trans m≡n n≡o)
-    transitive′ (this x≺y m≡n)   (next y≈z ys<zs) = this (≺-respʳ-≈ y≈z x≺y) (P.trans m≡n (length-equal ys<zs))
-    transitive′ (next x≈y xs<ys) (this y≺z n≡o)   = this (≺-respˡ-≈ (sym x≈y) y≺z) (P.trans (length-equal xs<ys) n≡o)
+    transitive′ (this x≺y m≡n)   (this y≺z n≡o)   = this (≺-trans x≺y y≺z) (≡.trans m≡n n≡o)
+    transitive′ (this x≺y m≡n)   (next y≈z ys<zs) = this (≺-respʳ-≈ y≈z x≺y) (≡.trans m≡n (length-equal ys<zs))
+    transitive′ (next x≈y xs<ys) (this y≺z n≡o)   = this (≺-respˡ-≈ (sym x≈y) y≺z) (≡.trans (length-equal xs<ys) n≡o)
     transitive′ (next x≈y xs<ys) (next y≈z ys<zs) = next (trans x≈y y≈z) (transitive′ xs<ys ys<zs)
 
     transitive : ∀ {m n o} → Trans (_<ₗₑₓ_ {m} {n}) (_<ₗₑₓ_ {n} {o}) _<ₗₑₓ_
diff --git a/src/Data/Vec/Relation/Binary/Lex/Strict.agda b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
index 83608e4456..52f529aec9 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Strict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Strict.agda
@@ -33,7 +33,7 @@ open import Relation.Binary.Definitions
   using (Irreflexive; _Respects₂_; _Respectsˡ_; _Respectsʳ_; Antisymmetric; Asymmetric; Symmetric; Trans; Decidable; Total; Trichotomous; Transitive; Irrelevant; tri≈; tri>; tri<)
 open import Relation.Binary.Consequences
 open import Relation.Binary.Construct.On as On using (wellFounded)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Level using (Level; _⊔_)
 
 private
@@ -103,8 +103,8 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
     <-cmp : ∀ {n} → Trichotomous _≋_ (_<_ {n})
     <-cmp [] [] = tri≈ ¬[]<[] [] ¬[]<[]
     <-cmp (x ∷ xs) (y ∷ ys) with ≺-cmp x y
-    ... | tri< x≺y x≉y x⊁y = tri< (this x≺y P.refl) (x≉y ∘ head) (≰-this (x≉y ∘ ≈-sym) x⊁y)
-    ... | tri> x⊀y x≉y x≻y = tri> (≰-this x≉y x⊀y) (x≉y ∘ head) (this x≻y P.refl)
+    ... | tri< x≺y x≉y x⊁y = tri< (this x≺y refl) (x≉y ∘ head) (≰-this (x≉y ∘ ≈-sym) x⊁y)
+    ... | tri> x⊀y x≉y x≻y = tri> (≰-this x≉y x⊀y) (x≉y ∘ head) (this x≻y refl)
     ... | tri≈ x⊀y x≈y x⊁y with <-cmp xs ys
     ...   | tri< xs<ys xs≋̸ys xs≯ys = tri< (next x≈y xs<ys) (xs≋̸ys ∘ tail) (≰-next x⊁y xs≯ys)
     ...   | tri≈ xs≮ys xs≋ys xs≯ys = tri≈ (≰-next x⊀y xs≮ys) (x≈y ∷ xs≋ys) (≰-next x⊁y xs≯ys)
@@ -237,8 +237,8 @@ module _ {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
     ≤-total : ∀ {n} → Total (_≤_ {n} {n})
     ≤-total [] [] = inj₁ (base tt)
     ≤-total (x ∷ xs) (y ∷ ys) with ≺-cmp x y
-    ... | tri< x≺y _   _   = inj₁ (this x≺y P.refl)
-    ... | tri> _   _   x≻y = inj₂ (this x≻y P.refl)
+    ... | tri< x≺y _   _   = inj₁ (this x≺y refl)
+    ... | tri> _   _   x≻y = inj₂ (this x≻y refl)
     ... | tri≈ _   x≈y _ with ≤-total xs ys
     ...   | inj₁ xs<ys = inj₁ (next x≈y xs<ys)
     ...   | inj₂ xs>ys = inj₂ (next (≈-sym x≈y) xs>ys)
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
index 5dcaa6b900..928bae5176 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Extensional.agda
@@ -22,7 +22,7 @@ open import Relation.Binary.Core using (Rel; REL; _⇒_; _=[_]⇒_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence; IsDecEquivalence)
 open import Relation.Binary.Definitions using (Reflexive; Sym; Trans; Decidable)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Construct.Closure.Transitive as Plus
   hiding (equivalent; map)
 open import Relation.Nullary
@@ -134,12 +134,12 @@ isDecEquivalence decEquiv = record
 -- Pointwise _≡_ is equivalent to _≡_.
 
 Pointwise-≡⇒≡ : ∀ {n} {xs ys : Vec A n} → Pointwise _≡_ xs ys → xs ≡ ys
-Pointwise-≡⇒≡ {xs = []}     {[]}     (ext app) = P.refl
+Pointwise-≡⇒≡ {xs = []}     {[]}     (ext app) = ≡.refl
 Pointwise-≡⇒≡ {xs = x ∷ xs} {y ∷ ys} xs∼ys     =
-  P.cong₂ _∷_ (head xs∼ys) (Pointwise-≡⇒≡ (tail xs∼ys))
+  ≡.cong₂ _∷_ (head xs∼ys) (Pointwise-≡⇒≡ (tail xs∼ys))
 
 ≡⇒Pointwise-≡ : ∀ {n} {xs ys : Vec A n} → xs ≡ ys → Pointwise _≡_ xs ys
-≡⇒Pointwise-≡ P.refl = refl P.refl
+≡⇒Pointwise-≡ ≡.refl = refl ≡.refl
 
 Pointwise-≡↔≡ : ∀ {n} {xs ys : Vec A n} → Pointwise _≡_ xs ys ⇔ xs ≡ ys
 Pointwise-≡↔≡ {ℓ} {A} = mk⇔ Pointwise-≡⇒≡ ≡⇒Pointwise-≡
diff --git a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
index c6835b73e1..17cc144319 100644
--- a/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
+++ b/src/Data/Vec/Relation/Binary/Pointwise/Inductive.agda
@@ -22,7 +22,7 @@ open import Relation.Binary.Structures
   using (IsEquivalence; IsDecEquivalence)
 open import Relation.Binary.Definitions
   using (Trans; Decidable; Reflexive; Sym)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Nullary.Decidable using (yes; no; _×-dec_; map′)
 open import Relation.Unary using (Pred)
 
@@ -52,8 +52,8 @@ data Pointwise {a b ℓ} {A : Set a} {B : Set b} (_∼_ : REL A B ℓ) :
 
 length-equal : ∀ {m n} {_∼_ : REL A B ℓ} {xs : Vec A m} {ys : Vec B n} →
                Pointwise _∼_ xs ys → m ≡ n
-length-equal []          = P.refl
-length-equal (_ ∷ xs∼ys) = P.cong suc (length-equal xs∼ys)
+length-equal []          = ≡.refl
+length-equal (_ ∷ xs∼ys) = ≡.cong suc (length-equal xs∼ys)
 
 ------------------------------------------------------------------------
 -- Operations
@@ -261,11 +261,11 @@ module _ {P : Pred A ℓ} where
 -- Pointwise _≡_ is equivalent to _≡_
 
 Pointwise-≡⇒≡ : ∀ {n} {xs ys : Vec A n} → Pointwise _≡_ xs ys → xs ≡ ys
-Pointwise-≡⇒≡ []               = P.refl
-Pointwise-≡⇒≡ (P.refl ∷ xs∼ys) = P.cong (_ ∷_) (Pointwise-≡⇒≡ xs∼ys)
+Pointwise-≡⇒≡ []               = ≡.refl
+Pointwise-≡⇒≡ (≡.refl ∷ xs∼ys) = ≡.cong (_ ∷_) (Pointwise-≡⇒≡ xs∼ys)
 
 ≡⇒Pointwise-≡ : ∀ {n} {xs ys : Vec A n} → xs ≡ ys → Pointwise _≡_ xs ys
-≡⇒Pointwise-≡ P.refl = refl P.refl
+≡⇒Pointwise-≡ ≡.refl = refl ≡.refl
 
 Pointwise-≡↔≡ : ∀ {n} {xs ys : Vec A n} → Pointwise _≡_ xs ys ⇔ xs ≡ ys
 Pointwise-≡↔≡ = mk⇔ Pointwise-≡⇒≡ ≡⇒Pointwise-≡
diff --git a/src/Data/Vec/Relation/Unary/All.agda b/src/Data/Vec/Relation/Unary/All.agda
index 4cc1805c25..1c20d9c478 100644
--- a/src/Data/Vec/Relation/Unary/All.agda
+++ b/src/Data/Vec/Relation/Unary/All.agda
@@ -21,7 +21,7 @@ open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no)
 open import Relation.Unary hiding (_∈_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (_Respects_)
-open import Relation.Binary.PropositionalEquality.Core as P using (subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 
 private
   variable
@@ -94,7 +94,7 @@ lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.loo
 lookupWith f pxs i = Product.uncurry f (lookupAny pxs i)
 
 lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
-lookup pxs = lookupWith (λ { px P.refl → px }) pxs
+lookup pxs = lookupWith (λ { px refl → px }) pxs
 
 module _(S : Setoid a ℓ) {P : Pred (Setoid.Carrier S) p} where
   open Setoid S renaming (sym to sym₁)
@@ -115,9 +115,9 @@ universal u []       = []
 universal u (x ∷ xs) = u x ∷ universal u xs
 
 irrelevant : Irrelevant P → ∀ {n} → Irrelevant (All P {n})
-irrelevant irr []           []           = P.refl
+irrelevant irr []           []           = refl
 irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =
-  P.cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)
+  cong₂ _∷_ (irr px₁ px₂) (irrelevant irr pxs₁ pxs₂)
 
 satisfiable : Satisfiable P → ∀ {n} → Satisfiable (All P {n})
 satisfiable (x , p) {zero}  = [] , []
diff --git a/src/Data/Vec/Relation/Unary/Any/Properties.agda b/src/Data/Vec/Relation/Unary/Any/Properties.agda
index d330ea5cef..0fb5512677 100644
--- a/src/Data/Vec/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Any/Properties.agda
@@ -30,7 +30,8 @@ open import Relation.Nullary.Negation using (¬_)
 open import Relation.Unary hiding (_∈_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (_Respects_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; trans; cong)
 
 private
   variable
@@ -93,15 +94,15 @@ module _ {P : Pred A p} where
 map-id : ∀ {P : Pred A p} (f : P ⊆ P) {n xs} →
          (∀ {x} (p : P x) → f p ≡ p) →
          (p : Any P {n} xs) → Any.map f p ≡ p
-map-id f hyp (here  p) = P.cong here (hyp p)
-map-id f hyp (there p) = P.cong there $ map-id f hyp p
+map-id f hyp (here  p) = cong here (hyp p)
+map-id f hyp (there p) = cong there $ map-id f hyp p
 
 map-∘ : ∀ {P : Pred A p} {Q : A → Set q} {R : A → Set r}
         (f : Q ⊆ R) (g : P ⊆ Q)
         {n xs} (p : Any P {n} xs) →
         Any.map (f ∘ g) p ≡ Any.map f (Any.map g p)
 map-∘ f g (here  p) = refl
-map-∘ f g (there p) = P.cong there $ map-∘ f g p
+map-∘ f g (there p) = cong there $ map-∘ f g p
 
 ------------------------------------------------------------------------
 -- Swapping
@@ -117,7 +118,7 @@ module _ {P : A → B → Set ℓ} where
   swap-there : ∀ {n m x xs ys} → (any : Any (λ x → Any (P x) {n} ys) {m} xs) →
                swap (Any.map (there {x = x}) any) ≡ there (swap any)
   swap-there (here  pys)  = refl
-  swap-there (there pxys) = P.cong (Any.map there) (swap-there pxys)
+  swap-there (there pxys) = cong (Any.map there) (swap-there pxys)
 
 module _ {P : A → B → Set ℓ} where
 
@@ -125,9 +126,9 @@ module _ {P : A → B → Set ℓ} where
                (any : Any (λ x → Any (P x) ys) xs) →
                swap (swap any) ≡ any
   swap-invol (here (here _)) = refl
-  swap-invol (here (there pys)) = P.cong (Any.map there) (swap-invol (here pys))
-  swap-invol (there pxys) = P.trans (swap-there (swap pxys))
-                          $ P.cong there (swap-invol pxys)
+  swap-invol (here (there pys)) = cong (Any.map there) (swap-invol (here pys))
+  swap-invol (there pxys) = trans (swap-there (swap pxys))
+                          $ cong there (swap-invol pxys)
 
 module _ {P : A → B → Set ℓ} where
 
@@ -237,12 +238,12 @@ module _ {f : A → B} where
   map⁺∘map⁻ : ∀ {P : Pred B p} {n} {xs : Vec A n} →
               (p : Any P (map f xs)) → map⁺ (map⁻ p) ≡ p
   map⁺∘map⁻ {xs = x ∷ xs} (here  p) = refl
-  map⁺∘map⁻ {xs = x ∷ xs} (there p) = P.cong there (map⁺∘map⁻ p)
+  map⁺∘map⁻ {xs = x ∷ xs} (there p) = cong there (map⁺∘map⁻ p)
 
   map⁻∘map⁺ : ∀ (P : Pred B p) {n} {xs : Vec A n} →
               (p : Any (P ∘ f) xs) → map⁻ {P = P} (map⁺ p) ≡ p
   map⁻∘map⁺ P (here  p) = refl
-  map⁻∘map⁺ P (there p) = P.cong there (map⁻∘map⁺ P p)
+  map⁻∘map⁺ P (there p) = cong there (map⁻∘map⁺ P p)
 
   map↔ : ∀ {P : Pred B p} {n} {xs : Vec A n} →
          Any (P ∘ f) xs ↔ Any P (map f xs)
@@ -271,8 +272,8 @@ module _ {P : Pred A p} where
   ++⁺∘++⁻ []       p         = refl
   ++⁺∘++⁻ (x ∷ xs) (here  p) = refl
   ++⁺∘++⁻ (x ∷ xs) (there p) with ++⁻ xs p | ++⁺∘++⁻ xs p
-  ++⁺∘++⁻ (x ∷ xs) (there p) | inj₁ p′ | ih = P.cong there ih
-  ++⁺∘++⁻ (x ∷ xs) (there p) | inj₂ p′ | ih = P.cong there ih
+  ++⁺∘++⁻ (x ∷ xs) (there p) | inj₁ p′ | ih = cong there ih
+  ++⁺∘++⁻ (x ∷ xs) (there p) | inj₂ p′ | ih = cong there ih
 
   ++⁻∘++⁺ : ∀ {n m} (xs : Vec A n) {ys : Vec A m} (p : Any P xs ⊎ Any P ys) →
             ++⁻ xs ([ ++⁺ˡ , ++⁺ʳ xs ]′ p) ≡ p
@@ -336,10 +337,10 @@ module _ {P : Pred A p} where
                    concat⁺ (concat⁻ xss p) ≡ p
   concat⁺∘concat⁻ (xs ∷ xss) p  with ++⁻ xs p in eq
   ... | inj₁ pxs
-    = P.trans (P.cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (P.sym eq))
+    = trans (cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (sym eq))
     $ ++⁺∘++⁻ xs p
   ... | inj₂ pxss rewrite concat⁺∘concat⁻ xss pxss
-    = P.trans (P.cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (P.sym eq))
+    = trans (cong [ ++⁺ˡ , ++⁺ʳ xs ]′ (sym eq))
     $ ++⁺∘++⁻ xs p
 
   concat⁻∘concat⁺ : ∀ {n m} {xss : Vec (Vec A n) m} (p : Any (Any P) xss) →
@@ -402,7 +403,7 @@ module _ {P : Pred B p} where
               mapWith∈⁺ f (mapWith∈⁻ xs f p) ≡ p
     to∘from (y ∷ xs) f (here  p) = refl
     to∘from (y ∷ xs) f (there p) =
-      P.cong there $ to∘from xs (f ∘ there) p
+      cong there $ to∘from xs (f ∘ there) p
 
 ------------------------------------------------------------------------
 -- _∷_
diff --git a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
index 692f2530bf..fa19dc3e21 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Setoid/Properties.agda
@@ -20,7 +20,7 @@ open import Function.Base using (_∘_; id)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Nullary.Negation using (contradiction; contraposition)
 
 private
@@ -71,7 +71,7 @@ module _ (S : Setoid a ℓ) where
   open Setoid S
 
   lookup-injective : ∀ {n xs} → Unique S {n} xs → ∀ i j → lookup xs i ≈ lookup xs j → i ≡ j
-  lookup-injective (px ∷ pxs) zero zero eq = P.refl
+  lookup-injective (px ∷ pxs) zero zero eq = ≡.refl
   lookup-injective (px ∷ pxs) zero (suc j) eq = contradiction eq (All.lookup⁺ px j)
   lookup-injective (px ∷ pxs) (suc i) zero eq = contradiction (sym eq) (All.lookup⁺ px i)
-  lookup-injective (px ∷ pxs) (suc i) (suc j) eq = P.cong suc (lookup-injective pxs i j eq)
+  lookup-injective (px ∷ pxs) (suc i) (suc j) eq = ≡.cong suc (lookup-injective pxs i j eq)
diff --git a/src/Effect/Applicative.agda b/src/Effect/Applicative.agda
index 95e2debda8..ef7dd5cc2f 100644
--- a/src/Effect/Applicative.agda
+++ b/src/Effect/Applicative.agda
@@ -21,7 +21,7 @@ open import Effect.Functor as Fun using (RawFunctor)
 
 open import Function.Base using (const; flip; _∘′_)
 open import Level using (Level; suc; _⊔_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 private
   variable
diff --git a/src/Effect/Applicative/Indexed.agda b/src/Effect/Applicative/Indexed.agda
index f547e68b27..d08a515086 100644
--- a/src/Effect/Applicative/Indexed.agda
+++ b/src/Effect/Applicative/Indexed.agda
@@ -15,7 +15,7 @@ open import Effect.Functor using (RawFunctor)
 open import Data.Product.Base using (_×_; _,_)
 open import Function.Base
 open import Level
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 
@@ -111,6 +111,6 @@ record Morphism {I : Set i} {F₁ F₂ : IFun I f}
            op (f A₁.<$> x) ≡ (f A₂.<$> op x)
   op-<$> f x = begin
     op (A₁._⊛_ (A₁.pure f) x)       ≡⟨ op-⊛ _ _ ⟩
-    A₂._⊛_ (op (A₁.pure f)) (op x)  ≡⟨ P.cong₂ A₂._⊛_ (op-pure _) P.refl ⟩
+    A₂._⊛_ (op (A₁.pure f)) (op x)  ≡⟨ cong₂ A₂._⊛_ (op-pure _) refl ⟩
     A₂._⊛_ (A₂.pure f) (op x)       ∎
     where open ≡-Reasoning
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index 44cf13464d..c0723f1370 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -27,8 +27,8 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Bundles
   using (Preorder; Setoid; Poset)
 import Relation.Binary.Properties.Setoid as SetoidProperties
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Relation.Nullary
 open import Relation.Nullary.Decidable hiding (map)
 open import Relation.Nullary.Negation
@@ -117,13 +117,13 @@ data Kind : Set where
 infix 4 _≟-Kind_
 
 _≟-Kind_ : Decidable (_≡_ {A = Kind})
-_≟-Kind_ strong       strong       = yes P.refl
+_≟-Kind_ strong       strong       = yes ≡.refl
 _≟-Kind_ strong       (other k)    = no λ()
 _≟-Kind_ (other k)    strong       = no λ()
-_≟-Kind_ (other geq)  (other geq)  = yes P.refl
+_≟-Kind_ (other geq)  (other geq)  = yes ≡.refl
 _≟-Kind_ (other geq)  (other weak) = no λ()
 _≟-Kind_ (other weak) (other geq)  = no λ()
-_≟-Kind_ (other weak) (other weak) = yes P.refl
+_≟-Kind_ (other weak) (other weak) = yes ≡.refl
 
 -- A predicate which is satisfied only for equalities. Note that, for
 -- concrete inputs, this predicate evaluates to ⊤ or ⊥.
@@ -303,14 +303,14 @@ module _ {A : Set a} {_∼_ : A → A → Set ℓ} where
     ; _≈_        = _≡_
     ; _≲_        = Rel k
     ; isPreorder = record
-      { isEquivalence = P.isEquivalence
+      { isEquivalence = ≡.isEquivalence
       ; reflexive     = refl′
       ; trans         = Equivalence.trans (IsPreorder.trans pre)
       }
     }
     where
     refl′ : ∀ {k} {x y : A ⊥} → x ≡ y → Rel k x y
-    refl′ P.refl = Equivalence.refl (IsPreorder.refl pre)
+    refl′ ≡.refl = Equivalence.refl (IsPreorder.refl pre)
 
   private
     preorder′ : IsEquivalence _∼_ → Kind → Preorder _ _ _
@@ -371,7 +371,7 @@ module _ {A : Set a} {_∼_ : A → A → Set ℓ} where
     infixr 2 _≡⟨_⟩_ _≅⟨_⟩_ _≳⟨_⟩_ _≈⟨_⟩_
 
     _≡⟨_⟩_ : ∀ {k} x {y z : A ⊥} → x ≡ y → Rel k y z → Rel k x z
-    _ ≡⟨ P.refl ⟩ y∼z = y∼z
+    _ ≡⟨ ≡.refl ⟩ y∼z = y∼z
 
     _≅⟨_⟩_ : ∀ {k} x {y z : A ⊥} → x ≅ y → Rel k y z → Rel k x z
     _ ≅⟨ x≅y ⟩ y∼z = Pre.trans (≅⇒ x≅y) y∼z
@@ -436,7 +436,7 @@ module _ {A : Set a} {_∼_ : A → A → Set ℓ} where
 
   ≡⇒ : Reflexive _∼_ →
        ∀ {k x y} → Equality.Rel _≡_ k x y → Rel k x y
-  ≡⇒ refl-∼ = map (flip (P.subst (_∼_ _)) refl-∼)
+  ≡⇒ refl-∼ = map (flip (≡.subst (_∼_ _)) refl-∼)
 
 ------------------------------------------------------------------------
 -- Steps
@@ -454,17 +454,17 @@ module _ {A : Set a} {_∼_ : A → A → Set ℓ} where
       ∀ {k x y} {z : A}
       (x≅y : x ≅ y) (y⇓z : y ⇓[ k ] z) →
       steps (Equivalence.trans trans-∼ (≅⇒ x≅y) y⇓z) ≡ steps y⇓z
-    left-identity (now _)     (now _)      = P.refl
+    left-identity (now _)     (now _)      = ≡.refl
     left-identity (later x≅y) (laterˡ y⇓z) =
-      P.cong suc $ left-identity (♭ x≅y) y⇓z
+      ≡.cong suc $ left-identity (♭ x≅y) y⇓z
 
     right-identity :
       ∀ {k x} {y z : A}
       (x⇓y : x ⇓[ k ] y) (y≈z : now y ⇓[ k ] z) →
       steps (Equivalence.trans trans-∼ x⇓y y≈z) ≡ steps x⇓y
-    right-identity (now x∼y)    (now y∼z) = P.refl
+    right-identity (now x∼y)    (now y∼z) = ≡.refl
     right-identity (laterˡ x∼y) (now y∼z) =
-      P.cong suc $ right-identity x∼y (now y∼z)
+      ≡.cong suc $ right-identity x∼y (now y∼z)
 
 ------------------------------------------------------------------------
 -- Laws related to bind
@@ -538,9 +538,9 @@ module _ {A B : Set s}
     ∃ λ z → ∃₂ λ (x⇓ : x ⇓[ k ]A z) (fz⇓ : f z ⇓[ k ]B y) →
                  steps x⇓ + steps fz⇓ ≡ steps x>>=f⇓
   >>=-inversion-⇓ refl (now x) fx⇓ =
-    (x , now refl , fx⇓ , P.refl)
+    (x , now refl , fx⇓ , ≡.refl)
   >>=-inversion-⇓ refl (later x) (laterˡ x>>=f⇓) =
-    Prod.map id (Prod.map laterˡ (Prod.map id (P.cong suc))) $
+    Prod.map id (Prod.map laterˡ (Prod.map id (≡.cong suc))) $
       >>=-inversion-⇓ refl (♭ x) x>>=f⇓
 
   >>=-inversion-⇑ : IsEquivalence _∼A_ →
@@ -589,7 +589,7 @@ module _ {A B : Set ℓ} {_∼_ : B → B → Set ℓ} where
     Rel _∼_ k (x₁ >>= f₁) (x₂ >>= f₂)
   _≡->>=-cong_ {k} {f₁ = f₁} {f₂} x₁≈x₂ f₁≈f₂ =
     x₁≈x₂ >>=-cong λ {x} x≡x′ →
-    P.subst (λ y → Rel _∼_ k (f₁ x) (f₂ y)) x≡x′ (f₁≈f₂ x)
+    ≡.subst (λ y → Rel _∼_ k (f₁ x) (f₂ y)) x≡x′ (f₁≈f₂ x)
 
 ------------------------------------------------------------------------
 -- Productivity checker workaround
@@ -641,7 +641,7 @@ module Workaround {a} where
 
     private
       open module Eq {A : Set a} = Equality  {A = A} _≡_
-      open module R  {A : Set a} = Reasoning (P.isEquivalence {A = A})
+      open module R  {A : Set a} = Reasoning (≡.isEquivalence {A = A})
 
     now-hom : (x : A) → ⟦ now x ⟧P ≅ now x
     now-hom x = now x ∎
@@ -820,7 +820,7 @@ module AlternativeEquality {a ℓ} where
     whnf≅ (x₁≅x₂ >>= f₁≅f₂)   = whnf≅ x₁≅x₂ >>=W λ xRy → whnf≅ (f₁≅f₂ xRy)
     whnf≅ (x ∎)               = reflW x
     whnf≅ (sym x≅y)           = symW _ (whnf≅ x≅y)
-    whnf≅ (x ≡⟨ P.refl ⟩ y≅z) = whnf≅ y≅z
+    whnf≅ (x ≡⟨ ≡.refl ⟩ y≅z) = whnf≅ y≅z
     whnf≅ (x ≅⟨ x≅y    ⟩ y≅z) = trans≅W (whnf≅ x≅y) (whnf≅ y≅z)
 
     -- More transitivity lemmas.
@@ -853,7 +853,7 @@ module AlternativeEquality {a ℓ} where
     whnf≳ (laterˡ x≲y)         = laterˡ (whnf≳ x≲y)
     whnf≳ (x₁∼x₂ >>= f₁∼f₂)    = whnf≳ x₁∼x₂ >>=W λ xRy → whnf≳ (f₁∼f₂ xRy)
     whnf≳ (x ∎)                = reflW x
-    whnf≳ (x ≡⟨ P.refl ⟩  y≳z) = whnf≳ y≳z
+    whnf≳ (x ≡⟨ ≡.refl ⟩  y≳z) = whnf≳ y≳z
     whnf≳ (x ≅⟨ x≅y    ⟩  y≳z) = trans≅∼W (whnf≅ x≅y) (whnf≳ y≳z)
     whnf≳ (x ≳⟨ x≳y    ⟩  y≳z) = trans≳-W        x≳y  (whnf≳ y≳z)
     whnf≳ (x ≳⟨ x≳y    ⟩≅ y≅z) = trans∼≅W (whnf≳ x≳y) (whnf≅ y≅z)
@@ -879,7 +879,7 @@ module AlternativeEquality {a ℓ} where
     whnf (x₁∼x₂ >>= f₁∼f₂)    = whnf x₁∼x₂ >>=W λ xRy → whnf (f₁∼f₂ xRy)
     whnf (x ∎)                = reflW x
     whnf (sym {eq = eq} x≈y)  = symW eq (whnf x≈y)
-    whnf (x ≡⟨ P.refl ⟩  y∼z) = whnf y∼z
+    whnf (x ≡⟨ ≡.refl ⟩  y∼z) = whnf y∼z
     whnf (x ≅⟨ x≅y    ⟩  y∼z) = trans≅∼W (whnf x≅y) (whnf y∼z)
     whnf (x ≳⟨ x≳y    ⟩  y≳z) = trans≳-W       x≳y  (whnf y≳z)
     whnf (x ≳⟨ x≳y    ⟩≅ y≅z) = trans∼≅W (whnf x≳y) (whnf y≅z)
@@ -930,7 +930,7 @@ idempotent {A = A} B x f = sound (idem x)
              (x >>= λ y′ → f y′ y′)
   idem (now   x) = f x x ∎
   idem (later x) = later (♯ (
-    (♭ x >>= λ y′ → later x >>= λ y″ → f y′ y″)  ≳⟨ (refl P.refl {x = ♭ x} ≡->>=-cong λ _ →
+    (♭ x >>= λ y′ → later x >>= λ y″ → f y′ y″)  ≳⟨ (refl ≡.refl {x = ♭ x} ≡->>=-cong λ _ →
                                                      laterˡ (refl (Setoid.refl B))) ⟩
     (♭ x >>= λ y′ →     ♭ x >>= λ y″ → f y′ y″)  ≳⟨ idem (♭ x) ⟩≅
     (♭ x >>= λ y′ → f y′ y′)                     ∎))
diff --git a/src/Effect/Monad/Partiality/All.agda b/src/Effect/Monad/Partiality/All.agda
index c5bf2da9ef..6f7b30ef41 100644
--- a/src/Effect/Monad/Partiality/All.agda
+++ b/src/Effect/Monad/Partiality/All.agda
@@ -15,7 +15,7 @@ open import Function.Base using (flip; _∘_)
 open import Level
 open import Relation.Binary.Definitions using (_Respects_)
 open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 open Partiality._⊥
 open Partiality.Equality using (Rel)
@@ -82,7 +82,7 @@ module Reasoning {P : A → Set p}
   infixr 2 _≡⟨_⟩_ _∼⟨_⟩_
 
   _≡⟨_⟩_ : ∀ x {y} → x ≡ y → All P y → All P x
-  _ ≡⟨ P.refl ⟩ p = p
+  _ ≡⟨ ≡.refl ⟩ p = p
 
   _∼⟨_⟩_ : ∀ {k} x {y} → Rel _∼_ k x y → All P y → All P x
   _ ∼⟨ x∼y ⟩ p = respects-flip resp (⇒≈ x∼y) p
@@ -98,7 +98,7 @@ module Reasoning {P : A → Set p}
 -- equality.
 
 module Reasoning-≡ {a p} {A : Set a} {P : A → Set p}
-  = Reasoning {P = P} {_∼_ = _≡_} (P.subst P ∘ P.sym)
+  = Reasoning {P = P} {_∼_ = _≡_} (≡.subst P ∘ ≡.sym)
 
 ------------------------------------------------------------------------
 -- An alternative, but equivalent, formulation of All
@@ -140,12 +140,12 @@ module Alternative {a p : Level} where
 
     trans-≅ : {P : A → Set p} {x y : A ⊥} →
               x ≅ y → AllW P y → AllW P x
-    trans-≅ (now P.refl) (now   p) = now p
+    trans-≅ (now ≡.refl) (now   p) = now p
     trans-≅ (later  x≅y) (later p) = later (_ ≅⟨ ♭ x≅y ⟩P p)
 
     trans-≳ : {P : A → Set p} {x y : A ⊥} →
               x ≳ y → AllW P y → AllW P x
-    trans-≳ (now P.refl) (now   p) = now p
+    trans-≳ (now ≡.refl) (now   p) = now p
     trans-≳ (later  x≳y) (later p) = later (_ ≳⟨ ♭ x≳y ⟩P p)
     trans-≳ (laterˡ x≳y)        p  = later (_ ≳⟨   x≳y ⟩P program p)
 
diff --git a/src/Function/Bijection.agda b/src/Function/Bijection.agda
index ac32159e7a..91974b6294 100644
--- a/src/Function/Bijection.agda
+++ b/src/Function/Bijection.agda
@@ -16,7 +16,7 @@ Use the standard function hierarchy in Function/Function.Bundles instead."
 
 open import Level
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality as ≡
 open import Function.Equality as F
   using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
 open import Function.Injection   as Inj  hiding (id; _∘_; injection)
@@ -90,7 +90,7 @@ Please use Function.(Bundles.)Bijection instead."
 infix 3 _⤖_
 
 _⤖_ : ∀ {f t} → Set f → Set t → Set _
-From ⤖ To = Bijection (P.setoid From) (P.setoid To)
+From ⤖ To = Bijection (≡.setoid From) (≡.setoid To)
 {-# WARNING_ON_USAGE _⤖_
 "Warning: _⤖_ was deprecated in v2.0.
 Please use Function.(Bundles.)mk⤖ instead."
diff --git a/src/Function/Endomorphism/Propositional.agda b/src/Function/Endomorphism/Propositional.agda
index c41305f9e9..c8dd685328 100644
--- a/src/Function/Endomorphism/Propositional.agda
+++ b/src/Function/Endomorphism/Propositional.agda
@@ -21,10 +21,10 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (id; _∘′_; _∋_)
 open import Function.Equality using (_⟨$⟩_)
 open import Relation.Binary.Core using (_Preserves_⟶_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
-import Function.Endomorphism.Setoid (P.setoid A) as Setoid
+import Function.Endomorphism.Setoid (≡.setoid A) as Setoid
 
 Endo : Set a
 Endo = A → A
@@ -38,7 +38,7 @@ fromSetoidEndo = _⟨$⟩_
 toSetoidEndo : Endo → Setoid.Endo
 toSetoidEndo f = record
   { _⟨$⟩_ = f
-  ; cong  = P.cong f
+  ; cong  = cong f
   }
 
 ------------------------------------------------------------------------
@@ -52,15 +52,15 @@ f ^ suc n = f ∘′ (f ^ n)
 
 ^-homo : ∀ f → Homomorphic₂ ℕ Endo _≡_ (f ^_) _+_ _∘′_
 ^-homo f zero    n = refl
-^-homo f (suc m) n = P.cong (f ∘′_) (^-homo f m n)
+^-homo f (suc m) n = cong (f ∘′_) (^-homo f m n)
 
 ------------------------------------------------------------------------
 -- Structures
 
 ∘-isMagma : IsMagma _≡_ (Op₂ Endo ∋ _∘′_)
 ∘-isMagma = record
-  { isEquivalence = P.isEquivalence
-  ; ∙-cong        = P.cong₂ _∘′_
+  { isEquivalence = ≡.isEquivalence
+  ; ∙-cong        = cong₂ _∘′_
   }
 
 ∘-magma : Magma _ _
@@ -89,7 +89,7 @@ f ^ suc n = f ∘′ (f ^ n)
 
 ^-isSemigroupMorphism : ∀ f → IsSemigroupMorphism +-semigroup ∘-semigroup (f ^_)
 ^-isSemigroupMorphism f = record
-  { ⟦⟧-cong = P.cong (f ^_)
+  { ⟦⟧-cong = cong (f ^_)
   ; ∙-homo  = ^-homo f
   }
 
diff --git a/src/Function/Equality.agda b/src/Function/Equality.agda
index 7451aaf3f7..647dca7416 100644
--- a/src/Function/Equality.agda
+++ b/src/Function/Equality.agda
@@ -21,8 +21,8 @@ open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid; _=[_]⇒_)
 import Relation.Binary.Indexed.Heterogeneous.Construct.Trivial
   as Trivial
-import Relation.Binary.PropositionalEquality.Core as P
-import Relation.Binary.PropositionalEquality.Properties as P
+import Relation.Binary.PropositionalEquality.Core as ≡
+import Relation.Binary.PropositionalEquality.Properties as ≡
 
 ------------------------------------------------------------------------
 -- Functions which preserve equality
@@ -147,8 +147,8 @@ flip {B = B} f = record
   }
 
 →-to-⟶ : ∀ {a b ℓ} {A : Set a} {B : Setoid b ℓ} →
-         (A → Setoid.Carrier B) → P.setoid A ⟶ B
+         (A → Setoid.Carrier B) → ≡.setoid A ⟶ B
 →-to-⟶ {B = B} to = record
   { _⟨$⟩_ = to
-  ; cong = λ { P.refl → Setoid.refl B }
+  ; cong = λ { ≡.refl → Setoid.refl B }
   }
diff --git a/src/Function/Equivalence.agda b/src/Function/Equivalence.agda
index 1e3652da7f..9556023b56 100644
--- a/src/Function/Equivalence.agda
+++ b/src/Function/Equivalence.agda
@@ -20,7 +20,7 @@ open import Function.Equality as F
 open import Level
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (Reflexive; TransFlip; Sym)
-import Relation.Binary.PropositionalEquality as P
+import Relation.Binary.PropositionalEquality as ≡
 
 ------------------------------------------------------------------------
 -- Setoid equivalence
@@ -43,7 +43,7 @@ Please use Function.(Bundles.)Equivalence instead."
 infix 3 _⇔_
 
 _⇔_ : ∀ {f t} → Set f → Set t → Set _
-From ⇔ To = Equivalence (P.setoid From) (P.setoid To)
+From ⇔ To = Equivalence (≡.setoid From) (≡.setoid To)
 {-# WARNING_ON_USAGE _⇔_
 "Warning: _⇔_ was deprecated in v2.0.
 Please use Function.(Bundles.)_⇔_ instead."
diff --git a/src/Function/Injection.agda b/src/Function/Injection.agda
index 462b8b3428..72785584f1 100644
--- a/src/Function/Injection.agda
+++ b/src/Function/Injection.agda
@@ -19,7 +19,7 @@ open import Level
 open import Relation.Binary.Bundles using (Setoid)
 open import Function.Equality as F
   using (_⟶_; _⟨$⟩_ ; Π) renaming (_∘_ to _⟪∘⟫_)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
 
 ------------------------------------------------------------------------
 -- Injective functions
@@ -58,7 +58,7 @@ Please use Function.(Bundles.)Injection instead."
 infix 3 _↣_
 
 _↣_ : ∀ {f t} → Set f → Set t → Set _
-From ↣ To = Injection (P.setoid From) (P.setoid To)
+From ↣ To = Injection (≡.setoid From) (≡.setoid To)
 {-# WARNING_ON_USAGE _↣_
 "Warning: _↣_ was deprecated in v2.0.
 Please use Function.(Bundles.)_↣_ instead."
@@ -69,7 +69,7 @@ injection : ∀ {f t} {From : Set f} {To : Set t} → (to : From → To) →
 injection to injective = record
   { to        = record
     { _⟨$⟩_ = to
-    ; cong = P.cong to
+    ; cong = ≡.cong to
     }
   ; injective = injective
   }
diff --git a/src/Function/Inverse.agda b/src/Function/Inverse.agda
index e9d796eddf..792382ca17 100644
--- a/src/Function/Inverse.agda
+++ b/src/Function/Inverse.agda
@@ -22,7 +22,7 @@ open import Function.Equality as F
 open import Function.LeftInverse as Left hiding (id; _∘_)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (Reflexive; TransFlip; Sym)
-open import Relation.Binary.PropositionalEquality as P using (_≗_; _≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≗_; _≡_)
 open import Relation.Unary using (Pred)
 
 ------------------------------------------------------------------------
@@ -90,7 +90,7 @@ Please use Function.(Bundles.)Inverse instead."
 infix 3 _↔_ _↔̇_
 
 _↔_ : ∀ {f t} → Set f → Set t → Set _
-From ↔ To = Inverse (P.setoid From) (P.setoid To)
+From ↔ To = Inverse (≡.setoid From) (≡.setoid To)
 {-# WARNING_ON_USAGE _↔_
 "Warning: _↔_ was deprecated in v2.0.
 Please use Function.(Bundles.)_↔_ instead."
diff --git a/src/Function/LeftInverse.agda b/src/Function/LeftInverse.agda
index 3824ca95d3..d9ccc9a806 100644
--- a/src/Function/LeftInverse.agda
+++ b/src/Function/LeftInverse.agda
@@ -21,7 +21,7 @@ open import Function.Equality as Eq
   using (_⟶_; _⟨$⟩_) renaming (_∘_ to _⟪∘⟫_)
 open import Function.Equivalence using (Equivalence)
 open import Function.Injection using (Injective; Injection)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
 
 ------------------------------------------------------------------------
 -- Left and right inverses.
@@ -106,7 +106,7 @@ Please use Function.(Bundles.)LeftInverse instead."
 infix 3 _↞_
 
 _↞_ : ∀ {f t} → Set f → Set t → Set _
-From ↞ To = LeftInverse (P.setoid From) (P.setoid To)
+From ↞ To = LeftInverse (≡.setoid From) (≡.setoid To)
 {-# WARNING_ON_USAGE _↞_
 "Warning: _↞_ was deprecated in v2.0.
 Please use Function.(Bundles.)_↪_ instead."
diff --git a/src/Function/Related.agda b/src/Function/Related.agda
index 9846b4d034..3bcb839237 100644
--- a/src/Function/Related.agda
+++ b/src/Function/Related.agda
@@ -27,8 +27,8 @@ open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Bundles using (Setoid; Preorder)
 open import Relation.Binary.Structures using (IsEquivalence; IsPreorder)
 open import Relation.Binary.Definitions using (Reflexive; Trans; Sym)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Data.Product.Base using (_,_; proj₁; proj₂; <_,_>)
 
 import Function.Related.Propositional as R
@@ -90,12 +90,12 @@ infix 4 _∼[_]_
 _∼[_]_ : ∀ {ℓ₁ ℓ₂} → Set ℓ₁ → Kind → Set ℓ₂ → Set _
 A ∼[ implication         ] B = A → B
 A ∼[ reverse-implication ] B = A ← B
-A ∼[ equivalence         ] B = Equivalence (P.setoid A) (P.setoid B)
-A ∼[ injection           ] B = Injection   (P.setoid A) (P.setoid B)
+A ∼[ equivalence         ] B = Equivalence (≡.setoid A) (≡.setoid B)
+A ∼[ injection           ] B = Injection   (≡.setoid A) (≡.setoid B)
 A ∼[ reverse-injection   ] B = A ↢ B
-A ∼[ left-inverse        ] B = LeftInverse (P.setoid A) (P.setoid B)
-A ∼[ surjection          ] B = Surjection  (P.setoid A) (P.setoid B)
-A ∼[ bijection           ] B = Inverse     (P.setoid A) (P.setoid B)
+A ∼[ left-inverse        ] B = LeftInverse (≡.setoid A) (≡.setoid B)
+A ∼[ surjection          ] B = Surjection  (≡.setoid A) (≡.setoid B)
+A ∼[ bijection           ] B = Inverse     (≡.setoid A) (≡.setoid B)
 
 -- A non-infix synonym.
 
@@ -125,7 +125,7 @@ fromRelated {K = left-inverse}        record { to = to ; from = from ; left-inve
   B.mk↪ {to = to ⟨$⟩_} {from = from ⟨$⟩_} (strictlyInverseʳ⇒inverseʳ (to ⟨$⟩_) left-inverse-of)
 fromRelated {K = surjection}          record { to = to ; surjective = surjective } with surjective
 ... | record { from = from ; right-inverse-of = right-inverse-of } =
-  B.mk↠ {to = to ⟨$⟩_} < from ⟨$⟩_ , (λ { x P.refl → right-inverse-of x }) >
+  B.mk↠ {to = to ⟨$⟩_} < from ⟨$⟩_ , (λ { x ≡.refl → right-inverse-of x }) >
 fromRelated {K = bijection}           rel = B.mk↔ₛ′ (to ⟨$⟩_) (from ⟨$⟩_) right-inverse-of left-inverse-of
   where open Inverse rel
 
@@ -145,7 +145,7 @@ fromRelated {K = bijection}           rel = B.mk↔ₛ′ (to ⟨$⟩_) (from 
 -- Actual equality also implies any kind of relatedness.
 
 ≡⇒ : ∀ {k ℓ} {X Y : Set ℓ} → X ≡ Y → X ∼[ k ] Y
-≡⇒ P.refl = ↔⇒ Inv.id
+≡⇒ ≡.refl = ↔⇒ Inv.id
 
 ------------------------------------------------------------------------
 -- Special kinds of kinds
@@ -313,7 +313,7 @@ K-refl {surjection}          = Surj.id
 K-refl {bijection}           = Inv.id
 
 K-reflexive : ∀ {k ℓ} → _≡_ ⇒ Related k {ℓ}
-K-reflexive P.refl = K-refl
+K-reflexive ≡.refl = K-refl
 
 K-trans : ∀ {k ℓ₁ ℓ₂ ℓ₃} → Trans (Related k {ℓ₁} {ℓ₂})
                                 (Related k {ℓ₂} {ℓ₃})
@@ -383,7 +383,7 @@ module EquationalReasoning where
 
   _≡˘⟨_⟩_ : ∀ {k ℓ z} (X : Set ℓ) {Y : Set ℓ} {Z : Set z} →
             Y ≡ X → Y ∼[ k ] Z → X ∼[ k ] Z
-  X ≡˘⟨ Y≡X ⟩ Y⇔Z = X ∼⟨ ≡⇒ (P.sym Y≡X) ⟩ Y⇔Z
+  X ≡˘⟨ Y≡X ⟩ Y⇔Z = X ∼⟨ ≡⇒ (≡.sym Y≡X) ⟩ Y⇔Z
 
   _≡⟨_⟩_ : ∀ {k ℓ z} (X : Set ℓ) {Y : Set ℓ} {Z : Set z} →
            X ≡ Y → Y ∼[ k ] Z → X ∼[ k ] Z
@@ -406,10 +406,10 @@ InducedPreorder₁ k S = record
   { _≈_        = _≡_
   ; _≲_        = InducedRelation₁ k S
   ; isPreorder = record
-    { isEquivalence = P.isEquivalence
+    { isEquivalence = ≡.isEquivalence
     ; reflexive     = reflexive ∘
                       K-reflexive ∘
-                      P.cong S
+                      ≡.cong S
     ; trans         = K-trans
     }
   } where open Preorder (K-preorder _ _)
@@ -439,11 +439,11 @@ InducedPreorder₂ k _S_ = record
   { _≈_        = _≡_
   ; _≲_        = InducedRelation₂ k _S_
   ; isPreorder = record
-    { isEquivalence = P.isEquivalence
+    { isEquivalence = ≡.isEquivalence
     ; reflexive     = λ x≡y {z} →
                         reflexive $
                         K-reflexive $
-                        P.cong (_S_ z) x≡y
+                        ≡.cong (_S_ z) x≡y
 
     ; trans         = λ i↝j j↝k → K-trans i↝j j↝k
     }
diff --git a/src/Function/Related/Propositional.agda b/src/Function/Related/Propositional.agda
index d7f0610e78..589c36a6b3 100644
--- a/src/Function/Related/Propositional.agda
+++ b/src/Function/Related/Propositional.agda
@@ -13,8 +13,8 @@ open import Relation.Binary
   using (Rel; REL; Sym; Reflexive; Trans; IsEquivalence; Setoid; IsPreorder; Preorder)
 open import Function.Bundles
 open import Function.Base
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Relation.Binary.Reasoning.Syntax
 
 open import Function.Properties.Surjection   using (↠⇒↪; ↠⇒⇔)
@@ -86,7 +86,7 @@ Related k A B = A ∼[ k ] B
 -- Propositional equality also implies any kind of relatedness.
 
 ≡⇒ : A ≡ B → A ∼[ k ] B
-≡⇒ P.refl = ↔⇒ (Identity.↔-id _)
+≡⇒ ≡.refl = ↔⇒ (Identity.↔-id _)
 
 ------------------------------------------------------------------------
 -- Special kinds of kinds
@@ -252,7 +252,7 @@ K-refl {k = surjection}         = Identity.↠-id _
 K-refl {k = bijection}          = Identity.↔-id _
 
 K-reflexive : _≡_ Relation.Binary.⇒ Related {a} k
-K-reflexive P.refl = K-refl
+K-reflexive ≡.refl = K-refl
 
 K-trans : Trans (Related {a} {b} k)
                 (Related {b} {c} k)
@@ -340,10 +340,10 @@ InducedPreorder₁ k P = record
   { _≈_        = _≡_
   ; _≲_        = InducedRelation₁ k P
   ; isPreorder = record
-    { isEquivalence = P.isEquivalence
+    { isEquivalence = ≡.isEquivalence
     ; reflexive     = reflexive ∘
                       K-reflexive ∘
-                      P.cong P
+                      ≡.cong P
     ; trans         = K-trans
     }
   } where open Preorder (K-preorder _ _)
@@ -370,11 +370,11 @@ InducedPreorder₂ k _S_ = record
   { _≈_        = _≡_
   ; _≲_        = InducedRelation₂ k _S_
   ; isPreorder = record
-    { isEquivalence = P.isEquivalence
+    { isEquivalence = ≡.isEquivalence
     ; reflexive     = λ x≡y {z} →
                         reflexive $
                         K-reflexive $
-                        P.cong (_S_ z) x≡y
+                        ≡.cong (_S_ z) x≡y
 
     ; trans         = λ i↝j j↝k → K-trans i↝j j↝k
     }
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index d8a78e052c..eafe0b50b6 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -28,7 +28,7 @@ open import Function.Bundles
 open import Function.Related.Propositional
 import Function.Construct.Identity as Identity
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 open import Relation.Nullary.Reflects using (invert)
@@ -47,20 +47,20 @@ private
 -- Σ is associative
 Σ-assoc : ∀ {A : Set a} {B : A → Set b} {C : (a : A) → B a → Set c} →
           Σ (Σ A B) (uncurry C) ↔ Σ A (λ a → Σ (B a) (C a))
-Σ-assoc = mk↔ₛ′ Product.assocʳ Product.assocˡ (λ _ → P.refl) (λ _ → P.refl)
+Σ-assoc = mk↔ₛ′ Product.assocʳ Product.assocˡ (λ _ → refl) (λ _ → refl)
 
 -- × is commutative
 
 ×-comm : ∀ (A : Set a) (B : Set b) → (A × B) ↔ (B × A)
-×-comm _ _ = mk↔ₛ′ Product.swap Product.swap (λ _ → P.refl) λ _ → P.refl
+×-comm _ _ = mk↔ₛ′ Product.swap Product.swap (λ _ → refl) λ _ → refl
 
 -- × has ⊤ as its identity
 
 ×-identityˡ : ∀ ℓ → LeftIdentity  {ℓ = ℓ} _↔_ ⊤ _×_
-×-identityˡ _ _ = mk↔ₛ′ proj₂ -,_ (λ _ → P.refl) (λ _ → P.refl)
+×-identityˡ _ _ = mk↔ₛ′ proj₂ -,_ (λ _ → refl) (λ _ → refl)
 
 ×-identityʳ : ∀ ℓ → RightIdentity  {ℓ = ℓ} _↔_ ⊤ _×_
-×-identityʳ _ _ = mk↔ₛ′ proj₁ (_, _) (λ _ → P.refl) (λ _ → P.refl)
+×-identityʳ _ _ = mk↔ₛ′ proj₁ (_, _) (λ _ → refl) (λ _ → refl)
 
 ×-identity : ∀ ℓ → Identity _↔_ ⊤ _×_
 ×-identity ℓ = ×-identityˡ ℓ , ×-identityʳ ℓ
@@ -68,10 +68,10 @@ private
 -- × has ⊥ has its zero
 
 ×-zeroˡ : ∀ ℓ → LeftZero {ℓ = ℓ} _↔_ ⊥ _×_
-×-zeroˡ ℓ A = mk↔ₛ′ proj₁ < id , ⊥ₚ-elim > (λ _ → P.refl) (λ { () })
+×-zeroˡ ℓ A = mk↔ₛ′ proj₁ < id , ⊥ₚ-elim > (λ _ → refl) (λ { () })
 
 ×-zeroʳ : ∀ ℓ → RightZero {ℓ = ℓ} _↔_ ⊥ _×_
-×-zeroʳ ℓ A = mk↔ₛ′ proj₂ < ⊥ₚ-elim , id > (λ _ → P.refl) (λ { () })
+×-zeroʳ ℓ A = mk↔ₛ′ proj₂ < ⊥ₚ-elim , id > (λ _ → refl) (λ { () })
 
 ×-zero : ∀ ℓ → Zero _↔_ ⊥ _×_
 ×-zero ℓ  = ×-zeroˡ ℓ , ×-zeroʳ ℓ
@@ -85,8 +85,8 @@ private
 ⊎-assoc ℓ _ _ _ = mk↔ₛ′
   [ [ inj₁ , inj₂ ∘′ inj₁ ]′ , inj₂ ∘′ inj₂ ]′
   [ inj₁ ∘′ inj₁ , [ inj₁ ∘′ inj₂ , inj₂ ]′ ]′
-  [ (λ _ → P.refl) , [ (λ _ → P.refl) , (λ _ → P.refl) ] ]
-  [ [ (λ _ → P.refl) , (λ _ → P.refl) ] , (λ _ → P.refl) ]
+  [ (λ _ → refl) , [ (λ _ → refl) , (λ _ → refl) ] ]
+  [ [ (λ _ → refl) , (λ _ → refl) ] , (λ _ → refl) ]
 
 -- ⊎ is commutative
 
@@ -96,12 +96,12 @@ private
 -- ⊎ has ⊥ as its identity
 
 ⊎-identityˡ : ∀ ℓ → LeftIdentity _↔_ (⊥ {ℓ}) _⊎_
-⊎-identityˡ _ _ = mk↔ₛ′ [ (λ ()) , id ]′ inj₂ (λ _ → P.refl)
-                          [ (λ ()) , (λ _ → P.refl) ]
+⊎-identityˡ _ _ = mk↔ₛ′ [ (λ ()) , id ]′ inj₂ (λ _ → refl)
+                          [ (λ ()) , (λ _ → refl) ]
 
 ⊎-identityʳ : ∀ ℓ → RightIdentity _↔_ (⊥ {ℓ}) _⊎_
-⊎-identityʳ _ _ = mk↔ₛ′ [ id , (λ ()) ]′ inj₁ (λ _ → P.refl)
-                          [ (λ _ → P.refl) , (λ ()) ]
+⊎-identityʳ _ _ = mk↔ₛ′ [ id , (λ ()) ]′ inj₁ (λ _ → refl)
+                          [ (λ _ → refl) , (λ ()) ]
 
 ⊎-identity : ∀ ℓ → Identity _↔_ ⊥ _⊎_
 ⊎-identity ℓ = ⊎-identityˡ ℓ , ⊎-identityʳ ℓ
@@ -115,15 +115,15 @@ private
 ×-distribˡ-⊎ ℓ _ _ _ = mk↔ₛ′
   (uncurry λ x → [ inj₁ ∘′ (x ,_) , inj₂ ∘′ (x ,_) ]′)
   [ Product.map₂ inj₁ , Product.map₂ inj₂ ]′
-  [ (λ _ → P.refl) , (λ _ → P.refl) ]
-  (uncurry λ _ → [ (λ _ → P.refl) , (λ _ → P.refl) ])
+  [ (λ _ → refl) , (λ _ → refl) ]
+  (uncurry λ _ → [ (λ _ → refl) , (λ _ → refl) ])
 
 ×-distribʳ-⊎ : ∀ ℓ → _DistributesOverʳ_ {ℓ = ℓ} _↔_ _×_ _⊎_
 ×-distribʳ-⊎ ℓ _ _ _ = mk↔ₛ′
   (uncurry [ curry inj₁ , curry inj₂ ]′)
   [ Product.map₁ inj₁ , Product.map₁ inj₂ ]′
-  [ (λ _ → P.refl) , (λ _ → P.refl) ]
-  (uncurry [ (λ _ _ → P.refl) , (λ _ _ → P.refl) ])
+  [ (λ _ → refl) , (λ _ → refl) ]
+  (uncurry [ (λ _ _ → refl) , (λ _ _ → refl) ])
 
 ×-distrib-⊎ : ∀ ℓ → _DistributesOver_ {ℓ = ℓ} _↔_ _×_ _⊎_
 ×-distrib-⊎ ℓ = ×-distribˡ-⊎ ℓ , ×-distribʳ-⊎ ℓ
@@ -244,11 +244,11 @@ private
 
 ΠΠ↔ΠΠ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →
         ((x : A) (y : B) → P x y) ↔ ((y : B) (x : A) → P x y)
-ΠΠ↔ΠΠ _ = mk↔ₛ′ flip flip (λ _ → P.refl) (λ _ → P.refl)
+ΠΠ↔ΠΠ _ = mk↔ₛ′ flip flip (λ _ → refl) (λ _ → refl)
 
 ∃∃↔∃∃ : ∀ {a b p} {A : Set a} {B : Set b} (P : A → B → Set p) →
         (∃₂ λ x y → P x y) ↔ (∃₂ λ y x → P x y)
-∃∃↔∃∃ P = mk↔ₛ′ to from (λ _ → P.refl) (λ _ → P.refl)
+∃∃↔∃∃ P = mk↔ₛ′ to from (λ _ → refl) (λ _ → refl)
   where
   to : (∃₂ λ x y → P x y) → (∃₂ λ y x → P x y)
   to (x , y , Pxy) = (y , x , Pxy)
@@ -261,7 +261,7 @@ private
 
 Π↔Π : ∀ {A : Set a} {B : A → Set b} →
       ((x : A) → B x) ↔ ({x : A} → B x)
-Π↔Π = mk↔ₛ′ _$- λ- (λ _ → P.refl) (λ _ → P.refl)
+Π↔Π = mk↔ₛ′ _$- λ- (λ _ → refl) (λ _ → refl)
 
 ------------------------------------------------------------------------
 -- _→_ preserves the symmetric relations
@@ -280,12 +280,12 @@ private
   (λ f → from C↔D ∘ f ∘ to A↔B)
   (λ f → ext₂ λ x → begin
     to C↔D (from C↔D (f (to A↔B (from A↔B x)))) ≡⟨ strictlyInverseˡ C↔D _ ⟩
-    f (to A↔B (from A↔B x))                     ≡⟨ P.cong f $ strictlyInverseˡ A↔B x ⟩
+    f (to A↔B (from A↔B x))                     ≡⟨ cong f $ strictlyInverseˡ A↔B x ⟩
     f x                                         ∎)
   (λ f → ext₁ λ x → begin
-    from C↔D (to C↔D (f (from A↔B (to A↔B x))))  ≡⟨ strictlyInverseʳ C↔D _ ⟩
-    f (from A↔B (to A↔B x))                        ≡⟨ P.cong f $ strictlyInverseʳ A↔B x ⟩
-    f x                                              ∎)
+    from C↔D (to C↔D (f (from A↔B (to A↔B x)))) ≡⟨ strictlyInverseʳ C↔D _ ⟩
+    f (from A↔B (to A↔B x))                     ≡⟨ cong f $ strictlyInverseʳ A↔B x ⟩
+    f x                                         ∎)
   where open Inverse; open ≡-Reasoning
 
 →-cong :  Extensionality a c → Extensionality b d →
@@ -303,7 +303,7 @@ private
 ¬-cong : Extensionality a 0ℓ → Extensionality b 0ℓ →
          ∀ {k} {A : Set a} {B : Set b} →
          A ∼[ ⌊ k ⌋ ] B → (¬ A) ∼[ ⌊ k ⌋ ] (¬ B)
-¬-cong extA extB A≈B = →-cong extA extB A≈B (K-reflexive P.refl)
+¬-cong extA extB A≈B = →-cong extA extB A≈B (K-reflexive refl)
 
 ------------------------------------------------------------------------
 -- _⇔_ preserves _⇔_
@@ -330,6 +330,6 @@ Related-cong {A = A} {B = B} {C = C} {D = D} A≈B C≈D = mk⇔
 True↔ : ∀ {p} {P : Set p}
         (dec : Dec P) → ((p₁ p₂ : P) → p₁ ≡ p₂) → True dec ↔ P
 True↔ ( true because  [p]) irr =
-  mk↔ₛ′ (λ _ → invert [p]) (λ _ → _) (irr _) (λ _ → P.refl)
+  mk↔ₛ′ (λ _ → invert [p]) (λ _ → _) (irr _) (λ _ → refl)
 True↔ (false because ofⁿ ¬p) _ =
   mk↔ₛ′ (λ()) (invert (ofⁿ ¬p)) (⊥-elim ∘ ¬p) (λ ())
diff --git a/src/Function/Surjection.agda b/src/Function/Surjection.agda
index 2fdfefa9af..684da0b3d7 100644
--- a/src/Function/Surjection.agda
+++ b/src/Function/Surjection.agda
@@ -21,7 +21,7 @@ open import Function.Equivalence using (Equivalence)
 open import Function.Injection           hiding (id; _∘_; injection)
 open import Function.LeftInverse as Left hiding (id; _∘_)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
 
 ------------------------------------------------------------------------
 -- Surjective functions.
@@ -100,7 +100,7 @@ Please use Function.(Properties.)RightInverse.RightInverse⇒Surjection instead.
 infix 3 _↠_
 
 _↠_ : ∀ {f t} → Set f → Set t → Set _
-From ↠ To = Surjection (P.setoid From) (P.setoid To)
+From ↠ To = Surjection (≡.setoid From) (≡.setoid To)
 {-# WARNING_ON_USAGE _↠_
 "Warning: _↠_ was deprecated in v2.0.
 Please use Function.(Bundles.)_↠_ instead."
diff --git a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
index 5d5cb2a214..c3b186c62a 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/NonStrict.agda
@@ -20,9 +20,8 @@ module Relation.Binary.Construct.Add.Infimum.NonStrict
 
 open import Level using (_⊔_)
 open import Data.Sum.Base as Sum
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 import Relation.Binary.Construct.Add.Infimum.Equality as Equality
 open import Relation.Nullary hiding (Irrelevant)
 open import Relation.Nullary.Construct.Add.Infimum
@@ -60,8 +59,8 @@ data _≤₋_ : Rel (A ₋) (a ⊔ ℓ) where
 ≤₋-total ≤-total [ k ] [ l ] = Sum.map [_] [_] (≤-total k l)
 
 ≤₋-irrelevant : Irrelevant _≤_ → Irrelevant _≤₋_
-≤₋-irrelevant ≤-irr (⊥₋≤ k) (⊥₋≤ k) = P.refl
-≤₋-irrelevant ≤-irr [ p ]   [ q ]   = P.cong _ (≤-irr p q)
+≤₋-irrelevant ≤-irr (⊥₋≤ k) (⊥₋≤ k) = refl
+≤₋-irrelevant ≤-irr [ p ]   [ q ]   = cong _ (≤-irr p q)
 
 ------------------------------------------------------------------------
 -- Relational properties + propositional equality
@@ -72,7 +71,7 @@ data _≤₋_ : Rel (A ₋) (a ⊔ ℓ) where
 
 ≤₋-antisym-≡ : Antisymmetric _≡_ _≤_ → Antisymmetric _≡_ _≤₋_
 ≤₋-antisym-≡ antisym (⊥₋≤ _) (⊥₋≤ _) = refl
-≤₋-antisym-≡ antisym [ p ] [ q ]       = P.cong [_] (antisym p q)
+≤₋-antisym-≡ antisym [ p ] [ q ]     = cong [_] (antisym p q)
 
 ------------------------------------------------------------------------
 -- Relational properties + setoid equality
@@ -94,7 +93,7 @@ module _ {e} {_≈_ : Rel A e} where
 
 ≤₋-isPreorder-≡ : IsPreorder _≡_ _≤_ → IsPreorder _≡_ _≤₋_
 ≤₋-isPreorder-≡ ≤-isPreorder = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; reflexive     = ≤₋-reflexive-≡ reflexive
   ; trans         = ≤₋-trans trans
   } where open IsPreorder ≤-isPreorder
diff --git a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
index 5f3bdd7622..c1312d7b14 100644
--- a/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Infimum/Strict.agda
@@ -21,9 +21,8 @@ module Relation.Binary.Construct.Add.Infimum.Strict
 open import Level using (_⊔_)
 open import Data.Product.Base using (_,_; map)
 open import Function.Base
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; subst)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 import Relation.Binary.Construct.Add.Infimum.Equality as Equality
 import Relation.Binary.Construct.Add.Infimum.NonStrict as NonStrict
 open import Relation.Nullary hiding (Irrelevant)
@@ -59,8 +58,8 @@ data _<₋_ : Rel (A ₋) (a ⊔ ℓ) where
 <₋-dec _<?_ [ k ] [ l ] = Dec.map′ [_] [<]-injective (k <? l)
 
 <₋-irrelevant : Irrelevant _<_ → Irrelevant _<₋_
-<₋-irrelevant <-irr ⊥₋<[ l ] ⊥₋<[ l ] = P.refl
-<₋-irrelevant <-irr [ p ]    [ q ]    = P.cong _ (<-irr p q)
+<₋-irrelevant <-irr ⊥₋<[ l ] ⊥₋<[ l ] = refl
+<₋-irrelevant <-irr [ p ]    [ q ]    = cong _ (<-irr p q)
 
 module _ {r} {_≤_ : Rel A r} where
 
@@ -91,10 +90,10 @@ module _ {r} {_≤_ : Rel A r} where
 <₋-irrefl-≡ <-irrefl refl [ x ] = <-irrefl refl x
 
 <₋-respˡ-≡ : _<₋_ Respectsˡ _≡_
-<₋-respˡ-≡ = P.subst (_<₋ _)
+<₋-respˡ-≡ = subst (_<₋ _)
 
 <₋-respʳ-≡ : _<₋_ Respectsʳ _≡_
-<₋-respʳ-≡ = P.subst (_ <₋_)
+<₋-respʳ-≡ = subst (_ <₋_)
 
 <₋-resp-≡ : _<₋_ Respects₂ _≡_
 <₋-resp-≡ = <₋-respʳ-≡ , <₋-respˡ-≡
@@ -136,7 +135,7 @@ module _ {e} {_≈_ : Rel A e} where
 <₋-isStrictPartialOrder-≡ : IsStrictPartialOrder _≡_ _<_ →
                             IsStrictPartialOrder _≡_ _<₋_
 <₋-isStrictPartialOrder-≡ strict = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; irrefl        = <₋-irrefl-≡ irrefl
   ; trans         = <₋-trans trans
   ; <-resp-≈      = <₋-resp-≡
diff --git a/src/Relation/Binary/Construct/Add/Point/Equality.agda b/src/Relation/Binary/Construct/Add/Point/Equality.agda
index 704d963123..b3f2ea6dcf 100644
--- a/src/Relation/Binary/Construct/Add/Point/Equality.agda
+++ b/src/Relation/Binary/Construct/Add/Point/Equality.agda
@@ -20,7 +20,7 @@ module Relation.Binary.Construct.Add.Point.Equality
 
 open import Level using (_⊔_)
 open import Function.Base
-import Relation.Binary.PropositionalEquality.Core as P
+import Relation.Binary.PropositionalEquality.Core as ≡
 open import Relation.Nullary hiding (Irrelevant)
 open import Relation.Nullary.Construct.Add.Point
 import Relation.Nullary.Decidable as Dec
@@ -59,8 +59,8 @@ data _≈∙_ : Rel (Pointed A) (a ⊔ ℓ) where
 ≈∙-dec _≟_ [ k ] [ l ] = Dec.map′ [_] [≈]-injective (k ≟ l)
 
 ≈∙-irrelevant : Irrelevant _≈_ → Irrelevant _≈∙_
-≈∙-irrelevant ≈-irr ∙≈∙   ∙≈∙   = P.refl
-≈∙-irrelevant ≈-irr [ p ] [ q ] = P.cong _ (≈-irr p q)
+≈∙-irrelevant ≈-irr ∙≈∙   ∙≈∙   = ≡.refl
+≈∙-irrelevant ≈-irr [ p ] [ q ] = ≡.cong _ (≈-irr p q)
 
 ≈∙-substitutive : ∀ {ℓ} → Substitutive _≈_ ℓ → Substitutive _≈∙_ ℓ
 ≈∙-substitutive ≈-subst P ∙≈∙   = id
diff --git a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
index b550dd4924..9eb4cc7189 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/NonStrict.agda
@@ -22,9 +22,9 @@ open import Level using (_⊔_)
 open import Data.Sum.Base as Sum
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Relation.Nullary.Construct.Add.Supremum
 import Relation.Binary.Construct.Add.Supremum.Equality as Equality
 
@@ -61,8 +61,8 @@ data _≤⁺_ : Rel (A ⁺) (a ⊔ ℓ) where
 ≤⁺-total ≤-total [ k ] [ l ] = Sum.map [_] [_] (≤-total k l)
 
 ≤⁺-irrelevant : Irrelevant _≤_ → Irrelevant _≤⁺_
-≤⁺-irrelevant ≤-irr [ p ]   [ q ]    = P.cong _ (≤-irr p q)
-≤⁺-irrelevant ≤-irr (k ≤⊤⁺) (k ≤⊤⁺) = P.refl
+≤⁺-irrelevant ≤-irr [ p ]   [ q ]   = cong _ (≤-irr p q)
+≤⁺-irrelevant ≤-irr (k ≤⊤⁺) (k ≤⊤⁺) = refl
 
 ------------------------------------------------------------------------
 -- Relational properties + propositional equality
@@ -73,7 +73,7 @@ data _≤⁺_ : Rel (A ⁺) (a ⊔ ℓ) where
 
 ≤⁺-antisym-≡ : Antisymmetric _≡_ _≤_ → Antisymmetric _≡_ _≤⁺_
 ≤⁺-antisym-≡ antisym (_ ≤⊤⁺) (_ ≤⊤⁺) = refl
-≤⁺-antisym-≡ antisym [ p ] [ q ]       = P.cong [_] (antisym p q)
+≤⁺-antisym-≡ antisym [ p ] [ q ]     = cong [_] (antisym p q)
 
 ------------------------------------------------------------------------
 -- Relation properties + setoid equality
@@ -95,7 +95,7 @@ module _ {e} {_≈_ : Rel A e} where
 
 ≤⁺-isPreorder-≡ : IsPreorder _≡_ _≤_ → IsPreorder _≡_ _≤⁺_
 ≤⁺-isPreorder-≡ ≤-isPreorder = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; reflexive     = ≤⁺-reflexive-≡ reflexive
   ; trans         = ≤⁺-trans trans
   } where open IsPreorder ≤-isPreorder
diff --git a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
index 733981deb0..5613be8112 100644
--- a/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Supremum/Strict.agda
@@ -23,9 +23,8 @@ open import Data.Product.Base using (_,_; map)
 open import Function.Base
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
-open import Relation.Binary.PropositionalEquality.Core as P
-  using (_≡_; refl)
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; subst)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Relation.Nullary.Construct.Add.Supremum
 import Relation.Binary.Construct.Add.Supremum.Equality as Equality
 import Relation.Binary.Construct.Add.Supremum.NonStrict as NonStrict
@@ -59,8 +58,8 @@ data _<⁺_ : Rel (A ⁺) (a ⊔ r) where
 <⁺-dec _<?_ ⊤⁺    ⊤⁺    = no (λ ())
 
 <⁺-irrelevant : Irrelevant _<_ → Irrelevant _<⁺_
-<⁺-irrelevant <-irr [ p ]    [ q ]    = P.cong _ (<-irr p q)
-<⁺-irrelevant <-irr [ k ]<⊤⁺ [ k ]<⊤⁺ = P.refl
+<⁺-irrelevant <-irr [ p ]    [ q ]    = cong _ (<-irr p q)
+<⁺-irrelevant <-irr [ k ]<⊤⁺ [ k ]<⊤⁺ = refl
 
 
 module _ {r} {_≤_ : Rel A r} where
@@ -92,10 +91,10 @@ module _ {r} {_≤_ : Rel A r} where
 <⁺-irrefl-≡ <-irrefl refl [ x ] = <-irrefl refl x
 
 <⁺-respˡ-≡ : _<⁺_ Respectsˡ _≡_
-<⁺-respˡ-≡ = P.subst (_<⁺ _)
+<⁺-respˡ-≡ = subst (_<⁺ _)
 
 <⁺-respʳ-≡ : _<⁺_ Respectsʳ _≡_
-<⁺-respʳ-≡ = P.subst (_ <⁺_)
+<⁺-respʳ-≡ = subst (_ <⁺_)
 
 <⁺-resp-≡ : _<⁺_ Respects₂ _≡_
 <⁺-resp-≡ = <⁺-respʳ-≡ , <⁺-respˡ-≡
@@ -137,7 +136,7 @@ module _ {e} {_≈_ : Rel A e} where
 <⁺-isStrictPartialOrder-≡ : IsStrictPartialOrder _≡_ _<_ →
                             IsStrictPartialOrder _≡_ _<⁺_
 <⁺-isStrictPartialOrder-≡ strict = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; irrefl        = <⁺-irrefl-≡ irrefl
   ; trans         = <⁺-trans trans
   ; <-resp-≈      = <⁺-resp-≡
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index ff04ed09a1..d06f81a0af 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -19,8 +19,8 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Definitions
   using (Symmetric; Transitive; Reflexive; Asymmetric; Antisymmetric; Trichotomous; Total; Decidable; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects_; _Respects₂_)
 open import Relation.Binary.Construct.Closure.Reflexive
-open import Relation.Binary.PropositionalEquality.Core as PropEq using (_≡_; refl)
-import Relation.Binary.PropositionalEquality.Properties as PropEq
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
+import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Relation.Nullary
 import Relation.Nullary.Decidable as Dec
 open import Relation.Unary using (Pred)
@@ -115,7 +115,7 @@ module _ {_~_ : Rel A ℓ} where
 
   isPreorder : Transitive _~_ → IsPreorder _≡_ _~ᵒ_
   isPreorder ~-trans = record
-    { isEquivalence = PropEq.isEquivalence
+    { isEquivalence = ≡.isEquivalence
     ; reflexive     = λ { refl → refl }
     ; trans         = trans ~-trans
     }
diff --git a/src/Relation/Binary/Construct/Closure/Transitive.agda b/src/Relation/Binary/Construct/Closure/Transitive.agda
index e4154d49cd..3c91ce2318 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive.agda
@@ -15,7 +15,7 @@ open import Level
 open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 private
   variable
@@ -102,7 +102,7 @@ data Plus {A : Set a} (_∼_ : Rel A ℓ) : Rel A (a ⊔ ℓ) where
 module _ {_∼_ : Rel A ℓ} where
 
  []-injective : ∀ {x y p q} → (x [ _∼_ ]⁺ y ∋ [ p ]) ≡ [ q ] → p ≡ q
- []-injective P.refl = P.refl
+ []-injective ≡.refl = ≡.refl
 
  -- See also ∼⁺⟨⟩-injectiveˡ and ∼⁺⟨⟩-injectiveʳ in
  -- Relation.Binary.Construct.Closure.Transitive.WithK.
diff --git a/src/Relation/Binary/HeterogeneousEquality.agda b/src/Relation/Binary/HeterogeneousEquality.agda
index 9181229eec..41d6b709a6 100644
--- a/src/Relation/Binary/HeterogeneousEquality.agda
+++ b/src/Relation/Binary/HeterogeneousEquality.agda
@@ -25,10 +25,10 @@ open import Relation.Binary.Indexed.Heterogeneous
   using (IndexedSetoid)
 open import Relation.Binary.Indexed.Heterogeneous.Construct.At
   using (_atₛ_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)
 open import Relation.Binary.Reasoning.Syntax
 
-import Relation.Binary.PropositionalEquality.Properties as P
+import Relation.Binary.PropositionalEquality.Properties as ≡
 import Relation.Binary.HeterogeneousEquality.Core as Core
 
 private
@@ -59,7 +59,7 @@ open Core public using (≅-to-≡; ≡-to-≅)
 ≅-to-type-≡ refl = refl
 
 ≅-to-subst-≡ : ∀ {A B : Set a} {x : A} {y : B} → (p : x ≅ y) →
-               P.subst (λ x → x) (≅-to-type-≡ p) x ≡ y
+               ≡.subst (λ x → x) (≅-to-type-≡ p) x ≡ y
 ≅-to-subst-≡ refl = refl
 
 ------------------------------------------------------------------------
@@ -89,7 +89,7 @@ subst₂-removable : ∀ (_∼_ : REL A B r) {x y u v} (eq₁ : x ≅ y) (eq₂
 subst₂-removable _∼_ refl refl z = refl
 
 ≡-subst-removable : ∀ (P : Pred A p) {x y} (eq : x ≡ y) (z : P x) →
-                    P.subst P eq z ≅ z
+                    ≡.subst P eq z ≅ z
 ≡-subst-removable P refl z = refl
 
 cong : ∀ {A : Set a} {B : A → Set b} {x y}
@@ -133,7 +133,7 @@ module _ {I : Set ℓ} (A : I → Set a) {B : {k : I} → A k → Set b} where
   icong-≡-subst-removable : {i j : I} (eq : i ≡ j)
                             (f : {k : I} → (z : A k) → B z)
                             (x : A i) →
-                            f (P.subst A eq x) ≅ f x
+                            f (≡.subst A eq x) ≅ f x
   icong-≡-subst-removable refl _ _ = refl
 
 ------------------------------------------------------------------------
@@ -183,13 +183,13 @@ indexedSetoid B = record
   }
 
 ≡↔≅ : ∀ {A : Set a} (B : A → Set b) {x : A} →
-      Inverse (P.setoid (B x)) ((indexedSetoid B) atₛ x)
+      Inverse (≡.setoid (B x)) ((indexedSetoid B) atₛ x)
 ≡↔≅ B = record
   { to         = id
   ; to-cong    = ≡-to-≅
   ; from       = id
   ; from-cong  = ≅-to-≡
-  ; inverse    = (λ { P.refl → refl }) , λ { refl → P.refl }
+  ; inverse    = (λ { ≡.refl → refl }) , λ { refl → ≡.refl }
   }
 
 decSetoid : Decidable {A = A} {B = A} (λ x y → x ≅ y) →
@@ -211,7 +211,7 @@ isPreorder = record
 
 isPreorder-≡ : IsPreorder {A = A} _≡_ (λ x y → x ≅ y)
 isPreorder-≡ = record
-  { isEquivalence = P.isEquivalence
+  { isEquivalence = ≡.isEquivalence
   ; reflexive     = reflexive
   ; trans         = trans
   }
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
index 29d4db5253..e1d8e53b07 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Bundles.agda
@@ -11,10 +11,7 @@
 
 module Relation.Binary.Indexed.Heterogeneous.Bundles where
 
-open import Function.Base
 open import Level using (suc; _⊔_)
-open import Relation.Binary.Core using (_⇒_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Binary.Indexed.Heterogeneous.Core
 open import Relation.Binary.Indexed.Heterogeneous.Structures
 
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Core.agda b/src/Relation/Binary/Indexed/Heterogeneous/Core.agda
index 7195bb9ec4..1015ef5449 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Core.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Core.agda
@@ -14,7 +14,6 @@ module Relation.Binary.Indexed.Heterogeneous.Core where
 open import Level
 import Relation.Binary.Core as B
 import Relation.Binary.Definitions as B
-import Relation.Binary.PropositionalEquality.Core as P
 
 ------------------------------------------------------------------------
 -- Indexed binary relations
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Definitions.agda b/src/Relation/Binary/Indexed/Heterogeneous/Definitions.agda
index f6657a6861..0fb09c6dd6 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Definitions.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Definitions.agda
@@ -12,9 +12,7 @@
 module Relation.Binary.Indexed.Heterogeneous.Definitions where
 
 open import Level
-import Relation.Binary.Core as B
 import Relation.Binary.Definitions as B
-import Relation.Binary.PropositionalEquality.Core as P
 open import Relation.Binary.Indexed.Heterogeneous.Core
 
 private
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda b/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
index 4569ef2fb0..c084dadc8c 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Structures.agda
@@ -18,7 +18,7 @@ module Relation.Binary.Indexed.Heterogeneous.Structures
 open import Function.Base
 open import Level using (suc; _⊔_)
 open import Relation.Binary.Core using (_⇒_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Indexed.Heterogeneous.Definitions
 
 ------------------------------------------------------------------------
@@ -31,7 +31,7 @@ record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where
     trans : Transitive A _≈_
 
   reflexive : ∀ {i} → _≡_ ⟨ _⇒_ ⟩ _≈_ {i}
-  reflexive P.refl = refl
+  reflexive ≡.refl = refl
 
 
 record IsIndexedPreorder {ℓ₂} (_≲_ : IRel A ℓ₂) : Set (i ⊔ a ⊔ ℓ ⊔ ℓ₂) where
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
index 4bbd118c13..aa56c0dcdb 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Bundles.agda
@@ -11,12 +11,9 @@
 
 module Relation.Binary.Indexed.Homogeneous.Bundles where
 
-open import Data.Product.Base using (_,_)
-open import Function.Base using (_⟨_⟩_)
-open import Level using (Level; _⊔_; suc)
-open import Relation.Binary.Core using (_⇒_; Rel)
+open import Level using (suc; _⊔_)
+open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles as B
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Binary.Indexed.Homogeneous.Core
 open import Relation.Binary.Indexed.Homogeneous.Structures
diff --git a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
index 77438dafdd..f5a941b65b 100644
--- a/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
+++ b/src/Relation/Binary/Indexed/Homogeneous/Structures.agda
@@ -23,7 +23,7 @@ open import Level using (Level; _⊔_; suc)
 open import Relation.Binary.Core using (_⇒_)
 import Relation.Binary.Definitions as B
 import Relation.Binary.Structures as B
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Indexed.Homogeneous.Definitions
 
 ------------------------------------------------------------------------
@@ -40,12 +40,12 @@ record IsIndexedEquivalence : Set (i ⊔ a ⊔ ℓ) where
     transᵢ : Transitive A _≈ᵢ_
 
   reflexiveᵢ : ∀ {i} → _≡_ ⟨ _⇒_ ⟩ _≈ᵢ_ {i}
-  reflexiveᵢ P.refl = reflᵢ
+  reflexiveᵢ ≡.refl = reflᵢ
 
   -- Lift properties
 
   reflexive : _≡_ ⇒ (Lift A _≈ᵢ_)
-  reflexive P.refl i = reflᵢ
+  reflexive ≡.refl i = reflᵢ
 
   refl : B.Reflexive (Lift A _≈ᵢ_)
   refl i = reflᵢ
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index 968256077f..f21eb2f4df 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -9,7 +9,7 @@
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; id; _$_; flip)
 open import Relation.Nullary.Negation.Core using (¬_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
 open import Relation.Binary.Definitions
   using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
@@ -26,9 +26,9 @@ open Setoid S
 isPreorder : IsPreorder _≡_ _≈_
 isPreorder = record
   { isEquivalence = record
-    { refl  = P.refl
-    ; sym   = P.sym
-    ; trans = P.trans
+    { refl  = ≡.refl
+    ; sym   = ≡.sym
+    ; trans = ≡.trans
     }
   ; reflexive     = reflexive
   ; trans         = trans
diff --git a/src/Relation/Binary/Structures.agda b/src/Relation/Binary/Structures.agda
index 0dc5187228..14426b4cff 100644
--- a/src/Relation/Binary/Structures.agda
+++ b/src/Relation/Binary/Structures.agda
@@ -18,7 +18,7 @@ module Relation.Binary.Structures
 open import Data.Product.Base using (proj₁; proj₂; _,_)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary.Negation.Core using (¬_)
-open import Relation.Binary.PropositionalEquality.Core as P using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Consequences
 open import Relation.Binary.Definitions
 
@@ -51,7 +51,7 @@ record IsEquivalence : Set (a ⊔ ℓ) where
     trans : Transitive _≈_
 
   reflexive : _≡_ ⇒ _≈_
-  reflexive P.refl = refl
+  reflexive ≡.refl = refl
 
   isPartialEquivalence : IsPartialEquivalence
   isPartialEquivalence = record
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 846fde2484..2c02ec6a4c 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -36,7 +36,7 @@ open import Data.Unit.Base            using (⊤)
 open import Data.Word.Base   as Word  using (toℕ)
 open import Data.Product.Base         using (_×_; map₁; _,_)
 
-open import Relation.Binary.PropositionalEquality.Core as Eq using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
 -- 'Data.String.Properties' defines this via 'Dec', so let's use the
 -- builtin for maximum speed.

From 3146ae576bb453e367089af87664ade66267dd1a Mon Sep 17 00:00:00 2001
From: Guilherme Silva <guilhermehas@hotmail.com>
Date: Sun, 25 Feb 2024 11:35:14 +0100
Subject: [PATCH 228/316] Added functional vector permutation (#2066)

* added functional vector permutation

* added one line to CHANGELOG

* added permutation properties

* Added Base to imports

Co-authored-by: Nathan van Doorn <nvd1234@gmail.com>

* Added Base to import

Co-authored-by: Nathan van Doorn <nvd1234@gmail.com>

* Added core to import

Co-authored-by: Nathan van Doorn <nvd1234@gmail.com>

* added definitions

* removed unnecessary zs

* renamed types in changelog

* removed unnecessary code

* added more to changelog

* added end of changelog

---------

Co-authored-by: Nathan van Doorn <nvd1234@gmail.com>
Co-authored-by: Guilherme <alvares@chalmers.se>
---
 CHANGELOG.md                                  | 13 ++++++
 .../Relation/Binary/Permutation.agda          | 26 +++++++++++
 .../Binary/Permutation/Properties.agda        | 45 +++++++++++++++++++
 3 files changed, 84 insertions(+)
 create mode 100644 src/Data/Vec/Functional/Relation/Binary/Permutation.agda
 create mode 100644 src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7fa21bf90e..d38675a2f4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -187,3 +187,16 @@ Additions to existing modules
   ```agda
   ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
   ```
+
+* Added module `Data.Vec.Functional.Relation.Binary.Permutation`:
+  ```agda
+  _↭_ : IRel (Vector A) _
+  ```
+
+* Added new file `Data.Vec.Functional.Relation.Binary.Permutation.Properties`:
+  ```agda
+  ↭-refl      : Reflexive (Vector A) _↭_
+  ↭-reflexive : xs ≡ ys → xs ↭ ys
+  ↭-sym       : Symmetric (Vector A) _↭_
+  ↭-trans     : Transitive (Vector A) _↭_
+  ```
diff --git a/src/Data/Vec/Functional/Relation/Binary/Permutation.agda b/src/Data/Vec/Functional/Relation/Binary/Permutation.agda
new file mode 100644
index 0000000000..58ee430d86
--- /dev/null
+++ b/src/Data/Vec/Functional/Relation/Binary/Permutation.agda
@@ -0,0 +1,26 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Permutation relations over Vector
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Functional.Relation.Binary.Permutation where
+
+open import Level using (Level)
+open import Data.Product.Base using (Σ-syntax)
+open import Data.Fin.Permutation using (Permutation; _⟨$⟩ʳ_)
+open import Data.Vec.Functional using (Vector)
+open import Relation.Binary.Indexed.Heterogeneous.Core using (IRel)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+
+private
+  variable
+    ℓ : Level
+    A : Set ℓ
+
+infix 3 _↭_
+
+_↭_ : IRel (Vector A) _
+xs ↭ ys = Σ[ ρ ∈ Permutation _ _ ] (∀ i → xs (ρ ⟨$⟩ʳ i) ≡ ys i)
diff --git a/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
new file mode 100644
index 0000000000..df5970dff3
--- /dev/null
+++ b/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
@@ -0,0 +1,45 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties of permutation
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Functional.Relation.Binary.Permutation.Properties where
+
+open import Level using (Level)
+open import Data.Product.Base using (_,_; proj₁; proj₂)
+open import Data.Nat.Base using (ℕ)
+open import Data.Fin.Permutation using (id; flip; _⟨$⟩ʳ_; inverseʳ; _∘ₚ_)
+open import Data.Vec.Functional
+open import Data.Vec.Functional.Relation.Binary.Permutation
+open import Relation.Binary.PropositionalEquality
+  using (refl; trans; _≡_; cong; module ≡-Reasoning)
+open import Relation.Binary.Indexed.Heterogeneous.Definitions
+
+open ≡-Reasoning
+
+private
+  variable
+    ℓ : Level
+    A : Set ℓ
+    n : ℕ
+    xs ys : Vector A n
+
+↭-refl : Reflexive (Vector A) _↭_
+↭-refl = id , λ _ → refl
+
+↭-reflexive : xs ≡ ys → xs ↭ ys
+↭-reflexive refl = ↭-refl
+
+↭-sym : Symmetric (Vector A) _↭_
+proj₁ (↭-sym (xs↭ys , _)) = flip xs↭ys
+proj₂ (↭-sym {x = xs} {ys} (xs↭ys , xs↭ys≡)) i = begin
+  ys (flip xs↭ys ⟨$⟩ʳ i)             ≡˘⟨ xs↭ys≡ _ ⟩
+  xs (xs↭ys ⟨$⟩ʳ (flip xs↭ys ⟨$⟩ʳ i)) ≡⟨ cong xs (inverseʳ xs↭ys) ⟩
+  xs i ∎
+
+↭-trans : Transitive (Vector A) _↭_
+proj₁ (↭-trans (xs↭ys , _) (ys↭zs , _))   = ys↭zs ∘ₚ xs↭ys
+proj₂ (↭-trans (_ , xs↭ys) (_ , ys↭zs)) _ = trans (xs↭ys _) (ys↭zs _)

From bb7b51b7a7bf4dadf45cec62e34d8e93cae1ea3f Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 25 Feb 2024 10:37:06 +0000
Subject: [PATCH 229/316] Nagata's "idealization of a module" construction
 (#2244)

* Nagata's construction

* removed redundant `zero`

* first round of Jacques' review comments

* proved the additional properties

* some of Matthew's suggestions, plus more vertical whitespace, less horizontal; more comments

* Matthew's suggestion: using `private` modules

* Matthew's suggestion: lifting out left-/right- sublemmas

* standardised names, as far as possible

* Matthew's suggestion: lifting out left-/right- sublemmas

* fixed constraint problem with ambiguous symbol; renamed ideal lemmas

* renamed module

* renamed module in `CHANGELOG`

* added generalised annihilation lemma

* typos

* use correct rexported names

* now as a paramterised module instead

* or did you intend this?

* fix whitespace

* aligned one step of reasoning

* more re-alignment of reasoning steps

* more re-alignment of reasoning steps

* Matthew's review comments

* blanklines
---
 CHANGELOG.md                                  |   5 +
 .../Module/Construct/Idealization.agda        | 194 ++++++++++++++++++
 2 files changed, 199 insertions(+)
 create mode 100644 src/Algebra/Module/Construct/Idealization.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d38675a2f4..1860e7a73d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -48,6 +48,11 @@ New modules
 
 * `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
 
+* Nagata's construction of the "idealization of a module":
+  ```agda
+  Algebra.Module.Construct.Idealization
+  ```
+
 Additions to existing modules
 -----------------------------
 
diff --git a/src/Algebra/Module/Construct/Idealization.agda b/src/Algebra/Module/Construct/Idealization.agda
new file mode 100644
index 0000000000..0526f059fd
--- /dev/null
+++ b/src/Algebra/Module/Construct/Idealization.agda
@@ -0,0 +1,194 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The non-commutative analogue of Nagata's construction of
+-- the "idealization of a module", (Local Rings, 1962; Wiley)
+-- defined here on R-R-*bi*modules M over a ring R, as used in
+-- "Forward- or reverse-mode automatic differentiation: What's the difference?"
+-- (Van den Berg, Schrijvers, McKinna, Vandenbroucke;
+-- Science of Computer Programming, Vol. 234, January 2024
+-- https://doi.org/10.1016/j.scico.2023.103010)
+--
+-- The construction N =def R ⋉ M , for which there is unfortunately
+-- no consistent notation in the literature, consists of:
+-- * carrier: pairs |R| × |M|
+-- * with additive structure that of the direct sum R ⊕ M _of modules_
+-- * but with multiplication _*_ such that M forms an _ideal_ of N
+-- * moreover satisfying 'm * m ≈ 0' for every m ∈ M ⊆ N
+--
+-- The fundamental lemma (proved here) is that N, in fact, defines a Ring:
+-- this ring is essentially the 'ring of dual numbers' construction R[M]
+-- (Clifford, 1874; generalised!) for an ideal M, and thus the synthetic/algebraic
+-- analogue of the tangent space of M (considered as a 'vector space' over R)
+-- in differential geometry, hence its application to Automatic Differentiation.
+--
+-- Nagata's more fundamental insight (not yet shown here) is that
+-- the lattice of R-submodules of M is in order-isomorphism with
+-- the lattice of _ideals_ of R ⋉ M , and hence that the study of
+-- modules can be reduced to that of ideals of a ring, and vice versa.
+--
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Algebra.Bundles using (AbelianGroup; Ring)
+open import Algebra.Module.Bundles using (Bimodule)
+
+module Algebra.Module.Construct.Idealization
+  {r ℓr m ℓm} (ring : Ring r ℓr) (bimodule : Bimodule ring ring m ℓm) where
+
+open import Algebra.Core
+import Algebra.Consequences.Setoid as Consequences
+import Algebra.Definitions as Definitions
+import Algebra.Module.Construct.DirectProduct as DirectProduct
+import Algebra.Module.Construct.TensorUnit as TensorUnit
+open import Algebra.Structures using (IsAbelianGroup; IsRing)
+open import Data.Product using (_,_; ∃-syntax)
+open import Level using (Level; _⊔_)
+open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
+
+------------------------------------------------------------------------
+-- Definitions
+
+private
+  open module R = Ring ring
+    using ()
+    renaming (Carrier to R)
+
+  open module M = Bimodule bimodule
+    renaming (Carrierᴹ to M)
+
+  +ᴹ-middleFour = Consequences.comm∧assoc⇒middleFour ≈ᴹ-setoid +ᴹ-cong +ᴹ-comm +ᴹ-assoc
+
+  open module N = Bimodule (DirectProduct.bimodule TensorUnit.bimodule bimodule)
+    using ()
+    renaming ( Carrierᴹ to N
+             ; _≈ᴹ_ to _≈_
+             ; _+ᴹ_ to _+_
+             ; 0ᴹ to 0#
+             ; -ᴹ_ to -_
+             ; +ᴹ-isAbelianGroup to +-isAbelianGroup
+             )
+
+open AbelianGroup M.+ᴹ-abelianGroup hiding (_≈_)
+open ≈-Reasoning ≈ᴹ-setoid
+open Definitions _≈_
+
+-- Injections ι from the components of the direct sum
+-- ιᴹ in fact exhibits M as an _ideal_ of R ⋉ M (see below)
+ιᴿ : R → N
+ιᴿ r = r , 0ᴹ
+
+ιᴹ : M → N
+ιᴹ m = R.0# , m
+
+-- Multiplicative unit
+
+1# : N
+1# = ιᴿ R.1#
+
+-- Multiplication
+
+infixl 7 _*_
+
+_*_ : Op₂ N
+(r₁ , m₁) * (r₂ , m₂) = r₁ R.* r₂ , r₁ *ₗ m₂ +ᴹ m₁ *ᵣ r₂
+
+-- Properties: because we work in the direct sum, every proof has
+-- * an 'R'-component, which inherits directly from R, and
+-- * an 'M'-component, where the work happens
+
+*-cong : Congruent₂ _*_
+*-cong (r₁ , m₁) (r₂ , m₂) = R.*-cong r₁ r₂ , +ᴹ-cong (*ₗ-cong r₁ m₂) (*ᵣ-cong m₁ r₂)
+
+*-identityˡ : LeftIdentity 1# _*_
+*-identityˡ (r , m) = R.*-identityˡ r , (begin
+  R.1# *ₗ m +ᴹ 0ᴹ *ᵣ r ≈⟨ +ᴹ-cong (*ₗ-identityˡ m) (*ᵣ-zeroˡ r) ⟩
+  m +ᴹ 0ᴹ              ≈⟨ +ᴹ-identityʳ m ⟩
+  m                    ∎)
+
+*-identityʳ : RightIdentity 1# _*_
+*-identityʳ (r , m) = R.*-identityʳ r , (begin
+  r *ₗ 0ᴹ +ᴹ m *ᵣ R.1# ≈⟨ +ᴹ-cong (*ₗ-zeroʳ r) (*ᵣ-identityʳ m) ⟩
+  0ᴹ +ᴹ m              ≈⟨ +ᴹ-identityˡ m ⟩
+  m                    ∎)
+
+*-identity : Identity 1# _*_
+*-identity = *-identityˡ , *-identityʳ
+
+*-assoc : Associative _*_
+*-assoc (r₁ , m₁) (r₂ , m₂) (r₃ , m₃) = R.*-assoc r₁ r₂ r₃ , (begin
+  (r₁ R.* r₂) *ₗ m₃ +ᴹ (r₁ *ₗ m₂ +ᴹ m₁ *ᵣ r₂) *ᵣ r₃
+    ≈⟨ +ᴹ-cong (*ₗ-assoc r₁ r₂ m₃) (*ᵣ-distribʳ r₃ (r₁ *ₗ m₂) (m₁ *ᵣ r₂)) ⟩
+  r₁ *ₗ (r₂ *ₗ m₃) +ᴹ ((r₁ *ₗ m₂) *ᵣ r₃ +ᴹ (m₁ *ᵣ r₂) *ᵣ r₃)
+    ≈⟨ +ᴹ-congˡ (+ᴹ-congʳ (*ₗ-*ᵣ-assoc r₁ m₂ r₃)) ⟩
+  r₁ *ₗ (r₂ *ₗ m₃) +ᴹ (r₁ *ₗ (m₂ *ᵣ r₃) +ᴹ (m₁ *ᵣ r₂) *ᵣ r₃)
+    ≈⟨ +ᴹ-assoc (r₁ *ₗ (r₂ *ₗ m₃)) (r₁ *ₗ (m₂ *ᵣ r₃)) ((m₁ *ᵣ r₂) *ᵣ r₃) ⟨
+  (r₁ *ₗ (r₂ *ₗ m₃) +ᴹ r₁ *ₗ (m₂ *ᵣ r₃)) +ᴹ (m₁ *ᵣ r₂) *ᵣ r₃
+    ≈⟨ +ᴹ-cong (≈ᴹ-sym (*ₗ-distribˡ r₁ (r₂ *ₗ m₃) (m₂ *ᵣ r₃))) (*ᵣ-assoc m₁ r₂ r₃) ⟩
+  r₁ *ₗ (r₂ *ₗ m₃ +ᴹ m₂ *ᵣ r₃) +ᴹ m₁ *ᵣ (r₂ R.* r₃) ∎)
+
+distribˡ : _*_ DistributesOverˡ _+_
+distribˡ (r₁ , m₁) (r₂ , m₂) (r₃ , m₃) = R.distribˡ r₁ r₂ r₃ , (begin
+  r₁ *ₗ (m₂ +ᴹ m₃) +ᴹ m₁ *ᵣ (r₂ R.+ r₃)
+    ≈⟨ +ᴹ-cong (*ₗ-distribˡ r₁ m₂ m₃) (*ᵣ-distribˡ m₁ r₂ r₃) ⟩
+  (r₁ *ₗ m₂ +ᴹ r₁ *ₗ m₃) +ᴹ (m₁ *ᵣ r₂ +ᴹ m₁ *ᵣ r₃)
+    ≈⟨ +ᴹ-middleFour (r₁ *ₗ m₂) (r₁ *ₗ m₃) (m₁ *ᵣ r₂) (m₁ *ᵣ r₃) ⟩
+  (r₁ *ₗ m₂ +ᴹ m₁ *ᵣ r₂) +ᴹ (r₁ *ₗ m₃ +ᴹ m₁ *ᵣ r₃) ∎)
+
+
+distribʳ : _*_ DistributesOverʳ _+_
+distribʳ (r₁ , m₁) (r₂ , m₂) (r₃ , m₃) = R.distribʳ r₁ r₂ r₃ , (begin
+  (r₂ R.+ r₃) *ₗ m₁ +ᴹ (m₂ +ᴹ m₃) *ᵣ r₁
+    ≈⟨ +ᴹ-cong (*ₗ-distribʳ m₁ r₂ r₃) (*ᵣ-distribʳ r₁ m₂ m₃) ⟩
+  (r₂ *ₗ m₁ +ᴹ r₃ *ₗ m₁) +ᴹ (m₂ *ᵣ r₁ +ᴹ m₃ *ᵣ r₁)
+    ≈⟨ +ᴹ-middleFour (r₂ *ₗ m₁) (r₃ *ₗ m₁) (m₂ *ᵣ r₁) (m₃ *ᵣ r₁) ⟩
+  (r₂ *ₗ m₁ +ᴹ m₂ *ᵣ r₁) +ᴹ (r₃ *ₗ m₁ +ᴹ m₃ *ᵣ r₁) ∎)
+
+distrib : _*_ DistributesOver _+_
+distrib = distribˡ , distribʳ
+
+
+------------------------------------------------------------------------
+-- The Fundamental Lemma
+
+-- Structure
+
+isRingᴺ : IsRing _≈_ _+_ _*_ -_ 0#  1#
+isRingᴺ = record
+  { +-isAbelianGroup = +-isAbelianGroup
+  ; *-cong = *-cong
+  ; *-assoc = *-assoc
+  ; *-identity = *-identity
+  ; distrib = distrib
+  }
+
+-- Bundle
+
+ringᴺ : Ring (r ⊔ m) (ℓr ⊔ ℓm)
+ringᴺ = record { isRing = isRingᴺ }
+
+------------------------------------------------------------------------
+-- M is an ideal of R ⋉ M satisfying m₁ * m₂ ≈ 0#
+
+ιᴹ-idealˡ : (n : N) (m : M) → ∃[ n*m ] n * ιᴹ m ≈ ιᴹ n*m
+ιᴹ-idealˡ n@(r , _) m = _ , R.zeroʳ r , ≈ᴹ-refl
+
+ιᴹ-idealʳ : (m : M) (n : N) → ∃[ m*n ] ιᴹ m * n ≈ ιᴹ m*n
+ιᴹ-idealʳ m n@(r , _) = _ , R.zeroˡ r , ≈ᴹ-refl
+
+*-annihilates-ιᴹ : (m₁ m₂ : M) → ιᴹ m₁ * ιᴹ m₂ ≈ 0#
+*-annihilates-ιᴹ m₁ m₂ = R.zeroˡ R.0# , (begin
+  R.0# *ₗ m₂ +ᴹ m₁ *ᵣ R.0# ≈⟨ +ᴹ-cong (*ₗ-zeroˡ m₂) (*ᵣ-zeroʳ m₁) ⟩
+  0ᴹ +ᴹ 0ᴹ                 ≈⟨ +ᴹ-identityˡ 0ᴹ ⟩
+  0ᴹ                       ∎)
+
+m*m≈0 : (m : M) → ιᴹ m * ιᴹ m ≈ 0#
+m*m≈0 m = *-annihilates-ιᴹ m m
+
+------------------------------------------------------------------------
+-- Infix notation for when opening the module unparameterised
+
+infixl 4 _⋉_
+_⋉_ = ringᴺ

From aebf296b42ed36141e701931eff28c0759bc2890 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 25 Feb 2024 10:39:05 +0000
Subject: [PATCH 230/316] Qualified import of `Data.Sum.Base` fixing #2280
 (#2290)

* Qualified import of `Data.Sum.Base as Sum`

* resolve merge conflict in favour of #2293
---
 src/Codata/Sized/Delay.agda                   |  6 ++--
 src/Data/Container/Combinator.agda            | 32 +++++++++----------
 src/Data/Container/Combinator/Properties.agda |  4 +--
 3 files changed, 21 insertions(+), 21 deletions(-)

diff --git a/src/Codata/Sized/Delay.agda b/src/Codata/Sized/Delay.agda
index b4549089e7..4219bae341 100644
--- a/src/Codata/Sized/Delay.agda
+++ b/src/Codata/Sized/Delay.agda
@@ -16,9 +16,9 @@ open import Data.Empty
 open import Relation.Nullary
 open import Data.Nat.Base
 open import Data.Maybe.Base hiding (map ; fromMaybe ; zipWith ; alignWith ; zip ; align)
-open import Data.Product.Base as P hiding (map ; zip)
-open import Data.Sum.Base as S hiding (map)
-open import Data.These.Base as T using (These; this; that; these)
+open import Data.Product.Base hiding (map ; zip)
+open import Data.Sum.Base hiding (map)
+open import Data.These.Base using (These; this; that; these)
 open import Function.Base using (id)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Container/Combinator.agda b/src/Data/Container/Combinator.agda
index 01e102cd5c..3d73977470 100644
--- a/src/Data/Container/Combinator.agda
+++ b/src/Data/Container/Combinator.agda
@@ -10,8 +10,8 @@ module Data.Container.Combinator where
 
 open import Level using (Level; _⊔_; lower)
 open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
-open import Data.Product.Base as P using (_,_; <_,_>; proj₁; proj₂; ∃)
-open import Data.Sum.Base as S using ([_,_]′)
+open import Data.Product.Base as Product using (_,_; <_,_>; proj₁; proj₂; ∃)
+open import Data.Sum.Base as Sum using ([_,_]′)
 open import Data.Unit.Polymorphic.Base using (⊤)
 import Function.Base as F
 
@@ -58,10 +58,10 @@ module _ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s
   _∘_ .Position = ◇ C₁ (Position C₂)
 
   to-∘ : ∀ {a} {A : Set a} → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A) → ⟦ _∘_ ⟧ A
-  to-∘ (s , f) = ((s , proj₁ F.∘ f) , P.uncurry (proj₂ F.∘ f) F.∘′ ◇.proof)
+  to-∘ (s , f) = ((s , proj₁ F.∘ f) , Product.uncurry (proj₂ F.∘ f) F.∘′ ◇.proof)
 
   from-∘ : ∀ {a} {A : Set a} → ⟦ _∘_ ⟧ A → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A)
-  from-∘ ((s , f) , g) = (s , < f , P.curry (g F.∘′ any) >)
+  from-∘ ((s , f) , g) = (s , < f , Product.curry (g F.∘′ any) >)
 
 -- Product. (Note that, up to isomorphism, this is a special case of
 -- indexed product.)
@@ -69,14 +69,14 @@ module _ {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s
   infixr 2 _×_
 
   _×_ : Container (s₁ ⊔ s₂) (p₁ ⊔ p₂)
-  _×_ .Shape    = Shape C₁ P.× Shape C₂
-  _×_ .Position = P.uncurry λ s₁ s₂ → (Position C₁ s₁) S.⊎ (Position C₂ s₂)
+  _×_ .Shape    = Shape C₁ Product.× Shape C₂
+  _×_ .Position = Product.uncurry λ s₁ s₂ → (Position C₁ s₁) Sum.⊎ (Position C₂ s₂)
 
-  to-× : ∀ {a} {A : Set a} → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A → ⟦ _×_ ⟧ A
+  to-× : ∀ {a} {A : Set a} → ⟦ C₁ ⟧ A Product.× ⟦ C₂ ⟧ A → ⟦ _×_ ⟧ A
   to-× ((s₁ , f₁) , (s₂ , f₂)) = ((s₁ , s₂) , [ f₁ , f₂ ]′)
 
-  from-× : ∀ {a} {A : Set a} → ⟦ _×_ ⟧ A → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A
-  from-× ((s₁ , s₂) , f) = ((s₁ , f F.∘ S.inj₁) , (s₂ , f F.∘ S.inj₂))
+  from-× : ∀ {a} {A : Set a} → ⟦ _×_ ⟧ A → ⟦ C₁ ⟧ A Product.× ⟦ C₂ ⟧ A
+  from-× ((s₁ , s₂) , f) = ((s₁ , f F.∘ Sum.inj₁) , (s₂ , f F.∘ Sum.inj₂))
 
 -- Indexed product.
 
@@ -87,7 +87,7 @@ module _ {i s p} (I : Set i) (Cᵢ : I → Container s p) where
   Π .Position = λ s → ∃ λ i → Position (Cᵢ i) (s i)
 
   to-Π : ∀ {a} {A : Set a} → (∀ i → ⟦ Cᵢ i ⟧ A) → ⟦ Π ⟧ A
-  to-Π f = (proj₁ F.∘ f , P.uncurry (proj₂ F.∘ f))
+  to-Π f = (proj₁ F.∘ f , Product.uncurry (proj₂ F.∘ f))
 
   from-Π : ∀ {a} {A : Set a} → ⟦ Π ⟧ A → ∀ i → ⟦ Cᵢ i ⟧ A
   from-Π (s , f) = λ i → (s i , λ p → f (i , p))
@@ -108,15 +108,15 @@ module _ {s₁ s₂ p} (C₁ : Container s₁ p) (C₂ : Container s₂ p) where
   infixr 1 _⊎_
 
   _⊎_ : Container (s₁ ⊔ s₂) p
-  _⊎_ .Shape    = (Shape C₁ S.⊎ Shape C₂)
+  _⊎_ .Shape    = (Shape C₁ Sum.⊎ Shape C₂)
   _⊎_ .Position = [ Position C₁ , Position C₂ ]′
 
-  to-⊎ : ∀ {a} {A : Set a} → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A → ⟦ _⊎_ ⟧ A
-  to-⊎ = [ P.map S.inj₁ F.id , P.map S.inj₂ F.id ]′
+  to-⊎ : ∀ {a} {A : Set a} → ⟦ C₁ ⟧ A Sum.⊎ ⟦ C₂ ⟧ A → ⟦ _⊎_ ⟧ A
+  to-⊎ = [ Product.map Sum.inj₁ F.id , Product.map Sum.inj₂ F.id ]′
 
-  from-⊎ : ∀ {a} {A : Set a} → ⟦ _⊎_ ⟧ A → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A
-  from-⊎ (S.inj₁ s₁ , f) = S.inj₁ (s₁ , f)
-  from-⊎ (S.inj₂ s₂ , f) = S.inj₂ (s₂ , f)
+  from-⊎ : ∀ {a} {A : Set a} → ⟦ _⊎_ ⟧ A → ⟦ C₁ ⟧ A Sum.⊎ ⟦ C₂ ⟧ A
+  from-⊎ (Sum.inj₁ s₁ , f) = Sum.inj₁ (s₁ , f)
+  from-⊎ (Sum.inj₂ s₂ , f) = Sum.inj₂ (s₂ , f)
 
 -- Indexed sum.
 
diff --git a/src/Data/Container/Combinator/Properties.agda b/src/Data/Container/Combinator/Properties.agda
index f4b9568eb2..13823bef75 100644
--- a/src/Data/Container/Combinator/Properties.agda
+++ b/src/Data/Container/Combinator/Properties.agda
@@ -13,7 +13,7 @@ open import Data.Container.Core
 open import Data.Container.Combinator
 open import Data.Container.Relation.Unary.Any
 open import Data.Empty using (⊥-elim)
-open import Data.Product.Base as Prod using (∃; _,_; proj₁; proj₂; <_,_>; uncurry; curry)
+open import Data.Product.Base as P using (∃; _,_; proj₁; proj₂; <_,_>; uncurry; curry)
 open import Data.Sum.Base as S using (inj₁; inj₂; [_,_]′; [_,_])
 open import Function.Base as F using (_∘′_)
 open import Function.Bundles
@@ -45,7 +45,7 @@ module Composition {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Co
 module Product (ext : ∀ {ℓ ℓ′} → Extensionality ℓ ℓ′)
        {s₁ s₂ p₁ p₂} (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) where
 
-  correct : ∀ {x} {X : Set x} →  ⟦ C₁ × C₂ ⟧ X ↔ (⟦ C₁ ⟧ X Prod.× ⟦ C₂ ⟧ X)
+  correct : ∀ {x} {X : Set x} →  ⟦ C₁ × C₂ ⟧ X ↔ (⟦ C₁ ⟧ X P.× ⟦ C₂ ⟧ X)
   correct {X = X} = mk↔ₛ′ (from-× C₁ C₂) (to-× C₁ C₂) (λ _ → refl) from∘to
     where
     from∘to : (to-× C₁ C₂) F.∘ (from-× C₁ C₂) ≗ F.id

From b601831c8cd70fa696a71c6055d4d0e1219d1a69 Mon Sep 17 00:00:00 2001
From: James Wood <laMudri@gmail.com>
Date: Mon, 26 Feb 2024 01:13:59 +0000
Subject: [PATCH 231/316] [ new ] associativity of Appending (#2023)

* [ new ] associativity of Appending

* Removed unneeded variable

* Renamed Product module

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                                  | 40 ++++++++
 .../List/Relation/Binary/Pointwise/Base.agda  | 11 ++-
 .../Ternary/Appending/Properties.agda         | 99 +++++++++++++------
 .../Ternary/Appending/Setoid/Properties.agda  | 35 +++++--
 src/Relation/Binary/Properties/Setoid.agda    | 11 +++
 5 files changed, 157 insertions(+), 39 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1860e7a73d..32258f0604 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -13,6 +13,9 @@ Bug-fixes
   was mistakenly applied to the level of the type `A` instead of the
   variable `x` of type `A`.
 
+* Module `Data.List.Relation.Ternary.Appending.Setoid.Properties` no longer
+  exports the `Setoid` module under the alias `S`.
+
 Non-backwards compatible changes
 --------------------------------
 
@@ -131,6 +134,37 @@ Additions to existing modules
   tabulate⁺-< : (i < j → R (f i) (f j)) → AllPairs R (tabulate f)
   ```
 
+* In `Data.List.Relation.Ternary.Appending.Setoid.Properties`:
+  ```agda
+  through→ : ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs → 
+             ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs
+  through← : ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs → 
+             ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs
+  assoc→   : ∃[ xs ] Appending as bs xs × Appending xs cs ds → 
+             ∃[ ys ] Appending bs cs ys × Appending as ys ds
+  ```
+
+* In `Data.List.Relation.Ternary.Appending.Properties`:
+  ```agda
+  through→ : (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) → 
+	         ∃[ xs ] Pointwise U as xs × Appending V R xs bs cs → 
+			 ∃[ ys ] Appending W S as bs ys × Pointwise T ys cs
+  through← : ((R ; S) ⇒ T) → ((U ; S) ⇒ (V ; W)) → 
+	         ∃[ ys ] Appending U R as bs ys × Pointwise S ys cs → 
+			 ∃[ xs ] Pointwise V as xs × Appending W T xs bs cs
+  assoc→ :   (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) → ((Y ; V) ⇒ X) → 
+		     ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds → 
+			 ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds
+  assoc← :   ((S ; T) ⇒ R) → ((W ; T) ⇒ (U ; V)) → (X ⇒ (Y ; V)) → 
+             ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds → 
+			 ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
+  ```
+
+* In `Data.List.Relation.Binary.Pointwise.Base`:
+  ```agda
+  unzip : Pointwise (R ; S) ⇒ (Pointwise R ; Pointwise S)
+  ```
+
 * In `Data.Maybe.Relation.Binary.Pointwise`:
   ```agda
   pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
@@ -176,6 +210,12 @@ Additions to existing modules
   Stable          : Pred A ℓ → Set _
   ```
 
+* Added new proofs in `Relation.Binary.Properties.Setoid`:
+  ```agda
+  ≈;≈⇒≈ : _≈_ ; _≈_ ⇒ _≈_
+  ≈⇒≈;≈ : _≈_ ⇒ _≈_ ; _≈_
+  ```
+
 * Added new definitions in `Relation.Nullary`
   ```
   Recomputable    : Set _
diff --git a/src/Data/List/Relation/Binary/Pointwise/Base.agda b/src/Data/List/Relation/Binary/Pointwise/Base.agda
index e547aea169..9a7a928742 100644
--- a/src/Data/List/Relation/Binary/Pointwise/Base.agda
+++ b/src/Data/List/Relation/Binary/Pointwise/Base.agda
@@ -8,16 +8,16 @@
 
 module Data.List.Relation.Binary.Pointwise.Base where
 
-open import Data.Product.Base using (_×_; <_,_>)
+open import Data.Product.Base as Product using (_×_; _,_; <_,_>; ∃-syntax)
 open import Data.List.Base using (List; []; _∷_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (REL; _⇒_)
+open import Relation.Binary.Construct.Composition using (_;_)
 
 private
   variable
     a b c ℓ : Level
-    A : Set a
-    B : Set b
+    A B : Set a
     x y : A
     xs ys : List A
     R S : REL A B ℓ
@@ -58,3 +58,8 @@ rec P c n (Rxy ∷ Rxsys) = c Rxy (rec P c n Rxsys)
 map : R ⇒ S → Pointwise R ⇒ Pointwise S
 map R⇒S []            = []
 map R⇒S (Rxy ∷ Rxsys) = R⇒S Rxy ∷ map R⇒S Rxsys
+
+unzip : Pointwise (R ; S) ⇒ (Pointwise R ; Pointwise S)
+unzip [] = [] , [] , []
+unzip ((y , r , s) ∷ xs∼ys) =
+  Product.map (y ∷_) (Product.map (r ∷_) (s ∷_)) (unzip xs∼ys)
diff --git a/src/Data/List/Relation/Ternary/Appending/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Properties.agda
index 4063d504a7..9ab5089f08 100644
--- a/src/Data/List/Relation/Ternary/Appending/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Properties.agda
@@ -10,48 +10,89 @@ module Data.List.Relation.Ternary.Appending.Properties where
 
 open import Data.List.Base using (List; [])
 open import Data.List.Relation.Ternary.Appending
+open import Data.List.Relation.Binary.Pointwise as Pw using (Pointwise; []; _∷_)
+open import Data.Product.Base as Product using (∃-syntax; _×_; _,_)
+open import Function.Base using (id)
 open import Data.List.Relation.Binary.Pointwise.Base as Pw using (Pointwise; []; _∷_)
 open import Data.List.Relation.Binary.Pointwise.Properties as Pw using (transitive)
 open import Level using (Level)
-open import Relation.Binary.Core using (REL; Rel)
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
 open import Relation.Binary.Definitions using (Trans)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
+open import Relation.Binary.Construct.Composition using (_;_)
 
 private
   variable
-    a a′ b b′ c c′ l r : Level
-    A : Set a
-    A′ : Set a′
-    B : Set b
-    B′ : Set b′
-    C : Set c
-    C′ : Set c′
-    L : REL A C l
-    R : REL B C r
-    as : List A
-    bs : List B
-    cs : List C
-
-module _  {e} {E : REL C C′ e} {L′ : REL A C′ l} {R′ : REL B C′ r}
-          (LEL′ : Trans L E L′) (RER′ : Trans R E R′)
-          where
+    a ℓ l r : Level
+    A A′ B B′ C C′ D D′ : Set a
+    R S T U V W X Y : REL A B ℓ
+    as bs cs ds : List A
+
+module _  (RST : Trans R S T) (USV : Trans U S V) where
 
-  respʳ-≋ : ∀ {cs′} → Appending L R as bs cs → Pointwise E cs cs′ → Appending L′ R′ as bs cs′
-  respʳ-≋ ([]++ rs) es       = []++ Pw.transitive RER′ rs es
-  respʳ-≋ (l ∷ lrs) (e ∷ es) = LEL′ l e ∷ respʳ-≋ lrs es
+  respʳ-≋ : Appending R U as bs cs → Pointwise S cs ds → Appending T V as bs ds
+  respʳ-≋ ([]++ rs) es       = []++ Pw.transitive USV rs es
+  respʳ-≋ (l ∷ lrs) (e ∷ es) = RST l e ∷ respʳ-≋ lrs es
 
-module _  {eᴬ eᴮ} {Eᴬ : REL A′ A eᴬ} {Eᴮ : REL B′ B eᴮ}
-          {L′ : REL A′ C l} (ELL′ : Trans Eᴬ L L′)
-          {R′ : REL B′ C r} (ERR′ : Trans Eᴮ R R′)
+module _  {T : REL A B l} (RST : Trans R S T)
+          {W : REL A B r} (ERW : Trans U V W)
           where
 
-  respˡ-≋ : ∀ {as′ bs′} → Pointwise Eᴬ as′ as → Pointwise Eᴮ bs′ bs →
-            Appending L R as bs cs → Appending L′ R′ as′ bs′ cs
-  respˡ-≋ []         esʳ ([]++ rs) = []++ Pw.transitive ERR′ esʳ rs
-  respˡ-≋ (eˡ ∷ esˡ) esʳ (l ∷ lrs) = ELL′ eˡ l ∷ respˡ-≋ esˡ esʳ lrs
+  respˡ-≋ : ∀ {as′ bs′} → Pointwise R as′ as → Pointwise U bs′ bs →
+            Appending S V as bs cs → Appending T W as′ bs′ cs
+  respˡ-≋ []         esʳ ([]++ rs) = []++ Pw.transitive ERW esʳ rs
+  respˡ-≋ (eˡ ∷ esˡ) esʳ (l ∷ lrs) = RST eˡ l ∷ respˡ-≋ esˡ esʳ lrs
 
-conicalˡ : Appending L R as bs [] → as ≡ []
+conicalˡ : Appending R S as bs [] → as ≡ []
 conicalˡ ([]++ rs) = refl
 
-conicalʳ : Appending L R as bs [] → bs ≡ []
+conicalʳ : Appending R S as bs [] → bs ≡ []
 conicalʳ ([]++ []) = refl
+
+through→ :
+  (R ⇒ (S ; T)) →
+  ((U ; V) ⇒ (W ; T)) →
+  ∃[ xs ] Pointwise U as xs × Appending V R xs bs cs →
+  ∃[ ys ] Appending W S as bs ys × Pointwise T ys cs
+through→ f g (_ , [] , []++ rs) =
+  let _ , rs′ , ps′ = Pw.unzip (Pw.map f rs) in
+  _ , []++ rs′ , ps′
+through→ f g (_ , p ∷ ps , l ∷ lrs) =
+  let _ , l′ , p′ = g (_ , p , l) in
+  Product.map _ (Product.map (l′ ∷_) (p′ ∷_)) (through→ f g (_ , ps , lrs))
+
+through← :
+  ((R ; S) ⇒ T) →
+  ((U ; S) ⇒ (V ; W)) →
+  ∃[ ys ] Appending U R as bs ys × Pointwise S ys cs →
+  ∃[ xs ] Pointwise V as xs × Appending W T xs bs cs
+through← f g (_ , []++ rs′ , ps′) =
+  _ , [] , []++ (Pw.transitive (λ r′ p′ → f (_ , r′ , p′)) rs′ ps′)
+through← f g (_ , l′ ∷ lrs′ , p′ ∷ ps′) =
+  let _ , p , l = g (_ , l′ , p′) in
+  Product.map _ (Product.map (p ∷_) (l ∷_)) (through← f g (_ , lrs′ , ps′))
+
+assoc→ :
+  (R ⇒ (S ; T)) →
+  ((U ; V) ⇒ (W ; T)) →
+  ((Y ; V) ⇒ X) →
+  ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds →
+  ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds
+assoc→ f g h (_ , []++ rs , lrs′) =
+  let _ , mss , ss′ = through→ f g (_ , rs , lrs′) in
+  _ , mss , []++ ss′
+assoc→ f g h (_ , l ∷ lrs , l′ ∷ lrs′) =
+  Product.map₂ (Product.map₂ (h (_ , l , l′) ∷_)) (assoc→ f g h (_ , lrs , lrs′))
+
+assoc← :
+  ((S ; T) ⇒ R) →
+  ((W ; T) ⇒ (U ; V)) →
+  (X ⇒ (Y ; V)) →
+  ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds →
+  ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
+assoc← f g h (_ , mss , []++ ss′) =
+  let _ , rs , lrs′ = through← f g (_ , mss , ss′) in
+  _ , []++ rs , lrs′
+assoc← f g h (_ , mss , m′ ∷ mss′) =
+  let _ , l , l′ = h m′ in
+  Product.map _ (Product.map (l ∷_) (l′ ∷_)) (assoc← f g h (_ , mss , mss′))
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
index 1e2ef9cd6a..a8b6464f66 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
@@ -8,27 +8,33 @@
 
 open import Relation.Binary.Bundles using (Setoid)
 
-module Data.List.Relation.Ternary.Appending.Setoid.Properties {c l} (S : Setoid c l) where
+module Data.List.Relation.Ternary.Appending.Setoid.Properties
+  {c l} (S : Setoid c l)
+  where
 
 open import Data.List.Base as List using (List; [])
 import Data.List.Properties as Listₚ
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; [])
 import Data.List.Relation.Ternary.Appending.Properties as Appendingₚ
-open import Data.Product.Base using (_,_)
+open import Data.Product using (∃-syntax; _×_; _,_)
+open import Function.Base using (id)
+open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
+open import Relation.Binary.Construct.Composition using (_;_)
 
+open Setoid S renaming (Carrier to A)
+open import Relation.Binary.Properties.Setoid S using (≈;≈⇒≈; ≈⇒≈;≈)  
 open import Data.List.Relation.Ternary.Appending.Setoid S
-module S = Setoid S; open S renaming (Carrier to A) using (_≈_)
 
 private
   variable
-    as bs cs : List A
+    as bs cs ds : List A
 
 ------------------------------------------------------------------------
 -- Re-exporting existing properties
 
 open Appendingₚ public
-  hiding (respʳ-≋; respˡ-≋)
+  using (conicalˡ; conicalʳ)
 
 ------------------------------------------------------------------------
 -- Proving setoid-specific ones
@@ -44,8 +50,23 @@ open Appendingₚ public
 
 respʳ-≋ : ∀ {cs′} → Appending as bs cs → Pointwise _≈_ cs cs′ →
           Appending as bs cs′
-respʳ-≋ = Appendingₚ.respʳ-≋ S.trans S.trans
+respʳ-≋ = Appendingₚ.respʳ-≋ trans trans
 
 respˡ-≋ : ∀ {as′ bs′} → Pointwise _≈_ as′ as → Pointwise _≈_ bs′ bs →
           Appending as bs cs → Appending as′ bs′ cs
-respˡ-≋ = Appendingₚ.respˡ-≋ S.trans S.trans
+respˡ-≋ = Appendingₚ.respˡ-≋ trans trans
+
+through→ :
+  ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs →
+  ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs
+through→ = Appendingₚ.through→ ≈⇒≈;≈ id
+
+through← :
+  ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs →
+  ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs
+through← = Appendingₚ.through← ≈;≈⇒≈ id
+
+assoc→ :
+  ∃[ xs ] Appending as bs xs × Appending xs cs ds →
+  ∃[ ys ] Appending bs cs ys × Appending as ys ds
+assoc→ = Appendingₚ.assoc→ ≈⇒≈;≈ id ≈;≈⇒≈
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index f21eb2f4df..c037352df4 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -9,11 +9,13 @@
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; id; _$_; flip)
 open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
 open import Relation.Binary.Definitions
   using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
+open import Relation.Binary.Construct.Composition using (_;_)
 
 module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where
 
@@ -77,6 +79,15 @@ preorder = record
 ≉-resp₂ : _≉_ Respects₂ _≈_
 ≉-resp₂ = ≉-respʳ , ≉-respˡ
 
+------------------------------------------------------------------------
+-- Equality is closed under composition
+
+≈;≈⇒≈ : _≈_ ; _≈_ ⇒ _≈_
+≈;≈⇒≈ (_ , p , q) = trans p q
+
+≈⇒≈;≈ : _≈_ ⇒ _≈_ ; _≈_
+≈⇒≈;≈ q = _ , q , refl
+
 ------------------------------------------------------------------------
 -- Other properties
 

From 6c925fafe83be41af9403791338a7388c5030b8e Mon Sep 17 00:00:00 2001
From: James Wood <laMudri@gmail.com>
Date: Mon, 26 Feb 2024 09:11:04 +0000
Subject: [PATCH 232/316] [ new ] symmetric core of a binary relation (#2071)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* [ new ] symmetric core of a binary relation

* [ fix ] name clashes

* [ more ] respond to McKinna's comments

* [ rename ] fields to lhs≤rhs and rhs≤lhs

* [ more ] incorporate parts of McKinna's review

* [ minor ] remove implicit argument application from transitive

* [ edit ] pull out isEquivalence and do some renaming

* [ minor ] respond to easy comments

* [ refactor ] remove IsProset, and move Proset to main hierarchy

* Eliminate Proset

* Fixed CHANGELOG typo

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                                  | 10 ++-
 src/Data/Fin/Subset/Properties.agda           |  2 +-
 .../Binary/Construct/Interior/Symmetric.agda  | 83 +++++++++++++++++++
 src/Relation/Binary/Definitions.agda          |  7 ++
 .../Unary/Polymorphic/Properties.agda         |  2 +-
 src/Relation/Unary/Properties.agda            |  3 +-
 6 files changed, 102 insertions(+), 5 deletions(-)
 create mode 100644 src/Relation/Binary/Construct/Interior/Symmetric.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 32258f0604..9b45f2409b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -49,6 +49,11 @@ Deprecated names
 New modules
 -----------
 
+* Symmetric interior of a binary relation
+  ```
+  Relation.Binary.Construct.Interior.Symmetric
+  ```
+
 * `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
 
 * Nagata's construction of the "idealization of a module":
@@ -205,9 +210,10 @@ Additions to existing modules
 * In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
   be used infix.
 
-* Added new definitions in `Relation.Binary`
+* Added new definitions in `Relation.Binary.Definitions`
   ```
-  Stable          : Pred A ℓ → Set _
+  Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)
+  Empty  _∼_ = ∀ {x y} → x ∼ y → ⊥
   ```
 
 * Added new proofs in `Relation.Binary.Properties.Setoid`:
diff --git a/src/Data/Fin/Subset/Properties.agda b/src/Data/Fin/Subset/Properties.agda
index fb230ea468..769c20ce06 100644
--- a/src/Data/Fin/Subset/Properties.agda
+++ b/src/Data/Fin/Subset/Properties.agda
@@ -35,7 +35,7 @@ open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsStrictPartialOrder; IsDecStrictPartialOrder)
 open import Relation.Binary.Bundles
   using (Preorder; Poset; StrictPartialOrder; DecStrictPartialOrder)
-open import Relation.Binary.Definitions as B hiding (Decidable)
+open import Relation.Binary.Definitions as B hiding (Decidable; Empty)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _⊎-dec_)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Relation/Binary/Construct/Interior/Symmetric.agda b/src/Relation/Binary/Construct/Interior/Symmetric.agda
new file mode 100644
index 0000000000..a954dc99ec
--- /dev/null
+++ b/src/Relation/Binary/Construct/Interior/Symmetric.agda
@@ -0,0 +1,83 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Symmetric interior of a binary relation
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Relation.Binary.Construct.Interior.Symmetric where
+
+open import Function.Base using (flip)
+open import Level
+open import Relation.Binary
+
+private
+  variable
+    a b c ℓ r s t : Level
+    A : Set a
+    R S T : Rel A r
+
+------------------------------------------------------------------------
+-- Definition
+
+record SymInterior (R : Rel A ℓ) (x y : A) : Set ℓ where
+  constructor _,_
+  field
+    lhs≤rhs : R x y
+    rhs≤lhs : R y x
+
+open SymInterior public
+
+------------------------------------------------------------------------
+-- Properties
+
+-- The symmetric interior is symmetric.
+symmetric : Symmetric (SymInterior R)
+symmetric (r , r′) = r′ , r
+
+-- The symmetric interior of R is greater than (or equal to) any other symmetric
+-- relation contained by R.
+unfold : Symmetric S → S ⇒ R → S ⇒ SymInterior R
+unfold sym f s = f s , f (sym s)
+
+-- SymInterior preserves various properties.
+reflexive : Reflexive R → Reflexive (SymInterior R)
+reflexive refl = refl , refl
+
+trans : Trans R S T → Trans S R T →
+  Trans (SymInterior R) (SymInterior S) (SymInterior T)
+trans trans-rs trans-sr (r , r′) (s , s′) = trans-rs r s , trans-sr s′ r′
+
+transitive : Transitive R → Transitive (SymInterior R)
+transitive tr = trans tr tr
+
+-- The symmetric interior of a strict relation is empty.
+asymmetric⇒empty : Asymmetric R → Empty (SymInterior R)
+asymmetric⇒empty asym (r , r′) = asym r r′
+
+-- A reflexive transitive relation _≤_ gives rise to a poset in which the
+-- equivalence relation is SymInterior _≤_.
+
+isEquivalence : Reflexive R → Transitive R → IsEquivalence (SymInterior R)
+isEquivalence refl trans = record
+  { refl = reflexive refl
+  ; sym = symmetric
+  ; trans = transitive trans
+  }
+
+isPartialOrder : Reflexive R → Transitive R → IsPartialOrder (SymInterior R) R
+isPartialOrder refl trans = record
+  { isPreorder = record
+    { isEquivalence = isEquivalence refl trans
+    ; reflexive = lhs≤rhs
+    ; trans = trans
+    }
+  ; antisym = _,_
+  }
+
+poset : ∀ {a} {A : Set a} {R : Rel A ℓ} → Reflexive R → Transitive R → Poset a ℓ ℓ
+poset {R = R} refl trans = record
+  { _≤_ = R
+  ; isPartialOrder = isPartialOrder refl trans
+  }
diff --git a/src/Relation/Binary/Definitions.agda b/src/Relation/Binary/Definitions.agda
index dd0e016c12..0e3f09e443 100644
--- a/src/Relation/Binary/Definitions.agda
+++ b/src/Relation/Binary/Definitions.agda
@@ -12,6 +12,8 @@ module Relation.Binary.Definitions where
 
 open import Agda.Builtin.Equality using (_≡_)
 
+open import Data.Empty using (⊥)
+open import Data.Maybe.Base using (Maybe)
 open import Data.Product.Base using (_×_; ∃-syntax)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_on_; flip)
@@ -243,6 +245,11 @@ DecidableEquality A = Decidable {A = A} _≡_
 Universal : REL A B ℓ → Set _
 Universal _∼_ = ∀ x y → x ∼ y
 
+-- Empty - no elements are related
+
+Empty : REL A B ℓ → Set _
+Empty _∼_ = ∀ {x y} → x ∼ y → ⊥
+
 -- Non-emptiness - at least one pair of elements are related.
 
 record NonEmpty {A : Set a} {B : Set b}
diff --git a/src/Relation/Unary/Polymorphic/Properties.agda b/src/Relation/Unary/Polymorphic/Properties.agda
index 3a59912e2b..91f332433f 100644
--- a/src/Relation/Unary/Polymorphic/Properties.agda
+++ b/src/Relation/Unary/Polymorphic/Properties.agda
@@ -10,7 +10,7 @@
 module Relation.Unary.Polymorphic.Properties where
 
 open import Level using (Level)
-open import Relation.Binary.Definitions hiding (Decidable; Universal)
+open import Relation.Binary.Definitions hiding (Decidable; Universal; Empty)
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Unary hiding (∅; U)
 open import Relation.Unary.Polymorphic
diff --git a/src/Relation/Unary/Properties.agda b/src/Relation/Unary/Properties.agda
index 520f5e93c8..874869e271 100644
--- a/src/Relation/Unary/Properties.agda
+++ b/src/Relation/Unary/Properties.agda
@@ -13,7 +13,8 @@ open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Unit.Base using (tt)
 open import Level using (Level)
 open import Relation.Binary.Core as Binary
-open import Relation.Binary.Definitions hiding (Decidable; Universal; Irrelevant)
+open import Relation.Binary.Definitions
+  hiding (Decidable; Universal; Irrelevant; Empty)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 open import Relation.Unary
 open import Relation.Nullary.Decidable using (yes; no; _⊎-dec_; _×-dec_; ¬?)

From 08a89c75805db1e1129583813c91646e7d73df72 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 27 Feb 2024 01:45:48 +0000
Subject: [PATCH 233/316] refactored proofs from #2023 (#2301)

---
 CHANGELOG.md                                  | 35 +++++++++++--------
 .../Binary/Construct/Composition.agda         |  3 ++
 src/Relation/Binary/Properties/Setoid.agda    |  7 ++--
 3 files changed, 27 insertions(+), 18 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9b45f2409b..b569f488e4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -141,28 +141,28 @@ Additions to existing modules
 
 * In `Data.List.Relation.Ternary.Appending.Setoid.Properties`:
   ```agda
-  through→ : ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs → 
+  through→ : ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs →
              ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs
-  through← : ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs → 
+  through← : ∃[ ys ] Appending as bs ys × Pointwise _≈_ ys cs →
              ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs
-  assoc→   : ∃[ xs ] Appending as bs xs × Appending xs cs ds → 
+  assoc→   : ∃[ xs ] Appending as bs xs × Appending xs cs ds →
              ∃[ ys ] Appending bs cs ys × Appending as ys ds
   ```
 
 * In `Data.List.Relation.Ternary.Appending.Properties`:
   ```agda
-  through→ : (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) → 
-	         ∃[ xs ] Pointwise U as xs × Appending V R xs bs cs → 
-			 ∃[ ys ] Appending W S as bs ys × Pointwise T ys cs
-  through← : ((R ; S) ⇒ T) → ((U ; S) ⇒ (V ; W)) → 
-	         ∃[ ys ] Appending U R as bs ys × Pointwise S ys cs → 
-			 ∃[ xs ] Pointwise V as xs × Appending W T xs bs cs
-  assoc→ :   (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) → ((Y ; V) ⇒ X) → 
-		     ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds → 
-			 ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds
-  assoc← :   ((S ; T) ⇒ R) → ((W ; T) ⇒ (U ; V)) → (X ⇒ (Y ; V)) → 
-             ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds → 
-			 ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
+  through→ : (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) →
+                 ∃[ xs ] Pointwise U as xs × Appending V R xs bs cs →
+                         ∃[ ys ] Appending W S as bs ys × Pointwise T ys cs
+  through← : ((R ; S) ⇒ T) → ((U ; S) ⇒ (V ; W)) →
+                 ∃[ ys ] Appending U R as bs ys × Pointwise S ys cs →
+                         ∃[ xs ] Pointwise V as xs × Appending W T xs bs cs
+  assoc→ :   (R ⇒ (S ; T)) → ((U ; V) ⇒ (W ; T)) → ((Y ; V) ⇒ X) →
+                     ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds →
+                         ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds
+  assoc← :   ((S ; T) ⇒ R) → ((W ; T) ⇒ (U ; V)) → (X ⇒ (Y ; V)) →
+             ∃[ ys ] Appending W S bs cs ys × Appending X T as ys ds →
+                         ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
   ```
 
 * In `Data.List.Relation.Binary.Pointwise.Base`:
@@ -210,6 +210,11 @@ Additions to existing modules
 * In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
   be used infix.
 
+* Added new proofs in `Relation.Binary.Construct.Composition`:
+  ```agda
+  transitive⇒≈;≈⊆≈ : Transitive ≈ → (≈ ; ≈) ⇒ ≈
+  ```
+
 * Added new definitions in `Relation.Binary.Definitions`
   ```
   Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)
diff --git a/src/Relation/Binary/Construct/Composition.agda b/src/Relation/Binary/Construct/Composition.agda
index 0f7a3cb02c..35df23947d 100644
--- a/src/Relation/Binary/Construct/Composition.agda
+++ b/src/Relation/Binary/Construct/Composition.agda
@@ -82,3 +82,6 @@ module _ (L : Rel A ℓ₁) (R : Rel A ℓ₂) (comm : R ; L ⇒ L ; R) where
     ; trans         = transitive Oˡ.trans Oʳ.trans
     }
     where module Oˡ = IsPreorder Oˡ; module Oʳ = IsPreorder Oʳ
+
+transitive⇒≈;≈⊆≈ : (≈ : Rel A ℓ) → Transitive ≈ → (≈ ; ≈) ⇒ ≈
+transitive⇒≈;≈⊆≈ _ trans (_ , l , r) = trans l r
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index c037352df4..1b56c32d2a 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -15,7 +15,8 @@ open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
 open import Relation.Binary.Definitions
   using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
-open import Relation.Binary.Construct.Composition using (_;_)
+open import Relation.Binary.Construct.Composition
+  using (_;_; impliesˡ; transitive⇒≈;≈⊆≈)
 
 module Relation.Binary.Properties.Setoid {a ℓ} (S : Setoid a ℓ) where
 
@@ -83,10 +84,10 @@ preorder = record
 -- Equality is closed under composition
 
 ≈;≈⇒≈ : _≈_ ; _≈_ ⇒ _≈_
-≈;≈⇒≈ (_ , p , q) = trans p q
+≈;≈⇒≈ = transitive⇒≈;≈⊆≈ _ trans
 
 ≈⇒≈;≈ : _≈_ ⇒ _≈_ ; _≈_
-≈⇒≈;≈ q = _ , q , refl
+≈⇒≈;≈ = impliesˡ _≈_ _≈_ refl id
 
 ------------------------------------------------------------------------
 -- Other properties

From 07da788268443881e1b7b26b70f58a5761574b44 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 3 Mar 2024 07:04:11 +0000
Subject: [PATCH 234/316] Qualified imports in `Data.Integer.Divisibility`
 fixing #2280 (#2294)

* fixing #2280

* re-export constructor via pattern synonym

* updated `README`

* refactor: better disambiguation; added a note in `CHANGELOG`
---
 CHANGELOG.md                              |  5 +++++
 doc/README/Data/Integer.agda              | 16 ++++++++--------
 src/Data/Integer/Divisibility.agda        | 20 ++++++++++----------
 src/Data/Integer/Divisibility/Signed.agda | 14 ++++++--------
 src/Data/Integer/GCD.agda                 |  3 +--
 src/Data/Integer/LCM.agda                 |  3 +--
 6 files changed, 31 insertions(+), 30 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b569f488e4..30b517fbc7 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -127,6 +127,11 @@ Additions to existing modules
   nonZeroIndex : Fin n → ℕ.NonZero n
   ```
 
+* In `Data.Integer.Divisisbility`: introduce `divides` as an explicit pattern synonym
+  ```agda
+  pattern divides k eq = Data.Nat.Divisibility.divides k eq
+  ```
+
 * In `Data.List.Relation.Unary.All.Properties`:
   ```agda
   All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)
diff --git a/doc/README/Data/Integer.agda b/doc/README/Data/Integer.agda
index 23619757d3..bb3f867137 100644
--- a/doc/README/Data/Integer.agda
+++ b/doc/README/Data/Integer.agda
@@ -33,29 +33,29 @@ ex₃ = + 1  +  + 3 * - + 2  -  + 4
 -- Propositional equality and some related properties can be found
 -- in Relation.Binary.PropositionalEquality.
 
-open import Relation.Binary.PropositionalEquality as P using (_≡_)
+open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
 
 ex₄ : ex₃ ≡ - + 9
-ex₄ = P.refl
+ex₄ = ≡.refl
 
 -- Data.Integer.Properties contains a number of properties related to
 -- integers. Algebra defines what a commutative ring is, among other
 -- things.
 
-import Data.Integer.Properties as ℤₚ
+import Data.Integer.Properties as ℤ
 
 ex₅ : ∀ i j → i * j ≡ j * i
-ex₅ i j = ℤₚ.*-comm i j
+ex₅ i j = ℤ.*-comm i j
 
 -- The module ≡-Reasoning in Relation.Binary.PropositionalEquality
 -- provides some combinators for equational reasoning.
 
-open P.≡-Reasoning
+open ≡.≡-Reasoning
 
 ex₆ : ∀ i j → i * (j + + 0) ≡ j * i
 ex₆ i j = begin
-  i * (j + + 0)  ≡⟨ P.cong (i *_) (ℤₚ.+-identityʳ j) ⟩
-  i * j          ≡⟨ ℤₚ.*-comm i j ⟩
+  i * (j + + 0)  ≡⟨ ≡.cong (i *_) (ℤ.+-identityʳ j) ⟩
+  i * j          ≡⟨ ℤ.*-comm i j ⟩
   j * i          ∎
 
 -- The module RingSolver in Data.Integer.Solver contains a solver
@@ -67,4 +67,4 @@ open +-*-Solver
 
 ex₇ : ∀ i j → i * - j - j * i ≡ - + 2 * i * j
 ex₇ = solve 2 (λ i j → i :* :- j :- j :* i  :=  :- con (+ 2) :* i :* j)
-              P.refl
+              ≡.refl
diff --git a/src/Data/Integer/Divisibility.agda b/src/Data/Integer/Divisibility.agda
index 0ad36ad1f4..28963c8d24 100644
--- a/src/Data/Integer/Divisibility.agda
+++ b/src/Data/Integer/Divisibility.agda
@@ -14,10 +14,10 @@ open import Function.Base using (_on_; _$_)
 open import Data.Integer.Base
 open import Data.Integer.Properties
 import Data.Nat.Base as ℕ
-import Data.Nat.Divisibility as ℕᵈ
+import Data.Nat.Divisibility as ℕ
 open import Level
 open import Relation.Binary.Core using (Rel; _Preserves_⟶_)
-open import Relation.Binary.PropositionalEquality
+
 
 ------------------------------------------------------------------------
 -- Divisibility
@@ -25,9 +25,9 @@ open import Relation.Binary.PropositionalEquality
 infix 4 _∣_
 
 _∣_ : Rel ℤ 0ℓ
-_∣_ = ℕᵈ._∣_ on ∣_∣
+_∣_ = ℕ._∣_ on ∣_∣
 
-open ℕᵈ public using (divides)
+pattern divides k eq  = ℕ.divides k eq
 
 ------------------------------------------------------------------------
 -- Properties of divisibility
@@ -35,21 +35,21 @@ open ℕᵈ public using (divides)
 *-monoʳ-∣ : ∀ k → (k *_) Preserves _∣_ ⟶ _∣_
 *-monoʳ-∣ k {i} {j} i∣j = begin
   ∣ k * i ∣       ≡⟨ abs-* k i ⟩
-  ∣ k ∣ ℕ.* ∣ i ∣ ∣⟨ ℕᵈ.*-monoʳ-∣ ∣ k ∣ i∣j ⟩
-  ∣ k ∣ ℕ.* ∣ j ∣ ≡⟨ sym (abs-* k j) ⟩
+  ∣ k ∣ ℕ.* ∣ i ∣ ∣⟨ ℕ.*-monoʳ-∣ ∣ k ∣ i∣j ⟩
+  ∣ k ∣ ℕ.* ∣ j ∣ ≡⟨ abs-* k j ⟨
   ∣ k * j ∣       ∎
-  where open ℕᵈ.∣-Reasoning
+  where open ℕ.∣-Reasoning
 
 *-monoˡ-∣ : ∀ k → (_* k) Preserves _∣_ ⟶ _∣_
 *-monoˡ-∣ k {i} {j} rewrite *-comm i k | *-comm j k = *-monoʳ-∣ k
 
 *-cancelˡ-∣ : ∀ k {i j} .{{_ : NonZero k}} → k * i ∣ k * j → i ∣ j
-*-cancelˡ-∣ k {i} {j} k*i∣k*j = ℕᵈ.*-cancelˡ-∣ ∣ k ∣ $ begin
-  ∣ k ∣ ℕ.* ∣ i ∣  ≡⟨ sym (abs-* k i) ⟩
+*-cancelˡ-∣ k {i} {j} k*i∣k*j = ℕ.*-cancelˡ-∣ ∣ k ∣ $ begin
+  ∣ k ∣ ℕ.* ∣ i ∣  ≡⟨ abs-* k i ⟨
   ∣ k * i ∣        ∣⟨ k*i∣k*j ⟩
   ∣ k * j ∣        ≡⟨ abs-* k j ⟩
   ∣ k ∣ ℕ.* ∣ j ∣  ∎
-  where open ℕᵈ.∣-Reasoning
+  where open ℕ.∣-Reasoning
 
 *-cancelʳ-∣ : ∀ k {i j} .{{_ : NonZero k}} → i * k ∣ j * k → i ∣ j
 *-cancelʳ-∣ k {i} {j} rewrite *-comm i k | *-comm j k = *-cancelˡ-∣ k
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index d171b7015e..15fa5a4c1e 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -11,9 +11,7 @@ module Data.Integer.Divisibility.Signed where
 open import Function.Base using (_⟨_⟩_; _$_; _$′_; _∘_; _∘′_)
 open import Data.Integer.Base
 open import Data.Integer.Properties
-open import Data.Integer.Divisibility as Unsigned
-  using (divides)
-  renaming (_∣_ to _∣ᵤ_)
+import Data.Integer.Divisibility as Unsigned
 import Data.Nat.Base as ℕ
 import Data.Nat.Divisibility as ℕ
 import Data.Nat.Coprimality as ℕ
@@ -45,9 +43,9 @@ open _∣_ using (quotient) public
 ------------------------------------------------------------------------
 -- Conversion between signed and unsigned divisibility
 
-∣ᵤ⇒∣ : ∀ {k i} → k ∣ᵤ i → k ∣ i
-∣ᵤ⇒∣ {k} {i} (divides 0           eq) = divides (+ 0) (∣i∣≡0⇒i≡0 eq)
-∣ᵤ⇒∣ {k} {i} (divides q@(ℕ.suc _) eq) with k ≟ +0
+∣ᵤ⇒∣ : ∀ {k i} → k Unsigned.∣ i → k ∣ i
+∣ᵤ⇒∣ {k} {i} (Unsigned.divides 0           eq) = divides (+ 0) (∣i∣≡0⇒i≡0 eq)
+∣ᵤ⇒∣ {k} {i} (Unsigned.divides q@(ℕ.suc _) eq) with k ≟ +0
 ... | yes refl = divides +0 (∣i∣≡0⇒i≡0 (trans eq (ℕ.*-zeroʳ q)))
 ... | no  neq  = divides (sign i Sign.* sign k ◃ q) (◃-cong sign-eq abs-eq)
   where
@@ -85,8 +83,8 @@ open _∣_ using (quotient) public
     ∣ i ∣              ∎
     where open ≡-Reasoning
 
-∣⇒∣ᵤ : ∀ {k i} → k ∣ i → k ∣ᵤ i
-∣⇒∣ᵤ {k} {i} (divides q eq) = divides ∣ q ∣ $′ begin
+∣⇒∣ᵤ : ∀ {k i} → k ∣ i → k Unsigned.∣ i
+∣⇒∣ᵤ {k} {i} (divides q eq) = Unsigned.divides ∣ q ∣ $′ begin
   ∣ i ∣           ≡⟨ cong ∣_∣ eq ⟩
   ∣ q * k ∣       ≡⟨ abs-* q k ⟩
   ∣ q ∣ ℕ.* ∣ k ∣ ∎
diff --git a/src/Data/Integer/GCD.agda b/src/Data/Integer/GCD.agda
index 3f46877f45..d34529ae16 100644
--- a/src/Data/Integer/GCD.agda
+++ b/src/Data/Integer/GCD.agda
@@ -11,10 +11,9 @@ module Data.Integer.GCD where
 open import Data.Integer.Base
 open import Data.Integer.Divisibility
 open import Data.Integer.Properties
-open import Data.Nat.Base
 import Data.Nat.GCD as ℕ
 open import Data.Product.Base using (_,_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 
 open import Algebra.Definitions {A = ℤ} _≡_ as Algebra
   using (Associative; Commutative; LeftIdentity; RightIdentity; LeftZero; RightZero; Zero)
diff --git a/src/Data/Integer/LCM.agda b/src/Data/Integer/LCM.agda
index ebcbc8ed73..065e0613b2 100644
--- a/src/Data/Integer/LCM.agda
+++ b/src/Data/Integer/LCM.agda
@@ -11,9 +11,8 @@ module Data.Integer.LCM where
 open import Data.Integer.Base
 open import Data.Integer.Divisibility
 open import Data.Integer.GCD
-open import Data.Nat.Base using (ℕ)
 import Data.Nat.LCM as ℕ
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 
 ------------------------------------------------------------------------
 -- Definition

From ed8d28db1ee10dd7da2fd281ffd61d8719886127 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 6 Mar 2024 02:36:57 +0000
Subject: [PATCH 235/316] guideline for `-Reasoning` module imports (#2309)

---
 doc/style-guide.md | 6 ++++++
 1 file changed, 6 insertions(+)

diff --git a/doc/style-guide.md b/doc/style-guide.md
index 0e3a54841a..ae50d2f9cd 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -146,6 +146,12 @@ automate most of this.
   as `Structures` etc.
   NB. Historical legacy means that these conventions have not always been observed!
 
+* Special case of the above for `*-Reasoning` (sub-)modules: by analogy with
+  `Relation.Binary.PropositionalEquality.≡-Reasoning`, when importing qualified
+  the `-Reasoning` (sub-)module associated with a given (canonical) choice of
+  symbol (eg. `≲` for `Preorder` reasoning), use the qualified name
+  `<symbol>-Reasoning`, ie. `≲-Reasoning` for the example given.
+
 * When using only a few items (i.e. < 5) from a module, it is a good practice to
   enumerate the items that will be used by declaring the import statement
   with the directive `using`. This makes the dependencies clearer, e.g.

From ce23ff589ab3923af46cb79a1d121ac4f6a50173 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Thu, 7 Mar 2024 10:13:20 +0100
Subject: [PATCH 236/316] Function setoid is back. (#2240)

* Function setoid is back.

* make all changes asked for in review

* fix indentation
---
 CHANGELOG.md                                  | 20 +++++---
 .../Relation/Binary/Setoid/Equality.agda      | 50 +++++++++++++++++++
 2 files changed, 64 insertions(+), 6 deletions(-)
 create mode 100644 src/Function/Relation/Binary/Setoid/Equality.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 30b517fbc7..bbc65af5d7 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -48,18 +48,26 @@ Deprecated names
 
 New modules
 -----------
-
-* Symmetric interior of a binary relation
-  ```
-  Relation.Binary.Construct.Interior.Symmetric
-  ```
-
 * `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
 
 * Nagata's construction of the "idealization of a module":
   ```agda
   Algebra.Module.Construct.Idealization
   ```
+  
+* `Function.Relation.Binary.Equality`
+  ```agda
+  setoid : Setoid a₁ a₂ → Setoid b₁ b₂ → Setoid _ _
+  ```
+  and a convenient infix version
+  ```agda
+  _⇨_ = setoid
+  ```
+
+* Symmetric interior of a binary relation
+  ```
+  Relation.Binary.Construct.Interior.Symmetric
+  ```
 
 Additions to existing modules
 -----------------------------
diff --git a/src/Function/Relation/Binary/Setoid/Equality.agda b/src/Function/Relation/Binary/Setoid/Equality.agda
new file mode 100644
index 0000000000..2631d83da9
--- /dev/null
+++ b/src/Function/Relation/Binary/Setoid/Equality.agda
@@ -0,0 +1,50 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Function Equality setoid
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Level using (Level; _⊔_)
+open import Relation.Binary.Bundles using (Setoid)
+
+module Function.Relation.Binary.Setoid.Equality {a₁ a₂ b₁ b₂ : Level}
+  (From : Setoid a₁ a₂) (To : Setoid b₁ b₂) where
+
+open import Function.Bundles using (Func; _⟨$⟩_)
+open import Relation.Binary.Definitions
+  using (Reflexive; Symmetric; Transitive)
+
+private
+  module To = Setoid To
+  module From = Setoid From
+
+infix 4 _≈_
+_≈_ : (f g : Func From To) → Set (a₁ ⊔ b₂)
+f ≈ g = {x : From.Carrier} → f ⟨$⟩ x To.≈ g ⟨$⟩ x
+
+refl : Reflexive _≈_
+refl = To.refl
+
+sym : Symmetric _≈_
+sym = λ f≈g → To.sym f≈g
+
+trans : Transitive _≈_
+trans = λ f≈g g≈h → To.trans f≈g g≈h
+
+setoid : Setoid _ _
+setoid = record
+  { Carrier = Func From To
+  ; _≈_ = _≈_
+  ; isEquivalence = record  -- need to η-expand else Agda gets confused
+    { refl = λ {f} → refl {f}
+    ; sym = λ {f} {g} → sym {f} {g}
+    ; trans = λ {f} {g} {h} → trans {f} {g} {h}
+    }
+  }
+
+-- most of the time, this infix version is nicer to use
+infixr 9 _⇨_
+_⇨_ : Setoid _ _
+_⇨_ = setoid

From ebbe65d4888a3f093d575222de3da505f885a068 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 7 Mar 2024 12:31:28 +0000
Subject: [PATCH 237/316] Style guide: avoid `renaming` on qualified import cf.
 #2294 (#2308)

* recommendation from #2294

* spellcheck

* comma

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 doc/style-guide.md | 12 ++++++++++--
 1 file changed, 10 insertions(+), 2 deletions(-)

diff --git a/doc/style-guide.md b/doc/style-guide.md
index ae50d2f9cd..b2ee87d8ee 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -134,7 +134,7 @@ automate most of this.
 * Naming conventions for qualified `import`s: if importing a module under
   a root of the form `Data.X` (e.g. the `Base` module for basic operations,
   or `Properties` for lemmas about them etc.) then conventionally, the
-  qualified name(s) for the import(s) should (all) share as qualfied name
+  qualified name(s) for the import(s) should (all) share as qualified name
   that of the name of the `X` datatype defined: i.e. `Data.Nat.Base`
   should be imported as `ℕ`, `Data.List.Properties` as `List`,  etc.
   In this spirit, the convention applies also to (the datatype defined by)
@@ -142,7 +142,7 @@ automate most of this.
   with the name `≡`.
   Other modules should be given a 'suitable' qualified name based on its 'long'
   path-derived name (such as `SetoidEquality` in the example above); commonly
-  occcurring examples such as `Algebra.Structures` should be imported qualified
+  occurring examples such as `Algebra.Structures` should be imported qualified
   as `Structures` etc.
   NB. Historical legacy means that these conventions have not always been observed!
 
@@ -152,6 +152,14 @@ automate most of this.
   symbol (eg. `≲` for `Preorder` reasoning), use the qualified name
   `<symbol>-Reasoning`, ie. `≲-Reasoning` for the example given.
 
+* Qualified `open import`s should, in general, avoid `renaming`
+  identifiers, in favour of using the long(er) qualified name,
+  although similar remarks about legacy failure to observe this
+  recommendation apply!
+  NB. `renaming` directives are, of course, permitted when a module is
+  imported qualified, in order to be *subsequently* `open`ed for
+  `public` export (see below).
+
 * When using only a few items (i.e. < 5) from a module, it is a good practice to
   enumerate the items that will be used by declaring the import statement
   with the directive `using`. This makes the dependencies clearer, e.g.

From 163b7e6929c23906a1205e18398045fb06966d5c Mon Sep 17 00:00:00 2001
From: Gan Shen <gan_shen@icloud.com>
Date: Sat, 9 Mar 2024 14:29:52 -0800
Subject: [PATCH 238/316] make `mkRawMonad` and `mkRawApplicative`
 universe-polymorphic (#2314)

* make `mkRawMonad` and `mkRawApplicative` universe-polymorphic

* fix unresolved metas

---------

Co-authored-by: Gan Shen <gan@kresge-100-64-83-91.resnet.ucsc.edu>
---
 src/Data/Product/Effectful/Left.agda  | 2 +-
 src/Data/Product/Effectful/Right.agda | 2 +-
 src/Effect/Applicative.agda           | 2 +-
 src/Effect/Monad.agda                 | 2 +-
 src/Reflection/AST/DeBruijn.agda      | 2 +-
 5 files changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/Data/Product/Effectful/Left.agda b/src/Data/Product/Effectful/Left.agda
index 279555b9c9..6fcfc7b266 100644
--- a/src/Data/Product/Effectful/Left.agda
+++ b/src/Data/Product/Effectful/Left.agda
@@ -49,7 +49,7 @@ monad = record
   }
 
 -- The monad instance also requires some mucking about with universe levels.
-monadT : RawMonadT (_∘′ Productₗ)
+monadT : ∀ {ℓ} → RawMonadT {g₁ = ℓ} (_∘′ Productₗ)
 monadT M = record
   { lift = (ε ,_) <$>_
   ; rawMonad = mkRawMonad _
diff --git a/src/Data/Product/Effectful/Right.agda b/src/Data/Product/Effectful/Right.agda
index fd5cb9b423..a10bbad006 100644
--- a/src/Data/Product/Effectful/Right.agda
+++ b/src/Data/Product/Effectful/Right.agda
@@ -48,7 +48,7 @@ monad = record
   ; _>>=_ = uncurry λ a w₁ f → map₂ (w₁ ∙_) (f a)
   }
 
-monadT : RawMonadT (_∘′ Productᵣ)
+monadT : ∀ {ℓ} → RawMonadT {g₁ = ℓ} (_∘′ Productᵣ)
 monadT M = record
   { lift = (_, ε) <$>_
   ; rawMonad = mkRawMonad _
diff --git a/src/Effect/Applicative.agda b/src/Effect/Applicative.agda
index ef7dd5cc2f..1fde40139f 100644
--- a/src/Effect/Applicative.agda
+++ b/src/Effect/Applicative.agda
@@ -77,7 +77,7 @@ module _ where
 
   -- Smart constructor
   mkRawApplicative :
-    (F : Set f → Set f) →
+    (F : Set f → Set g) →
     (pure : ∀ {A} → A → F A) →
     (app : ∀ {A B} → F (A → B) → F A → F B) →
     RawApplicative F
diff --git a/src/Effect/Monad.agda b/src/Effect/Monad.agda
index 0448f0d6cd..5979c48b43 100644
--- a/src/Effect/Monad.agda
+++ b/src/Effect/Monad.agda
@@ -74,7 +74,7 @@ module _ where
   open RawApplicative
 
   mkRawMonad :
-    (F : Set f → Set f) →
+    (F : Set f → Set g) →
     (pure : ∀ {A} → A → F A) →
     (bind : ∀ {A B} → F A → (A → F B) → F B) →
     RawMonad F
diff --git a/src/Reflection/AST/DeBruijn.agda b/src/Reflection/AST/DeBruijn.agda
index 2236138550..2973912d3f 100644
--- a/src/Reflection/AST/DeBruijn.agda
+++ b/src/Reflection/AST/DeBruijn.agda
@@ -116,7 +116,7 @@ module _ where
 module _ where
 
   private
-    anyApplicative : RawApplicative (λ _ → Bool)
+    anyApplicative : ∀ {ℓ} → RawApplicative {ℓ} (λ _ → Bool)
     anyApplicative = mkRawApplicative _ (λ _ → false)  _∨_
 
   open Traverse anyApplicative

From 242546be360ccd0f450095cef0cb11e844c4004c Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Tue, 12 Mar 2024 16:45:27 +0100
Subject: [PATCH 239/316] Some properties of upTo and downFrom (#2316)

* Some properties of upTo and downFrom

* Rename things per review comments

* Fix changelog typo
---
 CHANGELOG.md                  | 12 ++++++++++
 src/Data/List/Properties.agda | 41 ++++++++++++++++++++++++++++++++++-
 2 files changed, 52 insertions(+), 1 deletion(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index bbc65af5d7..82c677d186 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -140,6 +140,18 @@ Additions to existing modules
   pattern divides k eq = Data.Nat.Divisibility.divides k eq
   ```
 
+* In `Data.List.Properties`:
+  ```agda
+  applyUpTo-∷ʳ          : applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
+  applyDownFrom-∷ʳ      : applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
+  upTo-∷ʳ               : upTo n ∷ʳ n ≡ upTo (suc n)
+  downFrom-∷ʳ           : applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)
+  reverse-applyUpTo     : reverse (applyUpTo f n) ≡ applyDownFrom f n
+  reverse-upTo          : reverse (upTo n) ≡ downFrom n
+  reverse-applyDownFrom : reverse (applyDownFrom f n) ≡ applyUpTo f n
+  reverse-downFrom      : reverse (downFrom n) ≡ upTo n
+  ```
+
 * In `Data.List.Relation.Unary.All.Properties`:
   ```agda
   All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 0dd6574711..0cda410424 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -657,8 +657,12 @@ lookup-applyUpTo : ∀ (f : ℕ → A) n i → lookup (applyUpTo f n) i ≡ f (t
 lookup-applyUpTo f (suc n) zero    = refl
 lookup-applyUpTo f (suc n) (suc i) = lookup-applyUpTo (f ∘ suc) n i
 
+applyUpTo-∷ʳ : ∀ (f : ℕ → A) n → applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
+applyUpTo-∷ʳ f zero = refl
+applyUpTo-∷ʳ f (suc n) = cong (f 0 ∷_) (applyUpTo-∷ʳ (f ∘ suc) n)
+
 ------------------------------------------------------------------------
--- applyUpTo
+-- applyDownFrom
 
 module _ (f : ℕ → A) where
 
@@ -670,6 +674,10 @@ module _ (f : ℕ → A) where
   lookup-applyDownFrom (suc n) zero    = refl
   lookup-applyDownFrom (suc n) (suc i) = lookup-applyDownFrom n i
 
+  applyDownFrom-∷ʳ : ∀ n → applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
+  applyDownFrom-∷ʳ zero = refl
+  applyDownFrom-∷ʳ (suc n) = cong (f (suc n) ∷_) (applyDownFrom-∷ʳ n)
+
 ------------------------------------------------------------------------
 -- upTo
 
@@ -679,6 +687,9 @@ length-upTo = length-applyUpTo id
 lookup-upTo : ∀ n i → lookup (upTo n) i ≡ toℕ i
 lookup-upTo = lookup-applyUpTo id
 
+upTo-∷ʳ : ∀ n → upTo n ∷ʳ n ≡ upTo (suc n)
+upTo-∷ʳ = applyUpTo-∷ʳ id
+
 ------------------------------------------------------------------------
 -- downFrom
 
@@ -688,6 +699,9 @@ length-downFrom = length-applyDownFrom id
 lookup-downFrom : ∀ n i → lookup (downFrom n) i ≡ n ∸ (suc (toℕ i))
 lookup-downFrom = lookup-applyDownFrom id
 
+downFrom-∷ʳ : ∀ n → applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)
+downFrom-∷ʳ = applyDownFrom-∷ʳ id
+
 ------------------------------------------------------------------------
 -- tabulate
 
@@ -1173,6 +1187,31 @@ reverse-foldl : ∀ (f : B → A → B) b xs →
                 foldl f b (reverse xs) ≡ foldr (flip f) b xs
 reverse-foldl f b xs = foldl-ʳ++ f b xs
 
+------------------------------------------------------------------------
+-- reverse, applyUpTo, and applyDownFrom
+
+reverse-applyUpTo : ∀ (f : ℕ → A) n → reverse (applyUpTo f n) ≡ applyDownFrom f n
+reverse-applyUpTo f zero = refl
+reverse-applyUpTo f (suc n) = begin
+  reverse (f 0 ∷ applyUpTo (f ∘ suc) n)  ≡⟨ reverse-++ [ f 0 ] (applyUpTo (f ∘ suc) n) ⟩
+  reverse (applyUpTo (f ∘ suc) n) ∷ʳ f 0 ≡⟨ cong (_∷ʳ f 0) (reverse-applyUpTo (f ∘ suc) n) ⟩
+  applyDownFrom (f ∘ suc) n ∷ʳ f 0       ≡⟨ applyDownFrom-∷ʳ f n ⟩
+  applyDownFrom f (suc n)                ∎
+
+reverse-upTo : ∀ n → reverse (upTo n) ≡ downFrom n
+reverse-upTo = reverse-applyUpTo id
+
+reverse-applyDownFrom : ∀ (f : ℕ → A) n → reverse (applyDownFrom f n) ≡ applyUpTo f n
+reverse-applyDownFrom f zero = refl
+reverse-applyDownFrom f (suc n) = begin
+  reverse (f n ∷ applyDownFrom f n)  ≡⟨ reverse-++ [ f n ] (applyDownFrom f n) ⟩
+  reverse (applyDownFrom f n) ∷ʳ f n ≡⟨ cong (_∷ʳ f n) (reverse-applyDownFrom f n) ⟩
+  applyUpTo f n ∷ʳ f n               ≡⟨ applyUpTo-∷ʳ f n ⟩
+  applyUpTo f (suc n)                ∎
+
+reverse-downFrom : ∀ n → reverse (downFrom n) ≡ upTo n
+reverse-downFrom = reverse-applyDownFrom id
+
 ------------------------------------------------------------------------
 -- _∷ʳ_
 

From 2f5d88d7debfb886ec528d3ae4af1bfb7b17d819 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 13 Mar 2024 01:10:13 +0000
Subject: [PATCH 240/316] Tidy up `README` imports #2280 (#2313)

* tidying up `README` imports

* address Matthew's review comments
---
 doc/README/Data/Container/FreeMonad.agda      |  12 +-
 doc/README/Data/List/Membership.agda          |   8 +-
 .../Data/List/Relation/Binary/Equality.agda   |   4 +-
 doc/README/Data/Tree/AVL.agda                 |   6 +-
 doc/README/Data/Trie/NonDependent.agda        |  18 +--
 .../Vec/Relation/Binary/Equality/Cast.agda    | 146 +++++++++++-------
 doc/README/Function/Reasoning.agda            |   8 +-
 doc/README/Tactic/Cong.agda                   |  14 +-
 doc/README/Tactic/MonoidSolver.agda           |   4 +-
 9 files changed, 122 insertions(+), 98 deletions(-)

diff --git a/doc/README/Data/Container/FreeMonad.agda b/doc/README/Data/Container/FreeMonad.agda
index ab302c90f5..3d3b2d5ff4 100644
--- a/doc/README/Data/Container/FreeMonad.agda
+++ b/doc/README/Data/Container/FreeMonad.agda
@@ -16,12 +16,12 @@ open import Data.Unit
 open import Data.Bool.Base using (Bool; true)
 open import Data.Nat
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Data.Product.Base renaming (_×_ to _⟨×⟩_)
+open import Data.Product.Base
 open import Data.Container using (Container; _▷_)
-open import Data.Container.Combinator
-open import Data.Container.FreeMonad as FreeMonad
+open import Data.Container.Combinator hiding (_×_)
+open import Data.Container.FreeMonad
 open import Data.W
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.PropositionalEquality as ≡
 
 ------------------------------------------------------------------------
 -- Defining the signature of an effect and building trees describing
@@ -51,13 +51,13 @@ prog =
   where
   open RawMonad monad using (_>>_)
 
-runState : {S X : Set} → State S ⋆ X → (S → X ⟨×⟩ S)
+runState : {S X : Set} → State S ⋆ X → (S → X × S)
 runState (pure x)                = λ s → x , s
 runState (impure ((inj₁ _) , k)) = λ s → runState (k s) s
 runState (impure ((inj₂ s) , k)) = λ _ → runState (k _) s
 
 test : runState prog 0 ≡ (true , 1)
-test = P.refl
+test = ≡.refl
 
 -- It should be noted that @State S ⋆ X@ is not the state monad. If we
 -- could quotient @State S ⋆ X@ by the seven axioms of state (see
diff --git a/doc/README/Data/List/Membership.agda b/doc/README/Data/List/Membership.agda
index 9cb89a8159..38732fe631 100644
--- a/doc/README/Data/List/Membership.agda
+++ b/doc/README/Data/List/Membership.agda
@@ -7,11 +7,11 @@
 module README.Data.List.Membership where
 
 open import Data.Char.Base using (Char; fromℕ)
-open import Data.Char.Properties as CharProp hiding (setoid)
-open import Data.Nat as ℕ using (ℕ; _+_; _<_; s≤s; z≤n; _*_; _∸_; _≤_)
+open import Data.Char.Properties as Char hiding (setoid)
 open import Data.List.Base using (List; []; _∷_; map)
+open import Data.Nat as ℕ using (ℕ)
 open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; sym; cong; setoid)
+  using (_≡_; refl; cong; setoid)
 
 ------------------------------------------------------------------------
 -- Membership
@@ -66,7 +66,7 @@ import Data.List.Membership.Propositional.Properties as PropProperties
 -- following the first `∈` refers to lists of type `List ℕ` whereas
 -- the second `∈` refers to lists of type `List Char`.
 
-open DecPropMembership CharProp._≟_ renaming (_∈_ to _∈ᶜ_)
+open DecPropMembership Char._≟_ renaming (_∈_ to _∈ᶜ_)
 open SetoidProperties using (∈-map⁺)
 
 lem₂ : {v : ℕ} {xs : List ℕ} → v ∈ xs → fromℕ v ∈ᶜ map fromℕ xs
diff --git a/doc/README/Data/List/Relation/Binary/Equality.agda b/doc/README/Data/List/Relation/Binary/Equality.agda
index 4ce5fe2478..5820501564 100644
--- a/doc/README/Data/List/Relation/Binary/Equality.agda
+++ b/doc/README/Data/List/Relation/Binary/Equality.agda
@@ -9,10 +9,8 @@
 module README.Data.List.Relation.Binary.Equality where
 
 open import Data.Nat using (ℕ; _+_; _<_; s≤s; z≤n; _*_; _∸_; _≤_)
-open import Data.Nat.Properties as NatProp
+open import Data.Nat.Properties as ℕ
 open import Data.List.Base
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; sym; cong; setoid)
 
 ------------------------------------------------------------------------
 -- Pointwise equality
diff --git a/doc/README/Data/Tree/AVL.agda b/doc/README/Data/Tree/AVL.agda
index 0955887f23..7e29f09f17 100644
--- a/doc/README/Data/Tree/AVL.agda
+++ b/doc/README/Data/Tree/AVL.agda
@@ -20,7 +20,7 @@ import Data.Tree.AVL
 -- natural numbers as keys and vectors of strings as values.
 
 open import Data.Nat.Properties using (<-strictTotalOrder)
-open import Data.Product.Base as Prod using (_,_; _,′_)
+open import Data.Product.Base as Product using (_,_; _,′_)
 open import Data.String.Base using (String)
 open import Data.Vec.Base using (Vec; _∷_; [])
 open import Relation.Binary.PropositionalEquality
@@ -111,14 +111,14 @@ open import Function.Base using (id)
 v₆ : headTail t₀ ≡ nothing
 v₆ = refl
 
-v₇ : Maybe.map (Prod.map₂ toList) (headTail t₂) ≡
+v₇ : Maybe.map (Product.map₂ toList) (headTail t₂) ≡
      just ((1 , v₁) , ((2 , v₂) ∷ []))
 v₇ = refl
 
 v₈ : initLast t₀ ≡ nothing
 v₈ = refl
 
-v₉ : Maybe.map (Prod.map₁ toList) (initLast t₄) ≡
+v₉ : Maybe.map (Product.map₁ toList) (initLast t₄) ≡
      just (((1 , v₁) ∷ []) ,′ (2 , v₂))
 v₉ = refl
 
diff --git a/doc/README/Data/Trie/NonDependent.agda b/doc/README/Data/Trie/NonDependent.agda
index 0484e6005a..385e981880 100644
--- a/doc/README/Data/Trie/NonDependent.agda
+++ b/doc/README/Data/Trie/NonDependent.agda
@@ -51,15 +51,15 @@ module README.Data.Trie.NonDependent where
 open import Level
 open import Data.Unit
 open import Data.Bool
-open import Data.Char        as Char
-import Data.Char.Properties  as Char
-open import Data.List.Base   as List using (List; []; _∷_)
-open import Data.List.Fresh  as List# using (List#; []; _∷#_)
-open import Data.Maybe       as Maybe
-open import Data.Product.Base as Prod using (_×_; ∃; proj₁; _,_)
-open import Data.String.Base as String using (String)
+open import Data.Char              as Char
+import Data.Char.Properties        as Char
+open import Data.List.Base         as List using (List; []; _∷_)
+open import Data.List.Fresh        as List# using (List#; []; _∷#_)
+open import Data.Maybe             as Maybe
+open import Data.Product.Base      as Product using (_×_; ∃; proj₁; _,_)
+open import Data.String.Base       as String using (String)
 open import Data.String.Properties as String using (_≟_)
-open import Data.These       as These
+open import Data.These             as These
 
 open import Function.Base using (case_of_; _$_; _∘′_; id; _on_)
 open import Relation.Nary
@@ -127,7 +127,7 @@ module _ {t} (L : Lexer t) where
     -- characters one by one
 
     init : Keywords
-    init = fromList $ List.map (Prod.map₁ String.toList) $ proj₁ $ List#.toList keywords
+    init = fromList $ List.map (Product.map₁ String.toList) $ proj₁ $ List#.toList keywords
 
     -- Kickstart the tokeniser with an empty accumulator and the initial
     -- trie.
diff --git a/doc/README/Data/Vec/Relation/Binary/Equality/Cast.agda b/doc/README/Data/Vec/Relation/Binary/Equality/Cast.agda
index 42cd8bfd3d..e06cdb49f0 100644
--- a/doc/README/Data/Vec/Relation/Binary/Equality/Cast.agda
+++ b/doc/README/Data/Vec/Relation/Binary/Equality/Cast.agda
@@ -3,6 +3,9 @@
 --
 -- An equational reasoning library for propositional equality over
 -- vectors of different indices using cast.
+--
+-- To see example usages of this library, scroll to the `Combinators`
+-- section.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -10,15 +13,15 @@
 module README.Data.Vec.Relation.Binary.Equality.Cast where
 
 open import Agda.Primitive
-open import Data.List.Base as L using (List)
-import Data.List.Properties as Lₚ
+open import Data.List.Base as List using (List)
+import Data.List.Properties as List
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Data.Vec.Base
 open import Data.Vec.Properties
 open import Data.Vec.Relation.Binary.Equality.Cast
 open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; trans; sym; cong; subst; module ≡-Reasoning)
+  using (_≡_; refl; sym; cong; module ≡-Reasoning)
 
 private variable
   a : Level
@@ -27,10 +30,6 @@ private variable
   xs ys zs ws : Vec A n
 
 
--- To see example usages of this library, scroll to the combinators
--- section.
-
-
 ------------------------------------------------------------------------
 -- Motivation
 --
@@ -60,20 +59,20 @@ private variable
 -- Although `cast` makes it possible to prove vector identities by ind-
 -- uction, the explicit type-casting nature poses a significant barrier
 -- to code reuse in larger proofs. For example, consider the identity
--- ‘fromList (xs L.∷ʳ x) ≡ (fromList xs) ∷ʳ x’ where `L._∷ʳ_` is the
+-- ‘fromList (xs List.∷ʳ x) ≡ (fromList xs) ∷ʳ x’ where `List._∷ʳ_` is the
 -- snoc function of lists. We have
 --
---     fromList (xs L.∷ʳ x)            : Vec A (L.length (xs L.∷ʳ x))
+--     fromList (xs List.∷ʳ x)            : Vec A (List.length (xs List.∷ʳ x))
 --   =   {- by definition -}
---     fromList (xs L.++ L.[ x ])      : Vec A (L.length (xs L.++ L.[ x ]))
+--     fromList (xs List.++ List.[ x ])   : Vec A (List.length (xs List.++ List.[ x ]))
 --   =   {- by fromList-++ -}
---     fromList xs ++ fromList L.[ x ] : Vec A (L.length xs + L.length [ x ])
+--     fromList xs ++ fromList List.[ x ] : Vec A (List.length xs + List.length [ x ])
 --   =   {- by definition -}
---     fromList xs ++ [ x ]            : Vec A (L.length xs + 1)
+--     fromList xs ++ [ x ]               : Vec A (List.length xs + 1)
 --   =   {- by unfold-∷ʳ -}
---     fromList xs ∷ʳ x                : Vec A (suc (L.length xs))
+--     fromList xs ∷ʳ x                   : Vec A (suc (List.length xs))
 -- where
---     fromList-++ : cast _ (fromList (xs L.++ ys)) ≡ fromList xs ++ fromList ys
+--     fromList-++ : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
 --     unfold-∷ʳ   : cast _ (xs ∷ʳ x) ≡ xs ++ [ x ]
 --
 -- Although the identity itself is simple, the reasoning process changes
@@ -82,31 +81,42 @@ private variable
 -- rearrange (the Agda version of) the identity into one with two
 -- `cast`s, resulting in lots of boilerplate code as demonstrated by
 -- `example1a-fromList-∷ʳ`.
-example1a-fromList-∷ʳ : ∀ (x : A) xs .(eq : L.length (xs L.∷ʳ x) ≡ suc (L.length xs)) →
-                        cast eq (fromList (xs L.∷ʳ x)) ≡ fromList xs ∷ʳ x
+example1a-fromList-∷ʳ : ∀ (x : A) xs →
+                        .(eq : List.length (xs List.∷ʳ x) ≡ suc (List.length xs)) →
+                        cast eq (fromList (xs List.∷ʳ x)) ≡ fromList xs ∷ʳ x
 example1a-fromList-∷ʳ x xs eq = begin
-  cast eq (fromList (xs L.∷ʳ x))                   ≡⟨⟩
-  cast eq (fromList (xs L.++ L.[ x ]))             ≡⟨ cast-trans eq₁ eq₂ (fromList (xs L.++ L.[ x ])) ⟨
-  cast eq₂ (cast eq₁ (fromList (xs L.++ L.[ x ]))) ≡⟨ cong (cast eq₂) (fromList-++ xs) ⟩
-  cast eq₂ (fromList xs ++ [ x ])                  ≡⟨ ≈-sym (unfold-∷ʳ (sym eq₂) x (fromList xs)) ⟩
-  fromList xs ∷ʳ x                                 ∎
+  cast eq (fromList (xs List.∷ʳ x))
+    ≡⟨⟩
+  cast eq (fromList (xs List.++ List.[ x ]))
+    ≡⟨ cast-trans eq₁ eq₂ (fromList (xs List.++ List.[ x ])) ⟨
+  cast eq₂ (cast eq₁ (fromList (xs List.++ List.[ x ])))
+    ≡⟨ cong (cast eq₂) (fromList-++ xs) ⟩
+  cast eq₂ (fromList xs ++ [ x ])
+    ≡⟨ ≈-sym (unfold-∷ʳ (sym eq₂) x (fromList xs)) ⟩
+  fromList xs ∷ʳ x
+    ∎
   where
   open ≡-Reasoning
-  eq₁ = Lₚ.length-++ xs {L.[ x ]}
-  eq₂ = +-comm (L.length xs) 1
+  eq₁ = List.length-++ xs {List.[ x ]}
+  eq₂ = +-comm (List.length xs) 1
 
 -- The `cast`s are irrelevant to core of the proof. At the same time,
 -- they can be inferred from the lemmas used during the reasoning steps
 -- (e.g. `fromList-++` and `unfold-∷ʳ`). To eliminate the boilerplate,
 -- this library provides a set of equational reasoning combinators for
 -- equality of the form `cast eq xs ≡ ys`.
-example1b-fromList-∷ʳ : ∀ (x : A) xs .(eq : L.length (xs L.∷ʳ x) ≡ suc (L.length xs)) →
-                        cast eq (fromList (xs L.∷ʳ x)) ≡ fromList xs ∷ʳ x
+example1b-fromList-∷ʳ : ∀ (x : A) xs →
+                        .(eq : List.length (xs List.∷ʳ x) ≡ suc (List.length xs)) →
+                        cast eq (fromList (xs List.∷ʳ x)) ≡ fromList xs ∷ʳ x
 example1b-fromList-∷ʳ x xs eq = begin
-  fromList (xs L.∷ʳ x)       ≈⟨⟩
-  fromList (xs L.++ L.[ x ]) ≈⟨ fromList-++ xs ⟩
-  fromList xs ++ [ x ]       ≈⟨ unfold-∷ʳ (+-comm 1 (L.length xs)) x (fromList xs) ⟨
-  fromList xs ∷ʳ x           ∎
+  fromList (xs List.∷ʳ x)
+    ≈⟨⟩
+  fromList (xs List.++ List.[ x ])
+    ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ [ x ]
+    ≈⟨ unfold-∷ʳ (+-comm 1 (List.length xs)) x (fromList xs) ⟨
+  fromList xs ∷ʳ x
+    ∎
   where open CastReasoning
 
 
@@ -149,7 +159,7 @@ example2b eq xs a ys = begin
   (a ∷ xs) ʳ++ ys         ≂⟨ unfold-ʳ++ (a ∷ xs) ys ⟩          -- index: suc m + n
   reverse (a ∷ xs) ++ ys  ≂⟨ cong (_++ ys) (reverse-∷ a xs) ⟩  -- index: suc m + n
   (reverse xs ∷ʳ a) ++ ys ≈⟨ ∷ʳ-++ eq a (reverse xs) ⟩         -- index: suc m + n
-  reverse xs ++ (a ∷ ys)  ≂⟨ unfold-ʳ++ xs (a ∷ ys) ⟨         -- index: m + suc n
+  reverse xs ++ (a ∷ ys)  ≂⟨ unfold-ʳ++ xs (a ∷ ys) ⟨          -- index: m + suc n
   xs ʳ++ (a ∷ ys)         ∎                                    -- index: m + suc n
   where open CastReasoning
 
@@ -169,29 +179,35 @@ example2b eq xs a ys = begin
 -- Note. Technically, `A` and `B` should be vectors of different length
 -- and that `ys`, `zs` are vectors of non-definitionally equal index.
 example3a-fromList-++-++ : {xs ys zs : List A} →
-                           .(eq : L.length (xs L.++ ys L.++ zs) ≡
-                                  L.length xs + (L.length ys + L.length zs)) →
-                           cast eq (fromList (xs L.++ ys L.++ zs)) ≡
+                           .(eq : List.length (xs List.++ ys List.++ zs) ≡
+                                  List.length xs + (List.length ys + List.length zs)) →
+                           cast eq (fromList (xs List.++ ys List.++ zs)) ≡
                                    fromList xs ++ fromList ys ++ fromList zs
 example3a-fromList-++-++ {xs = xs} {ys} {zs} eq = begin
-  fromList (xs L.++ ys L.++ zs)             ≈⟨ fromList-++ xs ⟩
-  fromList xs ++ fromList (ys L.++ zs)      ≈⟨ ≈-cong (fromList xs ++_) (cast-++ʳ (Lₚ.length-++ ys) (fromList xs))
-                                                      (fromList-++ ys) ⟩
-  fromList xs ++ fromList ys ++ fromList zs ∎
+  fromList (xs List.++ ys List.++ zs)
+    ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ fromList (ys List.++ zs)
+    ≈⟨ ≈-cong (fromList xs ++_) (cast-++ʳ (List.length-++ ys) (fromList xs)) (fromList-++ ys) ⟩
+  fromList xs ++ fromList ys ++ fromList zs
+    ∎
   where open CastReasoning
 
 -- As an alternative, one can manually apply `cast-++ʳ` to expose `cast`
 -- in the subterm. However, this unavoidably duplicates the proof term.
 example3b-fromList-++-++′ : {xs ys zs : List A} →
-                            .(eq : L.length (xs L.++ ys L.++ zs) ≡
-                                   L.length xs + (L.length ys + L.length zs)) →
-                            cast eq (fromList (xs L.++ ys L.++ zs)) ≡
+                            .(eq : List.length (xs List.++ ys List.++ zs) ≡
+                                   List.length xs + (List.length ys + List.length zs)) →
+                            cast eq (fromList (xs List.++ ys List.++ zs)) ≡
                                     fromList xs ++ fromList ys ++ fromList zs
 example3b-fromList-++-++′ {xs = xs} {ys} {zs} eq = begin
-  fromList (xs L.++ ys L.++ zs)                 ≈⟨ fromList-++ xs ⟩
-  fromList xs ++ fromList (ys L.++ zs)          ≈⟨ cast-++ʳ (Lₚ.length-++ ys) (fromList xs) ⟩
-  fromList xs ++ cast _ (fromList (ys L.++ zs)) ≂⟨ cong (fromList xs ++_) (fromList-++ ys) ⟩
-  fromList xs ++ fromList ys ++ fromList zs     ∎
+  fromList (xs List.++ ys List.++ zs)
+    ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ fromList (ys List.++ zs)
+    ≈⟨ cast-++ʳ (List.length-++ ys) (fromList xs) ⟩
+  fromList xs ++ cast _ (fromList (ys List.++ zs))
+    ≂⟨ cong (fromList xs ++_) (fromList-++ ys) ⟩
+  fromList xs ++ fromList ys ++ fromList zs
+    ∎
   where open CastReasoning
 
 -- `≈-cong` can be chained together much like how `cong` can be nested.
@@ -201,12 +217,16 @@ example3b-fromList-++-++′ {xs = xs} {ys} {zs} eq = begin
 example4-cong² : ∀ .(eq : (m + 1) + n ≡ n + suc m) a (xs : Vec A m) ys →
           cast eq (reverse ((xs ++ [ a ]) ++ ys)) ≡ ys ʳ++ reverse (xs ∷ʳ a)
 example4-cong² {m = m} {n} eq a xs ys = begin
-  reverse ((xs ++ [ a ]) ++ ys)   ≈⟨ ≈-cong reverse (cast-reverse (cong (_+ n) (+-comm 1 m)) ((xs ∷ʳ a) ++ ys))
+  reverse ((xs ++ [ a ]) ++ ys)
+    ≈⟨ ≈-cong reverse (cast-reverse (cong (_+ n) (+-comm 1 m)) ((xs ∷ʳ a) ++ ys))
                                              (≈-cong (_++ ys) (cast-++ˡ (+-comm 1 m) (xs ∷ʳ a))
                                                      (unfold-∷ʳ _ a xs)) ⟨
-  reverse ((xs ∷ʳ a) ++ ys)       ≈⟨ reverse-++ (+-comm (suc m) n) (xs ∷ʳ a) ys ⟩
-  reverse ys ++ reverse (xs ∷ʳ a) ≂⟨ unfold-ʳ++ ys (reverse (xs ∷ʳ a)) ⟨
-  ys ʳ++ reverse (xs ∷ʳ a)        ∎
+  reverse ((xs ∷ʳ a) ++ ys)
+    ≈⟨ reverse-++ (+-comm (suc m) n) (xs ∷ʳ a) ys ⟩
+  reverse ys ++ reverse (xs ∷ʳ a)
+    ≂⟨ unfold-ʳ++ ys (reverse (xs ∷ʳ a)) ⟨
+  ys ʳ++ reverse (xs ∷ʳ a)
+    ∎
   where open CastReasoning
 
 ------------------------------------------------------------------------
@@ -222,25 +242,33 @@ example4-cong² {m = m} {n} eq a xs ys = begin
 -- reasoning system of `_≈[_]_` and switches back to the reasoning
 -- system of `_≡_`.
 example5-fromList-++-++′ : {xs ys zs : List A} →
-                           .(eq : L.length (xs L.++ ys L.++ zs) ≡
-                                  L.length xs + (L.length ys + L.length zs)) →
-                           cast eq (fromList (xs L.++ ys L.++ zs)) ≡
+                           .(eq : List.length (xs List.++ ys List.++ zs) ≡
+                                  List.length xs + (List.length ys + List.length zs)) →
+                           cast eq (fromList (xs List.++ ys List.++ zs)) ≡
                                    fromList xs ++ fromList ys ++ fromList zs
 example5-fromList-++-++′ {xs = xs} {ys} {zs} eq = begin
-  fromList (xs L.++ ys L.++ zs)                 ≈⟨ fromList-++ xs ⟩
-  fromList xs ++ fromList (ys L.++ zs)          ≃⟨ cast-++ʳ (Lₚ.length-++ ys) (fromList xs) ⟩
-  fromList xs ++ cast _ (fromList (ys L.++ zs)) ≡⟨ cong (fromList xs ++_) (fromList-++ ys) ⟩
-  fromList xs ++ fromList ys ++ fromList zs     ≡-∎
+  fromList (xs List.++ ys List.++ zs)
+    ≈⟨ fromList-++ xs ⟩
+  fromList xs ++ fromList (ys List.++ zs)
+    ≃⟨ cast-++ʳ (List.length-++ ys) (fromList xs) ⟩
+  fromList xs ++ cast _ (fromList (ys List.++ zs))
+    ≡⟨ cong (fromList xs ++_) (fromList-++ ys) ⟩
+  fromList xs ++ fromList ys ++ fromList zs
+    ≡-∎
   where open CastReasoning
 
 -- Of course, it is possible to start with the reasoning system of `_≡_`
 -- and then switch to the reasoning system of `_≈[_]_`.
 example6a-reverse-∷ʳ : ∀ x (xs : Vec A n) → reverse (xs ∷ʳ x) ≡ x ∷ reverse xs
 example6a-reverse-∷ʳ {n = n} x xs = begin-≡
-  reverse (xs ∷ʳ x)     ≡⟨ ≈-reflexive refl ⟨
-  reverse (xs ∷ʳ x)     ≈⟨ ≈-cong reverse (cast-reverse _ _) (unfold-∷ʳ (+-comm 1 n) x xs) ⟩
-  reverse (xs ++ [ x ]) ≈⟨ reverse-++ (+-comm n 1) xs [ x ] ⟩
-  x ∷ reverse xs        ∎
+  reverse (xs ∷ʳ x)
+    ≡⟨ ≈-reflexive refl ⟨
+  reverse (xs ∷ʳ x)
+    ≈⟨ ≈-cong reverse (cast-reverse _ _) (unfold-∷ʳ (+-comm 1 n) x xs) ⟩
+  reverse (xs ++ [ x ])
+    ≈⟨ reverse-++ (+-comm n 1) xs [ x ] ⟩
+  x ∷ reverse xs
+    ∎
   where open CastReasoning
 
 example6b-reverse-∷ʳ-by-induction : ∀ x (xs : Vec A n) → reverse (xs ∷ʳ x) ≡ x ∷ reverse xs
diff --git a/doc/README/Function/Reasoning.agda b/doc/README/Function/Reasoning.agda
index 0125fa9d99..810f048aa3 100644
--- a/doc/README/Function/Reasoning.agda
+++ b/doc/README/Function/Reasoning.agda
@@ -1,9 +1,9 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Some examples showing how the Function.Reasoning module
--- can be used to perform "functional reasoning" similar to what is being
--- described in: https://stackoverflow.com/q/22676703/3168666
+-- Some examples showing how the Function.Reasoning module can be used
+-- to perform "functional reasoning" similar to what is being described
+-- in: https://stackoverflow.com/q/22676703/3168666
 ------------------------------------------------------------------------
 
 {-# OPTIONS --with-K #-}
@@ -39,7 +39,7 @@ open import Data.String.Base as String using (String; toList; fromList)
 open import Data.String.Properties as String using (_==_)
 open import Function.Base using (_∘_)
 open import Data.Bool hiding (_≤?_)
-open import Data.Product.Base as P using (_×_; <_,_>; uncurry; proj₁)
+open import Data.Product.Base using (_×_; <_,_>; uncurry; proj₁)
 open import Agda.Builtin.Equality
 
 -- This can give us for instance this decomposition of a function
diff --git a/doc/README/Tactic/Cong.agda b/doc/README/Tactic/Cong.agda
index 20cb5ad7d9..2a775f06e7 100644
--- a/doc/README/Tactic/Cong.agda
+++ b/doc/README/Tactic/Cong.agda
@@ -5,8 +5,8 @@ module README.Tactic.Cong where
 open import Data.Nat
 open import Data.Nat.Properties
 
-open import Relation.Binary.PropositionalEquality as Eq
-import Relation.Binary.Reasoning.Preorder as PR
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; sym; cong; module ≡-Reasoning)
 
 open import Tactic.Cong using (cong!)
 
@@ -26,7 +26,7 @@ open import Tactic.Cong using (cong!)
 
 verbose-example : ∀ m n → m ≡ n → suc (suc (m + 0)) + m ≡ suc (suc n) + (n + 0)
 verbose-example m n eq =
-  let open Eq.≡-Reasoning in
+  let open ≡-Reasoning in
   begin
     suc (suc (m + 0)) + m
   ≡⟨ cong (λ ϕ → suc (suc (ϕ + m))) (+-identityʳ m) ⟩
@@ -44,7 +44,7 @@ verbose-example m n eq =
 
 succinct-example : ∀ m n → m ≡ n → suc (suc (m + 0)) + m ≡ suc (suc n) + (n + 0)
 succinct-example m n eq =
-  let open Eq.≡-Reasoning in
+  let open ≡-Reasoning in
   begin
     suc (suc (m + 0)) + m
   ≡⟨ cong! (+-identityʳ m) ⟩
@@ -117,7 +117,7 @@ module EquationalReasoningTests where
 
   test₁ : ∀ m n → m ≡ n → suc (suc (m + 0)) + m ≡ suc (suc n) + (n + 0)
   test₁ m n eq =
-    let open Eq.≡-Reasoning in
+    let open ≡-Reasoning in
     begin
       suc (suc (m + 0)) + m
     ≡⟨ cong! (+-identityʳ m) ⟩
@@ -130,11 +130,11 @@ module EquationalReasoningTests where
 
   test₂ : ∀ m n → m ≡ n → suc (m + m) ≤ suc (suc (n + n))
   test₂ m n eq =
-    let open PR ≤-preorder in
+    let open ≤-Reasoning in
     begin
       suc (m + m)
     ≡⟨ cong! eq ⟩
       suc (n + n)
-    ∼⟨ ≤-step ≤-refl ⟩
+    ≤⟨ n≤1+n _ ⟩
       suc (suc (n + n))
     ∎
diff --git a/doc/README/Tactic/MonoidSolver.agda b/doc/README/Tactic/MonoidSolver.agda
index 530091bc97..4631d67089 100644
--- a/doc/README/Tactic/MonoidSolver.agda
+++ b/doc/README/Tactic/MonoidSolver.agda
@@ -10,11 +10,9 @@ module README.Tactic.MonoidSolver {a ℓ} (M : Monoid a ℓ) where
 
 open Monoid M
 
-open import Data.Nat as Nat using (ℕ; suc; zero; _+_)
-open import Data.Nat.Properties as Properties using (+-0-monoid; +-comm)
 open import Relation.Binary.Reasoning.Setoid setoid
 
-open import Tactic.MonoidSolver using (solve; solve-macro)
+open import Tactic.MonoidSolver using (solve)
 
 -- The monoid solver is capable to of solving equations without having
 -- to specify the equation itself in the proof.

From 4529e73b3dbf4531a70c72f11eef3fc584a18f2f Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 16 Mar 2024 02:28:35 +0000
Subject: [PATCH 241/316] Add unique morphisms from/to `Initial` and `Terminal`
 algebras (#2296)

* added unique morphisms

* refactored for uniformity's sake

* exploit the uniformity

* add missing instances

* finish up, for now

* `CHANGELOG`

* `CHANGELOG`

* `TheMorphism`

* comment

* comment

* comment

* `The` to `Unique`

* lifted out istantiated `import`

* blank line

* note on instantiated `import`s

* parametrise on the `Raw` bundle

* parametrise on the `Raw` bundle

* Rerranged to get rid of lots of boilerplate

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                                 | 12 +++
 doc/style-guide.md                           |  4 +
 src/Algebra/Construct/Initial.agda           |  4 +-
 src/Algebra/Construct/Terminal.agda          | 12 ++-
 src/Algebra/Construct/Zero.agda              |  4 +-
 src/Algebra/Morphism/Construct/Initial.agda  | 62 ++++++++++++++
 src/Algebra/Morphism/Construct/Terminal.agda | 88 ++++++++++++++++++++
 7 files changed, 179 insertions(+), 7 deletions(-)
 create mode 100644 src/Algebra/Morphism/Construct/Initial.agda
 create mode 100644 src/Algebra/Morphism/Construct/Terminal.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 82c677d186..b780aa6e1f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -50,6 +50,12 @@ New modules
 -----------
 * `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
 
+* The unique morphism from the initial, resp. terminal, algebra:
+  ```agda
+  Algebra.Morphism.Construct.Initial
+  Algebra.Morphism.Construct.Terminal
+  ```
+
 * Nagata's construction of the "idealization of a module":
   ```agda
   Algebra.Module.Construct.Idealization
@@ -117,6 +123,12 @@ Additions to existing modules
   rawModule          : RawModule R c ℓ
   ```
 
+* In `Algebra.Construct.Terminal`:
+  ```agda
+  rawNearSemiring : RawNearSemiring c ℓ
+  nearSemiring    : NearSemiring c ℓ
+  ```
+
 * In `Algebra.Properties.Monoid.Mult`:
   ```agda
   ×-homo-0 : ∀ x → 0 × x ≈ 0#
diff --git a/doc/style-guide.md b/doc/style-guide.md
index b2ee87d8ee..254a7ec071 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -131,6 +131,10 @@ automate most of this.
   open SetoidEquality S
   ```
 
+* If importing a parametrised module, qualified or otherwise, with its
+  parameters instantiated, then such 'instantiated imports' should be placed
+  *after* the main block of `import`s, and *before* any `variable` declarations.
+
 * Naming conventions for qualified `import`s: if importing a module under
   a root of the form `Data.X` (e.g. the `Base` module for basic operations,
   or `Properties` for lemmas about them etc.) then conventionally, the
diff --git a/src/Algebra/Construct/Initial.agda b/src/Algebra/Construct/Initial.agda
index 99d80b98a3..57648d6735 100644
--- a/src/Algebra/Construct/Initial.agda
+++ b/src/Algebra/Construct/Initial.agda
@@ -4,8 +4,8 @@
 -- Instances of algebraic structures where the carrier is ⊥.
 -- In mathematics, this is usually called 0.
 --
--- From monoids up, these are zero-objects – i.e, the initial
--- object is the terminal object in the relevant category.
+-- From monoids up, these are zero-objects – i.e, the terminal
+-- object is *also* the initial object in the relevant category.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Algebra/Construct/Terminal.agda b/src/Algebra/Construct/Terminal.agda
index 4a6f1decfe..202982da50 100644
--- a/src/Algebra/Construct/Terminal.agda
+++ b/src/Algebra/Construct/Terminal.agda
@@ -1,8 +1,8 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Instances of algebraic structures where the carrier is ⊤.
--- In mathematics, this is usually called 0 (1 in the case of Group).
+-- Instances of algebraic structures where the carrier is ⊤. In
+-- mathematics, this is usually called 0 (1 in the case of Monoid, Group).
 --
 -- From monoids up, these are zero-objects – i.e, both the initial
 -- and the terminal object in the relevant category.
@@ -27,7 +27,7 @@ module 𝕆ne where
   Carrier : Set c
   Carrier = ⊤
 
-  _≈_     : Rel Carrier ℓ
+  _≈_   : Rel Carrier ℓ
   _ ≈ _ = ⊤
 
 ------------------------------------------------------------------------
@@ -42,6 +42,9 @@ rawMonoid = record { 𝕆ne }
 rawGroup : RawGroup c ℓ
 rawGroup = record { 𝕆ne }
 
+rawNearSemiring : RawNearSemiring c ℓ
+rawNearSemiring = record { 𝕆ne }
+
 rawSemiring : RawSemiring c ℓ
 rawSemiring = record { 𝕆ne }
 
@@ -78,6 +81,9 @@ group = record { 𝕆ne }
 abelianGroup : AbelianGroup c ℓ
 abelianGroup = record { 𝕆ne }
 
+nearSemiring : NearSemiring c ℓ
+nearSemiring = record { 𝕆ne }
+
 semiring : Semiring c ℓ
 semiring = record { 𝕆ne }
 
diff --git a/src/Algebra/Construct/Zero.agda b/src/Algebra/Construct/Zero.agda
index 912c3c1e8d..38bfb59b13 100644
--- a/src/Algebra/Construct/Zero.agda
+++ b/src/Algebra/Construct/Zero.agda
@@ -1,8 +1,8 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Instances of algebraic structures where the carrier is ⊤.
--- In mathematics, this is usually called 0 (1 in the case of Group).
+-- Instances of algebraic structures where the carrier is ⊤. In
+-- mathematics, this is usually called 0 (1 in the case of Monoid, Group).
 --
 -- From monoids up, these are are zero-objects – i.e, both the initial
 -- and the terminal object in the relevant category.
diff --git a/src/Algebra/Morphism/Construct/Initial.agda b/src/Algebra/Morphism/Construct/Initial.agda
new file mode 100644
index 0000000000..916e181030
--- /dev/null
+++ b/src/Algebra/Morphism/Construct/Initial.agda
@@ -0,0 +1,62 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The unique morphism from the initial object,
+-- for each of the relevant categories. Since
+-- `Semigroup` and `Band` are simply `Magma`s
+-- satisfying additional properties, it suffices to
+-- define the morphism on the underlying `RawMagma`.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Level using (Level)
+
+module Algebra.Morphism.Construct.Initial {c ℓ : Level} where
+
+open import Algebra.Bundles.Raw using (RawMagma)
+open import Algebra.Morphism.Structures
+open import Function.Definitions using (Injective)
+import Relation.Binary.Morphism.Definitions as Rel
+open import Relation.Binary.Morphism.Structures
+open import Relation.Binary.Core using (Rel)
+
+open import Algebra.Construct.Initial {c} {ℓ}
+
+private
+  variable
+    a m ℓm : Level
+    A : Set a
+    ≈ : Rel A ℓm
+
+------------------------------------------------------------------------
+-- The unique morphism
+
+zero : ℤero.Carrier → A
+zero ()
+
+------------------------------------------------------------------------
+-- Basic properties
+
+cong : (≈ : Rel A ℓm) → Rel.Homomorphic₂ ℤero.Carrier A ℤero._≈_ ≈ zero
+cong _ {x = ()} 
+
+injective : (≈ : Rel A ℓm) → Injective ℤero._≈_ ≈ zero
+injective _ {x = ()}
+
+------------------------------------------------------------------------
+-- Morphism structures
+
+isMagmaHomomorphism : (M : RawMagma m ℓm) →
+                      IsMagmaHomomorphism rawMagma M zero
+isMagmaHomomorphism M = record
+  { isRelHomomorphism = record { cong = cong (RawMagma._≈_ M) }
+  ; homo = λ()
+  } 
+
+isMagmaMonomorphism : (M : RawMagma m ℓm) →
+                      IsMagmaMonomorphism rawMagma M zero
+isMagmaMonomorphism M = record
+  { isMagmaHomomorphism = isMagmaHomomorphism M
+  ; injective = injective (RawMagma._≈_ M)
+  }
diff --git a/src/Algebra/Morphism/Construct/Terminal.agda b/src/Algebra/Morphism/Construct/Terminal.agda
new file mode 100644
index 0000000000..227ebf2887
--- /dev/null
+++ b/src/Algebra/Morphism/Construct/Terminal.agda
@@ -0,0 +1,88 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- The unique morphism to the terminal object,
+-- for each of the relevant categories. Since
+-- each terminal algebra builds on `Monoid`,
+-- possibly with additional (trivial) operations,
+-- satisfying additional properties, it suffices to
+-- define the morphism on the underlying `RawMonoid`
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Level using (Level)
+
+module Algebra.Morphism.Construct.Terminal {c ℓ : Level} where
+
+open import Algebra.Bundles.Raw
+  using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring; RawRing)
+open import Algebra.Morphism.Structures
+
+open import Data.Product.Base using (_,_)
+open import Function.Definitions using (StrictlySurjective)
+import Relation.Binary.Morphism.Definitions as Rel
+open import Relation.Binary.Morphism.Structures
+
+open import Algebra.Construct.Terminal {c} {ℓ}
+
+private
+  variable
+    a ℓa : Level
+    A : Set a
+
+------------------------------------------------------------------------
+-- The unique morphism
+
+one : A → 𝕆ne.Carrier
+one _ = _
+
+------------------------------------------------------------------------
+-- Basic properties
+
+strictlySurjective : A → StrictlySurjective 𝕆ne._≈_ one
+strictlySurjective x _ = x , _
+
+------------------------------------------------------------------------
+-- Homomorphisms
+
+isMagmaHomomorphism : (M : RawMagma a ℓa) →
+                      IsMagmaHomomorphism M rawMagma one
+isMagmaHomomorphism M = record
+  { isRelHomomorphism = record { cong = _ }
+  ; homo = _
+  }
+
+isMonoidHomomorphism : (M : RawMonoid a ℓa) →
+                       IsMonoidHomomorphism M rawMonoid one
+isMonoidHomomorphism M = record
+  { isMagmaHomomorphism = isMagmaHomomorphism (RawMonoid.rawMagma M)
+  ; ε-homo = _
+  }
+
+isGroupHomomorphism : (G : RawGroup a ℓa) →
+                      IsGroupHomomorphism G rawGroup one
+isGroupHomomorphism G = record
+  { isMonoidHomomorphism = isMonoidHomomorphism (RawGroup.rawMonoid G)
+  ; ⁻¹-homo = λ _ → _
+  }
+
+isNearSemiringHomomorphism : (N : RawNearSemiring a ℓa) →
+                             IsNearSemiringHomomorphism N rawNearSemiring one
+isNearSemiringHomomorphism N = record
+  { +-isMonoidHomomorphism = isMonoidHomomorphism (RawNearSemiring.+-rawMonoid N)
+  ; *-homo = λ _ _ → _
+  }
+
+isSemiringHomomorphism : (S : RawSemiring a ℓa) →
+                         IsSemiringHomomorphism S rawSemiring one
+isSemiringHomomorphism S = record
+  { isNearSemiringHomomorphism = isNearSemiringHomomorphism (RawSemiring.rawNearSemiring S)
+  ; 1#-homo = _
+  }
+
+isRingHomomorphism : (R : RawRing a ℓa) → IsRingHomomorphism R rawRing one
+isRingHomomorphism R = record
+  { isSemiringHomomorphism = isSemiringHomomorphism (RawRing.rawSemiring R)
+  ; -‿homo = λ _ → _
+  }

From dd5d38259465be8cd99a602f0dfd5264482efa47 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Sat, 16 Mar 2024 03:31:58 +0100
Subject: [PATCH 242/316] Setoid version of indexed containers. (#1511)

* Setoid version of indexed containers.

Following the structure for non-indexed containers.

* An example for indexed containers: multi-sorted algebras.

This tests the new setoid version of indexed containers.

* Brought code up to date

* Added CHANGELOG entry

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                                  |  12 +
 doc/README/Data.agda                          |   3 +-
 .../Indexed/MultiSortedAlgebraExample.agda    | 431 ++++++++++++++++++
 .../VectorExample.agda}                       |  14 +-
 src/Data/Container/Indexed/Core.agda          |  30 +-
 .../Relation/Binary/Equality/Setoid.agda      |  60 +++
 .../Indexed/Relation/Binary/Pointwise.agda    |  58 +++
 .../Relation/Binary/Pointwise/Properties.agda |  45 ++
 8 files changed, 637 insertions(+), 16 deletions(-)
 create mode 100644 doc/README/Data/Container/Indexed/MultiSortedAlgebraExample.agda
 rename doc/README/Data/Container/{Indexed.agda => Indexed/VectorExample.agda} (89%)
 create mode 100644 src/Data/Container/Indexed/Relation/Binary/Equality/Setoid.agda
 create mode 100644 src/Data/Container/Indexed/Relation/Binary/Pointwise.agda
 create mode 100644 src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index b780aa6e1f..343460e8c2 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -75,6 +75,13 @@ New modules
   Relation.Binary.Construct.Interior.Symmetric
   ```
 
+* Pointwise and equality relations over indexed containers:
+  ```agda
+  Data.Container.Indexed.Relation.Binary.Pointwise
+  Data.Container.Indexed.Relation.Binary.Pointwise.Properties
+  Data.Container.Indexed.Relation.Binary.Equality.Setoid
+  ```
+
 Additions to existing modules
 -----------------------------
 
@@ -142,6 +149,11 @@ Additions to existing modules
   idem-×-homo-* : (_*_ IdempotentOn x) → (m × x) * (n × x) ≈ (m ℕ.* n) × x
   ```
 
+* In `Data.Container.Indexed.Core`:
+  ```agda
+  Subtrees o c = (r : Response c) → X (next c r)
+  ```
+
 * In `Data.Fin.Properties`:
   ```agda
   nonZeroIndex : Fin n → ℕ.NonZero n
diff --git a/doc/README/Data.agda b/doc/README/Data.agda
index e7d8a13886..c1e900e541 100644
--- a/doc/README/Data.agda
+++ b/doc/README/Data.agda
@@ -218,7 +218,8 @@ import README.Data.Vec.Relation.Binary.Equality.Cast
 -- monad, least fixed point, etc.) can be used
 
 import README.Data.Container.FreeMonad
-import README.Data.Container.Indexed
+import README.Data.Container.Indexed.VectorExample
+import README.Data.Container.Indexed.MultiSortedAlgebraExample
 
 -- Wrapping n-ary relations into a record definition so type-inference
 -- remembers the things being related.
diff --git a/doc/README/Data/Container/Indexed/MultiSortedAlgebraExample.agda b/doc/README/Data/Container/Indexed/MultiSortedAlgebraExample.agda
new file mode 100644
index 0000000000..0c466c932c
--- /dev/null
+++ b/doc/README/Data/Container/Indexed/MultiSortedAlgebraExample.agda
@@ -0,0 +1,431 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Example of multi-sorted algebras as indexed containers
+------------------------------------------------------------------------
+
+{-# OPTIONS --safe --cubical-compatible #-}
+
+module README.Data.Container.Indexed.MultiSortedAlgebraExample where
+
+------------------------------------------------------------------------
+-- Preliminaries
+------------------------------------------------------------------------
+-- We import library content for indexed containers, standard types,
+-- and setoids.
+
+open import Level
+
+open import Data.Container.Indexed.Core using (Container; ⟦_⟧; _◃_/_)
+open import Data.Container.Indexed.FreeMonad using (_⋆C_)
+open import Data.W.Indexed using (W; sup)
+
+open import Data.Product using (Σ; _×_; _,_; Σ-syntax)
+open import Data.Sum using (_⊎_; inj₁; inj₂; [_,_])
+open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
+
+open import Function using (_∘_)
+open import Function.Bundles using (Func)
+
+open import Relation.Binary using (Setoid; IsEquivalence)
+open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+
+import Data.Container.Indexed.Relation.Binary.Equality.Setoid as ICSetoid
+import Relation.Binary.Reasoning.Setoid as SetoidReasoning
+
+open Setoid using (Carrier; _≈_; isEquivalence)
+open Func renaming (to to apply)
+
+-- Letter ℓ denotes universe levels.
+
+variable
+  ℓ ℓ' ℓˢ ℓᵒ ℓᵃ ℓᵐ ℓᵉ ℓⁱ : Level
+  I : Set ℓⁱ
+  S : Set ℓˢ
+
+------------------------------------------------------------------------
+-- The interpretation of a container (Op ◃ Ar / sort) is
+--
+--   ⟦ Op ◃ Ar / sort ⟧ X s = Σ[ o ∈ Op s ] ((i : Ar o) → X (sort o i))
+--
+-- which contains pairs consisting of an operator $o$ and its collection
+-- of arguments.  The least fixed point of (X ↦ ⟦ C ⟧ X) is the indexed
+-- W-type given by C, and it contains closed first-order terms of the
+-- multi-sorted algebra C.
+
+-- We need to interpret indexed containers on Setoids.
+-- This definition is missing from the standard library v1.7.
+
+⟦_⟧s : (C : Container I S ℓᵒ ℓᵃ) (ξ : I → Setoid ℓᵐ ℓᵉ) → S → Setoid _ _
+⟦ C ⟧s ξ = ICSetoid.setoid ξ C
+
+------------------------------------------------------------------------
+-- Multi-sorted algebras
+--------------------------------------------------------------------------
+-- A multi-sorted algebra is an indexed container.
+--
+-- * Sorts are indexes.
+--
+-- * Operators are commands/shapes.
+--
+-- * Arities/argument are responses/positions.
+--
+-- Closed terms (initial model) are given by the W type for a container,
+-- renamed to μ here (for least fixed-point).
+
+-- We assume a fixed signature (Sort, Ops).
+
+module _ (Sort : Set ℓˢ) (Ops : Container Sort Sort ℓᵒ ℓᵃ) where
+  open Container Ops renaming
+    ( Command   to  Op
+    ; Response  to  Arity
+    ; next      to  sort
+    )
+
+-- We let letter $s$ range over sorts and $\mathit{op}$ over operators.
+
+  variable
+    s s'    : Sort
+    op op'  : Op s
+
+------------------------------------------------------------------------
+-- Models
+
+  -- A model is given by an interpretation (Den $s$) for each sort $s$
+  -- plus an interpretation (den $o$) for each operator $o$.
+
+  record SetModel ℓᵐ : Set (ℓˢ ⊔ ℓᵒ ⊔ ℓᵃ ⊔ suc ℓᵐ) where
+    field
+      Den : Sort → Set ℓᵐ
+      den : {s : Sort} → ⟦ Ops ⟧ Den s → Den s
+
+  -- The setoid model requires operators to respect equality.
+  -- The Func record packs a function (apply) with a proof (cong)
+  -- that the function maps equals to equals.
+
+  record SetoidModel ℓᵐ ℓᵉ : Set (ℓˢ ⊔ ℓᵒ ⊔ ℓᵃ ⊔ suc (ℓᵐ ⊔ ℓᵉ)) where
+    field
+      Den  :  Sort → Setoid ℓᵐ ℓᵉ
+      den  :  {s : Sort} → Func (⟦ Ops ⟧s Den s) (Den s)
+
+
+------------------------------------------------------------------------
+-- Terms
+
+  -- To obtain terms with free variables, we add additional nullary
+  -- operators, each representing a variable.
+  --
+  -- These are covered in the standard library FreeMonad module,
+  -- albeit with the restriction that the operator and variable sets
+  -- have the same size.
+
+  Cxt = Sort → Set ℓᵒ
+
+  variable
+    Γ Δ : Cxt
+
+  -- Terms with free variables in Var.
+
+  module _ (Var : Cxt) where
+
+    -- We keep the same sorts, but add a nullary operator for each variable.
+
+    Ops⁺ : Container Sort Sort ℓᵒ ℓᵃ
+    Ops⁺ = Ops ⋆C Var
+
+    -- Terms with variables are then given by the W-type for the extended container.
+
+    Tm = W Ops⁺
+
+    -- We define nice constructors for variables and operator application
+    -- via pattern synonyms.
+    -- Note that the $f$ in constructor var' is a function from the empty set,
+    -- so it should be uniquely determined.  However, Agda's equality is
+    -- more intensional and will not identify all functions from the empty set.
+    -- Since we do not make use of the axiom of function extensionality,
+    -- we sometimes have to consult the extensional equality of the
+    -- function setoid.
+
+    pattern _∙_ op args  = sup (inj₂ op , args)
+    pattern var' x f     = sup (inj₁ x , f    )
+    pattern var x        = var' x _
+
+  -- Letter $t$ ranges over terms, and $\mathit{ts}$ over argument vectors.
+
+  variable
+    t t' t₁ t₂ t₃  :  Tm Γ s
+    ts ts'         :  (i : Arity op) → Tm Γ (sort _ i)
+
+------------------------------------------------------------------------
+-- Parallel substitutions
+
+  -- A substitution from Δ to Γ holds a term in Γ for each variable in Δ.
+
+  Sub : (Γ Δ : Cxt) → Set _
+  Sub Γ Δ = ∀{s} (x : Δ s) → Tm Γ s
+
+  -- Application of a substitution.
+
+  _[_] : (t : Tm Δ s) (σ : Sub Γ Δ) → Tm Γ s
+  (var x  )  [ σ ] = σ x
+  (op ∙ ts)  [ σ ] = op ∙ λ i → ts i [ σ ]
+
+  -- Letter $σ$ ranges over substitutions.
+
+  variable
+    σ σ' : Sub Γ Δ
+
+------------------------------------------------------------------------
+-- Interpretation of terms in a model
+------------------------------------------------------------------------
+
+  -- Given an algebra $M$ of set-size $ℓ^m$ and equality-size $ℓ^e$,
+  -- we define the interpretation of an $s$-sorted term $t$ as element
+  -- of $M(s)$ according to an environment $ρ$ that maps each variable
+  -- of sort $s'$ to an element of $M(s')$.
+
+  module _ {M : SetoidModel ℓᵐ ℓᵉ} where
+    open SetoidModel M
+
+    -- Equality in $M$'s interpretation of sort $s$.
+
+    _≃_ : Den s .Carrier → Den s .Carrier → Set _
+    _≃_ {s = s} = Den s ._≈_
+
+    -- An environment for Γ maps each variable $x : Γ(s)$ to an element of $M(s)$.
+    -- Equality of environments is defined pointwise.
+
+    Env : Cxt → Setoid _ _
+    Env Γ .Carrier   = {s : Sort} (x : Γ s) → Den s .Carrier
+    Env Γ ._≈_ ρ ρ'  = {s : Sort} (x : Γ s) → ρ x ≃ ρ' x
+    Env Γ .isEquivalence .IsEquivalence.refl   {s = s}   x = Den s .Setoid.refl
+    Env Γ .isEquivalence .IsEquivalence.sym       h {s}  x = Den s .Setoid.sym   (h x)
+    Env Γ .isEquivalence .IsEquivalence.trans  g  h {s}  x = Den s .Setoid.trans (g x) (h x)
+
+    -- Interpretation of terms is iteration on the W-type.
+    -- The standard library offers `iter' (on sets), but we need this to be a Func (on setoids).
+
+    ⦅_⦆ : ∀{s} (t : Tm Γ s) → Func (Env Γ) (Den s)
+    ⦅ var x      ⦆ .apply  ρ     = ρ x
+    ⦅ var x      ⦆ .cong   ρ=ρ'  = ρ=ρ' x
+    ⦅ op ∙ args  ⦆ .apply  ρ     = den .apply  (op    , λ i → ⦅ args i ⦆ .apply ρ)
+    ⦅ op ∙ args  ⦆ .cong   ρ=ρ'  = den .cong   (refl  , λ i → ⦅ args i ⦆ .cong ρ=ρ')
+
+    -- An equality between two terms holds in a model
+    -- if the two terms are equal under all valuations of their free variables.
+
+    Equal : ∀ {Γ s} (t t' : Tm Γ s) → Set _
+    Equal {Γ} {s} t t' = ∀ (ρ : Env Γ .Carrier) → ⦅ t ⦆ .apply ρ ≃ ⦅ t' ⦆ .apply ρ
+
+    -- This notion is an equivalence relation.
+
+    isEquiv : IsEquivalence (Equal {Γ = Γ} {s = s})
+    isEquiv {s = s} .IsEquivalence.refl  ρ       = Den s .Setoid.refl
+    isEquiv {s = s} .IsEquivalence.sym   e ρ     = Den s .Setoid.sym (e ρ)
+    isEquiv {s = s} .IsEquivalence.trans e e' ρ  = Den s .Setoid.trans (e ρ) (e' ρ)
+
+------------------------------------------------------------------------
+-- Substitution lemma
+
+    -- Evaluation of a substitution gives an environment.
+
+    ⦅_⦆s : Sub Γ Δ → Env Γ .Carrier → Env Δ .Carrier
+    ⦅ σ ⦆s ρ x = ⦅ σ x ⦆ .apply ρ
+
+    -- Substitution lemma: ⦅t[σ]⦆ρ ≃ ⦅t⦆⦅σ⦆ρ
+
+    substitution : (t : Tm Δ s) (σ : Sub Γ Δ) (ρ : Env Γ .Carrier) →
+      ⦅ t [ σ ] ⦆ .apply ρ ≃ ⦅ t ⦆ .apply (⦅ σ ⦆s ρ)
+    substitution (var x)    σ ρ = Den _ .Setoid.refl
+    substitution (op ∙ ts)  σ ρ = den .cong (refl , λ i → substitution (ts i) σ ρ)
+
+------------------------------------------------------------------------
+-- Equations
+
+  -- An equation is a pair $t ≐ t'$ of terms of the same sort in the same context.
+
+  record Eq : Set (ℓˢ ⊔ suc ℓᵒ ⊔ ℓᵃ) where
+    constructor _≐_
+    field
+      {cxt}  : Sort → Set ℓᵒ
+      {srt}  : Sort
+      lhs    : Tm cxt srt
+      rhs    : Tm cxt srt
+
+  -- Equation $t ≐ t'$ holding in model $M$.
+
+  _⊧_ : (M : SetoidModel ℓᵐ ℓᵉ) (eq : Eq) → Set _
+  M ⊧ (t ≐ t') = Equal {M = M} t t'
+
+  -- Sets of equations are presented as collection E : I → Eq
+  -- for some index set I : Set ℓⁱ.
+
+  -- An entailment/consequence $E ⊃ t ≐ t'$ is valid
+  -- if $t ≐ t'$ holds in all models that satify equations $E$.
+
+  module _ {ℓᵐ ℓᵉ} where
+
+    _⊃_ : (E : I → Eq) (eq : Eq) → Set _
+    E ⊃ eq = ∀ (M : SetoidModel ℓᵐ ℓᵉ) → (∀ i → M ⊧ E i) → M ⊧ eq
+
+  -- Derivations
+  --------------
+
+  -- Equalitional logic allows us to prove entailments via the
+  -- inference rules for the judgment $E ⊢ Γ ▹ t ≡ t'$.
+  -- This could be coined as equational theory over a given
+  -- set of equations $E$.
+  -- Relation $E ⊢ Γ ▹ \_ ≡ \_$ is the least congruence over the equations $E$.
+
+  data _⊢_▹_≡_ {I : Set ℓⁱ}
+    (E : I → Eq) : (Γ : Cxt) (t t' : Tm Γ s) → Set (ℓˢ ⊔ suc ℓᵒ ⊔ ℓᵃ ⊔ ℓⁱ) where
+
+    hyp    :  ∀ i → let t ≐ t' = E i in
+              E ⊢ _ ▹ t ≡ t'
+
+    base   :  ∀ (x : Γ s) {f f' : (i : ⊥) → Tm _ (⊥-elim i)} →
+              E ⊢ Γ ▹ var' x f ≡ var' x f'
+
+    app    :  (∀ i → E ⊢ Γ ▹ ts i ≡ ts' i) →
+              E ⊢ Γ ▹ (op ∙ ts) ≡ (op ∙ ts')
+
+    sub    :  E ⊢ Δ ▹ t ≡ t' →
+              ∀ (σ : Sub Γ Δ) →
+              E ⊢ Γ ▹ (t [ σ ]) ≡ (t' [ σ ])
+
+    refl   :  ∀ (t : Tm Γ s) →
+              E ⊢ Γ ▹ t ≡ t
+
+    sym    :  E ⊢ Γ ▹ t ≡ t' →
+              E ⊢ Γ ▹ t' ≡ t
+
+    trans  :  E ⊢ Γ ▹ t₁ ≡ t₂ →
+              E ⊢ Γ ▹ t₂ ≡ t₃ →
+              E ⊢ Γ ▹ t₁ ≡ t₃
+
+------------------------------------------------------------------------
+-- Soundness of the inference rules
+
+  -- We assume a model $M$ that validates all equations in $E$.
+
+  module Soundness {I : Set ℓⁱ} (E : I → Eq) (M : SetoidModel ℓᵐ ℓᵉ)
+    (V : ∀ i → M ⊧ E i) where
+    open SetoidModel M
+
+    -- In any model $M$ that satisfies the equations $E$,
+    -- derived equality is actual equality.
+
+    sound : E ⊢ Γ ▹ t ≡ t' → M ⊧ (t ≐ t')
+
+    sound (hyp i)                        =  V i
+    sound (app {op = op} es) ρ           =  den .cong (refl , λ i → sound (es i) ρ)
+    sound (sub {t = t} {t' = t'} e σ) ρ  =  begin
+      ⦅ t [ σ ]   ⦆ .apply ρ   ≈⟨ substitution {M = M} t σ ρ ⟩
+      ⦅ t         ⦆ .apply ρ'  ≈⟨ sound e ρ' ⟩
+      ⦅ t'        ⦆ .apply ρ'  ≈⟨ substitution {M = M} t' σ ρ ⟨
+      ⦅ t' [ σ ]  ⦆ .apply ρ   ∎
+      where
+      open SetoidReasoning (Den _)
+      ρ' = ⦅ σ ⦆s ρ
+
+    sound (base x {f} {f'})              =  isEquiv {M = M} .IsEquivalence.refl {var' x λ()}
+
+    sound (refl t)                       =  isEquiv {M = M} .IsEquivalence.refl {t}
+    sound (sym {t = t} {t' = t'} e)      =  isEquiv {M = M} .IsEquivalence.sym
+                                            {x = t} {y = t'} (sound e)
+    sound (trans  {t₁ = t₁} {t₂ = t₂}
+                  {t₃ = t₃} e e')        =  isEquiv {M = M} .IsEquivalence.trans
+                                            {i = t₁} {j = t₂} {k = t₃} (sound e) (sound e')
+
+
+------------------------------------------------------------------------
+-- Birkhoff's completeness theorem
+------------------------------------------------------------------------
+
+  -- Birkhoff proved that any equation $t ≐ t'$ is derivable from $E$
+  -- when it is valid in all models satisfying $E$.  His proof (for
+  -- single-sorted algebras) is a blue print for many more
+  -- completeness proofs.  They all proceed by constructing a
+  -- universal model aka term model.  In our case, it is terms
+  -- quotiented by derivable equality $E ⊢ Γ ▹ \_ ≡ \_$.  It then
+  -- suffices to prove that this model satisfies all equations in $E$.
+
+------------------------------------------------------------------------
+-- Universal model
+
+  -- A term model for $E$ and $Γ$ interprets sort $s$ by (Tm Γ s) quotiented by $E ⊢ Γ ▹ \_ ≡ \_$.
+
+  module TermModel {I : Set ℓⁱ} (E : I → Eq) where
+    open SetoidModel
+
+    -- Tm Γ s quotiented by E⊢Γ▹·≡·.
+
+    TmSetoid : Cxt → Sort → Setoid _ _
+    TmSetoid Γ s .Carrier                             = Tm Γ s
+    TmSetoid Γ s ._≈_                                 = E ⊢ Γ ▹_≡_
+    TmSetoid Γ s .isEquivalence .IsEquivalence.refl   = refl _
+    TmSetoid Γ s .isEquivalence .IsEquivalence.sym    = sym
+    TmSetoid Γ s .isEquivalence .IsEquivalence.trans  = trans
+
+    -- The interpretation of an operator is simply the operator.
+    -- This works because $E⊢Γ▹\_≡\_$ is a congruence.
+
+    tmInterp : ∀ {Γ s} → Func (⟦ Ops ⟧s (TmSetoid Γ) s) (TmSetoid Γ s)
+    tmInterp .apply (op , ts) = op ∙ ts
+    tmInterp .cong (refl , h) = app h
+
+    -- The term model per context Γ.
+
+    M : Cxt → SetoidModel _ _
+    M Γ .Den = TmSetoid Γ
+    M Γ .den = tmInterp
+
+    -- The identity substitution σ₀ maps variables to themselves.
+
+    σ₀ : {Γ : Cxt} → Sub Γ Γ
+    σ₀ x = var' x  λ()
+
+    -- σ₀ acts indeed as identity.
+
+    identity : (t : Tm Γ s) → E ⊢ Γ ▹ t [ σ₀ ] ≡ t
+    identity (var x)    = base x
+    identity (op ∙ ts)  = app λ i → identity (ts i)
+
+    -- Evaluation in the term model is substitution $E ⊢ Γ ▹ ⦅t⦆σ ≡ t[σ]$.
+    -- This would even hold "up to the nose" if we had function extensionality.
+
+    evaluation : (t : Tm Δ s) (σ : Sub Γ Δ) → E ⊢ Γ ▹ (⦅_⦆ {M = M Γ} t .apply σ) ≡ (t [ σ ])
+    evaluation (var x)    σ = refl (σ x)
+    evaluation (op ∙ ts)  σ = app (λ i → evaluation (ts i) σ)
+
+    -- The term model satisfies all the equations it started out with.
+
+    satisfies : ∀ i → M Γ ⊧ E i
+    satisfies i σ = begin
+      ⦅ tₗ ⦆ .apply σ  ≈⟨ evaluation tₗ σ ⟩
+      tₗ [ σ ]         ≈⟨ sub (hyp i) σ ⟩
+      tᵣ [ σ ]         ≈⟨ evaluation tᵣ σ ⟨
+      ⦅ tᵣ ⦆ .apply σ  ∎
+      where
+      open SetoidReasoning (TmSetoid _ _)
+      tₗ  = E i .Eq.lhs
+      tᵣ = E i .Eq.rhs
+
+------------------------------------------------------------------------
+-- Completeness
+
+-- Birkhoff's completeness theorem \citeyearpar{birkhoff:1935}:
+-- Any valid consequence is derivable in the equational theory.
+
+  module Completeness {I : Set ℓⁱ} (E : I → Eq) {Γ s} {t t' : Tm Γ s} where
+    open TermModel E
+
+    completeness : E ⊃ (t ≐ t') → E ⊢ Γ ▹ t ≡ t'
+    completeness V =     begin
+      t                  ≈˘⟨ identity t ⟩
+      t  [ σ₀ ]          ≈˘⟨ evaluation t σ₀ ⟩
+      ⦅ t   ⦆ .apply σ₀  ≈⟨ V (M Γ) satisfies σ₀ ⟩
+      ⦅ t'  ⦆ .apply σ₀  ≈⟨ evaluation t' σ₀ ⟩
+      t' [ σ₀ ]          ≈⟨ identity t' ⟩
+      t'                 ∎
+      where open SetoidReasoning (TmSetoid Γ s)
diff --git a/doc/README/Data/Container/Indexed.agda b/doc/README/Data/Container/Indexed/VectorExample.agda
similarity index 89%
rename from doc/README/Data/Container/Indexed.agda
rename to doc/README/Data/Container/Indexed/VectorExample.agda
index adf6cbc9e5..45d45cc740 100644
--- a/doc/README/Data/Container/Indexed.agda
+++ b/doc/README/Data/Container/Indexed/VectorExample.agda
@@ -6,7 +6,7 @@
 
 {-# OPTIONS --with-K --safe --guardedness #-}
 
-module README.Data.Container.Indexed where
+module README.Data.Container.Indexed.VectorExample where
 
 open import Data.Unit
 open import Data.Empty
@@ -27,12 +27,14 @@ module _ {a} (A : Set a) where
 -- 2. Responses the indexed container returns to these commands
 -- 3. Update of the index based on the command and the response issued.
 
--- For a vector, commands are constructors, responses are the number of subvectors
--- (0 if the vector is empty, 1 otherwise) and the update corresponds to setting the
--- size of the tail (if it exists). We can formalize these ideas like so:
+-- For a vector, commands are constructors, responses are the number
+-- of subvectors (0 if the vector is empty, 1 otherwise) and the
+-- update corresponds to setting the size of the tail (if it exists).
+-- We can formalize these ideas like so:
 
--- Depending on the size of the vector, we may have reached the end already (nil)
--- or we may specify what the head should be (cons). This is the type of commands.
+-- Depending on the size of the vector, we may have reached the end
+-- already (nil) or we may specify what the head should be (cons).
+-- This is the type of commands.
 
   data VecC : ℕ → Set a where
     nil  :           VecC zero
diff --git a/src/Data/Container/Indexed/Core.agda b/src/Data/Container/Indexed/Core.agda
index 75d1a0d986..04175781d9 100644
--- a/src/Data/Container/Indexed/Core.agda
+++ b/src/Data/Container/Indexed/Core.agda
@@ -12,34 +12,46 @@ open import Level
 open import Data.Product.Base using (Σ; Σ-syntax; _,_; ∃)
 open import Relation.Unary
 
+private variable
+  i o c r ℓ ℓ′ : Level
+  I : Set i
+  O : Set o
+
+------------------------------------------------------------------------
+-- Definition
+
 infix 5 _◃_/_
 
-record Container {i o} (I : Set i) (O : Set o) (c r : Level) :
-                 Set (i ⊔ o ⊔ suc c ⊔ suc r) where
+record Container (I : Set i) (O : Set o) c r : Set (i ⊔ o ⊔ suc c ⊔ suc r) where
   constructor _◃_/_
   field
     Command  : (o : O) → Set c
     Response : ∀ {o} → Command o → Set r
     next     : ∀ {o} (c : Command o) → Response c → I
 
+------------------------------------------------------------------------
 -- The semantics ("extension") of an indexed container.
 
-⟦_⟧ : ∀ {i o c r ℓ} {I : Set i} {O : Set o} → Container I O c r →
-      Pred I ℓ → Pred O _
-⟦ C ◃ R / n ⟧ X o = Σ[ c ∈ C o ] ((r : R c) → X (n c r))
+module _ (C : Container I O c r) (X : Pred I ℓ) where
+  open Container C
+
+  Subtrees : ∀ o → Command o → Set _
+  Subtrees o c = (r : Response c) → X (next c r)
+
+  ⟦_⟧ : Pred O _
+  ⟦_⟧ o = Σ (Command o) (Subtrees o)
 
 ------------------------------------------------------------------------
 -- All and any
 
-module _ {i o c r ℓ} {I : Set i} {O : Set o}
-         (C : Container I O c r) {X : Pred I ℓ} where
+module _ (C : Container I O c r) {X : Pred I ℓ} where
 
   -- All.
 
-  □ : ∀ {ℓ′} → Pred (Σ I X) ℓ′ → Pred (Σ O (⟦ C ⟧ X)) _
+  □ : Pred (Σ I X) ℓ′ → Pred (Σ O (⟦ C ⟧ X)) _
   □ P (_ , _ , k) = ∀ r → P (_ , k r)
 
   -- Any.
 
-  ◇ : ∀ {ℓ′} → Pred (Σ I X) ℓ′ → Pred (Σ O (⟦ C ⟧ X)) _
+  ◇ : Pred (Σ I X) ℓ′ → Pred (Σ O (⟦ C ⟧ X)) _
   ◇ P (_ , _ , k) = ∃ λ r → P (_ , k r)
diff --git a/src/Data/Container/Indexed/Relation/Binary/Equality/Setoid.agda b/src/Data/Container/Indexed/Relation/Binary/Equality/Setoid.agda
new file mode 100644
index 0000000000..8e259e90f7
--- /dev/null
+++ b/src/Data/Container/Indexed/Relation/Binary/Equality/Setoid.agda
@@ -0,0 +1,60 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Equality over indexed container extensions parametrised by a setoid
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary using (Setoid)
+
+module Data.Container.Indexed.Relation.Binary.Equality.Setoid
+  {ℓⁱ ℓᶜ ℓᵉ} {I : Set ℓⁱ} (S : I → Setoid ℓᶜ ℓᵉ)
+  where
+
+open import Function
+open import Level using (Level; _⊔_; suc)
+open import Relation.Binary
+
+open import Data.Container.Indexed.Core
+open import Data.Container.Indexed.Relation.Binary.Pointwise
+import Data.Container.Indexed.Relation.Binary.Pointwise.Properties
+  as Pointwise
+
+open Setoid using (Carrier; _≈_)
+
+private variable
+  ℓˢ ℓᵖ : Level
+  O : Set _
+
+------------------------------------------------------------------------
+-- Definition of equality
+
+module _ (C : Container I O ℓˢ ℓᵖ) (o : O) where
+
+  Eq : Rel (⟦ C ⟧ (Carrier ∘ S) o) (ℓᵉ ⊔ ℓˢ ⊔ ℓᵖ)
+  Eq = Pointwise C (_≈_ ∘ S) o
+
+------------------------------------------------------------------------
+-- Relational properties
+
+  refl : Reflexive Eq
+  refl = Pointwise.refl C _ (Setoid.refl ∘ S)
+
+  sym : Symmetric Eq
+  sym = Pointwise.sym C _ (Setoid.sym ∘ S)
+
+  trans : Transitive Eq
+  trans = Pointwise.trans C _ (Setoid.trans ∘ S)
+
+  isEquivalence : IsEquivalence Eq
+  isEquivalence = record
+    { refl  = refl
+    ; sym   = sym
+    ; trans = trans
+    }
+
+  setoid : Setoid (ℓˢ ⊔ ℓᵖ ⊔ ℓᶜ) (ℓˢ ⊔ ℓᵖ ⊔ ℓᵉ)
+  setoid = record
+    { isEquivalence = isEquivalence
+    }
diff --git a/src/Data/Container/Indexed/Relation/Binary/Pointwise.agda b/src/Data/Container/Indexed/Relation/Binary/Pointwise.agda
new file mode 100644
index 0000000000..afaba23f09
--- /dev/null
+++ b/src/Data/Container/Indexed/Relation/Binary/Pointwise.agda
@@ -0,0 +1,58 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Pointwise equality for indexed containers
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Container.Indexed.Relation.Binary.Pointwise where
+
+open import Data.Product using (_,_; Σ-syntax)
+open import Function
+open import Level using (Level; _⊔_)
+open import Relation.Binary using (REL; _⇒_)
+open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+
+open import Data.Container.Indexed.Core using (Container; Subtrees; ⟦_⟧)
+
+private variable
+  ℓᵉ ℓᵉ′ ℓᵖ ℓˢ ℓˣ ℓʸ : Level
+  I O : Set _
+
+------------------------------------------------------------------------
+-- Equality, parametrised on an underlying relation.
+
+-- Since ⟦_⟧ is a Σ-type, not a record, I'd say Pointwise should also be
+-- a Σ-type, not a record. Maybe we need to update module
+-- `Data.Container.Relation.Binary.Pointwise` accordingly...
+--
+-- record Pointwise  : Set (ℓˢ ⊔ ℓᵖ ⊔ ℓᵉ) where
+--   constructor _,_
+--   field shape    : c ≡ c'
+--         position : Eqs shape xs ys
+
+module _ (C : Container I O ℓˢ ℓᵖ)
+         {X : I → Set ℓˣ} {Y : I → Set ℓʸ} (R : (i : I) → REL (X i) (Y i) ℓᵉ)
+         (o : O)
+         ((c  , xs) : ⟦ C ⟧ X o)
+         ((c' , ys) : ⟦ C ⟧ Y o)
+  where
+  open Container C
+
+  Eqs : c ≡ c' → Subtrees C X o c → Subtrees C Y o c' → Set _
+  Eqs refl xs ys = (r : Response c) → R (next c r) (xs r) (ys r)
+
+  Pointwise = Σ[ eq ∈ c ≡ c' ] Eqs eq xs ys
+
+------------------------------------------------------------------------
+-- Operations
+
+module _ {C : Container I O ℓˢ ℓᵖ}
+         {X : I → Set ℓˣ} {Y : I → Set ℓʸ}
+         {R  : (i : I) → REL (X i) (Y i) ℓᵉ}
+         {R′ : (i : I) → REL (X i) (Y i) ℓᵉ′}
+         where
+
+  map : (R⇒R′ : ∀ i → R i ⇒ R′ i) {o : O} → Pointwise C R o ⇒ Pointwise C R′ o
+  map R⇒R′ (refl , f) = refl , R⇒R′ _ ∘ f
diff --git a/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
new file mode 100644
index 0000000000..58e6966154
--- /dev/null
+++ b/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
@@ -0,0 +1,45 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties of pointwise equality for indexed containers
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Container.Indexed.Relation.Binary.Pointwise.Properties where
+
+open import Axiom.Extensionality.Propositional
+open import Data.Container.Indexed.Core
+open import Data.Container.Indexed.Relation.Binary.Pointwise
+open import Data.Product using (_,_; Σ-syntax; -,_)
+open import Level using (Level; _⊔_)
+open import Relation.Binary
+open import Relation.Binary.PropositionalEquality as P
+  using (_≡_; subst; cong)
+
+private variable
+  ℓᵉ ℓᵖ ℓˢ ℓˣ : Level
+  I O : Set _
+
+module _
+  (C : Container I O ℓˢ ℓᵖ) {X : I → Set ℓˣ}
+  (R : (i : I) → Rel (X i) ℓᵉ)
+  {o : O}
+  where
+
+  refl : (∀ i → Reflexive (R i)) → Reflexive (Pointwise C R o)
+  refl R-refl = P.refl , λ p → R-refl _
+
+  sym : (∀ i → Symmetric (R i)) → Symmetric (Pointwise C R o)
+  sym R-sym (P.refl , f) = P.refl , λ p → R-sym _ (f p)
+
+  trans : (∀ i → Transitive (R i)) → Transitive (Pointwise C R o)
+  trans R-trans (P.refl , f) (P.refl , g) = P.refl , λ p → R-trans _ (f p) (g p)
+
+-- If propositional equality is extensional, then `Eq _≡_` and `_≡_` coincide.
+Eq⇒≡ : {C : Container I O ℓˢ ℓᵖ} {X : I → Set ℓˣ} {R : (i : I) → Rel (X i) ℓᵉ}
+       {o : O} {xs ys : ⟦ C ⟧ X o} →
+       Extensionality ℓᵖ ℓˣ →
+       Pointwise C (λ (i : I) → _≡_ {A = X i}) o xs ys →
+       xs ≡ ys
+Eq⇒≡ ext (P.refl , f≈f′) = cong -,_ (ext f≈f′)

From 56804582c010021d2719e07078c765eb6041e375 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 16 Mar 2024 03:40:38 +0000
Subject: [PATCH 243/316] 'No infinite descent' for (`Acc`essible elements of)
 `WellFounded` relations (#2126)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* basic result

* added missing lemma; is there a better name for this?

* renamed lemma

* final tidying up; `CHANGELOG`

* final tidying up

* missing `CHANGELOG` entry

* `InfiniteDescent` definition and proof

* revised `InfiniteDescent` definition and proof

* major revision: renames things, plus additional corollaries

* spacing

* added `NoSmallestCounterExample` principles for `Stable` predicates

* refactoring; move `Stable` to `Relation.Unary`

* refactoring; remove explicit definition of `CounterExample`

* refactoring; rename qualified `import`

* [fixes #2130] Moving `Properties.HeytingAlgebra` from `Relation.Binary` to `Relation.Binary.Lattice` (#2131)

* [fixes #2127] Fixes #1930 `import` bug (#2128)

* [fixes #1214] Add negated ordering relation symbols systematically to `Relation.Binary.*` (#2095)

* Refactoring (inversion) properties of `_<_` on `Nat`, plus consequences (#2000)

* Bump CI tests to Agda-2.6.4 (#2134)

* Remove `Algebra.Ordered` (#2133)

* [ fix ] missing name in LICENCE file (#2139)

* Add new blocking primitives to `Reflection.TCM` (#1972)

* Change definition of `IsStrictTotalOrder` (#2137)

* Add _<$>_ operator for Function bundle (#2144)

* [ fix #2086 ] new web deployment strategy (#2147)

* [ admin ] fix sorting logic (#2151)

With the previous script we were sorting entries of the form
html/vX.Y.Z/index.html but the order is such that vX.Y/ < vX.Y.Z/
and so we were ending up with v1.7 coming after v1.7.3.

This fixes that by using sed to get rid of the html/ prefix
and the /index.html suffix before the sorting phase.

* Add coincidence law to modules (#1898)

* Make reasoning modular by adding new `Reasoning.Syntax` module (#2152)

* Fixes typos identified in #2154 (#2158)

* tackles #2124 as regards `case_return_of_` (#2157)

* Rename preorder ~ relation reasoning combinators (#2156)

* Move ≡-Reasoning from Core to Properties and implement using syntax (#2159)

* Add consistent deprecation warnings to old function hierarchy (#2163)

* Rename symmetric reasoning combinators. Minimal set of changes (#2160)

* Restore 'return' as an alias for 'pure' (#2164)

* [ fix #2153 ] Properly re-export specialised combinators (#2161)

* Connect `LeftInverse` with (`Split`)`Surjection` (#2054)

* Added remaining flipped and negated relations to binary relation bundles (#2162)

* Tidy up CHANGELOG in preparation for release candidate (#2165)

* Spellcheck CHANGELOG (#2167)

* spellcheck; `fix-whitespace`; unfixed: a few alignment issues; typo `predications` to `predicates` in `Relation.Unary.Relation.Binary.Equality`

* Fixed Agda version typo in README (#2176)

* Fixed in deprecation warning for `<-transˡ` (#2173)

* Bump Haskell CI (original!) to latest minor GHC versions (#2187)

* [fixes #2175] Documentation misc. typos etc. for RC1  (#2183)

* missing comma!

* corrected reference to `README.Design.Decidability`

* typo: capitalisation

* updated `installation-guide`

* added word to `NonZero` section heading

* Run workflows on any PR

* Add merge-group trigger to workflows

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>

* [fixes #2178] regularise and specify/systematise, the conventions for symbol usage (#2185)

* regularise and specify/systematise, the conventions for symbol usage

* typos/revisions

* Move `T?` to `Relation.Nullary.Decidable.Core` (#2189)

* Move `T?` to `Relation.Nullary.Decidable.Core`

* Var name fix

* Some refactoring

* Fix failing tests and address remaining comments

* Fix style-guide

* Make decidable versions of sublist functions the default (#2186)

* Make decdable versions of sublist functions the default

* Remove imports Bool.Properties

* Address comments

* new `CHANGELOG`

* resolved merge conflict

* resolved merge conflict, this time

* revised in line with review comments

* tweaked `export`s; does that fix things now?

* Fix merge mistake

* Refactored to remove a indirection and nested modules

* Touch up names

* Reintroduce anonymous module for descent

* cosmetics asper final comment

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
Co-authored-by: Amélia <me@amelia.how>
Co-authored-by: Nathan van Doorn <nvd1234@gmail.com>
Co-authored-by: James Wood <laMudri@gmail.com>
Co-authored-by: Andreas Abel <andreas.abel@ifi.lmu.de>
---
 CHANGELOG.md                                  |  14 ++
 src/Induction/InfiniteDescent.agda            | 189 ++++++++++++++++++
 .../Binary/Construct/Closure/Transitive.agda  |   4 +
 src/Relation/Unary.agda                       |   5 +-
 4 files changed, 210 insertions(+), 2 deletions(-)
 create mode 100644 src/Induction/InfiniteDescent.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 343460e8c2..6ad8f21635 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -50,6 +50,11 @@ New modules
 -----------
 * `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
 
+* Consequences of 'infinite descent' for (accessible elements of) well-founded relations:
+  ```agda
+  Induction.InfiniteDescent
+  ```
+
 * The unique morphism from the initial, resp. terminal, algebra:
   ```agda
   Algebra.Morphism.Construct.Initial
@@ -254,6 +259,15 @@ Additions to existing modules
 * Added new functions in `Data.String.Base`:
   ```agda
   map : (Char → Char) → String → String
+
+* Added new definition in `Relation.Binary.Construct.Closure.Transitive`
+  ```
+  transitive⁻ : Transitive _∼_ → TransClosure _∼_ ⇒ _∼_
+  ```
+
+* Added new definition in `Relation.Unary`
+  ```
+  Stable : Pred A ℓ → Set _
   ```
 
 * In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
diff --git a/src/Induction/InfiniteDescent.agda b/src/Induction/InfiniteDescent.agda
new file mode 100644
index 0000000000..ed25c880d5
--- /dev/null
+++ b/src/Induction/InfiniteDescent.agda
@@ -0,0 +1,189 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- A standard consequence of accessibility/well-foundedness:
+-- the impossibility of 'infinite descent' from any (accessible)
+-- element x satisfying P to 'smaller' y also satisfying P
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Induction.InfiniteDescent where
+
+open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
+open import Data.Nat.Properties as ℕ
+open import Data.Product.Base using (_,_; proj₁; ∃-syntax; _×_)
+open import Function.Base using (_∘_)
+open import Induction.WellFounded
+  using (WellFounded; Acc; acc; acc-inverse; module Some)
+open import Level using (Level)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Construct.Closure.Transitive
+open import Relation.Binary.PropositionalEquality.Core
+open import Relation.Nullary.Negation.Core as Negation using (¬_)
+open import Relation.Unary
+  using (Pred; ∁; _∩_; _⊆_;  _⇒_; Universal; IUniversal; Stable; Empty)
+
+private
+  variable
+    a r ℓ : Level
+    A : Set a
+    f : ℕ → A
+    _<_ : Rel A r
+    P : Pred A ℓ
+
+------------------------------------------------------------------------
+-- Definitions
+
+InfiniteDescendingSequence : Rel A r → (ℕ → A) → Set _
+InfiniteDescendingSequence _<_ f = ∀ n → f (suc n) < f n
+
+InfiniteDescendingSequenceFrom : Rel A r → (ℕ → A) → Pred A _
+InfiniteDescendingSequenceFrom _<_ f x = f zero ≡ x × InfiniteDescendingSequence _<_ f
+
+InfiniteDescendingSequence⁺ : Rel A r → (ℕ → A) → Set _
+InfiniteDescendingSequence⁺ _<_ f = ∀ {m n} → m ℕ.< n → TransClosure _<_ (f n) (f m)
+
+InfiniteDescendingSequenceFrom⁺ : Rel A r → (ℕ → A) → Pred A _
+InfiniteDescendingSequenceFrom⁺ _<_ f x = f zero ≡ x × InfiniteDescendingSequence⁺ _<_ f
+
+DescentFrom : Rel A r → Pred A ℓ → Pred A _
+DescentFrom _<_ P x = P x → ∃[ y ] y < x × P y
+
+Descent : Rel A r → Pred A ℓ → Set _
+Descent _<_ P = ∀ {x} → DescentFrom _<_ P x
+
+InfiniteDescentFrom : Rel A r → Pred A ℓ → Pred A _
+InfiniteDescentFrom _<_ P x = P x → ∃[ f ] InfiniteDescendingSequenceFrom _<_ f x × ∀ n → P (f n)
+
+InfiniteDescent : Rel A r → Pred A ℓ →  Set _
+InfiniteDescent _<_ P = ∀ {x} → InfiniteDescentFrom _<_ P x
+
+InfiniteDescentFrom⁺ : Rel A r → Pred A ℓ → Pred A _
+InfiniteDescentFrom⁺ _<_ P x = P x → ∃[ f ] InfiniteDescendingSequenceFrom⁺ _<_ f x × ∀ n → P (f n)
+
+InfiniteDescent⁺ : Rel A r → Pred A ℓ → Set _
+InfiniteDescent⁺ _<_ P = ∀ {x} → InfiniteDescentFrom⁺ _<_ P x
+
+NoSmallestCounterExample : Rel A r → Pred A ℓ →  Set _
+NoSmallestCounterExample _<_ P = ∀ {x} → Acc _<_ x → DescentFrom (TransClosure _<_) (∁ P) x
+
+------------------------------------------------------------------------
+-- We can swap between transitively closed and non-transitively closed
+-- definitions
+
+sequence⁺ : InfiniteDescendingSequence (TransClosure _<_) f →
+            InfiniteDescendingSequence⁺ _<_ f
+sequence⁺ {_<_ = _<_} {f = f} seq[f] = seq⁺[f]′ ∘ ℕ.<⇒<′
+  where
+  seq⁺[f]′ : ∀ {m n} → m ℕ.<′ n → TransClosure _<_ (f n) (f m)
+  seq⁺[f]′ ℕ.<′-base        = seq[f] _
+  seq⁺[f]′ (ℕ.<′-step m<′n) = seq[f] _ ++ seq⁺[f]′ m<′n
+
+sequence⁻ : InfiniteDescendingSequence⁺ _<_ f →
+            InfiniteDescendingSequence (TransClosure _<_) f
+sequence⁻ seq[f] = seq[f] ∘ n<1+n
+
+------------------------------------------------------------------------
+-- Results about unrestricted descent
+
+module _ (descent : Descent _<_ P) where
+
+  descent∧acc⇒infiniteDescentFrom : (Acc _<_) ⊆ (InfiniteDescentFrom _<_ P)
+  descent∧acc⇒infiniteDescentFrom {x} =
+    Some.wfRec (InfiniteDescentFrom _<_ P) rec x
+    where
+    rec : _
+    rec y rec[y] py
+      with z , z<y , pz ← descent py
+      with g , (g0≡z , g<P) , Π[P∘g] ← rec[y] z<y pz
+         = h , (h0≡y , h<P) , Π[P∘h]
+      where
+      h : ℕ → _
+      h zero = y
+      h (suc n) = g n
+
+      h0≡y : h zero ≡ y
+      h0≡y = refl
+
+      h<P : ∀ n → h (suc n) < h n
+      h<P zero rewrite g0≡z = z<y
+      h<P (suc n)           = g<P n
+
+      Π[P∘h] : ∀ n →  P (h n)
+      Π[P∘h] zero rewrite g0≡z = py
+      Π[P∘h] (suc n)           = Π[P∘g] n
+
+  descent∧wf⇒infiniteDescent : WellFounded _<_ → InfiniteDescent _<_ P
+  descent∧wf⇒infiniteDescent wf = descent∧acc⇒infiniteDescentFrom (wf _)
+
+  descent∧acc⇒unsatisfiable : Acc _<_ ⊆ ∁ P
+  descent∧acc⇒unsatisfiable {x} = Some.wfRec (∁ P) rec x
+    where
+    rec : _
+    rec y rec[y] py = let z , z<y , pz = descent py in rec[y] z<y pz
+
+  descent∧wf⇒empty : WellFounded _<_ → Empty P
+  descent∧wf⇒empty wf x = descent∧acc⇒unsatisfiable (wf x)
+
+------------------------------------------------------------------------
+-- Results about descent only from accessible elements
+
+module _ (accDescent : Acc _<_ ⊆ DescentFrom _<_ P) where
+
+  private
+    descent∩ : Descent _<_ (P ∩ Acc _<_)
+    descent∩ (px , acc[x]) =
+      let y , y<x , py = accDescent acc[x] px
+      in  y , y<x , py , acc-inverse acc[x] y<x
+
+  accDescent∧acc⇒infiniteDescentFrom : Acc _<_ ⊆ InfiniteDescentFrom _<_ P
+  accDescent∧acc⇒infiniteDescentFrom acc[x] px =
+    let f , sequence[f] , Π[[P∩Acc]∘f] = descent∧acc⇒infiniteDescentFrom descent∩ acc[x] (px , acc[x])
+    in f , sequence[f] , proj₁ ∘ Π[[P∩Acc]∘f]
+
+  accDescent∧wf⇒infiniteDescent : WellFounded _<_ → InfiniteDescent _<_ P
+  accDescent∧wf⇒infiniteDescent wf = accDescent∧acc⇒infiniteDescentFrom (wf _)
+
+  accDescent∧acc⇒unsatisfiable : Acc _<_ ⊆ ∁ P
+  accDescent∧acc⇒unsatisfiable acc[x] px = descent∧acc⇒unsatisfiable descent∩ acc[x] (px , acc[x])
+
+  wf⇒empty : WellFounded _<_ → Empty P
+  wf⇒empty wf x = accDescent∧acc⇒unsatisfiable (wf x)
+
+------------------------------------------------------------------------
+-- Results about transitively-closed descent only from accessible elements
+
+module _ (accDescent⁺ : Acc _<_ ⊆ DescentFrom (TransClosure _<_) P) where
+
+  private
+    descent : Acc (TransClosure _<_) ⊆ DescentFrom (TransClosure _<_) P
+    descent = accDescent⁺  ∘ accessible⁻ _
+
+  accDescent⁺∧acc⇒infiniteDescentFrom⁺ : Acc _<_ ⊆ InfiniteDescentFrom⁺ _<_ P
+  accDescent⁺∧acc⇒infiniteDescentFrom⁺ acc[x] px
+    with f , (f0≡x , sequence[f]) , Π[P∘f]
+       ← accDescent∧acc⇒infiniteDescentFrom descent (accessible _ acc[x]) px
+       = f , (f0≡x , sequence⁺ sequence[f]) , Π[P∘f]
+
+  accDescent⁺∧wf⇒infiniteDescent⁺ : WellFounded _<_ → InfiniteDescent⁺ _<_ P
+  accDescent⁺∧wf⇒infiniteDescent⁺ wf = accDescent⁺∧acc⇒infiniteDescentFrom⁺ (wf _)
+
+  accDescent⁺∧acc⇒unsatisfiable : Acc _<_ ⊆ ∁ P
+  accDescent⁺∧acc⇒unsatisfiable = accDescent∧acc⇒unsatisfiable descent ∘ accessible _
+
+  accDescent⁺∧wf⇒empty : WellFounded _<_ → Empty P
+  accDescent⁺∧wf⇒empty = wf⇒empty descent ∘ (wellFounded _)
+
+------------------------------------------------------------------------
+-- Finally: the (classical) no smallest counterexample principle itself
+
+module _ (stable : Stable P) (noSmallest : NoSmallestCounterExample _<_ P) where
+
+  noSmallestCounterExample∧acc⇒satisfiable : Acc _<_ ⊆ P
+  noSmallestCounterExample∧acc⇒satisfiable =
+    stable _ ∘ accDescent⁺∧acc⇒unsatisfiable noSmallest
+
+  noSmallestCounterExample∧wf⇒universal : WellFounded _<_ → Universal P
+  noSmallestCounterExample∧wf⇒universal wf =
+    stable _ ∘ accDescent⁺∧wf⇒empty noSmallest wf
diff --git a/src/Relation/Binary/Construct/Closure/Transitive.agda b/src/Relation/Binary/Construct/Closure/Transitive.agda
index 3c91ce2318..fde3fe17a3 100644
--- a/src/Relation/Binary/Construct/Closure/Transitive.agda
+++ b/src/Relation/Binary/Construct/Closure/Transitive.agda
@@ -70,6 +70,10 @@ module _ (_∼_ : Rel A ℓ) where
   transitive : Transitive _∼⁺_
   transitive = _++_
 
+  transitive⁻ : Transitive _∼_ → _∼⁺_ ⇒ _∼_
+  transitive⁻ trans [ x∼y ]      = x∼y
+  transitive⁻ trans (x∼y ∷ x∼⁺y) = trans x∼y (transitive⁻ trans x∼⁺y)
+
   accessible⁻ : ∀ {x} → Acc _∼⁺_ x → Acc _∼_ x
   accessible⁻ = ∼⊆∼⁺.accessible
 
diff --git a/src/Relation/Unary.agda b/src/Relation/Unary.agda
index 357237a2be..9f3ea16633 100644
--- a/src/Relation/Unary.agda
+++ b/src/Relation/Unary.agda
@@ -14,7 +14,8 @@ open import Data.Product.Base using (_×_; _,_; Σ-syntax; ∃; uncurry; swap)
 open import Data.Sum.Base using (_⊎_; [_,_])
 open import Function.Base using (_∘_; _|>_)
 open import Level using (Level; _⊔_; 0ℓ; suc; Lift)
-open import Relation.Nullary as Nullary using (¬_; Dec; True)
+open import Relation.Nullary.Decidable.Core using (Dec; True)
+open import Relation.Nullary as Nullary using (¬_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 private
@@ -173,7 +174,7 @@ Irrelevant P = ∀ {x} → Nullary.Irrelevant (P x)
 Recomputable : Pred A ℓ → Set _
 Recomputable P = ∀ {x} → Nullary.Recomputable (P x)
 
--- Weak Decidability
+-- Stability - instances of P are stable wrt double negation
 
 Stable : Pred A ℓ → Set _
 Stable P = ∀ x → Nullary.Stable (P x)

From 9a4453e4b6e80057102f5e23ac93e76353564a66 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 16 Mar 2024 06:11:13 +0000
Subject: [PATCH 244/316] Add new operations (cf. `RawQuasigroup`) to `IsGroup`
 (#2251)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* added new operations to `IsGroup`

* fixed notations

* fixed `CHANGELOG`

* refactoring `Group` properties: added `isQuasigroup` and `isLoop`

* refactoring `*-helper` lemmas

* fixed `CHANGELOG`

* lemma relating quasigroup operations and `Commutative` property

* simplified proof

* added converse property to \¬Algebra.Properties.AbelianGroup`

* moved lemma

* indentation; congruence lemmas

* untangled operation definitions

* untangled operation definitions in `CHANGELOG`

* fixed names

* reflexive reasoning; tightened imports

* refactoring to push properties into `Loop`, and re-export them from there and `Quasigroup`

* further refactoring

* final refactoring

* Minor naming tweaks

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                           |  41 +++++
 src/Algebra/Properties/Group.agda      | 202 +++++++++++++++----------
 src/Algebra/Properties/Loop.agda       |  31 +++-
 src/Algebra/Properties/Quasigroup.agda |   8 +-
 src/Algebra/Structures.agda            |  17 ++-
 5 files changed, 202 insertions(+), 97 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6ad8f21635..cd29c450ad 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -41,6 +41,11 @@ Deprecated names
   1×-identityʳ  ↦  ×-homo-1
   ```
 
+* In `Algebra.Structures.IsGroup`:
+  ```agda
+  _-_  ↦  _//_
+  ```
+
 * In `Data.Nat.Divisibility.Core`:
   ```agda
   *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
@@ -135,6 +140,32 @@ Additions to existing modules
   rawModule          : RawModule R c ℓ
   ```
 
+* In `Algebra.Properties.Group`:
+  ```agda
+  isQuasigroup    : IsQuasigroup _∙_ _\\_ _//_
+  quasigroup      : Quasigroup _ _
+  isLoop          : IsLoop _∙_ _\\_ _//_ ε
+  loop            : Loop _ _
+  
+  \\-leftDividesˡ  : LeftDividesˡ _∙_ _\\_
+  \\-leftDividesʳ  : LeftDividesʳ _∙_ _\\_
+  \\-leftDivides   : LeftDivides _∙_ _\\_
+  //-rightDividesˡ : RightDividesˡ _∙_ _//_
+  //-rightDividesʳ : RightDividesʳ _∙_ _//_
+  //-rightDivides  : RightDivides _∙_ _//_
+
+  ⁻¹-selfInverse  : SelfInverse _⁻¹
+  \\≗flip-//⇒comm : (∀ x y → x \\ y ≈ y // x) → Commutative _∙_
+  comm⇒\\≗flip-// : Commutative _∙_ → ∀ x y → x \\ y ≈ y // x
+  ```
+
+* In `Algebra.Properties.Loop`:
+  ```agda
+  identityˡ-unique : x ∙ y ≈ y → x ≈ ε
+  identityʳ-unique : x ∙ y ≈ x → y ≈ ε
+  identity-unique  : Identity x _∙_ → x ≈ ε
+  ```
+ 
 * In `Algebra.Construct.Terminal`:
   ```agda
   rawNearSemiring : RawNearSemiring c ℓ
@@ -154,6 +185,16 @@ Additions to existing modules
   idem-×-homo-* : (_*_ IdempotentOn x) → (m × x) * (n × x) ≈ (m ℕ.* n) × x
   ```
 
+* In `Algebra.Structures.IsGroup`:
+  ```agda
+  infixl 6 _//_
+  _//_ : Op₂ A
+  x // y = x ∙ (y ⁻¹)
+  infixr 6 _\\_
+  _\\_ : Op₂ A
+  x \\ y = (x ⁻¹) ∙ y
+  ```
+ 
 * In `Data.Container.Indexed.Core`:
   ```agda
   Subtrees o c = (r : Response c) → X (next c r)
diff --git a/src/Algebra/Properties/Group.agda b/src/Algebra/Properties/Group.agda
index 31f9aedf39..3d4a7d7713 100644
--- a/src/Algebra/Properties/Group.agda
+++ b/src/Algebra/Properties/Group.agda
@@ -10,99 +10,133 @@ open import Algebra.Bundles
 
 module Algebra.Properties.Group {g₁ g₂} (G : Group g₁ g₂) where
 
+import Algebra.Properties.Loop as LoopProperties
+import Algebra.Properties.Quasigroup as QuasigroupProperties
+open import Data.Product.Base using (_,_)
+open import Function.Base using (_$_)
+open import Function.Definitions
+
 open Group G
+open import Algebra.Consequences.Setoid setoid
 open import Algebra.Definitions _≈_
+open import Algebra.Structures _≈_ using (IsLoop; IsQuasigroup)
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Function.Base using (_$_; _⟨_⟩_)
-open import Data.Product.Base using (_,_; proj₂)
 
-ε⁻¹≈ε : ε ⁻¹ ≈ ε
-ε⁻¹≈ε = begin
-  ε ⁻¹      ≈⟨ sym $ identityʳ (ε ⁻¹) ⟩
-  ε ⁻¹ ∙ ε  ≈⟨ inverseˡ ε ⟩
-  ε         ∎
-
-private
-
-  left-helper : ∀ x y → x ≈ (x ∙ y) ∙ y ⁻¹
-  left-helper x y = begin
-    x              ≈⟨ sym (identityʳ x) ⟩
-    x ∙ ε          ≈⟨ ∙-congˡ $ sym (inverseʳ y) ⟩
-    x ∙ (y ∙ y ⁻¹) ≈⟨ sym (assoc x y (y ⁻¹)) ⟩
-    (x ∙ y) ∙ y ⁻¹ ∎
-
-  right-helper : ∀ x y → y ≈ x ⁻¹ ∙ (x ∙ y)
-  right-helper x y = begin
-    y              ≈⟨ sym (identityˡ y) ⟩
-    ε          ∙ y ≈⟨ ∙-congʳ $ sym (inverseˡ x) ⟩
-    (x ⁻¹ ∙ x) ∙ y ≈⟨ assoc (x ⁻¹) x y ⟩
-    x ⁻¹ ∙ (x ∙ y) ∎
-
-∙-cancelˡ : LeftCancellative _∙_
-∙-cancelˡ x y z eq = begin
-              y  ≈⟨ right-helper x y ⟩
-  x ⁻¹ ∙ (x ∙ y) ≈⟨ ∙-congˡ eq ⟩
-  x ⁻¹ ∙ (x ∙ z) ≈⟨ right-helper x z ⟨
-              z  ∎
-
-∙-cancelʳ : RightCancellative _∙_
-∙-cancelʳ x y z eq = begin
-  y            ≈⟨ left-helper y x ⟩
-  y ∙ x ∙ x ⁻¹ ≈⟨ ∙-congʳ eq ⟩
-  z ∙ x ∙ x ⁻¹ ≈⟨ left-helper z x ⟨
-  z            ∎
-
-∙-cancel : Cancellative _∙_
-∙-cancel = ∙-cancelˡ , ∙-cancelʳ
-
-⁻¹-involutive : ∀ x → x ⁻¹ ⁻¹ ≈ x
-⁻¹-involutive x = begin
-  x ⁻¹ ⁻¹              ≈⟨ identityʳ _ ⟨
-  x ⁻¹ ⁻¹ ∙ ε          ≈⟨ ∙-congˡ $ inverseˡ _ ⟨
-  x ⁻¹ ⁻¹ ∙ (x ⁻¹ ∙ x) ≈⟨ right-helper (x ⁻¹) x ⟨
-  x                    ∎
-
-⁻¹-injective : ∀ {x y} → x ⁻¹ ≈ y ⁻¹ → x ≈ y
-⁻¹-injective {x} {y} eq = ∙-cancelʳ _ _ _ ( begin
-  x ∙ x ⁻¹ ≈⟨ inverseʳ x ⟩
-  ε        ≈⟨ inverseʳ y ⟨
-  y ∙ y ⁻¹ ≈⟨ ∙-congˡ eq ⟨
-  y ∙ x ⁻¹ ∎ )
+\\-cong₂ : Congruent₂ _\\_
+\\-cong₂ x≈y u≈v = ∙-cong (⁻¹-cong x≈y) u≈v
 
-⁻¹-anti-homo-∙ : ∀ x y → (x ∙ y) ⁻¹ ≈ y ⁻¹ ∙ x ⁻¹
-⁻¹-anti-homo-∙ x y = ∙-cancelˡ _ _ _ ( begin
-  x ∙ y ∙ (x ∙ y) ⁻¹    ≈⟨ inverseʳ _ ⟩
-  ε                     ≈⟨ inverseʳ _ ⟨
-  x ∙ x ⁻¹              ≈⟨ ∙-congʳ (left-helper x y) ⟩
-  (x ∙ y) ∙ y ⁻¹ ∙ x ⁻¹ ≈⟨ assoc (x ∙ y) (y ⁻¹) (x ⁻¹) ⟩
-  x ∙ y ∙ (y ⁻¹ ∙ x ⁻¹) ∎ )
+//-cong₂ : Congruent₂ _//_
+//-cong₂ x≈y u≈v = ∙-cong x≈y (⁻¹-cong u≈v)
+
+------------------------------------------------------------------------
+-- Groups are quasi-groups
+
+\\-leftDividesˡ : LeftDividesˡ _∙_ _\\_
+\\-leftDividesˡ x y = begin
+  x  ∙ (x \\ y)  ≈⟨ assoc x (x ⁻¹) y ⟨
+  x ∙ x ⁻¹ ∙ y   ≈⟨ ∙-congʳ (inverseʳ x) ⟩
+  ε ∙ y          ≈⟨ identityˡ y ⟩
+  y              ∎
+
+\\-leftDividesʳ : LeftDividesʳ _∙_ _\\_
+\\-leftDividesʳ x y = begin
+  x \\ x ∙ y     ≈⟨ assoc (x ⁻¹) x y ⟨
+  x ⁻¹ ∙ x ∙ y   ≈⟨ ∙-congʳ (inverseˡ x) ⟩
+  ε ∙ y          ≈⟨ identityˡ y ⟩
+  y              ∎
+
+
+\\-leftDivides : LeftDivides _∙_ _\\_
+\\-leftDivides = \\-leftDividesˡ , \\-leftDividesʳ
+
+//-rightDividesˡ : RightDividesˡ _∙_ _//_
+//-rightDividesˡ x y = begin
+  (y // x) ∙ x    ≈⟨ assoc y (x ⁻¹) x ⟩
+  y ∙ (x ⁻¹ ∙ x)  ≈⟨ ∙-congˡ (inverseˡ x) ⟩
+  y ∙ ε           ≈⟨ identityʳ y ⟩
+  y               ∎
+
+//-rightDividesʳ : RightDividesʳ _∙_ _//_
+//-rightDividesʳ x y = begin
+  y ∙ x // x    ≈⟨ assoc y x (x ⁻¹) ⟩
+  y ∙ (x // x)  ≈⟨ ∙-congˡ (inverseʳ x) ⟩
+  y ∙ ε         ≈⟨ identityʳ y ⟩
+  y             ∎
+
+//-rightDivides : RightDivides _∙_ _//_
+//-rightDivides = //-rightDividesˡ , //-rightDividesʳ
+
+isQuasigroup : IsQuasigroup _∙_ _\\_ _//_
+isQuasigroup = record
+  { isMagma = isMagma
+  ; \\-cong = \\-cong₂
+  ; //-cong = //-cong₂
+  ; leftDivides = \\-leftDivides
+  ; rightDivides = //-rightDivides
+  }
+
+quasigroup : Quasigroup _ _
+quasigroup = record { isQuasigroup = isQuasigroup }
+
+open QuasigroupProperties quasigroup public
+  using (x≈z//y; y≈x\\z)
+  renaming (cancelˡ to ∙-cancelˡ; cancelʳ to ∙-cancelʳ; cancel to ∙-cancel)
+
+------------------------------------------------------------------------
+-- Groups are loops
+
+isLoop : IsLoop _∙_ _\\_ _//_ ε
+isLoop = record { isQuasigroup = isQuasigroup ; identity = identity }
 
-identityˡ-unique : ∀ x y → x ∙ y ≈ y → x ≈ ε
-identityˡ-unique x y eq = begin
-  x              ≈⟨ left-helper x y ⟩
-  (x ∙ y) ∙ y ⁻¹ ≈⟨ ∙-congʳ eq ⟩
-       y  ∙ y ⁻¹ ≈⟨ inverseʳ y ⟩
-  ε              ∎
+loop : Loop _ _
+loop = record { isLoop = isLoop }
 
-identityʳ-unique : ∀ x y → x ∙ y ≈ x → y ≈ ε
-identityʳ-unique x y eq = begin
-  y              ≈⟨ right-helper x y ⟩
-  x ⁻¹ ∙ (x ∙ y) ≈⟨ refl ⟨ ∙-cong ⟩ eq ⟩
-  x ⁻¹ ∙  x      ≈⟨ inverseˡ x ⟩
-  ε              ∎
+open LoopProperties loop public
+  using (identityˡ-unique; identityʳ-unique; identity-unique)
 
-identity-unique : ∀ {x} → Identity x _∙_ → x ≈ ε
-identity-unique {x} id = identityˡ-unique x x (proj₂ id x)
+------------------------------------------------------------------------
+-- Other properties
 
 inverseˡ-unique : ∀ x y → x ∙ y ≈ ε → x ≈ y ⁻¹
-inverseˡ-unique x y eq = begin
-  x              ≈⟨ left-helper x y ⟩
-  (x ∙ y) ∙ y ⁻¹ ≈⟨ ∙-congʳ eq ⟩
-       ε  ∙ y ⁻¹ ≈⟨ identityˡ (y ⁻¹) ⟩
-            y ⁻¹ ∎
+inverseˡ-unique x y eq = trans (x≈z//y x y ε eq) (identityˡ _)
 
 inverseʳ-unique : ∀ x y → x ∙ y ≈ ε → y ≈ x ⁻¹
-inverseʳ-unique x y eq = begin
-  y       ≈⟨ sym (⁻¹-involutive y) ⟩
-  y ⁻¹ ⁻¹ ≈⟨ ⁻¹-cong (sym (inverseˡ-unique x y eq)) ⟩
-  x ⁻¹    ∎
+inverseʳ-unique x y eq = trans (y≈x\\z x y ε eq) (identityʳ _)
+
+ε⁻¹≈ε : ε ⁻¹ ≈ ε
+ε⁻¹≈ε = sym $ inverseˡ-unique _ _ (identityˡ ε)
+
+⁻¹-selfInverse : SelfInverse _⁻¹
+⁻¹-selfInverse {x} {y} eq = sym $ inverseˡ-unique x y $ begin
+  x ∙ y    ≈⟨ ∙-congˡ eq ⟨
+  x ∙ x ⁻¹ ≈⟨ inverseʳ x  ⟩
+  ε        ∎
+
+⁻¹-involutive : Involutive _⁻¹
+⁻¹-involutive = selfInverse⇒involutive ⁻¹-selfInverse
+
+⁻¹-injective : Injective _≈_ _≈_ _⁻¹
+⁻¹-injective = selfInverse⇒injective ⁻¹-selfInverse
+
+⁻¹-anti-homo-∙ : ∀ x y → (x ∙ y) ⁻¹ ≈ y ⁻¹ ∙ x ⁻¹
+⁻¹-anti-homo-∙ x y = ∙-cancelˡ _ _ _ $ begin
+  x ∙ y ∙ (x ∙ y) ⁻¹    ≈⟨ inverseʳ _ ⟩
+  ε                     ≈⟨ inverseʳ _ ⟨
+  x ∙ x ⁻¹              ≈⟨ ∙-congʳ (//-rightDividesʳ y x) ⟨
+  (x ∙ y) ∙ y ⁻¹ ∙ x ⁻¹ ≈⟨ assoc (x ∙ y) (y ⁻¹) (x ⁻¹) ⟩
+  x ∙ y ∙ (y ⁻¹ ∙ x ⁻¹) ∎
+
+\\≗flip-//⇒comm : (∀ x y → x \\ y ≈ y // x) → Commutative _∙_
+\\≗flip-//⇒comm \\≗//ᵒ x y = begin
+  x ∙ y                ≈⟨ ∙-congˡ (//-rightDividesˡ x y) ⟨
+  x ∙ ((y // x) ∙ x)   ≈⟨ ∙-congˡ (∙-congʳ (\\≗//ᵒ x y)) ⟨
+  x ∙ ((x \\ y) ∙ x)   ≈⟨ assoc x (x \\ y) x ⟨
+  x ∙ (x \\ y) ∙ x     ≈⟨ ∙-congʳ (\\-leftDividesˡ x y) ⟩
+  y ∙ x                ∎
+
+comm⇒\\≗flip-// : Commutative _∙_ → ∀ x y → x \\ y ≈ y // x
+comm⇒\\≗flip-// comm x y = begin
+  x \\ y    ≡⟨⟩
+  x ⁻¹ ∙ y  ≈⟨ comm _ _ ⟩
+  y ∙ x ⁻¹  ≡⟨⟩
+  y // x    ∎
diff --git a/src/Algebra/Properties/Loop.agda b/src/Algebra/Properties/Loop.agda
index 1ecb2c52f4..46ed1e7f7c 100644
--- a/src/Algebra/Properties/Loop.agda
+++ b/src/Algebra/Properties/Loop.agda
@@ -1,7 +1,7 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Some basic properties of Quasigroup
+-- Some basic properties of Loop
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -12,29 +12,46 @@ module Algebra.Properties.Loop {l₁ l₂} (L : Loop l₁ l₂) where
 
 open Loop L
 open import Algebra.Definitions _≈_
+open import Algebra.Properties.Quasigroup quasigroup
+open import Data.Product.Base using (proj₂)
 open import Relation.Binary.Reasoning.Setoid setoid
-open import Algebra.Properties.Quasigroup
 
 x//x≈ε : ∀ x → x // x ≈ ε
 x//x≈ε x = begin
-  x // x       ≈⟨ //-congʳ (sym (identityˡ x)) ⟩
+  x // x       ≈⟨ //-congʳ (identityˡ x) ⟨
   (ε ∙ x) // x ≈⟨ rightDividesʳ x ε ⟩
   ε            ∎
 
-x\\x≈ε : ∀ x → x \\ x ≈ ε
+x\\x≈ε : ∀ x → x \\ x ≈ ε
 x\\x≈ε x = begin
-  x \\ x       ≈⟨ \\-congˡ (sym (identityʳ x )) ⟩
+  x \\ x       ≈⟨ \\-congˡ (identityʳ x ) ⟨
   x \\ (x ∙ ε) ≈⟨ leftDividesʳ x ε ⟩
   ε            ∎
 
 ε\\x≈x : ∀ x → ε \\ x ≈ x
 ε\\x≈x x = begin
-  ε \\ x       ≈⟨ sym (identityˡ (ε \\ x)) ⟩
+  ε \\ x       ≈⟨ identityˡ (ε \\ x) ⟨
   ε ∙ (ε \\ x) ≈⟨ leftDividesˡ ε x ⟩
   x            ∎
 
 x//ε≈x : ∀ x → x // ε ≈ x
 x//ε≈x x = begin
- x // ε       ≈⟨ sym (identityʳ (x // ε)) ⟩
+ x // ε       ≈⟨ identityʳ (x // ε) ⟨
  (x // ε) ∙ ε ≈⟨ rightDividesˡ ε x ⟩
  x            ∎
+
+identityˡ-unique : ∀ x y → x ∙ y ≈ y → x ≈ ε
+identityˡ-unique x y eq = begin
+  x      ≈⟨ x≈z//y x y y eq ⟩
+  y // y ≈⟨ x//x≈ε y ⟩
+  ε      ∎
+
+identityʳ-unique : ∀ x y → x ∙ y ≈ x → y ≈ ε
+identityʳ-unique x y eq = begin
+  y       ≈⟨ y≈x\\z x y x eq ⟩
+  x \\ x  ≈⟨ x\\x≈ε x ⟩
+  ε       ∎
+
+identity-unique : ∀ {x} → Identity x _∙_ → x ≈ ε
+identity-unique {x} id = identityˡ-unique x x (proj₂ id x)
+
diff --git a/src/Algebra/Properties/Quasigroup.agda b/src/Algebra/Properties/Quasigroup.agda
index 335a9ea7a3..e8c7516978 100644
--- a/src/Algebra/Properties/Quasigroup.agda
+++ b/src/Algebra/Properties/Quasigroup.agda
@@ -17,14 +17,14 @@ open import Data.Product.Base using (_,_)
 
 cancelˡ : LeftCancellative _∙_
 cancelˡ x y z eq = begin
-  y             ≈⟨ sym( leftDividesʳ x y) ⟩
+  y             ≈⟨ leftDividesʳ x y ⟨
   x \\ (x ∙ y)  ≈⟨ \\-congˡ eq ⟩
   x \\ (x ∙ z)  ≈⟨ leftDividesʳ x z ⟩
   z             ∎
 
 cancelʳ : RightCancellative _∙_
 cancelʳ x y z eq = begin
-  y             ≈⟨ sym( rightDividesʳ x y) ⟩
+  y             ≈⟨ rightDividesʳ x y ⟨
   (y ∙ x) // x  ≈⟨ //-congʳ eq ⟩
   (z ∙ x) // x  ≈⟨ rightDividesʳ x z ⟩
   z             ∎
@@ -34,12 +34,12 @@ cancel = cancelˡ , cancelʳ
 
 y≈x\\z : ∀ x y z → x ∙ y ≈ z → y ≈ x \\ z
 y≈x\\z x y z eq = begin
-  y            ≈⟨ sym (leftDividesʳ x y) ⟩
+  y            ≈⟨ leftDividesʳ x y ⟨
   x \\ (x ∙ y) ≈⟨ \\-congˡ eq ⟩
   x \\ z       ∎
 
 x≈z//y : ∀ x y z → x ∙ y ≈ z → x ≈ z // y
 x≈z//y x y z eq = begin
-  x            ≈⟨ sym (rightDividesʳ y x) ⟩
+  x            ≈⟨ rightDividesʳ y x ⟨
   (x ∙ y) // y ≈⟨ //-congʳ eq ⟩
   z // y       ∎
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index 44208367ea..bb28e89882 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -253,9 +253,22 @@ record IsGroup (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) w
 
   open IsMonoid isMonoid public
 
+  infixr 6 _\\_
+  _\\_ : Op₂ A
+  x \\ y = (x ⁻¹) ∙ y
+
+  infixl 6 _//_
+  _//_ : Op₂ A
+  x // y = x ∙ (y ⁻¹)
+
+  -- Deprecated.
   infixl 6 _-_
   _-_ : Op₂ A
-  x - y = x ∙ (y ⁻¹)
+  _-_ = _//_
+  {-# WARNING_ON_USAGE _-_
+  "Warning: _-_ was deprecated in v2.1.
+  Please use _//_ instead. "
+  #-}
 
   inverseˡ : LeftInverse ε _⁻¹ _∙_
   inverseˡ = proj₁ inverse
@@ -291,7 +304,7 @@ record IsAbelianGroup (∙ : Op₂ A)
     isGroup : IsGroup ∙ ε ⁻¹
     comm    : Commutative ∙
 
-  open IsGroup isGroup public
+  open IsGroup isGroup public renaming (_//_ to _-_) hiding (_\\_; _-_)
 
   isCommutativeMonoid : IsCommutativeMonoid ∙ ε
   isCommutativeMonoid = record

From 6dcfbdd051014e381eaf139f4a943da2fbc1b0cc Mon Sep 17 00:00:00 2001
From: Nathan van Doorn <nvd1234@gmail.com>
Date: Sat, 16 Mar 2024 15:03:07 +0100
Subject: [PATCH 245/316] Add prime factorization and its properties (#1969)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* Add prime factorization and its properties

* Add missing header comment

I'd missed this when copy-pasting from my old code in a separate repo

* Remove completely trivial lemma

* Use equational reasoning in quotient|n proof

* Fix typo in module header

* Factorization => Factorisation

* Use Nat lemma in extend-|/

* [ cleanup ] part of the proof

* [ cleanup ] finishing up the job

* [ cleanup ] a little bit more

* [ cleanup ] the import list

* [ fix ] header style

* [ fix ] broken merge: missing import

* Move Data.Nat.Rough to Data.Nat.Primality.Rough

* Rename productPreserves↭⇒≡ to product-↭

* Use proof of Prime=>NonZero

* Open reasoning once in factoriseRec

* Rename Factorisation => PrimeFactorisation

* Move wheres around

* Tidy up Rough a bit

* Move quotient|n to top of file

* Replace factorisationPullToFront with slightly more generally useful factorisationHasAllPrimeFactors and a bit of logic

* Fix import after merge

* Clean up proof of 2-rough-n

* Make argument to 2-rough-n implicit

* Rename 2-rough-n to 2-rough

* Complete merge, rewrite factorisation logic a bit

Rewrite partially based on suggestions from James McKinna

* Short circuit when candidate is greater than square root of product

* Remove redefined lemma

* Minor simplifications

* Remove private pattern synonyms

* Change name of lemma

* Typo

* Remove using list from import

It feels like we're importing half the module anyway

* Clean up proof

* Fixes after merge

* Addressed some feedback

* Fix previous merge

---------

Co-authored-by: Guillaume Allais <guillaume.allais@ens-lyon.org>
Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                                  |  23 +-
 .../Permutation/Propositional/Properties.agda |  28 ++-
 src/Data/Nat/Primality.agda                   |  23 ++
 src/Data/Nat/Primality/Factorisation.agda     | 198 ++++++++++++++++++
 4 files changed, 267 insertions(+), 5 deletions(-)
 create mode 100644 src/Data/Nat/Primality/Factorisation.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index cd29c450ad..ee056ad8c9 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -53,7 +53,16 @@ Deprecated names
 
 New modules
 -----------
-* `Algebra.Module.Bundles.Raw`: raw bundles for module-like algebraic structures
+
+* Raw bundles for module-like algebraic structures:
+  ```
+  Algebra.Module.Bundles.Raw
+  ```
+
+* Prime factorisation of natural numbers.
+  ```
+  Data.Nat.Primality.Factorisation
+  ```
 
 * Consequences of 'infinite descent' for (accessible elements of) well-founded relations:
   ```agda
@@ -296,6 +305,18 @@ Additions to existing modules
   pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n
   pred-cancel-≡ : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n
   ```
+  
+* Added new proofs to `Data.Nat.Primality`:
+  ```agda
+  rough∧square>⇒prime : .{{NonTrivial n}} → m Rough n → m * m > n → Prime n
+  productOfPrimes≢0 : All Prime as → NonZero (product as)
+  productOfPrimes≥1 : All Prime as → product as ≥ 1
+  ```
+
+* Added new proofs to `Data.List.Relation.Binary.Permutation.Propositional.Properties`:
+  ```agda
+  product-↭ : product Preserves _↭_ ⟶ _≡_
+  ```
 
 * Added new functions in `Data.String.Base`:
   ```agda
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index 781de1cc36..d56bdb7fb3 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -12,7 +12,8 @@ open import Algebra.Bundles
 open import Algebra.Definitions
 open import Algebra.Structures
 open import Data.Bool.Base using (Bool; true; false)
-open import Data.Nat using (suc)
+open import Data.Nat.Base using (suc; _*_)
+open import Data.Nat.Properties using (*-assoc; *-comm)
 open import Data.Product.Base using (-,_; proj₂)
 open import Data.List.Base as List
 open import Data.List.Relation.Binary.Permutation.Propositional
@@ -25,14 +26,13 @@ open import Data.Product.Base using (_,_; _×_; ∃; ∃₂)
 open import Function.Base using (_∘_; _⟨_⟩_)
 open import Level using (Level)
 open import Relation.Unary using (Pred)
-open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_)
+open import Relation.Binary.Core using (Rel; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
 open import Relation.Binary.Definitions using (_Respects_; Decidable)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_ ; refl ; cong; cong₂; _≢_)
+open import Relation.Binary.PropositionalEquality.Properties using (module ≡-Reasoning)
 open import Relation.Nullary
 
-open PermutationReasoning
-
 private
   variable
     a b p : Level
@@ -172,6 +172,7 @@ shift v (x ∷ xs) ys = begin
   x ∷ (xs ++ [ v ] ++ ys) <⟨ shift v xs ys ⟩
   x ∷ v ∷ xs ++ ys        <<⟨ refl ⟩
   v ∷ x ∷ xs ++ ys        ∎
+  where open PermutationReasoning
 
 drop-mid-≡ : ∀ {x : A} ws xs {ys} {zs} →
              ws ++ [ x ] ++ ys ≡ xs ++ [ x ] ++ zs →
@@ -216,11 +217,13 @@ drop-mid {A = A} {x} ws xs p = drop-mid′ p ws xs refl refl
       _ ∷ (xs ++ _ ∷ _) <⟨ shift _ _ _ ⟩
       _ ∷ _ ∷ xs ++ _   <<⟨ refl ⟩
       _ ∷ _ ∷ xs ++ _   ∎
+      where open PermutationReasoning
   drop-mid′ (swap y z p) (y ∷ z ∷ ws) (z ∷ [])     refl refl = begin
       _ ∷ _ ∷ ws ++ _   <<⟨ refl ⟩
       _ ∷ (_ ∷ ws ++ _) <⟨ ↭-sym (shift _ _ _) ⟩
       _ ∷ (ws ++ _ ∷ _) <⟨ p ⟩
       _ ∷ _             ∎
+      where open PermutationReasoning
   drop-mid′ (swap y z p) (y ∷ z ∷ ws) (z ∷ y ∷ xs) refl refl = swap y z (drop-mid′ p _ _ refl refl)
   drop-mid′ (trans p₁ p₂) ws  xs refl refl with ∈-∃++ (∈-resp-↭ p₁ (∈-insert ws))
   ... | (h , t , refl) = trans (drop-mid′ p₁ ws h refl refl) (drop-mid′ p₂ h xs refl refl)
@@ -245,6 +248,7 @@ drop-mid {A = A} {x} ws xs p = drop-mid′ p ws xs refl refl
   x ∷ xs ++ ys   <⟨ ++-comm xs ys ⟩
   x ∷ ys ++ xs   ↭⟨ shift x ys xs ⟨
   ys ++ (x ∷ xs) ∎
+  where open PermutationReasoning
 
 ++-isMagma : IsMagma {A = List A} _↭_ _++_
 ++-isMagma = record
@@ -300,6 +304,7 @@ shifts xs ys {zs} = begin
   (xs ++ ys) ++ zs ↭⟨ ++⁺ʳ zs (++-comm xs ys) ⟩
   (ys ++ xs) ++ zs ↭⟨ ++-assoc ys xs zs ⟩
    ys ++ xs  ++ zs ∎
+   where open PermutationReasoning
 
 ------------------------------------------------------------------------
 -- _∷_
@@ -315,6 +320,7 @@ drop-∷ = drop-mid [] []
   xs ++ [ x ]   ↭⟨ shift x xs [] ⟩
   x ∷ xs ++ []  ≡⟨ Lₚ.++-identityʳ _ ⟩
   x ∷ xs        ∎)
+  where open PermutationReasoning
 
 ------------------------------------------------------------------------
 -- ʳ++
@@ -353,3 +359,17 @@ module _ {ℓ} {R : Rel A ℓ} (R? : Decidable R) where
     (x ∷ xs) ++ y ∷ ys       ≡⟨ Lₚ.++-assoc [ x ] xs (y ∷ ys) ⟨
     x ∷ xs ++ y ∷ ys         ∎
     where open PermutationReasoning
+
+------------------------------------------------------------------------
+-- product
+
+product-↭ : product Preserves _↭_ ⟶ _≡_
+product-↭ refl = refl
+product-↭ (prep x r) = cong (x *_) (product-↭ r)
+product-↭ (trans r s) = ≡.trans (product-↭ r) (product-↭ s)
+product-↭ (swap {xs} {ys} x y r) = begin
+  x * (y * product xs) ≡˘⟨ *-assoc x y (product xs) ⟩
+  (x * y) * product xs ≡⟨ cong₂ _*_ (*-comm x y) (product-↭ r) ⟩
+  (y * x) * product ys ≡⟨ *-assoc y x (product ys) ⟩
+  y * (x * product ys) ∎
+  where open ≡-Reasoning
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index 683c7d8a87..1a491fdf93 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -8,6 +8,8 @@
 
 module Data.Nat.Primality where
 
+open import Data.List.Base using ([]; _∷_; product)
+open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
 open import Data.Nat.GCD using (module GCD; module Bézout)
@@ -294,6 +296,17 @@ prime⇒rough (prime pr) = pr
 rough∧∣⇒prime : .{{NonTrivial p}} → p Rough n → p ∣ n → Prime p
 rough∧∣⇒prime r p∣n = prime (rough∧∣⇒rough r p∣n)
 
+-- If a number n is m-rough, and m * m > n, then n must be prime.
+rough∧square>⇒prime : .{{NonTrivial n}} → m Rough n → m * m > n → Prime n
+rough∧square>⇒prime rough m*m>n = prime ¬composite
+  where
+    ¬composite : ¬ Composite _
+    ¬composite (composite d<n d∣n) = contradiction (m∣n⇒n≡quotient*m d∣n)
+      (<⇒≢ (<-≤-trans m*m>n (*-mono-≤
+        (rough⇒≤ (rough∧∣⇒rough rough (quotient-∣ d∣n)))
+        (rough⇒≤ (rough∧∣⇒rough rough d∣n)))))
+      where instance _ = n>1⇒nonTrivial (quotient>1 d∣n d<n)
+
 -- Relationship between compositeness and primality.
 composite⇒¬prime : Composite n → ¬ Prime n
 composite⇒¬prime composite[d] (prime p) = p composite[d]
@@ -309,6 +322,16 @@ prime⇒¬composite (prime p) = p
 ¬prime⇒composite {n} ¬prime[n] =
   decidable-stable (composite? n) (¬prime[n] ∘′ ¬composite⇒prime)
 
+productOfPrimes≢0 : ∀ {as} → All Prime as → NonZero (product as)
+productOfPrimes≢0 pas = product≢0 (All.map prime⇒nonZero pas)
+  where
+  product≢0 : ∀ {ns} → All NonZero ns → NonZero (product ns)
+  product≢0 [] = _
+  product≢0 {n ∷ ns} (nzn ∷ nzns) = m*n≢0 n _ {{nzn}} {{product≢0 nzns}}
+
+productOfPrimes≥1 : ∀ {as} → All Prime as → product as ≥ 1
+productOfPrimes≥1 {as} pas = >-nonZero⁻¹ _ {{productOfPrimes≢0 pas}}
+
 ------------------------------------------------------------------------
 -- Basic (counter-)examples of Irreducible
 
diff --git a/src/Data/Nat/Primality/Factorisation.agda b/src/Data/Nat/Primality/Factorisation.agda
new file mode 100644
index 0000000000..fed89d7d45
--- /dev/null
+++ b/src/Data/Nat/Primality/Factorisation.agda
@@ -0,0 +1,198 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Prime factorisation of natural numbers and its properties
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Nat.Primality.Factorisation where
+
+open import Data.Empty using (⊥-elim)
+open import Data.Nat.Base
+open import Data.Nat.Divisibility
+open import Data.Nat.Properties
+open import Data.Nat.Induction using (<-Rec; <-rec; <-recBuilder)
+open import Data.Nat.Primality
+open import Data.Product as Π using (∃-syntax; _×_; _,_; proj₁; proj₂)
+open import Data.List.Base using (List; []; _∷_; _++_; product)
+open import Data.List.Membership.Propositional using (_∈_)
+open import Data.List.Membership.Propositional.Properties using (∈-∃++)
+open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
+open import Data.List.Relation.Unary.Any using (here; there)
+open import Data.List.Relation.Binary.Permutation.Propositional
+  using (_↭_; prep; swap; ↭-reflexive; ↭-refl; ↭-trans; refl; module PermutationReasoning)
+open import Data.List.Relation.Binary.Permutation.Propositional.Properties using (product-↭; All-resp-↭; shift)
+open import Data.Sum.Base using (inj₁; inj₂)
+open import Function.Base using (_$_; _∘_; _|>_; flip)
+open import Induction using (build)
+open import Induction.Lexicographic using (_⊗_; [_⊗_])
+open import Relation.Nullary.Decidable using (yes; no)
+open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; cong; module ≡-Reasoning)
+
+private
+  variable
+    n : ℕ
+    
+------------------------------------------------------------------------
+-- Core definition
+
+record PrimeFactorisation (n : ℕ) : Set where
+  field
+    factors : List ℕ
+    isFactorisation : n ≡ product factors
+    factorsPrime : All Prime factors
+
+open PrimeFactorisation public using (factors)
+open PrimeFactorisation
+
+------------------------------------------------------------------------
+-- Finding a factorisation
+
+primeFactorisation[1] : PrimeFactorisation 1
+primeFactorisation[1] = record
+  { factors = []
+  ; isFactorisation = refl
+  ; factorsPrime = []
+  }
+
+primeFactorisation[p] : Prime n → PrimeFactorisation n
+primeFactorisation[p] {n} pr = record
+  { factors = n ∷ []
+  ; isFactorisation = sym (*-identityʳ n)
+  ; factorsPrime = pr ∷ []
+  }
+
+-- This builds up three important things:
+-- * a proof that every number we've gotten to so far has increasingly higher
+--   possible least prime factor, so we don't have to repeat smaller factors
+--   over and over (this is the "m" and "rough" parameters)
+-- * a witness that this limit is getting closer to the number of interest, in a
+--   way that helps the termination checker (the "k" and "eq" parameters)
+-- * a proof that we can factorise any smaller number, which is useful when we
+--   encounter a factor, as we can then divide by that factor and continue from
+--   there without termination issues
+factorise : ∀ n → .{{NonZero n}} → PrimeFactorisation n
+factorise 1 = primeFactorisation[1]
+factorise n₀@(2+ _) = build [ <-recBuilder ⊗ <-recBuilder ] P facRec (n₀ , suc n₀ ∸ 4) 2-rough refl
+  where
+  P : ℕ × ℕ → Set
+  P (n , k) = ∀ {m} → .{{NonTrivial n}} → .{{NonTrivial m}} → m Rough n → suc n ∸ m * m ≡ k → PrimeFactorisation n
+
+  facRec : ∀ n×k → (<-Rec ⊗ <-Rec) P n×k → P n×k
+  facRec (n , zero) _ rough eq =
+  -- Case 1: m * m > n, ∴ Prime n
+    primeFactorisation[p] (rough∧square>⇒prime rough (m∸n≡0⇒m≤n eq))
+  facRec (n@(2+ _) , suc k) (recFactor , recQuotient) {m@(2+ _)} rough eq with m ∣? n
+  -- Case 2: m ∤ n, try larger m, reducing k accordingly
+  ... | no m∤n = recFactor (≤-<-trans (m∸n≤m k (m + m)) (n<1+n k)) {suc m} (∤⇒rough-suc m∤n rough) $ begin
+    suc n ∸ (suc m + m * suc m)   ≡⟨ cong (λ # → suc n ∸ (suc m + #)) (*-suc m m) ⟩
+    suc n ∸ (suc m + (m + m * m)) ≡⟨ cong (suc n ∸_) (+-assoc (suc m) m (m * m)) ⟨
+    suc n ∸ (suc (m + m) + m * m) ≡⟨ cong (suc n ∸_) (+-comm (suc (m + m)) (m * m)) ⟩
+    suc n ∸ (m * m + suc (m + m)) ≡⟨ ∸-+-assoc (suc n) (m * m) (suc (m + m)) ⟨
+    (suc n ∸ m * m) ∸ suc (m + m) ≡⟨ cong (_∸ suc (m + m)) eq ⟩
+    suc k ∸ suc (m + m)           ∎
+    where open ≡-Reasoning
+  -- Case 3: m ∣ n, record m and recurse on the quotient
+  ... | yes m∣n = record
+    { factors = m ∷ ps
+    ; isFactorisation = sym m*Πps≡n
+    ; factorsPrime = rough∧∣⇒prime rough m∣n ∷ primes
+    }
+    where
+      m<n : m < n
+      m<n = begin-strict
+        m            <⟨ s≤s (≤-trans (m≤n+m m _) (+-monoʳ-≤ _ (m≤m+n m _))) ⟩
+        pred (m * m) <⟨ s<s⁻¹ (m∸n≢0⇒n<m λ eq′ → 0≢1+n (trans (sym eq′) eq)) ⟩
+        n            ∎
+        where open ≤-Reasoning
+        
+      q = quotient m∣n
+      
+      instance _  = n>1⇒nonTrivial (quotient>1 m∣n m<n)
+      
+      factorisation[q] : PrimeFactorisation q
+      factorisation[q] = recQuotient (quotient-< m∣n) (suc q ∸ m * m) (rough∧∣⇒rough rough (quotient-∣ m∣n)) refl
+      
+      ps = factors factorisation[q]
+      
+      primes = factorsPrime factorisation[q]
+      
+      m*Πps≡n : m * product ps ≡ n
+      m*Πps≡n = begin
+        m * product ps ≡⟨ cong (m *_) (isFactorisation factorisation[q]) ⟨
+        m * q          ≡⟨ m∣n⇒n≡m*quotient m∣n ⟨
+        n              ∎
+        where open ≡-Reasoning
+
+------------------------------------------------------------------------
+-- Properties of a factorisation
+
+factorisationHasAllPrimeFactors : ∀ {as} {p} → Prime p → p ∣ product as → All Prime as → p ∈ as
+factorisationHasAllPrimeFactors {[]} {2+ p} pPrime p∣Πas [] = contradiction (∣1⇒≡1 p∣Πas) λ ()
+factorisationHasAllPrimeFactors {a ∷ as} {p} pPrime p∣aΠas (aPrime ∷ asPrime) with euclidsLemma a (product as) pPrime p∣aΠas
+... | inj₂ p∣Πas = there (factorisationHasAllPrimeFactors pPrime p∣Πas asPrime)
+... | inj₁ p∣a with prime⇒irreducible aPrime p∣a
+...   | inj₁ refl = contradiction pPrime ¬prime[1]
+...   | inj₂ refl = here refl
+
+private
+  factorisationUnique′ : (as bs : List ℕ) → product as ≡ product bs → All Prime as → All Prime bs → as ↭ bs
+  factorisationUnique′ [] [] Πas≡Πbs asPrime bsPrime = refl
+  factorisationUnique′ [] (b@(2+ _) ∷ bs) Πas≡Πbs prime[as] (_ ∷ prime[bs]) =
+    contradiction Πas≡Πbs (<⇒≢ Πas<Πbs)
+    where
+      Πas<Πbs : product [] < product (b ∷ bs)
+      Πas<Πbs = begin-strict
+        1                ≡⟨⟩
+        1 * 1            <⟨ *-monoˡ-< 1 {1} {b} sz<ss ⟩
+        b * 1            ≤⟨ *-monoʳ-≤ b (productOfPrimes≥1 prime[bs]) ⟩
+        b * product bs   ≡⟨⟩
+        product (b ∷ bs) ∎
+        where open ≤-Reasoning
+
+  factorisationUnique′ (a ∷ as) bs Πas≡Πbs (prime[a] ∷ prime[as]) prime[bs] = a∷as↭bs
+    where
+      a∣Πbs : a ∣ product bs
+      a∣Πbs = divides (product as) $ begin
+        product bs       ≡⟨ Πas≡Πbs ⟨
+        product (a ∷ as) ≡⟨⟩
+        a * product as   ≡⟨ *-comm a (product as) ⟩
+        product as * a   ∎
+        where open ≡-Reasoning
+
+      shuffle : ∃[ bs′ ] bs ↭ a ∷ bs′
+      shuffle with ys , zs , p ← ∈-∃++ (factorisationHasAllPrimeFactors prime[a] a∣Πbs prime[bs])
+        = ys ++ zs , ↭-trans (↭-reflexive p) (shift a ys zs)
+
+      bs′ = proj₁ shuffle
+      bs↭a∷bs′ = proj₂ shuffle
+
+      Πas≡Πbs′ : product as ≡ product bs′
+      Πas≡Πbs′ = *-cancelˡ-≡ (product as) (product bs′) a {{prime⇒nonZero prime[a]}} $ begin
+        a * product as  ≡⟨ Πas≡Πbs ⟩
+        product bs      ≡⟨ product-↭ bs↭a∷bs′ ⟩
+        a * product bs′ ∎
+        where open ≡-Reasoning
+
+      prime[bs'] : All Prime bs′
+      prime[bs'] = All.tail (All-resp-↭ bs↭a∷bs′ prime[bs])
+
+      a∷as↭bs : a ∷ as ↭ bs
+      a∷as↭bs = begin
+        a ∷ as  <⟨ factorisationUnique′ as bs′ Πas≡Πbs′ prime[as] prime[bs'] ⟩
+        a ∷ bs′ ↭⟨ bs↭a∷bs′ ⟨
+        bs      ∎
+        where open PermutationReasoning
+
+factorisationUnique : (f f′ : PrimeFactorisation n) → factors f ↭ factors f′
+factorisationUnique {n} f f′ =
+  factorisationUnique′ (factors f) (factors f′) Πf≡Πf′ (factorsPrime f) (factorsPrime f′)
+  where
+    Πf≡Πf′ : product (factors f) ≡ product (factors f′)
+    Πf≡Πf′ = begin
+      product (factors f)  ≡⟨ isFactorisation f ⟨
+      n                    ≡⟨ isFactorisation f′ ⟩
+      product (factors f′) ∎
+      where open ≡-Reasoning

From 33055417c04e64d4c6439ccb4702d1eab44a62a5 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 17 Mar 2024 11:46:49 +0000
Subject: [PATCH 246/316] Refactor
 `Data.List.Relation.Binary.BagAndSetEquality` (#2321)

* easy refactorings for better abstraction

* tweaking irrefutable `with`
---
 .../Relation/Binary/BagAndSetEquality.agda    | 89 +++++++++----------
 1 file changed, 43 insertions(+), 46 deletions(-)

diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 389721d4c5..f014df8b41 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -245,16 +245,15 @@ commutativeMonoid {a} k A = record
 
 empty-unique : ∀ {k} → xs ∼[ ⌊ k ⌋→ ] [] → xs ≡ []
 empty-unique {xs = []}    _    = refl
-empty-unique {xs = _ ∷ _} ∷∼[] with ⇒→ ∷∼[] (here refl)
-... | ()
+empty-unique {xs = _ ∷ _} ∷∼[] with () ← ⇒→ ∷∼[] (here refl)
 
 -- _++_ is idempotent (under set equality).
 
 ++-idempotent : Idempotent {A = List A} _∼[ set ]_ _++_
 ++-idempotent xs {x} =
-  x ∈ xs ++ xs  ∼⟨ mk⇔ ([ id , id ]′ ∘ (Inverse.from $ ++↔))
-                                  (Inverse.to ++↔ ∘ inj₁) ⟩
-  x ∈ xs        ∎
+  x ∈ xs ++ xs
+    ∼⟨ mk⇔ ([ id , id ]′ ∘ (Inverse.from $ ++↔)) (Inverse.to ++↔ ∘ inj₁) ⟩
+  x ∈ xs ∎
   where open Related.EquationalReasoning
 
 -- The list monad's bind distributes from the left over _++_.
@@ -538,62 +537,60 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
 
   lemma : ∀ {xs ys} (inv : x ∷ xs ∼[ bag ] x ∷ ys) {z} →
           Well-behaved (Inverse.to (∼→⊎↔⊎ inv {z}))
-  lemma {xs} inv {b = z∈xs} {a = p} {a′ = q} hyp₁ hyp₂ with
-    zero                                                                  ≡⟨⟩
-    index-of {xs = x ∷ xs} (here p)                                       ≡⟨⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₁ p)                         ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ ≡.sym $
-                                                                               to-from (∼→⊎↔⊎ inv) {x = inj₁ p} hyp₂ ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) $ (from (∼→⊎↔⊎ inv) $ inj₁ q))  ≡⟨ ≡.cong index-of $
-                                                                               strictlyInverseˡ (∷↔ _) (from inv (here q)) ⟩
-    index-of {xs = x ∷ xs} (to (SK-sym inv) $ here q)                   ≡⟨ index-equality-preserved (SK-sym inv) refl ⟩
-    index-of {xs = x ∷ xs} (to (SK-sym inv) $ here p)                   ≡⟨ ≡.cong index-of $ ≡.sym $
-                                                                               strictlyInverseˡ (∷↔ _) (from inv (here p)) ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) (from (∼→⊎↔⊎ inv) $ inj₁ p))  ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $
-                                                                               to-from (∼→⊎↔⊎ inv) {x = inj₂ z∈xs} hyp₁ ⟩
-    index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₂ z∈xs)                      ≡⟨⟩
-    index-of {xs = x ∷ xs} (there z∈xs)                                   ≡⟨⟩
-    suc (index-of {xs = xs} z∈xs)                                         ∎
+  lemma {xs} inv {b = z∈xs} {a = p} {a′ = q} hyp₁ hyp₂ = case contra of λ ()
     where
     open Inverse
     open ≡-Reasoning
-  ... | ()
+    contra : zero ≡ suc (index-of {xs = xs} z∈xs)
+    contra = begin
+      zero
+        ≡⟨⟩
+      index-of {xs = x ∷ xs} (here p)
+        ≡⟨⟩
+      index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₁ p)
+        ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ to-from (∼→⊎↔⊎ inv) {x = inj₁ p} hyp₂ ⟨
+      index-of {xs = x ∷ xs} (to (∷↔ _) $ (from (∼→⊎↔⊎ inv) $ inj₁ q))
+        ≡⟨ ≡.cong index-of $ strictlyInverseˡ (∷↔ _) (from inv (here q)) ⟩
+      index-of {xs = x ∷ xs} (to (SK-sym inv) $ here q)
+        ≡⟨ index-equality-preserved (SK-sym inv) refl ⟩
+      index-of {xs = x ∷ xs} (to (SK-sym inv) $ here p)
+        ≡⟨ ≡.cong index-of $ strictlyInverseˡ (∷↔ _) (from inv (here p)) ⟨
+      index-of {xs = x ∷ xs} (to (∷↔ _) (from (∼→⊎↔⊎ inv) $ inj₁ p))
+        ≡⟨ ≡.cong (index-of ∘ (to (∷↔ (_ ≡_)))) $ to-from (∼→⊎↔⊎ inv) {x = inj₂ z∈xs} hyp₁ ⟩
+      index-of {xs = x ∷ xs} (to (∷↔ _) $ inj₂ z∈xs)
+        ≡⟨⟩
+      index-of {xs = x ∷ xs} (there z∈xs)
+        ≡⟨⟩
+      suc (index-of {xs = xs} z∈xs)
+        ∎
+
 
 ------------------------------------------------------------------------
 -- Relationships to other relations
 
 ↭⇒∼bag : _↭_ {A = A} ⇒ _∼[ bag ]_
-↭⇒∼bag {A = A} xs↭ys {v} = mk↔ₛ′ (to xs↭ys) (from xs↭ys) (to∘from xs↭ys) (from∘to xs↭ys)
+↭⇒∼bag xs↭ys {v} = mk↔ₛ′ (to xs↭ys) (from xs↭ys) (to∘from xs↭ys) (from∘to xs↭ys)
   where
   to : ∀ {xs ys} → xs ↭ ys → v ∈ xs → v ∈ ys
-  to xs↭ys = Any-resp-↭ {A = A} xs↭ys
+  to xs↭ys = ∈-resp-↭ xs↭ys
 
   from : ∀ {xs ys} → xs ↭ ys → v ∈ ys → v ∈ xs
-  from xs↭ys = Any-resp-↭ (↭-sym xs↭ys)
+  from xs↭ys = ∈-resp-↭ (↭-sym xs↭ys)
 
   from∘to : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ xs) → from p (to p q) ≡ q
-  from∘to refl          v∈xs                 = refl
-  from∘to (prep _ _)    (here refl)          = refl
-  from∘to (prep _ p)    (there v∈xs)         = ≡.cong there (from∘to p v∈xs)
-  from∘to (swap x y p)  (here refl)          = refl
-  from∘to (swap x y p)  (there (here refl))  = refl
-  from∘to (swap x y p)  (there (there v∈xs)) = ≡.cong (there ∘ there) (from∘to p v∈xs)
-  from∘to (trans p₁ p₂) v∈xs
-    rewrite from∘to p₂ (Any-resp-↭ p₁ v∈xs)
-          | from∘to p₁ v∈xs                  = refl
+  from∘to = ∈-resp-[σ⁻¹∘σ]
 
   to∘from : ∀ {xs ys} (p : xs ↭ ys) (q : v ∈ ys) → to p (from p q) ≡ q
-  to∘from p with from∘to (↭-sym p)
-  ... | res rewrite ↭-sym-involutive p = res
+  to∘from p with res ← from∘to (↭-sym p) rewrite ↭-sym-involutive p = res
 
 ∼bag⇒↭ : _∼[ bag ]_ ⇒ _↭_ {A = A}
-∼bag⇒↭ {A = A} {[]} eq with empty-unique (↔-sym eq)
-... | refl = refl
-∼bag⇒↭ {A = A} {x ∷ xs} eq with ∈-∃++ (Inverse.to (eq {x}) (here ≡.refl))
-... | zs₁ , zs₂ , p rewrite p = begin
-  x ∷ xs           <⟨ ∼bag⇒↭ (drop-cons (↔-trans eq (comm zs₁ (x ∷ zs₂)))) ⟩
-  x ∷ (zs₂ ++ zs₁) <⟨ ++-comm zs₂ zs₁ ⟩
-  x ∷ (zs₁ ++ zs₂) ↭⟨ shift x zs₁ zs₂ ⟨
-  zs₁ ++ x ∷ zs₂   ∎
-  where
-  open CommutativeMonoid (commutativeMonoid bag A)
-  open PermutationReasoning
+∼bag⇒↭ {A = A} {[]} eq with refl ← empty-unique (↔-sym eq) = refl
+∼bag⇒↭ {A = A} {x ∷ xs} eq
+  with zs₁ , zs₂ , p ← ∈-∃++ (Inverse.to (eq {x}) (here ≡.refl)) rewrite p = begin
+    x ∷ xs           <⟨ ∼bag⇒↭ (drop-cons (↔-trans eq (comm zs₁ (x ∷ zs₂)))) ⟩
+    x ∷ (zs₂ ++ zs₁) <⟨ ++-comm zs₂ zs₁ ⟩
+    x ∷ (zs₁ ++ zs₂) ↭⟨ shift x zs₁ zs₂ ⟨
+    zs₁ ++ x ∷ zs₂   ∎
+    where
+    open CommutativeMonoid (commutativeMonoid bag A)
+    open PermutationReasoning

From 0c7a3d7457dbffef28cf4800f7464ef3d8399284 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 17 Mar 2024 17:10:01 +0000
Subject: [PATCH 247/316] Tidy up functional vector permutation #2066 (#2312)

* added structure and bundle; tidied up

* added structure and bundle; tidied up

* fix order of entries

* blank line

* standardise `variable` names

* alignment
---
 CHANGELOG.md                                  | 36 +++++++++--------
 .../Relation/Binary/Permutation.agda          |  4 +-
 .../Binary/Permutation/Properties.agda        | 40 +++++++++++++++----
 3 files changed, 53 insertions(+), 27 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ee056ad8c9..6bfc0cf99a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -79,7 +79,22 @@ New modules
   ```agda
   Algebra.Module.Construct.Idealization
   ```
-  
+
+* `Data.Vec.Functional.Relation.Binary.Permutation`, defining:
+  ```agda
+  _↭_ : IRel (Vector A) _
+  ```
+
+* `Data.Vec.Functional.Relation.Binary.Permutation.Properties` of the above:
+  ```agda
+  ↭-refl      : Reflexive (Vector A) _↭_
+  ↭-reflexive : xs ≡ ys → xs ↭ ys
+  ↭-sym       : Symmetric (Vector A) _↭_
+  ↭-trans     : Transitive (Vector A) _↭_
+  isIndexedEquivalence : {A : Set a} → IsIndexedEquivalence (Vector A) _↭_
+  indexedSetoid        : {A : Set a} → IndexedSetoid ℕ a _
+  ```
+
 * `Function.Relation.Binary.Equality`
   ```agda
   setoid : Setoid a₁ a₂ → Setoid b₁ b₂ → Setoid _ _
@@ -358,26 +373,13 @@ Additions to existing modules
   WeaklyDecidable : Set _
   ```
 
-* Added new definitions in `Relation.Unary`
-  ```
-  Stable          : Pred A ℓ → Set _
-  WeaklyDecidable : Pred A ℓ → Set _
-  ```
-
 * Added new proof in `Relation.Nullary.Decidable`:
   ```agda
   ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
   ```
 
-* Added module `Data.Vec.Functional.Relation.Binary.Permutation`:
-  ```agda
-  _↭_ : IRel (Vector A) _
+* Added new definitions in `Relation.Unary`
   ```
-
-* Added new file `Data.Vec.Functional.Relation.Binary.Permutation.Properties`:
-  ```agda
-  ↭-refl      : Reflexive (Vector A) _↭_
-  ↭-reflexive : xs ≡ ys → xs ↭ ys
-  ↭-sym       : Symmetric (Vector A) _↭_
-  ↭-trans     : Transitive (Vector A) _↭_
+  Stable          : Pred A ℓ → Set _
+  WeaklyDecidable : Pred A ℓ → Set _
   ```
diff --git a/src/Data/Vec/Functional/Relation/Binary/Permutation.agda b/src/Data/Vec/Functional/Relation/Binary/Permutation.agda
index 58ee430d86..bc4052f215 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Permutation.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Permutation.agda
@@ -17,8 +17,8 @@ open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 private
   variable
-    ℓ : Level
-    A : Set ℓ
+    a : Level
+    A : Set a
 
 infix 3 _↭_
 
diff --git a/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
index df5970dff3..445f96be90 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
@@ -15,18 +15,20 @@ open import Data.Fin.Permutation using (id; flip; _⟨$⟩ʳ_; inverseʳ; _∘
 open import Data.Vec.Functional
 open import Data.Vec.Functional.Relation.Binary.Permutation
 open import Relation.Binary.PropositionalEquality
-  using (refl; trans; _≡_; cong; module ≡-Reasoning)
-open import Relation.Binary.Indexed.Heterogeneous.Definitions
-
-open ≡-Reasoning
+  using (_≡_; refl; trans; cong; module ≡-Reasoning)
+open import Relation.Binary.Indexed.Heterogeneous
 
 private
   variable
-    ℓ : Level
-    A : Set ℓ
+    a : Level
+    A : Set a
     n : ℕ
     xs ys : Vector A n
 
+
+------------------------------------------------------------------------
+-- Basics
+
 ↭-refl : Reflexive (Vector A) _↭_
 ↭-refl = id , λ _ → refl
 
@@ -36,10 +38,32 @@ private
 ↭-sym : Symmetric (Vector A) _↭_
 proj₁ (↭-sym (xs↭ys , _)) = flip xs↭ys
 proj₂ (↭-sym {x = xs} {ys} (xs↭ys , xs↭ys≡)) i = begin
-  ys (flip xs↭ys ⟨$⟩ʳ i)             ≡˘⟨ xs↭ys≡ _ ⟩
+  ys (flip xs↭ys ⟨$⟩ʳ i)              ≡⟨ xs↭ys≡ _ ⟨
   xs (xs↭ys ⟨$⟩ʳ (flip xs↭ys ⟨$⟩ʳ i)) ≡⟨ cong xs (inverseʳ xs↭ys) ⟩
-  xs i ∎
+  xs i                                ∎
+  where open ≡-Reasoning
 
 ↭-trans : Transitive (Vector A) _↭_
 proj₁ (↭-trans (xs↭ys , _) (ys↭zs , _))   = ys↭zs ∘ₚ xs↭ys
 proj₂ (↭-trans (_ , xs↭ys) (_ , ys↭zs)) _ = trans (xs↭ys _) (ys↭zs _)
+
+------------------------------------------------------------------------
+-- Structure
+
+isIndexedEquivalence : IsIndexedEquivalence (Vector A) _↭_
+isIndexedEquivalence {A = A} = record
+  { refl = ↭-refl
+  ; sym = ↭-sym
+  ; trans = λ {n₁ n₂ n₃} {xs : Vector A n₁} {ys : Vector A n₂} {zs : Vector A n₃}
+              xs↭ys ys↭zs → ↭-trans {i = n₁} {i = xs} xs↭ys ys↭zs
+  }
+
+------------------------------------------------------------------------
+-- Bundle
+
+indexedSetoid : {A : Set a} → IndexedSetoid ℕ a _
+indexedSetoid {A = A} = record
+  { Carrier = Vector A
+  ; _≈_ = _↭_
+  ; isEquivalence = isIndexedEquivalence
+  }

From d8158c01fb2c175d6e255eba24524e24e56964c0 Mon Sep 17 00:00:00 2001
From: Uncle Betty <3210857+uncle-betty@users.noreply.github.com>
Date: Mon, 18 Mar 2024 09:40:03 +0100
Subject: [PATCH 248/316] A tweak for the cong! tactic (#2310)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* ⌞_⌟ for marking spots to rewrite.

* Added documentation.

* In case of failure, try swapping the RHS with the LHS. Fixes ≡˘⟨ ... ⟩.

* Added comments.

* More clarity. Less redundancy.

* Fixed performance regression.

* Changelog entry.

* cong!-specific combinators instead of catchTC.

* Support other reasoning modules.

* Clarifying comment.

* Slight performance boost.

* Go back to catchTC.

* Fix example after merge.

* Use renamed backward step.

* Language tweaks.

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                |  3 +++
 doc/README/Tactic/Cong.agda | 27 +++++++++++++++++++++++++--
 src/Tactic/Cong.agda        | 31 ++++++++++++++++++++++++++++---
 3 files changed, 56 insertions(+), 5 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 6bfc0cf99a..1cb95ff90b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -383,3 +383,6 @@ Additions to existing modules
   Stable          : Pred A ℓ → Set _
   WeaklyDecidable : Pred A ℓ → Set _
   ```
+
+* `Tactic.Cong` now provides a marker function, `⌞_⌟`, for user-guided
+  anti-unification. See README.Tactic.Cong for details.
diff --git a/doc/README/Tactic/Cong.agda b/doc/README/Tactic/Cong.agda
index 2a775f06e7..f6adfe9f9f 100644
--- a/doc/README/Tactic/Cong.agda
+++ b/doc/README/Tactic/Cong.agda
@@ -8,7 +8,7 @@ open import Data.Nat.Properties
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; sym; cong; module ≡-Reasoning)
 
-open import Tactic.Cong using (cong!)
+open import Tactic.Cong using (cong! ; ⌞_⌟)
 
 ----------------------------------------------------------------------
 -- Usage
@@ -56,7 +56,7 @@ succinct-example m n eq =
   ∎
 
 ----------------------------------------------------------------------
--- Limitations
+-- Explicit markings
 ----------------------------------------------------------------------
 
 -- The 'cong!' tactic can handle simple cases, but will
@@ -68,6 +68,29 @@ succinct-example m n eq =
 -- to deduce where to generalize. When presented with two sides
 -- of an equality like 'm + n ≡ n + m', it will anti-unify to
 -- 'ϕ + ϕ', which is too specific.
+--
+-- In cases like these, you may explicitly mark the subterms to
+-- be generalized by wrapping them in the marker function, ⌞_⌟.
+
+marker-example₁ : ∀ m n o p → m + n + (o + p) ≡ n + m + (p + o)
+marker-example₁ m n o p =
+  let open ≡-Reasoning in
+  begin
+    ⌞ m + n ⌟ + (o + p)
+  ≡⟨ cong! (+-comm m n) ⟩
+    n + m + ⌞ o + p ⌟
+  ≡⟨ cong! (+-comm p o) ⟨
+    n + m + (p + o)
+  ∎
+
+marker-example₂ : ∀ m n → m + n + (m + n) ≡ n + m + (n + m)
+marker-example₂ m n =
+  let open ≤-Reasoning in
+  begin-equality
+    ⌞ m + n ⌟ + ⌞ m + n ⌟
+  ≡⟨ cong! (+-comm m n) ⟩
+    n + m + (n + m)
+  ∎
 
 ----------------------------------------------------------------------
 -- Unit Tests
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 2c02ec6a4c..0259191800 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -18,6 +18,8 @@
 --   ≡⟨ cong! (+-identityʳ n) ⟨
 --     suc (suc n) + (n + 0)
 --   ∎
+--
+-- Please see README.Tactic.Cong for more details.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -55,6 +57,20 @@ open import Reflection.AST.Term                 as Term
 
 open import Reflection.TCM.Syntax
 
+-- Marker to keep anti-unification from descending into the wrapped
+-- subterm.
+--
+-- For instance, anti-unification of ⌞ a + b ⌟ + c and b + a + c
+-- yields λ ϕ → ϕ + c, as opposed to λ ϕ → ϕ + ϕ + c without ⌞_⌟.
+--
+-- The marker is only visible to the cong! tactic, which inhibits
+-- normalisation. Anywhere else, ⌞ a + b ⌟ reduces to a + b.
+--
+-- Thus, proving ⌞ a + b ⌟ + c ≡ b + a + c via cong! (+-comm a b)
+-- also proves a + b + c ≡ b + a + c.
+⌞_⌟ : ∀ {a} {A : Set a} → A → A
+⌞_⌟ x = x
+
 ------------------------------------------------------------------------
 -- Utilities
 ------------------------------------------------------------------------
@@ -136,6 +152,8 @@ private
   antiUnifyClauses : ℕ → Clauses → Clauses → Maybe Clauses
   antiUnifyClause  : ℕ → Clause → Clause → Maybe Clause
 
+  pattern apply-⌞⌟ t = (def (quote ⌞_⌟) (_ ∷ _ ∷ arg _ t ∷ []))
+
   antiUnify ϕ (var x args) (var y args') with x ℕ.≡ᵇ y | antiUnifyArgs ϕ args args'
   ... | _     | nothing    = var ϕ []
   ... | false | just uargs = var ϕ uargs
@@ -144,6 +162,7 @@ private
   ... | _     | nothing    = var ϕ []
   ... | false | just uargs = var ϕ []
   ... | true  | just uargs = con c uargs
+  antiUnify ϕ (def f args) (apply-⌞⌟ t) = antiUnify ϕ (def f args) t
   antiUnify ϕ (def f args) (def f' args') with f Name.≡ᵇ f' | antiUnifyArgs ϕ args args'
   ... | _     | nothing    = var ϕ []
   ... | false | just uargs = var ϕ []
@@ -217,6 +236,12 @@ macro
     withNormalisation false $ do
       goal ← inferType hole
       eqGoal ← destructEqualityGoal goal
-      let cong-lam = antiUnify 0 (EqualityGoal.lhs eqGoal) (EqualityGoal.rhs eqGoal)
-      cong-tm ← `cong eqGoal cong-lam x≡y
-      unify cong-tm hole
+      let uni = λ lhs rhs → do
+        let cong-lam = antiUnify 0 lhs rhs
+        cong-tm ← `cong eqGoal cong-lam x≡y
+        unify cong-tm hole
+      let lhs = EqualityGoal.lhs eqGoal
+      let rhs = EqualityGoal.rhs eqGoal
+      -- When using ⌞_⌟ with ≡⟨ ... ⟨, (uni lhs rhs) fails and
+      -- (uni rhs lhs) succeeds.
+      catchTC (uni lhs rhs) (uni rhs lhs)

From 204b2b251afd676c225bd12b5ad7b006ee0d3659 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 21 Mar 2024 01:07:27 +0000
Subject: [PATCH 249/316] Simplify `Data.List.Relation.Unary.Any.Properties`:
 remove dependency on `List.Properties` (#2323)

* simplifed proof; removed dependency on `List.Properties`

* corrected module long name in comment
---
 .../Propositional/Properties/Core.agda        |  4 ++--
 .../List/Relation/Unary/Any/Properties.agda   | 19 ++++++++-----------
 2 files changed, 10 insertions(+), 13 deletions(-)

diff --git a/src/Data/List/Membership/Propositional/Properties/Core.agda b/src/Data/List/Membership/Propositional/Properties/Core.agda
index 7ad4f94573..ca8ffbb1ea 100644
--- a/src/Data/List/Membership/Propositional/Properties/Core.agda
+++ b/src/Data/List/Membership/Propositional/Properties/Core.agda
@@ -5,8 +5,8 @@
 ------------------------------------------------------------------------
 
 -- This file is needed to break the cyclic dependency with the proof
--- `Any-cong` in `Data.Any.Properties` which relies on `Any↔` in this
--- file.
+-- `Any-cong` in `Data.List.Relation.Unary.Any.Properties` which relies
+-- on `Any↔` defined in this file.
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index a6eaa9d480..6383f15a7e 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -13,8 +13,7 @@ open import Data.Bool.Properties using (T-∨; T-≡)
 open import Data.Empty using (⊥)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List hiding (find)
-open import Data.List.Properties using (ʳ++-defn)
-open import Data.List.Effectful as Listₑ using (monad)
+open import Data.List.Effectful as List using (monad)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional
 open import Data.List.Membership.Propositional.Properties.Core
@@ -624,18 +623,16 @@ reverseAcc⁺ acc (x ∷ xs) (inj₂ (there y)) = reverseAcc⁺ (x ∷ acc) xs (
 
 reverseAcc⁻ : ∀ acc xs → Any P (reverseAcc acc xs) → Any P acc ⊎ Any P xs
 reverseAcc⁻ acc []       ps = inj₁ ps
-reverseAcc⁻ acc (x ∷ xs) ps rewrite ʳ++-defn xs {x ∷ acc} with ++⁻ (reverseAcc [] xs) ps
-... | inj₂ (here p') = inj₂ (here p')
-... | inj₂ (there ps') = inj₁ ps'
-... | inj₁ ps' with reverseAcc⁻ [] xs ps'
-...   | inj₂ ps'' = inj₂ (there ps'')
+reverseAcc⁻ acc (x ∷ xs) ps with reverseAcc⁻ (x ∷ acc) xs ps
+... | inj₁ (here px) = inj₂ (here px)
+... | inj₁ (there pxs) = inj₁ pxs
+... | inj₂ pxs = inj₂ (there pxs)
 
 reverse⁺ : Any P xs → Any P (reverse xs)
 reverse⁺ ps = reverseAcc⁺ [] _ (inj₂ ps)
 
 reverse⁻ : Any P (reverse xs) → Any P xs
-reverse⁻ ps with reverseAcc⁻ [] _ ps
-... | inj₂ ps' = ps'
+reverse⁻ ps with inj₂ pxs ← reverseAcc⁻ [] _ ps = pxs
 
 ------------------------------------------------------------------------
 -- pure
@@ -686,7 +683,7 @@ module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where
   Any (λ f → Any (P ∘ f) xs) fs                ↔⟨ Any-cong (λ _ → Any-cong (λ _ → pure↔) (_ ∎)) (_ ∎) ⟩
   Any (λ f → Any (Any P ∘ pure ∘ f) xs) fs     ↔⟨ Any-cong (λ _ → >>=↔ ) (_ ∎) ⟩
   Any (λ f → Any P (xs >>= pure ∘ f)) fs       ↔⟨ >>=↔ ⟩
-  Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡⟨ cong (Any P) (Listₑ.Applicative.unfold-⊛ fs xs) ⟨
+  Any P (fs >>= λ f → xs >>= λ x → pure (f x)) ≡⟨ cong (Any P) (List.Applicative.unfold-⊛ fs xs) ⟨
   Any P (fs ⊛ xs)                               ∎
   where open Related.EquationalReasoning
 
@@ -706,7 +703,7 @@ module _ {A B : Set ℓ} {P : B → Set p} {f : A → List B} where
   Any (λ x → Any (λ y → P (x , y)) ys) xs                           ↔⟨ pure↔ ⟩
   Any (λ _,_ → Any (λ x → Any (λ y → P (x , y)) ys) xs) (pure _,_)  ↔⟨ ⊛↔ ⟩
   Any (λ x, → Any (P ∘ x,) ys) (pure _,_ ⊛ xs)                      ↔⟨ ⊛↔ ⟩
-  Any P (pure _,_ ⊛ xs ⊛ ys)                                        ≡⟨ cong (Any P ∘′ (_⊛ ys)) (Listₑ.Applicative.unfold-<$> _,_ xs) ⟨
+  Any P (pure _,_ ⊛ xs ⊛ ys)                                        ≡⟨ cong (Any P ∘′ (_⊛ ys)) (List.Applicative.unfold-<$> _,_ xs) ⟨
   Any P (xs ⊗ ys)                                                   ∎
   where open Related.EquationalReasoning
 

From 5989a60e1e8cdf49929dfab60d806d0f04c73fdd Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 24 Mar 2024 07:49:12 +0000
Subject: [PATCH 250/316] Refactor `Data.Integer.Divisibility.Signed` (#2307)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* removed extra space

* refactor to introduce new `Data.Integer.Properties`

* refactor proofs to use new properties; streamline reasoning

* remove final blank line

* first review comment: missing annotation

* removed two new lemmas: `i*j≢0⇒i≢0` and `i*j≢0⇒j≢0`
---
 CHANGELOG.md                              |  9 ++++-
 src/Data/Integer/Divisibility.agda        |  2 +-
 src/Data/Integer/Divisibility/Signed.agda | 49 ++++++++++-------------
 src/Data/Integer/Properties.agda          | 24 +++++++----
 4 files changed, 47 insertions(+), 37 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 1cb95ff90b..186fe9d051 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -229,11 +229,18 @@ Additions to existing modules
   nonZeroIndex : Fin n → ℕ.NonZero n
   ```
 
-* In `Data.Integer.Divisisbility`: introduce `divides` as an explicit pattern synonym
+* In `Data.Integer.Divisibility`: introduce `divides` as an explicit pattern synonym
   ```agda
   pattern divides k eq = Data.Nat.Divisibility.divides k eq
   ```
 
+* In `Data.Integer.Properties`:
+  ```agda
+  ◃-nonZero : .{{_ : ℕ.NonZero n}} → NonZero (s ◃ n)
+  sign-*    : .{{NonZero (i * j)}} → sign (i * j) ≡ sign i Sign.* sign j
+  i*j≢0     : .{{_ : NonZero i}} .{{_ : NonZero j}} → NonZero (i * j)
+  ```
+
 * In `Data.List.Properties`:
   ```agda
   applyUpTo-∷ʳ          : applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
diff --git a/src/Data/Integer/Divisibility.agda b/src/Data/Integer/Divisibility.agda
index 28963c8d24..7cd8109fce 100644
--- a/src/Data/Integer/Divisibility.agda
+++ b/src/Data/Integer/Divisibility.agda
@@ -27,7 +27,7 @@ infix 4 _∣_
 _∣_ : Rel ℤ 0ℓ
 _∣_ = ℕ._∣_ on ∣_∣
 
-pattern divides k eq  = ℕ.divides k eq
+pattern divides k eq = ℕ.divides k eq
 
 ------------------------------------------------------------------------
 -- Properties of divisibility
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index 15fa5a4c1e..18df0c8a91 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -29,6 +29,7 @@ import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 open import Relation.Nullary.Decidable as Dec using (yes; no)
 open import Relation.Binary.Reasoning.Syntax
 
+
 ------------------------------------------------------------------------
 -- Type
 
@@ -44,43 +45,35 @@ open _∣_ using (quotient) public
 -- Conversion between signed and unsigned divisibility
 
 ∣ᵤ⇒∣ : ∀ {k i} → k Unsigned.∣ i → k ∣ i
-∣ᵤ⇒∣ {k} {i} (Unsigned.divides 0           eq) = divides (+ 0) (∣i∣≡0⇒i≡0 eq)
+∣ᵤ⇒∣ {k} {i} (Unsigned.divides 0           eq) = divides +0 (∣i∣≡0⇒i≡0 eq)
 ∣ᵤ⇒∣ {k} {i} (Unsigned.divides q@(ℕ.suc _) eq) with k ≟ +0
 ... | yes refl = divides +0 (∣i∣≡0⇒i≡0 (trans eq (ℕ.*-zeroʳ q)))
-... | no  neq  = divides (sign i Sign.* sign k ◃ q) (◃-cong sign-eq abs-eq)
+... | no  neq  = divides s[i*k]◃q (◃-cong sign-eq abs-eq)
   where
-  ikq = sign i Sign.* sign k ◃ q
-
-  *-nonZero : ∀ m n .{{_ : ℕ.NonZero m}} .{{_ : ℕ.NonZero n}} → ℕ.NonZero (m ℕ.* n)
-  *-nonZero (ℕ.suc _) (ℕ.suc _) = _
-
-  ◃-nonZero : ∀ s n .{{_ : ℕ.NonZero n}} → NonZero (s ◃ n)
-  ◃-nonZero Sign.- (ℕ.suc _) = _
-  ◃-nonZero Sign.+ (ℕ.suc _) = _
-
-  ikq≢0 : NonZero ikq
-  ikq≢0 = ◃-nonZero (sign i Sign.* sign k) q
+  s[i*k] = sign i Sign.* sign k
+  s[i*k]◃q = s[i*k] ◃ q
 
   instance
-    ikq*∣k∣≢0 : ℕ.NonZero (∣ ikq ∣ ℕ.* ∣ k ∣)
-    ikq*∣k∣≢0 = *-nonZero ∣ ikq ∣ ∣ k ∣ {{ikq≢0}} {{≢-nonZero neq}}
+    _ = ≢-nonZero neq
+    _ = ◃-nonZero s[i*k] q
+    _ = i*j≢0 s[i*k]◃q k
 
-  sign-eq : sign i ≡ sign (ikq * k)
+  sign-eq : sign i ≡ sign (s[i*k]◃q * k)
   sign-eq = sym $ begin
-    sign (ikq * k)                        ≡⟨ sign-◃ (sign ikq Sign.* sign k) (∣ ikq ∣ ℕ.* ∣ k ∣) ⟩
-    sign ikq Sign.* sign k                ≡⟨ cong (Sign._* sign k) (sign-◃ (sign i Sign.* sign k) q) ⟩
-    (sign i Sign.* sign k) Sign.* sign k  ≡⟨ Sign.*-assoc (sign i) (sign k) (sign k) ⟩
+    sign (s[i*k]◃q * k)                   ≡⟨ sign-* s[i*k]◃q k ⟩
+    sign s[i*k]◃q Sign.* sign k           ≡⟨ cong (Sign._* _) (sign-◃ s[i*k] q) ⟩
+    s[i*k] Sign.* sign k                  ≡⟨ Sign.*-assoc (sign i) (sign k) (sign k) ⟩
     sign i Sign.* (sign k Sign.* sign k)  ≡⟨ cong (sign i Sign.*_) (Sign.s*s≡+ (sign k)) ⟩
     sign i Sign.* Sign.+                  ≡⟨ Sign.*-identityʳ (sign i) ⟩
-    sign i                          ∎
+    sign i                                ∎
     where open ≡-Reasoning
 
-  abs-eq : ∣ i ∣ ≡ ∣ ikq * k ∣
+  abs-eq : ∣ i ∣ ≡ ∣ s[i*k]◃q * k ∣
   abs-eq = sym $ begin
-    ∣ ikq * k ∣        ≡⟨ ∣i*j∣≡∣i∣*∣j∣ ikq k ⟩
-    ∣ ikq ∣ ℕ.* ∣ k ∣  ≡⟨ cong (ℕ._* ∣ k ∣) (abs-◃ (sign i Sign.* sign k) q) ⟩
-    q ℕ.* ∣ k ∣        ≡⟨ sym eq ⟩
-    ∣ i ∣              ∎
+    ∣ s[i*k]◃q * k ∣        ≡⟨ abs-* s[i*k]◃q k ⟩
+    ∣ s[i*k]◃q ∣ ℕ.* ∣ k ∣  ≡⟨ cong (ℕ._* ∣ k ∣) (abs-◃ s[i*k] q) ⟩
+    q ℕ.* ∣ k ∣             ≡⟨ eq ⟨
+    ∣ i ∣                   ∎
     where open ≡-Reasoning
 
 ∣⇒∣ᵤ : ∀ {k i} → k ∣ i → k Unsigned.∣ i
@@ -148,7 +141,7 @@ m∣∣m∣ = ∣ᵤ⇒∣ ℕ.∣-refl
 ∣m⇒∣-m : ∀ {i m} → i ∣ m → i ∣ - m
 ∣m⇒∣-m {i} {m} i∣m = ∣ᵤ⇒∣ $′ begin
   ∣ i ∣   ∣⟨ ∣⇒∣ᵤ i∣m ⟩
-  ∣ m ∣   ≡⟨ sym (∣-i∣≡∣i∣ m) ⟩
+  ∣ m ∣   ≡⟨ ∣-i∣≡∣i∣ m ⟨
   ∣ - m ∣ ∎
   where open ℕ.∣-Reasoning
 
@@ -159,7 +152,7 @@ m∣∣m∣ = ∣ᵤ⇒∣ ℕ.∣-refl
 ∣m+n∣m⇒∣n {i} {m} {n} i∣m+n i∣m = begin
   i             ∣⟨ ∣m∣n⇒∣m-n i∣m+n i∣m ⟩
   m + n - m     ≡⟨ +-comm (m + n) (- m) ⟩
-  - m + (m + n) ≡⟨ sym (+-assoc (- m) m n) ⟩
+  - m + (m + n) ≡⟨ +-assoc (- m) m n ⟨
   - m + m + n   ≡⟨ cong (_+ n) (+-inverseˡ m) ⟩
   + 0 + n       ≡⟨ +-identityˡ n ⟩
   n             ∎
@@ -171,7 +164,7 @@ m∣∣m∣ = ∣ᵤ⇒∣ ℕ.∣-refl
 ∣n⇒∣m*n : ∀ {i} m {n} → i ∣ n → i ∣ m * n
 ∣n⇒∣m*n {i} m {n} (divides q eq) = divides (m * q) $′ begin
   m * n       ≡⟨ cong (m *_) eq ⟩
-  m * (q * i) ≡⟨ sym (*-assoc m q i) ⟩
+  m * (q * i) ≡⟨ *-assoc m q i ⟨
   m * q * i   ∎
   where open ≡-Reasoning
 
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 840b4302b5..d5706b5d53 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -23,7 +23,7 @@ import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
 open import Data.Product.Base using (proj₁; proj₂; _,_; _×_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
-open import Data.Sign as Sign using (Sign) renaming (_*_ to _𝕊*_)
+open import Data.Sign as Sign using (Sign)
 import Data.Sign.Properties as Sign
 open import Function.Base using (_∘_; _$_; id)
 open import Level using (0ℓ)
@@ -508,6 +508,10 @@ neg-cancel-< { -[1+ m ]} { -[1+ n ]} (+<+ m<n) = -<- (s<s⁻¹ m<n)
 ------------------------------------------------------------------------
 -- Properties of sign and _◃_
 
+◃-nonZero : ∀ s n .{{_ : ℕ.NonZero n}} → NonZero (s ◃ n)
+◃-nonZero Sign.- (ℕ.suc _) = _
+◃-nonZero Sign.+ (ℕ.suc _) = _
+
 ◃-inverse : ∀ i → sign i ◃ ∣ i ∣ ≡ i
 ◃-inverse -[1+ n ] = refl
 ◃-inverse +0       = refl
@@ -1348,7 +1352,7 @@ private
 *-assoc i j +0 rewrite
     ℕ.*-zeroʳ ∣ j ∣
   | ℕ.*-zeroʳ ∣ i ∣
-  | ℕ.*-zeroʳ ∣ sign i 𝕊* sign j ◃ ∣ i ∣ ℕ.* ∣ j ∣ ∣
+  | ℕ.*-zeroʳ ∣ sign i Sign.* sign j ◃ ∣ i ∣ ℕ.* ∣ j ∣ ∣
   = refl
 *-assoc -[1+ m ] -[1+ n ] +[1+ o ] = cong (+_ ∘ suc) (lemma m n o)
 *-assoc -[1+ m ] +[1+ n ] -[1+ o ] = cong (+_ ∘ suc) (lemma m n o)
@@ -1390,11 +1394,11 @@ private
         = refl
 *-distribʳ-+ x +0 z
   rewrite +-identityˡ z
-        | +-identityˡ (sign z 𝕊* sign x ◃ ∣ z ∣ ℕ.* ∣ x ∣)
+        | +-identityˡ (sign z Sign.* sign x ◃ ∣ z ∣ ℕ.* ∣ x ∣)
         = refl
 *-distribʳ-+ x y +0
   rewrite +-identityʳ y
-        | +-identityʳ (sign y 𝕊* sign x ◃ ∣ y ∣ ℕ.* ∣ x ∣)
+        | +-identityʳ (sign y Sign.* sign x ◃ ∣ y ∣ ℕ.* ∣ x ∣)
         = refl
 *-distribʳ-+ -[1+ m ] -[1+ n ] -[1+ o ] = cong (+_) $
   solve 3 (λ m n o → (con 2 :+ n :+ o) :* (con 1 :+ m)
@@ -1594,6 +1598,9 @@ private
 abs-* : ℤtoℕ.Homomorphic₂ ∣_∣ _*_ ℕ._*_
 abs-* i j = abs-◃ _ _
 
+sign-* : ∀ i j → .{{NonZero (i * j)}} → sign (i * j) ≡ sign i Sign.* sign j
+sign-* i j rewrite abs-* i j = sign-◃ (sign i Sign.* sign j) (∣ i ∣ ℕ.* ∣ j ∣)
+
 *-cancelʳ-≡ : ∀ i j k .{{_ : NonZero k}} → i * k ≡ j * k → i ≡ j
 *-cancelʳ-≡ i j k eq with sign-cong′ eq
 ... | inj₁ s[ik]≡s[jk] = ◃-cong
@@ -1631,6 +1638,9 @@ i*j≡0⇒i≡0∨j≡0 i p with ℕ.m*n≡0⇒m≡0∨n≡0 ∣ i ∣ (abs-cong
 ... | inj₁ ∣i∣≡0 = inj₁ (∣i∣≡0⇒i≡0 ∣i∣≡0)
 ... | inj₂ ∣j∣≡0 = inj₂ (∣i∣≡0⇒i≡0 ∣j∣≡0)
 
+i*j≢0 : ∀ i j .{{_ : NonZero i}} .{{_ : NonZero j}} → NonZero (i * j)
+i*j≢0 i j rewrite abs-* i j = ℕ.m*n≢0 ∣ i ∣ ∣ j ∣
+
 ------------------------------------------------------------------------
 -- Properties of _^_
 ------------------------------------------------------------------------
@@ -1704,7 +1714,7 @@ neg-distribʳ-* i j = begin
 ------------------------------------------------------------------------
 -- Properties of _*_ and _◃_
 
-◃-distrib-* :  ∀ s t m n → (s 𝕊* t) ◃ (m ℕ.* n) ≡ (s ◃ m) * (t ◃ n)
+◃-distrib-* :  ∀ s t m n → (s Sign.* t) ◃ (m ℕ.* n) ≡ (s ◃ m) * (t ◃ n)
 ◃-distrib-* s t zero    zero    = refl
 ◃-distrib-* s t zero    (suc n) = refl
 ◃-distrib-* s t (suc m) zero    =
@@ -1713,7 +1723,7 @@ neg-distribʳ-* i j = begin
     (*-comm (t ◃ zero) (s ◃ suc m))
 ◃-distrib-* s t (suc m) (suc n) =
   sym (cong₂ _◃_
-    (cong₂ _𝕊*_ (sign-◃ s (suc m)) (sign-◃ t (suc n)))
+    (cong₂ Sign._*_ (sign-◃ s (suc m)) (sign-◃ t (suc n)))
     (∣s◃m∣*∣t◃n∣≡m*n s t (suc m) (suc n)))
 
 ------------------------------------------------------------------------
@@ -1828,7 +1838,7 @@ neg-distribʳ-* i j = begin
 -- Properties of _*_ and ∣_∣
 
 ∣i*j∣≡∣i∣*∣j∣ : ∀ i j → ∣ i * j ∣ ≡ ∣ i ∣ ℕ.* ∣ j ∣
-∣i*j∣≡∣i∣*∣j∣ i j = abs-◃ (sign i 𝕊* sign j) (∣ i ∣ ℕ.* ∣ j ∣)
+∣i*j∣≡∣i∣*∣j∣ = abs-*
 
 ------------------------------------------------------------------------
 -- Properties of _⊓_ and _⊔_

From 014a069f9f572582d555df41fb190034a45f99f1 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Sun, 24 Mar 2024 11:18:45 +0100
Subject: [PATCH 251/316] Sublists: generalize disjoint-union-is-cospan to
 upper-bound-is-cospan (#2320)

* Sublists: generalize disjoint-union-is-cospan to upper-bound-is-cospan

The disjointness assumption is superfluous.

* Move sublist cospan stuff to new module SubList.Propositional.Slice

* CHANGELOG
---
 CHANGELOG.md                                  | 24 ++++--
 .../Sublist/Propositional/Disjoint.agda       | 53 +++++++------
 .../Binary/Sublist/Propositional/Slice.agda   | 79 +++++++++++++++++++
 .../List/Relation/Binary/Sublist/Setoid.agda  | 48 +++++++----
 standard-library.agda-lib                     |  3 +-
 5 files changed, 159 insertions(+), 48 deletions(-)
 create mode 100644 src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 186fe9d051..2b9d357511 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,7 +1,7 @@
 Version 2.1-dev
 ===============
 
-The library has been tested using Agda 2.6.4 and 2.6.4.1.
+The library has been tested using Agda 2.6.4, 2.6.4.1, and 2.6.4.3.
 
 Highlights
 ----------
@@ -33,6 +33,9 @@ Other major improvements
 Deprecated modules
 ------------------
 
+* `Data.List.Relation.Binary.Sublist.Propositional.Disjoint` deprecated in favour of
+  `Data.List.Relation.Binary.Sublist.Propositional.Slice`.
+
 Deprecated names
 ----------------
 
@@ -80,6 +83,12 @@ New modules
   Algebra.Module.Construct.Idealization
   ```
 
+* `Data.List.Relation.Binary.Sublist.Propositional.Slice`
+  replacing `Data.List.Relation.Binary.Sublist.Propositional.Disjoint` (*)
+  and adding new functions:
+  - `⊆-upper-bound-is-cospan` generalising `⊆-disjoint-union-is-cospan` from (*)
+  - `⊆-upper-bound-cospan` generalising `⊆-disjoint-union-cospan` from (*)
+
 * `Data.Vec.Functional.Relation.Binary.Permutation`, defining:
   ```agda
   _↭_ : IRel (Vector A) _
@@ -170,7 +179,7 @@ Additions to existing modules
   quasigroup      : Quasigroup _ _
   isLoop          : IsLoop _∙_ _\\_ _//_ ε
   loop            : Loop _ _
-  
+
   \\-leftDividesˡ  : LeftDividesˡ _∙_ _\\_
   \\-leftDividesʳ  : LeftDividesʳ _∙_ _\\_
   \\-leftDivides   : LeftDivides _∙_ _\\_
@@ -189,7 +198,7 @@ Additions to existing modules
   identityʳ-unique : x ∙ y ≈ x → y ≈ ε
   identity-unique  : Identity x _∙_ → x ≈ ε
   ```
- 
+
 * In `Algebra.Construct.Terminal`:
   ```agda
   rawNearSemiring : RawNearSemiring c ℓ
@@ -218,7 +227,7 @@ Additions to existing modules
   _\\_ : Op₂ A
   x \\ y = (x ⁻¹) ∙ y
   ```
- 
+
 * In `Data.Container.Indexed.Core`:
   ```agda
   Subtrees o c = (r : Response c) → X (next c r)
@@ -301,6 +310,11 @@ Additions to existing modules
   pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
   ```
 
+* In `Data.List.Relation.Binary.Sublist.Setoid`:
+  ```agda
+  ⊆-upper-bound : ∀ {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) → UpperBound τ σ
+  ```
+
 * In `Data.Nat.Divisibility`:
   ```agda
   quotient≢0       : m ∣ n → .{{NonZero n}} → NonZero quotient
@@ -327,7 +341,7 @@ Additions to existing modules
   pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n
   pred-cancel-≡ : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n
   ```
-  
+
 * Added new proofs to `Data.Nat.Primality`:
   ```agda
   rough∧square>⇒prime : .{{NonTrivial n}} → m Rough n → m * m > n → Prime n
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda
index a6a33cc43a..97557891a9 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Disjoint.agda
@@ -1,7 +1,9 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Sublist-related properties
+-- This module is DEPRECATED.
+-- Please use `Data.List.Relation.Binary.Sublist.Propositional.Slice`
+-- instead.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
@@ -9,26 +11,23 @@
 module Data.List.Relation.Binary.Sublist.Propositional.Disjoint
   {a} {A : Set a} where
 
+{-# WARNING_ON_IMPORT
+"Data.List.Relation.Binary.Sublist.Propositional.Disjoint was deprecated in v2.1.
+Use Data.List.Relation.Binary.Sublist.Propositional.Slice instead."
+#-}
+
 open import Data.List.Base using (List)
-open import Data.List.Relation.Binary.Sublist.Propositional
+open import Data.List.Relation.Binary.Sublist.Propositional using
+  ( _⊆_; _∷_; _∷ʳ_
+  ; Disjoint; ⊆-disjoint-union; _∷ₙ_; _∷ₗ_; _∷ᵣ_
+  )
+import Data.List.Relation.Binary.Sublist.Propositional.Slice as SPSlice
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 
-------------------------------------------------------------------------
--- A Union where the triangles commute is a
--- Cospan in the slice category (_ ⊆ zs).
-
-record IsCospan {xs ys zs : List A} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} (u : UpperBound τ₁ τ₂) : Set a where
-  field
-    tri₁ : ⊆-trans (UpperBound.inj₁ u) (UpperBound.sub u) ≡ τ₁
-    tri₂ : ⊆-trans (UpperBound.inj₂ u) (UpperBound.sub u) ≡ τ₂
+open SPSlice using (⊆-upper-bound-is-cospan; ⊆-upper-bound-cospan)
 
-record Cospan {xs ys zs : List A} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) : Set a where
-  field
-    upperBound : UpperBound τ₁ τ₂
-    isCospan   : IsCospan upperBound
-
-  open UpperBound upperBound public
-  open IsCospan isCospan public
+-- For backward compatibility reexport these:
+open SPSlice public using ( IsCospan; Cospan )
 
 open IsCospan
 open Cospan
@@ -57,14 +56,18 @@ module _
 
 ⊆-disjoint-union-is-cospan : ∀ {xs ys zs : List A} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →
   (d : Disjoint τ₁ τ₂) → IsCospan (⊆-disjoint-union d)
-⊆-disjoint-union-is-cospan [] = record { tri₁ = refl ; tri₂ = refl }
-⊆-disjoint-union-is-cospan (x    ∷ₙ d) = ∷ₙ-cospan d (⊆-disjoint-union-is-cospan d)
-⊆-disjoint-union-is-cospan (refl ∷ₗ d) = ∷ₗ-cospan d (⊆-disjoint-union-is-cospan d)
-⊆-disjoint-union-is-cospan (refl ∷ᵣ d) = ∷ᵣ-cospan d (⊆-disjoint-union-is-cospan d)
+⊆-disjoint-union-is-cospan {τ₁ = τ₁} {τ₂ = τ₂} _ = ⊆-upper-bound-is-cospan τ₁ τ₂
 
 ⊆-disjoint-union-cospan : ∀ {xs ys zs : List A} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} →
   Disjoint τ₁ τ₂ → Cospan τ₁ τ₂
-⊆-disjoint-union-cospan d = record
-  { upperBound = ⊆-disjoint-union d
-  ; isCospan   = ⊆-disjoint-union-is-cospan d
-  }
+⊆-disjoint-union-cospan {τ₁ = τ₁} {τ₂ = τ₂} _ = ⊆-upper-bound-cospan τ₁ τ₂
+
+{-# WARNING_ON_USAGE ⊆-disjoint-union-is-cospan
+"Warning: ⊆-disjoint-union-is-cospan was deprecated in v2.1.
+Please use `⊆-upper-bound-is-cospan` from `Data.List.Relation.Binary.Sublist.Propositional.Slice` instead."
+#-}
+
+{-# WARNING_ON_USAGE ⊆-disjoint-union-cospan
+"Warning: ⊆-disjoint-union-cospan was deprecated in v2.1.
+Please use `⊆-upper-bound-cospan` from `Data.List.Relation.Binary.Sublist.Propositional.Slice` instead."
+#-}
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda
new file mode 100644
index 0000000000..949fded0ae
--- /dev/null
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda
@@ -0,0 +1,79 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Slices in the propositional sublist category.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.Relation.Binary.Sublist.Propositional.Slice
+  {a} {A : Set a} where
+
+open import Data.List.Base using (List)
+open import Data.List.Relation.Binary.Sublist.Propositional
+open import Relation.Binary.PropositionalEquality
+
+------------------------------------------------------------------------
+-- A Union where the triangles commute is a
+-- Cospan in the slice category (_ ⊆ zs).
+
+record IsCospan {xs ys zs : List A} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs} (u : UpperBound τ₁ τ₂) : Set a where
+  field
+    tri₁ : ⊆-trans (UpperBound.inj₁ u) (UpperBound.sub u) ≡ τ₁
+    tri₂ : ⊆-trans (UpperBound.inj₂ u) (UpperBound.sub u) ≡ τ₂
+
+record Cospan {xs ys zs : List A} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) : Set a where
+  field
+    upperBound : UpperBound τ₁ τ₂
+    isCospan   : IsCospan upperBound
+
+  open UpperBound upperBound public
+  open IsCospan isCospan public
+
+open IsCospan
+open Cospan
+
+module _
+  {x : A} {xs ys zs : List A} {τ₁ : xs ⊆ zs} {τ₂ : ys ⊆ zs}
+  {u : UpperBound τ₁ τ₂} (c : IsCospan u) where
+  open UpperBound u
+  open IsCospan c
+
+  ∷ₙ-cospan : IsCospan (∷ₙ-ub u)
+  ∷ₙ-cospan = record
+    { tri₁ = cong (x ∷ʳ_) (c .tri₁)
+    ; tri₂ = cong (x ∷ʳ_) (c .tri₂)
+    }
+
+  ∷ₗ-cospan : IsCospan (refl {x = x} ∷ₗ-ub u)
+  ∷ₗ-cospan = record
+    { tri₁ = cong (refl ∷_) (c .tri₁)
+    ; tri₂ = cong (x   ∷ʳ_) (c .tri₂)
+    }
+
+  ∷ᵣ-cospan : IsCospan (refl {x = x} ∷ᵣ-ub u)
+  ∷ᵣ-cospan = record
+    { tri₁ = cong (x   ∷ʳ_) (c .tri₁)
+    ; tri₂ = cong (refl ∷_) (c .tri₂)
+    }
+
+  ∷-cospan : IsCospan (refl {x = x} , refl {x = x} ∷-ub u)
+  ∷-cospan = record
+   { tri₁ =  cong (refl ∷_) (c .tri₁)
+   ; tri₂ =  cong (refl ∷_) (c .tri₂)
+   }
+
+⊆-upper-bound-is-cospan : ∀ {xs ys zs : List A} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) →
+  IsCospan (⊆-upper-bound τ₁ τ₂)
+⊆-upper-bound-is-cospan [] [] = record { tri₁ = refl ; tri₂ = refl }
+⊆-upper-bound-is-cospan (z   ∷ʳ τ₁) (.z  ∷ʳ τ₂) = ∷ₙ-cospan (⊆-upper-bound-is-cospan τ₁ τ₂)
+⊆-upper-bound-is-cospan (z   ∷ʳ τ₁) (refl ∷ τ₂) = ∷ᵣ-cospan (⊆-upper-bound-is-cospan τ₁ τ₂)
+⊆-upper-bound-is-cospan (refl ∷ τ₁) (z   ∷ʳ τ₂) = ∷ₗ-cospan (⊆-upper-bound-is-cospan τ₁ τ₂)
+⊆-upper-bound-is-cospan (refl ∷ τ₁) (refl ∷ τ₂) = ∷-cospan (⊆-upper-bound-is-cospan τ₁ τ₂)
+
+⊆-upper-bound-cospan : ∀ {xs ys zs : List A} (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) →
+  Cospan τ₁ τ₂
+⊆-upper-bound-cospan τ₁ τ₂ = record
+  { upperBound = ⊆-upper-bound τ₁ τ₂
+  ; isCospan   = ⊆-upper-bound-is-cospan τ₁ τ₂
+  }
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid.agda b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
index 83a16444d0..2fc0373d67 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid.agda
@@ -162,7 +162,7 @@ record RawPushout {xs ys zs : List A} (τ : xs ⊆ ys) (σ : xs ⊆ zs) : Set (c
     leg₁         : ys ⊆ upperBound
     leg₂         : zs ⊆ upperBound
 
-open RawPushout
+open RawPushout using (leg₁; leg₂)
 
 ------------------------------------------------------------------------
 -- Extending corners of a raw pushout square
@@ -211,13 +211,25 @@ z ∷ʳ₂ rpo = record
 
 ⊆-joinˡ : ∀ {xs ys zs : List A} →
           (τ : xs ⊆ ys) (σ : xs ⊆ zs) → ∃ λ us → xs ⊆ us
-⊆-joinˡ τ σ = upperBound rpo , ⊆-trans τ (leg₁ rpo)
+⊆-joinˡ τ σ = RawPushout.upperBound rpo , ⊆-trans τ (leg₁ rpo)
   where
   rpo = ⊆-pushoutˡ τ σ
 
 
 ------------------------------------------------------------------------
 -- Upper bound of two sublists xs,ys ⊆ zs
+--
+-- Two sublists τ : xs ⊆ zs and σ : ys ⊆ zs
+-- can be joined in a unique way if τ and σ are respected.
+--
+-- For instance, if τ : [x] ⊆ [x,y,x] and σ : [y] ⊆ [x,y,x]
+-- then the union will be [x,y] or [y,x], depending on whether
+-- τ picks the first x or the second one.
+--
+-- NB: If the content of τ and σ were ignored then the union would not
+-- be unique.  Expressing uniqueness would require a notion of equality
+-- of sublist proofs, which we do not (yet) have for the setoid case
+-- (however, for the propositional case).
 
 record UpperBound {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) : Set (c ⊔ ℓ) where
   field
@@ -254,24 +266,26 @@ x≈y ∷ᵣ-ub u = record
   ; inj₂ = x≈y  ∷ u .inj₂
   }
 
+_,_∷-ub_ : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} {x y z} →
+         (x≈z : x ≈ z) (y≈z : y ≈ z) → UpperBound τ σ → UpperBound (x≈z ∷ τ) (y≈z ∷ σ)
+x≈z , y≈z ∷-ub u = record
+  { sub  =  refl ∷ u .sub
+  ; inj₁ =  x≈z  ∷ u .inj₁
+  ; inj₂ =  y≈z  ∷ u .inj₂
+  }
+
+⊆-upper-bound : ∀ {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) → UpperBound τ σ
+⊆-upper-bound []        []        = record { sub = [] ; inj₁ = [] ; inj₂ = [] }
+⊆-upper-bound (y ∷ʳ τ)  (.y ∷ʳ σ) = ∷ₙ-ub (⊆-upper-bound τ σ)
+⊆-upper-bound (y ∷ʳ τ)  (x≈y ∷ σ) = x≈y ∷ᵣ-ub ⊆-upper-bound τ σ
+⊆-upper-bound (x≈y ∷ τ) (y  ∷ʳ σ) = x≈y ∷ₗ-ub ⊆-upper-bound τ σ
+⊆-upper-bound (x≈z ∷ τ) (y≈z ∷ σ) = x≈z , y≈z ∷-ub ⊆-upper-bound τ σ
+
 ------------------------------------------------------------------------
 -- Disjoint union
 --
--- Two non-overlapping sublists τ : xs ⊆ zs and σ : ys ⊆ zs
--- can be joined in a unique way if τ and σ are respected.
---
--- For instance, if τ : [x] ⊆ [x,y,x] and σ : [y] ⊆ [x,y,x]
--- then the union will be [x,y] or [y,x], depending on whether
--- τ picks the first x or the second one.
---
--- NB: If the content of τ and σ were ignored then the union would not
--- be unique.  Expressing uniqueness would require a notion of equality
--- of sublist proofs, which we do not (yet) have for the setoid case
--- (however, for the propositional case).
+-- Upper bound of two non-overlapping sublists.
 
 ⊆-disjoint-union : ∀ {xs ys zs} {τ : xs ⊆ zs} {σ : ys ⊆ zs} →
                    Disjoint τ σ → UpperBound τ σ
-⊆-disjoint-union []         = record { sub = [] ; inj₁ = [] ; inj₂ = [] }
-⊆-disjoint-union (x   ∷ₙ d) = ∷ₙ-ub (⊆-disjoint-union d)
-⊆-disjoint-union (x≈y ∷ₗ d) = x≈y ∷ₗ-ub (⊆-disjoint-union d)
-⊆-disjoint-union (x≈y ∷ᵣ d) = x≈y ∷ᵣ-ub (⊆-disjoint-union d)
+⊆-disjoint-union {τ = τ} {σ = σ} _ = ⊆-upper-bound τ σ
diff --git a/standard-library.agda-lib b/standard-library.agda-lib
index ba7ae571b9..cedacd1041 100644
--- a/standard-library.agda-lib
+++ b/standard-library.agda-lib
@@ -1,4 +1,5 @@
-name: standard-library-2.1
+name: standard-library-tbt
 include: src
 flags:
   --warning=noUnsupportedIndexedMatch
+--   --type-based-termination

From 2055bb30a0c209546fa669b123804f4a0d1efc2e Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Mon, 25 Mar 2024 03:07:26 +0800
Subject: [PATCH 252/316] Include CHANGELOG in fix-whitespace and whitespace
 fixes (#2325)

---
 CHANGELOG/v2.0.md                             | 20 +++++++++----------
 HACKING.md                                    |  2 +-
 fix-whitespace.yaml                           |  3 +--
 src/Algebra/Module/Bundles/Raw.agda           |  2 +-
 src/Algebra/Morphism/Construct/Initial.agda   |  4 ++--
 .../Ternary/Appending/Setoid/Properties.agda  |  2 +-
 src/Data/Nat/Primality/Factorisation.agda     | 14 ++++++-------
 7 files changed, 23 insertions(+), 24 deletions(-)

diff --git a/CHANGELOG/v2.0.md b/CHANGELOG/v2.0.md
index 134309712c..ce3c85e979 100644
--- a/CHANGELOG/v2.0.md
+++ b/CHANGELOG/v2.0.md
@@ -38,7 +38,7 @@ Bug-fixes
   Consequently this field has been removed from the record, and the record
   `IsRingWithoutAnnihilatingZero` in `Algebra.Structures.Biased` has been
   deprecated as it is now identical to is `IsRing`.
-  
+
 * In `Algebra.Definitions.RawSemiring` the record `Prime` did not
   enforce that the number was not divisible by `1#`. To fix this
   `p∤1 : p ∤ 1#` has been added as a field.
@@ -212,7 +212,7 @@ Non-backwards compatible changes
 
 * Added new aliases `Is(Meet/Join)(Bounded)Semilattice` for `Is(Bounded)Semilattice`
   which can be used to indicate meet/join-ness of the original structures, and
-  the field names in `IsSemilattice` and `Semilattice` have been renamed from 
+  the field names in `IsSemilattice` and `Semilattice` have been renamed from
   `∧-cong` to `∙-cong`to indicate their undirected nature.
 
 * Finally, the following auxiliary files have been moved:
@@ -763,10 +763,10 @@ Non-backwards compatible changes
    - The records in  `Function.Structures` and `Function.Bundles` export proofs
          of these under the names `strictlySurjective`, `strictlyInverseˡ` and
          `strictlyInverseʳ`,
-   - Conversion functions for the definitions have been added in both directions 
-		 to `Function.Consequences(.Propositional/Setoid)`.
-   - Conversion functions for structures have been added in 
-		 `Function.Structures.Biased`.
+   - Conversion functions for the definitions have been added in both directions
+                 to `Function.Consequences(.Propositional/Setoid)`.
+   - Conversion functions for structures have been added in
+                 `Function.Structures.Biased`.
 
 ### New `Function.Strict`
 
@@ -852,9 +852,9 @@ Non-backwards compatible changes
 
   4. The modules `Relation.Nullary.(Product/Sum/Implication)` have been deprecated
          and their contents moved to `Relation.Nullary.(Negation/Reflects/Decidable)`.
-		 
+
   5. The proof `T?` has been moved from `Data.Bool.Properties` to `Relation.Nullary.Decidable.Core`
-	 (but is still re-exported by the former).
+         (but is still re-exported by the former).
 
   as well as the following breaking changes:
 
@@ -1214,7 +1214,7 @@ Other major improvements
 
 * We have then moved raw bundles defined in `Data.X.Properties` to `Data.X.Base` for
   `X` = `Nat`/`Nat.Binary`/`Integer`/`Rational`/`Rational.Unnormalised`.
-  
+
 ### Upgrades to `README` sub-library
 
 * The `README` sub-library has been moved to `doc/README` and a new `doc/standard-library-doc.agda-lib` has been added.
@@ -1223,7 +1223,7 @@ Other major improvements
   using an out-of-the-box standard Agda installation without altering the main
   `standard-library.agda-lib` file.
 
-* The second is that the `README` files are now their own first-class library 
+* The second is that the `README` files are now their own first-class library
   and can be imported like an other library.
 
 Deprecated modules
diff --git a/HACKING.md b/HACKING.md
index ff859c0507..4e8925648d 100644
--- a/HACKING.md
+++ b/HACKING.md
@@ -31,7 +31,7 @@ Note on contributions to related/'coupled' modules
 ==================================================
 
 Before making changes to a `Data` module please have a look at related modules
-and see if they have any content along similar lines. If so, then please 
+and see if they have any content along similar lines. If so, then please
 follow those conventions (e.g. naming, argument order).
 For example, if working on `Data.Rational`, please check `Data.Rational.Unnormalised`
 or if working on `Data.Vec` please check `Data.List` and vice versa.
diff --git a/fix-whitespace.yaml b/fix-whitespace.yaml
index a12022572e..aacf310d54 100644
--- a/fix-whitespace.yaml
+++ b/fix-whitespace.yaml
@@ -7,5 +7,4 @@ included-files:
   - "*.md"
 
 excluded-files:
-  - "README/Text/Tabular.agda"
-  - CHANGELOG.md
+  - "README/Text/Tabular.agda"
\ No newline at end of file
diff --git a/src/Algebra/Module/Bundles/Raw.agda b/src/Algebra/Module/Bundles/Raw.agda
index b9cc108915..2eb280875f 100644
--- a/src/Algebra/Module/Bundles/Raw.agda
+++ b/src/Algebra/Module/Bundles/Raw.agda
@@ -200,7 +200,7 @@ record RawBimodule (R : Set r) (S : Set s) m ℓm : Set (r ⊔ s ⊔ suc (m ⊔
     ; 0ᴹ = 0ᴹ
     ; -ᴹ_ = -ᴹ_
     }
- 
+
   rawRightModule : RawRightModule S m ℓm
   rawRightModule = record
     { _≈ᴹ_ = _≈ᴹ_
diff --git a/src/Algebra/Morphism/Construct/Initial.agda b/src/Algebra/Morphism/Construct/Initial.agda
index 916e181030..883b70109d 100644
--- a/src/Algebra/Morphism/Construct/Initial.agda
+++ b/src/Algebra/Morphism/Construct/Initial.agda
@@ -39,7 +39,7 @@ zero ()
 -- Basic properties
 
 cong : (≈ : Rel A ℓm) → Rel.Homomorphic₂ ℤero.Carrier A ℤero._≈_ ≈ zero
-cong _ {x = ()} 
+cong _ {x = ()}
 
 injective : (≈ : Rel A ℓm) → Injective ℤero._≈_ ≈ zero
 injective _ {x = ()}
@@ -52,7 +52,7 @@ isMagmaHomomorphism : (M : RawMagma m ℓm) →
 isMagmaHomomorphism M = record
   { isRelHomomorphism = record { cong = cong (RawMagma._≈_ M) }
   ; homo = λ()
-  } 
+  }
 
 isMagmaMonomorphism : (M : RawMagma m ℓm) →
                       IsMagmaMonomorphism rawMagma M zero
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
index a8b6464f66..fca4f2793d 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
@@ -23,7 +23,7 @@ open import Relation.Binary.PropositionalEquality.Core using (refl)
 open import Relation.Binary.Construct.Composition using (_;_)
 
 open Setoid S renaming (Carrier to A)
-open import Relation.Binary.Properties.Setoid S using (≈;≈⇒≈; ≈⇒≈;≈)  
+open import Relation.Binary.Properties.Setoid S using (≈;≈⇒≈; ≈⇒≈;≈)
 open import Data.List.Relation.Ternary.Appending.Setoid S
 
 private
diff --git a/src/Data/Nat/Primality/Factorisation.agda b/src/Data/Nat/Primality/Factorisation.agda
index fed89d7d45..6dda6bac9d 100644
--- a/src/Data/Nat/Primality/Factorisation.agda
+++ b/src/Data/Nat/Primality/Factorisation.agda
@@ -34,7 +34,7 @@ open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans
 private
   variable
     n : ℕ
-    
+
 ------------------------------------------------------------------------
 -- Core definition
 
@@ -107,18 +107,18 @@ factorise n₀@(2+ _) = build [ <-recBuilder ⊗ <-recBuilder ] P facRec (n₀ ,
         pred (m * m) <⟨ s<s⁻¹ (m∸n≢0⇒n<m λ eq′ → 0≢1+n (trans (sym eq′) eq)) ⟩
         n            ∎
         where open ≤-Reasoning
-        
+
       q = quotient m∣n
-      
+
       instance _  = n>1⇒nonTrivial (quotient>1 m∣n m<n)
-      
+
       factorisation[q] : PrimeFactorisation q
       factorisation[q] = recQuotient (quotient-< m∣n) (suc q ∸ m * m) (rough∧∣⇒rough rough (quotient-∣ m∣n)) refl
-      
+
       ps = factors factorisation[q]
-      
+
       primes = factorsPrime factorisation[q]
-      
+
       m*Πps≡n : m * product ps ≡ n
       m*Πps≡n = begin
         m * product ps ≡⟨ cong (m *_) (isFactorisation factorisation[q]) ⟨

From ec51abf83d7a4d554db3bc49e141314f5a59cdd0 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Sun, 24 Mar 2024 20:07:57 +0100
Subject: [PATCH 253/316] Fix standard-library.agda-lib (#2326)

---
 standard-library.agda-lib | 3 +--
 1 file changed, 1 insertion(+), 2 deletions(-)

diff --git a/standard-library.agda-lib b/standard-library.agda-lib
index cedacd1041..ba7ae571b9 100644
--- a/standard-library.agda-lib
+++ b/standard-library.agda-lib
@@ -1,5 +1,4 @@
-name: standard-library-tbt
+name: standard-library-2.1
 include: src
 flags:
   --warning=noUnsupportedIndexedMatch
---   --type-based-termination

From 662cc672c180ea52a6b2503e0c5e469eda85fc63 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 25 Mar 2024 01:20:37 +0000
Subject: [PATCH 254/316] Predicate transformers: Reconciling
 `Induction.RecStruct` with `Relation.Unary.PredicateTransformer.PT` (#2140)

* reconciling `Induction.RecStruct` with `Relation.Unary.PredicateTransformer.PT`

* relaxing `PredicateTransformer` levels

* `CHANGELOG`; `fix-whitespace`

* added `variable`s; plus tweaked comments`

* restored `FreeMonad`

* restored `Predicate`

* removed linebreaks
---
 src/Induction.agda | 33 ++++++++++++++++++---------------
 1 file changed, 18 insertions(+), 15 deletions(-)

diff --git a/src/Induction.agda b/src/Induction.agda
index 56f0ae8267..03d7750b9e 100644
--- a/src/Induction.agda
+++ b/src/Induction.agda
@@ -17,45 +17,48 @@ module Induction where
 
 open import Level
 open import Relation.Unary
+open import Relation.Unary.PredicateTransformer using (PT)
 
+private
+  variable
+    a ℓ ℓ₁ ℓ₂ : Level
+    A : Set a
+    Q : Pred A ℓ
+    Rec : PT A A ℓ₁ ℓ₂
+
+
+------------------------------------------------------------------------
 -- A RecStruct describes the allowed structure of recursion. The
 -- examples in Data.Nat.Induction should explain what this is all
 -- about.
 
-RecStruct : ∀ {a} → Set a → (ℓ₁ ℓ₂ : Level) → Set _
-RecStruct A ℓ₁ ℓ₂ = Pred A ℓ₁ → Pred A ℓ₂
+RecStruct : Set a → (ℓ₁ ℓ₂ : Level) → Set _
+RecStruct A = PT A A
 
 -- A recursor builder constructs an instance of a recursion structure
 -- for a given input.
 
-RecursorBuilder : ∀ {a ℓ₁ ℓ₂} {A : Set a} → RecStruct A ℓ₁ ℓ₂ → Set _
+RecursorBuilder : RecStruct A ℓ₁ ℓ₂ → Set _
 RecursorBuilder Rec = ∀ P → Rec P ⊆′ P → Universal (Rec P)
 
 -- A recursor can be used to actually compute/prove something useful.
 
-Recursor : ∀ {a ℓ₁ ℓ₂} {A : Set a} → RecStruct A ℓ₁ ℓ₂ → Set _
+Recursor : RecStruct A ℓ₁ ℓ₂ → Set _
 Recursor Rec = ∀ P → Rec P ⊆′ P → Universal P
 
 -- And recursors can be constructed from recursor builders.
 
-build : ∀ {a ℓ₁ ℓ₂} {A : Set a} {Rec : RecStruct A ℓ₁ ℓ₂} →
-        RecursorBuilder Rec →
-        Recursor Rec
+build : RecursorBuilder Rec → Recursor Rec
 build builder P f x = f x (builder P f x)
 
 -- We can repeat the exercise above for subsets of the type we are
 -- recursing over.
 
-SubsetRecursorBuilder : ∀ {a ℓ₁ ℓ₂ ℓ₃} {A : Set a} →
-                        Pred A ℓ₁ → RecStruct A ℓ₂ ℓ₃ → Set _
+SubsetRecursorBuilder : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _
 SubsetRecursorBuilder Q Rec = ∀ P → Rec P ⊆′ P → Q ⊆′ Rec P
 
-SubsetRecursor : ∀ {a ℓ₁ ℓ₂ ℓ₃} {A : Set a} →
-                 Pred A ℓ₁ → RecStruct A ℓ₂ ℓ₃ → Set _
+SubsetRecursor : Pred A ℓ → RecStruct A ℓ₁ ℓ₂ → Set _
 SubsetRecursor Q Rec = ∀ P → Rec P ⊆′ P → Q ⊆′ P
 
-subsetBuild : ∀ {a ℓ₁ ℓ₂ ℓ₃}
-                {A : Set a} {Q : Pred A ℓ₁} {Rec : RecStruct A ℓ₂ ℓ₃} →
-                SubsetRecursorBuilder Q Rec →
-                SubsetRecursor Q Rec
+subsetBuild : SubsetRecursorBuilder Q Rec → SubsetRecursor Q Rec
 subsetBuild builder P f x q = f x (builder P f x q)

From 16ff145dde36be8c0691317537ba3c7bb4b7fc77 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Mon, 25 Mar 2024 03:06:42 +0100
Subject: [PATCH 255/316] Github action to check for whitespace violations
 (#2328)

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 .github/workflows/whitespace.yml | 20 ++++++++++++++++++++
 1 file changed, 20 insertions(+)
 create mode 100644 .github/workflows/whitespace.yml

diff --git a/.github/workflows/whitespace.yml b/.github/workflows/whitespace.yml
new file mode 100644
index 0000000000..9e98c51ba4
--- /dev/null
+++ b/.github/workflows/whitespace.yml
@@ -0,0 +1,20 @@
+name: Check whitespace
+on:
+  push:
+    branches:
+      - master
+      - experimental
+  pull_request:
+    branches:
+      - master
+      - experimental
+  merge_group:
+
+jobs:
+  check-whitespace:
+    runs-on: ubuntu-latest
+    steps:
+
+    - uses: actions/checkout@v4
+
+    - uses: andreasabel/fix-whitespace-action@v1

From 6d51975b4899b6fa22d326b693dba95fbf1a7ce1 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Wed, 27 Mar 2024 02:57:18 +0100
Subject: [PATCH 256/316] [refactor] Relation.Nulllary.Decidable.Core (#2329)

* minor refactor: use Data.Unit.Polymorphic instead of an explicit Lift.

* additional tweak: use contradiction where possible.
---
 src/Relation/Nullary/Decidable/Core.agda | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index f77613ea65..83da26e51b 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -13,14 +13,14 @@ module Relation.Nullary.Decidable.Core where
 
 open import Level using (Level; Lift)
 open import Data.Bool.Base using (Bool; T; false; true; not; _∧_; _∨_)
-open import Data.Unit.Base using (⊤)
-open import Data.Empty using (⊥)
+open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Empty.Irrelevant using (⊥-elim)
 open import Data.Product.Base using (_×_)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_∘_; const; _$_; flip)
 open import Relation.Nullary.Reflects
 open import Relation.Nullary.Negation.Core
+  using (¬_; Stable; negated-stable; contradiction; DoubleNegation)
 
 private
   variable
@@ -58,11 +58,11 @@ module _ {A : Set a} where
 
   From-yes : Dec A → Set a
   From-yes (true  because _) = A
-  From-yes (false because _) = Lift a ⊤
+  From-yes (false because _) = ⊤
 
   From-no : Dec A → Set a
   From-no (false because _) = ¬ A
-  From-no (true  because _) = Lift a ⊤
+  From-no (true  because _) = ⊤
 
 ------------------------------------------------------------------------
 -- Recompute
@@ -171,7 +171,7 @@ proof (map′ A→B B→A (false because [¬a])) = ofⁿ (invert [¬a] ∘ B→A
 
 decidable-stable : Dec A → Stable A
 decidable-stable (yes a) ¬¬a = a
-decidable-stable (no ¬a) ¬¬a = ⊥-elim (¬¬a ¬a)
+decidable-stable (no ¬a) ¬¬a = contradiction ¬a ¬¬a
 
 ¬-drop-Dec : Dec (¬ ¬ A) → Dec (¬ A)
 ¬-drop-Dec ¬¬a? = map′ negated-stable contradiction (¬? ¬¬a?)

From ddb420fc8bb5d96560b9858064933aaed1c70a2b Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Wed, 27 Mar 2024 12:07:28 +0100
Subject: [PATCH 257/316] Some design documentation (here: level polymorphism)
 (#2322)

* add some design documentation corresponding to issue #312. (Redo of bad 957)

* more documentation about LevelPolymorphism

* more documentation precision

* document issues wrt Relation.Unary, both in design doc and in file itself.
---
 doc/README/Design/LevelPolymorphism.md | 15 +++++++++++++++
 src/Data/Empty.agda                    |  2 +-
 src/Data/Unit.agda                     |  2 +-
 src/Relation/Unary.agda                |  3 +++
 4 files changed, 20 insertions(+), 2 deletions(-)
 create mode 100644 doc/README/Design/LevelPolymorphism.md

diff --git a/doc/README/Design/LevelPolymorphism.md b/doc/README/Design/LevelPolymorphism.md
new file mode 100644
index 0000000000..59436fe4a3
--- /dev/null
+++ b/doc/README/Design/LevelPolymorphism.md
@@ -0,0 +1,15 @@
+* Level polymorphism
+
+`⊥` in `Data.Empty` and `⊤` in `Data.Unit` are not `Level`-polymorphic as that
+tends to lead to unsolved metas (see discussion at issue #312).  This is understandable
+as very often the level of (say) `⊤` is under-determined by its surrounding context,
+leading to unsolved metas. This is frequent enough that it makes sense for the default
+versions to be monomorphic (at Level 0).
+
+But there are other cases where exactly the opposite is needed.  for that purpose,
+there are level-polymorphic versions in `Data.Empty.Polymorphic` and
+`Data.Unit.Polymorphic` respectively.
+
+The same issue happens in `Relation.Unary` which defines `Ø` and `U` at `Level` 0, else
+a lot of unsolved metas appear (for example in `Relation.Unary.Properties`). For that
+purpose, `Relation.Unary.Polymorphic` exists.
diff --git a/src/Data/Empty.agda b/src/Data/Empty.agda
index 990a9a1632..19754e8c7d 100644
--- a/src/Data/Empty.agda
+++ b/src/Data/Empty.agda
@@ -1,7 +1,7 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Empty type, judgementally proof irrelevant
+-- Empty type, judgementally proof irrelevant, Level-monomorphic
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Data/Unit.agda b/src/Data/Unit.agda
index 2e3464c7f4..73f269d9d6 100644
--- a/src/Data/Unit.agda
+++ b/src/Data/Unit.agda
@@ -1,7 +1,7 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- The unit type
+-- The unit type, Level-monomorphic version
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
diff --git a/src/Relation/Unary.agda b/src/Relation/Unary.agda
index 9f3ea16633..0f8a7534fd 100644
--- a/src/Relation/Unary.agda
+++ b/src/Relation/Unary.agda
@@ -42,6 +42,8 @@ Pred A ℓ = A → Set ℓ
 -- Special sets
 
 -- The empty set.
+-- Explicitly not level polymorphic as this often causes unsolved metas;
+-- see `Relation.Unary.Polymorphic` for a level-polymorphic version.
 
 ∅ : Pred A 0ℓ
 ∅ = λ _ → ⊥
@@ -52,6 +54,7 @@ Pred A ℓ = A → Set ℓ
 ｛ x ｝ = x ≡_
 
 -- The universal set.
+-- Explicitly not level polymorphic (see comments for `∅` for more details)
 
 U : Pred A 0ℓ
 U = λ _ → ⊤

From b4c78e2671adc6178dd0975366f9ed97898bc1c5 Mon Sep 17 00:00:00 2001
From: Chris <cspollard@users.noreply.github.com>
Date: Wed, 27 Mar 2024 14:44:50 +0000
Subject: [PATCH 258/316] mark variables private in Data.Container.FreeMonad
 (#2332)

---
 src/Data/Container/FreeMonad.agda | 11 ++++++-----
 1 file changed, 6 insertions(+), 5 deletions(-)

diff --git a/src/Data/Container/FreeMonad.agda b/src/Data/Container/FreeMonad.agda
index d848915983..2bbce4a7f7 100644
--- a/src/Data/Container/FreeMonad.agda
+++ b/src/Data/Container/FreeMonad.agda
@@ -20,11 +20,12 @@ open import Effect.Applicative using (RawApplicative)
 open import Effect.Monad using (RawMonad)
 open import Function.Base as Function using (_$_; _∘_)
 
-variable
-  x y s p ℓ : Level
-  C : Container s p
-  X : Set x
-  Y : Set y
+private
+  variable
+    x y s p ℓ : Level
+    C : Container s p
+    X : Set x
+    Y : Set y
 
 infixl 1 _⋆C_
 infix  1 _⋆_

From e31b6c7d84561103a2c191fbe00baf664aebd616 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Mon, 1 Apr 2024 07:49:49 +0200
Subject: [PATCH 259/316] Move pointwise equality to `.Core` module (#2335)

* Closes #2331.

Also makes a few changes of imports that are related. Many of the places
which look like this improvement might also help use other things from
`Relation.Binary.PropositionalEquality` that should *not* be moved to `.Core`.
(But it might make sense to split them off into their own lighter weight module.)

* don't import Data.Nullary itself since its sub-modules were already imported, just redistribute the using properly.
---
 src/Algebra/Properties/Monoid/Sum.agda          |  2 +-
 src/Data/List/Effectful.agda                    | 15 ++++++++++-----
 .../Membership/Propositional/Properties.agda    |  2 +-
 .../List/Relation/Unary/All/Properties.agda     | 17 +++++++++--------
 src/Data/Product/Properties.agda                |  3 ++-
 src/Relation/Binary/PropositionalEquality.agda  |  5 -----
 .../Binary/PropositionalEquality/Core.agda      |  9 +++++++++
 7 files changed, 32 insertions(+), 21 deletions(-)

diff --git a/src/Algebra/Properties/Monoid/Sum.agda b/src/Algebra/Properties/Monoid/Sum.agda
index fb383f3768..ae64a5f4dc 100644
--- a/src/Algebra/Properties/Monoid/Sum.agda
+++ b/src/Algebra/Properties/Monoid/Sum.agda
@@ -13,7 +13,7 @@ open import Data.Fin.Base using (zero; suc)
 open import Data.Unit using (tt)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Core using (_Preserves_⟶_)
-open import Relation.Binary.PropositionalEquality as ≡ using (_≗_; _≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≗_; _≡_)
 
 module Algebra.Properties.Monoid.Sum {a ℓ} (M : Monoid a ℓ) where
 
diff --git a/src/Data/List/Effectful.agda b/src/Data/List/Effectful.agda
index b6259d7e61..e3855b26c5 100644
--- a/src/Data/List/Effectful.agda
+++ b/src/Data/List/Effectful.agda
@@ -10,14 +10,19 @@ module Data.List.Effectful where
 
 open import Data.Bool.Base using (false; true)
 open import Data.List.Base
+  using (List; map; [_]; ap; []; _∷_; _++_; concat; concatMap)
 open import Data.List.Properties
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
+  using (++-identityʳ; ++-assoc; map-cong; concatMap-cong; map-concatMap;
+    concatMap-pure)
+open import Effect.Choice using (RawChoice)
+open import Effect.Empty using (RawEmpty)
+open import Effect.Functor using (RawFunctor)
 open import Effect.Applicative
+  using (RawApplicative; RawApplicativeZero; RawAlternative)
 open import Effect.Monad
-open import Function.Base
-open import Level
+  using (RawMonad; module Join; RawMonadZero; RawMonadPlus)
+open import Function.Base using (flip; _∘_; const; _$_; id; _∘′_; _$′_)
+open import Level using (Level)
 open import Relation.Binary.PropositionalEquality as ≡
   using (_≡_; _≢_; _≗_; refl)
 open ≡.≡-Reasoning
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index c5ab013b61..6988f4f0c4 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -37,7 +37,7 @@ open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions as Binary hiding (Decidable)
 open import Relation.Binary.PropositionalEquality as ≡
-  using (_≡_; _≢_; refl; sym; trans; cong; subst; →-to-⟶; _≗_)
+  using (_≡_; _≢_; refl; sym; trans; cong; subst; _≗_)
 import Relation.Binary.Properties.DecTotalOrder as DTOProperties
 open import Relation.Unary using (_⟨×⟩_; Decidable)
 import Relation.Nullary.Reflects as Reflects
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index a36a07f3b1..c77df039d2 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -11,12 +11,13 @@ module Data.List.Relation.Unary.All.Properties where
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Bool.Base using (Bool; T; true; false)
 open import Data.Bool.Properties using (T-∧)
-open import Data.Empty
+open import Data.Empty using (⊥-elim)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List hiding (lookup; updateAt)
 open import Data.List.Properties as Listₚ using (partition-defn)
-open import Data.List.Membership.Propositional
+open import Data.List.Membership.Propositional using (_∈_; _≢∈_)
 open import Data.List.Membership.Propositional.Properties
+  using (there-injective-≢∈; ∈-filter⁻)
 import Data.List.Membership.Setoid as SetoidMembership
 open import Data.List.Relation.Unary.All as All using
   ( All; []; _∷_; lookup; updateAt
@@ -33,18 +34,18 @@ open import Data.Maybe.Relation.Unary.Any as Maybe using (just)
 open import Data.Nat.Base using (zero; suc; s≤s; _<_; z<s; s<s)
 open import Data.Nat.Properties using (≤-refl; m≤n⇒m≤1+n)
 open import Data.Product.Base as Product using (_×_; _,_; uncurry; uncurry′)
-open import Function.Base
-open import Function.Bundles
+open import Function.Base using (_∘_; _$_; id; case_of_; flip)
+open import Function.Bundles using (_↠_; mk↠ₛ; _⇔_; mk⇔; _↔_; mk↔ₛ′; Equivalence)
 open import Level using (Level)
 open import Relation.Binary.Core using (REL)
 open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.Definitions as B
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; cong; cong₂; _≗_)
-open import Relation.Nullary
 open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Nullary.Decidable using (¬?; decidable-stable)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
+open import Relation.Nullary.Decidable
+  using (Dec; does; yes; no; _because_; ¬?; decidable-stable)
 open import Relation.Unary
   using (Decidable; Pred; Universal; ∁; _∩_; _⟨×⟩_) renaming (_⊆_ to _⋐_)
 open import Relation.Unary.Properties using (∁?)
diff --git a/src/Data/Product/Properties.agda b/src/Data/Product/Properties.agda
index 5e8509b76e..ec3f05d018 100644
--- a/src/Data/Product/Properties.agda
+++ b/src/Data/Product/Properties.agda
@@ -14,7 +14,8 @@ open import Function.Base using (_∋_; _∘_; id)
 open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; _≗_; subst; cong; cong₂; cong′)
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no)
 
 private
diff --git a/src/Relation/Binary/PropositionalEquality.agda b/src/Relation/Binary/PropositionalEquality.agda
index 6c2eb19bed..55b7d30926 100644
--- a/src/Relation/Binary/PropositionalEquality.agda
+++ b/src/Relation/Binary/PropositionalEquality.agda
@@ -37,14 +37,9 @@ open import Relation.Binary.PropositionalEquality.Algebra public
 ------------------------------------------------------------------------
 -- Pointwise equality
 
-infix 4 _≗_
-
 _→-setoid_ : ∀ (A : Set a) (B : Set b) → Setoid _ _
 A →-setoid B = ≡-setoid A (Trivial.indexedSetoid (setoid B))
 
-_≗_ : (f g : A → B) → Set _
-_≗_ {A = A} {B = B} = Setoid._≈_ (A →-setoid B)
-
 :→-to-Π : ∀ {A : Set a} {B : IndexedSetoid A b ℓ} →
           ((x : A) → IndexedSetoid.Carrier B x) →
           Dependent.Func (setoid A) B
diff --git a/src/Relation/Binary/PropositionalEquality/Core.agda b/src/Relation/Binary/PropositionalEquality/Core.agda
index 848258348b..f52fed277f 100644
--- a/src/Relation/Binary/PropositionalEquality/Core.agda
+++ b/src/Relation/Binary/PropositionalEquality/Core.agda
@@ -32,6 +32,15 @@ infix 4 _≢_
 _≢_ : {A : Set a} → Rel A a
 x ≢ y = ¬ x ≡ y
 
+------------------------------------------------------------------------
+-- Pointwise equality
+
+infix 4 _≗_
+
+_≗_ : (f g : A → B) → Set _
+_≗_ {A = A} {B = B} f g = ∀ x → f x ≡ g x
+
+
 ------------------------------------------------------------------------
 -- A variant of `refl` where the argument is explicit
 

From 5d687f7d90ec8a6ddb96bf021905cc208154b9e1 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Thu, 4 Apr 2024 08:40:13 +0200
Subject: [PATCH 260/316] fix warning: it was pointing to a record that did not
 exist. (#2344)

---
 src/Algebra/Morphism.agda | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/Algebra/Morphism.agda b/src/Algebra/Morphism.agda
index 45b7a46a1d..e3bcb53e9b 100644
--- a/src/Algebra/Morphism.agda
+++ b/src/Algebra/Morphism.agda
@@ -185,7 +185,7 @@ module _ {c₁ ℓ₁ c₂ ℓ₂}
 
 {-# WARNING_ON_USAGE IsSemigroupMorphism
 "Warning: IsSemigroupMorphism was deprecated in v1.5.
-Please use IsSemigroupHomomorphism instead."
+Please use IsMagmaHomomorphism instead."
 #-}
 {-# WARNING_ON_USAGE IsMonoidMorphism
 "Warning: IsMonoidMorphism was deprecated in v1.5.

From 5cc487e02a25b82405decbf94484c629e3295394 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Thu, 4 Apr 2024 17:00:32 +0200
Subject: [PATCH 261/316] Tighten imports some more (#2343)

* no need to use Function when only import is from Function.Base

* Use Function.Base when it suffices

* use Data.Product.Base as much as possible. Make imports more precise in many of the files touched. Split an example off into a README instead of using more imports just for its sake.

* more tightening of imports.

* a few more tightening imports
---
 doc/README/Effect/Monad/Partiality.agda       | 27 +++++++++++++
 .../Module/Construct/Idealization.agda        |  6 ++-
 src/Algebra/Properties/CommutativeMonoid.agda |  6 +--
 src/Data/Container/Core.agda                  |  6 +--
 src/Data/Container/Indexed.agda               |  8 ++--
 .../Indexed/Relation/Binary/Pointwise.agda    |  6 +--
 src/Data/List/Properties.agda                 |  2 +-
 .../Ternary/Appending/Setoid/Properties.agda  |  6 +--
 .../List/Relation/Unary/First/Properties.agda | 23 ++++++-----
 src/Data/Nat/Primality/Factorisation.agda     | 13 +++++--
 .../Binary/Pointwise/NonDependent.agda        | 13 ++++---
 src/Data/Rational/Unnormalised/Base.agda      |  1 +
 src/Data/Sum/Algebra.agda                     | 12 ++++--
 src/Data/Sum/Function/Propositional.agda      |  9 +++--
 src/Data/Unit/Polymorphic/Properties.agda     |  1 +
 src/Effect/Monad/Partiality.agda              | 38 +++++--------------
 .../Inverse/HalfAdjointEquivalence.agda       |  8 ++--
 src/Function/Related/TypeIsomorphisms.agda    | 13 +++----
 .../Binary/Reasoning/Base/Apartness.agda      |  2 +-
 .../Binary/Reasoning/Base/Double.agda         |  2 +-
 .../Binary/Reasoning/Base/Partial.agda        |  6 +--
 .../Binary/Reasoning/Base/Triple.agda         |  2 +-
 22 files changed, 118 insertions(+), 92 deletions(-)
 create mode 100644 doc/README/Effect/Monad/Partiality.agda

diff --git a/doc/README/Effect/Monad/Partiality.agda b/doc/README/Effect/Monad/Partiality.agda
new file mode 100644
index 0000000000..7b96749cd5
--- /dev/null
+++ b/doc/README/Effect/Monad/Partiality.agda
@@ -0,0 +1,27 @@
+-----------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Example showing the use of the partiality Monad
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe --guardedness #-}
+
+module README.Effect.Monad.Partiality where
+
+open import Codata.Musical.Notation using (♯_)
+open import Data.Bool.Base using (false; true)
+open import Data.Nat using (ℕ; _+_; _∸_; _≤?_)
+open import Effect.Monad.Partiality
+open import Relation.Nullary.Decidable using (does)
+
+open Workaround
+
+-- McCarthy's f91:
+
+f91′ : ℕ → ℕ ⊥P
+f91′ n with does (n ≤? 100)
+... | true  = later (♯ (f91′ (11 + n) >>= f91′))
+... | false = now (n ∸ 10)
+
+f91 : ℕ → ℕ ⊥
+f91 n = ⟦ f91′ n ⟧P
diff --git a/src/Algebra/Module/Construct/Idealization.agda b/src/Algebra/Module/Construct/Idealization.agda
index 0526f059fd..4a393ea879 100644
--- a/src/Algebra/Module/Construct/Idealization.agda
+++ b/src/Algebra/Module/Construct/Idealization.agda
@@ -43,9 +43,11 @@ import Algebra.Definitions as Definitions
 import Algebra.Module.Construct.DirectProduct as DirectProduct
 import Algebra.Module.Construct.TensorUnit as TensorUnit
 open import Algebra.Structures using (IsAbelianGroup; IsRing)
-open import Data.Product using (_,_; ∃-syntax)
+open import Data.Product.Base using (_,_; ∃-syntax)
 open import Level using (Level; _⊔_)
-open import Relation.Binary using (Rel; Setoid; IsEquivalence)
+open import Relation.Binary.Bundles using  (Setoid)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures using (IsEquivalence)
 import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Properties/CommutativeMonoid.agda b/src/Algebra/Properties/CommutativeMonoid.agda
index e83be61ac7..cef85545a1 100644
--- a/src/Algebra/Properties/CommutativeMonoid.agda
+++ b/src/Algebra/Properties/CommutativeMonoid.agda
@@ -6,9 +6,9 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra.Bundles
-open import Algebra.Definitions
-open import Data.Product using (_,_; proj₂)
+open import Algebra.Bundles using (CommutativeMonoid)
+open import Algebra.Definitions using (LeftInvertible; RightInvertible; Invertible)
+open import Data.Product.Base using (_,_; proj₂)
 
 module Algebra.Properties.CommutativeMonoid
   {g₁ g₂} (M : CommutativeMonoid g₁ g₂) where
diff --git a/src/Data/Container/Core.agda b/src/Data/Container/Core.agda
index 006f433d2e..07afabe2eb 100644
--- a/src/Data/Container/Core.agda
+++ b/src/Data/Container/Core.agda
@@ -8,10 +8,10 @@
 
 module Data.Container.Core where
 
-open import Level
+open import Level using (Level; _⊔_; suc)
 open import Data.Product.Base as Product using (Σ-syntax)
-open import Function.Base
-open import Function using (Inverse; _↔_)
+open import Function.Base using (_∘_; _∘′_)
+open import Function.Bundles using (Inverse; _↔_)
 open import Relation.Unary using (Pred; _⊆_)
 
 -- Definition of Containers
diff --git a/src/Data/Container/Indexed.agda b/src/Data/Container/Indexed.agda
index b34657a4be..f99fef80dc 100644
--- a/src/Data/Container/Indexed.agda
+++ b/src/Data/Container/Indexed.agda
@@ -11,14 +11,14 @@
 
 module Data.Container.Indexed where
 
-open import Level
+open import Level using (Level; zero; _⊔_)
 open import Data.Product.Base as Prod hiding (map)
-open import Data.W.Indexed
+open import Data.W.Indexed using (W)
 open import Function.Base renaming (id to ⟨id⟩; _∘_ to _⟨∘⟩_)
-open import Function using (_↔_; Inverse)
+open import Function.Bundles using (_↔_; Inverse)
 open import Relation.Unary using (Pred; _⊆_)
 open import Relation.Binary.Core using (Rel; REL)
-open import Relation.Binary.PropositionalEquality using (_≡_; _≗_; refl; trans; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≗_; refl; trans; subst)
 
 ------------------------------------------------------------------------
 
diff --git a/src/Data/Container/Indexed/Relation/Binary/Pointwise.agda b/src/Data/Container/Indexed/Relation/Binary/Pointwise.agda
index afaba23f09..4afafefbb9 100644
--- a/src/Data/Container/Indexed/Relation/Binary/Pointwise.agda
+++ b/src/Data/Container/Indexed/Relation/Binary/Pointwise.agda
@@ -8,11 +8,11 @@
 
 module Data.Container.Indexed.Relation.Binary.Pointwise where
 
-open import Data.Product using (_,_; Σ-syntax)
-open import Function
+open import Data.Product.Base using (_,_; Σ-syntax)
+open import Function.Base using (_∘_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary using (REL; _⇒_)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 open import Data.Container.Indexed.Core using (Container; Subtrees; ⟦_⟧)
 
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 0cda410424..2d3c9e20e0 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -23,7 +23,7 @@ open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Nat.Base
-open import Data.Nat.Divisibility
+open import Data.Nat.Divisibility using (_∣_; divides; ∣n⇒∣m*n)
 open import Data.Nat.Properties
 open import Data.Product.Base as Product
   using (_×_; _,_; uncurry; uncurry′; proj₁; proj₂; <_,_>)
diff --git a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
index fca4f2793d..b4d2a97d90 100644
--- a/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Setoid/Properties.agda
@@ -13,10 +13,10 @@ module Data.List.Relation.Ternary.Appending.Setoid.Properties
   where
 
 open import Data.List.Base as List using (List; [])
-import Data.List.Properties as Listₚ
+import Data.List.Properties as List
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; [])
 import Data.List.Relation.Ternary.Appending.Properties as Appendingₚ
-open import Data.Product using (∃-syntax; _×_; _,_)
+open import Data.Product.Base using (∃-syntax; _×_; _,_)
 open import Function.Base using (id)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
@@ -45,7 +45,7 @@ open Appendingₚ public
 ++[]⁻¹ : Appending as [] cs → Pointwise _≈_ as cs
 ++[]⁻¹ {as} {cs} ls with break ls
 ... | cs₁ , cs₂ , refl , pw , []
-  rewrite Listₚ.++-identityʳ cs₁
+  rewrite List.++-identityʳ cs₁
   = pw
 
 respʳ-≋ : ∀ {cs′} → Appending as bs cs → Pointwise _≈_ cs cs′ →
diff --git a/src/Data/List/Relation/Unary/First/Properties.agda b/src/Data/List/Relation/Unary/First/Properties.agda
index d598f14469..0f70a7659c 100644
--- a/src/Data/List/Relation/Unary/First/Properties.agda
+++ b/src/Data/List/Relation/Unary/First/Properties.agda
@@ -8,7 +8,6 @@
 
 module Data.List.Relation.Unary.First.Properties where
 
-open import Data.Empty
 open import Data.Fin.Base using (suc)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
@@ -16,9 +15,9 @@ open import Data.List.Relation.Unary.Any as Any using (here; there)
 open import Data.List.Relation.Unary.First
 import Data.Sum as Sum
 open import Function.Base using (_∘′_; _$_; _∘_; id)
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; refl; _≗_)
-open import Relation.Unary
-open import Relation.Nullary.Negation
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl; _≗_)
+open import Relation.Unary using (Pred; _⊆_; ∁; Irrelevant; Decidable)
+open import Relation.Nullary.Negation using (contradiction)
 
 ------------------------------------------------------------------------
 -- map
@@ -52,7 +51,7 @@ module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
 module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
 
   All⇒¬First : P ⊆ ∁ Q → All P ⊆ ∁ (First P Q)
-  All⇒¬First p⇒¬q (px ∷ pxs) [ qx ]   = ⊥-elim (p⇒¬q px qx)
+  All⇒¬First p⇒¬q (px ∷ pxs) [ qx ]   = contradiction qx (p⇒¬q px)
   All⇒¬First p⇒¬q (_ ∷ pxs)  (_ ∷ hf) = All⇒¬First p⇒¬q pxs hf
 
   First⇒¬All : Q ⊆ ∁ P → First P Q ⊆ ∁ (All P)
@@ -64,16 +63,16 @@ module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
 
   unique-index : ∀ {xs} → P ⊆ ∁ Q → (f₁ f₂ : First P Q xs) → index f₁ ≡ index f₂
   unique-index p⇒¬q [ _ ]    [ _ ]    = refl
-  unique-index p⇒¬q [ qx ]   (px ∷ _) = ⊥-elim (p⇒¬q px qx)
-  unique-index p⇒¬q (px ∷ _) [ qx ]   = ⊥-elim (p⇒¬q px qx)
+  unique-index p⇒¬q [ qx ]   (px ∷ _) = contradiction qx (p⇒¬q px)
+  unique-index p⇒¬q (px ∷ _) [ qx ]   = contradiction qx (p⇒¬q px)
   unique-index p⇒¬q (_ ∷ f₁) (_ ∷ f₂) = ≡.cong suc (unique-index p⇒¬q f₁ f₂)
 
   irrelevant : P ⊆ ∁ Q → Irrelevant P → Irrelevant Q → Irrelevant (First P Q)
-  irrelevant p⇒¬q p-irr q-irr [ qx₁ ]    [ qx₂ ]    = ≡.cong [_] (q-irr qx₁ qx₂)
-  irrelevant p⇒¬q p-irr q-irr [ qx₁ ]    (px₂ ∷ f₂) = ⊥-elim (p⇒¬q px₂ qx₁)
-  irrelevant p⇒¬q p-irr q-irr (px₁ ∷ f₁) [ qx₂ ]    = ⊥-elim (p⇒¬q px₁ qx₂)
-  irrelevant p⇒¬q p-irr q-irr (px₁ ∷ f₁) (px₂ ∷ f₂) =
-    ≡.cong₂ _∷_ (p-irr px₁ px₂) (irrelevant p⇒¬q p-irr q-irr f₁ f₂)
+  irrelevant p⇒¬q p-irr q-irr [ px ]    [ qx ]    = ≡.cong [_] (q-irr px qx)
+  irrelevant p⇒¬q p-irr q-irr [ qx ]    (px ∷ _)  = contradiction qx (p⇒¬q px)
+  irrelevant p⇒¬q p-irr q-irr (px ∷ _)  [ qx ]    = contradiction qx (p⇒¬q px)
+  irrelevant p⇒¬q p-irr q-irr (px ∷ f)  (qx ∷ g) =
+    ≡.cong₂ _∷_ (p-irr px qx) (irrelevant p⇒¬q p-irr q-irr f g)
 
 ------------------------------------------------------------------------
 -- Decidability
diff --git a/src/Data/Nat/Primality/Factorisation.agda b/src/Data/Nat/Primality/Factorisation.agda
index 6dda6bac9d..66e75dd4df 100644
--- a/src/Data/Nat/Primality/Factorisation.agda
+++ b/src/Data/Nat/Primality/Factorisation.agda
@@ -8,13 +8,16 @@
 
 module Data.Nat.Primality.Factorisation where
 
-open import Data.Empty using (⊥-elim)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility
+  using (_∣?_; quotient; quotient>1; quotient-<; quotient-∣; m∣n⇒n≡m*quotient; _∣_; ∣1⇒≡1;
+        divides)
 open import Data.Nat.Properties
 open import Data.Nat.Induction using (<-Rec; <-rec; <-recBuilder)
 open import Data.Nat.Primality
-open import Data.Product as Π using (∃-syntax; _×_; _,_; proj₁; proj₂)
+  using (Prime; _Rough_; rough∧square>⇒prime; ∤⇒rough-suc; rough∧∣⇒rough; rough∧∣⇒prime;
+         2-rough; euclidsLemma; prime⇒irreducible; ¬prime[1]; productOfPrimes≥1; prime⇒nonZero)
+open import Data.Product.Base using (∃-syntax; _×_; _,_; proj₁; proj₂)
 open import Data.List.Base using (List; []; _∷_; _++_; product)
 open import Data.List.Membership.Propositional using (_∈_)
 open import Data.List.Membership.Propositional.Properties using (∈-∃++)
@@ -22,14 +25,16 @@ open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any using (here; there)
 open import Data.List.Relation.Binary.Permutation.Propositional
   using (_↭_; prep; swap; ↭-reflexive; ↭-refl; ↭-trans; refl; module PermutationReasoning)
-open import Data.List.Relation.Binary.Permutation.Propositional.Properties using (product-↭; All-resp-↭; shift)
+open import Data.List.Relation.Binary.Permutation.Propositional.Properties
+  using (product-↭; All-resp-↭; shift)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (_$_; _∘_; _|>_; flip)
 open import Induction using (build)
 open import Induction.Lexicographic using (_⊗_; [_⊗_])
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary.PropositionalEquality using (_≡_; refl; sym; trans; cong; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; sym; trans; cong; module ≡-Reasoning)
 
 private
   variable
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index 6c8004b559..1cba24520a 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -9,13 +9,16 @@
 module Data.Product.Relation.Binary.Pointwise.NonDependent where
 
 open import Data.Product.Base as Product
-open import Data.Product.Properties using (≡-dec)
-open import Data.Sum.Base
-open import Data.Unit.Base using (⊤)
+open import Data.Sum.Base using (inj₁; inj₂)
 open import Level using (Level; _⊔_; 0ℓ)
-open import Function
+open import Function.Base using (_on_; id)
+open import Function.Bundles using (Inverse)
 open import Relation.Nullary.Decidable using (_×-dec_)
-open import Relation.Binary
+open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Bundles
+  using (Setoid; DecSetoid; Preorder; Poset; StrictPartialOrder)
+open import Relation.Binary.Definitions
+open import Relation.Binary.Structures
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as ≡
 
diff --git a/src/Data/Rational/Unnormalised/Base.agda b/src/Data/Rational/Unnormalised/Base.agda
index c9fcb8d54d..41c236062c 100644
--- a/src/Data/Rational/Unnormalised/Base.agda
+++ b/src/Data/Rational/Unnormalised/Base.agda
@@ -9,6 +9,7 @@
 module Data.Rational.Unnormalised.Base where
 
 open import Algebra.Bundles.Raw
+  using (RawMagma; RawMonoid; RawGroup; RawNearSemiring; RawSemiring; RawRing)
 open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Integer.Base as ℤ
   using (ℤ; +_; +0; +[1+_]; -[1+_]; +<+; +≤+)
diff --git a/src/Data/Sum/Algebra.agda b/src/Data/Sum/Algebra.agda
index f128af8075..ab51bece88 100644
--- a/src/Data/Sum/Algebra.agda
+++ b/src/Data/Sum/Algebra.agda
@@ -8,12 +8,16 @@
 
 module Data.Sum.Algebra where
 
-open import Algebra
+open import Algebra.Bundles
+  using (Magma; Semigroup; Monoid; CommutativeMonoid)
+open import Algebra.Definitions
+open import Algebra.Structures
+  using (IsMagma; IsSemigroup; IsMonoid; IsCommutativeMonoid)
 open import Data.Empty.Polymorphic using (⊥)
 open import Data.Product.Base using (_,_)
-open import Data.Sum.Base
-open import Data.Sum.Properties
-open import Data.Unit.Polymorphic using (⊤; tt)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂; map; [_,_]; swap; assocʳ; assocˡ)
+open import Data.Sum.Properties using (swap-involutive)
+open import Data.Unit.Polymorphic.Base using (⊤; tt)
 open import Function.Base using (id; _∘_)
 open import Function.Properties.Inverse using (↔-isEquivalence)
 open import Function.Bundles using (_↔_; Inverse; mk↔ₛ′)
diff --git a/src/Data/Sum/Function/Propositional.agda b/src/Data/Sum/Function/Propositional.agda
index b7830bc895..25bb1a0907 100644
--- a/src/Data/Sum/Function/Propositional.agda
+++ b/src/Data/Sum/Function/Propositional.agda
@@ -8,15 +8,18 @@
 
 module Data.Sum.Function.Propositional where
 
-open import Data.Sum.Base
+open import Data.Sum.Base using (_⊎_)
 open import Data.Sum.Function.Setoid
 open import Data.Sum.Relation.Binary.Pointwise using (Pointwise-≡↔≡; _⊎ₛ_)
 open import Function.Construct.Composition as Compose
 open import Function.Related.Propositional
-open import Function
+open import Function.Base using (id)
+open import Function.Bundles
+  using (Inverse; _⟶_; _⇔_; _↣_; _↠_; _↩_; _↪_; _⤖_; _↔_)
 open import Function.Properties.Inverse as Inv
 open import Level using (Level; _⊔_)
-open import Relation.Binary using (REL; Setoid)
+open import Relation.Binary.Core using (REL)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality using (setoid)
 
 private
diff --git a/src/Data/Unit/Polymorphic/Properties.agda b/src/Data/Unit/Polymorphic/Properties.agda
index 87519aa9ee..ad2ccc3a05 100644
--- a/src/Data/Unit/Polymorphic/Properties.agda
+++ b/src/Data/Unit/Polymorphic/Properties.agda
@@ -23,6 +23,7 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Definitions
   using (Decidable; Antisymmetric; Total)
 open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; trans; decSetoid; setoid; isEquivalence)
 
 private
   variable
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index c0723f1370..3143d0d7ad 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -8,15 +8,15 @@
 
 module Effect.Monad.Partiality where
 
-open import Codata.Musical.Notation
-open import Effect.Functor
-open import Effect.Applicative
-open import Effect.Monad
+open import Codata.Musical.Notation using (∞; ♯_; ♭)
+open import Effect.Functor using (RawFunctor)
+open import Effect.Applicative using (RawApplicative)
+open import Effect.Monad using (RawMonad; module Join)
 open import Data.Bool.Base using (Bool; false; true)
-open import Data.Nat using (ℕ; zero; suc; _+_)
-open import Data.Product as Prod hiding (map)
+open import Data.Nat.Base using (ℕ; zero; suc; _+_)
+open import Data.Product as Prod using (∃; ∄; -,_; ∃₂; _,_; _×_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
-open import Function.Base
+open import Function.Base using (_∘′_; flip; id; _∘_; _$_; _⟨_⟩_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core as B hiding (Rel; _⇔_)
@@ -29,9 +29,8 @@ open import Relation.Binary.Bundles
 import Relation.Binary.Properties.Setoid as SetoidProperties
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 import Relation.Binary.PropositionalEquality.Properties as ≡
-open import Relation.Nullary
-open import Relation.Nullary.Decidable hiding (map)
-open import Relation.Nullary.Negation
+open import Relation.Nullary.Decidable using (yes; no; False; Dec; ¬¬-excluded-middle)
+open import Relation.Nullary.Negation using (¬_; ¬¬-Monad)
 
 private
   variable
@@ -934,22 +933,3 @@ idempotent {A = A} B x f = sound (idem x)
                                                      laterˡ (refl (Setoid.refl B))) ⟩
     (♭ x >>= λ y′ →     ♭ x >>= λ y″ → f y′ y″)  ≳⟨ idem (♭ x) ⟩≅
     (♭ x >>= λ y′ → f y′ y′)                     ∎))
-
-------------------------------------------------------------------------
--- Example
-
-private
- module Example where
-
-  open Data.Nat
-  open Workaround
-
-  -- McCarthy's f91:
-
-  f91′ : ℕ → ℕ ⊥P
-  f91′ n with does (n ≤? 100)
-  ... | true  = later (♯ (f91′ (11 + n) >>= f91′))
-  ... | false = now (n ∸ 10)
-
-  f91 : ℕ → ℕ ⊥
-  f91 n = ⟦ f91′ n ⟧P
diff --git a/src/Function/Properties/Inverse/HalfAdjointEquivalence.agda b/src/Function/Properties/Inverse/HalfAdjointEquivalence.agda
index 4ea2965f99..05c20b9682 100644
--- a/src/Function/Properties/Inverse/HalfAdjointEquivalence.agda
+++ b/src/Function/Properties/Inverse/HalfAdjointEquivalence.agda
@@ -8,10 +8,12 @@
 
 module Function.Properties.Inverse.HalfAdjointEquivalence where
 
-open import Function.Base
-open import Function
-open import Level
+open import Function.Base using (id; _∘_)
+open import Function.Bundles using (Inverse; _↔_; mk↔ₛ′)
+open import Level using (Level; _⊔_)
 open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; cong; sym; trans; trans-reflʳ; cong-≡id; cong-∘; naturality;
+         cong-id; trans-assoc; trans-symˡ; module ≡-Reasoning)
 
 private
   variable
diff --git a/src/Function/Related/TypeIsomorphisms.agda b/src/Function/Related/TypeIsomorphisms.agda
index eafe0b50b6..3386032654 100644
--- a/src/Function/Related/TypeIsomorphisms.agda
+++ b/src/Function/Related/TypeIsomorphisms.agda
@@ -13,15 +13,14 @@ open import Algebra
 open import Algebra.Structures.Biased using (isCommutativeSemiringˡ)
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Bool.Base using (true; false)
-open import Data.Empty using (⊥-elim)
-open import Data.Empty.Polymorphic using (⊥) renaming (⊥-elim to ⊥ₚ-elim)
+open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
 open import Data.Product.Base as Product
   using (_×_; Σ; curry; uncurry; _,_; -,_; <_,_>; proj₁; proj₂; ∃₂; ∃)
 open import Data.Product.Function.NonDependent.Propositional
 open import Data.Sum.Base as Sum
 open import Data.Sum.Properties using (swap-involutive)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
-open import Data.Unit.Polymorphic using (⊤)
+open import Data.Unit.Polymorphic.Base using (⊤)
 open import Level using (Level; Lift; 0ℓ; suc)
 open import Function.Base
 open import Function.Bundles
@@ -32,7 +31,7 @@ open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ)
+open import Relation.Nullary using (Dec; ¬_; _because_; ofⁿ; contradiction)
 import Relation.Nullary.Indexed as I
 open import Relation.Nullary.Decidable using (True)
 
@@ -68,10 +67,10 @@ private
 -- × has ⊥ has its zero
 
 ×-zeroˡ : ∀ ℓ → LeftZero {ℓ = ℓ} _↔_ ⊥ _×_
-×-zeroˡ ℓ A = mk↔ₛ′ proj₁ < id , ⊥ₚ-elim > (λ _ → refl) (λ { () })
+×-zeroˡ ℓ A = mk↔ₛ′ proj₁ < id , ⊥-elim > (λ _ → refl) (λ { () })
 
 ×-zeroʳ : ∀ ℓ → RightZero {ℓ = ℓ} _↔_ ⊥ _×_
-×-zeroʳ ℓ A = mk↔ₛ′ proj₂ < ⊥ₚ-elim , id > (λ _ → refl) (λ { () })
+×-zeroʳ ℓ A = mk↔ₛ′ proj₂ < ⊥-elim , id > (λ _ → refl) (λ { () })
 
 ×-zero : ∀ ℓ → Zero _↔_ ⊥ _×_
 ×-zero ℓ  = ×-zeroˡ ℓ , ×-zeroʳ ℓ
@@ -332,4 +331,4 @@ True↔ : ∀ {p} {P : Set p}
 True↔ ( true because  [p]) irr =
   mk↔ₛ′ (λ _ → invert [p]) (λ _ → _) (irr _) (λ _ → refl)
 True↔ (false because ofⁿ ¬p) _ =
-  mk↔ₛ′ (λ()) (invert (ofⁿ ¬p)) (⊥-elim ∘ ¬p) (λ ())
+  mk↔ₛ′ (λ()) (invert (ofⁿ ¬p)) (λ x → flip contradiction ¬p x) (λ ())
diff --git a/src/Relation/Binary/Reasoning/Base/Apartness.agda b/src/Relation/Binary/Reasoning/Base/Apartness.agda
index 22eee2d505..d3a850eca7 100644
--- a/src/Relation/Binary/Reasoning/Base/Apartness.agda
+++ b/src/Relation/Binary/Reasoning/Base/Apartness.agda
@@ -8,7 +8,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Level using (Level; _⊔_)
-open import Function using (case_of_)
+open import Function.Base using (case_of_)
 open import Relation.Nullary.Decidable using (Dec; yes; no)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Structures using (IsEquivalence)
diff --git a/src/Relation/Binary/Reasoning/Base/Double.agda b/src/Relation/Binary/Reasoning/Base/Double.agda
index b5fd9f66e4..0c1ae9b60c 100644
--- a/src/Relation/Binary/Reasoning/Base/Double.agda
+++ b/src/Relation/Binary/Reasoning/Base/Double.agda
@@ -11,7 +11,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Level using (_⊔_)
-open import Function using (case_of_)
+open import Function.Base using (case_of_)
 open import Relation.Nullary.Decidable.Core using (Dec; yes; no)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Definitions using (Reflexive; Trans)
diff --git a/src/Relation/Binary/Reasoning/Base/Partial.agda b/src/Relation/Binary/Reasoning/Base/Partial.agda
index 3d24edf501..5da6792f64 100644
--- a/src/Relation/Binary/Reasoning/Base/Partial.agda
+++ b/src/Relation/Binary/Reasoning/Base/Partial.agda
@@ -6,10 +6,10 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Function using (case_of_)
+open import Function.Base using (case_of_)
 open import Level using (_⊔_)
-open import Relation.Binary.Core
-open import Relation.Binary.Definitions
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Transitive; Trans; Reflexive)
 open import Relation.Nullary.Decidable using (Dec; yes; no)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Reasoning.Syntax
diff --git a/src/Relation/Binary/Reasoning/Base/Triple.agda b/src/Relation/Binary/Reasoning/Base/Triple.agda
index 6afb937497..3433e58063 100644
--- a/src/Relation/Binary/Reasoning/Base/Triple.agda
+++ b/src/Relation/Binary/Reasoning/Base/Triple.agda
@@ -12,7 +12,7 @@
 
 open import Data.Product.Base using (proj₁; proj₂)
 open import Level using (_⊔_)
-open import Function using (case_of_)
+open import Function.Base using (case_of_)
 open import Relation.Nullary.Decidable.Core
   using (Dec; yes; no)
 open import Relation.Binary.Core using (Rel; _⇒_)

From 03a2d71a6b2d9f01ccc7cccbc32effc1fbad9b82 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Thu, 4 Apr 2024 17:01:19 +0200
Subject: [PATCH 262/316] Tighten imports (#2334)

* tighten imports in some files, and make imports explicit in others. Driven by staring at a partial dependency graph.

* more tightening of imports

* and even more tightening of imports

* some explicit  for precision

---------

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
---
 src/Algebra/Definitions/RawMagma.agda                  |  6 +++---
 src/Algebra/Structures.agda                            |  2 +-
 src/Data/List/Relation/Unary/AllPairs.agda             |  2 +-
 src/Data/Maybe/Base.agda                               |  6 +++---
 src/Data/Product/Properties.agda                       |  4 ++--
 src/Function/Indexed/Relation/Binary/Equality.agda     |  2 +-
 src/Relation/Binary/Bundles.agda                       |  9 ++++-----
 src/Relation/Binary/PropositionalEquality/Algebra.agda | 10 ++++++----
 src/Relation/Binary/Reasoning/Setoid.agda              |  2 +-
 src/Relation/Binary/Reasoning/Syntax.agda              |  3 ++-
 10 files changed, 24 insertions(+), 22 deletions(-)

diff --git a/src/Algebra/Definitions/RawMagma.agda b/src/Algebra/Definitions/RawMagma.agda
index f235f5d622..2e1bc657c7 100644
--- a/src/Algebra/Definitions/RawMagma.agda
+++ b/src/Algebra/Definitions/RawMagma.agda
@@ -10,11 +10,11 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra.Bundles using (RawMagma)
+open import Algebra.Bundles.Raw using (RawMagma)
 open import Data.Product.Base using (_×_; ∃)
 open import Level using (_⊔_)
-open import Relation.Binary.Core
-open import Relation.Nullary.Negation using (¬_)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Nullary.Negation.Core using (¬_)
 
 module Algebra.Definitions.RawMagma
   {a ℓ} (M : RawMagma a ℓ)
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index bb28e89882..5b88230e00 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -21,7 +21,7 @@ module Algebra.Structures
 -- The file is divided into sections depending on the arities of the
 -- components of the algebraic structure.
 
-open import Algebra.Core
+open import Algebra.Core using (Op₁; Op₂)
 open import Algebra.Definitions _≈_
 import Algebra.Consequences.Setoid as Consequences
 open import Data.Product.Base using (_,_; proj₁; proj₂)
diff --git a/src/Data/List/Relation/Unary/AllPairs.agda b/src/Data/List/Relation/Unary/AllPairs.agda
index bed4a5972d..6036586822 100644
--- a/src/Data/List/Relation/Unary/AllPairs.agda
+++ b/src/Data/List/Relation/Unary/AllPairs.agda
@@ -18,7 +18,7 @@ open import Function.Base using (id; _∘_)
 open import Level using (_⊔_)
 open import Relation.Binary.Definitions as B
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (refl; cong₂)
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
 open import Relation.Nullary.Decidable as Dec using (_×-dec_; yes; no)
 
diff --git a/src/Data/Maybe/Base.agda b/src/Data/Maybe/Base.agda
index b0ae001f46..14cff5e9ba 100644
--- a/src/Data/Maybe/Base.agda
+++ b/src/Data/Maybe/Base.agda
@@ -15,9 +15,9 @@ open import Data.Bool.Base using (Bool; true; false; not)
 open import Data.Unit.Base using (⊤)
 open import Data.These.Base using (These; this; that; these)
 open import Data.Product.Base as Prod using (_×_; _,_)
-open import Function.Base
-open import Relation.Nullary.Reflects
-open import Relation.Nullary.Decidable.Core
+open import Function.Base using (const; _∘_; id)
+open import Relation.Nullary.Reflects using (invert)
+open import Relation.Nullary.Decidable.Core using (Dec; _because_)
 
 private
   variable
diff --git a/src/Data/Product/Properties.agda b/src/Data/Product/Properties.agda
index ec3f05d018..0bfa3166df 100644
--- a/src/Data/Product/Properties.agda
+++ b/src/Data/Product/Properties.agda
@@ -8,8 +8,8 @@
 
 module Data.Product.Properties where
 
-open import Axiom.UniquenessOfIdentityProofs
-open import Data.Product.Base
+open import Axiom.UniquenessOfIdentityProofs using (UIP; module Decidable⇒UIP)
+open import Data.Product.Base using (_,_; Σ; _×_; map; swap; ∃; ∃₂)
 open import Function.Base using (_∋_; _∘_; id)
 open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
diff --git a/src/Function/Indexed/Relation/Binary/Equality.agda b/src/Function/Indexed/Relation/Binary/Equality.agda
index ef9d6b3657..6f0bd7db8f 100644
--- a/src/Function/Indexed/Relation/Binary/Equality.agda
+++ b/src/Function/Indexed/Relation/Binary/Equality.agda
@@ -8,7 +8,7 @@
 
 module Function.Indexed.Relation.Binary.Equality where
 
-open import Relation.Binary using (Setoid)
+open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Indexed.Heterogeneous using (IndexedSetoid)
 
 -- A variant of setoid which uses the propositional equality setoid
diff --git a/src/Relation/Binary/Bundles.agda b/src/Relation/Binary/Bundles.agda
index 641803f8ef..5d29cabe57 100644
--- a/src/Relation/Binary/Bundles.agda
+++ b/src/Relation/Binary/Bundles.agda
@@ -11,11 +11,10 @@
 module Relation.Binary.Bundles where
 
 open import Function.Base using (flip)
-open import Level
-open import Relation.Nullary.Negation using (¬_)
-open import Relation.Binary.Core
-open import Relation.Binary.Definitions
-open import Relation.Binary.Structures
+open import Level using (Level; suc; _⊔_)
+open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Structures -- most of it
 
 ------------------------------------------------------------------------
 -- Setoids
diff --git a/src/Relation/Binary/PropositionalEquality/Algebra.agda b/src/Relation/Binary/PropositionalEquality/Algebra.agda
index 22ac67b298..7e0118bbcb 100644
--- a/src/Relation/Binary/PropositionalEquality/Algebra.agda
+++ b/src/Relation/Binary/PropositionalEquality/Algebra.agda
@@ -8,10 +8,12 @@
 
 module Relation.Binary.PropositionalEquality.Algebra where
 
-open import Algebra
-open import Level
-open import Relation.Binary.PropositionalEquality.Core
-open import Relation.Binary.PropositionalEquality.Properties
+open import Algebra.Bundles using (Magma)
+open import Algebra.Core using (Op₂)
+open import Algebra.Structures using (IsMagma)
+open import Level using (Level)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence)
 
 private
   variable
diff --git a/src/Relation/Binary/Reasoning/Setoid.agda b/src/Relation/Binary/Reasoning/Setoid.agda
index ba6be9f79e..d80b2546d7 100644
--- a/src/Relation/Binary/Reasoning/Setoid.agda
+++ b/src/Relation/Binary/Reasoning/Setoid.agda
@@ -21,7 +21,7 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.Reasoning.Syntax
+open import Relation.Binary.Reasoning.Syntax using (module ≈-syntax)
 
 module Relation.Binary.Reasoning.Setoid {s₁ s₂} (S : Setoid s₁ s₂) where
 
diff --git a/src/Relation/Binary/Reasoning/Syntax.agda b/src/Relation/Binary/Reasoning/Syntax.agda
index 809d238b39..3860f97b35 100644
--- a/src/Relation/Binary/Reasoning/Syntax.agda
+++ b/src/Relation/Binary/Reasoning/Syntax.agda
@@ -9,9 +9,10 @@
 open import Level using (Level; _⊔_; suc)
 open import Relation.Nullary.Decidable.Core
   using (Dec; True; toWitness)
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Negation.Core using (contradiction)
 open import Relation.Binary.Core using (Rel; REL; _⇒_)
 open import Relation.Binary.Definitions
+  using (_Respectsʳ_; Asymmetric; Trans; Sym; Reflexive)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_)
 

From b7bc4f7187bdb187976583fc7db70cf3490094bc Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 4 Apr 2024 17:01:48 +0200
Subject: [PATCH 263/316] [fixes #2273] Add `SuccessorSet` and associated
 boilerplate (#2277)

* added `RawNNO`

* [fix #2273] Add `NNO`-related modules

* start again, fail better...

* rectify names

* tighten `import`s

* rectify names: `CHANGELOG`

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                         | 33 ++++++++++++++----
 src/Algebra/Bundles.agda             | 24 +++++++++++--
 src/Algebra/Bundles/Raw.agda         | 14 ++++++++
 src/Algebra/Morphism/Structures.agda | 52 ++++++++++++++++++++++++++--
 src/Algebra/Structures.agda          | 14 ++++++++
 5 files changed, 126 insertions(+), 11 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 2b9d357511..469c926af8 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -128,6 +128,16 @@ New modules
 Additions to existing modules
 -----------------------------
 
+* In `Algebra.Bundles`
+  ```agda
+  record SuccessorSet c ℓ : Set (suc (c ⊔ ℓ))
+  ```
+
+* In `Algebra.Bundles.Raw`
+  ```agda
+  record RawSuccessorSet c ℓ : Set (suc (c ⊔ ℓ))
+  ```
+
 * Exporting more `Raw` substructures from `Algebra.Bundles.Ring`:
   ```agda
   rawNearSemiring   : RawNearSemiring _ _
@@ -135,6 +145,12 @@ Additions to existing modules
   +-rawGroup        : RawGroup _ _
   ```
 
+* In `Algebra.Construct.Terminal`:
+  ```agda
+  rawNearSemiring : RawNearSemiring c ℓ
+  nearSemiring    : NearSemiring c ℓ
+  ```
+
 * In `Algebra.Module.Bundles`, raw bundles are re-exported and the bundles expose their raw counterparts.
 
 * In `Algebra.Module.Construct.DirectProduct`:
@@ -173,6 +189,13 @@ Additions to existing modules
   rawModule          : RawModule R c ℓ
   ```
 
+* In `Algebra.Morphism.Structures`
+  ```agda
+  module SuccessorSetMorphisms (N₁ : RawSuccessorSet a ℓ₁) (N₂ : RawSuccessorSet b ℓ₂) where
+    record IsSuccessorSetHomomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
+    record IsSuccessorSetMonomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
+    record IsSuccessorSetIsomorphism  (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
+
 * In `Algebra.Properties.Group`:
   ```agda
   isQuasigroup    : IsQuasigroup _∙_ _\\_ _//_
@@ -199,12 +222,6 @@ Additions to existing modules
   identity-unique  : Identity x _∙_ → x ≈ ε
   ```
 
-* In `Algebra.Construct.Terminal`:
-  ```agda
-  rawNearSemiring : RawNearSemiring c ℓ
-  nearSemiring    : NearSemiring c ℓ
-  ```
-
 * In `Algebra.Properties.Monoid.Mult`:
   ```agda
   ×-homo-0 : ∀ x → 0 × x ≈ 0#
@@ -218,6 +235,10 @@ Additions to existing modules
   idem-×-homo-* : (_*_ IdempotentOn x) → (m × x) * (n × x) ≈ (m ℕ.* n) × x
   ```
 
+* In `Algebra.Structures`
+  ```agda
+  record IsSuccessorSet (suc# : Op₁ A) (zero# : A) : Set _
+
 * In `Algebra.Structures.IsGroup`:
   ```agda
   infixl 6 _//_
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index 1a31d16496..1969731c2c 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -15,19 +15,37 @@ import Algebra.Bundles.Raw as Raw
 open import Algebra.Core
 open import Algebra.Structures
 open import Relation.Binary.Core using (Rel)
-open import Function.Base
-import Relation.Nullary as N
 open import Level
 
 ------------------------------------------------------------------------
 -- Re-export definitions of 'raw' bundles
 
 open Raw public
-  using (RawMagma; RawMonoid; RawGroup
+  using ( RawSuccessorSet; RawMagma; RawMonoid; RawGroup
         ; RawNearSemiring; RawSemiring
         ; RawRingWithoutOne; RawRing
         ; RawQuasigroup; RawLoop; RawKleeneAlgebra)
 
+------------------------------------------------------------------------
+-- Bundles with 1 unary operation & 1 element
+------------------------------------------------------------------------
+
+record SuccessorSet c ℓ : Set (suc (c ⊔ ℓ)) where
+  infix  4 _≈_
+  field
+    Carrier          : Set c
+    _≈_              : Rel Carrier ℓ
+    suc#             : Op₁ Carrier
+    zero#            : Carrier
+    isSuccessorSet   : IsSuccessorSet _≈_ suc# zero#
+
+  open IsSuccessorSet isSuccessorSet public
+
+  rawSuccessorSet : RawSuccessorSet _ _
+  rawSuccessorSet = record { _≈_ = _≈_; suc# = suc#; zero# = zero# }
+
+  open RawSuccessorSet rawSuccessorSet public
+
 ------------------------------------------------------------------------
 -- Bundles with 1 binary operation
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Bundles/Raw.agda b/src/Algebra/Bundles/Raw.agda
index 727440d85e..5c99d02790 100644
--- a/src/Algebra/Bundles/Raw.agda
+++ b/src/Algebra/Bundles/Raw.agda
@@ -13,6 +13,20 @@ open import Relation.Binary.Core using (Rel)
 open import Level using (suc; _⊔_)
 open import Relation.Nullary.Negation.Core using (¬_)
 
+------------------------------------------------------------------------
+-- Raw bundles with 1 unary operation & 1 element
+------------------------------------------------------------------------
+
+-- A raw SuccessorSet is a SuccessorSet without any laws.
+
+record RawSuccessorSet c ℓ : Set (suc (c ⊔ ℓ)) where
+  infix  4 _≈_
+  field
+    Carrier : Set c
+    _≈_     : Rel Carrier ℓ
+    suc#    : Op₁ Carrier
+    zero#   : Carrier
+
 ------------------------------------------------------------------------
 -- Raw bundles with 1 binary operation
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Morphism/Structures.agda b/src/Algebra/Morphism/Structures.agda
index e73fa7e9e6..c352904be7 100644
--- a/src/Algebra/Morphism/Structures.agda
+++ b/src/Algebra/Morphism/Structures.agda
@@ -6,8 +6,6 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Core
-
 module Algebra.Morphism.Structures where
 
 open import Algebra.Core
@@ -15,12 +13,62 @@ open import Algebra.Bundles
 import Algebra.Morphism.Definitions as MorphismDefinitions
 open import Level using (Level; _⊔_)
 open import Function.Definitions
+open import Relation.Binary.Core
 open import Relation.Binary.Morphism.Structures
 
 private
   variable
     a b ℓ₁ ℓ₂ : Level
 
+------------------------------------------------------------------------
+-- Morphisms over SuccessorSet-like structures
+------------------------------------------------------------------------
+
+module SuccessorSetMorphisms
+  (N₁ : RawSuccessorSet a ℓ₁) (N₂ : RawSuccessorSet b ℓ₂)
+  where
+
+  open RawSuccessorSet N₁
+    renaming (Carrier to A; _≈_ to _≈₁_; suc# to suc#₁; zero# to zero#₁)
+  open RawSuccessorSet N₂
+    renaming (Carrier to B; _≈_ to _≈₂_; suc# to suc#₂; zero# to zero#₂)
+  open MorphismDefinitions A B _≈₂_
+
+
+  record IsSuccessorSetHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
+    field
+      isRelHomomorphism : IsRelHomomorphism _≈₁_ _≈₂_ ⟦_⟧
+      suc#-homo         : Homomorphic₁ ⟦_⟧ suc#₁ suc#₂
+      zero#-homo        : Homomorphic₀ ⟦_⟧ zero#₁ zero#₂
+
+  record IsSuccessorSetMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂) where
+    field
+      isSuccessorSetHomomorphism : IsSuccessorSetHomomorphism ⟦_⟧
+      injective                  : Injective _≈₁_ _≈₂_ ⟦_⟧
+
+    open IsSuccessorSetHomomorphism isSuccessorSetHomomorphism public
+
+    isRelMonomorphism : IsRelMonomorphism _≈₁_ _≈₂_ ⟦_⟧
+    isRelMonomorphism = record
+      { isHomomorphism = isRelHomomorphism
+      ; injective      = injective
+      }
+
+
+  record IsSuccessorSetIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where
+    field
+      isSuccessorSetMonomorphism : IsSuccessorSetMonomorphism ⟦_⟧
+      surjective                 : Surjective _≈₁_ _≈₂_ ⟦_⟧
+
+    open IsSuccessorSetMonomorphism isSuccessorSetMonomorphism public
+
+    isRelIsomorphism : IsRelIsomorphism _≈₁_ _≈₂_ ⟦_⟧
+    isRelIsomorphism = record
+      { isMonomorphism = isRelMonomorphism
+      ; surjective     = surjective
+      }
+
+
 ------------------------------------------------------------------------
 -- Morphisms over magma-like structures
 ------------------------------------------------------------------------
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index 5b88230e00..066363686f 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -27,6 +27,20 @@ import Algebra.Consequences.Setoid as Consequences
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Level using (_⊔_)
 
+------------------------------------------------------------------------
+-- Structures with 1 unary operation & 1 element
+------------------------------------------------------------------------
+
+record IsSuccessorSet (suc# : Op₁ A) (zero# : A) : Set (a ⊔ ℓ) where
+  field
+    isEquivalence : IsEquivalence _≈_
+    suc#-cong     : Congruent₁ suc#
+
+  open IsEquivalence isEquivalence public
+
+  setoid : Setoid a ℓ
+  setoid = record { isEquivalence = isEquivalence }
+
 ------------------------------------------------------------------------
 -- Structures with 1 binary operation
 ------------------------------------------------------------------------

From ab9e0e485345a24240fb0fe499f723ab2bb2c9d1 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Tue, 9 Apr 2024 05:48:30 -0400
Subject: [PATCH 264/316] Tighten imports, last big versoin (#2347)

* more tightening of imports. Mostly driven by Data.Bool, Data.Nat and Data.Char.

* more tightening of imports

* mostly Data.Unit

* slightly tighten imports.

* remove import of unused Lift)

* fix all 3 things noticed by @jamesmckinna
---
 src/Algebra/Construct/LexProduct.agda            |  9 +++++----
 src/Algebra/Properties/Monoid/Sum.agda           |  9 +++++----
 .../Solver/IdempotentCommutativeMonoid.agda      |  2 +-
 src/Codata/Sized/Conat/Literals.agda             |  6 +++---
 src/Codata/Sized/Delay/Properties.agda           |  2 +-
 src/Data/Container/Fixpoints/Sized.agda          |  8 ++++----
 .../Relation/Binary/Pointwise/Properties.agda    |  2 +-
 src/Data/Fin/Permutation.agda                    |  2 +-
 src/Data/Fin/Properties.agda                     |  2 +-
 src/Data/Fin/Substitution/Lemmas.agda            |  6 +++---
 src/Data/Integer.agda                            |  2 +-
 src/Data/Integer/DivMod.agda                     |  1 -
 src/Data/Integer/Divisibility/Signed.agda        |  2 +-
 src/Data/Integer/Literals.agda                   |  6 +++---
 src/Data/Integer/Properties.agda                 |  2 +-
 src/Data/Integer/Tactic/RingSolver.agda          |  2 +-
 src/Data/List/Fresh/Properties.agda              |  2 +-
 src/Data/List/Literals.agda                      |  8 ++++----
 src/Data/List/NonEmpty/Properties.agda           |  2 +-
 .../Binary/Permutation/Setoid/Properties.agda    |  2 +-
 .../Binary/Subset/Propositional/Properties.agda  |  2 +-
 src/Data/List/Relation/Unary/Any/Properties.agda |  7 +++----
 .../List/Relation/Unary/Grouped/Properties.agda  |  8 ++++----
 .../Unique/DecPropositional/Properties.agda      | 10 +++++-----
 src/Data/List/Sort/MergeSort.agda                | 15 ++++++++-------
 src/Data/Maybe.agda                              |  6 +++---
 src/Data/Nat/Combinatorics/Specification.agda    |  2 +-
 src/Data/Nat/Literals.agda                       |  6 +++---
 src/Data/Nat/Logarithm.agda                      |  2 +-
 src/Data/Nat/Properties.agda                     |  2 +-
 src/Data/Nat/PseudoRandom/LCG/Unsafe.agda        |  4 ++--
 src/Data/Nat/Tactic/RingSolver.agda              |  2 +-
 src/Data/Product/Nary/NonDependent.agda          |  2 +-
 src/Data/Rational/Literals.agda                  |  6 +++---
 src/Data/Rational/Properties.agda                |  3 +--
 src/Data/Star/BoundedVec.agda                    | 10 ++++++----
 src/Data/Star/Decoration.agda                    |  5 ++---
 src/Data/Star/Environment.agda                   | 14 +++++++-------
 src/Data/Star/Fin.agda                           |  4 ++--
 src/Data/Star/List.agda                          |  5 +++--
 src/Data/Star/Nat.agda                           |  2 +-
 src/Data/Star/Vec.agda                           | 11 ++++++-----
 src/Data/String.agda                             |  2 +-
 src/Data/String/Literals.agda                    |  6 +++---
 src/Data/Tree/AVL/Key.agda                       |  8 +++-----
 src/Data/Tree/AVL/NonEmpty.agda                  |  6 ++----
 src/Data/Tree/Binary/Zipper.agda                 |  2 +-
 src/Data/Vec/Bounded.agda                        |  2 +-
 src/Data/Vec/Functional/Properties.agda          |  2 +-
 .../Vec/Relation/Binary/Equality/Setoid.agda     |  9 ++++-----
 src/Data/Vec/Relation/Binary/Lex/Core.agda       |  4 ++--
 src/Data/Vec/Relation/Binary/Lex/NonStrict.agda  |  6 +++---
 .../Vec/Relation/Unary/AllPairs/Properties.agda  |  2 +-
 src/Effect/Monad/Reader/Indexed.agda             |  8 ++++----
 src/Effect/Monad/State/Indexed.agda              | 16 +++++++++-------
 src/System/Environment/Primitive.agda            |  2 +-
 src/Tactic/MonoidSolver.agda                     |  6 +++---
 src/Tactic/RingSolver/Core/NatSet.agda           |  4 ++--
 .../Core/Polynomial/Homomorphism/Addition.agda   |  2 +-
 .../Core/Polynomial/Homomorphism/Lemmas.agda     | 10 +++++-----
 .../Core/Polynomial/Homomorphism/Negation.agda   |  2 +-
 .../Core/Polynomial/Homomorphism/Variables.agda  |  2 +-
 .../RingSolver/Core/Polynomial/Semantics.agda    |  2 +-
 src/Test/Golden.agda                             |  2 +-
 src/Text/Format.agda                             | 10 +++++-----
 src/Text/Pretty/Core.agda                        |  5 +++--
 src/Text/Printf.agda                             |  2 +-
 67 files changed, 164 insertions(+), 163 deletions(-)

diff --git a/src/Algebra/Construct/LexProduct.agda b/src/Algebra/Construct/LexProduct.agda
index e2f5825824..65a6b80ba1 100644
--- a/src/Algebra/Construct/LexProduct.agda
+++ b/src/Algebra/Construct/LexProduct.agda
@@ -6,16 +6,17 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra
-open import Data.Bool using (true; false)
+open import Algebra.Bundles using (Magma)
+open import Algebra.Definitions
+open import Data.Bool.Base using (true; false)
 open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (Pointwise)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Nullary using (¬_; does; yes; no)
-open import Relation.Nullary.Negation using (contradiction; contradiction₂)
+open import Relation.Nullary.Decidable.Core using (does; yes; no)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction; contradiction₂)
 import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
 module Algebra.Construct.LexProduct
diff --git a/src/Algebra/Properties/Monoid/Sum.agda b/src/Algebra/Properties/Monoid/Sum.agda
index ae64a5f4dc..78a8d3b205 100644
--- a/src/Algebra/Properties/Monoid/Sum.agda
+++ b/src/Algebra/Properties/Monoid/Sum.agda
@@ -7,16 +7,17 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Algebra.Bundles using (Monoid)
+
+module Algebra.Properties.Monoid.Sum {a ℓ} (M : Monoid a ℓ) where
+
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; NonZero)
-open import Data.Vec.Functional as Vector
+open import Data.Vec.Functional as Vector using (Vector; replicate; init;
+  last; head; tail)
 open import Data.Fin.Base using (zero; suc)
-open import Data.Unit using (tt)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Core using (_Preserves_⟶_)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≗_; _≡_)
 
-module Algebra.Properties.Monoid.Sum {a ℓ} (M : Monoid a ℓ) where
-
 open Monoid M
   renaming
   ( _∙_       to _+_
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
index 35cfcc5ea6..eefb5f40ab 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
@@ -8,7 +8,7 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra
+open import Algebra.Bundles using (IdempotentCommutativeMonoid)
 
 open import Data.Bool as Bool using (Bool; true; false; if_then_else_; _∨_)
 open import Data.Fin.Base using (Fin; zero; suc)
diff --git a/src/Codata/Sized/Conat/Literals.agda b/src/Codata/Sized/Conat/Literals.agda
index 7f0cccb26b..932fd4ab88 100644
--- a/src/Codata/Sized/Conat/Literals.agda
+++ b/src/Codata/Sized/Conat/Literals.agda
@@ -8,9 +8,9 @@
 
 module Codata.Sized.Conat.Literals where
 
-open import Agda.Builtin.FromNat
-open import Data.Unit
-open import Codata.Sized.Conat
+open import Agda.Builtin.FromNat using (Number)
+open import Data.Unit.Base using (⊤)
+open import Codata.Sized.Conat using (Conat; fromℕ)
 
 number : ∀ {i} → Number (Conat i)
 number = record
diff --git a/src/Codata/Sized/Delay/Properties.agda b/src/Codata/Sized/Delay/Properties.agda
index de2b3c95b4..2936660718 100644
--- a/src/Codata/Sized/Delay/Properties.agda
+++ b/src/Codata/Sized/Delay/Properties.agda
@@ -10,7 +10,7 @@ module Codata.Sized.Delay.Properties where
 
 open import Size
 import Data.Sum.Base as Sum
-import Data.Nat as ℕ
+import Data.Nat.Base as ℕ
 open import Codata.Sized.Thunk using (Thunk; force)
 open import Codata.Sized.Conat
 open import Codata.Sized.Conat.Bisimilarity as Coℕ using (zero ; suc)
diff --git a/src/Data/Container/Fixpoints/Sized.agda b/src/Data/Container/Fixpoints/Sized.agda
index abfb877e00..47655af29a 100644
--- a/src/Data/Container/Fixpoints/Sized.agda
+++ b/src/Data/Container/Fixpoints/Sized.agda
@@ -8,10 +8,10 @@
 
 module Data.Container.Fixpoints.Sized where
 
-open import Level
-open import Size
-open import Codata.Sized.M
-open import Data.W.Sized
+open import Level using (Level; _⊔_)
+open import Size using (Size)
+open import Codata.Sized.M using (M)
+open import Data.W.Sized using (W)
 open import Data.Container hiding (μ) public
 
 private
diff --git a/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
index 58e6966154..a1fc21f44e 100644
--- a/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
@@ -11,7 +11,7 @@ module Data.Container.Indexed.Relation.Binary.Pointwise.Properties where
 open import Axiom.Extensionality.Propositional
 open import Data.Container.Indexed.Core
 open import Data.Container.Indexed.Relation.Binary.Pointwise
-open import Data.Product using (_,_; Σ-syntax; -,_)
+open import Data.Product.Base using (_,_; Σ-syntax; -,_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary
 open import Relation.Binary.PropositionalEquality as P
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 97412c094d..3fa0ebf92f 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -8,7 +8,7 @@
 
 module Data.Fin.Permutation where
 
-open import Data.Bool using (true; false)
+open import Data.Bool.Base using (true; false)
 open import Data.Empty using (⊥-elim)
 open import Data.Fin.Base
 open import Data.Fin.Patterns
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index ff9239359c..e58797ecd1 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -22,7 +22,7 @@ open import Data.Nat.Base as ℕ
   using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; s<s⁻¹; _∸_; _^_)
 import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
-open import Data.Unit using (⊤; tt)
+open import Data.Unit.Base using (⊤; tt)
 open import Data.Product.Base as Product
   using (∃; ∃₂; _×_; _,_; map; proj₁; proj₂; uncurry; <_,_>)
 open import Data.Product.Properties using (,-injective)
diff --git a/src/Data/Fin/Substitution/Lemmas.agda b/src/Data/Fin/Substitution/Lemmas.agda
index 05e177acf7..9dedec0ebb 100644
--- a/src/Data/Fin/Substitution/Lemmas.agda
+++ b/src/Data/Fin/Substitution/Lemmas.agda
@@ -9,11 +9,12 @@
 module Data.Fin.Substitution.Lemmas where
 
 open import Data.Fin.Substitution
-open import Data.Nat hiding (_⊔_; _/_)
+open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Fin.Base using (Fin; zero; suc; lift)
-open import Data.Vec.Base
+open import Data.Vec.Base using (lookup; []; _∷_; map)
 import Data.Vec.Properties as Vec
 open import Function.Base as Fun using (_∘_; _$_; flip)
+open import Level using (Level; _⊔_)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; refl; sym; cong; cong₂)
 open import Relation.Binary.PropositionalEquality.Properties
@@ -21,7 +22,6 @@ open import Relation.Binary.PropositionalEquality.Properties
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
   using (Star; ε; _◅_; _▻_)
 open ≡-Reasoning
-open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred)
 
 private
diff --git a/src/Data/Integer.agda b/src/Data/Integer.agda
index 66c8100242..6e4976f3df 100644
--- a/src/Data/Integer.agda
+++ b/src/Data/Integer.agda
@@ -31,7 +31,7 @@ open import Data.Integer.Properties public
 -- Show
 
 import Data.Nat.Show as ℕ using (show)
-open import Data.Sign as Sign using (Sign)
+open import Data.Sign.Base as Sign using (Sign)
 open import Data.String.Base using (String; _++_)
 
 show : ℤ → String
diff --git a/src/Data/Integer/DivMod.agda b/src/Data/Integer/DivMod.agda
index 687570ba1b..28d15d05fd 100644
--- a/src/Data/Integer/DivMod.agda
+++ b/src/Data/Integer/DivMod.agda
@@ -13,7 +13,6 @@ open import Data.Integer.Properties
 open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s; z<s; s<s)
 import Data.Nat.Properties as ℕ
 import Data.Nat.DivMod as ℕ
-import Data.Sign as S
 open import Function.Base using (_∘′_)
 open import Relation.Nullary.Decidable
 open import Relation.Binary.PropositionalEquality
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index 18df0c8a91..f88f3ba859 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -16,7 +16,7 @@ import Data.Nat.Base as ℕ
 import Data.Nat.Divisibility as ℕ
 import Data.Nat.Coprimality as ℕ
 import Data.Nat.Properties as ℕ
-import Data.Sign as Sign
+import Data.Sign.Base as Sign
 import Data.Sign.Properties as Sign
 open import Level
 open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
diff --git a/src/Data/Integer/Literals.agda b/src/Data/Integer/Literals.agda
index 5794f1ceef..d06be38eea 100644
--- a/src/Data/Integer/Literals.agda
+++ b/src/Data/Integer/Literals.agda
@@ -8,9 +8,9 @@
 
 module Data.Integer.Literals where
 
-open import Agda.Builtin.FromNat
-open import Agda.Builtin.FromNeg
-open import Data.Unit using (⊤)
+open import Agda.Builtin.FromNat using (Number)
+open import Agda.Builtin.FromNeg using (Negative)
+open import Data.Unit.Base using (⊤)
 open import Data.Integer.Base using (ℤ; -_; +_)
 
 number : Number ℤ
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index d5706b5d53..842058f62e 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -23,7 +23,7 @@ import Data.Nat.Properties as ℕ
 open import Data.Nat.Solver
 open import Data.Product.Base using (proj₁; proj₂; _,_; _×_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
-open import Data.Sign as Sign using (Sign)
+open import Data.Sign.Base as Sign using (Sign)
 import Data.Sign.Properties as Sign
 open import Function.Base using (_∘_; _$_; id)
 open import Level using (0ℓ)
diff --git a/src/Data/Integer/Tactic/RingSolver.agda b/src/Data/Integer/Tactic/RingSolver.agda
index affe9ecd2e..d4e1a43536 100644
--- a/src/Data/Integer/Tactic/RingSolver.agda
+++ b/src/Data/Integer/Tactic/RingSolver.agda
@@ -16,7 +16,7 @@ open import Data.Maybe.Base using (just; nothing)
 open import Data.Integer.Base
 open import Data.Integer.Properties
 open import Level using (0ℓ)
-open import Data.Unit using (⊤)
+open import Data.Unit.Base using (⊤)
 open import Relation.Binary.PropositionalEquality
 import Tactic.RingSolver as Solver
 import Tactic.RingSolver.Core.AlmostCommutativeRing as ACR
diff --git a/src/Data/List/Fresh/Properties.agda b/src/Data/List/Fresh/Properties.agda
index 6619fbb18d..49452e08ee 100644
--- a/src/Data/List/Fresh/Properties.agda
+++ b/src/Data/List/Fresh/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.List.Fresh.Properties where
 
-open import Level using (Level; _⊔_; Lift)
+open import Level using (Level; _⊔_)
 open import Data.Product.Base using (_,_)
 open import Relation.Nullary
 open import Relation.Unary as U using (Pred)
diff --git a/src/Data/List/Literals.agda b/src/Data/List/Literals.agda
index 6307fe4dc5..80d57901d1 100644
--- a/src/Data/List/Literals.agda
+++ b/src/Data/List/Literals.agda
@@ -8,10 +8,10 @@
 
 module Data.List.Literals where
 
-open import Agda.Builtin.FromString
-open import Data.Unit
-open import Agda.Builtin.Char
-open import Agda.Builtin.List
+open import Agda.Builtin.FromString using (IsString)
+open import Data.Unit.Base using (⊤)
+open import Agda.Builtin.Char using (Char)
+open import Agda.Builtin.List using (List)
 open import Data.String.Base using (toList)
 
 isString : IsString (List Char)
diff --git a/src/Data/List/NonEmpty/Properties.agda b/src/Data/List/NonEmpty/Properties.agda
index 45b27f46c3..68e9bfcf0e 100644
--- a/src/Data/List/NonEmpty/Properties.agda
+++ b/src/Data/List/NonEmpty/Properties.agda
@@ -12,7 +12,7 @@ open import Effect.Monad
 open import Data.Nat
 open import Data.Nat.Properties
 open import Data.Maybe.Properties using (just-injective)
-open import Data.Bool using (Bool; true; false)
+open import Data.Bool.Base using (Bool; true; false)
 open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.Effectful using () renaming (monad to listMonad)
 open import Data.List.NonEmpty.Effectful using () renaming (monad to list⁺Monad)
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index 3af753e826..57c9100e02 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -30,7 +30,7 @@ import Data.List.Relation.Unary.Unique.Setoid as Unique
 import Data.List.Membership.Setoid as Membership
 open import Data.List.Membership.Setoid.Properties using (∈-∃++; ∈-insert)
 import Data.List.Properties as Lₚ
-open import Data.Nat hiding (_⊔_)
+open import Data.Nat.Base using (ℕ; suc; _<_; z<s; _+_)
 open import Data.Nat.Induction
 open import Data.Nat.Properties
 open import Data.Product.Base using (_,_; _×_; ∃; ∃₂; proj₁; proj₂)
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index 3e91679bb2..f4556c6760 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -25,7 +25,7 @@ import Data.List.Relation.Binary.Subset.Setoid.Properties as Subset
 open import Data.List.Relation.Binary.Subset.Propositional
 open import Data.List.Relation.Binary.Permutation.Propositional
 import Data.List.Relation.Binary.Permutation.Propositional.Properties as Permutation
-open import Data.Nat using (ℕ; _≤_)
+open import Data.Nat.Base using (ℕ; _≤_)
 import Data.Product.Base as Product
 import Data.Sum.Base as Sum
 open import Effect.Monad
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index 6383f15a7e..c91a784ff7 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -20,20 +20,19 @@ open import Data.List.Membership.Propositional.Properties.Core
   using (Any↔; find∘map; map∘find; lose∘find)
 open import Data.List.Relation.Binary.Pointwise
   using (Pointwise; []; _∷_)
-open import Data.Nat using (zero; suc; _<_; z<s; s<s; s≤s)
+open import Data.Nat.Base using (zero; suc; _<_; z<s; s<s; s≤s)
 open import Data.Nat.Properties using (_≟_; ≤∧≢⇒<; ≤-refl; m<n⇒m<1+n)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.Any as MAny using (just)
 open import Data.Product.Base as Product
   using (_×_; _,_; ∃; ∃₂; proj₁; proj₂; uncurry′)
-open import Data.Product.Properties
 open import Data.Product.Function.NonDependent.Propositional
   using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
-open import Effect.Monad
-open import Function.Base
+open import Effect.Monad using (RawMonad)
+open import Function.Base using (_$_; _∘_; flip; const; id; _∘′_)
 open import Function.Bundles
 import Function.Properties.Inverse as Inverse
 open import Function.Related.Propositional as Related using (Kind; Related)
diff --git a/src/Data/List/Relation/Unary/Grouped/Properties.agda b/src/Data/List/Relation/Unary/Grouped/Properties.agda
index 2907b9a665..b0e4e42a4d 100644
--- a/src/Data/List/Relation/Unary/Grouped/Properties.agda
+++ b/src/Data/List/Relation/Unary/Grouped/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.List.Relation.Unary.Grouped.Properties where
 
-open import Data.Bool using (true; false)
+open import Data.Bool.Base using (true; false)
 open import Data.List.Base using ([]; [_]; _∷_; map; derun)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 import Data.List.Relation.Unary.All.Properties as All
@@ -16,12 +16,12 @@ open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _
 open import Data.List.Relation.Unary.Grouped
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; _on_)
+open import Level using (Level)
 open import Relation.Binary.Definitions as B
 open import Relation.Binary.Core using (_⇔_; REL; Rel)
 open import Relation.Unary as U using (Pred)
-open import Relation.Nullary using (¬_; does; yes; no)
-open import Relation.Nullary.Negation using (contradiction)
-open import Level
+open import Relation.Nullary.Decidable.Core using (does; yes; no)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 
 private
   variable
diff --git a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
index 4184647ae2..44039ab152 100644
--- a/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/DecPropositional/Properties.agda
@@ -6,17 +6,17 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.List
-import Data.List.Relation.Unary.Unique.DecSetoid.Properties as Setoid
-open import Data.List.Relation.Unary.All.Properties using (all-filter)
-open import Level
 open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 module Data.List.Relation.Unary.Unique.DecPropositional.Properties
   {a} {A : Set a} (_≟_ : DecidableEquality A) where
 
+open import Data.List.Base using (deduplicate)
+open import Data.List.Relation.Unary.All.Properties using (all-filter)
 open import Data.List.Relation.Unary.Unique.DecPropositional _≟_
+import Data.List.Relation.Unary.Unique.DecSetoid.Properties as Setoid
+open import Level
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 ------------------------------------------------------------------------
 -- Re-export propositional properties
diff --git a/src/Data/List/Sort/MergeSort.agda b/src/Data/List/Sort/MergeSort.agda
index f476d83b96..529d8623d3 100644
--- a/src/Data/List/Sort/MergeSort.agda
+++ b/src/Data/List/Sort/MergeSort.agda
@@ -10,8 +10,13 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Bool using (true; false)
+open import Relation.Binary.Bundles using (DecTotalOrder)
+
+module Data.List.Sort.MergeSort
+  {a ℓ₁ ℓ₂} (O : DecTotalOrder a ℓ₁ ℓ₂) where
+open import Data.Bool.Base using (true; false)
 open import Data.List.Base
+  using (List; []; _∷_; merge; length; map; [_]; concat; _++_)
 open import Data.List.Properties using (length-partition; ++-assoc; concat-[-])
 open import Data.List.Relation.Unary.Linked using ([]; [-])
 import Data.List.Relation.Unary.Sorted.TotalOrder.Properties as Sorted
@@ -20,17 +25,13 @@ import Data.List.Relation.Unary.All.Properties as All
 open import Data.List.Relation.Binary.Permutation.Propositional
 import Data.List.Relation.Binary.Permutation.Propositional.Properties as Perm
 open import Data.Maybe.Base using (just)
-open import Relation.Nullary.Decidable using (does)
 open import Data.Nat.Base using (_<_; _>_; z<s; s<s)
 open import Data.Nat.Induction
 open import Data.Nat.Properties using (m<n⇒m<1+n)
 open import Data.Product.Base as Product using (_,_)
 open import Function.Base using (_∘_)
-open import Relation.Binary.Bundles using (DecTotalOrder)
-open import Relation.Nullary.Negation using (¬_)
-
-module Data.List.Sort.MergeSort
-  {a ℓ₁ ℓ₂} (O : DecTotalOrder a ℓ₁ ℓ₂) where
+open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Nullary.Decidable.Core using (does)
 
 open DecTotalOrder O renaming (Carrier to A)
 
diff --git a/src/Data/Maybe.agda b/src/Data/Maybe.agda
index f39d35d2ff..d3f78a7d78 100644
--- a/src/Data/Maybe.agda
+++ b/src/Data/Maybe.agda
@@ -9,10 +9,10 @@
 module Data.Maybe where
 
 open import Data.Empty using (⊥)
-open import Data.Unit using (⊤)
+open import Data.Unit.Base using (⊤)
 open import Data.Bool.Base using (T)
-open import Data.Maybe.Relation.Unary.All
-open import Data.Maybe.Relation.Unary.Any
+open import Data.Maybe.Relation.Unary.All using (All)
+open import Data.Maybe.Relation.Unary.Any using (Any; just)
 open import Level using (Level)
 
 private
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index b217ea4143..ac1c6ca1b4 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -10,7 +10,7 @@
 
 module Data.Nat.Combinatorics.Specification where
 
-open import Data.Bool using (T; true; false)
+open import Data.Bool.Base using (T; true; false)
 open import Data.Nat.Base
 open import Data.Nat.DivMod
 open import Data.Nat.Divisibility
diff --git a/src/Data/Nat/Literals.agda b/src/Data/Nat/Literals.agda
index 2c18dc185c..22a53dedeb 100644
--- a/src/Data/Nat/Literals.agda
+++ b/src/Data/Nat/Literals.agda
@@ -8,9 +8,9 @@
 
 module Data.Nat.Literals where
 
-open import Agda.Builtin.FromNat
-open import Agda.Builtin.Nat
-open import Data.Unit
+open import Agda.Builtin.FromNat using (Number)
+open import Agda.Builtin.Nat using (Nat)
+open import Data.Unit.Base using (⊤)
 
 number : Number Nat
 number = record
diff --git a/src/Data/Nat/Logarithm.agda b/src/Data/Nat/Logarithm.agda
index 49cdc0f362..48155f1427 100644
--- a/src/Data/Nat/Logarithm.agda
+++ b/src/Data/Nat/Logarithm.agda
@@ -8,7 +8,7 @@
 
 module Data.Nat.Logarithm where
 
-open import Data.Nat
+open import Data.Nat.Base
 open import Data.Nat.Induction using (<-wellFounded)
 open import Data.Nat.Logarithm.Core
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index b09b46db7d..e58385f587 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -24,7 +24,7 @@ open import Data.Bool.Properties using (T?)
 open import Data.Nat.Base
 open import Data.Product.Base using (∃; _×_; _,_)
 open import Data.Sum.Base as Sum
-open import Data.Unit using (tt)
+open import Data.Unit.Base using (tt)
 open import Function.Base
 open import Function.Bundles using (_↣_)
 open import Function.Metric.Nat
diff --git a/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda b/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda
index de8427b6d4..545c30c85f 100644
--- a/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda
+++ b/src/Data/Nat/PseudoRandom/LCG/Unsafe.agda
@@ -9,8 +9,8 @@
 {-# OPTIONS --cubical-compatible --sized-types #-}
 
 open import Codata.Sized.Stream using (Stream; unfold)
-open import Data.Nat.PseudoRandom.LCG
-open import Data.Nat using (ℕ)
+open import Data.Nat.PseudoRandom.LCG using (Generator; step)
+open import Data.Nat.Base using (ℕ)
 open import Data.Product.Base using (<_,_>)
 open import Function.Base using (id)
 
diff --git a/src/Data/Nat/Tactic/RingSolver.agda b/src/Data/Nat/Tactic/RingSolver.agda
index af327ce06c..e55bf6c963 100644
--- a/src/Data/Nat/Tactic/RingSolver.agda
+++ b/src/Data/Nat/Tactic/RingSolver.agda
@@ -16,7 +16,7 @@ open import Data.Maybe.Base using (just; nothing)
 open import Data.Nat.Base using (zero; suc)
 open import Data.Nat.Properties
 open import Level using (0ℓ)
-open import Data.Unit using (⊤)
+open import Data.Unit.Base using (⊤)
 open import Relation.Binary.PropositionalEquality
 
 import Tactic.RingSolver as Solver
diff --git a/src/Data/Product/Nary/NonDependent.agda b/src/Data/Product/Nary/NonDependent.agda
index 29bb483964..6a04492a5d 100644
--- a/src/Data/Product/Nary/NonDependent.agda
+++ b/src/Data/Product/Nary/NonDependent.agda
@@ -14,7 +14,7 @@ module Data.Product.Nary.NonDependent where
 -- behind the design decisions.
 ------------------------------------------------------------------------
 
-open import Level as L using (Level; _⊔_; Lift; 0ℓ)
+open import Level using (Level)
 open import Agda.Builtin.Unit
 open import Data.Product.Base as Prod
 import Data.Product.Properties as Prodₚ
diff --git a/src/Data/Rational/Literals.agda b/src/Data/Rational/Literals.agda
index 089ed20db0..4162468058 100644
--- a/src/Data/Rational/Literals.agda
+++ b/src/Data/Rational/Literals.agda
@@ -8,9 +8,9 @@
 
 module Data.Rational.Literals where
 
-open import Agda.Builtin.FromNat
-open import Agda.Builtin.FromNeg
-open import Data.Unit using (⊤)
+open import Agda.Builtin.FromNat using (Number)
+open import Agda.Builtin.FromNeg using (Negative)
+open import Data.Unit.Base using (⊤)
 open import Data.Nat.Base using (ℕ; zero)
 open import Data.Nat.Coprimality using (sym; 1-coprimeTo)
 open import Data.Integer.Base using (ℤ; ∣_∣; +_; -_)
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 6ff6a8bc6d..43088288e2 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -44,8 +44,7 @@ open import Data.Rational.Unnormalised.Base as ℚᵘ
   )
 import Data.Rational.Unnormalised.Properties as ℚᵘ
 open import Data.Sum.Base as Sum
-open import Data.Unit using (tt)
-import Data.Sign as S
+import Data.Sign.Base as S
 open import Function.Base using (_∘_; _∘′_; _∘₂_; _$_; flip)
 open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
diff --git a/src/Data/Star/BoundedVec.agda b/src/Data/Star/BoundedVec.agda
index 5664cf847d..11f2585e40 100644
--- a/src/Data/Star/BoundedVec.agda
+++ b/src/Data/Star/BoundedVec.agda
@@ -11,15 +11,17 @@
 module Data.Star.BoundedVec where
 
 import Data.Maybe.Base as Maybe
-open import Data.Star.Nat
-open import Data.Star.Decoration
+open import Data.Star.Nat using (ℕ; suc; length)
+open import Data.Star.Decoration using (decoration)
 open import Data.Star.Pointer
+  using (Any; this; that; Pointer; step; done; init)
 open import Data.Star.List using (List)
-open import Data.Unit
+open import Data.Unit.Base using (⊤; tt)
 open import Function.Base using (const)
 open import Relation.Binary.Core using (_=[_]⇒_)
-open import Relation.Binary.Consequences
+open import Relation.Binary.Consequences using (map-NonEmpty)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
+  using (gmap; ε; _◅_)
 
 ------------------------------------------------------------------------
 -- The type
diff --git a/src/Data/Star/Decoration.agda b/src/Data/Star/Decoration.agda
index d4e968388b..ce3ef5f7cd 100644
--- a/src/Data/Star/Decoration.agda
+++ b/src/Data/Star/Decoration.agda
@@ -8,9 +8,9 @@
 
 module Data.Star.Decoration where
 
-open import Data.Unit
+open import Data.Unit.Base using (⊤; tt)
 open import Function.Base using (flip)
-open import Level
+open import Level using (Level; suc; _⊔_)
 open import Relation.Binary.Core using (Rel; _=[_]⇒_; _⇒_)
 open import Relation.Binary.Definitions using (NonEmpty; nonEmpty)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
@@ -20,7 +20,6 @@ open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 EdgePred : {ℓ r : Level} (p : Level) {I : Set ℓ} → Rel I r → Set (suc p ⊔ ℓ ⊔ r)
 EdgePred p T = ∀ {i j} → T i j → Set p
 
-
 data NonEmptyEdgePred {ℓ r p : Level} {I : Set ℓ} (T : Rel I r)
                       (P : EdgePred p T) : Set (ℓ ⊔ r ⊔ p) where
   nonEmptyEdgePred : ∀ {i j} {x : T i j}
diff --git a/src/Data/Star/Environment.agda b/src/Data/Star/Environment.agda
index c7ccb18ef4..6d30db7c3c 100644
--- a/src/Data/Star/Environment.agda
+++ b/src/Data/Star/Environment.agda
@@ -8,12 +8,12 @@
 
 module Data.Star.Environment {ℓ} (Ty : Set ℓ) where
 
-open import Level
-open import Data.Star.List
-open import Data.Star.Decoration
-open import Data.Star.Pointer as Pointer hiding (lookup)
-open import Data.Unit
-open import Function.Base hiding (_∋_)
+open import Level using (_⊔_)
+open import Data.Star.List using (List)
+open import Data.Star.Decoration using (All)
+open import Data.Star.Pointer as Pointer using (Any; this; that; result)
+open import Data.Unit.Polymorphic using (⊤)
+open import Function.Base using (const)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
 
@@ -27,7 +27,7 @@ Ctxt = List Ty
 infix 4 _∋_
 
 _∋_ : Ctxt → Ty → Set ℓ
-Γ ∋ σ = Any (const (Lift ℓ ⊤)) (σ ≡_) Γ
+Γ ∋ σ = Any (const (⊤ {ℓ})) (σ ≡_) Γ
 
 vz : ∀ {Γ σ} → Γ ▻ σ ∋ σ
 vz = this refl
diff --git a/src/Data/Star/Fin.agda b/src/Data/Star/Fin.agda
index 9ed437f95f..ded64da4bb 100644
--- a/src/Data/Star/Fin.agda
+++ b/src/Data/Star/Fin.agda
@@ -9,8 +9,8 @@
 module Data.Star.Fin where
 
 open import Data.Star.Nat as ℕ using (ℕ)
-open import Data.Star.Pointer
-open import Data.Unit
+open import Data.Star.Pointer using (Any; this; that)
+open import Data.Unit.Base using (⊤; tt)
 
 -- Finite sets are undecorated pointers into natural numbers.
 
diff --git a/src/Data/Star/List.agda b/src/Data/Star/List.agda
index 358ea7281e..bcc9870375 100644
--- a/src/Data/Star/List.agda
+++ b/src/Data/Star/List.agda
@@ -8,11 +8,12 @@
 
 module Data.Star.List where
 
-open import Data.Star.Nat
-open import Data.Unit
+open import Data.Star.Nat using (ℕ; _+_; zero)
+open import Data.Unit.Base using (tt)
 open import Relation.Binary.Construct.Always using (Always)
 open import Relation.Binary.Construct.Constant using (Const)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
+  using (Star; ε; _◅_; fold)
 
 -- Lists.
 
diff --git a/src/Data/Star/Nat.agda b/src/Data/Star/Nat.agda
index aa1eb9a55b..960537c593 100644
--- a/src/Data/Star/Nat.agda
+++ b/src/Data/Star/Nat.agda
@@ -8,7 +8,7 @@
 
 module Data.Star.Nat where
 
-open import Data.Unit
+open import Data.Unit.Base using (tt)
 open import Function.Base using (const)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
diff --git a/src/Data/Star/Vec.agda b/src/Data/Star/Vec.agda
index 108fbe7399..3677db147b 100644
--- a/src/Data/Star/Vec.agda
+++ b/src/Data/Star/Vec.agda
@@ -8,15 +8,16 @@
 
 module Data.Star.Vec where
 
-open import Data.Star.Nat
+open import Data.Star.Nat using (ℕ; zero; suc; 1#; _+_; length)
 open import Data.Star.Fin using (Fin)
-open import Data.Star.Decoration
-open import Data.Star.Pointer as Pointer hiding (lookup)
+open import Data.Star.Decoration using (All; ↦; _◅◅◅_; decoration)
+open import Data.Star.Pointer as Pointer using (result)
 open import Data.Star.List using (List)
 open import Level using (Level)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
-open import Function.Base
-open import Data.Unit
+  using (ε; _◅_; gmap)
+open import Function.Base using (const)
+open import Data.Unit.Base using (tt)
 
 private
   variable
diff --git a/src/Data/String.agda b/src/Data/String.agda
index e92054a673..98b01b3b5f 100644
--- a/src/Data/String.agda
+++ b/src/Data/String.agda
@@ -9,7 +9,7 @@
 module Data.String where
 
 open import Data.Bool.Base using (if_then_else_)
-open import Data.Char as Char using (Char)
+open import Data.Char.Base as Char using (Char)
 open import Function.Base using (_∘_; _$_)
 open import Data.Nat.Base as ℕ using (ℕ)
 import Data.Nat.Properties as ℕ
diff --git a/src/Data/String/Literals.agda b/src/Data/String/Literals.agda
index 9995ceb133..f376958da9 100644
--- a/src/Data/String/Literals.agda
+++ b/src/Data/String/Literals.agda
@@ -8,9 +8,9 @@
 
 module Data.String.Literals where
 
-open import Agda.Builtin.FromString
-open import Data.Unit
-open import Agda.Builtin.String
+open import Agda.Builtin.FromString using (IsString)
+open import Data.Unit.Base using (⊤)
+open import Agda.Builtin.String using (String)
 
 isString : IsString String
 isString = record
diff --git a/src/Data/Tree/AVL/Key.agda b/src/Data/Tree/AVL/Key.agda
index 6fd5a1b021..1c32ef88d7 100644
--- a/src/Data/Tree/AVL/Key.agda
+++ b/src/Data/Tree/AVL/Key.agda
@@ -9,18 +9,16 @@
 
 open import Relation.Binary.Bundles
   using (StrictTotalOrder; StrictPartialOrder)
-open import Relation.Binary.Definitions using (Reflexive)
 
 module Data.Tree.AVL.Key
   {a ℓ₁ ℓ₂} (sto : StrictTotalOrder a ℓ₁ ℓ₂)
   where
 
-open import Level
-open import Data.Empty
-open import Data.Unit
+open import Level using (Level; _⊔_)
 open import Data.Product.Base using (_×_; _,_)
+open import Relation.Binary.Definitions using (Reflexive)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_ ; refl)
-open import Relation.Nullary.Negation using (¬_)
+open import Relation.Nullary.Negation.Core using (¬_)
 open import Relation.Nullary.Construct.Add.Extrema
   as AddExtremaToSet using (_±)
 import Relation.Binary.Construct.Add.Extrema.Equality
diff --git a/src/Data/Tree/AVL/NonEmpty.agda b/src/Data/Tree/AVL/NonEmpty.agda
index 79fd4eb842..ac013e65a7 100644
--- a/src/Data/Tree/AVL/NonEmpty.agda
+++ b/src/Data/Tree/AVL/NonEmpty.agda
@@ -17,15 +17,13 @@ module Data.Tree.AVL.NonEmpty
   {a ℓ₁ ℓ₂} (strictTotalOrder : StrictTotalOrder a ℓ₁ ℓ₂) where
 
 open import Data.Bool.Base using (Bool)
-open import Data.Empty
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_; _++⁺_)
 open import Data.Maybe.Base hiding (map)
 open import Data.Nat.Base hiding (_<_; _⊔_; compare)
 open import Data.Product.Base hiding (map)
-open import Data.Unit
 open import Function.Base using (_$_; _∘′_)
-open import Level using (_⊔_; Lift; lift)
-open import Relation.Unary
+open import Level using (_⊔_)
+open import Relation.Unary using (IUniversal; _⇒_)
 
 open StrictTotalOrder strictTotalOrder renaming (Carrier to Key)
 open import Data.Tree.AVL.Value Eq.setoid
diff --git a/src/Data/Tree/Binary/Zipper.agda b/src/Data/Tree/Binary/Zipper.agda
index 5786ca7672..3024b125ca 100644
--- a/src/Data/Tree/Binary/Zipper.agda
+++ b/src/Data/Tree/Binary/Zipper.agda
@@ -11,7 +11,7 @@ module Data.Tree.Binary.Zipper where
 open import Level using (Level; _⊔_)
 open import Data.Tree.Binary as BT using (Tree; node; leaf)
 open import Data.List.Base as List using (List; []; _∷_; sum; _++_; [_])
-open import Data.Maybe using (Maybe; nothing; just)
+open import Data.Maybe.Base using (Maybe; nothing; just)
 open import Data.Nat.Base using (ℕ; suc; _+_)
 open import Function.Base using (_$_; _∘_)
 
diff --git a/src/Data/Vec/Bounded.agda b/src/Data/Vec/Bounded.agda
index 881e3b86e1..410e3bebcc 100644
--- a/src/Data/Vec/Bounded.agda
+++ b/src/Data/Vec/Bounded.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Bounded where
 
 open import Level using (Level)
-open import Data.Nat.Base
+open import Data.Nat.Base using (_≤_)
 open import Data.Vec.Base using (Vec)
 import Data.Vec as Vec using (filter; takeWhile; dropWhile)
 open import Function.Base using (id)
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index 39d9e25abb..addfc32678 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -10,7 +10,7 @@ module Data.Vec.Functional.Properties where
 
 open import Data.Empty using (⊥-elim)
 open import Data.Fin.Base
-open import Data.Nat as ℕ
+open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 import Data.Nat.Properties as ℕ
 open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.List.Base as List using (List)
diff --git a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
index 28f022454b..f0cefce5c7 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Setoid.agda
@@ -6,21 +6,20 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Core using (REL)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Binary.Definitions using (Reflexive; Sym; Trans)
 
 module Data.Vec.Relation.Binary.Equality.Setoid
   {a ℓ} (S : Setoid a ℓ) where
 
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
-open import Data.Fin using (zero; suc)
+open import Data.Fin.Base using (zero; suc)
 open import Data.Vec.Base
 open import Data.Vec.Relation.Binary.Pointwise.Inductive as PW
   using (Pointwise)
-open import Function.Base
 open import Level using (_⊔_)
+open import Relation.Binary.Core using (REL)
+open import Relation.Binary.Structures using (IsEquivalence)
+open import Relation.Binary.Definitions using (Reflexive; Sym; Trans)
 
 open Setoid S renaming (Carrier to A)
 
diff --git a/src/Data/Vec/Relation/Binary/Lex/Core.agda b/src/Data/Vec/Relation/Binary/Lex/Core.agda
index ed23c7cf6a..92237d4f18 100644
--- a/src/Data/Vec/Relation/Binary/Lex/Core.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/Core.agda
@@ -9,7 +9,7 @@
 module Data.Vec.Relation.Binary.Lex.Core {a} {A : Set a} where
 
 open import Data.Empty
-open import Data.Nat using (ℕ; suc)
+open import Data.Nat.Base using (ℕ; suc)
 import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Vec.Base using (Vec; []; _∷_)
@@ -136,7 +136,7 @@ module _ {P : Set} {_≈_ : Rel A ℓ₁} {_≺_ : Rel A ℓ₂} where
   module _ (P? : Dec P) (_≈?_ : Decidable _≈_) (_≺?_ : Decidable _≺_) where
 
     decidable : ∀ {m n} → Decidable (_<ₗₑₓ_ {m} {n})
-    decidable {m} {n} xs ys with m Data.Nat.≟ n
+    decidable {m} {n} xs ys with m ℕ.≟ n
     decidable {_} {_} []       []       | yes refl = Dec.map P⇔[]<[] P?
     decidable {_} {_} (x ∷ xs) (y ∷ ys) | yes refl = Dec.map ∷<∷-⇔ ((x ≺? y) ⊎-dec (x ≈? y) ×-dec (decidable xs ys))
     decidable {_} {_} _        _        | no  m≢n    = no (λ xs<ys → contradiction (length-equal xs<ys) m≢n)
diff --git a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
index b273029052..57f05fa211 100644
--- a/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
+++ b/src/Data/Vec/Relation/Binary/Lex/NonStrict.agda
@@ -12,14 +12,15 @@
 module Data.Vec.Relation.Binary.Lex.NonStrict where
 
 open import Data.Empty
-open import Data.Unit using (⊤; tt)
+open import Data.Unit.Base using (⊤; tt)
 open import Data.Product.Base using (proj₁; proj₂)
-open import Data.Nat using (ℕ)
+open import Data.Nat.Base using (ℕ)
 open import Data.Vec.Base using (Vec; []; _∷_)
 import Data.Vec.Relation.Binary.Lex.Strict as Strict
 open import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
   using (Pointwise; []; _∷_; head; tail)
 open import Function.Base using (id)
+open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (REL; Rel; _⇒_)
 open import Relation.Binary.Bundles
   using (Poset; StrictPartialOrder; DecPoset; DecStrictPartialOrder; DecTotalOrder; StrictTotalOrder; Preorder; TotalOrder)
@@ -29,7 +30,6 @@ open import Relation.Binary.Definitions
   using (Irreflexive; _Respects₂_; Antisymmetric; Asymmetric; Symmetric; Trans; Decidable; Total; Trichotomous)
 import Relation.Binary.Construct.NonStrictToStrict as Conv
 open import Relation.Nullary hiding (Irrelevant)
-open import Level using (Level; _⊔_)
 
 private
   variable
diff --git a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
index b13962b750..32eced4f6e 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs/Properties.agda
@@ -8,7 +8,7 @@
 
 module Data.Vec.Relation.Unary.AllPairs.Properties where
 
-open import Data.Vec
+open import Data.Vec.Base using (_∷_; map; _++_; concat; take; drop; tabulate)
 import Data.Vec.Properties as Vec
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 import Data.Vec.Relation.Unary.All.Properties as All
diff --git a/src/Effect/Monad/Reader/Indexed.agda b/src/Effect/Monad/Reader/Indexed.agda
index 6ece55aa68..0c787abcc9 100644
--- a/src/Effect/Monad/Reader/Indexed.agda
+++ b/src/Effect/Monad/Reader/Indexed.agda
@@ -6,16 +6,16 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Level
+open import Level using (Level; _⊔_; suc; Lift; lift)
 
 module Effect.Monad.Reader.Indexed {r} (R : Set r) (a : Level) where
 
 open import Function.Base using (const; flip; _∘_)
 open import Function.Identity.Effectful as Id using (Identity)
 open import Effect.Applicative.Indexed
-open import Effect.Monad.Indexed
-open import Effect.Monad
-open import Data.Unit
+  using (IFun; RawIApplicative; RawIApplicativeZero; RawIAlternative)
+open import Effect.Monad.Indexed using (RawIMonad; RawIMonadZero;
+  RawIMonadPlus)
 
 private
   variable
diff --git a/src/Effect/Monad/State/Indexed.agda b/src/Effect/Monad/State/Indexed.agda
index 4150554ac9..e8345c363b 100644
--- a/src/Effect/Monad/State/Indexed.agda
+++ b/src/Effect/Monad/State/Indexed.agda
@@ -9,13 +9,15 @@
 module Effect.Monad.State.Indexed where
 
 open import Effect.Applicative.Indexed
-open import Effect.Monad
+  using (IFun; RawIApplicative; RawIApplicativeZero; RawIAlternative)
+open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus)
 open import Function.Identity.Effectful as Id using (Identity)
-open import Effect.Monad.Indexed
+open import Effect.Monad.Indexed using (RawIMonad; RawIMonadZero;
+  RawIMonadPlus)
 open import Data.Product.Base using (_×_; _,_; uncurry)
-open import Data.Unit
-open import Function.Base
-open import Level
+open import Data.Unit.Polymorphic using (⊤)
+open import Function.Base using (const; _∘_)
+open import Level using (Level; _⊔_; suc)
 
 private
   variable
@@ -86,11 +88,11 @@ record RawIMonadState {I : Set i} (S : I → Set f)
   field
     monad : RawIMonad M
     get   : ∀ {i} → M i i (S i)
-    put   : ∀ {i j} → S j → M i j (Lift f ⊤)
+    put   : ∀ {i j} → S j → M i j ⊤
 
   open RawIMonad monad public
 
-  modify : ∀ {i j} → (S i → S j) → M i j (Lift f ⊤)
+  modify : ∀ {i j} → (S i → S j) → M i j ⊤
   modify f = get >>= put ∘ f
 
 StateTIMonadState : ∀ {i f} {I : Set i} (S : I → Set f) {M} →
diff --git a/src/System/Environment/Primitive.agda b/src/System/Environment/Primitive.agda
index 103afc2ac9..c1f8533e03 100644
--- a/src/System/Environment/Primitive.agda
+++ b/src/System/Environment/Primitive.agda
@@ -18,7 +18,7 @@ open import IO.Primitive using (IO)
 open import Data.List.Base using (List)
 open import Data.Maybe.Base using (Maybe)
 open import Data.String.Base using (String)
-open import Data.Unit using (⊤)
+open import Data.Unit.Base using (⊤)
 
 open import Foreign.Haskell.Pair using (Pair)
 
diff --git a/src/Tactic/MonoidSolver.agda b/src/Tactic/MonoidSolver.agda
index 4e370bea25..33cbf95001 100644
--- a/src/Tactic/MonoidSolver.agda
+++ b/src/Tactic/MonoidSolver.agda
@@ -75,10 +75,10 @@ module Tactic.MonoidSolver where
 open import Algebra
 open import Function.Base using (_⟨_⟩_)
 
-open import Data.Bool         as Bool    using (Bool; _∨_; if_then_else_)
-open import Data.Maybe        as Maybe   using (Maybe; just; nothing; maybe)
+open import Data.Bool.Base    as Bool    using (Bool; _∨_; if_then_else_)
+open import Data.Maybe.Base   as Maybe   using (Maybe; just; nothing; maybe)
 open import Data.List.Base    as List    using (List; _∷_; [])
-open import Data.Nat          as ℕ       using (ℕ; suc; zero)
+open import Data.Nat.Base     as ℕ       using (ℕ; suc; zero)
 open import Data.Product.Base as Product using (_×_; _,_)
 
 open import Reflection.AST
diff --git a/src/Tactic/RingSolver/Core/NatSet.agda b/src/Tactic/RingSolver/Core/NatSet.agda
index e297bedbc9..1f42ef708c 100644
--- a/src/Tactic/RingSolver/Core/NatSet.agda
+++ b/src/Tactic/RingSolver/Core/NatSet.agda
@@ -36,10 +36,10 @@
 
 module Tactic.RingSolver.Core.NatSet where
 
-open import Data.Nat        as ℕ     using (ℕ; suc; zero)
+open import Data.Nat.Base   as ℕ     using (ℕ; suc; zero)
 open import Data.List.Base  as List  using (List; _∷_; [])
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
-open import Data.Bool       as Bool  using (Bool)
+open import Data.Bool.Base  as Bool  using (Bool)
 open import Function.Base            using (const; _∘_)
 open import Relation.Binary.PropositionalEquality
 
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
index e71ad07419..32cbefe63a 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Addition.agda
@@ -13,7 +13,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Addition
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Nat            as ℕ using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
+open import Data.Nat.Base       as ℕ using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
 open import Data.Nat.Properties as ℕ using (≤′-trans)
 open import Data.Product.Base        using (_,_; _×_; proj₂)
 open import Data.List.Base           using (_∷_; [])
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
index c879bdef38..36c66422c7 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Lemmas.agda
@@ -13,19 +13,19 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Lemmas
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Bool                                       using (Bool;true;false)
+open import Data.Bool.Base                                  using (Bool;true;false)
 open import Data.Nat.Base as ℕ                              using (ℕ; suc; zero; compare; _≤′_; ≤′-step; ≤′-refl)
 open import Data.Nat.Properties as ℕ                        using (≤′-trans)
 open import Data.Vec.Base as Vec                            using (Vec; _∷_)
-open import Data.Fin                                        using (Fin; zero; suc)
+open import Data.Fin.Base                                   using (Fin; zero; suc)
 open import Data.List.Base                                  using (_∷_; [])
-open import Data.Unit using (tt)
+open import Data.Unit.Base using (tt)
 open import Data.List.Kleene
 open import Data.Product.Base                               using (_,_; proj₁; proj₂; map₁; _×_)
-open import Data.Maybe                                      using (nothing; just)
+open import Data.Maybe.Base                                 using (nothing; just)
 open import Function.Base                                   using (_⟨_⟩_)
 open import Level                                           using (lift)
-open import Relation.Nullary                                using (Dec; yes; no)
+open import Relation.Nullary.Decidable.Core                 using (Dec; yes; no)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
 open Homomorphism homo hiding (_^_)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
index b2a4486d6a..9f6aac7e58 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Negation.agda
@@ -15,7 +15,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Negation
 
 open import Data.Vec.Base         using (Vec)
 open import Data.Product.Base     using (_,_)
-open import Data.Nat              using (_<′_)
+open import Data.Nat.Base         using (_<′_)
 open import Data.Nat.Induction
 
 open import Function.Base using (_⟨_⟩_; flip)
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda
index 9fcaca2f66..553f761768 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Homomorphism/Variables.agda
@@ -15,7 +15,7 @@ module Tactic.RingSolver.Core.Polynomial.Homomorphism.Variables
 
 open import Data.Product.Base    using (_,_)
 open import Data.Vec.Base as Vec using (Vec)
-open import Data.Fin             using (Fin)
+open import Data.Fin.Base        using (Fin)
 open import Data.List.Kleene
 
 open Homomorphism homo
diff --git a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
index 4e3e672916..c73a8ac587 100644
--- a/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
+++ b/src/Tactic/RingSolver/Core/Polynomial/Semantics.agda
@@ -13,7 +13,7 @@ module Tactic.RingSolver.Core.Polynomial.Semantics
   (homo : Homomorphism r₁ r₂ r₃ r₄)
   where
 
-open import Data.Nat          using (ℕ; suc; zero; _≤′_; ≤′-step; ≤′-refl)
+open import Data.Nat.Base     using (ℕ; suc; zero; _≤′_; ≤′-step; ≤′-refl)
 open import Data.Vec.Base     using (Vec; []; _∷_; uncons)
 open import Data.List.Base    using ([]; _∷_)
 open import Data.Product.Base using (_,_; _×_)
diff --git a/src/Test/Golden.agda b/src/Test/Golden.agda
index 43bbd3037e..7e592a3ff2 100644
--- a/src/Test/Golden.agda
+++ b/src/Test/Golden.agda
@@ -97,7 +97,7 @@ open import Data.Unit.Base using (⊤)
 
 open import Function.Base using (id; _$_; case_of_)
 
-open import Relation.Nullary.Decidable using (does)
+open import Relation.Nullary.Decidable.Core using (does)
 
 open import Codata.Musical.Notation using (♯_)
 open import IO
diff --git a/src/Text/Format.agda b/src/Text/Format.agda
index b597c419cf..ffa6501b27 100644
--- a/src/Text/Format.agda
+++ b/src/Text/Format.agda
@@ -12,11 +12,11 @@ open import Data.Maybe.Base
 open import Text.Format.Generic
 
 -- Formatted types
-open import Data.Char.Base
-open import Data.Integer.Base
-open import Data.Float
-open import Data.Nat.Base
-open import Data.String.Base
+open import Data.Char.Base using (Char)
+open import Data.Integer.Base using (ℤ)
+open import Data.Float.Base using (Float)
+open import Data.Nat.Base using (ℕ)
+open import Data.String.Base using (String)
 
 ------------------------------------------------------------------------
 -- Basic types
diff --git a/src/Text/Pretty/Core.agda b/src/Text/Pretty/Core.agda
index 39910f4dd5..4a5a210fef 100644
--- a/src/Text/Pretty/Core.agda
+++ b/src/Text/Pretty/Core.agda
@@ -16,13 +16,14 @@ open import Data.Bool.Base using (Bool)
 open import Data.Irrelevant as Irrelevant using (Irrelevant) hiding (module Irrelevant)
 open import Data.List.Base as List   using (List; []; _∷_)
 open import Data.Nat.Base            using (ℕ; zero; suc; _+_; _⊔_; _≤_; z≤n)
-open import Data.Nat.Properties
+open import Data.Nat.Properties      using (≤-refl; ≤-trans; +-identityʳ;
+  module ≤-Reasoning; m≤n⊔m; +-monoʳ-≤; m≤m⊔n; +-comm; _≤?_)
 open import Data.Product.Base as Prod using (_×_; _,_; uncurry; proj₁; proj₂)
 import Data.Product.Relation.Unary.All as Allᴾ
 
 open import Data.Tree.Binary as Tree using (Tree; leaf; node; #nodes; mapₙ)
 open import Data.Tree.Binary.Relation.Unary.All as Allᵀ using (leaf; node)
-open import Data.Unit using (⊤; tt)
+open import Data.Unit.Base using (⊤; tt)
 import Data.Tree.Binary.Relation.Unary.All.Properties as Allᵀ
 import Data.Tree.Binary.Properties as Tree
 
diff --git a/src/Text/Printf.agda b/src/Text/Printf.agda
index b7f1330c89..caa59b8a8c 100644
--- a/src/Text/Printf.agda
+++ b/src/Text/Printf.agda
@@ -12,7 +12,7 @@ open import Data.String.Base using (String; fromChar; concat)
 open import Function.Base using (id)
 
 import Data.Integer.Show as ℤ
-import Data.Float        as Float
+import Data.Float.Base   as Float
 import Data.Nat.Show     as ℕ
 
 open import Text.Format as Format hiding (Error)

From 7820c3b6f548b9958069c7e9f016feb682c4706b Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 11 Apr 2024 03:31:05 +0100
Subject: [PATCH 265/316] Systematise use of
 `Relation.Binary.Definitions.DecidableEquality` (#2354)

* systematic use of `DecidableEquality`

* tweak
---
 src/Algebra/Solver/Monoid.agda                           | 4 ++--
 src/Axiom/UniquenessOfIdentityProofs.agda                | 2 +-
 src/Data/Bool/Properties.agda                            | 4 ++--
 src/Data/Char/Properties.agda                            | 4 ++--
 src/Data/List/Membership/DecPropositional.agda           | 8 ++++----
 src/Data/List/Membership/Propositional/Properties.agda   | 2 +-
 .../List/Relation/Binary/Equality/DecPropositional.agda  | 8 ++++----
 .../List/Relation/Binary/Sublist/DecPropositional.agda   | 9 ++++-----
 .../Relation/Binary/Sublist/DecPropositional/Solver.agda | 9 ++++-----
 src/Data/Maybe/Properties.agda                           | 4 ++--
 src/Data/Nat/Binary/Properties.agda                      | 2 +-
 src/Data/Record.agda                                     | 5 ++---
 src/Data/String/Properties.agda                          | 4 ++--
 src/Data/Sum/Properties.agda                             | 8 ++++----
 src/Data/These/Properties.agda                           | 4 ++--
 src/Data/Unit/Polymorphic/Properties.agda                | 4 ++--
 src/Data/Unit/Properties.agda                            | 4 ++--
 src/Data/Vec/Membership/DecPropositional.agda            | 8 ++++----
 .../Vec/Relation/Binary/Equality/DecPropositional.agda   | 6 +++---
 src/Effect/Monad/Partiality.agda                         | 4 ++--
 src/Reflection/AST/Term.agda                             | 4 ++--
 .../Binary/Construct/Closure/Reflexive/Properties.agda   | 4 ++--
 .../Binary/PropositionalEquality/Properties.agda         | 4 ++--
 23 files changed, 56 insertions(+), 59 deletions(-)

diff --git a/src/Algebra/Solver/Monoid.agda b/src/Algebra/Solver/Monoid.agda
index ba6aabee06..0928586c41 100644
--- a/src/Algebra/Solver/Monoid.agda
+++ b/src/Algebra/Solver/Monoid.agda
@@ -20,7 +20,7 @@ open import Data.Nat.Base using (ℕ)
 open import Data.Product.Base using (_×_; uncurry)
 open import Data.Vec.Base using (Vec; lookup)
 open import Function.Base using (_∘_; _$_)
-open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Definitions using (DecidableEquality)
 
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 import Relation.Binary.Reflection
@@ -114,7 +114,7 @@ open module R = Relation.Binary.Reflection
 
 infix 5 _≟_
 
-_≟_ : ∀ {n} → Decidable {A = Normal n} _≡_
+_≟_ : ∀ {n} → DecidableEquality (Normal n)
 nf₁ ≟ nf₂ = Dec.map′ ≋⇒≡ ≡⇒≋ (nf₁ ≋? nf₂)
   where open ListEq Fin._≟_
 
diff --git a/src/Axiom/UniquenessOfIdentityProofs.agda b/src/Axiom/UniquenessOfIdentityProofs.agda
index 1cefc1a6a1..74463fe557 100644
--- a/src/Axiom/UniquenessOfIdentityProofs.agda
+++ b/src/Axiom/UniquenessOfIdentityProofs.agda
@@ -60,7 +60,7 @@ module Constant⇒UIP
 -- proof produced by the decision procedure.
 
 module Decidable⇒UIP
-  {a} {A : Set a} (_≟_ : Decidable {A = A} _≡_)
+  {a} {A : Set a} (_≟_ : DecidableEquality A)
   where
 
   ≡-normalise : _≡_ {A = A} ⇒ _≡_
diff --git a/src/Data/Bool/Properties.agda b/src/Data/Bool/Properties.agda
index bb8cda9a0e..eb7ffb3869 100644
--- a/src/Data/Bool/Properties.agda
+++ b/src/Data/Bool/Properties.agda
@@ -25,7 +25,7 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Bundles
   using (Setoid; DecSetoid; Poset; Preorder; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
 open import Relation.Binary.Definitions
-  using (Decidable; Reflexive; Transitive; Antisymmetric; Minimum; Maximum; Total; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri<; tri>; _Respects₂_)
+  using (Decidable; DecidableEquality; Reflexive; Transitive; Antisymmetric; Minimum; Maximum; Total; Irrelevant; Irreflexive; Asymmetric; Trans; Trichotomous; tri≈; tri<; tri>; _Respects₂_)
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Binary.PropositionalEquality.Properties
 open import Relation.Nullary.Decidable.Core using (True; yes; no; fromWitness)
@@ -48,7 +48,7 @@ private
 
 infix 4 _≟_
 
-_≟_ : Decidable {A = Bool} _≡_
+_≟_ : DecidableEquality Bool
 true  ≟ true  = yes refl
 false ≟ false = yes refl
 true  ≟ false = no λ()
diff --git a/src/Data/Char/Properties.agda b/src/Data/Char/Properties.agda
index b9c7270881..36e6d37d3c 100644
--- a/src/Data/Char/Properties.agda
+++ b/src/Data/Char/Properties.agda
@@ -23,7 +23,7 @@ open import Relation.Binary.Bundles
 open import Relation.Binary.Structures
   using (IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsPreorder; IsPartialOrder; IsDecPartialOrder; IsEquivalence)
 open import Relation.Binary.Definitions
-  using (Decidable; Trichotomous; Irreflexive; Transitive; Asymmetric; Antisymmetric; Symmetric; Substitutive; Reflexive; tri<; tri≈; tri>)
+  using (Decidable; DecidableEquality; Trichotomous; Irreflexive; Transitive; Asymmetric; Antisymmetric; Symmetric; Substitutive; Reflexive; tri<; tri≈; tri>)
 import Relation.Binary.Construct.On as On
 import Relation.Binary.Construct.Subst.Equality as Subst
 import Relation.Binary.Construct.Closure.Reflexive as Refl
@@ -55,7 +55,7 @@ open import Agda.Builtin.Char.Properties
 -- Properties of _≡_
 
 infix 4 _≟_
-_≟_ : Decidable {A = Char} _≡_
+_≟_ : DecidableEquality Char
 x ≟ y = map′ ≈⇒≡ ≈-reflexive (toℕ x ℕ.≟ toℕ y)
 
 setoid : Setoid _ _
diff --git a/src/Data/List/Membership/DecPropositional.agda b/src/Data/List/Membership/DecPropositional.agda
index 71cb595375..3c81c0ff1e 100644
--- a/src/Data/List/Membership/DecPropositional.agda
+++ b/src/Data/List/Membership/DecPropositional.agda
@@ -6,12 +6,12 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality using (_≡_)
-open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
+open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Membership.DecPropositional
-  {a} {A : Set a} (_≟_ : Decidable (_≡_ {A = A})) where
+  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 ------------------------------------------------------------------------
 -- Re-export contents of propositional membership
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index 6988f4f0c4..12785eb59b 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -259,7 +259,7 @@ module _ {r} {R : Rel A r} (R? : Binary.Decidable R) where
   ∈-deduplicate⁻ : ∀ xs {z} → z ∈ deduplicate R? xs → z ∈ xs
   ∈-deduplicate⁻ xs z∈dedup[R,xs] = Membership.∈-deduplicate⁻ (≡.setoid A) R? xs z∈dedup[R,xs]
 
-module _ (_≈?_ : Binary.Decidable {A = A} _≡_) where
+module _ (_≈?_ : DecidableEquality A) where
 
   ∈-derun⁺ : ∀ {xs z} → z ∈ xs → z ∈ derun _≈?_ xs
   ∈-derun⁺ z∈xs = Membership.∈-derun⁺ (≡.setoid A) _≈?_ (flip trans) z∈xs
diff --git a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
index 0be92107c7..8f6b20ec99 100644
--- a/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Equality/DecPropositional.agda
@@ -10,16 +10,16 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Equality.DecPropositional
-  {a} {A : Set a} (_≟_ : Decidable {A = A} _≡_) where
+  {a} {A : Set a} (_≟_ : DecidableEquality A) where
 
 open import Data.List.Base using (List)
 open import Data.List.Properties using (≡-dec)
 import Data.List.Relation.Binary.Equality.Propositional as PropositionalEq
 import Data.List.Relation.Binary.Equality.DecSetoid as DecSetoidEq
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 ------------------------------------------------------------------------
 -- Publically re-export everything from decSetoid and propositional
@@ -34,5 +34,5 @@ open DecSetoidEq (decSetoid _≟_) public
 
 infix 4 _≡?_
 
-_≡?_ : Decidable (_≡_ {A = List A})
+_≡?_ : DecidableEquality (List A)
 _≡?_ = ≡-dec _≟_
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
index 2e1878e906..d509b12be0 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
@@ -8,19 +8,18 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Bundles using (DecPoset)
-open import Relation.Binary.Structures using (IsDecPartialOrder)
-open import Relation.Binary.Definitions using (Decidable)
-open import Agda.Builtin.Equality using (_≡_)
+open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Sublist.DecPropositional
-  {a} {A : Set a} (_≟_ : Decidable {A = A} _≡_)
+  {a} {A : Set a} (_≟_ : DecidableEquality A)
   where
 
 open import Data.List.Relation.Binary.Equality.DecPropositional _≟_
   using (_≡?_)
 import Data.List.Relation.Binary.Sublist.DecSetoid as DecSetoidSublist
 import Data.List.Relation.Binary.Sublist.Propositional as PropositionalSublist
+open import Relation.Binary.Bundles using (DecPoset)
+open import Relation.Binary.Structures using (IsDecPartialOrder)
 open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
index a2532bc63b..a68c5a2b2e 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional/Solver.agda
@@ -7,13 +7,12 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.List.Relation.Binary.Sublist.DecPropositional.Solver
-       {a} {A : Set a} (_≟_ : Decidable {A = A} _≡_)
+       {a} {A : Set a} (_≟_ : DecidableEquality A)
        where
 
-import Relation.Binary.PropositionalEquality.Properties as P
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
-open import Data.List.Relation.Binary.Sublist.DecSetoid.Solver (P.decSetoid _≟_) public
+open import Data.List.Relation.Binary.Sublist.DecSetoid.Solver (decSetoid _≟_) public
diff --git a/src/Data/Maybe/Properties.agda b/src/Data/Maybe/Properties.agda
index 4940da433f..212f1608ff 100644
--- a/src/Data/Maybe/Properties.agda
+++ b/src/Data/Maybe/Properties.agda
@@ -17,7 +17,7 @@ open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_; id; _∘_; _∘′_)
 open import Function.Definitions using (Injective)
 open import Level using (Level)
-open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (map′)
@@ -35,7 +35,7 @@ private
 just-injective : ∀ {x y} → (Maybe A ∋ just x) ≡ just y → x ≡ y
 just-injective refl = refl
 
-≡-dec : Decidable _≡_ → Decidable {A = Maybe A} _≡_
+≡-dec : DecidableEquality A → DecidableEquality (Maybe A)
 ≡-dec _≟_ nothing  nothing  = yes refl
 ≡-dec _≟_ (just x) nothing  = no λ()
 ≡-dec _≟_ nothing  (just y) = no λ()
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 625634af1d..949969033c 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -67,7 +67,7 @@ infix 4  _<?_ _≟_ _≤?_
 1+[2_]-injective : Injective _≡_ _≡_ 1+[2_]
 1+[2_]-injective refl = refl
 
-_≟_ : Decidable {A = ℕᵇ} _≡_
+_≟_ : DecidableEquality ℕᵇ
 zero     ≟ zero     =  yes refl
 zero     ≟ 2[1+ _ ] =  no λ()
 zero     ≟ 1+[2 _ ] =  no λ()
diff --git a/src/Data/Record.agda b/src/Data/Record.agda
index 6992294a5e..bd916a9005 100644
--- a/src/Data/Record.agda
+++ b/src/Data/Record.agda
@@ -16,15 +16,14 @@ open import Data.Product.Base hiding (proj₁; proj₂)
 open import Data.Unit.Polymorphic
 open import Function.Base using (id; _∘_)
 open import Level
-open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Nullary
 open import Relation.Nullary.Decidable
 
 -- The module is parametrised by the type of labels, which should come
 -- with decidable equality.
 
-module Data.Record {ℓ} (Label : Set ℓ) (_≟_ : Decidable {A = Label} _≡_) where
+module Data.Record {ℓ} (Label : Set ℓ) (_≟_ : DecidableEquality Label) where
 
 ------------------------------------------------------------------------
 -- A Σ-type with a manifest field
diff --git a/src/Data/String/Properties.agda b/src/Data/String/Properties.agda
index 1d2e9a9a1c..06406af94d 100644
--- a/src/Data/String/Properties.agda
+++ b/src/Data/String/Properties.agda
@@ -23,7 +23,7 @@ open import Relation.Binary.Bundles
 open import Relation.Binary.Structures
   using (IsEquivalence; IsDecEquivalence; IsStrictPartialOrder; IsStrictTotalOrder; IsDecPartialOrder; IsDecTotalOrder)
 open import Relation.Binary.Definitions
-  using (Reflexive; Symmetric; Transitive; Substitutive; Decidable)
+  using (Reflexive; Symmetric; Transitive; Substitutive; Decidable; DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core
 import Relation.Binary.Construct.On as On
 import Relation.Binary.PropositionalEquality.Properties as PropEq
@@ -89,7 +89,7 @@ x ≈? y = Pointwise.decidable Char._≟_ (toList x) (toList y)
 
 infix 4 _≟_
 
-_≟_ : Decidable _≡_
+_≟_ : DecidableEquality String
 x ≟ y = map′ ≈⇒≡ ≈-reflexive $ x ≈? y
 
 ≡-setoid : Setoid _ _
diff --git a/src/Data/Sum/Properties.agda b/src/Data/Sum/Properties.agda
index d0ba12c2fd..72dc3aa641 100644
--- a/src/Data/Sum/Properties.agda
+++ b/src/Data/Sum/Properties.agda
@@ -12,7 +12,7 @@ open import Level
 open import Data.Sum.Base
 open import Function.Base using (_∋_; _∘_; id)
 open import Function.Bundles using (mk↔ₛ′; _↔_)
-open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (map′)
@@ -34,10 +34,10 @@ inj₁-injective refl = refl
 inj₂-injective : ∀ {x y} → (A ⊎ B ∋ inj₂ x) ≡ inj₂ y → x ≡ y
 inj₂-injective refl = refl
 
-module _ (dec₁ : Decidable {A = A} {B = A} _≡_)
-         (dec₂ : Decidable {A = B} {B = B} _≡_) where
+module _ (dec₁ : DecidableEquality A)
+         (dec₂ : DecidableEquality B) where
 
-  ≡-dec : Decidable {A = A ⊎ B} _≡_
+  ≡-dec : DecidableEquality (A ⊎ B)
   ≡-dec (inj₁ x) (inj₁ y) = map′ (cong inj₁) inj₁-injective (dec₁ x y)
   ≡-dec (inj₁ x) (inj₂ y) = no λ()
   ≡-dec (inj₂ x) (inj₁ y) = no λ()
diff --git a/src/Data/These/Properties.agda b/src/Data/These/Properties.agda
index e7111cfce1..9f6cb55ce1 100644
--- a/src/Data/These/Properties.agda
+++ b/src/Data/These/Properties.agda
@@ -11,7 +11,7 @@ module Data.These.Properties where
 open import Data.Product.Base using (_×_; _,_; <_,_>; uncurry)
 open import Data.These.Base
 open import Function.Base using (_∘_)
-open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Binary.Definitions using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no; map′; _×-dec_)
 
@@ -35,7 +35,7 @@ module _ {a b} {A : Set a} {B : Set b} where
   these-injective : ∀ {x y : A} {a b : B} → these x a ≡ these y b → x ≡ y × a ≡ b
   these-injective = < these-injectiveˡ , these-injectiveʳ >
 
-  ≡-dec : Decidable _≡_ → Decidable _≡_ → Decidable {A = These A B} _≡_
+  ≡-dec : DecidableEquality A → DecidableEquality B → DecidableEquality (These A B)
   ≡-dec dec₁ dec₂ (this x)    (this y) =
     map′ (cong this) this-injective (dec₁ x y)
   ≡-dec dec₁ dec₂ (this x)    (that y)    = no λ()
diff --git a/src/Data/Unit/Polymorphic/Properties.agda b/src/Data/Unit/Polymorphic/Properties.agda
index ad2ccc3a05..088758ab39 100644
--- a/src/Data/Unit/Polymorphic/Properties.agda
+++ b/src/Data/Unit/Polymorphic/Properties.agda
@@ -21,7 +21,7 @@ open import Relation.Binary.Bundles
 open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
 open import Relation.Binary.Definitions
-  using (Decidable; Antisymmetric; Total)
+  using (DecidableEquality; Antisymmetric; Total)
 open import Relation.Binary.PropositionalEquality
   using (_≡_; refl; trans; decSetoid; setoid; isEquivalence)
 
@@ -35,7 +35,7 @@ private
 
 infix 4 _≟_
 
-_≟_ : Decidable {A = ⊤ {ℓ}} _≡_
+_≟_ : DecidableEquality (⊤ {ℓ})
 _ ≟ _ = yes refl
 
 ≡-setoid : ∀ ℓ → Setoid ℓ ℓ
diff --git a/src/Data/Unit/Properties.agda b/src/Data/Unit/Properties.agda
index 5b4c0bee51..576e24c7dc 100644
--- a/src/Data/Unit/Properties.agda
+++ b/src/Data/Unit/Properties.agda
@@ -16,7 +16,7 @@ open import Relation.Binary.Bundles
   using (Setoid; DecSetoid; Poset; DecTotalOrder)
 open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
-open import Relation.Binary.Definitions using (Decidable; Total; Antisymmetric)
+open import Relation.Binary.Definitions using (DecidableEquality; Total; Antisymmetric)
 open import Relation.Binary.PropositionalEquality
 
 ------------------------------------------------------------------------
@@ -30,7 +30,7 @@ open import Relation.Binary.PropositionalEquality
 
 infix 4 _≟_
 
-_≟_ : Decidable {A = ⊤} _≡_
+_≟_ : DecidableEquality ⊤
 _ ≟ _ = yes refl
 
 ≡-setoid : Setoid 0ℓ 0ℓ
diff --git a/src/Data/Vec/Membership/DecPropositional.agda b/src/Data/Vec/Membership/DecPropositional.agda
index 8004af2720..f7b028e11f 100644
--- a/src/Data/Vec/Membership/DecPropositional.agda
+++ b/src/Data/Vec/Membership/DecPropositional.agda
@@ -6,12 +6,12 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_)
-open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
+open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.Vec.Membership.DecPropositional
-  {a} {A : Set a} (_≟_ : Decidable (_≡_ {A = A})) where
+  {a} {A : Set a} (_≟_ : DecidableEquality A) where
+
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 ------------------------------------------------------------------------
 -- Re-export contents of propositional membership
diff --git a/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda b/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
index 06f8d50e7d..e8a288b338 100644
--- a/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/DecPropositional.agda
@@ -6,14 +6,14 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.Definitions using (DecidableEquality)
 
 module Data.Vec.Relation.Binary.Equality.DecPropositional
-  {a} {A : Set a} (_≟_ : Decidable {A = A} _≡_) where
+  {a} {A : Set a} (_≟_ : DecidableEquality A) where
 
 import Data.Vec.Relation.Binary.Equality.Propositional as PEq
 import Data.Vec.Relation.Binary.Equality.DecSetoid as DSEq
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 ------------------------------------------------------------------------
 -- Publicly re-export everything from decSetoid and propositional
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index 3143d0d7ad..ba4bae8be1 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -21,7 +21,7 @@ open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core as B hiding (Rel; _⇔_)
 open import Relation.Binary.Definitions
-  using (Decidable; Reflexive; Symmetric; Transitive)
+  using (DecidableEquality; Reflexive; Symmetric; Transitive)
 open import Relation.Binary.Structures
   using (IsPreorder; IsEquivalence)
 open import Relation.Binary.Bundles
@@ -115,7 +115,7 @@ data Kind : Set where
 
 infix 4 _≟-Kind_
 
-_≟-Kind_ : Decidable (_≡_ {A = Kind})
+_≟-Kind_ : DecidableEquality Kind
 _≟-Kind_ strong       strong       = yes ≡.refl
 _≟-Kind_ strong       (other k)    = no λ()
 _≟-Kind_ (other k)    strong       = no λ()
diff --git a/src/Reflection/AST/Term.agda b/src/Reflection/AST/Term.agda
index cdcefb55fc..14381c3ba6 100644
--- a/src/Reflection/AST/Term.agda
+++ b/src/Reflection/AST/Term.agda
@@ -141,8 +141,8 @@ _≟-Clause_   : DecidableEquality Clause
 _≟-Clauses_  : DecidableEquality Clauses
 _≟_          : DecidableEquality Term
 _≟-Sort_     : DecidableEquality Sort
-_≟-Patterns_ : Decidable (_≡_ {A = Args Pattern})
-_≟-Pattern_  : Decidable (_≡_ {A = Pattern})
+_≟-Patterns_ : DecidableEquality (Args Pattern)
+_≟-Pattern_  : DecidableEquality Pattern
 
 -- Decidable equality 'transformers'
 -- We need to inline these because the terms are not sized so
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index d06f81a0af..6b14e91d09 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -17,7 +17,7 @@ open import Relation.Binary.Core using (Rel; REL; _=[_]⇒_)
 open import Relation.Binary.Structures
   using (IsPreorder; IsStrictPartialOrder; IsPartialOrder; IsDecStrictPartialOrder; IsDecPartialOrder; IsStrictTotalOrder; IsTotalOrder; IsDecTotalOrder)
 open import Relation.Binary.Definitions
-  using (Symmetric; Transitive; Reflexive; Asymmetric; Antisymmetric; Trichotomous; Total; Decidable; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects_; _Respects₂_)
+  using (Symmetric; Transitive; Reflexive; Asymmetric; Antisymmetric; Trichotomous; Total; Decidable; DecidableEquality; tri<; tri≈; tri>; _Respectsˡ_; _Respectsʳ_; _Respects_; _Respects₂_)
 open import Relation.Binary.Construct.Closure.Reflexive
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.PropositionalEquality.Properties as ≡
@@ -78,7 +78,7 @@ module _ {_~_ : Rel A ℓ} where
   ... | tri≈ _ refl _ = inj₁ refl
   ... | tri> _ _    c = inj₂ [ c ]
 
-  dec : Decidable {A = A} _≡_ → Decidable _~_ → Decidable _~ᵒ_
+  dec : DecidableEquality A → Decidable _~_ → Decidable _~ᵒ_
   dec ≡-dec ~-dec a b = Dec.map ⊎⇔Refl (≡-dec a b ⊎-dec ~-dec a b)
 
   decidable : Trichotomous _≡_ _~_ → Decidable _~ᵒ_
diff --git a/src/Relation/Binary/PropositionalEquality/Properties.agda b/src/Relation/Binary/PropositionalEquality/Properties.agda
index a420e45b64..4aba0ba4c8 100644
--- a/src/Relation/Binary/PropositionalEquality/Properties.agda
+++ b/src/Relation/Binary/PropositionalEquality/Properties.agda
@@ -19,7 +19,7 @@ open import Relation.Binary.Bundles
 open import Relation.Binary.Structures
   using (IsEquivalence; IsDecEquivalence; IsPreorder; IsPartialOrder)
 open import Relation.Binary.Definitions
-  using (Decidable; DecidableEquality)
+  using (DecidableEquality)
 import Relation.Binary.Properties.Setoid as Setoid
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Unary using (Pred)
@@ -157,7 +157,7 @@ isEquivalence = record
   ; trans = trans
   }
 
-isDecEquivalence : Decidable _≡_ → IsDecEquivalence {A = A} _≡_
+isDecEquivalence : DecidableEquality A → IsDecEquivalence _≡_
 isDecEquivalence _≟_ = record
   { isEquivalence = isEquivalence
   ; _≟_           = _≟_

From eafe34e6857920d1903035eb51bfe7abd086de48 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Thu, 11 Apr 2024 11:43:43 +0800
Subject: [PATCH 266/316] Added missing v1.7.3 CHANGELOG (#2356)

---
 CHANGELOG/v1.7.3.md | 18 ++++++++++++++++++
 1 file changed, 18 insertions(+)
 create mode 100644 CHANGELOG/v1.7.3.md

diff --git a/CHANGELOG/v1.7.3.md b/CHANGELOG/v1.7.3.md
new file mode 100644
index 0000000000..7226a01e8a
--- /dev/null
+++ b/CHANGELOG/v1.7.3.md
@@ -0,0 +1,18 @@
+Version 1.7.3
+=============
+
+The library has been tested using Agda 2.6.3 & 2.6.4.
+
+* To avoid _large indices_ that are by default no longer allowed in Agda 2.6.4,
+  universe levels have been increased in the following definitions:
+  - `Data.Star.Decoration.DecoratedWith`
+  - `Data.Star.Pointer.Pointer`
+  - `Reflection.AnnotatedAST.Typeₐ`
+  - `Reflection.AnnotatedAST.AnnotationFun`
+
+* The following aliases have been added:
+  - `IO.Primitive.pure` as alias for `IO.Primitive.return`
+  - modules `Effect.*` as aliases for modules `Category.*`
+
+  These allow to address said objects with the new name they will have in v2.0 of the library,
+  to ease the transition from v1.7.3 to v2.0.

From 1d0231b695cc716e62a16bf02b15ee3d7f63e78d Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Fri, 19 Apr 2024 13:15:30 +0100
Subject: [PATCH 267/316] Improve `Data.List.Base.mapMaybe` (#2359; deprecate
 use of `with` #2123) (#2361)

* `with`-free definitions plus tests

* `CHANGELOG`

* use `foldr` on @JacquesCarette 's solution

* tidied up unsolved metas

* factrored out comparison as removable module `MapMaybeTest`

* tidied up; removed `mapMaybeTest`

* tidied up; removed v2.1 deprecation section

* tidy up long line

* Update src/Data/List/Base.agda

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

* @gallais 's comments

* Update src/Data/List/Base.agda

Oops!

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

---------

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
---
 CHANGELOG.md                  |  1 +
 src/Data/List/Base.agda       | 25 ++++++-------
 src/Data/List/Properties.agda | 70 ++++++++++++++++++-----------------
 3 files changed, 49 insertions(+), 47 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 469c926af8..e7d776393a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -273,6 +273,7 @@ Additions to existing modules
 
 * In `Data.List.Properties`:
   ```agda
+  length-catMaybes      : length (catMaybes xs) ≤ length xs
   applyUpTo-∷ʳ          : applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
   applyDownFrom-∷ʳ      : applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
   upTo-∷ʳ               : upTo n ∷ʳ n ≡ upTo (suc n)
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 8b988c646d..03e5d4ae9a 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -49,15 +49,6 @@ map : (A → B) → List A → List B
 map f []       = []
 map f (x ∷ xs) = f x ∷ map f xs
 
-mapMaybe : (A → Maybe B) → List A → List B
-mapMaybe p []       = []
-mapMaybe p (x ∷ xs) with p x
-... | just y  = y ∷ mapMaybe p xs
-... | nothing = mapMaybe p xs
-
-catMaybes : List (Maybe A) → List A
-catMaybes = mapMaybe id
-
 infixr 5 _++_
 
 _++_ : List A → List A → List A
@@ -123,11 +114,11 @@ partitionSums : List (A ⊎ B) → List A × List B
 partitionSums = partitionSumsWith id
 
 merge : {R : Rel A ℓ} → B.Decidable R → List A → List A → List A
-merge R? []       ys       = ys
-merge R? xs       []       = xs
-merge R? (x ∷ xs) (y ∷ ys) = if does (R? x y)
-  then x ∷ merge R? xs (y ∷ ys)
-  else y ∷ merge R? (x ∷ xs) ys
+merge R? []           ys           = ys
+merge R? xs           []           = xs
+merge R? x∷xs@(x ∷ xs) y∷ys@(y ∷ ys) = if does (R? x y)
+  then x ∷ merge R? xs   y∷ys
+  else y ∷ merge R? x∷xs ys
 
 ------------------------------------------------------------------------
 -- Operations for reducing lists
@@ -149,6 +140,12 @@ concatMap f = concat ∘ map f
 ap : List (A → B) → List A → List B
 ap fs as = concatMap (flip map as) fs
 
+catMaybes : List (Maybe A) → List A
+catMaybes = foldr (maybe′ _∷_ id) []
+
+mapMaybe : (A → Maybe B) → List A → List B
+mapMaybe p = catMaybes ∘ map p
+
 null : List A → Bool
 null []       = true
 null (x ∷ xs) = false
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 2d3c9e20e0..1dcf0d1a20 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -21,7 +21,7 @@ open import Data.List.Base as List
 open import Data.List.Membership.Propositional using (_∈_)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
-open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
+open import Data.Maybe.Base as Maybe using (Maybe; just; nothing; maybe)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility using (_∣_; divides; ∣n⇒∣m*n)
 open import Data.Nat.Properties
@@ -56,23 +56,23 @@ private
     C : Set c
     D : Set d
     E : Set e
+    x y z w : A
+    xs ys zs ws : List A
 
 ------------------------------------------------------------------------
 -- _∷_
 
-module _ {x y : A} {xs ys : List A} where
+∷-injective : x ∷ xs ≡ y List.∷ ys → x ≡ y × xs ≡ ys
+∷-injective refl = refl , refl
 
-  ∷-injective : x ∷ xs ≡ y List.∷ ys → x ≡ y × xs ≡ ys
-  ∷-injective refl = (refl , refl)
+∷-injectiveˡ : x ∷ xs ≡ y List.∷ ys → x ≡ y
+∷-injectiveˡ refl = refl
 
-  ∷-injectiveˡ : x ∷ xs ≡ y List.∷ ys → x ≡ y
-  ∷-injectiveˡ refl = refl
+∷-injectiveʳ : x ∷ xs ≡ y List.∷ ys → xs ≡ ys
+∷-injectiveʳ refl = refl
 
-  ∷-injectiveʳ : x ∷ xs ≡ y List.∷ ys → xs ≡ ys
-  ∷-injectiveʳ refl = refl
-
-  ∷-dec : Dec (x ≡ y) → Dec (xs ≡ ys) → Dec (x List.∷ xs ≡ y ∷ ys)
-  ∷-dec x≟y xs≟ys = Decidable.map′ (uncurry (cong₂ _∷_)) ∷-injective (x≟y ×-dec xs≟ys)
+∷-dec : Dec (x ≡ y) → Dec (xs ≡ ys) → Dec (x ∷ xs ≡ y List.∷ ys)
+∷-dec x≟y xs≟ys = Decidable.map′ (uncurry (cong₂ _∷_)) ∷-injective (x≟y ×-dec xs≟ys)
 
 ≡-dec : DecidableEquality A → DecidableEquality (List A)
 ≡-dec _≟_ []       []       = yes refl
@@ -122,28 +122,34 @@ map-injective finj {x ∷ xs} {y ∷ ys} eq =
 ------------------------------------------------------------------------
 -- mapMaybe
 
+length-catMaybes : ∀ xs → length (catMaybes {A = A} xs) ≤ length xs
+length-catMaybes []        = ≤-refl
+length-catMaybes (just x ∷ xs) = s≤s (length-catMaybes xs)
+length-catMaybes (nothing ∷ xs) = m≤n⇒m≤1+n (length-catMaybes xs)
+
 mapMaybe-just : (xs : List A) → mapMaybe just xs ≡ xs
 mapMaybe-just []       = refl
 mapMaybe-just (x ∷ xs) = cong (x ∷_) (mapMaybe-just xs)
 
 mapMaybe-nothing : (xs : List A) →
-                   mapMaybe {B = A} (λ _ → nothing) xs ≡ []
+                   mapMaybe {B = B} (λ _ → nothing) xs ≡ []
 mapMaybe-nothing []       = refl
 mapMaybe-nothing (x ∷ xs) = mapMaybe-nothing xs
 
 module _ (f : A → Maybe B) where
 
   mapMaybe-concatMap : mapMaybe f ≗ concatMap (fromMaybe ∘ f)
-  mapMaybe-concatMap [] = refl
+  mapMaybe-concatMap []       = refl
   mapMaybe-concatMap (x ∷ xs) with ih ← mapMaybe-concatMap xs | f x
   ... | just y  = cong (y ∷_) ih
   ... | nothing = ih
 
   length-mapMaybe : ∀ xs → length (mapMaybe f xs) ≤ length xs
-  length-mapMaybe []       = z≤n
-  length-mapMaybe (x ∷ xs) with ih ← length-mapMaybe xs | f x
-  ... | just y  = s≤s ih
-  ... | nothing = m≤n⇒m≤1+n ih
+  length-mapMaybe xs = ≤-begin
+    length (mapMaybe f xs)      ≤⟨ length-catMaybes (map f xs) ⟩
+    length (map f xs)           ≤⟨ ≤-reflexive (length-map f xs) ⟩
+    length xs                   ≤-∎
+    where open ≤-Reasoning renaming (begin_ to ≤-begin_; _∎ to _≤-∎)
 
 ------------------------------------------------------------------------
 -- _++_
@@ -175,14 +181,14 @@ module _ {A : Set a} where
   ++-identityʳ-unique : ∀ (xs : List A) {ys} → xs ≡ xs ++ ys → ys ≡ []
   ++-identityʳ-unique []       refl = refl
   ++-identityʳ-unique (x ∷ xs) eq   =
-    ++-identityʳ-unique xs (proj₂ (∷-injective eq))
+    ++-identityʳ-unique xs (∷-injectiveʳ eq)
 
   ++-identityˡ-unique : ∀ {xs} (ys : List A) → xs ≡ ys ++ xs → ys ≡ []
   ++-identityˡ-unique               []       _  = refl
   ++-identityˡ-unique {xs = x ∷ xs} (y ∷ ys) eq
     with ++-identityˡ-unique (ys ++ [ x ]) (begin
-         xs                  ≡⟨ proj₂ (∷-injective eq) ⟩
-         ys ++ x ∷ xs        ≡⟨ sym (++-assoc ys [ x ] xs) ⟩
+         xs                  ≡⟨ ∷-injectiveʳ eq ⟩
+         ys ++ x ∷ xs        ≡⟨ ++-assoc ys [ x ] xs ⟨
          (ys ++ [ x ]) ++ xs ∎)
   ++-identityˡ-unique {xs = x ∷ xs} (y ∷ []   ) eq | ()
   ++-identityˡ-unique {xs = x ∷ xs} (y ∷ _ ∷ _) eq | ()
@@ -1215,22 +1221,20 @@ reverse-downFrom = reverse-applyDownFrom id
 ------------------------------------------------------------------------
 -- _∷ʳ_
 
-module _ {x y : A} where
-
-  ∷ʳ-injective : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
-  ∷ʳ-injective []          []          refl = (refl , refl)
-  ∷ʳ-injective (x ∷ xs)    (y  ∷ ys)   eq   with refl , eq′  ← ∷-injective eq
-    = Product.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
-  ∷ʳ-injective []          (_ ∷ _ ∷ _) ()
-  ∷ʳ-injective (_ ∷ _ ∷ _) []          ()
+∷ʳ-injective : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
+∷ʳ-injective []          []          refl = refl , refl
+∷ʳ-injective (x ∷ xs)    (y  ∷ ys)   eq   with refl , eq′  ← ∷-injective eq
+  = Product.map (cong (x ∷_)) id (∷ʳ-injective xs ys eq′)
+∷ʳ-injective []          (_ ∷ _ ∷ _) ()
+∷ʳ-injective (_ ∷ _ ∷ _) []          ()
 
-  ∷ʳ-injectiveˡ : ∀ (xs ys : List A) → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys
-  ∷ʳ-injectiveˡ xs ys eq = proj₁ (∷ʳ-injective xs ys eq)
+∷ʳ-injectiveˡ : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys
+∷ʳ-injectiveˡ xs ys eq = proj₁ (∷ʳ-injective xs ys eq)
 
-  ∷ʳ-injectiveʳ : ∀ (xs ys : List A) → xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y
-  ∷ʳ-injectiveʳ xs ys eq = proj₂ (∷ʳ-injective xs ys eq)
+∷ʳ-injectiveʳ : ∀ xs ys → xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y
+∷ʳ-injectiveʳ xs ys eq = proj₂ (∷ʳ-injective xs ys eq)
 
-∷ʳ-++ : ∀ (xs : List A) (a : A) (ys : List A) → xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
+∷ʳ-++ : ∀ xs (a : A) ys → xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
 ∷ʳ-++ xs a ys = ++-assoc xs [ a ] ys
 
 

From 01e20a891b08eb9ef7673e67211ef39e254c9311 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Fri, 19 Apr 2024 23:24:03 -0400
Subject: [PATCH 268/316] Add a consequence of definition straight to
 IsCancellativeCommutativeSemiring (#2370)

* Closes #2315

* whitespace
---
 CHANGELOG.md                | 6 ++++++
 src/Algebra/Structures.agda | 4 ++++
 2 files changed, 10 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e7d776393a..0b42e8d890 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -249,6 +249,12 @@ Additions to existing modules
   x \\ y = (x ⁻¹) ∙ y
   ```
 
+* In `Algebra.Structures.IsCancellativeCommutativeSemiring` add the
+  extra property as an exposed definition:
+  ```agda
+    *-cancelʳ-nonZero : AlmostRightCancellative 0# *
+  ```
+
 * In `Data.Container.Indexed.Core`:
   ```agda
   Subtrees o c = (r : Response c) → X (next c r)
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index 066363686f..3cdc55a4b5 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -573,6 +573,10 @@ record IsCancellativeCommutativeSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a 
 
   open IsCommutativeSemiring isCommutativeSemiring public
 
+  *-cancelʳ-nonZero : AlmostRightCancellative 0# *
+  *-cancelʳ-nonZero = Consequences.comm∧almostCancelˡ⇒almostCancelʳ setoid
+      *-comm *-cancelˡ-nonZero
+
 record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
   field
     isSemiring     : IsSemiring + * 0# 1#

From 22de3ef492b7cf6319d3c7d50c8ce5e4ee92bd80 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Sat, 20 Apr 2024 04:29:57 +0100
Subject: [PATCH 269/316] [ new ] System.Random bindings (#2368)

* [ new ] System.Random bindings

* [ more ] Show functions, test

* [ fix ] Nat bug, more random generators, test case

* [ fix ] missing file + local gitignore

* [ fix ] forgot to update the CHANGELOG
---
 CHANGELOG.md                        |  25 ++++
 GenerateEverything.hs               |   2 +
 src/Data/Float/Base.agda            |   8 +-
 src/Data/List/Show.agda             |  16 +++
 src/Data/String/Base.agda           |   7 +-
 src/Data/Vec/Bounded/Show.agda      |  17 +++
 src/Data/Vec/Show.agda              |  17 +++
 src/Data/Word/Base.agda             |   4 +
 src/System/Random.agda              | 199 ++++++++++++++++++++++++++++
 src/System/Random/Primitive.agda    |  42 ++++++
 tests/runtests.agda                 |   1 +
 tests/system/random/.gitignore      |   1 +
 tests/system/random/Main.agda       |  53 ++++++++
 tests/system/random/expected        |   0
 tests/system/random/random.agda-lib |   2 +
 tests/system/random/run             |   6 +
 16 files changed, 397 insertions(+), 3 deletions(-)
 create mode 100644 src/Data/List/Show.agda
 create mode 100644 src/Data/Vec/Bounded/Show.agda
 create mode 100644 src/Data/Vec/Show.agda
 create mode 100644 src/System/Random.agda
 create mode 100644 src/System/Random/Primitive.agda
 create mode 100644 tests/system/random/.gitignore
 create mode 100644 tests/system/random/Main.agda
 create mode 100644 tests/system/random/expected
 create mode 100644 tests/system/random/random.agda-lib
 create mode 100644 tests/system/random/run

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 0b42e8d890..7cb74659bc 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -125,6 +125,19 @@ New modules
   Data.Container.Indexed.Relation.Binary.Equality.Setoid
   ```
 
+* `System.Random` bindings:
+  ```agda
+  System.Random.Primitive
+  System.Random
+  ```
+
+* Show modules:
+  ```agda
+  Data.List.Show
+  Data.Vec.Show
+  Data.Vec.Bounded.Show
+  ```
+
 Additions to existing modules
 -----------------------------
 
@@ -265,6 +278,11 @@ Additions to existing modules
   nonZeroIndex : Fin n → ℕ.NonZero n
   ```
 
+* In `Data.Float.Base`:
+  ```agda
+  _≤_ : Rel Float _
+  ```
+
 * In `Data.Integer.Divisibility`: introduce `divides` as an explicit pattern synonym
   ```agda
   pattern divides k eq = Data.Nat.Divisibility.divides k eq
@@ -385,6 +403,13 @@ Additions to existing modules
 * Added new functions in `Data.String.Base`:
   ```agda
   map : (Char → Char) → String → String
+  between : String → String → String → String
+  ```
+
+* In `Data.Word.Base`:
+  ```agda
+  _≤_ : Rel Word64 zero
+  ```
 
 * Added new definition in `Relation.Binary.Construct.Closure.Transitive`
   ```
diff --git a/GenerateEverything.hs b/GenerateEverything.hs
index f6add73649..c3aa8ed710 100644
--- a/GenerateEverything.hs
+++ b/GenerateEverything.hs
@@ -76,6 +76,8 @@ unsafeModules = map modToFile
   , "System.FilePath.Posix.Primitive"
   , "System.Process"
   , "System.Process.Primitive"
+  , "System.Random"
+  , "System.Random.Primitive"
   , "Test.Golden"
   , "Text.Pretty.Core"
   , "Text.Pretty"
diff --git a/src/Data/Float/Base.agda b/src/Data/Float/Base.agda
index d09a5a811b..fdc7dab2db 100644
--- a/src/Data/Float/Base.agda
+++ b/src/Data/Float/Base.agda
@@ -8,11 +8,12 @@
 
 module Data.Float.Base where
 
-open import Relation.Binary.Core using (Rel)
+open import Data.Bool.Base using (T)
 import Data.Word.Base as Word
 import Data.Maybe.Base as Maybe
 open import Function.Base using (_on_; _∘_)
 open import Agda.Builtin.Equality
+open import Relation.Binary.Core using (Rel)
 
 ------------------------------------------------------------------------
 -- Re-export built-ins publically
@@ -69,3 +70,8 @@ infix 4 _≈_
 
 _≈_ : Rel Float _
 _≈_ = _≡_ on Maybe.map Word.toℕ ∘ toWord
+
+
+infix 4 _≤_
+_≤_ : Rel Float _
+x ≤ y = T (x ≤ᵇ y)
diff --git a/src/Data/List/Show.agda b/src/Data/List/Show.agda
new file mode 100644
index 0000000000..116bd82061
--- /dev/null
+++ b/src/Data/List/Show.agda
@@ -0,0 +1,16 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Showing lists
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.Show where
+
+open import Data.List.Base using (List; map)
+open import Data.String.Base using (String; between; intersperse)
+open import Function.Base using (_∘_)
+
+show : ∀ {a} {A : Set a} → (A → String) → (List A → String)
+show s = between "[" "]" ∘ intersperse ", " ∘ map s
diff --git a/src/Data/String/Base.agda b/src/Data/String/Base.agda
index fb0ebc1526..6eb10f3554 100644
--- a/src/Data/String/Base.agda
+++ b/src/Data/String/Base.agda
@@ -102,11 +102,14 @@ unwords = intersperse " "
 unlines : List String → String
 unlines = intersperse "\n"
 
+between : String → String → String → String
+between left right middle = left ++ middle ++ right
+
 parens : String → String
-parens s = "(" ++ s ++ ")"
+parens = between "(" ")"
 
 braces : String → String
-braces s = "{" ++ s ++ "}"
+braces = between "{" "}"
 
 -- append that also introduces spaces, if necessary
 infixr 5 _<+>_
diff --git a/src/Data/Vec/Bounded/Show.agda b/src/Data/Vec/Bounded/Show.agda
new file mode 100644
index 0000000000..dae5f6ec3d
--- /dev/null
+++ b/src/Data/Vec/Bounded/Show.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Showing bounded vectors
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Bounded.Show where
+
+open import Data.String.Base using (String)
+open import Data.Vec.Bounded.Base using (Vec≤)
+import Data.Vec.Show as Vec
+open import Function.Base using (_∘_)
+
+show : ∀ {a} {A : Set a} {n} → (A → String) → (Vec≤ A n → String)
+show s = Vec.show s ∘ Vec≤.vec
diff --git a/src/Data/Vec/Show.agda b/src/Data/Vec/Show.agda
new file mode 100644
index 0000000000..99daee3639
--- /dev/null
+++ b/src/Data/Vec/Show.agda
@@ -0,0 +1,17 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Showing vectors
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Vec.Show where
+
+import Data.List.Show as List
+open import Data.String.Base using (String)
+open import Data.Vec.Base using (Vec; toList)
+open import Function.Base using (_∘_)
+
+show : ∀ {a} {A : Set a} {n} → (A → String) → (Vec A n → String)
+show s = List.show s ∘ toList
diff --git a/src/Data/Word/Base.agda b/src/Data/Word/Base.agda
index 81a7ae6666..f0dddc1e2c 100644
--- a/src/Data/Word/Base.agda
+++ b/src/Data/Word/Base.agda
@@ -31,3 +31,7 @@ _≈_ = _≡_ on toℕ
 infix 4 _<_
 _<_ : Rel Word64 zero
 _<_ = ℕ._<_ on toℕ
+
+infix 4 _≤_
+_≤_ : Rel Word64 zero
+_≤_ = ℕ._≤_ on toℕ
diff --git a/src/System/Random.agda b/src/System/Random.agda
new file mode 100644
index 0000000000..fa327939d6
--- /dev/null
+++ b/src/System/Random.agda
@@ -0,0 +1,199 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- *Pseudo-random* number generation
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --guardedness #-}
+
+module System.Random where
+
+import System.Random.Primitive as Prim
+
+open import Data.Bool.Base using (T)
+open import Data.Nat.Base using (ℕ; z≤n) hiding (module ℕ)
+open import Foreign.Haskell.Pair using (_,_)
+open import Function.Base using (_$_; _∘_)
+open import IO.Base using (IO; lift; lift!; _<$>_; _>>=_; pure)
+import IO.Effectful as IO
+open import Level using (0ℓ; suc; _⊔_; lift)
+open import Relation.Binary.Core using (Rel)
+
+------------------------------------------------------------------------
+-- Ranged generation shall return proofs
+
+record InBounds {a r} {A : Set a} (_≤_ : Rel A r) (lo hi : A) : Set (a ⊔ r) where
+  constructor _∈[_,_]
+  field
+    value : A
+    .isLowerBound : lo ≤ value
+    .isUpperBound : value ≤ hi
+
+RandomRIO : ∀ {a r} {A : Set a} (_≤_ : Rel A r) → Set (suc (a ⊔ r))
+RandomRIO {A = A} _≤_ = (lo hi : A) → .(lo ≤ hi) → IO (InBounds _≤_ lo hi)
+
+------------------------------------------------------------------------
+-- Instances
+
+module Char where
+
+  open import Data.Char.Base using (Char; _≤_)
+
+  randomIO : IO Char
+  randomIO = lift Prim.randomIO-Char
+
+  randomRIO : RandomRIO _≤_
+  randomRIO lo hi _ = do
+    value ← lift (Prim.randomRIO-Char (lo , hi))
+    pure (value ∈[ trustMe , trustMe ])
+    where postulate trustMe : ∀ {A} → A
+
+module Float where
+
+  open import Data.Float.Base using (Float; _≤_)
+
+  randomIO : IO Float
+  randomIO = lift Prim.randomIO-Float
+
+  randomRIO : RandomRIO _≤_
+  randomRIO lo hi _ = do
+    value ← lift (Prim.randomRIO-Float (lo , hi))
+    pure (value ∈[ trustMe , trustMe ])
+    where postulate trustMe : ∀ {A} → A
+
+module ℤ where
+
+  open import Data.Integer.Base using (ℤ; _≤_)
+
+  randomIO : IO ℤ
+  randomIO = lift Prim.randomIO-Int
+
+  randomRIO : RandomRIO _≤_
+  randomRIO lo hi _ = do
+    value ← lift (Prim.randomRIO-Int (lo , hi))
+    pure (value ∈[ trustMe , trustMe ])
+    where postulate trustMe : ∀ {A} → A
+
+module ℕ where
+
+  open import Data.Nat.Base using (ℕ; _≤_)
+
+  randomIO : IO ℕ
+  randomIO = lift Prim.randomIO-Nat
+
+  randomRIO : RandomRIO _≤_
+  randomRIO lo hi _ = do
+    value ← lift (Prim.randomRIO-Nat (lo , hi))
+    pure (value ∈[ trustMe , trustMe ])
+    where postulate trustMe : ∀ {A} → A
+
+module Word64 where
+
+  open import Data.Word.Base using (Word64; _≤_)
+
+  randomIO : IO Word64
+  randomIO = lift Prim.randomIO-Word64
+
+  randomRIO : RandomRIO _≤_
+  randomRIO lo hi _ = do
+    value ← lift (Prim.randomRIO-Word64 (lo , hi))
+    pure (value ∈[ trustMe , trustMe ])
+    where postulate trustMe : ∀ {A} → A
+
+module Fin where
+
+  open import Data.Nat.Base as ℕ using (suc; NonZero; z≤n; s≤s)
+  import Data.Nat.Properties as ℕ
+  open import Data.Fin.Base using (Fin; _≤_; fromℕ<; toℕ)
+  import Data.Fin.Properties as Fin
+
+  randomIO : ∀ {n} → .{{NonZero n}} → IO (Fin n)
+  randomIO {n = n@(suc _)} = do
+    suc k ∈[ lo≤k , k≤hi ] ← ℕ.randomRIO 1 n (s≤s z≤n)
+    pure (fromℕ< k≤hi)
+
+  toℕ-cancel-InBounds : ∀ {n} {lo hi : Fin n} →
+                        InBounds ℕ._≤_ (toℕ lo) (toℕ hi) →
+                        InBounds _≤_ lo hi
+  toℕ-cancel-InBounds {n} {lo} {hi} (k ∈[ toℕlo≤k , k≤toℕhi ]) =
+    let
+      .k<n  : k ℕ.< n
+      k<n = ℕ.≤-<-trans k≤toℕhi (Fin.toℕ<n hi)
+
+      .lo≤k : lo ≤ fromℕ< k<n
+      lo≤k = Fin.toℕ-cancel-≤ $ let open ℕ.≤-Reasoning in begin
+        toℕ lo           ≤⟨ toℕlo≤k ⟩
+        k                ≡⟨ Fin.toℕ-fromℕ< k<n ⟨
+        toℕ (fromℕ< k<n) ∎
+
+      .k≤hi : fromℕ< k<n ≤ hi
+      k≤hi = Fin.toℕ-cancel-≤ $ let open ℕ.≤-Reasoning in begin
+        toℕ (fromℕ< k<n) ≡⟨ Fin.toℕ-fromℕ< k<n ⟩
+        k                ≤⟨ k≤toℕhi ⟩
+        toℕ hi           ∎
+
+    in fromℕ< k<n ∈[ lo≤k , k≤hi ]
+
+  randomRIO : ∀ {n} → RandomRIO {A = Fin n} _≤_
+  randomRIO {n} lo hi p = do
+    k ← ℕ.randomRIO (toℕ lo) (toℕ hi) (Fin.toℕ-mono-≤ p)
+    pure (toℕ-cancel-InBounds k)
+
+module List {a} {A : Set a} (rIO : IO A)  where
+
+  open import Data.List.Base using (List; replicate)
+  open import Data.List.Effectful using (module TraversableA)
+
+  -- Careful: this can generate very long lists!
+  -- You may want to use Vec≤ instead.
+  randomIO : IO (List A)
+  randomIO = do
+    lift n ← lift! ℕ.randomIO
+    TraversableA.sequenceA IO.applicative $ replicate n rIO
+
+module Vec {a} {A : Set a} (rIO : IO A) (n : ℕ) where
+
+  open import Data.Vec.Base using (Vec; replicate)
+  open import Data.Vec.Effectful using (module TraversableA)
+
+  randomIO : IO (Vec A n)
+  randomIO = TraversableA.sequenceA IO.applicative $ replicate n rIO
+
+module Vec≤ {a} {A : Set a} (rIO : IO A) (n : ℕ) where
+
+  open import Data.Vec.Bounded.Base using (Vec≤; _,_)
+
+  randomIO : IO (Vec≤ A n)
+  randomIO = do
+    lift (len ∈[ _ , len≤n ]) ← lift! (ℕ.randomRIO 0 n z≤n)
+    vec ← Vec.randomIO rIO len
+    pure (vec , len≤n)
+
+module String where
+
+  open import Data.String.Base using (String; fromList)
+
+  -- Careful: this can generate very long lists!
+  -- You may want to use String≤ instead.
+  randomIO : IO String
+  randomIO = fromList <$> List.randomIO Char.randomIO
+
+module String≤ (n : ℕ) where
+
+  import Data.Vec.Bounded.Base as Vec≤
+  open import Data.String.Base using (String; fromList)
+
+  randomIO : IO String
+  randomIO = fromList ∘ Vec≤.toList <$> Vec≤.randomIO Char.randomIO n
+
+open import Data.Char.Base using (Char; _≤_)
+
+module RangedString≤ (a b : Char)  .(a≤b : a ≤ b) (n : ℕ) where
+
+  import Data.Vec.Bounded.Base as Vec≤
+  open import Data.String.Base using (String; fromList)
+
+  randomIO : IO String
+  randomIO =
+    fromList ∘ Vec≤.toList ∘ Vec≤.map InBounds.value
+    <$> Vec≤.randomIO (Char.randomRIO a b a≤b) n
diff --git a/src/System/Random/Primitive.agda b/src/System/Random/Primitive.agda
new file mode 100644
index 0000000000..b93ee91fcb
--- /dev/null
+++ b/src/System/Random/Primitive.agda
@@ -0,0 +1,42 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Primitive System.Random simple bindings to Haskell functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module System.Random.Primitive where
+
+open import Agda.Builtin.IO using (IO)
+open import Agda.Builtin.Char using (Char)
+open import Agda.Builtin.Float using (Float)
+open import Agda.Builtin.Int using (Int)
+open import Agda.Builtin.Nat using (Nat)
+open import Agda.Builtin.Word using (Word64)
+open import Foreign.Haskell.Pair using (Pair)
+
+postulate
+  randomIO-Char : IO Char
+  randomRIO-Char : Pair Char Char → IO Char
+  randomIO-Int : IO Int
+  randomRIO-Int : Pair Int Int → IO Int
+  randomIO-Float : IO Float
+  randomRIO-Float : Pair Float Float → IO Float
+  randomIO-Nat : IO Nat
+  randomRIO-Nat : Pair Nat Nat → IO Nat
+  randomIO-Word64 : IO Word64
+  randomRIO-Word64 : Pair Word64 Word64 → IO Word64
+
+{-# FOREIGN GHC import System.Random #-}
+
+{-# COMPILE GHC randomIO-Char = randomIO #-}
+{-# COMPILE GHC randomRIO-Char = randomRIO #-}
+{-# COMPILE GHC randomIO-Int = randomIO #-}
+{-# COMPILE GHC randomRIO-Int = randomRIO #-}
+{-# COMPILE GHC randomIO-Float = randomIO #-}
+{-# COMPILE GHC randomRIO-Float = randomRIO #-}
+{-# COMPILE GHC randomIO-Nat = abs <$> randomIO #-}
+{-# COMPILE GHC randomRIO-Nat = randomRIO #-}
+{-# COMPILE GHC randomIO-Word64 = randomIO #-}
+{-# COMPILE GHC randomRIO-Word64 = randomRIO #-}
diff --git a/tests/runtests.agda b/tests/runtests.agda
index cf9af60245..a7840838b5 100644
--- a/tests/runtests.agda
+++ b/tests/runtests.agda
@@ -23,6 +23,7 @@ systemTests = mkTestPool "System modules"
   $ "ansi"
   ∷ "directory"
   ∷ "environment"
+  ∷ "random"
   ∷ []
 
 showTests : TestPool
diff --git a/tests/system/random/.gitignore b/tests/system/random/.gitignore
new file mode 100644
index 0000000000..3ef413e285
--- /dev/null
+++ b/tests/system/random/.gitignore
@@ -0,0 +1 @@
+random_values
diff --git a/tests/system/random/Main.agda b/tests/system/random/Main.agda
new file mode 100644
index 0000000000..379b14ab44
--- /dev/null
+++ b/tests/system/random/Main.agda
@@ -0,0 +1,53 @@
+{-# OPTIONS --guardedness #-}
+
+module Main where
+
+open import Level using (0ℓ)
+open import Data.Unit.Polymorphic.Base using (⊤)
+import Data.Char.Base as Char
+import Data.Fin.Base as Fin
+import Data.Fin.Show as Fin
+import Data.Float.Base as Float
+open import Data.Integer.Base using (-[1+_]; +_; -≤+)
+import Data.Integer.Show as ℤ
+open import Data.List.Base as List using ([]; _∷_)
+import Data.List.Show as List
+import Data.Nat.Base
+import Data.Nat.Properties as ℕ
+import Data.Nat.Show as ℕ
+open import Data.String.Base as String
+import Data.Vec.Show as Vec
+import Data.Word.Base as Word64
+open import IO.Base
+open import IO.Finite
+open import Function.Base using (_$_; _∘_)
+open import System.Random
+import Data.Vec.Bounded.Show as Vec≤
+
+open import Relation.Binary.Construct.Closure.Reflexive
+
+asA : String → String → IO {0ℓ} ⊤
+asA ty str = putStrLn (ty ++ ": " ++ str)
+
+main : Main
+main = run $ do
+  asA "Char" ∘ Char.show =<< Char.randomIO
+  asA "Char" ∘ Char.show ∘ InBounds.value =<< Char.randomRIO ' ' '~' [ ℕ.≤ᵇ⇒≤ _ _ _ ]
+  asA "Float" ∘ Float.show =<< Float.randomIO
+  asA "Float" ∘ Float.show ∘ InBounds.value =<< Float.randomRIO 0.0 1.0 _
+  asA "Integer" ∘ ℤ.show =<< ℤ.randomIO
+  asA "Integer" ∘ ℤ.show ∘ InBounds.value =<< ℤ.randomRIO -[1+ 2 ] (+ 5) -≤+
+  asA "Nat" ∘ ℕ.show =<< ℕ.randomIO
+  asA "Nat" ∘ ℕ.show ∘ InBounds.value =<< ℕ.randomRIO 1 10 (ℕ.≤ᵇ⇒≤ _ _ _)
+  asA "Word" ∘ ℕ.show ∘ Word64.toℕ =<< Word64.randomIO
+  asA "Word" ∘ ℕ.show ∘ Word64.toℕ ∘ InBounds.value =<<
+    Word64.randomRIO (Word64.fromℕ 10) (Word64.fromℕ 20) (ℕ.≤ᵇ⇒≤ _ _ _)
+  asA "Fin 10" ∘ Fin.show =<< Fin.randomIO {n = 10}
+  asA "Fin 10" ∘ Fin.show ∘ InBounds.value =<<
+    Fin.randomRIO {n = 10}
+      (Fin.fromℕ< {m = 3} (ℕ.≤ᵇ⇒≤ _ _ _))
+      (Fin.fromℕ< {m = 8} (ℕ.≤ᵇ⇒≤ _ _ _))
+      (ℕ.≤ᵇ⇒≤ _ _ _)
+  asA "Vec≤ Integer 10" ∘ Vec≤.show (ℤ.show ∘ InBounds.value) =<< Vec≤.randomIO (ℤ.randomRIO -[1+ 10 ] (+ 11) -≤+) 10
+  asA "Vec Float 5" ∘ Vec.show (Float.show ∘ InBounds.value) =<< Vec.randomIO (Float.randomRIO -20.0 20.0 _) 5
+  asA "String≤ 20" ∘ String.show =<< RangedString≤.randomIO ' ' '~' [ ℕ.≤ᵇ⇒≤ _ _ _ ] 20
diff --git a/tests/system/random/expected b/tests/system/random/expected
new file mode 100644
index 0000000000..e69de29bb2
diff --git a/tests/system/random/random.agda-lib b/tests/system/random/random.agda-lib
new file mode 100644
index 0000000000..367219e74b
--- /dev/null
+++ b/tests/system/random/random.agda-lib
@@ -0,0 +1,2 @@
+name: ansi
+include: ../../../src/ .
diff --git a/tests/system/random/run b/tests/system/random/run
new file mode 100644
index 0000000000..79772d89db
--- /dev/null
+++ b/tests/system/random/run
@@ -0,0 +1,6 @@
+$1 --compile-dir=../../_build -c Main.agda > log
+./../../_build/Main > random_values
+touch output
+
+rm ../../_build/Main
+rm ../../_build/MAlonzo/Code/Main*
\ No newline at end of file

From d6ea9bc224642781a90ff010cc799a11baf42468 Mon Sep 17 00:00:00 2001
From: Uncle Betty <3210857+uncle-betty@users.noreply.github.com>
Date: Sat, 20 Apr 2024 05:41:12 +0200
Subject: [PATCH 270/316] Another tweak to cong! (#2340)

* Disallow meta variables in goals - they break anti-unification.

* Postpone checking for metas.

* CHANGELOG entry, unit tests, code tweak.

* Move blockOnMetas to a new Reflection.TCM.Utilities module.
---
 CHANGELOG.md                      |  6 ++++--
 doc/README/Tactic/Cong.agda       | 33 +++++++++++++++++++++++++++++++
 src/Reflection/TCM/Utilities.agda | 33 +++++++++++++++++++++++++++++++
 src/Tactic/Cong.agda              | 33 ++++++++++++++++++++++++-------
 4 files changed, 96 insertions(+), 9 deletions(-)
 create mode 100644 src/Reflection/TCM/Utilities.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7cb74659bc..81fc7f7f0b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -458,5 +458,7 @@ Additions to existing modules
   WeaklyDecidable : Pred A ℓ → Set _
   ```
 
-* `Tactic.Cong` now provides a marker function, `⌞_⌟`, for user-guided
-  anti-unification. See README.Tactic.Cong for details.
+* Enhancements to `Tactic.Cong` - see `README.Tactic.Cong` for details.
+  - Provide a marker function, `⌞_⌟`, for user-guided anti-unification.
+  - Improved support for equalities between terms with instance arguments,
+    such as terms that contain `_/_` or `_%_`.
diff --git a/doc/README/Tactic/Cong.agda b/doc/README/Tactic/Cong.agda
index f6adfe9f9f..01365aee96 100644
--- a/doc/README/Tactic/Cong.agda
+++ b/doc/README/Tactic/Cong.agda
@@ -3,6 +3,7 @@
 module README.Tactic.Cong where
 
 open import Data.Nat
+open import Data.Nat.DivMod
 open import Data.Nat.Properties
 
 open import Relation.Binary.PropositionalEquality
@@ -161,3 +162,35 @@ module EquationalReasoningTests where
     ≤⟨ n≤1+n _ ⟩
       suc (suc (n + n))
     ∎
+
+module MetaTests where
+
+  test₁ : ∀ m n o → .⦃ _ : NonZero o ⦄ → (m + n) / o ≡ (n + m) / o
+  test₁ m n o =
+    let open ≤-Reasoning in
+    begin-equality
+      ⌞ m + n ⌟ / o
+     ≡⟨ cong! (+-comm m n) ⟩
+      (n + m) / o
+    ∎
+
+  test₂ : ∀ m n o p q r → .⦃ _ : NonZero o ⦄ → .⦃ _ : NonZero p ⦄ →
+          .⦃ _ : NonZero q ⦄ → p ≡ q ^ r → (m + n) % o % p ≡ (n + m) % o % p
+  test₂ m n o p q r eq =
+    let
+      open ≤-Reasoning
+      instance q^r≢0 = m^n≢0 q r
+    in
+    begin-equality
+      (m + n) % o % p
+     ≡⟨ %-congʳ eq ⟩
+      ⌞ m + n ⌟ % o % q ^ r
+     ≡⟨ cong! (+-comm m n) ⟩
+      ⌞ n + m ⌟ % o % q ^ r
+     ≡⟨ cong! (+-comm m n) ⟨
+      ⌞ m + n ⌟ % o % q ^ r
+     ≡⟨ cong! (+-comm m n) ⟩
+      (n + m) % o % q ^ r
+     ≡⟨ %-congʳ eq ⟨
+      (n + m) % o % p
+    ∎
diff --git a/src/Reflection/TCM/Utilities.agda b/src/Reflection/TCM/Utilities.agda
new file mode 100644
index 0000000000..d61db6350e
--- /dev/null
+++ b/src/Reflection/TCM/Utilities.agda
@@ -0,0 +1,33 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Reflection utilities
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Reflection.TCM.Utilities where
+
+open import Data.List using (List; []; _∷_; _++_; map)
+open import Data.Unit using (⊤; tt)
+open import Effect.Applicative using (RawApplicative; mkRawApplicative)
+open import Function using (case_of_)
+open import Reflection.AST.Meta using (Meta)
+open import Reflection.AST.Term using (Term)
+open import Reflection.TCM using (TC; pure; blockTC; blockerAll; blockerMeta)
+
+import Reflection.AST.Traversal as Traversal
+
+blockOnMetas : Term → TC ⊤
+blockOnMetas t =
+  case traverseTerm actions (0 , []) t of λ where
+    []         → pure tt
+    xs@(_ ∷ _) → blockTC (blockerAll (map blockerMeta xs))
+  where
+  applicative : ∀ {ℓ} → RawApplicative {ℓ} (λ _ → List Meta)
+  applicative = mkRawApplicative (λ _ → List Meta) (λ _ → []) _++_
+
+  open Traversal applicative
+
+  actions : Actions
+  actions = record defaultActions { onMeta = λ _ x → x ∷ [] }
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 0259191800..80cc7ec6d4 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -37,8 +37,10 @@ open import Data.Nat.Base    as ℕ     using (ℕ; zero; suc; _≡ᵇ_; _+_)
 open import Data.Unit.Base            using (⊤)
 open import Data.Word.Base   as Word  using (toℕ)
 open import Data.Product.Base         using (_×_; map₁; _,_)
+open import Function                  using (flip; case_of_)
 
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
+open import Relation.Nullary.Decidable.Core            using (yes; no)
 
 -- 'Data.String.Properties' defines this via 'Dec', so let's use the
 -- builtin for maximum speed.
@@ -54,8 +56,10 @@ open import Reflection.AST.Literal              as Literal
 open import Reflection.AST.Meta                 as Meta
 open import Reflection.AST.Name                 as Name
 open import Reflection.AST.Term                 as Term
+import Reflection.AST.Traversal                 as Traversal
 
 open import Reflection.TCM.Syntax
+open import Reflection.TCM.Utilities
 
 -- Marker to keep anti-unification from descending into the wrapped
 -- subterm.
@@ -102,6 +106,16 @@ private
   notEqualityError : ∀ {A : Set} Term → TC A
   notEqualityError goal = typeError (strErr "Cannot rewrite a goal that is not equality: " ∷ termErr goal ∷ [])
 
+  unificationError : ∀ {A : Set} → TC Term → TC Term → TC A
+  unificationError term symTerm = do
+    term' ← term
+    symTerm' ← symTerm
+    -- Don't show the same term twice.
+    let symErr = case term' Term.≟ symTerm' of λ where
+      (yes _) → []
+      (no _) → strErr "\n" ∷ termErr symTerm' ∷ []
+    typeError (strErr "cong! failed, tried:\n" ∷ termErr term' ∷ symErr)
+
   record EqualityGoal : Set where
     constructor equals
     field
@@ -236,12 +250,17 @@ macro
     withNormalisation false $ do
       goal ← inferType hole
       eqGoal ← destructEqualityGoal goal
-      let uni = λ lhs rhs → do
-        let cong-lam = antiUnify 0 lhs rhs
-        cong-tm ← `cong eqGoal cong-lam x≡y
-        unify cong-tm hole
+      let makeTerm = λ lhs rhs → `cong eqGoal (antiUnify 0 lhs rhs) x≡y
       let lhs = EqualityGoal.lhs eqGoal
       let rhs = EqualityGoal.rhs eqGoal
-      -- When using ⌞_⌟ with ≡⟨ ... ⟨, (uni lhs rhs) fails and
-      -- (uni rhs lhs) succeeds.
-      catchTC (uni lhs rhs) (uni rhs lhs)
+      let term = makeTerm lhs rhs
+      let symTerm = makeTerm rhs lhs
+      let uni = _>>= flip unify hole
+      -- When using ⌞_⌟ with ≡⟨ ... ⟨, (uni term) fails and
+      -- (uni symTerm) succeeds.
+      catchTC (uni term) $
+        catchTC (uni symTerm) $ do
+          -- If we failed because of unresolved metas, restart.
+          blockOnMetas goal
+          -- If we failed for a different reason, show an error.
+          unificationError term symTerm

From 9148fa597f7a871df856356478c91f7684b489bf Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 20 Apr 2024 05:19:23 +0100
Subject: [PATCH 271/316] Improve `Data.List.Base.unfold` (#2359; deprecate use
 of `with` #2123) (#2364)

* `with`-free definition of `unfold`

* fixed previous commit
---
 src/Data/List/Base.agda | 4 +---
 1 file changed, 1 insertion(+), 3 deletions(-)

diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 03e5d4ae9a..bc31ca0f9e 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -250,9 +250,7 @@ unfold : ∀ (P : ℕ → Set b)
          (f : ∀ {n} → P (suc n) → Maybe (A × P n)) →
          ∀ {n} → P n → List A
 unfold P f {n = zero}  s = []
-unfold P f {n = suc n} s with f s
-... | nothing       = []
-... | just (x , s′) = x ∷ unfold P f s′
+unfold P f {n = suc n} s = maybe′ (λ (x , s′) → x ∷ unfold P f s′) [] (f s)
 
 ------------------------------------------------------------------------
 -- Operations for reversing lists

From e54ffecf890b13a446548f0fb1e501a0278b4b30 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sat, 20 Apr 2024 08:48:51 +0100
Subject: [PATCH 272/316] fix #2253 (#2357)

---
 CHANGELOG.md                       |  6 ++++++
 src/Algebra/Structures/Biased.agda | 12 ++++++++++++
 2 files changed, 18 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 81fc7f7f0b..804bcbd8c8 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -49,6 +49,12 @@ Deprecated names
   _-_  ↦  _//_
   ```
 
+* In `Algebra.Structures.Biased`:
+  ```agda
+  IsRing*  ↦  Algebra.Structures.IsRing
+  isRing*  ↦  Algebra.Structures.isRing
+  ```
+
 * In `Data.Nat.Divisibility.Core`:
   ```agda
   *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
diff --git a/src/Algebra/Structures/Biased.agda b/src/Algebra/Structures/Biased.agda
index 8218a1ee1b..59e47da553 100644
--- a/src/Algebra/Structures/Biased.agda
+++ b/src/Algebra/Structures/Biased.agda
@@ -262,5 +262,17 @@ Please use the standard `IsRing` instead."
 #-}
 {-# WARNING_ON_USAGE isRingWithoutAnnihilatingZero
 "Warning: isRingWithoutAnnihilatingZero was deprecated in v2.0.
+Please use the standard `isRing` instead."
+#-}
+
+-- Version 2.1
+
+-- issue #2253
+{-# WARNING_ON_USAGE IsRing*
+"Warning: IsRing* was deprecated in v2.1.
 Please use the standard `IsRing` instead."
 #-}
+{-# WARNING_ON_USAGE isRing*
+"Warning: isRing* was deprecated in v2.1.
+Please use the standard `isRing` instead."
+#-}

From 05c938c41cdb46c3ff5528434e8c6d4e1e3c135d Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Sat, 20 Apr 2024 09:35:30 -0400
Subject: [PATCH 273/316] Remove uses of `Data.Nat.Solver` from a number of
 places (#2337)

* tighten imports in some files, and make imports explicit in others. Driven by staring at a partial dependency graph.

* more tightening of imports

* and even more tightening of imports

* some explicit  for precision

* new Nat Lemmas to use instead of Solve in Data.Integer.Properties

* use direct proofs instead of solver for all Nat proofs in these files

* two more uses of Data.Nat.Solver for rather simple proofs, now made direct.

* fix whitespace violations

* remove some unneeded parens

* minor fixups based on review
---
 src/Data/Digit.agda                           |  50 ++--
 src/Data/Fin/Properties.agda                  |  19 +-
 src/Data/Integer/Properties.agda              |  52 +---
 src/Data/Integer/Properties/NatLemmas.agda    |  69 ++++++
 src/Data/Nat/GCD/Lemmas.agda                  | 226 ++++++++++++------
 .../Rational/Unnormalised/Properties.agda     |  11 +-
 6 files changed, 286 insertions(+), 141 deletions(-)
 create mode 100644 src/Data/Integer/Properties/NatLemmas.agda

diff --git a/src/Data/Digit.agda b/src/Data/Digit.agda
index 49370776da..1fde8d22d4 100644
--- a/src/Data/Digit.agda
+++ b/src/Data/Digit.agda
@@ -9,20 +9,24 @@
 module Data.Digit where
 
 open import Data.Nat.Base
-open import Data.Nat.Properties using (_≤?_; _<?_; ≤⇒≤′; module ≤-Reasoning; m≤m+n)
-open import Data.Nat.Solver using (module +-*-Solver)
+  using (ℕ; zero; suc; _<_; _/_; _%_; sz<ss; _+_; _*_; 2+; _≤′_)
+open import Data.Nat.Properties
+  using (_≤?_; _<?_; ≤⇒≤′; module ≤-Reasoning; m≤m+n; +-comm; +-assoc;
+    *-distribˡ-+; *-identityʳ)
 open import Data.Fin.Base as Fin using (Fin; zero; suc; toℕ)
 open import Data.Bool.Base using (Bool; true; false)
 open import Data.Char.Base using (Char)
-open import Data.List.Base
+open import Data.List.Base using (List; replicate; [_]; _∷_; [])
 open import Data.Product.Base using (∃; _,_)
 open import Data.Vec.Base as Vec using (Vec; _∷_; [])
-open import Data.Nat.DivMod
+open import Data.Nat.DivMod using (m/n<m; _divMod_; result)
 open import Data.Nat.Induction
+  using (Acc; acc; <-wellFounded-fast; <′-Rec; <′-rec)
+open import Function.Base using (_$_)
 open import Relation.Nullary.Decidable using (True; does; toWitness)
 open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; sym; cong)
-open import Function.Base using (_$_)
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; refl; sym; cong; cong₂; module ≡-Reasoning)
 
 ------------------------------------------------------------------------
 -- Digits
@@ -87,20 +91,32 @@ toDigits base@(suc (suc k)) n = <′-rec Pred helper n
   cons : ∀ {m} (r : Digit base) → Pred m → Pred (toℕ r + m * base)
   cons r (ds , eq) = (r ∷ ds , cong (λ i → toℕ r + i * base) eq)
 
-  open ≤-Reasoning
-  open +-*-Solver
+  lem′ : ∀ x k → x + x + (k + x * k) ≡ k + x * 2+ k
+  lem′ x k = begin
+    x + x + (k + x * k)   ≡⟨ +-assoc (x + x) k _ ⟨
+    x + x + k + x * k     ≡⟨ cong (_+ x * k) (+-comm _ k) ⟩
+    k + (x + x) + x * k   ≡⟨ +-assoc k (x + x) _ ⟩
+    k + ((x + x) + x * k) ≡⟨ cong (k +_) (begin
+      (x + x) + x * k       ≡⟨ +-assoc x x _ ⟩
+      x + (x + x * k)       ≡⟨ cong (x +_) (cong (_+ x * k) (*-identityʳ x)) ⟨
+      x + (x * 1 + x * k)   ≡⟨ cong₂ _+_ (*-identityʳ x) (*-distribˡ-+ x 1 k ) ⟨
+      x * 1 + (x * suc k)   ≡⟨ *-distribˡ-+ x 1 (1 + k) ⟨
+      x * 2+ k              ∎) ⟩
+    k + x * 2+ k          ∎
+    where open ≡-Reasoning
 
   lem : ∀ x k r → 2 + x ≤′ r + (1 + x) * (2 + k)
   lem x k r = ≤⇒≤′ $ begin
-    2 + x
-      ≤⟨ m≤m+n _ _ ⟩
-    2 + x + (x + (1 + x) * k + r)
-      ≡⟨ solve 3 (λ x r k → con 2 :+ x :+ (x :+ (con 1 :+ x) :* k :+ r)
-                              :=
-                            r :+ (con 1 :+ x) :* (con 2 :+ k))
-                 refl x r k ⟩
-    r + (1 + x) * (2 + k)
-      ∎
+    2 + x                             ≤⟨ m≤m+n _ _ ⟩
+    2 + x + (x + (1 + x) * k + r)     ≡⟨ cong ((2 + x) +_) (+-comm _ r) ⟩
+    2 + x + (r + (x + (1 + x) * k))   ≡⟨ +-assoc (2 + x) r _ ⟨
+    2 + x + r + (x + (1 + x) * k)     ≡⟨ cong (_+ (x + (1 + x) * k)) (+-comm (2 + x) r) ⟩
+    r + (2 + x) + (x + (1 + x) * k)   ≡⟨ +-assoc r (2 + x) _ ⟩
+    r + ((2 + x) + (x + (1 + x) * k)) ≡⟨ cong (r +_) (cong 2+ (+-assoc x x _)) ⟨
+    r + (2 + (x + x + (1 + x) * k))   ≡⟨ cong (λ z → r + (2+ z)) (lem′ x k) ⟩
+    r + (2 + (k + x * (2 + k)))       ≡⟨⟩
+    r + (1 + x) * (2 + k)             ∎
+    where open ≤-Reasoning
 
   helper : ∀ n → <′-Rec Pred n → Pred n
   helper n                       rec with n divMod base
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index e58797ecd1..6243e016eb 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -6,7 +6,7 @@
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
-{-# OPTIONS --warn=noUserWarning #-} -- for deprecated _≺_ (issue #1726)
+{-# OPTIONS --warn=noUserWarning #-} -- for deprecated _≺_ and _≻toℕ_ (issue #1726)
 
 module Data.Fin.Properties where
 
@@ -21,7 +21,6 @@ open import Data.Fin.Patterns
 open import Data.Nat.Base as ℕ
   using (ℕ; zero; suc; s≤s; z≤n; z<s; s<s; s<s⁻¹; _∸_; _^_)
 import Data.Nat.Properties as ℕ
-open import Data.Nat.Solver
 open import Data.Unit.Base using (⊤; tt)
 open import Data.Product.Base as Product
   using (∃; ∃₂; _×_; _,_; map; proj₁; proj₂; uncurry; <_,_>)
@@ -669,12 +668,14 @@ toℕ-combine {suc m} {n} i@0F j = begin
   n ℕ.* toℕ i ℕ.+ toℕ j      ∎
   where open ≡-Reasoning
 toℕ-combine {suc m} {n} (suc i) j = begin
-  toℕ (combine (suc i) j)        ≡⟨⟩
-  toℕ (n ↑ʳ combine i j)         ≡⟨ toℕ-↑ʳ n (combine i j) ⟩
-  n ℕ.+ toℕ (combine i j)        ≡⟨ cong (n ℕ.+_) (toℕ-combine i j) ⟩
-  n ℕ.+ (n ℕ.* toℕ i ℕ.+ toℕ j)  ≡⟨ solve 3 (λ n i j → n :+ (n :* i :+ j) := n :* (con 1 :+ i) :+ j) refl n (toℕ i) (toℕ j) ⟩
-  n ℕ.* toℕ (suc i) ℕ.+ toℕ j    ∎
-  where open ≡-Reasoning; open +-*-Solver
+  toℕ (combine (suc i) j)            ≡⟨⟩
+  toℕ (n ↑ʳ combine i j)             ≡⟨ toℕ-↑ʳ n (combine i j) ⟩
+  n ℕ.+ toℕ (combine i j)            ≡⟨ cong (n ℕ.+_) (toℕ-combine i j) ⟩
+  n ℕ.+ (n ℕ.* toℕ i ℕ.+ toℕ j)     ≡⟨ ℕ.+-assoc n _ (toℕ j) ⟨
+  n ℕ.+ n ℕ.* toℕ i ℕ.+ toℕ j       ≡⟨ cong (λ z → z ℕ.+ n ℕ.* toℕ i ℕ.+ toℕ j) (ℕ.*-identityʳ n) ⟨
+  n ℕ.* 1 ℕ.+ n ℕ.* toℕ i ℕ.+ toℕ j ≡⟨ cong (ℕ._+ toℕ j) (ℕ.*-distribˡ-+ n 1 (toℕ i) ) ⟨
+  n ℕ.* toℕ (suc i) ℕ.+ toℕ j       ∎
+  where open ≡-Reasoning
 
 combine-monoˡ-< : ∀ {i j : Fin m} (k l : Fin n) →
                   i < j → combine i k < combine j l
@@ -688,7 +689,7 @@ combine-monoˡ-< {m} {n} {i} {j} k l i<j = begin-strict
   n ℕ.* toℕ j            ≤⟨ ℕ.m≤m+n (n ℕ.* toℕ j) (toℕ l) ⟩
   n ℕ.* toℕ j ℕ.+ toℕ l  ≡⟨ toℕ-combine j l ⟨
   toℕ (combine j l)      ∎
-  where open ℕ.≤-Reasoning; open +-*-Solver
+  where open ℕ.≤-Reasoning
 
 combine-injectiveˡ : ∀ (i : Fin m) (j : Fin n) (k : Fin m) (l : Fin n) →
                      combine i j ≡ combine k l → i ≡ k
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 842058f62e..61477c9acc 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -16,11 +16,11 @@ import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
 import Algebra.Properties.AbelianGroup
 open import Data.Bool.Base using (T; true; false)
 open import Data.Integer.Base renaming (suc to sucℤ)
+open import Data.Integer.Properties.NatLemmas
 open import Data.Nat.Base as ℕ
   using (ℕ; suc; zero; _∸_; s≤s; z≤n; s<s; z<s; s≤s⁻¹; s<s⁻¹)
   hiding (module ℕ)
 import Data.Nat.Properties as ℕ
-open import Data.Nat.Solver
 open import Data.Product.Base using (proj₁; proj₂; _,_; _×_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Data.Sign.Base as Sign using (Sign)
@@ -45,7 +45,6 @@ open import Algebra.Consequences.Propositional
 open import Algebra.Structures {A = ℤ} _≡_
 module ℤtoℕ = Morphism.Definitions ℤ ℕ _≡_
 module ℕtoℤ = Morphism.Definitions ℕ ℤ _≡_
-open +-*-Solver
 
 private
   variable
@@ -1337,15 +1336,6 @@ pred-mono (+≤+ m≤n)         = ⊖-monoˡ-≤ 1 m≤n
 *-zero : Zero 0ℤ _*_
 *-zero = *-zeroˡ , *-zeroʳ
 
-private
-  lemma : ∀ m n o → o ℕ.+ (n ℕ.+ m ℕ.* suc n) ℕ.* suc o
-                  ≡ o ℕ.+ n ℕ.* suc o ℕ.+ m ℕ.* suc (o ℕ.+ n ℕ.* suc o)
-  lemma =
-    solve 3 (λ m n o → o :+ (n :+ m :* (con 1 :+ n)) :* (con 1 :+ o)
-                    := o :+ n :* (con 1 :+ o) :+
-                       m :* (con 1 :+ (o :+ n :* (con 1 :+ o))))
-            refl
-
 *-assoc : Associative _*_
 *-assoc +0 _ _ = refl
 *-assoc i +0 _ rewrite ℕ.*-zeroʳ ∣ i ∣ = refl
@@ -1354,14 +1344,14 @@ private
   | ℕ.*-zeroʳ ∣ i ∣
   | ℕ.*-zeroʳ ∣ sign i Sign.* sign j ◃ ∣ i ∣ ℕ.* ∣ j ∣ ∣
   = refl
-*-assoc -[1+ m ] -[1+ n ] +[1+ o ] = cong (+_ ∘ suc) (lemma m n o)
-*-assoc -[1+ m ] +[1+ n ] -[1+ o ] = cong (+_ ∘ suc) (lemma m n o)
-*-assoc +[1+ m ] +[1+ n ] +[1+ o ] = cong (+_ ∘ suc) (lemma m n o)
-*-assoc +[1+ m ] -[1+ n ] -[1+ o ] = cong (+_ ∘ suc) (lemma m n o)
-*-assoc -[1+ m ] -[1+ n ] -[1+ o ] = cong -[1+_] (lemma m n o)
-*-assoc -[1+ m ] +[1+ n ] +[1+ o ] = cong -[1+_] (lemma m n o)
-*-assoc +[1+ m ] -[1+ n ] +[1+ o ] = cong -[1+_] (lemma m n o)
-*-assoc +[1+ m ] +[1+ n ] -[1+ o ] = cong -[1+_] (lemma m n o)
+*-assoc -[1+ m ] -[1+ n ] +[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)
+*-assoc -[1+ m ] +[1+ n ] -[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)
+*-assoc +[1+ m ] +[1+ n ] +[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)
+*-assoc +[1+ m ] -[1+ n ] -[1+ o ] = cong (+_ ∘ suc) (inner-assoc m n o)
+*-assoc -[1+ m ] -[1+ n ] -[1+ o ] = cong -[1+_] (inner-assoc m n o)
+*-assoc -[1+ m ] +[1+ n ] +[1+ o ] = cong -[1+_] (inner-assoc m n o)
+*-assoc +[1+ m ] -[1+ n ] +[1+ o ] = cong -[1+_] (inner-assoc m n o)
+*-assoc +[1+ m ] +[1+ n ] -[1+ o ] = cong -[1+_] (inner-assoc m n o)
 
 private
 
@@ -1400,26 +1390,10 @@ private
   rewrite +-identityʳ y
         | +-identityʳ (sign y Sign.* sign x ◃ ∣ y ∣ ℕ.* ∣ x ∣)
         = refl
-*-distribʳ-+ -[1+ m ] -[1+ n ] -[1+ o ] = cong (+_) $
-  solve 3 (λ m n o → (con 2 :+ n :+ o) :* (con 1 :+ m)
-                  := (con 1 :+ n) :* (con 1 :+ m) :+
-                     (con 1 :+ o) :* (con 1 :+ m))
-          refl m n o
-*-distribʳ-+ +[1+ m ] +[1+ n ] +[1+ o ] = cong (+_) $
-  solve 3 (λ m n o → (con 1 :+ n :+ (con 1 :+ o)) :* (con 1 :+ m)
-                  := (con 1 :+ n) :* (con 1 :+ m) :+
-                     (con 1 :+ o) :* (con 1 :+ m))
-        refl m n o
-*-distribʳ-+ -[1+ m ] +[1+ n ] +[1+ o ] = cong -[1+_] $
-  solve 3 (λ m n o → m :+ (n :+ (con 1 :+ o)) :* (con 1 :+ m)
-                   := (con 1 :+ n) :* (con 1 :+ m) :+
-                      (m :+ o :* (con 1 :+ m)))
-         refl m n o
-*-distribʳ-+ +[1+ m ] -[1+ n ] -[1+ o ] = cong -[1+_] $
-  solve 3 (λ m n o → m :+ (con 1 :+ m :+ (n :+ o) :* (con 1 :+ m))
-                  := (con 1 :+ n) :* (con 1 :+ m) :+
-                     (m :+ o :* (con 1 :+ m)))
-         refl m n o
+*-distribʳ-+ -[1+ m ] -[1+ n ] -[1+ o ] = cong (+_) $ assoc₁ m n o
+*-distribʳ-+ +[1+ m ] +[1+ n ] +[1+ o ] = cong +[1+_] $ ℕ.suc-injective (assoc₂ m n o)
+*-distribʳ-+ -[1+ m ] +[1+ n ] +[1+ o ] = cong -[1+_] $ assoc₃ m n o
+*-distribʳ-+ +[1+ m ] -[1+ n ] -[1+ o ] = cong -[1+_] $ assoc₄ m n o
 *-distribʳ-+ -[1+ m ] -[1+ n ] +[1+ o ] = begin
   (suc o ⊖ suc n) * -[1+ m ]                ≡⟨ cong (_* -[1+ m ]) ([1+m]⊖[1+n]≡m⊖n o n) ⟩
   (o ⊖ n) * -[1+ m ]                        ≡⟨ distrib-lemma m n o ⟩
diff --git a/src/Data/Integer/Properties/NatLemmas.agda b/src/Data/Integer/Properties/NatLemmas.agda
new file mode 100644
index 0000000000..5aa44bb889
--- /dev/null
+++ b/src/Data/Integer/Properties/NatLemmas.agda
@@ -0,0 +1,69 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Some extra lemmas about natural numbers only needed for
+-- Data.Integer.Properties (for distributivity)
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Integer.Properties.NatLemmas where
+
+open import Data.Nat.Base using (ℕ; _+_; _*_; suc)
+open import Data.Nat.Properties
+  using (*-distribʳ-+; *-assoc; +-assoc; +-comm; +-suc)
+open import Function.Base using (_∘_)
+open import Relation.Binary.PropositionalEquality
+  using (_≡_; cong; cong₂; sym; module ≡-Reasoning)
+
+open ≡-Reasoning
+
+inner-assoc : ∀ m n o → o + (n + m * suc n) * suc o
+                ≡ o + n * suc o + m * suc (o + n * suc o)
+inner-assoc m n o = begin
+  o + (n + m * suc n) * suc o                ≡⟨ cong (o +_) (begin
+    (n + m * suc n) * suc o             ≡⟨ *-distribʳ-+ (suc o) n (m * suc n) ⟩
+    n * suc o + m * suc n * suc o       ≡⟨ cong (n * suc o +_) (*-assoc m (suc n) (suc o)) ⟩
+    n * suc o + m * suc (o + n * suc o) ∎) ⟩
+  o + (n * suc o + m * suc (o + n * suc o))  ≡⟨ +-assoc o _ _ ⟨
+  o + n * suc o + m * suc (o + n * suc o)    ∎
+
+private
+  assoc-comm : ∀ a b c d → a + b + c + d ≡ (a + c) + (b + d)
+  assoc-comm a b c d = begin
+    a + b + c + d     ≡⟨ cong (_+ d) (+-assoc a b c) ⟩
+    a + (b + c) + d   ≡⟨ cong (λ z → a + z + d) (+-comm b c) ⟩
+    a + (c + b) + d   ≡⟨ cong (_+ d) (+-assoc a c b) ⟨
+    (a + c) + b + d   ≡⟨ +-assoc (a + c) b d ⟩
+    (a + c) + (b + d) ∎
+
+  assoc-comm′ : ∀ a b c d → a + (b + (c + d)) ≡ a + c + (b + d)
+  assoc-comm′ a b c d = begin
+    a + (b + (c + d)) ≡⟨ cong (a +_) (+-assoc b c d) ⟨
+    a + (b + c + d)   ≡⟨ cong (λ z → a + (z + d)) (+-comm b c) ⟩
+    a + (c + b + d)   ≡⟨ cong (a +_) (+-assoc c b d) ⟩
+    a + (c + (b + d)) ≡⟨ +-assoc a c _ ⟨
+    a + c + (b + d)   ∎
+
+assoc₁ : ∀ m n o → (2 + n + o) * (1 + m) ≡ (1 + n) * (1 + m) + (1 + o) * (1 + m)
+assoc₁ m n o = begin
+  (2 + n + o) * (1 + m)                  ≡⟨ cong (_* (1 + m)) (assoc-comm 1 1 n o) ⟩
+  ((1 + n) + (1 + o)) * (1 + m)          ≡⟨ *-distribʳ-+ (1 + m) (1 + n) (1 + o) ⟩
+  (1 + n) * (1 + m) + (1 + o) * (1 + m)   ∎
+
+assoc₂ : ∀ m n o → (1 + n + (1 + o)) * (1 + m) ≡ (1 + n) * (1 + m) + (1 + o) * (1 + m)
+assoc₂ m n o = *-distribʳ-+ (1 + m) (1 + n) (1 + o)
+
+assoc₃ : ∀ m n o → m + (n + (1 + o)) * (1 + m) ≡ (1 + n) * (1 + m) + (m + o * (1 + m))
+assoc₃ m n o = begin
+  m + (n + (1 + o)) * (1 + m)           ≡⟨ cong (m +_) (*-distribʳ-+ (1 + m) n (1 + o)) ⟩
+  m + (n * (1 + m) + (1 + o) * (1 + m)) ≡⟨ +-assoc m _ _ ⟨
+  (m + n * (1 + m)) + (1 + o) * (1 + m) ≡⟨ +-suc _ _ ⟩
+  (1 + n) * (1 + m) + (m + o * (1 + m)) ∎
+
+assoc₄ : ∀ m n o → m + (1 + m + (n + o) * (1 + m)) ≡ (1 + n) * (1 + m) + (m + o * (1 + m))
+assoc₄ m n o = begin
+  m + (1 + m + (n + o) * (1 + m))           ≡⟨ +-suc _ _ ⟩
+  1 + m + (m + (n + o) * (1 + m))           ≡⟨ cong (λ z → suc (m + (m + z))) (*-distribʳ-+ (suc m) n o) ⟩
+  1 + m + (m + (n * (1 + m) + o * (1 + m))) ≡⟨ cong suc (assoc-comm′ m m _ _) ⟩
+  (1 + n) * (1 + m) + (m + o * (1 + m))     ∎
diff --git a/src/Data/Nat/GCD/Lemmas.agda b/src/Data/Nat/GCD/Lemmas.agda
index 3f9a4b6d42..21788f0d88 100644
--- a/src/Data/Nat/GCD/Lemmas.agda
+++ b/src/Data/Nat/GCD/Lemmas.agda
@@ -10,21 +10,23 @@ module Data.Nat.GCD.Lemmas where
 
 open import Data.Nat.Base
 open import Data.Nat.Properties
-open import Data.Nat.Solver
 open import Function.Base using (_$_)
 open import Relation.Binary.PropositionalEquality
 
-open +-*-Solver
 open ≡-Reasoning
 
 private
-  distrib-comm : ∀ x k n → x * k + x * n ≡ x * (n + k)
-  distrib-comm =
-    solve 3 (λ x k n → x :* k :+ x :* n  :=  x :* (n :+ k)) refl
+  comm-factor : ∀ x k n → x * k + x * n ≡ x * (n + k)
+  comm-factor x k n = begin
+    x * k + x * n ≡⟨ +-comm (x * k) _ ⟩
+    x * n + x * k ≡⟨ *-distribˡ-+ x n k ⟨
+    x * (n + k)   ∎
 
   distrib-comm₂ : ∀ d x k n → d + x * (n + k) ≡ d + x * k + x * n
-  distrib-comm₂ =
-    solve 4 (λ d x k n → d :+ x :* (n :+ k) :=  d :+ x :* k :+ x :* n) refl
+  distrib-comm₂ d x k n = begin
+    d + x * (n + k)     ≡⟨ cong (d +_) (comm-factor x k n) ⟨
+    d + (x * k + x * n) ≡⟨ +-assoc d _ _ ⟨
+    d + x * k + x * n   ∎
 
 -- Other properties
 -- TODO: Can this proof be simplified? An automatic solver which can
@@ -40,26 +42,112 @@ lem₀ i j m n eq = begin
 lem₁ : ∀ i j → 2 + i ≤′ 2 + j + i
 lem₁ i j = ≤⇒≤′ $ s≤s $ s≤s $ m≤n+m i j
 
+private
+  times2 : ∀ x → x + x ≡ 2 * x
+  times2 x = cong (x +_) (sym (+-identityʳ x))
+
+  times2′ : ∀ x y → x * y + x * y ≡ 2 * x * y
+  times2′ x y = begin
+    x * y + x * y ≡⟨ times2 (x * y) ⟩
+    2 * (x * y)   ≡⟨ *-assoc 2 x y ⟨
+    2 * x * y     ∎
+
 lem₂ : ∀ d x {k n} →
        d + x * k ≡ x * n → d + x * (n + k) ≡ 2 * x * n
 lem₂ d x {k} {n} eq = begin
   d + x * (n + k)    ≡⟨ distrib-comm₂ d x k n ⟩
-  d + x * k + x * n  ≡⟨ cong₂ _+_ eq refl ⟩
-  x * n + x * n      ≡⟨ solve 3 (λ x n k → x :* n :+ x :* n
-                                       :=  con 2 :* x :* n)
-                                 refl x n k ⟩
+  d + x * k + x * n  ≡⟨ cong (_+ x * n) eq ⟩
+  x * n + x * n      ≡⟨ times2′ x n ⟩
   2 * x * n          ∎
 
+private
+  distrib₃ : ∀ a b c x → (a + b + c) * x ≡ a * x + b * x + c * x
+  distrib₃ a b c x = begin
+    (a + b + c) * x       ≡⟨ *-distribʳ-+ x (a + b) c ⟩
+    (a + b) * x + c * x   ≡⟨ cong (_+ c * x) (*-distribʳ-+ x a b) ⟩
+    a * x + b * x + c * x ∎
+
+  lem₃₁ : ∀ a b c → a + (b + c) ≡ b + a + c
+  lem₃₁ a b c = begin
+    a + (b + c) ≡⟨ +-assoc a b c ⟨
+    (a + b) + c ≡⟨ cong (_+ c) (+-comm a b) ⟩
+    b + a + c   ∎
+
+  +-assoc-comm : ∀ a b c d → a + (b + c + d) ≡ (a + c) + (b + d)
+  +-assoc-comm a b c d = begin
+    a + (b + c + d)   ≡⟨ cong (a +_) (cong (_+ d) (+-comm b c)) ⟩
+    a + (c + b + d)   ≡⟨ cong (a +_) (+-assoc c b d) ⟩
+    a + (c + (b + d)) ≡⟨ +-assoc a c _ ⟨
+    (a + c) + (b + d) ∎
+
+  *-on-right : ∀ a b c {d} → b * c ≡ d → a * b * c ≡ a * d
+  *-on-right a b c {d} eq = begin
+    a * b * c   ≡⟨ *-assoc a b c ⟩
+    a * (b * c) ≡⟨ cong (a *_) eq ⟩
+    a * d       ∎
+
+  *-on-left : ∀ a b c {d} → a * b ≡ d → a * (b * c) ≡ d * c
+  *-on-left a b c {d} eq = begin
+    a * (b * c) ≡⟨ *-assoc a b c ⟨
+    a * b * c   ≡⟨ cong (_* c) eq ⟩
+    d * c       ∎
+
+  +-on-right : ∀ a b c {d} → b + c ≡ d → a + b + c ≡ a + d
+  +-on-right a b c {d} eq = begin
+    a + b + c   ≡⟨ +-assoc a b c ⟩
+    a + (b + c) ≡⟨ cong (a +_) eq ⟩
+    a + d       ∎
+
+  +-on-left : ∀ a b c d → a + b ≡ d → a + (b + c) ≡ d + c
+  +-on-left a b c d eq = begin
+    a + (b + c) ≡⟨ +-assoc a b c ⟨
+    a + b + c   ≡⟨ cong (_+ c) eq ⟩
+    d + c       ∎
+
+  +-focus-mid : ∀ a b c d → a + b + c + d ≡ a + (b + c) + d
+  +-focus-mid a b c d = begin
+    a + b + c + d     ≡⟨ cong (_+ d) (+-assoc a b c) ⟩
+    a + (b + c) + d   ∎
+
+  +-assoc-comm′ : ∀ a b c d → a + b + c + d ≡ a + (b + d) + c
+  +-assoc-comm′ a b c d = begin
+    a + b + c + d     ≡⟨ +-on-left a (b + c) d (a + b + c) (sym $ +-assoc a b c) ⟨
+    a + (b + c + d)   ≡⟨ cong (a +_) (+-on-right b c d (+-comm c d)) ⟩
+    a + (b + (d + c)) ≡⟨ cong (a +_) (+-assoc b d c) ⟨
+    a + (b + d + c)   ≡⟨ +-assoc a _ c ⟨
+    a + (b + d) + c   ∎
+
+  lem₃₂ : ∀ a b c n → a * n + (b * n + a * n + c * n) ≡ (a + a + (b + c)) * n
+  lem₃₂ a b c n = begin
+    a * n + (b * n + a * n + c * n) ≡⟨ cong (a * n +_) (distrib₃ b a c n) ⟨
+    a * n + (b + a + c) * n         ≡⟨ *-distribʳ-+ n a _ ⟨
+    (a + (b + a + c)) * n           ≡⟨ cong (_* n) (+-assoc-comm a b a c) ⟩
+    (a + a + (b + c)) * n           ∎
+
+  mid-to-right : ∀ a b c → a + 2 * b + c ≡ (a + b + c) + b
+  mid-to-right a b c = begin
+    a + 2 * b + c   ≡⟨ cong (λ x → a + x + c) (times2 b) ⟨
+    a + (b + b) + c ≡⟨ +-assoc-comm′ a b c b ⟨
+    a + b + c + b   ∎
+
+  mid-to-left : ∀ a b c → a + 2 * b + c ≡ b + (a + b + c)
+  mid-to-left a b c = begin
+    a + 2 * b + c   ≡⟨ cong (λ x → a + x + c) (times2 b) ⟨
+    a + (b + b) + c ≡⟨ cong (_+ c) (+-on-left a b b _ (+-comm a b)) ⟩
+    b + a + b + c   ≡⟨ +-on-left b (a + b) c (b + a + b) (sym $ +-assoc b a b) ⟨
+    b + (a + b + c) ∎
+
 lem₃ : ∀ d x {i k n} →
        d + (1 + x + i) * k ≡ x * n →
        d + (1 + x + i) * (n + k) ≡ (1 + 2 * x + i) * n
 lem₃ d x {i} {k} {n} eq = begin
-  d + y * (n + k)      ≡⟨ distrib-comm₂ d y k n ⟩
-  d + y * k + y * n    ≡⟨ cong₂ _+_ eq refl ⟩
-  x * n + y * n        ≡⟨ solve 3 (λ x n i → x :* n :+ (con 1 :+ x :+ i) :* n
-                                         :=  (con 1 :+ con 2 :* x :+ i) :* n)
-                                  refl x n i ⟩
-  (1 + 2 * x + i) * n  ∎
+  d + y * (n + k)                     ≡⟨ distrib-comm₂ d y k n ⟩
+  d + y * k + y * n                   ≡⟨ cong (_+ y * n) eq ⟩
+  x * n + y * n                       ≡⟨ cong (x * n +_) (distrib₃ 1 x i n) ⟩
+  x * n + (1 * n + x * n + i * n)     ≡⟨ lem₃₂ x 1 i n  ⟩
+  (x + x + (1 + i)) * n               ≡⟨ cong (_* n) (cong (_+ (1 + i)) (times2 x)) ⟩
+  (2 * x + (1 + i)) * n               ≡⟨ cong (_* n) (lem₃₁ (2 * x) 1 i) ⟩
+  (1 + 2 * x + i) * n                 ∎
   where y = 1 + x + i
 
 lem₄ : ∀ d y {k i} n →
@@ -67,32 +155,29 @@ lem₄ : ∀ d y {k i} n →
        d + y * (n + k) ≡ (1 + 2 * y + i) * n
 lem₄ d y {k} {i} n eq = begin
   d + y * (n + k)          ≡⟨ distrib-comm₂ d y k n ⟩
-  d + y * k + y * n        ≡⟨ cong₂ _+_ eq refl ⟩
-  (1 + y + i) * n + y * n  ≡⟨ solve 3 (λ y i n → (con 1 :+ y :+ i) :* n :+ y :* n
-                                             :=  (con 1 :+ con 2 :* y :+ i) :* n)
-                                      refl y i n ⟩
+  d + y * k + y * n        ≡⟨ cong (_+ y * n) eq ⟩
+  (1 + y + i) * n + y * n  ≡⟨ *-distribʳ-+ n (1 + y + i) y ⟨
+  (1 + y + i + y) * n      ≡⟨ cong (_* n) (mid-to-right 1 y i) ⟨
   (1 + 2 * y + i) * n      ∎
 
 lem₅ : ∀ d x {n k} →
        d + x * n ≡ x * k →
        d + 2 * x * n ≡ x * (n + k)
 lem₅ d x {n} {k} eq = begin
-  d + 2 * x * n      ≡⟨ solve 3 (λ d x n → d :+ con 2 :* x :* n
-                                       :=  d :+ x :* n :+ x :* n)
-                                refl d x n ⟩
-  d + x * n + x * n  ≡⟨ cong₂ _+_ eq refl ⟩
-  x * k + x * n      ≡⟨ distrib-comm x k n ⟩
-  x * (n + k)        ∎
+  d + 2 * x * n       ≡⟨ cong (d +_) (times2′ x n) ⟨
+  d + (x * n + x * n) ≡⟨ +-assoc d (x * n) _ ⟨
+  d + x * n + x * n   ≡⟨ cong (_+ x * n) eq ⟩
+  x * k + x * n       ≡⟨ comm-factor x k n ⟩
+  x * (n + k)         ∎
 
 lem₆ : ∀ d x {n i k} →
        d + x * n ≡ (1 + x + i) * k →
        d + (1 + 2 * x + i) * n ≡ (1 + x + i) * (n + k)
 lem₆ d x {n} {i} {k} eq = begin
-  d + (1 + 2 * x + i) * n  ≡⟨ solve 4 (λ d x i n → d :+ (con 1 :+ con 2 :* x :+ i) :* n
-                                               :=  d :+ x :* n :+ (con 1 :+ x :+ i) :* n)
-                                      refl d x i n ⟩
-  d + x * n + y * n        ≡⟨ cong₂ _+_ eq refl ⟩
-  y * k + y * n            ≡⟨ distrib-comm y k n ⟩
+  d + (1 + 2 * x + i) * n  ≡⟨ cong (λ z → d + z * n) (mid-to-left 1 x i) ⟩
+  d + (x + y) * n          ≡⟨ cong (d +_) (*-distribʳ-+ n x y) ⟩
+  d + (x * n + y * n)      ≡⟨ +-on-left d _ _ _ eq ⟩
+  y * k + y * n            ≡⟨ comm-factor y k n ⟩
   y * (n + k)              ∎
   where y = 1 + x + i
 
@@ -100,12 +185,11 @@ lem₇ : ∀ d y {i} n {k} →
        d + (1 + y + i) * n ≡ y * k →
        d + (1 + 2 * y + i) * n ≡ y * (n + k)
 lem₇ d y {i} n {k} eq = begin
-  d + (1 + 2 * y + i) * n      ≡⟨ solve 4 (λ d y i n → d :+ (con 1 :+ con 2 :* y :+ i) :* n
-                                                   :=  d :+ (con 1 :+ y :+ i) :* n :+ y :* n)
-                                          refl d y i n ⟩
-  d + (1 + y + i) * n + y * n  ≡⟨ cong₂ _+_ eq refl ⟩
-  y * k + y * n                ≡⟨ distrib-comm y k n ⟩
-  y * (n + k)                  ∎
+  d + (1 + 2 * y + i) * n       ≡⟨ cong (λ z → d + z * n) (mid-to-right 1 y i) ⟩
+  d + (1 + y + i + y) * n       ≡⟨ cong (d +_) (*-distribʳ-+ n (1 + y + i) y) ⟩
+  d + ((1 + y + i) * n + y * n) ≡⟨ +-on-left d _ _ _ eq ⟩
+  y * k + y * n                 ≡⟨ comm-factor y k n ⟩
+  y * (n + k)                   ∎
 
 lem₈ : ∀ {i j k q} x y →
        1 + y * j ≡ x * i → j * k ≡ q * i →
@@ -114,17 +198,12 @@ lem₈ {i} {j} {k} {q} x y eq eq′ =
   sym (lem₀ (x * k) (y * q) i k lemma)
   where
   lemma = begin
-    x * k * i        ≡⟨ solve 3 (λ x k i → x :* k :* i
-                                       :=  x :* i :* k)
-                                refl x k i ⟩
-    x * i * k        ≡⟨ cong (_* k) (sym eq) ⟩
-    (1 + y * j) * k  ≡⟨ solve 3 (λ y j k → (con 1 :+ y :* j) :* k
-                                       :=  y :* (j :* k) :+ k)
-                                refl y j k ⟩
+    x * k * i        ≡⟨ *-on-right x k i (*-comm k i) ⟩
+    x * (i * k)      ≡⟨ *-on-left x i k (sym eq) ⟩
+    (1 + y * j) * k  ≡⟨ +-comm k _ ⟩
+    (y * j) * k + k  ≡⟨ cong (_+ k) (*-assoc y j k) ⟩
     y * (j * k) + k  ≡⟨ cong (λ n → y * n + k) eq′ ⟩
-    y * (q * i) + k  ≡⟨ solve 4 (λ y q i k → y :* (q :* i) :+ k
-                                         :=  y :*  q :* i  :+ k)
-                                refl y q i k ⟩
+    y * (q * i) + k  ≡⟨ cong (_+ k) (*-assoc y q i)  ⟨
     y *  q * i  + k  ∎
 
 lem₉ : ∀ {i j k q} x y →
@@ -133,15 +212,19 @@ lem₉ : ∀ {i j k q} x y →
 lem₉ {i} {j} {k} {q} x y eq eq′ =
   sym (lem₀ (y * q) (x * k) i k lemma)
   where
-  lem   = solve 3 (λ a b c → a :* b :* c  :=  b :* c :* a) refl
+  lem : ∀ a b c → a * b * c ≡ b * c * a
+  lem a b c = begin
+    a * b * c   ≡⟨ *-assoc a b c ⟩
+    a * (b * c) ≡⟨ *-comm a _ ⟩
+    b * c * a   ∎
   lemma = begin
     y * q * i        ≡⟨ lem y q i ⟩
-    q * i * y        ≡⟨ cong (λ n → n * y) (sym eq′) ⟩
-    j * k * y        ≡⟨ sym (lem y j k) ⟩
-    y * j * k        ≡⟨ cong (λ n → n * k) (sym eq) ⟩
-    (1 + x * i) * k  ≡⟨ solve 3 (λ x i k → (con 1 :+ x :* i) :* k
-                                       :=  x :* k :* i :+ k)
-                                refl x i k ⟩
+    q * i * y        ≡⟨ cong (λ n → n * y) eq′ ⟨
+    j * k * y        ≡⟨ lem y j k ⟨
+    y * j * k        ≡⟨ cong (λ n → n * k) eq ⟨
+    (1 + x * i) * k  ≡⟨ +-comm k _ ⟩
+    x * i * k + k    ≡⟨ cong (_+ k) (*-on-right x i k (*-comm i k)) ⟩
+    x * (k * i) + k  ≡⟨ cong (_+ k) (*-assoc x k i) ⟨
     x * k * i + k    ∎
 
 lem₁₀ : ∀ {a′} b c {d} e f → let a = suc a′ in
@@ -151,28 +234,29 @@ lem₁₀ {a′} b c {d} e f eq =
   m*n≡1⇒n≡1 (e * f ∸ b * c) d
     (lem₀ (e * f) (b * c) d 1
        (*-cancelʳ-≡ (e * f * d) (b * c * d + 1) _ (begin
-          e * f * d * a        ≡⟨ solve 4 (λ e f d a → e :* f :* d :* a
-                                                   :=  e :* (f :* d :* a))
-                                          refl e f d a ⟩
-          e * (f * d * a)      ≡⟨ sym eq ⟩
-          a + b * (c * d * a)  ≡⟨ solve 4 (λ a b c d → a :+ b :* (c :* d :* a)
-                                                   :=  (b :* c :* d :+ con 1) :* a)
-                                          refl a b c d ⟩
-          (b * c * d + 1) * a  ∎)))
+          e * f * d * a           ≡⟨ *-assoc₄₃ e f d a ⟩
+          e * (f * d * a)         ≡⟨ eq ⟨
+          a + b * (c * d * a)     ≡⟨ cong (a +_) (*-assoc₄₃ b c d a) ⟨
+          a + b * c * d * a       ≡⟨ +-comm a _ ⟩
+          b * c * d * a + a       ≡⟨ cong (b * c * d * a +_) (+-identityʳ a) ⟨
+          b * c * d * a + (a + 0) ≡⟨ *-distribʳ-+ a (b * c * d) 1 ⟨
+          (b * c * d + 1) * a     ∎)))
   where a = suc a′
+        *-assoc₄₃ : ∀ w x y z → w * x * y * z ≡ w * (x * y * z)
+        *-assoc₄₃ w x y z = begin
+          w * x * y * z   ≡⟨ cong (_* z) (*-assoc w x y) ⟩
+          w * (x * y) * z ≡⟨ *-assoc w _ z ⟩
+          w * (x * y * z) ∎
 
 lem₁₁ : ∀ {i j m n k d} x y →
        1 + y * j ≡ x * i → i * k ≡ m * d → j * k ≡ n * d →
        k ≡ (x * m ∸ y * n) * d
 lem₁₁ {i} {j} {m} {n} {k} {d} x y eq eq₁ eq₂ =
   sym (lem₀ (x * m) (y * n) d k (begin
-    x * m * d        ≡⟨ *-assoc x m d ⟩
-    x * (m * d)      ≡⟨ cong (x *_) (sym eq₁) ⟩
-    x * (i * k)      ≡⟨ sym (*-assoc x i k) ⟩
-    x * i * k        ≡⟨ cong₂ _*_ (sym eq) refl ⟩
-    (1 + y * j) * k  ≡⟨ solve 3 (λ y j k → (con 1 :+ y :* j) :* k
-                                       :=  y :* (j :* k) :+ k)
-                                refl y j k ⟩
+    x * m * d        ≡⟨ *-on-right x m d (sym eq₁) ⟩
+    x * (i * k)      ≡⟨ *-on-left x i k (sym eq) ⟩
+    (1 + y * j) * k  ≡⟨ +-comm k _ ⟩
+    y * j * k + k    ≡⟨ cong (_+ k) (*-assoc y j k) ⟩
     y * (j * k) + k  ≡⟨ cong (λ p → y * p + k) eq₂ ⟩
-    y * (n * d) + k  ≡⟨ cong₂ _+_ (sym $ *-assoc y n d) refl ⟩
+    y * (n * d) + k  ≡⟨ cong (_+ k) (*-assoc y n d) ⟨
     y * n * d + k    ∎))
diff --git a/src/Data/Rational/Unnormalised/Properties.agda b/src/Data/Rational/Unnormalised/Properties.agda
index 2174d29c74..181e4a6189 100644
--- a/src/Data/Rational/Unnormalised/Properties.agda
+++ b/src/Data/Rational/Unnormalised/Properties.agda
@@ -20,7 +20,6 @@ import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
 open import Data.Bool.Base using (T; true; false)
 open import Data.Nat.Base as ℕ using (suc; pred)
 import Data.Nat.Properties as ℕ
-open import Data.Nat.Solver renaming (module +-*-Solver to ℕ-solver)
 open import Data.Integer.Base as ℤ using (ℤ; +0; +[1+_]; -[1+_]; 0ℤ; 1ℤ; -1ℤ)
 open import Data.Integer.Solver renaming (module +-*-Solver to ℤ-solver)
 import Data.Integer.Properties as ℤ
@@ -1123,12 +1122,14 @@ p≤q⇒0≤q-p {p} {q} p≤q = begin
 *-inverseˡ p@(mkℚᵘ -[1+ n ] d) = *-inverseˡ (mkℚᵘ +[1+ n ] d)
 *-inverseˡ p@(mkℚᵘ +[1+ n ] d) = *≡* $ cong +[1+_] $ begin
   (n ℕ.+ d ℕ.* suc n) ℕ.* 1 ≡⟨ ℕ.*-identityʳ _ ⟩
-  (n ℕ.+ d ℕ.* suc n)       ≡⟨ cong (n ℕ.+_) (ℕ.*-suc d n) ⟩
-  (n ℕ.+ (d ℕ.+ d ℕ.* n))   ≡⟨ solve 2 (λ n d → n :+ (d :+ d :* n) := d :+ (n :+ n :* d)) refl n d ⟩
-  (d ℕ.+ (n ℕ.+ n ℕ.* d))   ≡⟨ cong (d ℕ.+_) (sym (ℕ.*-suc n d)) ⟩
+  n ℕ.+ d ℕ.* suc n         ≡⟨ cong (n ℕ.+_) (ℕ.*-suc d n) ⟩
+  n ℕ.+ (d ℕ.+ d ℕ.* n)     ≡⟨ trans (sym $ ℕ.+-assoc n d _) (trans
+                                      (cong₂ ℕ._+_ (ℕ.+-comm n d) (ℕ.*-comm d n))
+                                      (ℕ.+-assoc d n _)) ⟩
+  d ℕ.+ (n ℕ.+ n ℕ.* d)     ≡⟨ cong (d ℕ.+_) (sym (ℕ.*-suc n d)) ⟩
   d ℕ.+ n ℕ.* suc d         ≡⟨ ℕ.+-identityʳ _ ⟨
   d ℕ.+ n ℕ.* suc d ℕ.+ 0   ∎
-  where open ≡-Reasoning; open ℕ-solver
+  where open ≡-Reasoning
 
 *-inverseʳ : ∀ p .{{_ : NonZero p}} → p * 1/ p ≃ 1ℚᵘ
 *-inverseʳ p = ≃-trans (*-comm p (1/ p)) (*-inverseˡ p)

From 1442ee7a80d8eb35fdf0840fe3fadf7967dcde20 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Sat, 20 Apr 2024 12:00:30 -0400
Subject: [PATCH 274/316] Relation.Binary.PropositionalEquality over-use,
 (#2345)

* Relation.Binary.PropositionalEquality over-use, can usually be .Core and/or .Properties

* unused import

* another large batch of dependency cleanup, mostly cenetered on Relation.Binary.PropositionalEquality.

* another big batch of making imports more precise.

* last big batch. After this will come some tweaks based on reviews.

* fix things pointed out in review; add CHANGELOG.

* Update DecPropositional.agda

Bad merge

* proper merge
---
 CHANGELOG.md                                  |  6 ++++
 src/Algebra/Consequences/Propositional.agda   |  5 ++-
 src/Algebra/Definitions/RawSemiring.agda      |  2 +-
 .../Properties/BooleanAlgebra/Expression.agda | 14 ++++----
 .../Solver/IdempotentCommutativeMonoid.agda   |  3 +-
 .../IdempotentCommutativeMonoid/Example.agda  |  7 ++--
 src/Codata/Sized/Cofin.agda                   | 10 +++---
 src/Data/Container/Combinator/Properties.agda |  7 ++--
 src/Data/Container/Indexed/Combinator.agda    |  7 ++--
 .../Relation/Binary/Pointwise/Properties.agda | 18 +++++-----
 src/Data/Container/Membership.agda            |  6 ++--
 src/Data/Container/Morphism/Properties.agda   |  2 +-
 .../Relation/Binary/Pointwise/Properties.agda |  5 +--
 .../Relation/Unary/Any/Properties.agda        | 12 ++++---
 src/Data/Fin/Induction.agda                   | 19 +++++++----
 src/Data/Fin/Permutation.agda                 | 18 +++++-----
 src/Data/Fin/Permutation/Components.agda      |  8 +++--
 .../Fin/Permutation/Transposition/List.agda   |  8 +++--
 src/Data/Fin/Properties.agda                  |  6 ++--
 src/Data/Fin/Subset/Properties.agda           | 13 ++++---
 src/Data/Fin/Substitution/List.agda           | 11 +++---
 src/Data/Float/Properties.agda                |  5 ++-
 src/Data/Integer/DivMod.agda                  |  9 ++---
 src/Data/Integer/Divisibility/Signed.agda     |  9 +++--
 src/Data/Integer/Properties.agda              |  9 +++--
 src/Data/Integer/Tactic/RingSolver.agda       |  6 ++--
 src/Data/List/Effectful.agda                  |  4 ++-
 .../List/Membership/DecPropositional.agda     |  2 ++
 .../Membership/Propositional/Properties.agda  | 32 +++++++++--------
 src/Data/List/NonEmpty/Properties.agda        | 17 ++++++----
 src/Data/List/Properties.agda                 |  3 +-
 .../Relation/Binary/BagAndSetEquality.agda    | 34 ++++++++++++-------
 .../List/Relation/Binary/Equality/Setoid.agda | 16 ++++-----
 .../Binary/Sublist/DecPropositional.agda      |  3 +-
 .../Binary/Sublist/Heterogeneous/Solver.agda  | 12 +++----
 .../Binary/Sublist/Propositional.agda         |  5 ++-
 .../Example/UniqueBoundVariables.agda         |  6 ++--
 .../Sublist/Propositional/Properties.agda     |  7 ++--
 .../Binary/Sublist/Propositional/Slice.agda   |  5 ++-
 .../Subset/Propositional/Properties.agda      | 21 +++++++-----
 .../Ternary/Appending/Propositional.agda      |  6 ++--
 .../Relation/Unary/AllPairs/Properties.agda   |  7 ++--
 src/Data/List/Relation/Unary/Linked.agda      |  2 +-
 .../Unique/Propositional/Properties.agda      | 12 ++++---
 src/Data/List/Reverse.agda                    |  6 ++--
 src/Data/List/Zipper/Properties.agda          |  7 +++-
 src/Data/Maybe/Properties.agda                | 10 ++++--
 src/Data/Nat/Binary/Properties.agda           | 26 +++++++++-----
 src/Data/Nat/Binary/Subtraction.agda          | 15 +++++---
 src/Data/Nat/Combinatorics.agda               | 12 ++++---
 src/Data/Nat/Combinatorics/Specification.agda | 15 ++++----
 src/Data/Nat/DivMod.agda                      |  3 +-
 src/Data/Nat/DivMod/Core.agda                 |  6 ++--
 src/Data/Nat/DivMod/WithK.agda                | 11 +++---
 src/Data/Nat/GCD/Lemmas.agda                  |  5 ++-
 src/Data/Nat/GeneralisedArithmetic.agda       | 10 ++++--
 src/Data/Nat/InfinitelyOften.agda             |  4 +--
 src/Data/Nat/Logarithm/Core.agda              |  9 +++--
 src/Data/Nat/Primality.agda                   |  2 +-
 src/Data/Nat/Primality/Factorisation.agda     |  6 ++--
 src/Data/Nat/Properties.agda                  | 15 +++++---
 src/Data/Nat/Tactic/RingSolver.agda           |  8 ++---
 src/Data/Parity/Properties.agda               | 20 ++++++-----
 .../Function/Dependent/Propositional.agda     | 19 ++++++-----
 .../Function/NonDependent/Propositional.agda  | 15 +++++---
 .../Product/Function/NonDependent/Setoid.agda |  8 +++--
 src/Data/Product/Properties/WithK.agda        |  3 +-
 src/Data/Rational/Properties.agda             | 27 ++++++++-------
 src/Data/Record.agda                          | 11 +++---
 src/Data/Sign/Properties.agda                 | 16 +++++----
 src/Data/Star/Environment.agda                |  5 +--
 src/Data/Sum/Function/Propositional.agda      |  4 ++-
 src/Data/Sum/Properties.agda                  | 12 +++----
 src/Data/These/Properties.agda                |  5 +--
 .../AVL/Map/Membership/Propositional.agda     |  2 +-
 .../Membership/Propositional/Properties.agda  |  2 +-
 src/Data/Tree/Binary/Properties.agda          |  5 +--
 src/Data/Tree/Binary/Zipper/Properties.agda   | 17 ++++++----
 src/Data/Tree/Rose/Properties.agda            | 11 +++---
 src/Data/Unit/Polymorphic/Properties.agda     | 10 +++---
 src/Data/Unit/Properties.agda                 | 11 +++---
 src/Data/Vec/Functional/Properties.agda       | 13 ++++---
 .../Binary/Permutation/Properties.agda        | 11 +++---
 src/Data/Vec/Properties.agda                  | 11 +++---
 .../Binary/Equality/Propositional.agda        |  5 +--
 .../Binary/Equality/Propositional/WithK.agda  |  9 +++--
 .../Vec/Relation/Unary/All/Properties.agda    |  7 ++--
 src/Data/Vec/Relation/Unary/AllPairs.agda     |  2 +-
 src/Data/Vec/Relation/Unary/Linked.agda       |  4 +--
 .../Unique/Propositional/Properties.agda      | 15 ++++----
 src/Data/Word/Base.agda                       |  2 +-
 src/Data/Word/Properties.agda                 | 10 +++---
 src/Effect/Monad/Predicate.agda               | 11 +++---
 src/Function/Construct/Symmetry.agda          |  4 +--
 src/Function/Metric/Nat/Bundles.agda          |  8 ++---
 src/Function/Metric/Rational/Bundles.agda     | 13 ++++---
 src/Function/Nary/NonDependent.agda           |  4 ++-
 src/Function/Properties/Bijection.agda        |  7 ++--
 src/Function/Properties/Surjection.agda       | 11 +++---
 .../Binary/Construct/Add/Extrema/Strict.agda  | 13 ++++---
 .../ReflexiveTransitive/Properties/WithK.agda |  4 +--
 .../Binary/Indexed/Heterogeneous/Core.agda    |  1 -
 src/Relation/Binary/Rewriting.agda            | 16 ++++-----
 src/Tactic/RingSolver/Core/NatSet.agda        |  3 +-
 src/Text/Pretty/Core.agda                     | 21 +++++++-----
 src/Text/Regex/Properties/Core.agda           |  4 +--
 106 files changed, 602 insertions(+), 399 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 804bcbd8c8..24fb3493a4 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -30,6 +30,12 @@ Non-backwards compatible changes
 Other major improvements
 ------------------------
 
+Minor improvements
+------------------
+The size of the dependency graph for many modules has been
+reduced. This may lead to speed ups for first-time loading of some
+modules.
+
 Deprecated modules
 ------------------
 
diff --git a/src/Algebra/Consequences/Propositional.agda b/src/Algebra/Consequences/Propositional.agda
index 5c10a9e4cc..29490cbd0c 100644
--- a/src/Algebra/Consequences/Propositional.agda
+++ b/src/Algebra/Consequences/Propositional.agda
@@ -14,7 +14,10 @@ open import Data.Sum.Base using (inj₁; inj₂)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (Symmetric; Total)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; cong₂; subst)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid)
 open import Relation.Unary using (Pred)
 
 open import Algebra.Core
diff --git a/src/Algebra/Definitions/RawSemiring.agda b/src/Algebra/Definitions/RawSemiring.agda
index 01aebc2d18..b1819d856d 100644
--- a/src/Algebra/Definitions/RawSemiring.agda
+++ b/src/Algebra/Definitions/RawSemiring.agda
@@ -8,7 +8,7 @@
 
 open import Algebra.Bundles using (RawSemiring)
 open import Data.Sum.Base using (_⊎_)
-open import Data.Nat using (ℕ; zero; suc)
+open import Data.Nat.Base using (ℕ; zero; suc)
 open import Level using (_⊔_)
 open import Relation.Binary.Core using (Rel)
 
diff --git a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
index 5583f22a5b..96e6b69103 100644
--- a/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
+++ b/src/Algebra/Lattice/Properties/BooleanAlgebra/Expression.agda
@@ -6,7 +6,8 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra.Lattice
+open import Algebra.Lattice using (BooleanAlgebra; isBooleanAlgebraʳ;
+  isDistributiveLatticeʳʲᵐ)
 
 module Algebra.Lattice.Properties.BooleanAlgebra.Expression
   {b} (B : BooleanAlgebra b b)
@@ -14,10 +15,8 @@ module Algebra.Lattice.Properties.BooleanAlgebra.Expression
 
 open BooleanAlgebra B
 
-open import Effect.Applicative as Applicative
-open import Effect.Monad
 open import Data.Fin.Base using (Fin)
-open import Data.Nat.Base
+open import Data.Nat.Base using (ℕ)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Data.Vec.Base as Vec using (Vec)
 import Data.Vec.Effectful as Vec
@@ -25,8 +24,11 @@ import Function.Identity.Effectful as Identity
 open import Data.Vec.Properties using (lookup-map)
 open import Data.Vec.Relation.Binary.Pointwise.Extensional as PW
   using (Pointwise; ext)
+open import Effect.Applicative as Applicative
 open import Function.Base using (_∘_; _$_; flip)
-open import Relation.Binary.PropositionalEquality as ≡ using (_≗_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≗_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 import Relation.Binary.Reflection as Reflection
 
 -- Expressions made up of variables and the operations of a boolean
@@ -68,7 +70,7 @@ module Naturality
   (f : Applicative.Morphism A₁ A₂)
   where
 
-  open ≡.≡-Reasoning
+  open ≡-Reasoning
   open Applicative.Morphism f
   open Semantics A₁ renaming (⟦_⟧ to ⟦_⟧₁)
   open Semantics A₂ renaming (⟦_⟧ to ⟦_⟧₂)
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
index eefb5f40ab..9ce8836bd4 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
@@ -25,7 +25,8 @@ import Relation.Binary.Reflection            as Reflection
 import Relation.Nullary.Decidable            as Dec
 import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; decSetoid)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 open import Relation.Nullary.Decidable using (Dec)
 
 module Algebra.Solver.IdempotentCommutativeMonoid
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid/Example.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid/Example.agda
index 7a70fb8cfc..962abd6ebc 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid/Example.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid/Example.agda
@@ -9,7 +9,7 @@
 
 module Algebra.Solver.IdempotentCommutativeMonoid.Example where
 
-open import Relation.Binary.PropositionalEquality using (_≡_)
+import Algebra.Solver.IdempotentCommutativeMonoid as ICM-Solver
 
 open import Data.Bool.Base using (_∨_)
 open import Data.Bool.Properties using (∨-idempotentCommutativeMonoid)
@@ -17,8 +17,9 @@ open import Data.Bool.Properties using (∨-idempotentCommutativeMonoid)
 open import Data.Fin.Base using (zero; suc)
 open import Data.Vec.Base using ([]; _∷_)
 
-open import Algebra.Solver.IdempotentCommutativeMonoid
-  ∨-idempotentCommutativeMonoid
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+
+open ICM-Solver ∨-idempotentCommutativeMonoid
 
 test : ∀ x y z → (x ∨ y) ∨ (x ∨ z) ≡ (z ∨ y) ∨ x
 test a b c = let _∨_ = _⊕_ in
diff --git a/src/Codata/Sized/Cofin.agda b/src/Codata/Sized/Cofin.agda
index a7f74afb5f..e04be324de 100644
--- a/src/Codata/Sized/Cofin.agda
+++ b/src/Codata/Sized/Cofin.agda
@@ -8,15 +8,15 @@
 
 module Codata.Sized.Cofin where
 
-open import Size
-open import Codata.Sized.Thunk
+open import Size using (∞)
+open import Codata.Sized.Thunk using (force)
 open import Codata.Sized.Conat as Conat
   using (Conat; zero; suc; infinity; _ℕ<_; sℕ≤s; _ℕ≤infinity)
 open import Codata.Sized.Conat.Bisimilarity as Bisim using (_⊢_≲_ ; s≲s)
-open import Data.Nat.Base
-open import Data.Fin.Base as Fin hiding (fromℕ; fromℕ<; toℕ)
+open import Data.Nat.Base using (ℕ; zero; suc)
+open import Data.Fin.Base using (Fin; zero; suc)
 open import Function.Base using (_∋_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- The type
diff --git a/src/Data/Container/Combinator/Properties.agda b/src/Data/Container/Combinator/Properties.agda
index 13823bef75..4a91a903a2 100644
--- a/src/Data/Container/Combinator/Properties.agda
+++ b/src/Data/Container/Combinator/Properties.agda
@@ -9,16 +9,15 @@
 module Data.Container.Combinator.Properties where
 
 open import Axiom.Extensionality.Propositional using (Extensionality)
-open import Data.Container.Core
+open import Data.Container.Core using (Container; ⟦_⟧)
 open import Data.Container.Combinator
-open import Data.Container.Relation.Unary.Any
 open import Data.Empty using (⊥-elim)
 open import Data.Product.Base as P using (∃; _,_; proj₁; proj₂; <_,_>; uncurry; curry)
 open import Data.Sum.Base as S using (inj₁; inj₂; [_,_]′; [_,_])
 open import Function.Base as F using (_∘′_)
-open import Function.Bundles
+open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (_⊔_; lower)
-open import Relation.Binary.PropositionalEquality using (_≡_; _≗_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≗_; refl; cong)
 
 -- I have proved some of the correctness statements under the
 -- assumption of functional extensionality. I could have reformulated
diff --git a/src/Data/Container/Indexed/Combinator.agda b/src/Data/Container/Indexed/Combinator.agda
index 825d173db7..bc5bd194c7 100644
--- a/src/Data/Container/Indexed/Combinator.agda
+++ b/src/Data/Container/Indexed/Combinator.agda
@@ -9,7 +9,8 @@
 module Data.Container.Indexed.Combinator where
 
 open import Axiom.Extensionality.Propositional using (Extensionality)
-open import Data.Container.Indexed
+open import Data.Container.Indexed using (Container; _◃_/_; ⟦_⟧;
+  Command; Response; ◇; next)
 open import Data.Empty.Polymorphic using (⊥; ⊥-elim)
 open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Product.Base as Prod hiding (Σ) renaming (_×_ to _⟨×⟩_)
@@ -18,10 +19,10 @@ open import Data.Sum.Relation.Unary.All as All using (All)
 open import Function.Base as F hiding (id; const) renaming (_∘_ to _⟨∘⟩_)
 open import Function.Bundles using (mk↔ₛ′)
 open import Function.Indexed.Bundles using (_↔ᵢ_)
-open import Level
+open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred; _⊆_; _∪_; _∩_; ⋃; ⋂)
   renaming (_⟨×⟩_ to _⟪×⟫_; _⟨⊙⟩_ to _⟪⊙⟫_; _⟨⊎⟩_ to _⟪⊎⟫_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
   using (_≗_; refl; cong)
 
 private
diff --git a/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
index a1fc21f44e..eecd3af3d3 100644
--- a/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Container/Indexed/Relation/Binary/Pointwise/Properties.agda
@@ -8,13 +8,15 @@
 
 module Data.Container.Indexed.Relation.Binary.Pointwise.Properties where
 
-open import Axiom.Extensionality.Propositional
-open import Data.Container.Indexed.Core
+open import Axiom.Extensionality.Propositional using (Extensionality)
+open import Data.Container.Indexed.Core using (Container; ⟦_⟧)
 open import Data.Container.Indexed.Relation.Binary.Pointwise
+  using (Pointwise)
 open import Data.Product.Base using (_,_; Σ-syntax; -,_)
 open import Level using (Level; _⊔_)
-open import Relation.Binary
-open import Relation.Binary.PropositionalEquality as P
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; subst; cong)
 
 private variable
@@ -28,13 +30,13 @@ module _
   where
 
   refl : (∀ i → Reflexive (R i)) → Reflexive (Pointwise C R o)
-  refl R-refl = P.refl , λ p → R-refl _
+  refl R-refl = ≡.refl , λ p → R-refl _
 
   sym : (∀ i → Symmetric (R i)) → Symmetric (Pointwise C R o)
-  sym R-sym (P.refl , f) = P.refl , λ p → R-sym _ (f p)
+  sym R-sym (≡.refl , f) = ≡.refl , λ p → R-sym _ (f p)
 
   trans : (∀ i → Transitive (R i)) → Transitive (Pointwise C R o)
-  trans R-trans (P.refl , f) (P.refl , g) = P.refl , λ p → R-trans _ (f p) (g p)
+  trans R-trans (≡.refl , f) (≡.refl , g) = ≡.refl , λ p → R-trans _ (f p) (g p)
 
 -- If propositional equality is extensional, then `Eq _≡_` and `_≡_` coincide.
 Eq⇒≡ : {C : Container I O ℓˢ ℓᵖ} {X : I → Set ℓˣ} {R : (i : I) → Rel (X i) ℓᵉ}
@@ -42,4 +44,4 @@ Eq⇒≡ : {C : Container I O ℓˢ ℓᵖ} {X : I → Set ℓˣ} {R : (i : I) 
        Extensionality ℓᵖ ℓˣ →
        Pointwise C (λ (i : I) → _≡_ {A = X i}) o xs ys →
        xs ≡ ys
-Eq⇒≡ ext (P.refl , f≈f′) = cong -,_ (ext f≈f′)
+Eq⇒≡ ext (≡.refl , f≈f′) = cong -,_ (ext f≈f′)
diff --git a/src/Data/Container/Membership.agda b/src/Data/Container/Membership.agda
index 3ace9cdc06..6a04f397e8 100644
--- a/src/Data/Container/Membership.agda
+++ b/src/Data/Container/Membership.agda
@@ -10,10 +10,10 @@ module Data.Container.Membership where
 
 open import Level using (_⊔_)
 open import Relation.Unary using (Pred)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
-open import Data.Container.Core
-open import Data.Container.Relation.Unary.Any
+open import Data.Container.Core using (Container; ⟦_⟧)
+open import Data.Container.Relation.Unary.Any using (◇)
 
 module _ {s p} {C : Container s p} {x} {X : Set x} where
 
diff --git a/src/Data/Container/Morphism/Properties.agda b/src/Data/Container/Morphism/Properties.agda
index 39471b5e48..ca2e4e5b55 100644
--- a/src/Data/Container/Morphism/Properties.agda
+++ b/src/Data/Container/Morphism/Properties.agda
@@ -12,7 +12,7 @@ open import Level using (_⊔_; suc)
 open import Function.Base as F using (_$_)
 open import Data.Product.Base using (∃; proj₁; proj₂; _,_)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_; _≗_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; _≗_)
 
 open import Data.Container.Core
 open import Data.Container.Morphism
diff --git a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
index 978bfc55e4..9bdf815092 100644
--- a/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Container/Relation/Binary/Pointwise/Properties.agda
@@ -8,9 +8,10 @@
 
 module Data.Container.Relation.Binary.Pointwise.Properties where
 
-open import Axiom.Extensionality.Propositional
-open import Data.Container.Core
+open import Axiom.Extensionality.Propositional using (Extensionality)
+open import Data.Container.Core using (Container; ⟦_⟧)
 open import Data.Container.Relation.Binary.Pointwise
+  using (Pointwise; _,_)
 open import Data.Product.Base using (_,_; Σ-syntax; -,_)
 open import Level using (_⊔_)
 open import Relation.Binary.Core using (Rel)
diff --git a/src/Data/Container/Relation/Unary/Any/Properties.agda b/src/Data/Container/Relation/Unary/Any/Properties.agda
index 601d922a56..6e643a6031 100644
--- a/src/Data/Container/Relation/Unary/Any/Properties.agda
+++ b/src/Data/Container/Relation/Unary/Any/Properties.agda
@@ -8,23 +8,25 @@
 
 module Data.Container.Relation.Unary.Any.Properties where
 
-open import Level
-open import Algebra
+open import Algebra.Bundles using (CommutativeSemiring)
 open import Data.Product.Base using (∃; _×_; ∃₂; _,_; proj₂)
 open import Data.Product.Properties using (Σ-≡,≡→≡)
 open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_])
-open import Function.Base
-open import Function.Bundles
+open import Function.Base using (_∘_; _∘′_; id; flip; _$_)
+open import Function.Bundles using (_↔_; mk↔ₛ′; Inverse)
 open import Function.Properties.Inverse using (↔-refl)
 open import Function.Properties.Inverse.HalfAdjointEquivalence using (_≃_; ↔⇒≃)
 open import Function.Related.Propositional as Related using (Related; SK-sym)
 open import Function.Related.TypeIsomorphisms
+  using (×-⊎-commutativeSemiring; ∃∃↔∃∃; Σ-assoc; ×-comm)
+open import Level using (Level; _⊔_)
 open import Relation.Unary using (Pred ; _∪_ ; _∩_)
 open import Relation.Binary.Core using (REL)
-open import Relation.Binary.PropositionalEquality as ≡
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≗_; refl)
+open import Relation.Binary.PropositionalEquality.Properties as ≡
 
 private
   module ×⊎ {k ℓ} = CommutativeSemiring (×-⊎-commutativeSemiring k ℓ)
diff --git a/src/Data/Fin/Induction.agda b/src/Data/Fin/Induction.agda
index 97ca14015b..398c1a3966 100644
--- a/src/Data/Fin/Induction.agda
+++ b/src/Data/Fin/Induction.agda
@@ -7,8 +7,13 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 {-# OPTIONS --warn=noUserWarning #-} -- for deprecated _≺_ (issue #1726)
 
-open import Data.Fin.Base
-open import Data.Fin.Properties
+module Data.Fin.Induction where
+
+open import Data.Fin.Base using (Fin; zero; suc; _<_; toℕ; inject₁;
+  _≥_; _>_; fromℕ; _≺_)
+open import Data.Fin.Properties using (toℕ-inject₁; ≤-refl; <-cmp;
+  toℕ≤n; toℕ-injective; toℕ-fromℕ; toℕ-lower₁; inject₁-lower₁;
+  pigeonhole; ≺⇒<′)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _∸_; s≤s)
 open import Data.Nat.Properties using (n<1+n; ≤⇒≯)
 import Data.Nat.Induction as ℕ
@@ -18,8 +23,9 @@ open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.Vec.Relation.Unary.Linked as Linked using (Linked; [-]; _∷_)
 import Data.Vec.Relation.Unary.Linked.Properties as Linked
 open import Function.Base using (flip; _$_)
-open import Induction
-open import Induction.WellFounded as WF
+open import Induction using (RecStruct)
+open import Induction.WellFounded as WF using (WellFounded; WfRec;
+  module Subrelation)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (StrictPartialOrder)
@@ -30,13 +36,12 @@ import Relation.Binary.Construct.Flip.Ord as Ord
 import Relation.Binary.Construct.NonStrictToStrict as ToStrict
 import Relation.Binary.Construct.On as On
 open import Relation.Binary.Definitions using (Tri; tri<; tri≈; tri>)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; subst; trans; cong)
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Unary using (Pred)
 
-module Data.Fin.Induction where
-
 private
   variable
     ℓ : Level
diff --git a/src/Data/Fin/Permutation.agda b/src/Data/Fin/Permutation.agda
index 3fa0ebf92f..794b93f76b 100644
--- a/src/Data/Fin/Permutation.agda
+++ b/src/Data/Fin/Permutation.agda
@@ -9,10 +9,10 @@
 module Data.Fin.Permutation where
 
 open import Data.Bool.Base using (true; false)
-open import Data.Empty using (⊥-elim)
-open import Data.Fin.Base
-open import Data.Fin.Patterns
-open import Data.Fin.Properties
+open import Data.Fin.Base using (Fin; suc; opposite; punchIn; punchOut)
+open import Data.Fin.Patterns using (0F)
+open import Data.Fin.Properties using (punchInᵢ≢i; punchOut-punchIn;
+  punchOut-cong; punchOut-cong′; punchIn-punchOut; _≟_; ¬Fin0)
 import Data.Fin.Permutation.Components as PC
 open import Data.Nat.Base using (ℕ; suc; zero)
 open import Data.Product.Base using (_,_; proj₂)
@@ -20,7 +20,7 @@ open import Function.Bundles using (_↔_; Injection; Inverse; mk↔ₛ′)
 open import Function.Construct.Composition using (_↔-∘_)
 open import Function.Construct.Identity using (↔-id)
 open import Function.Construct.Symmetry using (↔-sym)
-open import Function.Definitions
+open import Function.Definitions using (StrictlyInverseˡ; StrictlyInverseʳ)
 open import Function.Properties.Inverse using (↔⇒↣)
 open import Function.Base using (_∘_)
 open import Level using (0ℓ)
@@ -28,8 +28,10 @@ open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary using (does; ¬_; yes; no)
 open import Relation.Nullary.Decidable using (dec-yes; dec-no)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; _≢_; refl; sym; trans; subst; →-to-⟶; cong; cong₂; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≢_; refl; sym; trans; subst; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 
 private
@@ -241,7 +243,7 @@ module _ (π : Permutation (suc m) (suc n)) where
   lift₀-remove p (suc i) = punchOut-zero (πʳ (suc i)) p
     where
     punchOut-zero : ∀ {i} (j : Fin (suc n)) {neq} → i ≡ 0F → suc (punchOut {i = i} {j} neq) ≡ j
-    punchOut-zero 0F {neq} p = ⊥-elim (neq p)
+    punchOut-zero 0F {neq} p = contradiction p neq
     punchOut-zero (suc j) refl = refl
 
 ↔⇒≡ : Permutation m n → m ≡ n
diff --git a/src/Data/Fin/Permutation/Components.agda b/src/Data/Fin/Permutation/Components.agda
index d345018778..2d2f34deb7 100644
--- a/src/Data/Fin/Permutation/Components.agda
+++ b/src/Data/Fin/Permutation/Components.agda
@@ -9,15 +9,19 @@
 module Data.Fin.Permutation.Components where
 
 open import Data.Bool.Base using (Bool; true; false)
-open import Data.Fin.Base
+open import Data.Fin.Base using (Fin; suc; opposite; toℕ)
 open import Data.Fin.Properties
+  using (_≟_; opposite-prop; opposite-involutive; opposite-suc)
 open import Data.Nat.Base as ℕ using (zero; suc; _∸_)
 open import Data.Product.Base using (proj₂)
 open import Function.Base using (_∘_)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (does; _because_; yes; no)
 open import Relation.Nullary.Decidable using (dec-true; dec-false)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; trans)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Algebra.Definitions using (Involutive)
 open ≡-Reasoning
 
diff --git a/src/Data/Fin/Permutation/Transposition/List.agda b/src/Data/Fin/Permutation/Transposition/List.agda
index 4da8f08d37..3d14d36b05 100644
--- a/src/Data/Fin/Permutation/Transposition/List.agda
+++ b/src/Data/Fin/Permutation/Transposition/List.agda
@@ -8,7 +8,7 @@
 
 module Data.Fin.Permutation.Transposition.List where
 
-open import Data.Fin.Base
+open import Data.Fin.Base using (Fin; suc)
 open import Data.Fin.Patterns using (0F)
 open import Data.Fin.Permutation as P hiding (lift₀)
 import Data.Fin.Permutation.Components as PC
@@ -16,8 +16,10 @@ open import Data.List.Base using (List; []; _∷_; map)
 open import Data.Nat.Base using (ℕ; suc; zero)
 open import Data.Product.Base using (_×_; _,_)
 open import Function.Base using (_∘_)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; sym; cong; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; sym; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 
 private
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index 6243e016eb..a1db966f92 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -40,8 +40,10 @@ open import Relation.Binary.Bundles
   using (Preorder; Setoid; DecSetoid; Poset; TotalOrder; DecTotalOrder; StrictPartialOrder; StrictTotalOrder)
 open import Relation.Binary.Structures
   using (IsDecEquivalence; IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
-open import Relation.Binary.PropositionalEquality as ≡
-  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core as ≡
+  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_)
+open import Relation.Binary.PropositionalEquality.Properties as ≡
+  using (module ≡-Reasoning)
 open import Relation.Nullary.Decidable as Dec
   using (Dec; _because_; yes; no; _×-dec_; _⊎-dec_; map′)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
diff --git a/src/Data/Fin/Subset/Properties.agda b/src/Data/Fin/Subset/Properties.agda
index 769c20ce06..03467dcee7 100644
--- a/src/Data/Fin/Subset/Properties.agda
+++ b/src/Data/Fin/Subset/Properties.agda
@@ -11,8 +11,10 @@ module Data.Fin.Subset.Properties where
 import Algebra.Definitions as AlgebraicDefinitions
 import Algebra.Structures as AlgebraicStructures
 import Algebra.Lattice.Structures as AlgebraicLatticeStructures
-open import Algebra.Bundles
-open import Algebra.Lattice.Bundles
+open import Algebra.Bundles using (Magma; Semigroup; Monoid; Band;
+  CommutativeMonoid; IdempotentCommutativeMonoid)
+open import Algebra.Lattice.Bundles using (Semilattice; Lattice;
+  DistributiveLattice; BooleanAlgebra)
 import Algebra.Lattice.Properties.Lattice as L
 import Algebra.Lattice.Properties.DistributiveLattice as DL
 import Algebra.Lattice.Properties.BooleanAlgebra as BA
@@ -25,7 +27,7 @@ open import Data.Nat.Base hiding (∣_-_∣)
 import Data.Nat.Properties as ℕ
 open import Data.Product as Product using (∃; ∄; _×_; _,_; proj₁)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
-open import Data.Vec.Base
+open import Data.Vec.Base using (Vec; []; _∷_; here; there)
 open import Data.Vec.Properties
 open import Function.Base using (_∘_; const; id; case_of_)
 open import Function.Bundles using (_⇔_; mk⇔)
@@ -36,7 +38,10 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Bundles
   using (Preorder; Poset; StrictPartialOrder; DecStrictPartialOrder)
 open import Relation.Binary.Definitions as B hiding (Decidable; Empty)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; subst; _≢_; sym)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning; isEquivalence)
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _⊎-dec_)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Unary using (Pred; Decidable; Satisfiable)
diff --git a/src/Data/Fin/Substitution/List.agda b/src/Data/Fin/Substitution/List.agda
index ce5ff3e27c..179f806a22 100644
--- a/src/Data/Fin/Substitution/List.agda
+++ b/src/Data/Fin/Substitution/List.agda
@@ -9,16 +9,17 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Fin.Substitution.Lemmas
+open import Data.Fin.Substitution.Lemmas using (Lemmas₄; AppLemmas)
 open import Data.Nat.Base using (ℕ)
 
 module Data.Fin.Substitution.List {ℓ} {T : ℕ → Set ℓ} (lemmas₄ : Lemmas₄ T) where
 
-open import Data.List.Base
-open import Data.List.Properties
-open import Data.Fin.Substitution
+open import Data.List.Base using (List; map)
+open import Data.List.Properties using (map-id; map-cong; map-∘)
+open import Data.Fin.Substitution using (Sub)
 import Function.Base as Fun
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
diff --git a/src/Data/Float/Properties.agda b/src/Data/Float/Properties.agda
index 526f71f6c4..fab340ea0e 100644
--- a/src/Data/Float/Properties.agda
+++ b/src/Data/Float/Properties.agda
@@ -24,7 +24,10 @@ open import Relation.Binary.Structures
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive; Substitutive; Decidable; DecidableEquality)
 import Relation.Binary.Construct.On as On
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; sym; trans; subst)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; decSetoid)
 
 ------------------------------------------------------------------------
 -- Primitive properties
diff --git a/src/Data/Integer/DivMod.agda b/src/Data/Integer/DivMod.agda
index 28d15d05fd..639188c180 100644
--- a/src/Data/Integer/DivMod.agda
+++ b/src/Data/Integer/DivMod.agda
@@ -8,15 +8,16 @@
 
 module Data.Integer.DivMod where
 
-open import Data.Integer.Base
+open import Data.Integer.Base using (+_; -[1+_]; +[1+_]; NonZero; _%_; ∣_∣;
+  _%ℕ_; _/ℕ_; _+_; _*_; -_; _-_; pred; -1ℤ; 0ℤ; _⊖_; _≤_; _<_; +≤+; suc;
+  +<+)
 open import Data.Integer.Properties
 open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s; z<s; s<s)
 import Data.Nat.Properties as ℕ
 import Data.Nat.DivMod as ℕ
 open import Function.Base using (_∘′_)
-open import Relation.Nullary.Decidable
-open import Relation.Binary.PropositionalEquality
-
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; cong; sym; subst)
 open ≤-Reasoning
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Integer/Divisibility/Signed.agda b/src/Data/Integer/Divisibility/Signed.agda
index f88f3ba859..8272ded259 100644
--- a/src/Data/Integer/Divisibility/Signed.agda
+++ b/src/Data/Integer/Divisibility/Signed.agda
@@ -9,7 +9,8 @@
 module Data.Integer.Divisibility.Signed where
 
 open import Function.Base using (_⟨_⟩_; _$_; _$′_; _∘_; _∘′_)
-open import Data.Integer.Base
+open import Data.Integer.Base using (ℤ; _*_; +0; sign; _◃_; ≢-nonZero;
+  ∣_∣; 0ℤ; +_; _+_; _-_; -_; NonZero)
 open import Data.Integer.Properties
 import Data.Integer.Divisibility as Unsigned
 import Data.Nat.Base as ℕ
@@ -18,13 +19,15 @@ import Data.Nat.Coprimality as ℕ
 import Data.Nat.Properties as ℕ
 import Data.Sign.Base as Sign
 import Data.Sign.Properties as Sign
-open import Level
 open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
 open import Relation.Binary.Bundles using (Preorder)
 open import Relation.Binary.Structures using (IsPreorder)
 open import Relation.Binary.Definitions
   using (Reflexive; Transitive; Decidable)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; trans; sym; cong; refl)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning; isEquivalence)
 import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 open import Relation.Nullary.Decidable as Dec using (yes; no)
 open import Relation.Binary.Reasoning.Syntax
diff --git a/src/Data/Integer/Properties.agda b/src/Data/Integer/Properties.agda
index 61477c9acc..f89d3164f6 100644
--- a/src/Data/Integer/Properties.agda
+++ b/src/Data/Integer/Properties.agda
@@ -34,10 +34,13 @@ open import Relation.Binary.Structures
   using (IsPreorder; IsTotalPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder; IsStrictPartialOrder; IsStrictTotalOrder)
 open import Relation.Binary.Definitions
   using (DecidableEquality; Reflexive; Transitive; Antisymmetric; Total; Decidable; Irrelevant; Irreflexive; Asymmetric; LeftTrans; RightTrans; Trichotomous; tri≈; tri<; tri>)
-open import Relation.Binary.PropositionalEquality
-open import Relation.Nullary using (yes; no; ¬_)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; sym; _≢_; subst; resp₂; trans)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning; setoid; decSetoid; isEquivalence)
+open import Relation.Nullary.Decidable.Core using (yes; no)
 import Relation.Nullary.Reflects as Reflects
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 import Relation.Nullary.Decidable as Dec
 
 open import Algebra.Definitions {A = ℤ} _≡_
diff --git a/src/Data/Integer/Tactic/RingSolver.agda b/src/Data/Integer/Tactic/RingSolver.agda
index d4e1a43536..e673c0f3d7 100644
--- a/src/Data/Integer/Tactic/RingSolver.agda
+++ b/src/Data/Integer/Tactic/RingSolver.agda
@@ -13,11 +13,11 @@ module Data.Integer.Tactic.RingSolver where
 open import Agda.Builtin.Reflection
 
 open import Data.Maybe.Base using (just; nothing)
-open import Data.Integer.Base
-open import Data.Integer.Properties
+open import Data.Integer.Base using (+0)
+open import Data.Integer.Properties using (+-*-commutativeRing)
 open import Level using (0ℓ)
 open import Data.Unit.Base using (⊤)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 import Tactic.RingSolver as Solver
 import Tactic.RingSolver.Core.AlmostCommutativeRing as ACR
 
diff --git a/src/Data/List/Effectful.agda b/src/Data/List/Effectful.agda
index e3855b26c5..0bbcad4973 100644
--- a/src/Data/List/Effectful.agda
+++ b/src/Data/List/Effectful.agda
@@ -23,8 +23,10 @@ open import Effect.Monad
   using (RawMonad; module Join; RawMonadZero; RawMonadPlus)
 open import Function.Base using (flip; _∘_; const; _$_; id; _∘′_; _$′_)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality as ≡
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≢_; _≗_; refl)
+open import Relation.Binary.PropositionalEquality.Properties as ≡
+
 open ≡.≡-Reasoning
 
 private
diff --git a/src/Data/List/Membership/DecPropositional.agda b/src/Data/List/Membership/DecPropositional.agda
index 3c81c0ff1e..57bbb2cca0 100644
--- a/src/Data/List/Membership/DecPropositional.agda
+++ b/src/Data/List/Membership/DecPropositional.agda
@@ -13,6 +13,8 @@ module Data.List.Membership.DecPropositional
 
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
+
 ------------------------------------------------------------------------
 -- Re-export contents of propositional membership
 
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index 12785eb59b..47ee739817 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -8,36 +8,40 @@
 
 module Data.List.Membership.Propositional.Properties where
 
-open import Algebra using (Op₂; Selective)
+open import Algebra.Core using (Op₂)
+open import Algebra.Definitions using (Selective)
 open import Effect.Monad using (RawMonad)
 open import Data.Bool.Base using (Bool; false; true; T)
 open import Data.Fin.Base using (Fin)
 open import Data.List.Base as List
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Relation.Unary.Any.Properties
+  using (map↔; concat↔; >>=↔; ⊛↔; Any-cong; ⊗↔′; ¬Any[])
 open import Data.List.Membership.Propositional
+  using (_∈_; _∉_; mapWith∈; _≢∈_)
 import Data.List.Membership.Setoid.Properties as Membership
 open import Data.List.Relation.Binary.Equality.Propositional
   using (_≋_; ≡⇒≋; ≋⇒≡)
 open import Data.List.Effectful using (monad)
 open import Data.Nat.Base using (ℕ; zero; suc; pred; s≤s; _≤_; _<_; _≤ᵇ_)
-open import Data.Nat.Properties
-open import Data.Product.Base hiding (map)
+open import Data.Nat.Properties using (_≤?_; m≤n⇒m≤1+n; ≤ᵇ-reflects-≤; <⇒≢; ≰⇒>)
+open import Data.Product.Base using (∃; ∃₂; _×_; _,_)
 open import Data.Product.Properties using (×-≡,≡↔≡)
 open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
-open import Function.Base
-open import Function.Definitions
+open import Function.Base using (_∘_; _∘′_; _$_; id; flip; _⟨_⟩_)
+open import Function.Definitions using (Injective)
 import Function.Related.Propositional as Related
-open import Function.Bundles
-open import Function.Related.TypeIsomorphisms
+open import Function.Bundles using (_↔_; _↣_; Injection)
+open import Function.Related.TypeIsomorphisms using (×-comm; ∃∃↔∃∃)
 open import Function.Construct.Identity using (↔-id)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions as Binary hiding (Decidable)
-open import Relation.Binary.PropositionalEquality as ≡
-  using (_≡_; _≢_; refl; sym; trans; cong; subst; _≗_)
+open import Relation.Binary.PropositionalEquality.Core as ≡
+  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_)
+open import Relation.Binary.PropositionalEquality.Properties as ≡ using (setoid)
 import Relation.Binary.Properties.DecTotalOrder as DTOProperties
 open import Relation.Unary using (_⟨×⟩_; Decidable)
 import Relation.Nullary.Reflects as Reflects
@@ -74,7 +78,7 @@ mapWith∈-cong : ∀ (xs : List A) → (f g : ∀ {x} → x ∈ xs → B) →
                 (∀ {x} → (x∈xs : x ∈ xs) → f x∈xs ≡ g x∈xs) →
                 mapWith∈ xs f ≡ mapWith∈ xs g
 mapWith∈-cong []       f g cong = refl
-mapWith∈-cong (x ∷ xs) f g cong = ≡.cong₂ _∷_ (cong (here refl))
+mapWith∈-cong (x ∷ xs) f g cong = cong₂ _∷_ (cong (here refl))
   (mapWith∈-cong xs (f ∘ there) (g ∘ there) (cong ∘ there))
 
 mapWith∈≗map : ∀ (f : A → B) xs → mapWith∈ xs (λ {x} _ → f x) ≡ map f xs
@@ -94,7 +98,7 @@ map-mapWith∈ = Membership.map-mapWith∈ (≡.setoid _)
 module _ (f : A → B) where
 
   ∈-map⁺ : ∀ {x xs} → x ∈ xs → f x ∈ map f xs
-  ∈-map⁺ = Membership.∈-map⁺ (≡.setoid A) (≡.setoid B) (≡.cong f)
+  ∈-map⁺ = Membership.∈-map⁺ (≡.setoid A) (≡.setoid B) (cong f)
 
   ∈-map⁻ : ∀ {y xs} → y ∈ map f xs → ∃ λ x → x ∈ xs × y ≡ f x
   ∈-map⁻ = Membership.∈-map⁻ (≡.setoid A) (≡.setoid B)
@@ -163,7 +167,7 @@ module _ (f : A → B → C) where
   ∈-cartesianProductWith⁺ : ∀ {xs ys a b} → a ∈ xs → b ∈ ys →
                             f a b ∈ cartesianProductWith f xs ys
   ∈-cartesianProductWith⁺ = Membership.∈-cartesianProductWith⁺
-    (≡.setoid A) (≡.setoid B) (≡.setoid C) (≡.cong₂ f)
+    (≡.setoid A) (≡.setoid B) (≡.setoid C) (cong₂ f)
 
   ∈-cartesianProductWith⁻ : ∀ xs ys {v} → v ∈ cartesianProductWith f xs ys →
                             ∃₂ λ a b → a ∈ xs × b ∈ ys × v ≡ f a b
@@ -243,10 +247,10 @@ module _ {n} {f : Fin n → A} where
 module _ {p} {P : A → Set p} (P? : Decidable P) where
 
   ∈-filter⁺ : ∀ {x xs} → x ∈ xs → P x → x ∈ filter P? xs
-  ∈-filter⁺ = Membership.∈-filter⁺ (≡.setoid A) P? (≡.subst P)
+  ∈-filter⁺ = Membership.∈-filter⁺ (≡.setoid A) P? (subst P)
 
   ∈-filter⁻ : ∀ {v xs} → v ∈ filter P? xs → v ∈ xs × P v
-  ∈-filter⁻ = Membership.∈-filter⁻ (≡.setoid A) P? (≡.subst P)
+  ∈-filter⁻ = Membership.∈-filter⁻ (≡.setoid A) P? (subst P)
 
 ------------------------------------------------------------------------
 -- derun and deduplicate
diff --git a/src/Data/List/NonEmpty/Properties.agda b/src/Data/List/NonEmpty/Properties.agda
index 68e9bfcf0e..c20f024c4a 100644
--- a/src/Data/List/NonEmpty/Properties.agda
+++ b/src/Data/List/NonEmpty/Properties.agda
@@ -8,24 +8,29 @@
 
 module Data.List.NonEmpty.Properties where
 
-open import Effect.Monad
-open import Data.Nat
-open import Data.Nat.Properties
+open import Effect.Monad using (RawMonad)
+open import Data.Nat.Base using (suc; _+_)
+open import Data.Nat.Properties using (suc-injective)
 open import Data.Maybe.Properties using (just-injective)
 open import Data.Bool.Base using (Bool; true; false)
 open import Data.List.Base as List using (List; []; _∷_; _++_)
 open import Data.List.Effectful using () renaming (monad to listMonad)
 open import Data.List.NonEmpty.Effectful using () renaming (monad to list⁺Monad)
 open import Data.List.NonEmpty
-open import Data.List.NonEmpty.Relation.Unary.All
+  using (List⁺; _∷_; tail; head; toList; _⁺++_; _⁺++⁺_; _++⁺_; length; fromList;
+    drop+; map; groupSeqs; ungroupSeqs)
+open import Data.List.NonEmpty.Relation.Unary.All using (All; toList⁺; _∷_)
 open import Data.List.Relation.Unary.All using ([]; _∷_) renaming (All to ListAll)
 import Data.List.Properties as List
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Data.Sum.Relation.Unary.All using (inj₁; inj₂)
 import Data.Sum.Relation.Unary.All as Sum using (All; inj₁; inj₂)
 open import Level using (Level)
-open import Function.Base
-open import Relation.Binary.PropositionalEquality
+open import Function.Base using (_∘_; _$_)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; _≗_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Unary using (Pred; Decidable; ∁)
 open import Relation.Nullary using (¬_; does; yes; no)
 
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 1dcf0d1a20..20ac353903 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -36,7 +36,8 @@ open import Function.Definitions using (Injective)
 open import Level using (Level)
 open import Relation.Binary.Definitions as B using (DecidableEquality)
 import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
-open import Relation.Binary.PropositionalEquality as ≡ hiding ([_])
+open import Relation.Binary.PropositionalEquality.Core as ≡
+open import Relation.Binary.PropositionalEquality.Properties as ≡
 open import Relation.Binary.Core using (Rel)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Nullary using (¬_; Dec; does; _because_; yes; no; contradiction)
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index f014df8b41..9c400fdcff 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -8,33 +8,39 @@
 
 module Data.List.Relation.Binary.BagAndSetEquality where
 
-open import Algebra using (Idempotent; CommutativeMonoid)
+open import Algebra.Bundles using (CommutativeMonoid)
+open import Algebra.Definitions using (Idempotent)
 open import Algebra.Structures.Biased using (isCommutativeMonoidˡ)
 open import Effect.Monad using (RawMonad)
-open import Data.Empty
-open import Data.Fin.Base
+open import Data.Empty using (⊥; ⊥-elim)
+open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base
+  using (List; []; _∷_; map; _++_; concat; [_]; lookup; length)
 open import Data.List.Effectful using (monad; module Applicative; module MonadProperties)
 import Data.List.Properties as List
 open import Data.List.Relation.Unary.Any using (Any; here; there)
-open import Data.List.Relation.Unary.Any.Properties hiding (++-comm)
+open import Data.List.Relation.Unary.Any.Properties
+  using (∷↔; map↔; Any-cong; ++↔; concat↔; >>=↔; ++↔++; ⊎↔; ⊥↔Any[])
 open import Data.List.Membership.Propositional using (_∈_)
 open import Data.List.Membership.Propositional.Properties
+  using (∈-∃++)
 open import Data.List.Relation.Binary.Subset.Propositional.Properties
   using (⊆-preorder)
 open import Data.List.Relation.Binary.Permutation.Propositional
+  using (_↭_; ↭-sym; refl; module PermutationReasoning)
 open import Data.List.Relation.Binary.Permutation.Propositional.Properties
-open import Data.Product.Base as Prod hiding (map)
+  using (∈-resp-↭; ∈-resp-[σ⁻¹∘σ]; ↭-sym-involutive; shift; ++-comm)
+open import Data.Product.Base as Prod using (∃; _,_; proj₁; proj₂; _×_)
 import Data.Product.Function.Dependent.Propositional as Σ
-open import Data.Sum.Base as Sum hiding (map)
-open import Data.Sum.Properties hiding (map-cong)
+open import Data.Sum.Base as Sum using (_⊎_; [_,_]′; inj₁; inj₂)
+open import Data.Sum.Properties using (inj₂-injective; inj₁-injective)
 open import Data.Sum.Function.Propositional using (_⊎-cong_)
-open import Data.Unit.Polymorphic.Base
-open import Function.Base
+open import Data.Unit.Polymorphic.Base using (⊤)
+open import Function.Base using (_∘_; _$_; id; _⟨_⟩_; case_of_)
 open import Function.Bundles using (_↔_; Inverse; Equivalence; mk↔ₛ′; mk⇔)
 open import Function.Related.Propositional as Related
   using (↔⇒; ⌊_⌋; ⌊_⌋→; ⇒→; K-refl; SK-sym)
-open import Function.Related.TypeIsomorphisms
+open import Function.Related.TypeIsomorphisms using (×-identityʳ; ∃∃↔∃∃)
 open import Function.Properties.Inverse using (↔-sym; ↔-trans; to-from)
 open import Level using (Level)
 open import Relation.Binary.Core using (_⇒_)
@@ -42,10 +48,12 @@ open import Relation.Binary.Definitions using (Trans)
 open import Relation.Binary.Bundles using (Preorder; Setoid)
 import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
-open import Relation.Binary.PropositionalEquality as ≡
-  using (_≡_; _≢_; _≗_; refl; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core as ≡
+  using (_≡_; _≢_; _≗_; refl)
+open import Relation.Binary.PropositionalEquality.Properties as ≡
+  using (module ≡-Reasoning)
 open import Relation.Binary.Reasoning.Syntax
-open import Relation.Nullary.Negation using (¬_)
+open import Relation.Nullary.Negation.Core using (¬_)
 
 private
   variable
diff --git a/src/Data/List/Relation/Binary/Equality/Setoid.agda b/src/Data/List/Relation/Binary/Equality/Setoid.agda
index af78f43c2c..0253c55460 100644
--- a/src/Data/List/Relation/Binary/Equality/Setoid.agda
+++ b/src/Data/List/Relation/Binary/Equality/Setoid.agda
@@ -6,22 +6,22 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Algebra.Core using (Op₂)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.Definitions using (Transitive; Symmetric; Reflexive; _Respects_)
-open import Relation.Binary.Structures using (IsEquivalence)
 
 module Data.List.Relation.Binary.Equality.Setoid {a ℓ} (S : Setoid a ℓ) where
-
+open import Algebra.Core using (Op₂)
 open import Data.Fin.Base using (Fin)
-open import Data.List.Base
+open import Data.List.Base using (List; length; map; foldr; _++_; concat;
+  tabulate; filter; _ʳ++_; reverse)
 open import Data.List.Relation.Binary.Pointwise as PW using (Pointwise)
 open import Data.List.Relation.Unary.Unique.Setoid S using (Unique)
 open import Function.Base using (_∘_)
-open import Level
-open import Relation.Binary.Core renaming (Rel to Rel₂)
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
+open import Level using (Level; _⊔_)
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_) renaming (Rel to Rel₂)
+open import Relation.Binary.Definitions using (Transitive; Symmetric; Reflexive; _Respects_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Properties.Setoid S using (≉-resp₂)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Unary as U using (Pred)
 
 open Setoid S renaming (Carrier to A)
diff --git a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
index d509b12be0..a1df8dfd6e 100644
--- a/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
+++ b/src/Data/List/Relation/Binary/Sublist/DecPropositional.agda
@@ -20,7 +20,8 @@ import Data.List.Relation.Binary.Sublist.DecSetoid as DecSetoidSublist
 import Data.List.Relation.Binary.Sublist.Propositional as PropositionalSublist
 open import Relation.Binary.Bundles using (DecPoset)
 open import Relation.Binary.Structures using (IsDecPartialOrder)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
 
 ------------------------------------------------------------------------
 -- Re-export core definitions and operations
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index 86194916f8..57ae74303b 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -25,16 +25,16 @@ open import Data.Maybe.Base as Maybe
 open import Data.Nat.Base as ℕ using (ℕ)
 open import Data.Product.Base using (Σ-syntax; _,_)
 open import Data.Vec.Base as Vec using (Vec ; lookup)
-open import Data.List.Base hiding (lookup)
-open import Data.List.Properties
+open import Data.List.Base using (List; []; _∷_; [_]; _++_)
+open import Data.List.Properties using (++-assoc; ++-identityʳ)
 open import Data.List.Relation.Binary.Sublist.Heterogeneous
-  hiding (lookup)
+  using (Sublist; minimum; _∷_)
 open import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
-open import Function
+open import Function.Base using (_$_; case_of_)
 
-open import Relation.Binary.PropositionalEquality as ≡
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≗_; sym; cong; cong₂; subst₂)
-open import Relation.Nullary
+open import Relation.Binary.PropositionalEquality.Properties as ≡
 
 open ≡.≡-Reasoning
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional.agda b/src/Data/List/Relation/Binary/Sublist/Propositional.agda
index a0f816ad84..b423548af2 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional.agda
@@ -18,7 +18,10 @@ open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Bundles using (Preorder; Poset)
 open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
 open import Relation.Binary.Definitions using (Antisymmetric)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (subst; _≡_; refl)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; isEquivalence)
 open import Relation.Unary using (Pred)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
index 163818b6e8..4235daf001 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Example/UniqueBoundVariables.agda
@@ -10,8 +10,10 @@
 
 module Data.List.Relation.Binary.Sublist.Propositional.Example.UniqueBoundVariables (Base : Set) where
 
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; sym; cong; subst; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; cong; subst)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 
 open import Data.List.Base using (List; []; _∷_; [_])
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
index c9d2ea6724..94ccb9038b 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
@@ -20,10 +20,13 @@ open import Data.List.Relation.Binary.Sublist.Propositional
 import Data.List.Relation.Binary.Sublist.Setoid.Properties
   as SetoidProperties
 open import Data.Product.Base using (∃; _,_; proj₂)
-open import Function.Base
+open import Function.Base using (id; _∘_; _∘′_)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (_Respects_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; _≗_; trans)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; subst-injective)
 open import Relation.Unary using (Pred)
 
 private
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda
index 949fded0ae..cf16aa0f30 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Slice.agda
@@ -11,7 +11,10 @@ module Data.List.Relation.Binary.Sublist.Propositional.Slice
 
 open import Data.List.Base using (List)
 open import Data.List.Relation.Binary.Sublist.Propositional
-open import Relation.Binary.PropositionalEquality
+ using (_⊆_; UpperBound; ⊆-trans; ∷ₙ-ub; _∷ʳ_; _∷ₗ-ub_; _∷_; _∷ᵣ-ub_;
+        _,_∷-ub_; ⊆-upper-bound; [])
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong)
 
 ------------------------------------------------------------------------
 -- A Union where the triangles commute is a
diff --git a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
index f4556c6760..75e1a5b00a 100644
--- a/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Propositional/Properties.agda
@@ -6,24 +6,25 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Definitions hiding (Decidable)
-open import Relation.Binary.Structures using (IsPreorder)
-
 module Data.List.Relation.Binary.Subset.Propositional.Properties
   where
 
 open import Data.Bool.Base using (Bool; true; false; T)
-open import Data.List.Base
+open import Data.List.Base using (List; map; _∷_; _++_; concat; applyUpTo;
+  any; filter)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Data.List.Relation.Unary.All using (All)
 import Data.List.Relation.Unary.Any.Properties as Any hiding (filter⁺)
-open import Data.List.Effectful
+open import Data.List.Effectful using (monad)
 open import Data.List.Relation.Unary.Any using (Any)
-open import Data.List.Membership.Propositional
+open import Data.List.Membership.Propositional using (_∈_; mapWith∈)
 open import Data.List.Membership.Propositional.Properties
+  using (map-∈↔; concat-∈↔; >>=-∈↔; ⊛-∈↔; ⊗-∈↔)
 import Data.List.Relation.Binary.Subset.Setoid.Properties as Subset
 open import Data.List.Relation.Binary.Subset.Propositional
+  using (_⊆_; _⊇_)
 open import Data.List.Relation.Binary.Permutation.Propositional
+  using (_↭_; ↭-sym; ↭-isEquivalence)
 import Data.List.Relation.Binary.Permutation.Propositional.Properties as Permutation
 open import Data.Nat.Base using (ℕ; _≤_)
 import Data.Product.Base as Product
@@ -36,8 +37,12 @@ open import Relation.Nullary using (¬_; yes; no)
 open import Relation.Unary using (Decidable; Pred) renaming (_⊆_ to _⋐_)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.Bundles using (Preorder)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; _≗_; isEquivalence; subst; resp; refl; setoid; module ≡-Reasoning)
+open import Relation.Binary.Definitions hiding (Decidable)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≗_; subst; resp; refl)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (isEquivalence; setoid; module ≡-Reasoning)
+open import Relation.Binary.Structures using (IsPreorder)
 import Relation.Binary.Reasoning.Preorder as ≲-Reasoning
 
 private
diff --git a/src/Data/List/Relation/Ternary/Appending/Propositional.agda b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
index f807a0d649..1d1ad63509 100644
--- a/src/Data/List/Relation/Ternary/Appending/Propositional.agda
+++ b/src/Data/List/Relation/Ternary/Appending/Propositional.agda
@@ -15,8 +15,10 @@ open import Data.Product.Base using (_,_)
 
 import Data.List.Properties as List
 import Data.List.Relation.Binary.Pointwise as Pw using (≡⇒Pointwise-≡; Pointwise-≡⇒≡)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; setoid; refl; trans; cong₂; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; trans; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning; setoid)
 
 import Data.List.Relation.Ternary.Appending.Setoid (setoid A) as General
 import Data.List.Relation.Ternary.Appending.Setoid.Properties (setoid A)
diff --git a/src/Data/List/Relation/Unary/AllPairs/Properties.agda b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
index 9b145c3834..173dd87b62 100644
--- a/src/Data/List/Relation/Unary/AllPairs/Properties.agda
+++ b/src/Data/List/Relation/Unary/AllPairs/Properties.agda
@@ -8,12 +8,13 @@
 
 module Data.List.Relation.Unary.AllPairs.Properties where
 
-open import Data.List.Base hiding (any)
+open import Data.List.Base using (List; []; _∷_; map; _++_; concat; take; drop;
+  applyUpTo; applyDownFrom; tabulate; filter; catMaybes)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.All.Properties as All using (Any-catMaybes⁺)
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs; []; _∷_)
 open import Data.Bool.Base using (true; false)
-open import Data.Maybe using (Maybe; nothing; just)
+open import Data.Maybe.Base using (Maybe; nothing; just)
 open import Data.Maybe.Relation.Binary.Pointwise using (pointwise⊆any; Pointwise)
 open import Data.Fin.Base as F using (Fin)
 open import Data.Fin.Properties using (suc-injective; <⇒≢)
@@ -25,7 +26,7 @@ open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (DecSetoid)
 open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Unary using (Pred; Decidable; _⊆_)
-open import Relation.Nullary.Decidable using (does)
+open import Relation.Nullary.Decidable.Core using (does)
 
 private
   variable
diff --git a/src/Data/List/Relation/Unary/Linked.agda b/src/Data/List/Relation/Unary/Linked.agda
index 38fcdd1507..dc1b7fb87d 100644
--- a/src/Data/List/Relation/Unary/Linked.agda
+++ b/src/Data/List/Relation/Unary/Linked.agda
@@ -19,7 +19,7 @@ open import Level using (Level; _⊔_)
 open import Relation.Binary.Definitions as B
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (refl; cong₂)
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
 open import Relation.Nullary.Decidable using (yes; no; map′; _×-dec_)
 
diff --git a/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda b/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
index e12b5cf47b..bcd7df1176 100644
--- a/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Unary/Unique/Propositional/Properties.agda
@@ -8,11 +8,14 @@
 
 module Data.List.Relation.Unary.Unique.Propositional.Properties where
 
-open import Data.List.Base
+open import Data.List.Base using (map; _++_; concat; cartesianProductWith;
+  cartesianProduct; drop; take; applyUpTo; upTo; applyDownFrom; downFrom;
+  tabulate; allFin; filter)
 open import Data.List.Relation.Binary.Disjoint.Propositional
+  using (Disjoint)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.AllPairs as AllPairs using (AllPairs)
-open import Data.List.Relation.Unary.Unique.Propositional
+open import Data.List.Relation.Unary.Unique.Propositional using (Unique)
 import Data.List.Relation.Unary.Unique.Setoid.Properties as Setoid
 open import Data.Fin.Base using (Fin)
 open import Data.Nat.Base using (_<_)
@@ -23,8 +26,9 @@ open import Function.Base using (id; _∘_)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality
-  using (refl; _≡_; _≢_; sym; setoid)
+open import Relation.Binary.PropositionalEquality.Core
+  using (refl; _≡_; _≢_; sym)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Negation using (¬_)
 
diff --git a/src/Data/List/Reverse.agda b/src/Data/List/Reverse.agda
index 0464181932..a197142c3f 100644
--- a/src/Data/List/Reverse.agda
+++ b/src/Data/List/Reverse.agda
@@ -8,10 +8,12 @@
 
 module Data.List.Reverse where
 
-open import Data.List.Base as L hiding (reverse)
+open import Data.List.Base as L using (List; []; _∷_; _∷ʳ_)
 open import Data.List.Properties
+  using (unfold-reverse; reverse-involutive)
 open import Function.Base using (_$_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (subst; sym)
 
 -- If you want to traverse a list from the end, then you can use the
 -- reverse view of it.
diff --git a/src/Data/List/Zipper/Properties.agda b/src/Data/List/Zipper/Properties.agda
index 5c4ce7c7d8..1548b5bde4 100644
--- a/src/Data/List/Zipper/Properties.agda
+++ b/src/Data/List/Zipper/Properties.agda
@@ -11,9 +11,14 @@ module Data.List.Zipper.Properties where
 open import Data.List.Base as List using (List ; [] ; _∷_)
 open import Data.List.Properties
 open import Data.List.Zipper
+  using (Zipper; toList; left; right; mkZipper; reverse; _ˡ++_; _ʳ++_;
+         _++ˡ_; _++ʳ_; map; foldr)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; sym)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 open import Function.Base using (_on_; _$′_; _$_; flip)
 
diff --git a/src/Data/Maybe/Properties.agda b/src/Data/Maybe/Properties.agda
index 212f1608ff..2b4f1567e8 100644
--- a/src/Data/Maybe/Properties.agda
+++ b/src/Data/Maybe/Properties.agda
@@ -8,17 +8,21 @@
 
 module Data.Maybe.Properties where
 
-open import Algebra.Bundles
+open import Algebra.Bundles using (Semigroup; Monoid)
 import Algebra.Structures as Structures
 import Algebra.Definitions as Definitions
-open import Data.Maybe.Base
+open import Data.Maybe.Base using (Maybe; just; nothing; map; _<∣>_;
+  maybe; maybe′)
 open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∋_; id; _∘_; _∘′_)
 open import Function.Definitions using (Injective)
 open import Level using (Level)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; _≗_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (isEquivalence)
 open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Decidable using (map′)
 
diff --git a/src/Data/Nat/Binary/Properties.agda b/src/Data/Nat/Binary/Properties.agda
index 949969033c..4669e479d9 100644
--- a/src/Data/Nat/Binary/Properties.agda
+++ b/src/Data/Nat/Binary/Properties.agda
@@ -11,30 +11,40 @@ module Data.Nat.Binary.Properties where
 open import Algebra.Bundles
 open import Algebra.Morphism.Structures
 import Algebra.Morphism.MonoidMonomorphism as MonoidMonomorphism
-open import Algebra.Consequences.Propositional
+open import Algebra.Consequences.Propositional using (comm∧distrˡ⇒distrʳ)
 open import Data.Bool.Base using (if_then_else_; Bool; true; false)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Nat.Binary.Base
-open import Data.Nat as ℕ using (ℕ; z≤n; s≤s; s<s⁻¹)
+open import Data.Nat.Base as ℕ using (ℕ; z≤n; s≤s; s<s⁻¹)
 open import Data.Nat.DivMod using (_%_; _/_; m/n≤m; +-distrib-/-∣ˡ)
 open import Data.Nat.Divisibility using (∣-refl)
-import Data.Nat.Base as ℕ
 import Data.Nat.Properties as ℕ
-open import Data.Nat.Solver
+open import Data.Nat.Solver using (module +-*-Solver)
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; ∃)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; _$_; id)
-open import Function.Definitions
+open import Function.Definitions using (Injective; Surjective;
+  Inverseˡ; Inverseʳ; Inverseᵇ)
 open import Function.Consequences.Propositional
+ using (strictlySurjective⇒surjective; strictlyInverseˡ⇒inverseˡ;
+        strictlyInverseʳ⇒inverseʳ)
 open import Level using (0ℓ)
 open import Relation.Binary
-open import Relation.Binary.Consequences
+open import Relation.Binary.Consequences using (trans∧irr⇒asym; tri⇒dec<)
 open import Relation.Binary.Morphism
+ using (IsRelHomomorphism; IsOrderHomomorphism; IsOrderMonomorphism)
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphism
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Algebra
+  using (magma; isMagma)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≢_; refl; cong; cong₂; sym; _≗_; trans; ≢-sym; subst₂;
+        subst; resp₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (isDecEquivalence; setoid; decSetoid; module ≡-Reasoning;
+         isEquivalence)
 open import Relation.Nullary using (¬_; yes; no)
 import Relation.Nullary.Decidable as Dec
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Negation.Core using (contradiction)
 
 open import Algebra.Definitions {A = ℕᵇ} _≡_
 open import Algebra.Structures {A = ℕᵇ} _≡_
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index 079f138ffa..5ab67ce7c1 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -8,12 +8,15 @@
 
 module Data.Nat.Binary.Subtraction where
 
-open import Algebra using (Op₂; Magma)
+open import Algebra.Core using (Op₂)
+open import Algebra.Bundles using (Magma)
 open import Algebra.Consequences.Propositional using (comm∧distrˡ⇒distrʳ)
 open import Algebra.Morphism.Consequences using (homomorphic₂-inv)
 open import Data.Bool.Base using (true; false)
-open import Data.Nat as ℕ using (ℕ)
-open import Data.Nat.Binary.Base
+open import Data.Nat.Base as ℕ using (ℕ)
+open import Data.Nat.Binary.Base using (ℕᵇ; 0ᵇ; 2[1+_]; 1+[2_]; double;
+  pred; toℕ; fromℕ; even<odd; odd<even; _≥_; _>_; _≤_; _<_; _+_; zero; suc; 1ᵇ;
+  _*_)
 open import Data.Nat.Binary.Properties
 import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂; ∃)
@@ -23,7 +26,11 @@ open import Function.Base using (_∘_; _$_)
 open import Level using (0ℓ)
 open import Relation.Binary
   using (Tri; tri<; tri≈; tri>; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Algebra using (magma)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; sym; trans; subst; _≢_)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Nullary using (Dec; yes; no; does)
 open import Relation.Nullary.Negation using (contradiction)
 
diff --git a/src/Data/Nat/Combinatorics.agda b/src/Data/Nat/Combinatorics.agda
index d07d22db47..9a1cae638f 100644
--- a/src/Data/Nat/Combinatorics.agda
+++ b/src/Data/Nat/Combinatorics.agda
@@ -8,12 +8,14 @@
 
 module Data.Nat.Combinatorics where
 
-open import Data.Nat.Base
-open import Data.Nat.DivMod
-open import Data.Nat.Divisibility
+open import Data.Nat.Base using (ℕ; zero; suc; _!; _∸_; z≤n; s≤s; _≤_;
+  _+_; _*_; _<_; s<s; ≢-nonZero)
+open import Data.Nat.DivMod using (_/_; n/1≡n; /-congˡ; /-congʳ; m*n/n≡m;
+  m/n/o≡m/[n*o]; n/n≡1; +-distrib-/-∣ˡ; m*n/m*o≡n/o)
+open import Data.Nat.Divisibility using (_∣_; *-monoʳ-∣)
 open import Data.Nat.Properties
-open import Relation.Binary.Definitions
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.Definitions using (tri>; tri≈; tri<)
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; sym; cong; subst)
 
 import Data.Nat.Combinatorics.Base          as Base
diff --git a/src/Data/Nat/Combinatorics/Specification.agda b/src/Data/Nat/Combinatorics/Specification.agda
index ac1c6ca1b4..cf7f826522 100644
--- a/src/Data/Nat/Combinatorics/Specification.agda
+++ b/src/Data/Nat/Combinatorics/Specification.agda
@@ -11,18 +11,19 @@
 module Data.Nat.Combinatorics.Specification where
 
 open import Data.Bool.Base using (T; true; false)
-open import Data.Nat.Base
-open import Data.Nat.DivMod
-open import Data.Nat.Divisibility
+open import Data.Nat.Base using (zero; suc; _≤ᵇ_; _≤_; _!; _∸_; pred;
+  >-nonZero; _*_; NonZero; _+_; s≤s; z≤n; _>_)
+open import Data.Nat.DivMod using (_/_; n/n≡1; /-congʳ; m*n/m!≡n/[m∸1]!;
+  *-/-assoc; n/1≡n; m/n/o≡m/[n*o]; /-congˡ)
+open import Data.Nat.Divisibility using (m≤n⇒m!∣n!; _∣_; ∣-refl;
+  ∣-reflexive; module ∣-Reasoning; ∣m∣n⇒∣m+n; *-monoʳ-∣; m∣n/o⇒o*m∣n)
 open import Data.Nat.Properties
 open import Data.Nat.Combinatorics.Base
 open import Data.Sum.Base using (inj₁; inj₂)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; trans; _≢_)
 open import Relation.Nullary.Decidable using (yes; no; does)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary.PropositionalEquality
-  using (subst; refl; sym; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; trans; _≢_; subst; refl; sym; cong; cong₂)
 
 import Algebra.Properties.CommutativeSemigroup *-commutativeSemigroup as *-CS
 
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index 5ff23e4c72..cddb2768f7 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -19,7 +19,8 @@ open import Data.Nat.Induction
 open import Data.Nat.Properties
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (_$_; _∘_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; cong; cong₂; refl; trans; _≢_; sym)
 open import Relation.Nullary.Negation using (contradiction)
 
 open ≤-Reasoning
diff --git a/src/Data/Nat/DivMod/Core.agda b/src/Data/Nat/DivMod/Core.agda
index a49318fbcd..7e627f4bfa 100644
--- a/src/Data/Nat/DivMod/Core.agda
+++ b/src/Data/Nat/DivMod/Core.agda
@@ -11,11 +11,13 @@ module Data.Nat.DivMod.Core where
 open import Agda.Builtin.Nat using ()
   renaming (div-helper to divₕ; mod-helper to modₕ)
 
-open import Data.Nat.Base
+open import Data.Nat.Base using (zero; suc; _+_; _*_; _∸_; _≤_; _<_;
+ _≥_; z≤n; s≤s)
 open import Data.Nat.Properties
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Product.Base using (_×_; _,_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; sym; subst; trans)
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 
diff --git a/src/Data/Nat/DivMod/WithK.agda b/src/Data/Nat/DivMod/WithK.agda
index 10ef4c4f21..c161d7ab3e 100644
--- a/src/Data/Nat/DivMod/WithK.agda
+++ b/src/Data/Nat/DivMod/WithK.agda
@@ -8,16 +8,17 @@
 
 module Data.Nat.DivMod.WithK where
 
-open import Data.Nat using (ℕ; NonZero; _+_; _*_)
+open import Data.Nat.Base using (ℕ; NonZero; _+_; _*_)
 open import Data.Nat.DivMod hiding (_mod_; _divMod_)
 open import Data.Nat.Properties using (≤⇒≤″)
-open import Data.Nat.WithK
+open import Data.Nat.WithK using (≤″-erase)
 open import Data.Fin.Base using (Fin; toℕ; fromℕ<″)
 open import Data.Fin.Properties using (toℕ-fromℕ<″)
 open import Function.Base using (_$_)
-open import Relation.Binary.PropositionalEquality
-  using (cong; module ≡-Reasoning)
-open import Relation.Binary.PropositionalEquality.WithK
+open import Relation.Binary.PropositionalEquality.Core using (cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.WithK using (≡-erase)
 
 open ≡-Reasoning
 
diff --git a/src/Data/Nat/GCD/Lemmas.agda b/src/Data/Nat/GCD/Lemmas.agda
index 21788f0d88..5f358e6cf3 100644
--- a/src/Data/Nat/GCD/Lemmas.agda
+++ b/src/Data/Nat/GCD/Lemmas.agda
@@ -11,7 +11,10 @@ module Data.Nat.GCD.Lemmas where
 open import Data.Nat.Base
 open import Data.Nat.Properties
 open import Function.Base using (_$_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; sym)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 open ≡-Reasoning
 
diff --git a/src/Data/Nat/GeneralisedArithmetic.agda b/src/Data/Nat/GeneralisedArithmetic.agda
index 38d9472092..a5b1d7700f 100644
--- a/src/Data/Nat/GeneralisedArithmetic.agda
+++ b/src/Data/Nat/GeneralisedArithmetic.agda
@@ -8,11 +8,15 @@
 
 module Data.Nat.GeneralisedArithmetic where
 
-open import Data.Nat.Base
-open import Data.Nat.Properties
+open import Data.Nat.Base using (ℕ; zero; suc; _+_; _*_; _^_)
+open import Data.Nat.Properties using (+-comm; +-assoc; *-identityˡ;
+  *-assoc)
 open import Function.Base using (_∘′_; _∘_; id)
 open import Level using (Level)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; sym)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 
 private
diff --git a/src/Data/Nat/InfinitelyOften.agda b/src/Data/Nat/InfinitelyOften.agda
index 6d86336193..38a4c3eecd 100644
--- a/src/Data/Nat/InfinitelyOften.agda
+++ b/src/Data/Nat/InfinitelyOften.agda
@@ -11,12 +11,12 @@ module Data.Nat.InfinitelyOften where
 open import Effect.Monad using (RawMonad)
 open import Level using (Level; 0ℓ)
 open import Data.Empty using (⊥-elim)
-open import Data.Nat.Base
+open import Data.Nat.Base using (ℕ; _≤_; _⊔_; _+_; suc; zero; s≤s)
 open import Data.Nat.Properties
 open import Data.Product.Base as Prod hiding (map)
 open import Data.Sum.Base using (inj₁; inj₂; _⊎_)
 open import Function.Base using (_∘_; id)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≢_)
 open import Relation.Nullary.Negation using (¬_)
 open import Relation.Nullary.Negation using (¬¬-Monad; call/cc)
 open import Relation.Unary using (Pred; _∪_; _⊆_)
diff --git a/src/Data/Nat/Logarithm/Core.agda b/src/Data/Nat/Logarithm/Core.agda
index 397dd3b3b9..fea77bf42d 100644
--- a/src/Data/Nat/Logarithm/Core.agda
+++ b/src/Data/Nat/Logarithm/Core.agda
@@ -8,12 +8,15 @@
 
 module Data.Nat.Logarithm.Core where
 
-open import Data.Nat.Base
+open import Data.Nat.Base using (ℕ; _<_; zero; suc; _+_; ⌊_/2⌋; ⌈_/2⌉;
+  _≤_; z≤n; s≤s; _∸_; NonZero; _*_; _^_)
 open import Data.Nat.Properties
 open import Data.Nat.Induction using (<-wellFounded)
 open import Induction.WellFounded using (Acc; acc)
-open import Relation.Binary.PropositionalEquality
-open import Data.Unit
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; sym)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 ------------------------------------------------------------------------
 -- Logarithm base 2
diff --git a/src/Data/Nat/Primality.agda b/src/Data/Nat/Primality.agda
index 1a491fdf93..b07883ce9b 100644
--- a/src/Data/Nat/Primality.agda
+++ b/src/Data/Nat/Primality.agda
@@ -23,7 +23,7 @@ open import Relation.Nullary.Decidable as Dec
 open import Relation.Nullary.Negation using (¬_; contradiction; contradiction₂)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; _≢_; refl; cong)
 
 private
diff --git a/src/Data/Nat/Primality/Factorisation.agda b/src/Data/Nat/Primality/Factorisation.agda
index 66e75dd4df..4a91348a66 100644
--- a/src/Data/Nat/Primality/Factorisation.agda
+++ b/src/Data/Nat/Primality/Factorisation.agda
@@ -33,8 +33,10 @@ open import Induction using (build)
 open import Induction.Lexicographic using (_⊗_; [_⊗_])
 open import Relation.Nullary.Decidable using (yes; no)
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; sym; trans; cong; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; trans; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
 private
   variable
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index e58385f587..9bca65bacc 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -11,11 +11,14 @@
 
 module Data.Nat.Properties where
 
-open import Axiom.UniquenessOfIdentityProofs
-open import Algebra.Bundles
+open import Axiom.UniquenessOfIdentityProofs using (module Decidable⇒UIP)
+open import Algebra.Bundles using (Magma; Semigroup; CommutativeSemigroup;
+  CommutativeMonoid; Monoid; Semiring; CommutativeSemiring; CommutativeSemiringWithoutOne)
 open import Algebra.Morphism
 open import Algebra.Consequences.Propositional
+  using (comm+cancelˡ⇒cancelʳ; comm∧distrʳ⇒distrˡ; comm∧distrˡ⇒distrʳ)
 open import Algebra.Construct.NaturalChoice.Base
+  using (MinOperator; MaxOperator)
 import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp
 import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
 import Algebra.Properties.CommutativeSemigroup as CommSemigroupProperties
@@ -23,11 +26,13 @@ open import Data.Bool.Base using (Bool; false; true; T)
 open import Data.Bool.Properties using (T?)
 open import Data.Nat.Base
 open import Data.Product.Base using (∃; _×_; _,_)
-open import Data.Sum.Base as Sum
+open import Data.Sum.Base as Sum using (inj₁; inj₂; _⊎_; [_,_]′)
 open import Data.Unit.Base using (tt)
-open import Function.Base
+open import Function.Base using (_∘_; flip; _$_; id; _∘′_; _$′_)
 open import Function.Bundles using (_↣_)
-open import Function.Metric.Nat
+open import Function.Metric.Nat using (TriangleInequality; IsProtoMetric; IsPreMetric;
+  IsQuasiSemiMetric; IsSemiMetric; IsMetric; PreMetric; QuasiSemiMetric;
+  SemiMetric; Metric)
 open import Level using (0ℓ)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary.Core
diff --git a/src/Data/Nat/Tactic/RingSolver.agda b/src/Data/Nat/Tactic/RingSolver.agda
index e55bf6c963..1d5c2036f5 100644
--- a/src/Data/Nat/Tactic/RingSolver.agda
+++ b/src/Data/Nat/Tactic/RingSolver.agda
@@ -10,14 +10,14 @@
 
 module Data.Nat.Tactic.RingSolver where
 
-open import Agda.Builtin.Reflection
+open import Agda.Builtin.Reflection using (Term; TC)
 
 open import Data.Maybe.Base using (just; nothing)
-open import Data.Nat.Base using (zero; suc)
-open import Data.Nat.Properties
+open import Data.Nat.Base using (zero)
+open import Data.Nat.Properties using (+-*-commutativeSemiring)
 open import Level using (0ℓ)
 open import Data.Unit.Base using (⊤)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (refl)
 
 import Tactic.RingSolver as Solver
 import Tactic.RingSolver.Core.AlmostCommutativeRing as ACR
diff --git a/src/Data/Parity/Properties.agda b/src/Data/Parity/Properties.agda
index 079257482b..ebca1ef000 100644
--- a/src/Data/Parity/Properties.agda
+++ b/src/Data/Parity/Properties.agda
@@ -9,21 +9,23 @@
 module Data.Parity.Properties where
 
 open import Algebra.Bundles
-open import Data.Empty
 open import Data.Nat.Base as ℕ using (zero; suc; parity)
 open import Data.Parity.Base as ℙ using (Parity; 0ℙ; 1ℙ; _⁻¹; toSign; fromSign)
 open import Data.Product.Base using (_,_)
-open import Data.Sign.Base as 𝕊
+import Data.Sign.Base as 𝕊
 open import Function.Base using (_$_; id)
 open import Function.Definitions
 open import Function.Consequences.Propositional
+  using (inverseʳ⇒injective; inverseˡ⇒surjective)
 open import Level using (0ℓ)
 open import Relation.Binary
   using (Decidable; DecidableEquality; Setoid; DecSetoid; IsDecEquivalence)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; _≢_; refl; sym; cong; cong₂; module ≡-Reasoning
-        ; setoid; isEquivalence; decSetoid; isDecEquivalence)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≢_; refl; sym; cong; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning; setoid; isEquivalence; decSetoid; isDecEquivalence)
 open import Relation.Nullary using (yes; no)
+open import Relation.Nullary.Negation.Core using (contradiction)
 
 open import Algebra.Structures {A = Parity} _≡_
 open import Algebra.Definitions {A = Parity} _≡_
@@ -129,8 +131,8 @@ p+p≡0ℙ 1ℙ = refl
 
 +-cancelʳ-≡ : RightCancellative ℙ._+_
 +-cancelʳ-≡ _ 1ℙ 1ℙ _  = refl
-+-cancelʳ-≡ _ 1ℙ 0ℙ eq = ⊥-elim (p≢p⁻¹ _ $ sym eq)
-+-cancelʳ-≡ _ 0ℙ 1ℙ eq = ⊥-elim (p≢p⁻¹ _ eq)
++-cancelʳ-≡ _ 1ℙ 0ℙ eq = contradiction (sym eq) (p≢p⁻¹ _)
++-cancelʳ-≡ _ 0ℙ 1ℙ eq = contradiction eq (p≢p⁻¹ _)
 +-cancelʳ-≡ _ 0ℙ 0ℙ _  = refl
 
 +-cancelˡ-≡ : LeftCancellative ℙ._+_
@@ -407,8 +409,8 @@ toSign-inverseʳ {0ℙ} refl = refl
 toSign-inverseʳ {1ℙ} refl = refl
 
 toSign-inverseˡ : Inverseˡ _≡_ _≡_ toSign fromSign
-toSign-inverseˡ { + }  refl = refl
-toSign-inverseˡ { - } refl = refl
+toSign-inverseˡ { 𝕊.+ } refl = refl
+toSign-inverseˡ { 𝕊.- } refl = refl
 
 toSign-injective : Injective _≡_ _≡_ toSign
 toSign-injective = inverseʳ⇒injective toSign toSign-inverseʳ
diff --git a/src/Data/Product/Function/Dependent/Propositional.agda b/src/Data/Product/Function/Dependent/Propositional.agda
index 2063e1cfbb..cb8a8d99a1 100644
--- a/src/Data/Product/Function/Dependent/Propositional.agda
+++ b/src/Data/Product/Function/Dependent/Propositional.agda
@@ -9,22 +9,25 @@
 
 module Data.Product.Function.Dependent.Propositional where
 
-open import Data.Product.Base as Product
-open import Data.Product.Function.NonDependent.Setoid using ()
-open import Data.Product.Relation.Binary.Pointwise.NonDependent using ()
+open import Data.Product.Base as Product using (Σ; map; proj₂; _,_)
 open import Data.Product.Properties using (Σ-≡,≡→≡; Σ-≡,≡↔≡; Σ-≡,≡←≡)
 open import Level using (Level; 0ℓ)
-open import Function.Related.TypeIsomorphisms
 open import Function.Related.Propositional
-open import Function.Base
-open import Function.Properties.Inverse
-open import Function.Properties.RightInverse
+  using (_∼[_]_; module EquationalReasoning; K-reflexive;
+         implication; reverseImplication; equivalence; injection;
+         reverseInjection; leftInverse; surjection; bijection)
+open import Function.Base using (_$_; _∘_; _∘′_)
+open import Function.Properties.Inverse using (↔⇒↠; ↔⇒⟶; ↔⇒⟵; ↔-sym; ↔⇒↩)
+open import Function.Properties.RightInverse using (↩⇒↪; ↪⇒↩)
 open import Function.Properties.Inverse.HalfAdjointEquivalence
+  using (↔⇒≃; _≃_; ≃⇒↔)
 open import Function.Consequences.Propositional
   using (inverseʳ⇒injective; strictlySurjective⇒surjective)
 open import Function.Definitions using (Inverseˡ; Inverseʳ; Injective; StrictlySurjective)
 open import Function.Bundles
-open import Relation.Binary.PropositionalEquality as ≡ using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties as ≡
+  using (module ≡-Reasoning)
 
 private
   variable
diff --git a/src/Data/Product/Function/NonDependent/Propositional.agda b/src/Data/Product/Function/NonDependent/Propositional.agda
index 5e7f3d1496..1552564ed0 100644
--- a/src/Data/Product/Function/NonDependent/Propositional.agda
+++ b/src/Data/Product/Function/NonDependent/Propositional.agda
@@ -12,13 +12,20 @@ module Data.Product.Function.NonDependent.Propositional where
 open import Data.Product.Base using (_×_; map)
 open import Data.Product.Function.NonDependent.Setoid
 open import Data.Product.Relation.Binary.Pointwise.NonDependent
-open import Function
+  using (_×ₛ_; Pointwise-≡↔≡)
+open import Function.Base using (id)
+open import Function.Bundles
+  using (Inverse; _⟶_; _⇔_; _↣_; _↠_; _⤖_; _↩_; _↪_; _↔_)
 open import Function.Properties.Inverse as Inv
+  using (Inverse⇒Equivalence; Inverse⇒Injection; Inverse⇒Surjection;
+         Inverse⇒Bijection)
 open import Function.Related.Propositional
-open import Function.Construct.Composition as Compose
+  using (_∼[_]_; implication; reverseImplication; equivalence; injection;
+         reverseInjection; leftInverse; surjection; bijection)
+import Function.Construct.Composition as Compose
 open import Level using (Level; _⊔_)
 open import Relation.Binary hiding (_⇔_)
-open import Relation.Binary.PropositionalEquality using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 private
   variable
@@ -41,7 +48,7 @@ private
 ------------------------------------------------------------------------
 -- Combinators for various function types
 
-infixr 2 _×-⇔_ _×-↣_ _×-↠_ _×-⤖_ _×-↩_ _×-↪_ _×-↔_
+infixr 2 _×-⟶_ _×-⇔_ _×-↣_ _×-↠_ _×-⤖_ _×-↩_ _×-↪_ _×-↔_
 
 _×-⟶_ : A ⟶ B → C ⟶ D → (A × C) ⟶ (B × D)
 _×-⟶_ = liftViaInverse Compose.function Inv.toFunction _×-function_
diff --git a/src/Data/Product/Function/NonDependent/Setoid.agda b/src/Data/Product/Function/NonDependent/Setoid.agda
index 95951951b4..861b960c36 100644
--- a/src/Data/Product/Function/NonDependent/Setoid.agda
+++ b/src/Data/Product/Function/NonDependent/Setoid.agda
@@ -10,10 +10,12 @@
 module Data.Product.Function.NonDependent.Setoid where
 
 open import Data.Product.Base as Product
-open import Data.Product.Relation.Binary.Pointwise.NonDependent
+open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 open import Level using (Level)
-open import Relation.Binary
-open import Function
+open import Relation.Binary.Bundles using (Setoid)
+open import Function.Bundles
+  using (Func; Equivalence; Injection; Surjection; Bijection; LeftInverse;
+         RightInverse; Inverse)
 
 private
   variable
diff --git a/src/Data/Product/Properties/WithK.agda b/src/Data/Product/Properties/WithK.agda
index 329cdaf675..593f18ab37 100644
--- a/src/Data/Product/Properties/WithK.agda
+++ b/src/Data/Product/Properties/WithK.agda
@@ -10,7 +10,8 @@ module Data.Product.Properties.WithK where
 
 open import Data.Product.Base using (Σ; _,_)
 open import Function.Base using (_∋_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- Equality
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index 43088288e2..a00ef23f82 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -30,7 +30,7 @@ open import Data.Integer.Solver renaming (module +-*-Solver to ℤ-solver)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 import Data.Nat.Properties as ℕ
 open import Data.Nat.Coprimality as C using (Coprime; coprime?)
-open import Data.Nat.Divisibility
+open import Data.Nat.Divisibility using (_∣_; divides; ∣-antisym; *-pres-∣)
 import Data.Nat.GCD as ℕ
 import Data.Nat.DivMod as ℕ
 open import Data.Product.Base using (proj₁; proj₂; _×_; _,_; uncurry)
@@ -43,19 +43,22 @@ open import Data.Rational.Unnormalised.Base as ℚᵘ
   ; _+_ to _+ᵘ_
   )
 import Data.Rational.Unnormalised.Properties as ℚᵘ
-open import Data.Sum.Base as Sum
-import Data.Sign.Base as S
+open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′; _⊎_)
+import Data.Sign.Base as Sign
 open import Function.Base using (_∘_; _∘′_; _∘₂_; _$_; flip)
 open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
 open import Relation.Binary
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; sym; trans; _≢_; subst; subst₂; resp₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; decSetoid; module ≡-Reasoning; isEquivalence)
 open import Relation.Binary.Morphism.Structures
 import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms
 open import Relation.Nullary.Decidable.Core as Dec
   using (yes; no; recompute; map′; _×-dec_)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
-open import Relation.Binary.Reasoning.Syntax
+open import Relation.Binary.Reasoning.Syntax using (module ≃-syntax)
 
 open import Algebra.Definitions {A = ℚ} _≡_
 open import Algebra.Structures  {A = ℚ} _≡_
@@ -254,8 +257,8 @@ normalize-coprime {n} {d-1} c = begin
 ↥-normalize i n = begin
   ↥ (normalize i n) ℤ.* + g  ≡⟨ cong (ℤ._* + g) (↥-mkℚ+ _ (n ℕ./ g)) ⟩
   + i/g     ℤ.* + g          ≡⟨⟩
-  S.+ ◃ i/g ℕ.* g            ≡⟨ cong (S.+ ◃_) (ℕ.m/n*n≡m (ℕ.gcd[m,n]∣m i n)) ⟩
-  S.+ ◃ i                    ≡⟨ ℤ.+◃n≡+n i ⟩
+  Sign.+ ◃ i/g ℕ.* g         ≡⟨ cong (Sign.+ ◃_) (ℕ.m/n*n≡m (ℕ.gcd[m,n]∣m i n)) ⟩
+  Sign.+ ◃ i                 ≡⟨ ℤ.+◃n≡+n i ⟩
   + i                        ∎
   where
   open ≡-Reasoning
@@ -268,8 +271,8 @@ normalize-coprime {n} {d-1} c = begin
 ↧-normalize i n = begin
   ↧ (normalize i n) ℤ.* + g  ≡⟨ cong (ℤ._* + g) (↧-mkℚ+ _ (n ℕ./ g)) ⟩
   + (n ℕ./ g)       ℤ.* + g  ≡⟨⟩
-  S.+ ◃ n ℕ./ g     ℕ.* g    ≡⟨ cong (S.+ ◃_) (ℕ.m/n*n≡m (ℕ.gcd[m,n]∣n i n)) ⟩
-  S.+ ◃ n                    ≡⟨ ℤ.+◃n≡+n n ⟩
+  Sign.+ ◃ n ℕ./ g     ℕ.* g ≡⟨ cong (Sign.+ ◃_) (ℕ.m/n*n≡m (ℕ.gcd[m,n]∣n i n)) ⟩
+  Sign.+ ◃ n                 ≡⟨ ℤ.+◃n≡+n n ⟩
   + n                        ∎
   where
   open ≡-Reasoning
@@ -352,7 +355,7 @@ normalize-injective-≃ m n c d eq = ℕ./-cancelʳ-≡
   ↥ (- norm)   ℤ.* + g  ≡⟨ cong (ℤ._* + g) (↥-neg norm) ⟩
   ℤ.- (↥ norm) ℤ.* + g  ≡⟨ sym (ℤ.neg-distribˡ-* (↥ norm) (+ g)) ⟩
   ℤ.- (↥ norm  ℤ.* + g) ≡⟨ cong (ℤ.-_) (↥-normalize (suc m) n) ⟩
-  S.- ◃ suc m           ≡⟨⟩
+  Sign.- ◃ suc m        ≡⟨⟩
   -[1+ m ]              ∎
   where
   open ℤ.≤-Reasoning
@@ -401,13 +404,13 @@ private
 
 /-injective-≃ : ∀ p q → ↥ᵘ p / ↧ₙᵘ p ≡ ↥ᵘ q / ↧ₙᵘ q → p ≃ᵘ q
 /-injective-≃ (mkℚᵘ (+ m)    c-1) (mkℚᵘ (+ n)    d-1) eq =
-  *≡* (cong (S.+ ◃_) (normalize-injective-≃ m n _ _ eq))
+  *≡* (cong (Sign.+ ◃_) (normalize-injective-≃ m n _ _ eq))
 /-injective-≃ (mkℚᵘ (+ m)    c-1) (mkℚᵘ -[1+ n ] d-1) eq =
   ℚᵘ.≃-sym (/-injective-≃-helper (sym eq))
 /-injective-≃ (mkℚᵘ -[1+ m ] c-1) (mkℚᵘ (+ n)    d-1) eq =
   /-injective-≃-helper eq
 /-injective-≃ (mkℚᵘ -[1+ m ] c-1) (mkℚᵘ -[1+ n ] d-1) eq =
-  *≡* (cong (S.- ◃_) (normalize-injective-≃ (suc m) (suc n) _ _ (neg-injective eq)))
+  *≡* (cong (Sign.- ◃_) (normalize-injective-≃ (suc m) (suc n) _ _ (neg-injective eq)))
 
 ------------------------------------------------------------------------
 -- Properties of toℚ/fromℚ
diff --git a/src/Data/Record.agda b/src/Data/Record.agda
index bd916a9005..709ce69a67 100644
--- a/src/Data/Record.agda
+++ b/src/Data/Record.agda
@@ -10,15 +10,14 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Data.Bool.Base using (true; false; if_then_else_)
-open import Data.Empty
-open import Data.List.Base
+open import Data.Empty using (⊥)
+open import Data.List.Base using (List; []; _∷_; foldr)
 open import Data.Product.Base hiding (proj₁; proj₂)
-open import Data.Unit.Polymorphic
+open import Data.Unit.Polymorphic using (⊤)
 open import Function.Base using (id; _∘_)
-open import Level
+open import Level using (suc; _⊔_)
 open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Nullary
-open import Relation.Nullary.Decidable
+open import Relation.Nullary.Decidable using (does)
 
 -- The module is parametrised by the type of labels, which should come
 -- with decidable equality.
diff --git a/src/Data/Sign/Properties.agda b/src/Data/Sign/Properties.agda
index 7ac07167e7..f72f5ce476 100644
--- a/src/Data/Sign/Properties.agda
+++ b/src/Data/Sign/Properties.agda
@@ -8,17 +8,21 @@
 
 module Data.Sign.Properties where
 
-open import Algebra.Bundles
-open import Data.Empty
-open import Data.Sign.Base
+open import Algebra.Bundles using (Magma; Semigroup; CommutativeSemigroup;
+  Monoid; CommutativeMonoid; Group; AbelianGroup)
+open import Data.Sign.Base using (Sign; opposite; _*_; +; -)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_$_; id)
 open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
 open import Relation.Binary
   using (Decidable; DecidableEquality; Setoid; DecSetoid; IsDecEquivalence)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; _≢_; sym; cong₂)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; decSetoid; isDecEquivalence; isEquivalence)
 open import Relation.Nullary.Decidable using (yes; no)
+open import Relation.Nullary.Negation.Core using (contradiction)
 
 open import Algebra.Structures {A = Sign} _≡_
 open import Algebra.Definitions {A = Sign} _≡_
@@ -102,8 +106,8 @@ s*s≡+ - = refl
 
 *-cancelʳ-≡ : RightCancellative _*_
 *-cancelʳ-≡ _ - - _  = refl
-*-cancelʳ-≡ _ - + eq = ⊥-elim (s≢opposite[s] _ $ sym eq)
-*-cancelʳ-≡ _ + - eq = ⊥-elim (s≢opposite[s] _ eq)
+*-cancelʳ-≡ _ - + eq = contradiction (sym eq) (s≢opposite[s] _)
+*-cancelʳ-≡ _ + - eq = contradiction eq (s≢opposite[s] _)
 *-cancelʳ-≡ _ + + _  = refl
 
 *-cancelˡ-≡ : LeftCancellative _*_
diff --git a/src/Data/Star/Environment.agda b/src/Data/Star/Environment.agda
index 6d30db7c3c..67ac8e8abd 100644
--- a/src/Data/Star/Environment.agda
+++ b/src/Data/Star/Environment.agda
@@ -12,10 +12,11 @@ open import Level using (_⊔_)
 open import Data.Star.List using (List)
 open import Data.Star.Decoration using (All)
 open import Data.Star.Pointer as Pointer using (Any; this; that; result)
-open import Data.Unit.Polymorphic using (⊤)
+open import Data.Unit.Polymorphic.Base using (⊤)
 open import Function.Base using (const)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
+  using (_▻_)
 
 -- Contexts, listing the types of all the elements in an environment.
 
diff --git a/src/Data/Sum/Function/Propositional.agda b/src/Data/Sum/Function/Propositional.agda
index 25bb1a0907..e2b1b10000 100644
--- a/src/Data/Sum/Function/Propositional.agda
+++ b/src/Data/Sum/Function/Propositional.agda
@@ -13,6 +13,8 @@ open import Data.Sum.Function.Setoid
 open import Data.Sum.Relation.Binary.Pointwise using (Pointwise-≡↔≡; _⊎ₛ_)
 open import Function.Construct.Composition as Compose
 open import Function.Related.Propositional
+  using (_∼[_]_; implication; reverseImplication; equivalence; injection;
+    reverseInjection; leftInverse; surjection; bijection)
 open import Function.Base using (id)
 open import Function.Bundles
   using (Inverse; _⟶_; _⇔_; _↣_; _↠_; _↩_; _↪_; _⤖_; _↔_)
@@ -20,7 +22,7 @@ open import Function.Properties.Inverse as Inv
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (REL)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 private
   variable
diff --git a/src/Data/Sum/Properties.agda b/src/Data/Sum/Properties.agda
index 72dc3aa641..a19a1daae6 100644
--- a/src/Data/Sum/Properties.agda
+++ b/src/Data/Sum/Properties.agda
@@ -8,15 +8,15 @@
 
 module Data.Sum.Properties where
 
-open import Level
-open import Data.Sum.Base
+open import Level using (Level)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂; swap; [_,_]; map;
+  map₁; map₂; assocˡ; assocʳ)
 open import Function.Base using (_∋_; _∘_; id)
 open import Function.Bundles using (mk↔ₛ′; _↔_)
 open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality
-open import Relation.Nullary.Decidable using (yes; no)
-open import Relation.Nullary.Decidable using (map′)
-
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≗_; refl; cong)
+open import Relation.Nullary.Decidable.Core using (yes; no; map′)
 
 private
   variable
diff --git a/src/Data/These/Properties.agda b/src/Data/These/Properties.agda
index 9f6cb55ce1..79d35e53a6 100644
--- a/src/Data/These/Properties.agda
+++ b/src/Data/These/Properties.agda
@@ -9,10 +9,11 @@
 module Data.These.Properties where
 
 open import Data.Product.Base using (_×_; _,_; <_,_>; uncurry)
-open import Data.These.Base
+open import Data.These.Base using (These; this; that; these)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂)
 open import Relation.Nullary.Decidable using (yes; no; map′; _×-dec_)
 
 ------------------------------------------------------------------------
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional.agda b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
index 0a3ecd2337..7f5151600f 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional.agda
@@ -24,7 +24,7 @@ open import Level using (Level)
 open import Relation.Binary.Definitions using (Transitive; Symmetric; _Respectsˡ_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Intersection using (_∩_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; cong) renaming (refl to ≡-refl; sym to ≡-sym; trans to ≡-trans)
 open import Relation.Nullary using (Reflects; ¬_; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
index 3c98800c66..fefb27764f 100644
--- a/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
+++ b/src/Data/Tree/AVL/Map/Membership/Propositional/Properties.agda
@@ -24,7 +24,7 @@ open import Level using (Level)
 open import Relation.Binary.Definitions using (Transitive; Symmetric; _Respectsˡ_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Construct.Intersection using (_∩_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; cong) renaming (refl to ≡-refl; sym to ≡-sym; trans to ≡-trans)
 open import Relation.Nullary using (Reflects; ¬_; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
diff --git a/src/Data/Tree/Binary/Properties.agda b/src/Data/Tree/Binary/Properties.agda
index b523b1e6f1..270e6c66a2 100644
--- a/src/Data/Tree/Binary/Properties.agda
+++ b/src/Data/Tree/Binary/Properties.agda
@@ -13,8 +13,9 @@ open import Function.Nary.NonDependent using (congₙ)
 open import Level using (Level)
 open import Data.Nat.Base using (suc; _+_)
 open import Data.Tree.Binary
-open import Function.Base
-open import Relation.Binary.PropositionalEquality
+open import Function.Base using (id; flip)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂; _≗_)
 
 private
   variable
diff --git a/src/Data/Tree/Binary/Zipper/Properties.agda b/src/Data/Tree/Binary/Zipper/Properties.agda
index fa2f09c190..03a88b09d2 100644
--- a/src/Data/Tree/Binary/Zipper/Properties.agda
+++ b/src/Data/Tree/Binary/Zipper/Properties.agda
@@ -8,17 +8,22 @@
 
 module Data.Tree.Binary.Zipper.Properties where
 
-open import Data.Tree.Binary as BT using (Tree; node; leaf)
-open import Data.Tree.Binary.Zipper
-open import Data.Nat using (ℕ; suc; _+_)
-open import Data.Nat.Properties
 open import Data.List.Base as List using (List ; [] ; _∷_; sum)
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.All using (All; just; nothing)
-open import Relation.Binary.PropositionalEquality
-open ≡-Reasoning
+open import Data.Nat.Base using (ℕ; suc; _+_)
+open import Data.Nat.Properties using (+-identityʳ; +-comm; +-assoc)
+open import Data.Tree.Binary as BT using (Tree; node; leaf)
+open import Data.Tree.Binary.Zipper using (Zipper; toTree; up; mkZipper;
+  leftBranch; rightBranch; left; right; #nodes; Crumb; getTree; #leaves;
+  map; foldr; _⟪_⟫ˡ_; _⟪_⟫ʳ_)
 open import Function.Base using (_on_; _∘_; _$_)
 open import Level using (Level)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
+open ≡-Reasoning
 
 private
   variable
diff --git a/src/Data/Tree/Rose/Properties.agda b/src/Data/Tree/Rose/Properties.agda
index 91600e68c8..da86ca31ad 100644
--- a/src/Data/Tree/Rose/Properties.agda
+++ b/src/Data/Tree/Rose/Properties.agda
@@ -11,12 +11,15 @@ module Data.Tree.Rose.Properties where
 open import Level using (Level)
 open import Size
 open import Data.List.Base as List using (List)
-open import Data.List.Extrema.Nat
+open import Data.List.Extrema.Nat using (max)
 import Data.List.Properties as List
 open import Data.Nat.Base using (ℕ; zero; suc)
-open import Data.Tree.Rose
-open import Function.Base
-open import Relation.Binary.PropositionalEquality
+open import Data.Tree.Rose using (Rose; node; map; depth; foldr)
+open import Function.Base using (_∘′_; _$_; _∘_)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≗_; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open ≡-Reasoning
 
 private
diff --git a/src/Data/Unit/Polymorphic/Properties.agda b/src/Data/Unit/Polymorphic/Properties.agda
index 088758ab39..f5e9fb9be0 100644
--- a/src/Data/Unit/Polymorphic/Properties.agda
+++ b/src/Data/Unit/Polymorphic/Properties.agda
@@ -9,21 +9,23 @@
 
 module Data.Unit.Polymorphic.Properties where
 
-open import Level
+open import Level using (Level)
 open import Function.Bundles using (_↔_; mk↔)
 open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (inj₁)
 open import Data.Unit.Base renaming (⊤ to ⊤*)
 open import Data.Unit.Polymorphic.Base using (⊤; tt)
-open import Relation.Nullary
+open import Relation.Nullary.Decidable using (yes)
 open import Relation.Binary.Bundles
   using (Setoid; DecSetoid; Preorder; Poset; TotalOrder; DecTotalOrder)
 open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
 open import Relation.Binary.Definitions
   using (DecidableEquality; Antisymmetric; Total)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; trans; decSetoid; setoid; isEquivalence)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; trans)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (decSetoid; setoid; isEquivalence)
 
 private
   variable
diff --git a/src/Data/Unit/Properties.agda b/src/Data/Unit/Properties.agda
index 576e24c7dc..05cf0fbb94 100644
--- a/src/Data/Unit/Properties.agda
+++ b/src/Data/Unit/Properties.agda
@@ -8,16 +8,19 @@
 
 module Data.Unit.Properties where
 
-open import Data.Sum.Base
-open import Data.Unit.Base
+open import Data.Sum.Base using (inj₁)
+open import Data.Unit.Base using (⊤)
 open import Level using (0ℓ)
-open import Relation.Nullary
+open import Relation.Nullary using (Irrelevant; yes)
 open import Relation.Binary.Bundles
   using (Setoid; DecSetoid; Poset; DecTotalOrder)
 open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsTotalOrder; IsDecTotalOrder)
 open import Relation.Binary.Definitions using (DecidableEquality; Total; Antisymmetric)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; trans)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; decSetoid; isEquivalence)
 
 ------------------------------------------------------------------------
 -- Irrelevancy
diff --git a/src/Data/Vec/Functional/Properties.agda b/src/Data/Vec/Functional/Properties.agda
index addfc32678..954f56c0d3 100644
--- a/src/Data/Vec/Functional/Properties.agda
+++ b/src/Data/Vec/Functional/Properties.agda
@@ -9,7 +9,8 @@
 module Data.Vec.Functional.Properties where
 
 open import Data.Empty using (⊥-elim)
-open import Data.Fin.Base
+open import Data.Fin.Base using (Fin; zero; suc; toℕ; fromℕ<; reduce≥;
+  _↑ˡ_; _↑ʳ_; punchIn; punchOut)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 import Data.Nat.Properties as ℕ
 open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
@@ -18,11 +19,15 @@ import Data.List.Properties as List
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Data.Vec.Base as Vec using (Vec)
 import Data.Vec.Properties as Vec
-open import Data.Vec.Functional
-open import Function.Base
+open import Data.Vec.Functional using (Vector; head; tail; updateAt;
+  map; _++_; insertAt; removeAt; toVec; fromVec; toList; fromList)
+open import Function.Base using (_∘_; id)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (DecidableEquality; Decidable)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≗_; refl; _≢_; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Nullary.Decidable
   using (Dec; does; yes; no; map′; _×-dec_)
 
diff --git a/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
index 445f96be90..83d65ab5bb 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Permutation/Properties.agda
@@ -12,11 +12,14 @@ open import Level using (Level)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Data.Nat.Base using (ℕ)
 open import Data.Fin.Permutation using (id; flip; _⟨$⟩ʳ_; inverseʳ; _∘ₚ_)
-open import Data.Vec.Functional
-open import Data.Vec.Functional.Relation.Binary.Permutation
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; trans; cong; module ≡-Reasoning)
+open import Data.Vec.Functional using (Vector)
+open import Data.Vec.Functional.Relation.Binary.Permutation using (_↭_)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; trans; cong)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Binary.Indexed.Heterogeneous
+  using (Reflexive; Symmetric; Transitive; IsIndexedEquivalence; IndexedSetoid)
 
 private
   variable
diff --git a/src/Data/Vec/Properties.agda b/src/Data/Vec/Properties.agda
index 66752c65e7..bea3cfea95 100644
--- a/src/Data/Vec/Properties.agda
+++ b/src/Data/Vec/Properties.agda
@@ -14,7 +14,8 @@ open import Data.Fin.Base as Fin
   using (Fin; zero; suc; toℕ; fromℕ<; _↑ˡ_; _↑ʳ_)
 open import Data.List.Base as List using (List)
 import Data.List.Properties as List
-open import Data.Nat.Base
+open import Data.Nat.Base using (ℕ; zero; suc; _+_; _≤_; _<_; s≤s; pred; s<s⁻¹; _≥_;
+  s≤s⁻¹; z≤n)
 open import Data.Nat.Properties
   using (+-assoc; m≤n⇒m≤1+n; m≤m+n; ≤-refl; ≤-trans; ≤-irrelevant; ≤⇒≤″; suc-injective; +-comm; +-suc)
 open import Data.Product.Base as Product
@@ -24,12 +25,14 @@ open import Data.Sum.Properties using ([,]-map)
 open import Data.Vec.Base
 open import Data.Vec.Relation.Binary.Equality.Cast as VecCast
   using (_≈[_]_; ≈-sym; module CastReasoning)
-open import Function.Base
+open import Function.Base using (_∘_; id; _$_; const; _ˢ_; flip)
 open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (DecidableEquality)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; _≢_; _≗_; refl; sym; trans; cong; cong₂; subst; module ≡-Reasoning)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≢_; _≗_; refl; sym; trans; cong; cong₂; subst)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 open import Relation.Unary using (Pred; Decidable)
 open import Relation.Nullary.Decidable.Core using (Dec; does; yes; _×-dec_; map′)
 open import Relation.Nullary.Negation.Core using (contradiction)
diff --git a/src/Data/Vec/Relation/Binary/Equality/Propositional.agda b/src/Data/Vec/Relation/Binary/Equality/Propositional.agda
index 93b837fed6..09107d6b3e 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Propositional.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Propositional.agda
@@ -9,11 +9,12 @@
 module Data.Vec.Relation.Binary.Equality.Propositional {a} {A : Set a} where
 
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
-open import Data.Vec.Base
+open import Data.Vec.Base using (Vec)
 open import Data.Vec.Relation.Binary.Pointwise.Inductive
   using (Pointwise-≡⇒≡; ≡⇒Pointwise-≡)
 import Data.Vec.Relation.Binary.Equality.Setoid as SEq
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 ------------------------------------------------------------------------
 -- Publically re-export everything from setoid equality
diff --git a/src/Data/Vec/Relation/Binary/Equality/Propositional/WithK.agda b/src/Data/Vec/Relation/Binary/Equality/Propositional/WithK.agda
index a86a09e942..61763b3c49 100644
--- a/src/Data/Vec/Relation/Binary/Equality/Propositional/WithK.agda
+++ b/src/Data/Vec/Relation/Binary/Equality/Propositional/WithK.agda
@@ -10,12 +10,11 @@
 module Data.Vec.Relation.Binary.Equality.Propositional.WithK
   {a} {A : Set a} where
 
-open import Data.Vec.Base
+open import Data.Vec.Base using (Vec)
 open import Data.Vec.Relation.Binary.Equality.Propositional {A = A}
-open import Relation.Binary.PropositionalEquality
-open import Relation.Binary.HeterogeneousEquality as H using (_≅_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
+open import Relation.Binary.HeterogeneousEquality using (_≅_; ≡-to-≅)
 
 ≋⇒≅ : ∀ {m n} {xs : Vec A m} {ys : Vec A n} →
       xs ≋ ys → xs ≅ ys
-≋⇒≅ p with length-equal p
-... | refl = H.≡-to-≅ (≋⇒≡ p)
+≋⇒≅ p with refl <- length-equal p = ≡-to-≅ (≋⇒≡ p)
diff --git a/src/Data/Vec/Relation/Unary/All/Properties.agda b/src/Data/Vec/Relation/Unary/All/Properties.agda
index 3c2a126230..3bd7a53d73 100644
--- a/src/Data/Vec/Relation/Unary/All/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/All/Properties.agda
@@ -13,15 +13,16 @@ open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base using ([]; _∷_)
 open import Data.List.Relation.Unary.All as List using ([]; _∷_)
 open import Data.Product.Base as Product using (_×_; _,_; uncurry; uncurry′)
-open import Data.Vec.Base as Vec
+open import Data.Vec.Base as Vec using (Vec; []; _∷_; map; _++_; concat;
+  tabulate; drop; take; toList; fromList)
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 open import Level using (Level)
 open import Function.Base using (_∘_; id)
 open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Relation.Unary using (Pred) renaming (_⊆_ to _⋐_)
 open import Relation.Binary.Core using (REL)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; refl; cong; cong₂; →-to-⟶)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; cong₂)
 
 private
   variable
diff --git a/src/Data/Vec/Relation/Unary/AllPairs.agda b/src/Data/Vec/Relation/Unary/AllPairs.agda
index a702a1dbf9..9fe7ff703f 100644
--- a/src/Data/Vec/Relation/Unary/AllPairs.agda
+++ b/src/Data/Vec/Relation/Unary/AllPairs.agda
@@ -19,7 +19,7 @@ open import Function.Base using (id; _∘_)
 open import Level using (_⊔_)
 open import Relation.Binary.Definitions as B
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (refl; cong₂)
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
 open import Relation.Nullary.Decidable as Dec using (yes; no; _×-dec_)
 
diff --git a/src/Data/Vec/Relation/Unary/Linked.agda b/src/Data/Vec/Relation/Unary/Linked.agda
index c0f6b5fe89..c821a21263 100644
--- a/src/Data/Vec/Relation/Unary/Linked.agda
+++ b/src/Data/Vec/Relation/Unary/Linked.agda
@@ -15,9 +15,9 @@ open import Data.Product.Base as Product using (_,_; _×_; uncurry; <_,_>)
 open import Function.Base using (id; _∘_)
 open import Level using (Level; _⊔_)
 open import Relation.Binary.Core using (Rel; _⇒_)
-open import Relation.Binary.Definitions as B
+import Relation.Binary.Definitions as B
 open import Relation.Binary.Construct.Intersection renaming (_∩_ to _∩ᵇ_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong₂)
 open import Relation.Unary as U renaming (_∩_ to _∩ᵘ_) hiding (_⇒_)
 open import Relation.Nullary.Decidable as Dec using (yes; no; _×-dec_; map′)
 
diff --git a/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda b/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
index 8e1fe54b4f..dc78ff30d3 100644
--- a/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
+++ b/src/Data/Vec/Relation/Unary/Unique/Propositional/Properties.agda
@@ -8,13 +8,13 @@
 
 module Data.Vec.Relation.Unary.Unique.Propositional.Properties where
 
-open import Data.Vec.Base
+open import Data.Vec.Base using (Vec; map; take; drop; tabulate; lookup)
 open import Data.Vec.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.Vec.Relation.Unary.AllPairs as AllPairs using (AllPairs)
-open import Data.Vec.Relation.Unary.Unique.Propositional
+open import Data.Vec.Relation.Unary.Unique.Propositional using (Unique)
 import Data.Vec.Relation.Unary.Unique.Setoid.Properties as Setoid
-open import Data.Fin.Base using(Fin)
-open import Data.Nat.Base
+open import Data.Fin.Base using (Fin)
+open import Data.Nat.Base using (_+_)
 open import Data.Nat.Properties using (<⇒≢)
 open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (≡⇒≡×≡)
@@ -22,10 +22,11 @@ open import Function.Base using (id; _∘_)
 open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality
-  using (refl; _≡_; _≢_; sym; setoid)
+open import Relation.Binary.PropositionalEquality.Core
+  using (refl; _≡_; _≢_; sym)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 open import Relation.Unary using (Pred; Decidable)
-open import Relation.Nullary.Negation using (¬_)
+open import Relation.Nullary.Negation.Core using (¬_)
 
 private
   variable
diff --git a/src/Data/Word/Base.agda b/src/Data/Word/Base.agda
index f0dddc1e2c..161502e4e9 100644
--- a/src/Data/Word/Base.agda
+++ b/src/Data/Word/Base.agda
@@ -12,7 +12,7 @@ open import Level using (zero)
 import Data.Nat.Base as ℕ
 open import Function.Base using (_on_)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
 
 ------------------------------------------------------------------------
 -- Re-export built-ins publicly
diff --git a/src/Data/Word/Properties.agda b/src/Data/Word/Properties.agda
index e0c2308909..2681d6ce15 100644
--- a/src/Data/Word/Properties.agda
+++ b/src/Data/Word/Properties.agda
@@ -10,16 +10,18 @@ module Data.Word.Properties where
 
 import Data.Nat.Base as ℕ
 open import Data.Bool.Base using (Bool)
-open import Data.Word.Base
+open import Data.Word.Base using (_≈_; toℕ; Word64; _<_)
 import Data.Nat.Properties as ℕ
-open import Function.Base
-open import Relation.Nullary.Decidable using (map′; ⌊_⌋)
+open import Relation.Nullary.Decidable.Core using (map′; ⌊_⌋)
 open import Relation.Binary
   using ( _⇒_; Reflexive; Symmetric; Transitive; Substitutive
         ; Decidable; DecidableEquality; IsEquivalence; IsDecEquivalence
         ; Setoid; DecSetoid; StrictTotalOrder)
 import Relation.Binary.Construct.On as On
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; sym; trans; subst)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; decSetoid)
 
 ------------------------------------------------------------------------
 -- Primitive properties
diff --git a/src/Effect/Monad/Predicate.agda b/src/Effect/Monad/Predicate.agda
index 7a46fb38a8..6e4ed0ffb0 100644
--- a/src/Effect/Monad/Predicate.agda
+++ b/src/Effect/Monad/Predicate.agda
@@ -10,15 +10,12 @@
 
 module Effect.Monad.Predicate where
 
-open import Effect.Applicative.Indexed
-open import Effect.Monad
-open import Effect.Monad.Indexed
-open import Data.Unit
+open import Effect.Monad.Indexed using (RawIMonad)
 open import Data.Product.Base using (_,_)
 open import Function.Base using (const; id; _∘_)
-open import Level
-open import Relation.Binary.PropositionalEquality
-open import Relation.Unary
+open import Level using (Level; suc; _⊔_)
+open import Relation.Binary.PropositionalEquality.Core using (refl)
+open import Relation.Unary using (_⊆_; _⇒_; _∈_; _∩_; ｛_｝)
 open import Relation.Unary.PredicateTransformer using (Pt)
 
 private
diff --git a/src/Function/Construct/Symmetry.agda b/src/Function/Construct/Symmetry.agda
index 5506f93dd4..e7e9f13db9 100644
--- a/src/Function/Construct/Symmetry.agda
+++ b/src/Function/Construct/Symmetry.agda
@@ -20,8 +20,8 @@ open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Reflexive; Symmetric; Transitive)
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; cong; setoid)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 private
   variable
diff --git a/src/Function/Metric/Nat/Bundles.agda b/src/Function/Metric/Nat/Bundles.agda
index 57ff945ee0..6623fac24f 100644
--- a/src/Function/Metric/Nat/Bundles.agda
+++ b/src/Function/Metric/Nat/Bundles.agda
@@ -15,11 +15,11 @@ module Function.Metric.Nat.Bundles where
 open import Data.Nat.Base hiding (suc; _⊔_)
 open import Function.Base using (const)
 open import Level using (Level; suc; _⊔_)
-open import Relation.Binary.Core
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; isEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence)
 
-open import Function.Metric.Nat.Core
+open import Function.Metric.Nat.Core using (DistanceFunction)
 open import Function.Metric.Nat.Structures
 open import Function.Metric.Bundles as Base
   using (GeneralMetric)
diff --git a/src/Function/Metric/Rational/Bundles.agda b/src/Function/Metric/Rational/Bundles.agda
index f578a8a8a2..018d26c6df 100644
--- a/src/Function/Metric/Rational/Bundles.agda
+++ b/src/Function/Metric/Rational/Bundles.agda
@@ -10,20 +10,19 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Data.Rational.Base hiding (_⊔_)
+module Function.Metric.Rational.Bundles where
+
 open import Function.Base using (const)
 open import Level using (Level; suc; _⊔_)
-open import Relation.Binary.Core
-open import Relation.Binary.PropositionalEquality
-  using (_≡_; isEquivalence)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Binary.PropositionalEquality.Properties using (isEquivalence)
 
-open import Function.Metric.Rational.Core
+open import Function.Metric.Rational.Core using (DistanceFunction)
 open import Function.Metric.Rational.Structures
 open import Function.Metric.Bundles as Base
   using (GeneralMetric)
 
-module Function.Metric.Rational.Bundles where
-
 ------------------------------------------------------------------------
 -- Proto-metric
 
diff --git a/src/Function/Nary/NonDependent.agda b/src/Function/Nary/NonDependent.agda
index aa9c35a0f5..b1962a68c4 100644
--- a/src/Function/Nary/NonDependent.agda
+++ b/src/Function/Nary/NonDependent.agda
@@ -19,9 +19,11 @@ open import Data.Nat.Base using (ℕ; zero; suc)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Nary.NonDependent
+  using (Product; uncurryₙ; Equalₙ; curryₙ; fromEqualₙ; toEqualₙ)
 open import Function.Base using (_∘′_; _$′_; const; flip)
 open import Relation.Unary using (IUniversal)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; cong)
 
 private
   variable
diff --git a/src/Function/Properties/Bijection.agda b/src/Function/Properties/Bijection.agda
index 50c84ce3c9..21e295e807 100644
--- a/src/Function/Properties/Bijection.agda
+++ b/src/Function/Properties/Bijection.agda
@@ -8,12 +8,13 @@
 
 module Function.Properties.Bijection where
 
-open import Function.Bundles
+open import Function.Bundles using (Bijection; Inverse; Equivalence;
+  _⤖_; _↔_; _⇔_)
 open import Level using (Level)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Binary.Definitions using (Reflexive; Trans)
-open import Relation.Binary.PropositionalEquality as ≡ using (setoid)
+open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 open import Data.Product.Base using (_,_; proj₁; proj₂)
 open import Function.Base using (_∘_)
 open import Function.Properties.Surjection using (injective⇒to⁻-cong)
@@ -37,7 +38,7 @@ refl = Identity.bijection _
 
 -- Can't prove full symmetry as we have no proof that the witness
 -- produced by the surjection proof preserves equality
-sym-≡ : Bijection S (≡.setoid B) → Bijection (≡.setoid B) S
+sym-≡ : Bijection S (setoid B) → Bijection (setoid B) S
 sym-≡ = Symmetry.bijection-≡
 
 trans : Trans (Bijection {a} {ℓ₁} {b} {ℓ₂}) (Bijection {b} {ℓ₂} {c} {ℓ₃}) Bijection
diff --git a/src/Function/Properties/Surjection.agda b/src/Function/Properties/Surjection.agda
index 99e13c218d..d5e6b8e75d 100644
--- a/src/Function/Properties/Surjection.agda
+++ b/src/Function/Properties/Surjection.agda
@@ -8,15 +8,16 @@
 
 module Function.Properties.Surjection where
 
-open import Function.Base
-open import Function.Definitions
-open import Function.Bundles
+open import Function.Base using (_∘_; _$_)
+open import Function.Definitions using (Surjective; Injective; Congruent)
+open import Function.Bundles using (Func; Surjection; _↠_; _⟶_; _↪_; mk↪;
+  _⇔_; mk⇔)
 import Function.Construct.Identity as Identity
 import Function.Construct.Composition as Compose
 open import Level using (Level)
 open import Data.Product.Base using (proj₁; proj₂)
-import Relation.Binary.PropositionalEquality as ≡
-open import Relation.Binary.Definitions
+import Relation.Binary.PropositionalEquality.Core as ≡
+open import Relation.Binary.Definitions using (Reflexive; Trans)
 open import Relation.Binary.Bundles using (Setoid)
 import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
 
diff --git a/src/Relation/Binary/Construct/Add/Extrema/Strict.agda b/src/Relation/Binary/Construct/Add/Extrema/Strict.agda
index 4ed89565ae..496ac3d617 100644
--- a/src/Relation/Binary/Construct/Add/Extrema/Strict.agda
+++ b/src/Relation/Binary/Construct/Add/Extrema/Strict.agda
@@ -10,26 +10,25 @@
 -- Relation.Nullary.Construct.Add.Extrema
 
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.Structures
-  using (IsStrictPartialOrder; IsDecStrictPartialOrder; IsStrictTotalOrder)
-open import Relation.Binary.Definitions
-  using (Asymmetric; Transitive; Decidable; Irrelevant; Trichotomous; Irreflexive; Trans; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
 
 module Relation.Binary.Construct.Add.Extrema.Strict
   {a ℓ} {A : Set a} (_<_ : Rel A ℓ) where
 
 open import Level
-open import Function.Base
-open import Relation.Nullary hiding (Irrelevant)
+open import Function.Base using (_∘′_)
 
 import Relation.Nullary.Construct.Add.Infimum as I
 open import Relation.Nullary.Construct.Add.Extrema
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl)
 import Relation.Binary.Construct.Add.Infimum.Strict as AddInfimum
 import Relation.Binary.Construct.Add.Supremum.Strict as AddSupremum
 import Relation.Binary.Construct.Add.Extrema.Equality as Equality
 import Relation.Binary.Construct.Add.Extrema.NonStrict as NonStrict
+open import Relation.Binary.Definitions
+  using (Asymmetric; Transitive; Decidable; Irrelevant; Trichotomous; Irreflexive; Trans; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
+open import Relation.Binary.Structures
+  using (IsStrictPartialOrder; IsDecStrictPartialOrder; IsStrictTotalOrder)
 
 ------------------------------------------------------------------------
 -- Definition
diff --git a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
index ee4078a75c..899be634a1 100644
--- a/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
+++ b/src/Relation/Binary/Construct/Closure/ReflexiveTransitive/Properties/WithK.agda
@@ -13,8 +13,8 @@ module
 
 open import Function.Base using (_∋_)
 open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.Construct.Closure.ReflexiveTransitive
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.Construct.Closure.ReflexiveTransitive using (Star; _◅_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- Equality
diff --git a/src/Relation/Binary/Indexed/Heterogeneous/Core.agda b/src/Relation/Binary/Indexed/Heterogeneous/Core.agda
index 1015ef5449..a58276ad32 100644
--- a/src/Relation/Binary/Indexed/Heterogeneous/Core.agda
+++ b/src/Relation/Binary/Indexed/Heterogeneous/Core.agda
@@ -13,7 +13,6 @@ module Relation.Binary.Indexed.Heterogeneous.Core where
 
 open import Level
 import Relation.Binary.Core as B
-import Relation.Binary.Definitions as B
 
 ------------------------------------------------------------------------
 -- Indexed binary relations
diff --git a/src/Relation/Binary/Rewriting.agda b/src/Relation/Binary/Rewriting.agda
index 903ab9018e..de63347546 100644
--- a/src/Relation/Binary/Rewriting.agda
+++ b/src/Relation/Binary/Rewriting.agda
@@ -11,19 +11,19 @@ module Relation.Binary.Rewriting where
 
 open import Agda.Builtin.Equality using (_≡_ ; refl)
 open import Data.Product.Base using (_×_ ; ∃ ; -,_; _,_ ; proj₁ ; proj₂)
-open import Data.Empty
+open import Data.Empty using (⊥-elim)
 open import Function.Base using (flip)
-open import Induction.WellFounded
-open import Level
-open import Relation.Binary.Core
+open import Induction.WellFounded using (WellFounded; Acc; acc)
+open import Relation.Binary.Core using (REL; Rel)
 open import Relation.Binary.Construct.Closure.Equivalence using (EqClosure)
 open import Relation.Binary.Construct.Closure.Equivalence.Properties
-open import Relation.Binary.Construct.Closure.Reflexive
+  using (a—↠b&a—↠c⇒b↔c)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
-open import Relation.Binary.Construct.Closure.Symmetric
+  using (Star; _◅_; ε; _◅◅_)
+open import Relation.Binary.Construct.Closure.Symmetric using (fwd; bwd)
 open import Relation.Binary.Construct.Closure.Transitive
-open import Relation.Binary.PropositionalEquality
-open import Relation.Nullary
+  using (Plus; [_]; _∼⁺⟨_⟩_)
+open import Relation.Nullary.Negation.Core using (¬_)
 
 -- The following definitions are taken from Klop [5]
 module _ {a ℓ} {A : Set a} (_⟶_ : Rel A ℓ) where
diff --git a/src/Tactic/RingSolver/Core/NatSet.agda b/src/Tactic/RingSolver/Core/NatSet.agda
index 1f42ef708c..1b27fae60a 100644
--- a/src/Tactic/RingSolver/Core/NatSet.agda
+++ b/src/Tactic/RingSolver/Core/NatSet.agda
@@ -41,7 +41,8 @@ open import Data.List.Base  as List  using (List; _∷_; [])
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Bool.Base  as Bool  using (Bool)
 open import Function.Base            using (const; _∘_)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl)
 
 ------------------------------------------------------------------------
 -- Helper methods
diff --git a/src/Text/Pretty/Core.agda b/src/Text/Pretty/Core.agda
index 4a5a210fef..9a157ddda2 100644
--- a/src/Text/Pretty/Core.agda
+++ b/src/Text/Pretty/Core.agda
@@ -32,13 +32,16 @@ open import Data.Maybe.Relation.Unary.All as Allᴹ using (nothing; just)
 
 open import Data.String.Base as String
   using (String; length; replicate; _++_; unlines)
-open import Data.String.Unsafe as String
-open import Function.Base
-open import Relation.Nullary.Decidable using (Dec)
+import Data.String.Unsafe as String
+open import Function.Base using (_∘_; flip; _on_; id; _∘′_; _$_)
+open import Relation.Nullary.Decidable.Core using (Dec)
 open import Relation.Unary using (IUniversal; _⇒_; U)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; sym; cong; cong₂; subst)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (module ≡-Reasoning)
 
-open import Data.Refinement hiding (map)
+open import Data.Refinement using (Refinement-syntax; _,_; value; proof)
 import Data.Refinement.Relation.Unary.All as Allᴿ
 
 ------------------------------------------------------------------------
@@ -148,7 +151,7 @@ private
     size-pad : maybe′ length 0 pad ≡ x.lastWidth
     size-pad with x.lastWidth
     ... | 0         = refl
-    ... | l@(suc _) = length-replicate l
+    ... | l@(suc _) = String.length-replicate l
 
     indent : Maybe String → String → String
     indent = maybe′ _++_ id
@@ -156,7 +159,7 @@ private
     size-indent : ∀ ma str → length (indent ma str)
                 ≡ maybe′ length 0 ma + length str
     size-indent nothing    str = refl
-    size-indent (just pad) str = length-++ pad str
+    size-indent (just pad) str = String.length-++ pad str
 
     indents : Maybe String → Tree String ⊤ → Tree String ⊤
     indents = maybe′ (mapₙ ∘ _++_) id
@@ -217,7 +220,7 @@ private
                  length vLast ≡ lastWidth
     isLastLine ∣x∣ ∣y∣ with blocky
     ... | nothing        = begin
-      length (lastx ++ lasty)       ≡⟨ length-++ lastx lasty ⟩
+      length (lastx ++ lasty)       ≡⟨ String.length-++ lastx lasty ⟩
       length lastx + length lasty   ≡⟨ cong₂ _+_ ∣x∣ ∣y∣ ⟩
       x.lastWidth + y.lastWidth     ∎ where open ≡-Reasoning
     ... | just (hd , tl) = begin
@@ -256,7 +259,7 @@ private
 
       middle : length (lastx ++ hd) ≤ vMaxWidth
       middle = begin
-        length (lastx ++ hd)     ≡⟨ length-++ lastx hd ⟩
+        length (lastx ++ hd)     ≡⟨ String.length-++ lastx hd ⟩
         length lastx + length hd ≡⟨ cong (_+ _) ∣x∣≡ ⟩
         x.lastWidth + length hd  ≤⟨ +-monoʳ-≤ x.lastWidth ∣hd∣ ⟩
         x.lastWidth + widthy     ≤⟨ m≤n⊔m _ _ ⟩
diff --git a/src/Text/Regex/Properties/Core.agda b/src/Text/Regex/Properties/Core.agda
index 51097a7656..40e1cdf15b 100644
--- a/src/Text/Regex/Properties/Core.agda
+++ b/src/Text/Regex/Properties/Core.agda
@@ -23,13 +23,13 @@ open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Relation.Nullary using (¬_; Dec; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Unary using (Pred)
-open import Relation.Binary.PropositionalEquality
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 open Preorder P using (_≈_) renaming (Carrier to A; _∼_ to _≤_)
 open import Text.Regex.Base P
 
-open import Data.List.Relation.Ternary.Appending.Propositional {A = A}
 open import Data.List.Relation.Ternary.Appending.Propositional.Properties {A = A}
+  using (++[]⁻¹; []++⁻¹; conicalˡ; conicalʳ)
 
 ------------------------------------------------------------------------
 -- Views

From c234c72693a2b41b88eed088ffbe6169305fe1bf Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Sun, 21 Apr 2024 14:05:56 +0100
Subject: [PATCH 275/316] [ new ] IO buffering, and loops (#2367)

* [ new ] IO conditionals & loops

* [ new ] stdin, stdout, stderr, buffering, etc.

* [ fix ] and test for the new IO primitives

* [ fix ] compilation

* [ fix ] more import fixing

* [ done (?) ] with compilation fixes

* [ test ] add test to runner

* [ doc ] explain the loop

* [ compat ] add deprecated stub

* [ fix ] my silly mistake

* [ doc ] list re-exports in CHANGELOG
---
 CHANGELOG.md                             | 35 +++++++++++++-
 GenerateEverything.hs                    |  3 ++
 doc/README/IO.agda                       | 24 ++++++++--
 src/Foreign/Haskell/Coerce.agda          |  2 +-
 src/IO.agda                              |  3 +-
 src/IO/Base.agda                         | 45 +++++++++++++++--
 src/IO/Finite.agda                       |  2 +-
 src/IO/Handle.agda                       | 41 ++++++++++++++++
 src/IO/Infinite.agda                     |  2 +-
 src/IO/Primitive.agda                    | 33 ++-----------
 src/IO/Primitive/Core.agda               | 38 +++++++++++++++
 src/IO/Primitive/Handle.agda             | 61 ++++++++++++++++++++++++
 src/System/Clock/Primitive.agda          |  2 +-
 src/System/Environment/Primitive.agda    |  2 +-
 src/System/FilePath/Posix/Primitive.agda |  2 +-
 src/System/Process/Primitive.agda        |  2 +-
 tests/runtests.agda                      |  1 +
 tests/system/io/Main.agda                | 24 ++++++++++
 tests/system/io/expected                 |  6 +++
 tests/system/io/input                    |  6 +++
 tests/system/io/io.agda-lib              |  2 +
 tests/system/io/run                      |  5 ++
 22 files changed, 295 insertions(+), 46 deletions(-)
 create mode 100644 src/IO/Handle.agda
 create mode 100644 src/IO/Primitive/Core.agda
 create mode 100644 src/IO/Primitive/Handle.agda
 create mode 100644 tests/system/io/Main.agda
 create mode 100644 tests/system/io/expected
 create mode 100644 tests/system/io/input
 create mode 100644 tests/system/io/io.agda-lib
 create mode 100644 tests/system/io/run

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 24fb3493a4..dc3c702779 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -26,6 +26,7 @@ Non-backwards compatible changes
   parametrized by _raw_ bundles, and as such take a proof of transitivity.
 * The definitions in `Algebra.Module.Morphism.Construct.Identity` are now
   parametrized by _raw_ bundles, and as such take a proof of reflexivity.
+* The module `IO.Primitive` was moved to `IO.Primitive.Core`.
 
 Other major improvements
 ------------------------
@@ -66,6 +67,11 @@ Deprecated names
   *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
   ```
 
+* In `IO.Base`:
+  ```agda
+  untilRight  ↦  untilInj₂
+  ```
+
 New modules
 -----------
 
@@ -137,6 +143,12 @@ New modules
   Data.Container.Indexed.Relation.Binary.Equality.Setoid
   ```
 
+* New IO primitives to handle buffering
+  ```agda
+  IO.Primitive.Handle
+  IO.Handle
+  ```
+
 * `System.Random` bindings:
   ```agda
   System.Random.Primitive
@@ -418,13 +430,34 @@ Additions to existing modules
   between : String → String → String → String
   ```
 
+* Re-exported new types and functions in `IO`:
+  ```agda
+  BufferMode : Set
+  noBuffering : BufferMode
+  lineBuffering : BufferMode
+  blockBuffering : Maybe ℕ → BufferMode
+  Handle : Set
+  stdin : Handle
+  stdout : Handle
+  stderr : Handle
+  hSetBuffering : Handle → BufferMode → IO ⊤
+  hGetBuffering : Handle → IO BufferMode
+  hFlush : Handle → IO ⊤
+  ```
+
+* Added new functions in `IO.Base`:
+  ```agda
+  whenInj₂ : E ⊎ A → (A → IO ⊤) → IO ⊤
+  forever : IO ⊤ → IO ⊤
+  ```
+
 * In `Data.Word.Base`:
   ```agda
   _≤_ : Rel Word64 zero
   ```
 
 * Added new definition in `Relation.Binary.Construct.Closure.Transitive`
-  ```
+  ```agda
   transitive⁻ : Transitive _∼_ → TransClosure _∼_ ⇒ _∼_
   ```
 
diff --git a/GenerateEverything.hs b/GenerateEverything.hs
index c3aa8ed710..d7895f96db 100644
--- a/GenerateEverything.hs
+++ b/GenerateEverything.hs
@@ -56,11 +56,14 @@ unsafeModules = map modToFile
   , "IO"
   , "IO.Base"
   , "IO.Categorical"
+  , "IO.Handle"
   , "IO.Infinite"
   , "IO.Instances"
   , "IO.Effectful"
   , "IO.Finite"
   , "IO.Primitive"
+  , "IO.Primitive.Core"
+  , "IO.Primitive.Handle"
   , "IO.Primitive.Infinite"
   , "IO.Primitive.Finite"
   , "Relation.Binary.PropositionalEquality.TrustMe"
diff --git a/doc/README/IO.agda b/doc/README/IO.agda
index 37553edd1d..bf913bbfdd 100644
--- a/doc/README/IO.agda
+++ b/doc/README/IO.agda
@@ -13,6 +13,7 @@ open import Data.Nat.Base
 open import Data.Nat.Show using (show)
 open import Data.String.Base using (String; _++_; lines)
 open import Data.Unit.Polymorphic
+open import Function.Base using (_$_)
 open import IO
 
 ------------------------------------------------------------------------
@@ -86,10 +87,21 @@ cat fp = do
   let ls = lines content
   List.mapM′ putStrLn ls
 
-open import Codata.Musical.Notation
-open import Codata.Musical.Colist
-open import Data.Bool
-open import Data.Unit.Polymorphic.Base
+------------------------------------------------------------------------
+-- TOP-LEVEL LOOP
+
+-- If you simply want to repeat the same action over and over again, you
+-- can use `forever` e.g. the following defines a REPL that echos whatever
+-- the user types
+
+echo : IO ⊤
+echo = do
+  hSetBuffering stdout noBuffering
+  forever $ do
+    putStr "echo< "
+    str ← getLine
+    putStrLn ("echo> " ++ str)
+
 
 ------------------------------------------------------------------------
 -- GUARDEDNESS
@@ -101,6 +113,10 @@ open import Data.Unit.Polymorphic.Base
 -- In this case you cannot use the convenient combinators that make `do`-notations
 -- and have to revert back to the underlying coinductive constructors.
 
+open import Codata.Musical.Notation
+open import Codata.Musical.Colist using (Colist; []; _∷_)
+open import Data.Bool
+open import Data.Unit.Polymorphic.Base
 
 -- Whether a colist is finite is semi decidable: just let the user wait until
 -- you reach the end!
diff --git a/src/Foreign/Haskell/Coerce.agda b/src/Foreign/Haskell/Coerce.agda
index 3eb411c325..4383a1ac86 100644
--- a/src/Foreign/Haskell/Coerce.agda
+++ b/src/Foreign/Haskell/Coerce.agda
@@ -35,7 +35,7 @@ open import Level using (Level; _⊔_)
 open import Agda.Builtin.Nat
 open import Agda.Builtin.Int
 
-import IO.Primitive            as STD
+import IO.Primitive.Core       as STD
 import Data.List.Base          as STD
 import Data.List.NonEmpty.Base as STD
 import Data.Maybe.Base         as STD
diff --git a/src/IO.agda b/src/IO.agda
index 96f7db2760..0462d9b8b9 100644
--- a/src/IO.agda
+++ b/src/IO.agda
@@ -14,7 +14,7 @@ open import Data.Unit.Polymorphic.Base
 open import Data.String.Base using (String)
 import Data.Unit.Base as Unit0
 open import Function.Base using (_∘_; flip)
-import IO.Primitive as Prim
+import IO.Primitive.Core as Prim
 open import Level
 
 private
@@ -27,6 +27,7 @@ private
 -- Re-exporting the basic type and functions
 
 open import IO.Base public
+open import IO.Handle public
 
 ------------------------------------------------------------------------
 -- Utilities
diff --git a/src/IO/Base.agda b/src/IO/Base.agda
index 8c44474935..54aff90dfc 100644
--- a/src/IO/Base.agda
+++ b/src/IO/Base.agda
@@ -16,13 +16,14 @@ open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 import Agda.Builtin.Unit as Unit0
 open import Data.Unit.Polymorphic.Base
 open import Function.Base using (_∘′_; const; flip)
-import IO.Primitive as Prim
+import IO.Primitive.Core as Prim
 
 private
   variable
-    a b : Level
+    a b e : Level
     A : Set a
     B : Set b
+    E : Set e
 
 ------------------------------------------------------------------------
 -- The IO monad
@@ -106,18 +107,37 @@ lift′ io = lift (io Prim.>>= λ _ → Prim.pure _)
 ignore : IO A → IO ⊤
 ignore io = io >> pure _
 
+
+------------------------------------------------------------------------
 -- Conditional executions
+
+-- Only run the action if the boolean is true
 when : Bool → IO {a} ⊤ → IO ⊤
 when true m = m
 when false _ = pure _
 
+-- Only run the action if the boolean is false
 unless : Bool → IO {a} ⊤ → IO ⊤
 unless = when ∘′ not
 
+-- Run the action if the `Maybe` computation was successful
 whenJust : Maybe A → (A → IO {a} ⊤) → IO ⊤
 whenJust (just a) k = k a
 whenJust nothing  _ = pure _
 
+-- Run the action if the `E ⊎_` computation was successful
+whenInj₂ : E ⊎ A → (A → IO {a} ⊤) → IO ⊤
+whenInj₂ (inj₂ a) k = k a
+whenInj₂ (inj₁ _) _ = pure _
+
+
+------------------------------------------------------------------------
+-- Loops
+
+-- Keep running the action forever
+forever : IO {a} ⊤ → IO {a} ⊤
+forever act = seq (♯ act) (♯ forever act)
+
 -- Keep running an IO action until we get a value. Convenient when user
 -- input is involved and it may be malformed.
 untilJust : IO (Maybe A) → IO A
@@ -127,7 +147,22 @@ untilJust m = bind (♯ m) λ where
   nothing  → ♯ untilJust m
   (just a) → ♯ pure a
 
-untilRight : {A B : Set a} → (A → IO (A ⊎ B)) → A → IO B
-untilRight f x = bind (♯ f x) λ where
-  (inj₁ x′) → ♯ untilRight f x′
+untilInj₂ : {A B : Set a} → (A → IO (A ⊎ B)) → A → IO B
+untilInj₂ f x = bind (♯ f x) λ where
+  (inj₁ x′) → ♯ untilInj₂ f x′
   (inj₂ y) → ♯ pure y
+
+
+
+
+
+
+
+------------------------------------------------------------------------
+-- DEPRECATIONS
+
+untilRight = untilInj₂
+{-# WARNING_ON_USAGE untilRight
+"Warning: untilRight was deprecated in v2.1.
+Please use untilInj₂ instead."
+#-}
diff --git a/src/IO/Finite.agda b/src/IO/Finite.agda
index 44b5edc050..1827d471b9 100644
--- a/src/IO/Finite.agda
+++ b/src/IO/Finite.agda
@@ -12,7 +12,7 @@ open import Data.Unit.Polymorphic.Base
 open import Agda.Builtin.String
 import Data.Unit.Base as Unit0
 open import IO.Base
-import IO.Primitive as Prim
+import IO.Primitive.Core as Prim
 import IO.Primitive.Finite as Prim
 open import Level
 
diff --git a/src/IO/Handle.agda b/src/IO/Handle.agda
new file mode 100644
index 0000000000..0c15906201
--- /dev/null
+++ b/src/IO/Handle.agda
@@ -0,0 +1,41 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- IO handles: simple bindings to Haskell types and functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --guardedness #-}
+
+module IO.Handle where
+
+open import Level using (Level)
+open import Data.Unit.Polymorphic.Base using (⊤)
+open import IO.Base using (IO; lift; lift′)
+
+private variable a : Level
+
+------------------------------------------------------------------------
+-- Re-exporting types and constructors
+
+open import IO.Primitive.Handle as Prim public
+  using ( BufferMode
+        ; noBuffering
+        ; lineBuffering
+        ; blockBuffering
+        ; Handle
+        ; stdin
+        ; stdout
+        ; stderr
+        )
+
+------------------------------------------------------------------------
+-- Wrapping functions
+
+hSetBuffering : Handle → BufferMode → IO {a} ⊤
+hSetBuffering hdl bm = lift′ (Prim.hSetBuffering hdl bm)
+
+hGetBuffering : Handle → IO BufferMode
+hGetBuffering hdl = lift (Prim.hGetBuffering hdl)
+
+hFlush : Handle → IO {a} ⊤
+hFlush hdl = lift′ (Prim.hFlush hdl)
diff --git a/src/IO/Infinite.agda b/src/IO/Infinite.agda
index f2137cc4c6..90afd8989d 100644
--- a/src/IO/Infinite.agda
+++ b/src/IO/Infinite.agda
@@ -13,7 +13,7 @@ open import Agda.Builtin.String
 open import Data.Unit.Polymorphic.Base
 import Data.Unit.Base as Unit0
 open import IO.Base
-import IO.Primitive as Prim
+import IO.Primitive.Core as Prim
 import IO.Primitive.Infinite as Prim
 open import Level
 
diff --git a/src/IO/Primitive.agda b/src/IO/Primitive.agda
index 8014a2bd7c..324a32b643 100644
--- a/src/IO/Primitive.agda
+++ b/src/IO/Primitive.agda
@@ -1,38 +1,15 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Primitive IO: simple bindings to Haskell types and functions
+-- This module is DEPRECATED. Please use IO.Primitive.Core instead
 ------------------------------------------------------------------------
 
--- NOTE: the contents of this module should be accessed via `IO`.
-
 {-# OPTIONS --cubical-compatible #-}
 
 module IO.Primitive where
 
-open import Level using (Level)
-private
-  variable
-    a : Level
-    A B : Set a
-
-------------------------------------------------------------------------
--- The IO monad
-
-open import Agda.Builtin.IO public
-  using (IO)
-
-infixl 1 _>>=_
-
-postulate
-  pure : A → IO A
-  _>>=_  : IO A → (A → IO B) → IO B
-
-{-# COMPILE GHC pure = \_ _ -> return    #-}
-{-# COMPILE GHC _>>=_  = \_ _ _ _ -> (>>=) #-}
-{-# COMPILE UHC pure = \_ _ x -> UHC.Agda.Builtins.primReturn x #-}
-{-# COMPILE UHC _>>=_  = \_ _ _ _ x y -> UHC.Agda.Builtins.primBind x y #-}
+open import IO.Primitive.Core public
 
--- Haskell-style alternative syntax
-return : A → IO A
-return = pure
+{-# WARNING_ON_IMPORT
+"IO.Primitive was deprecated in v2.1. Use IO.Primitive.Core instead."
+#-}
diff --git a/src/IO/Primitive/Core.agda b/src/IO/Primitive/Core.agda
new file mode 100644
index 0000000000..6fd22a03fb
--- /dev/null
+++ b/src/IO/Primitive/Core.agda
@@ -0,0 +1,38 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Primitive IO: simple bindings to Haskell types and functions
+------------------------------------------------------------------------
+
+-- NOTE: the contents of this module should be accessed via `IO`.
+
+{-# OPTIONS --cubical-compatible #-}
+
+module IO.Primitive.Core where
+
+open import Level using (Level)
+private
+  variable
+    a : Level
+    A B : Set a
+
+------------------------------------------------------------------------
+-- The IO monad
+
+open import Agda.Builtin.IO public
+  using (IO)
+
+infixl 1 _>>=_
+
+postulate
+  pure : A → IO A
+  _>>=_  : IO A → (A → IO B) → IO B
+
+{-# COMPILE GHC pure = \_ _ -> return    #-}
+{-# COMPILE GHC _>>=_  = \_ _ _ _ -> (>>=) #-}
+{-# COMPILE UHC pure = \_ _ x -> UHC.Agda.Builtins.primReturn x #-}
+{-# COMPILE UHC _>>=_  = \_ _ _ _ x y -> UHC.Agda.Builtins.primBind x y #-}
+
+-- Haskell-style alternative syntax
+return : A → IO A
+return = pure
diff --git a/src/IO/Primitive/Handle.agda b/src/IO/Primitive/Handle.agda
new file mode 100644
index 0000000000..f6b01a70be
--- /dev/null
+++ b/src/IO/Primitive/Handle.agda
@@ -0,0 +1,61 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Primitive IO handles: simple bindings to Haskell types and functions
+------------------------------------------------------------------------
+
+-- NOTE: the contents of this module should be accessed via `IO`.
+
+{-# OPTIONS --cubical-compatible #-}
+
+module IO.Primitive.Handle where
+
+open import Data.Maybe.Base using (Maybe)
+open import Data.Nat.Base using (ℕ)
+
+data BufferMode : Set where
+  noBuffering lineBuffering : BufferMode
+  blockBuffering : Maybe ℕ → BufferMode
+{-# FOREIGN GHC import System.IO #-}
+{-# FOREIGN GHC
+    data AgdaBufferMode
+      = AgdaNoBuffering
+      | AgdaLineBuffering
+      | AgdaBlockBuffering (Maybe Integer)
+    toBufferMode :: AgdaBufferMode -> BufferMode
+    toBufferMode x = case x of
+      AgdaNoBuffering       -> NoBuffering
+      AgdaLineBuffering     -> LineBuffering
+      AgdaBlockBuffering mi -> BlockBuffering (fromIntegral <$> mi)
+    fromBufferMode :: BufferMode -> AgdaBufferMode
+    fromBufferMode x = case x of
+      NoBuffering       -> AgdaNoBuffering
+      LineBuffering     -> AgdaLineBuffering
+      BlockBuffering mi -> AgdaBlockBuffering (fromIntegral <$> mi)
+#-}
+
+{-# COMPILE GHC BufferMode = data AgdaBufferMode
+                           ( AgdaNoBuffering
+                           | AgdaLineBuffering
+                           | AgdaBlockBuffering
+                           )
+#-}
+
+open import Data.Unit.Base using (⊤)
+open import IO.Primitive.Core using (IO)
+
+postulate
+  Handle : Set
+  stdin stdout stderr : Handle
+  hSetBuffering : Handle → BufferMode → IO ⊤
+  hGetBuffering : Handle → IO BufferMode
+  hFlush : Handle → IO ⊤
+
+{-# FOREIGN GHC import System.IO #-}
+{-# COMPILE GHC Handle = type Handle #-}
+{-# COMPILE GHC stdin = stdin #-}
+{-# COMPILE GHC stdout = stdout #-}
+{-# COMPILE GHC stderr = stderr #-}
+{-# COMPILE GHC hSetBuffering = \ h -> hSetBuffering h . toBufferMode #-}
+{-# COMPILE GHC hGetBuffering = fmap fromBufferMode . hGetBuffering #-}
+{-# COMPILE GHC hFlush = hFlush #-}
diff --git a/src/System/Clock/Primitive.agda b/src/System/Clock/Primitive.agda
index 330b9e36c4..9e394af190 100644
--- a/src/System/Clock/Primitive.agda
+++ b/src/System/Clock/Primitive.agda
@@ -9,7 +9,7 @@
 module System.Clock.Primitive where
 
 open import Agda.Builtin.Nat using (Nat)
-open import IO.Primitive using (IO)
+open import IO.Primitive.Core using (IO)
 open import Foreign.Haskell using (Pair)
 
 data Clock : Set where
diff --git a/src/System/Environment/Primitive.agda b/src/System/Environment/Primitive.agda
index c1f8533e03..9408613641 100644
--- a/src/System/Environment/Primitive.agda
+++ b/src/System/Environment/Primitive.agda
@@ -14,7 +14,7 @@
 
 module System.Environment.Primitive where
 
-open import IO.Primitive using (IO)
+open import IO.Primitive.Core using (IO)
 open import Data.List.Base using (List)
 open import Data.Maybe.Base using (Maybe)
 open import Data.String.Base using (String)
diff --git a/src/System/FilePath/Posix/Primitive.agda b/src/System/FilePath/Posix/Primitive.agda
index ed4cd56792..8226666ba6 100644
--- a/src/System/FilePath/Posix/Primitive.agda
+++ b/src/System/FilePath/Posix/Primitive.agda
@@ -14,7 +14,7 @@ open import Agda.Builtin.List using (List)
 open import Agda.Builtin.Maybe using (Maybe)
 open import Agda.Builtin.String using (String)
 open import Foreign.Haskell as FFI using (Pair; Either)
-open import IO.Primitive using (IO)
+open import IO.Primitive.Core using (IO)
 
 -- A filepath has a nature: it can be either relative or absolute.
 -- We postulate this nature rather than defining it as an inductive
diff --git a/src/System/Process/Primitive.agda b/src/System/Process/Primitive.agda
index aa2bdfcbf2..828a275433 100644
--- a/src/System/Process/Primitive.agda
+++ b/src/System/Process/Primitive.agda
@@ -13,7 +13,7 @@ open import Agda.Builtin.List
 open import Agda.Builtin.String
 open import Agda.Builtin.Unit
 open import Foreign.Haskell using (Pair)
-open import IO.Primitive using (IO)
+open import IO.Primitive.Core using (IO)
 open import System.Exit.Primitive using (ExitCode)
 
 postulate
diff --git a/tests/runtests.agda b/tests/runtests.agda
index a7840838b5..c6ad22e0ce 100644
--- a/tests/runtests.agda
+++ b/tests/runtests.agda
@@ -23,6 +23,7 @@ systemTests = mkTestPool "System modules"
   $ "ansi"
   ∷ "directory"
   ∷ "environment"
+  ∷ "io"
   ∷ "random"
   ∷ []
 
diff --git a/tests/system/io/Main.agda b/tests/system/io/Main.agda
new file mode 100644
index 0000000000..4d2697ea27
--- /dev/null
+++ b/tests/system/io/Main.agda
@@ -0,0 +1,24 @@
+{-# OPTIONS --guardedness #-}
+
+module Main where
+
+open import Data.String.Base
+open import Function.Base using (_$_; case_of_)
+open import IO
+open import System.Exit using (exitSuccess)
+
+main : Main
+main = run $ do
+  -- Ensure no buffering so that the prompt "echo< "
+  -- gets outputed immediately
+  hSetBuffering stdout noBuffering
+  forever $ do
+    putStr "echo< "
+    -- Get a line from the user and immediately inspect it
+    -- If it's a magic "exit" keyword then we exit, otherwise
+    -- we print the echo'd message and let the `forever` action
+    -- continue
+    str ← getLine
+    case str of λ where
+      "exit" → exitSuccess
+      _ → putStrLn ("echo> " ++ str)
diff --git a/tests/system/io/expected b/tests/system/io/expected
new file mode 100644
index 0000000000..361d15957e
--- /dev/null
+++ b/tests/system/io/expected
@@ -0,0 +1,6 @@
+echo< echo> hello
+echo< echo> my
+echo< echo> name
+echo< echo> is
+echo< echo> Agda
+echo< 
\ No newline at end of file
diff --git a/tests/system/io/input b/tests/system/io/input
new file mode 100644
index 0000000000..bba046a9d5
--- /dev/null
+++ b/tests/system/io/input
@@ -0,0 +1,6 @@
+hello
+my
+name
+is
+Agda
+exit
\ No newline at end of file
diff --git a/tests/system/io/io.agda-lib b/tests/system/io/io.agda-lib
new file mode 100644
index 0000000000..74c23d5ccc
--- /dev/null
+++ b/tests/system/io/io.agda-lib
@@ -0,0 +1,2 @@
+name: io
+include: ../../../src/ .
diff --git a/tests/system/io/run b/tests/system/io/run
new file mode 100644
index 0000000000..ad42dc90b2
--- /dev/null
+++ b/tests/system/io/run
@@ -0,0 +1,5 @@
+$1 --compile-dir=../../_build -c Main.agda > log
+./../../_build/Main hello world < input > output
+
+rm ../../_build/Main
+rm ../../_build/MAlonzo/Code/Main*
\ No newline at end of file

From 7199c18aff0c367bd14b9e3de84274fbf6597175 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Sun, 21 Apr 2024 16:35:57 +0100
Subject: [PATCH 276/316] [ cleanup ] imports in the Effect.Monad modules
 (#2371)

---
 src/Effect/Monad/Continuation.agda       |  9 ++++-----
 src/Effect/Monad/Error/Transformer.agda  | 10 +++-------
 src/Effect/Monad/IO.agda                 |  8 +++-----
 src/Effect/Monad/Identity.agda           | 10 +++++-----
 src/Effect/Monad/Partiality.agda         |  6 +++---
 src/Effect/Monad/Predicate.agda          |  6 +++---
 src/Effect/Monad/Reader.agda             | 10 +++-------
 src/Effect/Monad/Reader/Transformer.agda | 10 +++++-----
 src/Effect/Monad/State.agda              | 12 ++++--------
 src/Effect/Monad/State/Transformer.agda  | 19 +++++++++----------
 src/Effect/Monad/Writer.agda             |  9 +++------
 src/Effect/Monad/Writer/Transformer.agda | 12 ++++++------
 12 files changed, 51 insertions(+), 70 deletions(-)

diff --git a/src/Effect/Monad/Continuation.agda b/src/Effect/Monad/Continuation.agda
index 2ff9dd4bb9..a5764f8898 100644
--- a/src/Effect/Monad/Continuation.agda
+++ b/src/Effect/Monad/Continuation.agda
@@ -8,13 +8,12 @@
 
 module Effect.Monad.Continuation where
 
-open import Effect.Applicative
-open import Effect.Applicative.Indexed
-open import Effect.Monad
+open import Effect.Applicative.Indexed using (IFun)
+open import Effect.Monad using (RawMonad)
 open import Function.Identity.Effectful as Id using (Identity)
-open import Effect.Monad.Indexed
+open import Effect.Monad.Indexed using (RawIMonad)
 open import Function.Base using (flip)
-open import Level
+open import Level using (Level; _⊔_; suc)
 
 private
   variable
diff --git a/src/Effect/Monad/Error/Transformer.agda b/src/Effect/Monad/Error/Transformer.agda
index 308a20bbbd..bb8128ff4e 100644
--- a/src/Effect/Monad/Error/Transformer.agda
+++ b/src/Effect/Monad/Error/Transformer.agda
@@ -6,16 +6,12 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Level
+open import Level using (Level; _⊔_; suc)
 
 module Effect.Monad.Error.Transformer {e} (E : Set e) (a : Level) where
 
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
-open import Effect.Applicative
-open import Effect.Monad
-open import Function.Base
+open import Effect.Monad using (RawMonad)
+open import Function.Base using (_∘′_; _$_)
 
 private
   variable
diff --git a/src/Effect/Monad/IO.agda b/src/Effect/Monad/IO.agda
index f52c0a3eff..c0f662d6ef 100644
--- a/src/Effect/Monad/IO.agda
+++ b/src/Effect/Monad/IO.agda
@@ -6,15 +6,13 @@
 
 {-# OPTIONS --cubical-compatible --guardedness #-}
 
-open import Level
-
 module Effect.Monad.IO where
 
 open import Data.Product.Base using (_,_)
-open import Function.Base
+open import Effect.Functor using (RawFunctor)
+open import Function.Base using (id)
 open import IO.Base using (IO)
-open import Effect.Functor
-open import Effect.Monad
+open import Level using (Level; suc)
 
 private
   variable
diff --git a/src/Effect/Monad/Identity.agda b/src/Effect/Monad/Identity.agda
index b828d00fcc..e3b9b81069 100644
--- a/src/Effect/Monad/Identity.agda
+++ b/src/Effect/Monad/Identity.agda
@@ -8,12 +8,12 @@
 
 module Effect.Monad.Identity where
 
-open import Effect.Functor
-open import Effect.Applicative
-open import Effect.Monad
-open import Effect.Comonad
+open import Effect.Functor using (RawFunctor)
+open import Effect.Applicative using (RawApplicative)
+open import Effect.Monad using (RawMonad; module Join)
+open import Effect.Comonad using (RawComonad)
 open import Function.Base using (id; _∘′_; _|>′_; _$′_; flip)
-open import Level
+open import Level using (Level)
 
 private
   variable
diff --git a/src/Effect/Monad/Partiality.agda b/src/Effect/Monad/Partiality.agda
index ba4bae8be1..d2bf58cac5 100644
--- a/src/Effect/Monad/Partiality.agda
+++ b/src/Effect/Monad/Partiality.agda
@@ -9,13 +9,13 @@
 module Effect.Monad.Partiality where
 
 open import Codata.Musical.Notation using (∞; ♯_; ♭)
-open import Effect.Functor using (RawFunctor)
-open import Effect.Applicative using (RawApplicative)
-open import Effect.Monad using (RawMonad; module Join)
 open import Data.Bool.Base using (Bool; false; true)
 open import Data.Nat.Base using (ℕ; zero; suc; _+_)
 open import Data.Product as Prod using (∃; ∄; -,_; ∃₂; _,_; _×_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
+open import Effect.Functor using (RawFunctor)
+open import Effect.Applicative using (RawApplicative)
+open import Effect.Monad using (RawMonad; module Join)
 open import Function.Base using (_∘′_; flip; id; _∘_; _$_; _⟨_⟩_)
 open import Function.Bundles using (_⇔_; mk⇔)
 open import Level using (Level; _⊔_)
diff --git a/src/Effect/Monad/Predicate.agda b/src/Effect/Monad/Predicate.agda
index 6e4ed0ffb0..cd97826c33 100644
--- a/src/Effect/Monad/Predicate.agda
+++ b/src/Effect/Monad/Predicate.agda
@@ -10,10 +10,10 @@
 
 module Effect.Monad.Predicate where
 
-open import Effect.Monad.Indexed using (RawIMonad)
 open import Data.Product.Base using (_,_)
+open import Effect.Monad.Indexed using (RawIMonad)
 open import Function.Base using (const; id; _∘_)
-open import Level using (Level; suc; _⊔_)
+open import Level using (Level; _⊔_; suc)
 open import Relation.Binary.PropositionalEquality.Core using (refl)
 open import Relation.Unary using (_⊆_; _⇒_; _∈_; _∩_; ｛_｝)
 open import Relation.Unary.PredicateTransformer using (Pt)
@@ -24,7 +24,7 @@ private
 
 ------------------------------------------------------------------------
 
-record RawPMonad {I : Set i} (M : Pt I (i ⊔ ℓ)) : Set (suc i ⊔ suc ℓ) where
+record RawPMonad {I : Set i} (M : Pt I (i ⊔ ℓ)) : Set (suc (i ⊔ ℓ)) where
 
   infixl 1 _?>=_ _?>_ _>?>_ _?>=′_
   infixr 1 _=<?_ _<?<_
diff --git a/src/Effect/Monad/Reader.agda b/src/Effect/Monad/Reader.agda
index 42e6091d00..e4ef78da0d 100644
--- a/src/Effect/Monad/Reader.agda
+++ b/src/Effect/Monad/Reader.agda
@@ -6,15 +6,11 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Level
-
 module Effect.Monad.Reader where
 
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
-open import Effect.Applicative
-open import Effect.Monad
+open import Effect.Functor using (RawFunctor)
+open import Effect.Applicative using (RawApplicative)
+open import Effect.Monad using (RawMonad; module Join)
 open import Effect.Monad.Identity as Id using (Identity; runIdentity)
 open import Level using (Level)
 
diff --git a/src/Effect/Monad/Reader/Transformer.agda b/src/Effect/Monad/Reader/Transformer.agda
index 04121c6938..e13e3f4df1 100644
--- a/src/Effect/Monad/Reader/Transformer.agda
+++ b/src/Effect/Monad/Reader/Transformer.agda
@@ -10,11 +10,11 @@
 module Effect.Monad.Reader.Transformer where
 
 open import Algebra using (RawMonoid)
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
-open import Effect.Applicative
-open import Effect.Monad
+open import Effect.Choice using (RawChoice)
+open import Effect.Empty using (RawEmpty)
+open import Effect.Functor using (RawFunctor)
+open import Effect.Applicative using (RawApplicative; RawApplicativeZero; RawAlternative)
+open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus; RawMonadT)
 open import Function.Base using (_∘′_; const; _$_)
 open import Level using (Level; _⊔_)
 
diff --git a/src/Effect/Monad/State.agda b/src/Effect/Monad/State.agda
index 2b06c8b087..4af0b026c1 100644
--- a/src/Effect/Monad/State.agda
+++ b/src/Effect/Monad/State.agda
@@ -6,18 +6,14 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-
 module Effect.Monad.State where
 
 open import Data.Product.Base using (_×_)
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
-open import Effect.Applicative
-open import Effect.Monad
+open import Effect.Functor using (RawFunctor)
+open import Effect.Applicative using (RawApplicative)
+open import Effect.Monad using (RawMonad; module Join)
 open import Effect.Monad.Identity as Id using (Identity; runIdentity)
-open import Function.Base
-open import Level
+open import Level using (Level)
 
 import Effect.Monad.State.Transformer as Trans
 
diff --git a/src/Effect/Monad/State/Transformer.agda b/src/Effect/Monad/State/Transformer.agda
index 7c6fc1156c..3a2f272f34 100644
--- a/src/Effect/Monad/State/Transformer.agda
+++ b/src/Effect/Monad/State/Transformer.agda
@@ -6,21 +6,20 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Level using (Level; suc; _⊔_)
 
 module Effect.Monad.State.Transformer where
 
 open import Algebra using (RawMonoid)
 open import Data.Product.Base using (_×_; _,_; map₂; proj₁; proj₂)
-open import Data.Unit.Polymorphic.Base
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
-open import Effect.Applicative
-open import Effect.Monad
-open import Function.Base
-
-open import Effect.Monad.Reader.Transformer using (RawMonadReader)
+open import Data.Unit.Polymorphic.Base using (tt)
+open import Effect.Choice using (RawChoice)
+open import Effect.Empty using (RawEmpty)
+open import Effect.Functor using (RawFunctor)
+open import Effect.Applicative using (RawApplicative; RawApplicativeZero; RawAlternative)
+open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus; RawMonadT; RawMonadTd)
+open import Function.Base using (_∘′_; _$_; const)
+open import Level using (Level; _⊔_)
+
 
 private
   variable
diff --git a/src/Effect/Monad/Writer.agda b/src/Effect/Monad/Writer.agda
index 60f4dc2b72..92fbad4086 100644
--- a/src/Effect/Monad/Writer.agda
+++ b/src/Effect/Monad/Writer.agda
@@ -10,13 +10,10 @@ module Effect.Monad.Writer where
 
 open import Algebra using (RawMonoid)
 open import Data.Product.Base using (_×_)
-open import Effect.Applicative
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
-open import Effect.Monad
+open import Effect.Applicative using (RawApplicative)
+open import Effect.Functor using (RawFunctor)
+open import Effect.Monad using (RawMonad; module Join)
 open import Effect.Monad.Identity as Id using (Identity; runIdentity)
-open import Function.Base using (_∘′_)
 open import Level using (Level)
 
 import Effect.Monad.Writer.Transformer as Trans
diff --git a/src/Effect/Monad/Writer/Transformer.agda b/src/Effect/Monad/Writer/Transformer.agda
index d0924cfe29..46f1e20776 100644
--- a/src/Effect/Monad/Writer/Transformer.agda
+++ b/src/Effect/Monad/Writer/Transformer.agda
@@ -11,13 +11,13 @@ module Effect.Monad.Writer.Transformer where
 
 open import Algebra using (RawMonoid)
 open import Data.Product.Base using (_×_; _,_; proj₂; map₂)
-open import Effect.Applicative
-open import Effect.Choice
-open import Effect.Empty
-open import Effect.Functor
-open import Effect.Monad
+open import Effect.Applicative using (RawApplicative; RawApplicativeZero; RawAlternative)
+open import Effect.Choice using (RawChoice)
+open import Effect.Empty using (RawEmpty)
+open import Effect.Functor using (RawFunctor)
+open import Effect.Monad using (RawMonad; RawMonadZero; RawMonadPlus; RawMonadT)
 open import Function.Base using (_∘′_; const; _$_)
-open import Level using (Level; _⊔_; suc)
+open import Level using (Level)
 
 private
   variable

From ef2bcb8ac685602f6541c75723ab8d2380862c51 Mon Sep 17 00:00:00 2001
From: Alba Mendez <me@alba.sh>
Date: Sun, 21 Apr 2024 18:12:11 +0200
Subject: [PATCH 277/316] Add proofs to Data.List.Properties (#2355)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* Add map commutation with other operations

map commutes with most list operations in some way,
and I initially made a section just for these proofs,
but later decided to spread them into each section
for consistency.

* Add congruence to operations that miss it

* Add flip & comm proofs to align[With] & [un]zip[With]

For the case of zipWith, the existing comm proof
can be provided in terms of cong and flip.

For the case of unzip[With], the comm proof has
little use and the flip proof is better named "swap".

* Remove unbound parameter

The alignWith section begins with a
module _ {f g : These A B → C} where
but g is only used by the first function.

* Add properties for take

* Proof of zip[With] and unzip[With] being inverses

* fixup! don't put list parameters in modules

* fixup! typo in changelog entry

* fixup! use equational reasoning in place of trans

* Add interaction with ++ in two more operations

* fixup! foldl-cong: don't take d ≡ e

* prove properties about catMaybes

now that catMaybes is no longer defined in
terms of mapMaybe, properties about mapMaybe
need to be proven for catMaybe as well

* move mapMaybe properties down

see next commit (given that mapMaybe-concatMap
now depends on map-concatMap, it has to be
moved down)

* simplify mapMaybe proofs

since mapMaybe is now defined in terms
of catMaybes and map, we can derive its
proofs from the proofs of those rather
than using induction directly

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
---
 CHANGELOG.md                  |  34 ++++
 src/Data/List/Properties.agda | 319 +++++++++++++++++++++++++++++-----
 2 files changed, 312 insertions(+), 41 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index dc3c702779..ac44c5e869 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -16,6 +16,9 @@ Bug-fixes
 * Module `Data.List.Relation.Ternary.Appending.Setoid.Properties` no longer
   exports the `Setoid` module under the alias `S`.
 
+* Remove unbound parameter from `Data.List.Properties.length-alignWith`,
+  `alignWith-map` and `map-alignWith`.
+
 Non-backwards compatible changes
 --------------------------------
 
@@ -330,6 +333,37 @@ Additions to existing modules
   reverse-upTo          : reverse (upTo n) ≡ downFrom n
   reverse-applyDownFrom : reverse (applyDownFrom f n) ≡ applyUpTo f n
   reverse-downFrom      : reverse (downFrom n) ≡ upTo n
+  mapMaybe-map          : mapMaybe f ∘ map g ≗ mapMaybe (f ∘ g)
+  map-mapMaybe          : map g ∘ mapMaybe f ≗ mapMaybe (Maybe.map g ∘ f)
+  align-map             : align (map f xs) (map g ys) ≡ map (map f g) (align xs ys)
+  zip-map               : zip (map f xs) (map g ys) ≡ map (map f g) (zip xs ys)
+  unzipWith-map         : unzipWith f ∘ map g ≗ unzipWith (f ∘ g)
+  map-unzipWith         : map (map g) (map h) ∘ unzipWith f ≗ unzipWith (map g h ∘ f)
+  unzip-map             : unzip ∘ map (map f g) ≗ map (map f) (map g) ∘ unzip
+  splitAt-map           : splitAt n ∘ map f ≗ map (map f) (map f) ∘ splitAt n
+  uncons-map            : uncons ∘ map f ≗ map (map f (map f)) ∘ uncons
+  last-map              : last ∘ map f ≗ map f ∘ last
+  tail-map              : tail ∘ map f ≗ map (map f) ∘ tail
+  mapMaybe-cong         : f ≗ g → mapMaybe f ≗ mapMaybe g
+  zipWith-cong          : (∀ a b → f a b ≡ g a b) → ∀ as → zipWith f as ≗ zipWith g as
+  unzipWith-cong        : f ≗ g → unzipWith f ≗ unzipWith g
+  foldl-cong            : (∀ x y → f x y ≡ g x y) → ∀ x → foldl f x ≗ foldl g x
+  alignWith-flip        : alignWith f xs ys ≡ alignWith (f ∘ swap) ys xs
+  alignWith-comm        : f ∘ swap ≗ f → alignWith f xs ys ≡ alignWith f ys xs
+  align-flip            : align xs ys ≡ map swap (align ys xs)
+  zip-flip              : zip xs ys ≡ map swap (zip ys xs)
+  unzipWith-swap        : unzipWith (swap ∘ f) ≗ swap ∘ unzipWith f
+  unzip-swap            : unzip ∘ map swap ≗ swap ∘ unzip
+  take-take             : take n (take m xs) ≡ take (n ⊓ m) xs
+  take-drop             : take n (drop m xs) ≡ drop m (take (m + n) xs)
+  zip-unzip             : uncurry′ zip ∘ unzip ≗ id
+  unzipWith-zipWith     : f ∘ uncurry′ g ≗ id → length xs ≡ length ys → unzipWith f (zipWith g xs ys) ≡ (xs , ys)
+  unzip-zip             : length xs ≡ length ys → unzip (zip xs ys) ≡ (xs , ys)
+  mapMaybe-++           : mapMaybe f (xs ++ ys) ≡ mapMaybe f xs ++ mapMaybe f ys
+  unzipWith-++          : unzipWith f (xs ++ ys) ≡ zip _++_ _++_ (unzipWith f xs) (unzipWith f ys)
+  catMaybes-concatMap   : catMaybes ≗ concatMap fromMaybe
+  catMaybes-++          : catMaybes (xs ++ ys) ≡ catMaybes xs ++ catMaybes ys
+  map-catMaybes         : map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
   ```
 
 * In `Data.List.Relation.Unary.All.Properties`:
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index 20ac353903..d11777cd0d 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -121,36 +121,30 @@ map-injective finj {x ∷ xs} {y ∷ ys} eq =
   cong₂ _∷_ (finj fx≡fy) (map-injective finj fxs≡fys)
 
 ------------------------------------------------------------------------
--- mapMaybe
+-- catMaybes
+
+catMaybes-concatMap : catMaybes {A = A} ≗ concatMap fromMaybe
+catMaybes-concatMap []             = refl
+catMaybes-concatMap (just x  ∷ xs) = cong (x ∷_) (catMaybes-concatMap xs)
+catMaybes-concatMap (nothing ∷ xs) = catMaybes-concatMap xs
 
 length-catMaybes : ∀ xs → length (catMaybes {A = A} xs) ≤ length xs
-length-catMaybes []        = ≤-refl
-length-catMaybes (just x ∷ xs) = s≤s (length-catMaybes xs)
+length-catMaybes []             = ≤-refl
+length-catMaybes (just x  ∷ xs) = s≤s (length-catMaybes xs)
 length-catMaybes (nothing ∷ xs) = m≤n⇒m≤1+n (length-catMaybes xs)
 
-mapMaybe-just : (xs : List A) → mapMaybe just xs ≡ xs
-mapMaybe-just []       = refl
-mapMaybe-just (x ∷ xs) = cong (x ∷_) (mapMaybe-just xs)
-
-mapMaybe-nothing : (xs : List A) →
-                   mapMaybe {B = B} (λ _ → nothing) xs ≡ []
-mapMaybe-nothing []       = refl
-mapMaybe-nothing (x ∷ xs) = mapMaybe-nothing xs
+catMaybes-++ : (xs ys : List (Maybe A)) →
+               catMaybes (xs ++ ys) ≡ catMaybes xs ++ catMaybes ys
+catMaybes-++ []             ys = refl
+catMaybes-++ (just x  ∷ xs) ys = cong (x ∷_) (catMaybes-++ xs ys)
+catMaybes-++ (nothing ∷ xs) ys = catMaybes-++ xs ys
 
-module _ (f : A → Maybe B) where
+module _ (f : A → B) where
 
-  mapMaybe-concatMap : mapMaybe f ≗ concatMap (fromMaybe ∘ f)
-  mapMaybe-concatMap []       = refl
-  mapMaybe-concatMap (x ∷ xs) with ih ← mapMaybe-concatMap xs | f x
-  ... | just y  = cong (y ∷_) ih
-  ... | nothing = ih
-
-  length-mapMaybe : ∀ xs → length (mapMaybe f xs) ≤ length xs
-  length-mapMaybe xs = ≤-begin
-    length (mapMaybe f xs)      ≤⟨ length-catMaybes (map f xs) ⟩
-    length (map f xs)           ≤⟨ ≤-reflexive (length-map f xs) ⟩
-    length xs                   ≤-∎
-    where open ≤-Reasoning renaming (begin_ to ≤-begin_; _∎ to _≤-∎)
+  map-catMaybes : map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
+  map-catMaybes []             = refl
+  map-catMaybes (just x  ∷ xs) = cong (f x ∷_) (map-catMaybes xs)
+  map-catMaybes (nothing ∷ xs) = map-catMaybes xs
 
 ------------------------------------------------------------------------
 -- _++_
@@ -302,6 +296,8 @@ module _ {f g : These A B → C} where
   alignWith-cong f≗g (a ∷ as)   (b ∷ bs) =
     cong₂ _∷_ (f≗g (these a b)) (alignWith-cong f≗g as bs)
 
+module _ {f : These A B → C} where
+
   length-alignWith : ∀ xs ys →
                    length (alignWith f xs ys) ≡ length xs ⊔ length ys
   length-alignWith []         ys       = length-map (f ∘′ that) ys
@@ -324,18 +320,50 @@ module _ {f g : These A B → C} where
   map-alignWith g (x ∷ xs) (y ∷ ys) =
     cong₂ _∷_ refl (map-alignWith g xs ys)
 
+  alignWith-flip : ∀ xs ys →
+                 alignWith f xs ys ≡ alignWith (f ∘ These.swap) ys xs
+  alignWith-flip []       []       = refl
+  alignWith-flip []       (y ∷ ys) = refl
+  alignWith-flip (x ∷ xs) []       = refl
+  alignWith-flip (x ∷ xs) (y ∷ ys) = cong (_ ∷_) (alignWith-flip xs ys)
+
+module _ {f : These A A → B} where
+
+  alignWith-comm : f ∘ These.swap ≗ f →
+                 ∀ xs ys → alignWith f xs ys ≡ alignWith f ys xs
+  alignWith-comm f-comm xs ys = begin
+    alignWith f xs ys                ≡⟨ alignWith-flip xs ys ⟩
+    alignWith (f ∘ These.swap) ys xs ≡⟨ alignWith-cong f-comm ys xs ⟩
+    alignWith f ys xs                ∎
+
+------------------------------------------------------------------------
+-- align
+
+align-map : ∀ (f : A → B) (g : C → D) →
+            ∀ xs ys → align (map f xs) (map g ys) ≡
+                      map (These.map f g) (align xs ys)
+align-map f g xs ys = begin
+  align (map f xs) (map g ys)       ≡⟨ alignWith-map f g xs ys ⟩
+  alignWith (These.map f g) xs ys   ≡⟨ sym (map-alignWith (These.map f g) xs ys) ⟩
+  map (These.map f g) (align xs ys) ∎
+
+align-flip : ∀ (xs : List A) (ys : List B) →
+             align xs ys ≡ map These.swap (align ys xs)
+align-flip xs ys = begin
+  align xs ys                  ≡⟨ alignWith-flip xs ys ⟩
+  alignWith These.swap ys xs   ≡⟨ sym (map-alignWith These.swap ys xs) ⟩
+  map These.swap (align ys xs) ∎
+
 ------------------------------------------------------------------------
 -- zipWith
 
-module _ (f : A → A → B) where
+module _ {f g : A → B → C} where
 
-  zipWith-comm : (∀ x y → f x y ≡ f y x) →
-                 ∀ xs ys → zipWith f xs ys ≡ zipWith f ys xs
-  zipWith-comm f-comm []       []       = refl
-  zipWith-comm f-comm []       (x ∷ ys) = refl
-  zipWith-comm f-comm (x ∷ xs) []       = refl
-  zipWith-comm f-comm (x ∷ xs) (y ∷ ys) =
-    cong₂ _∷_ (f-comm x y) (zipWith-comm f-comm xs ys)
+  zipWith-cong : (∀ a b → f a b ≡ g a b) → ∀ as → zipWith f as ≗ zipWith g as
+  zipWith-cong f≗g []         bs       = refl
+  zipWith-cong f≗g as@(_ ∷ _) []       = refl
+  zipWith-cong f≗g (a ∷ as)   (b ∷ bs) =
+    cong₂ _∷_ (f≗g a b) (zipWith-cong f≗g as bs)
 
 module _ (f : A → B → C) where
 
@@ -354,7 +382,7 @@ module _ (f : A → B → C) where
   length-zipWith (x ∷ xs) []       = refl
   length-zipWith (x ∷ xs) (y ∷ ys) = cong suc (length-zipWith xs ys)
 
-  zipWith-map : ∀ {d e} {D : Set d} {E : Set e} (g : D → A) (h : E → B) →
+  zipWith-map : ∀ (g : D → A) (h : E → B) →
                 ∀ xs ys → zipWith f (map g xs) (map h ys) ≡
                           zipWith (λ x y → f (g x) (h y)) xs ys
   zipWith-map g h []       []       = refl
@@ -363,7 +391,7 @@ module _ (f : A → B → C) where
   zipWith-map g h (x ∷ xs) (y ∷ ys) =
     cong₂ _∷_ refl (zipWith-map g h xs ys)
 
-  map-zipWith : ∀ {d} {D : Set d} (g : C → D) → ∀ xs ys →
+  map-zipWith : ∀ (g : C → D) → ∀ xs ys →
                 map g (zipWith f xs ys) ≡
                 zipWith (λ x y → g (f x y)) xs ys
   map-zipWith g []       []       = refl
@@ -372,6 +400,39 @@ module _ (f : A → B → C) where
   map-zipWith g (x ∷ xs) (y ∷ ys) =
     cong₂ _∷_ refl (map-zipWith g xs ys)
 
+  zipWith-flip : ∀ xs ys → zipWith f xs ys ≡ zipWith (flip f) ys xs
+  zipWith-flip []       []       = refl
+  zipWith-flip []       (x ∷ ys) = refl
+  zipWith-flip (x ∷ xs) []       = refl
+  zipWith-flip (x ∷ xs) (y ∷ ys) = cong (f x y ∷_) (zipWith-flip xs ys)
+
+module _ (f : A → A → B) where
+
+  zipWith-comm : (∀ x y → f x y ≡ f y x) →
+                 ∀ xs ys → zipWith f xs ys ≡ zipWith f ys xs
+  zipWith-comm f-comm xs ys = begin
+    zipWith f xs ys        ≡⟨ zipWith-flip f xs ys ⟩
+    zipWith (flip f) ys xs ≡⟨ zipWith-cong (flip f-comm) ys xs ⟩
+    zipWith f ys xs        ∎
+
+------------------------------------------------------------------------
+-- zip
+
+zip-map : ∀ (f : A → B) (g : C → D) →
+          ∀ xs ys → zip (map f xs) (map g ys) ≡
+                    map (Product.map f g) (zip xs ys)
+zip-map f g xs ys = begin
+  zip (map f xs) (map g ys)         ≡⟨ zipWith-map _,_ f g xs ys ⟩
+  zipWith (λ x y → f x , g y) xs ys ≡⟨ sym (map-zipWith _,_ (Product.map f g) xs ys) ⟩
+  map (Product.map f g) (zip xs ys) ∎
+
+zip-flip : ∀ (xs : List A) (ys : List B) →
+           zip xs ys ≡ map Product.swap (zip ys xs)
+zip-flip xs ys = begin
+  zip xs ys                    ≡⟨ zipWith-flip _,_ xs ys ⟩
+  zipWith (flip _,_) ys xs     ≡⟨ sym (map-zipWith _,_ Product.swap ys xs) ⟩
+  map Product.swap (zip ys xs) ∎
+
 ------------------------------------------------------------------------
 -- unalignWith
 
@@ -429,6 +490,13 @@ module _ (f : C → These A B) where
 ------------------------------------------------------------------------
 -- unzipWith
 
+module _ {f g : A → B × C} where
+
+  unzipWith-cong : f ≗ g → unzipWith f ≗ unzipWith g
+  unzipWith-cong f≗g [] = refl
+  unzipWith-cong f≗g (x ∷ xs) =
+    cong₂ (Product.zip _∷_ _∷_) (f≗g x) (unzipWith-cong f≗g xs)
+
 module _ (f : A → B × C) where
 
   length-unzipWith₁ : ∀ xys →
@@ -443,9 +511,66 @@ module _ (f : A → B × C) where
 
   zipWith-unzipWith : (g : B → C → A) → uncurry′ g ∘ f ≗ id →
                       uncurry′ (zipWith g) ∘ (unzipWith f)  ≗ id
-  zipWith-unzipWith g f∘g≗id []       = refl
-  zipWith-unzipWith g f∘g≗id (x ∷ xs) =
-    cong₂ _∷_ (f∘g≗id x) (zipWith-unzipWith g f∘g≗id xs)
+  zipWith-unzipWith g g∘f≗id []       = refl
+  zipWith-unzipWith g g∘f≗id (x ∷ xs) =
+    cong₂ _∷_ (g∘f≗id x) (zipWith-unzipWith g g∘f≗id xs)
+
+  unzipWith-zipWith : (g : B → C → A) → f ∘ uncurry′ g ≗ id →
+                      ∀ xs ys → length xs ≡ length ys →
+                      unzipWith f (zipWith g xs ys) ≡ (xs , ys)
+  unzipWith-zipWith g f∘g≗id []       []       l≡l = refl
+  unzipWith-zipWith g f∘g≗id (x ∷ xs) (y ∷ ys) l≡l  =
+    cong₂ (Product.zip _∷_ _∷_) (f∘g≗id (x , y))
+          (unzipWith-zipWith g f∘g≗id xs ys (suc-injective l≡l))
+
+  unzipWith-map : (g : D → A) → unzipWith f ∘ map g ≗ unzipWith (f ∘ g)
+  unzipWith-map g []       = refl
+  unzipWith-map g (x ∷ xs) =
+    cong (Product.zip _∷_ _∷_ (f (g x))) (unzipWith-map g xs)
+
+  map-unzipWith : (g : B → D) (h : C → E) →
+                  Product.map (map g) (map h) ∘ unzipWith f ≗
+                  unzipWith (Product.map g h ∘ f)
+  map-unzipWith g h []       = refl
+  map-unzipWith g h (x ∷ xs) =
+    cong (Product.zip _∷_ _∷_ _) (map-unzipWith g h xs)
+
+  unzipWith-swap : unzipWith (Product.swap ∘ f) ≗
+                   Product.swap ∘ unzipWith f
+  unzipWith-swap []       = refl
+  unzipWith-swap (x ∷ xs) =
+    cong (Product.zip _∷_ _∷_ _) (unzipWith-swap xs)
+
+  unzipWith-++ : ∀ xs ys →
+                 unzipWith f (xs ++ ys) ≡
+                 Product.zip _++_ _++_ (unzipWith f xs) (unzipWith f ys)
+  unzipWith-++ []       ys = refl
+  unzipWith-++ (x ∷ xs) ys =
+    cong (Product.zip _∷_ _∷_ (f x)) (unzipWith-++ xs ys)
+
+------------------------------------------------------------------------
+-- unzip
+
+unzip-map : ∀ (f : A → B) (g : C → D) →
+            unzip ∘ map (Product.map f g) ≗
+            Product.map (map f) (map g) ∘ unzip
+unzip-map f g xs = begin
+  unzip (map (Product.map f g) xs)       ≡⟨ unzipWith-map id (Product.map f g) xs ⟩
+  unzipWith (Product.map f g) xs         ≡⟨ sym (map-unzipWith id f g xs) ⟩
+  Product.map (map f) (map g) (unzip xs) ∎
+
+unzip-swap : unzip ∘ map Product.swap ≗ Product.swap ∘ unzip {A = A} {B = B}
+unzip-swap xs = begin
+  unzip (map Product.swap xs) ≡⟨ unzipWith-map id Product.swap xs ⟩
+  unzipWith Product.swap xs   ≡⟨ unzipWith-swap id xs ⟩
+  Product.swap (unzip xs)     ∎
+
+zip-unzip : uncurry′ zip ∘ unzip ≗ id {A = List (A × B)}
+zip-unzip = zipWith-unzipWith id _,_ λ _ → refl
+
+unzip-zip : ∀ (xs : List A) (ys : List B) →
+            length xs ≡ length ys → unzip (zip xs ys) ≡ (xs , ys)
+unzip-zip = unzipWith-zipWith id _,_ λ _ → refl
 
 ------------------------------------------------------------------------
 -- foldr
@@ -534,6 +659,11 @@ module _ {P : Pred A p} {f : A → A → A} where
 ------------------------------------------------------------------------
 -- foldl
 
+foldl-cong : ∀ {f g : B → A → B} → (∀ x y → f x y ≡ g x y) →
+             ∀ x → foldl f x ≗ foldl g x
+foldl-cong f≗g x []       = refl
+foldl-cong f≗g x (y ∷ xs) rewrite f≗g x y = foldl-cong f≗g _ xs
+
 foldl-++ : ∀ (f : A → B → A) x ys zs →
            foldl f x (ys ++ zs) ≡ foldl f (foldl f x ys) zs
 foldl-++ f x []       zs = refl
@@ -608,6 +738,58 @@ map-concatMap f g xs = begin
   concatMap (map f ∘′ g) xs
     ∎
 
+------------------------------------------------------------------------
+-- mapMaybe
+
+module _ {f g : A → Maybe B} where
+
+  mapMaybe-cong : f ≗ g → mapMaybe f ≗ mapMaybe g
+  mapMaybe-cong f≗g = cong catMaybes ∘ map-cong f≗g
+
+mapMaybe-just : (xs : List A) → mapMaybe just xs ≡ xs
+mapMaybe-just []       = refl
+mapMaybe-just (x ∷ xs) = cong (x ∷_) (mapMaybe-just xs)
+
+mapMaybe-nothing : (xs : List A) →
+                   mapMaybe {B = B} (λ _ → nothing) xs ≡ []
+mapMaybe-nothing []       = refl
+mapMaybe-nothing (x ∷ xs) = mapMaybe-nothing xs
+
+module _ (f : A → Maybe B) where
+
+  mapMaybe-concatMap : mapMaybe f ≗ concatMap (fromMaybe ∘ f)
+  mapMaybe-concatMap xs = begin
+    catMaybes (map f xs)            ≡⟨ catMaybes-concatMap (map f xs) ⟩
+    concatMap fromMaybe (map f xs)  ≡⟨ concatMap-map fromMaybe f xs ⟩
+    concatMap (fromMaybe ∘ f) xs    ∎
+
+  length-mapMaybe : ∀ xs → length (mapMaybe f xs) ≤ length xs
+  length-mapMaybe xs = ≤-begin
+    length (mapMaybe f xs)      ≤⟨ length-catMaybes (map f xs) ⟩
+    length (map f xs)           ≤⟨ ≤-reflexive (length-map f xs) ⟩
+    length xs                   ≤-∎
+    where open ≤-Reasoning renaming (begin_ to ≤-begin_; _∎ to _≤-∎)
+
+  mapMaybe-++ : ∀ xs ys →
+                mapMaybe f (xs ++ ys) ≡ mapMaybe f xs ++ mapMaybe f ys
+  mapMaybe-++ xs ys = begin
+    catMaybes (map f (xs ++ ys))     ≡⟨ cong catMaybes (map-++ f xs ys) ⟩
+    catMaybes (map f xs ++ map f ys) ≡⟨ catMaybes-++ (map f xs) (map f ys) ⟩
+    mapMaybe f xs ++ mapMaybe f ys   ∎
+
+module _ (f : B → Maybe C) (g : A → B) where
+
+  mapMaybe-map : mapMaybe f ∘ map g ≗ mapMaybe (f ∘ g)
+  mapMaybe-map = cong catMaybes ∘ sym ∘ map-∘
+
+module _ (g : B → C) (f : A → Maybe B) where
+
+  map-mapMaybe : map g ∘ mapMaybe f ≗ mapMaybe (Maybe.map g ∘ f)
+  map-mapMaybe xs = begin
+    map g (catMaybes (map f xs))       ≡⟨ map-catMaybes g (map f xs) ⟩
+    mapMaybe (Maybe.map g) (map f xs)  ≡⟨ mapMaybe-map _ f xs ⟩
+    mapMaybe (Maybe.map g ∘ f) xs      ∎
+
 ------------------------------------------------------------------------
 -- sum
 
@@ -829,6 +1011,21 @@ take-[] : ∀ m → take {A = A} m [] ≡ []
 take-[] zero = refl
 take-[] (suc m) = refl
 
+-- Taking twice takes the minimum of both counts.
+take-take : ∀ n m (xs : List A) → take n (take m xs) ≡ take (n ⊓ m) xs
+take-take zero    m       xs       = refl
+take-take (suc n) zero    xs       = refl
+take-take (suc n) (suc m) []       = refl
+take-take (suc n) (suc m) (x ∷ xs) = cong (x ∷_) (take-take n m xs)
+
+-- Dropping m elements and then taking n is the same as
+-- taking n + m elements and then dropping m.
+take-drop : ∀ n m (xs : List A) →
+            take n (drop m xs) ≡ drop m (take (m + n) xs)
+take-drop n zero    xs       = refl
+take-drop n (suc m) []       = take-[] n
+take-drop n (suc m) (x ∷ xs) = take-drop n m xs
+
 ------------------------------------------------------------------------
 -- drop
 
@@ -919,6 +1116,15 @@ splitAt-defn zero    xs       = refl
 splitAt-defn (suc n) []       = refl
 splitAt-defn (suc n) (x ∷ xs) = cong (Product.map (x ∷_) id) (splitAt-defn n xs)
 
+module _ (f : A → B) where
+
+  splitAt-map : ∀ n → splitAt n ∘ map f ≗
+                      Product.map (map f) (map f) ∘ splitAt n
+  splitAt-map zero    xs       = refl
+  splitAt-map (suc n) []       = refl
+  splitAt-map (suc n) (x ∷ xs) =
+    cong (Product.map₁ (f x ∷_)) (splitAt-map n xs)
+
 ------------------------------------------------------------------------
 -- takeWhile, dropWhile, and span
 
@@ -1238,15 +1444,46 @@ reverse-downFrom = reverse-applyDownFrom id
 ∷ʳ-++ : ∀ xs (a : A) ys → xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys
 ∷ʳ-++ xs a ys = ++-assoc xs [ a ] ys
 
+------------------------------------------------------------------------
+-- uncons
+
+module _ (f : A → B) where
 
+  -- 'commute' List.uncons and List.map to obtain a Maybe.map and List.uncons.
+  uncons-map : uncons ∘ map f ≗ Maybe.map (Product.map f (map f)) ∘ uncons
+  uncons-map []       = refl
+  uncons-map (x ∷ xs) = refl
 
 ------------------------------------------------------------------------
 -- head
 
--- 'commute' List.head and List.map to obtain a Maybe.map and List.head.
-head-map : ∀ {f : A → B} xs → head (map f xs) ≡ Maybe.map f (head xs)
-head-map [] = refl
-head-map (_ ∷ _) = refl
+module _ {f : A → B} where
+
+  -- 'commute' List.head and List.map to obtain a Maybe.map and List.head.
+  head-map : head ∘ map f ≗ Maybe.map f ∘ head
+  head-map []      = refl
+  head-map (_ ∷ _) = refl
+
+------------------------------------------------------------------------
+-- last
+
+module _ (f : A → B) where
+
+  -- 'commute' List.last and List.map to obtain a Maybe.map and List.last.
+  last-map : last ∘ map f ≗ Maybe.map f ∘ last
+  last-map []               = refl
+  last-map (x ∷ [])         = refl
+  last-map (x ∷ xs@(_ ∷ _)) = last-map xs
+
+------------------------------------------------------------------------
+-- tail
+
+module _ (f : A → B) where
+
+  -- 'commute' List.tail and List.map to obtain a Maybe.map and List.tail.
+  tail-map : tail ∘ map f ≗ Maybe.map (map f) ∘ tail
+  tail-map []       = refl
+  tail-map (x ∷ xs) = refl
 
 
 

From aad1d1926c96110d31eae995ffe7a32f13fdca13 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 22 Apr 2024 06:37:06 +0100
Subject: [PATCH 278/316] Improve `Data.List.Base` (fix #2359; deprecate use of
 `with` #2123) (#2365)

* refactor towards `if_then_else_` and away from `yes`/`no`

* reverted chnages to `derun` proofs

* undo last reversion!

* layout
---
 src/Data/List/Base.agda       | 46 ++++++++++++++++++++---------------
 src/Data/List/Properties.agda | 30 +++++++++++------------
 2 files changed, 41 insertions(+), 35 deletions(-)

diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index bc31ca0f9e..c3c894cf65 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -364,45 +364,49 @@ removeAt (x ∷ xs) (suc i)  = x ∷ removeAt xs i
 
 takeWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
 takeWhile P? []       = []
-takeWhile P? (x ∷ xs) with does (P? x)
-... | true  = x ∷ takeWhile P? xs
-... | false = []
+takeWhile P? (x ∷ xs) = if does (P? x)
+  then x ∷ takeWhile P? xs
+  else []
 
 takeWhileᵇ : (A → Bool) → List A → List A
 takeWhileᵇ p = takeWhile (T? ∘ p)
 
 dropWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
 dropWhile P? []       = []
-dropWhile P? (x ∷ xs) with does (P? x)
-... | true  = dropWhile P? xs
-... | false = x ∷ xs
+dropWhile P? (x ∷ xs) = if does (P? x)
+  then dropWhile P? xs
+  else x ∷ xs
 
 dropWhileᵇ : (A → Bool) → List A → List A
 dropWhileᵇ p = dropWhile (T? ∘ p)
 
 filter : ∀ {P : Pred A p} → Decidable P → List A → List A
 filter P? [] = []
-filter P? (x ∷ xs) with does (P? x)
-... | false = filter P? xs
-... | true  = x ∷ filter P? xs
+filter P? (x ∷ xs) =
+  let xs′ = filter P? xs in
+  if does (P? x)
+    then x ∷ xs′
+    else xs′
 
 filterᵇ : (A → Bool) → List A → List A
 filterᵇ p = filter (T? ∘ p)
 
 partition : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
-partition P? []       = ([] , [])
-partition P? (x ∷ xs) with does (P? x) | partition P? xs
-... | true  | (ys , zs) = (x ∷ ys , zs)
-... | false | (ys , zs) = (ys , x ∷ zs)
+partition P? []       = [] , []
+partition P? (x ∷ xs) =
+  let ys , zs = partition P? xs in
+  if does (P? x)
+    then (x ∷ ys , zs)
+    else (ys , x ∷ zs)
 
 partitionᵇ : (A → Bool) → List A → List A × List A
 partitionᵇ p = partition (T? ∘ p)
 
 span : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
-span P? []       = ([] , [])
-span P? ys@(x ∷ xs) with does (P? x)
-... | true  = Product.map (x ∷_) id (span P? xs)
-... | false = ([] , ys)
+span P? []          = [] , []
+span P? ys@(x ∷ xs) = if does (P? x)
+  then Product.map (x ∷_) id (span P? xs)
+  else ([] , ys)
 
 
 spanᵇ : (A → Bool) → List A → List A × List A
@@ -453,9 +457,11 @@ wordsByᵇ p = wordsBy (T? ∘ p)
 derun : ∀ {R : Rel A p} → B.Decidable R → List A → List A
 derun R? [] = []
 derun R? (x ∷ []) = x ∷ []
-derun R? (x ∷ xs@(y ∷ _)) with does (R? x y) | derun R? xs
-... | true  | ys = ys
-... | false | ys = x ∷ ys
+derun R? (x ∷ xs@(y ∷ _)) =
+  let ys = derun R? xs in
+  if does (R? x y)
+    then ys
+    else x ∷ ys
 
 derunᵇ : (A → A → Bool) → List A → List A
 derunᵇ r = derun (T? ∘₂ r)
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index d11777cd0d..dc7e8e5831 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -1156,13 +1156,13 @@ module _ {P : Pred A p} (P? : Decidable P) where
   filter-all : ∀ {xs} → All P xs → filter P? xs ≡ xs
   filter-all {[]}     []         = refl
   filter-all {x ∷ xs} (px ∷ pxs) with P? x
-  ... | no          ¬px = contradiction px ¬px
-  ... | true  because _ = cong (x ∷_) (filter-all pxs)
+  ... | false because [¬px] = contradiction px (invert [¬px])
+  ... | true  because _     = cong (x ∷_) (filter-all pxs)
 
   filter-notAll : ∀ xs → Any (∁ P) xs → length (filter P? xs) < length xs
   filter-notAll (x ∷ xs) (here ¬px) with P? x
-  ... | false because _ = s≤s (length-filter xs)
-  ... | yes          px = contradiction px ¬px
+  ... | false because _    = s≤s (length-filter xs)
+  ... | true  because [px] = contradiction (invert [px]) ¬px
   filter-notAll (x ∷ xs) (there any) with ih ← filter-notAll xs any | does (P? x)
   ... | false = m≤n⇒m≤1+n ih
   ... | true  = s≤s ih
@@ -1178,8 +1178,8 @@ module _ {P : Pred A p} (P? : Decidable P) where
   filter-none : ∀ {xs} → All (∁ P) xs → filter P? xs ≡ []
   filter-none {[]}     []           = refl
   filter-none {x ∷ xs} (¬px ∷ ¬pxs) with P? x
-  ... | false because _ = filter-none ¬pxs
-  ... | yes          px = contradiction px ¬px
+  ... | false because _    = filter-none ¬pxs
+  ... | true  because [px] = contradiction (invert [px]) ¬px
 
   filter-complete : ∀ {xs} → length (filter P? xs) ≡ length xs →
                     filter P? xs ≡ xs
@@ -1190,13 +1190,13 @@ module _ {P : Pred A p} (P? : Decidable P) where
 
   filter-accept : ∀ {x xs} → P x → filter P? (x ∷ xs) ≡ x ∷ (filter P? xs)
   filter-accept {x} Px with P? x
-  ... | true because _ = refl
-  ... | no         ¬Px = contradiction Px ¬Px
+  ... | true  because _     = refl
+  ... | false because [¬Px] = contradiction Px (invert [¬Px])
 
   filter-reject : ∀ {x xs} → ¬ P x → filter P? (x ∷ xs) ≡ filter P? xs
   filter-reject {x} ¬Px with P? x
-  ... | yes          Px = contradiction Px ¬Px
-  ... | false because _ = refl
+  ... | true  because [Px] = contradiction (invert [Px]) ¬Px
+  ... | false because _    = refl
 
   filter-idem : filter P? ∘ filter P? ≗ filter P?
   filter-idem []       = refl
@@ -1234,13 +1234,13 @@ module _ {R : Rel A p} (R? : B.Decidable R) where
 
   derun-reject : ∀ {x y} xs → R x y → derun R? (x ∷ y ∷ xs) ≡ derun R? (y ∷ xs)
   derun-reject {x} {y} xs Rxy with R? x y
-  ... | yes _    = refl
-  ... | no  ¬Rxy = contradiction Rxy ¬Rxy
+  ... | true  because _      = refl
+  ... | false because [¬Rxy] = contradiction Rxy (invert [¬Rxy])
 
   derun-accept : ∀ {x y} xs → ¬ R x y → derun R? (x ∷ y ∷ xs) ≡ x ∷ derun R? (y ∷ xs)
   derun-accept {x} {y} xs ¬Rxy with R? x y
-  ... | yes Rxy = contradiction Rxy ¬Rxy
-  ... | no  _   = refl
+  ... | true  because [Rxy] = contradiction (invert [Rxy]) ¬Rxy
+  ... | false because  _    = refl
 
 ------------------------------------------------------------------------
 -- partition
@@ -1253,7 +1253,7 @@ module _ {P : Pred A p} (P? : Decidable P) where
   ...  | true  = cong (Product.map (x ∷_) id) ih
   ...  | false = cong (Product.map id (x ∷_)) ih
 
-  length-partition : ∀ xs → (let (ys , zs) = partition P? xs) →
+  length-partition : ∀ xs → (let ys , zs = partition P? xs) →
                      length ys ≤ length xs × length zs ≤ length xs
   length-partition []       = z≤n , z≤n
   length-partition (x ∷ xs) with ih ← length-partition xs | does (P? x)

From fd82f5ce24e45c58a41169c832d1ed984d5de937 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Tue, 23 Apr 2024 11:45:21 -0400
Subject: [PATCH 279/316] A number of `with` are not really needed (#2363)

* a number of 'with' are not really needed. Many, many more were, so left as is.

* whitespace

* deal with a few outstanding comments.

* actually re-introduce a 'with' as it is actually clearer.

* with fits better here too.

* tweak things a little from review by James.

* Update src/Codata/Sized/Cowriter.agda

layout for readability

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

* Update src/Data/Fin/Base.agda

layout for readability

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

* Update src/Data/List/Fresh/Relation/Unary/All.agda

layout for readability

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

---------

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
---
 src/Algebra/Construct/LexProduct.agda         |  6 ++---
 src/Algebra/Construct/LiftedChoice.agda       | 13 +++++------
 .../Construct/NaturalChoice/MinOp.agda        |  4 ++--
 src/Codata/Sized/Cowriter.agda                |  7 +++---
 src/Data/Fin/Base.agda                        | 11 +++++-----
 src/Data/List/Fresh.agda                      | 19 ++++++++--------
 src/Data/List/Fresh/Relation/Unary/All.agda   | 10 +++++----
 src/Data/List/Kleene/Base.agda                |  8 +++----
 src/Data/List/NonEmpty/Base.agda              |  7 ++++--
 .../Relation/Binary/Infix/Heterogeneous.agda  |  6 ++---
 .../Relation/Binary/Prefix/Heterogeneous.agda |  3 +--
 .../Relation/Binary/Suffix/Heterogeneous.agda |  3 +--
 .../Suffix/Heterogeneous/Properties.agda      |  6 ++---
 .../Unary/Enumerates/Setoid/Properties.agda   |  4 ++--
 src/Data/Nat/Binary/Subtraction.agda          | 12 ++++------
 src/Data/Rational/Base.agda                   |  8 ++-----
 src/Data/Rational/Unnormalised/Base.agda      |  8 ++-----
 src/Data/Star/Environment.agda                |  5 ++---
 src/Data/Star/Pointer.agda                    |  6 ++---
 src/Data/Star/Vec.agda                        |  5 ++---
 src/Data/String/Base.agda                     | 22 +++++++++----------
 src/Data/Tree/AVL.agda                        |  8 +++----
 src/Data/Vec.agda                             | 15 +++++--------
 src/Data/Vec/Base.agda                        |  6 ++---
 src/Data/Vec/Membership/Setoid.agda           |  5 ++---
 .../Binary/Construct/NaturalOrder/Left.agda   |  6 ++---
 .../Binary/Construct/NaturalOrder/Right.agda  |  7 +++---
 src/Text/Pretty/Core.agda                     |  4 ++--
 28 files changed, 98 insertions(+), 126 deletions(-)

diff --git a/src/Algebra/Construct/LexProduct.agda b/src/Algebra/Construct/LexProduct.agda
index 65a6b80ba1..3fbcf86c80 100644
--- a/src/Algebra/Construct/LexProduct.agda
+++ b/src/Algebra/Construct/LexProduct.agda
@@ -11,7 +11,7 @@ open import Algebra.Definitions
 open import Data.Bool.Base using (true; false)
 open import Data.Product.Base using (_×_; _,_)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (Pointwise)
-open import Data.Sum.Base using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂; map)
 open import Function.Base using (_∘_)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (Decidable)
@@ -107,6 +107,4 @@ sel ∙-sel ◦-sel (a , x) (b , y) with (a ∙ b) ≟₁ a | (a ∙ b) ≟₁ b
 ... | no  ab≉a | no  ab≉b  = contradiction₂ (∙-sel a b) ab≉a ab≉b
 ... | yes ab≈a | no  _     = inj₁ (ab≈a , ≈₂-refl)
 ... | no  _    | yes ab≈b  = inj₂ (ab≈b , ≈₂-refl)
-... | yes ab≈a | yes ab≈b  with ◦-sel x y
-...   | inj₁ xy≈x = inj₁ (ab≈a , xy≈x)
-...   | inj₂ xy≈y = inj₂ (ab≈b , xy≈y)
+... | yes ab≈a | yes ab≈b  = map (ab≈a ,_) (ab≈b ,_) (◦-sel x y)
diff --git a/src/Algebra/Construct/LiftedChoice.agda b/src/Algebra/Construct/LiftedChoice.agda
index 2377650cf8..e3d37009ed 100644
--- a/src/Algebra/Construct/LiftedChoice.agda
+++ b/src/Algebra/Construct/LiftedChoice.agda
@@ -11,13 +11,14 @@ open import Algebra
 module Algebra.Construct.LiftedChoice where
 
 open import Algebra.Consequences.Base
-open import Relation.Binary.Core using (Rel; _⇒_; _Preserves_⟶_)
-open import Relation.Binary.Structures using (IsEquivalence)
-open import Relation.Nullary using (¬_; yes; no)
-open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
+open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]; [_,_]′)
 open import Data.Product.Base using (_×_; _,_)
+open import Function.Base using (const; _$_)
 open import Level using (Level; _⊔_)
+open import Relation.Binary.Core using (Rel; _⇒_; _Preserves_⟶_)
+open import Relation.Nullary using (¬_; yes; no)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+open import Relation.Binary.Structures using (IsEquivalence)
 open import Relation.Unary using (Pred)
 
 import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
@@ -34,9 +35,7 @@ private
 module _ (_≈_ : Rel B ℓ) (_•_ : Op₂ B) where
 
   Lift : Selective _≈_ _•_ → (A → B) → Op₂ A
-  Lift ∙-sel f x y with ∙-sel (f x) (f y)
-  ... | inj₁ _ = x
-  ... | inj₂ _ = y
+  Lift ∙-sel f x y = [ const x , const y ]′ $ ∙-sel (f x) (f y)
 
 ------------------------------------------------------------------------
 -- Algebraic properties
diff --git a/src/Algebra/Construct/NaturalChoice/MinOp.agda b/src/Algebra/Construct/NaturalChoice/MinOp.agda
index 22d0686f8f..7ad8983a1f 100644
--- a/src/Algebra/Construct/NaturalChoice/MinOp.agda
+++ b/src/Algebra/Construct/NaturalChoice/MinOp.agda
@@ -39,13 +39,13 @@ open import Relation.Binary.Reasoning.Preorder preorder
 
 x⊓y≤x : ∀ x y → x ⊓ y ≤ x
 x⊓y≤x x y with total x y
-... | inj₁ x≤y = ≤-respˡ-≈ (Eq.sym (x≤y⇒x⊓y≈x x≤y)) refl
+... | inj₁ x≤y = reflexive (x≤y⇒x⊓y≈x x≤y)
 ... | inj₂ y≤x = ≤-respˡ-≈ (Eq.sym (x≥y⇒x⊓y≈y y≤x)) y≤x
 
 x⊓y≤y : ∀ x y → x ⊓ y ≤ y
 x⊓y≤y x y with total x y
 ... | inj₁ x≤y = ≤-respˡ-≈ (Eq.sym (x≤y⇒x⊓y≈x x≤y)) x≤y
-... | inj₂ y≤x = ≤-respˡ-≈ (Eq.sym (x≥y⇒x⊓y≈y y≤x)) refl
+... | inj₂ y≤x = reflexive (x≥y⇒x⊓y≈y y≤x)
 
 ------------------------------------------------------------------------
 -- Algebraic properties
diff --git a/src/Codata/Sized/Cowriter.agda b/src/Codata/Sized/Cowriter.agda
index 03d24d3d3c..fd4ba91787 100644
--- a/src/Codata/Sized/Cowriter.agda
+++ b/src/Codata/Sized/Cowriter.agda
@@ -113,6 +113,7 @@ _>>=_ : ∀ {i} → Cowriter W A i → (A → Cowriter W X i) → Cowriter W X i
 -- Construction.
 
 unfold : ∀ {i} → (X → (W × X) ⊎ A) → X → Cowriter W A i
-unfold next seed with next seed
-... | inj₁ (w , seed′) = w ∷ λ where .force → unfold next seed′
-... | inj₂ a           = [ a ]
+unfold next seed =
+  Sum.[ (λ where (w , seed′) → w ∷ λ where .force → unfold next seed′)
+      , [_]
+      ] (next seed)
diff --git a/src/Data/Fin/Base.agda b/src/Data/Fin/Base.agda
index d762e10007..a328a85a57 100644
--- a/src/Data/Fin/Base.agda
+++ b/src/Data/Fin/Base.agda
@@ -15,7 +15,7 @@ open import Data.Bool.Base using (Bool; T)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc)
 open import Data.Product.Base as Product using (_×_; _,_; proj₁; proj₂)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_]′)
-open import Function.Base using (id; _∘_; _on_; flip)
+open import Function.Base using (id; _∘_; _on_; flip; _$_)
 open import Level using (0ℓ)
 open import Relation.Binary.Core
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; refl; cong)
@@ -134,7 +134,7 @@ strengthen (suc i) = suc (strengthen i)
 splitAt : ∀ m {n} → Fin (m ℕ.+ n) → Fin m ⊎ Fin n
 splitAt zero    i       = inj₂ i
 splitAt (suc m) zero    = inj₁ zero
-splitAt (suc m) (suc i) = Sum.map suc id (splitAt m i)
+splitAt (suc m) (suc i) = Sum.map₁ suc (splitAt m i)
 
 -- inverse of above function
 join : ∀ m n → Fin m ⊎ Fin n → Fin (m ℕ.+ n)
@@ -144,9 +144,10 @@ join m n = [ _↑ˡ n , m ↑ʳ_ ]′
 -- This is dual to group from Data.Vec.
 
 quotRem : ∀ n → Fin (m ℕ.* n) → Fin n × Fin m
-quotRem {suc m} n i with splitAt n i
-... | inj₁ j = j , zero
-... | inj₂ j = Product.map₂ suc (quotRem {m} n j)
+quotRem {suc m} n i =
+  [ (_, zero)
+  , Product.map₂ suc ∘ quotRem {m} n
+  ]′ $ splitAt n i
 
 -- a variant of quotRem the type of whose result matches the order of multiplication
 remQuot : ∀ n → Fin (m ℕ.* n) → Fin m × Fin n
diff --git a/src/Data/List/Fresh.agda b/src/Data/List/Fresh.agda
index 3375b0d54a..028e186567 100644
--- a/src/Data/List/Fresh.agda
+++ b/src/Data/List/Fresh.agda
@@ -14,7 +14,7 @@
 module Data.List.Fresh where
 
 open import Level using (Level; _⊔_)
-open import Data.Bool.Base using (true; false)
+open import Data.Bool.Base using (true; false; if_then_else_)
 open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Product.Base using (∃; _×_; _,_; -,_; proj₁; proj₂)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
@@ -161,10 +161,10 @@ module _ {P : Pred A p} (P? : U.Decidable P) where
   takeWhile-# : ∀ {R : Rel A r} a (as : List# A R) → a # as → a # takeWhile as
 
   takeWhile []             = []
-  takeWhile (cons a as ps) with does (P? a)
-  ... | true  = cons a (takeWhile as) (takeWhile-# a as ps)
-  ... | false = []
+  takeWhile (cons a as ps) =
+    if does (P? a) then cons a (takeWhile as) (takeWhile-# a as ps) else []
 
+  -- this 'with' is needed to cause reduction in the type of 'takeWhile (a ∷# as)'
   takeWhile-# a []        _        = _
   takeWhile-# a (x ∷# xs) (p , ps) with does (P? x)
   ... | true  = p , takeWhile-# a xs ps
@@ -172,18 +172,17 @@ module _ {P : Pred A p} (P? : U.Decidable P) where
 
   dropWhile : {R : Rel A r} → List# A R → List# A R
   dropWhile []            = []
-  dropWhile aas@(a ∷# as) with does (P? a)
-  ... | true  = dropWhile as
-  ... | false = aas
+  dropWhile aas@(a ∷# as)  = if does (P? a) then dropWhile as else aas
 
   filter   : {R : Rel A r} → List# A R → List# A R
   filter-# : ∀ {R : Rel A r} a (as : List# A R) → a # as → a # filter as
 
   filter []             = []
-  filter (cons a as ps) with does (P? a)
-  ... | true  = cons a (filter as) (filter-# a as ps)
-  ... | false = filter as
+  filter (cons a as ps) =
+    let l = filter as in
+    if does (P? a) then cons a l (filter-# a as ps) else l
 
+  -- this 'with' is needed to cause reduction in the type of 'filter-# a (x ∷# xs)'
   filter-# a []        _        = _
   filter-# a (x ∷# xs) (p , ps) with does (P? x)
   ... | true  = p , filter-# a xs ps
diff --git a/src/Data/List/Fresh/Relation/Unary/All.agda b/src/Data/List/Fresh/Relation/Unary/All.agda
index 2f560e82c2..365bbf1a70 100644
--- a/src/Data/List/Fresh/Relation/Unary/All.agda
+++ b/src/Data/List/Fresh/Relation/Unary/All.agda
@@ -10,7 +10,8 @@ module Data.List.Fresh.Relation.Unary.All where
 
 open import Level using (Level; _⊔_; Lift)
 open import Data.Product.Base using (_×_; _,_; proj₁; uncurry)
-open import Data.Sum.Base as Sum using (inj₁; inj₂)
+open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′)
+open import Function.Base using (_∘_; _$_)
 open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _×-dec_)
 open import Relation.Unary  as U
 open import Relation.Binary.Core using (Rel)
@@ -74,6 +75,7 @@ module _ {R : Rel A r} {P : Pred A p} {Q : Pred A q} where
 
   decide :  Π[ P ∪ Q ] → Π[ All {R = R} P ∪ Any Q ]
   decide p∪q [] = inj₁ []
-  decide p∪q (x ∷# xs) with p∪q x
-  ... | inj₂ qx = inj₂ (here qx)
-  ... | inj₁ px = Sum.map (px ∷_) there (decide p∪q xs)
+  decide p∪q (x ∷# xs) =
+    [ (λ px → Sum.map (px ∷_) there (decide p∪q xs))
+    , inj₂ ∘ here
+    ]′ $ p∪q x
diff --git a/src/Data/List/Kleene/Base.agda b/src/Data/List/Kleene/Base.agda
index 8eb83ba864..ef0ca0b0a2 100644
--- a/src/Data/List/Kleene/Base.agda
+++ b/src/Data/List/Kleene/Base.agda
@@ -11,7 +11,7 @@ module Data.List.Kleene.Base where
 open import Data.Product.Base as Product
   using (_×_; _,_; map₂; map₁; proj₁; proj₂)
 open import Data.Nat.Base as ℕ using (ℕ; suc; zero)
-open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
+open import Data.Maybe.Base as Maybe using (Maybe; just; nothing; maybe′)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Level using (Level)
 
@@ -137,9 +137,7 @@ module _ (f : A → Maybe B) where
   mapMaybe+ : A + → B *
   mapMaybe* : A * → B *
 
-  mapMaybe+ (x & xs) with f x
-  ... | just y  = ∹ y & mapMaybe* xs
-  ... | nothing = mapMaybe* xs
+  mapMaybe+ (x & xs) = maybe′ (λ y z → ∹ y & z) id (f x) $ mapMaybe* xs
 
   mapMaybe* []     = []
   mapMaybe* (∹ xs) = mapMaybe+ xs
@@ -268,7 +266,7 @@ module _ (f : A → Maybe B → B) where
   foldrMaybe* []     = nothing
   foldrMaybe* (∹ xs) = just (foldrMaybe+ xs)
 
-  foldrMaybe+ xs = f (head xs) (foldrMaybe* (tail xs))
+  foldrMaybe+ (x & xs) = f x (foldrMaybe* xs)
 
 ------------------------------------------------------------------------
 -- Indexing
diff --git a/src/Data/List/NonEmpty/Base.agda b/src/Data/List/NonEmpty/Base.agda
index 82f16c270a..0b1793eaf7 100644
--- a/src/Data/List/NonEmpty/Base.agda
+++ b/src/Data/List/NonEmpty/Base.agda
@@ -203,9 +203,12 @@ snocView (x ∷ .(xs List.∷ʳ y)) | xs List.∷ʳ′ y = (x ∷ xs) ∷ʳ′ y
 
 -- The last element in the list.
 
+private
+  last′ : ∀ {l} → SnocView {A = A} l → A
+  last′ (_ ∷ʳ′ y) = y
+
 last : List⁺ A → A
-last xs with snocView xs
-last .(ys ∷ʳ y) | ys ∷ʳ′ y = y
+last = last′ ∘ snocView
 
 -- Groups all contiguous elements for which the predicate returns the
 -- same result into lists. The left sums are the ones for which the
diff --git a/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda
index b09af201e0..8a5b239eb0 100644
--- a/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Infix/Heterogeneous.agda
@@ -49,10 +49,8 @@ map R⇒S (here pref) = here (Prefix.map R⇒S pref)
 map R⇒S (there inf) = there (map R⇒S inf)
 
 toView : ∀ {as bs} → Infix R as bs → View R as bs
-toView (here p) with Prefix.toView p
-...| inf Prefix.++ suff = MkView [] inf suff
-toView (there p) with toView p
-... | MkView pref inf suff = MkView (_ ∷ pref) inf suff
+toView (here p)  with inf Prefix.++ suff ← Prefix.toView p = MkView [] inf suff
+toView (there p) with MkView pref inf suff ← toView p      = MkView (_ ∷ pref) inf suff
 
 fromView : ∀ {as bs} → View R as bs → Infix R as bs
 fromView (MkView []         inf suff) = here (Prefix.fromView (inf Prefix.++ suff))
diff --git a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
index 75104bd39e..29a2fb5d02 100644
--- a/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Prefix/Heterogeneous.agda
@@ -53,8 +53,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   toView : ∀ {as bs} → Prefix R as bs → PrefixView R as bs
   toView []       = [] ++ _
-  toView (r ∷ rs) with toView rs
-  ... | rs′ ++ ds = (r ∷ rs′) ++ ds
+  toView (r ∷ rs) with rs′ ++ ds ← toView rs = (r ∷ rs′) ++ ds
 
   fromView : ∀ {as bs} → PrefixView R as bs → Prefix R as bs
   fromView ([]       ++ ds) = []
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
index 441956829f..f15737ada3 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous.agda
@@ -47,8 +47,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   toView : ∀ {as bs} → Suffix R as bs → SuffixView R as bs
   toView (here rs) = [] ++ rs
-  toView (there {c} suf) with toView suf
-  ... | cs ++ rs = (c ∷ cs) ++ rs
+  toView (there {c} suf) with cs ++ rs ← toView suf = (c ∷ cs) ++ rs
 
   fromView : ∀ {as bs} → SuffixView R as bs → Suffix R as bs
   fromView ([]       ++ rs) = here rs
diff --git a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
index abacb301d0..3506d0000b 100644
--- a/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Suffix/Heterogeneous/Properties.agda
@@ -40,8 +40,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   fromPrefix : ∀ {as bs} → Prefix R as bs →
                Suffix R (reverse as) (reverse bs)
-  fromPrefix {as} {bs} p with Prefix.toView p
-  ... | Prefix._++_ {cs} rs ds =
+  fromPrefix {as} {bs} p with Prefix._++_ {cs} rs ds ← Prefix.toView p =
     subst (Suffix R (reverse as))
       (sym (List.reverse-++ cs ds))
       (Suffix.fromView (reverse ds Suffix.++ Pw.reverse⁺ rs))
@@ -56,8 +55,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 
   toPrefix-rev : ∀ {as bs} → Suffix R as bs →
                  Prefix R (reverse as) (reverse bs)
-  toPrefix-rev {as} {bs} s with Suffix.toView s
-  ... | Suffix._++_ cs {ds} rs =
+  toPrefix-rev {as} {bs} s with Suffix._++_ cs {ds} rs ← Suffix.toView s =
     subst (Prefix R (reverse as))
       (sym (List.reverse-++ cs ds))
       (Prefix.fromView (Pw.reverse⁺ rs Prefix.++ reverse cs))
diff --git a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
index f1a95b1e14..0f374ea1b5 100644
--- a/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Unary/Enumerates/Setoid/Properties.agda
@@ -40,8 +40,8 @@ module _ (S : Setoid a ℓ₁) (T : Setoid b ℓ₂) (surj : Surjection S T) whe
   open Surjection surj
 
   map⁺ : ∀ {xs} → IsEnumeration S xs → IsEnumeration T (map to xs)
-  map⁺ _∈xs y with strictlySurjective y
-  ... | (x , fx≈y) = ∈-resp-≈ T fx≈y (∈-map⁺ S T cong (x ∈xs))
+  map⁺ _∈xs y with (x , fx≈y) ← strictlySurjective y =
+      ∈-resp-≈ T fx≈y (∈-map⁺ S T cong (x ∈xs))
 
 ------------------------------------------------------------------------
 -- _++_
diff --git a/src/Data/Nat/Binary/Subtraction.agda b/src/Data/Nat/Binary/Subtraction.agda
index 5ab67ce7c1..1e38de460e 100644
--- a/src/Data/Nat/Binary/Subtraction.agda
+++ b/src/Data/Nat/Binary/Subtraction.agda
@@ -12,8 +12,8 @@ open import Algebra.Core using (Op₂)
 open import Algebra.Bundles using (Magma)
 open import Algebra.Consequences.Propositional using (comm∧distrˡ⇒distrʳ)
 open import Algebra.Morphism.Consequences using (homomorphic₂-inv)
-open import Data.Bool.Base using (true; false)
-open import Data.Nat.Base as ℕ using (ℕ)
+open import Data.Bool.Base using (true; false; if_then_else_)
+open import Data.Nat as ℕ using (ℕ)
 open import Data.Nat.Binary.Base using (ℕᵇ; 0ᵇ; 2[1+_]; 1+[2_]; double;
   pred; toℕ; fromℕ; even<odd; odd<even; _≥_; _>_; _≤_; _<_; _+_; zero; suc; 1ᵇ;
   _*_)
@@ -53,12 +53,8 @@ zero     ∸ _        = 0ᵇ
 x        ∸ zero     = x
 2[1+ x ] ∸ 2[1+ y ] = double (x ∸ y)
 1+[2 x ] ∸ 1+[2 y ] = double (x ∸ y)
-2[1+ x ] ∸ 1+[2 y ] with does (x <? y)
-... | true  = 0ᵇ
-... | false = 1+[2 (x ∸ y) ]
-1+[2 x ] ∸ 2[1+ y ] with does (x ≤? y)
-... | true  = 0ᵇ
-... | false = pred (double (x ∸ y))
+2[1+ x ] ∸ 1+[2 y ] = if does (x <? y) then 0ᵇ else 1+[2 (x ∸ y) ]
+1+[2 x ] ∸ 2[1+ y ] = if does (x ≤? y) then 0ᵇ else pred (double (x ∸ y))
 
 ------------------------------------------------------------------------
 -- Properties of _∸_ and _≡_
diff --git a/src/Data/Rational/Base.agda b/src/Data/Rational/Base.agda
index 32fdfc3ae0..cfde351ac1 100644
--- a/src/Data/Rational/Base.agda
+++ b/src/Data/Rational/Base.agda
@@ -269,15 +269,11 @@ ceiling p@record{} = ℤ.- floor (- p)
 
 -- Truncate  (round towards 0)
 truncate : ℚ → ℤ
-truncate p with p ≤ᵇ 0ℚ
-... | true  = ceiling p
-... | false = floor p
+truncate p = if p ≤ᵇ 0ℚ then ceiling p else floor p
 
 -- Round (to nearest integer)
 round : ℚ → ℤ
-round p with p ≤ᵇ 0ℚ
-... | true  = ceiling (p - ½)
-... | false = floor (p + ½)
+round p = if p ≤ᵇ 0ℚ then ceiling (p - ½) else floor (p + ½)
 
 -- Fractional part (remainder after floor)
 fracPart : ℚ → ℚ
diff --git a/src/Data/Rational/Unnormalised/Base.agda b/src/Data/Rational/Unnormalised/Base.agda
index 41c236062c..5714ed2181 100644
--- a/src/Data/Rational/Unnormalised/Base.agda
+++ b/src/Data/Rational/Unnormalised/Base.agda
@@ -276,15 +276,11 @@ ceiling p@record{} = ℤ.- floor (- p)
 
 -- Truncate  (round towards 0)
 truncate : ℚᵘ → ℤ
-truncate p with p ≤ᵇ 0ℚᵘ
-... | true  = ceiling p
-... | false = floor p
+truncate p = if p ≤ᵇ 0ℚᵘ then ceiling p else floor p
 
 -- Round (to nearest integer)
 round : ℚᵘ → ℤ
-round p with p ≤ᵇ 0ℚᵘ
-... | true  = ceiling (p - ½)
-... | false = floor (p + ½)
+round p = if p ≤ᵇ 0ℚᵘ then ceiling (p - ½) else floor (p + ½)
 
 -- Fractional part (remainder after floor)
 fracPart : ℚᵘ → ℚᵘ
diff --git a/src/Data/Star/Environment.agda b/src/Data/Star/Environment.agda
index 67ac8e8abd..77c7eff384 100644
--- a/src/Data/Star/Environment.agda
+++ b/src/Data/Star/Environment.agda
@@ -13,7 +13,7 @@ open import Data.Star.List using (List)
 open import Data.Star.Decoration using (All)
 open import Data.Star.Pointer as Pointer using (Any; this; that; result)
 open import Data.Unit.Polymorphic.Base using (⊤)
-open import Function.Base using (const)
+open import Function.Base using (const; case_of_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
   using (_▻_)
@@ -44,5 +44,4 @@ Env T Γ = All T Γ
 -- A safe lookup function for environments.
 
 lookup : ∀ {Γ σ} {T : Ty → Set} → Env T Γ → Γ ∋ σ → T σ
-lookup ρ i with Pointer.lookup ρ i
-... | result refl x = x
+lookup ρ i with result refl x ← Pointer.lookup ρ i = x
diff --git a/src/Data/Star/Pointer.agda b/src/Data/Star/Pointer.agda
index bc86786f74..c193cd776e 100644
--- a/src/Data/Star/Pointer.agda
+++ b/src/Data/Star/Pointer.agda
@@ -11,7 +11,7 @@ module Data.Star.Pointer {ℓ} {I : Set ℓ} where
 open import Data.Maybe.Base using (Maybe; nothing; just)
 open import Data.Star.Decoration
 open import Data.Unit.Base
-open import Function.Base using (const)
+open import Function.Base using (const; case_of_)
 open import Level
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions using (NonEmpty; nonEmpty)
@@ -92,5 +92,5 @@ module _ {T : Rel I r} {P : EdgePred p T} {Q : EdgePred q T} where
 
   last : ∀ {i j} {xs : Star T i j} →
          Any P Q xs → NonEmptyEdgePred T Q
-  last ps with lookup {r = p} (decorate (const (lift tt)) _) ps
-  ... | result q _ = nonEmptyEdgePred q
+  last ps with result q _ ← lookup {r = p} (decorate (const (lift tt)) _) ps =
+    nonEmptyEdgePred q
diff --git a/src/Data/Star/Vec.agda b/src/Data/Star/Vec.agda
index 3677db147b..b7187ae36a 100644
--- a/src/Data/Star/Vec.agda
+++ b/src/Data/Star/Vec.agda
@@ -16,7 +16,7 @@ open import Data.Star.List using (List)
 open import Level using (Level)
 open import Relation.Binary.Construct.Closure.ReflexiveTransitive
   using (ε; _◅_; gmap)
-open import Function.Base using (const)
+open import Function.Base using (const; case_of_)
 open import Data.Unit.Base using (tt)
 
 private
@@ -58,8 +58,7 @@ _++_ = _◅◅◅_
 -- Safe lookup.
 
 lookup : ∀ {n} → Vec A n → Fin n → A
-lookup xs i with Pointer.lookup xs i
-... | result _ x = x
+lookup xs i with result _ x ← Pointer.lookup xs i = x
 
 ------------------------------------------------------------------------
 -- Conversions
diff --git a/src/Data/String/Base.agda b/src/Data/String/Base.agda
index 6eb10f3554..43ff59c732 100644
--- a/src/Data/String/Base.agda
+++ b/src/Data/String/Base.agda
@@ -8,14 +8,14 @@
 
 module Data.String.Base where
 
-open import Data.Bool.Base using (Bool; true; false)
+open import Data.Bool.Base using (Bool; true; false; if_then_else_)
 open import Data.Char.Base as Char using (Char)
 open import Data.List.Base as List using (List; [_]; _∷_; [])
 open import Data.List.NonEmpty.Base as NE using (List⁺)
 open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise)
 open import Data.List.Relation.Binary.Lex.Core using (Lex-<; Lex-≤)
 open import Data.Maybe.Base as Maybe using (Maybe)
-open import Data.Nat.Base using (ℕ; _∸_; ⌊_/2⌋; ⌈_/2⌉)
+open import Data.Nat.Base using (ℕ; _∸_; ⌊_/2⌋; ⌈_/2⌉; _≡ᵇ_)
 open import Data.Product.Base using (proj₁; proj₂)
 open import Function.Base using (_on_; _∘′_; _∘_)
 open import Level using (Level; 0ℓ)
@@ -132,19 +132,19 @@ a <+> b = a ++ " " ++ b
 -- it is equal to 0.
 
 padLeft : Char → ℕ → String → String
-padLeft c n str with n ∸ length str
-... | 0 = str
-... | l = replicate l c ++ str
+padLeft c n str =
+  let l = n ∸ length str in
+  if l ≡ᵇ 0 then str else replicate l c ++ str
 
 padRight : Char → ℕ → String → String
-padRight c n str with n ∸ length str
-... | 0 = str
-... | l = str ++ replicate l c
+padRight c n str =
+  let l = n ∸ length str in
+  if l ≡ᵇ 0 then str else str ++ replicate l c
 
 padBoth : Char → Char → ℕ → String → String
-padBoth cₗ cᵣ n str with n ∸ length str
-... | 0 = str
-... | l = replicate ⌊ l /2⌋ cₗ ++ str ++ replicate ⌈ l /2⌉ cᵣ
+padBoth cₗ cᵣ n str =
+  let l = n ∸ length str in
+  if l ≡ᵇ 0 then str else replicate ⌊ l /2⌋ cₗ ++ str ++ replicate ⌈ l /2⌉ cᵣ
 
 ------------------------------------------------------------------------
 -- Alignment
diff --git a/src/Data/Tree/AVL.agda b/src/Data/Tree/AVL.agda
index 895a194085..63d6067a9d 100644
--- a/src/Data/Tree/AVL.agda
+++ b/src/Data/Tree/AVL.agda
@@ -92,13 +92,13 @@ module _ {v} {V : Value v} where
 
   headTail : Tree V → Maybe (K& V × Tree V)
   headTail (tree (Indexed.leaf _)) = nothing
-  headTail (tree {h = suc _} t)    with Indexed.headTail t
-  ... | (k , _ , _ , t′) = just (k , tree (Indexed.castˡ ⊥⁺<[ _ ] t′))
+  headTail (tree {h = suc _} t) with (k , _ , _ , t′) ← Indexed.headTail t
+    = just (k , tree (Indexed.castˡ ⊥⁺<[ _ ] t′))
 
   initLast : Tree V → Maybe (Tree V × K& V)
   initLast (tree (Indexed.leaf _)) = nothing
-  initLast (tree {h = suc _} t)    with Indexed.initLast t
-  ... | (k , _ , _ , t′) = just (tree (Indexed.castʳ t′ [ _ ]<⊤⁺) , k)
+  initLast (tree {h = suc _} t) with (k , _ , _ , t′) ← Indexed.initLast t
+    = just (tree (Indexed.castʳ t′ [ _ ]<⊤⁺) , k)
 
   foldr : (∀ {k} → Val k → A → A) → A → Tree V → A
   foldr cons nil (tree t) = Indexed.foldr cons nil t
diff --git a/src/Data/Vec.agda b/src/Data/Vec.agda
index 2e293acdf9..aa6435e445 100644
--- a/src/Data/Vec.agda
+++ b/src/Data/Vec.agda
@@ -21,6 +21,7 @@ open import Data.Bool.Base
 import Data.Nat.Properties as ℕ
 open import Data.Vec.Bounded.Base as Vec≤
   using (Vec≤; ≤-cast; fromVec)
+open import Function.Base using (_$_)
 open import Relation.Nullary
 open import Relation.Unary
 
@@ -41,18 +42,14 @@ module _ {P : A → Set p} (P? : Decidable P) where
 
   filter : ∀ {n} → Vec A n → Vec≤ A n
   filter []       = Vec≤.[]
-  filter (a ∷ as) with does (P? a)
-  ... | true  = a Vec≤.∷ filter as
-  ... | false = ≤-cast (ℕ.n≤1+n _) (filter as)
+  filter (a ∷ as) = if does (P? a) then a Vec≤.∷_ else ≤-cast (ℕ.n≤1+n _) $ filter as
 
   takeWhile : ∀ {n} → Vec A n → Vec≤ A n
   takeWhile []       = Vec≤.[]
-  takeWhile (a ∷ as) with does (P? a)
-  ... | true  = a Vec≤.∷ takeWhile as
-  ... | false = Vec≤.[]
+  takeWhile (a ∷ as) = if does (P? a) then a Vec≤.∷ takeWhile as else Vec≤.[]
 
   dropWhile : ∀ {n} → Vec A n → Vec≤ A n
   dropWhile Vec.[]       = Vec≤.[]
-  dropWhile (a Vec.∷ as) with does (P? a)
-  ... | true  = ≤-cast (ℕ.n≤1+n _) (dropWhile as)
-  ... | false = fromVec (a Vec.∷ as)
+  dropWhile (a Vec.∷ as) =
+    if does (P? a) then ≤-cast (ℕ.n≤1+n _) (dropWhile as)
+    else fromVec (a Vec.∷ as)
diff --git a/src/Data/Vec/Base.agda b/src/Data/Vec/Base.agda
index 5f57c22b2f..8c03fb8a83 100644
--- a/src/Data/Vec/Base.agda
+++ b/src/Data/Vec/Base.agda
@@ -14,7 +14,7 @@ open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List)
 open import Data.Product.Base as Product using (∃; ∃₂; _×_; _,_; proj₁; proj₂)
 open import Data.These.Base as These using (These; this; that; these)
-open import Function.Base using (const; _∘′_; id; _∘_)
+open import Function.Base using (const; _∘′_; id; _∘_; _$_)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; trans; cong)
 open import Relation.Nullary.Decidable.Core using (does; T?)
@@ -232,9 +232,7 @@ sum = foldr _ _+_ 0
 
 count : ∀ {P : Pred A p} → Decidable P → Vec A n → ℕ
 count P? []       = zero
-count P? (x ∷ xs) with does (P? x)
-... | true  = suc (count P? xs)
-... | false = count P? xs
+count P? (x ∷ xs) = if does (P? x) then suc else id $ count P? xs
 
 countᵇ : (A → Bool) → Vec A n → ℕ
 countᵇ p = count (T? ∘ p)
diff --git a/src/Data/Vec/Membership/Setoid.agda b/src/Data/Vec/Membership/Setoid.agda
index 72e57e15f3..cfa69e591b 100644
--- a/src/Data/Vec/Membership/Setoid.agda
+++ b/src/Data/Vec/Membership/Setoid.agda
@@ -51,9 +51,8 @@ _∷=_ {xs = xs} x∈xs v = xs Vec.[ index x∈xs ]≔ v
 module _ {p} {P : Pred A p} where
 
   find : ∀ {n} {xs : Vec A n} → Any P xs → ∃ λ x → x ∈ xs × P x
-  find (here px)   = (_ , here refl , px)
-  find (there pxs) with find pxs
-  ... | (x , x∈xs , px) = (x , there x∈xs , px)
+  find (here px)   = _ , here refl , px
+  find (there pxs) = let x , x∈xs , px = find pxs in x , there x∈xs , px
 
   lose : P Respects _≈_ → ∀ {x n} {xs : Vec A n} → x ∈ xs → P x → Any P xs
   lose resp x∈xs px = Any.map (flip resp px) x∈xs
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Left.agda b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
index 47465e8926..cd8864af6e 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Left.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Left.agda
@@ -9,7 +9,7 @@
 
 open import Algebra.Core
 open import Data.Product.Base using (_,_; _×_)
-open import Data.Sum.Base using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂; map)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Bundles
   using (Preorder; Poset; DecPoset; TotalOrder; DecTotalOrder)
@@ -58,9 +58,7 @@ antisym isEq comm {x} {y} x≤y y≤x = begin
   where open IsEquivalence isEq; open ≈-Reasoning (record { isEquivalence = isEq })
 
 total : Symmetric _≈_ → Transitive _≈_ → Selective _∙_ → Commutative _∙_ → Total _≤_
-total sym trans sel comm x y with sel x y
-... | inj₁ x∙y≈x = inj₁ (sym x∙y≈x)
-... | inj₂ x∙y≈y = inj₂ (sym (trans (comm y x) x∙y≈y))
+total sym trans sel comm x y = map sym (λ x∙y≈y → trans (sym x∙y≈y) (comm x y)) (sel x y)
 
 trans : IsSemigroup _∙_ → Transitive _≤_
 trans semi {x} {y} {z} x≤y y≤z = begin
diff --git a/src/Relation/Binary/Construct/NaturalOrder/Right.agda b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
index 76529350e2..024b946cc8 100644
--- a/src/Relation/Binary/Construct/NaturalOrder/Right.agda
+++ b/src/Relation/Binary/Construct/NaturalOrder/Right.agda
@@ -9,7 +9,7 @@
 
 open import Algebra.Core using (Op₂)
 open import Data.Product.Base using (_,_; _×_)
-open import Data.Sum.Base using (inj₁; inj₂)
+open import Data.Sum.Base using (inj₁; inj₂; map)
 open import Relation.Binary.Core using (Rel; _⇒_)
 open import Relation.Binary.Bundles
   using (Preorder; Poset; DecPoset; TotalOrder; DecTotalOrder)
@@ -58,9 +58,8 @@ antisym isEq comm {x} {y} x≤y y≤x = begin
   where open ≈-Reasoning (record { isEquivalence = isEq })
 
 total : Symmetric _≈_ → Transitive _≈_ → Selective _∙_ → Commutative _∙_ → Total _≤_
-total sym trans sel comm x y with sel x y
-... | inj₁ x∙y≈x = inj₁ (trans (sym x∙y≈x) (comm x y) )
-... | inj₂ x∙y≈y = inj₂ (sym x∙y≈y)
+total sym trans sel comm x y =
+  map (λ x∙y≈x → trans (sym x∙y≈x) (comm x y)) sym (sel x y)
 
 trans : IsSemigroup _∙_ → Transitive _≤_
 trans semi {x} {y} {z} x≤y y≤z = begin
diff --git a/src/Text/Pretty/Core.agda b/src/Text/Pretty/Core.agda
index 9a157ddda2..6883054ae8 100644
--- a/src/Text/Pretty/Core.agda
+++ b/src/Text/Pretty/Core.agda
@@ -145,8 +145,8 @@ private
 
     pad : Maybe String
     pad with x.lastWidth
-    ... | 0 = nothing
-    ... | l = just (replicate l ' ')
+    ... | 0         = nothing
+    ... | l@(suc _) = just (replicate l ' ')
 
     size-pad : maybe′ length 0 pad ≡ x.lastWidth
     size-pad with x.lastWidth

From 3d540425539f3ee90a6f9561e8161c230f14a540 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Wed, 24 Apr 2024 13:49:35 +0100
Subject: [PATCH 280/316] =?UTF-8?q?[=20new=20]=20=E2=81=BB=C2=B9-anti-homo?=
 =?UTF-8?q?-(//=20|=20\\)=20(#2349)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* [ new ] ¹-anti-homo‿-

* Update CHANGELOG.md

Co-authored-by: Nathan van Doorn <nvd1234@gmail.com>

* Update src/Algebra/Properties/Group.agda

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>

* Update CHANGELOG.md

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>

* [ more ] symmetric lemma

* [ new ] ⁻¹-anti-homo‿- : (x - y) ⁻¹ ≈ y - x

---------

Co-authored-by: MatthewDaggitt <matthewdaggitt@gmail.com>
Co-authored-by: Nathan van Doorn <nvd1234@gmail.com>
Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
---
 CHANGELOG.md                             |  7 +++++++
 src/Algebra/Properties/AbelianGroup.agda |  3 +++
 src/Algebra/Properties/Group.agda        | 16 ++++++++++++++++
 3 files changed, 26 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index ac44c5e869..895a3ad3bd 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -236,6 +236,11 @@ Additions to existing modules
     record IsSuccessorSetMonomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
     record IsSuccessorSetIsomorphism  (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
 
+* In `Algebra.Properties.AbelianGroup`:
+  ```
+  ⁻¹-anti-homo‿- : (x - y) ⁻¹ ≈ y - x
+  ```
+
 * In `Algebra.Properties.Group`:
   ```agda
   isQuasigroup    : IsQuasigroup _∙_ _\\_ _//_
@@ -253,6 +258,8 @@ Additions to existing modules
   ⁻¹-selfInverse  : SelfInverse _⁻¹
   \\≗flip-//⇒comm : (∀ x y → x \\ y ≈ y // x) → Commutative _∙_
   comm⇒\\≗flip-// : Commutative _∙_ → ∀ x y → x \\ y ≈ y // x
+  ⁻¹-anti-homo-// : (x // y) ⁻¹ ≈ y // x
+  ⁻¹-anti-homo-\\ : (x \\ y) ⁻¹ ≈ y \\ x
   ```
 
 * In `Algebra.Properties.Loop`:
diff --git a/src/Algebra/Properties/AbelianGroup.agda b/src/Algebra/Properties/AbelianGroup.agda
index 87c6b7c6b9..b5eb290081 100644
--- a/src/Algebra/Properties/AbelianGroup.agda
+++ b/src/Algebra/Properties/AbelianGroup.agda
@@ -23,6 +23,9 @@ open import Algebra.Properties.Group group public
 ------------------------------------------------------------------------
 -- Properties of abelian groups
 
+⁻¹-anti-homo‿- : ∀ x y → (x - y) ⁻¹ ≈ y - x
+⁻¹-anti-homo‿- = ⁻¹-anti-homo-//
+
 xyx⁻¹≈y : ∀ x y → x ∙ y ∙ x ⁻¹ ≈ y
 xyx⁻¹≈y x y = begin
   x ∙ y ∙ x ⁻¹    ≈⟨ ∙-congʳ $ comm _ _ ⟩
diff --git a/src/Algebra/Properties/Group.agda b/src/Algebra/Properties/Group.agda
index 3d4a7d7713..bac2c84d31 100644
--- a/src/Algebra/Properties/Group.agda
+++ b/src/Algebra/Properties/Group.agda
@@ -126,6 +126,22 @@ inverseʳ-unique x y eq = trans (y≈x\\z x y ε eq) (identityʳ _)
   (x ∙ y) ∙ y ⁻¹ ∙ x ⁻¹ ≈⟨ assoc (x ∙ y) (y ⁻¹) (x ⁻¹) ⟩
   x ∙ y ∙ (y ⁻¹ ∙ x ⁻¹) ∎
 
+⁻¹-anti-homo-// : ∀ x y → (x // y) ⁻¹ ≈ y // x
+⁻¹-anti-homo-// x y = begin
+  (x // y) ⁻¹      ≡⟨⟩
+  (x ∙ y ⁻¹) ⁻¹    ≈⟨ ⁻¹-anti-homo-∙ x (y ⁻¹) ⟩
+  (y ⁻¹) ⁻¹ ∙ x ⁻¹ ≈⟨ ∙-congʳ (⁻¹-involutive y) ⟩
+  y ∙ x ⁻¹         ≡⟨⟩
+  y // x ∎
+
+⁻¹-anti-homo-\\ : ∀ x y → (x \\ y) ⁻¹ ≈ y \\ x
+⁻¹-anti-homo-\\ x y = begin
+  (x \\ y) ⁻¹      ≡⟨⟩
+  (x ⁻¹ ∙ y) ⁻¹    ≈⟨ ⁻¹-anti-homo-∙ (x ⁻¹) y ⟩
+  y ⁻¹ ∙ (x ⁻¹) ⁻¹ ≈⟨ ∙-congˡ (⁻¹-involutive x) ⟩
+  y ⁻¹ ∙ x         ≡⟨⟩
+  y \\ x ∎
+
 \\≗flip-//⇒comm : (∀ x y → x \\ y ≈ y // x) → Commutative _∙_
 \\≗flip-//⇒comm \\≗//ᵒ x y = begin
   x ∙ y                ≈⟨ ∙-congˡ (//-rightDividesˡ x y) ⟨

From 618d83849b6d21e10955e30d4356e40adbd1892c Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Na=C3=AFm=20Favier?= <n@monade.li>
Date: Thu, 2 May 2024 09:12:59 +0200
Subject: [PATCH 281/316] Add `_>>_` for `IO.Primitive.Core` (#2374)

This has to be in scope for desugaring `do` blocks that don't bind
intermediate results.
---
 CHANGELOG.md               | 5 +++++
 src/IO/Primitive/Core.agda | 3 +++
 2 files changed, 8 insertions(+)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 895a3ad3bd..5c9cc9037b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -492,6 +492,11 @@ Additions to existing modules
   forever : IO ⊤ → IO ⊤
   ```
 
+* In `IO.Primitive.Core`:
+  ```agda
+  _>>_ : IO A → IO B → IO B
+  ```
+
 * In `Data.Word.Base`:
   ```agda
   _≤_ : Rel Word64 zero
diff --git a/src/IO/Primitive/Core.agda b/src/IO/Primitive/Core.agda
index 6fd22a03fb..04fba8c6f8 100644
--- a/src/IO/Primitive/Core.agda
+++ b/src/IO/Primitive/Core.agda
@@ -36,3 +36,6 @@ postulate
 -- Haskell-style alternative syntax
 return : A → IO A
 return = pure
+
+_>>_ : IO A → IO B → IO B
+a >> b = a >>= λ _ → b

From 5515a8c53de1249fc7068988c42f124168e4e084 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 8 May 2024 11:25:54 +0100
Subject: [PATCH 282/316] refactored to lift out `isEquivalence` (#2382)

---
 .../Relation/Binary/Setoid/Equality.agda      | 19 ++++++++++---------
 1 file changed, 10 insertions(+), 9 deletions(-)

diff --git a/src/Function/Relation/Binary/Setoid/Equality.agda b/src/Function/Relation/Binary/Setoid/Equality.agda
index 2631d83da9..9d0e9823da 100644
--- a/src/Function/Relation/Binary/Setoid/Equality.agda
+++ b/src/Function/Relation/Binary/Setoid/Equality.agda
@@ -15,6 +15,8 @@ module Function.Relation.Binary.Setoid.Equality {a₁ a₂ b₁ b₂ : Level}
 open import Function.Bundles using (Func; _⟨$⟩_)
 open import Relation.Binary.Definitions
   using (Reflexive; Symmetric; Transitive)
+open import Relation.Binary.Structures
+  using (IsEquivalence)
 
 private
   module To = Setoid To
@@ -33,17 +35,16 @@ sym = λ f≈g → To.sym f≈g
 trans : Transitive _≈_
 trans = λ f≈g g≈h → To.trans f≈g g≈h
 
-setoid : Setoid _ _
-setoid = record
-  { Carrier = Func From To
-  ; _≈_ = _≈_
-  ; isEquivalence = record  -- need to η-expand else Agda gets confused
-    { refl = λ {f} → refl {f}
-    ; sym = λ {f} {g} → sym {f} {g}
-    ; trans = λ {f} {g} {h} → trans {f} {g} {h}
-    }
+isEquivalence : IsEquivalence _≈_
+isEquivalence = record  -- need to η-expand else Agda gets confused
+  { refl = λ {f} → refl {f}
+  ; sym = λ {f} {g} → sym {f} {g}
+  ; trans = λ {f} {g} {h} → trans {f} {g} {h}
   }
 
+setoid : Setoid _ _
+setoid = record { isEquivalence = isEquivalence }
+
 -- most of the time, this infix version is nicer to use
 infixr 9 _⇨_
 _⇨_ : Setoid _ _

From b378d4b2950f44af7d947847e6924cb866cc3d65 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 13 May 2024 17:56:48 +0100
Subject: [PATCH 283/316] =?UTF-8?q?`Data.List.Base.reverse`=20is=20self=20?=
 =?UTF-8?q?adjoint=20wrt=20`Data.List.Relation.Binary.Subset.Setoid.=5F?=
 =?UTF-8?q?=E2=8A=86=5F`=20(#2378)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* `reverse` is `SelfInverse`

* use `Injective` for `reverse-injective`

* fixes #2353

* combinatory form

* removed redundant implicits

* added comment on unit/counit of the adjunction
---
 CHANGELOG.md                                  | 14 +++++++
 .../List/Membership/Setoid/Properties.agda    | 13 ++++++
 src/Data/List/Properties.agda                 | 29 ++++++++-----
 .../Binary/Subset/Setoid/Properties.agda      | 42 ++++++++++++++++++-
 4 files changed, 86 insertions(+), 12 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 5c9cc9037b..d74d751803 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -329,6 +329,12 @@ Additions to existing modules
   i*j≢0     : .{{_ : NonZero i}} .{{_ : NonZero j}} → NonZero (i * j)
   ```
 
+* In `Data.List.Membership.Setoid.Properties`:
+  ```agda
+  reverse⁺ : x ∈ xs → x ∈ reverse xs
+  reverse⁻ : x ∈ reverse xs → x ∈ xs
+  ```
+
 * In `Data.List.Properties`:
   ```agda
   length-catMaybes      : length (catMaybes xs) ≤ length xs
@@ -336,6 +342,7 @@ Additions to existing modules
   applyDownFrom-∷ʳ      : applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
   upTo-∷ʳ               : upTo n ∷ʳ n ≡ upTo (suc n)
   downFrom-∷ʳ           : applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)
+  reverse-selfInverse   : SelfInverse {A = List A} _≡_ reverse
   reverse-applyUpTo     : reverse (applyUpTo f n) ≡ applyDownFrom f n
   reverse-upTo          : reverse (upTo n) ≡ downFrom n
   reverse-applyDownFrom : reverse (applyDownFrom f n) ≡ applyUpTo f n
@@ -373,6 +380,13 @@ Additions to existing modules
   map-catMaybes         : map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
   ```
 
+* In `Data.List.Relation.Binary.Subset.Setoid.Properties`:
+  ```agda
+  reverse-selfAdjoint : as ⊆ reverse bs → reverse as ⊆ bs
+  reverse⁺            : as ⊆ bs → reverse as ⊆ reverse bs
+  reverse⁻            : reverse as ⊆ reverse bs → as ⊆ bs
+  ```
+
 * In `Data.List.Relation.Unary.All.Properties`:
   ```agda
   All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index c06897367b..a0fcb4881b 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -238,6 +238,19 @@ module _ (S : Setoid c ℓ) where
   ∈-concat⁻′ xss v∈c[xss] with find (∈-concat⁻ xss v∈c[xss])
   ... | xs , t , s = xs , s , t
 
+------------------------------------------------------------------------
+-- reverse
+
+module _ (S : Setoid c ℓ) where
+
+  open Membership S using (_∈_)
+
+  reverse⁺ : ∀ {x xs} → x ∈ xs → x ∈ reverse xs
+  reverse⁺ = Any.reverse⁺
+
+  reverse⁻ : ∀ {x xs} → x ∈ reverse xs → x ∈ xs
+  reverse⁻ = Any.reverse⁻
+
 ------------------------------------------------------------------------
 -- cartesianProductWith
 
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index dc7e8e5831..ab9b0241e1 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -12,7 +12,9 @@
 module Data.List.Properties where
 
 open import Algebra.Bundles
-open import Algebra.Definitions as AlgebraicDefinitions using (Involutive)
+open import Algebra.Consequences.Propositional
+ using (selfInverse⇒involutive; selfInverse⇒injective)
+open import Algebra.Definitions as AlgebraicDefinitions using (SelfInverse; Involutive)
 open import Algebra.Morphism.Structures using (IsMagmaHomomorphism; IsMonoidHomomorphism)
 import Algebra.Structures as AlgebraicStructures
 open import Data.Bool.Base using (Bool; false; true; not; if_then_else_)
@@ -1350,7 +1352,7 @@ foldl-ʳ++ f b (x ∷ xs) {ys} = begin
 unfold-reverse : ∀ (x : A) xs → reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
 unfold-reverse x xs = ʳ++-defn xs
 
--- reverse is an involution with respect to append.
+-- reverse is an anti-homomorphism with respect to append.
 
 reverse-++ : (xs ys : List A) →
                      reverse (xs ++ ys) ≡ reverse ys ++ reverse xs
@@ -1361,20 +1363,27 @@ reverse-++ xs ys = begin
   ys ʳ++ reverse xs          ≡⟨ ʳ++-defn ys ⟩
   reverse ys ++ reverse xs   ∎
 
+-- reverse is self-inverse.
+
+reverse-selfInverse : SelfInverse {A = List A} _≡_ reverse
+reverse-selfInverse {x = xs} {y = ys} xs⁻¹≈ys = begin
+  reverse ys         ≡⟨⟩
+  ys ʳ++ []          ≡⟨ cong (_ʳ++ []) xs⁻¹≈ys ⟨
+  reverse xs ʳ++ []  ≡⟨⟩
+  (xs ʳ++ []) ʳ++ [] ≡⟨ ʳ++-ʳ++ xs ⟩
+  [] ʳ++ xs ++ []    ≡⟨⟩
+  xs ++ []           ≡⟨ ++-identityʳ xs ⟩
+  xs                 ∎
+
 -- reverse is involutive.
 
 reverse-involutive : Involutive {A = List A} _≡_ reverse
-reverse-involutive xs = begin
-  reverse (reverse xs)  ≡⟨⟩
-  (xs ʳ++ []) ʳ++ []    ≡⟨ ʳ++-ʳ++ xs ⟩
-  [] ʳ++  xs ++ []      ≡⟨⟩
-  xs ++ []              ≡⟨ ++-identityʳ xs ⟩
-  xs                    ∎
+reverse-involutive = selfInverse⇒involutive reverse-selfInverse
 
 -- reverse is injective.
 
-reverse-injective : ∀ {xs ys : List A} → reverse xs ≡ reverse ys → xs ≡ ys
-reverse-injective = subst₂ _≡_ (reverse-involutive _) (reverse-involutive _) ∘ cong reverse
+reverse-injective : Injective {A = List A} _≡_ _≡_ reverse
+reverse-injective = selfInverse⇒injective reverse-selfInverse
 
 -- reverse preserves length.
 
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index bc1a693a2f..f42bcd0012 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -10,10 +10,11 @@ module Data.List.Relation.Binary.Subset.Setoid.Properties where
 
 open import Data.Bool.Base using (Bool; true; false)
 open import Data.List.Base hiding (_∷ʳ_; find)
+import Data.List.Properties as List
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Relation.Unary.All as All using (All)
 import Data.List.Membership.Setoid as Membership
-open import Data.List.Membership.Setoid.Properties
+import Data.List.Membership.Setoid.Properties as Membershipₚ
 open import Data.Nat.Base using (ℕ; s≤s; _≤_)
 import Data.List.Relation.Binary.Subset.Setoid as Subset
 import Data.List.Relation.Binary.Sublist.Setoid as Sublist
@@ -21,7 +22,7 @@ import Data.List.Relation.Binary.Equality.Setoid as Equality
 import Data.List.Relation.Binary.Permutation.Setoid as Permutation
 import Data.List.Relation.Binary.Permutation.Setoid.Properties as Permutationₚ
 open import Data.Product.Base using (_,_)
-open import Function.Base using (_∘_; _∘₂_)
+open import Function.Base using (_∘_; _∘₂_; _$_)
 open import Level using (Level)
 open import Relation.Nullary using (¬_; does; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
@@ -49,6 +50,7 @@ module _ (S : Setoid a ℓ) where
   open Subset S
   open Equality S
   open Membership S
+  open Membershipₚ
 
   ⊆-reflexive : _≋_ ⇒ _⊆_
   ⊆-reflexive xs≋ys = ∈-resp-≋ S xs≋ys
@@ -167,6 +169,7 @@ module _ (S : Setoid a ℓ) where
   open Setoid S
   open Subset S
   open Membership S
+  open Membershipₚ
 
   xs⊆x∷xs : ∀ xs x → xs ⊆ x ∷ xs
   xs⊆x∷xs xs x = there
@@ -186,6 +189,7 @@ module _ (S : Setoid a ℓ) where
 
   open Subset S
   open Membership S
+  open Membershipₚ
 
   xs⊆xs++ys : ∀ xs ys → xs ⊆ xs ++ ys
   xs⊆xs++ys xs ys = ∈-++⁺ˡ S
@@ -206,6 +210,39 @@ module _ (S : Setoid a ℓ) where
   ++⁺ : ∀ {ws xs ys zs} → ws ⊆ xs → ys ⊆ zs → ws ++ ys ⊆ xs ++ zs
   ++⁺ ws⊆xs ys⊆zs = ⊆-trans S (++⁺ˡ _ ws⊆xs) (++⁺ʳ _ ys⊆zs)
 
+------------------------------------------------------------------------
+-- reverse
+
+module _ (S : Setoid a ℓ) where
+
+  open Setoid S renaming (Carrier to A)
+  open Subset S
+
+  reverse-selfAdjoint : ∀ {as bs} → as ⊆ reverse bs → reverse as ⊆ bs
+  reverse-selfAdjoint rs = reverse⁻ ∘ rs ∘ reverse⁻
+    where reverse⁻ = Membershipₚ.reverse⁻ S
+
+-- NB. the unit and counit of this adjunction are given by:
+-- reverse-η : ∀ {xs} → xs ⊆ reverse xs
+-- reverse-η = Membershipₚ.reverse⁺ S
+-- reverse-ε : ∀ {xs} → reverse xs ⊆ xs
+-- reverse-ε = Membershipₚ.reverse⁻ S
+
+  reverse⁺ : ∀ {as bs} → as ⊆ bs → reverse as ⊆ reverse bs
+  reverse⁺ {as} {bs} rs = reverse-selfAdjoint $ begin
+    as                   ⊆⟨ rs ⟩
+    bs                   ≡⟨ List.reverse-involutive bs ⟨
+    reverse (reverse bs) ∎
+    where open ⊆-Reasoning S
+
+  reverse⁻ : ∀ {as bs} → reverse as ⊆ reverse bs → as ⊆ bs
+  reverse⁻ {as} {bs} rs = begin
+    as                   ≡⟨ List.reverse-involutive as ⟨
+    reverse (reverse as) ⊆⟨ reverse-selfAdjoint rs ⟩
+    bs                   ∎
+    where open ⊆-Reasoning S
+
+
 ------------------------------------------------------------------------
 -- filter
 
@@ -213,6 +250,7 @@ module _ (S : Setoid a ℓ) where
 
   open Setoid S renaming (Carrier to A)
   open Subset S
+  open Membershipₚ
 
   filter-⊆ : ∀ {P : Pred A p} (P? : Decidable P) →
              ∀ xs → filter P? xs ⊆ xs

From 507fcf804e8108391c74758bb49d325f59a6fb62 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 14 May 2024 13:50:26 +0100
Subject: [PATCH 284/316] fixes #2375 (#2377)

* fixes #2375

* removed redundant `Membership` instances

* fix merge conflict error
---
 CHANGELOG.md                                  |  2 ++
 .../Binary/Subset/Setoid/Properties.agda      | 26 ++++++++++++++++---
 2 files changed, 25 insertions(+), 3 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d74d751803..7732f9ea1f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -382,6 +382,8 @@ Additions to existing modules
 
 * In `Data.List.Relation.Binary.Subset.Setoid.Properties`:
   ```agda
+  map⁺ : f Preserves _≈_ ⟶ _≈′_ → as ⊆ bs → map f as ⊆′ map f bs
+
   reverse-selfAdjoint : as ⊆ reverse bs → reverse as ⊆ bs
   reverse⁺            : as ⊆ bs → reverse as ⊆ reverse bs
   reverse⁻            : reverse as ⊆ reverse bs → as ⊆ bs
diff --git a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
index f42bcd0012..0d0d685a27 100644
--- a/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Subset/Setoid/Properties.agda
@@ -27,7 +27,7 @@ open import Level using (Level)
 open import Relation.Nullary using (¬_; does; yes; no)
 open import Relation.Nullary.Negation using (contradiction)
 open import Relation.Unary using (Pred; Decidable) renaming (_⊆_ to _⋐_)
-open import Relation.Binary.Core using (_⇒_)
+open import Relation.Binary.Core using (_⇒_; _Preserves_⟶_)
 open import Relation.Binary.Definitions
   using (Reflexive; Transitive; _Respectsʳ_; _Respectsˡ_; _Respects_)
 open import Relation.Binary.Bundles using (Setoid; Preorder)
@@ -39,7 +39,7 @@ open Setoid using (Carrier)
 
 private
   variable
-    a p q ℓ : Level
+    a b p q r ℓ : Level
 
 ------------------------------------------------------------------------
 -- Relational properties with _≋_ (pointwise equality)
@@ -210,6 +210,27 @@ module _ (S : Setoid a ℓ) where
   ++⁺ : ∀ {ws xs ys zs} → ws ⊆ xs → ys ⊆ zs → ws ++ ys ⊆ xs ++ zs
   ++⁺ ws⊆xs ys⊆zs = ⊆-trans S (++⁺ˡ _ ws⊆xs) (++⁺ʳ _ ys⊆zs)
 
+------------------------------------------------------------------------
+-- map
+
+module _ (S : Setoid a ℓ) (R : Setoid b r) where
+
+  private
+    module S = Setoid S
+    module R = Setoid R
+
+    module S⊆ = Subset S
+    module R⊆ = Subset R
+
+  open Membershipₚ
+
+  map⁺ : ∀ {as bs} {f : S.Carrier → R.Carrier} →
+         f Preserves S._≈_ ⟶ R._≈_ →
+         as S⊆.⊆ bs → map f as R⊆.⊆ map f bs
+  map⁺ {f = f} f-pres as⊆bs v∈f[as] =
+    let x , x∈as , v≈f[x] = ∈-map⁻ S R v∈f[as] in
+    ∈-resp-≈ R (R.sym v≈f[x]) (∈-map⁺ S R f-pres (as⊆bs x∈as))
+
 ------------------------------------------------------------------------
 -- reverse
 
@@ -242,7 +263,6 @@ module _ (S : Setoid a ℓ) where
     bs                   ∎
     where open ⊆-Reasoning S
 
-
 ------------------------------------------------------------------------
 -- filter
 

From 85d1a6ba3748d61c5437bd286eda0bffe90f1472 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 15 May 2024 05:59:37 +0100
Subject: [PATCH 285/316] Add `Data.List.Relation.Binary.Sublist.Setoid`
 categorical properties (#2385)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* refactor: `variable` declarations and `contradiction`

* refactor: one more `contradiction`

* left- and right-unit lemmas

* left- and right-unit lemmas

* `CHANGELOG`

* `CHANGELOG`

* associativity; fixes #816

* Use cong2 to save rewrites

* Make splits for ⊆-assoc exact, simplifying the [] case

* Simplify ⊆-assoc not using rewrite

* Remove now unused private helper

* fix up names and `assoc` orientation; misc. cleanup

* new proofs can now move upwards

* delegate proofs to `Setoid.Properties`

---------

Co-authored-by: Andreas Abel <andreas.abel@ifi.lmu.de>
---
 CHANGELOG.md                                  |  12 +
 .../Sublist/Heterogeneous/Properties.agda     | 214 +++++++++---------
 .../Sublist/Propositional/Properties.agda     |  22 +-
 .../Binary/Sublist/Setoid/Properties.agda     | 102 ++++++---
 4 files changed, 197 insertions(+), 153 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7732f9ea1f..9d1bb7c58b 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -380,6 +380,18 @@ Additions to existing modules
   map-catMaybes         : map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
   ```
 
+* In `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
+  ```agda
+  ⊆-trans-idˡ   : (trans-reflˡ : ∀ {x y} (p : x ≈ y) → trans ≈-refl p ≡ p) →
+                  (pxs : xs ⊆ ys) → ⊆-trans ⊆-refl pxs ≡ pxs
+  ⊆-trans-idʳ   : (trans-reflʳ : ∀ {x y} (p : x ≈ y) → trans p ≈-refl ≡ p) →
+                  (pxs : xs ⊆ ys) → ⊆-trans pxs ⊆-refl ≡ pxs
+  ⊆-trans-assoc : (≈-assoc : ∀ {w x y z} (p : w ≈ x) (q : x ≈ y) (r : y ≈ z) →
+                             trans p (trans q r) ≡ trans (trans p q) r) →
+                  (ps : as ⊆ bs) (qs : bs ⊆ cs) (rs : cs ⊆ ds) →
+                  ⊆-trans ps (⊆-trans qs rs) ≡ ⊆-trans (⊆-trans ps qs) rs
+  ```
+
 * In `Data.List.Relation.Binary.Subset.Setoid.Properties`:
   ```agda
   map⁺ : f Preserves _≈_ ⟶ _≈′_ → as ⊆ bs → map f as ⊆′ map f bs
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
index b760e0287f..8dd94d8b3b 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Properties.agda
@@ -11,7 +11,6 @@ module Data.List.Relation.Binary.Sublist.Heterogeneous.Properties where
 open import Level
 
 open import Data.Bool.Base using (true; false)
-open import Data.Empty
 open import Data.List.Relation.Unary.All using (Null; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
 open import Data.List.Base as List hiding (map; _∷ʳ_)
@@ -29,9 +28,9 @@ open import Data.Product.Base using (∃₂; _×_; _,_; <_,_>; proj₂; uncurry)
 open import Function.Base
 open import Function.Bundles using (_⤖_; _⇔_ ; mk⤖; mk⇔)
 open import Function.Consequences.Propositional using (strictlySurjective⇒surjective)
+open import Relation.Nullary.Decidable as Dec using (Dec; does; _because_; yes; no; ¬?)
+open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary using (Dec; does; _because_; yes; no; ¬_)
-open import Relation.Nullary.Decidable as Dec using (¬?)
 open import Relation.Unary as U using (Pred)
 open import Relation.Binary.Core using (Rel; REL; _⇒_)
 open import Relation.Binary.Bundles using (Preorder; Poset; DecPoset)
@@ -41,26 +40,41 @@ open import Relation.Binary.Structures
   using (IsPreorder; IsPartialOrder; IsDecPartialOrder)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 
+private
+  variable
+    a b c d e p q r s t : Level
+    A : Set a
+    B : Set b
+    C : Set c
+    D : Set d
+    x : A
+    y : B
+    as cs xs : List A
+    bs ds ys : List B
+    ass : List (List A)
+    bss : List (List B)
+    m n : ℕ
+
 ------------------------------------------------------------------------
 -- Injectivity of constructors
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
+module _ {R : REL A B r} where
 
-  ∷-injectiveˡ : ∀ {x y xs ys} {px qx : R x y} {pxs qxs : Sublist R xs ys} →
+  ∷-injectiveˡ : {px qx : R x y} {pxs qxs : Sublist R xs ys} →
                  (Sublist R (x ∷ xs) (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → px ≡ qx
   ∷-injectiveˡ ≡.refl = ≡.refl
 
-  ∷-injectiveʳ : ∀ {x y xs ys} {px qx : R x y} {pxs qxs : Sublist R xs ys} →
+  ∷-injectiveʳ : {px qx : R x y} {pxs qxs : Sublist R xs ys} →
                  (Sublist R (x ∷ xs) (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → pxs ≡ qxs
   ∷-injectiveʳ ≡.refl = ≡.refl
 
-  ∷ʳ-injective : ∀ {y xs ys} {pxs qxs : Sublist R xs ys} →
+  ∷ʳ-injective : {pxs qxs : Sublist R xs ys} →
                  (Sublist R xs (y ∷ ys) ∋ y ∷ʳ pxs) ≡ (y ∷ʳ qxs) → pxs ≡ qxs
   ∷ʳ-injective ≡.refl = ≡.refl
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
+module _ {R : REL A B r} where
 
-  length-mono-≤ : ∀ {as bs} → Sublist R as bs → length as ≤ length bs
+  length-mono-≤ : Sublist R as bs → length as ≤ length bs
   length-mono-≤ []        = z≤n
   length-mono-≤ (y ∷ʳ rs) = ℕ.m≤n⇒m≤1+n (length-mono-≤ rs)
   length-mono-≤ (r ∷ rs)  = s≤s (length-mono-≤ rs)
@@ -72,12 +86,12 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   fromPointwise []       = []
   fromPointwise (p ∷ ps) = p ∷ fromPointwise ps
 
-  toPointwise : ∀ {as bs} → length as ≡ length bs →
+  toPointwise : length as ≡ length bs →
                 Sublist R as bs → Pointwise R as bs
   toPointwise {bs = []}     eq []         = []
   toPointwise {bs = b ∷ bs} eq (r ∷ rs)   = r ∷ toPointwise (ℕ.suc-injective eq) rs
   toPointwise {bs = b ∷ bs} eq (b ∷ʳ rs) =
-    ⊥-elim $ ℕ.<-irrefl eq (s≤s (length-mono-≤ rs))
+    contradiction (s≤s (length-mono-≤ rs)) (ℕ.<-irrefl eq)
 
 ------------------------------------------------------------------------
 -- Various functions' outputs are sublists
@@ -87,15 +101,15 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 -- indeed obtain `f xs ⊆ xs` from `xs ⊆ ys → f xs ⊆ ys`. The other
 -- direction is only true if R is both reflexive and transitive.
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
+module _ {R : REL A B r} where
 
-  tail-Sublist : ∀ {as bs} → Sublist R as bs →
+  tail-Sublist : Sublist R as bs →
                  Maybe.All (λ as → Sublist R as bs) (tail as)
   tail-Sublist []        = nothing
   tail-Sublist (b ∷ʳ ps) = Maybe.map (b ∷ʳ_) (tail-Sublist ps)
   tail-Sublist (p ∷ ps)  = just (_ ∷ʳ ps)
 
-  take-Sublist : ∀ {as bs} n → Sublist R as bs → Sublist R (take n as) bs
+  take-Sublist : ∀ n → Sublist R as bs → Sublist R (take n as) bs
   take-Sublist n       (y ∷ʳ rs) = y ∷ʳ take-Sublist n rs
   take-Sublist zero    rs        = minimum _
   take-Sublist (suc n) []        = []
@@ -107,24 +121,23 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
   drop-Sublist (suc n) []        = []
   drop-Sublist (suc n) (r ∷ rs)  = _ ∷ʳ drop-Sublist n rs
 
-module _ {a b r p} {A : Set a} {B : Set b}
-         {R : REL A B r} {P : Pred A p} (P? : U.Decidable P) where
+module _ {R : REL A B r} {P : Pred A p} (P? : U.Decidable P) where
 
-  takeWhile-Sublist : ∀ {as bs} → Sublist R as bs → Sublist R (takeWhile P? as) bs
+  takeWhile-Sublist : Sublist R as bs → Sublist R (takeWhile P? as) bs
   takeWhile-Sublist []        = []
   takeWhile-Sublist (y ∷ʳ rs) = y ∷ʳ takeWhile-Sublist rs
   takeWhile-Sublist {a ∷ as} (r ∷ rs) with does (P? a)
   ... | true  = r ∷ takeWhile-Sublist rs
   ... | false = minimum _
 
-  dropWhile-Sublist : ∀ {as bs} → Sublist R as bs → Sublist R (dropWhile P? as) bs
+  dropWhile-Sublist : Sublist R as bs → Sublist R (dropWhile P? as) bs
   dropWhile-Sublist []        = []
   dropWhile-Sublist (y ∷ʳ rs) = y ∷ʳ dropWhile-Sublist rs
   dropWhile-Sublist {a ∷ as} (r ∷ rs) with does (P? a)
   ... | true  = _ ∷ʳ dropWhile-Sublist rs
   ... | false = r ∷ rs
 
-  filter-Sublist : ∀ {as bs} → Sublist R as bs → Sublist R (filter P? as) bs
+  filter-Sublist : Sublist R as bs → Sublist R (filter P? as) bs
   filter-Sublist []        = []
   filter-Sublist (y ∷ʳ rs) = y ∷ʳ filter-Sublist rs
   filter-Sublist {a ∷ as} (r ∷ rs) with does (P? a)
@@ -137,65 +150,64 @@ module _ {a b r p} {A : Set a} {B : Set b}
 -- We write f⁺ for the proof that `xs ⊆ ys → f xs ⊆ f ys`
 -- and f⁻ for the one that `f xs ⊆ f ys → xs ⊆ ys`.
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
+module _ {R : REL A B r} where
 
 ------------------------------------------------------------------------
 -- _∷_
 
-  ∷ˡ⁻ : ∀ {a as bs} → Sublist R (a ∷ as) bs → Sublist R as bs
+  ∷ˡ⁻ : Sublist R (x ∷ xs) ys → Sublist R xs ys
   ∷ˡ⁻ (y ∷ʳ rs) = y ∷ʳ ∷ˡ⁻ rs
   ∷ˡ⁻ (r ∷  rs) = _ ∷ʳ rs
 
-  ∷ʳ⁻ : ∀ {a as b bs} → ¬ R a b → Sublist R (a ∷ as) (b ∷ bs) →
-        Sublist R (a ∷ as) bs
+  ∷ʳ⁻ : ¬ R x y → Sublist R (x ∷ xs) (y ∷ ys) →
+        Sublist R (x ∷ xs) ys
   ∷ʳ⁻ ¬r (y ∷ʳ rs) = rs
-  ∷ʳ⁻ ¬r (r ∷  rs) = ⊥-elim (¬r r)
+  ∷ʳ⁻ ¬r (r ∷  rs) = contradiction r ¬r
 
-  ∷⁻ : ∀ {a as b bs} → Sublist R (a ∷ as) (b ∷ bs) → Sublist R as bs
+  ∷⁻ : Sublist R (x ∷ xs) (y ∷ ys) → Sublist R xs ys
   ∷⁻ (y ∷ʳ rs) = ∷ˡ⁻ rs
   ∷⁻ (x ∷  rs) = rs
 
-module _ {a b c d r} {A : Set a} {B : Set b} {C : Set c} {D : Set d}
-         {R : REL C D r} where
+module _ {R : REL C D r} where
 
 ------------------------------------------------------------------------
 -- map
 
-  map⁺ : ∀ {as bs} (f : A → C) (g : B → D) →
+  map⁺ : (f : A → C) (g : B → D) →
          Sublist (λ a b → R (f a) (g b)) as bs →
          Sublist R (List.map f as) (List.map g bs)
   map⁺ f g []        = []
   map⁺ f g (y ∷ʳ rs) = g y ∷ʳ map⁺ f g rs
   map⁺ f g (r ∷ rs)  = r ∷ map⁺ f g rs
 
-  map⁻ : ∀ {as bs} (f : A → C) (g : B → D) →
+  map⁻ : (f : A → C) (g : B → D) →
          Sublist R (List.map f as) (List.map g bs) →
          Sublist (λ a b → R (f a) (g b)) as bs
-  map⁻ {[]}     {bs}     f g rs        = minimum _
-  map⁻ {a ∷ as} {b ∷ bs} f g (_ ∷ʳ rs) = b ∷ʳ map⁻ f g rs
-  map⁻ {a ∷ as} {b ∷ bs} f g (r ∷ rs)  = r ∷ map⁻ f g rs
+  map⁻ {as = []}     {bs}     f g rs        = minimum _
+  map⁻ {as = a ∷ as} {b ∷ bs} f g (_ ∷ʳ rs) = b ∷ʳ map⁻ f g rs
+  map⁻ {as = a ∷ as} {b ∷ bs} f g (r ∷ rs)  = r ∷ map⁻ f g rs
 
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
+module _ {R : REL A B r} where
 
 ------------------------------------------------------------------------
 -- _++_
 
-  ++⁺ : ∀ {as bs cs ds} → Sublist R as bs → Sublist R cs ds →
+  ++⁺ : Sublist R as bs → Sublist R cs ds →
         Sublist R (as ++ cs) (bs ++ ds)
   ++⁺ []         cds = cds
   ++⁺ (y ∷ʳ abs) cds = y ∷ʳ ++⁺ abs cds
   ++⁺ (ab ∷ abs) cds = ab ∷ ++⁺ abs cds
 
-  ++⁻ : ∀ {as bs cs ds} → length as ≡ length bs →
+  ++⁻ : length as ≡ length bs →
         Sublist R (as ++ cs) (bs ++ ds) → Sublist R cs ds
-  ++⁻ {[]}     {[]}     eq rs = rs
-  ++⁻ {a ∷ as} {b ∷ bs} eq rs = ++⁻ (ℕ.suc-injective eq) (∷⁻ rs)
+  ++⁻ {as = []}     {[]}     eq rs = rs
+  ++⁻ {as = a ∷ as} {b ∷ bs} eq rs = ++⁻ (ℕ.suc-injective eq) (∷⁻ rs)
 
-  ++ˡ : ∀ {as bs} (cs : List B) → Sublist R as bs → Sublist R as (cs ++ bs)
+  ++ˡ : (cs : List B) → Sublist R as bs → Sublist R as (cs ++ bs)
   ++ˡ zs = ++⁺ (minimum zs)
 
-  ++ʳ : ∀ {as bs} (cs : List B) → Sublist R as bs → Sublist R as (bs ++ cs)
+  ++ʳ : (cs : List B) → Sublist R as bs → Sublist R as (bs ++ cs)
   ++ʳ cs []        = minimum cs
   ++ʳ cs (y ∷ʳ rs) = y ∷ʳ ++ʳ cs rs
   ++ʳ cs (r ∷ rs)  = r ∷ ++ʳ cs rs
@@ -204,7 +216,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 ------------------------------------------------------------------------
 -- concat
 
-  concat⁺ : ∀ {ass bss} → Sublist (Sublist R) ass bss →
+  concat⁺ : Sublist (Sublist R) ass bss →
             Sublist R (concat ass) (concat bss)
   concat⁺ []          = []
   concat⁺ (y  ∷ʳ rss) = ++ˡ y (concat⁺ rss)
@@ -213,52 +225,51 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 ------------------------------------------------------------------------
 -- take / drop
 
-  take⁺ : ∀ {m n as bs} → m ≤ n → Pointwise R as bs →
+  take⁺ : m ≤ n → Pointwise R as bs →
           Sublist R (take m as) (take n bs)
   take⁺ z≤n       ps        = minimum _
   take⁺ (s≤s m≤n) []        = []
   take⁺ (s≤s m≤n) (p ∷  ps) = p ∷ take⁺ m≤n ps
 
-  drop⁺ : ∀ {m n as bs} → m ≥ n → Sublist R as bs →
+  drop⁺ : m ≥ n → Sublist R as bs →
           Sublist R (drop m as) (drop n bs)
-  drop⁺ {m} z≤n       rs        = drop-Sublist m rs
-  drop⁺     (s≤s m≥n) []        = []
-  drop⁺     (s≤s m≥n) (y ∷ʳ rs) = drop⁺ (ℕ.m≤n⇒m≤1+n m≥n) rs
-  drop⁺     (s≤s m≥n) (r ∷ rs)  = drop⁺ m≥n rs
+  drop⁺ (z≤n {m}) rs        = drop-Sublist m rs
+  drop⁺ (s≤s m≥n) []        = []
+  drop⁺ (s≤s m≥n) (y ∷ʳ rs) = drop⁺ (ℕ.m≤n⇒m≤1+n m≥n) rs
+  drop⁺ (s≤s m≥n) (r ∷ rs)  = drop⁺ m≥n rs
 
-  drop⁺-≥ : ∀ {m n as bs} → m ≥ n → Pointwise R as bs →
+  drop⁺-≥ : m ≥ n → Pointwise R as bs →
             Sublist R (drop m as) (drop n bs)
   drop⁺-≥ m≥n pw = drop⁺ m≥n (fromPointwise pw)
 
-  drop⁺-⊆ : ∀ {as bs} m → Sublist R as bs →
+  drop⁺-⊆ : ∀ m → Sublist R as bs →
             Sublist R (drop m as) (drop m bs)
   drop⁺-⊆ m = drop⁺ (ℕ.≤-refl {m})
 
-module _ {a b r p q} {A : Set a} {B : Set b}
-         {R : REL A B r} {P : Pred A p} {Q : Pred B q}
+module _ {R : REL A B r} {P : Pred A p} {Q : Pred B q}
          (P? : U.Decidable P) (Q? : U.Decidable Q) where
 
-  ⊆-takeWhile-Sublist : ∀ {as bs} →
-    (∀ {a b} → R a b → P a → Q b) →
-    Pointwise R as bs → Sublist R (takeWhile P? as) (takeWhile Q? bs)
+  ⊆-takeWhile-Sublist : (∀ {a b} → R a b → P a → Q b) →
+                        Pointwise R as bs →
+                        Sublist R (takeWhile P? as) (takeWhile Q? bs)
   ⊆-takeWhile-Sublist rp⇒q [] = []
   ⊆-takeWhile-Sublist {a ∷ as} {b ∷ bs} rp⇒q (p ∷ ps) with P? a | Q? b
   ... | false because _ | _               = minimum _
   ... | true  because _ | true  because _ = p ∷ ⊆-takeWhile-Sublist rp⇒q ps
-  ... | yes pa          | no ¬qb          = ⊥-elim $ ¬qb $ rp⇒q p pa
+  ... | yes pa          | no ¬qb          = contradiction (rp⇒q p pa) ¬qb
 
-  ⊇-dropWhile-Sublist : ∀ {as bs} →
-    (∀ {a b} → R a b → Q b → P a) →
-    Pointwise R as bs → Sublist R (dropWhile P? as) (dropWhile Q? bs)
+  ⊇-dropWhile-Sublist : (∀ {a b} → R a b → Q b → P a) →
+                        Pointwise R as bs →
+                        Sublist R (dropWhile P? as) (dropWhile Q? bs)
   ⊇-dropWhile-Sublist rq⇒p [] = []
   ⊇-dropWhile-Sublist {a ∷ as} {b ∷ bs} rq⇒p (p ∷ ps) with P? a | Q? b
   ... | true  because _ | true  because _ = ⊇-dropWhile-Sublist rq⇒p ps
   ... | true  because _ | false because _ =
     b ∷ʳ dropWhile-Sublist P? (fromPointwise ps)
   ... | false because _ | false because _ = p ∷ fromPointwise ps
-  ... | no ¬pa          | yes qb          = ⊥-elim $ ¬pa $ rq⇒p p qb
+  ... | no ¬pa          | yes qb          = contradiction (rq⇒p p qb) ¬pa
 
-  ⊆-filter-Sublist : ∀ {as bs} → (∀ {a b} → R a b → P a → Q b) →
+  ⊆-filter-Sublist : (∀ {a b} → R a b → P a → Q b) →
                      Sublist R as bs → Sublist R (filter P? as) (filter Q? bs)
   ⊆-filter-Sublist rp⇒q [] = []
   ⊆-filter-Sublist rp⇒q (y ∷ʳ rs) with does (Q? y)
@@ -268,18 +279,18 @@ module _ {a b r p q} {A : Set a} {B : Set b}
   ... | true  because _ | true  because _ = r ∷ ⊆-filter-Sublist rp⇒q rs
   ... | false because _ | true  because _ = _ ∷ʳ ⊆-filter-Sublist rp⇒q rs
   ... | false because _ | false because _ = ⊆-filter-Sublist rp⇒q rs
-  ... | yes pa          | no ¬qb          = ⊥-elim $ ¬qb $ rp⇒q r pa
+  ... | yes pa          | no ¬qb          = contradiction (rp⇒q r pa) ¬qb
 
-module _ {a r p} {A : Set a} {R : Rel A r} {P : Pred A p} (P? : U.Decidable P) where
+module _ {R : Rel A r} {P : Pred A p} (P? : U.Decidable P) where
 
-  takeWhile-filter : ∀ {as} → Pointwise R as as →
+  takeWhile-filter : Pointwise R as as →
                      Sublist R (takeWhile P? as) (filter P? as)
   takeWhile-filter [] = []
   takeWhile-filter {a ∷ as} (p ∷ ps) with does (P? a)
   ... | true  = p ∷ takeWhile-filter ps
   ... | false = minimum _
 
-  filter-dropWhile : ∀ {as} → Pointwise R as as →
+  filter-dropWhile : Pointwise R as as →
                      Sublist R (filter P? as) (dropWhile (¬? ∘ P?) as)
   filter-dropWhile [] = []
   filter-dropWhile {a ∷ as} (p ∷ ps) with does (P? a)
@@ -289,42 +300,40 @@ module _ {a r p} {A : Set a} {R : Rel A r} {P : Pred A p} (P? : U.Decidable P) w
 ------------------------------------------------------------------------
 -- reverse
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
+module _ {R : REL A B r} where
 
-  reverseAcc⁺ : ∀ {as bs cs ds} → Sublist R as bs → Sublist R cs ds →
+  reverseAcc⁺ : Sublist R as bs → Sublist R cs ds →
                 Sublist R (reverseAcc cs as) (reverseAcc ds bs)
   reverseAcc⁺ []         cds = cds
   reverseAcc⁺ (y ∷ʳ abs) cds = reverseAcc⁺ abs (y ∷ʳ cds)
   reverseAcc⁺ (r ∷ abs)  cds = reverseAcc⁺ abs (r ∷ cds)
 
-  ʳ++⁺ : ∀ {as bs cs ds} →
-         Sublist R as bs →
-         Sublist R cs ds →
+  ʳ++⁺ : Sublist R as bs → Sublist R cs ds →
          Sublist R (as ʳ++ cs) (bs ʳ++ ds)
   ʳ++⁺ = reverseAcc⁺
 
-  reverse⁺ : ∀ {as bs} → Sublist R as bs → Sublist R (reverse as) (reverse bs)
+  reverse⁺ : Sublist R as bs → Sublist R (reverse as) (reverse bs)
   reverse⁺ rs = reverseAcc⁺ rs []
 
-  reverse⁻ : ∀ {as bs} → Sublist R (reverse as) (reverse bs) → Sublist R as bs
-  reverse⁻ {as} {bs} p = cast (reverse⁺ p) where
+  reverse⁻ : Sublist R (reverse as) (reverse bs) → Sublist R as bs
+  reverse⁻ {as = as} {bs = bs} p = cast (reverse⁺ p) where
     cast = ≡.subst₂ (Sublist R) (List.reverse-involutive as) (List.reverse-involutive bs)
 
 ------------------------------------------------------------------------
 -- Inversion lemmas
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} {a as b bs} where
+module _ {R : REL A B r} where
 
-  ∷⁻¹ : R a b → Sublist R as bs ⇔ Sublist R (a ∷ as) (b ∷ bs)
+  ∷⁻¹ : R x y → Sublist R xs ys ⇔ Sublist R (x ∷ xs) (y ∷ ys)
   ∷⁻¹ r = mk⇔ (r ∷_) ∷⁻
 
-  ∷ʳ⁻¹ : ¬ R a b → Sublist R (a ∷ as) bs ⇔ Sublist R (a ∷ as) (b ∷ bs)
+  ∷ʳ⁻¹ : ¬ R x y → Sublist R (x ∷ xs) ys ⇔ Sublist R (x ∷ xs) (y ∷ ys)
   ∷ʳ⁻¹ ¬r = mk⇔ (_ ∷ʳ_) (∷ʳ⁻ ¬r)
 
 ------------------------------------------------------------------------
 -- Irrelevant special case
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
+module _ {R : REL A B r} where
 
   Sublist-[]-irrelevant : U.Irrelevant (Sublist R [])
   Sublist-[]-irrelevant []       []        = ≡.refl
@@ -333,30 +342,30 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} where
 ------------------------------------------------------------------------
 -- (to/from)Any is a bijection
 
-  toAny-injective : ∀ {xs x} {p q : Sublist R [ x ] xs} → toAny p ≡ toAny q → p ≡ q
+  toAny-injective :{p q : Sublist R [ x ] xs} → toAny p ≡ toAny q → p ≡ q
   toAny-injective {p = y ∷ʳ p} {y ∷ʳ q} =
     ≡.cong (y ∷ʳ_) ∘′ toAny-injective ∘′ there-injective
   toAny-injective {p = _ ∷ p}  {_ ∷ q}  =
     ≡.cong₂ (flip _∷_) (Sublist-[]-irrelevant p q) ∘′ here-injective
 
-  fromAny-injective : ∀ {xs x} {p q : Any (R x) xs} →
+  fromAny-injective : {p q : Any (R x) xs} →
                       fromAny {R = R} p ≡ fromAny q → p ≡ q
   fromAny-injective {p = here px} {here qx} = ≡.cong here ∘′ ∷-injectiveˡ
   fromAny-injective {p = there p} {there q} =
     ≡.cong there ∘′ fromAny-injective ∘′ ∷ʳ-injective
 
-  toAny∘fromAny≗id : ∀ {xs x} (p : Any (R x) xs) → toAny (fromAny {R = R} p) ≡ p
+  toAny∘fromAny≗id : (p : Any (R x) xs) → toAny (fromAny {R = R} p) ≡ p
   toAny∘fromAny≗id (here px) = ≡.refl
   toAny∘fromAny≗id (there p) = ≡.cong there (toAny∘fromAny≗id p)
 
-  Sublist-[x]-bijection : ∀ {x xs} → (Sublist R [ x ] xs) ⤖ (Any (R x) xs)
+  Sublist-[x]-bijection : (Sublist R [ x ] xs) ⤖ (Any (R x) xs)
   Sublist-[x]-bijection = mk⤖ (toAny-injective , strictlySurjective⇒surjective < fromAny , toAny∘fromAny≗id >)
 
 ------------------------------------------------------------------------
 -- Relational properties
 
 module Reflexivity
-    {a r} {A : Set a} {R : Rel A r}
+    {R : Rel A r}
     (R-refl : Reflexive R) where
 
   reflexive : _≡_ ⇒ Sublist R
@@ -368,7 +377,6 @@ module Reflexivity
 open Reflexivity public
 
 module Transitivity
-    {a b c r s t} {A : Set a} {B : Set b} {C : Set c}
     {R : REL A B r} {S : REL B C s} {T : REL A C t}
     (rs⇒t : Trans R S T) where
 
@@ -381,7 +389,6 @@ module Transitivity
 open Transitivity public
 
 module Antisymmetry
-    {a b r s e} {A : Set a} {B : Set b}
     {R : REL A B r} {S : REL B A s} {E : REL A B e}
     (rs⇒e : Antisym R S E) where
 
@@ -392,25 +399,25 @@ module Antisymmetry
   antisym (r ∷ rs)  (s ∷ ss)  = rs⇒e r s ∷ antisym rs ss
   -- impossible cases
   antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷ʳ_ {ys₂} {zs} z ss) =
-    ⊥-elim $ ℕ.<-irrefl ≡.refl $ begin
+    contradiction (begin
     length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩
     length zs        ≤⟨ ℕ.n≤1+n (length zs) ⟩
     length (z ∷ zs)  ≤⟨ length-mono-≤ rs ⟩
-    length ys₁       ∎
+    length ys₁       ∎) $ ℕ.<-irrefl ≡.refl
   antisym (_∷ʳ_ {xs} {ys₁} y rs) (_∷_ {y} {ys₂} {z} {zs} s ss)  =
-    ⊥-elim $ ℕ.<-irrefl ≡.refl $ begin
+    contradiction (begin
     length (z ∷ zs) ≤⟨ length-mono-≤ rs ⟩
     length ys₁      ≤⟨ length-mono-≤ ss ⟩
-    length zs       ∎
+    length zs       ∎) $ ℕ.<-irrefl ≡.refl
   antisym (_∷_ {x} {xs} {y} {ys₁} r rs)  (_∷ʳ_ {ys₂} {zs} z ss) =
-    ⊥-elim $ ℕ.<-irrefl ≡.refl $ begin
+    contradiction (begin
     length (y ∷ ys₁) ≤⟨ length-mono-≤ ss ⟩
     length xs        ≤⟨ length-mono-≤ rs ⟩
-    length ys₁       ∎
+    length ys₁       ∎) $ ℕ.<-irrefl ≡.refl
 
 open Antisymmetry public
 
-module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} (R? : Decidable R) where
+module _ {R : REL A B r} (R? : Decidable R) where
 
   sublist? : Decidable (Sublist R)
   sublist? []       ys       = yes (minimum ys)
@@ -421,7 +428,7 @@ module _ {a b r} {A : Set a} {B : Set b} {R : REL A B r} (R? : Decidable R) wher
   ... | false because [¬r] =
     Dec.map (∷ʳ⁻¹ (invert [¬r])) (sublist? (x ∷ xs) ys)
 
-module _ {a e r} {A : Set a} {E : Rel A e} {R : Rel A r} where
+module _ {E : Rel A e} {R : Rel A r} where
 
   isPreorder : IsPreorder E R → IsPreorder (Pointwise E) (Sublist R)
   isPreorder ER-isPreorder = record
@@ -444,22 +451,20 @@ module _ {a e r} {A : Set a} {E : Rel A e} {R : Rel A r} where
     ; _≤?_           = sublist?       ER._≤?_
     } where module ER = IsDecPartialOrder ER-isDecPartialOrder
 
-module _ {a e r} where
-
-  preorder : Preorder a e r → Preorder _ _ _
-  preorder ER-preorder = record
-    { isPreorder = isPreorder ER.isPreorder
-    } where module ER = Preorder ER-preorder
+preorder : Preorder a e r → Preorder _ _ _
+preorder ER-preorder = record
+  { isPreorder = isPreorder ER.isPreorder
+  } where module ER = Preorder ER-preorder
 
-  poset : Poset a e r → Poset _ _ _
-  poset ER-poset = record
-    { isPartialOrder = isPartialOrder ER.isPartialOrder
-    } where module ER = Poset ER-poset
+poset : Poset a e r → Poset _ _ _
+poset ER-poset = record
+  { isPartialOrder = isPartialOrder ER.isPartialOrder
+  } where module ER = Poset ER-poset
 
-  decPoset : DecPoset a e r → DecPoset _ _ _
-  decPoset ER-poset = record
-    { isDecPartialOrder = isDecPartialOrder ER.isDecPartialOrder
-    } where module ER = DecPoset ER-poset
+decPoset : DecPoset a e r → DecPoset _ _ _
+decPoset ER-poset = record
+  { isDecPartialOrder = isDecPartialOrder ER.isDecPartialOrder
+  } where module ER = DecPoset ER-poset
 
 ------------------------------------------------------------------------
 -- Properties of disjoint sublists
@@ -670,7 +675,6 @@ open Disjointness public
 -- Monotonicity of disjointness.
 
 module DisjointnessMonotonicity
-    {a b c r s t} {A : Set a} {B : Set b} {C : Set c}
     {R : REL A B r} {S : REL B C s} {T : REL A C t}
     (rs⇒t : Trans R S T) where
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
index 94ccb9038b..aeb7c74e23 100644
--- a/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Propositional/Properties.agda
@@ -23,10 +23,10 @@ open import Data.Product.Base using (∃; _,_; proj₂)
 open import Function.Base using (id; _∘_; _∘′_)
 open import Level using (Level)
 open import Relation.Binary.Definitions using (_Respects_)
-open import Relation.Binary.PropositionalEquality.Core
+open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; refl; cong; _≗_; trans)
 open import Relation.Binary.PropositionalEquality.Properties
-  using (setoid; subst-injective)
+  using (setoid; subst-injective; trans-reflʳ; trans-assoc)
 open import Relation.Unary using (Pred)
 
 private
@@ -38,7 +38,7 @@ private
 -- Re-exporting setoid properties
 
 open SetoidProperties (setoid A) public
-  hiding (map⁺)
+  hiding (map⁺; ⊆-trans-idˡ; ⊆-trans-idʳ; ⊆-trans-assoc)
 
 map⁺ : ∀ {as bs} (f : A → B) → as ⊆ bs → map f as ⊆ map f bs
 map⁺ {B = B} f = SetoidProperties.map⁺ (setoid A) (setoid B) (cong f)
@@ -48,16 +48,11 @@ map⁺ {B = B} f = SetoidProperties.map⁺ (setoid A) (setoid B) (cong f)
 
 ⊆-trans-idˡ : ∀ {xs ys : List A} {τ : xs ⊆ ys} →
               ⊆-trans ⊆-refl τ ≡ τ
-⊆-trans-idˡ {_}     {τ = []    } = refl
-⊆-trans-idˡ {_}     {τ = _ ∷  _} = cong (_ ∷_ ) ⊆-trans-idˡ
-⊆-trans-idˡ {[]}    {τ = _ ∷ʳ _} = cong (_ ∷ʳ_) ⊆-trans-idˡ
-⊆-trans-idˡ {_ ∷ _} {τ = _ ∷ʳ _} = cong (_ ∷ʳ_) ⊆-trans-idˡ
+⊆-trans-idˡ {τ = τ} = SetoidProperties.⊆-trans-idˡ (setoid A) (λ _ → refl) τ
 
 ⊆-trans-idʳ : ∀ {xs ys : List A} {τ : xs ⊆ ys} →
               ⊆-trans τ ⊆-refl ≡ τ
-⊆-trans-idʳ {τ = []      } = refl
-⊆-trans-idʳ {τ = _ ∷ʳ _  } = cong (_  ∷ʳ_ ) ⊆-trans-idʳ
-⊆-trans-idʳ {τ = refl ∷ _} = cong (refl ∷_) ⊆-trans-idʳ
+⊆-trans-idʳ {τ = τ} = SetoidProperties.⊆-trans-idʳ (setoid A) trans-reflʳ τ
 
 -- Note: The associativity law is oriented such that rewriting with it
 -- may trigger reductions of ⊆-trans, which matches first on its
@@ -66,11 +61,8 @@ map⁺ {B = B} f = SetoidProperties.map⁺ (setoid A) (setoid B) (cong f)
 ⊆-trans-assoc : ∀ {ws xs ys zs : List A}
                 {τ₁ : ws ⊆ xs} {τ₂ : xs ⊆ ys} {τ₃ : ys ⊆ zs} →
                 ⊆-trans τ₁ (⊆-trans τ₂ τ₃) ≡ ⊆-trans (⊆-trans τ₁ τ₂) τ₃
-⊆-trans-assoc {τ₁ = _}        {_}      {_ ∷ʳ _} = cong (_ ∷ʳ_) ⊆-trans-assoc
-⊆-trans-assoc {τ₁ = _}        {_ ∷ʳ _} {_ ∷  _} = cong (_ ∷ʳ_) ⊆-trans-assoc
-⊆-trans-assoc {τ₁ = _ ∷ʳ _  } {_ ∷  _} {_ ∷  _} = cong (_ ∷ʳ_) ⊆-trans-assoc
-⊆-trans-assoc {τ₁ = refl ∷ _} {_ ∷  _} {_ ∷  _} = cong (_ ∷_ ) ⊆-trans-assoc
-⊆-trans-assoc {τ₁ = []}       {[]}     {[]}     = refl
+⊆-trans-assoc {τ₁ = τ₁} {τ₂ = τ₂} {τ₃ = τ₃} =
+  SetoidProperties.⊆-trans-assoc (setoid A) (λ p _ _ → ≡.sym (trans-assoc p)) τ₁ τ₂ τ₃
 
 ------------------------------------------------------------------------
 -- Laws concerning ⊆-trans and ∷ˡ⁻
diff --git a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
index cf12328323..60be70aac1 100644
--- a/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Setoid/Properties.agda
@@ -15,14 +15,14 @@ module Data.List.Relation.Binary.Sublist.Setoid.Properties
 open import Data.List.Base hiding (_∷ʳ_)
 open import Data.List.Relation.Unary.Any using (Any)
 import Data.Maybe.Relation.Unary.All as Maybe
-open import Data.Nat.Base using (_≤_; _≥_)
+open import Data.Nat.Base using (ℕ; _≤_; _≥_)
 import Data.Nat.Properties as ℕ
 open import Data.Product.Base using (∃; _,_; proj₂)
 open import Function.Base
 open import Function.Bundles using (_⇔_; _⤖_)
 open import Level
 open import Relation.Binary.Definitions using () renaming (Decidable to Decidable₂)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl; cong; cong₂)
 open import Relation.Binary.Structures using (IsDecTotalOrder)
 open import Relation.Unary using (Pred; Decidable; Irrelevant)
 open import Relation.Nullary.Negation using (¬_)
@@ -39,23 +39,63 @@ open SetoidEquality S using (_≋_; ≋-refl)
 open SetoidSublist S hiding (map)
 open SetoidMembership S using (_∈_)
 
+private
+  variable
+    p q r s t : Level
+    a b x y : A
+    as bs cs ds xs ys : List A
+    P : Pred A p
+    Q : Pred A q
+    m n : ℕ
+
+
 ------------------------------------------------------------------------
 -- Injectivity of constructors
 ------------------------------------------------------------------------
 
-module _ {xs ys : List A} where
+module _ where
 
-  ∷-injectiveˡ : ∀ {x y} {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →
+  ∷-injectiveˡ : ∀ {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →
                  ((x ∷ xs) ⊆ (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → px ≡ qx
   ∷-injectiveˡ refl = refl
 
-  ∷-injectiveʳ : ∀ {x y} {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →
+  ∷-injectiveʳ : ∀ {px qx : x ≈ y} {pxs qxs : xs ⊆ ys} →
                  ((x ∷ xs) ⊆ (y ∷ ys) ∋ px ∷ pxs) ≡ (qx ∷ qxs) → pxs ≡ qxs
   ∷-injectiveʳ refl = refl
 
-  ∷ʳ-injective : ∀ {y} {pxs qxs : xs ⊆ ys} → y ∷ʳ pxs ≡ y ∷ʳ qxs → pxs ≡ qxs
+  ∷ʳ-injective : ∀ {pxs qxs : xs ⊆ ys} → y ∷ʳ pxs ≡ y ∷ʳ qxs → pxs ≡ qxs
   ∷ʳ-injective refl = refl
 
+------------------------------------------------------------------------
+-- Categorical properties
+------------------------------------------------------------------------
+
+module _ (trans-reflˡ : ∀ {x y} (p : x ≈ y) → trans ≈-refl p ≡ p) where
+
+  ⊆-trans-idˡ : (pxs : xs ⊆ ys) → ⊆-trans ⊆-refl pxs ≡ pxs
+  ⊆-trans-idˡ [] = refl
+  ⊆-trans-idˡ (y ∷ʳ pxs) = cong (y ∷ʳ_) (⊆-trans-idˡ pxs)
+  ⊆-trans-idˡ (x ∷ pxs) = cong₂ _∷_ (trans-reflˡ x) (⊆-trans-idˡ pxs)
+
+module _ (trans-reflʳ : ∀ {x y} (p : x ≈ y) → trans p ≈-refl ≡ p) where
+
+  ⊆-trans-idʳ : (pxs : xs ⊆ ys) → ⊆-trans pxs ⊆-refl ≡ pxs
+  ⊆-trans-idʳ [] = refl
+  ⊆-trans-idʳ (y ∷ʳ pxs) = cong (y ∷ʳ_) (⊆-trans-idʳ pxs)
+  ⊆-trans-idʳ (x ∷ pxs) = cong₂ _∷_ (trans-reflʳ x) (⊆-trans-idʳ pxs)
+
+module _ (≈-assoc : ∀ {w x y z} (p : w ≈ x) (q : x ≈ y) (r : y ≈ z) →
+                    trans p (trans q r) ≡ trans (trans p q) r) where
+
+  ⊆-trans-assoc : (ps : as ⊆ bs) (qs : bs ⊆ cs) (rs : cs ⊆ ds) →
+            ⊆-trans ps (⊆-trans qs rs) ≡ ⊆-trans (⊆-trans ps qs) rs
+  ⊆-trans-assoc ps qs (_ ∷ʳ rs) = cong (_ ∷ʳ_) (⊆-trans-assoc ps qs rs)
+  ⊆-trans-assoc ps (_ ∷ʳ qs) (_ ∷ rs) = cong (_ ∷ʳ_) (⊆-trans-assoc ps qs rs)
+  ⊆-trans-assoc (_ ∷ʳ ps) (_ ∷ qs) (_ ∷ rs) = cong (_ ∷ʳ_) (⊆-trans-assoc ps qs rs)
+  ⊆-trans-assoc (p ∷ ps) (q ∷ qs) (r ∷ rs) = cong₂ _∷_ (≈-assoc p q r) (⊆-trans-assoc ps qs rs)
+  ⊆-trans-assoc [] [] [] = refl
+
+
 ------------------------------------------------------------------------
 -- Various functions' outputs are sublists
 ------------------------------------------------------------------------
@@ -69,7 +109,7 @@ take-⊆ n xs = HeteroProperties.take-Sublist n ⊆-refl
 drop-⊆ : ∀ n xs → drop n xs ⊆ xs
 drop-⊆ n xs = HeteroProperties.drop-Sublist n ⊆-refl
 
-module _ {p} {P : Pred A p} (P? : Decidable P) where
+module _ (P? : Decidable P) where
 
   takeWhile-⊆ : ∀ xs → takeWhile P? xs ⊆ xs
   takeWhile-⊆ xs = HeteroProperties.takeWhile-Sublist P? ⊆-refl
@@ -80,7 +120,7 @@ module _ {p} {P : Pred A p} (P? : Decidable P) where
   filter-⊆ : ∀ xs → filter P? xs ⊆ xs
   filter-⊆ xs = HeteroProperties.filter-Sublist P? ⊆-refl
 
-module _ {p} {P : Pred A p} (P? : Decidable P) where
+module _ (P? : Decidable P) where
 
   takeWhile⊆filter : ∀ xs → takeWhile P? xs ⊆ filter P? xs
   takeWhile⊆filter xs = HeteroProperties.takeWhile-filter P? {xs} ≋-refl
@@ -94,15 +134,15 @@ module _ {p} {P : Pred A p} (P? : Decidable P) where
 -- We write f⁺ for the proof that `xs ⊆ ys → f xs ⊆ f ys`
 -- and f⁻ for the one that `f xs ⊆ f ys → xs ⊆ ys`.
 
-module _ {as bs : List A} where
+module _ where
 
-  ∷ˡ⁻ : ∀ {a} → a ∷ as ⊆ bs → as ⊆ bs
+  ∷ˡ⁻ : a ∷ as ⊆ bs → as ⊆ bs
   ∷ˡ⁻ = HeteroProperties.∷ˡ⁻
 
-  ∷ʳ⁻ : ∀ {a b} → ¬ (a ≈ b) → a ∷ as ⊆ b ∷ bs → a ∷ as ⊆ bs
+  ∷ʳ⁻ : ¬ (a ≈ b) → a ∷ as ⊆ b ∷ bs → a ∷ as ⊆ bs
   ∷ʳ⁻ = HeteroProperties.∷ʳ⁻
 
-  ∷⁻ : ∀ {a b} → a ∷ as ⊆ b ∷ bs → as ⊆ bs
+  ∷⁻ : a ∷ as ⊆ b ∷ bs → as ⊆ bs
   ∷⁻ = HeteroProperties.∷⁻
 
 ------------------------------------------------------------------------
@@ -121,7 +161,7 @@ module _ {b ℓ} (R : Setoid b ℓ) where
 ------------------------------------------------------------------------
 -- _++_
 
-module _ {as bs : List A} where
+module _ where
 
   ++⁺ˡ : ∀ cs → as ⊆ bs → as ⊆ cs ++ bs
   ++⁺ˡ = HeteroProperties.++ˡ
@@ -129,16 +169,16 @@ module _ {as bs : List A} where
   ++⁺ʳ : ∀ cs → as ⊆ bs → as ⊆ bs ++ cs
   ++⁺ʳ = HeteroProperties.++ʳ
 
-  ++⁺ : ∀ {cs ds} → as ⊆ bs → cs ⊆ ds → as ++ cs ⊆ bs ++ ds
+  ++⁺ : as ⊆ bs → cs ⊆ ds → as ++ cs ⊆ bs ++ ds
   ++⁺ = HeteroProperties.++⁺
 
-  ++⁻ : ∀ {cs ds} → length as ≡ length bs → as ++ cs ⊆ bs ++ ds → cs ⊆ ds
+  ++⁻ : length as ≡ length bs → as ++ cs ⊆ bs ++ ds → cs ⊆ ds
   ++⁻ = HeteroProperties.++⁻
 
 ------------------------------------------------------------------------
 -- take
 
-module _ {m n} {xs} where
+module _ where
 
   take⁺ : m ≤ n → take m xs ⊆ take n xs
   take⁺ m≤n = HeteroProperties.take⁺ m≤n ≋-refl
@@ -146,17 +186,17 @@ module _ {m n} {xs} where
 ------------------------------------------------------------------------
 -- drop
 
-module _ {m n} {xs ys : List A} where
+module _ where
 
   drop⁺ : m ≥ n → xs ⊆ ys → drop m xs ⊆ drop n ys
   drop⁺ = HeteroProperties.drop⁺
 
-module _ {m n} {xs : List A} where
+module _ where
 
   drop⁺-≥ : m ≥ n → drop m xs ⊆ drop n xs
   drop⁺-≥ m≥n = drop⁺ m≥n ⊆-refl
 
-module _ {xs ys : List A} where
+module _ where
 
   drop⁺-⊆ : ∀ n → xs ⊆ ys → drop n xs ⊆ drop n ys
   drop⁺-⊆ n xs⊆ys = drop⁺ {n} ℕ.≤-refl xs⊆ys
@@ -164,8 +204,7 @@ module _ {xs ys : List A} where
 ------------------------------------------------------------------------
 -- takeWhile / dropWhile
 
-module _ {p q} {P : Pred A p} {Q : Pred A q}
-         (P? : Decidable P) (Q? : Decidable Q) where
+module _ (P? : Decidable P) (Q? : Decidable Q) where
 
   takeWhile⁺ : ∀ {xs} → (∀ {a b} → a ≈ b → P a → Q b) →
                takeWhile P? xs ⊆ takeWhile Q? xs
@@ -178,25 +217,22 @@ module _ {p q} {P : Pred A p} {Q : Pred A q}
 ------------------------------------------------------------------------
 -- filter
 
-module _ {p q} {P : Pred A p} {Q : Pred A q}
-         (P? : Decidable P) (Q? : Decidable Q) where
+module _ (P? : Decidable P) (Q? : Decidable Q) where
 
-  filter⁺ : ∀ {as bs} → (∀ {a b} → a ≈ b → P a → Q b) →
+  filter⁺ : (∀ {a b} → a ≈ b → P a → Q b) →
             as ⊆ bs → filter P? as ⊆ filter Q? bs
   filter⁺ = HeteroProperties.⊆-filter-Sublist P? Q?
 
 ------------------------------------------------------------------------
 -- reverse
 
-module _ {as bs : List A} where
+module _ where
 
-  reverseAcc⁺ : ∀ {cs ds} → as ⊆ bs → cs ⊆ ds →
+  reverseAcc⁺ : as ⊆ bs → cs ⊆ ds →
                 reverseAcc cs as ⊆ reverseAcc ds bs
   reverseAcc⁺ = HeteroProperties.reverseAcc⁺
 
-  ʳ++⁺ : ∀ {cs ds} →
-         as ⊆ bs →
-         cs ⊆ ds →
+  ʳ++⁺ : as ⊆ bs → cs ⊆ ds →
          as ʳ++ cs ⊆ bs ʳ++ ds
   ʳ++⁺ = reverseAcc⁺
 
@@ -233,7 +269,7 @@ module _ {ℓ′} {_≤_ : Rel A ℓ′} (_≤?_ : Decidable₂ _≤_) where
 -- Inversion lemmas
 ------------------------------------------------------------------------
 
-module _ {a as b bs} where
+module _ where
 
   ∷⁻¹ : a ≈ b → as ⊆ bs ⇔ a ∷ as ⊆ b ∷ bs
   ∷⁻¹ = HeteroProperties.∷⁻¹
@@ -247,13 +283,13 @@ module _ {a as b bs} where
 
 module _ where
 
-  length-mono-≤ : ∀ {as bs} → as ⊆ bs → length as ≤ length bs
+  length-mono-≤ : as ⊆ bs → length as ≤ length bs
   length-mono-≤ = HeteroProperties.length-mono-≤
 
 ------------------------------------------------------------------------
 -- Conversion to and from list equality
 
-  to-≋ : ∀ {as bs} → length as ≡ length bs → as ⊆ bs → as ≋ bs
+  to-≋ : length as ≡ length bs → as ⊆ bs → as ≋ bs
   to-≋ = HeteroProperties.toPointwise
 
 ------------------------------------------------------------------------
@@ -265,7 +301,7 @@ module _ where
 ------------------------------------------------------------------------
 -- (to/from)∈ is a bijection
 
-module _ {x xs} where
+module _ where
 
   to∈-injective : ∀ {p q : [ x ] ⊆ xs} → to∈ p ≡ to∈ q → p ≡ q
   to∈-injective = HeteroProperties.toAny-injective

From c5538a8d1ee50e9c77e34c871c84b9eedd4f3a07 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 15 May 2024 07:08:14 +0100
Subject: [PATCH 286/316] Pointwise `Algebra` (#2381)

* Pointwise `Algebra`

* temporary commit

* better `CHANGELOG` entry?

* begin removing redundant `module` implementation

* finish removing redundant `module` implementation

* make `liftRel` implicitly quantified
---
 CHANGELOG.md                         |   6 +
 src/Algebra/Construct/Pointwise.agda | 184 +++++++++++++++++++++++++++
 2 files changed, 190 insertions(+)
 create mode 100644 src/Algebra/Construct/Pointwise.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9d1bb7c58b..30dc56b3d0 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -78,6 +78,12 @@ Deprecated names
 New modules
 -----------
 
+* Pointwise lifting of algebraic structures `IsX` and bundles `X` from
+  carrier set `C` to function space `A → C`:
+  ```
+  Algebra.Construct.Pointwise
+  ```
+
 * Raw bundles for module-like algebraic structures:
   ```
   Algebra.Module.Bundles.Raw
diff --git a/src/Algebra/Construct/Pointwise.agda b/src/Algebra/Construct/Pointwise.agda
new file mode 100644
index 0000000000..842e36b1ed
--- /dev/null
+++ b/src/Algebra/Construct/Pointwise.agda
@@ -0,0 +1,184 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- For each `IsX` algebraic structure `S`, lift the structure to the
+-- 'pointwise' function space `A → S`: categorically, this is the
+-- *power* object in the relevant category of `X` objects and morphisms
+--
+-- NB the module is parametrised only wrt `A`
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Algebra.Construct.Pointwise {a} (A : Set a) where
+
+open import Algebra.Bundles
+open import Algebra.Core using (Op₁; Op₂)
+open import Algebra.Structures
+open import Data.Product.Base using (_,_)
+open import Function.Base using (id; _∘′_; const)
+open import Level
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.Bundles using (Setoid)
+open import Relation.Binary.Structures using (IsEquivalence)
+
+
+private
+
+  variable
+    c ℓ : Level
+    C : Set c
+    _≈_ : Rel C ℓ
+    ε 0# 1# : C
+    _⁻¹ -_ : Op₁ C
+    _∙_ _+_ _*_ : Op₂ C
+
+  lift₀ : C → A → C
+  lift₀ = const
+
+  lift₁ : Op₁ C → Op₁ (A → C)
+  lift₁ = _∘′_
+
+  lift₂ : Op₂ C → Op₂ (A → C)
+  lift₂ _∙_ g h x = (g x) ∙ (h x)
+
+  liftRel : Rel C ℓ → Rel (A → C) (a ⊔ ℓ)
+  liftRel _≈_ g h = ∀ {x} → (g x) ≈ (h x)
+
+
+------------------------------------------------------------------------
+-- Setoid structure: here rather than elsewhere? (could be imported?)
+
+isEquivalence : IsEquivalence _≈_ → IsEquivalence (liftRel _≈_)
+isEquivalence isEquivalence = record
+  { refl = λ {f x} → refl {f x}
+  ; sym = λ f≈g → sym f≈g
+  ; trans = λ f≈g g≈h → trans f≈g g≈h
+  }
+  where open IsEquivalence isEquivalence
+
+------------------------------------------------------------------------
+-- Structures
+
+isMagma : IsMagma _≈_ _∙_ → IsMagma (liftRel _≈_) (lift₂ _∙_)
+isMagma isMagma = record
+  { isEquivalence = isEquivalence M.isEquivalence
+  ; ∙-cong = λ g h → M.∙-cong g h
+  }
+  where module M = IsMagma isMagma
+
+isSemigroup : IsSemigroup _≈_ _∙_ → IsSemigroup (liftRel _≈_) (lift₂ _∙_)
+isSemigroup isSemigroup = record
+  { isMagma = isMagma M.isMagma
+  ; assoc = λ f g h → M.assoc (f _) (g _) (h _)
+  }
+  where module M = IsSemigroup isSemigroup
+
+isBand : IsBand _≈_ _∙_ → IsBand (liftRel _≈_) (lift₂ _∙_)
+isBand isBand = record
+  { isSemigroup = isSemigroup M.isSemigroup
+  ; idem = λ f → M.idem (f _)
+  }
+  where module M = IsBand isBand
+
+isCommutativeSemigroup : IsCommutativeSemigroup _≈_ _∙_ →
+                         IsCommutativeSemigroup (liftRel _≈_) (lift₂ _∙_)
+isCommutativeSemigroup isCommutativeSemigroup = record
+  { isSemigroup = isSemigroup M.isSemigroup
+  ; comm = λ f g → M.comm (f _) (g _)
+  }
+  where module M = IsCommutativeSemigroup isCommutativeSemigroup
+
+isMonoid : IsMonoid _≈_ _∙_ ε → IsMonoid (liftRel _≈_) (lift₂ _∙_) (lift₀ ε)
+isMonoid isMonoid = record
+  { isSemigroup = isSemigroup M.isSemigroup
+  ; identity = (λ f → M.identityˡ (f _)) , λ f → M.identityʳ (f _)
+  }
+  where module M = IsMonoid isMonoid
+
+isCommutativeMonoid : IsCommutativeMonoid _≈_ _∙_ ε →
+                      IsCommutativeMonoid (liftRel _≈_) (lift₂ _∙_) (lift₀ ε)
+isCommutativeMonoid isCommutativeMonoid = record
+  { isMonoid = isMonoid M.isMonoid
+  ; comm = λ f g → M.comm (f _) (g _)
+  }
+  where module M = IsCommutativeMonoid isCommutativeMonoid
+
+isGroup : IsGroup _≈_ _∙_ ε _⁻¹ →
+          IsGroup (liftRel _≈_) (lift₂ _∙_) (lift₀ ε) (lift₁ _⁻¹)
+isGroup isGroup = record
+  { isMonoid = isMonoid M.isMonoid
+  ; inverse = (λ f → M.inverseˡ (f _)) , λ f → M.inverseʳ (f _)
+  ; ⁻¹-cong = λ f → M.⁻¹-cong f
+  }
+  where module M = IsGroup isGroup
+
+isAbelianGroup : IsAbelianGroup _≈_ _∙_ ε _⁻¹ →
+                 IsAbelianGroup (liftRel _≈_) (lift₂ _∙_) (lift₀ ε) (lift₁ _⁻¹)
+isAbelianGroup isAbelianGroup = record
+  { isGroup = isGroup M.isGroup
+  ; comm = λ f g → M.comm (f _) (g _)
+  }
+  where module M = IsAbelianGroup isAbelianGroup
+
+isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _≈_ _+_ _*_ 0# 1# →
+  IsSemiringWithoutAnnihilatingZero (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
+isSemiringWithoutAnnihilatingZero isSemiringWithoutAnnihilatingZero = record
+  { +-isCommutativeMonoid = isCommutativeMonoid M.+-isCommutativeMonoid
+  ; *-cong =  λ g h → M.*-cong g h
+  ; *-assoc =  λ f g h → M.*-assoc (f _) (g _) (h _)
+  ; *-identity = (λ f → M.*-identityˡ (f _)) , λ f → M.*-identityʳ (f _)
+  ; distrib = (λ f g h → M.distribˡ (f _) (g _) (h _)) , λ f g h → M.distribʳ (f _) (g _) (h _)
+  }
+  where module M = IsSemiringWithoutAnnihilatingZero isSemiringWithoutAnnihilatingZero
+
+isSemiring : IsSemiring _≈_ _+_ _*_ 0# 1# →
+             IsSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
+isSemiring isSemiring = record
+  { isSemiringWithoutAnnihilatingZero = isSemiringWithoutAnnihilatingZero M.isSemiringWithoutAnnihilatingZero
+  ; zero = (λ f → M.zeroˡ (f _)) , λ f → M.zeroʳ (f _)
+  }
+  where module M = IsSemiring isSemiring
+
+isRing : IsRing _≈_ _+_ _*_ -_ 0# 1# →
+         IsRing (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ -_) (lift₀ 0#) (lift₀ 1#)
+isRing isRing = record
+  { +-isAbelianGroup = isAbelianGroup M.+-isAbelianGroup
+  ; *-cong = λ g h → M.*-cong g h
+  ; *-assoc = λ f g h → M.*-assoc (f _) (g _) (h _)
+  ; *-identity = (λ f → M.*-identityˡ (f _)) , λ f → M.*-identityʳ (f _)
+  ; distrib = (λ f g h → M.distribˡ (f _) (g _) (h _)) , λ f g h → M.distribʳ (f _) (g _) (h _)
+  }
+  where module M = IsRing isRing
+
+
+------------------------------------------------------------------------
+-- Bundles
+
+magma : Magma c ℓ → Magma (a ⊔ c) (a ⊔ ℓ)
+magma m = record { isMagma = isMagma (Magma.isMagma m) }
+
+semigroup : Semigroup c ℓ → Semigroup (a ⊔ c) (a ⊔ ℓ)
+semigroup m = record { isSemigroup = isSemigroup (Semigroup.isSemigroup m) }
+
+band : Band c ℓ → Band (a ⊔ c) (a ⊔ ℓ)
+band m = record { isBand = isBand (Band.isBand m) }
+
+commutativeSemigroup : CommutativeSemigroup c ℓ → CommutativeSemigroup (a ⊔ c) (a ⊔ ℓ)
+commutativeSemigroup m = record { isCommutativeSemigroup = isCommutativeSemigroup (CommutativeSemigroup.isCommutativeSemigroup m) }
+
+monoid : Monoid c ℓ → Monoid (a ⊔ c) (a ⊔ ℓ)
+monoid m = record { isMonoid = isMonoid (Monoid.isMonoid m) }
+
+group : Group c ℓ → Group (a ⊔ c) (a ⊔ ℓ)
+group m = record { isGroup = isGroup (Group.isGroup m) }
+
+abelianGroup : AbelianGroup c ℓ → AbelianGroup (a ⊔ c) (a ⊔ ℓ)
+abelianGroup m = record { isAbelianGroup = isAbelianGroup (AbelianGroup.isAbelianGroup m) }
+
+semiring : Semiring c ℓ → Semiring (a ⊔ c) (a ⊔ ℓ)
+semiring m = record { isSemiring = isSemiring (Semiring.isSemiring m) }
+
+ring : Ring c ℓ → Ring (a ⊔ c) (a ⊔ ℓ)
+ring m = record { isRing = isRing (Ring.isRing m) }
+

From 8373463655ef2adecd4602122de1ba0ae52b4858 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Thu, 16 May 2024 02:06:00 -0400
Subject: [PATCH 287/316] Algebra fixity (#2386)

* put the fixity recommendations into the style guide

* mixing fixity declaration

* was missing a case

* use the new case

* add fixities for everything in Algebra

* incorporate some suggestions for extra cases
---
 doc/README/Design/Fixity.agda                 |  1 +
 doc/style-guide.md                            | 60 +++++++++++++++++++
 src/Algebra/Bundles.agda                      |  1 +
 src/Algebra/Definitions.agda                  |  4 ++
 src/Algebra/Definitions/RawMagma.agda         |  2 +-
 src/Algebra/Module/Bundles.agda               |  1 +
 src/Algebra/Module/Definitions/Left.agda      |  2 +
 src/Algebra/Module/Definitions/Right.agda     |  2 +
 .../Properties/Monoid/Divisibility.agda       |  2 +
 .../Properties/Semiring/Divisibility.agda     |  2 +
 .../Solver/Ring/AlmostCommutativeRing.agda    |  2 +
 11 files changed, 78 insertions(+), 1 deletion(-)

diff --git a/doc/README/Design/Fixity.agda b/doc/README/Design/Fixity.agda
index 85153c1ec4..52394ff864 100644
--- a/doc/README/Design/Fixity.agda
+++ b/doc/README/Design/Fixity.agda
@@ -34,3 +34,4 @@ module README.Design.Fixity where
 -- type formers:
 -- product-like infixr 2 _×_ _-×-_ _-,-_
 -- sum-like infixr 1 _⊎_
+-- binary properties infix 4 _Absorbs_
diff --git a/doc/style-guide.md b/doc/style-guide.md
index 254a7ec071..2562f76cfd 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -525,6 +525,66 @@ word within a compound word is capitalized except for the first word.
 * Any exceptions to these conventions should be flagged on the GitHub
   `agda-stdlib` issue tracker in the usual way.
 
+#### Fixity
+
+All functions and operators that are not purely prefix (typically
+anything that has a `_` in its name) should have an explicit fixity
+declared for it. The guidelines for these are as follows:
+
+General operations and relations:
+
+* binary relations of all kinds are `infix 4`
+
+* unary prefix relations `infix 4 ε∣_`
+
+* unary postfix relations `infixr 8 _∣0`
+
+* multiplication-like: `infixl 7 _*_`
+
+* addition-like  `infixl 6 _+_`
+
+* arithmetic prefix minus-like  `infix  8 -_`
+
+* arithmetic infix binary minus-like `infixl 6 _-_`
+
+* and-like  `infixr 7 _∧_`
+
+* or-like  `infixr 6 _∨_`
+
+* negation-like `infix 3 ¬_`
+
+* post-fix inverse  `infix  8 _⁻¹`
+
+* bind `infixl 1 _>>=_`
+
+* list concat-like `infixr 5 _∷_`
+
+* ternary reasoning `infix 1 _⊢_≈_`
+
+* composition `infixr 9 _∘_`
+
+* application `infixr -1 _$_ _$!_`
+
+* combinatorics `infixl 6.5 _P_ _P′_ _C_ _C′_`
+
+* pair `infixr 4 _,_`
+
+Reasoning:
+
+* QED  `infix  3 _∎`
+
+* stepping  `infixr 2 _≡⟨⟩_ step-≡ step-≡˘`
+
+* begin  `infix  1 begin_`
+
+Type formers:
+
+* product-like `infixr 2 _×_ _-×-_ _-,-_`
+
+* sum-like `infixr 1 _⊎_`
+
+*  binary properties `infix 4 _Absorbs_`
+
 #### Functions and relations over specific datatypes
 
 * When defining a new relation `P` over a datatype `X` in a `Data.X.Relation` module,
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index 1969731c2c..a86529f5cd 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -917,6 +917,7 @@ record NonAssociativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
     using () renaming (magma to *-magma; identity to *-identity)
 
 record Nearring c ℓ : Set (suc (c ⊔ ℓ)) where
+  infix  8 -_
   infixl 7 _*_
   infixl 6 _+_
   infix  4 _≈_
diff --git a/src/Algebra/Definitions.agda b/src/Algebra/Definitions.agda
index ae0566a4bf..62528e5b70 100644
--- a/src/Algebra/Definitions.agda
+++ b/src/Algebra/Definitions.agda
@@ -97,6 +97,8 @@ RightConical e _∙_ = ∀ x y → (x ∙ y) ≈ e → y ≈ e
 Conical : A → Op₂ A → Set _
 Conical e ∙ = (LeftConical e ∙) × (RightConical e ∙)
 
+infix 4 _DistributesOverˡ_ _DistributesOverʳ_ _DistributesOver_
+
 _DistributesOverˡ_ : Op₂ A → Op₂ A → Set _
 _*_ DistributesOverˡ _+_ =
   ∀ x y z → (x * (y + z)) ≈ ((x * y) + (x * z))
@@ -108,6 +110,8 @@ _*_ DistributesOverʳ _+_ =
 _DistributesOver_ : Op₂ A → Op₂ A → Set _
 * DistributesOver + = (* DistributesOverˡ +) × (* DistributesOverʳ +)
 
+infix 4 _MiddleFourExchange_ _IdempotentOn_ _Absorbs_
+
 _MiddleFourExchange_ : Op₂ A → Op₂ A → Set _
 _*_ MiddleFourExchange _+_ =
   ∀ w x y z → ((w + x) * (y + z)) ≈ ((w + y) * (x + z))
diff --git a/src/Algebra/Definitions/RawMagma.agda b/src/Algebra/Definitions/RawMagma.agda
index 2e1bc657c7..b8591e7ad2 100644
--- a/src/Algebra/Definitions/RawMagma.agda
+++ b/src/Algebra/Definitions/RawMagma.agda
@@ -25,7 +25,7 @@ open RawMagma M renaming (Carrier to A)
 ------------------------------------------------------------------------
 -- Divisibility
 
-infix 5 _∣ˡ_ _∤ˡ_ _∣ʳ_ _∤ʳ_ _∣_ _∤_
+infix 5 _∣ˡ_ _∤ˡ_ _∣ʳ_ _∤ʳ_ _∣_ _∤_ _∣∣_ _∤∤_
 
 -- Divisibility from the left.
 --
diff --git a/src/Algebra/Module/Bundles.agda b/src/Algebra/Module/Bundles.agda
index 33e6bfa731..5df96258d6 100644
--- a/src/Algebra/Module/Bundles.agda
+++ b/src/Algebra/Module/Bundles.agda
@@ -279,6 +279,7 @@ record Bimodule (R-ring : Ring r ℓr) (S-ring : Ring s ℓs) m ℓm
     module R = Ring R-ring
     module S = Ring S-ring
 
+  infix 8 -ᴹ_
   infixr 7 _*ₗ_
   infixl 7 _*ᵣ_
   infixl 6 _+ᴹ_
diff --git a/src/Algebra/Module/Definitions/Left.agda b/src/Algebra/Module/Definitions/Left.agda
index e8aa153507..0bb7d530c3 100644
--- a/src/Algebra/Module/Definitions/Left.agda
+++ b/src/Algebra/Module/Definitions/Left.agda
@@ -31,6 +31,8 @@ LeftIdentity a _∙ᴮ_ = ∀ m → (a ∙ᴮ m) ≈ m
 Associative : Op₂ A → Opₗ A B → Set _
 Associative _∙ᴬ_ _∙ᴮ_ = ∀ x y m → ((x ∙ᴬ y) ∙ᴮ m) ≈ (x ∙ᴮ (y ∙ᴮ m))
 
+infix 4 _DistributesOverˡ_ _DistributesOverʳ_⟶_
+
 _DistributesOverˡ_ : Opₗ A B → Op₂ B → Set _
 _*_ DistributesOverˡ _+_ =
   ∀ x m n → (x * (m + n)) ≈ ((x * m) + (x * n))
diff --git a/src/Algebra/Module/Definitions/Right.agda b/src/Algebra/Module/Definitions/Right.agda
index d1a7a878ea..a023b09e52 100644
--- a/src/Algebra/Module/Definitions/Right.agda
+++ b/src/Algebra/Module/Definitions/Right.agda
@@ -31,6 +31,8 @@ RightIdentity a _∙ᴮ_ = ∀ m → (m ∙ᴮ a) ≈ m
 Associative : Op₂ A → Opᵣ A B → Set _
 Associative _∙ᴬ_ _∙ᴮ_ = ∀ m x y → ((m ∙ᴮ x) ∙ᴮ y) ≈ (m ∙ᴮ (x ∙ᴬ y))
 
+infix 4 _DistributesOverʳ_ _DistributesOverˡ_⟶_
+
 _DistributesOverˡ_⟶_ : Opᵣ A B → Op₂ A → Op₂ B → Set _
 _*_ DistributesOverˡ _+ᴬ_ ⟶ _+ᴮ_ =
   ∀ m x y → (m * (x +ᴬ y)) ≈ ((m * x) +ᴮ (m * y))
diff --git a/src/Algebra/Properties/Monoid/Divisibility.agda b/src/Algebra/Properties/Monoid/Divisibility.agda
index 588f6ef375..c0f4c40af9 100644
--- a/src/Algebra/Properties/Monoid/Divisibility.agda
+++ b/src/Algebra/Properties/Monoid/Divisibility.agda
@@ -26,6 +26,8 @@ open import Algebra.Properties.Semigroup.Divisibility semigroup public
 ------------------------------------------------------------------------
 -- Additional properties
 
+infix 4 ε∣_
+
 ε∣_ : ∀ x → ε ∣ x
 ε∣ x = x , identityʳ x
 
diff --git a/src/Algebra/Properties/Semiring/Divisibility.agda b/src/Algebra/Properties/Semiring/Divisibility.agda
index 54a352d782..8cdcbe7e62 100644
--- a/src/Algebra/Properties/Semiring/Divisibility.agda
+++ b/src/Algebra/Properties/Semiring/Divisibility.agda
@@ -25,6 +25,8 @@ open MonoidDivisibility *-monoid public
 ------------------------------------------------------------------------
 -- Divisibility properties specific to semirings.
 
+infixr 8 _∣0
+
 _∣0 : ∀ x → x ∣ 0#
 x ∣0 = 0# , zeroˡ x
 
diff --git a/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda b/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
index 4c33c7b337..9659dfc6ca 100644
--- a/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
+++ b/src/Algebra/Solver/Ring/AlmostCommutativeRing.agda
@@ -76,6 +76,8 @@ record AlmostCommutativeRing c ℓ : Set (suc (c ⊔ ℓ)) where
 ------------------------------------------------------------------------
 -- Homomorphisms
 
+infix 4 _-Raw-AlmostCommutative⟶_
+
 record _-Raw-AlmostCommutative⟶_
   {r₁ r₂ r₃ r₄}
   (From : RawRing r₁ r₄)

From 2ac71e51d2fa41902a235b27f0ccc3ed15dbf99f Mon Sep 17 00:00:00 2001
From: Chris <cspollard@users.noreply.github.com>
Date: Wed, 22 May 2024 02:58:28 +0100
Subject: [PATCH 288/316] Decidable Setoid -> Apartness Relation and Rational
 Heyting Field (#2194)

* new stuff

* forgot to add bundles for rational instance of Heyting*

* one more (obvious) simplification

* a few more simplifications

* comments on DecSetoid properties from jamesmckinna

* not working, but partially there

* woo!

* fix anonymous instance

* fix errant whitespace

* `fix-whitespace`

* rectified wrt `contradiction`

* rectified `DecSetoid` properties

* rectified `Group` properties

* inlined lemma; tidied up

---------

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Co-authored-by: James McKinna <J.McKinna@hw.ac.uk>
---
 CHANGELOG.md                                  | 23 ++++++-
 src/Algebra/Properties/Group.agda             | 12 ++++
 src/Data/Rational/Properties.agda             | 60 ++++++++++++++++++-
 src/Relation/Binary/Properties/DecSetoid.agda | 43 +++++++++++++
 src/Relation/Binary/Properties/Setoid.agda    |  7 ++-
 5 files changed, 139 insertions(+), 6 deletions(-)
 create mode 100644 src/Relation/Binary/Properties/DecSetoid.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 30dc56b3d0..29839632c7 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -145,6 +145,11 @@ New modules
   Relation.Binary.Construct.Interior.Symmetric
   ```
 
+* Properties of `Setoid`s with decidable equality relation:
+  ```
+  Relation.Binary.Properties.DecSetoid
+  ```
+
 * Pointwise and equality relations over indexed containers:
   ```agda
   Data.Container.Indexed.Relation.Binary.Pointwise
@@ -262,6 +267,8 @@ Additions to existing modules
   //-rightDivides  : RightDivides _∙_ _//_
 
   ⁻¹-selfInverse  : SelfInverse _⁻¹
+  x∙y⁻¹≈ε⇒x≈y     : ∀ x y → (x ∙ y ⁻¹) ≈ ε → x ≈ y
+  x≈y⇒x∙y⁻¹≈ε     : ∀ {x y} → x ≈ y → (x ∙ y ⁻¹) ≈ ε
   \\≗flip-//⇒comm : (∀ x y → x \\ y ≈ y // x) → Commutative _∙_
   comm⇒\\≗flip-// : Commutative _∙_ → ∀ x y → x \\ y ≈ y // x
   ⁻¹-anti-homo-// : (x // y) ⁻¹ ≈ y // x
@@ -499,9 +506,22 @@ Additions to existing modules
   product-↭ : product Preserves _↭_ ⟶ _≡_
   ```
 
+* In `Data.Rational.Properties`:
+  ```agda
+  1≢0 : 1ℚ ≢ 0ℚ
+
+  #⇒invertible : p ≢ q → Invertible 1ℚ _*_ (p - q)
+  invertible⇒# : Invertible 1ℚ _*_ (p - q) → p ≢ q
+
+  isHeytingCommutativeRing : IsHeytingCommutativeRing _≡_ _≢_ _+_ _*_ -_ 0ℚ 1ℚ
+  isHeytingField           : IsHeytingField _≡_ _≢_ _+_ _*_ -_ 0ℚ 1ℚ
+  heytingCommutativeRing   : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
+  heytingField             : HeytingField 0ℓ 0ℓ 0ℓ
+  ```
+
 * Added new functions in `Data.String.Base`:
   ```agda
-  map : (Char → Char) → String → String
+  map     : (Char → Char) → String → String
   between : String → String → String → String
   ```
 
@@ -562,6 +582,7 @@ Additions to existing modules
 
 * Added new proofs in `Relation.Binary.Properties.Setoid`:
   ```agda
+  ≉-irrefl : Irreflexive _≈_ _≉_
   ≈;≈⇒≈ : _≈_ ; _≈_ ⇒ _≈_
   ≈⇒≈;≈ : _≈_ ⇒ _≈_ ; _≈_
   ```
diff --git a/src/Algebra/Properties/Group.agda b/src/Algebra/Properties/Group.agda
index bac2c84d31..bade624b3c 100644
--- a/src/Algebra/Properties/Group.agda
+++ b/src/Algebra/Properties/Group.agda
@@ -115,6 +115,18 @@ inverseʳ-unique x y eq = trans (y≈x\\z x y ε eq) (identityʳ _)
 ⁻¹-involutive : Involutive _⁻¹
 ⁻¹-involutive = selfInverse⇒involutive ⁻¹-selfInverse
 
+x∙y⁻¹≈ε⇒x≈y : ∀ x y → (x ∙ y ⁻¹) ≈ ε → x ≈ y
+x∙y⁻¹≈ε⇒x≈y x y x∙y⁻¹≈ε = begin
+  x         ≈⟨ inverseˡ-unique x (y ⁻¹) x∙y⁻¹≈ε ⟩
+  y ⁻¹ ⁻¹   ≈⟨ ⁻¹-involutive y ⟩
+  y         ∎
+
+x≈y⇒x∙y⁻¹≈ε : ∀ {x y} → x ≈ y → (x ∙ y ⁻¹) ≈ ε
+x≈y⇒x∙y⁻¹≈ε {x} {y} x≈y = begin
+  x ∙ y ⁻¹ ≈⟨ ∙-congʳ x≈y ⟩
+  y ∙ y ⁻¹ ≈⟨ inverseʳ y ⟩
+  ε        ∎
+
 ⁻¹-injective : Injective _≈_ _≈_ _⁻¹
 ⁻¹-injective = selfInverse⇒injective ⁻¹-selfInverse
 
diff --git a/src/Data/Rational/Properties.agda b/src/Data/Rational/Properties.agda
index a00ef23f82..fb186be08d 100644
--- a/src/Data/Rational/Properties.agda
+++ b/src/Data/Rational/Properties.agda
@@ -9,6 +9,7 @@
 
 module Data.Rational.Properties where
 
+open import Algebra.Apartness
 open import Algebra.Construct.NaturalChoice.Base
 import Algebra.Construct.NaturalChoice.MinMaxOp as MinMaxOp
 import Algebra.Lattice.Construct.NaturalChoice.MinMaxOp as LatticeMinMaxOp
@@ -21,6 +22,7 @@ import Algebra.Morphism.GroupMonomorphism  as GroupMonomorphisms
 import Algebra.Morphism.RingMonomorphism   as RingMonomorphisms
 import Algebra.Lattice.Morphism.LatticeMonomorphism as LatticeMonomorphisms
 import Algebra.Properties.CommutativeSemigroup as CommSemigroupProperties
+import Algebra.Properties.Group as GroupProperties
 open import Data.Bool.Base using (T; true; false)
 open import Data.Integer.Base as ℤ using (ℤ; +_; -[1+_]; +[1+_]; +0; 0ℤ; 1ℤ; _◃_)
 open import Data.Integer.Coprimality using (coprime-divisor)
@@ -49,16 +51,18 @@ open import Function.Base using (_∘_; _∘′_; _∘₂_; _$_; flip)
 open import Function.Definitions using (Injective)
 open import Level using (0ℓ)
 open import Relation.Binary
+open import Relation.Binary.Morphism.Structures
+import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms
+import Relation.Binary.Properties.DecSetoid as DecSetoidProperties
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; cong; cong₂; sym; trans; _≢_; subst; subst₂; resp₂)
 open import Relation.Binary.PropositionalEquality.Properties
   using (setoid; decSetoid; module ≡-Reasoning; isEquivalence)
-open import Relation.Binary.Morphism.Structures
-import Relation.Binary.Morphism.OrderMonomorphism as OrderMonomorphisms
+import Relation.Binary.Reasoning.Setoid as ≈-Reasoning
+open import Relation.Binary.Reasoning.Syntax using (module ≃-syntax)
 open import Relation.Nullary.Decidable.Core as Dec
   using (yes; no; recompute; map′; _×-dec_)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
-open import Relation.Binary.Reasoning.Syntax using (module ≃-syntax)
 
 open import Algebra.Definitions {A = ℚ} _≡_
 open import Algebra.Structures  {A = ℚ} _≡_
@@ -97,6 +101,9 @@ mkℚ n₁ d₁ _ ≟ mkℚ n₂ d₂ _ = map′
 ≡-decSetoid : DecSetoid 0ℓ 0ℓ
 ≡-decSetoid = decSetoid _≟_
 
+1≢0 : 1ℚ ≢ 0ℚ
+1≢0 = λ ()
+
 ------------------------------------------------------------------------
 -- mkℚ+
 ------------------------------------------------------------------------
@@ -1239,6 +1246,53 @@ neg-distribʳ-* = +-*-Monomorphism.neg-distribʳ-* ℚᵘ.+-0-isGroup ℚᵘ.*-i
   { isCommutativeRing = +-*-isCommutativeRing
   }
 
+
+------------------------------------------------------------------------
+-- HeytingField structures and bundles
+
+module _ where
+  open CommutativeRing +-*-commutativeRing
+    using (+-group; zeroˡ; *-congʳ; isCommutativeRing)
+
+  open GroupProperties +-group
+  open DecSetoidProperties ≡-decSetoid
+
+  #⇒invertible : p ≢ q → Invertible 1ℚ _*_ (p - q)
+  #⇒invertible {p} {q} p≢q = let r = p - q in 1/ r , *-inverseˡ r , *-inverseʳ r
+    where instance _ = ≢-nonZero (p≢q ∘ (x∙y⁻¹≈ε⇒x≈y p q))
+
+  invertible⇒# : Invertible 1ℚ _*_ (p - q) → p ≢ q
+  invertible⇒# {p} {q} (1/[p-q] , _ , [p-q]/[p-q]≡1) p≡q = contradiction 1≡0 1≢0
+    where
+    open ≈-Reasoning ≡-setoid
+    1≡0 : 1ℚ ≡ 0ℚ
+    1≡0 = begin
+      1ℚ                 ≈⟨ [p-q]/[p-q]≡1 ⟨
+      (p - q) * 1/[p-q]  ≈⟨ *-congʳ (x≈y⇒x∙y⁻¹≈ε p≡q) ⟩
+      0ℚ * 1/[p-q]       ≈⟨ zeroˡ 1/[p-q] ⟩
+      0ℚ                 ∎
+
+  isHeytingCommutativeRing : IsHeytingCommutativeRing _≡_ _≢_ _+_ _*_ -_ 0ℚ 1ℚ
+  isHeytingCommutativeRing = record
+    { isCommutativeRing = isCommutativeRing
+    ; isApartnessRelation = ≉-isApartnessRelation
+    ; #⇒invertible = #⇒invertible
+    ; invertible⇒# = invertible⇒#
+    }
+
+  isHeytingField : IsHeytingField _≡_ _≢_ _+_ _*_ -_ 0ℚ 1ℚ
+  isHeytingField = record
+    { isHeytingCommutativeRing = isHeytingCommutativeRing
+    ; tight = ≉-tight
+    }
+
+  heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
+  heytingCommutativeRing = record { isHeytingCommutativeRing = isHeytingCommutativeRing }
+
+  heytingField : HeytingField 0ℓ 0ℓ 0ℓ
+  heytingField = record { isHeytingField = isHeytingField }
+
+
 ------------------------------------------------------------------------
 -- Properties of _*_ and _≤_
 
diff --git a/src/Relation/Binary/Properties/DecSetoid.agda b/src/Relation/Binary/Properties/DecSetoid.agda
new file mode 100644
index 0000000000..7f9b323d85
--- /dev/null
+++ b/src/Relation/Binary/Properties/DecSetoid.agda
@@ -0,0 +1,43 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Every decidable setoid induces tight apartness relation.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Bundles using (DecSetoid; ApartnessRelation)
+
+module Relation.Binary.Properties.DecSetoid {c ℓ} (S : DecSetoid c ℓ) where
+
+open import Data.Product using (_,_)
+open import Data.Sum using (inj₁; inj₂)
+open import Relation.Binary.Definitions
+  using (Cotransitive; Tight)
+import Relation.Binary.Properties.Setoid as SetoidProperties
+open import Relation.Binary.Structures
+  using (IsApartnessRelation; IsDecEquivalence)
+open import Relation.Nullary.Decidable.Core
+  using (yes; no; decidable-stable)
+
+open DecSetoid S using (_≈_; _≉_; _≟_; setoid; trans)
+open SetoidProperties setoid
+
+≉-cotrans : Cotransitive _≉_
+≉-cotrans {x} {y} x≉y z with x ≟ z | z ≟ y
+... | _ | no z≉y = inj₂ z≉y
+... | no x≉z | _ = inj₁ x≉z
+... | yes x≈z | yes z≈y = inj₁ λ _ → x≉y (trans x≈z z≈y)
+
+≉-isApartnessRelation : IsApartnessRelation _≈_ _≉_
+≉-isApartnessRelation = record
+  { irrefl = ≉-irrefl
+  ; sym = ≉-sym
+  ; cotrans = ≉-cotrans
+  }
+
+≉-apartnessRelation : ApartnessRelation c ℓ ℓ
+≉-apartnessRelation = record { isApartnessRelation = ≉-isApartnessRelation }
+
+≉-tight : Tight _≈_ _≉_
+≉-tight x y = decidable-stable (x ≟ y) , ≉-irrefl
diff --git a/src/Relation/Binary/Properties/Setoid.agda b/src/Relation/Binary/Properties/Setoid.agda
index 1b56c32d2a..43dc7e67d0 100644
--- a/src/Relation/Binary/Properties/Setoid.agda
+++ b/src/Relation/Binary/Properties/Setoid.agda
@@ -8,12 +8,12 @@
 
 open import Data.Product.Base using (_,_)
 open import Function.Base using (_∘_; id; _$_; flip)
-open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Binary.Core using (_⇒_)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.Bundles using (Setoid; Preorder; Poset)
 open import Relation.Binary.Definitions
-  using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_)
+  using (Symmetric; _Respectsˡ_; _Respectsʳ_; _Respects₂_; Irreflexive)
 open import Relation.Binary.Structures using (IsPreorder; IsPartialOrder)
 open import Relation.Binary.Construct.Composition
   using (_;_; impliesˡ; transitive⇒≈;≈⊆≈)
@@ -80,6 +80,9 @@ preorder = record
 ≉-resp₂ : _≉_ Respects₂ _≈_
 ≉-resp₂ = ≉-respʳ , ≉-respˡ
 
+≉-irrefl : Irreflexive _≈_ _≉_
+≉-irrefl x≈y x≉y = contradiction x≈y x≉y
+
 ------------------------------------------------------------------------
 -- Equality is closed under composition
 

From db84b730c261f0fb49b2a8ed6ddd444b0673c19c Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Wed, 29 May 2024 18:30:12 +0200
Subject: [PATCH 289/316] CI bumps: ghc 9.10, action versions, Agda to 2.6.4.3
 (#2398)

* Haskell CI (for GenerateEverything) and dependencies bumped to GHC 9.10.1

* CI: bump some versions, satisfy some shellcheck complaints
---
 .github/workflows/ci-ubuntu.yml  | 24 +++++-----
 .github/workflows/haskell-ci.yml | 78 +++++++++++++-------------------
 agda-stdlib-utils.cabal          | 11 +++--
 3 files changed, 49 insertions(+), 64 deletions(-)

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index 095f28b1f8..ec9ae5c501 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -75,17 +75,17 @@ jobs:
           if [[ '${{ github.ref }}' == 'refs/heads/experimental' \
              || '${{ github.base_ref }}' == 'experimental' ]]; then
             # Pick Agda version for experimental
-            echo "AGDA_COMMIT=4d36cb37f8bfb765339b808b13356d760aa6f0ec" >> $GITHUB_ENV;
-            echo "AGDA_HTML_DIR=html/experimental" >> $GITHUB_ENV
+            echo "AGDA_COMMIT=4d36cb37f8bfb765339b808b13356d760aa6f0ec" >> "${GITHUB_ENV}";
+            echo "AGDA_HTML_DIR=html/experimental" >> "${GITHUB_ENV}"
           else
             # Pick Agda version for master
-            echo "AGDA_COMMIT=tags/v2.6.4.1" >> $GITHUB_ENV;
-            echo "AGDA_HTML_DIR=html/master" >> $GITHUB_ENV
+            echo "AGDA_COMMIT=tags/v2.6.4.3" >> "${GITHUB_ENV}";
+            echo "AGDA_HTML_DIR=html/master" >> "${GITHUB_ENV}"
           fi
 
           if [[ '${{ github.ref }}' == 'refs/heads/master' \
              || '${{ github.ref }}' == 'refs/heads/experimental' ]]; then
-             echo "AGDA_DEPLOY=true" >> $GITHUB_ENV
+             echo "AGDA_DEPLOY=true" >> "${GITHUB_ENV}"
           fi
 
 ########################################################################
@@ -98,7 +98,7 @@ jobs:
       # i.e. if we change either the version of Agda, ghc, or cabal that we want
       # to use for the build.
       - name: Cache ~/.cabal directories
-        uses: actions/cache@v2
+        uses: actions/cache@v4
         id: cache-cabal
         with:
           path: |
@@ -113,16 +113,14 @@ jobs:
 ########################################################################
 
       - name: Install ghc & cabal
-        uses: haskell/actions/setup@v1
+        uses: haskell-actions/setup@v2
         with:
           ghc-version: ${{ env.GHC_VERSION }}
           cabal-version: ${{ env.CABAL_VERSION }}
+          cabal-update: true
 
       - name: Put cabal programs in PATH
-        run: echo "~/.cabal/bin" >> $GITHUB_PATH
-
-      - name: Cabal update
-        run: cabal update
+        run: echo "~/.cabal/bin" >> "${GITHUB_PATH}"
 
       - name: Install alex & happy
         if: steps.cache-cabal.outputs.cache-hit != 'true'
@@ -147,7 +145,7 @@ jobs:
 
       # By default github actions do not pull the repo
       - name: Checkout stdlib
-        uses: actions/checkout@v2
+        uses: actions/checkout@v4
 
       - name: Test stdlib
         run: |
@@ -191,7 +189,7 @@ jobs:
 
       - name: Deploy HTML
         uses: JamesIves/github-pages-deploy-action@4.1.3
-        if: ${{ success() && env.AGDA_DEPLOY }}
+        if: success() && env.AGDA_DEPLOY
 
         with:
           branch: gh-pages
diff --git a/.github/workflows/haskell-ci.yml b/.github/workflows/haskell-ci.yml
index 5f9fa86b50..391a975ad6 100644
--- a/.github/workflows/haskell-ci.yml
+++ b/.github/workflows/haskell-ci.yml
@@ -8,9 +8,9 @@
 #
 # For more information, see https://github.com/haskell-CI/haskell-ci
 #
-# version: 0.17.20231010
+# version: 0.19.20240514
 #
-# REGENDATA ("0.17.20231010",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
+# REGENDATA ("0.19.20240514",["github","--no-cabal-check","agda-stdlib-utils.cabal"])
 #
 name: Haskell-CI
 on:
@@ -40,24 +40,29 @@ jobs:
     timeout-minutes:
       60
     container:
-      image: buildpack-deps:bionic
+      image: buildpack-deps:jammy
     continue-on-error: ${{ matrix.allow-failure }}
     strategy:
       matrix:
         include:
-          - compiler: ghc-9.8.1
+          - compiler: ghc-9.10.1
             compilerKind: ghc
-            compilerVersion: 9.8.1
+            compilerVersion: 9.10.1
             setup-method: ghcup
             allow-failure: false
-          - compiler: ghc-9.6.3
+          - compiler: ghc-9.8.2
             compilerKind: ghc
-            compilerVersion: 9.6.3
+            compilerVersion: 9.8.2
             setup-method: ghcup
             allow-failure: false
-          - compiler: ghc-9.4.7
+          - compiler: ghc-9.6.5
             compilerKind: ghc
-            compilerVersion: 9.4.7
+            compilerVersion: 9.6.5
+            setup-method: ghcup
+            allow-failure: false
+          - compiler: ghc-9.4.8
+            compilerKind: ghc
+            compilerVersion: 9.4.8
             setup-method: ghcup
             allow-failure: false
           - compiler: ghc-9.2.8
@@ -78,12 +83,12 @@ jobs:
           - compiler: ghc-8.8.4
             compilerKind: ghc
             compilerVersion: 8.8.4
-            setup-method: hvr-ppa
+            setup-method: ghcup
             allow-failure: false
           - compiler: ghc-8.6.5
             compilerKind: ghc
             compilerVersion: 8.6.5
-            setup-method: hvr-ppa
+            setup-method: ghcup
             allow-failure: false
       fail-fast: false
     steps:
@@ -91,21 +96,11 @@ jobs:
         run: |
           apt-get update
           apt-get install -y --no-install-recommends gnupg ca-certificates dirmngr curl git software-properties-common libtinfo5
-          if [ "${{ matrix.setup-method }}" = ghcup ]; then
-            mkdir -p "$HOME/.ghcup/bin"
-            curl -sL https://downloads.haskell.org/ghcup/0.1.19.5/x86_64-linux-ghcup-0.1.19.5 > "$HOME/.ghcup/bin/ghcup"
-            chmod a+x "$HOME/.ghcup/bin/ghcup"
-            "$HOME/.ghcup/bin/ghcup" install ghc "$HCVER" || (cat "$HOME"/.ghcup/logs/*.* && false)
-            "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
-          else
-            apt-add-repository -y 'ppa:hvr/ghc'
-            apt-get update
-            apt-get install -y "$HCNAME"
-            mkdir -p "$HOME/.ghcup/bin"
-            curl -sL https://downloads.haskell.org/ghcup/0.1.19.5/x86_64-linux-ghcup-0.1.19.5 > "$HOME/.ghcup/bin/ghcup"
-            chmod a+x "$HOME/.ghcup/bin/ghcup"
-            "$HOME/.ghcup/bin/ghcup" install cabal 3.10.1.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
-          fi
+          mkdir -p "$HOME/.ghcup/bin"
+          curl -sL https://downloads.haskell.org/ghcup/0.1.20.0/x86_64-linux-ghcup-0.1.20.0 > "$HOME/.ghcup/bin/ghcup"
+          chmod a+x "$HOME/.ghcup/bin/ghcup"
+          "$HOME/.ghcup/bin/ghcup" install ghc "$HCVER" || (cat "$HOME"/.ghcup/logs/*.* && false)
+          "$HOME/.ghcup/bin/ghcup" install cabal 3.10.2.0 || (cat "$HOME"/.ghcup/logs/*.* && false)
         env:
           HCKIND: ${{ matrix.compilerKind }}
           HCNAME: ${{ matrix.compiler }}
@@ -117,22 +112,13 @@ jobs:
           echo "CABAL_DIR=$HOME/.cabal" >> "$GITHUB_ENV"
           echo "CABAL_CONFIG=$HOME/.cabal/config" >> "$GITHUB_ENV"
           HCDIR=/opt/$HCKIND/$HCVER
-          if [ "${{ matrix.setup-method }}" = ghcup ]; then
-            HC=$("$HOME/.ghcup/bin/ghcup" whereis ghc "$HCVER")
-            HCPKG=$(echo "$HC" | sed 's#ghc$#ghc-pkg#')
-            HADDOCK=$(echo "$HC" | sed 's#ghc$#haddock#')
-            echo "HC=$HC" >> "$GITHUB_ENV"
-            echo "HCPKG=$HCPKG" >> "$GITHUB_ENV"
-            echo "HADDOCK=$HADDOCK" >> "$GITHUB_ENV"
-            echo "CABAL=$HOME/.ghcup/bin/cabal-3.10.1.0 -vnormal+nowrap" >> "$GITHUB_ENV"
-          else
-            HC=$HCDIR/bin/$HCKIND
-            echo "HC=$HC" >> "$GITHUB_ENV"
-            echo "HCPKG=$HCDIR/bin/$HCKIND-pkg" >> "$GITHUB_ENV"
-            echo "HADDOCK=$HCDIR/bin/haddock" >> "$GITHUB_ENV"
-            echo "CABAL=$HOME/.ghcup/bin/cabal-3.10.1.0 -vnormal+nowrap" >> "$GITHUB_ENV"
-          fi
-
+          HC=$("$HOME/.ghcup/bin/ghcup" whereis ghc "$HCVER")
+          HCPKG=$(echo "$HC" | sed 's#ghc$#ghc-pkg#')
+          HADDOCK=$(echo "$HC" | sed 's#ghc$#haddock#')
+          echo "HC=$HC" >> "$GITHUB_ENV"
+          echo "HCPKG=$HCPKG" >> "$GITHUB_ENV"
+          echo "HADDOCK=$HADDOCK" >> "$GITHUB_ENV"
+          echo "CABAL=$HOME/.ghcup/bin/cabal-3.10.2.0 -vnormal+nowrap" >> "$GITHUB_ENV"
           HCNUMVER=$(${HC} --numeric-version|perl -ne '/^(\d+)\.(\d+)\.(\d+)(\.(\d+))?$/; print(10000 * $1 + 100 * $2 + ($3 == 0 ? $5 != 1 : $3))')
           echo "HCNUMVER=$HCNUMVER" >> "$GITHUB_ENV"
           echo "ARG_TESTS=--enable-tests" >> "$GITHUB_ENV"
@@ -189,7 +175,7 @@ jobs:
           chmod a+x $HOME/.cabal/bin/cabal-plan
           cabal-plan --version
       - name: checkout
-        uses: actions/checkout@v3
+        uses: actions/checkout@v4
         with:
           path: source
       - name: initial cabal.project for sdist
@@ -217,7 +203,7 @@ jobs:
           echo "    ghc-options: -Werror=missing-methods" >> cabal.project
           cat >> cabal.project <<EOF
           EOF
-          $HCPKG list --simple-output --names-only | perl -ne 'for (split /\s+/) { print "constraints: $_ installed\n" unless /^(agda-stdlib-utils)$/; }' >> cabal.project.local
+          $HCPKG list --simple-output --names-only | perl -ne 'for (split /\s+/) { print "constraints: any.$_ installed\n" unless /^(agda-stdlib-utils)$/; }' >> cabal.project.local
           cat cabal.project
           cat cabal.project.local
       - name: dump install plan
@@ -225,7 +211,7 @@ jobs:
           $CABAL v2-build $ARG_COMPILER $ARG_TESTS $ARG_BENCH --dry-run all
           cabal-plan
       - name: restore cache
-        uses: actions/cache/restore@v3
+        uses: actions/cache/restore@v4
         with:
           key: ${{ runner.os }}-${{ matrix.compiler }}-${{ github.sha }}
           path: ~/.cabal/store
@@ -248,7 +234,7 @@ jobs:
           rm -f cabal.project.local
           $CABAL v2-build $ARG_COMPILER --disable-tests --disable-benchmarks all
       - name: save cache
-        uses: actions/cache/save@v3
+        uses: actions/cache/save@v4
         if: always()
         with:
           key: ${{ runner.os }}-${{ matrix.compiler }}-${{ github.sha }}
diff --git a/agda-stdlib-utils.cabal b/agda-stdlib-utils.cabal
index 3accc6972e..bbd7e4f118 100644
--- a/agda-stdlib-utils.cabal
+++ b/agda-stdlib-utils.cabal
@@ -6,9 +6,10 @@ description:     Helper programs for setting up the Agda standard library.
 license:         MIT
 
 tested-with:
-  GHC == 9.8.1
-  GHC == 9.6.3
-  GHC == 9.4.7
+  GHC == 9.10.1
+  GHC == 9.8.2
+  GHC == 9.6.5
+  GHC == 9.4.8
   GHC == 9.2.8
   GHC == 9.0.2
   GHC == 8.10.7
@@ -23,7 +24,7 @@ common common-build-parameters
     PatternSynonyms
 
   build-depends:
-      base          >= 4.12.0.0 && < 4.20
+      base          >= 4.12.0.0 && < 4.21
     , filemanip     >= 0.3.6.2  && < 0.4
 
 executable GenerateEverything
@@ -33,7 +34,7 @@ executable GenerateEverything
 
   build-depends:
       directory     >= 1.0.0.0  && < 1.4
-    , filepath      >= 1.4.1.0  && < 1.5
+    , filepath      >= 1.4.1.0  && < 1.6
     , mtl           >= 2.2.2    && < 2.4
 
 executable AllNonAsciiChars

From c8a11e7b743ec28e8e7c28208b999d352e900613 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 30 May 2024 13:35:31 +0100
Subject: [PATCH 290/316] Refactor `Data.List.Base.scan*` and their properties
 (#2395)

* refactor `scanr` etc.

* restore `inits` etc. but lazier; plus knock-on consequences

* more refactoring; plus knock-on consequences

* tidy up

* refactored into `Base`

* ... and `Properties`

* fix-up `inits` and `tails`

* fix up `import`s

* knock-ons

* Andreas's suggestions

* further, better, refactoring is possible

* yet further, better, refactoring is possible: remove explicit equational reasoning!

* Update CHANGELOG.md

Fix deprecated names

* Update Base.agda

Fix deprecations

* Update Properties.agda

Fix deprecations

* Update CHANGELOG.md

Fix deprecated names
---
 CHANGELOG.md                                  | 42 +++++++++++++
 src/Codata/Sized/Colist/Properties.agda       |  3 +-
 src/Data/List.agda                            |  4 ++
 src/Data/List/Base.agda                       | 46 +++++++++-----
 .../Membership/Propositional/Properties.agda  |  3 +-
 src/Data/List/NonEmpty/Base.agda              | 10 +++
 src/Data/List/NonEmpty/Properties.agda        | 14 ++++-
 src/Data/List/Properties.agda                 | 61 ++++++++++---------
 src/Data/List/Scans/Base.agda                 | 49 +++++++++++++++
 src/Data/List/Scans/Properties.agda           | 59 ++++++++++++++++++
 src/Tactic/RingSolver/Core/NatSet.agda        |  3 +-
 11 files changed, 245 insertions(+), 49 deletions(-)
 create mode 100644 src/Data/List/Scans/Base.agda
 create mode 100644 src/Data/List/Scans/Properties.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 29839632c7..88dbe3c075 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -65,6 +65,30 @@ Deprecated names
   isRing*  ↦  Algebra.Structures.isRing
   ```
 
+* In `Data.List.Base`:
+  ```agda
+  scanr  ↦  Data.List.Scans.Base.scanr
+  scanl  ↦  Data.List.Scans.Base.scanl
+  ```
+
+* In `Data.List.Properties`:
+  ```agda
+  scanr-defn  ↦  Data.List.Scans.Properties.scanr-defn
+  scanl-defn  ↦  Data.List.Scans.Properties.scanl-defn
+  ```
+
+* In `Data.List.NonEmpty.Base`:
+  ```agda
+  inits : List A → List⁺ (List A)
+  tails : List A → List⁺ (List A)
+  ```
+
+* In `Data.List.NonEmpty.Properties`:
+  ```agda
+  toList-inits : toList ∘ List⁺.inits ≗ List.inits
+  toList-tails : toList ∘ List⁺.tails ≗ List.tails
+  ```
+
 * In `Data.Nat.Divisibility.Core`:
   ```agda
   *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
@@ -89,6 +113,12 @@ New modules
   Algebra.Module.Bundles.Raw
   ```
 
+* Refactoring of `Data.List.Base.{scanr|scanl}` and their properties:
+  ```
+  Data.List.Scans.Base
+  Data.List.Scans.Properties
+  ```
+
 * Prime factorisation of natural numbers.
   ```
   Data.Nat.Primality.Factorisation
@@ -342,12 +372,24 @@ Additions to existing modules
   i*j≢0     : .{{_ : NonZero i}} .{{_ : NonZero j}} → NonZero (i * j)
   ```
 
+* In `Data.List.Base` redefine `inits` and `tails` in terms of:
+  ```agda
+  tail∘inits : List A → List (List A)
+  tail∘tails : List A → List (List A)
+  ```
+
 * In `Data.List.Membership.Setoid.Properties`:
   ```agda
   reverse⁺ : x ∈ xs → x ∈ reverse xs
   reverse⁻ : x ∈ reverse xs → x ∈ xs
   ```
 
+* In `Data.List.NonEmpty.Base`:
+  ```agda
+  inits : List A → List⁺ (List A)
+  tails : List A → List⁺ (List A)
+  ```
+
 * In `Data.List.Properties`:
   ```agda
   length-catMaybes      : length (catMaybes xs) ≤ length xs
diff --git a/src/Codata/Sized/Colist/Properties.agda b/src/Codata/Sized/Colist/Properties.agda
index c03b3c2096..3a9958b715 100644
--- a/src/Codata/Sized/Colist/Properties.agda
+++ b/src/Codata/Sized/Colist/Properties.agda
@@ -22,6 +22,7 @@ open import Codata.Sized.Stream as Stream using (Stream; _∷_)
 open import Data.Vec.Bounded as Vec≤ using (Vec≤)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.NonEmpty as List⁺ using (List⁺; _∷_)
+import Data.List.Scans.Base as Scans
 open import Data.List.Relation.Binary.Equality.Propositional using (≋-refl)
 open import Data.Maybe.Base as Maybe using (Maybe; nothing; just)
 import Data.Maybe.Properties as Maybe
@@ -283,7 +284,7 @@ fromList-++ []       bs = refl
 fromList-++ (a ∷ as) bs = ≡.refl ∷ λ where .force → fromList-++ as bs
 
 fromList-scanl : ∀ (c : B → A → B) n as →
-                 i ⊢ fromList (List.scanl c n as) ≈ scanl c n (fromList as)
+                 i ⊢ fromList (Scans.scanl c n as) ≈ scanl c n (fromList as)
 fromList-scanl c n []       = ≡.refl ∷ λ where .force → refl
 fromList-scanl c n (a ∷ as) =
   ≡.refl ∷ λ where .force → fromList-scanl c (c n a) as
diff --git a/src/Data/List.agda b/src/Data/List.agda
index 7747cbbd74..70e8519440 100644
--- a/src/Data/List.agda
+++ b/src/Data/List.agda
@@ -8,6 +8,7 @@
 -- lists.
 
 {-# OPTIONS --cubical-compatible --safe #-}
+{-# OPTIONS --warn=noUserWarning #-} -- for deprecated scans
 
 module Data.List where
 
@@ -15,3 +16,6 @@ module Data.List where
 -- Types and basic operations
 
 open import Data.List.Base public
+  hiding (scanr; scanl)
+open import Data.List.Scans.Base public
+  using (scanr; scanl)
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index c3c894cf65..9ea8b551b9 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -189,13 +189,19 @@ iterate : (A → A) → A → ℕ → List A
 iterate f e zero    = []
 iterate f e (suc n) = e ∷ iterate f (f e) n
 
+tail∘inits : List A → List (List A)
+tail∘inits []       = []
+tail∘inits (x ∷ xs) = [ x ] ∷ map (x ∷_) (tail∘inits xs)
+
 inits : List A → List (List A)
-inits []       = [] ∷ []
-inits (x ∷ xs) = [] ∷ map (x ∷_) (inits xs)
+inits xs = [] ∷ tail∘inits xs
+
+tail∘tails : List A → List (List A)
+tail∘tails []       = []
+tail∘tails (_ ∷ xs) = xs ∷ tail∘tails xs
 
 tails : List A → List (List A)
-tails []       = [] ∷ []
-tails (x ∷ xs) = (x ∷ xs) ∷ tails xs
+tails xs = xs ∷ tail∘tails xs
 
 insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
 insertAt xs       zero    v = v ∷ xs
@@ -205,18 +211,6 @@ updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
 updateAt (x ∷ xs) zero    f = f x ∷ xs
 updateAt (x ∷ xs) (suc i) f = x ∷ updateAt xs i f
 
--- Scans
-
-scanr : (A → B → B) → B → List A → List B
-scanr f e []       = e ∷ []
-scanr f e (x ∷ xs) with scanr f e xs
-... | []     = []                -- dead branch
-... | y ∷ ys = f x y ∷ y ∷ ys
-
-scanl : (A → B → A) → A → List B → List A
-scanl f e []       = e ∷ []
-scanl f e (x ∷ xs) = e ∷ scanl f (f e x) xs
-
 -- Tabulation
 
 applyUpTo : (ℕ → A) → ℕ → List A
@@ -571,3 +565,23 @@ _─_ = removeAt
 "Warning: _─_ was deprecated in v2.0.
 Please use removeAt instead."
 #-}
+
+-- Version 2.1
+
+scanr : (A → B → B) → B → List A → List B
+scanr f e []       = e ∷ []
+scanr f e (x ∷ xs) with scanr f e xs
+... | []         = []                -- dead branch
+... | ys@(y ∷ _) = f x y ∷ ys
+{-# WARNING_ON_USAGE scanr
+"Warning: scanr was deprecated in v2.1.
+Please use Data.List.Scans.Base.scanr instead."
+#-}
+
+scanl : (A → B → A) → A → List B → List A
+scanl f e []       = e ∷ []
+scanl f e (x ∷ xs) = e ∷ scanl f (f e x) xs
+{-# WARNING_ON_USAGE scanl
+"Warning: scanl was deprecated in v2.1.
+Please use Data.List.Scans.Base.scanl instead."
+#-}
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index 47ee739817..f93bf81f4a 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -338,8 +338,7 @@ module _ {_•_ : Op₂ A} where
 -- inits
 
 []∈inits : ∀ {a} {A : Set a} (as : List A) → [] ∈ inits as
-[]∈inits []       = here refl
-[]∈inits (a ∷ as) = here refl
+[]∈inits _ = here refl
 
 ------------------------------------------------------------------------
 -- Other properties
diff --git a/src/Data/List/NonEmpty/Base.agda b/src/Data/List/NonEmpty/Base.agda
index 0b1793eaf7..f332dcc162 100644
--- a/src/Data/List/NonEmpty/Base.agda
+++ b/src/Data/List/NonEmpty/Base.agda
@@ -143,6 +143,16 @@ concatMap f = concat ∘′ map f
 ap : List⁺ (A → B) → List⁺ A → List⁺ B
 ap fs as = concatMap (λ f → map f as) fs
 
+-- Inits
+
+inits : List A → List⁺ (List A)
+inits xs = [] ∷ List.tail∘inits xs
+
+-- Tails
+
+tails : List A → List⁺ (List A)
+tails xs = xs ∷ List.tail∘tails xs
+
 -- Reverse
 
 reverse : List⁺ A → List⁺ A
diff --git a/src/Data/List/NonEmpty/Properties.agda b/src/Data/List/NonEmpty/Properties.agda
index c20f024c4a..9469cd2b87 100644
--- a/src/Data/List/NonEmpty/Properties.agda
+++ b/src/Data/List/NonEmpty/Properties.agda
@@ -18,7 +18,7 @@ open import Data.List.Effectful using () renaming (monad to listMonad)
 open import Data.List.NonEmpty.Effectful using () renaming (monad to list⁺Monad)
 open import Data.List.NonEmpty
   using (List⁺; _∷_; tail; head; toList; _⁺++_; _⁺++⁺_; _++⁺_; length; fromList;
-    drop+; map; groupSeqs; ungroupSeqs)
+    drop+; map; inits; tails; groupSeqs; ungroupSeqs)
 open import Data.List.NonEmpty.Relation.Unary.All using (All; toList⁺; _∷_)
 open import Data.List.Relation.Unary.All using ([]; _∷_) renaming (All to ListAll)
 import Data.List.Properties as List
@@ -118,6 +118,18 @@ map-cong f≗g (x ∷ xs) = cong₂ _∷_ (f≗g x) (List.map-cong f≗g xs)
 map-∘ : {g : B → C} {f : A → B} → map (g ∘ f) ≗ map g ∘ map f
 map-∘ (x ∷ xs) = cong (_ ∷_) (List.map-∘ xs)
 
+------------------------------------------------------------------------
+-- inits
+
+toList-inits : toList ∘ inits ≗ List.inits {A = A}
+toList-inits _ = refl
+
+------------------------------------------------------------------------
+-- tails
+
+toList-tails : toList ∘ tails ≗ List.tails {A = A}
+toList-tails _ = refl
+
 ------------------------------------------------------------------------
 -- groupSeqs
 
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index ab9b0241e1..e280be132b 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -8,6 +8,7 @@
 -- equalities than _≡_.
 
 {-# OPTIONS --cubical-compatible --safe #-}
+{-# OPTIONS --warn=noUserWarning #-} -- for deprecated scans
 
 module Data.List.Properties where
 
@@ -809,34 +810,6 @@ sum-++ (x ∷ xs) ys = begin
 ∈⇒∣product {n} {n ∷ ns} (here  refl) = divides (product ns) (*-comm n (product ns))
 ∈⇒∣product {n} {m ∷ ns} (there n∈ns) = ∣n⇒∣m*n m (∈⇒∣product n∈ns)
 
-------------------------------------------------------------------------
--- scanr
-
-scanr-defn : ∀ (f : A → B → B) (e : B) →
-             scanr f e ≗ map (foldr f e) ∘ tails
-scanr-defn f e []             = refl
-scanr-defn f e (x ∷ [])       = refl
-scanr-defn f e (x ∷ y∷xs@(_ ∷ _))
-  with eq ← scanr-defn f e y∷xs
-  with z ∷ zs ← scanr f e y∷xs
-  = let z≡fy⦇f⦈xs , _ = ∷-injective eq in cong₂ (λ z → f x z ∷_) z≡fy⦇f⦈xs eq
-
-------------------------------------------------------------------------
--- scanl
-
-scanl-defn : ∀ (f : A → B → A) (e : A) →
-             scanl f e ≗ map (foldl f e) ∘ inits
-scanl-defn f e []       = refl
-scanl-defn f e (x ∷ xs) = cong (e ∷_) (begin
-   scanl f (f e x) xs
- ≡⟨ scanl-defn f (f e x) xs ⟩
-   map (foldl f (f e x)) (inits xs)
- ≡⟨ refl ⟩
-   map (foldl f e ∘ (x ∷_)) (inits xs)
- ≡⟨ map-∘ (inits xs) ⟩
-   map (foldl f e) (map (x ∷_) (inits xs))
- ∎)
-
 ------------------------------------------------------------------------
 -- applyUpTo
 
@@ -1582,3 +1555,35 @@ map-─ = map-removeAt
 "Warning: map-─ was deprecated in v2.0.
 Please use map-removeAt instead."
 #-}
+
+-- Version 2.1
+
+scanr-defn : ∀ (f : A → B → B) (e : B) →
+             scanr f e ≗ map (foldr f e) ∘ tails
+scanr-defn f e []             = refl
+scanr-defn f e (x ∷ [])       = refl
+scanr-defn f e (x ∷ xs@(_ ∷ _))
+  with eq ← scanr-defn f e xs
+  with ys@(_ ∷ _) ← scanr f e xs
+  = cong₂ (λ z → f x z ∷_) (∷-injectiveˡ eq) eq
+{-# WARNING_ON_USAGE scanr-defn
+"Warning: scanr-defn was deprecated in v2.1.
+Please use Data.List.Scans.Properties.scanr-defn instead."
+#-}
+
+scanl-defn : ∀ (f : A → B → A) (e : A) →
+             scanl f e ≗ map (foldl f e) ∘ inits
+scanl-defn f e []       = refl
+scanl-defn f e (x ∷ xs) = cong (e ∷_) (begin
+   scanl f (f e x) xs
+ ≡⟨ scanl-defn f (f e x) xs ⟩
+   map (foldl f (f e x)) (inits xs)
+ ≡⟨ refl ⟩
+   map (foldl f e ∘ (x ∷_)) (inits xs)
+ ≡⟨ map-∘ (inits xs) ⟩
+   map (foldl f e) (map (x ∷_) (inits xs))
+ ∎)
+{-# WARNING_ON_USAGE scanl-defn
+"Warning: scanl-defn was deprecated in v2.1.
+Please use Data.List.Scans.Properties.scanl-defn instead."
+#-}
diff --git a/src/Data/List/Scans/Base.agda b/src/Data/List/Scans/Base.agda
new file mode 100644
index 0000000000..73e7d794ad
--- /dev/null
+++ b/src/Data/List/Scans/Base.agda
@@ -0,0 +1,49 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- List scans: definitions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.Scans.Base where
+
+open import Data.List.Base as List using (List; []; _∷_)
+open import Data.List.NonEmpty.Base as List⁺ using (List⁺; _∷_; toList)
+open import Function.Base using (_∘_)
+open import Level using (Level)
+
+private
+  variable
+    a b : Level
+    A : Set a
+    B : Set b
+
+
+------------------------------------------------------------------------
+-- Definitions
+
+-- Scanr
+
+module _ (f : A → B → B) where
+
+  scanr⁺ : (e : B) → List A → List⁺ B
+  scanr⁺ e []       = e ∷ []
+  scanr⁺ e (x ∷ xs) = let y ∷ ys = scanr⁺ e xs in f x y ∷ y ∷ ys
+
+  scanr : (e : B) → List A → List B
+  scanr e = toList ∘ scanr⁺ e
+
+-- Scanl
+
+module _ (f : A → B → A) where
+
+  scanl⁺ : A → List B → List⁺ A
+  scanl⁺ e xs = e ∷ go e xs
+    where
+    go : A → List B → List A
+    go _ []       = []
+    go e (x ∷ xs) = let fex = f e x in fex ∷ go fex xs
+
+  scanl : A → List B → List A
+  scanl e = toList ∘ scanl⁺ e
diff --git a/src/Data/List/Scans/Properties.agda b/src/Data/List/Scans/Properties.agda
new file mode 100644
index 0000000000..01b2229684
--- /dev/null
+++ b/src/Data/List/Scans/Properties.agda
@@ -0,0 +1,59 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- List scans: properties
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.Scans.Properties where
+
+open import Data.List.Base as List using (List; []; _∷_)
+open import Data.List.NonEmpty.Base as List⁺ using (List⁺; _∷_; toList)
+import Data.List.Properties as List
+import Data.List.NonEmpty.Properties as List⁺
+open import Data.List.Scans.Base
+open import Function.Base using (_∘_; _$_)
+open import Level using (Level)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≗_; refl; trans; cong; cong₂)
+
+private
+  variable
+    a b : Level
+    A : Set a
+    B : Set b
+
+
+------------------------------------------------------------------------
+-- Properties
+
+-- scanr⁺ and scanr
+
+module _ (f : A → B → B) (e : B) where
+
+  private
+    h = List.foldr f e
+
+  scanr⁺-defn : scanr⁺ f e ≗ List⁺.map h ∘ List⁺.tails
+  scanr⁺-defn []       = refl
+  scanr⁺-defn (x ∷ xs) = let eq = scanr⁺-defn xs
+    in cong₂ (λ z → f x z ∷_) (cong List⁺.head eq) (cong toList eq)
+
+  scanr-defn : scanr f e ≗ List.map h ∘ List.tails
+  scanr-defn xs = cong toList (scanr⁺-defn xs)
+
+-- scanl⁺ and scanl
+
+module _ (f : A → B → A) where
+
+  private
+    h = List.foldl f
+
+  scanl⁺-defn : ∀ e → scanl⁺ f e ≗ List⁺.map (h e) ∘ List⁺.inits
+  scanl⁺-defn e []       = refl
+  scanl⁺-defn e (x ∷ xs) = let eq = scanl⁺-defn (f e x) xs in
+    cong (e ∷_) $ cong (f e x ∷_) $ trans (cong List⁺.tail eq) (List.map-∘ _)
+
+  scanl-defn : ∀ e → scanl f e ≗ List.map (h e) ∘ List.inits
+  scanl-defn e xs = cong toList (scanl⁺-defn e xs)
diff --git a/src/Tactic/RingSolver/Core/NatSet.agda b/src/Tactic/RingSolver/Core/NatSet.agda
index 1b27fae60a..53af68c511 100644
--- a/src/Tactic/RingSolver/Core/NatSet.agda
+++ b/src/Tactic/RingSolver/Core/NatSet.agda
@@ -38,6 +38,7 @@ module Tactic.RingSolver.Core.NatSet where
 
 open import Data.Nat.Base   as ℕ     using (ℕ; suc; zero)
 open import Data.List.Base  as List  using (List; _∷_; [])
+open import Data.List.Scans.Base as Scans using (scanl)
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Bool.Base  as Bool  using (Bool)
 open import Function.Base            using (const; _∘_)
@@ -98,7 +99,7 @@ fromList : List ℕ → NatSet
 fromList = List.foldr insert []
 
 toList : NatSet → List ℕ
-toList = List.drop 1 ∘ List.map ℕ.pred ∘ List.scanl (λ x y → suc (y ℕ.+ x)) 0
+toList = List.drop 1 ∘ List.map ℕ.pred ∘ Scans.scanl (λ x y → suc (y ℕ.+ x)) 0
 
 ------------------------------------------------------------------------
 -- Tests

From aef9afc295233ffd386ad4ca44bef45bd7d4a693 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 30 May 2024 13:45:04 +0100
Subject: [PATCH 291/316] fixes #2390 (#2392)

---
 doc/style-guide.md | 28 ++++++++++++++++++++++++++++
 1 file changed, 28 insertions(+)

diff --git a/doc/style-guide.md b/doc/style-guide.md
index 2562f76cfd..c4b9af1bf8 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -229,6 +229,34 @@ automate most of this.
     }
   ```
 
+#### Layout of initial `private` block
+
+* Since the introduction of generalizable `variable`s (see below),
+  this block provides a very useful way to 'fix'/standardise notation
+  for the rest of the module, as well as introducing local
+  instantiations of parameterised `module` definitions, again for the
+  sake of fixing notation via qualified names.
+
+* It should typically follow the `import` and `open` declarations, as
+  above, separated by one blankline, and be followed by *two* blank
+  lines ahead of the main module body.
+
+* The current preferred layout is to use successive indentation by two spaces, eg.
+  ```agda
+  private
+    variable
+      a : Level
+      A : Set a
+  ```
+  rather than to use the more permissive 'stacked' style, available
+  since [agda/agda#5319](https://github.com/agda/agda/pull/5319).
+
+* A possible exception to the above rule is when a *single* declaration
+  is made, such as eg.
+  ```agda
+  private open module M = ...
+  ```
+
 #### Layout of `where` blocks
 
 * `where` blocks are preferred rather than the `let` construction.

From d992900179fb4c96787c0904dedc36610a19e32b Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Thu, 30 May 2024 15:39:18 +0100
Subject: [PATCH 292/316] Add the `Setoid`-based `Monoid` on `(List, [], _++_)`
 (#2393)

* refactored from #2350

* enriched the `module` contents in response to review comments
---
 CHANGELOG.md                                  |  5 ++
 .../Binary/Equality/Setoid/Properties.agda    | 59 +++++++++++++++++++
 2 files changed, 64 insertions(+)
 create mode 100644 src/Data/List/Relation/Binary/Equality/Setoid/Properties.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 88dbe3c075..567782c261 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -113,6 +113,11 @@ New modules
   Algebra.Module.Bundles.Raw
   ```
 
+* Properties of `List` modulo `Setoid` equality (currently only the ([],++) monoid):
+  ```
+  Data.List.Relation.Binary.Equality.Setoid.Properties
+  ```
+
 * Refactoring of `Data.List.Base.{scanr|scanl}` and their properties:
   ```
   Data.List.Scans.Base
diff --git a/src/Data/List/Relation/Binary/Equality/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Equality/Setoid/Properties.agda
new file mode 100644
index 0000000000..fea2638f66
--- /dev/null
+++ b/src/Data/List/Relation/Binary/Equality/Setoid/Properties.agda
@@ -0,0 +1,59 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties of List modulo ≋
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Bundles using (Setoid)
+
+module Data.List.Relation.Binary.Equality.Setoid.Properties
+  {c ℓ} (S : Setoid c ℓ)
+  where
+
+open import Algebra.Bundles using (Magma; Semigroup; Monoid)
+import Algebra.Structures as Structures
+open import Data.List.Base using (List; []; _++_)
+import Data.List.Properties as List
+import Data.List.Relation.Binary.Equality.Setoid as ≋
+open import Data.Product.Base using (_,_)
+open import Function.Base using (_∘_)
+open import Level using (_⊔_)
+
+open ≋ S using (_≋_; ≋-refl; ≋-reflexive; ≋-isEquivalence; ++⁺)
+open Structures _≋_ using (IsMagma; IsSemigroup; IsMonoid)
+
+------------------------------------------------------------------------
+-- The []-++-Monoid
+
+-- Structures
+
+isMagma : IsMagma _++_
+isMagma = record
+  { isEquivalence = ≋-isEquivalence
+  ; ∙-cong = ++⁺
+  }
+
+isSemigroup : IsSemigroup _++_
+isSemigroup = record
+  { isMagma = isMagma
+  ; assoc = λ xs ys zs → ≋-reflexive (List.++-assoc xs ys zs)
+  }
+
+isMonoid : IsMonoid _++_ []
+isMonoid = record
+  { isSemigroup = isSemigroup
+  ; identity = (λ _ → ≋-refl) , ≋-reflexive ∘ List.++-identityʳ
+  }
+
+-- Bundles
+
+magma : Magma c (c ⊔ ℓ)
+magma = record { isMagma = isMagma }
+
+semigroup : Semigroup c (c ⊔ ℓ)
+semigroup = record { isSemigroup = isSemigroup }
+
+monoid : Monoid c (c ⊔ ℓ)
+monoid = record { isMonoid = isMonoid }

From 10fd6b95d633dbbe2937a5c97af32c0493e45115 Mon Sep 17 00:00:00 2001
From: Orestis Melkonian <melkon.or@gmail.com>
Date: Sat, 1 Jun 2024 16:10:32 +0100
Subject: [PATCH 293/316] Lists: decidability of Subset+Disjoint relations
 (#2399)

---
 CHANGELOG.md                                  | 12 +++++++
 src/Data/List/Membership/DecSetoid.agda       |  5 +--
 .../Binary/Disjoint/DecPropositional.agda     | 21 ++++++++++++
 .../Relation/Binary/Disjoint/DecSetoid.agda   | 29 ++++++++++++++++
 .../Binary/Subset/DecPropositional.agda       | 21 ++++++++++++
 .../Relation/Binary/Subset/DecSetoid.agda     | 33 +++++++++++++++++++
 6 files changed, 119 insertions(+), 2 deletions(-)
 create mode 100644 src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda
 create mode 100644 src/Data/List/Relation/Binary/Disjoint/DecSetoid.agda
 create mode 100644 src/Data/List/Relation/Binary/Subset/DecPropositional.agda
 create mode 100644 src/Data/List/Relation/Binary/Subset/DecSetoid.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 567782c261..fabdf73d9d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -211,6 +211,18 @@ New modules
   Data.Vec.Bounded.Show
   ```
 
+* Decidability for the subset relation on lists:
+  ```agda
+  Data.List.Relation.Binary.Subset.DecSetoid (_⊆?_)
+  Data.List.Relation.Binary.Subset.DecPropositional
+  ```
+
+* Decidability for the disjoint relation on lists:
+  ```agda
+  Data.List.Relation.Binary.Disjoint.DecSetoid (disjoint?)
+  Data.List.Relation.Binary.Disjoint.DecPropositional
+  ```
+
 Additions to existing modules
 -----------------------------
 
diff --git a/src/Data/List/Membership/DecSetoid.agda b/src/Data/List/Membership/DecSetoid.agda
index 8f34aab9f6..29fe88adb6 100644
--- a/src/Data/List/Membership/DecSetoid.agda
+++ b/src/Data/List/Membership/DecSetoid.agda
@@ -6,13 +6,14 @@
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
-open import Relation.Binary.Definitions using (Decidable)
 open import Relation.Binary.Bundles using (DecSetoid)
-open import Relation.Nullary.Decidable using (¬?)
 
 module Data.List.Membership.DecSetoid {a ℓ} (DS : DecSetoid a ℓ) where
 
 open import Data.List.Relation.Unary.Any using (any?)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Nullary.Decidable using (¬?)
+
 open DecSetoid DS
 
 ------------------------------------------------------------------------
diff --git a/src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda b/src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda
new file mode 100644
index 0000000000..82d39c261f
--- /dev/null
+++ b/src/Data/List/Relation/Binary/Disjoint/DecPropositional.agda
@@ -0,0 +1,21 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Decidability of the disjoint relation over propositional equality.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Definitions using (DecidableEquality)
+
+module Data.List.Relation.Binary.Disjoint.DecPropositional
+  {a} {A : Set a} (_≟_ : DecidableEquality A)
+  where
+
+------------------------------------------------------------------------
+-- Re-export core definitions and operations
+
+open import Data.List.Relation.Binary.Disjoint.Propositional {A = A} public
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
+open import Data.List.Relation.Binary.Disjoint.DecSetoid (decSetoid _≟_) public
+  using (disjoint?)
diff --git a/src/Data/List/Relation/Binary/Disjoint/DecSetoid.agda b/src/Data/List/Relation/Binary/Disjoint/DecSetoid.agda
new file mode 100644
index 0000000000..68b0f2b0a6
--- /dev/null
+++ b/src/Data/List/Relation/Binary/Disjoint/DecSetoid.agda
@@ -0,0 +1,29 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Decidability of the disjoint relation over setoid equality.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Bundles using (DecSetoid)
+
+module Data.List.Relation.Binary.Disjoint.DecSetoid {c ℓ} (S : DecSetoid c ℓ) where
+
+open import Data.Product.Base using (_,_)
+open import Data.List.Relation.Unary.Any using (map)
+open import Data.List.Relation.Unary.All using (all?; lookupₛ)
+open import Data.List.Relation.Unary.All.Properties using (¬All⇒Any¬)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Nullary using (yes; no; decidable-stable)
+open DecSetoid S
+open import Data.List.Relation.Binary.Equality.DecSetoid S
+open import Data.List.Relation.Binary.Disjoint.Setoid setoid public
+open import Data.List.Membership.DecSetoid S
+
+disjoint? : Decidable Disjoint
+disjoint? xs ys with all? (_∉? ys) xs
+... | yes xs♯ys = yes λ (v∈ , v∈′) →
+  lookupₛ setoid (λ x≈y ∉ys ∈ys → ∉ys (map (trans x≈y) ∈ys)) xs♯ys v∈ v∈′
+... | no ¬xs♯ys = let (x , x∈ , ¬∉ys) = find (¬All⇒Any¬ (_∉? _) _ ¬xs♯ys) in
+  no λ p → p (x∈ , decidable-stable (_ ∈? _) ¬∉ys)
diff --git a/src/Data/List/Relation/Binary/Subset/DecPropositional.agda b/src/Data/List/Relation/Binary/Subset/DecPropositional.agda
new file mode 100644
index 0000000000..a183d298f3
--- /dev/null
+++ b/src/Data/List/Relation/Binary/Subset/DecPropositional.agda
@@ -0,0 +1,21 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Decidability of the subset relation over propositional equality.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Definitions using (DecidableEquality)
+
+module Data.List.Relation.Binary.Subset.DecPropositional
+  {a} {A : Set a} (_≟_ : DecidableEquality A)
+  where
+
+------------------------------------------------------------------------
+-- Re-export core definitions and operations
+
+open import Data.List.Relation.Binary.Subset.Propositional {A = A} public
+open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
+open import Data.List.Relation.Binary.Subset.DecSetoid (decSetoid _≟_) public
+  using (_⊆?_)
diff --git a/src/Data/List/Relation/Binary/Subset/DecSetoid.agda b/src/Data/List/Relation/Binary/Subset/DecSetoid.agda
new file mode 100644
index 0000000000..fcff0abacc
--- /dev/null
+++ b/src/Data/List/Relation/Binary/Subset/DecSetoid.agda
@@ -0,0 +1,33 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Decidability of the subset relation over setoid equality.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Bundles using (DecSetoid)
+
+module Data.List.Relation.Binary.Subset.DecSetoid {c ℓ} (S : DecSetoid c ℓ) where
+
+open import Function.Base using (_∘_)
+open import Data.List.Base using ([]; _∷_)
+open import Data.List.Relation.Unary.Any using (here; there; map)
+open import Relation.Binary.Definitions using (Decidable)
+open import Relation.Nullary using (yes; no)
+open DecSetoid S
+open import Data.List.Relation.Binary.Equality.DecSetoid S
+open import Data.List.Membership.DecSetoid S
+
+-- Re-export definitions
+open import Data.List.Relation.Binary.Subset.Setoid setoid public
+
+infix 4 _⊆?_
+_⊆?_ : Decidable _⊆_
+[]       ⊆? _   = yes λ ()
+(x ∷ xs) ⊆? ys  with x ∈? ys
+... | no  x∉ys  = no λ xs⊆ys → x∉ys (xs⊆ys (here refl))
+... | yes x∈ys  with xs ⊆? ys
+...   | no  xs⊈ys = no λ xs⊆ys → xs⊈ys (xs⊆ys ∘ there)
+...   | yes xs⊆ys = yes λ where (here refl) → map (trans refl) x∈ys
+                                (there x∈)  → xs⊆ys x∈

From 30ef4718c7159e3910918682cf7911178f447228 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Tue, 4 Jun 2024 16:34:45 +0100
Subject: [PATCH 294/316] fixes #906 (#2401)

---
 CHANGELOG.md                                          | 11 ++++++++---
 .../Relation/Binary/Pointwise/NonDependent.agda       | 10 +++++-----
 .../Relation/Binary/Pointwise/Properties.agda         |  6 ++----
 3 files changed, 15 insertions(+), 12 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index fabdf73d9d..3db3923def 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -36,9 +36,14 @@ Other major improvements
 
 Minor improvements
 ------------------
-The size of the dependency graph for many modules has been
-reduced. This may lead to speed ups for first-time loading of some
-modules.
+
+* The size of the dependency graph for many modules has been
+  reduced. This may lead to speed ups for first-time loading of some
+  modules.
+
+* The definition of the `Pointwise` relational combinator in
+  `Data.Product.Relation.Binary.Pointwise.NonDependent.Pointwise`
+  has been generalised to take heterogeneous arguments in `REL`.
 
 Deprecated modules
 ------------------
diff --git a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
index 1cba24520a..5e1c5c5c80 100644
--- a/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
+++ b/src/Data/Product/Relation/Binary/Pointwise/NonDependent.agda
@@ -11,10 +11,10 @@ module Data.Product.Relation.Binary.Pointwise.NonDependent where
 open import Data.Product.Base as Product
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Level using (Level; _⊔_; 0ℓ)
-open import Function.Base using (_on_; id)
+open import Function.Base using (id)
 open import Function.Bundles using (Inverse)
 open import Relation.Nullary.Decidable using (_×-dec_)
-open import Relation.Binary.Core using (Rel; _⇒_)
+open import Relation.Binary.Core using (REL; Rel; _⇒_)
 open import Relation.Binary.Bundles
   using (Setoid; DecSetoid; Preorder; Poset; StrictPartialOrder)
 open import Relation.Binary.Definitions
@@ -25,14 +25,14 @@ import Relation.Binary.PropositionalEquality.Properties as ≡
 private
   variable
     a b ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level
-    A B : Set a
+    A B C D : Set a
     R S ≈₁ ≈₂ : Rel A ℓ₁
 
 ------------------------------------------------------------------------
 -- Definition
 
-Pointwise : Rel A ℓ₁ → Rel B ℓ₂ → Rel (A × B) (ℓ₁ ⊔ ℓ₂)
-Pointwise R S = (R on proj₁) -×- (S on proj₂)
+Pointwise : REL A B ℓ₁ → REL C D ℓ₂ → REL (A × C) (B × D) (ℓ₁ ⊔ ℓ₂)
+Pointwise R S (a , c) (b , d) = (R a b) × (S c d)
 
 ------------------------------------------------------------------------
 -- Pointwise preserves many relational properties
diff --git a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
index ced1999e3a..777c19d745 100644
--- a/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
+++ b/src/Data/Vec/Functional/Relation/Binary/Pointwise/Properties.agda
@@ -172,12 +172,10 @@ module _ {R : REL A B r} {S : REL A′ B′ s} {T : REL A″ B″ t} where
 module _ {R : REL A B r} {S : REL A′ B′ s} {n xs ys xs′ ys′} where
 
   zip⁺ : Pointwise R xs ys → Pointwise S xs′ ys′ →
-         Pointwise (λ xx yy → R (proj₁ xx) (proj₁ yy) × S (proj₂ xx) (proj₂ yy))
-                   (zip xs xs′) (zip {n = n} ys ys′)
+         Pointwise (×-Pointwise R S) (zip xs xs′) (zip {n = n} ys ys′)
   zip⁺ rs ss i = rs i , ss i
 
-  zip⁻ : Pointwise (λ xx yy → R (proj₁ xx) (proj₁ yy) × S (proj₂ xx) (proj₂ yy))
-                   (zip xs xs′) (zip {n = n} ys ys′) →
+  zip⁻ : Pointwise (×-Pointwise R S) (zip xs xs′) (zip {n = n} ys ys′) →
          Pointwise R xs ys × Pointwise S xs′ ys′
   zip⁻ rss = proj₁ ∘ rss , proj₂ ∘ rss
 

From 47499055df0769c26a04e1ea7db60f3d91f5053d Mon Sep 17 00:00:00 2001
From: Orestis Melkonian <melkon.or@gmail.com>
Date: Wed, 5 Jun 2024 14:56:05 +0100
Subject: [PATCH 295/316] More list properties about `catMaybes/mapMaybe`
 (#2389)

* More list properties about `catMaybes/mapMaybe`

- Follow-up to PR #2361
- Ported from my [formal-prelude](https://github.com/omelkonian/formal-prelude/tree/c10fe94876a7378088f2e4cf68d9b18858dcc256)

* Revert irrefutable `with`s for inductive hypotheses.
---
 CHANGELOG.md                                  | 100 ++++++++++--------
 src/Data/List/Properties.agda                 | 100 ++++++++++++------
 .../Permutation/Propositional/Properties.agda |  23 +++-
 .../Binary/Permutation/Setoid/Properties.agda |   2 +-
 .../Permutation/Setoid/Properties/Maybe.agda  |  70 ++++++++++++
 5 files changed, 217 insertions(+), 78 deletions(-)
 create mode 100644 src/Data/List/Relation/Binary/Permutation/Setoid/Properties/Maybe.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 3db3923def..7b8ca2683a 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -414,47 +414,55 @@ Additions to existing modules
 
 * In `Data.List.Properties`:
   ```agda
-  length-catMaybes      : length (catMaybes xs) ≤ length xs
-  applyUpTo-∷ʳ          : applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
-  applyDownFrom-∷ʳ      : applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
-  upTo-∷ʳ               : upTo n ∷ʳ n ≡ upTo (suc n)
-  downFrom-∷ʳ           : applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)
-  reverse-selfInverse   : SelfInverse {A = List A} _≡_ reverse
-  reverse-applyUpTo     : reverse (applyUpTo f n) ≡ applyDownFrom f n
-  reverse-upTo          : reverse (upTo n) ≡ downFrom n
-  reverse-applyDownFrom : reverse (applyDownFrom f n) ≡ applyUpTo f n
-  reverse-downFrom      : reverse (downFrom n) ≡ upTo n
-  mapMaybe-map          : mapMaybe f ∘ map g ≗ mapMaybe (f ∘ g)
-  map-mapMaybe          : map g ∘ mapMaybe f ≗ mapMaybe (Maybe.map g ∘ f)
-  align-map             : align (map f xs) (map g ys) ≡ map (map f g) (align xs ys)
-  zip-map               : zip (map f xs) (map g ys) ≡ map (map f g) (zip xs ys)
-  unzipWith-map         : unzipWith f ∘ map g ≗ unzipWith (f ∘ g)
-  map-unzipWith         : map (map g) (map h) ∘ unzipWith f ≗ unzipWith (map g h ∘ f)
-  unzip-map             : unzip ∘ map (map f g) ≗ map (map f) (map g) ∘ unzip
-  splitAt-map           : splitAt n ∘ map f ≗ map (map f) (map f) ∘ splitAt n
-  uncons-map            : uncons ∘ map f ≗ map (map f (map f)) ∘ uncons
-  last-map              : last ∘ map f ≗ map f ∘ last
-  tail-map              : tail ∘ map f ≗ map (map f) ∘ tail
-  mapMaybe-cong         : f ≗ g → mapMaybe f ≗ mapMaybe g
-  zipWith-cong          : (∀ a b → f a b ≡ g a b) → ∀ as → zipWith f as ≗ zipWith g as
-  unzipWith-cong        : f ≗ g → unzipWith f ≗ unzipWith g
-  foldl-cong            : (∀ x y → f x y ≡ g x y) → ∀ x → foldl f x ≗ foldl g x
-  alignWith-flip        : alignWith f xs ys ≡ alignWith (f ∘ swap) ys xs
-  alignWith-comm        : f ∘ swap ≗ f → alignWith f xs ys ≡ alignWith f ys xs
-  align-flip            : align xs ys ≡ map swap (align ys xs)
-  zip-flip              : zip xs ys ≡ map swap (zip ys xs)
-  unzipWith-swap        : unzipWith (swap ∘ f) ≗ swap ∘ unzipWith f
-  unzip-swap            : unzip ∘ map swap ≗ swap ∘ unzip
-  take-take             : take n (take m xs) ≡ take (n ⊓ m) xs
-  take-drop             : take n (drop m xs) ≡ drop m (take (m + n) xs)
-  zip-unzip             : uncurry′ zip ∘ unzip ≗ id
-  unzipWith-zipWith     : f ∘ uncurry′ g ≗ id → length xs ≡ length ys → unzipWith f (zipWith g xs ys) ≡ (xs , ys)
-  unzip-zip             : length xs ≡ length ys → unzip (zip xs ys) ≡ (xs , ys)
-  mapMaybe-++           : mapMaybe f (xs ++ ys) ≡ mapMaybe f xs ++ mapMaybe f ys
-  unzipWith-++          : unzipWith f (xs ++ ys) ≡ zip _++_ _++_ (unzipWith f xs) (unzipWith f ys)
-  catMaybes-concatMap   : catMaybes ≗ concatMap fromMaybe
-  catMaybes-++          : catMaybes (xs ++ ys) ≡ catMaybes xs ++ catMaybes ys
-  map-catMaybes         : map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
+  length-catMaybes       : length (catMaybes xs) ≤ length xs
+  applyUpTo-∷ʳ           : applyUpTo f n ∷ʳ f n ≡ applyUpTo f (suc n)
+  applyDownFrom-∷ʳ       : applyDownFrom (f ∘ suc) n ∷ʳ f 0 ≡ applyDownFrom f (suc n)
+  upTo-∷ʳ                : upTo n ∷ʳ n ≡ upTo (suc n)
+  downFrom-∷ʳ            : applyDownFrom suc n ∷ʳ 0 ≡ downFrom (suc n)
+  reverse-selfInverse    : SelfInverse {A = List A} _≡_ reverse
+  reverse-applyUpTo      : reverse (applyUpTo f n) ≡ applyDownFrom f n
+  reverse-upTo           : reverse (upTo n) ≡ downFrom n
+  reverse-applyDownFrom  : reverse (applyDownFrom f n) ≡ applyUpTo f n
+  reverse-downFrom       : reverse (downFrom n) ≡ upTo n
+  mapMaybe-map           : mapMaybe f ∘ map g ≗ mapMaybe (f ∘ g)
+  map-mapMaybe           : map g ∘ mapMaybe f ≗ mapMaybe (Maybe.map g ∘ f)
+  align-map              : align (map f xs) (map g ys) ≡ map (map f g) (align xs ys)
+  zip-map                : zip (map f xs) (map g ys) ≡ map (map f g) (zip xs ys)
+  unzipWith-map          : unzipWith f ∘ map g ≗ unzipWith (f ∘ g)
+  map-unzipWith          : map (map g) (map h) ∘ unzipWith f ≗ unzipWith (map g h ∘ f)
+  unzip-map              : unzip ∘ map (map f g) ≗ map (map f) (map g) ∘ unzip
+  splitAt-map            : splitAt n ∘ map f ≗ map (map f) (map f) ∘ splitAt n
+  uncons-map             : uncons ∘ map f ≗ map (map f (map f)) ∘ uncons
+  last-map               : last ∘ map f ≗ map f ∘ last
+  tail-map               : tail ∘ map f ≗ map (map f) ∘ tail
+  mapMaybe-cong          : f ≗ g → mapMaybe f ≗ mapMaybe g
+  zipWith-cong           : (∀ a b → f a b ≡ g a b) → ∀ as → zipWith f as ≗ zipWith g as
+  unzipWith-cong         : f ≗ g → unzipWith f ≗ unzipWith g
+  foldl-cong             : (∀ x y → f x y ≡ g x y) → ∀ x → foldl f x ≗ foldl g x
+  alignWith-flip         : alignWith f xs ys ≡ alignWith (f ∘ swap) ys xs
+  alignWith-comm         : f ∘ swap ≗ f → alignWith f xs ys ≡ alignWith f ys xs
+  align-flip             : align xs ys ≡ map swap (align ys xs)
+  zip-flip               : zip xs ys ≡ map swap (zip ys xs)
+  unzipWith-swap         : unzipWith (swap ∘ f) ≗ swap ∘ unzipWith f
+  unzip-swap             : unzip ∘ map swap ≗ swap ∘ unzip
+  take-take              : take n (take m xs) ≡ take (n ⊓ m) xs
+  take-drop              : take n (drop m xs) ≡ drop m (take (m + n) xs)
+  zip-unzip              : uncurry′ zip ∘ unzip ≗ id
+  unzipWith-zipWith      : f ∘ uncurry′ g ≗ id →
+                           length xs ≡ length ys →
+                           unzipWith f (zipWith g xs ys) ≡ (xs , ys)
+  unzip-zip              : length xs ≡ length ys → unzip (zip xs ys) ≡ (xs , ys)
+  mapMaybe-++            : mapMaybe f (xs ++ ys) ≡ mapMaybe f xs ++ mapMaybe f ys
+  unzipWith-++           : unzipWith f (xs ++ ys) ≡
+                           zip _++_ _++_ (unzipWith f xs) (unzipWith f ys)
+  catMaybes-concatMap    : catMaybes ≗ concatMap fromMaybe
+  catMaybes-++           : catMaybes (xs ++ ys) ≡ catMaybes xs ++ catMaybes ys
+  map-catMaybes          : map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
+  Any-catMaybes⁺         : Any (M.Any P) xs → Any P (catMaybes xs)
+  mapMaybeIsInj₁∘mapInj₁ : mapMaybe isInj₁ (map inj₁ xs) ≡ xs
+  mapMaybeIsInj₁∘mapInj₂ : mapMaybe isInj₁ (map inj₂ xs) ≡ []
+  mapMaybeIsInj₂∘mapInj₂ : mapMaybe isInj₂ (map inj₂ xs) ≡ xs
+  mapMaybeIsInj₂∘mapInj₁ : mapMaybe isInj₂ (map inj₁ xs) ≡ []
   ```
 
 * In `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
@@ -567,7 +575,15 @@ Additions to existing modules
 
 * Added new proofs to `Data.List.Relation.Binary.Permutation.Propositional.Properties`:
   ```agda
-  product-↭ : product Preserves _↭_ ⟶ _≡_
+  product-↭   : product Preserves _↭_ ⟶ _≡_
+  catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys
+  mapMaybe-↭  : xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
+  ```
+
+* Added new proofs to `Data.List.Relation.Binary.Permutation.Setoid.Properties.Maybe`:
+  ```agda
+  catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys
+  mapMaybe-↭  : xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
   ```
 
 * In `Data.Rational.Properties`:
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index e280be132b..d5bd6624f9 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -24,14 +24,15 @@ open import Data.List.Base as List
 open import Data.List.Membership.Propositional using (_∈_)
 open import Data.List.Relation.Unary.All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any using (Any; here; there)
-open import Data.Maybe.Base as Maybe using (Maybe; just; nothing; maybe)
+open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
+open import Data.Maybe.Relation.Unary.Any using (just) renaming (Any to MAny)
 open import Data.Nat.Base
 open import Data.Nat.Divisibility using (_∣_; divides; ∣n⇒∣m*n)
 open import Data.Nat.Properties
 open import Data.Product.Base as Product
   using (_×_; _,_; uncurry; uncurry′; proj₁; proj₂; <_,_>)
 import Data.Product.Relation.Unary.All as Product using (All)
-open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
+open import Data.Sum using (_⊎_; inj₁; inj₂; isInj₁; isInj₂)
 open import Data.These.Base as These using (These; this; that; these)
 open import Data.Fin.Properties using (toℕ-cast)
 open import Function.Base using (id; _∘_; _∘′_; _∋_; _-⟨_∣; ∣_⟩-_; _$_; const; flip)
@@ -49,12 +50,11 @@ open import Relation.Unary using (Pred; Decidable; ∁)
 open import Relation.Unary.Properties using (∁?)
 import Data.Nat.GeneralisedArithmetic as ℕ
 
-
 open ≡-Reasoning
 
 private
   variable
-    a b c d e p : Level
+    a b c d e p ℓ : Level
     A : Set a
     B : Set b
     C : Set c
@@ -123,32 +123,6 @@ map-injective finj {x ∷ xs} {y ∷ ys} eq =
   let fx≡fy , fxs≡fys = ∷-injective eq in
   cong₂ _∷_ (finj fx≡fy) (map-injective finj fxs≡fys)
 
-------------------------------------------------------------------------
--- catMaybes
-
-catMaybes-concatMap : catMaybes {A = A} ≗ concatMap fromMaybe
-catMaybes-concatMap []             = refl
-catMaybes-concatMap (just x  ∷ xs) = cong (x ∷_) (catMaybes-concatMap xs)
-catMaybes-concatMap (nothing ∷ xs) = catMaybes-concatMap xs
-
-length-catMaybes : ∀ xs → length (catMaybes {A = A} xs) ≤ length xs
-length-catMaybes []             = ≤-refl
-length-catMaybes (just x  ∷ xs) = s≤s (length-catMaybes xs)
-length-catMaybes (nothing ∷ xs) = m≤n⇒m≤1+n (length-catMaybes xs)
-
-catMaybes-++ : (xs ys : List (Maybe A)) →
-               catMaybes (xs ++ ys) ≡ catMaybes xs ++ catMaybes ys
-catMaybes-++ []             ys = refl
-catMaybes-++ (just x  ∷ xs) ys = cong (x ∷_) (catMaybes-++ xs ys)
-catMaybes-++ (nothing ∷ xs) ys = catMaybes-++ xs ys
-
-module _ (f : A → B) where
-
-  map-catMaybes : map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
-  map-catMaybes []             = refl
-  map-catMaybes (just x  ∷ xs) = cong (f x ∷_) (map-catMaybes xs)
-  map-catMaybes (nothing ∷ xs) = map-catMaybes xs
-
 ------------------------------------------------------------------------
 -- _++_
 
@@ -742,12 +716,40 @@ map-concatMap f g xs = begin
     ∎
 
 ------------------------------------------------------------------------
--- mapMaybe
+-- catMaybes
+
+catMaybes-concatMap : catMaybes {A = A} ≗ concatMap fromMaybe
+catMaybes-concatMap []             = refl
+catMaybes-concatMap (just x  ∷ xs) = cong (x ∷_) $ catMaybes-concatMap xs
+catMaybes-concatMap (nothing ∷ xs) = catMaybes-concatMap xs
+
+length-catMaybes : ∀ xs → length (catMaybes {A = A} xs) ≤ length xs
+length-catMaybes []             = ≤-refl
+length-catMaybes (just _  ∷ xs) = s≤s $ length-catMaybes xs
+length-catMaybes (nothing ∷ xs) = m≤n⇒m≤1+n $ length-catMaybes xs
 
-module _ {f g : A → Maybe B} where
+catMaybes-++ : (xs ys : List (Maybe A)) →
+               catMaybes (xs ++ ys) ≡ catMaybes xs ++ catMaybes ys
+catMaybes-++ []             _  = refl
+catMaybes-++ (just x  ∷ xs) ys = cong (x ∷_) $ catMaybes-++ xs ys
+catMaybes-++ (nothing ∷ xs) ys = catMaybes-++ xs ys
 
-  mapMaybe-cong : f ≗ g → mapMaybe f ≗ mapMaybe g
-  mapMaybe-cong f≗g = cong catMaybes ∘ map-cong f≗g
+map-catMaybes : (f : A → B) → map f ∘ catMaybes ≗ catMaybes ∘ map (Maybe.map f)
+map-catMaybes _ []             = refl
+map-catMaybes f (just x  ∷ xs) = cong (f x ∷_) $ map-catMaybes f xs
+map-catMaybes f (nothing ∷ xs) = map-catMaybes f xs
+
+Any-catMaybes⁺ : ∀ {P : Pred A ℓ} {xs : List (Maybe A)} →
+                 Any (MAny P) xs → Any P (catMaybes xs)
+Any-catMaybes⁺ {xs = nothing ∷ xs} (there x∈)       = Any-catMaybes⁺ x∈
+Any-catMaybes⁺ {xs = just x  ∷ xs} (here (just px)) = here px
+Any-catMaybes⁺ {xs = just x  ∷ xs} (there x∈)       = there $ Any-catMaybes⁺ x∈
+
+------------------------------------------------------------------------
+-- mapMaybe
+
+mapMaybe-cong : {f g : A → Maybe B} → f ≗ g → mapMaybe f ≗ mapMaybe g
+mapMaybe-cong f≗g = cong catMaybes ∘ map-cong f≗g
 
 mapMaybe-just : (xs : List A) → mapMaybe just xs ≡ xs
 mapMaybe-just []       = refl
@@ -793,6 +795,36 @@ module _ (g : B → C) (f : A → Maybe B) where
     mapMaybe (Maybe.map g) (map f xs)  ≡⟨ mapMaybe-map _ f xs ⟩
     mapMaybe (Maybe.map g ∘ f) xs      ∎
 
+-- embedding-projection pairs
+module _ {proj : B → Maybe A} {emb : A → B} where
+  mapMaybe-map-retract : proj ∘ emb ≗ just → mapMaybe proj ∘ map emb ≗ id
+  mapMaybe-map-retract retract xs = begin
+    mapMaybe proj (map emb xs) ≡⟨ mapMaybe-map _ _ xs ⟩
+    mapMaybe (proj ∘ emb) xs   ≡⟨ mapMaybe-cong retract xs ⟩
+    mapMaybe just xs           ≡⟨ mapMaybe-just _ ⟩
+    xs                         ∎
+
+module _ {proj : C → Maybe B} {emb : A → C} where
+  mapMaybe-map-none : proj ∘ emb ≗ const nothing → mapMaybe proj ∘ map emb ≗ const []
+  mapMaybe-map-none retract xs = begin
+    mapMaybe proj (map emb xs)  ≡⟨ mapMaybe-map _ _ xs ⟩
+    mapMaybe (proj ∘ emb) xs    ≡⟨ mapMaybe-cong retract xs ⟩
+    mapMaybe (const nothing) xs ≡⟨ mapMaybe-nothing xs ⟩
+    []                          ∎
+
+-- embedding-projection pairs on sums
+mapMaybeIsInj₁∘mapInj₁ : (xs : List A) → mapMaybe (isInj₁ {B = B}) (map inj₁ xs) ≡ xs
+mapMaybeIsInj₁∘mapInj₁ = mapMaybe-map-retract λ _ → refl
+
+mapMaybeIsInj₁∘mapInj₂ : (xs : List B) → mapMaybe (isInj₁ {A = A}) (map inj₂ xs) ≡ []
+mapMaybeIsInj₁∘mapInj₂ = mapMaybe-map-none λ _ → refl
+
+mapMaybeIsInj₂∘mapInj₂ : (xs : List B) → mapMaybe (isInj₂ {A = A}) (map inj₂ xs) ≡ xs
+mapMaybeIsInj₂∘mapInj₂ = mapMaybe-map-retract λ _ → refl
+
+mapMaybeIsInj₂∘mapInj₁ : (xs : List A) → mapMaybe (isInj₂ {B = B}) (map inj₁ xs) ≡ []
+mapMaybeIsInj₂∘mapInj₁ = mapMaybe-map-none λ _ → refl
+
 ------------------------------------------------------------------------
 -- sum
 
diff --git a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
index d56bdb7fb3..6fe439ed03 100644
--- a/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Propositional/Properties.agda
@@ -23,7 +23,8 @@ open import Data.List.Membership.Propositional
 open import Data.List.Membership.Propositional.Properties
 import Data.List.Properties as Lₚ
 open import Data.Product.Base using (_,_; _×_; ∃; ∃₂)
-open import Function.Base using (_∘_; _⟨_⟩_)
+open import Data.Maybe.Base using (Maybe; just; nothing)
+open import Function.Base using (_∘_; _⟨_⟩_; _$_)
 open import Level using (Level)
 open import Relation.Unary using (Pred)
 open import Relation.Binary.Core using (Rel; _Preserves_⟶_; _Preserves₂_⟶_⟶_)
@@ -38,6 +39,7 @@ private
     a b p : Level
     A : Set a
     B : Set b
+    xs ys : List A
 
 ------------------------------------------------------------------------
 -- Permutations of empty and singleton lists
@@ -373,3 +375,22 @@ product-↭ (swap {xs} {ys} x y r) = begin
   (y * x) * product ys ≡⟨ *-assoc y x (product ys) ⟩
   y * (x * product ys) ∎
   where open ≡-Reasoning
+
+------------------------------------------------------------------------
+-- catMaybes
+
+catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys
+catMaybes-↭ refl                         = refl
+catMaybes-↭ (trans xs↭ ↭ys)              = trans (catMaybes-↭ xs↭) (catMaybes-↭ ↭ys)
+catMaybes-↭ (prep nothing  xs↭)          = catMaybes-↭ xs↭
+catMaybes-↭ (prep (just x) xs↭)          = prep x $ catMaybes-↭ xs↭
+catMaybes-↭ (swap nothing  nothing  xs↭) = catMaybes-↭ xs↭
+catMaybes-↭ (swap nothing  (just y) xs↭) = prep y $ catMaybes-↭ xs↭
+catMaybes-↭ (swap (just x) nothing  xs↭) = prep x $ catMaybes-↭ xs↭
+catMaybes-↭ (swap (just x) (just y) xs↭) = swap x y $ catMaybes-↭ xs↭
+
+------------------------------------------------------------------------
+-- mapMaybe
+
+mapMaybe-↭ : (f : A → Maybe B) → xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
+mapMaybe-↭ f = catMaybes-↭ ∘ map⁺ f
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
index 57c9100e02..0440e3a3b6 100644
--- a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties.agda
@@ -144,7 +144,7 @@ steps-respʳ ys≋xs           (trans ys↭ws ws↭zs) = cong (steps ys↭ws +_)
 ------------------------------------------------------------------------
 -- map
 
-module _ (T : Setoid b ℓ) where
+module _ {ℓ} (T : Setoid b ℓ) where
 
   open Setoid T using () renaming (_≈_ to _≈′_)
   open Permutation T using () renaming (_↭_ to _↭′_)
diff --git a/src/Data/List/Relation/Binary/Permutation/Setoid/Properties/Maybe.agda b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties/Maybe.agda
new file mode 100644
index 0000000000..e453a599bb
--- /dev/null
+++ b/src/Data/List/Relation/Binary/Permutation/Setoid/Properties/Maybe.agda
@@ -0,0 +1,70 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties of permutations using setoid equality (on Maybe elements)
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.List.Relation.Binary.Permutation.Setoid.Properties.Maybe where
+
+open import Relation.Binary.Core using (_Preserves_⟶_)
+open import Relation.Binary.Bundles using (Setoid)
+
+open import Level using (Level)
+open import Function.Base using (_∘_; _$_)
+
+open import Data.List.Base using (catMaybes; mapMaybe)
+open import Data.List.Relation.Binary.Pointwise using (Pointwise; []; _∷_)
+import Data.List.Relation.Binary.Permutation.Setoid as Permutation
+open import Data.List.Relation.Binary.Permutation.Setoid.Properties
+
+open import Data.Maybe using (Maybe; nothing; just)
+open import Data.Maybe.Relation.Binary.Pointwise using (nothing; just)
+  renaming (setoid to setoidᴹ)
+
+private
+  variable
+    a b ℓ ℓ′ : Level
+
+------------------------------------------------------------------------
+-- catMaybes
+
+module _ (sᴬ : Setoid a ℓ) where
+  open Setoid sᴬ using (_≈_)
+  open Permutation sᴬ
+
+  private sᴹ = setoidᴹ sᴬ
+  open Setoid sᴹ using () renaming (_≈_ to _≈ᴹ_)
+  open Permutation sᴹ using () renaming (_↭_ to _↭ᴹ_)
+
+  catMaybes-↭ : ∀ {xs ys} → xs ↭ᴹ ys → catMaybes xs ↭ catMaybes ys
+  catMaybes-↭ (refl p) = refl (pointwise p)
+    where
+    pointwise : ∀ {xs ys} → Pointwise _≈ᴹ_ xs ys →
+                Pointwise _≈_ (catMaybes xs) (catMaybes ys)
+    pointwise []             = []
+    pointwise (just p  ∷ ps) = p ∷ pointwise ps
+    pointwise (nothing ∷ ps) = pointwise ps
+  catMaybes-↭ (trans xs↭ ↭ys)              = trans (catMaybes-↭ xs↭) (catMaybes-↭ ↭ys)
+  catMaybes-↭ (prep nothing    xs↭)        = catMaybes-↭ xs↭
+  catMaybes-↭ (prep (just x~y) xs↭)        = prep x~y $ catMaybes-↭ xs↭
+  catMaybes-↭ (swap nothing  nothing  xs↭) = catMaybes-↭ xs↭
+  catMaybes-↭ (swap nothing  (just y) xs↭) = prep y $ catMaybes-↭ xs↭
+  catMaybes-↭ (swap (just x) nothing  xs↭) = prep x $ catMaybes-↭ xs↭
+  catMaybes-↭ (swap (just x) (just y) xs↭) = swap x y $ catMaybes-↭ xs↭
+
+------------------------------------------------------------------------
+-- mapMaybe
+
+module _ (sᴬ : Setoid a ℓ) (sᴮ : Setoid b ℓ′) where
+  open Setoid sᴬ using () renaming (_≈_ to _≈ᴬ_)
+  open Permutation sᴬ using () renaming (_↭_ to _↭ᴬ_)
+  open Permutation sᴮ using () renaming (_↭_ to _↭ᴮ_)
+
+  private sᴹᴮ = setoidᴹ sᴮ
+  open Setoid sᴹᴮ using () renaming (_≈_ to _≈ᴹᴮ_)
+
+  mapMaybe-↭ : ∀ {f} → f Preserves _≈ᴬ_ ⟶ _≈ᴹᴮ_ →
+               ∀ {xs ys} → xs ↭ᴬ ys → mapMaybe f xs ↭ᴮ mapMaybe f ys
+  mapMaybe-↭ pres = catMaybes-↭ sᴮ ∘ map⁺ sᴬ sᴹᴮ pres

From d06c432b4d513257fc656b9e535502f6f8e2b1a7 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Wed, 5 Jun 2024 10:07:31 -0400
Subject: [PATCH 296/316] Very dependent map (#2373)

* add some 'very dependent' maps to Data.Product. More documentation to come later.

* and document

* make imports more precise

* minimal properties of very-dependen map and zipWith. Structured to make it easy to add more.

* whitespace

* implement new names and suggestions about using pattern-matching in the type

* whitespace in CHANGELOG

* small cleanup based on latest round of comments

* and fix the names in the comments too.

---------

Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
---
 CHANGELOG.md                               | 17 +++++++
 src/Data/Product.agda                      | 30 ++++++++++--
 src/Data/Product/Properties/Dependent.agda | 53 ++++++++++++++++++++++
 3 files changed, 95 insertions(+), 5 deletions(-)
 create mode 100644 src/Data/Product/Properties/Dependent.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7b8ca2683a..94a0fd3263 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -586,6 +586,23 @@ Additions to existing modules
   mapMaybe-↭  : xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
   ```
 
+* Added some very-dependent map and zipWith to `Data.Product`.
+  ```agda
+  map-Σ : {B : A → Set b} {P : A → Set p} {Q : {x : A} → P x → B x → Set q} →
+   (f : (x : A) → B x) → (∀ {x} → (y : P x) → Q y (f x)) →
+   ((x , y) : Σ A P) → Σ (B x) (Q y)
+
+  map-Σ′ : {B : A → Set b} {P : Set p} {Q : P → Set q} →
+    (f : (x : A) → B x) → ((x : P) → Q x) → ((x , y) : A × P) → B x × Q y
+
+  zipWith : {P : A → Set p} {Q : B → Set q} {R : C → Set r} {S : (x : C) → R x → Set s}
+    (_∙_ : A → B → C) → (_∘_ : ∀ {x y} → P x → Q y → R (x ∙ y)) →
+    (_*_ : (x : C) → (y : R x) → S x y) →
+    ((a , p) : Σ A P) → ((b , q) : Σ B Q) →
+        S (a ∙ b) (p ∘ q)
+
+  ```
+
 * In `Data.Rational.Properties`:
   ```agda
   1≢0 : 1ℚ ≢ 0ℚ
diff --git a/src/Data/Product.agda b/src/Data/Product.agda
index 02bf53a97b..d0f6db0cde 100644
--- a/src/Data/Product.agda
+++ b/src/Data/Product.agda
@@ -8,20 +8,40 @@
 
 module Data.Product where
 
-open import Function.Base
-open import Level
-open import Relation.Nullary.Negation.Core
+open import Level using (Level; _⊔_)
+open import Relation.Nullary.Negation.Core using (¬_)
 
 private
   variable
-    a b c ℓ : Level
-    A B : Set a
+    a b c ℓ p q r s : Level
+    A B C : Set a
 
 ------------------------------------------------------------------------
 -- Definition of dependent products
 
 open import Data.Product.Base public
 
+-- These are here as they are not 'basic' but instead "very dependent",
+-- i.e. the result type depends on the full input 'point' (x , y).
+map-Σ : {B : A → Set b} {P : A → Set p} {Q : {x : A} → P x → B x → Set q} →
+   (f : (x : A) → B x) → (∀ {x} → (y : P x) → Q y (f x)) →
+   ((x , y) : Σ A P) → Σ (B x) (Q y)
+map-Σ f g (x , y) = (f x , g y)
+
+-- This is a "non-dependent" version of map-Σ whereby the input is actually
+-- a pair (i.e. _×_ ) but the output type still depends on the input 'point' (x , y).
+map-Σ′ : {B : A → Set b} {P : Set p} {Q : P → Set q} →
+  (f : (x : A) → B x) → ((x : P) → Q x) → ((x , y) : A × P) → B x × Q y
+map-Σ′ f g (x , y) = (f x , g y)
+
+-- This is a generic zipWith for Σ where different functions are applied to each
+-- component pair, and recombined.
+zipWith : {P : A → Set p} {Q : B → Set q} {R : C → Set r} {S : (x : C) → R x → Set s}
+  (_∙_ : A → B → C) → (_∘_ : ∀ {x y} → P x → Q y → R (x ∙ y)) →
+  (_*_ : (x : C) → (y : R x) → S x y) →
+  ((a , p) : Σ A P) → ((b , q) : Σ B Q) → S (a ∙ b) (p ∘ q)
+zipWith _∙_ _∘_ _*_ (a , p) (b , q) = (a ∙ b) * (p ∘ q)
+
 ------------------------------------------------------------------------
 -- Negation of existential quantifier
 
diff --git a/src/Data/Product/Properties/Dependent.agda b/src/Data/Product/Properties/Dependent.agda
new file mode 100644
index 0000000000..122b4e3f12
--- /dev/null
+++ b/src/Data/Product/Properties/Dependent.agda
@@ -0,0 +1,53 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties of 'very dependent' map / zipWith over products
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Product.Properties.Dependent where
+
+open import Data.Product using (Σ; _×_; _,_; map-Σ; map-Σ′; zipWith)
+open import Function.Base using (id; flip)
+open import Level using (Level)
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; _≗_; cong₂; refl)
+
+private
+  variable
+    a b p q r s : Level
+    A B C : Set a
+
+------------------------------------------------------------------------
+-- map-Σ
+
+module _ {B : A → Set b} {P : A → Set p} {Q : {x : A} → P x → B x → Set q} where
+
+  map-Σ-cong : {f g : (x : A) → B x} → {h k : ∀ {x} → (y : P x) → Q y (f x)} →
+     (∀ x → f x ≡ g x) →
+     (∀ {x} → (y : P x) → h y ≡ k y) →
+     (v : Σ A P) → map-Σ f h v ≡ map-Σ g k v
+  map-Σ-cong f≗g h≗k (x , y) = cong₂ _,_ (f≗g x) (h≗k y)
+
+------------------------------------------------------------------------
+-- map-Σ′
+
+module _ {B : A → Set b} {P : Set p} {Q : P → Set q} where
+
+  map-Σ′-cong : {f g : (x : A) → B x} → {h k : (x : P) → Q x} →
+     (∀ x → f x ≡ g x) →
+     ((y : P) → h y ≡ k y) →
+     (v : A × P) → map-Σ′ f h v ≡ map-Σ′ g k v
+  map-Σ′-cong f≗g h≗k (x , y) = cong₂ _,_ (f≗g x) (h≗k y)
+
+------------------------------------------------------------------------
+-- zipWith
+
+module _ {P : A → Set p} {Q : B → Set q} {R : C → Set r} {S : (x : C) → R x → Set s} where
+
+  zipWith-flip : (_∙_ : A → B → C) → (_∘_ : ∀ {x y} → P x → Q y → R (x ∙ y)) →
+    (_*_ : (x : C) → (y : R x) → S x y) →
+    {x : Σ A P} → {y : Σ B Q} →
+    zipWith _∙_ _∘_ _*_ x y ≡ zipWith (flip _∙_) (flip _∘_) _*_ y x
+  zipWith-flip _∙_ _∘_ _*_ = refl

From 9ac0bd34d4b8b878a577491068663cf0f5d04980 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 5 Jun 2024 15:20:23 +0100
Subject: [PATCH 297/316] Improve `Data.List.Base` (fix #2359; deprecate use of
 with #2123) (#2366)

* refactor towards `if_then_else_`

* layout

* `let` into `where`
---
 src/Data/List/Base.agda | 13 +++++++------
 1 file changed, 7 insertions(+), 6 deletions(-)

diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 9ea8b551b9..f707729af7 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -421,9 +421,10 @@ linesBy {A = A} P? = go nothing where
 
   go : Maybe (List A) → List A → List (List A)
   go acc []       = maybe′ ([_] ∘′ reverse) [] acc
-  go acc (c ∷ cs) with does (P? c)
-  ... | true  = reverse (Maybe.fromMaybe [] acc) ∷ go nothing cs
-  ... | false = go (just (c ∷ Maybe.fromMaybe [] acc)) cs
+  go acc (c ∷ cs) = if does (P? c)
+    then reverse acc′ ∷ go nothing cs
+    else go (just (c ∷ acc′)) cs
+    where acc′ = Maybe.fromMaybe [] acc
 
 linesByᵇ : (A → Bool) → List A → List (List A)
 linesByᵇ p = linesBy (T? ∘ p)
@@ -441,9 +442,9 @@ wordsBy {A = A} P? = go [] where
 
   go : List A → List A → List (List A)
   go acc []       = cons acc []
-  go acc (c ∷ cs) with does (P? c)
-  ... | true  = cons acc (go [] cs)
-  ... | false = go (c ∷ acc) cs
+  go acc (c ∷ cs) = if does (P? c)
+    then cons acc (go [] cs)
+    else go (c ∷ acc) cs
 
 wordsByᵇ : (A → Bool) → List A → List (List A)
 wordsByᵇ p = wordsBy (T? ∘ p)

From 4018fef524156c5b427a45368529380243452a46 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 5 Jun 2024 16:28:11 +0100
Subject: [PATCH 298/316] Adds `Relation.Nullary.Recomputable` plus
 consequences (#2243)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* prototype for fixing #2199

* delegate to `Relation.Nullary.Negation.Core.weak-contradiction`

* refactoring: lift out `Recomputable` in its own right

* forgot to add this module

* refactoring: tweak

* tidying up; added `CHANGELOG`

* more tidying up

* streamlined `import`s

* removed `Recomputable` from `Relation.Nullary`

* fixed multiple definitions in `Relation.Unary`

* reorder `CHANGELOG` entries after merge

* `contradiciton` via `weak-contradiction`

* irrefutable `with`

* use `of`

* only use *irrelevant* `⊥-elim-irr`

* tightened `import`s

* removed `irr-contradiction`

* inspired by #2329

* conjecture: all uses of `contradiction` can be made weak

* second thoughts: reverted last round of chnages

* lazier pattern analysis cf. #2055

* dependencies: uncoupled from `Recomputable`

* moved `⊥` and `¬ A` properties to this one place

* removed `contradictionᵒ` and rephrased everything in terms of `weak-contradiction`

* knock-on consequences; simplified `import`s

* narrow `import`s

* narrow `import`s; irrefutable `with`

* narrow `import`s; `CHANGELOG`

* narrow `import`s

* response to review comments

* irrelevance of `recompute`

* knock-on, plus proof of `UIP` from #2149

* knock-ons from renaming

* knock-on from renaming

* pushed proof `recompute-constant` to `Recomputable`
---
 CHANGELOG.md                              | 78 +++++++++++++++--------
 src/Axiom/UniquenessOfIdentityProofs.agda | 35 +++++-----
 src/Data/Empty.agda                       |  3 +
 src/Relation/Binary/Definitions.agda      |  4 +-
 src/Relation/Nullary.agda                 | 10 +--
 src/Relation/Nullary/Decidable.agda       | 24 +++----
 src/Relation/Nullary/Decidable/Core.agda  | 23 ++++---
 src/Relation/Nullary/Negation.agda        | 11 ++--
 src/Relation/Nullary/Negation/Core.agda   | 15 +++--
 src/Relation/Nullary/Recomputable.agda    | 63 ++++++++++++++++++
 src/Relation/Nullary/Reflects.agda        | 45 ++++++++-----
 src/Relation/Unary.agda                   |  3 +-
 12 files changed, 207 insertions(+), 107 deletions(-)
 create mode 100644 src/Relation/Nullary/Recomputable.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 94a0fd3263..e02ff2dc30 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -118,25 +118,9 @@ New modules
   Algebra.Module.Bundles.Raw
   ```
 
-* Properties of `List` modulo `Setoid` equality (currently only the ([],++) monoid):
-  ```
-  Data.List.Relation.Binary.Equality.Setoid.Properties
-  ```
-
-* Refactoring of `Data.List.Base.{scanr|scanl}` and their properties:
-  ```
-  Data.List.Scans.Base
-  Data.List.Scans.Properties
-  ```
-
-* Prime factorisation of natural numbers.
-  ```
-  Data.Nat.Primality.Factorisation
-  ```
-
-* Consequences of 'infinite descent' for (accessible elements of) well-founded relations:
+* Nagata's construction of the "idealization of a module":
   ```agda
-  Induction.InfiniteDescent
+  Algebra.Module.Construct.Idealization
   ```
 
 * The unique morphism from the initial, resp. terminal, algebra:
@@ -145,9 +129,22 @@ New modules
   Algebra.Morphism.Construct.Terminal
   ```
 
-* Nagata's construction of the "idealization of a module":
+* Pointwise and equality relations over indexed containers:
   ```agda
-  Algebra.Module.Construct.Idealization
+  Data.Container.Indexed.Relation.Binary.Pointwise
+  Data.Container.Indexed.Relation.Binary.Pointwise.Properties
+  Data.Container.Indexed.Relation.Binary.Equality.Setoid
+  ```
+
+* Refactoring of `Data.List.Base.{scanr|scanl}` and their properties:
+  ```
+  Data.List.Scans.Base
+  Data.List.Scans.Properties
+  ```
+
+* Properties of `List` modulo `Setoid` equality (currently only the ([],++) monoid):
+  ```
+  Data.List.Relation.Binary.Equality.Setoid.Properties
   ```
 
 * `Data.List.Relation.Binary.Sublist.Propositional.Slice`
@@ -155,6 +152,12 @@ New modules
   and adding new functions:
   - `⊆-upper-bound-is-cospan` generalising `⊆-disjoint-union-is-cospan` from (*)
   - `⊆-upper-bound-cospan` generalising `⊆-disjoint-union-cospan` from (*)
+  ```
+
+* Prime factorisation of natural numbers.
+  ```
+  Data.Nat.Primality.Factorisation
+  ```
 
 * `Data.Vec.Functional.Relation.Binary.Permutation`, defining:
   ```agda
@@ -180,6 +183,11 @@ New modules
   _⇨_ = setoid
   ```
 
+* Consequences of 'infinite descent' for (accessible elements of) well-founded relations:
+  ```agda
+  Induction.InfiniteDescent
+  ```
+
 * Symmetric interior of a binary relation
   ```
   Relation.Binary.Construct.Interior.Symmetric
@@ -190,12 +198,11 @@ New modules
   Relation.Binary.Properties.DecSetoid
   ```
 
-* Pointwise and equality relations over indexed containers:
+* Systematise the use of `Recomputable A = .A → A`:
   ```agda
-  Data.Container.Indexed.Relation.Binary.Pointwise
-  Data.Container.Indexed.Relation.Binary.Pointwise.Properties
-  Data.Container.Indexed.Relation.Binary.Equality.Setoid
+  Relation.Nullary.Recomputable
   ```
+  with `Recomputable` exported publicly from `Relation.Nullary`.
 
 * New IO primitives to handle buffering
   ```agda
@@ -372,6 +379,11 @@ Additions to existing modules
   Subtrees o c = (r : Response c) → X (next c r)
   ```
 
+* In `Data.Empty`:
+  ```agda
+  ⊥-elim-irr  : .⊥ → Whatever
+  ```
+
 * In `Data.Fin.Properties`:
   ```agda
   nonZeroIndex : Fin n → ℕ.NonZero n
@@ -674,7 +686,7 @@ Additions to existing modules
 * Added new definitions in `Relation.Binary.Definitions`
   ```
   Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)
-  Empty  _∼_ = ∀ {x y} → x ∼ y → ⊥
+  Empty  _∼_ = ∀ {x y} → ¬ (x ∼ y)
   ```
 
 * Added new proofs in `Relation.Binary.Properties.Setoid`:
@@ -690,11 +702,27 @@ Additions to existing modules
   WeaklyDecidable : Set _
   ```
 
+* Added new proof in `Relation.Nullary.Decidable.Core`:
+  ```agda
+  recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q
+  ```
+
 * Added new proof in `Relation.Nullary.Decidable`:
   ```agda
   ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
   ```
 
+* Added new definitions in `Relation.Nullary.Negation.Core`:
+  ```agda
+  contradiction-irr : .A → ¬ A → Whatever
+  ```
+
+* Added new definitions in `Relation.Nullary.Reflects`:
+  ```agda
+  recompute          : Reflects A b → Recomputable A
+  recompute-constant : (r : Reflects A b) (p q : A) → recompute r p ≡ recompute r q
+  ```
+
 * Added new definitions in `Relation.Unary`
   ```
   Stable          : Pred A ℓ → Set _
diff --git a/src/Axiom/UniquenessOfIdentityProofs.agda b/src/Axiom/UniquenessOfIdentityProofs.agda
index 74463fe557..670941afb1 100644
--- a/src/Axiom/UniquenessOfIdentityProofs.agda
+++ b/src/Axiom/UniquenessOfIdentityProofs.agda
@@ -8,15 +8,19 @@
 
 module Axiom.UniquenessOfIdentityProofs where
 
-open import Data.Bool.Base using (true; false)
-open import Data.Empty
-open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary hiding (Irrelevant)
+open import Level using (Level)
+open import Relation.Nullary.Decidable.Core using (recompute; recompute-constant)
 open import Relation.Binary.Core
 open import Relation.Binary.Definitions
 open import Relation.Binary.PropositionalEquality.Core
 open import Relation.Binary.PropositionalEquality.Properties
 
+private
+  variable
+    a : Level
+    A : Set a
+    x y : A
+
 ------------------------------------------------------------------------
 -- Definition
 --
@@ -24,7 +28,7 @@ open import Relation.Binary.PropositionalEquality.Properties
 -- equality are themselves equal. In other words, the equality relation
 -- is irrelevant. Here we define UIP relative to a given type.
 
-UIP : ∀ {a} (A : Set a) → Set a
+UIP : (A : Set a) → Set a
 UIP A = Irrelevant {A = A} _≡_
 
 ------------------------------------------------------------------------
@@ -39,11 +43,11 @@ UIP A = Irrelevant {A = A} _≡_
 -- the image of any other proof.
 
 module Constant⇒UIP
-  {a} {A : Set a} (f : _≡_ {A = A} ⇒ _≡_)
-  (f-constant : ∀ {a b} (p q : a ≡ b) → f p ≡ f q)
+  (f : _≡_ {A = A} ⇒ _≡_)
+  (f-constant : ∀ {x y} (p q : x ≡ y) → f p ≡ f q)
   where
 
-  ≡-canonical : ∀ {a b} (p : a ≡ b) → trans (sym (f refl)) (f p) ≡ p
+  ≡-canonical : (p : x ≡ y) → trans (sym (f refl)) (f p) ≡ p
   ≡-canonical refl = trans-symˡ (f refl)
 
   ≡-irrelevant : UIP A
@@ -59,19 +63,14 @@ module Constant⇒UIP
 -- function over proofs of equality which is constant: it returns the
 -- proof produced by the decision procedure.
 
-module Decidable⇒UIP
-  {a} {A : Set a} (_≟_ : DecidableEquality A)
+module Decidable⇒UIP (_≟_ : DecidableEquality A)
   where
 
   ≡-normalise : _≡_ {A = A} ⇒ _≡_
-  ≡-normalise {a} {b} a≡b with a ≟ b
-  ... | true  because  [p] = invert [p]
-  ... | false because [¬p] = ⊥-elim (invert [¬p] a≡b)
-
-  ≡-normalise-constant : ∀ {a b} (p q : a ≡ b) → ≡-normalise p ≡ ≡-normalise q
-  ≡-normalise-constant {a} {b} p q with a ≟ b
-  ... | true  because   _  = refl
-  ... | false because [¬p] = ⊥-elim (invert [¬p] p)
+  ≡-normalise {x} {y} x≡y = recompute (x ≟ y) x≡y
+
+  ≡-normalise-constant : (p q : x ≡ y) → ≡-normalise p ≡ ≡-normalise q
+  ≡-normalise-constant {x = x} {y = y} = recompute-constant (x ≟ y)
 
   ≡-irrelevant : UIP A
   ≡-irrelevant = Constant⇒UIP.≡-irrelevant ≡-normalise ≡-normalise-constant
diff --git a/src/Data/Empty.agda b/src/Data/Empty.agda
index 19754e8c7d..d2647a1f18 100644
--- a/src/Data/Empty.agda
+++ b/src/Data/Empty.agda
@@ -35,3 +35,6 @@ private
 
 ⊥-elim : ∀ {w} {Whatever : Set w} → ⊥ → Whatever
 ⊥-elim ()
+
+⊥-elim-irr : ∀ {w} {Whatever : Set w} → .⊥ → Whatever
+⊥-elim-irr ()
diff --git a/src/Relation/Binary/Definitions.agda b/src/Relation/Binary/Definitions.agda
index 0e3f09e443..167b63a8cb 100644
--- a/src/Relation/Binary/Definitions.agda
+++ b/src/Relation/Binary/Definitions.agda
@@ -12,8 +12,6 @@ module Relation.Binary.Definitions where
 
 open import Agda.Builtin.Equality using (_≡_)
 
-open import Data.Empty using (⊥)
-open import Data.Maybe.Base using (Maybe)
 open import Data.Product.Base using (_×_; ∃-syntax)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_on_; flip)
@@ -248,7 +246,7 @@ Universal _∼_ = ∀ x y → x ∼ y
 -- Empty - no elements are related
 
 Empty : REL A B ℓ → Set _
-Empty _∼_ = ∀ {x y} → x ∼ y → ⊥
+Empty _∼_ = ∀ {x y} → ¬ (x ∼ y)
 
 -- Non-emptiness - at least one pair of elements are related.
 
diff --git a/src/Relation/Nullary.agda b/src/Relation/Nullary.agda
index 92e0607f4d..5ea302fd98 100644
--- a/src/Relation/Nullary.agda
+++ b/src/Relation/Nullary.agda
@@ -23,8 +23,9 @@ private
 ------------------------------------------------------------------------
 -- Re-exports
 
+open import Relation.Nullary.Recomputable public using (Recomputable)
 open import Relation.Nullary.Negation.Core public
-open import Relation.Nullary.Reflects public
+open import Relation.Nullary.Reflects public hiding (recompute; recompute-constant)
 open import Relation.Nullary.Decidable.Core public
 
 ------------------------------------------------------------------------
@@ -33,13 +34,6 @@ open import Relation.Nullary.Decidable.Core public
 Irrelevant : Set p → Set p
 Irrelevant P = ∀ (p₁ p₂ : P) → p₁ ≡ p₂
 
-------------------------------------------------------------------------
--- Recomputability - we can rebuild a relevant proof given an
--- irrelevant one.
-
-Recomputable : Set p → Set p
-Recomputable P = .P → P
-
 ------------------------------------------------------------------------
 -- Weak decidability
 -- `nothing` is 'don't know'/'give up'; `just` is `yes`/`definitely`
diff --git a/src/Relation/Nullary/Decidable.agda b/src/Relation/Nullary/Decidable.agda
index 4335a5df5a..4b28ed43f1 100644
--- a/src/Relation/Nullary/Decidable.agda
+++ b/src/Relation/Nullary/Decidable.agda
@@ -9,15 +9,14 @@
 module Relation.Nullary.Decidable where
 
 open import Level using (Level)
-open import Data.Bool.Base using (true; false; if_then_else_)
-open import Data.Empty using (⊥-elim)
+open import Data.Bool.Base using (true; false)
 open import Data.Product.Base using (∃; _,_)
-open import Function.Base
 open import Function.Bundles using
   (Injection; module Injection; module Equivalence; _⇔_; _↔_; mk↔ₛ′)
 open import Relation.Binary.Bundles using (Setoid; module Setoid)
 open import Relation.Binary.Definitions using (Decidable)
-open import Relation.Nullary
+open import Relation.Nullary using (Irrelevant)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Nullary.Reflects using (invert)
 open import Relation.Binary.PropositionalEquality.Core
   using (_≡_; refl; sym; trans; cong′)
@@ -52,8 +51,8 @@ via-injection inj _≟_ x y = map′ injective cong (to x ≟ to y)
 -- A lemma relating True and Dec
 
 True-↔ : (a? : Dec A) → Irrelevant A → True a? ↔ A
-True-↔ (true  because [a]) irr = mk↔ₛ′ (λ _ → invert [a]) _ (irr (invert [a])) cong′
-True-↔ (false because ofⁿ ¬a) _ = mk↔ₛ′ (λ ()) (invert (ofⁿ ¬a)) (⊥-elim ∘ ¬a) λ ()
+True-↔ (true  because [a]) irr = let a = invert [a] in mk↔ₛ′ (λ _ → a) _ (irr a) cong′
+True-↔ (false because [¬a]) _  = let ¬a = invert [¬a] in mk↔ₛ′ (λ ()) ¬a (λ a → contradiction a ¬a) λ ()
 
 ------------------------------------------------------------------------
 -- Result of decidability
@@ -64,23 +63,20 @@ isYes≗does (false because _) = refl
 
 dec-true : (a? : Dec A) → A → does a? ≡ true
 dec-true (true  because   _ ) a = refl
-dec-true (false because [¬a]) a = ⊥-elim (invert [¬a] a)
+dec-true (false because [¬a]) a = contradiction a (invert [¬a])
 
 dec-false : (a? : Dec A) → ¬ A → does a? ≡ false
 dec-false (false because  _ ) ¬a = refl
-dec-false (true  because [a]) ¬a = ⊥-elim (¬a (invert [a]))
+dec-false (true  because [a]) ¬a = contradiction (invert [a]) ¬a
 
 dec-yes : (a? : Dec A) → A → ∃ λ a → a? ≡ yes a
-dec-yes a? a with dec-true a? a
-dec-yes (yes a′) a | refl = a′ , refl
+dec-yes a? a with yes a′ ← a? | refl ← dec-true a? a = a′ , refl
 
 dec-no : (a? : Dec A) (¬a : ¬ A) → a? ≡ no ¬a
-dec-no a? ¬a with dec-false a? ¬a
-dec-no (no _) _ | refl = refl
+dec-no a? ¬a with no _ ← a? | refl ← dec-false a? ¬a = refl
 
 dec-yes-irr : (a? : Dec A) → Irrelevant A → (a : A) → a? ≡ yes a
-dec-yes-irr a? irr a with dec-yes a? a
-... | a′ , eq rewrite irr a a′ = eq
+dec-yes-irr a? irr a with a′ , eq ← dec-yes a? a rewrite irr a a′ = eq
 
 ⌊⌋-map′ : ∀ t f (a? : Dec A) → ⌊ map′ {B = B} t f a? ⌋ ≡ ⌊ a? ⌋
 ⌊⌋-map′ t f a? = trans (isYes≗does (map′ t f a?)) (sym (isYes≗does a?))
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index 83da26e51b..9af6cc0078 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -11,17 +11,19 @@
 
 module Relation.Nullary.Decidable.Core where
 
+open import Agda.Builtin.Equality using (_≡_)
 open import Level using (Level; Lift)
 open import Data.Bool.Base using (Bool; T; false; true; not; _∧_; _∨_)
 open import Data.Unit.Polymorphic.Base using (⊤)
-open import Data.Empty.Irrelevant using (⊥-elim)
 open import Data.Product.Base using (_×_)
 open import Data.Sum.Base using (_⊎_)
 open import Function.Base using (_∘_; const; _$_; flip)
-open import Relation.Nullary.Reflects
+open import Relation.Nullary.Recomputable as Recomputable hiding (recompute-constant)
+open import Relation.Nullary.Reflects as Reflects hiding (recompute; recompute-constant)
 open import Relation.Nullary.Negation.Core
   using (¬_; Stable; negated-stable; contradiction; DoubleNegation)
 
+
 private
   variable
     a b : Level
@@ -69,9 +71,12 @@ module _ {A : Set a} where
 
 -- Given an irrelevant proof of a decidable type, a proof can
 -- be recomputed and subsequently used in relevant contexts.
-recompute : Dec A → .A → A
-recompute (yes a) _ = a
-recompute (no ¬a) a = ⊥-elim (¬a a)
+
+recompute : Dec A → Recomputable A
+recompute = Reflects.recompute ∘ proof
+
+recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q
+recompute-constant = Recomputable.recompute-constant ∘ recompute
 
 ------------------------------------------------------------------------
 -- Interaction with negation, sum, product etc.
@@ -161,8 +166,8 @@ from-no (true  because   _ ) = _
 
 map′ : (A → B) → (B → A) → Dec A → Dec B
 does  (map′ A→B B→A a?)                   = does a?
-proof (map′ A→B B→A (true  because  [a])) = ofʸ (A→B (invert [a]))
-proof (map′ A→B B→A (false because [¬a])) = ofⁿ (invert [¬a] ∘ B→A)
+proof (map′ A→B B→A (true  because  [a])) = of (A→B (invert [a]))
+proof (map′ A→B B→A (false because [¬a])) = of (invert [¬a] ∘ B→A)
 
 ------------------------------------------------------------------------
 -- Relationship with double-negation
@@ -170,8 +175,8 @@ proof (map′ A→B B→A (false because [¬a])) = ofⁿ (invert [¬a] ∘ B→A
 -- Decidable predicates are stable.
 
 decidable-stable : Dec A → Stable A
-decidable-stable (yes a) ¬¬a = a
-decidable-stable (no ¬a) ¬¬a = contradiction ¬a ¬¬a
+decidable-stable (true  because  [a]) ¬¬a = invert [a]
+decidable-stable (false because [¬a]) ¬¬a = contradiction (invert [¬a]) ¬¬a
 
 ¬-drop-Dec : Dec (¬ ¬ A) → Dec (¬ A)
 ¬-drop-Dec ¬¬a? = map′ negated-stable contradiction (¬? ¬¬a?)
diff --git a/src/Relation/Nullary/Negation.agda b/src/Relation/Nullary/Negation.agda
index b9116504ec..5372ac5f5e 100644
--- a/src/Relation/Nullary/Negation.agda
+++ b/src/Relation/Nullary/Negation.agda
@@ -8,11 +8,10 @@
 
 module Relation.Nullary.Negation where
 
-open import Effect.Monad using (RawMonad; mkRawMonad)
-open import Data.Bool.Base using (Bool; false; true; if_then_else_; not)
-open import Data.Empty using (⊥-elim)
+open import Data.Bool.Base using (Bool; false; true; if_then_else_)
 open import Data.Product.Base as Product using (_,_; Σ; Σ-syntax; ∃; curry; uncurry)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂; [_,_])
+open import Effect.Monad using (RawMonad; mkRawMonad)
 open import Function.Base using (flip; _∘_; const; _∘′_)
 open import Level using (Level)
 open import Relation.Nullary.Decidable.Core using (Dec; yes; no; ¬¬-excluded-middle)
@@ -73,7 +72,7 @@ open import Relation.Nullary.Negation.Core public
 -- ⊥).
 
 call/cc : ((A → Whatever) → DoubleNegation A) → DoubleNegation A
-call/cc hyp ¬a = hyp (λ a → ⊥-elim (¬a a)) ¬a
+call/cc hyp ¬a = hyp (flip contradiction ¬a) ¬a
 
 -- The "independence of premise" rule, in the double-negation monad.
 -- It is assumed that the index set (A) is inhabited.
@@ -83,7 +82,7 @@ independence-of-premise {A = A} {B = B} {P = P} q f = ¬¬-map helper ¬¬-exclu
   where
   helper : Dec B → Σ[ x ∈ A ] (B → P x)
   helper (yes p) = Product.map₂ const (f p)
-  helper (no ¬p) = (q , ⊥-elim ∘′ ¬p)
+  helper (no ¬p) = (q , flip contradiction ¬p)
 
 -- The independence of premise rule for binary sums.
 
@@ -92,7 +91,7 @@ independence-of-premise-⊎ {A = A} {B = B} {C = C} f = ¬¬-map helper ¬¬-exc
   where
   helper : Dec A → (A → B) ⊎ (A → C)
   helper (yes p) = Sum.map const const (f p)
-  helper (no ¬p) = inj₁ (⊥-elim ∘′ ¬p)
+  helper (no ¬p) = inj₁ (flip contradiction ¬p)
 
 private
 
diff --git a/src/Relation/Nullary/Negation/Core.agda b/src/Relation/Nullary/Negation/Core.agda
index cbe9ac6da9..702fb16353 100644
--- a/src/Relation/Nullary/Negation/Core.agda
+++ b/src/Relation/Nullary/Negation/Core.agda
@@ -8,10 +8,9 @@
 
 module Relation.Nullary.Negation.Core where
 
-open import Data.Bool.Base using (not)
-open import Data.Empty using (⊥; ⊥-elim)
+open import Data.Empty using (⊥; ⊥-elim-irr)
 open import Data.Sum.Base using (_⊎_; [_,_]; inj₁; inj₂)
-open import Function.Base using (flip; _$_; _∘_; const)
+open import Function.Base using (flip; _∘_; const)
 open import Level
 
 private
@@ -48,8 +47,11 @@ _¬-⊎_ = [_,_]
 ------------------------------------------------------------------------
 -- Uses of negation
 
+contradiction-irr : .A → ¬ A → Whatever
+contradiction-irr a ¬a = ⊥-elim-irr (¬a a)
+
 contradiction : A → ¬ A → Whatever
-contradiction a ¬a = ⊥-elim (¬a a)
+contradiction a = contradiction-irr a
 
 contradiction₂ : A ⊎ B → ¬ A → ¬ B → Whatever
 contradiction₂ (inj₁ a) ¬a ¬b = contradiction a ¬a
@@ -60,11 +62,11 @@ contraposition f ¬b a = contradiction (f a) ¬b
 
 -- Everything is stable in the double-negation monad.
 stable : ¬ ¬ Stable A
-stable ¬[¬¬a→a] = ¬[¬¬a→a] (λ ¬¬a → ⊥-elim (¬¬a (¬[¬¬a→a] ∘ const)))
+stable ¬[¬¬a→a] = ¬[¬¬a→a] (contradiction (¬[¬¬a→a] ∘ const))
 
 -- Negated predicates are stable.
 negated-stable : Stable (¬ A)
-negated-stable ¬¬¬a a = ¬¬¬a (_$ a)
+negated-stable ¬¬¬a a = ¬¬¬a (contradiction a)
 
 ¬¬-map : (A → B) → ¬ ¬ A → ¬ ¬ B
 ¬¬-map f = contraposition (contraposition f)
@@ -73,3 +75,4 @@ negated-stable ¬¬¬a a = ¬¬¬a (_$ a)
 private
   note : (A → ¬ B) → B → ¬ A
   note = flip
+
diff --git a/src/Relation/Nullary/Recomputable.agda b/src/Relation/Nullary/Recomputable.agda
new file mode 100644
index 0000000000..cd0b83f113
--- /dev/null
+++ b/src/Relation/Nullary/Recomputable.agda
@@ -0,0 +1,63 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Recomputable types and their algebra as Harrop formulas
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Relation.Nullary.Recomputable where
+
+open import Agda.Builtin.Equality using (_≡_; refl)
+open import Data.Empty using (⊥)
+open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
+open import Level using (Level)
+open import Relation.Nullary.Negation.Core using (¬_)
+
+private
+  variable
+    a b : Level
+    A : Set a
+    B : Set b
+
+------------------------------------------------------------------------
+-- Definition
+--
+-- The idea of being 'recomputable' is that, given an *irrelevant* proof
+-- of a proposition `A` (signalled by being a value of type `.A`, all of
+-- whose inhabitants are identified up to definitional equality, and hence
+-- do *not* admit pattern-matching), one may 'promote' such a value to a
+-- 'genuine' value of `A`, available for subsequent eg. pattern-matching.
+
+Recomputable : (A : Set a) → Set a
+Recomputable A = .A → A
+
+------------------------------------------------------------------------
+-- Fundamental property: 'promotion' is a constant function
+
+recompute-constant : (r : Recomputable A) (p q : A) → r p ≡ r q
+recompute-constant r p q = refl
+
+------------------------------------------------------------------------
+-- Constructions
+
+⊥-recompute : Recomputable ⊥
+⊥-recompute ()
+
+_×-recompute_ : Recomputable A → Recomputable B → Recomputable (A × B)
+(rA ×-recompute rB) p = rA (p .proj₁) , rB (p .proj₂)
+
+_→-recompute_ : (A : Set a) → Recomputable B → Recomputable (A → B)
+(A →-recompute rB) f a = rB (f a)
+
+Π-recompute : (B : A → Set b) → (∀ x → Recomputable (B x)) → Recomputable (∀ x → B x)
+Π-recompute B rB f a = rB a (f a)
+
+∀-recompute : (B : A → Set b) → (∀ {x} → Recomputable (B x)) → Recomputable (∀ {x} → B x)
+∀-recompute B rB f = rB f
+
+-- corollary: negated propositions are Recomputable
+
+¬-recompute : Recomputable (¬ A)
+¬-recompute {A = A} = A →-recompute ⊥-recompute
+
diff --git a/src/Relation/Nullary/Reflects.agda b/src/Relation/Nullary/Reflects.agda
index 3b301fdbce..00cee309a7 100644
--- a/src/Relation/Nullary/Reflects.agda
+++ b/src/Relation/Nullary/Reflects.agda
@@ -11,14 +11,13 @@ module Relation.Nullary.Reflects where
 open import Agda.Builtin.Equality
 
 open import Data.Bool.Base
-open import Data.Unit.Base using (⊤)
-open import Data.Empty
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Data.Product.Base using (_×_; _,_; proj₁; proj₂)
 open import Level using (Level)
 open import Function.Base using (_$_; _∘_; const; id)
-
 open import Relation.Nullary.Negation.Core
+  using (¬_; contradiction-irr; contradiction; _¬-⊎_)
+open import Relation.Nullary.Recomputable as Recomputable using (Recomputable)
 
 private
   variable
@@ -52,6 +51,20 @@ invert : ∀ {b} → Reflects A b → if b then A else ¬ A
 invert (ofʸ  a) = a
 invert (ofⁿ ¬a) = ¬a
 
+------------------------------------------------------------------------
+-- recompute
+
+-- Given an irrelevant proof of a reflected type, a proof can
+-- be recomputed and subsequently used in relevant contexts.
+
+recompute : ∀ {b} → Reflects A b → Recomputable A
+recompute (ofʸ  a) _ = a
+recompute (ofⁿ ¬a) a = contradiction-irr a ¬a
+
+recompute-constant : ∀ {b} (r : Reflects A b) (p q : A) →
+                     recompute r p ≡ recompute r q
+recompute-constant = Recomputable.recompute-constant ∘ recompute
+
 ------------------------------------------------------------------------
 -- Interaction with negation, product, sums etc.
 
@@ -64,34 +77,34 @@ T-reflects false = of id
 
 -- If we can decide A, then we can decide its negation.
 ¬-reflects : ∀ {b} → Reflects A b → Reflects (¬ A) (not b)
-¬-reflects (ofʸ  a) = ofⁿ (_$ a)
-¬-reflects (ofⁿ ¬a) = ofʸ ¬a
+¬-reflects (ofʸ  a) = of (_$ a)
+¬-reflects (ofⁿ ¬a) = of ¬a
 
 -- If we can decide A and Q then we can decide their product
 _×-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
                Reflects (A × B) (a ∧ b)
-ofʸ  a ×-reflects ofʸ  b = ofʸ (a , b)
-ofʸ  a ×-reflects ofⁿ ¬b = ofⁿ (¬b ∘ proj₂)
-ofⁿ ¬a ×-reflects _      = ofⁿ (¬a ∘ proj₁)
+ofʸ  a ×-reflects ofʸ  b = of (a , b)
+ofʸ  a ×-reflects ofⁿ ¬b = of (¬b ∘ proj₂)
+ofⁿ ¬a ×-reflects _      = of (¬a ∘ proj₁)
 
 _⊎-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
                Reflects (A ⊎ B) (a ∨ b)
-ofʸ  a ⊎-reflects      _ = ofʸ (inj₁ a)
-ofⁿ ¬a ⊎-reflects ofʸ  b = ofʸ (inj₂ b)
-ofⁿ ¬a ⊎-reflects ofⁿ ¬b = ofⁿ (¬a ¬-⊎ ¬b)
+ofʸ  a ⊎-reflects      _ = of (inj₁ a)
+ofⁿ ¬a ⊎-reflects ofʸ  b = of (inj₂ b)
+ofⁿ ¬a ⊎-reflects ofⁿ ¬b = of (¬a ¬-⊎ ¬b)
 
 _→-reflects_ : ∀ {a b} → Reflects A a → Reflects B b →
                 Reflects (A → B) (not a ∨ b)
-ofʸ  a →-reflects ofʸ  b = ofʸ (const b)
-ofʸ  a →-reflects ofⁿ ¬b = ofⁿ (¬b ∘ (_$ a))
-ofⁿ ¬a →-reflects _      = ofʸ (⊥-elim ∘ ¬a)
+ofʸ  a →-reflects ofʸ  b = of (const b)
+ofʸ  a →-reflects ofⁿ ¬b = of (¬b ∘ (_$ a))
+ofⁿ ¬a →-reflects _      = of (λ a → contradiction a ¬a)
 
 ------------------------------------------------------------------------
 -- Other lemmas
 
 fromEquivalence : ∀ {b} → (T b → A) → (A → T b) → Reflects A b
-fromEquivalence {b = true}  sound complete = ofʸ (sound _)
-fromEquivalence {b = false} sound complete = ofⁿ complete
+fromEquivalence {b = true}  sound complete = of (sound _)
+fromEquivalence {b = false} sound complete = of complete
 
 -- `Reflects` is deterministic.
 det : ∀ {b b′} → Reflects A b → Reflects A b′ → b ≡ b′
diff --git a/src/Relation/Unary.agda b/src/Relation/Unary.agda
index 0f8a7534fd..3abc070ace 100644
--- a/src/Relation/Unary.agda
+++ b/src/Relation/Unary.agda
@@ -14,9 +14,8 @@ open import Data.Product.Base using (_×_; _,_; Σ-syntax; ∃; uncurry; swap)
 open import Data.Sum.Base using (_⊎_; [_,_])
 open import Function.Base using (_∘_; _|>_)
 open import Level using (Level; _⊔_; 0ℓ; suc; Lift)
-open import Relation.Nullary.Decidable.Core using (Dec; True)
-open import Relation.Nullary as Nullary using (¬_)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Relation.Nullary as Nullary using (¬_; Dec; True)
 
 private
   variable

From 891d50fe4f60923df9907df7c2c1b865dcf7341c Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 5 Jun 2024 16:46:06 +0100
Subject: [PATCH 299/316] Refactor `List.Membership.*` and
 `List.Relation.Unary.Any` (#2324)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* `contradiction` against `⊥-elim`

* tightened `import`s

* added two operations `∃∈` and `∀∈`

* added to `CHANGELOG`

* knock-on for the `Propositional` case

* refactor `lose∘find` and `find∘lose` in terms of new lemmas `∃∈-Any∘find` and `find∘∃∈-Any`

* `CHANGELOG`

* reorder

* knock-on viscosity

* knock-on viscosity; plus refactoring of `×↔` for intelligibility

* knock-on viscosity

* tightened `import`s

* misc. import and other tweaks

* misc. import and other tweaks

* misc. import and other tweaks

* removed spurious module name

* better definition of `find`

* make intermediate terms explicit in proof of `to∘from`

* tweaks

* Update Setoid.agda

Remove redundant import of `index` from `Any`

* Update Setoid.agda

Removed more redundant `import`ed operations

* Update Properties.agda

Another redundant `import`

* Update Properties.agda

Another redundant `import`ed operation

* `with` into `let`

* `with` into `let`

* `with` into `let`

* `with` into `let`

* indentation

* fix `universal-U`

* added `map-cong`

* deprecate `map-compose` in favour of `map-∘`

* explicit `let` in statement of `find∘map`

* knock-on viscosity: hide `map-cong`

* use of `Product.map₁`

* revert use of `Product.map₁`

* inlined lemmas!

* alpha conversion and further inlining!

* correctted use of `Product.map₂`

* added `syntax` declarations for the new combinators

* remove`⊥-elim`

* reordered `CHANGELOG`

* revise combinator names

* tighten `import`s

* tighten `import`s

* fixed merge conflict bug

* removed new syntax

* knock-on
---
 CHANGELOG.md                                  |  64 +++++---
 src/Data/List/Membership/Propositional.agda   |   4 +-
 .../Membership/Propositional/Properties.agda  |  79 +++++----
 .../Propositional/Properties/Core.agda        |  87 +++++-----
 src/Data/List/Membership/Setoid.agda          |  14 +-
 .../List/Membership/Setoid/Properties.agda    |  64 ++++----
 .../Relation/Binary/BagAndSetEquality.agda    |  24 +--
 src/Data/List/Relation/Unary/All.agda         |  23 +--
 .../List/Relation/Unary/All/Properties.agda   |  77 +++++----
 src/Data/List/Relation/Unary/Any.agda         |   5 +-
 .../List/Relation/Unary/Any/Properties.agda   | 154 ++++++++++--------
 11 files changed, 319 insertions(+), 276 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index e02ff2dc30..156756ff00 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -82,16 +82,9 @@ Deprecated names
   scanl-defn  ↦  Data.List.Scans.Properties.scanl-defn
   ```
 
-* In `Data.List.NonEmpty.Base`:
-  ```agda
-  inits : List A → List⁺ (List A)
-  tails : List A → List⁺ (List A)
-  ```
-
-* In `Data.List.NonEmpty.Properties`:
+* In `Data.List.Relation.Unary.All.Properties`:
   ```agda
-  toList-inits : toList ∘ List⁺.inits ≗ List.inits
-  toList-tails : toList ∘ List⁺.tails ≗ List.tails
+  map-compose  ↦  map-∘
   ```
 
 * In `Data.Nat.Divisibility.Core`:
@@ -412,6 +405,12 @@ Additions to existing modules
   tail∘tails : List A → List (List A)
   ```
 
+* In `Data.List.Membership.Propositional.Properties.Core`:
+  ```agda
+  find∘∃∈-Any : (p : ∃ λ x → x ∈ xs × P x) → find (∃∈-Any p) ≡ p
+  ∃∈-Any∘find : (p : Any P xs) → ∃∈-Any (find p) ≡ p
+  ```
+
 * In `Data.List.Membership.Setoid.Properties`:
   ```agda
   reverse⁺ : x ∈ xs → x ∈ reverse xs
@@ -424,6 +423,12 @@ Additions to existing modules
   tails : List A → List⁺ (List A)
   ```
 
+* In `Data.List.NonEmpty.Properties`:
+  ```agda
+  toList-inits : toList ∘ List⁺.inits ≗ List.inits
+  toList-tails : toList ∘ List⁺.tails ≗ List.tails
+  ```
+
 * In `Data.List.Properties`:
   ```agda
   length-catMaybes       : length (catMaybes xs) ≤ length xs
@@ -477,6 +482,21 @@ Additions to existing modules
   mapMaybeIsInj₂∘mapInj₁ : mapMaybe isInj₂ (map inj₁ xs) ≡ []
   ```
 
+* In `Data.List.Relation.Binary.Pointwise.Base`:
+  ```agda
+  unzip : Pointwise (R ; S) ⇒ (Pointwise R ; Pointwise S)
+  ```
+
+* In `Data.Maybe.Relation.Binary.Pointwise`:
+  ```agda
+  pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
+  ```
+
+* In `Data.List.Relation.Binary.Sublist.Setoid`:
+  ```agda
+  ⊆-upper-bound : ∀ {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) → UpperBound τ σ
+  ```
+
 * In `Data.List.Relation.Binary.Sublist.Setoid.Properties`:
   ```agda
   ⊆-trans-idˡ   : (trans-reflˡ : ∀ {x y} (p : x ≈ y) → trans ≈-refl p ≡ p) →
@@ -498,6 +518,11 @@ Additions to existing modules
   reverse⁻            : reverse as ⊆ reverse bs → as ⊆ bs
   ```
 
+* In `Data.List.Relation.Unary.All`:
+  ```agda
+  universal-U : Universal (All U)
+  ```
+
 * In `Data.List.Relation.Unary.All.Properties`:
   ```agda
   All-catMaybes⁺ : All (Maybe.All P) xs → All P (catMaybes xs)
@@ -510,6 +535,12 @@ Additions to existing modules
   tabulate⁺-< : (i < j → R (f i) (f j)) → AllPairs R (tabulate f)
   ```
 
+* In `Data.List.Relation.Unary.Any.Properties`:
+  ```agda
+  map-cong : (f g : P ⋐ Q) → (∀ {x} (p : P x) → f p ≡ g p) →
+             (p : Any P xs) → Any.map f p ≡ Any.map g p
+  ```
+
 * In `Data.List.Relation.Ternary.Appending.Setoid.Properties`:
   ```agda
   through→ : ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs →
@@ -536,21 +567,6 @@ Additions to existing modules
                          ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
   ```
 
-* In `Data.List.Relation.Binary.Pointwise.Base`:
-  ```agda
-  unzip : Pointwise (R ; S) ⇒ (Pointwise R ; Pointwise S)
-  ```
-
-* In `Data.Maybe.Relation.Binary.Pointwise`:
-  ```agda
-  pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
-  ```
-
-* In `Data.List.Relation.Binary.Sublist.Setoid`:
-  ```agda
-  ⊆-upper-bound : ∀ {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) → UpperBound τ σ
-  ```
-
 * In `Data.Nat.Divisibility`:
   ```agda
   quotient≢0       : m ∣ n → .{{NonZero n}} → NonZero quotient
diff --git a/src/Data/List/Membership/Propositional.agda b/src/Data/List/Membership/Propositional.agda
index c1ae08761e..2f5edc2d7e 100644
--- a/src/Data/List/Membership/Propositional.agda
+++ b/src/Data/List/Membership/Propositional.agda
@@ -10,7 +10,7 @@
 module Data.List.Membership.Propositional {a} {A : Set a} where
 
 open import Data.List.Relation.Unary.Any using (Any)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; subst)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; _≢_; resp; subst)
 open import Relation.Binary.PropositionalEquality.Properties using (setoid)
 
 import Data.List.Membership.Setoid as SetoidMembership
@@ -32,4 +32,4 @@ _≢∈_ x∈xs y∈xs = ∀ x≡y → subst (_∈ _) x≡y x∈xs ≢ y∈xs
 -- Other operations
 
 lose : ∀ {p} {P : A → Set p} {x xs} → x ∈ xs → P x → Any P xs
-lose = SetoidMembership.lose (setoid A) (subst _)
+lose = SetoidMembership.lose (setoid A) (resp _)
diff --git a/src/Data/List/Membership/Propositional/Properties.agda b/src/Data/List/Membership/Propositional/Properties.agda
index f93bf81f4a..03e31d8aae 100644
--- a/src/Data/List/Membership/Propositional/Properties.agda
+++ b/src/Data/List/Membership/Propositional/Properties.agda
@@ -10,26 +10,26 @@ module Data.List.Membership.Propositional.Properties where
 
 open import Algebra.Core using (Op₂)
 open import Algebra.Definitions using (Selective)
-open import Effect.Monad using (RawMonad)
-open import Data.Bool.Base using (Bool; false; true; T)
 open import Data.Fin.Base using (Fin)
 open import Data.List.Base as List
-open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
-open import Data.List.Relation.Unary.Any.Properties
-  using (map↔; concat↔; >>=↔; ⊛↔; Any-cong; ⊗↔′; ¬Any[])
+open import Data.List.Effectful using (monad)
 open import Data.List.Membership.Propositional
   using (_∈_; _∉_; mapWith∈; _≢∈_)
 import Data.List.Membership.Setoid.Properties as Membership
 open import Data.List.Relation.Binary.Equality.Propositional
   using (_≋_; ≡⇒≋; ≋⇒≡)
-open import Data.List.Effectful using (monad)
-open import Data.Nat.Base using (ℕ; zero; suc; pred; s≤s; _≤_; _<_; _≤ᵇ_)
-open import Data.Nat.Properties using (_≤?_; m≤n⇒m≤1+n; ≤ᵇ-reflects-≤; <⇒≢; ≰⇒>)
+open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
+open import Data.List.Relation.Unary.Any.Properties
+  using (map↔; concat↔; >>=↔; ⊛↔; Any-cong; ⊗↔′; ¬Any[])
+open import Data.Nat.Base using (ℕ; suc; s≤s; _≤_; _<_; _≰_)
+open import Data.Nat.Properties
+  using (suc-injective; m≤n⇒m≤1+n; _≤?_; <⇒≢; ≰⇒>)
 open import Data.Product.Base using (∃; ∃₂; _×_; _,_)
 open import Data.Product.Properties using (×-≡,≡↔≡)
 open import Data.Product.Function.NonDependent.Propositional using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
+open import Effect.Monad using (RawMonad)
 open import Function.Base using (_∘_; _∘′_; _$_; id; flip; _⟨_⟩_)
 open import Function.Definitions using (Injective)
 import Function.Related.Propositional as Related
@@ -40,15 +40,14 @@ open import Level using (Level)
 open import Relation.Binary.Core using (Rel)
 open import Relation.Binary.Definitions as Binary hiding (Decidable)
 open import Relation.Binary.PropositionalEquality.Core as ≡
-  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; subst; _≗_)
+  using (_≡_; _≢_; refl; sym; trans; cong; cong₂; resp; _≗_)
 open import Relation.Binary.PropositionalEquality.Properties as ≡ using (setoid)
 import Relation.Binary.Properties.DecTotalOrder as DTOProperties
-open import Relation.Unary using (_⟨×⟩_; Decidable)
-import Relation.Nullary.Reflects as Reflects
+open import Relation.Nullary.Decidable.Core
+  using (Dec; yes; no; ¬¬-excluded-middle)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary using (¬_; Dec; does; yes; no; _because_)
-open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Nullary.Decidable using (¬¬-excluded-middle)
+open import Relation.Unary using (_⟨×⟩_; Decidable)
 
 private
   open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})
@@ -128,8 +127,9 @@ module _ {v : A} where
   ∈-insert xs = Membership.∈-insert (≡.setoid A) xs refl
 
   ∈-∃++ : ∀ {xs} → v ∈ xs → ∃₂ λ ys zs → xs ≡ ys ++ [ v ] ++ zs
-  ∈-∃++ v∈xs with Membership.∈-∃++ (≡.setoid A) v∈xs
-  ... | ys , zs , _ , refl , eq = ys , zs , ≋⇒≡ eq
+  ∈-∃++ v∈xs
+    with ys , zs , _ , refl , eq ← Membership.∈-∃++ (≡.setoid A) v∈xs
+    = ys , zs , ≋⇒≡ eq
 
 ------------------------------------------------------------------------
 -- concat
@@ -147,8 +147,9 @@ module _ {v : A} where
     Membership.∈-concat⁺′ (≡.setoid A) v∈vs (Any.map ≡⇒≋ vs∈xss)
 
   ∈-concat⁻′ : ∀ xss → v ∈ concat xss → ∃ λ xs → v ∈ xs × xs ∈ xss
-  ∈-concat⁻′ xss v∈c with Membership.∈-concat⁻′ (≡.setoid A) xss v∈c
-  ... | xs , v∈xs , xs∈xss = xs , v∈xs , Any.map ≋⇒≡ xs∈xss
+  ∈-concat⁻′ xss v∈c =
+    let xs , v∈xs , xs∈xss = Membership.∈-concat⁻′ (≡.setoid A) xss v∈c
+    in xs , v∈xs , Any.map ≋⇒≡ xs∈xss
 
   concat-∈↔ : ∀ {xss : List (List A)} →
               (∃ λ xs → v ∈ xs × xs ∈ xss) ↔ v ∈ concat xss
@@ -183,8 +184,9 @@ module _ (f : A → B → C) where
 
 ∈-cartesianProduct⁻ : ∀ xs ys {xy@(x , y) : A × B} →
                       xy ∈ cartesianProduct xs ys → x ∈ xs × y ∈ ys
-∈-cartesianProduct⁻ xs ys xy∈p[xs,ys] with ∈-cartesianProductWith⁻ _,_ xs ys xy∈p[xs,ys]
-... | (x , y , x∈xs , y∈ys , refl) = x∈xs , y∈ys
+∈-cartesianProduct⁻ xs ys xy∈p[xs,ys]
+  with _ , _ , x∈xs , y∈ys , refl ← ∈-cartesianProductWith⁻ _,_ xs ys xy∈p[xs,ys]
+  = x∈xs , y∈ys
 
 ------------------------------------------------------------------------
 -- applyUpTo
@@ -205,8 +207,7 @@ module _ (f : ℕ → A) where
 ∈-upTo⁺ = ∈-applyUpTo⁺ id
 
 ∈-upTo⁻ : ∀ {n i} → i ∈ upTo n → i < n
-∈-upTo⁻ p with ∈-applyUpTo⁻ id p
-... | _ , i<n , refl = i<n
+∈-upTo⁻ p with _ , i<n , refl ← ∈-applyUpTo⁻ id p = i<n
 
 ------------------------------------------------------------------------
 -- applyDownFrom
@@ -227,8 +228,7 @@ module _ (f : ℕ → A) where
 ∈-downFrom⁺ i<n = ∈-applyDownFrom⁺ id i<n
 
 ∈-downFrom⁻ : ∀ {n i} → i ∈ downFrom n → i < n
-∈-downFrom⁻ p with ∈-applyDownFrom⁻ id p
-... | _ , i<n , refl = i<n
+∈-downFrom⁻ p with _ , i<n , refl ← ∈-applyDownFrom⁻ id p = i<n
 
 ------------------------------------------------------------------------
 -- tabulate
@@ -247,10 +247,10 @@ module _ {n} {f : Fin n → A} where
 module _ {p} {P : A → Set p} (P? : Decidable P) where
 
   ∈-filter⁺ : ∀ {x xs} → x ∈ xs → P x → x ∈ filter P? xs
-  ∈-filter⁺ = Membership.∈-filter⁺ (≡.setoid A) P? (subst P)
+  ∈-filter⁺ = Membership.∈-filter⁺ (≡.setoid A) P? (≡.resp P)
 
   ∈-filter⁻ : ∀ {v xs} → v ∈ filter P? xs → v ∈ xs × P v
-  ∈-filter⁻ = Membership.∈-filter⁻ (≡.setoid A) P? (subst P)
+  ∈-filter⁻ = Membership.∈-filter⁻ (≡.setoid A) P? (≡.resp P)
 
 ------------------------------------------------------------------------
 -- derun and deduplicate
@@ -310,7 +310,7 @@ module _ (_≈?_ : DecidableEquality A) where
 ------------------------------------------------------------------------
 -- length
 
-∈-length : ∀ {x : A} {xs} → x ∈ xs → 1 ≤ length xs
+∈-length : ∀ {x : A} {xs} → x ∈ xs → 0 < length xs
 ∈-length = Membership.∈-length (≡.setoid _)
 
 ------------------------------------------------------------------------
@@ -365,28 +365,27 @@ finite inj (x ∷ xs) fᵢ∈x∷xs = ¬¬-excluded-middle helper
   helper (yes (i , fᵢ≡x)) = finite f′-inj xs f′ⱼ∈xs
     where
     f′ : ℕ → _
-    f′ j with does (i ≤? j)
-    ... | true  = f (suc j)
-    ... | false = f j
+    f′ j with i ≤? j
+    ... | yes _ = f (suc j)
+    ... | no  _ = f j
 
     ∈-if-not-i : ∀ {j} → i ≢ j → f j ∈ xs
     ∈-if-not-i i≢j = not-x (i≢j ∘ f-inj ∘ trans fᵢ≡x ∘ sym)
 
-    lemma : ∀ {k j} → i ≤ j → ¬ (i ≤ k) → suc j ≢ k
+    lemma : ∀ {k j} → i ≤ j → i ≰ k → suc j ≢ k
     lemma i≤j i≰1+j refl = i≰1+j (m≤n⇒m≤1+n i≤j)
 
     f′ⱼ∈xs : ∀ j → f′ j ∈ xs
-    f′ⱼ∈xs j with i ≤ᵇ j | Reflects.invert (≤ᵇ-reflects-≤ i j)
-    ... | true  | p = ∈-if-not-i (<⇒≢ (s≤s p))
-    ... | false | p = ∈-if-not-i (<⇒≢ (≰⇒> p) ∘ sym)
+    f′ⱼ∈xs j with i ≤? j
+    ... | yes i≤j = ∈-if-not-i (<⇒≢ (s≤s i≤j))
+    ... | no  i≰j = ∈-if-not-i (<⇒≢ (≰⇒> i≰j) ∘ sym)
 
     f′-injective′ : Injective _≡_ _≡_ f′
-    f′-injective′ {j} {k} eq with i ≤ᵇ j | Reflects.invert (≤ᵇ-reflects-≤ i j)
-                                | i ≤ᵇ k | Reflects.invert (≤ᵇ-reflects-≤ i k)
-    ... | true  | p | true  | q = ≡.cong pred (f-inj eq)
-    ... | true  | p | false | q = contradiction (f-inj eq) (lemma p q)
-    ... | false | p | true  | q = contradiction (f-inj eq) (lemma q p ∘ sym)
-    ... | false | p | false | q = f-inj eq
+    f′-injective′ {j} {k} eq with i ≤? j | i ≤? k
+    ... | yes i≤j | yes i≤k = suc-injective (f-inj eq)
+    ... | yes i≤j | no  i≰k = contradiction (f-inj eq) (lemma i≤j i≰k)
+    ... | no  i≰j | yes i≤k = contradiction (f-inj eq) (lemma i≤k i≰j ∘ sym)
+    ... | no  i≰j | no  i≰k = f-inj eq
 
     f′-inj : ℕ ↣ _
     f′-inj = record
diff --git a/src/Data/List/Membership/Propositional/Properties/Core.agda b/src/Data/List/Membership/Propositional/Properties/Core.agda
index ca8ffbb1ea..5c77e40b96 100644
--- a/src/Data/List/Membership/Propositional/Properties/Core.agda
+++ b/src/Data/List/Membership/Propositional/Properties/Core.agda
@@ -12,75 +12,74 @@
 
 module Data.List.Membership.Propositional.Properties.Core where
 
-open import Function.Base using (flip; id; _∘_)
-open import Function.Bundles
 open import Data.List.Base using (List)
-open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional
-open import Data.Product.Base as Product
-  using (_,_; proj₁; proj₂; uncurry′; ∃; _×_)
+open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
+open import Data.Product.Base as Product using (_,_; ∃; _×_)
+open import Function.Base using (flip; id; _∘_)
+open import Function.Bundles using (_↔_; mk↔ₛ′)
 open import Level using (Level)
 open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; cong; subst)
+  using (_≡_; refl; cong; resp)
 open import Relation.Unary using (Pred; _⊆_)
 
 private
   variable
     a p q : Level
     A : Set a
-
-------------------------------------------------------------------------
--- Lemmas relating map and find.
-
-map∘find : ∀ {P : Pred A p} {xs}
-           (p : Any P xs) → let p′ = find p in
-           {f : _≡_ (proj₁ p′) ⊆ P} →
-           f refl ≡ proj₂ (proj₂ p′) →
-           Any.map f (proj₁ (proj₂ p′)) ≡ p
-map∘find (here  p) hyp = cong here  hyp
-map∘find (there p) hyp = cong there (map∘find p hyp)
-
-find∘map : ∀ {P : Pred A p} {Q : Pred A q}
-           {xs : List A} (p : Any P xs) (f : P ⊆ Q) →
-           find (Any.map f p) ≡ Product.map id (Product.map id f) (find p)
-find∘map (here  p) f = refl
-find∘map (there p) f rewrite find∘map p f = refl
+    x : A
+    xs : List A
 
 ------------------------------------------------------------------------
 -- find satisfies a simple equality when the predicate is a
 -- propositional equality.
 
-find-∈ : ∀ {x : A} {xs : List A} (x∈xs : x ∈ xs) →
-         find x∈xs ≡ (x , x∈xs , refl)
+find-∈ : (x∈xs : x ∈ xs) → find x∈xs ≡ (x , x∈xs , refl)
 find-∈ (here refl)  = refl
 find-∈ (there x∈xs) rewrite find-∈ x∈xs = refl
 
 ------------------------------------------------------------------------
--- find and lose are inverses (more or less).
+-- Lemmas relating map and find.
 
-lose∘find : ∀ {P : Pred A p} {xs : List A}
-            (p : Any P xs) →
-            uncurry′ lose (proj₂ (find p)) ≡ p
-lose∘find p = map∘find p refl
+module _ {P : Pred A p} where
 
-find∘lose : ∀ (P : Pred A p) {x xs}
-            (x∈xs : x ∈ xs) (pp : P x) →
-            find {P = P} (lose x∈xs pp) ≡ (x , x∈xs , pp)
-find∘lose P x∈xs p
-  rewrite find∘map x∈xs (flip (subst P) p)
-        | find-∈ x∈xs
-        = refl
+  map∘find : (p : Any P xs) → let x , x∈xs , px = find p in
+             {f : (x ≡_) ⊆ P} → f refl ≡ px →
+             Any.map f x∈xs ≡ p
+  map∘find (here  p) hyp = cong here  hyp
+  map∘find (there p) hyp = cong there (map∘find p hyp)
+
+  find∘map : ∀ {Q : Pred A q} {xs} (p : Any P xs) (f : P ⊆ Q) →
+             let x , x∈xs , px = find p in
+             find (Any.map f p) ≡ (x , x∈xs , f px)
+  find∘map (here  p) f = refl
+  find∘map (there p) f rewrite find∘map p f = refl
 
 ------------------------------------------------------------------------
 -- Any can be expressed using _∈_
 
 module _ {P : Pred A p} where
 
-  ∃∈-Any : ∀  {xs} → (∃ λ x → x ∈ xs × P x) → Any P xs
-  ∃∈-Any = uncurry′ lose ∘ proj₂
+  ∃∈-Any : (∃ λ x → x ∈ xs × P x) → Any P xs
+  ∃∈-Any (x , x∈xs , px) = lose {P = P} x∈xs px
+
+  ∃∈-Any∘find : (p : Any P xs) → ∃∈-Any (find p) ≡ p
+  ∃∈-Any∘find p = map∘find p refl
+
+  find∘∃∈-Any : (p : ∃ λ x → x ∈ xs × P x) → find (∃∈-Any p) ≡ p
+  find∘∃∈-Any p@(x , x∈xs , px)
+    rewrite find∘map x∈xs (flip (resp P) px) | find-∈ x∈xs = refl
 
-  Any↔ : ∀ {xs} → (∃ λ x → x ∈ xs × P x) ↔ Any P xs
-  Any↔ = mk↔ₛ′ ∃∈-Any find lose∘find from∘to
-    where
-    from∘to : ∀ v → find (∃∈-Any v) ≡ v
-    from∘to p = find∘lose _ (proj₁ (proj₂ p)) (proj₂ (proj₂ p))
+  Any↔ : (∃ λ x → x ∈ xs × P x) ↔ Any P xs
+  Any↔ = mk↔ₛ′ ∃∈-Any find ∃∈-Any∘find find∘∃∈-Any
+
+------------------------------------------------------------------------
+-- Hence, find and lose are inverses (more or less).
+
+lose∘find : ∀ {P : Pred A p} {xs} (p : Any P xs) → ∃∈-Any (find p) ≡ p
+lose∘find = ∃∈-Any∘find
+
+find∘lose : ∀ (P : Pred A p) {x xs}
+            (x∈xs : x ∈ xs) (px : P x) →
+            find (lose {P = P} x∈xs px) ≡ (x , x∈xs , px)
+find∘lose P {x} x∈xs px = find∘∃∈-Any (x , x∈xs , px)
diff --git a/src/Data/List/Membership/Setoid.agda b/src/Data/List/Membership/Setoid.agda
index cadc600646..b1ff4f2773 100644
--- a/src/Data/List/Membership/Setoid.agda
+++ b/src/Data/List/Membership/Setoid.agda
@@ -7,17 +7,17 @@
 {-# OPTIONS --cubical-compatible --safe #-}
 
 open import Relation.Binary.Bundles using (Setoid)
-open import Relation.Binary.Definitions using (_Respects_)
 
 module Data.List.Membership.Setoid {c ℓ} (S : Setoid c ℓ) where
 
-open import Function.Base using (_∘_; id; flip)
-open import Data.List.Base as List using (List; []; _∷_; length; lookup)
+open import Data.List.Base using (List; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any
-  using (Any; index; map; here; there)
+  using (Any; map; here; there)
 open import Data.Product.Base as Product using (∃; _×_; _,_)
-open import Relation.Unary using (Pred)
+open import Function.Base using (_∘_; flip; const)
+open import Relation.Binary.Definitions using (_Respects_)
 open import Relation.Nullary.Negation using (¬_)
+open import Relation.Unary using (Pred)
 
 open Setoid S renaming (Carrier to A)
 
@@ -49,8 +49,8 @@ mapWith∈ (x ∷ xs) f = f (here refl) ∷ mapWith∈ xs (f ∘ there)
 module _ {p} {P : Pred A p} where
 
   find : ∀ {xs} → Any P xs → ∃ λ x → x ∈ xs × P x
-  find (here px)   = (_ , here refl , px)
-  find (there pxs) = Product.map id (Product.map there id) (find pxs)
+  find (here px)   = _ , here refl , px
+  find (there pxs) = let x , x∈xs , px = find pxs in x , there x∈xs , px
 
   lose : P Respects _≈_ →  ∀ {x xs} → x ∈ xs → P x → Any P xs
   lose resp x∈xs px = map (flip resp px) x∈xs
diff --git a/src/Data/List/Membership/Setoid/Properties.agda b/src/Data/List/Membership/Setoid/Properties.agda
index a0fcb4881b..48303dec37 100644
--- a/src/Data/List/Membership/Setoid/Properties.agda
+++ b/src/Data/List/Membership/Setoid/Properties.agda
@@ -13,15 +13,14 @@ open import Data.Bool.Base using (true; false)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Fin.Properties using (suc-injective)
 open import Data.List.Base hiding (find)
-open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
-open import Data.List.Relation.Unary.All as All using (All)
-import Data.List.Relation.Unary.Any.Properties as Any
 import Data.List.Membership.Setoid as Membership
 import Data.List.Relation.Binary.Equality.Setoid as Equality
+open import Data.List.Relation.Unary.All as All using (All)
+open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
+import Data.List.Relation.Unary.Any.Properties as Any
 import Data.List.Relation.Unary.Unique.Setoid as Unique
-open import Data.Nat.Base using (suc; z≤n; s≤s; _≤_; _<_)
-open import Data.Nat.Properties using (≤-trans; n≤1+n)
-open import Data.Product.Base as Product using (∃; _×_; _,_ ; ∃₂; proj₁; proj₂)
+open import Data.Nat.Base using (suc; z<s; _<_)
+open import Data.Product.Base as Product using (∃; _×_; _,_ ; ∃₂)
 open import Data.Product.Relation.Binary.Pointwise.NonDependent using (_×ₛ_)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
 open import Function.Base using (_$_; flip; _∘_; _∘′_; id)
@@ -31,11 +30,11 @@ open import Relation.Binary.Core using (Rel; _Preserves₂_⟶_⟶_; _Preserves_
 open import Relation.Binary.Definitions as Binary hiding (Decidable)
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
-open import Relation.Unary as Unary using (Decidable; Pred)
-open import Relation.Nullary using (¬_; does; _because_; yes; no)
+open import Relation.Nullary.Decidable using (does; _because_; yes; no)
+open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Nullary.Decidable using (¬?)
+open import Relation.Unary as Unary using (Decidable; Pred)
+
 open Setoid using (Carrier)
 
 private
@@ -148,24 +147,24 @@ module _ (S : Setoid c ℓ) where
 
 module _ (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) where
 
-  open Setoid S₁ renaming (Carrier to A₁; _≈_ to _≈₁_; refl to refl₁)
-  open Setoid S₂ renaming (Carrier to A₂; _≈_ to _≈₂_)
+  open Setoid S₁ renaming (_≈_ to _≈₁_)
+  open Setoid S₂ renaming (_≈_ to _≈₂_)
   private module M₁ = Membership S₁; open M₁ using (find) renaming (_∈_ to _∈₁_)
   private module M₂ = Membership S₂; open M₂ using () renaming (_∈_ to _∈₂_)
 
-  ∈-map⁺ : ∀ {f} → f Preserves _≈₁_ ⟶ _≈₂_ → ∀ {v xs} →
-            v ∈₁ xs → f v ∈₂ map f xs
+  ∈-map⁺ : ∀ {f} → f Preserves _≈₁_ ⟶ _≈₂_ →
+           ∀ {v xs} → v ∈₁ xs → f v ∈₂ map f xs
   ∈-map⁺ pres x∈xs = Any.map⁺ (Any.map pres x∈xs)
 
   ∈-map⁻ : ∀ {v xs f} → v ∈₂ map f xs →
            ∃ λ x → x ∈₁ xs × v ≈₂ f x
   ∈-map⁻ x∈map = find (Any.map⁻ x∈map)
 
-  map-∷= : ∀ {f} (f≈ : f Preserves _≈₁_ ⟶ _≈₂_)
-           {xs x v} → (x∈xs : x ∈₁ xs) →
-           map f (x∈xs M₁.∷= v) ≡ ∈-map⁺ f≈ x∈xs M₂.∷= f v
-  map-∷= f≈ (here x≈y)   = ≡.refl
-  map-∷= f≈ (there x∈xs) = ≡.cong (_ ∷_) (map-∷= f≈ x∈xs)
+  map-∷= : ∀ {f} (pres : f Preserves _≈₁_ ⟶ _≈₂_) →
+           ∀ {xs x v} → (x∈xs : x ∈₁ xs) →
+           map f (x∈xs M₁.∷= v) ≡ ∈-map⁺ pres x∈xs M₂.∷= f v
+  map-∷= pres (here x≈y)   = ≡.refl
+  map-∷= pres (there x∈xs) = ≡.cong (_ ∷_) (map-∷= pres x∈xs)
 
 ------------------------------------------------------------------------
 -- _++_
@@ -211,9 +210,10 @@ module _ (S : Setoid c ℓ) where
 
   ∈-∃++ : ∀ {v xs} → v ∈ xs → ∃₂ λ ys zs → ∃ λ w →
           v ≈ w × xs ≋ ys ++ [ w ] ++ zs
-  ∈-∃++ (here px)                  = [] , _ , _ , px , ≋-refl
-  ∈-∃++ (there {d} v∈xs) with ∈-∃++ v∈xs
-  ... | hs , _ , _ , v≈v′ , eq = d ∷ hs , _ , _ , v≈v′ , refl ∷ eq
+  ∈-∃++ (here px)        = [] , _ , _ , px , ≋-refl
+  ∈-∃++ (there {d} v∈xs) =
+    let hs , _ , _ , v≈v′ , eq = ∈-∃++ v∈xs
+    in d ∷ hs , _ , _ , v≈v′ , refl ∷ eq
 
 ------------------------------------------------------------------------
 -- concat
@@ -235,8 +235,8 @@ module _ (S : Setoid c ℓ) where
   ∈-concat⁺′ v∈vs = ∈-concat⁺ ∘ Any.map (flip (∈-resp-≋ S) v∈vs)
 
   ∈-concat⁻′ : ∀ {v} xss → v ∈ concat xss → ∃ λ xs → v ∈ xs × xs ∈ₗ xss
-  ∈-concat⁻′ xss v∈c[xss] with find (∈-concat⁻ xss v∈c[xss])
-  ... | xs , t , s = xs , s , t
+  ∈-concat⁻′ xss v∈c[xss] =
+    let xs , xs∈xss , v∈xs = find (∈-concat⁻ xss v∈c[xss]) in xs , v∈xs , xs∈xss
 
 ------------------------------------------------------------------------
 -- reverse
@@ -271,10 +271,12 @@ module _ (S₁ : Setoid c₁ ℓ₁) (S₂ : Setoid c₂ ℓ₂) (S₃ : Setoid
   ∈-cartesianProductWith⁻ : ∀ f xs ys {v} → v ∈₃ cartesianProductWith f xs ys →
                             ∃₂ λ a b → a ∈₁ xs × b ∈₂ ys × v ≈₃ f a b
   ∈-cartesianProductWith⁻ f (x ∷ xs) ys v∈c with ∈-++⁻ S₃ (map (f x) ys) v∈c
-  ∈-cartesianProductWith⁻ f (x ∷ xs) ys v∈c | inj₁ v∈map with ∈-map⁻ S₂ S₃ v∈map
-  ... | (b , b∈ys , v≈fxb) = x , b , here refl₁ , b∈ys , v≈fxb
-  ∈-cartesianProductWith⁻ f (x ∷ xs) ys v∈c | inj₂ v∈com with ∈-cartesianProductWith⁻ f xs ys v∈com
-  ... | (a , b , a∈xs , b∈ys , v≈fab) = a , b , there a∈xs , b∈ys , v≈fab
+  ... | inj₁ v∈map =
+    let b , b∈ys , v≈fxb = ∈-map⁻ S₂ S₃ v∈map
+    in x , b , here refl₁ , b∈ys , v≈fxb
+  ... | inj₂ v∈com =
+    let a , b , a∈xs , b∈ys , v≈fab = ∈-cartesianProductWith⁻ f xs ys v∈com
+    in  a , b , there a∈xs , b∈ys , v≈fab
 
 ------------------------------------------------------------------------
 -- cartesianProduct
@@ -388,9 +390,9 @@ module _ (S : Setoid c ℓ) where
 
   open Membership S using (_∈_)
 
-  ∈-length : ∀ {x xs} → x ∈ xs → 1 ≤ length xs
-  ∈-length (here px)    = s≤s z≤n
-  ∈-length (there x∈xs) = ≤-trans (∈-length x∈xs) (n≤1+n _)
+  ∈-length : ∀ {x xs} → x ∈ xs → 0 < length xs
+  ∈-length (here px)    = z<s
+  ∈-length (there x∈xs) = z<s
 
 ------------------------------------------------------------------------
 -- lookup
diff --git a/src/Data/List/Relation/Binary/BagAndSetEquality.agda b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
index 9c400fdcff..815b27db89 100644
--- a/src/Data/List/Relation/Binary/BagAndSetEquality.agda
+++ b/src/Data/List/Relation/Binary/BagAndSetEquality.agda
@@ -30,7 +30,7 @@ open import Data.List.Relation.Binary.Permutation.Propositional
   using (_↭_; ↭-sym; refl; module PermutationReasoning)
 open import Data.List.Relation.Binary.Permutation.Propositional.Properties
   using (∈-resp-↭; ∈-resp-[σ⁻¹∘σ]; ↭-sym-involutive; shift; ++-comm)
-open import Data.Product.Base as Prod using (∃; _,_; proj₁; proj₂; _×_)
+open import Data.Product.Base as Product using (∃; _,_; proj₁; proj₂; _×_)
 import Data.Product.Function.Dependent.Propositional as Σ
 open import Data.Sum.Base as Sum using (_⊎_; [_,_]′; inj₁; inj₂)
 open import Data.Sum.Properties using (inj₂-injective; inj₁-injective)
@@ -396,13 +396,13 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
     index-of (to xs≈ys (proj₂
       (from (Fin-length xs) (to (Fin-length xs) (z , p)))))   ≡⟨⟩
 
-    index-of (proj₂ (Prod.map id (to xs≈ys)
-      (from (Fin-length xs) (to (Fin-length xs) (z , p)))))  ≡⟨⟩
+    index-of (proj₂ (Product.map₂ (to xs≈ys)
+      (from (Fin-length xs) (to (Fin-length xs) (z , p)))))   ≡⟨⟩
 
-    to (Fin-length ys) (Prod.map id (to xs≈ys)
-      (from (Fin-length xs) (index-of p)))                          ≡⟨⟩
+    to (Fin-length ys) (Product.map₂ (to xs≈ys)
+      (from (Fin-length xs) (index-of p)))                    ≡⟨⟩
 
-    to (Fin-length-cong xs≈ys) (index-of p)                        ∎
+    to (Fin-length-cong xs≈ys) (index-of p)                   ∎
     where
     open ≡-Reasoning
     open Inverse
@@ -429,10 +429,10 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
     index-of (Inverse.to xs≈ys p) ≡
     index-of (Inverse.to xs≈ys q)
   index-equality-preserved {p = p} {q} xs≈ys eq =
-    index-of (Inverse.to xs≈ys p)                  ≡⟨ index-of-commutes xs≈ys p ⟩
-    Inverse.to (Fin-length-cong xs≈ys) (index-of p)  ≡⟨ ≡.cong (Inverse.to (Fin-length-cong xs≈ys)) eq ⟩
-    Inverse.to (Fin-length-cong xs≈ys) (index-of q)  ≡⟨ ≡.sym $ index-of-commutes xs≈ys q ⟩
-    index-of (Inverse.to xs≈ys q)                  ∎
+    index-of (Inverse.to xs≈ys p)                   ≡⟨ index-of-commutes xs≈ys p ⟩
+    Inverse.to (Fin-length-cong xs≈ys) (index-of p) ≡⟨ ≡.cong (Inverse.to (Fin-length-cong xs≈ys)) eq ⟩
+    Inverse.to (Fin-length-cong xs≈ys) (index-of q) ≡⟨ ≡.sym $ index-of-commutes xs≈ys q ⟩
+    index-of (Inverse.to xs≈ys q)                   ∎
     where
     open ≡-Reasoning
 
@@ -520,13 +520,13 @@ drop-cons {x = x} {xs} {ys} x∷xs≈x∷ys =
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₁ a″ , eq₄ = ⊥-elim $ from-hyp eq₃ eq₄
       g∘g′ | inj₁ a , eq₁ | inj₂ c  , eq₂ | inj₁ a′ , eq₃ | inj₂ b′ , eq₄ = inj₂-injective (
         let lemma =
-              inj₁ a′             ≡⟨ ≡.sym eq₃ ⟩
+              inj₁ a′            ≡⟨ ≡.sym eq₃ ⟩
               from f (inj₂ c)    ≡⟨ to-from f eq₂ ⟩
               inj₁ a             ∎
         in
         inj₂ b′             ≡⟨ ≡.sym eq₄ ⟩
         from f (inj₁ a′)    ≡⟨ ≡.cong (from f ∘ inj₁) $ inj₁-injective lemma ⟩
-        from f (inj₁ a)    ≡⟨ to-from f eq₁ ⟩
+        from f (inj₁ a)     ≡⟨ to-from f eq₁ ⟩
         inj₂ b              ∎)
 
   -- Some final lemmas.
diff --git a/src/Data/List/Relation/Unary/All.agda b/src/Data/List/Relation/Unary/All.agda
index bd8677630b..ea2ab858df 100644
--- a/src/Data/List/Relation/Unary/All.agda
+++ b/src/Data/List/Relation/Unary/All.agda
@@ -8,9 +8,6 @@
 
 module Data.List.Relation.Unary.All where
 
-open import Effect.Applicative
-open import Effect.Monad
-open import Data.Empty using (⊥)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
 open import Data.List.Membership.Propositional renaming (_∈_ to _∈ₚ_)
@@ -18,11 +15,14 @@ import Data.List.Membership.Setoid as SetoidMembership
 open import Data.Product.Base as Product
   using (∃; -,_; _×_; _,_; proj₁; proj₂; uncurry)
 open import Data.Sum.Base as Sum using (inj₁; inj₂)
+open import Effect.Applicative
+open import Effect.Monad
 open import Function.Base using (_∘_; _∘′_; id; const)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary hiding (Irrelevant)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Unary hiding (_∈_)
+import Relation.Unary.Properties as Unary
 open import Relation.Binary.Bundles using (Setoid)
 open import Relation.Binary.Definitions using (_Respects_)
 open import Relation.Binary.PropositionalEquality.Core as ≡
@@ -68,7 +68,7 @@ data _[_]=_ {A : Set a} {P : Pred A p} :
 -- A list is empty if having an element is impossible.
 
 Null : Pred (List A) _
-Null = All (λ _ → ⊥)
+Null = All ∅
 
 ------------------------------------------------------------------------
 -- Operations on All
@@ -116,17 +116,17 @@ unzip : All (P ∩ Q) ⊆ All P ∩ All Q
 unzip = unzipWith id
 
 module _(S : Setoid a ℓ) {P : Pred (Setoid.Carrier S) p} where
-  open Setoid S renaming (Carrier to C; refl to refl₁)
+  open Setoid S renaming (refl to ≈-refl)
   open SetoidMembership S
 
   tabulateₛ : (∀ {x} → x ∈ xs → P x) → All P xs
-  tabulateₛ {[]}     hyp = []
-  tabulateₛ {x ∷ xs} hyp = hyp (here refl₁) ∷ tabulateₛ (hyp ∘ there)
+  tabulateₛ {[]}    hyp = []
+  tabulateₛ {_ ∷ _} hyp = hyp (here ≈-refl) ∷ tabulateₛ (hyp ∘ there)
 
 tabulate : (∀ {x} → x ∈ₚ xs → P x) → All P xs
 tabulate = tabulateₛ (≡.setoid _)
 
-self : ∀ {xs : List A} → All (const A) xs
+self : ∀ {xs} → All (const A) xs
 self = tabulate (λ {x} _ → x)
 
 ------------------------------------------------------------------------
@@ -195,11 +195,11 @@ lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
 lookup pxs = lookupWith (λ { px refl → px }) pxs
 
 module _(S : Setoid a ℓ) {P : Pred (Setoid.Carrier S) p} where
-  open Setoid S renaming (sym to sym₁)
+  open Setoid S renaming (sym to ≈-sym)
   open SetoidMembership S
 
   lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)
-  lookupₛ resp pxs = lookupWith (λ py x=y → resp (sym₁ x=y) py) pxs
+  lookupₛ resp pxs = lookupWith (λ py x≈y → resp (≈-sym x≈y) py) pxs
 
 ------------------------------------------------------------------------
 -- Properties of predicates preserved by All
@@ -212,6 +212,9 @@ universal : Universal P → Universal (All P)
 universal u []       = []
 universal u (x ∷ xs) = u x ∷ universal u xs
 
+universal-U : Universal (All {A = A} U)
+universal-U = universal Unary.U-Universal
+
 irrelevant : Irrelevant P → Irrelevant (All P)
 irrelevant irr []           []           = ≡.refl
 irrelevant irr (px₁ ∷ pxs₁) (px₂ ∷ pxs₂) =
diff --git a/src/Data/List/Relation/Unary/All/Properties.agda b/src/Data/List/Relation/Unary/All/Properties.agda
index c77df039d2..9e3c4968c8 100644
--- a/src/Data/List/Relation/Unary/All/Properties.agda
+++ b/src/Data/List/Relation/Unary/All/Properties.agda
@@ -11,23 +11,22 @@ module Data.List.Relation.Unary.All.Properties where
 open import Axiom.Extensionality.Propositional using (Extensionality)
 open import Data.Bool.Base using (Bool; T; true; false)
 open import Data.Bool.Properties using (T-∧)
-open import Data.Empty using (⊥-elim)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List hiding (lookup; updateAt)
-open import Data.List.Properties as Listₚ using (partition-defn)
 open import Data.List.Membership.Propositional using (_∈_; _≢∈_)
 open import Data.List.Membership.Propositional.Properties
   using (there-injective-≢∈; ∈-filter⁻)
 import Data.List.Membership.Setoid as SetoidMembership
+import Data.List.Properties as List
+import Data.List.Relation.Binary.Equality.Setoid as ≋
+open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
+open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_)
 open import Data.List.Relation.Unary.All as All using
   ( All; []; _∷_; lookup; updateAt
   ; _[_]=_; here; there
   ; Null
   )
 open import Data.List.Relation.Unary.Any as Any using (Any; here; there)
-import Data.List.Relation.Binary.Equality.Setoid as ListEq using (_≋_; []; _∷_)
-open import Data.List.Relation.Binary.Pointwise.Base using (Pointwise; []; _∷_)
-open import Data.List.Relation.Binary.Subset.Propositional using (_⊆_)
 open import Data.Maybe.Base as Maybe using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.All as Maybe using (just; nothing; fromAny)
 open import Data.Maybe.Relation.Unary.Any as Maybe using (just)
@@ -98,7 +97,7 @@ All¬⇒¬Any (¬p ∷ _)  (here  p) = ¬p p
 All¬⇒¬Any (_  ∷ ¬p) (there p) = All¬⇒¬Any ¬p p
 
 ¬All⇒Any¬ : Decidable P → ∀ xs → ¬ All P xs → Any (¬_ ∘ P) xs
-¬All⇒Any¬ dec []       ¬∀ = ⊥-elim (¬∀ [])
+¬All⇒Any¬ dec []       ¬∀ = contradiction [] ¬∀
 ¬All⇒Any¬ dec (x ∷ xs) ¬∀ with dec x
 ... |  true because  [p] = there (¬All⇒Any¬ dec xs (¬∀ ∘ _∷_ (invert [p])))
 ... | false because [¬p] = here (invert [¬p])
@@ -133,7 +132,7 @@ private
   -- equivalence could be strengthened to a surjection.
   to∘from : Extensionality _ _ → (dec : Decidable P) →
             (¬∀ : ¬ All P xs) → Any¬⇒¬All (¬All⇒Any¬ dec xs ¬∀) ≡ ¬∀
-  to∘from ext P ¬∀ = ext (⊥-elim ∘ ¬∀)
+  to∘from ext P ¬∀ = ext λ ∀P → contradiction ∀P ¬∀
 
 module _ {_~_ : REL A B ℓ} where
 
@@ -177,19 +176,19 @@ lookup⇒[]= pxs i refl = []=lookup pxs i
 ------------------------------------------------------------------------
 -- map
 
-map-id : ∀ (pxs : All P xs) → All.map id pxs ≡ pxs
-map-id []         = refl
-map-id (px ∷ pxs) = cong (px ∷_)  (map-id pxs)
-
 map-cong : ∀ {f : P ⋐ Q} {g : P ⋐ Q} (pxs : All P xs) →
            (∀ {x} → f {x} ≗ g) → All.map f pxs ≡ All.map g pxs
 map-cong []         _   = refl
 map-cong (px ∷ pxs) feq = cong₂ _∷_ (feq px) (map-cong pxs feq)
 
-map-compose : ∀ {f : P ⋐ Q} {g : Q ⋐ R} (pxs : All P xs) →
-              All.map g (All.map f pxs) ≡ All.map (g ∘ f) pxs
-map-compose []         = refl
-map-compose (px ∷ pxs) = cong (_ ∷_) (map-compose pxs)
+map-id : ∀ (pxs : All P xs) → All.map id pxs ≡ pxs
+map-id []         = refl
+map-id (px ∷ pxs) = cong (px ∷_)  (map-id pxs)
+
+map-∘ : ∀ {f : P ⋐ Q} {g : Q ⋐ R} (pxs : All P xs) →
+        All.map g (All.map f pxs) ≡ All.map (g ∘ f) pxs
+map-∘ []         = refl
+map-∘ (px ∷ pxs) = cong (_ ∷_) (map-∘ pxs)
 
 lookup-map : ∀ {f : P ⋐ Q} (pxs : All P xs) (i : x ∈ xs) →
              lookup (All.map f pxs) i ≡ f (lookup pxs i)
@@ -218,7 +217,7 @@ updateAt-minimal : ∀ (i : x ∈ xs) (j : y ∈ xs) →
                    pxs              [ i ]= px →
                    updateAt j f pxs [ i ]= px
 updateAt-minimal (here .refl) (here refl) (px ∷ pxs) i≢j here        =
-  ⊥-elim (i≢j refl refl)
+  contradiction refl (i≢j refl)
 updateAt-minimal (here .refl) (there j)   (px ∷ pxs) i≢j here        = here
 updateAt-minimal (there i)    (here refl) (px ∷ pxs) i≢j (there val) = there val
 updateAt-minimal (there i)    (there j)   (px ∷ pxs) i≢j (there val) =
@@ -311,7 +310,7 @@ updateAt-commutes : ∀ (i : x ∈ xs) (j : y ∈ xs) →
                     i ≢∈ j →
                     updateAt {P = P} i f ∘ updateAt j g ≗ updateAt j g ∘ updateAt i f
 updateAt-commutes (here refl) (here refl) i≢j (px ∷ pxs) =
-  ⊥-elim (i≢j refl refl)
+  contradiction refl (i≢j refl)
 updateAt-commutes (here refl) (there j)   i≢j (px ∷ pxs) = refl
 updateAt-commutes (there i)   (here refl) i≢j (px ∷ pxs) = refl
 updateAt-commutes (there i)   (there j)   i≢j (px ∷ pxs) =
@@ -387,8 +386,7 @@ mapMaybe⁺ : ∀ {f : A → Maybe B} →
 mapMaybe⁺ {xs = []}     {f = f} []         = []
 mapMaybe⁺ {xs = x ∷ xs} {f = f} (px ∷ pxs) with f x
 ... | nothing = mapMaybe⁺ pxs
-... | just v with px
-...   | just pv = pv ∷ mapMaybe⁺ pxs
+... | just v with just pv ← px = pv ∷ mapMaybe⁺ pxs
 
 ------------------------------------------------------------------------
 -- catMaybes
@@ -524,7 +522,7 @@ takeWhile⁺ {xs = x ∷ xs} Q? (px ∷ pxs) with does (Q? x)
 takeWhile⁻ : (P? : Decidable P) → takeWhile P? xs ≡ xs → All P xs
 takeWhile⁻ {xs = []}     P? eq = []
 takeWhile⁻ {xs = x ∷ xs} P? eq with P? x
-... | yes px = px ∷ takeWhile⁻ P? (Listₚ.∷-injectiveʳ eq)
+... | yes px = px ∷ takeWhile⁻ P? (List.∷-injectiveʳ eq)
 ... | no ¬px = case eq of λ ()
 
 all-takeWhile : (P? : Decidable P) → ∀ xs → All P (takeWhile P? xs)
@@ -596,7 +594,7 @@ module _ (P? : Decidable P) where
   all-filter : ∀ xs → All P (filter P? xs)
   all-filter []       = []
   all-filter (x ∷ xs) with P? x
-  ... |  true because [Px] = invert [Px] ∷ all-filter xs
+  ... | true  because [Px] = invert [Px] ∷ all-filter xs
   ... | false because  _   = all-filter xs
 
   filter⁺ : All Q xs → All Q (filter P? xs)
@@ -606,12 +604,12 @@ module _ (P? : Decidable P) where
   ... | true  = Qx ∷ filter⁺ Qxs
 
   filter⁻ : All Q (filter P? xs) → All Q (filter (¬? ∘ P?) xs) → All Q xs
-  filter⁻ {xs = []}           []          []                           = []
-  filter⁻ {xs = x ∷ xs}       all⁺        all⁻ with P? x  | ¬? (P? x)
-  filter⁻ {xs = x ∷ xs}       all⁺        all⁻  | yes  Px | yes  ¬Px = contradiction Px ¬Px
-  filter⁻ {xs = x ∷ xs} (qx ∷ all⁺)       all⁻  | yes  Px | no  ¬¬Px = qx ∷ filter⁻ all⁺ all⁻
-  filter⁻ {xs = x ∷ xs}       all⁺  (qx ∷ all⁻) | no    _ | yes  ¬Px = qx ∷ filter⁻ all⁺ all⁻
-  filter⁻ {xs = x ∷ xs}       all⁺        all⁻  | no  ¬Px | no  ¬¬Px = contradiction ¬Px ¬¬Px
+  filter⁻ {xs = []}          []          []                         = []
+  filter⁻ {xs = x ∷ _}       all⁺        all⁻ with P? x  | ¬? (P? x)
+  filter⁻ {xs = x ∷ _}       all⁺        all⁻  | yes  Px | yes  ¬Px = contradiction Px ¬Px
+  filter⁻ {xs = x ∷ _} (qx ∷ all⁺)       all⁻  | yes  Px | no  ¬¬Px = qx ∷ filter⁻ all⁺ all⁻
+  filter⁻ {xs = x ∷ _}       all⁺  (qx ∷ all⁻) | no    _ | yes  ¬Px = qx ∷ filter⁻ all⁺ all⁻
+  filter⁻ {xs = x ∷ _}       all⁺        all⁻  | no  ¬Px | no  ¬¬Px = contradiction ¬Px ¬¬Px
 
 ------------------------------------------------------------------------
 -- partition
@@ -620,7 +618,7 @@ module _ {P : A → Set p} (P? : Decidable P) where
 
   partition-All : ∀ xs → (let ys , zs = partition P? xs) →
                   All P ys × All (∁ P) zs
-  partition-All xs rewrite partition-defn P? xs =
+  partition-All xs rewrite List.partition-defn P? xs =
     all-filter P? xs , all-filter (∁? P?) xs
 
 ------------------------------------------------------------------------
@@ -636,7 +634,7 @@ module _ {R : A → A → Set q} (R? : B.Decidable R) where
   ... | true  = derun⁺ all[P,y∷xs]
 
   deduplicate⁺ : All P xs → All P (deduplicate R? xs)
-  deduplicate⁺ []               = []
+  deduplicate⁺ []         = []
   deduplicate⁺ (px ∷ pxs) = px ∷ filter⁺ (¬? ∘ R? _) (deduplicate⁺ pxs)
 
   derun⁻ : P B.Respects (flip R) → ∀ xs → All P (derun R? xs) → All P xs
@@ -646,8 +644,8 @@ module _ {R : A → A → Set q} (R? : B.Decidable R) where
     aux : ∀ x xs → All P (derun R? (x ∷ xs)) → All P (x ∷ xs)
     aux x []       (px ∷ []) = px ∷ []
     aux x (y ∷ xs) all[P,x∷y∷xs] with R? x y
-    aux x (y ∷ xs) all[P,y∷xs]        | yes Rxy with aux y xs all[P,y∷xs]
-    aux x (y ∷ xs) all[P,y∷xs]        | yes Rxy | r@(py ∷ _) = P-resp-R Rxy py ∷ r
+    aux x (y ∷ xs) all[P,y∷xs]        | yes Rxy
+      with r@(py ∷ _) ← aux y xs all[P,y∷xs] = P-resp-R Rxy py ∷ r
     aux x (y ∷ xs) (px ∷ all[P,y∷xs]) | no _ = px ∷ aux y xs all[P,y∷xs]
 
   deduplicate⁻ : P B.Respects R → ∀ xs → All P (deduplicate R? xs) → All P xs
@@ -720,9 +718,10 @@ tails⁻ (x ∷ xs) (pxxs ∷ _) = pxxs
 module _ (p : A → Bool) where
 
   all⁺ : ∀ xs → T (all p xs) → All (T ∘ p) xs
-  all⁺ []       _     = []
-  all⁺ (x ∷ xs) px∷xs with Equivalence.to (T-∧ {p x}) px∷xs
-  ... | (px , pxs) = px ∷ all⁺ xs pxs
+  all⁺ []       _      = []
+  all⁺ (x ∷ xs) px∷pxs =
+    let px , pxs = Equivalence.to (T-∧ {p x}) px∷pxs
+    in px ∷ all⁺ xs pxs
 
   all⁻ : All (T ∘ p) xs → T (all p xs)
   all⁻ []         = _
@@ -744,7 +743,7 @@ all-anti-mono p xs⊆ys = all⁻ p ∘ anti-mono xs⊆ys ∘ all⁺ p _
 module _ (S : Setoid c ℓ) where
 
   open Setoid S
-  open ListEq S
+  open ≋ S
 
   respects : P B.Respects _≈_ → (All P) B.Respects _≋_
   respects p≈ []            []         = []
@@ -795,3 +794,11 @@ gmap = gmap⁺
 "Warning: gmap was deprecated in v2.0.
 Please use gmap⁺ instead."
 #-}
+
+-- Version 2.1
+
+map-compose = map-∘
+{-# WARNING_ON_USAGE map-compose
+"Warning: map-compose was deprecated in v2.1.
+Please use map-∘ instead."
+#-}
diff --git a/src/Data/List/Relation/Unary/Any.agda b/src/Data/List/Relation/Unary/Any.agda
index 1b6093cadc..ad9707cd77 100644
--- a/src/Data/List/Relation/Unary/Any.agda
+++ b/src/Data/List/Relation/Unary/Any.agda
@@ -8,7 +8,6 @@
 
 module Data.List.Relation.Unary.Any where
 
-open import Data.Empty
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.List.Base as List using (List; []; [_]; _∷_; removeAt)
 open import Data.Product.Base as Product using (∃; _,_)
@@ -17,7 +16,7 @@ open import Level using (Level; _⊔_)
 open import Relation.Nullary using (¬_; yes; no; _⊎-dec_)
 import Relation.Nullary.Decidable as Dec
 open import Relation.Nullary.Negation using (contradiction)
-open import Relation.Unary hiding (_∈_)
+open import Relation.Unary using (Pred; _⊆_; Decidable; Satisfiable)
 
 private
   variable
@@ -46,7 +45,7 @@ head ¬pxs (here px)   = px
 head ¬pxs (there pxs) = contradiction pxs ¬pxs
 
 tail : ¬ P x → Any P (x ∷ xs) → Any P xs
-tail ¬px (here  px)  = ⊥-elim (¬px px)
+tail ¬px (here  px)  = contradiction px ¬px
 tail ¬px (there pxs) = pxs
 
 map : P ⊆ Q → Any P ⊆ Any Q
diff --git a/src/Data/List/Relation/Unary/Any/Properties.agda b/src/Data/List/Relation/Unary/Any/Properties.agda
index c91a784ff7..836e73d4a4 100644
--- a/src/Data/List/Relation/Unary/Any/Properties.agda
+++ b/src/Data/List/Relation/Unary/Any/Properties.agda
@@ -25,7 +25,7 @@ open import Data.Nat.Properties using (_≟_; ≤∧≢⇒<; ≤-refl; m<n⇒m<1
 open import Data.Maybe.Base using (Maybe; just; nothing)
 open import Data.Maybe.Relation.Unary.Any as MAny using (just)
 open import Data.Product.Base as Product
-  using (_×_; _,_; ∃; ∃₂; proj₁; proj₂; uncurry′)
+  using (_×_; _,_; ∃; ∃₂; proj₁; proj₂)
 open import Data.Product.Function.NonDependent.Propositional
   using (_×-cong_)
 import Data.Product.Function.Dependent.Propositional as Σ
@@ -40,14 +40,15 @@ open import Level using (Level)
 open import Relation.Binary.Core using (Rel; REL)
 open import Relation.Binary.Definitions as B
 open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; sym; trans; cong; cong₂; subst)
+  using (_≡_; refl; sym; trans; cong; cong₂; resp)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 open import Relation.Unary as U
   using (Pred; _⟨×⟩_; _⟨→⟩_) renaming (_⊆_ to _⋐_)
-open import Relation.Nullary using (¬_; _because_; does; ofʸ; ofⁿ; yes; no)
-open import Relation.Nullary.Decidable using (¬?; decidable-stable)
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Decidable.Core
+  using (¬?; _because_; does; yes; no; decidable-stable)
+open import Relation.Nullary.Negation.Core using (¬_; contradiction)
+open import Relation.Nullary.Reflects using (invert)
 
 private
   open module ListMonad {ℓ} = RawMonad (monad {ℓ = ℓ})
@@ -96,6 +97,11 @@ Any-cong {P = P} {Q = Q} {xs = xs} {ys} P↔Q xs≈ys =
 ------------------------------------------------------------------------
 -- Any.map
 
+map-cong : (f g : P ⋐ Q) → (∀ {x} (p : P x) → f p ≡ g p) →
+          (p : Any P xs) → Any.map f p ≡ Any.map g p
+map-cong f g hyp (here  p) = cong here (hyp p)
+map-cong f g hyp (there p) = cong there $ map-cong f g hyp p
+
 map-id : ∀ (f : P ⋐ P) → (∀ {x} (p : P x) → f p ≡ p) →
          (p : Any P xs) → Any.map f p ≡ p
 map-id f hyp (here  p) = cong here (hyp p)
@@ -122,29 +128,31 @@ lookup-index (there pxs) = lookup-index pxs
 
 -- Nested occurrences of Any can sometimes be swapped. See also ×↔.
 
-swap : ∀ {P : A → B → Set ℓ} →
-       Any (λ x → Any (P x) ys) xs → Any (λ y → Any (flip P y) xs) ys
-swap (here  pys)  = Any.map here pys
-swap (there pxys) = Any.map there (swap pxys)
+module _ {R : REL A B ℓ} where
+
+  swap : Any (λ x → Any (R x) ys) xs → Any (λ y → Any (flip R y) xs) ys
+  swap (here  pys)  = Any.map here pys
+  swap (there pxys) = Any.map there (swap pxys)
+
+  swap-there : (any : Any (λ x → Any (R x) ys) xs) →
+               swap (Any.map (there {x = x}) any) ≡ there (swap any)
+  swap-there (here  pys)  = refl
+  swap-there (there pxys) = cong (Any.map there) (swap-there pxys)
 
-swap-there : ∀ {P : A → B → Set ℓ} →
-             (any : Any (λ x → Any (P x) ys) xs) →
-             swap (Any.map (there {x = x}) any) ≡ there (swap any)
-swap-there (here  pys)  = refl
-swap-there (there pxys) = cong (Any.map there) (swap-there pxys)
+module _ {R : REL A B ℓ} where
 
-swap-invol : ∀ {P : A → B → Set ℓ} →
-             (any : Any (λ x → Any (P x) ys) xs) →
-             swap (swap any) ≡ any
-swap-invol (here (here px))   = refl
-swap-invol (here (there pys)) =
-  cong (Any.map there) (swap-invol (here pys))
-swap-invol (there pxys)       =
-  trans (swap-there (swap pxys)) (cong there (swap-invol pxys))
+  swap-invol : (any : Any (λ x → Any (R x) ys) xs) →
+               swap (swap any) ≡ any
+  swap-invol (here (here px))   = refl
+  swap-invol (here (there pys)) =
+    cong (Any.map there) (swap-invol (here pys))
+  swap-invol (there pxys)       =
+    trans (swap-there (swap pxys)) (cong there (swap-invol pxys))
 
-swap↔ : ∀ {P : A → B → Set ℓ} →
-       Any (λ x → Any (P x) ys) xs ↔ Any (λ y → Any (flip P y) xs) ys
-swap↔ = mk↔ₛ′ swap swap swap-invol swap-invol
+module _ {R : REL A B ℓ} where
+
+  swap↔ : Any (λ x → Any (R x) ys) xs ↔ Any (λ y → Any (flip R y) xs) ys
+  swap↔ = mk↔ₛ′ swap swap swap-invol swap-invol
 
 ------------------------------------------------------------------------
 -- Lemmas relating Any to ⊥
@@ -212,8 +220,9 @@ Any-×⁺ (p , q) = Any.map (λ p → Any.map (λ q → (p , q)) q) p
 
 Any-×⁻ : Any (λ x → Any (λ y → P x × Q y) ys) xs →
          Any P xs × Any Q ys
-Any-×⁻ pq with Product.map₂ (Product.map₂ find) (find pq)
-... | (x , x∈xs , y , y∈ys , p , q) = lose x∈xs p , lose y∈ys q
+Any-×⁻ pq = let x , x∈xs , pq′ = find pq in
+            let y , y∈ys , p , q = find pq′ in
+            lose x∈xs p , lose y∈ys q
 
 ×↔ : ∀ {xs ys} →
      (Any P xs × Any Q ys) ↔ Any (λ x → Any (λ y → P x × Q y) ys) xs
@@ -222,7 +231,7 @@ Any-×⁻ pq with Product.map₂ (Product.map₂ find) (find pq)
   open ≡-Reasoning
 
   from∘to : ∀ pq → Any-×⁻ (Any-×⁺ pq) ≡ pq
-  from∘to (p , q) =
+  from∘to (p , q) = let x , x∈xs , px = find p in
 
     Any-×⁻ (Any-×⁺ (p , q))
 
@@ -241,10 +250,10 @@ Any-×⁻ pq with Product.map₂ (Product.map₂ find) (find pq)
          (y , y∈ys , p , q) = find (Any.map (p ,_) q)
      in  lose x∈xs p , lose y∈ys q)
 
-     ≡⟨ cong (λ • → let (x , x∈xs , p)     = find p
+     ≡⟨ cong (λ • → let (x , x∈xs , _)     = find p
                         (y , y∈ys , p , q) = •
                     in  lose x∈xs p , lose y∈ys q)
-             (find∘map q (proj₂ (proj₂ (find p)) ,_)) ⟩
+             (find∘map q (px ,_)) ⟩
 
     (let (x , x∈xs , p) = find p
          (y , y∈ys , q) = find q
@@ -254,24 +263,32 @@ Any-×⁻ pq with Product.map₂ (Product.map₂ find) (find pq)
 
     (p , q) ∎
 
-  to∘from : ∀ pq → Any-×⁺ {xs = xs} (Any-×⁻ pq) ≡ pq
-  to∘from pq with find pq
-      | (λ (f : (proj₁ (find pq) ≡_) ⋐ _) → map∘find pq {f})
-  ... | (x , x∈xs , pq′) | lem₁
-    with find pq′
-      | (λ (f : (proj₁ (find pq′) ≡_) ⋐ _) → map∘find pq′ {f})
-  ... | (y , y∈ys , p , q) | lem₂
-    rewrite sym $ map-∘ {R = λ x → Any (λ y → P x × Q y) ys}
-                        (λ p → Any.map (λ q → p , q) (lose y∈ys q))
-                        (λ y → subst P y p)
-                        x∈xs
-            = lem₁ _ helper
-    where
-    helper : Any.map (λ q → p , q) (lose y∈ys q) ≡ pq′
-    helper rewrite sym $ map-∘ (λ q → p , q)
-                               (λ y → subst Q y q)
-                               y∈ys
-           = lem₂ _ refl
+  to∘from : ∀ pq → Any-×⁺ (Any-×⁻ pq) ≡ pq
+  to∘from pq =
+    let x , x∈xs , pq′ = find pq
+        y , y∈ys , px , qy = find pq′
+
+        h : P ⋐ λ x → Any (λ y → (P x) × (Q y)) ys
+        h p = Any.map (p ,_) (lose y∈ys qy)
+
+        helper : h px ≡ pq′
+        helper = begin
+          Any.map (px ,_) (lose y∈ys qy)
+            ≡⟨ map-∘ (px ,_) (λ z → resp Q z qy) y∈ys ⟨
+          Any.map (λ z → px , resp Q z qy) y∈ys
+            ≡⟨ map∘find pq′ refl ⟩
+          pq′
+            ∎
+
+    in  begin
+      Any-×⁺ (Any-×⁻ pq)
+        ≡⟨⟩
+      Any.map h (lose x∈xs px)
+        ≡⟨ map-∘ h (λ z → resp P z px) x∈xs ⟨
+      Any.map (λ z → Any.map (resp P z px ,_) (lose y∈ys qy)) x∈xs
+        ≡⟨ map∘find pq helper ⟩
+      pq
+        ∎
 
 ------------------------------------------------------------------------
 -- Half-applied product commutes with Any.
@@ -360,9 +377,9 @@ module _ {P : A → Set p} where
   ++⁺∘++⁻ : ∀ xs {ys} (p : Any P (xs ++ ys)) → [ ++⁺ˡ , ++⁺ʳ xs ]′ (++⁻ xs p) ≡ p
   ++⁺∘++⁻ []       p         = refl
   ++⁺∘++⁻ (x ∷ xs) (here  p) = refl
-  ++⁺∘++⁻ (x ∷ xs) (there p) with ++⁻ xs p | ++⁺∘++⁻ xs p
-  ... | inj₁ p′ | ih = cong there ih
-  ... | inj₂ p′ | ih = cong there ih
+  ++⁺∘++⁻ (x ∷ xs) (there p) with ih ← ++⁺∘++⁻ xs p | ++⁻ xs p
+  ... | inj₁ _ = cong there ih
+  ... | inj₂ _ = cong there ih
 
   ++⁻∘++⁺ : ∀ xs {ys} (p : Any P xs ⊎ Any P ys) →
             ++⁻ xs ([ ++⁺ˡ , ++⁺ʳ xs ]′ p) ≡ p
@@ -459,10 +476,10 @@ module _ (f : A → B → C) where
                           Any R (cartesianProductWith f xs ys) →
                           Any P xs × Any Q ys
   cartesianProductWith⁻ resp (x ∷ xs) ys Rxsys with ++⁻ (map (f x) ys) Rxsys
-  cartesianProductWith⁻ resp (x ∷ xs) ys Rxsys | inj₁ Rfxys with map⁻ Rfxys
-  ... | Rxys = here (proj₁ (resp (proj₂ (Any.satisfied Rxys)))) , Any.map (proj₂ ∘ resp) Rxys
-  cartesianProductWith⁻ resp (x ∷ xs) ys Rxsys | inj₂ Rc with cartesianProductWith⁻ resp xs ys Rc
-  ... | pxs , qys = there pxs , qys
+  ... | inj₁ Rfxys = let Rxys = map⁻ Rfxys
+    in here (proj₁ (resp (proj₂ (Any.satisfied Rxys)))) , Any.map (proj₂ ∘ resp) Rxys
+  ... | inj₂ Rc    = let pxs , qys = cartesianProductWith⁻ resp xs ys Rc
+    in there pxs , qys
 
 ------------------------------------------------------------------------
 -- cartesianProduct
@@ -486,8 +503,8 @@ applyUpTo⁺ f p (s<s i<n@(s≤s _)) =
 applyUpTo⁻ : ∀ f {n} → Any P (applyUpTo f n) →
              ∃ λ i → i < n × P (f i)
 applyUpTo⁻ f {suc n} (here p)  = zero , z<s , p
-applyUpTo⁻ f {suc n} (there p) with applyUpTo⁻ (f ∘ suc) p
-... | i , i<n , q = suc i , s<s i<n , q
+applyUpTo⁻ f {suc n} (there p) =
+  let i , i<n , q = applyUpTo⁻ (f ∘ suc) p in suc i , s<s i<n , q
 
 ------------------------------------------------------------------------
 -- applyDownFrom
@@ -495,13 +512,13 @@ applyUpTo⁻ f {suc n} (there p) with applyUpTo⁻ (f ∘ suc) p
 applyDownFrom⁺ : ∀ f {i n} → P (f i) → i < n → Any P (applyDownFrom f n)
 applyDownFrom⁺ f {i} {suc n} p (s≤s i≤n) with i ≟ n
 ... | yes refl = here p
-... | no  i≢n    = there (applyDownFrom⁺ f p (≤∧≢⇒< i≤n i≢n))
+... | no  i≢n  = there (applyDownFrom⁺ f p (≤∧≢⇒< i≤n i≢n))
 
 applyDownFrom⁻ : ∀ f {n} → Any P (applyDownFrom f n) →
                  ∃ λ i → i < n × P (f i)
 applyDownFrom⁻ f {suc n} (here p)  = n , ≤-refl , p
-applyDownFrom⁻ f {suc n} (there p) with applyDownFrom⁻ f p
-... | i , i<n , pf = i , m<n⇒m<1+n i<n , pf
+applyDownFrom⁻ f {suc n} (there p) =
+  let i , i<n , q = applyDownFrom⁻ f p in i , m<n⇒m<1+n i<n , q
 
 ------------------------------------------------------------------------
 -- tabulate
@@ -521,8 +538,8 @@ module _ (Q? : U.Decidable Q) where
 
   filter⁺ : (p : Any P xs) → Any P (filter Q? xs) ⊎ ¬ Q (Any.lookup p)
   filter⁺ {xs = x ∷ _} (here px) with Q? x
-  ... | true  because _       = inj₁ (here px)
-  ... | false because ofⁿ ¬Qx = inj₂ ¬Qx
+  ... | true  because _     = inj₁ (here px)
+  ... | false because [¬Qx] = inj₂ (invert [¬Qx])
   filter⁺ {xs = x ∷ _} (there p) with does (Q? x)
   ... | true  = Sum.map₁ there (filter⁺ p)
   ... | false = filter⁺ p
@@ -536,14 +553,14 @@ module _ (Q? : U.Decidable Q) where
 ------------------------------------------------------------------------
 -- derun and deduplicate
 
-module _ {R : A → A → Set r} (R? : B.Decidable R) where
+module _ {R : Rel A r} (R? : B.Decidable R) where
 
   private
     derun⁺-aux : ∀ x xs → P Respects R → P x → Any P (derun R? (x ∷ xs))
     derun⁺-aux x [] resp Px = here Px
     derun⁺-aux x (y ∷ xs) resp Px with R? x y
-    ... | true  because ofʸ Rxy = derun⁺-aux y xs resp (resp Rxy Px)
-    ... | false because _       = here Px
+    ... | true  because [Rxy] = derun⁺-aux y xs resp (resp (invert [Rxy]) Px)
+    ... | false because _     = here Px
 
   derun⁺ : P Respects R → Any P xs → Any P (derun R? xs)
   derun⁺ {xs = x ∷ xs}     resp (here px)   = derun⁺-aux x xs resp px
@@ -555,9 +572,10 @@ module _ {R : A → A → Set r} (R? : B.Decidable R) where
   deduplicate⁺ {xs = x ∷ xs} resp (here px)   = here px
   deduplicate⁺ {xs = x ∷ xs} resp (there pxs)
     with filter⁺ (¬? ∘ R? x) (deduplicate⁺ resp pxs)
-  ... | inj₁ p = there p
-  ... | inj₂ ¬¬q with decidable-stable (R? x (Any.lookup (deduplicate⁺ resp pxs))) ¬¬q
-  ...  | q = here (resp q (lookup-result (deduplicate⁺ resp pxs)))
+  ... | inj₁ p   = there p
+  ... | inj₂ ¬¬q =
+    let q = decidable-stable (R? x (Any.lookup (deduplicate⁺ resp pxs))) ¬¬q
+    in  here (resp q (lookup-result (deduplicate⁺ resp pxs)))
 
   private
     derun⁻-aux : Any P (derun R? (x ∷ xs)) → Any P (x ∷ xs)

From e89fa26a5d476d423dd2945fdecd7c08de40270f Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Fri, 7 Jun 2024 07:14:13 -0400
Subject: [PATCH 300/316] Port two Endomorphism submodules over to new Function
 hierarchy (#2342)

* port over two modules

* and add to CHANGELOG

* fix whitespace

* fix warning: it was pointing to a record that did not exist.

* fix things as per Matthew's review - though this remains a breaking change.

* take care of comments from James.

* adjust CHANGELOG for what will be implemented shortly

* Revert "take care of comments from James."

This reverts commit 93e9e0fdf482164c0fb7e709ef8799fbc3fa0385.

* Revert "fix things as per Matthew's review - though this remains a breaking change."

This reverts commit d1cae72dcb3b44d4ba3c7290f3535544be32ab1e.

* Revert "fix whitespace"

This reverts commit 81230ecf0c8ff634433c9d17d53c9d2e8b3c1fd2.

* Revert "port over two modules"

This reverts commit 6619f114346e86ead3cf62b1170ab23d31056b48.

* rename these

* fix tiny merge issue

* get deprecations right (remove where not needed, make more global where needed)

* style guide - missing blank lines

* fix a bad merge

* fixed deprecations

* fix #2394

* minor tweaks

---------

Co-authored-by: James McKinna <J.McKinna@hw.ac.uk>
---
 CHANGELOG.md                                 |  18 ++-
 src/Function/Endo/Propositional.agda         | 121 +++++++++++++++++++
 src/Function/Endo/Setoid.agda                | 117 ++++++++++++++++++
 src/Function/Endomorphism/Propositional.agda |  10 +-
 src/Function/Endomorphism/Setoid.agda        |  10 +-
 src/Function/Equality.agda                   |   1 -
 6 files changed, 266 insertions(+), 11 deletions(-)
 create mode 100644 src/Function/Endo/Propositional.agda
 create mode 100644 src/Function/Endo/Setoid.agda

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 156756ff00..7d1e9b4e6c 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -51,6 +51,11 @@ Deprecated modules
 * `Data.List.Relation.Binary.Sublist.Propositional.Disjoint` deprecated in favour of
   `Data.List.Relation.Binary.Sublist.Propositional.Slice`.
 
+* The modules `Function.Endomorphism.Propositional` and
+  `Function.Endomorphism.Setoid` that use the old `Function`
+  hierarchy. Use `Function.Endo.Propositional` and
+  `Function.Endo.Setoid` instead.
+
 Deprecated names
 ----------------
 
@@ -167,6 +172,11 @@ New modules
   indexedSetoid        : {A : Set a} → IndexedSetoid ℕ a _
   ```
 
+* The modules `Function.Endo.Propositional` and
+  `Function.Endo.Setoid` are new but are actually proper ports of
+  `Function.Endomorphism.Propositional` and
+  `Function.Endomorphism.Setoid`.
+
 * `Function.Relation.Binary.Equality`
   ```agda
   setoid : Setoid a₁ a₂ → Setoid b₁ b₂ → Setoid _ _
@@ -292,13 +302,17 @@ Additions to existing modules
   rawModule          : RawModule R c ℓ
   ```
 
-* In `Algebra.Morphism.Structures`
+* In `Algebra.Morphism.Structures`:
   ```agda
   module SuccessorSetMorphisms (N₁ : RawSuccessorSet a ℓ₁) (N₂ : RawSuccessorSet b ℓ₂) where
     record IsSuccessorSetHomomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
     record IsSuccessorSetMonomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
-    record IsSuccessorSetIsomorphism  (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
+    record IsSuccessorSetIsomorphism  (⟦_⟧ : N₁.Carrier →  N₂.Carrier) : Set _
 
+  IsSemigroupHomomorphism : (A → B) → Set _
+  IsSemigroupMonomorphism : (A → B) → Set _
+  IsSemigroupIsomorphism : (A → B) → Set _
+  ```
 * In `Algebra.Properties.AbelianGroup`:
   ```
   ⁻¹-anti-homo‿- : (x - y) ⁻¹ ≈ y - x
diff --git a/src/Function/Endo/Propositional.agda b/src/Function/Endo/Propositional.agda
new file mode 100644
index 0000000000..4d1bb21cbb
--- /dev/null
+++ b/src/Function/Endo/Propositional.agda
@@ -0,0 +1,121 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Endomorphisms on a Set
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Function.Endo.Propositional {a} (A : Set a) where
+
+open import Algebra using (Semigroup; Magma; RawMagma; Monoid; RawMonoid)
+open import Algebra.Core
+import Algebra.Definitions.RawMonoid as RawMonoidDefinitions
+import Algebra.Properties.Monoid.Mult as MonoidMultProperties
+open import Algebra.Structures using (IsMagma; IsSemigroup; IsMonoid)
+open import Algebra.Morphism
+  using (module Definitions; IsMagmaHomomorphism; IsMonoidHomomorphism)
+open Definitions using (Homomorphic₂)
+
+open import Data.Nat.Base using (ℕ; zero; suc; _+_; +-rawMagma; +-0-rawMonoid)
+open import Data.Nat.Properties using (+-0-monoid; +-semigroup)
+open import Data.Product.Base using (_,_)
+
+open import Function.Base using (id; _∘′_; _∋_; flip)
+open import Function.Bundles using (Func; _⟶ₛ_; _⟨$⟩_)
+open import Relation.Binary.Core using (_Preserves_⟶_)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
+import Relation.Binary.PropositionalEquality.Properties as ≡
+
+import Function.Endo.Setoid (≡.setoid A) as Setoid
+
+------------------------------------------------------------------------
+-- Basic type and raw bundles
+
+Endo : Set a
+Endo = A → A
+
+private
+
+  _∘_ : Op₂ Endo
+  _∘_ = _∘′_
+
+  ∘-id-rawMonoid : RawMonoid a a
+  ∘-id-rawMonoid = record { Carrier = Endo; _≈_ = _≡_ ; _∙_ = _∘_ ; ε = id }
+
+  open RawMonoid ∘-id-rawMonoid
+    using ()
+    renaming (rawMagma to ∘-rawMagma)
+
+
+------------------------------------------------------------------------
+-- Conversion back and forth with the Setoid-based notion of Endomorphism
+
+fromSetoidEndo : Setoid.Endo → Endo
+fromSetoidEndo = _⟨$⟩_
+
+toSetoidEndo : Endo → Setoid.Endo
+toSetoidEndo f = record
+  { to = f
+  ; cong  = cong f
+  }
+
+------------------------------------------------------------------------
+-- Structures
+
+∘-isMagma : IsMagma _≡_ _∘_
+∘-isMagma = record
+  { isEquivalence = ≡.isEquivalence
+  ; ∙-cong        = cong₂ _∘_
+  }
+
+∘-magma : Magma _ _
+∘-magma = record { isMagma = ∘-isMagma }
+
+∘-isSemigroup : IsSemigroup _≡_ _∘_
+∘-isSemigroup = record
+  { isMagma = ∘-isMagma
+  ; assoc   = λ _ _ _ → refl
+  }
+
+∘-semigroup : Semigroup _ _
+∘-semigroup = record { isSemigroup = ∘-isSemigroup }
+
+∘-id-isMonoid : IsMonoid _≡_ _∘_ id
+∘-id-isMonoid = record
+  { isSemigroup = ∘-isSemigroup
+  ; identity    = (λ _ → refl) , (λ _ → refl)
+  }
+
+∘-id-monoid : Monoid _ _
+∘-id-monoid = record { isMonoid = ∘-id-isMonoid }
+
+------------------------------------------------------------------------
+-- n-th iterated composition
+
+infixr 8 _^_
+
+_^_ : Endo → ℕ → Endo
+_^_ = flip _×_ where open RawMonoidDefinitions ∘-id-rawMonoid using (_×_)
+
+------------------------------------------------------------------------
+-- Homomorphism
+
+module _ (f : Endo) where
+
+  open MonoidMultProperties ∘-id-monoid using (×-homo-+)
+
+  ^-homo : Homomorphic₂ ℕ Endo _≡_ (f ^_) _+_ _∘_
+  ^-homo = ×-homo-+ f
+
+  ^-isMagmaHomomorphism : IsMagmaHomomorphism +-rawMagma ∘-rawMagma (f ^_)
+  ^-isMagmaHomomorphism = record
+    { isRelHomomorphism = record { cong = cong (f ^_) }
+    ; homo = ^-homo
+    }
+
+  ^-isMonoidHomomorphism : IsMonoidHomomorphism +-0-rawMonoid ∘-id-rawMonoid (f ^_)
+  ^-isMonoidHomomorphism = record
+    { isMagmaHomomorphism = ^-isMagmaHomomorphism
+    ; ε-homo = refl
+    }
diff --git a/src/Function/Endo/Setoid.agda b/src/Function/Endo/Setoid.agda
new file mode 100644
index 0000000000..99716b35db
--- /dev/null
+++ b/src/Function/Endo/Setoid.agda
@@ -0,0 +1,117 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Endomorphisms on a Setoid
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+open import Relation.Binary.Bundles using (Setoid)
+
+module Function.Endo.Setoid {c e} (S : Setoid c e) where
+
+open import Agda.Builtin.Equality using (_≡_)
+
+open import Algebra using (Semigroup; Magma; RawMagma; Monoid; RawMonoid)
+import Algebra.Definitions.RawMonoid as RawMonoidDefinitions
+import Algebra.Properties.Monoid.Mult as MonoidMultProperties
+open import Algebra.Structures using (IsMagma; IsSemigroup; IsMonoid)
+open import Algebra.Morphism
+  using (module Definitions; IsMagmaHomomorphism; IsMonoidHomomorphism)
+open Definitions using (Homomorphic₂)
+open import Data.Nat.Base using (ℕ; zero; suc; _+_; +-rawMagma; +-0-rawMonoid)
+open import Data.Nat.Properties using (+-semigroup; +-identityʳ)
+open import Data.Product.Base using (_,_)
+open import Function.Bundles using (Func; _⟶ₛ_; _⟨$⟩_)
+open import Function.Construct.Identity using () renaming (function to identity)
+open import Function.Construct.Composition using () renaming (function to _∘_)
+open import Function.Relation.Binary.Setoid.Equality as Eq using (_⇨_)
+open import Level using (Level; _⊔_)
+open import Relation.Binary.Core using (_Preserves_⟶_)
+
+private
+  open module E = Setoid (S ⇨ S) hiding (refl)
+  module S = Setoid S
+  open Func using (cong)
+
+
+------------------------------------------------------------------------
+-- Basic type and raw bundles
+
+Endo : Set _
+Endo = S ⟶ₛ S
+
+private
+  id : Endo
+  id = identity S
+
+  ∘-id-rawMonoid : RawMonoid (c ⊔ e) (c ⊔ e)
+  ∘-id-rawMonoid = record { Carrier = Endo; _≈_ = _≈_ ; _∙_ = _∘_ ; ε = id }
+
+  open RawMonoid ∘-id-rawMonoid
+    using ()
+    renaming (rawMagma to ∘-rawMagma)
+
+--------------------------------------------------------------
+-- Structures
+
+∘-isMagma : IsMagma _≈_ _∘_
+∘-isMagma = record
+  { isEquivalence = isEquivalence
+  ; ∙-cong        = λ {_} {_} {_} {v} x≈y u≈v → S.trans u≈v (cong v x≈y)
+  }
+
+∘-magma : Magma (c ⊔ e) (c ⊔ e)
+∘-magma = record { isMagma = ∘-isMagma }
+
+∘-isSemigroup : IsSemigroup _≈_ _∘_
+∘-isSemigroup = record
+  { isMagma = ∘-isMagma
+  ; assoc   = λ _ _ _ → S.refl
+  }
+
+∘-semigroup : Semigroup (c ⊔ e) (c ⊔ e)
+∘-semigroup = record { isSemigroup = ∘-isSemigroup }
+
+∘-id-isMonoid : IsMonoid _≈_ _∘_ id
+∘-id-isMonoid = record
+  { isSemigroup = ∘-isSemigroup
+  ; identity    = (λ _ → S.refl) , (λ _ → S.refl)
+  }
+
+∘-id-monoid : Monoid (c ⊔ e) (c ⊔ e)
+∘-id-monoid = record { isMonoid = ∘-id-isMonoid }
+
+------------------------------------------------------------------------
+-- -- n-th iterated composition
+
+infixr 8 _^_
+
+_^_ : Endo → ℕ → Endo
+f ^ n = n × f where open RawMonoidDefinitions ∘-id-rawMonoid using (_×_)
+
+------------------------------------------------------------------------
+-- Homomorphism
+
+module _ (f : Endo) where
+
+  open MonoidMultProperties ∘-id-monoid using (×-congˡ; ×-homo-+)
+
+  ^-cong₂ : (f ^_) Preserves _≡_ ⟶ _≈_
+  ^-cong₂ = ×-congˡ {f}
+
+  ^-homo : Homomorphic₂ ℕ Endo _≈_ (f ^_) _+_ _∘_
+  ^-homo = ×-homo-+ f
+
+  ^-isMagmaHomomorphism : IsMagmaHomomorphism +-rawMagma ∘-rawMagma (f ^_)
+  ^-isMagmaHomomorphism = record
+    { isRelHomomorphism = record { cong = ^-cong₂ }
+    ; homo = ^-homo
+    }
+
+  ^-isMonoidHomomorphism : IsMonoidHomomorphism +-0-rawMonoid ∘-id-rawMonoid (f ^_)
+  ^-isMonoidHomomorphism = record
+    { isMagmaHomomorphism = ^-isMagmaHomomorphism
+    ; ε-homo = S.refl
+    }
+
diff --git a/src/Function/Endomorphism/Propositional.agda b/src/Function/Endomorphism/Propositional.agda
index c8dd685328..7f9d99a155 100644
--- a/src/Function/Endomorphism/Propositional.agda
+++ b/src/Function/Endomorphism/Propositional.agda
@@ -1,16 +1,18 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Endomorphisms on a Set
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
--- Disabled to prevent warnings from deprecated names
-{-# OPTIONS --warn=noUserWarning #-}
-
 module Function.Endomorphism.Propositional {a} (A : Set a) where
 
+{-# WARNING_ON_IMPORT
+"Function.Endomorphism.Propositional was deprecated in v2.1.
+Use Function.Endo.Propositional instead."
+#-}
+
 open import Algebra
 open import Algebra.Morphism; open Definitions
 
diff --git a/src/Function/Endomorphism/Setoid.agda b/src/Function/Endomorphism/Setoid.agda
index a715ef5262..5d4993ecf6 100644
--- a/src/Function/Endomorphism/Setoid.agda
+++ b/src/Function/Endomorphism/Setoid.agda
@@ -1,19 +1,21 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Endomorphisms on a Setoid
+-- This module is DEPRECATED.
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
--- Disabled to prevent warnings from deprecated names
-{-# OPTIONS --warn=noUserWarning #-}
-
 open import Relation.Binary.Core using (_Preserves_⟶_)
 open import Relation.Binary.Bundles using (Setoid)
 
 module Function.Endomorphism.Setoid {c e} (S : Setoid c e) where
 
+{-# WARNING_ON_IMPORT
+"Function.Endomorphism.Setoid was deprecated in v2.1.
+Use Function.Endo.Setoid instead."
+#-}
+
 open import Agda.Builtin.Equality
 
 open import Algebra
diff --git a/src/Function/Equality.agda b/src/Function/Equality.agda
index 647dca7416..4748834a83 100644
--- a/src/Function/Equality.agda
+++ b/src/Function/Equality.agda
@@ -5,7 +5,6 @@
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
-{-# OPTIONS --warn=noUserWarning #-}
 
 module Function.Equality where
 

From fd903e34924b0ec24699b577eebc578d52f28fee Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Fri, 7 Jun 2024 14:34:49 +0100
Subject: [PATCH 301/316] Add `IsIdempotentMonoid` and `IsCommutativeBand` to
 `Algebra.Structures` (#2402)

* fix #2138

* fixed `Structures`; added `Bundles`

* added fields; plus redefined `IsSemilattice` and `IsBoundedSemilattice` as aliases

* `fix-whitespace`

* reexported `comm`

* fixed `Lattice.Bundles`

* knock-on

* added text about redefinition of `Is(Bounded)Semilattice`

* add manifest fields to `IsIdempotentSemiring`

* final missing `Bundles`

* `CHANGELOG`
---
 CHANGELOG.md                                  | 21 +++++++
 src/Algebra/Bundles.agda                      | 63 +++++++++++++++++++
 src/Algebra/Lattice/Bundles.agda              |  2 +-
 .../Lattice/Construct/Subst/Equality.agda     |  2 +-
 src/Algebra/Lattice/Structures.agda           | 16 ++---
 src/Algebra/Structures.agda                   | 48 +++++++++++++-
 6 files changed, 137 insertions(+), 15 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 7d1e9b4e6c..9f8d6de9bd 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -45,6 +45,11 @@ Minor improvements
   `Data.Product.Relation.Binary.Pointwise.NonDependent.Pointwise`
   has been generalised to take heterogeneous arguments in `REL`.
 
+* The structures `IsSemilattice` and `IsBoundedSemilattice` in
+  `Algebra.Lattice.Structures` have been redefined as aliases of
+  `IsCommutativeBand` and `IsIdempotentMonoid` in `Algebra.Structures`.
+
+
 Deprecated modules
 ------------------
 
@@ -244,8 +249,16 @@ Additions to existing modules
 * In `Algebra.Bundles`
   ```agda
   record SuccessorSet c ℓ : Set (suc (c ⊔ ℓ))
+  record CommutativeBand c ℓ : Set (suc (c ⊔ ℓ))
+  record IdempotentMonoid c ℓ : Set (suc (c ⊔ ℓ))
+  ```
+  and additional manifest fields for sub-bundles arising from these in:
+  ```agda
+  IdempotentCommutativeMonoid
+  IdempotentSemiring
   ```
 
+
 * In `Algebra.Bundles.Raw`
   ```agda
   record RawSuccessorSet c ℓ : Set (suc (c ⊔ ℓ))
@@ -364,6 +377,14 @@ Additions to existing modules
 * In `Algebra.Structures`
   ```agda
   record IsSuccessorSet (suc# : Op₁ A) (zero# : A) : Set _
+  record IsCommutativeBand (∙ : Op₂ A) : Set _
+  record IsIdempotentMonoid (∙ : Op₂ A) (ε : A) : Set _
+  ```
+  and additional manifest fields for substructures arising from these in:
+  ```agda
+  IsIdempotentCommutativeMonoid
+  IsIdempotentSemiring
+  ```
 
 * In `Algebra.Structures.IsGroup`:
   ```agda
diff --git a/src/Algebra/Bundles.agda b/src/Algebra/Bundles.agda
index a86529f5cd..d96584a733 100644
--- a/src/Algebra/Bundles.agda
+++ b/src/Algebra/Bundles.agda
@@ -243,6 +243,30 @@ record CommutativeSemigroup c ℓ : Set (suc (c ⊔ ℓ)) where
   commutativeMagma : CommutativeMagma c ℓ
   commutativeMagma = record { isCommutativeMagma = isCommutativeMagma }
 
+record CommutativeBand c ℓ : Set (suc (c ⊔ ℓ)) where
+  infixl 7 _∙_
+  infix  4 _≈_
+  field
+    Carrier            : Set c
+    _≈_                : Rel Carrier ℓ
+    _∙_                : Op₂ Carrier
+    isCommutativeBand  : IsCommutativeBand _≈_ _∙_
+
+  open IsCommutativeBand isCommutativeBand public
+
+  band : Band _ _
+  band = record { isBand = isBand }
+
+  open Band band public
+    using (_≉_; magma; rawMagma; semigroup)
+
+  commutativeSemigroup : CommutativeSemigroup c ℓ
+  commutativeSemigroup = record { isCommutativeSemigroup = isCommutativeSemigroup }
+
+  open CommutativeSemigroup commutativeSemigroup public
+    using (commutativeMagma)
+
+
 ------------------------------------------------------------------------
 -- Bundles with 1 binary operation & 1 element
 ------------------------------------------------------------------------
@@ -315,6 +339,27 @@ record CommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
   open CommutativeSemigroup commutativeSemigroup public
     using (commutativeMagma)
 
+record IdempotentMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
+  infixl 7 _∙_
+  infix  4 _≈_
+  field
+    Carrier            : Set c
+    _≈_                : Rel Carrier ℓ
+    _∙_                : Op₂ Carrier
+    ε                  : Carrier
+    isIdempotentMonoid : IsIdempotentMonoid _≈_ _∙_ ε
+
+  open IsIdempotentMonoid isIdempotentMonoid public
+
+  monoid : Monoid _ _
+  monoid = record { isMonoid = isMonoid }
+
+  open Monoid monoid public
+    using (_≉_; rawMagma; magma; semigroup; unitalMagma; rawMonoid)
+
+  band : Band _ _
+  band = record { isBand = isBand }
+
 
 record IdempotentCommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
   infixl 7 _∙_
@@ -331,6 +376,12 @@ record IdempotentCommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
   commutativeMonoid : CommutativeMonoid _ _
   commutativeMonoid = record { isCommutativeMonoid = isCommutativeMonoid }
 
+  idempotentMonoid : IdempotentMonoid _ _
+  idempotentMonoid = record { isIdempotentMonoid = isIdempotentMonoid }
+
+  commutativeBand : CommutativeBand _ _
+  commutativeBand = record { isCommutativeBand = isCommutativeBand }
+
   open CommutativeMonoid commutativeMonoid public
     using
     ( _≉_; rawMagma; magma; unitalMagma; commutativeMagma
@@ -338,6 +389,8 @@ record IdempotentCommutativeMonoid c ℓ : Set (suc (c ⊔ ℓ)) where
     ; rawMonoid; monoid
     )
 
+  open CommutativeBand commutativeBand public
+    using (band)
 
 -- Idempotent commutative monoids are also known as bounded lattices.
 -- Note that the BoundedLattice necessarily uses the notation inherited
@@ -778,6 +831,16 @@ record IdempotentSemiring c ℓ : Set (suc (c ⊔ ℓ)) where
     ; rawSemiring
     )
 
+  +-idempotentCommutativeMonoid : IdempotentCommutativeMonoid _ _
+  +-idempotentCommutativeMonoid = record { isIdempotentCommutativeMonoid = +-isIdempotentCommutativeMonoid }
+
+  open IdempotentCommutativeMonoid +-idempotentCommutativeMonoid public
+    using ()
+    renaming ( band to +-band
+             ; commutativeBand to +-commutativeBand
+             ; idempotentMonoid to +-idempotentMonoid
+             )
+
 record KleeneAlgebra c ℓ : Set (suc (c ⊔ ℓ)) where
   infix  8 _⋆
   infixl 7 _*_
diff --git a/src/Algebra/Lattice/Bundles.agda b/src/Algebra/Lattice/Bundles.agda
index 28fd07d5b3..51318bf65c 100644
--- a/src/Algebra/Lattice/Bundles.agda
+++ b/src/Algebra/Lattice/Bundles.agda
@@ -43,7 +43,7 @@ record Semilattice c ℓ : Set (suc (c ⊔ ℓ)) where
     _∙_           : Op₂ Carrier
     isSemilattice : IsSemilattice _≈_ _∙_
 
-  open IsSemilattice isSemilattice public
+  open IsSemilattice _≈_ isSemilattice public
 
   band : Band c ℓ
   band = record { isBand = isBand }
diff --git a/src/Algebra/Lattice/Construct/Subst/Equality.agda b/src/Algebra/Lattice/Construct/Subst/Equality.agda
index 8a498afbb2..48393f2b64 100644
--- a/src/Algebra/Lattice/Construct/Subst/Equality.agda
+++ b/src/Algebra/Lattice/Construct/Subst/Equality.agda
@@ -35,7 +35,7 @@ isSemilattice : IsSemilattice ≈₁ ∧ → IsSemilattice ≈₂ ∧
 isSemilattice S = record
   { isBand = isBand S.isBand
   ; comm   = comm S.comm
-  } where module S = IsSemilattice S
+  } where module S = IsSemilattice ≈₁ S
 
 isLattice : IsLattice ≈₁ ∨ ∧ → IsLattice ≈₂ ∨ ∧
 isLattice {∨} {∧} S = record
diff --git a/src/Algebra/Lattice/Structures.agda b/src/Algebra/Lattice/Structures.agda
index 116ed85271..c82ed2db9e 100644
--- a/src/Algebra/Lattice/Structures.agda
+++ b/src/Algebra/Lattice/Structures.agda
@@ -31,11 +31,10 @@ open import Algebra.Structures _≈_
 ------------------------------------------------------------------------
 -- Structures with 1 binary operation
 
-record IsSemilattice (∙ : Op₂ A) : Set (a ⊔ ℓ) where
-  field
-    isBand : IsBand ∙
-    comm   : Commutative ∙
-
+IsSemilattice = IsCommutativeBand
+module IsSemilattice {∙} (L : IsSemilattice ∙) where
+  open IsCommutativeBand L public
+    using (isBand; comm)
   open IsBand isBand public
 
 -- Used to bring names appropriate for a meet semilattice into scope.
@@ -67,12 +66,7 @@ IsBoundedSemilattice = IsIdempotentCommutativeMonoid
 module IsBoundedSemilattice {∙ ε} (L : IsBoundedSemilattice ∙ ε) where
 
   open IsIdempotentCommutativeMonoid L public
-
-  isSemilattice : IsSemilattice ∙
-  isSemilattice = record
-    { isBand = isBand
-    ; comm   = comm
-    }
+    renaming (isCommutativeBand to isSemilattice)
 
 
 -- Used to bring names appropriate for a bounded meet semilattice
diff --git a/src/Algebra/Structures.agda b/src/Algebra/Structures.agda
index 3cdc55a4b5..b58245e6df 100644
--- a/src/Algebra/Structures.agda
+++ b/src/Algebra/Structures.agda
@@ -153,6 +153,20 @@ record IsCommutativeSemigroup (∙ : Op₂ A) : Set (a ⊔ ℓ) where
     ; comm    = comm
     }
 
+
+record IsCommutativeBand (∙ : Op₂ A) : Set (a ⊔ ℓ) where
+  field
+    isBand : IsBand ∙
+    comm   : Commutative ∙
+
+  open IsBand isBand public
+
+  isCommutativeSemigroup : IsCommutativeSemigroup ∙
+  isCommutativeSemigroup = record { isSemigroup = isSemigroup ; comm = comm }
+
+  open IsCommutativeSemigroup isCommutativeSemigroup public
+    using (isCommutativeMagma)
+
 ------------------------------------------------------------------------
 -- Structures with 1 binary operation & 1 element
 ------------------------------------------------------------------------
@@ -208,6 +222,17 @@ record IsCommutativeMonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
     using (isCommutativeMagma)
 
 
+record IsIdempotentMonoid (∙ : Op₂ A) (ε : A) : Set (a ⊔ ℓ) where
+  field
+    isMonoid : IsMonoid ∙ ε
+    idem     : Idempotent ∙
+
+  open IsMonoid isMonoid public
+
+  isBand : IsBand ∙
+  isBand = record { isSemigroup = isSemigroup ; idem = idem }
+
+
 record IsIdempotentCommutativeMonoid (∙ : Op₂ A)
                                      (ε : A) : Set (a ⊔ ℓ) where
   field
@@ -216,9 +241,14 @@ record IsIdempotentCommutativeMonoid (∙ : Op₂ A)
 
   open IsCommutativeMonoid isCommutativeMonoid public
 
-  isBand : IsBand ∙
-  isBand = record { isSemigroup = isSemigroup ; idem = idem }
+  isIdempotentMonoid : IsIdempotentMonoid ∙ ε
+  isIdempotentMonoid = record { isMonoid = isMonoid ; idem = idem }
 
+  open IsIdempotentMonoid isIdempotentMonoid public
+    using (isBand)
+
+  isCommutativeBand : IsCommutativeBand ∙
+  isCommutativeBand = record { isBand = isBand ; comm = comm }
 
 ------------------------------------------------------------------------
 -- Structures with 1 binary operation, 1 unary operation & 1 element
@@ -584,6 +614,20 @@ record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
 
   open IsSemiring isSemiring public
 
+  +-isIdempotentCommutativeMonoid : IsIdempotentCommutativeMonoid + 0#
+  +-isIdempotentCommutativeMonoid = record
+    { isCommutativeMonoid = +-isCommutativeMonoid
+    ; idem = +-idem
+    }
+
+  open IsIdempotentCommutativeMonoid +-isIdempotentCommutativeMonoid public
+    using ()
+    renaming ( isCommutativeBand to +-isCommutativeBand
+             ; isBand to +-isBand
+             ; isIdempotentMonoid to +-isIdempotentMonoid
+             )
+
+
 record IsKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ) where
   field
     isIdempotentSemiring  : IsIdempotentSemiring + * 0# 1#

From ccd432da71f04c48d3ff3753661efbfbdd338191 Mon Sep 17 00:00:00 2001
From: "G. Allais" <guillaume.allais@ens-lyon.org>
Date: Thu, 13 Jun 2024 01:07:06 +0100
Subject: [PATCH 302/316] [ new ] Word8, Bytestring, Bytestring builder (#2376)

* [ admin ] deprecate Word -> Word64

* [ new ] a type of bytes

* [ new ] Bytestrings and builders

* [ test ] for bytestrings, builders, word conversion, show, etc.

* [ ci ] bump ghc & cabal numbers

* [ fix ] actually set the bits ya weapon

* [ ci ] bump options to match makefile

* [ ci ] more heap

* [ more ] add hexadecimal show functions too
---
 .github/workflows/ci-ubuntu.yml            |   6 +-
 CHANGELOG.md                               |  45 ++++++++-
 GenerateEverything.hs                      |  12 +++
 src/Data/Bool/Show.agda                    |   5 +
 src/Data/Bytestring/Base.agda              |  73 ++++++++++++++
 src/Data/Bytestring/Builder/Base.agda      |  72 ++++++++++++++
 src/Data/Bytestring/Builder/Primitive.agda |  38 ++++++++
 src/Data/Bytestring/IO.agda                |  22 +++++
 src/Data/Bytestring/IO/Primitive.agda      |  24 +++++
 src/Data/Bytestring/Primitive.agda         |  44 +++++++++
 src/Data/Float/Base.agda                   |  16 ++-
 src/Data/Float/Properties.agda             |  22 +++--
 src/Data/Word.agda                         |  10 +-
 src/Data/Word/Base.agda                    |  32 +-----
 src/Data/Word/Instances.agda               |  11 +--
 src/Data/Word/Properties.agda              | 103 +-------------------
 src/Data/Word64.agda                       |  15 +++
 src/Data/Word64/Base.agda                  |  41 ++++++++
 src/Data/Word64/Instances.agda             |  16 +++
 src/Data/Word64/Literals.agda              |  19 ++++
 src/Data/Word64/Primitive.agda             |  29 ++++++
 src/Data/Word64/Properties.agda            | 108 +++++++++++++++++++++
 src/Data/Word64/Show.agda                  |  36 +++++++
 src/Data/Word64/Unsafe.agda                |  45 +++++++++
 src/Data/Word8/Base.agda                   |  51 ++++++++++
 src/Data/Word8/Literals.agda               |  19 ++++
 src/Data/Word8/Primitive.agda              |  36 +++++++
 src/Data/Word8/Show.agda                   |  33 +++++++
 src/Reflection/AST/Literal.agda            |   4 +-
 src/Reflection/AST/Name.agda               |   2 +-
 src/Reflection/AST/Show.agda               |   4 +-
 src/System/Directory/Primitive.agda        |   4 +-
 src/System/Random.agda                     |   2 +-
 src/Tactic/Cong.agda                       |   2 +-
 tests/data/bytestring/Main.agda            |  66 +++++++++++++
 tests/data/bytestring/bytestring.agda-lib  |   2 +
 tests/data/bytestring/expected             |  20 ++++
 tests/data/bytestring/run                  |   5 +
 tests/runtests.agda                        |   1 +
 39 files changed, 937 insertions(+), 158 deletions(-)
 create mode 100644 src/Data/Bytestring/Base.agda
 create mode 100644 src/Data/Bytestring/Builder/Base.agda
 create mode 100644 src/Data/Bytestring/Builder/Primitive.agda
 create mode 100644 src/Data/Bytestring/IO.agda
 create mode 100644 src/Data/Bytestring/IO/Primitive.agda
 create mode 100644 src/Data/Bytestring/Primitive.agda
 create mode 100644 src/Data/Word64.agda
 create mode 100644 src/Data/Word64/Base.agda
 create mode 100644 src/Data/Word64/Instances.agda
 create mode 100644 src/Data/Word64/Literals.agda
 create mode 100644 src/Data/Word64/Primitive.agda
 create mode 100644 src/Data/Word64/Properties.agda
 create mode 100644 src/Data/Word64/Show.agda
 create mode 100644 src/Data/Word64/Unsafe.agda
 create mode 100644 src/Data/Word8/Base.agda
 create mode 100644 src/Data/Word8/Literals.agda
 create mode 100644 src/Data/Word8/Primitive.agda
 create mode 100644 src/Data/Word8/Show.agda
 create mode 100644 tests/data/bytestring/Main.agda
 create mode 100644 tests/data/bytestring/bytestring.agda-lib
 create mode 100644 tests/data/bytestring/expected
 create mode 100644 tests/data/bytestring/run

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index ec9ae5c501..5086e96696 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -47,11 +47,11 @@ on:
 ########################################################################
 
 env:
-  GHC_VERSION: 8.10.7
-  CABAL_VERSION: 3.6.2.0
+  GHC_VERSION: 9.8.2
+  CABAL_VERSION: 3.10.3.0
   CABAL_INSTALL: cabal v1-install --ghc-options='-O1 +RTS -M6G -RTS'
   # CABAL_INSTALL: cabal install --overwrite-policy=always --ghc-options='-O1 +RTS -M6G -RTS'
-  AGDA: agda -Werror +RTS -M3.5G -H3.5G -A128M -RTS -i. -isrc -idoc
+  AGDA: agda -Werror +RTS -M5G -H3.5G -A128M -RTS -i. -isrc -idoc
 
 jobs:
   test-stdlib:
diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9f8d6de9bd..a243edfd5f 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -30,6 +30,8 @@ Non-backwards compatible changes
 * The definitions in `Algebra.Module.Morphism.Construct.Identity` are now
   parametrized by _raw_ bundles, and as such take a proof of reflexivity.
 * The module `IO.Primitive` was moved to `IO.Primitive.Core`.
+* The modules in the `Data.Word` hierarchy were moved to the `Data.Word64`
+  one instead.
 
 Other major improvements
 ------------------------
@@ -107,6 +109,16 @@ Deprecated names
   untilRight  ↦  untilInj₂
   ```
 
+* In `Data.Float.Base`:
+  ```agda
+  toWord ↦ toWord64
+  ```
+
+* In `Data.Float.Properties`:
+  ```agda
+  toWord-injective ↦ toWord64-injective
+  ```
+
 New modules
 -----------
 
@@ -231,6 +243,31 @@ New modules
   Data.Vec.Bounded.Show
   ```
 
+* Word64 literals and bit-based functions:
+  ```agda
+  Data.Word64.Literals
+  Data.Word64.Unsafe
+  Data.Word64.Show
+  ```
+
+* A type of bytes:
+  ```agda
+  Data.Word8.Primitive
+  Data.Word8.Base
+  Data.Word8.Literals
+  Data.Word8.Show
+  ```
+
+* Bytestrings and builders:
+  ```agda
+  Data.Bytestring.Base
+  Data.Bytestring.Builder.Base
+  Data.Bytestring.Builder.Primitive
+  Data.Bytestring.IO
+  Data.Bytestring.IO.Primitive
+  Data.Bytestring.Primitive
+  ```
+
 * Decidability for the subset relation on lists:
   ```agda
   Data.List.Relation.Binary.Subset.DecSetoid (_⊆?_)
@@ -402,6 +439,11 @@ Additions to existing modules
     *-cancelʳ-nonZero : AlmostRightCancellative 0# *
   ```
 
+* In `Data.Bool.Show`:
+  ```agda
+  showBit : Bool → Char
+  ```
+
 * In `Data.Container.Indexed.Core`:
   ```agda
   Subtrees o c = (r : Response c) → X (next c r)
@@ -711,9 +753,10 @@ Additions to existing modules
   _>>_ : IO A → IO B → IO B
   ```
 
-* In `Data.Word.Base`:
+* In `Data.Word64.Base`:
   ```agda
   _≤_ : Rel Word64 zero
+  show : Word64 → String
   ```
 
 * Added new definition in `Relation.Binary.Construct.Closure.Transitive`
diff --git a/GenerateEverything.hs b/GenerateEverything.hs
index d7895f96db..801c1b7e35 100644
--- a/GenerateEverything.hs
+++ b/GenerateEverything.hs
@@ -44,6 +44,18 @@ unsafeModules = map modToFile
   , "Codata.Musical.Covec"
   , "Codata.Musical.Conversion"
   , "Codata.Musical.Stream"
+  , "Data.Bytestring.Base"
+  , "Data.Bytestring.Builder.Base"
+  , "Data.Bytestring.Builder.Primitive"
+  , "Data.Bytestring.IO"
+  , "Data.Bytestring.IO.Primitive"
+  , "Data.Bytestring.Primitive"
+  , "Data.Word8.Base"
+  , "Data.Word8.Literals"
+  , "Data.Word8.Primitive"
+  , "Data.Word64.Primitive"
+  , "Data.Word8.Show"
+  , "Data.Word64.Show"
   , "Debug.Trace"
   , "Effect.Monad.IO"
   , "Effect.Monad.IO.Instances"
diff --git a/src/Data/Bool/Show.agda b/src/Data/Bool/Show.agda
index 21f2ec1a55..075efaa63b 100644
--- a/src/Data/Bool/Show.agda
+++ b/src/Data/Bool/Show.agda
@@ -9,8 +9,13 @@
 module Data.Bool.Show where
 
 open import Data.Bool.Base using (Bool; false; true)
+open import Data.Char.Base using (Char)
 open import Data.String.Base using (String)
 
 show : Bool → String
 show true  = "true"
 show false = "false"
+
+showBit : Bool → Char
+showBit true = '1'
+showBit false = '0'
diff --git a/src/Data/Bytestring/Base.agda b/src/Data/Bytestring/Base.agda
new file mode 100644
index 0000000000..69bdb23db4
--- /dev/null
+++ b/src/Data/Bytestring/Base.agda
@@ -0,0 +1,73 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bytestrings: simple types and functions
+-- Note that these functions do not perform bound checks.
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Bytestring.Base where
+
+open import Data.Nat.Base using (ℕ; _+_; _*_; _^_)
+open import Agda.Builtin.String using (String)
+import Data.Fin.Base as Fin
+open import Data.Vec.Base as Vec using (Vec)
+open import Data.Word8.Base as Word8 using (Word8)
+open import Data.Word64.Base as Word64 using (Word64)
+
+open import Function.Base using (const; _$_)
+
+------------------------------------------------------------------------------
+-- Re-export type and operations
+
+open import Data.Bytestring.Primitive as Prim public
+  using ( Bytestring
+        ; length
+        ; take
+        ; drop
+        ; show
+        )
+  renaming (index to getWord8)
+
+------------------------------------------------------------------------------
+-- Operations
+
+slice : Word64 → Word64 → Bytestring → Bytestring
+slice start chunk buf = take chunk (drop start buf)
+
+------------------------------------------------------------------------------
+-- Generic combinators for fixed-size encodings
+
+getWord8s : (n : ℕ) → Bytestring → Word64 → Vec Word8 n
+getWord8s n buf idx
+  = let idx = Word64.toℕ idx in
+  Vec.map (λ k → getWord8 buf (Word64.fromℕ (idx + Fin.toℕ k)))
+  $ Vec.allFin n
+
+-- Little endian representation:
+-- Low place values first
+fromWord8sLE : ∀ {n w} {W : Set w} → (fromℕ : ℕ → W) → Vec Word8 n → W
+fromWord8sLE f ws = f (Vec.foldr′ (λ w acc → Word8.toℕ w + acc * (2 ^ 8)) 0 ws)
+
+-- Big endian representation
+-- Big place values first
+fromWord8sBE : ∀ {n w} {W : Set w} → (fromℕ : ℕ → W) → Vec Word8 n → W
+fromWord8sBE f ws = f (Vec.foldl′ (λ acc w → acc * (2 ^ 8) + Word8.toℕ w) 0 ws)
+
+------------------------------------------------------------------------------
+-- Decoding to a vector of bytes
+
+toWord8s : (b : Bytestring) → Vec Word8 (length b)
+toWord8s b = getWord8s _ b (Word64.fromℕ 0)
+
+------------------------------------------------------------------------------
+-- Getting Word64
+
+getWord64LE : Bytestring → Word64 → Word64
+getWord64LE buf idx
+  = fromWord8sLE Word64.fromℕ (getWord8s 8 buf idx)
+
+getWord64BE : Bytestring → Word64 → Word64
+getWord64BE buf idx
+  = fromWord8sBE Word64.fromℕ (getWord8s 8 buf idx)
diff --git a/src/Data/Bytestring/Builder/Base.agda b/src/Data/Bytestring/Builder/Base.agda
new file mode 100644
index 0000000000..728ceb3896
--- /dev/null
+++ b/src/Data/Bytestring/Builder/Base.agda
@@ -0,0 +1,72 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bytestrings: builder type and functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Bytestring.Builder.Base where
+
+open import Data.Nat.Base using (ℕ; zero; suc; _/_; _%_; _^_)
+open import Data.Word8.Base as Word8 using (Word8)
+open import Data.Word64.Base as Word64 using (Word64)
+open import Function.Base using (_∘′_)
+
+------------------------------------------------------------------------------
+-- Re-export type and operations
+
+open import Data.Bytestring.Builder.Primitive as Prim public
+  using ( Builder
+        ; toBytestring
+        ; bytestring
+        ; word8
+        ; empty
+        ; _<>_
+        )
+
+------------------------------------------------------------------------------
+-- High-level combinators
+
+module List where
+
+  open import Data.List.Base as List using (List)
+
+  concat : List Builder → Builder
+  concat = List.foldr _<>_ empty
+
+module Vec where
+
+  open import Data.Vec.Base as Vec using (Vec)
+
+  concat : ∀ {n} → Vec Builder n → Builder
+  concat = Vec.foldr′ _<>_ empty
+open Vec
+
+------------------------------------------------------------------------------
+-- Generic word-specific combinators
+
+open import Data.Vec.Base as Vec using (Vec; []; _∷_)
+
+wordN : ∀ {n} → Vec Word8 n → Builder
+wordN = concat ∘′ Vec.map word8
+
+toWord8sLE : ∀ {w} {W : Set w} (n : ℕ) (toℕ : W → ℕ) → W → Vec Word8 n
+toWord8sLE n toℕ w = loop (toℕ w) n where
+
+  loop : ℕ → (n : ℕ) → Vec Word8 n
+  loop acc 0 = []
+  loop acc 1 = Word8.fromℕ acc ∷ []
+  loop acc (suc n) = Word8.fromℕ (acc % 2 ^ 8) ∷ loop (acc / 2 ^ 8) n
+
+toWord8sBE : ∀ {w} {W : Set w} (n : ℕ) (toℕ : W → ℕ) → W → Vec Word8 n
+toWord8sBE n toℕ w = Vec.reverse (toWord8sLE n toℕ w)
+
+------------------------------------------------------------------------------
+-- Builders for Word64
+
+word64LE : Word64 → Builder
+word64LE w = wordN (toWord8sLE 8 Word64.toℕ w)
+
+word64BE : Word64 → Builder
+word64BE w = wordN (toWord8sBE 8 Word64.toℕ w)
diff --git a/src/Data/Bytestring/Builder/Primitive.agda b/src/Data/Bytestring/Builder/Primitive.agda
new file mode 100644
index 0000000000..b9f2bd8937
--- /dev/null
+++ b/src/Data/Bytestring/Builder/Primitive.agda
@@ -0,0 +1,38 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Primitive Bytestrings: builder type and functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Bytestring.Builder.Primitive where
+
+open import Agda.Builtin.Nat
+open import Agda.Builtin.String
+
+open import Data.Word8.Primitive
+open import Data.Bytestring.Primitive using (Bytestring)
+
+infixr 6 _<>_
+
+postulate
+  -- Builder and execution
+  Builder : Set
+  toBytestring : Builder → Bytestring
+
+  -- Assembling a builder
+  bytestring : Bytestring → Builder
+  word8 : Word8 → Builder
+  empty : Builder
+  _<>_ : Builder → Builder → Builder
+
+{-# FOREIGN GHC import qualified Data.ByteString.Builder as Builder #-}
+{-# FOREIGN GHC import qualified Data.ByteString.Lazy as Lazy #-}
+{-# FOREIGN GHC import qualified Data.Text as T #-}
+{-# COMPILE GHC Builder = type Builder.Builder #-}
+{-# COMPILE GHC toBytestring = Lazy.toStrict . Builder.toLazyByteString #-}
+{-# COMPILE GHC bytestring = Builder.byteString #-}
+{-# COMPILE GHC word8 = Builder.word8 #-}
+{-# COMPILE GHC empty = mempty #-}
+{-# COMPILE GHC _<>_ = mappend #-}
diff --git a/src/Data/Bytestring/IO.agda b/src/Data/Bytestring/IO.agda
new file mode 100644
index 0000000000..e79bc2c8a6
--- /dev/null
+++ b/src/Data/Bytestring/IO.agda
@@ -0,0 +1,22 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bytestrings: IO operations
+------------------------------------------------------------------------
+
+{-# OPTIONS --guardedness --cubical-compatible #-}
+
+module Data.Bytestring.IO where
+
+open import Agda.Builtin.String
+open import IO using (IO; lift)
+open import Data.Bytestring.Base using (Bytestring)
+open import Data.Unit.Base using (⊤)
+
+import Data.Bytestring.IO.Primitive as Prim
+
+readFile : String → IO Bytestring
+readFile fp = lift (Prim.readFile fp)
+
+writeFile : String → Bytestring → IO ⊤
+writeFile fp str = lift (Prim.writeFile fp str)
diff --git a/src/Data/Bytestring/IO/Primitive.agda b/src/Data/Bytestring/IO/Primitive.agda
new file mode 100644
index 0000000000..113ef19e86
--- /dev/null
+++ b/src/Data/Bytestring/IO/Primitive.agda
@@ -0,0 +1,24 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Primitive Bytestrings: IO operations
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Bytestring.IO.Primitive where
+
+open import Agda.Builtin.String using (String)
+open import Agda.Builtin.Unit using (⊤)
+open import IO.Primitive.Core using (IO)
+
+open import Data.Bytestring.Primitive
+
+postulate
+  readFile : String → IO Bytestring
+  writeFile : String → Bytestring → IO ⊤
+
+{-# FOREIGN GHC import qualified Data.Text as T #-}
+{-# FOREIGN GHC import Data.ByteString #-}
+{-# COMPILE GHC readFile = Data.ByteString.readFile . T.unpack #-}
+{-# COMPILE GHC writeFile = Data.ByteString.writeFile . T.unpack #-}
diff --git a/src/Data/Bytestring/Primitive.agda b/src/Data/Bytestring/Primitive.agda
new file mode 100644
index 0000000000..94e0d64b2c
--- /dev/null
+++ b/src/Data/Bytestring/Primitive.agda
@@ -0,0 +1,44 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Primitive bytestrings: simple bindings to Haskell types and functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Bytestring.Primitive where
+
+open import Agda.Builtin.Nat using (Nat)
+open import Agda.Builtin.String using (String)
+open import Agda.Builtin.Word using (Word64)
+open import Data.Word8.Primitive using (Word8)
+
+-- NB: the bytestring package uses `Int` as the indexing type which
+-- Haskell's base specifies as:
+--
+-- > A fixed-precision integer type with at least the range
+-- > [-2^29 .. 2^29-1]. The exact range for a given implementation
+-- > can be determined by using minBound and maxBound from the
+-- > Bounded class.
+--
+-- There is no ergonomic way to encode that in a type-safe manner.
+-- For now we use `Word64` with the understanding that using indices
+-- greater than 2^29-1 may lead to undefined behaviours...
+
+postulate
+  Bytestring : Set
+  index : Bytestring → Word64 → Word8
+  length : Bytestring → Nat
+  take : Word64 → Bytestring → Bytestring
+  drop : Word64 → Bytestring → Bytestring
+  show : Bytestring → String
+
+{-# FOREIGN GHC import qualified Data.ByteString as B #-}
+{-# FOREIGN GHC import qualified Data.Text as T #-}
+
+{-# COMPILE GHC Bytestring = type B.ByteString #-}
+{-# COMPILE GHC index = \ buf idx -> B.index buf (fromIntegral idx) #-}
+{-# COMPILE GHC length = \ buf -> fromIntegral (B.length buf) #-}
+{-# COMPILE GHC take = B.take . fromIntegral #-}
+{-# COMPILE GHC drop = B.drop . fromIntegral #-}
+{-# COMPILE GHC show = T.pack . Prelude.show #-}
diff --git a/src/Data/Float/Base.agda b/src/Data/Float/Base.agda
index fdc7dab2db..a991c119a9 100644
--- a/src/Data/Float/Base.agda
+++ b/src/Data/Float/Base.agda
@@ -9,7 +9,7 @@
 module Data.Float.Base where
 
 open import Data.Bool.Base using (T)
-import Data.Word.Base as Word
+import Data.Word64.Base as Word64
 import Data.Maybe.Base as Maybe
 open import Function.Base using (_on_; _∘_)
 open import Agda.Builtin.Equality
@@ -30,7 +30,7 @@ open import Agda.Builtin.Float public
   ; primFloatIsNegativeZero    to isNegativeZero
   ; primFloatIsSafeInteger     to isSafeInteger
   -- Conversions
-  ; primFloatToWord64          to toWord
+  ; primFloatToWord64          to toWord64
   ; primNatToFloat             to fromℕ
   ; primIntToFloat             to fromℤ
   ; primFloatRound             to round
@@ -69,9 +69,19 @@ open import Agda.Builtin.Float public
 infix 4 _≈_
 
 _≈_ : Rel Float _
-_≈_ = _≡_ on Maybe.map Word.toℕ ∘ toWord
+_≈_ = _≡_ on Maybe.map Word64.toℕ ∘ toWord64
 
 
 infix 4 _≤_
 _≤_ : Rel Float _
 x ≤ y = T (x ≤ᵇ y)
+
+
+------------------------------------------------------------------------
+-- DEPRECATIONS
+
+toWord = toWord64
+{-# WARNING_ON_USAGE toWord
+"Warning: toWord was deprecated in v2.1.
+Please use toWord64 instead."
+#-}
diff --git a/src/Data/Float/Properties.agda b/src/Data/Float/Properties.agda
index fab340ea0e..6975804193 100644
--- a/src/Data/Float/Properties.agda
+++ b/src/Data/Float/Properties.agda
@@ -13,8 +13,8 @@ open import Data.Float.Base
 import Data.Maybe.Base as Maybe
 import Data.Maybe.Properties as Maybe
 import Data.Nat.Properties as ℕ
-import Data.Word.Base as Word
-import Data.Word.Properties as Word
+import Data.Word64.Base as Word64
+import Data.Word64.Properties as Word64
 open import Function.Base using (_∘_)
 open import Relation.Nullary.Decidable as RN using (map′)
 open import Relation.Binary.Core using (_⇒_)
@@ -33,17 +33,17 @@ open import Relation.Binary.PropositionalEquality.Properties
 -- Primitive properties
 
 open import Agda.Builtin.Float.Properties
-  renaming (primFloatToWord64Injective to toWord-injective)
+  renaming (primFloatToWord64Injective to toWord64-injective)
   public
 
 ------------------------------------------------------------------------
 -- Properties of _≈_
 
 ≈⇒≡ : _≈_ ⇒ _≡_
-≈⇒≡ eq = toWord-injective _ _ (Maybe.map-injective Word.≈⇒≡ eq)
+≈⇒≡ eq = toWord64-injective _ _ (Maybe.map-injective Word64.≈⇒≡ eq)
 
 ≈-reflexive : _≡_ ⇒ _≈_
-≈-reflexive eq = cong (Maybe.map Word.toℕ ∘ toWord) eq
+≈-reflexive eq = cong (Maybe.map Word64.toℕ ∘ toWord64) eq
 
 ≈-refl : Reflexive _≈_
 ≈-refl = refl
@@ -59,7 +59,7 @@ open import Agda.Builtin.Float.Properties
 
 infix 4 _≈?_
 _≈?_ : Decidable _≈_
-_≈?_ = On.decidable (Maybe.map Word.toℕ ∘ toWord) _≡_ (Maybe.≡-dec ℕ._≟_)
+_≈?_ = On.decidable (Maybe.map Word64.toℕ ∘ toWord64) _≡_ (Maybe.≡-dec ℕ._≟_)
 
 ≈-isEquivalence : IsEquivalence _≈_
 ≈-isEquivalence = record
@@ -95,3 +95,13 @@ x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)
 
 ≡-decSetoid : DecSetoid _ _
 ≡-decSetoid = decSetoid _≟_
+
+
+------------------------------------------------------------------------
+-- DEPRECATIONS
+
+toWord-injective = toWord64-injective
+{-# WARNING_ON_USAGE toWord-injective
+"Warning: toWord-injective was deprecated in v2.1.
+Please use toWord64-injective instead."
+#-}
diff --git a/src/Data/Word.agda b/src/Data/Word.agda
index 8c07cac9c7..df6594641c 100644
--- a/src/Data/Word.agda
+++ b/src/Data/Word.agda
@@ -1,15 +1,15 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Machine words
+-- This module is DEPRECATED. Please use Data.Word64 instead
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module Data.Word where
 
-------------------------------------------------------------------------
--- Re-export base definitions and decidability of equality
+open import Data.Word64 public
 
-open import Data.Word.Base public
-open import Data.Word.Properties using (_≈?_; _<?_; _≟_; _==_) public
+{-# WARNING_ON_IMPORT
+"Data.Word was deprecated in v2.1. Use Data.Word64 instead."
+#-}
diff --git a/src/Data/Word/Base.agda b/src/Data/Word/Base.agda
index 161502e4e9..f1f766885b 100644
--- a/src/Data/Word/Base.agda
+++ b/src/Data/Word/Base.agda
@@ -1,37 +1,15 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Machine words: basic type and conversion functions
+-- This module is DEPRECATED. Please use Data.Word64.Base instead
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module Data.Word.Base where
 
-open import Level using (zero)
-import Data.Nat.Base as ℕ
-open import Function.Base using (_on_)
-open import Relation.Binary.Core using (Rel)
-open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+open import Data.Word64.Base public
 
-------------------------------------------------------------------------
--- Re-export built-ins publicly
-
-open import Agda.Builtin.Word public
-  using (Word64)
-  renaming
-  ( primWord64ToNat   to toℕ
-  ; primWord64FromNat to fromℕ
-  )
-
-infix 4 _≈_
-_≈_ : Rel Word64 zero
-_≈_ = _≡_ on toℕ
-
-infix 4 _<_
-_<_ : Rel Word64 zero
-_<_ = ℕ._<_ on toℕ
-
-infix 4 _≤_
-_≤_ : Rel Word64 zero
-_≤_ = ℕ._≤_ on toℕ
+{-# WARNING_ON_IMPORT
+"Data.Word.Base was deprecated in v2.1. Use Data.Word64.Base instead."
+#-}
diff --git a/src/Data/Word/Instances.agda b/src/Data/Word/Instances.agda
index 20b630ec4b..e2df252dda 100644
--- a/src/Data/Word/Instances.agda
+++ b/src/Data/Word/Instances.agda
@@ -1,16 +1,15 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Instances for words
+-- This module is DEPRECATED. Please use Data.Word64.Instances instead
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module Data.Word.Instances where
 
-open import Data.Word.Properties
-open import Relation.Binary.PropositionalEquality.Properties
-  using (isDecEquivalence)
+open import Data.Word64.Instances public
 
-instance
-  Word-≡-isDecEquivalence = isDecEquivalence _≟_
+{-# WARNING_ON_IMPORT
+"Data.Word.Instances was deprecated in v2.1. Use Data.Word64.Instances instead."
+#-}
diff --git a/src/Data/Word/Properties.agda b/src/Data/Word/Properties.agda
index 2681d6ce15..b053aee190 100644
--- a/src/Data/Word/Properties.agda
+++ b/src/Data/Word/Properties.agda
@@ -1,108 +1,15 @@
 ------------------------------------------------------------------------
 -- The Agda standard library
 --
--- Properties of operations on machine words
+-- This module is DEPRECATED. Please use Data.Word64.Properties instead
 ------------------------------------------------------------------------
 
 {-# OPTIONS --cubical-compatible --safe #-}
 
 module Data.Word.Properties where
 
-import Data.Nat.Base as ℕ
-open import Data.Bool.Base using (Bool)
-open import Data.Word.Base using (_≈_; toℕ; Word64; _<_)
-import Data.Nat.Properties as ℕ
-open import Relation.Nullary.Decidable.Core using (map′; ⌊_⌋)
-open import Relation.Binary
-  using ( _⇒_; Reflexive; Symmetric; Transitive; Substitutive
-        ; Decidable; DecidableEquality; IsEquivalence; IsDecEquivalence
-        ; Setoid; DecSetoid; StrictTotalOrder)
-import Relation.Binary.Construct.On as On
-open import Relation.Binary.PropositionalEquality.Core
-  using (_≡_; refl; cong; sym; trans; subst)
-open import Relation.Binary.PropositionalEquality.Properties
-  using (setoid; decSetoid)
+open import Data.Word64.Properties public
 
-------------------------------------------------------------------------
--- Primitive properties
-
-open import Agda.Builtin.Word.Properties
-  renaming (primWord64ToNatInjective to toℕ-injective)
-  public
-
-------------------------------------------------------------------------
--- Properties of _≈_
-
-≈⇒≡ : _≈_ ⇒ _≡_
-≈⇒≡ = toℕ-injective _ _
-
-≈-reflexive : _≡_ ⇒ _≈_
-≈-reflexive = cong toℕ
-
-≈-refl : Reflexive _≈_
-≈-refl = refl
-
-≈-sym : Symmetric _≈_
-≈-sym = sym
-
-≈-trans : Transitive _≈_
-≈-trans = trans
-
-≈-subst : ∀ {ℓ} → Substitutive _≈_ ℓ
-≈-subst P x≈y p = subst P (≈⇒≡ x≈y) p
-
-infix 4 _≈?_
-_≈?_ : Decidable _≈_
-x ≈? y = toℕ x ℕ.≟ toℕ y
-
-≈-isEquivalence : IsEquivalence _≈_
-≈-isEquivalence = record
-  { refl  = λ {i} → ≈-refl {i}
-  ; sym   = λ {i j} → ≈-sym {i} {j}
-  ; trans = λ {i j k} → ≈-trans {i} {j} {k}
-  }
-
-≈-setoid : Setoid _ _
-≈-setoid = record
-  { isEquivalence = ≈-isEquivalence
-  }
-
-≈-isDecEquivalence : IsDecEquivalence _≈_
-≈-isDecEquivalence = record
-  { isEquivalence = ≈-isEquivalence
-  ; _≟_           = _≈?_
-  }
-
-≈-decSetoid : DecSetoid _ _
-≈-decSetoid = record
-  { isDecEquivalence = ≈-isDecEquivalence
-  }
-------------------------------------------------------------------------
--- Properties of _≡_
-
-infix 4 _≟_
-_≟_ : DecidableEquality Word64
-x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)
-
-≡-setoid : Setoid _ _
-≡-setoid = setoid Word64
-
-≡-decSetoid : DecSetoid _ _
-≡-decSetoid = decSetoid _≟_
-
-------------------------------------------------------------------------
--- Boolean equality test.
-
-infix 4 _==_
-_==_ : Word64 → Word64 → Bool
-w₁ == w₂ = ⌊ w₁ ≟ w₂ ⌋
-
-------------------------------------------------------------------------
--- Properties of _<_
-
-infix 4 _<?_
-_<?_ : Decidable _<_
-_<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
-
-<-strictTotalOrder-≈ : StrictTotalOrder _ _ _
-<-strictTotalOrder-≈ = On.strictTotalOrder ℕ.<-strictTotalOrder toℕ
+{-# WARNING_ON_IMPORT
+"Data.Word.Properties was deprecated in v2.1. Use Data.Word64.Properties instead."
+#-}
diff --git a/src/Data/Word64.agda b/src/Data/Word64.agda
new file mode 100644
index 0000000000..5c5004a1a4
--- /dev/null
+++ b/src/Data/Word64.agda
@@ -0,0 +1,15 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Machine words
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Word64 where
+
+------------------------------------------------------------------------
+-- Re-export base definitions and decidability of equality
+
+open import Data.Word64.Base public
+open import Data.Word64.Properties using (_≈?_; _<?_; _≟_; _==_) public
diff --git a/src/Data/Word64/Base.agda b/src/Data/Word64/Base.agda
new file mode 100644
index 0000000000..8ce6daa615
--- /dev/null
+++ b/src/Data/Word64/Base.agda
@@ -0,0 +1,41 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Machine words: basic type and conversion functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Word64.Base where
+
+open import Algebra.Core using (Op₂)
+open import Data.Nat.Base as ℕ using (ℕ)
+open import Function.Base using (_on_; _∘₂′_)
+open import Level using (zero)
+open import Relation.Binary.Core using (Rel)
+open import Relation.Binary.PropositionalEquality.Core using (_≡_)
+
+------------------------------------------------------------------------
+-- Re-export built-ins publicly
+
+open import Agda.Builtin.Word public
+  using (Word64)
+  renaming
+  ( primWord64ToNat   to toℕ
+  ; primWord64FromNat to fromℕ
+  )
+
+liftOp₂ : Op₂ ℕ → Op₂ Word64
+liftOp₂ op = fromℕ ∘₂′ op on toℕ
+
+infix 4 _≈_
+_≈_ : Rel Word64 zero
+_≈_ = _≡_ on toℕ
+
+infix 4 _<_
+_<_ : Rel Word64 zero
+_<_ = ℕ._<_ on toℕ
+
+infix 4 _≤_
+_≤_ : Rel Word64 zero
+_≤_ = ℕ._≤_ on toℕ
diff --git a/src/Data/Word64/Instances.agda b/src/Data/Word64/Instances.agda
new file mode 100644
index 0000000000..b771529d13
--- /dev/null
+++ b/src/Data/Word64/Instances.agda
@@ -0,0 +1,16 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Instances for words
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Word64.Instances where
+
+open import Data.Word64.Properties
+open import Relation.Binary.PropositionalEquality.Properties
+  using (isDecEquivalence)
+
+instance
+  Word64-≡-isDecEquivalence = isDecEquivalence _≟_
diff --git a/src/Data/Word64/Literals.agda b/src/Data/Word64/Literals.agda
new file mode 100644
index 0000000000..70d1365e5f
--- /dev/null
+++ b/src/Data/Word64/Literals.agda
@@ -0,0 +1,19 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Word64 Literals
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Word64.Literals where
+
+open import Agda.Builtin.FromNat using (Number)
+open import Data.Unit.Base using (⊤)
+open import Data.Word64.Base using (Word64; fromℕ)
+
+number : Number Word64
+number = record
+  { Constraint = λ _ → ⊤
+  ; fromNat    = λ w → fromℕ w
+  }
diff --git a/src/Data/Word64/Primitive.agda b/src/Data/Word64/Primitive.agda
new file mode 100644
index 0000000000..ff92d306fa
--- /dev/null
+++ b/src/Data/Word64/Primitive.agda
@@ -0,0 +1,29 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Word64: simple bindings to Haskell types and functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Word64.Primitive where
+
+open import Agda.Builtin.Bool using (Bool)
+open import Agda.Builtin.Nat using (Nat)
+open import Agda.Builtin.String using (String)
+open import Agda.Builtin.Word using (Word64)
+
+postulate
+  testBit : Word64 → Nat → Bool
+  setBit : Word64 → Nat → Word64
+  clearBit : Word64 → Nat → Word64
+  show : Word64 → String
+
+{-# FOREIGN GHC import GHC.Word #-}
+{-# FOREIGN GHC import qualified Data.Bits as B #-}
+{-# FOREIGN GHC import qualified Data.Text as T #-}
+
+{-# COMPILE GHC testBit = \ w -> B.testBit w . fromIntegral #-}
+{-# COMPILE GHC setBit = \ w -> B.setBit w . fromIntegral #-}
+{-# COMPILE GHC clearBit = \ w -> B.clearBit  w . fromIntegral #-}
+{-# COMPILE GHC show = T.pack . Prelude.show #-}
diff --git a/src/Data/Word64/Properties.agda b/src/Data/Word64/Properties.agda
new file mode 100644
index 0000000000..0f5b4d8fe5
--- /dev/null
+++ b/src/Data/Word64/Properties.agda
@@ -0,0 +1,108 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Properties of operations on machine words
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible --safe #-}
+
+module Data.Word64.Properties where
+
+import Data.Nat.Base as ℕ
+open import Data.Bool.Base using (Bool)
+open import Data.Word64.Base using (_≈_; toℕ; Word64; _<_)
+import Data.Nat.Properties as ℕ
+open import Relation.Nullary.Decidable.Core using (map′; ⌊_⌋)
+open import Relation.Binary
+  using ( _⇒_; Reflexive; Symmetric; Transitive; Substitutive
+        ; Decidable; DecidableEquality; IsEquivalence; IsDecEquivalence
+        ; Setoid; DecSetoid; StrictTotalOrder)
+import Relation.Binary.Construct.On as On
+open import Relation.Binary.PropositionalEquality.Core
+  using (_≡_; refl; cong; sym; trans; subst)
+open import Relation.Binary.PropositionalEquality.Properties
+  using (setoid; decSetoid)
+
+------------------------------------------------------------------------
+-- Primitive properties
+
+open import Agda.Builtin.Word.Properties
+  renaming (primWord64ToNatInjective to toℕ-injective)
+  public
+
+------------------------------------------------------------------------
+-- Properties of _≈_
+
+≈⇒≡ : _≈_ ⇒ _≡_
+≈⇒≡ = toℕ-injective _ _
+
+≈-reflexive : _≡_ ⇒ _≈_
+≈-reflexive = cong toℕ
+
+≈-refl : Reflexive _≈_
+≈-refl = refl
+
+≈-sym : Symmetric _≈_
+≈-sym = sym
+
+≈-trans : Transitive _≈_
+≈-trans = trans
+
+≈-subst : ∀ {ℓ} → Substitutive _≈_ ℓ
+≈-subst P x≈y p = subst P (≈⇒≡ x≈y) p
+
+infix 4 _≈?_
+_≈?_ : Decidable _≈_
+x ≈? y = toℕ x ℕ.≟ toℕ y
+
+≈-isEquivalence : IsEquivalence _≈_
+≈-isEquivalence = record
+  { refl  = λ {i} → ≈-refl {i}
+  ; sym   = λ {i j} → ≈-sym {i} {j}
+  ; trans = λ {i j k} → ≈-trans {i} {j} {k}
+  }
+
+≈-setoid : Setoid _ _
+≈-setoid = record
+  { isEquivalence = ≈-isEquivalence
+  }
+
+≈-isDecEquivalence : IsDecEquivalence _≈_
+≈-isDecEquivalence = record
+  { isEquivalence = ≈-isEquivalence
+  ; _≟_           = _≈?_
+  }
+
+≈-decSetoid : DecSetoid _ _
+≈-decSetoid = record
+  { isDecEquivalence = ≈-isDecEquivalence
+  }
+------------------------------------------------------------------------
+-- Properties of _≡_
+
+infix 4 _≟_
+_≟_ : DecidableEquality Word64
+x ≟ y = map′ ≈⇒≡ ≈-reflexive (x ≈? y)
+
+≡-setoid : Setoid _ _
+≡-setoid = setoid Word64
+
+≡-decSetoid : DecSetoid _ _
+≡-decSetoid = decSetoid _≟_
+
+------------------------------------------------------------------------
+-- Boolean equality test.
+
+infix 4 _==_
+_==_ : Word64 → Word64 → Bool
+w₁ == w₂ = ⌊ w₁ ≟ w₂ ⌋
+
+------------------------------------------------------------------------
+-- Properties of _<_
+
+infix 4 _<?_
+_<?_ : Decidable _<_
+_<?_ = On.decidable toℕ ℕ._<_ ℕ._<?_
+
+<-strictTotalOrder-≈ : StrictTotalOrder _ _ _
+<-strictTotalOrder-≈ = On.strictTotalOrder ℕ.<-strictTotalOrder toℕ
diff --git a/src/Data/Word64/Show.agda b/src/Data/Word64/Show.agda
new file mode 100644
index 0000000000..cdf9a0cf98
--- /dev/null
+++ b/src/Data/Word64/Show.agda
@@ -0,0 +1,36 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bytes: showing bit patterns
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Word64.Show where
+
+open import Agda.Builtin.String using (String)
+
+open import Data.Bool.Show using (showBit)
+open import Data.Fin.Base as Fin using (Fin)
+import Data.Nat.Show as ℕ
+open import Data.String using (_++_; fromVec; padLeft)
+open import Data.Vec.Base as Vec using (Vec; []; _∷_)
+
+open import Data.Word64.Base
+open import Data.Word64.Unsafe
+
+open import Function.Base using (_$_)
+
+showBits : Word64 → String
+showBits w
+  = "0b" ++_
+  $ fromVec
+  $ Vec.reverse
+  $ Vec.map showBit
+  $ toBits w
+
+showHexa : Word64 → String
+showHexa w
+  = "0x" ++_
+  $ padLeft '0' 8
+  $ ℕ.showInBase 16 (toℕ w)
diff --git a/src/Data/Word64/Unsafe.agda b/src/Data/Word64/Unsafe.agda
new file mode 100644
index 0000000000..dcae0d6c61
--- /dev/null
+++ b/src/Data/Word64/Unsafe.agda
@@ -0,0 +1,45 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Machine words: unsafe functions using the FFI
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Word64.Unsafe where
+
+open import Agda.Builtin.Bool using (Bool; true; false)
+open import Data.Fin.Base as Fin using (Fin)
+open import Data.Product.Base using (proj₁)
+open import Data.Vec.Base as Vec using (Vec)
+open import Data.Word8.Base as Word8 using (Word8)
+open import Data.Word64.Base
+open import Function.Base using (_$_; _|>_)
+
+------------------------------------------------------------------------
+-- Re-export primitives publicly
+
+open import Data.Word64.Primitive as Prim public
+  using ( show )
+
+testBit : Word64 → Fin 64 → Bool
+testBit w i = Prim.testBit w (Fin.toℕ i)
+
+_[_]≔_ : Word64 → Fin 64 → Bool → Word64
+w [ i ]≔ false = Prim.clearBit w (Fin.toℕ i)
+w [ i ]≔ true  = Prim.setBit w (Fin.toℕ i)
+
+------------------------------------------------------------------------
+-- Convert to its components
+
+toBits : Word64 → Vec Bool 64
+toBits w = Vec.tabulate (testBit w)
+
+fromBits : Vec Bool 64 → Word64
+fromBits bs = Vec.foldl′ _|>_ (fromℕ 0)
+            $ Vec.zipWith (λ i b → _[ i ]≔ b) (Vec.allFin 64) bs
+
+toWord64s : Word64 → Vec Word8 8
+toWord64s w =
+  let ws = proj₁ (Vec.group 8 8 (toBits w)) in
+  Vec.map Word8.fromBits ws
diff --git a/src/Data/Word8/Base.agda b/src/Data/Word8/Base.agda
new file mode 100644
index 0000000000..142b544d78
--- /dev/null
+++ b/src/Data/Word8/Base.agda
@@ -0,0 +1,51 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bytes: base type and functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Word8.Base where
+
+open import Agda.Builtin.Bool using (Bool; true; false)
+open import Agda.Builtin.Char using (Char)
+open import Agda.Builtin.Nat using (Nat; _==_)
+open import Agda.Builtin.Unit using (⊤)
+
+open import Data.Fin.Base as Fin using (Fin)
+open import Data.Vec.Base as Vec using (Vec; []; _∷_)
+
+open import Function.Base using (_$_; _|>_)
+
+------------------------------------------------------------------------------
+-- Re-export type and operations
+
+open import Data.Word8.Primitive as Prim public
+  using ( Word8
+        ; _+_
+        ; show
+        )
+  renaming ( fromNat to fromℕ
+           ; toNat to toℕ
+           )
+
+testBit : Word8 → Fin 8 → Bool
+testBit w i = Prim.testBit w (Fin.toℕ i)
+
+_[_]≔_ : Word8 → Fin 8 → Bool → Word8
+w [ i ]≔ false = Prim.clearBit w (Fin.toℕ i)
+w [ i ]≔ true  = Prim.setBit w (Fin.toℕ i)
+
+------------------------------------------------------------------------------
+-- Basic functions
+
+toBits : Word8 → Vec Bool 8
+toBits w = Vec.tabulate (testBit w)
+
+fromBits : Vec Bool 8 → Word8
+fromBits bs = Vec.foldl′ _|>_ (fromℕ 0)
+            $ Vec.zipWith (λ i b → _[ i ]≔ b) (Vec.allFin 8) bs
+
+_≡ᵇ_ : Word8 → Word8 → Bool
+w ≡ᵇ w′ = toℕ w == toℕ w′
diff --git a/src/Data/Word8/Literals.agda b/src/Data/Word8/Literals.agda
new file mode 100644
index 0000000000..acc608f899
--- /dev/null
+++ b/src/Data/Word8/Literals.agda
@@ -0,0 +1,19 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Byte Literals
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Word8.Literals where
+
+open import Agda.Builtin.FromNat using (Number)
+open import Data.Unit.Base using (⊤)
+open import Data.Word8.Base using (Word8; fromℕ)
+
+number : Number Word8
+number = record
+  { Constraint = λ _ → ⊤
+  ; fromNat    = λ w → fromℕ w
+  }
diff --git a/src/Data/Word8/Primitive.agda b/src/Data/Word8/Primitive.agda
new file mode 100644
index 0000000000..e0d2ac87c7
--- /dev/null
+++ b/src/Data/Word8/Primitive.agda
@@ -0,0 +1,36 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bytes: simple bindings to Haskell types and functions
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Word8.Primitive where
+
+open import Agda.Builtin.Bool using (Bool)
+open import Agda.Builtin.Nat using (Nat)
+open import Agda.Builtin.String using (String)
+
+postulate
+  Word8 : Set
+  testBit : Word8 → Nat → Bool
+  setBit : Word8 → Nat → Word8
+  clearBit : Word8 → Nat → Word8
+  fromNat : Nat → Word8
+  toNat : Word8 → Nat
+  _+_ : Word8 → Word8 → Word8
+  show : Word8 → String
+
+{-# FOREIGN GHC import GHC.Word #-}
+{-# FOREIGN GHC import qualified Data.Bits as B #-}
+{-# FOREIGN GHC import qualified Data.Text as T #-}
+
+{-# COMPILE GHC Word8 = type Word8 #-}
+{-# COMPILE GHC testBit = \ w -> B.testBit w . fromIntegral #-}
+{-# COMPILE GHC setBit = \ w -> B.setBit w . fromIntegral #-}
+{-# COMPILE GHC clearBit = \ w -> B.clearBit  w . fromIntegral #-}
+{-# COMPILE GHC fromNat = fromIntegral #-}
+{-# COMPILE GHC toNat = fromIntegral #-}
+{-# COMPILE GHC _+_ = (+) #-}
+{-# COMPILE GHC show = T.pack . Prelude.show #-}
diff --git a/src/Data/Word8/Show.agda b/src/Data/Word8/Show.agda
new file mode 100644
index 0000000000..0cc5b5d7b5
--- /dev/null
+++ b/src/Data/Word8/Show.agda
@@ -0,0 +1,33 @@
+------------------------------------------------------------------------
+-- The Agda standard library
+--
+-- Bytes: showing bit patterns
+------------------------------------------------------------------------
+
+{-# OPTIONS --cubical-compatible #-}
+
+module Data.Word8.Show where
+
+open import Agda.Builtin.String using (String)
+
+open import Data.Bool.Show using (showBit)
+open import Data.Fin.Base as Fin using (Fin)
+import Data.Nat.Show as ℕ
+open import Data.String using (_++_; fromVec; padLeft)
+open import Data.Vec.Base as Vec using (Vec; []; _∷_)
+open import Data.Word8.Base
+open import Function.Base using (_$_)
+
+showBits : Word8 → String
+showBits w
+  = "0b" ++_
+  $ fromVec
+  $ Vec.reverse
+  $ Vec.map showBit
+  $ toBits w
+
+showHexa : Word8 → String
+showHexa w
+  = "0x" ++_
+  $ padLeft '0' 2
+  $ ℕ.showInBase 16 (toℕ w)
diff --git a/src/Reflection/AST/Literal.agda b/src/Reflection/AST/Literal.agda
index abdd2b08a6..540e150169 100644
--- a/src/Reflection/AST/Literal.agda
+++ b/src/Reflection/AST/Literal.agda
@@ -13,7 +13,7 @@ import Data.Char.Properties as Char
 import Data.Float.Properties as Float
 import Data.Nat.Properties as ℕ
 import Data.String.Properties as String
-import Data.Word.Properties as Word
+import Data.Word64.Properties as Word64
 import Reflection.AST.Meta as Meta
 import Reflection.AST.Name as Name
 open import Relation.Nullary.Decidable.Core            using (yes; no; map′; isYes)
@@ -60,7 +60,7 @@ nat x ≟ char x₁ = no (λ ())
 nat x ≟ string x₁ = no (λ ())
 nat x ≟ name x₁ = no (λ ())
 nat x ≟ meta x₁ = no (λ ())
-word64 x ≟ word64 x₁ = map′ (cong word64) word64-injective (x Word.≟ x₁)
+word64 x ≟ word64 x₁ = map′ (cong word64) word64-injective (x Word64.≟ x₁)
 word64 x ≟ nat x₁ = no (λ ())
 word64 x ≟ float x₁ = no (λ ())
 word64 x ≟ char x₁ = no (λ ())
diff --git a/src/Reflection/AST/Name.agda b/src/Reflection/AST/Name.agda
index c386912f08..5ceb025126 100644
--- a/src/Reflection/AST/Name.agda
+++ b/src/Reflection/AST/Name.agda
@@ -10,7 +10,7 @@ module Reflection.AST.Name where
 
 open import Data.List.Base              using (List)
 import Data.Product.Properties as Prodₚ using (≡-dec)
-import Data.Word.Properties as Wₚ       using (_≟_)
+import Data.Word64.Properties as Wₚ       using (_≟_)
 open import Function.Base               using (_on_)
 open import Relation.Nullary.Decidable.Core            using (map′)
 open import Relation.Binary.Core                       using (Rel)
diff --git a/src/Reflection/AST/Show.agda b/src/Reflection/AST/Show.agda
index 5d748e089b..a9de9a2993 100644
--- a/src/Reflection/AST/Show.agda
+++ b/src/Reflection/AST/Show.agda
@@ -18,7 +18,7 @@ import Data.Nat.Show as ℕ
 open import Data.String.Base as String using (String; _++_; intersperse; braces; parens; _<+>_)
 open import Data.String as String using (parensIfSpace)
 open import Data.Product.Base using (_×_; _,_)
-import Data.Word.Base as Word
+import Data.Word64.Base as Word64
 open import Function.Base using (id; _∘′_; case_of_)
 open import Relation.Nullary.Decidable.Core using (yes; no)
 
@@ -60,7 +60,7 @@ showVisibility instance′ = "instance"
 
 showLiteral : Literal → String
 showLiteral (nat x)    = ℕ.show x
-showLiteral (word64 x) = ℕ.show (Word.toℕ x)
+showLiteral (word64 x) = ℕ.show (Word64.toℕ x)
 showLiteral (float x)  = Float.show x
 showLiteral (char x)   = Char.show x
 showLiteral (string x) = String.show x
diff --git a/src/System/Directory/Primitive.agda b/src/System/Directory/Primitive.agda
index 57da7461bb..d5bae96742 100644
--- a/src/System/Directory/Primitive.agda
+++ b/src/System/Directory/Primitive.agda
@@ -21,12 +21,12 @@ open import System.FilePath.Posix.Primitive
 {-# FOREIGN GHC import Data.Text        #-}
 
 data XdgDirectory : Set where
-  XdgData XdgConfig XdgCache : XdgDirectory
+  XdgData XdgConfig XdgCache XdgState : XdgDirectory
 
 data XdgDirectoryList : Set where
   XdgDataDirs XdgConfigDirs : XdgDirectoryList
 
-{-# COMPILE GHC XdgDirectory     = data XdgDirectory (XdgData|XdgConfig|XdgCache)    #-}
+{-# COMPILE GHC XdgDirectory     = data XdgDirectory (XdgData|XdgConfig|XdgCache|XdgState) #-}
 {-# COMPILE GHC XdgDirectoryList = data XdgDirectoryList (XdgDataDirs|XdgConfigDirs) #-}
 
 private
diff --git a/src/System/Random.agda b/src/System/Random.agda
index fa327939d6..3587a0b927 100644
--- a/src/System/Random.agda
+++ b/src/System/Random.agda
@@ -89,7 +89,7 @@ module ℕ where
 
 module Word64 where
 
-  open import Data.Word.Base using (Word64; _≤_)
+  open import Data.Word64.Base using (Word64; _≤_)
 
   randomIO : IO Word64
   randomIO = lift Prim.randomIO-Word64
diff --git a/src/Tactic/Cong.agda b/src/Tactic/Cong.agda
index 80cc7ec6d4..6fea3dd1e0 100644
--- a/src/Tactic/Cong.agda
+++ b/src/Tactic/Cong.agda
@@ -35,7 +35,7 @@ open import Data.List.Base   as List  using ([]; _∷_)
 open import Data.Maybe.Base  as Maybe using (Maybe; just; nothing)
 open import Data.Nat.Base    as ℕ     using (ℕ; zero; suc; _≡ᵇ_; _+_)
 open import Data.Unit.Base            using (⊤)
-open import Data.Word.Base   as Word  using (toℕ)
+open import Data.Word64.Base   as Word64  using (toℕ)
 open import Data.Product.Base         using (_×_; map₁; _,_)
 open import Function                  using (flip; case_of_)
 
diff --git a/tests/data/bytestring/Main.agda b/tests/data/bytestring/Main.agda
new file mode 100644
index 0000000000..85721598da
--- /dev/null
+++ b/tests/data/bytestring/Main.agda
@@ -0,0 +1,66 @@
+{-# OPTIONS --guardedness #-}
+
+module Main where
+
+open import Agda.Builtin.FromNat
+
+open import Data.Bytestring.Base as Bytestring
+open import Data.Bytestring.Builder.Base
+open import Data.List.Base using ([]; _∷_)
+import Data.Nat.Literals; instance numberNat = Data.Nat.Literals.number
+open import Data.Product.Base using (_×_; _,_)
+open import Data.String using (String; _++_; fromVec)
+open import Data.Unit.Base using (⊤; tt)
+import Data.Vec.Base as Vec
+open import Data.Word8.Base as Word8
+import Data.Word8.Show as Word8
+import Data.Word8.Literals; instance numberWord8 = Data.Word8.Literals.number
+open import Data.Word64.Base as Word64 using (Word64)
+import Data.Word64.Unsafe as Word64
+import Data.Word64.Show as Word64
+import Data.Word64.Literals; instance numberWord64 = Data.Word64.Literals.number
+
+open import Function.Base using (_$_)
+
+open import IO.Base
+open import IO.Finite
+
+1⋯3 : Bytestring
+1⋯3 = toBytestring
+    $ List.concat
+    $ word8 1
+    ∷ word64LE 2
+    ∷ word64BE 3
+    ∷ []
+
+1,⋯,3 : Word8 × Word64 × Word64
+1,⋯,3 = getWord8 1⋯3 0
+      , getWord64LE 1⋯3 1
+      , getWord64BE 1⋯3 9
+
+main : Main
+main = run $ do
+  let separation = fromVec (Vec.replicate 72 '-')
+  putStrLn (Bytestring.show 1⋯3)
+  putStrLn separation
+  let (one , two , three) = 1,⋯,3
+
+  let word8test : Word8 → IO _
+      word8test w = do
+        putStrLn (Word8.show w ++ " = " ++ Word8.showBits w)
+        putStrLn (Word8.show w ++ " = " ++ Word8.showHexa w)
+        putStrLn (Word8.show w ++ " = " ++ Word8.show (Word8.fromBits (Word8.toBits w)))
+
+
+  let word64test : Word64 → IO _
+      word64test w = do
+        putStrLn separation
+        putStrLn (Word64.show w ++ " = " ++ Word64.showBits w)
+        putStrLn (Word64.show w ++ " = " ++ Word64.showHexa w)
+        putStrLn (Word64.show w ++ " = " ++ Word64.show (Word64.fromBits (Word64.toBits w)))
+
+  word8test one
+  word8test (Word8.fromℕ 144)
+  word64test two
+  word64test three
+  word64test (Word64.fromℕ 2024)
diff --git a/tests/data/bytestring/bytestring.agda-lib b/tests/data/bytestring/bytestring.agda-lib
new file mode 100644
index 0000000000..325830a199
--- /dev/null
+++ b/tests/data/bytestring/bytestring.agda-lib
@@ -0,0 +1,2 @@
+name: bytestring
+include: ../../../src/ .
diff --git a/tests/data/bytestring/expected b/tests/data/bytestring/expected
new file mode 100644
index 0000000000..240520e413
--- /dev/null
+++ b/tests/data/bytestring/expected
@@ -0,0 +1,20 @@
+"\SOH\STX\NUL\NUL\NUL\NUL\NUL\NUL\NUL\NUL\NUL\NUL\NUL\NUL\NUL\NUL\ETX"
+------------------------------------------------------------------------
+1 = 0b00000001
+1 = 0x01
+1 = 1
+144 = 0b10010000
+144 = 0x90
+144 = 144
+------------------------------------------------------------------------
+2 = 0b0000000000000000000000000000000000000000000000000000000000000010
+2 = 0x00000002
+2 = 2
+------------------------------------------------------------------------
+3 = 0b0000000000000000000000000000000000000000000000000000000000000011
+3 = 0x00000003
+3 = 3
+------------------------------------------------------------------------
+2024 = 0b0000000000000000000000000000000000000000000000000000011111101000
+2024 = 0x000007e8
+2024 = 2024
diff --git a/tests/data/bytestring/run b/tests/data/bytestring/run
new file mode 100644
index 0000000000..f6e7c9b3c0
--- /dev/null
+++ b/tests/data/bytestring/run
@@ -0,0 +1,5 @@
+$1 --compile-dir=../../_build -c Main.agda > log
+./../../_build/Main > output
+
+rm ../../_build/Main
+rm ../../_build/MAlonzo/Code/Main*
\ No newline at end of file
diff --git a/tests/runtests.agda b/tests/runtests.agda
index c6ad22e0ce..480b890d72 100644
--- a/tests/runtests.agda
+++ b/tests/runtests.agda
@@ -16,6 +16,7 @@ dataTests = mkTestPool "Data structures"
   ∷ "rational"
   ∷ "rational-unnormalised"
   ∷ "trie"
+  ∷ "bytestring"
   ∷ []
 
 systemTests : TestPool

From 25b3e875b3e205958e56d832357ade7f5c3b301e Mon Sep 17 00:00:00 2001
From: Lex van der Stoep <lex.vanderstoep@gmail.com>
Date: Sun, 16 Jun 2024 14:01:58 +0200
Subject: [PATCH 303/316] Update LICENSE (#2409)

* Update LICENSE year

* Remove extraneous 'i'

---------

Co-authored-by: Lex van der Stoep <lex.vanderstoep@imc.com>
---
 LICENCE | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/LICENCE b/LICENCE
index 3a9d1c3624..3cd3c4395b 100644
--- a/LICENCE
+++ b/LICENCE
@@ -1,4 +1,4 @@
-Copyright (c) 2007-2023 Nils Anders Danielsson, Ulf Norell, Shin-Cheng
+Copyright (c) 2007-2024 Nils Anders Danielsson, Ulf Norell, Shin-Cheng
 Mu, Bradley Hardy, Samuel Bronson, Dan Doel, Patrik Jansson,
 Liang-Ting Chen, Jean-Philippe Bernardy, Andrés Sicard-Ramírez,
 Nicolas Pouillard, Darin Morrison, Peter Berry, Daniel Brown,

From bf7c7459b4b13d86cb065dde5143f95580659144 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 17 Jun 2024 09:53:37 +0100
Subject: [PATCH 304/316] =?UTF-8?q?Remove=20(almost!)=20all=20external=20u?=
 =?UTF-8?q?se=20of=20`=5F=E2=89=A4=E2=80=B3=5F`=20beyond=20`Data.Nat.*`=20?=
 =?UTF-8?q?(#2262)?=
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* additional proofs and patterns in `Data.Nat.Properties`

* added two projections; refactored `pad*` operations

* `CHANGELOG`

* removed one more use

* removed final uses of `<″-offset` outside `Data.Nat.Base|Properties`

* rename `≤-proof` to `m≤n⇒∃[o]m+o≡n`

* removed new pattern synonyms

* interim commit: deprecate everything!

* add guarded monus; make arguments more irrelevant

* fixed up `CHANGELOG`

* propagate use of irrelevance

* tidy up deprecations; reinstate `s<″s⁻¹` for `Data.Fin.Properties`

* tightened up the deprecation story

* paragraph on use of `pattern` synonyms

* removed `import`

* Update CHANGELOG.md

Fix refs to Algebra.Definitions.RawMagma

* Update Base.agda

Fix refs. to Algebra.Definitions.RawMagma

* inlined guarded monus definition in property

* remove comment about guarded monus
---
 CHANGELOG.md                   | 29 +++++++++++++---
 doc/style-guide.md             |  8 +++++
 src/Data/Fin/Properties.agda   |  6 ++--
 src/Data/Nat/Base.agda         | 37 +++++++++++++-------
 src/Data/Nat/DivMod.agda       |  3 +-
 src/Data/Nat/Properties.agda   | 62 +++++++++++++++++++++-------------
 src/Data/Nat/WithK.agda        |  3 +-
 src/Data/Vec/Bounded/Base.agda | 19 +++++++----
 8 files changed, 114 insertions(+), 53 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index a243edfd5f..732ff78e7e 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -99,6 +99,15 @@ Deprecated names
   map-compose  ↦  map-∘
   ```
 
+* In `Data.Nat.Base`: the following pattern synonyms and definitions are all
+  deprecated in favour of direct pattern matching on `Algebra.Definitions.RawMagma._∣ˡ_._,_`
+  ```agda
+  pattern less-than-or-equal {k} eq = k , eq
+  pattern ≤″-offset k = k , refl
+  pattern <″-offset k = k , refl
+  s≤″s⁻¹
+ ```
+
 * In `Data.Nat.Divisibility.Core`:
   ```agda
   *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
@@ -663,12 +672,16 @@ Additions to existing modules
 
 * Added new proofs in `Data.Nat.Properties`:
   ```agda
-  m≤n+o⇒m∸n≤o : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o
-  m<n+o⇒m∸n<o : ∀ m n {o} → .{{NonZero o}} → m < n + o → m ∸ n < o
-  pred-cancel-≤ : pred m ≤ pred n → (m ≡ 1 × n ≡ 0) ⊎ m ≤ n
-  pred-cancel-< : pred m < pred n → m < n
+  m≤n+o⇒m∸n≤o    : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o
+  m<n+o⇒m∸n<o    : ∀ m n {o} → .{{NonZero o}} → m < n + o → m ∸ n < o
+  pred-cancel-≤  : pred m ≤ pred n → (m ≡ 1 × n ≡ 0) ⊎ m ≤ n
+  pred-cancel-<  : pred m < pred n → m < n
   pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n
-  pred-cancel-≡ : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n
+  pred-cancel-≡  : pred m ≡ pred n → ((m ≡ 0 × n ≡ 1) ⊎ (m ≡ 1 × n ≡ 0)) ⊎ m ≡ n
+
+  <⇒<″          : _<_ ⇒ _<″_
+  m≤n⇒∃[o]m+o≡n : .(m ≤ n) → ∃ λ k → m + k ≡ n
+  guarded-∸≗∸   : .(m≤n : m ≤ n) → let k , _ = m≤n⇒∃[o]m+o≡n m≤n in k ≡ n ∸ m
   ```
 
 * Added new proofs to `Data.Nat.Primality`:
@@ -769,6 +782,12 @@ Additions to existing modules
   Stable : Pred A ℓ → Set _
   ```
 
+* Added new functions in `Data.Vec.Bounded.Base`:
+  ```agda
+  isBounded : (as : Vec≤ A n) → Vec≤.length as ≤ n
+  toVec     : (as : Vec≤ A n) → Vec A (Vec≤.length as)
+  ```
+
 * In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
   be used infix.
 
diff --git a/doc/style-guide.md b/doc/style-guide.md
index c4b9af1bf8..1a8b883f3a 100644
--- a/doc/style-guide.md
+++ b/doc/style-guide.md
@@ -657,6 +657,14 @@ Type formers:
 
 #### Specific pragmatics/idiomatic patterns
 
+## Use of `pattern` synonyms
+
+In general, these are intended to be used to provide specialised
+constructors for `Data` types (and sometimes, inductive
+families/binary relations such as `Data.Nat.Divisibility._∣_`), and as
+such, their use should be restricted to `Base` or `Core` modules, and
+not pollute the namespaces of `Properties` or other modules.
+
 ## Use of `with` notation
 
 Thinking on this has changed since the early days of the library, with
diff --git a/src/Data/Fin/Properties.agda b/src/Data/Fin/Properties.agda
index a1db966f92..e1f2d25d24 100644
--- a/src/Data/Fin/Properties.agda
+++ b/src/Data/Fin/Properties.agda
@@ -247,9 +247,9 @@ fromℕ<-injective (suc _) (suc _) {o = suc _} m<n n<o r
 
 fromℕ<≡fromℕ<″ : ∀ (m<n : m ℕ.< n) (m<″n : m ℕ.<″ n) →
                  fromℕ< m<n ≡ fromℕ<″ m m<″n
-fromℕ<≡fromℕ<″ {m = zero}  m<n (ℕ.<″-offset _) = refl
-fromℕ<≡fromℕ<″ {m = suc m} m<n (ℕ.<″-offset _)
-  = cong suc (fromℕ<≡fromℕ<″ (ℕ.s<s⁻¹ m<n) (ℕ.<″-offset _))
+fromℕ<≡fromℕ<″ {m = zero}  {n = suc _} _ _ = refl
+fromℕ<≡fromℕ<″ {m = suc m} {n = suc _} m<n m<″n
+  = cong suc (fromℕ<≡fromℕ<″ (ℕ.s<s⁻¹ m<n) (ℕ.s<″s⁻¹ m<″n))
 
 toℕ-fromℕ<″ : ∀ (m<n : m ℕ.<″ n) → toℕ (fromℕ<″ m m<n) ≡ m
 toℕ-fromℕ<″ {m} {n} m<n = begin
diff --git a/src/Data/Nat/Base.agda b/src/Data/Nat/Base.agda
index 478b71f783..db46274f24 100644
--- a/src/Data/Nat/Base.agda
+++ b/src/Data/Nat/Base.agda
@@ -363,8 +363,6 @@ infix 4 _≤″_ _<″_ _≥″_ _>″_
 _≤″_ : (m n : ℕ)  → Set
 _≤″_ = _∣ˡ_ +-rawMagma
 
-pattern less-than-or-equal {k} proof = k , proof
-
 _<″_ : Rel ℕ 0ℓ
 m <″ n = suc m ≤″ n
 
@@ -374,18 +372,10 @@ m ≥″ n = n ≤″ m
 _>″_ : Rel ℕ 0ℓ
 m >″ n = n <″ m
 
--- Smart constructors of _≤″_ and _<″_
-
-pattern ≤″-offset k = less-than-or-equal {k = k} refl
-pattern <″-offset k = ≤″-offset k
-
--- Smart destructors of _<″_
-
-s≤″s⁻¹ : ∀ {m n} → suc m ≤″ suc n → m ≤″ n
-s≤″s⁻¹ (≤″-offset k) = ≤″-offset k
+-- Smart destructor of _<″_
 
 s<″s⁻¹ : ∀ {m n} → suc m <″ suc n → m <″ n
-s<″s⁻¹ (<″-offset k) = <″-offset k
+s<″s⁻¹ (k , refl) = k , refl
 
 -- _≤‴_: this definition is useful for induction with an upper bound.
 
@@ -429,5 +419,26 @@ compare (suc m) (suc n) with compare m n
 -- Please use the new names as continuing support for the old names is
 -- not guaranteed.
 
--- Version 2.0
+-- Version 2.1
+
+-- Smart constructors of _≤″_ and _<″_
+pattern less-than-or-equal {k} eq = k , eq
+{-# WARNING_ON_USAGE less-than-or-equal
+"Warning: less-than-or-equal was deprecated in v2.1. Please match directly on proofs of ≤″ using constructor Algebra.Definitions.RawMagma._∣ˡ_._,_ instead. "
+#-}
+pattern ≤″-offset k = k , refl
+{-# WARNING_ON_USAGE ≤″-offset
+"Warning: ≤″-offset was deprecated in v2.1. Please match directly on proofs of ≤″ using pattern (_, refl) from Algebra.Definitions.RawMagma._∣ˡ_ instead. "
+#-}
+pattern <″-offset k = k , refl
+{-# WARNING_ON_USAGE <″-offset
+"Warning: <″-offset was deprecated in v2.1. Please match directly on proofs of ≤″ using pattern (_, refl) from Algebra.Definitions.RawMagma._∣ˡ_ instead. "
+#-}
+
+-- Smart destructors of _<″_
 
+s≤″s⁻¹ : ∀ {m n} → suc m ≤″ suc n → m ≤″ n
+s≤″s⁻¹ (k , refl) = k , refl
+{-# WARNING_ON_USAGE s≤″s⁻¹
+"Warning: s≤″s⁻¹ was deprecated in v2.1. Please match directly on proofs of ≤″ using pattern (_, refl) from Algebra.Definitions.RawMagma._∣ˡ_ instead. "
+#-}
diff --git a/src/Data/Nat/DivMod.agda b/src/Data/Nat/DivMod.agda
index cddb2768f7..86bc8d3168 100644
--- a/src/Data/Nat/DivMod.agda
+++ b/src/Data/Nat/DivMod.agda
@@ -17,6 +17,7 @@ open import Data.Nat.DivMod.Core
 open import Data.Nat.Divisibility.Core
 open import Data.Nat.Induction
 open import Data.Nat.Properties
+open import Data.Product.Base using (_,_)
 open import Data.Sum.Base using (inj₁; inj₂)
 open import Function.Base using (_$_; _∘_)
 open import Relation.Binary.PropositionalEquality.Core
@@ -105,7 +106,7 @@ m%n≤m : ∀ m n .{{_ : NonZero n}} → m % n ≤ m
 m%n≤m m (suc n-1) = a[modₕ]n≤a 0 m n-1
 
 m≤n⇒m%n≡m : m ≤ n → m % suc n ≡ m
-m≤n⇒m%n≡m {m = m} m≤n with ≤″-offset k ← ≤⇒≤″ m≤n
+m≤n⇒m%n≡m {m = m} m≤n with k , refl ← m≤n⇒∃[o]m+o≡n m≤n
   = a≤n⇒a[modₕ]n≡a 0 (m + k) m k
 
 m<n⇒m%n≡m : .{{_ : NonZero n}} → m < n → m % n ≡ m
diff --git a/src/Data/Nat/Properties.agda b/src/Data/Nat/Properties.agda
index 9bca65bacc..21ea925467 100644
--- a/src/Data/Nat/Properties.agda
+++ b/src/Data/Nat/Properties.agda
@@ -14,6 +14,7 @@ module Data.Nat.Properties where
 open import Axiom.UniquenessOfIdentityProofs using (module Decidable⇒UIP)
 open import Algebra.Bundles using (Magma; Semigroup; CommutativeSemigroup;
   CommutativeMonoid; Monoid; Semiring; CommutativeSemiring; CommutativeSemiringWithoutOne)
+open import Algebra.Definitions.RawMagma using (_,_)
 open import Algebra.Morphism
 open import Algebra.Consequences.Propositional
   using (comm+cancelˡ⇒cancelʳ; comm∧distrʳ⇒distrˡ; comm∧distrˡ⇒distrʳ)
@@ -41,7 +42,7 @@ open import Relation.Binary
 open import Relation.Binary.Consequences using (flip-Connex)
 open import Relation.Binary.PropositionalEquality
 open import Relation.Nullary hiding (Irrelevant)
-open import Relation.Nullary.Decidable using (True; via-injection; map′)
+open import Relation.Nullary.Decidable using (True; via-injection; map′; recompute)
 open import Relation.Nullary.Negation.Core using (¬_; contradiction)
 open import Relation.Nullary.Reflects using (fromEquivalence)
 
@@ -2106,31 +2107,46 @@ n≤′m+n (suc m) n = ≤′-step (n≤′m+n m n)
 -- Properties of _≤″_ and _<″_
 ------------------------------------------------------------------------
 
-m<ᵇn⇒1+m+[n-1+m]≡n : ∀ m n → T (m <ᵇ n) → suc m + (n ∸ suc m) ≡ n
-m<ᵇn⇒1+m+[n-1+m]≡n m n lt = m+[n∸m]≡n (<ᵇ⇒< m n lt)
+-- equivalence of  _≤″_ to _≤_
 
-m<ᵇ1+m+n : ∀ m {n} → T (m <ᵇ suc (m + n))
-m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)
+≤⇒≤″ : _≤_ ⇒ _≤″_
+≤⇒≤″ = (_ ,_) ∘ m+[n∸m]≡n
 
-<ᵇ⇒<″ : T (m <ᵇ n) → m <″ n
-<ᵇ⇒<″ {m} {n} leq = less-than-or-equal (m+[n∸m]≡n (<ᵇ⇒< m n leq))
+<⇒<″ : _<_ ⇒ _<″_
+<⇒<″ = ≤⇒≤″
 
-<″⇒<ᵇ : ∀ {m n} → m <″ n → T (m <ᵇ n)
-<″⇒<ᵇ {m} (<″-offset k) = <⇒<ᵇ (m≤m+n (suc m) k)
+≤″⇒≤ : _≤″_ ⇒ _≤_
+≤″⇒≤ (k , refl) = m≤m+n _ k
 
 -- equivalence to the old definition of _≤″_
 
-≤″-proof : ∀ {m n} (le : m ≤″ n) → let less-than-or-equal {k} _ = le in m + k ≡ n
-≤″-proof (less-than-or-equal prf) = prf
+≤″-proof : (le : m ≤″ n) → let k , _ = le in m + k ≡ n
+≤″-proof (_ , prf) = prf
 
--- equivalence to _≤_
+-- yielding analogous proof for _≤_
 
-≤″⇒≤ : _≤″_ ⇒ _≤_
-≤″⇒≤ {zero}  (≤″-offset k) = z≤n {k}
-≤″⇒≤ {suc m} (≤″-offset k) = s≤s (≤″⇒≤ (≤″-offset k))
+m≤n⇒∃[o]m+o≡n : .(m ≤ n) → ∃ λ k → m + k ≡ n
+m≤n⇒∃[o]m+o≡n m≤n = _ , m+[n∸m]≡n (recompute (_ ≤? _) m≤n)
 
-≤⇒≤″ : _≤_ ⇒ _≤″_
-≤⇒≤″ = less-than-or-equal ∘ m+[n∸m]≡n
+-- whose witness is equal to monus
+
+guarded-∸≗∸ : ∀ {m n} → .(m≤n : m ≤ n) →
+              let k , _ = m≤n⇒∃[o]m+o≡n m≤n in k ≡ n ∸ m
+guarded-∸≗∸ m≤n = refl
+
+-- equivalence of _<″_ to _<ᵇ_
+
+m<ᵇn⇒1+m+[n-1+m]≡n : ∀ m n → T (m <ᵇ n) → suc m + (n ∸ suc m) ≡ n
+m<ᵇn⇒1+m+[n-1+m]≡n m n lt = m+[n∸m]≡n (<ᵇ⇒< m n lt)
+
+m<ᵇ1+m+n : ∀ m {n} → T (m <ᵇ suc (m + n))
+m<ᵇ1+m+n m = <⇒<ᵇ (m≤m+n (suc m) _)
+
+<ᵇ⇒<″ : T (m <ᵇ n) → m <″ n
+<ᵇ⇒<″ {m} {n} = <⇒<″ ∘ (<ᵇ⇒< m n)
+
+<″⇒<ᵇ : ∀ {m n} → m <″ n → T (m <ᵇ n)
+<″⇒<ᵇ {m} (k , refl) = <⇒<ᵇ (m≤m+n (suc m) k)
 
 -- NB: we use the builtin function `_<ᵇ_ : (m n : ℕ) → Bool` here so
 -- that the function quickly decides whether to return `yes` or `no`.
@@ -2144,7 +2160,7 @@ _<″?_ : Decidable _<″_
 m <″? n = map′ <ᵇ⇒<″ <″⇒<ᵇ (T? (m <ᵇ n))
 
 _≤″?_ : Decidable _≤″_
-zero  ≤″? n = yes (≤″-offset n)
+zero  ≤″? n = yes (n , refl)
 suc m ≤″? n = m <″? n
 
 _≥″?_ : Decidable _≥″_
@@ -2154,10 +2170,9 @@ _>″?_ : Decidable _>″_
 _>″?_ = flip _<″?_
 
 ≤″-irrelevant : Irrelevant _≤″_
-≤″-irrelevant {m} (less-than-or-equal eq₁)
-                  (less-than-or-equal eq₂)
+≤″-irrelevant {m} (_ , eq₁) (_ , eq₂)
   with refl ← +-cancelˡ-≡ m _ _ (trans eq₁ (sym eq₂))
-  = cong less-than-or-equal (≡-irrelevant eq₁ eq₂)
+  = cong (_ ,_) (≡-irrelevant eq₁ eq₂)
 
 <″-irrelevant : Irrelevant _<″_
 <″-irrelevant = ≤″-irrelevant
@@ -2173,8 +2188,8 @@ _>″?_ = flip _<″?_
 ------------------------------------------------------------------------
 
 ≤‴⇒≤″ : ∀{m n} → m ≤‴ n → m ≤″ n
-≤‴⇒≤″ {m = m} ≤‴-refl       = less-than-or-equal {k = 0} (+-identityʳ m)
-≤‴⇒≤″ {m = m} (≤‴-step m≤n) = less-than-or-equal (trans (+-suc m _) (≤″-proof (≤‴⇒≤″ m≤n)))
+≤‴⇒≤″ {m = m} ≤‴-refl       = 0 , +-identityʳ m
+≤‴⇒≤″ {m = m} (≤‴-step m≤n) = _ , trans (+-suc m _) (≤″-proof (≤‴⇒≤″ m≤n))
 
 m≤‴m+k : ∀{m n k} → m + k ≡ n → m ≤‴ n
 m≤‴m+k {m} {k = zero}  refl = subst (λ z → m ≤‴ z) (sym (+-identityʳ m)) (≤‴-refl {m})
@@ -2397,4 +2412,3 @@ open Data.Nat.Base public
 {-# WARNING_ON_USAGE <-transˡ
 "Warning: <-transˡ was deprecated in v2.0. Please use <-≤-trans instead. "
 #-}
-
diff --git a/src/Data/Nat/WithK.agda b/src/Data/Nat/WithK.agda
index 49966a93f0..bb6a97a7aa 100644
--- a/src/Data/Nat/WithK.agda
+++ b/src/Data/Nat/WithK.agda
@@ -8,8 +8,9 @@
 
 module Data.Nat.WithK where
 
+open import Algebra.Definitions.RawMagma using (_,_)
 open import Data.Nat.Base
 open import Relation.Binary.PropositionalEquality.WithK
 
 ≤″-erase : ∀ {m n} → m ≤″ n → m ≤″ n
-≤″-erase (less-than-or-equal eq) = less-than-or-equal (≡-erase eq)
+≤″-erase (_ , eq) = _ , ≡-erase eq
diff --git a/src/Data/Vec/Bounded/Base.agda b/src/Data/Vec/Bounded/Base.agda
index 7476afb2d4..fa7eb758ed 100644
--- a/src/Data/Vec/Bounded/Base.agda
+++ b/src/Data/Vec/Bounded/Base.agda
@@ -21,7 +21,7 @@ open import Data.These.Base as These using (These)
 open import Function.Base using (_∘_; id; _$_)
 open import Level using (Level)
 open import Relation.Nullary.Decidable.Core using (recompute)
-open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
+open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl)
 open import Relation.Binary.PropositionalEquality.Properties
   using (module ≡-Reasoning)
 
@@ -43,20 +43,27 @@ record Vec≤ (A : Set a) (n : ℕ) : Set a where
         vec      : Vec A length
         .bound   : length ≤ n
 
+-- projection to recompute irrelevant field
+isBounded : (as : Vec≤ A n) → Vec≤.length as ≤ n
+isBounded as@(_ , m≤n) = recompute (_ ℕ.≤? _) m≤n
+
 ------------------------------------------------------------------------
 -- Conversion functions
 
+toVec : (as : Vec≤ A n) → Vec A (Vec≤.length as)
+toVec as@(vs , _) = vs
+
 fromVec : Vec A n → Vec≤ A n
 fromVec v = v , ℕ.≤-refl
 
 padRight : A → Vec≤ A n → Vec A n
-padRight a (vs , m≤n)
-  with ≤″-offset k ← recompute (_ ℕ.≤″? _) (ℕ.≤⇒≤″ m≤n)
+padRight a as@(vs , m≤n)
+  with k , refl ← ℕ.m≤n⇒∃[o]m+o≡n m≤n
   = vs Vec.++ Vec.replicate k a
 
 padLeft : A → Vec≤ A n → Vec A n
 padLeft a record { length = m ; vec = vs ; bound = m≤n }
-  with ≤″-offset k ← recompute (_ ℕ.≤″? _) (ℕ.≤⇒≤″ m≤n)
+  with k , refl ← ℕ.m≤n⇒∃[o]m+o≡n m≤n
   rewrite ℕ.+-comm m k
   = Vec.replicate k a Vec.++ vs
 
@@ -72,7 +79,7 @@ private
 
 padBoth : A → A → Vec≤ A n → Vec A n
 padBoth aₗ aᵣ record { length = m ; vec = vs ; bound = m≤n }
-  with ≤″-offset k ← recompute (_ ℕ.≤″? _) (ℕ.≤⇒≤″ m≤n)
+  with k , refl ← ℕ.m≤n⇒∃[o]m+o≡n m≤n
   rewrite split m k
   = Vec.replicate ⌊ k /2⌋ aₗ
       Vec.++ vs
@@ -83,7 +90,7 @@ fromList : (as : List A) → Vec≤ A (List.length as)
 fromList = fromVec ∘ Vec.fromList
 
 toList : Vec≤ A n → List A
-toList = Vec.toList ∘ Vec≤.vec
+toList = Vec.toList ∘ toVec
 
 ------------------------------------------------------------------------
 -- Creating new Vec≤ vectors

From 3908d7cb8e26dc3b9e2f2da2e92ce1c88dcbe705 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Mon, 17 Jun 2024 09:53:55 +0100
Subject: [PATCH 305/316] Refactor `Data.Sum.{to|from}Dec` via move+deprecate
 in `Relation.Nullary.Decidable.Core` (#2405)

* fixes #2059 on the model of #2336

* fixed `import` dependencies

* simplified `import` dependencies

* final tweak

* `CHANGELOG`
---
 CHANGELOG.md                                  |  8 +++++
 src/Data/List/Fresh/Relation/Unary/Any.agda   |  9 +++---
 src/Data/List/Relation/Unary/Any.agda         |  5 ++--
 .../List/Relation/Unary/First/Properties.agda | 19 ++++++------
 src/Data/Sum.agda                             | 24 +++++++--------
 src/Data/Vec/Relation/Unary/Any.agda          |  5 ++--
 .../Closure/Reflexive/Properties.agda         |  6 ++--
 src/Relation/Nullary/Decidable/Core.agda      | 29 +++++++++++++++++--
 8 files changed, 66 insertions(+), 39 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 732ff78e7e..089957ef38 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -113,6 +113,12 @@ Deprecated names
   *-pres-∣  ↦  Data.Nat.Divisibility.*-pres-∣
   ```
 
+* In `Data.Sum`:
+  ```agda
+  fromDec  ↦  Relation.Nullary.Decidable.Core.toSum
+  toDec    ↦  Relation.Nullary.Decidable.Core.fromSum
+  ```
+
 * In `IO.Base`:
   ```agda
   untilRight  ↦  untilInj₂
@@ -818,6 +824,8 @@ Additions to existing modules
 * Added new proof in `Relation.Nullary.Decidable.Core`:
   ```agda
   recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q
+  toSum              : Dec A → A ⊎ ¬ A
+  fromSum            : A ⊎ ¬ A → Dec A
   ```
 
 * Added new proof in `Relation.Nullary.Decidable`:
diff --git a/src/Data/List/Fresh/Relation/Unary/Any.agda b/src/Data/List/Fresh/Relation/Unary/Any.agda
index dbb9dd0dec..5d10af8e81 100644
--- a/src/Data/List/Fresh/Relation/Unary/Any.agda
+++ b/src/Data/List/Fresh/Relation/Unary/Any.agda
@@ -9,12 +9,11 @@
 module Data.List.Fresh.Relation.Unary.Any where
 
 open import Level using (Level; _⊔_; Lift)
-open import Data.Empty
 open import Data.Product.Base using (∃; _,_; -,_)
 open import Data.Sum.Base using (_⊎_; [_,_]′; inj₁; inj₂)
 open import Function.Bundles using (_⇔_; mk⇔)
-open import Relation.Nullary.Negation using (¬_)
-open import Relation.Nullary.Decidable as Dec using (Dec; yes; no; _⊎-dec_)
+open import Relation.Nullary.Negation using (¬_; contradiction)
+open import Relation.Nullary.Decidable as Dec using (Dec; no; _⊎-dec_)
 open import Relation.Unary  as U
 open import Relation.Binary.Core using (Rel)
 
@@ -35,10 +34,10 @@ module _ {R : Rel A r} {P : Pred A p} {x} {xs : List# A R} {pr} where
 
   head : ¬ Any P xs → Any P (cons x xs pr) → P x
   head ¬tail (here p)   = p
-  head ¬tail (there ps) = ⊥-elim (¬tail ps)
+  head ¬tail (there ps) = contradiction ps ¬tail
 
   tail : ¬ P x → Any P (cons x xs pr) → Any P xs
-  tail ¬head (here p)   = ⊥-elim (¬head p)
+  tail ¬head (here p)   = contradiction p ¬head
   tail ¬head (there ps) = ps
 
   toSum : Any P (cons x xs pr) → P x ⊎ Any P xs
diff --git a/src/Data/List/Relation/Unary/Any.agda b/src/Data/List/Relation/Unary/Any.agda
index ad9707cd77..361cbb4160 100644
--- a/src/Data/List/Relation/Unary/Any.agda
+++ b/src/Data/List/Relation/Unary/Any.agda
@@ -13,9 +13,8 @@ open import Data.List.Base as List using (List; []; [_]; _∷_; removeAt)
 open import Data.Product.Base as Product using (∃; _,_)
 open import Data.Sum.Base as Sum using (_⊎_; inj₁; inj₂)
 open import Level using (Level; _⊔_)
-open import Relation.Nullary using (¬_; yes; no; _⊎-dec_)
-import Relation.Nullary.Decidable as Dec
-open import Relation.Nullary.Negation using (contradiction)
+open import Relation.Nullary.Decidable.Core as Dec using (no; _⊎-dec_)
+open import Relation.Nullary.Negation using (¬_; contradiction)
 open import Relation.Unary using (Pred; _⊆_; Decidable; Satisfiable)
 
 private
diff --git a/src/Data/List/Relation/Unary/First/Properties.agda b/src/Data/List/Relation/Unary/First/Properties.agda
index 0f70a7659c..4a68a544af 100644
--- a/src/Data/List/Relation/Unary/First/Properties.agda
+++ b/src/Data/List/Relation/Unary/First/Properties.agda
@@ -13,11 +13,12 @@ open import Data.List.Base as List using (List; []; _∷_)
 open import Data.List.Relation.Unary.All as All using (All; []; _∷_)
 open import Data.List.Relation.Unary.Any as Any using (here; there)
 open import Data.List.Relation.Unary.First
-import Data.Sum as Sum
-open import Function.Base using (_∘′_; _$_; _∘_; id)
+import Data.Sum.Base as Sum
+open import Function.Base using (_∘′_; _∘_; id)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_; refl; _≗_)
+import Relation.Nullary.Decidable.Core as Dec
+open import Relation.Nullary.Negation.Core using (contradiction)
 open import Relation.Unary using (Pred; _⊆_; ∁; Irrelevant; Decidable)
-open import Relation.Nullary.Negation using (contradiction)
 
 ------------------------------------------------------------------------
 -- map
@@ -80,14 +81,14 @@ module _ {a p q} {A : Set a} {P : Pred A p} {Q : Pred A q} where
 module _ {a p} {A : Set a} {P : Pred A p} where
 
   first? : Decidable P → Decidable (First P (∁ P))
-  first? P? xs = Sum.toDec
-               $ Sum.map₂ (All⇒¬First contradiction)
-               $ first (Sum.fromDec ∘ P?) xs
+  first? P? = Dec.fromSum
+            ∘ Sum.map₂ (All⇒¬First contradiction)
+            ∘ first (Dec.toSum ∘ P?)
 
   cofirst? : Decidable P → Decidable (First (∁ P) P)
-  cofirst? P? xs = Sum.toDec
-                 $ Sum.map₂ (All⇒¬First id)
-                 $ first (Sum.swap ∘ Sum.fromDec ∘ P?) xs
+  cofirst? P? = Dec.fromSum
+              ∘ Sum.map₂ (All⇒¬First id)
+              ∘ first (Sum.swap ∘ Dec.toSum ∘ P?)
 
 ------------------------------------------------------------------------
 -- Conversion to Any
diff --git a/src/Data/Sum.agda b/src/Data/Sum.agda
index 99cdbd3620..41e46fa8ae 100644
--- a/src/Data/Sum.agda
+++ b/src/Data/Sum.agda
@@ -8,15 +8,10 @@
 
 module Data.Sum where
 
-open import Agda.Builtin.Equality
-
-open import Data.Bool.Base using (true; false)
 open import Data.Unit.Polymorphic.Base using (⊤; tt)
 open import Data.Maybe.Base using (Maybe; just; nothing)
-open import Function.Base
 open import Level
-open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary using (Dec; yes; no; _because_; ¬_)
+import Relation.Nullary.Decidable.Core as Dec
 
 private
   variable
@@ -31,7 +26,7 @@ open import Data.Sum.Base public
 ------------------------------------------------------------------------
 -- Additional functions
 
-module _ {a b} {A : Set a} {B : Set b} where
+module _ {A : Set a} {B : Set b} where
 
   isInj₁ : A ⊎ B → Maybe A
   isInj₁ (inj₁ x) = just x
@@ -57,12 +52,13 @@ module _ {a b} {A : Set a} {B : Set b} where
   from-inj₂ (inj₁ _) = _
   from-inj₂ (inj₂ x) = x
 
--- Conversion back and forth with Dec
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.1
 
-fromDec : Dec A → A ⊎ ¬ A
-fromDec ( true because  [p]) = inj₁ (invert  [p])
-fromDec (false because [¬p]) = inj₂ (invert [¬p])
+open Dec public using (fromDec; toDec)
 
-toDec : A ⊎ ¬ A → Dec A
-toDec (inj₁ p)  = yes p
-toDec (inj₂ ¬p) = no ¬p
diff --git a/src/Data/Vec/Relation/Unary/Any.agda b/src/Data/Vec/Relation/Unary/Any.agda
index b4d857a69c..bf30b01076 100644
--- a/src/Data/Vec/Relation/Unary/Any.agda
+++ b/src/Data/Vec/Relation/Unary/Any.agda
@@ -8,7 +8,6 @@
 
 module Data.Vec.Relation.Unary.Any {a} {A : Set a} where
 
-open import Data.Empty
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Nat.Base using (ℕ; zero; suc; NonZero)
 open import Data.Sum.Base using (_⊎_; inj₁; inj₂; [_,_]′)
@@ -16,7 +15,7 @@ open import Data.Vec.Base as Vec using (Vec; []; [_]; _∷_)
 open import Data.Product.Base as Product using (∃; _,_)
 open import Level using (Level; _⊔_)
 open import Relation.Nullary.Negation using (¬_; contradiction)
-open import Relation.Nullary.Decidable as Dec using (yes; no; _⊎-dec_)
+open import Relation.Nullary.Decidable as Dec using (no; _⊎-dec_)
 open import Relation.Unary
 
 private
@@ -44,7 +43,7 @@ head ¬pxs (there pxs) = contradiction pxs ¬pxs
 
 -- If the head does not satisfy the predicate, then the tail will.
 tail : ∀ {x} → ¬ P x → Any P (x ∷ xs) → Any P xs
-tail ¬px (here  px)  = ⊥-elim (¬px px)
+tail ¬px (here  px)  = contradiction px ¬px
 tail ¬px (there pxs) = pxs
 
 -- Convert back and forth with sum
diff --git a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
index 6b14e91d09..c19304d464 100644
--- a/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
+++ b/src/Relation/Binary/Construct/Closure/Reflexive/Properties.agda
@@ -21,8 +21,8 @@ open import Relation.Binary.Definitions
 open import Relation.Binary.Construct.Closure.Reflexive
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 import Relation.Binary.PropositionalEquality.Properties as ≡
-open import Relation.Nullary
-import Relation.Nullary.Decidable as Dec
+open import Relation.Nullary.Negation.Core using (contradiction)
+open import Relation.Nullary.Decidable as Dec using (yes; no)
 open import Relation.Unary using (Pred)
 
 private
@@ -79,7 +79,7 @@ module _ {_~_ : Rel A ℓ} where
   ... | tri> _ _    c = inj₂ [ c ]
 
   dec : DecidableEquality A → Decidable _~_ → Decidable _~ᵒ_
-  dec ≡-dec ~-dec a b = Dec.map ⊎⇔Refl (≡-dec a b ⊎-dec ~-dec a b)
+  dec ≡-dec ~-dec a b = Dec.map ⊎⇔Refl (≡-dec a b Dec.⊎-dec ~-dec a b)
 
   decidable : Trichotomous _≡_ _~_ → Decidable _~ᵒ_
   decidable ~-tri a b with ~-tri a b
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index 9af6cc0078..bb6a30dccb 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -12,11 +12,11 @@
 module Relation.Nullary.Decidable.Core where
 
 open import Agda.Builtin.Equality using (_≡_)
-open import Level using (Level; Lift)
+open import Level using (Level)
 open import Data.Bool.Base using (Bool; T; false; true; not; _∧_; _∨_)
 open import Data.Unit.Polymorphic.Base using (⊤)
 open import Data.Product.Base using (_×_)
-open import Data.Sum.Base using (_⊎_)
+open import Data.Sum.Base using (_⊎_; inj₁; inj₂)
 open import Function.Base using (_∘_; const; _$_; flip)
 open import Relation.Nullary.Recomputable as Recomputable hiding (recompute-constant)
 open import Relation.Nullary.Reflects as Reflects hiding (recompute; recompute-constant)
@@ -103,6 +103,17 @@ _→-dec_ : Dec A → Dec B → Dec (A → B)
 does  (a? →-dec b?) = not (does a?) ∨ does b?
 proof (a? →-dec b?) = proof a? →-reflects proof b?
 
+------------------------------------------------------------------------
+-- Relationship with Sum
+
+toSum : Dec A → A ⊎ ¬ A
+toSum ( true because  [p]) = inj₁ (invert  [p])
+toSum (false because [¬p]) = inj₂ (invert [¬p])
+
+fromSum : A ⊎ ¬ A → Dec A
+fromSum (inj₁ p)  = yes p
+fromSum (inj₂ ¬p) = no ¬p
+
 ------------------------------------------------------------------------
 -- Relationship with booleans
 
@@ -201,3 +212,17 @@ excluded-middle = ¬¬-excluded-middle
 "Warning: excluded-middle was deprecated in v2.0.
 Please use ¬¬-excluded-middle instead."
 #-}
+
+-- Version 2.1
+
+fromDec = toSum
+{-# WARNING_ON_USAGE fromDec
+"Warning: fromDec was deprecated in v2.1.
+Please use Relation.Nullary.Decidable.Core.toSum instead."
+#-}
+
+toDec = fromSum
+{-# WARNING_ON_USAGE toDec
+"Warning: toDec was deprecated in v2.1.
+Please use Relation.Nullary.Decidable.Core.fromSum instead."
+#-}

From f89b4155ae7be688e2e8d6a966822d848241ce75 Mon Sep 17 00:00:00 2001
From: Jacques Carette <carette@mcmaster.ca>
Date: Mon, 17 Jun 2024 21:20:49 -0400
Subject: [PATCH 306/316] Implement move-via-deprecate of `decToMaybe` as per
 #2330 (#2336)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* Make the main home of `decToMaybe` be `Relation.Nullary.Decidable.Core`, but
deprecate the one in `Data.Maybe.Base` instead of removing it entirely.
Fix the library accordingly.

Note that this forces `Relation.Nullary.Decidable.Core` to use `Agda.Builtin.Maybe`
to avoid a cyclic import. This can be fixed once the deprecation is done.

* Update src/Relation/Nullary/Decidable/Core.agda

Good idea.

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>

* simplified the deprecation

* `CHANGELOG`

* narrowed import too far

* tweak a couple of the solvers for consistency, as per suggestions.

* chnage to `public` re-export of `Relation.Nullary.Decidable.Core.decToMaybe`

* Revert "chnage to `public` re-export of `Relation.Nullary.Decidable.Core.decToMaybe`"

This reverts commit 256a50589dacab913c69f4db5b4570b46cf440d7.

* `fix-whitespace`

* simplify `import`s

* make consistent with `Idempotent` case

* tidy up

* instead of an alias, open public so that Agda knows these really are the same. This is a better deprecation strategy.

* rename(provisional)+deprecate

* knock-on

* knock-on: refactor via `dec⇒maybe`

* deprecation via delegation

* fix `CHANGELOG`

---------

Co-authored-by: G. Allais <guillaume.allais@ens-lyon.org>
Co-authored-by: James McKinna <J.McKinna@hw.ac.uk>
Co-authored-by: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
---
 CHANGELOG.md                                  | 10 +++++++--
 src/Algebra/Solver/CommutativeMonoid.agda     |  7 +++---
 .../Solver/IdempotentCommutativeMonoid.agda   |  9 ++++----
 src/Algebra/Solver/Monoid.agda                |  8 +++----
 src/Data/Graph/Acyclic.agda                   |  8 +++----
 .../Binary/Sublist/Heterogeneous/Solver.agda  |  7 +++---
 src/Data/Maybe/Base.agda                      | 22 ++++++++++++-------
 src/Relation/Binary/Consequences.agda         | 12 +++++-----
 src/Relation/Nullary/Decidable/Core.agda      | 17 ++++++++++++++
 9 files changed, 66 insertions(+), 34 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 089957ef38..9eb3d703eb 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -76,6 +76,11 @@ Deprecated names
   _-_  ↦  _//_
   ```
 
+* In `Data.Maybe.Base`:
+  ```agda
+  decToMaybe  ↦  Relation.Nullary.Decidable.Core.dec⇒maybe
+  ```
+
 * In `Algebra.Structures.Biased`:
   ```agda
   IsRing*  ↦  Algebra.Structures.IsRing
@@ -821,8 +826,9 @@ Additions to existing modules
   WeaklyDecidable : Set _
   ```
 
-* Added new proof in `Relation.Nullary.Decidable.Core`:
+* Added new definition and proof in `Relation.Nullary.Decidable.Core`:
   ```agda
+  dec⇒maybe : Dec A → Maybe A
   recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q
   toSum              : Dec A → A ⊎ ¬ A
   fromSum            : A ⊎ ¬ A → Dec A
@@ -842,7 +848,7 @@ Additions to existing modules
   ```agda
   recompute          : Reflects A b → Recomputable A
   recompute-constant : (r : Reflects A b) (p q : A) → recompute r p ≡ recompute r q
-  ```
+ ```
 
 * Added new definitions in `Relation.Unary`
   ```
diff --git a/src/Algebra/Solver/CommutativeMonoid.agda b/src/Algebra/Solver/CommutativeMonoid.agda
index f15f83824f..04d4d372e6 100644
--- a/src/Algebra/Solver/CommutativeMonoid.agda
+++ b/src/Algebra/Solver/CommutativeMonoid.agda
@@ -14,7 +14,7 @@ module Algebra.Solver.CommutativeMonoid {m₁ m₂} (M : CommutativeMonoid m₁
 
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Maybe.Base as Maybe
-  using (Maybe; decToMaybe; From-just; from-just)
+  using (Maybe; From-just; from-just)
 open import Data.Nat as ℕ using (ℕ; zero; suc; _+_)
 open import Data.Nat.GeneralisedArithmetic using (fold)
 open import Data.Product.Base using (_×_; uncurry)
@@ -27,8 +27,9 @@ import Relation.Binary.Reflection as Reflection
 import Relation.Nullary.Decidable as Dec
 import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 
+open import Relation.Binary.Consequences using (dec⇒weaklyDec)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
-open import Relation.Nullary.Decidable using (Dec)
+open import Relation.Nullary.Decidable as Dec using (Dec)
 
 open CommutativeMonoid M
 open ≈-Reasoning setoid
@@ -184,7 +185,7 @@ nf₁ ≟ nf₂ = Dec.map Pointwise-≡↔≡ (decidable ℕ._≟_ nf₁ nf₂)
 
 prove′ : (e₁ e₂ : Expr n) → Maybe (∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
 prove′ e₁ e₂ =
-  Maybe.map lemma (decToMaybe (normalise e₁ ≟ normalise e₂))
+  Maybe.map lemma (dec⇒weaklyDec _≟_ (normalise e₁) (normalise e₂))
   where
   lemma : normalise e₁ ≡ normalise e₂ → ∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ
   lemma eq ρ =
diff --git a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
index 9ce8836bd4..00b8824975 100644
--- a/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
+++ b/src/Algebra/Solver/IdempotentCommutativeMonoid.agda
@@ -13,7 +13,7 @@ open import Algebra.Bundles using (IdempotentCommutativeMonoid)
 open import Data.Bool as Bool using (Bool; true; false; if_then_else_; _∨_)
 open import Data.Fin.Base using (Fin; zero; suc)
 open import Data.Maybe.Base as Maybe
-  using (Maybe; decToMaybe; From-just; from-just)
+  using (Maybe; From-just; from-just)
 open import Data.Nat.Base as ℕ using (ℕ; zero; suc; _+_)
 open import Data.Product.Base using (_×_; uncurry)
 open import Data.Vec.Base using (Vec; []; _∷_; lookup; replicate)
@@ -22,12 +22,13 @@ open import Function.Base using (_∘_)
 
 import Relation.Binary.Reasoning.Setoid  as ≈-Reasoning
 import Relation.Binary.Reflection            as Reflection
-import Relation.Nullary.Decidable            as Dec
 import Data.Vec.Relation.Binary.Pointwise.Inductive as Pointwise
 
+open import Relation.Binary.Consequences using (dec⇒weaklyDec)
 open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_)
 open import Relation.Binary.PropositionalEquality.Properties using (decSetoid)
-open import Relation.Nullary.Decidable using (Dec)
+open import Relation.Nullary.Decidable as Dec using (Dec)
+
 
 module Algebra.Solver.IdempotentCommutativeMonoid
   {m₁ m₂} (M : IdempotentCommutativeMonoid m₁ m₂) where
@@ -198,7 +199,7 @@ nf₁ ≟ nf₂ = Dec.map Pointwise-≡↔≡ (decidable Bool._≟_ nf₁ nf₂)
 
 prove′ : (e₁ e₂ : Expr n) → Maybe (∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
 prove′ e₁ e₂ =
-  Maybe.map lemma (decToMaybe (normalise e₁ ≟ normalise e₂))
+  Maybe.map lemma (dec⇒weaklyDec _≟_ (normalise e₁) (normalise e₂))
   where
   lemma : normalise e₁ ≡ normalise e₂ → ∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ
   lemma eq ρ =
diff --git a/src/Algebra/Solver/Monoid.agda b/src/Algebra/Solver/Monoid.agda
index 0928586c41..7debc4e1aa 100644
--- a/src/Algebra/Solver/Monoid.agda
+++ b/src/Algebra/Solver/Monoid.agda
@@ -15,17 +15,17 @@ import Data.Fin.Properties as Fin
 open import Data.List.Base hiding (lookup)
 import Data.List.Relation.Binary.Equality.DecPropositional as ListEq
 open import Data.Maybe.Base as Maybe
-  using (Maybe; decToMaybe; From-just; from-just)
+  using (Maybe; From-just; from-just)
 open import Data.Nat.Base using (ℕ)
 open import Data.Product.Base using (_×_; uncurry)
 open import Data.Vec.Base using (Vec; lookup)
 open import Function.Base using (_∘_; _$_)
+open import Relation.Binary.Consequences using (dec⇒weaklyDec)
 open import Relation.Binary.Definitions using (DecidableEquality)
 
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; cong)
 import Relation.Binary.Reflection
-open import Relation.Nullary
-import Relation.Nullary.Decidable as Dec
+import Relation.Nullary.Decidable.Core as Dec
 
 open Monoid M
 open import Relation.Binary.Reasoning.Setoid setoid
@@ -123,7 +123,7 @@ nf₁ ≟ nf₂ = Dec.map′ ≋⇒≡ ≡⇒≋ (nf₁ ≋? nf₂)
 
 prove′ : ∀ {n} (e₁ e₂ : Expr n) → Maybe (∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ)
 prove′ e₁ e₂ =
-  Maybe.map lemma $ decToMaybe (normalise e₁ ≟ normalise e₂)
+  Maybe.map lemma $ dec⇒weaklyDec _≟_ (normalise e₁) (normalise e₂)
   where
   lemma : normalise e₁ ≡ normalise e₂ → ∀ ρ → ⟦ e₁ ⟧ ρ ≈ ⟦ e₂ ⟧ ρ
   lemma eq ρ =
diff --git a/src/Data/Graph/Acyclic.agda b/src/Data/Graph/Acyclic.agda
index bf4c7267b7..451581bf4e 100644
--- a/src/Data/Graph/Acyclic.agda
+++ b/src/Data/Graph/Acyclic.agda
@@ -20,13 +20,13 @@ open import Data.Fin as Fin
   using (Fin; Fin′; zero; suc; #_; toℕ; _≟_; opposite) renaming (_ℕ-ℕ_ to _-_)
 import Data.Fin.Properties as Fin
 open import Data.Product.Base as Prod using (∃; _×_; _,_)
-open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; decToMaybe)
-open import Data.Empty
+open import Data.Maybe.Base as Maybe using (Maybe)
+open import Data.Empty using (⊥)
 open import Data.Unit.Base using (⊤; tt)
 open import Data.Vec.Base as Vec using (Vec; []; _∷_)
 open import Data.List.Base as List using (List; []; _∷_)
 open import Function.Base using (_$_; _∘′_; _∘_; id)
-open import Relation.Nullary
+open import Relation.Binary.Consequences using (dec⇒weaklyDec)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl)
 
 ------------------------------------------------------------------------
@@ -238,7 +238,7 @@ preds (c & g) (suc i) =
             (List.map (Prod.map suc id) $ preds g i)
   where
   p : ∀ {e} {E : Set e} {n} (i : Fin n) → E × Fin n → Maybe (Fin′ (suc i) × E)
-  p i (e , j) = Maybe.map (λ{ refl → zero , e }) (decToMaybe (i ≟ j))
+  p i (e , j) = Maybe.map (λ{ refl → zero , e }) (dec⇒weaklyDec _≟_ i j)
 
 private
 
diff --git a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
index 57ae74303b..11db87fa52 100644
--- a/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
+++ b/src/Data/List/Relation/Binary/Sublist/Heterogeneous/Solver.agda
@@ -21,7 +21,7 @@ module Data.List.Relation.Binary.Sublist.Heterogeneous.Solver
 
 open import Level using (_⊔_)
 open import Data.Fin as Fin
-open import Data.Maybe.Base as Maybe
+open import Data.Maybe.Base as Maybe using (Maybe; nothing; just; From-just; from-just)
 open import Data.Nat.Base as ℕ using (ℕ)
 open import Data.Product.Base using (Σ-syntax; _,_)
 open import Data.Vec.Base as Vec using (Vec ; lookup)
@@ -32,6 +32,7 @@ open import Data.List.Relation.Binary.Sublist.Heterogeneous
 open import Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
 open import Function.Base using (_$_; case_of_)
 
+open import Relation.Binary.Consequences using (dec⇒weaklyDec)
 open import Relation.Binary.PropositionalEquality.Core as ≡
   using (_≡_; _≗_; sym; cong; cong₂; subst₂)
 open import Relation.Binary.PropositionalEquality.Properties as ≡
@@ -124,8 +125,8 @@ private
 
 -- Solver for items
 solveI : ∀ {n} (a b : Item n) → Maybe (a ⊆I b)
-solveI (var k) (var l) = Maybe.map var $ decToMaybe (k Fin.≟ l)
-solveI (val a) (val b) = Maybe.map val $ decToMaybe (R? a b)
+solveI (var k) (var l) = Maybe.map var $ dec⇒weaklyDec Fin._≟_ k  l
+solveI (val a) (val b) = Maybe.map val $ dec⇒weaklyDec R? a b
 solveI _ _ = nothing
 
 -- Solver for linearised expressions
diff --git a/src/Data/Maybe/Base.agda b/src/Data/Maybe/Base.agda
index 14cff5e9ba..447280b9dd 100644
--- a/src/Data/Maybe/Base.agda
+++ b/src/Data/Maybe/Base.agda
@@ -10,14 +10,13 @@
 
 module Data.Maybe.Base where
 
-open import Level
+open import Level using (Level; Lift)
 open import Data.Bool.Base using (Bool; true; false; not)
 open import Data.Unit.Base using (⊤)
 open import Data.These.Base using (These; this; that; these)
 open import Data.Product.Base as Prod using (_×_; _,_)
-open import Function.Base using (const; _∘_; id)
-open import Relation.Nullary.Reflects using (invert)
-open import Relation.Nullary.Decidable.Core using (Dec; _because_)
+open import Function.Base using (_∘_; id; const)
+import Relation.Nullary.Decidable.Core as Dec
 
 private
   variable
@@ -46,10 +45,6 @@ is-just nothing  = false
 is-nothing : Maybe A → Bool
 is-nothing = not ∘ is-just
 
-decToMaybe : Dec A → Maybe A
-decToMaybe ( true because [a]) = just (invert [a])
-decToMaybe (false because  _ ) = nothing
-
 -- A dependent eliminator.
 
 maybe : ∀ {A : Set a} {B : Maybe A → Set b} →
@@ -137,3 +132,14 @@ thisM a = maybe′ (these a) (this a)
 
 thatM : Maybe A → B → These A B
 thatM = maybe′ these that
+
+------------------------------------------------------------------------
+-- DEPRECATED NAMES
+------------------------------------------------------------------------
+-- Please use the new names as continuing support for the old names is
+-- not guaranteed.
+
+-- Version 2.1
+-- decToMaybe
+
+open Dec using (decToMaybe) public
diff --git a/src/Relation/Binary/Consequences.agda b/src/Relation/Binary/Consequences.agda
index cd560ec995..3cdb4db356 100644
--- a/src/Relation/Binary/Consequences.agda
+++ b/src/Relation/Binary/Consequences.agda
@@ -8,16 +8,16 @@
 
 module Relation.Binary.Consequences where
 
-open import Data.Maybe.Base using (just; nothing; decToMaybe)
-open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′)
+open import Data.Empty using (⊥-elim)
 open import Data.Product.Base using (_,_)
-open import Data.Empty.Irrelevant using (⊥-elim)
+open import Data.Sum.Base as Sum using (inj₁; inj₂; [_,_]′)
 open import Function.Base using (_∘_; _∘₂_; _$_; flip)
 open import Level using (Level)
 open import Relation.Binary.Core
 open import Relation.Binary.Definitions
-open import Relation.Nullary using (yes; no; recompute; ¬_)
-open import Relation.Nullary.Decidable.Core using (map′)
+open import Relation.Nullary.Negation.Core using (¬_)
+open import Relation.Nullary.Decidable.Core
+  using (yes; no; recompute; map′; dec⇒maybe)
 open import Relation.Unary using (∁; Pred)
 
 private
@@ -193,7 +193,7 @@ module _  {_R_ : Rel A ℓ₁} {Q : Rel A ℓ₂} where
 module _ {R : REL A B p} where
 
   dec⇒weaklyDec : Decidable R → WeaklyDecidable R
-  dec⇒weaklyDec dec x y = decToMaybe (dec x y)
+  dec⇒weaklyDec dec x y = dec⇒maybe (dec x y)
 
   dec⇒recomputable : Decidable R → Recomputable R
   dec⇒recomputable dec {a} {b} = recompute $ dec a b
diff --git a/src/Relation/Nullary/Decidable/Core.agda b/src/Relation/Nullary/Decidable/Core.agda
index bb6a30dccb..3715c248e9 100644
--- a/src/Relation/Nullary/Decidable/Core.agda
+++ b/src/Relation/Nullary/Decidable/Core.agda
@@ -11,6 +11,10 @@
 
 module Relation.Nullary.Decidable.Core where
 
+-- decToMaybe was deprecated in v2.1 #2330/#2336
+-- this can go through `Data.Maybe.Base` once that deprecation is fully done.
+open import Agda.Builtin.Maybe using (Maybe; just; nothing)
+
 open import Agda.Builtin.Equality using (_≡_)
 open import Level using (Level)
 open import Data.Bool.Base using (Bool; T; false; true; not; _∧_; _∨_)
@@ -103,6 +107,13 @@ _→-dec_ : Dec A → Dec B → Dec (A → B)
 does  (a? →-dec b?) = not (does a?) ∨ does b?
 proof (a? →-dec b?) = proof a? →-reflects proof b?
 
+------------------------------------------------------------------------
+-- Relationship with Maybe
+
+dec⇒maybe : Dec A → Maybe A
+dec⇒maybe ( true because [a]) = just (invert [a])
+dec⇒maybe (false because  _ ) = nothing
+
 ------------------------------------------------------------------------
 -- Relationship with Sum
 
@@ -215,6 +226,12 @@ Please use ¬¬-excluded-middle instead."
 
 -- Version 2.1
 
+decToMaybe = dec⇒maybe
+{-# WARNING_ON_USAGE decToMaybe
+"Warning: decToMaybe was deprecated in v2.1.
+Please use Relation.Nullary.Decidable.Core.dec⇒maybe instead."
+#-}
+
 fromDec = toSum
 {-# WARNING_ON_USAGE fromDec
 "Warning: fromDec was deprecated in v2.1.

From 14fc21114003148c4111008d1dd0f8ba6b3d3091 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Wed, 19 Jun 2024 08:35:27 +0100
Subject: [PATCH 307/316] fixes #2411 (#2413)

* fixes #2411

* now via local -defined auxiliaries
---
 CHANGELOG.md                     |  4 ++--
 src/Data/List/Base.agda          | 20 ++++++++++----------
 src/Data/List/NonEmpty/Base.agda |  4 ++--
 3 files changed, 14 insertions(+), 14 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 9eb3d703eb..95de9d08dd 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -498,8 +498,8 @@ Additions to existing modules
 
 * In `Data.List.Base` redefine `inits` and `tails` in terms of:
   ```agda
-  tail∘inits : List A → List (List A)
-  tail∘tails : List A → List (List A)
+  Inits.tail : List A → List (List A)
+  Tails.tail : List A → List (List A)
   ```
 
 * In `Data.List.Membership.Propositional.Properties.Core`:
diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index f707729af7..5c98372dc5 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -189,19 +189,19 @@ iterate : (A → A) → A → ℕ → List A
 iterate f e zero    = []
 iterate f e (suc n) = e ∷ iterate f (f e) n
 
-tail∘inits : List A → List (List A)
-tail∘inits []       = []
-tail∘inits (x ∷ xs) = [ x ] ∷ map (x ∷_) (tail∘inits xs)
-
 inits : List A → List (List A)
-inits xs = [] ∷ tail∘inits xs
-
-tail∘tails : List A → List (List A)
-tail∘tails []       = []
-tail∘tails (_ ∷ xs) = xs ∷ tail∘tails xs
+inits {A = A} = λ xs → [] ∷ tail xs
+  module Inits where
+    tail : List A → List (List A)
+    tail []       = []
+    tail (x ∷ xs) = [ x ] ∷ map (x ∷_) (tail xs)
 
 tails : List A → List (List A)
-tails xs = xs ∷ tail∘tails xs
+tails {A = A} = λ xs → xs ∷ tail xs
+  module Tails where
+    tail : List A → List (List A)
+    tail []       = []
+    tail (_ ∷ xs) = xs ∷ tail xs
 
 insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
 insertAt xs       zero    v = v ∷ xs
diff --git a/src/Data/List/NonEmpty/Base.agda b/src/Data/List/NonEmpty/Base.agda
index f332dcc162..35a3608a0f 100644
--- a/src/Data/List/NonEmpty/Base.agda
+++ b/src/Data/List/NonEmpty/Base.agda
@@ -146,12 +146,12 @@ ap fs as = concatMap (λ f → map f as) fs
 -- Inits
 
 inits : List A → List⁺ (List A)
-inits xs = [] ∷ List.tail∘inits xs
+inits xs = [] ∷ List.Inits.tail xs
 
 -- Tails
 
 tails : List A → List⁺ (List A)
-tails xs = xs ∷ List.tail∘tails xs
+tails xs = xs ∷ List.Tails.tail xs
 
 -- Reverse
 

From 949e06526f7a96f4a48cd194efb252da342056e2 Mon Sep 17 00:00:00 2001
From: MatthewDaggitt <matthewdaggitt@gmail.com>
Date: Wed, 19 Jun 2024 17:23:22 +0900
Subject: [PATCH 308/316] Tidy up CHANGELOG in preparation for v2.1 release
 candidate (#2412)

* Tidy up CHANGELOG in preparation for v2.1 release candidate

* Fixed WHITESPACE

* Fixed James' feedback and improved alphabetical order
---
 CHANGELOG.md            | 464 +++++++++++++++++++---------------------
 agda-stdlib-utils.cabal |   2 +-
 2 files changed, 220 insertions(+), 246 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index 95de9d08dd..d5851dde0d 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -1,48 +1,54 @@
 Version 2.1-dev
 ===============
 
-The library has been tested using Agda 2.6.4, 2.6.4.1, and 2.6.4.3.
+The library has been tested using Agda 2.6.4.3.
 
 Highlights
 ----------
 
+* The size of the dependency graph for many modules has been
+  reduced. This may lead to speed ups for first-time loading of some
+  modules.
+
+* Added bindings for file handles in `IO.Handle`.
+
+* Added bindings for random number generation in `System.Random`
+
+* Added support for 8-bit words and bytestrings in `Data.Word8` and `Data.ByteString`.
+
 Bug-fixes
 ---------
 
-* Fix statement of `Data.Vec.Properties.toList-replicate`, where `replicate`
+* Fixed type of `toList-replicate` in `Data.Vec.Properties`, where `replicate`
   was mistakenly applied to the level of the type `A` instead of the
   variable `x` of type `A`.
 
 * Module `Data.List.Relation.Ternary.Appending.Setoid.Properties` no longer
-  exports the `Setoid` module under the alias `S`.
+  incorrectly publicly exports the `Setoid` module under the alias `S`.
 
-* Remove unbound parameter from `Data.List.Properties.length-alignWith`,
-  `alignWith-map` and `map-alignWith`.
+* Removed unbound parameter from `length-alignWith`,
+  `alignWith-map` and `map-alignWith` in `Data.List.Properties`.
 
 Non-backwards compatible changes
 --------------------------------
 
-* The modules and morphisms in `Algebra.Module.Morphism.Structures` are now
-  parametrized by _raw_ bundles rather than lawful bundles, in line with other
-  modules defining morphism structures.
-* The definitions in `Algebra.Module.Morphism.Construct.Composition` are now
-  parametrized by _raw_ bundles, and as such take a proof of transitivity.
-* The definitions in `Algebra.Module.Morphism.Construct.Identity` are now
-  parametrized by _raw_ bundles, and as such take a proof of reflexivity.
-* The module `IO.Primitive` was moved to `IO.Primitive.Core`.
-* The modules in the `Data.Word` hierarchy were moved to the `Data.Word64`
-  one instead.
+* The recently added modules and (therefore their contents) in:
+  ```agda
+  Algebra.Module.Morphism.Structures
+  Algebra.Module.Morphism.Construct.Composition
+  Algebra.Module.Morphism.Construct.Identity
+  ```
+  have been changed so they are now parametrized by _raw_ bundles rather
+  than lawful bundles.
+  This is in line with other modules that define morphisms.
+  As a result many of the `Composition` lemmas now take a proof of
+  transitivity and the `Identity` lemmas now take a proof of reflexivity.
 
-Other major improvements
-------------------------
+* The module `IO.Primitive` was moved to `IO.Primitive.Core`.
 
 Minor improvements
 ------------------
 
-* The size of the dependency graph for many modules has been
-  reduced. This may lead to speed ups for first-time loading of some
-  modules.
-
 * The definition of the `Pointwise` relational combinator in
   `Data.Product.Relation.Binary.Pointwise.NonDependent.Pointwise`
   has been generalised to take heterogeneous arguments in `REL`.
@@ -51,17 +57,25 @@ Minor improvements
   `Algebra.Lattice.Structures` have been redefined as aliases of
   `IsCommutativeBand` and `IsIdempotentMonoid` in `Algebra.Structures`.
 
-
 Deprecated modules
 ------------------
 
-* `Data.List.Relation.Binary.Sublist.Propositional.Disjoint` deprecated in favour of
-  `Data.List.Relation.Binary.Sublist.Propositional.Slice`.
+* All modules in the `Data.Word` hierarchy have been deprecated in favour
+  of their newly introduced counterparts in `Data.Word64`.
+
+* The module `Data.List.Relation.Binary.Sublist.Propositional.Disjoint`
+  has been deprecated in favour of `Data.List.Relation.Binary.Sublist.Propositional.Slice`.
 
-* The modules `Function.Endomorphism.Propositional` and
-  `Function.Endomorphism.Setoid` that use the old `Function`
-  hierarchy. Use `Function.Endo.Propositional` and
-  `Function.Endo.Setoid` instead.
+* The modules
+  ```
+  Function.Endomorphism.Propositional
+  Function.Endomorphism.Setoid
+  ```
+  that used the old `Function` hierarchy have been deprecated in favour of:
+  ```
+  Function.Endo.Propositional
+  Function.Endo.Setoid
+  ```
 
 Deprecated names
 ----------------
@@ -76,17 +90,22 @@ Deprecated names
   _-_  ↦  _//_
   ```
 
-* In `Data.Maybe.Base`:
-  ```agda
-  decToMaybe  ↦  Relation.Nullary.Decidable.Core.dec⇒maybe
-  ```
-
 * In `Algebra.Structures.Biased`:
   ```agda
   IsRing*  ↦  Algebra.Structures.IsRing
   isRing*  ↦  Algebra.Structures.isRing
   ```
 
+* In `Data.Float.Base`:
+  ```agda
+  toWord ↦ toWord64
+  ```
+
+* In `Data.Float.Properties`:
+  ```agda
+  toWord-injective ↦ toWord64-injective
+  ```
+
 * In `Data.List.Base`:
   ```agda
   scanr  ↦  Data.List.Scans.Base.scanr
@@ -104,6 +123,11 @@ Deprecated names
   map-compose  ↦  map-∘
   ```
 
+* In `Data.Maybe.Base`:
+  ```agda
+  decToMaybe  ↦  Relation.Nullary.Decidable.Core.dec⇒maybe
+  ```
+
 * In `Data.Nat.Base`: the following pattern synonyms and definitions are all
   deprecated in favour of direct pattern matching on `Algebra.Definitions.RawMagma._∣ˡ_._,_`
   ```agda
@@ -111,7 +135,7 @@ Deprecated names
   pattern ≤″-offset k = k , refl
   pattern <″-offset k = k , refl
   s≤″s⁻¹
- ```
+  ```
 
 * In `Data.Nat.Divisibility.Core`:
   ```agda
@@ -129,16 +153,6 @@ Deprecated names
   untilRight  ↦  untilInj₂
   ```
 
-* In `Data.Float.Base`:
-  ```agda
-  toWord ↦ toWord64
-  ```
-
-* In `Data.Float.Properties`:
-  ```agda
-  toWord-injective ↦ toWord64-injective
-  ```
-
 New modules
 -----------
 
@@ -164,6 +178,16 @@ New modules
   Algebra.Morphism.Construct.Terminal
   ```
 
+* Bytestrings and builders:
+  ```agda
+  Data.Bytestring.Base
+  Data.Bytestring.Builder.Base
+  Data.Bytestring.Builder.Primitive
+  Data.Bytestring.IO
+  Data.Bytestring.IO.Primitive
+  Data.Bytestring.Primitive
+  ```
+
 * Pointwise and equality relations over indexed containers:
   ```agda
   Data.Container.Indexed.Relation.Binary.Pointwise
@@ -177,72 +201,66 @@ New modules
   Data.List.Scans.Properties
   ```
 
+* Various show modules for lists and vector types:
+  ```agda
+  Data.List.Show
+  Data.Vec.Show
+  Data.Vec.Bounded.Show
+  ```
+
 * Properties of `List` modulo `Setoid` equality (currently only the ([],++) monoid):
   ```
   Data.List.Relation.Binary.Equality.Setoid.Properties
   ```
 
-* `Data.List.Relation.Binary.Sublist.Propositional.Slice`
-  replacing `Data.List.Relation.Binary.Sublist.Propositional.Disjoint` (*)
-  and adding new functions:
-  - `⊆-upper-bound-is-cospan` generalising `⊆-disjoint-union-is-cospan` from (*)
-  - `⊆-upper-bound-cospan` generalising `⊆-disjoint-union-cospan` from (*)
-  ```
-
-* Prime factorisation of natural numbers.
-  ```
-  Data.Nat.Primality.Factorisation
+* Decidability for the subset relation on lists:
+  ```agda
+  Data.List.Relation.Binary.Subset.DecSetoid (_⊆?_)
+  Data.List.Relation.Binary.Subset.DecPropositional
   ```
 
-* `Data.Vec.Functional.Relation.Binary.Permutation`, defining:
+* Decidability for the disjoint relation on lists:
   ```agda
-  _↭_ : IRel (Vector A) _
+  Data.List.Relation.Binary.Disjoint.DecSetoid (disjoint?)
+  Data.List.Relation.Binary.Disjoint.DecPropositional
   ```
 
-* `Data.Vec.Functional.Relation.Binary.Permutation.Properties` of the above:
+* Prime factorisation of natural numbers.
   ```agda
-  ↭-refl      : Reflexive (Vector A) _↭_
-  ↭-reflexive : xs ≡ ys → xs ↭ ys
-  ↭-sym       : Symmetric (Vector A) _↭_
-  ↭-trans     : Transitive (Vector A) _↭_
-  isIndexedEquivalence : {A : Set a} → IsIndexedEquivalence (Vector A) _↭_
-  indexedSetoid        : {A : Set a} → IndexedSetoid ℕ a _
+  Data.Nat.Primality.Factorisation
   ```
 
-* The modules `Function.Endo.Propositional` and
-  `Function.Endo.Setoid` are new but are actually proper ports of
-  `Function.Endomorphism.Propositional` and
-  `Function.Endomorphism.Setoid`.
-
-* `Function.Relation.Binary.Equality`
+* Permutations of vectors as functions:
   ```agda
-  setoid : Setoid a₁ a₂ → Setoid b₁ b₂ → Setoid _ _
+  Data.Vec.Functional.Relation.Binary.Permutation
+  Data.Vec.Functional.Relation.Binary.Permutation.Properties
   ```
-  and a convenient infix version
+
+* A type of bytes:
   ```agda
-  _⇨_ = setoid
+  Data.Word8.Primitive
+  Data.Word8.Base
+  Data.Word8.Literals
+  Data.Word8.Show
   ```
 
-* Consequences of 'infinite descent' for (accessible elements of) well-founded relations:
+* Word64 literals and bit-based functions:
   ```agda
-  Induction.InfiniteDescent
+  Data.Word64.Literals
+  Data.Word64.Unsafe
+  Data.Word64.Show
   ```
 
-* Symmetric interior of a binary relation
-  ```
-  Relation.Binary.Construct.Interior.Symmetric
-  ```
 
-* Properties of `Setoid`s with decidable equality relation:
+* Pointwise equality over functions
   ```
-  Relation.Binary.Properties.DecSetoid
+  Function.Relation.Binary.Equality`
   ```
 
-* Systematise the use of `Recomputable A = .A → A`:
+* Consequences of 'infinite descent' for (accessible elements of) well-founded relations:
   ```agda
-  Relation.Nullary.Recomputable
+  Induction.InfiniteDescent
   ```
-  with `Recomputable` exported publicly from `Relation.Nullary`.
 
 * New IO primitives to handle buffering
   ```agda
@@ -250,60 +268,32 @@ New modules
   IO.Handle
   ```
 
-* `System.Random` bindings:
-  ```agda
-  System.Random.Primitive
-  System.Random
-  ```
-
-* Show modules:
-  ```agda
-  Data.List.Show
-  Data.Vec.Show
-  Data.Vec.Bounded.Show
+* Symmetric interior of a binary relation
   ```
-
-* Word64 literals and bit-based functions:
-  ```agda
-  Data.Word64.Literals
-  Data.Word64.Unsafe
-  Data.Word64.Show
+  Relation.Binary.Construct.Interior.Symmetric
   ```
 
-* A type of bytes:
-  ```agda
-  Data.Word8.Primitive
-  Data.Word8.Base
-  Data.Word8.Literals
-  Data.Word8.Show
+* Properties of `Setoid`s with decidable equality relation:
   ```
-
-* Bytestrings and builders:
-  ```agda
-  Data.Bytestring.Base
-  Data.Bytestring.Builder.Base
-  Data.Bytestring.Builder.Primitive
-  Data.Bytestring.IO
-  Data.Bytestring.IO.Primitive
-  Data.Bytestring.Primitive
+  Relation.Binary.Properties.DecSetoid
   ```
 
-* Decidability for the subset relation on lists:
+* Collection of results about recomputability in
   ```agda
-  Data.List.Relation.Binary.Subset.DecSetoid (_⊆?_)
-  Data.List.Relation.Binary.Subset.DecPropositional
+  Relation.Nullary.Recomputable
   ```
+  with the main definition `Recomputable` exported publicly from `Relation.Nullary`.
 
-* Decidability for the disjoint relation on lists:
+* New bindings to random numbers:
   ```agda
-  Data.List.Relation.Binary.Disjoint.DecSetoid (disjoint?)
-  Data.List.Relation.Binary.Disjoint.DecPropositional
+  System.Random.Primitive
+  System.Random
   ```
 
 Additions to existing modules
 -----------------------------
 
-* In `Algebra.Bundles`
+* Added new definitions in `Algebra.Bundles`:
   ```agda
   record SuccessorSet c ℓ : Set (suc (c ⊔ ℓ))
   record CommutativeBand c ℓ : Set (suc (c ⊔ ℓ))
@@ -315,28 +305,21 @@ Additions to existing modules
   IdempotentSemiring
   ```
 
-
-* In `Algebra.Bundles.Raw`
+* Added new definition in `Algebra.Bundles.Raw`
   ```agda
   record RawSuccessorSet c ℓ : Set (suc (c ⊔ ℓ))
   ```
 
-* Exporting more `Raw` substructures from `Algebra.Bundles.Ring`:
-  ```agda
-  rawNearSemiring   : RawNearSemiring _ _
-  rawRingWithoutOne : RawRingWithoutOne _ _
-  +-rawGroup        : RawGroup _ _
-  ```
-
-* In `Algebra.Construct.Terminal`:
+* Added new proofs in `Algebra.Construct.Terminal`:
   ```agda
   rawNearSemiring : RawNearSemiring c ℓ
   nearSemiring    : NearSemiring c ℓ
   ```
 
-* In `Algebra.Module.Bundles`, raw bundles are re-exported and the bundles expose their raw counterparts.
+* In `Algebra.Module.Bundles`, raw bundles are now re-exported and bundles
+  consistently expose their raw counterparts.
 
-* In `Algebra.Module.Construct.DirectProduct`:
+* Added proofs in `Algebra.Module.Construct.DirectProduct`:
   ```agda
   rawLeftSemimodule  : RawLeftSemimodule R m ℓm → RawLeftSemimodule m′ ℓm′ → RawLeftSemimodule R (m ⊔ m′) (ℓm ⊔ ℓm′)
   rawLeftModule      : RawLeftModule R m ℓm → RawLeftModule m′ ℓm′ → RawLeftModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
@@ -348,7 +331,7 @@ Additions to existing modules
   rawModule          : RawModule R m ℓm → RawModule m′ ℓm′ → RawModule R (m ⊔ m′) (ℓm ⊔ ℓm′)
   ```
 
-* In `Algebra.Module.Construct.TensorUnit`:
+* Added proofs in `Algebra.Module.Construct.TensorUnit`:
   ```agda
   rawLeftSemimodule  : RawLeftSemimodule _ c ℓ
   rawLeftModule      : RawLeftModule _ c ℓ
@@ -360,7 +343,7 @@ Additions to existing modules
   rawModule          : RawModule _ c ℓ
   ```
 
-* In `Algebra.Module.Construct.Zero`:
+* Added proofs in `Algebra.Module.Construct.Zero`:
   ```agda
   rawLeftSemimodule  : RawLeftSemimodule R c ℓ
   rawLeftModule      : RawLeftModule R c ℓ
@@ -372,23 +355,23 @@ Additions to existing modules
   rawModule          : RawModule R c ℓ
   ```
 
-* In `Algebra.Morphism.Structures`:
+* Added definitions in `Algebra.Morphism.Structures`:
   ```agda
-  module SuccessorSetMorphisms (N₁ : RawSuccessorSet a ℓ₁) (N₂ : RawSuccessorSet b ℓ₂) where
-    record IsSuccessorSetHomomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
-    record IsSuccessorSetMonomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
-    record IsSuccessorSetIsomorphism  (⟦_⟧ : N₁.Carrier →  N₂.Carrier) : Set _
+  record IsSuccessorSetHomomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
+  record IsSuccessorSetMonomorphism (⟦_⟧ : N₁.Carrier → N₂.Carrier) : Set _
+  record IsSuccessorSetIsomorphism  (⟦_⟧ : N₁.Carrier →  N₂.Carrier) : Set _
 
   IsSemigroupHomomorphism : (A → B) → Set _
   IsSemigroupMonomorphism : (A → B) → Set _
   IsSemigroupIsomorphism : (A → B) → Set _
   ```
-* In `Algebra.Properties.AbelianGroup`:
+
+* Added proof in `Algebra.Properties.AbelianGroup`:
   ```
   ⁻¹-anti-homo‿- : (x - y) ⁻¹ ≈ y - x
   ```
 
-* In `Algebra.Properties.Group`:
+* Added proofs in `Algebra.Properties.Group`:
   ```agda
   isQuasigroup    : IsQuasigroup _∙_ _\\_ _//_
   quasigroup      : Quasigroup _ _
@@ -411,52 +394,42 @@ Additions to existing modules
   ⁻¹-anti-homo-\\ : (x \\ y) ⁻¹ ≈ y \\ x
   ```
 
-* In `Algebra.Properties.Loop`:
+* Added new proofs in `Algebra.Properties.Loop`:
   ```agda
   identityˡ-unique : x ∙ y ≈ y → x ≈ ε
   identityʳ-unique : x ∙ y ≈ x → y ≈ ε
   identity-unique  : Identity x _∙_ → x ≈ ε
   ```
 
-* In `Algebra.Properties.Monoid.Mult`:
+* Added new proofs in `Algebra.Properties.Monoid.Mult`:
   ```agda
-  ×-homo-0 : ∀ x → 0 × x ≈ 0#
-  ×-homo-1 : ∀ x → 1 × x ≈ x
+  ×-homo-0 : 0 × x ≈ 0#
+  ×-homo-1 : 1 × x ≈ x
   ```
 
-* In `Algebra.Properties.Semiring.Mult`:
+* Added new proofs in `Algebra.Properties.Semiring.Mult`:
   ```agda
-  ×-homo-0#     : ∀ x → 0 × x ≈ 0# * x
-  ×-homo-1#     : ∀ x → 1 × x ≈ 1# * x
+  ×-homo-0#     : 0 × x ≈ 0# * x
+  ×-homo-1#     : 1 × x ≈ 1# * x
   idem-×-homo-* : (_*_ IdempotentOn x) → (m × x) * (n × x) ≈ (m ℕ.* n) × x
   ```
 
-* In `Algebra.Structures`
+* Added new definitions to `Algebra.Structures`:
   ```agda
   record IsSuccessorSet (suc# : Op₁ A) (zero# : A) : Set _
   record IsCommutativeBand (∙ : Op₂ A) : Set _
   record IsIdempotentMonoid (∙ : Op₂ A) (ε : A) : Set _
   ```
-  and additional manifest fields for substructures arising from these in:
-  ```agda
-  IsIdempotentCommutativeMonoid
-  IsIdempotentSemiring
-  ```
 
-* In `Algebra.Structures.IsGroup`:
+* Added new definitions in `IsGroup` record in `Algebra.Structures`:
   ```agda
-  infixl 6 _//_
-  _//_ : Op₂ A
   x // y = x ∙ (y ⁻¹)
-  infixr 6 _\\_
-  _\\_ : Op₂ A
   x \\ y = (x ⁻¹) ∙ y
   ```
 
-* In `Algebra.Structures.IsCancellativeCommutativeSemiring` add the
-  extra property as an exposed definition:
+* In `Algebra.Structures` added new proof to `IsCancellativeCommutativeSemiring` record:
   ```agda
-    *-cancelʳ-nonZero : AlmostRightCancellative 0# *
+  *-cancelʳ-nonZero : AlmostRightCancellative 0# *
   ```
 
 * In `Data.Bool.Show`:
@@ -484,7 +457,7 @@ Additions to existing modules
   _≤_ : Rel Float _
   ```
 
-* In `Data.Integer.Divisibility`: introduce `divides` as an explicit pattern synonym
+* In `Data.Integer.Divisibility` introduced `divides` as an explicit pattern synonym
   ```agda
   pattern divides k eq = Data.Nat.Divisibility.divides k eq
   ```
@@ -496,11 +469,12 @@ Additions to existing modules
   i*j≢0     : .{{_ : NonZero i}} .{{_ : NonZero j}} → NonZero (i * j)
   ```
 
-* In `Data.List.Base` redefine `inits` and `tails` in terms of:
+* In `Data.List.Base` added two new functions:
   ```agda
   Inits.tail : List A → List (List A)
   Tails.tail : List A → List (List A)
   ```
+  and redefined `inits` and `tails` in terms of them.
 
 * In `Data.List.Membership.Propositional.Properties.Core`:
   ```agda
@@ -514,18 +488,6 @@ Additions to existing modules
   reverse⁻ : x ∈ reverse xs → x ∈ xs
   ```
 
-* In `Data.List.NonEmpty.Base`:
-  ```agda
-  inits : List A → List⁺ (List A)
-  tails : List A → List⁺ (List A)
-  ```
-
-* In `Data.List.NonEmpty.Properties`:
-  ```agda
-  toList-inits : toList ∘ List⁺.inits ≗ List.inits
-  toList-tails : toList ∘ List⁺.tails ≗ List.tails
-  ```
-
 * In `Data.List.Properties`:
   ```agda
   length-catMaybes       : length (catMaybes xs) ≤ length xs
@@ -584,11 +546,6 @@ Additions to existing modules
   unzip : Pointwise (R ; S) ⇒ (Pointwise R ; Pointwise S)
   ```
 
-* In `Data.Maybe.Relation.Binary.Pointwise`:
-  ```agda
-  pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
-  ```
-
 * In `Data.List.Relation.Binary.Sublist.Setoid`:
   ```agda
   ⊆-upper-bound : ∀ {xs ys zs} (τ : xs ⊆ zs) (σ : ys ⊆ zs) → UpperBound τ σ
@@ -606,15 +563,6 @@ Additions to existing modules
                   ⊆-trans ps (⊆-trans qs rs) ≡ ⊆-trans (⊆-trans ps qs) rs
   ```
 
-* In `Data.List.Relation.Binary.Subset.Setoid.Properties`:
-  ```agda
-  map⁺ : f Preserves _≈_ ⟶ _≈′_ → as ⊆ bs → map f as ⊆′ map f bs
-
-  reverse-selfAdjoint : as ⊆ reverse bs → reverse as ⊆ bs
-  reverse⁺            : as ⊆ bs → reverse as ⊆ reverse bs
-  reverse⁻            : reverse as ⊆ reverse bs → as ⊆ bs
-  ```
-
 * In `Data.List.Relation.Unary.All`:
   ```agda
   universal-U : Universal (All U)
@@ -638,6 +586,34 @@ Additions to existing modules
              (p : Any P xs) → Any.map f p ≡ Any.map g p
   ```
 
+* Added new proofs to `Data.List.Relation.Binary.Permutation.Propositional.Properties`:
+  ```agda
+  product-↭   : product Preserves _↭_ ⟶ _≡_
+  catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys
+  mapMaybe-↭  : xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
+  ```
+
+* Added new proofs to `Data.List.Relation.Binary.Permutation.Setoid.Properties.Maybe`:
+  ```agda
+  catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys
+  mapMaybe-↭  : xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
+  ```
+
+* In `Data.List.Relation.Binary.Subset.Setoid.Properties`:
+  ```agda
+  map⁺ : f Preserves _≈_ ⟶ _≈′_ → as ⊆ bs → map f as ⊆′ map f bs
+
+  reverse-selfAdjoint : as ⊆ reverse bs → reverse as ⊆ bs
+  reverse⁺            : as ⊆ bs → reverse as ⊆ reverse bs
+  reverse⁻            : reverse as ⊆ reverse bs → as ⊆ bs
+  ```
+
+* Added new proofs to `Data.List.Relation.Binary.Sublist.Propositional.Slice`:
+  ```agda
+  ⊆-upper-bound-is-cospan : (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) → IsCospan (⊆-upper-bound τ₁ τ₂)
+  ⊆-upper-bound-cospan    : (τ₁ : xs ⊆ zs) (τ₂ : ys ⊆ zs) → Cospan τ₁ τ₂
+  ```
+
 * In `Data.List.Relation.Ternary.Appending.Setoid.Properties`:
   ```agda
   through→ : ∃[ xs ] Pointwise _≈_ as xs × Appending xs bs cs →
@@ -664,6 +640,23 @@ Additions to existing modules
                          ∃[ xs ] Appending Y U as bs xs × Appending V R xs cs ds
   ```
 
+* In `Data.List.NonEmpty.Base`:
+  ```agda
+  inits : List A → List⁺ (List A)
+  tails : List A → List⁺ (List A)
+  ```
+
+* In `Data.List.NonEmpty.Properties`:
+  ```agda
+  toList-inits : toList ∘ List⁺.inits ≗ List.inits
+  toList-tails : toList ∘ List⁺.tails ≗ List.tails
+  ```
+
+* In `Data.Maybe.Relation.Binary.Pointwise`:
+  ```agda
+  pointwise⊆any : Pointwise R (just x) ⊆ Any (R x)
+  ```
+
 * In `Data.Nat.Divisibility`:
   ```agda
   quotient≢0       : m ∣ n → .{{NonZero n}} → NonZero quotient
@@ -681,10 +674,17 @@ Additions to existing modules
   nonZeroDivisor : DivMod dividend divisor → NonZero divisor
   ```
 
+* Added new proofs to `Data.Nat.Primality`:
+  ```agda
+  rough∧square>⇒prime : .{{NonTrivial n}} → m Rough n → m * m > n → Prime n
+  productOfPrimes≢0 : All Prime as → NonZero (product as)
+  productOfPrimes≥1 : All Prime as → product as ≥ 1
+  ```
+
 * Added new proofs in `Data.Nat.Properties`:
   ```agda
-  m≤n+o⇒m∸n≤o    : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o
-  m<n+o⇒m∸n<o    : ∀ m n {o} → .{{NonZero o}} → m < n + o → m ∸ n < o
+  m≤n+o⇒m∸n≤o    : m ≤ n + o → m ∸ n ≤ o
+  m<n+o⇒m∸n<o    : .{{NonZero o}} → m < n + o → m ∸ n < o
   pred-cancel-≤  : pred m ≤ pred n → (m ≡ 1 × n ≡ 0) ⊎ m ≤ n
   pred-cancel-<  : pred m < pred n → m < n
   pred-injective : .{{NonZero m}} → .{{NonZero n}} → pred m ≡ pred n → m ≡ n
@@ -695,26 +695,6 @@ Additions to existing modules
   guarded-∸≗∸   : .(m≤n : m ≤ n) → let k , _ = m≤n⇒∃[o]m+o≡n m≤n in k ≡ n ∸ m
   ```
 
-* Added new proofs to `Data.Nat.Primality`:
-  ```agda
-  rough∧square>⇒prime : .{{NonTrivial n}} → m Rough n → m * m > n → Prime n
-  productOfPrimes≢0 : All Prime as → NonZero (product as)
-  productOfPrimes≥1 : All Prime as → product as ≥ 1
-  ```
-
-* Added new proofs to `Data.List.Relation.Binary.Permutation.Propositional.Properties`:
-  ```agda
-  product-↭   : product Preserves _↭_ ⟶ _≡_
-  catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys
-  mapMaybe-↭  : xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
-  ```
-
-* Added new proofs to `Data.List.Relation.Binary.Permutation.Setoid.Properties.Maybe`:
-  ```agda
-  catMaybes-↭ : xs ↭ ys → catMaybes xs ↭ catMaybes ys
-  mapMaybe-↭  : xs ↭ ys → mapMaybe f xs ↭ mapMaybe f ys
-  ```
-
 * Added some very-dependent map and zipWith to `Data.Product`.
   ```agda
   map-Σ : {B : A → Set b} {P : A → Set p} {Q : {x : A} → P x → B x → Set q} →
@@ -729,7 +709,6 @@ Additions to existing modules
     (_*_ : (x : C) → (y : R x) → S x y) →
     ((a , p) : Σ A P) → ((b , q) : Σ B Q) →
         S (a ∙ b) (p ∘ q)
-
   ```
 
 * In `Data.Rational.Properties`:
@@ -751,6 +730,21 @@ Additions to existing modules
   between : String → String → String → String
   ```
 
+* Added new functions in `Data.Vec.Bounded.Base`:
+  ```agda
+  isBounded : (as : Vec≤ A n) → Vec≤.length as ≤ n
+  toVec     : (as : Vec≤ A n) → Vec A (Vec≤.length as)
+  ```
+
+* In `Data.Word64.Base`:
+  ```agda
+  _≤_ : Rel Word64 zero
+  show : Word64 → String
+  ```
+
+* In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
+  be used infix.
+
 * Re-exported new types and functions in `IO`:
   ```agda
   BufferMode : Set
@@ -777,38 +771,18 @@ Additions to existing modules
   _>>_ : IO A → IO B → IO B
   ```
 
-* In `Data.Word64.Base`:
-  ```agda
-  _≤_ : Rel Word64 zero
-  show : Word64 → String
-  ```
-
 * Added new definition in `Relation.Binary.Construct.Closure.Transitive`
   ```agda
   transitive⁻ : Transitive _∼_ → TransClosure _∼_ ⇒ _∼_
   ```
 
-* Added new definition in `Relation.Unary`
-  ```
-  Stable : Pred A ℓ → Set _
-  ```
-
-* Added new functions in `Data.Vec.Bounded.Base`:
-  ```agda
-  isBounded : (as : Vec≤ A n) → Vec≤.length as ≤ n
-  toVec     : (as : Vec≤ A n) → Vec A (Vec≤.length as)
-  ```
-
-* In `Function.Bundles`, added `_⟶ₛ_` as a synonym for `Func` that can
-  be used infix.
-
 * Added new proofs in `Relation.Binary.Construct.Composition`:
   ```agda
   transitive⇒≈;≈⊆≈ : Transitive ≈ → (≈ ; ≈) ⇒ ≈
   ```
 
 * Added new definitions in `Relation.Binary.Definitions`
-  ```
+  ```agda
   Stable _∼_ = ∀ x y → Nullary.Stable (x ∼ y)
   Empty  _∼_ = ∀ {x y} → ¬ (x ∼ y)
   ```
@@ -821,22 +795,22 @@ Additions to existing modules
   ```
 
 * Added new definitions in `Relation.Nullary`
-  ```
+  ```agda
   Recomputable    : Set _
   WeaklyDecidable : Set _
   ```
 
-* Added new definition and proof in `Relation.Nullary.Decidable.Core`:
+* Added new proof in `Relation.Nullary.Decidable`:
   ```agda
-  dec⇒maybe : Dec A → Maybe A
-  recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q
-  toSum              : Dec A → A ⊎ ¬ A
-  fromSum            : A ⊎ ¬ A → Dec A
+  ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
   ```
 
-* Added new proof in `Relation.Nullary.Decidable`:
+* Added new definitions and proofs in `Relation.Nullary.Decidable.Core`:
   ```agda
-  ⌊⌋-map′ : (a? : Dec A) → ⌊ map′ t f a? ⌋ ≡ ⌊ a? ⌋
+  dec⇒maybe          : Dec A → Maybe A
+  recompute-constant : (a? : Dec A) (p q : A) → recompute a? p ≡ recompute a? q
+  toSum              : Dec A → A ⊎ ¬ A
+  fromSum            : A ⊎ ¬ A → Dec A
   ```
 
 * Added new definitions in `Relation.Nullary.Negation.Core`:
@@ -851,7 +825,7 @@ Additions to existing modules
  ```
 
 * Added new definitions in `Relation.Unary`
-  ```
+  ```agda
   Stable          : Pred A ℓ → Set _
   WeaklyDecidable : Pred A ℓ → Set _
   ```
diff --git a/agda-stdlib-utils.cabal b/agda-stdlib-utils.cabal
index bbd7e4f118..d9fe1dc676 100644
--- a/agda-stdlib-utils.cabal
+++ b/agda-stdlib-utils.cabal
@@ -1,6 +1,6 @@
 cabal-version:   2.4
 name:            agda-stdlib-utils
-version:         2.0
+version:         2.1
 build-type:      Simple
 description:     Helper programs for setting up the Agda standard library.
 license:         MIT

From 5a4482e536a7759abc0d891d2075fee29b6d97aa Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Fri, 5 Jul 2024 10:02:13 +0200
Subject: [PATCH 309/316] [v2.1-rc1] fixes #2400: use explicit quantification
 instead (#2429)

* fixes #2400: use explicit quantification

* fix knock-ons
---
 src/Function/Endo/Setoid.agda                     | 8 ++++----
 src/Function/Relation/Binary/Setoid/Equality.agda | 8 ++++----
 2 files changed, 8 insertions(+), 8 deletions(-)

diff --git a/src/Function/Endo/Setoid.agda b/src/Function/Endo/Setoid.agda
index 99716b35db..583ba2f4c6 100644
--- a/src/Function/Endo/Setoid.agda
+++ b/src/Function/Endo/Setoid.agda
@@ -58,7 +58,7 @@ private
 ∘-isMagma : IsMagma _≈_ _∘_
 ∘-isMagma = record
   { isEquivalence = isEquivalence
-  ; ∙-cong        = λ {_} {_} {_} {v} x≈y u≈v → S.trans u≈v (cong v x≈y)
+  ; ∙-cong        = λ {_} {_} {_} {k} f≈g h≈k x → S.trans (h≈k _) (cong k (f≈g x))
   }
 
 ∘-magma : Magma (c ⊔ e) (c ⊔ e)
@@ -67,7 +67,7 @@ private
 ∘-isSemigroup : IsSemigroup _≈_ _∘_
 ∘-isSemigroup = record
   { isMagma = ∘-isMagma
-  ; assoc   = λ _ _ _ → S.refl
+  ; assoc   = λ _ _ _ _ → S.refl
   }
 
 ∘-semigroup : Semigroup (c ⊔ e) (c ⊔ e)
@@ -76,7 +76,7 @@ private
 ∘-id-isMonoid : IsMonoid _≈_ _∘_ id
 ∘-id-isMonoid = record
   { isSemigroup = ∘-isSemigroup
-  ; identity    = (λ _ → S.refl) , (λ _ → S.refl)
+  ; identity    = (λ _ _ → S.refl) , (λ _ _ → S.refl)
   }
 
 ∘-id-monoid : Monoid (c ⊔ e) (c ⊔ e)
@@ -112,6 +112,6 @@ module _ (f : Endo) where
   ^-isMonoidHomomorphism : IsMonoidHomomorphism +-0-rawMonoid ∘-id-rawMonoid (f ^_)
   ^-isMonoidHomomorphism = record
     { isMagmaHomomorphism = ^-isMagmaHomomorphism
-    ; ε-homo = S.refl
+    ; ε-homo = λ _ → S.refl
     }
 
diff --git a/src/Function/Relation/Binary/Setoid/Equality.agda b/src/Function/Relation/Binary/Setoid/Equality.agda
index 9d0e9823da..6c530b0e9b 100644
--- a/src/Function/Relation/Binary/Setoid/Equality.agda
+++ b/src/Function/Relation/Binary/Setoid/Equality.agda
@@ -24,16 +24,16 @@ private
 
 infix 4 _≈_
 _≈_ : (f g : Func From To) → Set (a₁ ⊔ b₂)
-f ≈ g = {x : From.Carrier} → f ⟨$⟩ x To.≈ g ⟨$⟩ x
+f ≈ g = ∀ x → f ⟨$⟩ x To.≈ g ⟨$⟩ x
 
 refl : Reflexive _≈_
-refl = To.refl
+refl _ = To.refl
 
 sym : Symmetric _≈_
-sym = λ f≈g → To.sym f≈g
+sym f≈g x = To.sym (f≈g x)
 
 trans : Transitive _≈_
-trans = λ f≈g g≈h → To.trans f≈g g≈h
+trans f≈g g≈h x = To.trans (f≈g x) (g≈h x)
 
 isEquivalence : IsEquivalence _≈_
 isEquivalence = record  -- need to η-expand else Agda gets confused

From 0c755fc7db269f2c3435b357de883c853810c273 Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Fri, 5 Jul 2024 10:03:33 +0200
Subject: [PATCH 310/316] Revert "Improve `Data.List.Base` (fix #2359;
 deprecate use of `with` #2123) (#2365)" (#2423)

This reverts commit 438f9ed4c30751be35718297bbe0c8ec7d3fb669.

Specifically, it restores `with`-based definitions of the
`Decidable`-definable functions on `List`s, incl. `filter`
Fixes #2415
---
 src/Data/List/Base.agda       | 46 +++++++++++++++--------------------
 src/Data/List/Properties.agda | 30 +++++++++++------------
 2 files changed, 35 insertions(+), 41 deletions(-)

diff --git a/src/Data/List/Base.agda b/src/Data/List/Base.agda
index 5c98372dc5..0c63e34940 100644
--- a/src/Data/List/Base.agda
+++ b/src/Data/List/Base.agda
@@ -358,49 +358,45 @@ removeAt (x ∷ xs) (suc i)  = x ∷ removeAt xs i
 
 takeWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
 takeWhile P? []       = []
-takeWhile P? (x ∷ xs) = if does (P? x)
-  then x ∷ takeWhile P? xs
-  else []
+takeWhile P? (x ∷ xs) with does (P? x)
+... | true  = x ∷ takeWhile P? xs
+... | false = []
 
 takeWhileᵇ : (A → Bool) → List A → List A
 takeWhileᵇ p = takeWhile (T? ∘ p)
 
 dropWhile : ∀ {P : Pred A p} → Decidable P → List A → List A
 dropWhile P? []       = []
-dropWhile P? (x ∷ xs) = if does (P? x)
-  then dropWhile P? xs
-  else x ∷ xs
+dropWhile P? (x ∷ xs) with does (P? x)
+... | true  = dropWhile P? xs
+... | false = x ∷ xs
 
 dropWhileᵇ : (A → Bool) → List A → List A
 dropWhileᵇ p = dropWhile (T? ∘ p)
 
 filter : ∀ {P : Pred A p} → Decidable P → List A → List A
 filter P? [] = []
-filter P? (x ∷ xs) =
-  let xs′ = filter P? xs in
-  if does (P? x)
-    then x ∷ xs′
-    else xs′
+filter P? (x ∷ xs) with does (P? x)
+... | false = filter P? xs
+... | true  = x ∷ filter P? xs
 
 filterᵇ : (A → Bool) → List A → List A
 filterᵇ p = filter (T? ∘ p)
 
 partition : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
-partition P? []       = [] , []
-partition P? (x ∷ xs) =
-  let ys , zs = partition P? xs in
-  if does (P? x)
-    then (x ∷ ys , zs)
-    else (ys , x ∷ zs)
+partition P? []       = ([] , [])
+partition P? (x ∷ xs) with does (P? x) | partition P? xs
+... | true  | (ys , zs) = (x ∷ ys , zs)
+... | false | (ys , zs) = (ys , x ∷ zs)
 
 partitionᵇ : (A → Bool) → List A → List A × List A
 partitionᵇ p = partition (T? ∘ p)
 
 span : ∀ {P : Pred A p} → Decidable P → List A → (List A × List A)
-span P? []          = [] , []
-span P? ys@(x ∷ xs) = if does (P? x)
-  then Product.map (x ∷_) id (span P? xs)
-  else ([] , ys)
+span P? []       = ([] , [])
+span P? ys@(x ∷ xs) with does (P? x)
+... | true  = Product.map (x ∷_) id (span P? xs)
+... | false = ([] , ys)
 
 
 spanᵇ : (A → Bool) → List A → List A × List A
@@ -452,11 +448,9 @@ wordsByᵇ p = wordsBy (T? ∘ p)
 derun : ∀ {R : Rel A p} → B.Decidable R → List A → List A
 derun R? [] = []
 derun R? (x ∷ []) = x ∷ []
-derun R? (x ∷ xs@(y ∷ _)) =
-  let ys = derun R? xs in
-  if does (R? x y)
-    then ys
-    else x ∷ ys
+derun R? (x ∷ xs@(y ∷ _)) with does (R? x y) | derun R? xs
+... | true  | ys = ys
+... | false | ys = x ∷ ys
 
 derunᵇ : (A → A → Bool) → List A → List A
 derunᵇ r = derun (T? ∘₂ r)
diff --git a/src/Data/List/Properties.agda b/src/Data/List/Properties.agda
index d5bd6624f9..b9936f2a3b 100644
--- a/src/Data/List/Properties.agda
+++ b/src/Data/List/Properties.agda
@@ -1163,13 +1163,13 @@ module _ {P : Pred A p} (P? : Decidable P) where
   filter-all : ∀ {xs} → All P xs → filter P? xs ≡ xs
   filter-all {[]}     []         = refl
   filter-all {x ∷ xs} (px ∷ pxs) with P? x
-  ... | false because [¬px] = contradiction px (invert [¬px])
-  ... | true  because _     = cong (x ∷_) (filter-all pxs)
+  ... | no          ¬px = contradiction px ¬px
+  ... | true  because _ = cong (x ∷_) (filter-all pxs)
 
   filter-notAll : ∀ xs → Any (∁ P) xs → length (filter P? xs) < length xs
   filter-notAll (x ∷ xs) (here ¬px) with P? x
-  ... | false because _    = s≤s (length-filter xs)
-  ... | true  because [px] = contradiction (invert [px]) ¬px
+  ... | false because _ = s≤s (length-filter xs)
+  ... | yes          px = contradiction px ¬px
   filter-notAll (x ∷ xs) (there any) with ih ← filter-notAll xs any | does (P? x)
   ... | false = m≤n⇒m≤1+n ih
   ... | true  = s≤s ih
@@ -1185,8 +1185,8 @@ module _ {P : Pred A p} (P? : Decidable P) where
   filter-none : ∀ {xs} → All (∁ P) xs → filter P? xs ≡ []
   filter-none {[]}     []           = refl
   filter-none {x ∷ xs} (¬px ∷ ¬pxs) with P? x
-  ... | false because _    = filter-none ¬pxs
-  ... | true  because [px] = contradiction (invert [px]) ¬px
+  ... | false because _ = filter-none ¬pxs
+  ... | yes          px = contradiction px ¬px
 
   filter-complete : ∀ {xs} → length (filter P? xs) ≡ length xs →
                     filter P? xs ≡ xs
@@ -1197,13 +1197,13 @@ module _ {P : Pred A p} (P? : Decidable P) where
 
   filter-accept : ∀ {x xs} → P x → filter P? (x ∷ xs) ≡ x ∷ (filter P? xs)
   filter-accept {x} Px with P? x
-  ... | true  because _     = refl
-  ... | false because [¬Px] = contradiction Px (invert [¬Px])
+  ... | true because _ = refl
+  ... | no         ¬Px = contradiction Px ¬Px
 
   filter-reject : ∀ {x xs} → ¬ P x → filter P? (x ∷ xs) ≡ filter P? xs
   filter-reject {x} ¬Px with P? x
-  ... | true  because [Px] = contradiction (invert [Px]) ¬Px
-  ... | false because _    = refl
+  ... | yes          Px = contradiction Px ¬Px
+  ... | false because _ = refl
 
   filter-idem : filter P? ∘ filter P? ≗ filter P?
   filter-idem []       = refl
@@ -1241,13 +1241,13 @@ module _ {R : Rel A p} (R? : B.Decidable R) where
 
   derun-reject : ∀ {x y} xs → R x y → derun R? (x ∷ y ∷ xs) ≡ derun R? (y ∷ xs)
   derun-reject {x} {y} xs Rxy with R? x y
-  ... | true  because _      = refl
-  ... | false because [¬Rxy] = contradiction Rxy (invert [¬Rxy])
+  ... | yes _    = refl
+  ... | no  ¬Rxy = contradiction Rxy ¬Rxy
 
   derun-accept : ∀ {x y} xs → ¬ R x y → derun R? (x ∷ y ∷ xs) ≡ x ∷ derun R? (y ∷ xs)
   derun-accept {x} {y} xs ¬Rxy with R? x y
-  ... | true  because [Rxy] = contradiction (invert [Rxy]) ¬Rxy
-  ... | false because  _    = refl
+  ... | yes Rxy = contradiction Rxy ¬Rxy
+  ... | no  _   = refl
 
 ------------------------------------------------------------------------
 -- partition
@@ -1260,7 +1260,7 @@ module _ {P : Pred A p} (P? : Decidable P) where
   ...  | true  = cong (Product.map (x ∷_) id) ih
   ...  | false = cong (Product.map id (x ∷_)) ih
 
-  length-partition : ∀ xs → (let ys , zs = partition P? xs) →
+  length-partition : ∀ xs → (let (ys , zs) = partition P? xs) →
                      length ys ≤ length xs × length zs ≤ length xs
   length-partition []       = z≤n , z≤n
   length-partition (x ∷ xs) with ih ← length-partition xs | does (P? x)

From 1071dc5040285d3f6d50d0c2ad36a58397f1332d Mon Sep 17 00:00:00 2001
From: jamesmckinna <31931406+jamesmckinna@users.noreply.github.com>
Date: Sun, 7 Jul 2024 04:23:47 +0200
Subject: [PATCH 311/316] implicit to explicit in `liftRel` (#2433)

---
 src/Algebra/Construct/Pointwise.agda | 44 ++++++++++++++--------------
 1 file changed, 22 insertions(+), 22 deletions(-)

diff --git a/src/Algebra/Construct/Pointwise.agda b/src/Algebra/Construct/Pointwise.agda
index 842e36b1ed..5b984dbc19 100644
--- a/src/Algebra/Construct/Pointwise.agda
+++ b/src/Algebra/Construct/Pointwise.agda
@@ -43,7 +43,7 @@ private
   lift₂ _∙_ g h x = (g x) ∙ (h x)
 
   liftRel : Rel C ℓ → Rel (A → C) (a ⊔ ℓ)
-  liftRel _≈_ g h = ∀ {x} → (g x) ≈ (h x)
+  liftRel _≈_ g h = ∀ x → (g x) ≈ (h x)
 
 
 ------------------------------------------------------------------------
@@ -51,9 +51,9 @@ private
 
 isEquivalence : IsEquivalence _≈_ → IsEquivalence (liftRel _≈_)
 isEquivalence isEquivalence = record
-  { refl = λ {f x} → refl {f x}
-  ; sym = λ f≈g → sym f≈g
-  ; trans = λ f≈g g≈h → trans f≈g g≈h
+  { refl = λ {f} _ → refl {f _}
+  ; sym = λ f≈g _ → sym (f≈g _)
+  ; trans = λ f≈g g≈h _ → trans (f≈g _) (g≈h _)
   }
   where open IsEquivalence isEquivalence
 
@@ -63,21 +63,21 @@ isEquivalence isEquivalence = record
 isMagma : IsMagma _≈_ _∙_ → IsMagma (liftRel _≈_) (lift₂ _∙_)
 isMagma isMagma = record
   { isEquivalence = isEquivalence M.isEquivalence
-  ; ∙-cong = λ g h → M.∙-cong g h
+  ; ∙-cong = λ g h _ → M.∙-cong (g _) (h _)
   }
   where module M = IsMagma isMagma
 
 isSemigroup : IsSemigroup _≈_ _∙_ → IsSemigroup (liftRel _≈_) (lift₂ _∙_)
 isSemigroup isSemigroup = record
   { isMagma = isMagma M.isMagma
-  ; assoc = λ f g h → M.assoc (f _) (g _) (h _)
+  ; assoc = λ f g h _ → M.assoc (f _) (g _) (h _)
   }
   where module M = IsSemigroup isSemigroup
 
 isBand : IsBand _≈_ _∙_ → IsBand (liftRel _≈_) (lift₂ _∙_)
 isBand isBand = record
   { isSemigroup = isSemigroup M.isSemigroup
-  ; idem = λ f → M.idem (f _)
+  ; idem = λ f _ → M.idem (f _)
   }
   where module M = IsBand isBand
 
@@ -85,14 +85,14 @@ isCommutativeSemigroup : IsCommutativeSemigroup _≈_ _∙_ →
                          IsCommutativeSemigroup (liftRel _≈_) (lift₂ _∙_)
 isCommutativeSemigroup isCommutativeSemigroup = record
   { isSemigroup = isSemigroup M.isSemigroup
-  ; comm = λ f g → M.comm (f _) (g _)
+  ; comm = λ f g _ → M.comm (f _) (g _)
   }
   where module M = IsCommutativeSemigroup isCommutativeSemigroup
 
 isMonoid : IsMonoid _≈_ _∙_ ε → IsMonoid (liftRel _≈_) (lift₂ _∙_) (lift₀ ε)
 isMonoid isMonoid = record
   { isSemigroup = isSemigroup M.isSemigroup
-  ; identity = (λ f → M.identityˡ (f _)) , λ f → M.identityʳ (f _)
+  ; identity = (λ f _ → M.identityˡ (f _)) , λ f _ → M.identityʳ (f _)
   }
   where module M = IsMonoid isMonoid
 
@@ -100,7 +100,7 @@ isCommutativeMonoid : IsCommutativeMonoid _≈_ _∙_ ε →
                       IsCommutativeMonoid (liftRel _≈_) (lift₂ _∙_) (lift₀ ε)
 isCommutativeMonoid isCommutativeMonoid = record
   { isMonoid = isMonoid M.isMonoid
-  ; comm = λ f g → M.comm (f _) (g _)
+  ; comm = λ f g _ → M.comm (f _) (g _)
   }
   where module M = IsCommutativeMonoid isCommutativeMonoid
 
@@ -108,8 +108,8 @@ isGroup : IsGroup _≈_ _∙_ ε _⁻¹ →
           IsGroup (liftRel _≈_) (lift₂ _∙_) (lift₀ ε) (lift₁ _⁻¹)
 isGroup isGroup = record
   { isMonoid = isMonoid M.isMonoid
-  ; inverse = (λ f → M.inverseˡ (f _)) , λ f → M.inverseʳ (f _)
-  ; ⁻¹-cong = λ f → M.⁻¹-cong f
+  ; inverse = (λ f _ → M.inverseˡ (f _)) , λ f _ → M.inverseʳ (f _)
+  ; ⁻¹-cong = λ f _ → M.⁻¹-cong (f _)
   }
   where module M = IsGroup isGroup
 
@@ -117,7 +117,7 @@ isAbelianGroup : IsAbelianGroup _≈_ _∙_ ε _⁻¹ →
                  IsAbelianGroup (liftRel _≈_) (lift₂ _∙_) (lift₀ ε) (lift₁ _⁻¹)
 isAbelianGroup isAbelianGroup = record
   { isGroup = isGroup M.isGroup
-  ; comm = λ f g → M.comm (f _) (g _)
+  ; comm = λ f g _ → M.comm (f _) (g _)
   }
   where module M = IsAbelianGroup isAbelianGroup
 
@@ -125,10 +125,10 @@ isSemiringWithoutAnnihilatingZero : IsSemiringWithoutAnnihilatingZero _≈_ _+_
   IsSemiringWithoutAnnihilatingZero (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
 isSemiringWithoutAnnihilatingZero isSemiringWithoutAnnihilatingZero = record
   { +-isCommutativeMonoid = isCommutativeMonoid M.+-isCommutativeMonoid
-  ; *-cong =  λ g h → M.*-cong g h
-  ; *-assoc =  λ f g h → M.*-assoc (f _) (g _) (h _)
-  ; *-identity = (λ f → M.*-identityˡ (f _)) , λ f → M.*-identityʳ (f _)
-  ; distrib = (λ f g h → M.distribˡ (f _) (g _) (h _)) , λ f g h → M.distribʳ (f _) (g _) (h _)
+  ; *-cong =  λ g h _ → M.*-cong (g _) (h _)
+  ; *-assoc =  λ f g h _ → M.*-assoc (f _) (g _) (h _)
+  ; *-identity = (λ f _ → M.*-identityˡ (f _)) , λ f _ → M.*-identityʳ (f _)
+  ; distrib = (λ f g h _ → M.distribˡ (f _) (g _) (h _)) , λ f g h _ → M.distribʳ (f _) (g _) (h _)
   }
   where module M = IsSemiringWithoutAnnihilatingZero isSemiringWithoutAnnihilatingZero
 
@@ -136,7 +136,7 @@ isSemiring : IsSemiring _≈_ _+_ _*_ 0# 1# →
              IsSemiring (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₀ 0#) (lift₀ 1#)
 isSemiring isSemiring = record
   { isSemiringWithoutAnnihilatingZero = isSemiringWithoutAnnihilatingZero M.isSemiringWithoutAnnihilatingZero
-  ; zero = (λ f → M.zeroˡ (f _)) , λ f → M.zeroʳ (f _)
+  ; zero = (λ f _ → M.zeroˡ (f _)) , λ f _ → M.zeroʳ (f _)
   }
   where module M = IsSemiring isSemiring
 
@@ -144,10 +144,10 @@ isRing : IsRing _≈_ _+_ _*_ -_ 0# 1# →
          IsRing (liftRel _≈_) (lift₂ _+_) (lift₂ _*_) (lift₁ -_) (lift₀ 0#) (lift₀ 1#)
 isRing isRing = record
   { +-isAbelianGroup = isAbelianGroup M.+-isAbelianGroup
-  ; *-cong = λ g h → M.*-cong g h
-  ; *-assoc = λ f g h → M.*-assoc (f _) (g _) (h _)
-  ; *-identity = (λ f → M.*-identityˡ (f _)) , λ f → M.*-identityʳ (f _)
-  ; distrib = (λ f g h → M.distribˡ (f _) (g _) (h _)) , λ f g h → M.distribʳ (f _) (g _) (h _)
+  ; *-cong = λ g h _ → M.*-cong (g _) (h _)
+  ; *-assoc = λ f g h _ → M.*-assoc (f _) (g _) (h _)
+  ; *-identity = (λ f _ → M.*-identityˡ (f _)) , λ f _ → M.*-identityʳ (f _)
+  ; distrib = (λ f g h _ → M.distribˡ (f _) (g _) (h _)) , λ f g h _ → M.distribʳ (f _) (g _) (h _)
   }
   where module M = IsRing isRing
 

From c1a984175446982c14504b130d6cb4df49d7fe9c Mon Sep 17 00:00:00 2001
From: Alba Mendez <me@alba.sh>
Date: Mon, 8 Jul 2024 12:00:42 +0200
Subject: [PATCH 312/316] fix changelog (#2435)

couple fixes for changelog rendering
---
 CHANGELOG.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/CHANGELOG.md b/CHANGELOG.md
index d5851dde0d..bb6e91c887 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -822,9 +822,9 @@ Additions to existing modules
   ```agda
   recompute          : Reflects A b → Recomputable A
   recompute-constant : (r : Reflects A b) (p q : A) → recompute r p ≡ recompute r q
- ```
+  ```
 
-* Added new definitions in `Relation.Unary`
+* Added new definitions in `Relation.Unary`:
   ```agda
   Stable          : Pred A ℓ → Set _
   WeaklyDecidable : Pred A ℓ → Set _

From ec69137d2b425850850bac1fbfcf315d37a531dd Mon Sep 17 00:00:00 2001
From: Jesper Cockx <jesper@sikanda.be>
Date: Fri, 5 Jul 2024 19:48:42 +0200
Subject: [PATCH 313/316] Export missing IsDecEquivalence instance for Quantity
 from Reflection.AST.Instances

---
 src/Reflection/AST/Instances.agda | 2 ++
 1 file changed, 2 insertions(+)

diff --git a/src/Reflection/AST/Instances.agda b/src/Reflection/AST/Instances.agda
index cb7cc014d5..27a766076d 100644
--- a/src/Reflection/AST/Instances.agda
+++ b/src/Reflection/AST/Instances.agda
@@ -17,6 +17,7 @@ import Reflection.AST.Abstraction as Abstraction
 import Reflection.AST.Argument as Argument
 import Reflection.AST.Argument.Visibility as Visibility
 import Reflection.AST.Argument.Relevance as Relevance
+import Reflection.AST.Argument.Quantity as Quantity
 import Reflection.AST.Argument.Information as Information
 import Reflection.AST.Pattern as Pattern
 import Reflection.AST.Term as Term
@@ -37,6 +38,7 @@ instance
   Meta-≡-isDecEquivalence = isDecEquivalence Meta._≟_
   Visibility-≡-isDecEquivalence = isDecEquivalence Visibility._≟_
   Relevance-≡-isDecEquivalence = isDecEquivalence Relevance._≟_
+  Quantity-≡-isDecEquivalence = isDecEquivalence Quantity._≟_
   ArgInfo-≡-isDecEquivalence = isDecEquivalence Information._≟_
   Pattern-≡-isDecEquivalence = isDecEquivalence Pattern._≟_
   Clause-≡-isDecEquivalence = isDecEquivalence Term._≟-Clause_

From 586f56ade1574c83e11f63c4a5139fd24becc95c Mon Sep 17 00:00:00 2001
From: Jesper Cockx <jesper@sikanda.be>
Date: Fri, 5 Jul 2024 19:49:22 +0200
Subject: [PATCH 314/316] Add missing `showQuantity` function to
 Reflection.AST.Show

---
 src/Reflection/AST/Show.agda | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/src/Reflection/AST/Show.agda b/src/Reflection/AST/Show.agda
index a9de9a2993..467120b4ba 100644
--- a/src/Reflection/AST/Show.agda
+++ b/src/Reflection/AST/Show.agda
@@ -24,6 +24,7 @@ open import Relation.Nullary.Decidable.Core using (yes; no)
 
 open import Reflection.AST.Abstraction hiding (map)
 open import Reflection.AST.Argument hiding (map)
+open import Reflection.AST.Argument.Quantity
 open import Reflection.AST.Argument.Relevance
 open import Reflection.AST.Argument.Visibility
 open import Reflection.AST.Argument.Modality
@@ -58,6 +59,10 @@ showVisibility visible   = "visible"
 showVisibility hidden    = "hidden"
 showVisibility instance′ = "instance"
 
+showQuantity : Quantity → String
+showQuantity quantity-0 = "quantity-0"
+showQuantity quantity-ω = "quantity-ω"
+
 showLiteral : Literal → String
 showLiteral (nat x)    = ℕ.show x
 showLiteral (word64 x) = ℕ.show (Word64.toℕ x)

From c7d65e0c40fb257979316c321a4ae0730c1764c1 Mon Sep 17 00:00:00 2001
From: Jesper Cockx <jesper@sikanda.be>
Date: Fri, 5 Jul 2024 19:49:44 +0200
Subject: [PATCH 315/316] Compatibility with Agda PR #7322: add quantity to
 reflected syntax of constructor

---
 src/Reflection/AST/Definition.agda | 42 ++++++++++++++++++------------
 src/Reflection/AST/Show.agda       |  2 +-
 2 files changed, 26 insertions(+), 18 deletions(-)

diff --git a/src/Reflection/AST/Definition.agda b/src/Reflection/AST/Definition.agda
index 3f06fef070..dab8ab2505 100644
--- a/src/Reflection/AST/Definition.agda
+++ b/src/Reflection/AST/Definition.agda
@@ -15,9 +15,10 @@ open import Relation.Nullary.Decidable.Core            using (map′; _×-dec_;
 open import Relation.Binary.Definitions                using (DecidableEquality)
 open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl; cong; cong₂)
 
-import Reflection.AST.Argument as Arg
-import Reflection.AST.Name     as Name
-import Reflection.AST.Term     as Term
+import Reflection.AST.Argument          as Arg
+import Reflection.AST.Argument.Quantity as Quantity
+import Reflection.AST.Name              as Name
+import Reflection.AST.Term              as Term
 
 ------------------------------------------------------------------------
 -- Re-exporting type publically
@@ -56,8 +57,14 @@ record′-injective₂ refl = refl
 record′-injective : ∀ {c c′ fs fs′} → record′ c fs ≡ record′ c′ fs′ → c ≡ c′ × fs ≡ fs′
 record′-injective = < record′-injective₁ , record′-injective₂ >
 
-constructor′-injective : ∀ {c c′} → constructor′ c ≡ constructor′ c′ → c ≡ c′
-constructor′-injective refl = refl
+constructor′-injective₁ : ∀ {c c′ q q′} → constructor′ c q ≡ constructor′ c′ q′ → c ≡ c′
+constructor′-injective₁ refl = refl
+
+constructor′-injective₂ : ∀ {c c′ q q′} → constructor′ c q ≡ constructor′ c′ q′ → q ≡ q′
+constructor′-injective₂ refl = refl
+
+constructor′-injective : ∀ {c c′ q q′} → constructor′ c q ≡ constructor′ c′ q′ → c ≡ c′ × q ≡ q′
+constructor′-injective = < constructor′-injective₁ , constructor′-injective₂ >
 
 infix 4 _≟_
 
@@ -70,39 +77,40 @@ data-type pars cs ≟ data-type pars′ cs′ =
 record′ c fs      ≟ record′ c′ fs′      =
   map′ (uncurry (cong₂ record′)) record′-injective
            (c Name.≟ c′ ×-dec List.≡-dec (Arg.≡-dec Name._≟_) fs fs′)
-constructor′ d    ≟ constructor′ d′     =
-  map′ (cong constructor′) constructor′-injective (d Name.≟ d′)
+constructor′ d q  ≟ constructor′ d′ q′  =
+  map′ (uncurry (cong₂ constructor′)) constructor′-injective
+           (d Name.≟ d′ ×-dec q Quantity.≟ q′)
 axiom             ≟ axiom               = yes refl
 primitive′        ≟ primitive′          = yes refl
 
 -- antidiagonal
 function cs ≟ data-type pars cs₁ = no (λ ())
 function cs ≟ record′ c fs = no (λ ())
-function cs ≟ constructor′ d = no (λ ())
+function cs ≟ constructor′ d q = no (λ ())
 function cs ≟ axiom = no (λ ())
 function cs ≟ primitive′ = no (λ ())
 data-type pars cs ≟ function cs₁ = no (λ ())
 data-type pars cs ≟ record′ c fs = no (λ ())
-data-type pars cs ≟ constructor′ d = no (λ ())
+data-type pars cs ≟ constructor′ d q = no (λ ())
 data-type pars cs ≟ axiom = no (λ ())
 data-type pars cs ≟ primitive′ = no (λ ())
 record′ c fs ≟ function cs = no (λ ())
 record′ c fs ≟ data-type pars cs = no (λ ())
-record′ c fs ≟ constructor′ d = no (λ ())
+record′ c fs ≟ constructor′ d q = no (λ ())
 record′ c fs ≟ axiom = no (λ ())
 record′ c fs ≟ primitive′ = no (λ ())
-constructor′ d ≟ function cs = no (λ ())
-constructor′ d ≟ data-type pars cs = no (λ ())
-constructor′ d ≟ record′ c fs = no (λ ())
-constructor′ d ≟ axiom = no (λ ())
-constructor′ d ≟ primitive′ = no (λ ())
+constructor′ d q ≟ function cs = no (λ ())
+constructor′ d q ≟ data-type pars cs = no (λ ())
+constructor′ d q ≟ record′ c fs = no (λ ())
+constructor′ d q ≟ axiom = no (λ ())
+constructor′ d q ≟ primitive′ = no (λ ())
 axiom ≟ function cs = no (λ ())
 axiom ≟ data-type pars cs = no (λ ())
 axiom ≟ record′ c fs = no (λ ())
-axiom ≟ constructor′ d = no (λ ())
+axiom ≟ constructor′ d q = no (λ ())
 axiom ≟ primitive′ = no (λ ())
 primitive′ ≟ function cs = no (λ ())
 primitive′ ≟ data-type pars cs = no (λ ())
 primitive′ ≟ record′ c fs = no (λ ())
-primitive′ ≟ constructor′ d = no (λ ())
+primitive′ ≟ constructor′ d q = no (λ ())
 primitive′ ≟ axiom = no (λ ())
diff --git a/src/Reflection/AST/Show.agda b/src/Reflection/AST/Show.agda
index 467120b4ba..8757e3aef1 100644
--- a/src/Reflection/AST/Show.agda
+++ b/src/Reflection/AST/Show.agda
@@ -149,6 +149,6 @@ showDefinition (data-type pars cs) =
  "datatype" <+> ℕ.show pars <+> braces (intersperse ", " (map showName cs))
 showDefinition (record′ c fs)      =
  "record" <+> showName c <+> braces (intersperse ", " (map (showName ∘′ unArg) fs))
-showDefinition (constructor′ d)    = "constructor" <+> showName d
+showDefinition (constructor′ d q)  = "constructor" <+> showName d <+> showQuantity q
 showDefinition axiom               = "axiom"
 showDefinition primitive′          = "primitive"

From 96d44779bdd34e61624c2fe5ae161df223748238 Mon Sep 17 00:00:00 2001
From: Andreas Abel <andreas.abel@ifi.lmu.de>
Date: Wed, 10 Jul 2024 09:12:10 +0200
Subject: [PATCH 316/316] Bump CI for experimental to latest Agda master

---
 .github/workflows/ci-ubuntu.yml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/ci-ubuntu.yml b/.github/workflows/ci-ubuntu.yml
index 5086e96696..dbcdb2b247 100644
--- a/.github/workflows/ci-ubuntu.yml
+++ b/.github/workflows/ci-ubuntu.yml
@@ -75,7 +75,7 @@ jobs:
           if [[ '${{ github.ref }}' == 'refs/heads/experimental' \
              || '${{ github.base_ref }}' == 'experimental' ]]; then
             # Pick Agda version for experimental
-            echo "AGDA_COMMIT=4d36cb37f8bfb765339b808b13356d760aa6f0ec" >> "${GITHUB_ENV}";
+            echo "AGDA_COMMIT=f25a71a1c1d22c8702514df22e4f206982c15f40" >> "${GITHUB_ENV}";
             echo "AGDA_HTML_DIR=html/experimental" >> "${GITHUB_ENV}"
           else
             # Pick Agda version for master